Primordial Cosmology (Oxford Graduate Texts)

For a null geodesic, we have also the extra condition

(1.141 )

.

kl-'kl-' =

0, which we can write as

Using the constants (1.141), we can deduce that the equation of the traj ectory is

= (E)2 e~l/(r)~I-'(r) ( ~)2 d

_

r2

e~l/(r) r2

(1.142)

The functions v(r) and J-i(r) can be identified with the corresponding terms in (1.140) . After introducing y = G N M/c 2 r and b = L /E, we therefore have

d2 y _ + y d
2

G M2 = 3,PPN y 2 + (l_,PPN)_N_ . b2 c4

To second order in M / b, the solution of this equation is given by the relation y = (G NM/bc) sin 00, we get

roc

which reduces to D.

~ iro

Actual source position

Observer

Fig. 1.12 Geometry of the trajectory of a photon d eflect ed by a massive body.

1.6.2.4

Shapiro Effect

The Shapiro effect , discovered in 1964, is an abnormal time delay when electromagnetic waves are emitted from Earth and reflected back by a satellite or a planet. Using the null geodesic equations derived in the previous section, we obtain

(r ~:

=

el-'(r) ~l/(r)

[1 -

el-'(r)

~~] ,

62

General relativity

the coordinate time a photon takes to go from r to ro is t herefore

t(r, ro) =

f

ro

r

Integrating this expression from rEarth to of change of the travel proper time dl:l.T

- - ::: -(1

dT

rplan et,

back and forth, we get that the r

+ ,PPN) IlS . h - 1

(1.1

assuming ro ::: 7 x 108 m. 1.6.2.5

Precession of the perihelion of Mercury

The planet trajectories are obtained using the time-like geodesics, such that

e-~(r) E2 = 1 + ev(r) (~ ) (:~ ) + (~ ) . 2

2

Introducing y = l / r and neglecting terms of order (G N M / rc 2)2 , we can integrate t equation to obtain the approximate solution

y(cp) =

G~M

(1

+ e cos {[ 1 _

(2

+ 2,PPN _

jJPPN)

G~~2] cp} ) ,

where e is the eccentricity. At each orbit, the perihelion advances by

l:I.cp =

27r(2

+ 2,

G 2 M2 = 27rG NM2 (2 £2 a(l - e )

PPN _ jJPPN)_N_

+ 2,PPN

_ jJPPN).

(1.1

For Mercury, we get 42.98" per century if , PPN = jJPPN = 1, which is consistent w observations. 1.6.2.6 Summary

The tests performed in the laboratory and in the Solar System give a bound to violation of the equivalence principle and to the deviations from the theory of gene relativity. We have only presented here some of these parameters, but the param terized post-Newtonian (PPN) formalism brings into play other parameters that beyond the purely Newtonian theory. This formalism has been developed in order take into account all kinds of possible deviations. The following table and Fig. 1 give a summary of the main constraints obtained in the Solar System (see Ref. [ for more details).

1.6.3

Gravitational radiation

Just as any charged object emits electromagnetic waves, a massive body can radi gravitational waves. In what follows, we review the properties of this radiation , with giving any proof. The existence and the properties of gravitational waves can be u to t est the theory of general relativity.

Tests of general relativity

63

Table 1.1 Solar-system constraints on the post-Newtonian parameters.

Parameters rJ (weak)

Constraints 10 - :3 5 x 10- 9 (-1.9 ±2 .5) x 10- 12 5.5 x 10(1 ± 15) x x l0 10

rJ (strong)

a

- ,j

- <:

2 x 10- 4 1.7 x 10 · 1(i eV

15m') . I

10- 23 10- 22 5 x 10- 18

6'

1.6.3.1

13

Details Pendulum Torsion balance Torsion balance (Be-Cu) Torsion balance (Earth-Moon) LLR Photon emitted (Moss bauer effect) H-Maser in orbit Lithium Strong interaction Electromagnetic interaction Weak interaction

Reference Newton (1686) [1 4] [19] [20] [21] [22] [23] [15]

Radiation of gravitational waves

The gravitational radiation emitted by an object is given by (TT)

hi)'

(x, t)

=

2

T) ,

2G N d Q kl ( -4-Pijkld 2 t - C T t c

where Pijk1(n) = (6 i k - n ink )(6jl - nknl) - ~(6ij - ninj)(6kl - nknt) and where Qkl is the quadrupole of the mass distribution Qkl(t)

=

J

p(x,t) ( XkXl -

~X26kl) d 3x .

The radiation is therefore sourced by the quadrupole of the matter distribution, and there is no monopole contribution (due to the mass conservation) or dipole (from the equivalence principle). The total power of the radiation is given by the Einstein quadrupole formula dE G N d 3Qkl d 3Q kl dt 5c 5 dt3 dt3 ' 1.6.3.2

Binary pulsar PSR 1913+16

Discovered in 1974 by Taylor and Hulse, this pulsar made it possible to test the formula for the gravitational radiation and prove the existence of gravitational radiation. Using Kepler' s law, it can be shown that if the energy of the binary system decreases , its period must t hen increase as

jdP) \dt

=

_1927r (27rG N 5c2 P

)5/3 m1m2

4 2 1+73e / 24 + 37e /96 m1+m2 (l _e 2 )7/ 2 '

where m1 and m2 are the two masses and e is the eccentricity. For the pulsar PSR 1913+16, m1 = 1.44M8 and m1 = 1.38M8 , e = 0.617 and P = 7 h 40 min so that

64

General relativity (( Lunar

\l Laser

Ranging

Y PPN

Precession of the perihelion of Mercury

1.004

Mars Radar Ranging

1.002 - - - - - - - - - - - - -

1

0.998

0.996

0.996

0.998

1

1.002

1.004

i3 PPN

Fig. 1.13 Summary of the constraints on the post-Newtonia n parameters in the Solar tem . The current constraints are 12,PPN - j3PPN - 11 < 3 x 10- 3 form the precession o perihelion of Mercury, 4j3PPN - , PPN - 3 = (-0.7 ± 1) x 10- 3 from the Lunar laser ran (LLR) experiment , b PPN - 11 < 4 x 10- 4 from the light bending (VLBI: Very Long Bas Interferometer) and b PPN - 11 = (2.1 ±2 .3) x 10- 5 from the measure of the time d elay o Cassini space probe signal.

(dP / dt ) = -2 .4 x 10- 12 . We may note that there is, however, a real discrepanc 14 (J" between the observed decrease and the theoretical prediction. This is because Doppler effect coming from the acceleration of the pulsar towards the centre o Galaxy should be taken into account . The theoretical works in Ref. [24] have ma possible to compute the equations of motion up to order v 5 / c5 and have led to idea how to t est general relativity in the strong-field limit [25] (see Ref. [26] for a revi They also prove that gravity propagates at the speed of light. 1.6.3.3

Direct detection

Many experiments are now in progress to detect gr avitational waves, either from i ferometry (LIGO, VIRGO, LISA, . . . ) or using Weber bars (see Ref. [27] for a revi In order to understand how difficult it is to det ect them, let us give an estima the amplitude of the gravitational waves. Consider for that a bar of length L and m M rotating with angular velocity w. The characteristic amplitude of the quadru is Q = lVI L2, so that dnQ/dt n rv lVI L 2w n We therefore have a typical amplit ude dE

dt

rv

G N M2L4 w 6 c5

Tests of general relativity

65

For a 500-ton mass and 20-m long bar rotating at 5 rad/s, we get h ~ 10- 38 at 50 m, and dE / dt ~ 10- 32 W! The radiation of ordinary objects is therefore undetectable. For a binary system of typical mass 1Mo, with an orbit of 10 6 km and an orbital period of order 10 h, we get dE/dt ~ 10 22 W. This power can even be increased and reach a typical value of dE/dt ~ c 5 /G N ~ 3.6 X 10 52 W for an ultrarelativistic source. 1.6.4

Need for tests at astrophysical scales

Most of the tests presented here have a characteristic scale of the order of the size of the Solar System or at most, of the size of our Galaxy for binary pulsars. In cosmology, we will extrapolate the law of gravitation to the Universe itself. This extrapolation requires some checks that we will present in detail in the rest of this book. Let us, nevertheless, remind ourselves of some facts worth keeping in mind (see Ref. [28] for more details): 1. At galaxy scales, the rotation curves indicate that the gravitational potential goes as ¢ ex In r on large scales. This behaviour is usually explained by the presence of dark matter, which has not been detected yet. Beyond a given scale, everything goes effectively as if gravity had a bidimensional behaviour. The study of the rotation curves, however, shows that if such a transition existed, this scale should depend on the galaxy mass (see Chapter 7 for more details on these relations). 2. For galaxy clusters of around 2 Mpc, the effects of gravitational lensing, which measures the gravitational potential from light bending, and the X-ray emission, which traces out the gas temperature, can be compared. This comparison has shown that the Poisson equation is valid up to a factor 2. Hence , it provides a test of general relativity (testing both the light bending and Einstein's equations in the weak-field limit). 3. At astrophysical and cosmological scales, there are no direct tests of the law of gravity. The growth of the large-scale structure of the Universe provides a test, but it mixes the properties of gravity with those of matter. For the theory to be consistent with the observations, dark matter should be present. 4. The observed recent accelerated expansion of the Universe (see Chapter 4) cannot be explained within the framework of general relativity with ordinary matter (among other conditions, matter with positive pressure; this follows from the Raychaudhuri equation). This observation is explained either through exotic matter (some candidates such as the cosmological constant, the quintessence , ... will be described in Chapter 12) or by modifying the laws of gravity at large scales. 5. Two tests are currently proposed. On the one hand , the Einstein equivalence principle can be tested by checking how constant the 'constants of Nature' are on large scales. On the other hand, the Poisson law can be tested up to scales of a hundred megaparsecs , by comparing the matter distribution in the galaxy catalogues with the weak lensing by large-scale structures (see Ref. [29]).

This shows the importance of testing gravity either in order to validate the standard approach that adds new exotic matter sectors (dark matter, dark energy), or in order to give new explanations for the Universe. We will come back to all these issues in Chapter 12.

References

[1] R HAKIM , An introduction to relativistic gravitation, Cambridge Univers Press, 1999. [2] H. STEPHANI, General relativity, Cambridge University Press, 1990. [3] S. WEINBERG, Gravitation and cosmology, John Willey and Sons, 1972. [4] R WALD , General relativity, Chicago University Press, 1984. [5] L. LANDAU and E. LIFSCHITZ, Course of theoretical physics, Volume 2, P ergam Press , 1976. [6] J.D . JACKSON , Classical electrody namics, Wiley, 1998. [7] J. EISENSTAEDT, Einstein et la relativite generale, CNRS Editions, 2003. [8] C .M. WILL , Theory and experiments in gravitational physics, Cambridge Univ sity Press, 1984. [9] G.F.R ELLIS, 'Relativistic cosmology' , in Proc. Inti. School of Physics 'Enri Fermi', Corso XLVII , R.K. Sachs (ed.), pp. 104- 179, Academic Press, 1971. [10] S.W. HAWKING and G .F.R ELLIS , The large scale struct ure of space-time, Ca bridge University Press, 1973. [11] J. BARDEEN, 'Gauge-invariant cosmologica l perturbations' , Phys. Rev. D 2 1882, 1981. [12] C.M. WILL , 'The confrontation between general relativity and experiment ', Livi Rev. Relativity 4, 4, 200l. [13] 1. CIUFOLINI and J.A. WHEELER, Gravitation and inertia, Princeton Univers Press, 1995. [14] RV. EOTvos, V. PEKAR and E. FEKETE, 'Beitrage zum Gesetze der Prop o tionalitat von Tragheit und Gravitat ', Ann. Physik 68 , 11, 1922. [15] V.W. HUGHES et al. , 'Upper limit for the anisotropy of inertial mass from nucle resonance experiments', Phys. R ev. Lett. 4, 342, 1960; RW. DREVER, 'A sear for the anisotropy of inertial mass using a free precession technique' , Phil. Ma 6, 683, 1961. [16] J.P. UZAN , 'The fundamental constants and their variation: observational a theoretical status', Rev. Mod . Phys. 75,403, 2003. [17] E. FISCHBACH and C . TALMADGE, 'Ten years of the fifth force' , in Dark matter cosmology, quantum measurem ents, experimental gravitation , Moriond procee ings, 443 (1996) , arXiv:hep - ph/9606249; E. ADELBERGER, B. HECKEL , and NELSON, 'Tests of the gravitational inverse square law', Ann. Rev. Nuc. Phy 53 , 77, 2003. [18J C.D. HOYLE et al. , 'Sub-millimeter tests of the gravitational inverse square law Phys. Rev. D 70 , 042004, 2004 . [19] Y. Su et al. , 'New tests of the universality of free fall ', Phys. Rev. D 50, 361 1994.

References

67

[20] S. BAESSLER et al., 'Improved test of the equivalence principle for gravitational self-energy', Phys. Rev. Lett. 83, 3585, 1999. [2 1] J .O. DICKEY et al., 'Lunar laser ranging: a continuous legacy of the Apollo program', Science 265 , 482, 1994. [22] R.V. POUND and G.A. REBKA , 'Apparent weight of photons', Phys. Rev. Lett. 4 , 337, 1960. [23] R. VESSOT and M. LEVINE , 'A test of the equivalence principle using a spaceborne clock', J. Gen. Rel. Grav. 10, 181, 1979. [24] T. DAMOUR and N. D ERUELLE, 'General relativistic celestial mechanics of binary systems' , Ann. Inst. H. Poincare 43, 107, 1985; 44, 263 , 1986. [25] T. DAMOUR and J .H. TAYLOR, 'Strong field tests of relativistic gravity and binary pulsars ', Phys. R ev. D 45 , 1840, 1992. [26] I.H. STAIRS, 'Testing relativity with pulsar timing' , Living Rev. R elativity 5 , 1, 2003. [2 7] I. BLAIR, 'Detection of gravitational waves', Rep. Prog. Phys. 63, 1317, 2000. [28] J.-P. UZAN, 'Tests of gravity on large scales and variation of the constants', Int . J. Theor. Phys. 42, 1163, 2003. [29] J.-P. UZAN and F. BERNARDEAU, 'Lensing at cosmological scales: a test of higher dimensional gravity', Phys. Rev. D 64, 083004, 2000.

2

Overview of particle physics and the Standard Model

Although gravity structures space-time on large scales, numerous physical processes rely on the description of t he three other interactions, namely electromagnetism , the weak and strong nuclear forces. This chapter is dedicated to t he description of these interactions and of the particles on which they act . This standard model of particle physics describes ordinary matter , as observed in the laboratory. We shall see lat er that the cosmological observations concerning the existence of dark matter call for an extension of this framework. It is hence necessary to present it in detail. For this, we will work in this chapter in a flat space-time. Section 2.1 recalls the basis of analytical mechanics and the steps from classica mechanics to quantum mechanics. Section 2.2 then focuses on the special case of a scalar field. Being the simplest prototypal model, the free or self-interacting scala field illustrates in a complete way what can be learnt from quantum field theory. The following sections sket ch a general picture of the standard model of particle physics. Section 2.3 presents the spectrum of particles detected in accelerators, and Section 2.4 describes how , relying on the notion of symmetry, a classification can be sketched out. The Higgs mechanism will t hen be described in Section 2.5. Applying this mechanism to electroweak interactions will a llow us to obtain the standard mode of particle physics , presented in Section 2.6. F inally, Section 2.7 presents a study o discrete symmetries, followed by an overview on the current state of t his standard model. The ideas introduced in this chapter will be used repeatedly in the rest of the book. For instance, t he aspects of quantization and in particular t hose of scalar fields will be central to our discussion of inflation (Chapter 8) , and also when we touch on the cosmological constant problem (Chapter 12); the notion of symmetry breaking plays a crucial role in the formation of topological defects (Chapter 11 ) and in the grand unified theories. This chapter therefore lays the necessary basis for the study of cosmologically relevant topics such as inflation of course, but also of topologica defects, the origin of dark matter , baryogenesis, the cosmological constant problem etc.

2.1

From classical to quantum

It is now a little more t han a century since the principles of quantum mechanics started being used to describe the laws of Nature. This section summarizes t he steps from analytical classical mechanics to quantum mechanics.

From classical to quantum

2.1.1

69

Analytical classical mechanics

Formally, quantum mechanics can be seen as an extension of classical mechanics as formalized finally, e.g., by Lagrange in the second half of the eighteenth century and Hamilton at the beginning of the nineteenth century. The classical laws can be transformed into their quantum counterparts by applying the correspondence principle.

2.1.1.1

Analytical classical mechanics

The essence of classical mechanics (all the details of which can be found in numerous works , in Refs. [1,2] among others) can be summarized by the existence of a Lagrangian function L from which all the dynamics of a system can be derived. For a set of N interacting particles , this Lagrangian depends on the coordinates Xi (i = 1, ' " , N) of these particles, and on their velocities, Xi . We merge together the set of all these coordinates, which are not necessarily Cartesian, under the designation qi (generalized coordinates) , with the index i now varying from 1 to 3N, and the generalized velocities qi. The dynamics of this system can be obtained by minimizing the action

S=

ltfinal L [qi(t), qi(t), t] dt ,

tinitial , tfinal

= const. ,

(2.1)

tinitial

assuming that the value of the coordinates is fixed at the boundary of the integration domain [JS = 0 with Jqi(tinitial) = Jqi(tfinal) = 0]. The Euler- Lagrange equations are then obtained as (2.2) The first term of this equation brings into play the quantities pi == 8L / 8qi, which are the generalized momenta. It is possible to invert these relations to determine the generalized velocities in terms of the coordinates and momenta. Thanks to a Legendre transformation , all the dynamics can then be described by changing variables and using the generalized momenta instead of the velocities. For that, let us define the Hamilton function , or Hamiltonian H , by 3N

H [qi(t) ,pi(t) , t]

== LpiiJi - L {qi(t), qi [r (t)] ,t} ,

(2.3)

i=l

which allows us to write the second-order differential equations (2.2) as a first-order system, .i 8H . 8H

p = - 8qi '

qi = 8pi'

(2.4)

This system allows us to easily obtain the time derivative of any physical quantity by using the Poisson brackets defined by

(2.5)

70

Overview of particle physics and the Sta ndard Model

A and B being two quantities depending on qi and pi. Substituting for B the Hami nian H in the definition (2.5) , and using the equations of motion (2.4), the total t derivative of A can easily be seen to be dA

ill

=

8A

at + Li

(8A ' i fiiP

8A.)

+ ~qi q,

P

=

8A

at + {A , H}.

(

Finally, the Poisson brackets of the coordinates and momenta satisfy

(

This formalism introduces an antisymmetric operator between two quantities, such that {A,B} = - {B , A}. This property will allow us to make the transi formally but very easily to the quantum description. 2.1.1.2

Functional derivative

Classical field theory, which can be illustrat ed, for instance, by general relativity, be written using the same formalism as the analytical theory of point particles, i rely on field functionals rather than on functions. A functional Z [J], is defined as a map from a normalized linear space offunctio .c = {f (x); x E IR}, onto the set of complex (or real in some cases) numbers , Z : .c f-----+ C (or IR). The functional derivative is then defined in a similar way the partial derivative for a function of many variables. A functional can indeed understood as a function with an infinite number of variables. Its infinitesimal varia is

<5Z[J] =

<5Z[J] <5f(x) <5f(x)dx,

J

(

introducing the dummy variable x , this reduces to a definition similar to the on normal analysis , that is

<5Z[J(x)] 1 <5f() == lim - {Z[f(x) y

<--+ 0 E

+ E<5(X -

y) ]- Z[J(x)]},

(

where <5 is the Dirac distribution in one dimension. Note that for all computations, functions are supposed to be integrable and to vanish fast enough at large x in or for the surface terms to vanish in the integration by parts. The two definitions (2. 8) and (2.9) are equivalent and can both easily be shown satisfy the rules required for a derivative, i.e. the Leibniz rule

<5 <5f(x) (Z[J]W[J])

=

and the functional composition law ITechnically, this is called a Banach space.

<5Z[J] <5f(x) W[f]

<5W[J]

+ Z[J]<5f(x)

,

(2

From classical to quantum

5Z {W[j]} = 5f(x)

J

5Z[W] 5W[f] d 5W(y) 5f(x) y.

71

(2.11)

From the definition (2.9), it can easily be shown that

5f(x) 5f(y) = 5(x - y),

(2.12)

where the first function, f(x), is seen here as the trivial functional that maps any function into itself. We also have the useful property

_5 5f(y)

J

g

[df(X)] d = _~ I [df(y)] dx x dyg dy ,

where the prime indicates a simple derivative of the function with respect to its argument. Finally, it is interesting to show that , from the functional point of view, a function and its derivative are not independent. By choosing Z[j(x)] = f'(x), and using the definition (2 .8), we find that

5f I (y) =

J

5f'(y) 5f(x) 5f(x)dx =

J

f (y)] d [55f(x) dy 5f(x)dx

=

J

d [5(y - x)] 5f(x)dx, dy

and we can hence make the identification

5f'(y) = ~ [5(y - x)]. 5f(x) dy All of these relations can easily be generalized to an arbitrary number of dimensions. Most importantly, they allow us to write a classical field theory in a manner that bears close resemblance to the Hamiltonian formulation of a classical particle.

2.1.1.3

Lagrangian of a classical field

The equivalent of the action (2.1) for a scalar field is obtained from a Lagrangian density L[¢(x!")], as Sfield

=

J

L(t) dt =

JJ dt

d3 xL [¢(x, t), o!"¢(x , t) ].

(2.13)

Let us first consider L as a function of x and t. Using an ordinary derivative followed by an integration by parts in four dimensions gives rise to the Euler- Lagrange equations, equivalent to (2.2) for a scalar field ,

oL o¢(x a

)

o oL ox!" o[o!"¢(xa) ] '

(2.14)

This relation can also be obtained using a functional differentiation, i.e. viewing L as a functional of the field and its time derivative , in the following way. The time and spatial dep endencies of the Lagrangian can be separated explicit ly before the variation

72

Overview of particle physics and the Standard Model

oL =

J

3

d x

({

8L 8L } 8L . ) 8¢>(x, t) - V' 8 [V' ¢>(x, t)] o¢>(x , t) + 8¢(x, t) o¢>(x , t) ,

where an integration by parts was p erformed to get the second term. The definitio (2.8) for the functional derivative then implies t hat

oL =

J

d3 x

[

. t) ] , O¢>(o.c /¢>(X, t) + . OL O¢>(X, x,t 8¢>(x,t)

and t herefore, by identification, we obtain

o.c O¢>(X, t)

8L _ V' 8L 8¢>(x, t) 8 [V' ¢>(x, t)]

o.c

and

o¢(x, t)

8L 8¢(x, t)

Integrating these relations by parts, assuming that t he variation at the bounda vanishes, t his reduces to (2. 14).

2.1.1.4

Hamiltonian and neld Poisson brackets

Starting from the Lagrangian (2. 13), and using an approach completely similar t hat used in point-particle mechanics, we can first define the canonically conjuga field 7r(x, t) , which is equivalent to the generalized momentum [see (2.2)], by

_

oL(t) , o¢>(x, t)

7r(x , t) = .

(2. 1

so that the equation of motion is of t he same form as for point particles, name

ir(x , t) = oLjo¢>(x, t). In order to generalize (2 .3) to the scalar field case, t he Hamiltonian can be writte in the following form

H(t) =

J

7r(x, t)¢(x , t) d 3 x - L(t)

==

J

H (x , t) d 3 x,

(2.1

which hence defines t he Hamiltonian density, related to the Lagrangian density L , b the relation

H(x , t)

=

7r(x, t) ¢(x , t) - L(x , t).

(2.17

In t his for malism, the Euler- Lagrange equations (2. 14) get replaced by t he set Hamilton equations

(2 .1

This system of first-order equations is the equivalent of (2.4) for a scalar field.

From classical to quantum

73

The Poisson bracket of two functionals is constructed in the same way as for two classical quantities via the straightforward generalization of (2.5)

_J

{W[¢(x),7f(X)] ,Z[¢(x),7f(X)]} =

3

d x

[OW oZ oW OZ] O¢(x) 07f(X) - 07f(X) O¢(x) .

(2.19)

The Hamilton equations can hence provide the time evolution of any functional W by dW = oW dt

at

The conjugate momentum

7f

+

{WH}.

,

and the field ¢ obey the relations

{¢(X ,t) ,¢(X',t)} {¢(x , t) , 7f(X' , t)}

= =

{7f(x,t) ,7f(X',t)} 0(3) (x - x') ,

=

0,

(2 .20)

which generalizes (2.7). 2.1.1.5

Nrether Theorem and conservation laws

In 1918, E. Nrether enunciated the theorem according to which a conserved quantity can be associated to any transformation that leaves the action of a theory invariant. More precisely, this theorem is enunciated in the following way:

If the action of a given theory is invariant under the infinitesimal transformations of a symmetry group , then there exist as many conserved quantities (constant in time) as infinitesimal transformations associated to this symmetry group. Furthermore, these conserved quantities can be obtained from the Lagrangian density. As an example of this theorem , as well as for its own interest, let us mention the conservation of energy that results from the time translation invariance of the laws of physics (one invariance, one conserved quantity) , as well as the conservation of momentum and angular momentum, which arise, respectively, from the translation and the rotation invariances of space-time. In these two last cases, the invariance is vectorlike (three coordinates for the translation vector, or three angles for the rotation), and the conserved quantity remains of the same nature. All these invariances, which are usual in classical mechanics, follow from the conservation of a unique quantity, the energy-momentum tensor. This construction can be generalized to the case of general relativity (Chapter 1, Section 1.4). 2.1.1.6

Case of classical point particles

The classical mechanics of point particles is completely described by specifying a Lagrangian L(qi, (ii , t). If the physics that corresponds to this Lagrangian is timetranslation invariant, i. e. such that the result of any experiment involving this theory does not depend on the time at which it is made , it is then expected that

74

Overview of particle physics and the Standard Model

L(qi, qi, t - to) = L(qi, qi, t), for any arbitrary initial time to. This implies that t Lagrangian cannot explicitly depend on time , i. e. L(qi' qi, t) = L(qi, qi); it depen on time only through the variables qi. It follows that aLlot = 0, so that its to derivative with respect to time is

(2.2 The equat ions of motion (2.2) then imply that the function E, defined as

. oL

E= " q ·- - L , ~ to' i qi

(2.2

is conserved in time, i. e. dE I dt = O. One can easily be convinced that E is indeed energy if we take the example of a set of N point particles of mass m in a potential for which 3N

L ex =

1

2:= 2mqi2 -

(2.2

V (qj).

i= 1

The relation (2.22) then implies that

which corresponds to the classical expression for the total energy (kinetic plus p tential) of a system of point particles . Note that (2.22) is simply the Hamiltoni (2 .3). The inva riance under infinitesimal coordinates transformations and generalized locities, is obtained using the transformation qi f------+ Qi = qi + Eli, and qi f------+ Qi qi + Eji' where E is a small parameter (E « 1) and Ii is a vector that will be specifi later. The invariance under this transformation can be written as

Expanding the left-hand side of this relation to leading order in

E,

we get

Once again, the equations of motion (2.2) imply that

(2.2

and hence to the existence of a new conserved quantity. Equation (2.24) can be illustrated on the example (2.23) if one chooses translati in space as the coordinat e transformation (i.e. Ii = £i constants in Cartesian coor nates). For each of the three possible independent quantities £i (i.e. along the thr

From classical to quantum

75

possible axes), the quantity p = mq is conserved: this is the momentum. One can clearly see here that for a vector-like invariance (three independent quantities form a vector) corresponds to a conservation law that is itself vector-like. This is the meaning of Nocther's theorem. 2.1.1.7 Field theory case Let us now come back to the case of field theory, an example of which is the action (2. 13) , and let us consider a general coordinate transformation (2.25) Then the scalar field ¢ transforms as

which is a local variation. One can also define a global variation d through

which only takes into consideration the modification of the field itself at the point xc> , thus defining a total variation after the local variation is brought back to its original point. To first order in E, these two variations are related by (2.26 ) Note that the variation

d commutes with the derivative, unlike the variation 6. Indeed,

while

still to first order in E. The theory will be said to be invariant under the transformation x f--+ x' (to simplify, we write x == xC> when there is no ambiguity) if the action , i. e. the integration of t he Lagrangian density over a space-time volume V4 , remains unchanged during the operation. In other words , (2.27) where the integration is performed over the same volume of space-time, expressed in terms of the new coordinates (hence the notation VI in the second equality). The volume element is modified by the determinant of the J acobian matrix

76

Overview of particle physics and th e Standard Model

where the second equality comes from the expansion to first order in then b ecomes

E.

Equation (2.2

using the relation (2.26) between the operators 6 and d, this leads to

(2.2 Furthermore, we can expand d£ in this expression as

which is possible since d is an exact differential (d and expression and using the field equations (2.14), we get

1 V4

d 4 x d£ =

1 V4

d 4 x 0J.L

an commute). Integrating th

(0£J.L ¢> - ) ~ d ¢>

,

such that (2.28) b ecomes

Since V4 is arbitrary, this implies that the integrand vanishes, or in other word that the term in parenthesis is a conserved current JJ.L , i.e. that satisfies oJ.LJJ.L = T his current is explicitly given , using (2.26 ), by

(2 .2

for a transformation of the form (2.25). In this expression we have reintroduced th local variation 6¢>, which is often better controlled than the total variation d¢>. The previous discussion does not depend on the fact that ¢> is a scalar , i.e. on th way it transforms under a change of coordinates, nor does it depend on the fact th we have considered only one field. More generically, for a theory with N fields ¢ where a E [I , N ], the relation (2 .29) can be generalized to

(2.3

This relation is still valid if the indices of ¢>a form , for instance, a vector (¢>a or a tensor of higher rank.

->

AJ

From classical to quantum

77

The conservation of t he current J implies the existence of a quantity constant in t ime. To see t his , it is sufficient to evaluat e the integral of 0l-' JI-' on an arbitrary space volume V3 , namely

before using the theorem of Stokes- Ostrogradski to eliminate the second term

It t hus follows that

r

JV

oo J od 3 x

i.

=

dt

3

r

JV

JOd 3 x

= 0,

3

and t he integral of the time comp onent of the current is indeed conserved (see the generalization of Section 1. 4 from Chapter 1) . 2.1 .1.8 Field en ergy and momentum

Just as t ime and space t r anslation invariance imply the conservation of energy and moment um in point-particle mechanics , t he same invariance, expressed in t erms of the current (2.30) , generat es the conservation of the energy-momentum tensor. Under an infinitesimal t ransformat ion X'I-' = xl-' + dJ' in Minkowskian coordinat es, where the fl-' are fo ur constant quantit ies, we can see that the field remains unchanged , i.e. b¢ = O. Note that t his remains true for any vect or-like or tensor quantity of arbitrary rank , since aX' l-' / ox v = b~ for this specific case. Since the quantities fa are supposed to be constant, it is possible to read them off from the definition (2.30) . The conserved quantity is then a t ensor of r ank 2, since

which allows us to define a canonical energy-momentum t ensor

8j.J.a

(2 .31 ) This t ensor is conserved by construction , i.e. (2.32) For each value of a = 0, . . . , 3, corresponding to independent variations of fa, is associated a conserved quantity: they are the components of the energy-momentum vector p o. = (E / c, P) , given by

p o. =

~ c

which is hence conserved in time.

r d x8o<> (xf3) , 3

JV

3

(2 .33)

78

Overview of particle physics and the Standard Model

2. 1.1 .9 Symmetrization of the energy-momentum tensor

T he energy-momentum tensor (2.31 ) is not necessarily symmetric and thus does not necessarily correspond to t he t ensor (1.83) that appears in the Einst ein equations (1.85) However , it is always possible t o make it symmetric in an appropriate way, such that it remains conserved and such that the physical quantities we can derive from it are not affect ed. This allows us t o use its symmetric version as the source for the E inst ein equations. Let us define TP,cx == GP,cx + opGPP,cx .

This t ensor is conserved by construction if GPP,cx is antisymmetric in its first two indices, i.e. GP,pcx = _ GPp,cx . Furthermore, TP,cx leads to t he same conserved quantity as long as the fields decrease fast enough at infinity. We indeed have

(2.34) The second term of the last equality vanishes identically since GPP,cx is antisymmetric The last term can be expressed as an integral over the surface that limits V3 , i.e. oV3 which we send to infinity, where the fields are supposed to vanish . It follows that

(2 .35)

which we can identify with the momentum vector. This can b e illustrat ed by the example of free electromagnetism. In that case, the field ¢ get s replaced by the vect or potential AI-'" Its dynamics is determined by the Lagrangian density [ef. (1.94)]

which only contains a kinetic term , with Pcx(3 == ocx A (3 - o(3 A cx . The part of the energy-momentum tensor that is proportional to the metric is already symmetric. The second t erm of (2.31) is given by GP,cx - _ o£e. m. oC< A _ ~ p,cx F p cx(3 e .m . 00 A P 47) cx(3 p,

P

,

which can be expressed , after a simple rearrangement , as

Here again, only the last t erm prevents the tensor from being symmetric (note, furthermore, that it is not even gauge invariant).

From classical to quantum

79

Let us now look for a t ensor with t hree indices G PWY., ant isymmet ric in its first two indices, which will allow us t o remove the last t erm. In other words , it should satisfy opGPJ.LQ = FPJ.L o pAn Using Maxwell 's equations, t his can be easily solved to obtain GPJ.LQ = FPJ.L A The symmetrized energy-moment um t ensor is hence Q .

TJ.LV e .J11 .

=

FVP F J.L _ P

~",J.LV F FQ{3 4 '/ Q{3 ,

(2. 37)

which turns out to be the same as t he one obtained by variation wit h respect to t he metric (1.83) . 2.1.2

2.1 .2. 1

Quantum physics

Uncertain ty principle and Hilbert spaces

The great concept ual revolut ion of quantum mechanics [3J arose from its st atement that one cannot determine simultaneously with arbitrary precision the positions and momenta of any obj ects. T his means, for inst ance, t hat if one measures t he posit ion and moment um of a particle with precisions 6.q and 6.p , then , at best (in one dimension), 6.q. 6.p >

1 -n

- 2 '

(2. 38)

where n is Planck 's constant. T he existence of such an intrinsic limitation in t he possibility of knowing the stat e of a physical system obliges us to give up the not ion of a classical t rajectory. (See, however , Ref [4], which discusses t he Bohm- de Broglie ontological interpretation fo r which such notions are still meaningfuL ) T he quantum descript ion replaces the classical hypotheses, by assuming t hat any physical system is described by an element of a complex vector space, a space of states, i.e. configurations, the dimension of which depends on t he system. Let us imagine, for instance, that we would like to study a light beam polarized linearly, either vertically or horizont ally. If t his is t he only prop erty we describe on the syst em, we then have a system wit h two possible st ates, denoted as I f-T) and 11), in the 'kets' not ation introduced by Dirac; t he reverse notation allows us t o define a scalar product , fo r example (11 represents a ' bra', so that the set of t hem forms a 'bra-c-ket '. The vector space, in which t his system evolves, can have these two st ates as a basis, thus it is a two-dimensional space. Any element of this space can always be decomposed as 11lf ) = 01 f-T) + .81 1) with 0, f3 E
80

Overview of particle physics and the Standard Model

unaccountably infinite dimension. In this case (infinite dimension and endowed with a norm), it is called a Hilbert space.2 2.1.2.2

Operators and the superposition principle

In the quantum framework, the physical variables are replaced by operators that act on the vector space of states. For instance, for the stat e !q), which describes a particle at the position q, there exists a position operator, denoted as ij such that

(2.39)

In other words, the position state becomes an eigenstate of the posit ion operator. To avoid any risk of confusion and error in what follows, we will systematically denote a quantum quantity corresponding to a classical one by the same letter with the symbol , ' ': q f-+ ij, P f-+ p, etc. Any operator A should satisfy some conditions in order to correspond to a measurable quantity. In particular, since the space of states is complex, the eigenvalues o an observable, which can potentially be measured, should be required to be real. This implies that A is Hermitian , i. e. At :==TA* = A. Furthermore, since the eigenvalues Ok of this operator correspond to all possible values of the concerned property, the eigenstates !ak) of A should form an orthonormal basis of the space of states. In other words , any arbitrary physical state !\[! ) can b e decomposed as

(2.40)

where the discrete sum should be replaced by an integral in the case of a continuous infinity of degrees of freedom i.e. in the case of a Hilbert space of states. An operator A that satisfies these conditions is called an observable. The relation (2.40) directly leads to the central property of quant um theory, namely t he superposition principle, according to which several physical configurations can be superposed in a linear combination that is still a physical state; this is the case in (2.40). Moreover, a measurement of the observable A performed on the state !\[!) will necessarily give a unique eigenvalue Ok of this operator. Based on this principle, we can see that the dynamical equation satisfied by the physical state is necessarily linear. 2. 1.2.3 Non-commutativity

Any given classical physical quantity corresponds to an operator in the space of states, and the equations that control the dynamics must reproduce those of classical physics in the limit where we know they are valid. The easiest way to obtain the quantum laws is thus to derive them from the corresponding classical laws. Now, unlike the functions that define classical physics, operators act on a vector space and do not always commute . It is this property that is used to realize the correspondence.

We define the commutator between two operators A and B as [A, B] :== AB - BA. This obj ect has the same symmetry properties as the Poisson brackets introduced in

2The phrase ' Hilbert space' is a lso very often used for finite-dim ensiona l spaces, although this is mathemat ically incorrect.

From classical to quantum

81

(2.5). It turns out that we obtain an appropriate prescription if we make the substitution (2.41 ) {A , B} f-----+ - ~ [A , 13] .

In particular, t he position and momentum variables qi and pi are now variables that no longer commute and (2 .7) becomes

I [tii,qj]

= 0=

[pi,pJ]

and

(2.42)

This relation can be generalized to numerous cases (see for example the case of a scalar field later on). One can explicitly check here that in the representation where the position operator is expressed by (2.39), t he momentum operator then becomes

pi =

- infJ/fJqi .

2. 1.2.4

Predictability and statistics

The predictions of quantum mechanics are essentially probabilistic. Formally, t his translates into saying t hat if a physical system is in a state I"'), then t he probability Pw(<1» for it to be measured in a state 1<1» is proportional to the square of the scalar product (<1>1 "' ). Since only these probabilit ies are effectively predicted by the t heory, one can choose to normalize t hese physical states, which will always be assumed in what follows, i. e. ("'I"') = 1. With this convention, we have (2.43) Note that the normalization of a state is always possible, also because the scalar product of a state with itself is by definition positive in a vector space with positivedefinite norm (which is a pre-requisite here to define probabilities)

For an observable operator A, we have seen that the only values a measurement could give were those belonging to the spectrum (the set of possible eigenvalues) of t his operator. Suppose that we have a great number of particles prepared in the same physical state, for definiteness, we choose, for instance, the state I"') of the decomposition (2.40). The measurement of A on this set of particles will be given by the average of t he operator in t he state I"'), namely

with a variance around this average value given by

As we would expect, this variance vanishes when I"') is an eigenstate of A. The uncertainty relation for the observables A and 13 is obtained in a similar way [5]

by considering the operator

C = 613 (A - (A) ) + i6A (13 - (B) ) , where

from now

82

Overview of particle physics and the Standard Model

on (unless specified otherwise) the average is taken in an arbitrary state 11l1), an the redundant index has been suppressed. It is sufficient to notice that (ot 0) = II011l1)11 2 2: 0 and to expand this inequality to obtain

llAll13 2: ~ 1\ i [A, 13] ) I'

(2.44

which, once applied to the position and momentum operators, reduces to the Heisen berg inequality (2.38) , with an extra factor of, arising from t he fact that the coordi nates and momentum in orthogonal directions commute.

2.1 .2.5 Heisenberg and Schrodinger representations

Any physical state can always be normalized to unity. This allows for a probabilisti interpretation, which makes sense only if the dynamical evolution conserves the nor malization of the states. Denoting by (;(t) the t ime evolut ion operator, the fact tha the evolution is linear implies that the state 11l1(t)) evolves as

1111 (t ))

= (; (t

- to) 1111 (to) ) ,

(2.45

where (; satisfies

N (t) 1111 (t)) = N (to) 11l1(to) ) ¢=? (;t (; =

l.

(2.46

Thus the evolution operator (; is a unitary operator. In order to be physical, an quantum theory must satisfy this unitarity principle; this criterion allows us to rule ou some theories, usually very speculative, without even needing to resort to experiments Unitary transformations actually allow us to go further and to define some rep

d)

resentations i. e. some pairs ({An}, {11l1 ) t hat contain all possible operators and states of the theory. These representations are related to each other by

I ~h

{ An

-t

I~) k =, ~1 1l1 ) ~ ,

-t

An

= U An U

(2.4 7

.

These rela tions guarantee the conservation of the norm of t he states, and t hus of th probabilities: all predictions made in a given representation will be identical to thos obtained in another representation. There exist many useful representations, in particular, the Heisenberg picture i which the operators are time dependent but where the basis vectors, t he states, ar time independent. In this representation, thanks to (2.41), the classical equation i directly transposed to

dA

[ ,, ]

illcit = A , H ,

(2.48

where fl is the operator that corresponds to the Hamiltonian function of the classi cal system. To obtain (2.48) , we have assumed that the operator A did not depend explicitly on time. Such an explicit dependence can be taken into account by simpl adding the required term with a partial derivative. The Schrodinger picture corresponds to t he other extreme in the possibility o choices. In t his representation t he operators do not depend on time, and only t h

From classical to quantum

83

states are time dependent. The difference between these two representations is similar to the choice one can make when describing rotations. Either one considers an active rotation of the system , leaving the axis unchanged , or one can describe the same rotations in a passive way, considering that it is the axes that change (in the opposite direction) while the system itself does not move.

2.1.2.6

Schrodinger equation

In the Schrodinger representation, the quantum dynamics of the state, now depending on time, simply follows from the requirement that predictions, and thus probabilities, should be time-translation invariant. Indeed, (2.45) implies that the time propagator U(t - to) satisfies d d ' dt 1\IF(t)) = dt U(t - to) 1\IF(to )) . However, if the equation describing the dynamics of a state is linear and independent of the initial state and time to, then the right-hand side term in the previous relation should be proportional to the vector 1\IF(t) ). We can hence conclude that d ' dt U(t - to)

,

<X

U(t - to ).

This differential equation can be solved very easily to give

U(t - to) = exp

[- ~iI x (t -

to)] ,

(2.49)

where iI is a Hermitian operator. From t he time-translation invariance, this operator should be conserved , and should therefore be related to the energy. Actually it can only be the Hamiltonian. The form of (2.49) guarantees that the operator U is indeed unitary. Let us now consider again (2.45) with the solution (2.49), we obtain the Schrodinger equation

ifi~ I\IF (t)) dt

=

iI I\IF (t)) .

(2 .50)

Moreover, we can note t hat U is precisely the unitary operator one needs to shift from the Schrodinger representation to the Heisenberg one. The vector 1\IF(to)), which is effectively time independent, forms the basis of this representation. Equation (2.50) can also be understood as a formal correspondence rule between the Hamilton func tion , i.e. t he energy, and the operator ifidj dt.

2.1.2.7 Harmonic oscillator Before attacking the subj ects of relativistic quantum mechanics and field theory, let us first recall the results of a very simple system, the harmonic oscillator. This will allow us to put into practice the notions discussed previously. Furthermore, this syst em will also be useful in field theory since all t he fields , whose properties we shall use, can be expanded as sometimes infinite sets of such harmonic oscillators.

84

Overview of particle physics and the Standard Model

The Hamiltonian of a one-dimensional harmonic oscillator of mass m and angu frequency w is

(2.5

The Schrodinger equation (2 .50) for this Hamiltonian is obtained from the correspo dences discussed previously, namely H --+ ilid/ dt and p --+ - ilia / aq, . d'if; 2lidt

li2 a 2 'if; - 2m aq2

=--

+

1 2

2

- mw 'if;

(2.5

'

where we have defined the wavefunction 'if;(q , t) == (ql'if; (t)). This function is usua complex. Given the interpretation (2.43) of quantum mechanics, the square of modulus, 1'if;1 2 , represents the density of probability to find the particle at position q t ime t. The normalization of the state I'if;(t)) simply implies that the total probabili integrated over the whole space, is unity, so that 1'if;1 2 dq = l. The spectrum of the Hamiltonian, i.e. the set of the eigenvalues associated wi the operator il , is here obtained by solving the eigenvalue equation il'if; = E 'if; , equivalently by looking for the oscillation modes of the wavefunction of the fo 'if; = f(q) exp (-iEt/li). The equat ion we obtain is well known and its solutions a normalizable only if the energy is positive, which is physically acceptable. Furthermo t he derivative of t he solut ion at q = 0 is continuous only for a discrete set of values the energy, En. We define the operators a and at, called annihilation and creation operators, respe tively, (understood as being associated with each oscillation mode), by the relation

J

, Y2ii fmW(,q+2.ft) a== mw '

a, t -

f{#w ('q-2. ft ) . -

2li

mw'

(2.5

t hey are conjugate to each other but not Hermitian: they are not observables. Inverti these relations and using the commutation rules (2.42) , we find that [a , at ] = l. T Hamiltonian can then be expressed as

In this form, we can understand why the energies are positive-definite since

by the definition of the norm in a Hilbert space. Once we obtain a state In) of energy En, we can determine all t he others applying t he creation operator:

which shows that In + 1) = (n + 1) - 1/2a t ln ) is an eigenstate of the Hamiltonian wi the eigenvalue E n+ 1 = En + fiw. Similarly, In - 1) = n- 1 / 2 aln) is the eigenstate of

From classical to quantum

85

with the energy E n - 1 = En - nw. The state with lowest energy, the vacuum state 10), satisfies alO) = 0, and all the states can be obtained by

In) =

~ (att 10).

vn!

Thus, the vacuum state has a non-vanishing energy, Eo = ~nw , and the full spectrum is En = (n + ~) nw.

2.1.2.8

General potential

Let us now come back to the three-dimensional space. A particle evolving in a general potential V (x) can be expanded in a similar way in the basis of the eigenstates of the Hamiltonian, with eigenfunctions, '0E(XI-') satisfying (2.54) where the eigenvalue of the Hamiltonian E, is the energy of the state. For a harmonic oscillator , Vi1.o. ex x 2 , and (2.54) becomes invariant under the transformations x -> - x and p -> p. Therefore, the eigenstates then have vanishing average values of the position and momentum, i. e. (nlqln) = 0 = (nlpln). This reflects the fact that the potential has a minimum at the origin , increases with the distance (quantum analogue of a restoring force) and is symmetric. Thus, the wavefunctions are essentially localized at the origin and the requirement that they are normalizable demands that they vanish asymptotically. A general potential V(x) does not necessarily respect these conditions. In particular, it can be non-vanishing only in a finite region of space. This is the case, for instance, for the Coulomb potential of the electric interaction between an electron and a proton. This attractive potential is, in addition , negative. We therefore deduce that the eigenvalues are negative energies forming the different possible states of the hydrogen atom, spread on a discrete spectrum of normalizable wavefunctions. There is also a continuous component formed by all the positive energies, which corresponds to nonnormalizable wavefunctions. Actually, they are the wavefunctions of the momentum operator that can be expressed as

'0 p (xl-') ex

e i( p."' - wt ) ,

i.e. plane waves . We usually normalize them by artificially putting the particle in a box of finite volume V3 , and dividing the wavefunction by V3. Once all computations are performed nothing should depend any longer on V3 and one can take the limit V3 ---> 00. If this is not the case, using the wavefunctions of the Hamiltonian is delicate and one should resort to other methods and different representations. An important point, valid both in classical and in quantum mechanics, is that the potential V (x) must be determined by experiments, or should be imposed in one way or another. Its functional form is in the theoretical context completely arbitrary. The description of some interactions in t he framework of field theory will allow us to compute the form of this potential for many cases .

86

Overview of particle physics and the Sta ndard Model

The eigenstates of the momentum can also be used as a basis for the representatio of free particles, i. e. without a potential. In this case, the creation operator at (p) act on the vacuum state to create a particle with moment um p , and the wavefunction i obtained using the Fourier transform of these states. It follows that

which allows us to define the field operator If, by

,'1-,( ) 'I'

X

=

J

3

d p i p·x (27f)3/ 2e ap , A

(2 .56

such that 1f;(x, t) = (O!If,(x)!1f; (t)). The operator If,(x) annihilates a particle at th point x, while its Hermitian conjugate If,t produces a particle at the same point. The generalized commutation relations for the creation and annihilation operator for momentum eigenstates, namely

(2.57

translate into

(2.58

Note that it is also possible to define a conjugate momentum IT( x) and a Hamiltonia formalism. We will come back to this in the context of the relativistic theory. 2.1.3

Quantum and relativistic mechanics

The application of the correspondence principle allows the immediate formulation o a relativistic quantum theory. Instead of using the classical expressions relating th total energy, the Hamiltonian , to the kinetic and potential energies, it is sufficient t consider the relativistic momentum , pi-' = mui-' , and the invariant Pi-'Pi-' = _ m 2 c2 which can be expressed as a function of the energy, E , and of the momentum , p , a Pi-'pi-' = _ E2/C 2 + p2 = - m 2 c 2 , to get E2

=

p

2 2 C

+ m 2 c4 .

The Klein- Gordon equation for the wave function ¢ can then be obtained using th correspondence E ---4 inOt and p ---4 - inV

(2.59

It will be shown in what follows how it is possible to take into account the propertie of special relativity to describe quantum field theory. An important part of thes developments can be generalized to curved space-times, in particular by replacing th partial derivatives oe. by covariant ones \7 e.' This is how we can consider the influenc

From classical to quantum

87

of gravity by means of general relativity to a large degree: the quantum field theory in curved space-time assumes that gravitation alters field theory only through the metric in which the (test) quantum fields evolve. From now on, natural units will be considered and we therefore fix 'It = c = l ' (see appendix A).

2.1.3. 1 Klein- Gordon and Dirac equations

In natural units , the Klein- Gordon equation , which describes the dynamics of a free scalar field , becomes (2.60) For a general self-interacting potential, the Lagrangian from which this equation is derived is given by (l.106). In the case of a massive free scalar field, we will take (2.61 ) i.e. a simple mass term. Its quantization will be discussed in detail in Section 2.2. A problem immediately appears in (2.60), that of the existence of negative-energy solut ions . Indeed, the classical relation E = p 2 + m 2 assumes that E > O. But the solutions of (2 .60) are simply the eigenstates of the momentum and the energy, which are ex: exp [i(p . x - Et)], but nothing indicates what the sign of the energy should be. In other words, both choices E = p 2 + m 2 are possible. But if the wavefunction rP corresponds to one of these states, then it contains less energy than the vacuum state itself. Thus, a theory containing such states is necessarily unstable. This is why we should go to the level of quantum field theory, for which the solution of (2.60) is not a wavefunction, as in the case of 'lj; for the Schrodinger equation , but an operator , similar to >it of (2.56). The difficulty we have just mentioned for the scalar field comes from the fact that its dynamical equation, unlike in the classical case, is second order in time, which is normal to respect the relativistic invariance. Another way to proceed is to postulate a first-order equation in time, and hence also in space. In order to respect the relativistic invariance, one should be able to write it as

J

±J

(2.62) which we can obtain by varying the Dirac Lagrangian (2.63) Since this equation , known as the Dirac equation, describes the dynamics of a relativistic particle its square should also reproduce the correspondence observed in the Klein- Gordon equation. We should therefore also have ('T)J.LI/ 8J.L81/ - m 2 )'lj; = 0, which implies that

88

Overview of particle physics and the Standard Model

This is only compatible with t he Klein- Gordon equation if the coefficients , " satisf

(2.64

T hus, the '" cannot be simple numbers. Actually, t hey should be matrices of rank a least equal to four . A useful representation for these ," is

,o=

(0 f) fO

a")

0 ( (j" 0

'

where each entry of these matrices is itself a 2 x 2 matrix, f = two-dimensional identity matrix, i. e. f = is 1

a =

(01) 10 '

a

2

=

(0 -i) 0 ' i

(~ ~).

aD

'

(2 .65

standing for th

T he a i are the Pauli matrices, tha

and

a

3

(1 0)

= 0-1 .

(2 .66

Finally, we have defined the vectors a" == (I , a i ) and (j" == (I , -ai ). For reasons t ha will become clear later , the matrices of (2.65) are called the chiral representation o the Clifford algebra, defined by (2.64). It is interesting to note that (2 .62) can also be obtained from the action

(2.67 Assuming that 'I/J , defined by

(2.68

and 'I/J are considered as two independent quantities , then t he variations must b performed simultaneously. T his action introduces the operator f--7

FoG == Fo(G) - o(F)G,

(2.69

for any two quantities F and G . In the case of (2.67) , F and G are spinors, but th definition applies also for any kind of field, for instance scalars.

2.1.3.2

Maxwell 's and Praca's theories

The scalar field ¢ is of spin 0 and the spinor field 'I/J of spin 1/2. T he fields know experimentally, among which we can find all the particles listed in Section 2.3, als include particles of spin 1 such as the photon, of zero mass, or the W ± and ZO vector of t he electroweak interaction. Such fields are described by a vector A" , the dynamic of which is governed by P roca's Lagrangian

(2.70

for a real field of mass M A . The Faraday tensor has the usual definition , F"v == 20I"Av

Canonical decompositions

89

Varying (2.70) using (2.14) for each component Ao< of the vector field, we find

and hence (2.71 ) This equation does not seem to be equivalent to the Klein- Gordon equation for each component of the field due to the right-hand side term. However, if we take the divergence 80< of this equation, the right-hand side term becomes a d'Alembertian , which simplifies with the one on the left, so that

Note that things are more complicated in the relativistic case, where the covariant derivatives do not commute [ef. (1.97)]. As can be seen here, if the vector field is massive, i.e. MA of. 0, then t he Lorentz gauge condition 8J.L AJ.L = 0 is automatically satisfied [the fact that Proca's theory (2.70) is not gauge invariant precisely because of this term is no surprise], and (2. 71) leads to a Klein- Gordon equation for each component, which is to be exp ected from the classical-quantum correspondence principle discussed around (2.59). The energy-momentum tensor fo r the massive vector field, once it has been suitably symmetrized, is identical to the one for the gauge field of electromagnetism (2.37) up to the mass term that has to be added. This leads to (2.72)

2.2

Canonical decompositions

The particular case of a scalar field, let it be real or complex, is discussed in detail in most textbooks on field theory; see, for instance, Refs. [6- 9]. We briefly review it in this section. For completeness, we also briefly describe the case of the vector and spinor fields. 2.2.1

Technical clarification

In what follows, we will systematically choose a symmetric definition of the Fourier transform and of its inverse that are defined by (B.58) and (B.59). 2.2.1.1

Lorentz invariant measure

One of the important properties of t he Dirac distribution is that

df

o [j(x) - a] = ~ ( dx a

where the

Xa

)-

1

o(x - xa),

(2.73)

X = Xa

represent the zeros of the argument, i.e. the solutions of f(xa) = a.

90

Overview of particle physics and the Standard Model

This property allows us to replace the Fourier integrals in 4 dimensions by three dimensional integrals when the fields over which we perform the integration are 'o shell ', i. e. their Fourier components satisfy the relation E2 == k;5 = k 2 - m 2 , the energ being posit ive-definite . In that case, it is possible to perform the integration over th time component ko with the previous relation by setting f(k o) = k;5 . This gives

with the definit ion

(2.75

and where e is the Heaviside step function. T he relation (2.74) defines a Lorent invariant measure despite being three-dimensional. Note that to obtain this measure it is necessary to impose the positive energy condition in order to keep only one term of (2 .73).

2.2.2

Real free fie ld

Equation (2.60) can be obtained by varying t he Lagrangian (l.106) in which we pos tulate an arbitrary potential V(¢) for t he scalar field ¢ . The conjugate field is then

7r(x) = so that the Hamiltonian density, H =

Of:

~ . -

J¢(x)

.

= ¢(x),

(2 .76

7r¢ - L , is given by

(2.77

General potentials V(¢) are useful in cosmology, for instance, when addressing t h inflation scenarios (Chapter 8).

2.2.2.1

Mode expansion

The eigenstates of the momentum operator are t he basis of every expansion in quantum field theory. To deduce that, we should first know the energy-momentum tensor , tha is 8 J.'
_ ~8
= 8J.'¢8
[~ (8¢)2 + V(¢) ] = TJ.'n

(2 .78

The last equality follows from the fact that the energy-momentum tensor of a scala field is already symmetric. Moreover, it coincides with t he tensor (l.1 07) obtained by varying (l.84) with respect to the metric.

Canonica l decompositions

91

The tensor (2.78) allows us to construct the four-momentum vector that is obtained by po. == TOO. , from which we indeed get, first the Hamiltonian density, which corresponds to the purely time-like component, i.e. pO == TOO = 1i, and then the associated momentum to the scalar field P = 7rV ¢. Because the presence of a general potential V (¢) can lead to some potential nonlinearities that can imply some unnecessary complications, here and in the following sections we restrict ourselves to the case of the free massive field for which V = ~m2¢2. Having found the momentum P, it is sufficient to perform the substitution of the correspondence principle and to replace P by P -+ -iV. The eigenmodes Uk(X) with eigenvalue k of the momentum operator thus satisfy -iVUk (x) = kUk (x). They are plane waves Uk(X) = Nkeik.x,

N k being a normalization factor to be determined later. It is now sufficient to expand the field on the basis of these eigenstates as

(2.79) Since ¢ satisfies the Klein- Gordon equation (2.60), we find that the coefficients ak(t) evolve as adt) + (k 2 + m 2 )ak(t) = 0, where we used the fact that V 2Uk (x) of this equation is simply

=

2 - k uk (x). Setting

wk = k 2 + m 2, the solution

with b;" = a - k since ¢ E lR. After some algebraic manipulations, and using the relations between the coefficients of the expansion, we find a more explicit four-dimensional expression, that is (2.80) where we recall that k . x == k . x - wkt. From (2.80) and from the invariant measure (2.74) we see that the field expansion, even though it was realized on the basis of three-dimensional states, is invariant, i.e. that ¢ is effectively a Lorentz scalar, as long as we choose the normalization (2.81) Moreover, this relation allows us to recover the usual commutation relations after quantization. Note that the conjugate momentum of the field is simply (2 .82) which we obtain in a similar manner as before. The factor ( - iWk) in the integrand of (2.82) implies that the latter is not explicitly a Lorentz scalar. This is not a problem since only the field should absolutely satisfy this property.

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Overview of particle physics and the Standard Model

2.2.2.2

Quantization

The field and its conjugate momentum can now be 'promoted' to the status of oper tors , which can be done by replacing the coefficients ak and a"k by the annihilation a creation operators, respectively, ak and The field Poisson brackets then becom the equal time commutations relations. If we impose

al .

(2.8 and all others to vanish, i. e. the usual commutation relations , we indeed obtain t canonical relations for the field

[¢(X,t),7T(X',t) ] =i5(3) (x-x') ,

(2 .8

a computation that is recommended to be performed explicitly. The relations (2.8 and (2.84) can only be respected if the normalization of the field modes is given (2.81) . The eigenmodes of the field operator are now defined by

u k (x , t)

=

N k ei( k .x- w /c t) ,

(2.8

so that the field operator is expressed as

(2.8

This relation can be inverted to express the creation and annihilation operators as function of the field , namely

(2.8 f--7

where the derivative 8 is defined by (2.69) . This analysis is then complet e as soon as we know the Hamiltonian, which is giv by

(2.8 We then get

(2.8

where we have used the commutation relation (2 .83) and defined Nk as the 'number particles in the state with momentum k ' operator, the significance of which will le us to the definition of the Fock space associated with this theory. A remark concerning (2.89) is necessary at this point. The second term, propo tional to 5(3) (0 ), is clearly divergent , even in the case where the momentum space finite, i. e. if there exists a cutoff k m ax above which one can no longer perform the com putation. In field theory, as discussed here, this infinite energy is not embarrassing

Canonical decompositions

93

it has no physical meaning since in practice only differences of energy are measurable. In practice, we define a normal ordering that automatically puts the creation operators on the left and the annihilation ones on the right, or by 'renormalizing ' the energy by simply suppressing the vacuum energy obtained this way. This possibility is unfortunately no longer an option as soon as the theory is coupled to gravity. The vacuum energy then assumes all its meaning and acts as a cosmological constant , which is in any case considerably more significant than the one we measure (Chapter 12). It is, in the current state of knowledge, the most serious problem that theoretical cosmology has to confront, and it comes from quantum fi eld theory!

2.2.2.3

Fock space

The complete quantum theory is obtained once we know the state of the particles. For that, we st art by defining the vacuum 10) as being the state annihilated by all the annihilation operators, i. e. aklO) = 0 'tIk E JR3 . We consider, furthermore, that this state is normalized, namely (010) = 1. A state with one particle is then constructed from this vacuum state by applying a creation operator to it. We thus have

(2.90)

Ik ) == 0,110) , normalized by

(klk' ) = (ola k al,10) =

5(3)

(k - k') ,

and we construct in this way a complete space of states, each state having n( k) particles in the state of momentum k . The definition (2.90) shows that a state with n(k) is an eigenstate of the operator Nk , since (2.91) It follows that any state describing a set of particles will be an element of

called a Fo ck space and having an unaccountable infinity of states. Finally, note that starting from the vacuum state 10) , it is possible to produce a particle at the point x simply by applying the field operator itself

lx,t ) =

q} (x , t) IO) =

J

d3k e- ik . x (271")3/2 J2Wk Ik),

(2.92)

which appears essentially as the Fourier transform of the one particle st ate with given momentum, hence its interpretation. Note that this equation uses the operator q) to generate a particle, which plays no important role here since the field is real and hence cfyt = cfy, but in the more general case of a complex field , the distinction is important.

2.2.2.4

Spherical waves exp ansion

The previous expansion is only a special case of the expansion in the basis of the momentum eigenstates expressed in Cartesian coordinates (x , y, z) . It is also possible

94

Overview of particle physics and the Standard Model

to perform this expansion in spherical coordinates (r, 0, rp) using spherical harmon In that case, we split the Laplacian into radial and angular parts t. = t.r + r - 2 t with t.r = 8; + (2 /r)8r and t.11 = 8~ + sin 2 08~. We express the eigenvalues

JP2

the Laplacian using the amplitude p == and two integer numbers £ E Na m E [-£, +£j . The scalar field ¢(x, t) can then be decomposed into the eigenmod ¢pem(x), of the Laplacian (2. which are given by

¢pem(x , t)

=

upe(r)Yem (0, rp).

Since the spherical harmonics are solutions of (B.8), (2.93) leads to an ordin differential equation for the radial part upe(r) , namely

2

d [ dr2

2d

+ ~ dr -

£(£+ 1)

r2

2]

(2.

+ P upe = 0,

the solution of which is a spherical Bessel function upe(r) ex: je(pr), (Appendix B). T normalization relations of the spherical harmonics (B.15) together with those of spherical Bessel functions (B .51) show that choosing

the modes of the scalar field are normalized , i.e.

(2 . Also note t hat with these convent ions, we have the completeness relation

roo

in

o

+e

L L 00

dp

Upe(r)Upe(r')y£~(O , rp) Yfm(e', rp')

= 5(3)

(x - x').

(2.

e=o m=-e

The scalar field operator is now expressed as

(2.

where the operators iipfm depend explicitly on time. Inserting the relation (2.97) the Klein- Gordon equation (2.60) , we get

22 (:t

w;

+ w~)

iipem(t)

=

0,

(2.

where = p2 + m 2 for each independent mode. The relation (2.98) shows that creation and annihilation operators, which could a priori depend on the three numb p , £ and m , have a time dependence that is only a function of the magnitude p of

Canonical decompositions

95

momentum. The solutions of equation (2.98) are complex exponentials, so that the total field takes the form

1 z= 00

¢(x, t) =

dp

o

Npupe(r) [Y£m(B,
+ Yi::n(B,
(2 .99)

e,m

and the momentum conjugate

(2.100) The remaining thing to do is to use the commutation relations (2.84) at equal time and to impose the same normalization as previously, namely Np = (2wp) ~ 1 / 2, to get the commutation relations for the creation and annihilation operators (2.101 )

The link between the Cartesian and spherical representations is obtained by expanding the plane wave in the basis of spherical waves (B.21). By identifying the coefficients of the term e ~ iwpt in both expansions (2.86) and (2.99), we obtain (2.102) where (Bp,
The complex scalar field

The generalization to the case of a complex scalar field, ¢ E
¢ = 3le(¢)

+ i~m(¢) ==

1

yf2(¢1

+ i¢2),

and we assume t hat the real and imaginary parts are independent. In a perfectly equivalent way, and this is the point of view we will adopt, we can also consider ¢ and ¢* as two independent fields. The Lagrangian obtained in both cases should be real, so that for a free massive field (2.103) Note the absence of the factor ~ in the mass term, compared with the Lagrangian (2 .61 ). This is due to the fact that since the two degrees of freedom of the field are

96

Overview of particle physics and the Standard Model

independent, the variation with respect to ¢ or ¢* does not give the factor 2 of real scalar field. The two conjugate momentums are

7T =

~~

= ¢* and

7T* = : ; = ¢,

which are indeed complex conjugates of one another and are hence independent. theory describing a complex scalar field is simply that of two real scalar fields of same mass, related to each other by the fact that they are the real and imagi parts of a complex number. The energy-momentum tensor is obtained by summing (2.31) over the indepen degrees of freedom

which gives

(2.

We find again the same result as the one obtained using the definition that comes f the variation with respect to the metric. The commutation relations at equal time between the field operators and t conjugate momentum are now

(2. The mode expansion is of t he form

(2. where we have introduced two different sets of operators, complex. Their commutation relations are now

ELk

and

bk , since the fie

(2.

all the other commutators vanish. With the normal ordering for the operators discussed in the case of a real sc field, the Hamiltonian takes a form equivalent to (2.89), that is

(2 .

which shows t hat a complex scalar field can be interpreted as a collection of kinds of different particles, but with the same mass, corresponding to the two of operators ELk and bk . The Fock space is then the tensor product of the two F spaces generated by these operators, with one mutual state, which is the vacuum annihilated by both ELk and bk .

Canonical decompositions

2.2.3.1

97

M axwell or Praca field

The Lagrangian of a massive complex vector field is obtained by generalizing Proca's Lagrangian (2. 71 ) as LP roca

=

-~FI.wpi.w + M~A:A" ,

(2.109)

where A" now has four complex components. As for the scalar field , the mass term no longer has the factor ~ characteristic of the real field. These types of fields can describe charged vector particles, such as t he W± bosons mediating the weak inter action. The expansion into creation and annihilation operators is analogous to (2.106), up to the fact that due to the gauge invariance for the Maxwell case with vanishing mass , or for the gauge constraint for the massive case of Proca [see the discussion below (2.71)], t he number of degrees of freedom is not equal to the number of field components (i.e. four ), but three for the massive field and two otherwise. This point leads to subtleties in the quantization that are discussed in Refs. [7- 9] to which we direct the reader. 2.2.3.2

Internal symmetries

The Lagrangian (2. 103) not only represents two independent scalar fields of t he same mass. Indeed, these two fields are the components of a complex number and the Lagrangian is thus invariant under the phase shift and

(2.110)

where ex E IR is an arbitrary constant. Expressed in terms of t he real and imaginary parts of ¢, the phase shift (2.110) corresponds to a global internal rotation. It is this invariance that imposes the stat es 'a' and ' b' to have the same mass. From Ncether's theorem , the existence of such an invariance should translate into the existence of a conserved quantity. To determine it let us work in the limit ex « 1, such that we can expand t he exponential in (2.110), leading to the infinitesimal transformation ¢' = (1 + iex)¢. This transformation being independent of the spatial coordinates, x, we have i" = 0 in (2 .29) , and the current

J

" _ . 8L . J. - 2 88,,¢ 'f'

-

. 8L ..J..* 88,,¢* 'f' ,

2

(2.111)

is conserved since in that case, we have o¢ = iex¢ and o¢* = -iex¢* (we have removed the irrelevant normalisation factor ex in this definition). The conserved 'charge ' is obtained by direct integration, i.e. (2. 112) After quantization , this charge leads to the operator (2. 11 3)

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Overview of particle physics and the Standard Model

which is indeed time independent since its equation of motion , using the form (2.1 of the Hamiltonian , is

dQ dt

. [,Q, He , ]=

= -~

O.

Equation (2.113) can be easily interpreted by saying that particles associated w the operators a and b are particles of opposite charges; in reality they are antipartic of one another. The complex scalar field is only an example of internal symmetry, in this case of invariance under transformations of a U( l) group. The associated charge is, in gene called hypercharge, and corresponds to the electric charge only in the case where U(l) group is that of electromagnetism. The different existing particles of which will make an inventory in Section 2.3, indicate the existence of more symmetries, ba on larger symmetry groups.

2.2.3.3 Fermions and anticommutation relations

It is possible to expand the Dirac field (2.63) in a similar way as in the exp sion (2 .106) , with an important difference however: the field operators, and hen those associated to the annihilation and creation of particles, must respect antico mutation relations instead of commutation relations of the type (2.83) or (2.84). T consist ency problem related to this point is made explicit in Refs. [7,8], and is beyo the scope of this presentation. It will be sufficient to know that for fermions we ne to replace the Poisson brackets by anticommutation relations

and

(2. 1

where we used the fact that , from the Lagrangian (2.63), the conjugate momentum the spinor is given by

=

7r7j; -

O£Dirae _ .

o1/J

-

'"I,t

~'f/

.

(2.11

Note that these anticommutators, similarly to the commutators for bosonic fiel are taken at equal time. At this point, it is interesting to note that if we place oursel in t he context of field t heory at finite temperature (thermal field theory, see e Ref. [10]) , then these anticommutation relations are necessary so that the fermio satisfy the Pauli exclusion principle; moreover, they lead to the correct statisti distribution law, i. e. the Fermi- Dirac law (just as the commutation relations for bosons lead to the Bose- Einstein distribution) .

2.3

Classification and properties of the elementary particles

The general properties of elementary particles are gathered in Table 2.1 (see Ref. [11 When one looks at these properties, one finds structures, such as illustrated , for stance, in Fig. 2.1 showing the particular case of baryons of spin ~ and of posit parity (parity is defined later , here it is only an illustration).

Classification and properties of the elementary particles I

I

I

I

t,-

t, 0

t,+

t, ++

or-

0

0

0

o

2: -

2: 0

2:+

-1

I-

0

0

0

-2

r--

0

-3

-

~

99

-

1236 MeV

- 1385 MeV

so - 1532 MeV

0

Q-

0 I

-1

- 1672 MeV I

o

I

I

1

2

Fig. 2.1 An example of the structure observed between the elementary particles : the baryons of spin ~ and of positive parity. Table 2.1 Nomenclature of the elementary particles. The families based on quarks (q) have several hundreds of particles identified so far , whereas the leptons and the gauge fields are here presented in an exhaustive way in the context of the standard model. The gauge fields are written in the following way: I is the photon, the ZO and W ± bosons are the mediators of the weak interaction, and ga represent the 8 gluons of the strong interaction . Spin

Name

Examples

B aryons = qqq Ha drons

{

Mesons = qij

n

7r°'±, Ko'± ,J/'Ij;, Do , Bo , 1'/ , '

Leptons Gauge fields

This kind of diagram has made it possible to understand the existence of a very precise structure, the history of which is discussed in Ref. [12]. In what follows, we will only indicate the result of numerous years of experimental research that led to the so-called 'standard ' model of particle physics, its theory is the subject of Section 2.6.

2.3.0.4

Exchange bosons

The known particles come m three forms. First, we find the bosons of unit spin, mediating fundamental interactions: they are the gauge bosons, namely the photon " which mediates the electromagnetic interaction, the eight gluons ga, which mediate the

100

Overview of particle physics and the Standard Model

strong nuclear interaction , and the vectors ZO and W ± . The latter have the proper of being massive and are hence described by Proca's Lagrangian (2.109) , at least long as we ignore their interactions. The vectors ZO and W ± allow the weak nucle interaction to propagate, W ± are in addition electrically charged . According to the current theoretical ideas, to be complete, one should add to th table the graviton, hl-'v, massless 'particle' (in the framework of a quantum theo of gravity equivalent to that of the other interactions) of spin 2 responsible for t gravitational interaction (Chapter 1). In the absence of any satisfactory quantu theory of gravity, we do not consider at the moment the graviton as an elementa particle belonging to the standard model.

2.3. 0.5

Leptons

Of spin ~ , t he leptons, among which t he electron is the best known , are of negati electrical charge (e - , p,- and T- ) or vanishing (v e , vI-' and VT which are the associat neutrinos). The evolution of these particles is dict at ed by the Dirac equation, coupl or not to electromagnetism , depending on the case. They are also subject to the we interaction and can be grouped into three 'doublets' and

(::)

which have exactly the same properties with respect to this weak interaction. Actual the only characteristic that allows us to distinguish between these different double is the mass of the particles that constitute them. Moreover, among them, only t doublet of the electron is stable, the others, of higher mass, are subject to the weak a electromagnetic interactions, and can decay into, for instance, electrons and electro neutrinos. For a long time, the neutrinos have been considered to have a vanishing mass, b some recent experiments and measurements (see Ref. [2] of Chapter 9) have show that there must be a mixing between t he different species of neutrinos, which is on possible for massive particles.

2.3.0.6

Hadrons

Only one category of particles , albeit t he most numerous one, is subj ect to the stro interaction: it is the hadrons. They can be of half-integer spin, such as the proton a the neutron , and then belong to the category of the baryons. The mesons, of integ or vanishing spin, count among them , for instance, the 71"0 and its charged partne 71"±, which are described by the Klein- Gordon equation for the real scalar fields ( the 71"0) or the complex (for the 71"±). The surprising profusion of the hadrons, as well as the observed properties symmetry between the different elements of the families sharing some properties (sp and parity, for instance, as in the illustration of Fig. 2.1) have led to the understandi that these particles, unlike what we currently think of gauge bosons and lepton are not elementary particles, but are constit uted of 'quarks' a The latter are for t

3 According to the legend , this word was proposed by M urray C ell- Mann (Nobel Prize 1969) w would h ave read it in the book ' Finnegan's Wake ' by J ames J oyce.

Internal symmetries

101

moment seen as effectively elementary, of spin ~, and of electric charge + ~ for the quarks u, c and t, and for the quarks d, sand b. We can also group them into doublets for the weak interaction, with a similar structure to the one for the leptons , more precisely

-1

a nd This description, which is satisfying from the theoretical point of view, currently does not have any generally accepted explanation (even if many models exist) . The names of the quarks themselves come from this structure: at the time where only the first family was experimentally accessible, the chosen names were simply up (u) and down (d). Then , a third quark was needed to b e introduced, which was considered as strange (s). The second family was then completed by the charm (c). In the same philosophy, when the presence of the fifth quark started to appear, the name of beauty (b) was proposed, soon replaced by bottom, allowing the first family to be reproduced with the top (t). Just as for the leptons, the only difference between the quark families comes from their masses, as well as the fact that only the members of the first family are stable. The ones from the second and third families decay much faster than their leptonic counterparts since they do so through the strong interaction instead of the weak interaction, the latter having a much weaker amplitude , as its name suggests. All these particles have been produced in accelerators, but have never b een observed alone, which can be explained by the notion of confinement of strong interactions. This is how it is only possible to observe states containing three quarks , baryonic states of half-integer spin, and states composed of one quark and one antiquark, of integer spin: a meson. This point is not yet completely understood theoretically. Table 2.1 summarizes the nomenclature used to describe the diversity of elementary particles. The origin of the mass of all these particles, together with the important disparities observed between them , still pose many questions that are not all yet resolved. As we can see, the known particles reflect, or seem to reflect, some elements of symmetry. These are described by group theory, to which we now turn.

2.4

Internal symmetries

Contrary to leptons , which all appear in doublets , hadrons are found in numerous forms , suggesting that they are not elem entary. In reality, using a much smaller number of elementary particles, in this case quarks , and placing them in a relevant way in group representations, it is actually possible to combine them so that the other representations are perceived as new elementary particles. The role of group theory is precisely to lead us to this conclusion.

2.4.1

2.4.1.1

Group theory

Definitions

A group g is defined as a collection of objects {g;} , where the index i is not necessarily discrete , satisfying the following rules:

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Overview of particle physics and the Standard Model

- There exists a composition law, denoted by 0, which can combine any two eleme of the group , the resulting combination being an element of the group: if gl E and g2 E g, then g3 = gl 0 g2 E g. Furthermore, this law is associative, gl 0 (g2 0 g3) = (gl 0 g2) 0 g3 = gl 0 g2 0 g3· - There exists a neutral element, denoted by 1, such that Vg E g, l og = go 1 = - Each element 9 Eg is associated with an inverse element , denoted by g - l, sat fying go g- l = g- l 0 9 = l.

Note that the law 0 in not necessarily commutative: when it is, i.e. if Vgl , g2 E gl 0 g2 = g2 0 gl, the group is then said to be Abelian. In the opposite case, the gro is usually said to be non-Abelian. The simplest group one can think of, besides the trivial group that only conta one element, is the discrete group Z2 that contains two elements, denoted by 1 (wh is the identity) and -1 with the ordinary multiplication law. This is an Abelian gro for which each element is its own inverse. For discrete groups, one can construct t complete table of all possible operations. 2.4.1.2

Lie Groups

Another well-known example of a group is that of translations that transforms a po in space x into x + a. In this case, any element of the group can be directly associat with a translation vector a: the composition law is vector addition, the inverse eleme of a is simply - a, and the identity is the translation by the null vector o. It is ag an Abelian group. Such a group contains an infinite number of elements, which can all b e identified the knowledge of three real numbers (the three coordinates of the translation vecto More generally, if it is possible to describe the elements of a group by a set of numb varying in a continuous interval (but not necessarily infinite), it is then a Lie grou A Lie group is also a manifold. The best-known non-Abelian group is probably that of rotations in three dime sions. Each rotation is determined by, for instance, the three Euler angles , or equiv lently, a unit vector (rotation axis) and the angle of rotation around this rotation ax The composition of two rotations is equivalent to a third rotation, which makes i group (since a rotation can always be inverted by performing the same rotation in t other way, and so a rotation of angle zero is the identity), but the order in which t rotations are performed is crucial and this group is not Abelian. This group, denot by SO(3) , is identical to the group formed by the orthogonal 3 x 3 matrices with u determinant. We can generalize it to the group SO(n) , which contains the orthogo n x n matrices with unit determinant. Each group is associated with an invariance. For instance , in the case of a rotatio the norm of a vector on which the rotation has been performed remains unchang during the transformation. A rotation in a generalized Minkowski space, i.e. havi p spatial dimensions and q time-like dimensions, must keep the form up.up. inva ant; it contains not only rotations in the p-dimensional space, but also pure Loren transformations (some ' boosts') . This group, generalizing SO(n) for arbitrary met signatures, describe p space-like coordinates and q time-like ones, and is denoted SO(p , q). The so-called Lorentz group, is then simply SO(3, 1). The translation gro

Internal symmetries

103

can be associated with the Lorentz group to form the Poincare group on which special relativity is based (see Chapter 1) . 2.4.2

2.4.2.1

Generators

Infinitesimal transformations

An infinitesimal transformation, T€, of a Lie group is a transformation close to the identity and hence can always be written as T€ '::' 1 + iEaAa. This defines the generators Aa , which are Hermitian matrices and depend on the nature of the performed transformation. We suppose here that the Lie group is described by p parameters, i. e. a = 1" " , po A finite transformation can be obtained from an infinitesimal transformation by exponent iation. Such a fini te transformation can always be written as the composition of a large number, tending to infinity, of infinitesimal transformations. If a represent the p necessary numbers for the finite transformation, then we can always write E = a lN, where N » 1, and performing N transformations will indeed lead to the finite transformation. In other words, we will have (2.117)

which is the general form of a finite transformation of the group. 2.4.2.2

Lie Algebra

In order for TOt the property

0

Tf3 to be a group transformation, T-y for instance, one should have

(2.118)

For infinitesimal transformations this gives to second order in the expansion of the exponentials eiOt Aeif3

A

= 1 + i (aa + f3a) Aa - ~ (aaab + f3af3b + 2a af3b) Aa Ab,

the logarithm of which, a lso expanded to second order [using the relation In(l E - ~ E2 + ... ], should be proportional to Aa . We find explicitly

a+

i (a a + f3a) A

~aaf3b

a [A , Ab]

=

+ E)

=

hc Ac .

This equality is in general only satisfied if the commutator of the>. satisfies (2.119)

This relation defines the algebr a associated with the Lie group , i.e. the set of operators of the form aa Aa. The numbers rabe are the structure constants of the group , and the Aa form its algebra. Note that since the generators are Hermitian operators, the structure constants are real numbers .

104

Overview of particle physics and the Standard Model

2.4.2.3

Representations

In order to be able to classify the particles in terms of symmetry groups , we need sort them depending on the representations of the group of the considered invarian An n-dimensional representation being a set {n a} of n X n matrices satisfying t commutation relations (2.119) of the algebra, the fields that describe the particles o given representation will be gathered in multiplets q. = {iP a}, a = 1,' . . ,n on whi the action of the group is

(2. 12

It is usual to say that iPa transforms under the representation n a , or more direct that iPa is 'in the representation n a'. If two representations are related to each oth by a similarity transformation , i. e.

na = unau t ,

where U is unitary (U t = U - 1 ) , then we say that they are equivalent. In other wor for all practical purposes, it is the same representation , such that the similarity tran formation does not provide any complementary information. If there is an invaria subspace in the representation , i. e. if the action of any element of the group on t vectors of this subspace leads to an element of the same subspace, then the represe tation is said to be reducible. In the opposite case, it is called irreducible. Only t irreducible representations are used to classify the particles in the representations the invariance groups. Moreover , if {na} forms a representation, then so does { _ (na)*}, which is cal the 'conjugate representation ' , [one can easily be convinced that these matrices a respect the algebra closure relation (2.119) since the structure constants of the gro are real] . If a representation and its conjugate are equivalent then we are deali with a so-called real representation. Otherwise, we say the representation is comple In order for a group to be able to potentially reproduce the internal symmetries particle physics, it is often required that it has some complex representations in whi fermions can be placed in a simple way. The representations that are the most relevant for quantum physics are those th preserve the norm defined by liP I2 == iP tiP,

of the multiplet iP . In order for the norm to be conserved during an arbitrary tran formation n a , we see that we need to have

(iP)tiP = iP'tiP' = (iP)t(TaYTa iP ,

for all transformations Ta. We should therefore require that (Ta)tTa = 1, i. e. th it is a unitary transformation. A representation satisfying this condition is a unita representation. Finally, the adjoint representation of a Lie group is formed using its structu constants. It is a representation of the same dimension as the order of the group , which the matrix elements of n a are

(2 .12

Internal symmetries

2.4.2.4

105

Caftan classification

Besides the space-time symmetries (Poincare group) , the invariance groups that are useful from the point of view of the classification of particles are the compact ones, i.e. for which the parameters vary on a finite interval, as if we had to deal with generalized rotations in an internal space (and in many cases, it is actually precisely the required invariance). Moreover, we wish to classify the particles along 'fundament al' invariances, i.e. with no substructure (we will see later that only one coupling constant comes into play in these groups, which makes them even more interesting). This is why we are interested in simple groups: if H is a subgroup of 9 and if Vg E 9 and Vh E H , the element go h 0 g - l is still in H, then H is an invariant subgroup of g. 9 is simple if there is no invariant subgroup in g, and semi-simple if it does not contain any Abelian invariant subgroup. One can show that a compact and semi-simple group is thus either forced to be simple or is the direct product of two or more simple groups. Using this philosophy, studying simple Lie groups is therefore sufficient to classify all semi-simple compact groups . Note also that the groups with structure S x [U(l)]n with S a semi-simple group and n an arbitrary integer, which are called reducible groups, have representations that can all be decomposed as a product of irreducible representations; t his property makes them especially interesting for the classification of elementary particles. 8ince the simple Lie groups are also subj ect to the constraint (2.119), they can be classified in terms of their structure constants . This classification was performed by E. Cartan at the beginning of the twentieth century. He proved that the simple Lie groups could be classified into 4 large categories, namely 80(2n) , 80(2n + 1), 8U(n + I) , and 8p(2n), to which one should add the so-called exceptional groups , which can not be classified into any of these categories, called G 2 , F 4, E 6 , E7 and Es· For the four infinite sequences, n is the rank of the group, i. e. the maximal number of generators that can be simultaneously diagonalised. For the exceptional groups, the rank is given by the index in the notation. For each group , the order, i.e. the number of generators, is indicated in Table 2.2.

Table 2.2 Order (number of generators) or dimension of the simple Lie groups of rank n according to the Cartan classification (the index of the exceptional groups is the rank). Group SU(n + 1) SO(2n + 1) Sp(2n) SO(2n)

G2

F4 E6 E7 Es

N generators

(n (n (n (n

~ ~

~ ~

1) 2) 3) 4)

n(n+2) n(2n + 1) n(2n + 1) n(2n -1) 14 52 78 133 248

106

Overview of particle physics and the Standard Model

The groups of the four infinite sequences can be understood using their fundamenta representation, i.e. the one of smallest dimension (from which, by direct product, a the other representations can be constructed) . The set of all complex unitary matrice of dimension n x n forms the group U(n) ('U' for 'unitary'), the subgroup restricted t the unit determinant matrices is 8U(n) ('8' for 'special'). It is the set of transformation that leave invariant the inner product of two vectors of a complex space of dimensio n. The transformations of this group leave invariant, among other things , the norm o the complex vectors and can be used in quantum mechanics to assure the conservatio of probabilities. The sequence 80(n) is constructed from the matrices that leave invariant the inne product of two real vectors. They are orthogonal matrices (hence the '0') with uni determinant (special) of dimension n x n. Finally, the group 8p(2n) regroups all th matrices M of dimension 2n x 2n that leave invariant the antisymmetric matrix

o

1

- 10

s=

o

1

-10

o

1

- 10

These matrices therefore satisfy MT S M = S, where MT is the transpose matrix of M Thus , the antisymmetric and quadratic form y T Sx for two real vectors of dimension 2 is conserved. The matrices M defined in this way are called the symplectic matrices thus the name of the group. The only transformations that allow us to conserve probabilities in quantum me chanics are those that are unitary. Can we hence deduce that only the groups of th type 8U (n) can playa role in physics, and that hence all theories should be based o such a group? Unfortunately, the answer to this question is negative: some representa tions of the groups classified by Cartan are unitary, as well as respecting the rules o their classification. We can therefore exclude a group as a possible internal invarianc only if we can show that it has no unitary representation. This is , for instance, th case of 80(11) , which is hence excluded as a grand unification group candidate. Thi is why it is indispensable to study in detail the representations of the possible Li groups.

2.5

Symmetry breaking

Not all the symmetries existing in Nature are manifest , and some underlying ones ar not immediately visible at low energy. This is , for instance, the case for the doublet of the type (e - , ve). If these particles really belong to the same representation of th gauge group of the weak interaction, why do they not have the same characteristic as well (mass , electric charge, for instance)? The answer to this question relies on th mechanism that enables us to break symmetries, and in particular the gauged ones.

Symmetry breaking

2.5.1

107

Gauge group

We are generally confronted with 'gauge' invariances, i.e. local invariances. This leads to the existence of a gauge boson, through which some interactions are performed: the presence of a gauged (local) symmetry in particle physics, as in gravity, implies the existence of an interaction.

2.5.1.1

Local symmetries

The simplest illustration of a local symmetry is given by electromagnetism. Its action is invariant under the transformation of the group U(l), which leads to the prediction of the existence of the photon, i. e. the term (1.94) of Chapter 1 in the total action. Consider a complex scalar field , cp(xl-') E C, invariant under the phase transformations cp - 7 cp' = e ic 4>C
(0l-'cp)*ol-'cp

-7

(ol-'cp')*ol-'cp'

= =

[(01-' - iC¢0l-'a) cp*] [(01-' + ic¢ol-'a) cp] (2.122) (0l-'cp)*ol-'cp + ic¢JI-'0l-'a + c~lcpI20I-'aol-'a ,

where the conserved current JI-' = -i(cp*ol-' cp - cpol-'cp*) is given by (2.111). We can note here that this term, proportional to the conserved current, leads, after integration by parts, to a surface term in the action and thus has no physical effects. Despite this fact, the action is still not invariant since the last term is remaining.

2.5.1.2

Gauge boson

The form (2 .122) suggests the replacement of the partial derivative by a gauge covariant derivative by introducing a new vector field GI-'. If we define (2.123) with 9 an arbitrary coupling constant, we find that this term transforms as

We can therefore simply impose GI-' to transform as

GI-'

-7

G'I-'

=

GI-' -

~ol-'a 9 ,

(2.124)

in order for the action to be again invariant , whatever the value of c¢. Since the action (l.94) with the Minkowski metric T) c
108

Overview of particle physics and the Standard Model

where GI-'V = ol-'CV - ovCI-" and we have generalized the potential of the scalar field for it to be an a priori arbitrary function of the field norm I¢I == j[¢f2. The action (2.125) is the basis of the scalar electrodynamics, an example of gauge theory. We should note that (2.125) does not have any mass term for the vector field CI-'. This comes from the fact that such a term, described by the Lagrangian of Proca (2.71) , is not invariant under the transformation (2.124). The action (2.125) may be generalized to the case where the gauge group is not Abelian, this is what is done in the context of the standard model of Section 2.6 Note already here that the gauge bosons introduced in the framework of non-Abelian theories can neither be massive, for the same reason that a mass term a la Proca is not gauge invariant.

2.5.1.3

Fermions

In a similar way, the action for the fermions (2.67) is made invariant under the local transformation 'Ij; ----> 'Ij;' = eicrcx'lj; by replacing the partial derivative by the gauge covariant derivative (2.126)

VIle can also see that an extra coupling term with the scalar field can be introduced. Finally, the action for the fermions, can be written in a general form as

(2.127)

with f an arbitrary constant, which, in addition of (2.125), allows us to describe a theory containing a fermion and a complex scalar field coupled via an intermediate vector. We can also see that the fermion can have a purely massive term in addition to its coupling with the scalar field , while respecting the invariance. 2.5.2

Higgs mechanism

The Higgs mechanism allows us to build a theory invariant under the transformations of a given group, while this invariance is not manifest. In other words, it allows us to impose an invariance with some properties to theories that do not seem to have it, as is, for instance, the case for theories having massive gauge bosons.

2.5.2.1

Higgs potential

We consider a complex scalar field ¢ and a gauge field AI-', the dynamics of which is described by the action (2. 125). For technical reasons that we will not discuss here, (of renormalization, see, for instance, Ref. [7]), the potential V(I ¢I ) cannot introduce powers of I¢I higher t han 4. We generally choose a quadratic term, i.e. we postulate a mass for the scalar field, and a quartic term, in AI¢1 4 . In the case of the Higgs mechanism, the chosen potential is given by

VHiggs (¢) = A ( I¢I 2 - Ti 2)2 ,

(2.128)

depicted in Fig. 2.2. The parameter Ti, having the dimension of a mass, is called a 'vacuum expectation value' (VEV).

Symmetry breaking

109

!Jle (1/1 / 17)

'i5'rn

(1/1 /17)

Fig. 2.2 The Higgs potential as a fun ction of the components (real and imaginary parts) of the Higgs field . The circle 11>12 = TJ2 indicates the set of possible minima.

For the theory to make sense, one should first define the vacuum. It can be obtained , at the classical level, by demanding the energy, i.e. the Hamiltonian, to be a minimum. A simple generalization of (2.88) gives in that case H(x , t)

= ~ [1>2 + (V¢) 2 + 2V(¢)]

,

(2.129)

which is the sum of two positive-definite terms and of the potential. As a consequence, the minimal energy is reached when the field is invariant with respect to the transformation of space and time, so that 1> = V ¢ = O. Furthermore, the potential should also be minimized , which leads to

(2.130) As illustrated in Fig. 2.2 , one of the solutions, here ¢ = 0, represents a local maximum of the potential, thus it is an unstable state of the theory. The second solution, i¢i = T), on the other hand, is the absolute minimum of the potential. This is the value the field will hence naturally take and that describes the vacuum st ate of the theory (thus the name of t he parameter T)) .

2.5.2.2 Goldstone boson The expansion of the field ¢ around its VEV is

¢ = [T) +

~h(X, t)]

ei8(X,t )/V2rJ,

(2.131)

110

Overview of particle physics and the Standard Model

e

where h and are two real fields. Noting be written as ,

mh

==

2,j),'rJ , the potential for h and

e

(2 .

and is independent of e. This can easily be understood by examining the poten the fluctuations h and e around the expectation value 'rJ of the field correspon fluctuations either along the radial direction (h) , thus changing the energy of configuration , or along the circle of the minima (e) , which by definition do not any energy. The field h is called the Higgs field , whereas e is the Goldstone boson

2.5.2. 3

Masses of the bosons and fermions

The degree of freedom corresponding to the Goldstone boson is not physical. Actu it is always possible to choose a gauge in which it is suppressed , since the only p where it comes in is in the kinematic term of the field cp, which can be expanded

(2.

Choosing the gauge in which CI-' ----> CI-' + (0l-' e) / (V2'rJ9 Cf ) , the field e comple disappears. After expanding the kinematic part (2.133) ofthe scalar field cp, one can see app ing a term similar t o Proca's one (2.70) for the vector field C w This implies that field has gained a mass during this process. After identification, we find Me = V2g The appear ance of a massive vector field can be explained by the loss of the G stone boson. Its corresp onding degree of freedom cannot have completely disappea and it has actually been completely absorbed ('eat en ', as it is sometimes writ by the gauge field. A massless vector field has indeed only two degrees of freed corresponding in t he classical theory to t he two t ransverse polarization modes ther circular or linear. On the other hand , a massive vector field evolves such its angular moment um has a vanishing projection , and therefore has three poss modes. The coupling of the scalar cp to the fermion in (2.127) leads to a modification of fermion 's mass, which is increased (or decreased , depending on the sign of the coup const ant f), by 6.mf = i'rJ. Thus, if the fermion was by symmetry initially mass as is the case in t he electroweak and strong st andard model, it will t hus acqui mass, in the same way as the gauge field. This is why fermions of extremely diffe masses can be placed in the same multiplet . We will see later , in the context of st andard model of the electroweak and strong interactions , how this process is app in a precise case, and is checked experimentally. 2.5.3

Non-Abelian case

T he Higgs mechanism can easily be generalized to an arbitrary group Q, which coul non-Abelian , but t hat we suppose is simple. We now consider a field in a representa

Symmetry breaking

111

of dimension n , i.e. a column vector ¢ composed of n real scalar fields ¢a, with a = 1, ... , n. The VEV is now a vector

~~

(Oi¢ iO)

~

C) ·

(2.134)

with 10) the vacuum st at e of the scalar fields. The individual VEVs , 7]a = (OI ¢a IO), satisfy L a7]; = 7]2 . The potential of ¢ is always of the form (2.128) , with I¢I = L a¢; , so that once again we have oV(¢) I =0

O¢a

=TJ

'

and we define the quantum fields ¢a == ¢a - 7]a. The mass term then t akes the general form 1 02V ' , Lm ass

= - "2 O¢aO¢b¢a ¢b·

If we perform a global infinitesimal transformation with paramet er a c tential transforms as V(¢) --> V (¢) + SV(¢) , with

«

1, the po-

OV .oV C b SV (¢) = o¢a S¢a = Z o¢a (ac R) a ¢b, where we assume that ¢ transforms under the given representation with the matrices

RC For the pot ential to be invariant under the previous transformation, we need OV (¢ ) = 0, implying that

oV (Rc)b A. = 0 O¢a a 'f' b . Differentiating with respect to ¢d for ¢ = ry (for which t he first partial derivative of the potential cancels), leads to

I

0 2V o¢ o¢ a

d

=TJ

(RCry)a = O.

There are now two possible ways to make these terms vanish , according to whether or not (R Cry) a vanishes. The set of the generators satisfying (RCry )a = 0 forms a subgroup 'H of 9 under which the vacuum is still invariant. Thus, the scheme of the symmetry breaking is 9 --> H. The expansion (2.131) around the VEV can be generalized t o 7]1

+ hI (x) (2.1 35)

o

112

Overview of particle physics and the Standard Model

assuming that only the p first values of
so that

(2 .

o

which defines the p physical Higgs fields of the theory. Under a general transformation with parameters aa, the gauge bosons Cal" tr form as the generalization of (2.124) to the non-Abelian case, more precisely

(2.

where 9 gives the coupling of the group and the r bca its structure constants. We sh stress t hat there is only one coupling constant , valid for all the fields submitted to required invariance. This property is only valid for a simple group or a simple g risen to any power with a transverse symmetry, which allows us to identify the diffe groups. In t he case of the Higgs mechanism, we use this relation for the transformat t hat only bring into play the n - (p + 1) Goldstone fields (h, i = p + 1, . . . ,n ins of the general paramet ers aa' To first order in Bi , the relation (2 .137) gives

Substituting t his expression, together with (2.136) in the kinetic term of
2.6

The standard model SU(3)c x SU(2) L x U(l)y

The standard model of non-gravitational interactions accounts for all the interact mentioned in Section 2.3 classified dep ending on the representations of the gr

The standard model SU(3)c X SU(2) L X U(1)y

113

SU(3)c for t he strong interactions, and SU(2)L x U(l)y for the electroweak ones. This last symmetry is broken by t he Higgs mechanism described in Section 2.5.2 along the following symmetry-breaking process SU(2)L x U(l)y

--+

U(l) elec,

leaving the photon m assless, as observed. 2.6.1

The strong interaction (QeD)

The description of the strong interaction is based on an invariance under the transformations of the group SU(3). Since t his group has 8 generators, its joint represent ation, in which t he gauge bosons G~ are placed , is of dimension 8: a = I , . . ·8. T he leptons are not subj ect to these interactions , and t hus appear as singlets of SU(3), which do not couple to the gauge boson. The quarks exist in three possible states, and are thus placed in a three-dimensional representation, denoted by 3. The quarks are therefore represented in column vectors with these three elements , Q ==

(ur) (dr) (sr) (Cr) (tr) (br) ug

'ub

,

dg

,

db

Sg,

cg

sb

cb

,

tg

,

tb

bg

.

(2.138)

bb

The index is called colour ,4 which has nothing to do with any actual coloured appearance. Inst ead , this terminology comes from an analogy: due to the fundamental properties of the strong interaction , a quark can never be observed alone but only in pairs of quark- antiquarks, or in sets of t hree quarks. In the latter combinations, the total colour cancels, in the same way as it is possible to combine the three fundamental colours to give white, which can be seen as t he absence of any colour. This is why t his SU(3)c group is called the colour group. The covariant derivative of each quark Q, with respect to the strong inter action is given by (2.139) where t he Aa are 8 m atrices forming the algebra of SU(3). We often take the Gell-Mann matrices (see, for instance, Refs. [7, 13] for more details) which satisfy the appropriate commutation relations of SU(3) , i.e. (2 .119) with the structure constants of SU(3) , denoted as r~. The coupling constant g3 of the group measures the intensity of the interaction. Experimentally, it has been measured that this coupling constant has a higher numerical value than those for the other interactions, t hus the name of strong interaction. By analogy with the colour states, this interaction is also called quantum chromo dynamics 'QCD'. The Lagrangian of QCD takes t he form .cQCD

=-

L

Q'"YJ1. D (3 )Q -

~GaJ1.vGaJ1.v,

(2.140)

quarks

4The index tha t t a kes the va lue r, g and b represents the ' fund amental ' colours red, green and blue.

114

Overview of particle physics and the Standard Model

2.6.2

Electroweak interaction

The electroweak interaction combines the properties of the weak interaction , com from the invariance under transformations of the group SU(2)L' and of electrom netism , based on U(l)elec, brought together as a unique interaction. All particles combined in singlets or doublets of SU(2)L' 2.6.2.1

Quarks

The quarks of each generation are supposed to be equivalent , so that the doublets

(2.1 where each fermion appears as an eigenstate of chirality with the definition 1

+

,5)'I/J,

(2.1

'l/JR== ( - 2where the matrix

,5is defined as - 10) ( 01'

One can explicitly check that (,5) 2 = 1, so that the operators P R,L == ~ (1 ± are proj ectors orthogonal to each other , i. e. they satisfy P ; ,L = PR,L and PRPL = Another useful property is to do with the anticommutation relations: anticommu with all , l-' , or (2.1 {,5"I-'} = 0,

,5

relation that will be very useful in the following sections. Finally, the quarks with right chirality are defined as singlets of SU(2)L:

2.6.2.2

Chirality

The proj ectors PR,L on the state of definite chirality 5 come into playas soon as we interested in the properties of the spinor with respect to the Lorentz transformatio In terms of the so-called chiral representation (2.65), t he projectors are found to t the following explicit form

1_,5 (10)

P L = -2- =

00

'

and

P = R

1+2,5 = (001' 0)

(2.1

In this form, the proj ection properties of these operators become transparent. N by the way that due to the fact that and anticommute (2.143), the joint spin 7jjL,R are defined using opposite proj ectors

,5

,0

5The op erator 'Y 5 is called chirality b ecause it indicates how its eigenstates transform unde parity transformation [6 , 9].

The standard model SU(3)c X SU(2)L X U(l)y

~L = ~ (I ~15 ) = ~PR

and

115

(2.145)

Thus, they satisfy (2.146) and and

(2 .147)

which confirms that a spinor existing only in one chirality state, cannot have any mass term in the Lagrangian. From the projectors (2.1 44) , one can deduce that the eigenstat es of the chirality are quadri-spinors and where ~ and X are two bi-spinors representing respectively the left and right components of the total spinor. The Dirac equation (2.62) can then be written as two coupled equations, the Weyl equations, for the bi-spinors

(at + ( j . V ) ~ = -imx, (at - ( j . V )X = -im~ ,

(2.148) (2.149)

where the mass m should now be understood as a coupling t erm between the two components. For a massless spinor , both equations decouple and each bi-spinor , ~ and X, called Weyl spinors, evolve independently. We will see lat er that these kinds of spinors have an especially important role to play in the formalism of supersymmetry.

2.6.2.3

Leptons

For a long time, neutrinos were thought to be represented by some Weyl tensors. Experimentally, t hey are only observed in the left state V L , leading to the conclusion that they should be massless. Since they only have two degrees of freedom , in order to put them in similar doublets as in (2.141), it is only possible to pair them with the left component of the corresponding lepton, leaving the right component in a singlet of SU(2)L' thus the index 'L' of the group . The structure is thus

(:eLL) , R, (~LL ) , e

~R '

(V;LL) ,

TR

,

(2.150)

which gathers in a similar scheme, the electron, the muon and the tau with their associated neutrinos. The covariant derivatives are defined as

D12) (:e L

)

== (01" -

ig2 B il"ai )

L

(:e L

)

•

(2. 151 )

L

(Ji (i = 1,2, 3) are the Pauli matrices (2.66), generators of the group SU(2) since the satisfy the algebra

[a i ,a j

]

= 2i Ei\ a k .

(2 .152 )

Similar to the strong interaction, the vector bosons B il" are in the adjoint representation of SU(2). The coupling constant of the group is here called g2 . The coefficients

116

Overview of particle physics and the Standard Model

ij E k

take numerical values +1 (resp. - 1) if (i,j,k) is an even (resp. odd) permutat of (1,2 , 3), and 0 otherwise (i .e. if two indices are the same) . The structure constant the group SU(2) are given by the rank 3 Levi-Civita tensor 6 , and this group is loc equivalent to that of rotations. Moreover, a charge with respect to the phase transformations is associated to e particle. This 'hypercharge' , to distinguish it from the electric charge Q, is deno by Y and we have YR = -2 for the right leptons, YL = -1 for the left lepto Y(qJ = ~ for the quark doublets (2.141) , of left chirality, and for the right chiral Y(UR , cR , t R) =! and Y(dR , SR , bR ) = -~. The doublet structure of (2.141) and (2.150) reminds us of that of spin ~ w its components 11) and 11)· This is why the doublet classification of SU(2)L is a called weak isospin, denoted by T to avoid any confusion with the spin S. The val T3 = ±~ are respectively given to the top and bottom component of the doub Using this analogy and bearing in mind that the electric charges of the quarks Qu ,c,t = + ~ and Qd,s ,b = -~, the ones for the massive leptons Qe,/-"T = - 1 and t neutrinos have no electric charge, we obtain the Gell-Mann- Nishijima relation

(2.1 Note that for this relation to hold for all particles, we need to set T = 0 for singlets. We can then add a covariant derivative t erm U(l)y for each field ,

where gl is the coupling constant associated with the group U(l)y. 2.6.3

A complete model

Putting together all the previous building blocks, a complete Lagrangian can be writ down to describe the three non-gravitational interactions and all their invariances

2.6.3.1

Kinetic terms

The kinetic terms of the standard model for the electroweak and strong interacti are thus given by

.c kin

=

- i-;!ry/-'

(a/-' -

- 4"1 (o/-,Gv -

- ~ (O/-,G av -

ig 3 G a /-,)...a

ov G/-,)

-

ig2Bi/-,ai - ig1YG/-,) 'ljJ

2

OvGa/-,

+ ifbcaGb/-,Gcv)

2

(2.154

6The Levi-Civita tensor components El , ... ,n = + 1

€

in n dimensions is the completely antisymmetric rank n tensor w

The standard model SU(3)c X SU(2)L X U(l)y

117

where t he action of the covariant derivative depends on which particle it acts. For inst ance, for the lepton doublet of the first family, using the hyper charges obtained earlier , t his gives,

since the leptons are not subj ect to the strong interaction. 2.6.3. 2 W± and ZO bosons and photon ; Weak angle Equation (2. 155) can explicit ly be expanded in left and right components as .c~ in

= - iv eL, I-'81-'ve L

-

ie,l-' (81-'

+ 2ig2 sin Bw A I-') e

- V2g2 (veL,I-'W:e R + eR,I-'W;veJ - L ' I-'Zl-' e L - - g2 -B - (-VeL' I-'ZI-' VeL - COS 2B WVe cos w

-

2 SIn . 2 BWV - e R' I-'Zl-'e ) , R

(2.156) where t he weak a ngle Bw is defined by t he relations

g1

-

g2

=

tanB w ·

(2. 157)

As for the vector fields, they are defined by

(2.158) and

Bw - sinBw) (B31-') ~ (B31-' ) == ( CO~Bw sinBw ) (AZ~,.. ) == (C~S sm Bw cos Bw CI-' CI-' - smBw cos Bw

(AZ~ )

,

,.. (2. 159)

AI-' being t he electromagnetic field. The Lagrangian (2.1 56) can be understood in the following way. First , we notice that the kinet ic term of the electron leads to a gauge coupling of the U( l ) type with the photon , since it can be written in the form

with

q = g2 sin Bw the electromagnetic coupling const ant and Q the electric charge operator , with value Qe = - 1 for the electron. This coupling is the same fo r the left and right degrees of

118

Overview of particle physics and the Standard Model

freedom of the electron. In addition to the coupling terms, an intermediate neut boson ZJ1. has appeared , coupling both right and left components of the electron a different way, and also adding a self-coupling term for the (left-handed) neutri Finally, charged intermediate bosons make interactions between neutrinos a electrons possible (the charges of t he W± come from the fact that each term of final Lagrangian should preserve the electric charge, which between a neutral neutr and an electron is only possible wit h the indicated signs for the charges). For a quark doublet , one can perform the same expansion, which gives us electric charges of the quarks, but leads to 36 different interaction terms for e generation of quarks once the triplet of colour is made explicit .

W:

2.6.3.3

Symmetry breaking

The Lagrangian (2.154) counts almost all possible terms that satisfy the invariance t he standard model and bringing into play t he particles known experimentally. D to its invariances, and in particular due to the weak isospin, it is not possible to wr mass terms for the leptons. As for the quarks, all elements of the same doublet SU(2)L should have the same mass, like for instance both quarks u and d, which is contradiction with the measurements. Thus t he model can only be made compati with experiments if the SU(2)L symmetry is broken. For this , we introduce a complex field doublet

the dynamics of which is governed by the potential (2.128)

The minimum of this potential is 14>112 + 14>2 12 = ry2. Assuming that the Higgs field not subject to the strong interaction, the kinetic term becomes

(2. 1 which can be expanded as

1&4>11 2 + 1&4>21 2 + v2g2 (.JJ1.- W+J1. + .JJ1.+ W - J1.) + (g2 B 3J1. + gl Yq, C J1.).Ji - (g2 B 3J1. - gl Yq, C J1.).Ji: + (g2B3J1. + glYq,CJ1.)214>112 + (g2B3J1. - gl Yq,CJ1.)2 14>21 2 + v2g2 (4)~4>2 + 4>14>;) [w-J1. (g2B3/l- + glYq,C/l-) - W+J1. (g2 B 3/l- + 2g2 (14)11 2 + 14>212) W:W-J1..

ID12 =

glYq, CJ1.)]

(2.16

In (2 .160) , we have introduced the Uy(l) hypercharge Yq, of the Higgs doublet; determine its value below.

The standard model SU(3)c X SU(2)L X U(1)y

119

The currents are defined by

and

JI-'- == i (cP~0l-'cP2 - cP20l-'cP~) , with J + = (J- )t = i (cPZ0l-'cPl - (h0l-'cPz) . We now work around the minimum of the potential, and perform a gauge transformation to suppress any of the three redundant degrees of freedom . In the unitary gauge, for which cPl = 0, and cP2 = 7) + hlJ2, i.e.

+ O
with


(~)

and

(2.162)

we are left with only the Higgs field, h, which is real. We then get

c :r =

- ~Ol-'hOl-'h - (7)2+ ~h2 + V27)h)

[(92B31-' - glY<jJ CI-')2

+ 29~W+I-'W;],

(2.163) where we can recognize the intermediate vector boson ZI-' provided that we fix the hypercharge of the Higgs scalar doublet to Yq, = 1. It is the only value that will preserve the electromagnetic invariance after the symmetry breaking, ensuring the photon defined by the relation (2.159) remains massless.

2.6.3.4

Masses of the intermedia te bosons

Equation (2.163) contains interaction terms between the Higgs boson, h, and the intermediate bosons, ZO and W±. The mass term of the latter can be extracted. Taking h = 0 in (2. 163) , we get

.c masses --

-

2g27) 2 2W+ W - 1-' I-'

-

7) 2 cosg22 Bw

Z I-' ZI-' =-

-

M2W W+W-I-' I-'

-

-21 M Z2ZI-' ZI-' ,

(2.164) so that we can make the identification

Mz =

and

..)2g27)

= Mw .

cosBw

cos Bw

(2.165)

This last relation allows us to derive the value of the weak angle Bw from the masses of the intermediate bosons. This has been done following their discovery at CERN in 1982 . Their actual masses are [11]

Mw = 80.419 ±0.056 GeV

and

Mz = 91.1882 ±0.0022 GeV,

which gives the weak mixing angle, namely sin 2 Bw ,...., 0.22 , that is approximatively ()w ~ 280. Note that no term of the type AI-'AI-' appears in the Lagrangian (2.164). The photon remains massless, in agreement with Maxwell 's electromagnetism.

120

Overview of particle physics an d the Sta ndard Model

2.6.3.5

Masses of the fermions

T he particles of right and left chirality belong to different representations of SUL (2 thus it is not possible to construct mass terms in a simple way just as for t he gau fields, one should impose that all the fermions are massless before the symmetry brea ing. Moreover , since the right neutrino should be absent from this theory, one shou postulate that the neutrino is massless, even after symmetry breaking. The most general coupling terms between the fermions and the Higgs field th satisfy the required invariances, are given by 7

(fleptonsfL . cpe R

+ f qT ii L

. ;PU R

+ fqlii L

.

cpd R

+ h. c. )

,

generations

(2.16 with ;;,. 'f'

. 2 A.*

=W

'f'

(

¢2 )

= -¢r '

where the matrix (j2 is defined by (2 .66), with hypercharge YJ) = -Y¢ = -1 , an where the sum is over the three different generations , so that the fields e R , u R an dR represent, respectively, (e, j.L , T)R ' (u , c, t) R and (d , s, b)R' In (2.166), the numbe iteptons, fqT and fq l, called the Yukawa coefficients, not only represent the coupling the quarks and leptons with the Higgs fields, but, once the symmetry is broken , als give the values of the m asses of the different fermions. One can see that the interactio of a given particle with the Higgs field is all the more important that it is massive. 2.6.3.6

Electromagnetic symmetry

Working in the unitary gauge defined by (2.162), we see that the vacuum configuratio CPo is identically annihilated by the electric charge operator Q given by the Gell-Mann Nishijima relation (2.153). Indeed, with

T3 =

~2(j 3 = ~2

(1 0) 0-1

'

and Y¢ = 1, we have

The annihilation of the vacuum state by the generator of electromagnetism impli that this state satisfies (2.167

and hence it is invariant under a transformation U(I)elec. Thus, the symmetry breakin is not total and the electromagnetic invariance is recovered, as required for the theor to be compatible with experiments .

70 ne can explicitly check that with the hypercharges given previously for all particles, these term are the only possible ones t h at are invariants under the tran sformation of U(1)y.

Discrete invariances

121

Note that even if here the relation (2.167) has been explicitly obtained in a unitary gauge, from the invariance of the initial gauge of the theory we can conclude , independently of the gauge , that this result will apply for a ny choice of gauge, i.e. that for any choice of the configuration of the Higgs doublet, as long as it is at the minimum of the potential.

2.7

Discrete invariances

In addition to the continuous symmetries, usually gauged ones, there exist discrete symmetries such as parity, charge conjugation and time-reversal invariance. The first one appears to be maximally violated by the weak inter actions, and the last one weakly, for instance , in the system J(o - Ko · In what follows, we will limit ourselves to the study of the scalar field, all results being generalizable to the fermionic and vector sectors [8]. For a classical field, a symmetry operation can be obtained in the following way. For a given transformation x f---7 f( x , a), where a represents a set of parameters defining the transformation , the field ¢( x) transforms as ¢'(X')

= A(a)¢(x) ,

(2. 168)

where A is a matrix if the scalar field ¢ has several components. A state I'lj;) in the Fock space, corresponding to the field operator ¢, transforms linearly i'lj;') = U(a)I'lj;), where ut = u - 1, enssuring that ('lj;I'lj;) = ('lj;' i'lj;'). We require that the matrix elements of the field operator satisfy the relation (2.168) ,

between two arbitrary states I'lj;l) and 1'lj;2) and the transformed states I'lj;D and which leads to ut(a)¢(x')U(a) = A(a)¢,

14)~) ,

i.e. to where the dummy variable x' has been replaced by x and expressed using the inverse transformation law. We can now see how these general transformations can be applied to different particular cases, useful in particle physics. 2.7.1

Parity

Let us assume that we perform a reflection transformation on the spatial coordinates

{

X

f---7

x' = -x,

t

f---7

t'

=

t.

(2.169)

If the physics remains unchanged under t his transformation (as is the case, for instance, for the Klein- Gordon equation t hat is of second order in the spatial derivatives) , t hen a classical field can only be modified by at most a phase factor

122

Overview of particle physics and the Standard Model

¢'(x') = sp¢(x), with Isp 12

=

1. For the field operator this translates into

p - l ¢(X , t)p =

sp¢( - x , t),

(2.1

where P is a unitary operator representing the parity. If ¢ E C, then the variation can be performed via a phase, i. e. sp = e iQ , a E However, for a real field (¢ E ~), the corresponding operator should be Hermitian, ¢t = ¢, and hence

pt ¢t (x , t)

(P-l) t =

p - l ¢ (X , t) p = s;¢( - x , t) ,

s;

from which we deduce that = sp. This, together with the relation Isp 12 = 1, imp t hat sp = ± 1. In this special case, the particle associated with the field will be s to be scalar if sp = + 1 and pseudo-scalar for Sp = -1 , i.e. when it has a negat intrinsic parity. The action of the parity operator on the Hilbert space can be obtained by expand the field in the basis (2 .106) (we consider from now on a complex field) and rewrit t he equation. We find

J

d3k [Pa pt ei(kx-wktl (21T)3 / 2V2wk k =

s p

J

3

d k

(21T)3 / 2V2Wk

+ Pb t pt e- i(k'X - wktl] k

[a e-i(k,x +Wktl k

+ bt ei(k'X -wktl] k

,

(2.1

where the sign of x in the plane-wave exponentials of the right-hand side have b inverted, in accordance with (2.170). Since t his last operation is equivalent, in exponentials , to the one that changes the sign of k , it gives the transformation la of the creation and annihilation operators

Pakpt = spa_ k and

Pblpt

= Spb~k'

(2.1

T hus, the parity operator reverses the sign of the momenta, which had to be expec since t hey are 3-vectors. For the case of a free complex scalar field, i.e. massive but with no interact with any other field , the Hamiltonian, given by the relation (2.108), transforms un parity as

(2. 1

since the integral is p erformed on all values of the vector k and Isp 12 = 1 whicheve the parity of the states we consider . Similarly, the field momentum, equal to

(2.1 transforms as

ppp-l =

_P,

which corresponds to the transformation law of a vector. At this point, it is interest to note from (2.173) that the parity operator commutes with the Hamiltonian, and th

Discrete invariances

123

the parity of a physical state is a constant of motion for the free complex scalar field. Some interactions can modify this conclusion: this is the case of the weak interactions. A parity can also be attributed to the vacuum, and we choose PIO) = 10) (the vacuum must remain identical to itself if the coordinates are inverted). Note that we can explicitly construct an operator P satisfying (2.172) for which this relation is verified, but it is also possible to give a negative intrinsic parity to the vacuum: this only complicates things a little and simply inverts all the fields parities. In other words , this is only a matter of convention. 2.7.2

Charge conjugation

Another discrete transformation as simple as parity can also be performed. This is charge conjugation C, which consists in replacing each particle in a given physical state by its antiparticle. In terms of operators, this amounts to transforming (p into (Pt and vice versa. We can therefore write (2 .175)

and

the second relation of (2. 175) , being simply the Hermitian conjugate of the first one. Just as with parity, in (2.175) , t he eigenvalue of Sc has a unit modulus since transforming twice in a row all the particles and their antiparticles is in the end equivalent to no transformation at all. This is equivalent to saying that C2 = 1. Furthermore, note that C should be unitary, so that finally, we have = C- 1 = C. In terms of the annihilation operators, we find

ct

(2.176) and the Hermitian conjugate relations for the creation operators . Assuming, as before, that the vacuum is an eigenstat e of the charge conjugation, with eigenvalue + 1, i.e. CIO) = 10), we can then determine the charge C of any physical state created with a determined number of particles. Here again, we find t hat for a free scalar field , the charge conjugation is a constant of motion, leaving the Hamiltonian (as well as the momentum) unchanged. However, the conserved charge Q of (2.11 3) changes sign, as one could expect,

CQC- 1 = - Q, so that an eigenstate of the conserved charge Qcannot simultaneously be an eigenstate of the charge conjugation. The time conservation of the latter is thus only useful for physical states with a global vanishing charge. = 0, and the ones that violate parity conserMost of the interactions satisfy

[iI,C]

vation, such as t he weak interactions, more often than not, leave the combination CP invariant . In the section dedicated to baryogenesis in Chapter 9, we will discuss the cases where even this constraint is not valid. However, in all field theories analogous to the one presented here, a discrete symmetry is satisfied, it is the combination CPT, where the last symmetry, the t ime-reversal invariance, requires some technical details that are given in what follows .

124

2.7.3

Overview of particle physics and the Standard Model

Time reversal and CPT theorem

The last discrete invariance, valid for instance for a free scalar field, is that related t time reversal T: {

X

f---+

x' = x ,

t

f---+

t' = - to

(2 .177

Unlike the two previous symmetries, which are unitary operators, this symmetry ca be implemented in the Fock sp~ce of states only in the form of an antiunitary operator An operator T is antiunitary if it is antilinear, i.e. such that

(2.178

and if it satisfies TTt = TtT = l. Thanks to this last operator , one can prove [14] the following very general theorem If a local quantum field theory can be represented by a Hermitian Lagrangian .c(XD that is invariant under proper Lorentz transformations (i.e. without the space an time inversions) and if its field operators satisfy the spin-statistics theorem,9 then w have

(2.179

Due to this relation (2.179), the action integral S = J d 4 x..c is invariant under th transformation CPT. Thus, the equations of motion, which are obtained by varyin the action, and the canonical commutation relations, are also unchanged under th transformation. We can also deduce that the Hamiltonian of the given theory mu commute with CPT,

[CPT, iT] = 0,

and hence any realistic theory, i.e. satisfying the hypothesis of this theorem, has th invariance. This result is often written in the symbolic form 'CPT = 1' . A non-trivial consequence of this theorem is the following: for any theory satisfyin it , i.e. for any realistic theory, the particles and antiparticles should have exactly th same masses and the same lifetimes. This consequence is verified with a very hig precision since, for example, for the Ko/ Ko system, we find [ll]iMKo - MRoi/MKo 10- 18 . For most of the other particles, we find typical values of order of ib.M i/M 10- 5 , and constraints on the differences of lifetimes of order of ib.Ti/T < 10- 3 .

8This happens b ecause of technical reasons related to the fact that the time-reversal operator mu also reverse the roles of the initial and final configurations [8] . 9The spin-statistics theorem , itself also very general in field theory, specifies that the particl associated to integer-spin fields (for instance bosons, scalars and vectors) have a statistics describe by the Bose- Einstein distribution function , whereas the ones corresponding to half-integer fiel (fermions, Dirac spinor, for instance) , are d escr ibed by Fermi- Dirac statistics. This theorem on relies on microcausality. It is discussed in detail in Ref. [15] .

Discrete invariances

125

The standard model of particle physics presented in this chapter has so far never displayed any flaws from the point of view of experimental physics. However no one sincerely thinks that it describes all interactions between the elementary particles in a final way. There are many reasons for this, which are as much experimental as theoretical. From an experimenta l point of view, the recent discovery of the existence of a mass for the neutrinos and the discussion concerning the chirality (Section 2.6.2) indicate that they cannot uniquely exist in the left chirality as attributed by the standard model. There must therefore exist a right neutrino , which has different properties from the left one since it would have otherwise already been detected experimentally. They therefore need to have a very large mass, in order to explain why no accelerator has already produced it. At the experimental level, there is still an unknown . Since the discovery of t he 'top' quark , all particles of the standard model are identified except one , which is maybe the most important since the principle of symmetry breaking relies on it, namely the Higgs boson . As long as it has not been produced directly in an accelerator , it will remain hypothetical, and the model will not have been fully verified. Now from a theoretical point of view, t he situation is even worse. First, this model, although it unifies in a quantum framework the three non-gravitational interactions (electromagnetism , weak and strong) , makes no mention of the last interaction, gravity, despite t he fact t hat one of its pillars, the Higgs mechanism, is supposed to be at the origin of the mass of elementary particles. Interestingly, these masses are also the gravitational charges of these particles . Moreover, t he standard model relies on three different symmetry groups, wit h t hree coupling constants having no links between them, which is not justified by any fundamental principle. Finally, and this may be the most serious, there are 19 free parameters in the theory, which can only be measured. T his proliferation of parameters is often considered as t he sign t hat it is not a fundamental theory but actually a low-energy effective theory. We will see later that in many extensions, the number of free parameters is seriously reduced, and that it is in principle possible to calculate t he 19 arbitrary coefficients that appear in t he general Lagrangian in terms of a much sm aller set of parameters.

References

[1] L. D. LANDAU and E. LIFSCHITZ, Mechanics, Pergamon Press, 1976. [2] N. D ERUELLE and J. P. UZAN , Mecanique et gravitation newtoniennes, Vuibe 2006 . [3] C. COHEN-TANNOUDJI , B. DIU and F. LALOE, Quantum mechanics, John Wil 1992. [4] P. R. HOLLAND, The quantum theory of motion, Cambridge University Pre 1993. [5] J. SCHWINGER, Quantum mechanics, Springer, 200l. [6] F. HALZEN and A. D. MARTIN, Quarks & leptons, John Wiley and Sons, 198 [7] M. A . PESKIN and D . V. SCHROEDER, An introduction to quantum field theo Addison-Wesley, 1995. [8] W . GREINER and J. REINHARDT , Field quantization, Springer-Verlag, 1996. [9] L. H. RYDER, Quantum field theory, Cambridge University Press, 1987. [10] J. 1. Kapusta, Finite-temperature field theory, Cambridge Monographs on Ma ematical Physics, 1994 [11] W. -M. YAO et al., 'The review of particle physics', Journal of Physics, G 33 2006; see also the website http ://pdg .lbl. gov / that gives the same informati Updated almost every two years to take into account new experiments and th results. [12] E. SEGRE, From X-Rays to quarks: modern physicists and their discoveries, Fr man , 1980. [13] H. GEORGI, Lie Algebras in particle physics, Westview Press, 1999. [14] W. PAULI, in Niels Bohr and the development of physics, Pergamon Press, 19 G. LUDERS, Ann . Phys. (NY) 2 , 1, 1957. [15] B. DIU et ai. , Physique statistique, Hermann, 1996.

Part II The modern standard cosmological model

3 The homogeneous Universe The first cosmological solution of Einstein's equation was given by Einstein himself in 1917. At the expense of introducing a cosmological constant, he was able to construct a closed and static space. The general cosmological solutions were discovered independently by Alexandre Friedmann and Georges Lemaitre in 1922 and 1927. This chapter details t he derivation of these solutions from Einstein's equations and studies their kinematics and dynamics before defining the various observational quantities that will be used in this book. It ends on some considerations on lesssymmetric space-times. For complementary developments , see Refs . [1-7].

3.1 3.1.1

The cosmological solution of Friedmann-Lemaitre Constructing a Universe model

Within the framework of general relativity, one can try to answer the question of which solution of Einstein's equation describes the Universe we observe, or more modestly, which solution is an idealized (but good) model of our Universe. To carry out this program, we need enough input from astrophysical and cosmological observations. This program is indeed difficult to tackle and its solution will, in particular, depend on the amount of available data. But , as we noted in the introduction, even with ideal data, there are limitations to the answers one can give to this problem. These limitations arise mainly because we observe only a finite part of one Universe (it is a unique object and we cannot discuss its probable nature by comparing it to similar objects) from a given space-time position (and we are unable to choose this point) . Astrophysical data are mainly localized on our past light cone and around our past worldline for geophysical data. They are thus localized on (just a portion of) a 3-dimensional hyp ersurface. First, it may be that various space-times are compatible with the same data and second, the interpretation of these data is not independent of the space-time structure. Thus , we must distinguish between the observable Universe (for which , by definition , we have data) and the Universe, which includes regions we cannot directly influence or experiment. The inference of the geometrical properties of the Universe from the observable Universe cannot b e achieved without hypothesis and philosophical prejudices. The standard cosmological model, in its simplest form, relies on four central hypotheses. HI Gravitation is well described by general relativity. Indeed , as discussed in Chapter 1, general relativity is well t ested on small scales (e.g. , Solar System scales),

130

The homogeneous Universe

but we cannot exclude that it is not t he case on large scales or at early times (se e.g., Chapter 12 for examples of theories that dynamically become more and mo similar to general relat ivity during t he evolut ion of the Universe). In particula t he E instein equivalence principle implies that the laws of physics can be extrap lated at all times (see Chapter 1). We will thus keep applying, as long as possibl the standard laws of physics, in particular for non-gravitational interactions. H2 Hypothesis on the nature of matter. We do not examine the space-time itself or i matter distribution. Rather , we observe particular objects (e.g. , galaxies, ... ) t h are supposed to be representative of t he total matter content . The influence of th variations of the properties of individual objects on the inference of the matte content of the Universe has to be quantified. On large scales, we will assume t h matter is described by a mixture of a pressureless fluid and radiation, allowin also for a cosmological constant. In particular , we assume that there is no matt outside the standard model of particle physics described in Chapter 2. The perfe fluid hypothesis will be shown to derive from the hypothesis H3. H3 Uniformity principles. The previous hypotheses are not sufficient to solve th E instein equations. We must assume some symmetries for t he space-time. Th hypothesis and their consequences are discussed in Section 3. 1.2 H4 Global struct ure. Hypotheses H1- H3 enable us, as we shall see, to solve Einstein equation and to determine t he lo cal structure of the Universe model. Howeve there may exist various manifolds with identical local geometry and differe global topology, which correspond to the choice of various boundary condition For a globally hyperbolic space-time, the choice of the topology reduces to th choice of the topology of the spatial sections.

As we shall see, t hese hypotheses allow us to construct very successful Univer models but there are reasons to challenge everyone of t hese four assumptions. Th will be discussed mainly in Chapter 12. 3.1.2

Cosmological and Copernican principles

Galaxies seem to be spread isotropically around us, just as the cosmic microwav background radiation. This indicates that the space-time describing the observab Universe should have a spherical symmetry around us. T here are two possible expl nations: either our Galaxy lies in a special place in the Universe so t hat it is no homogeneous and must have a centre. Or the space-time is homogeneous; every poin is then similar and the Universe is isotropic around each point (see F ig. 3.1 ). To construct cosmological models, we can distinguish two uniformity principle T he cosmological principle supposes t hat t he Universe is spatially isotropic and homo geneous. In particular, this implies that any observer sees an isotropic Universe aroun him. We can distinguish it from t he Copernican principle that merely states t hat w do not lie in a special place (the centre) of the Universe. This principle, together wit the isotropic hypothesis, actually implies t he cosmological principle. The cosmological principle makes definite predictions about all unobservable r gions beyond the observable Universe. It completely determines t he entire structu of the Universe, even for regions that cannot be observed. From this point of view this hypothesis, which cannot be tested, is very strong. T he Copernican principle ha

The cosmological solution of Friedmann-Lemaltre

{:] p

{:] Q

131

.......... .. .... ..-. ... .. .. .. ...... ........ .... ··· .. . .. •.. . .... ··.. . -.. U .- ....... . . . . . . .. -. -. .. .... ....... .. ........ {:] p

• • Q.

•

Fig. 3.1 A point distribution , statistically isotropic around every point (left) and around a unique point (P) (right). In the second version, P and Q are not equivalent. The cosmological principle excludes such kinds of solutions, which would assume that we lie in a sp ecial place in the Universe . From Ref. [l J of the introduction.

more modest consequences and leads to the same conclusions but only for the observable Universe where isotropy has been verified. It does not make any prediction on the structure of the Universe for unobserved regions (in particular, space could be homogeneous and nOn isotropic On scales larger than the observable Universe). Some cosmological spaces that do not satisfy this cosmological principle are described in Section 3.6. From a pragmatic point of view, these uniformity principles should be considered in a statistical sense, i. e. for a Universe smoothed On dist ances greater than its larger structures. Thus, they implicitly involve a smoothing scale and a smoothing procedure to which we shall return. They lead to the Friedmann- Lemaitre solut ions that are at the basis of modern cosmology. Recently, a test of the Copernican principle was proposed [8J. 3.1.3

Cosmological principle and metric

Homogeneity and isotropy are two distinct notions that we should define more precisely. The homogeneity of space means t hat at every moment, each point of space is similar to any other One. More precisely, a space-time is (spatially) homogeneous if there exists a one-parameter family of space-like hypersurfaces, I: t , foliating the spacetime such that for any t and for any pair of points (P, Q) of I: t , t here exists an isometry of the space-t ime metric taking Pinto Q (see Fig. 3.2). Isotropy means that at every point the Universe can be seen as isotropic. More precisely, a space-time is (spatially) isotropic at each point if there exists a congruence of time-like worldlines with tangent vector uJ.L (called isotropic observers) such that at any point P and for any pairs of unit spatial tangent vectors ei, e~ E Vp (see Chapter 1), there is an isometry of the metric that leaves P and uJ.L at P fixed and that rotates the vectors ei into e~. Thus, it is impossible to construct a preferred tangent vector perpendicular to uJ.L.

132

The homogeneous Universe

/ /

~

... Q

p. III

..

..

·x7 ./

p

Fig. 3.2 (left): A homogeneous space-time can be foliated by a family of homogeneous spaces (right) : for an isotropic space, there is an isometry that rotates any unit spatial tangent vecto ei to any other e~, while the vector u lL remains invariant . From Ref. [9J.

We recagnize that for a hamageneaus and isatropic space-time, the surfaces of ho mogeneity, ~t, must be orthogonal to the tangent vector uj.t of the worldline of the isotropic observers , that are thus called fundamental observers. (Proof If it were no the case; then, assuming that the homogeneous space and the cangruence of isotropi observers are unique, the failure of the tangent subspace perpendicular to uj.t to coin cide with ~t wauld enable us ta construct a geometrically privileged spatial direction This is in cantradiction with isotropy.) The metric gj.tv induces a spatial metric :rj.tv(t) on each hypersurface ~t · As w have just seen, there must exist an isametry of :rj.tv bringing any point of ~t into another point of ~t and it must be impossible to construct a preferred vector from :r,w. As one can show, this implies that the Riemann tensor canstructed from :rj.tv fo the three-dimensianal spaces ~t must be ..of the farm I [2, 9, 10]

where K has , because ..of homogeneity, the same value at any paint ..of ~t, and can ..onl be a functian ..of t, K(t) . A space satisfying this property is called a maximally sym metric space (as we shall see it has the maximum number ..of Killing vectars allawed by its dimensian) and is a space of constant curvature. Two spaces ..of canst ant curvatur with the same dimensian, signature and the same value ..of K are isametric. Thus, it) sufficient ta enumerate the three-dimensianal spaces ~orresJ::>0nding ta all values ..of K We classify these spaces according ta the sign of K . If K is pasitive, ~t is a three dimensional sphere ..of radius R(t) , with embedding equatian x 2 + y2 + z2 + w 2 = R2(t in a faur-dimensianal Euclidean space. In spherical caordinates,

1 As shown in Ref. [9], this conclusion can be drawn from the isotropy a rgument. Consider (3) R ILv cx { as the components of a linear map, L, of the vector space of 2-forms into itself. Since L is symmetric it can b e diagonalized. If its eigenvalues were not equal, then one could construct a preferred 2-form and then a preferred direction. Thus , isotropy implies that L is proportional to the identity and thu (3) R cx{3 = 2K b cx b{3 by symmetry. ILl.'

[IL

vi

The cosmological solution of Friedmann-Lemai'tre

x

=

RcosX,

y

=

Rsin xcos B,

z

=

R sin x sinB cos
w

133

Rsin xsinBsin
=

t he E uclidean metric ds 2 = dx 2 + dy2 + dz 2 + dw 2 induces the three-dimensional metric on I;t If K

= 0, I;t is a three-dimensional Euclidean space with metric

If K is negative, I;t is a t hree-dimensional hyperboloid , with embedding equation _w 2 + x 2 + y2 + Z2 = _ R 2(t). In spherical coordinates ,

x

=

R cosh X,

y

=

R sinh xcos B,

z

=

R sinh xsinB cos

w

=

R sinh xsinB sin
the Minkowski metric ds 2 = - dw 2 + dx 2 + dy2 + d z 2 induces the three-dimensional metric on I; t dS(3) = R 2(t ) [dX2 + sinh2 X (dB 2 + sin 2 Bd
3.1. 3.1

Four-dimensional m etric

The worldlines of the fundamental observers are orthogonal to the hypersurfaces I;t. The coordinates syst em can be transported from one hypersurface to another by means of this congruence of worldlines. The general form of the space-time metric is imposed by these symmetry considerations 91.1.1/ = - uJ.L UV + ~J.Lv(t), where, for each value of t , ~J.LV is the metric of a three-dimensional sphere, Euclidean space, or hyperboloid. It follows that the line-element of the cosmological space t akes the form ds 2 = - (U J.L dxJ.L)2 +~J.L v( t)dxJ.Ldxv . Setting uJ.LdxJ.L == dt , t he coordinat e t is t he proper time measured by the fundamental observers (i. e. co moving wit h uJ.L) and we get the general form of the space-t ime metric as (3.1 )

t being the cosmic time, a(t ) t he scale factor and

lij the metric of constant t ime hypersurfaces, I;t , in comoving coordinat es. 2 Latin indices i, j , . .. run from 1 t o 3. The conformal t ime 7) , defined by

2This implies t hat ;Yij (xk, t ) = a 2 'Yij (x k ), so that the sp atia l met r ic satisfies (3) R ijkl

and K(t )

= K/a 2 (t).

= 2K 'Yk(i'Yj]I ,

134

The homogeneous Universe

(3.

dt = a(t)d7] , can also be introduced to recast the metric (3.1) in the form

(3. T he spatial metric, in comoving spherical coordinates, takes the general for m

(3.

where X is a radial coordinate and dn 2 = dB 2 + sin 2 Bd'P2 is the infinitesimal so angle, 'P runs from 0 to 27r and B from - 7r to 7r. T he quantity runs from 0 to when K > 0, and from 0 to +00 otherwise. The function f K depends on the curvatu K (which now is a pure number) as

vTKTx

K - l / 2sin

fK( X) =

(VRx)

K

>

0

X

K =O { (- K) -1/ 2 sinh(v'- Kx) K < O.

(3.

It relates the surface S(X) of a comoving sphere to its radius X by S(X) = 47rf'k( that reduces to t he E uclidean expression for distances small compared to t he curvatu (i.e. for X « II Using the radial coordinate r = fK( X), (3 .1) can also take t for m

vTKT)·

(3.

This metric, or the equivalent forms (3 .1) or (3.3) , is called the Friedmann- Lemait metric. T his construction shows how the cosmological principle, and thus the symmet assumptions that follow, has allowed us to reduce the ten arbitrary functions of t space-time metric to a single function of one variable, a(t ), and a pure number , K.

3.1 .3.2

Geom etrical quantities

The expression for the Christoffel symbols can be obtained , using the definitions Chapter 1 and the metric (3.1). Their only non-vanishing components are f

o ij

= H a2'Yij,

fjo. = HOj,.

i

_

f jk -

(3)

rijk ,

(3 .

where the (3)fjk are the Christoffel symbols of the spatial metric 'Yij . The quanti H = ala is t he Hubble parameter , and a dot represents the derivative wit h respect the cosmic time t. The explicit form of (3)fjk is not needed to compute the component of the Ric t ensor because the spatial metric 'Yij satisfies (3) R ij kl = K hik'Yjl - 'Yinjk) , so th

The cosmological solution of Friedmann-Lemai'tre

135

(3) Rij = 2K lij and (3) R = 6K. It follows that the only non-vanishing components of the Ricci tensor are 3

Roo =

a a

-3~,

R ij

=

(

2H

2

K) a

+ ~ii + 2 a 2

2

(3.8)

lij,

from which we deduce the expression for the scalar curvature

R=6 ( H 2

K) ,

ii a

(3.9)

+~+2

a

and for the non-vanishing components of the Einstein tensor

Goo 3.1.3.3

=

3 ( H2

+

~) ,

G ij

=-

(

H

2

+ 2~ii +

K) a

a2

2

lij.

(3.10)

Killing vectors

As explained in Chapter I, to each symmetry of a space-time is associated a Killing vector field,~ , satisfying (1.58). We have seen that for a space-time of dimension n , the maximal number of independent Killing vectors, and thus of symmetries, is n(n + 1)/2. For the Friedmann- Lemaitre space-times, the spatial sections are maximally symmetric so that we expect to obtain 6 Killing vectors. They can be explicitly determined by considering the Weinberg coordinates

in terms of which the spatial metric, and its inverse , take the form

(3.11) with r2 == 6mn xmxn. The Killing equation, \7 I-'~V + \7 V~I-' = 0, gives for the OO-component, ~o = 0, so that ~o is a function of the spatial coordinates only, ~o = F(xk). Then, the Oi and ij components take the form ~i = gki Dk~O and likDj~k + IjkDie + 2H lij~O = 0, where Di is the covariant derivative associated to lij. They can be combined to get

Only iI a 2 may possibly depend on time in this equation. If this is the case, iI a 2 is not a pure number , then this implies that F = O. We thus have six solutions to the Killing equations: 3 As

fbpf?j

an example, consider the computation of R ij . It can b e decomposed as Rij = oor?j - OJ

~

r3jrf", + (3) R ij , where

r?o +

all terms involving no p, = 0 component have been gathered to

identify the Ricci tensor of the spatial metric. We then use that

(3) Ri j =

2K "Iij = 2(K / a 2 )gij.

136

The homogeneous Universe

• three Killing vectors associat ed to spatial translations

r = 1, . . . , 3,

(3.12)

• three Killing vectors associated to spatial rotations

r , S = 1 ... , 3.

(3.13)

When Ha 2 does not depend on time, Ha 2 = K, as we shall see from the Friedman equations. This occurs for the Minkowski, de Sitter and anti-de Sitter spaces. It follow that there are four additional solutions associat ed to time translation and Loren boost s. They are , respectively, given by

(3 .14 and

L /-L[rJ ..

L o[rJ -_

for K = 0, r . X ,

(3.15 for K = ±1 ,

in t erms of the conformal time TJ. Because of the expansion of the Universe , these four vectors are no longer Killin vectors of the Friedmann- Lemaitre space-time. They now satisfy the equation V /-L ~lI V lI~/-L = (V (X~(X)g/-L lI /2 and are called conformal Killing vectors. The symmetries in volving t ime (i.e. time translation and Lorentz boosts) are lost due to the fact tha the cosmological space-times are not static but they are still conformal to a stat space-time, a property related to the existence of these four conformal Killing vector 3.1.4

Kinematics

Without knowing the evolution law for the scale factor, a(t) , a few general consequence can be obtained simply from the form of the Friedmann-Lemait re metric. 3.1.4.1

Fundam en tal observers

Notice first of all that in the metric (3.1), the coordinates xi are comoving. Th worldline of a fund amental observer with traj ect ory orthogonal to the hypersurface I;t is given by x O = t and X i =constant (one can check that this is indeed a time-lik geodesic). As a consequence , this observer has a four-velocity

(3 .16 As seen earlier , this metric can be re-expressed in the form

(3 .17

The cosmological solution of Friedmann- Lemaitre

137

where 9 /-Lv = g/-LV + u/-LU V is the projector t ensor introduced in Section l.2.5. The expansion of the Universe thus allows a comoving observer to naturally decompose any vector p/-L into a time-like and space-like component as (3 .18) In particular, all tools introduced in sections l.2.5 and l. 3.2 of the first chapter can be used. 3.1 .4.2 Hubble law Let us now consider two comoving observers with traj ectories given by x = x = X2. In the physical space, their separation is given by r12

Xl

and

= a(t)(xl - X2)'

It follows that

(3 .19) The farther the observers are from each other , the greater is their relative velocity. T his is what is usually called the Hubble law. Notice t hat H depends a priori on time and can only be approximat ed as a constant for obj ect s whose distance is small in comparison to the Hubble radius. Note that this law is a special case of the generalized Hubble law (l. 119) with e = 3H and (J/-LV = 0, as imposed by homogeneity and isotropy. 3. 1.4.3

Recession velocity and proper velocity

Since the physical distance r is related to t he co moving distance x wit h the relation r = a(t)x, we deduce that

r = Hr + a(t)x ==

V rec

+ vpro p er '

(3.20)

The recession velocity, V rec , is a t ime-dependent quantity and represents t he component of the velocity relat ed to the Hubble flow. As for v proper, it is the proper velocity of the object with respect to the cosmological reference frame privileged by the expansion. 3.1.4.4

Properties of ligh t

In the geometrical optics limit , it was shown that the traj ectory of a light ray, X"' (A) , is given by a null geodesic with wavevector k'" = dx'" / dA satisfying (3 .21 ) Using the general decomposition (3. 18), we obtain

where E = - k/-Lu/-L represents the energy of a photon measured by the comoving observer and p/-L its momentum (p/-Lu/-L = 0). It follows that

138

Th e homogeneous Universe _E2

+ a2iij p i p j = 0,

which is simply the generalization of the well-known relation between the energy of obj ect , its momentum and its mass. Moreover, the projection along ul-' of the geode equation gives . Oij . 2 i' EE + f ijP P = EE + H a iij P rY = 0,

so that t/E = - H and hence E = ko la , ko being a constant. The energy, a hence t he frequency 1/ oc E , of any photon varies as the inverse of t he scale fact Its wavelength varies as the scale factor. In an expanding space, wavelengths are th redshifted achromatically. T he redshift defined by the general equation (l.122) is t herefore given by 1+ z

=

(ul-'kl-')emission

(ul-' kl-' )observation

a( t o b servation) a( temission) .

(3. 22

The redshift can be measured from the comparison of an observed spectrum to laboratory spectrum , from which one can deduce how much the Universe has expand since it was emitted. We stress at this point that most observations give an angu position on the celestial sphere and a redshift. The radial distance cannot be measur directly but is obtained from the redshift and requires the knowledge of the expansi law of the Universe, that is a(t). 3.1.4.5

Beh aviour of a m assive test particle

Consider now the trajectory of a massive t est particle with 4-velocity vI-' . It satisfie

(3.2

can be split as vI-' = aul-' + pI-', where ul-' is t he 4-velocity of t he fundamen observers and pI-' satisfies ul-'pl-' = O. It follows from ul-'ul-' = vl-'vl-' = - 1 t hat pl-'pl-' p2 = - 1 +a 2 . The geodesic equation fo r vI-' then leads to -aa = (8/3)11-'''PI-'P'' = H so that aa + H( - 1 + ( 2 ) = 0 from which we obtain that - 1 + a 2 oc a- 2 . We conclude that p2 oc a - 2 so that a --+ 1 and vI-' --+ ul-'. The proper velocity any massive t est part icle is damped as a -I and its worldline tends to adjust to t Hubble flow , that is to the worldlines of the fundamental observers. vI-'

3.1.5

Space-time dynamics

Up to now we have described the implications of the Robertson- Walker geometry. investigate the dyna mics of these space-times, we must obtain the equations of moti deriving from the E instein equation. When these equations are fulfilled , we refer these solutions as Friedmann- Lemaitre space-t imes. 3.1. 5.1

Energy-moment um tensor

The energy-momentum tensor, TI-'''' is a symmetric tensor of rank 2. The possi forms this tensor can take are reduced by t he space-time symmetries. Indeed , t most general form compatible with symmetries is

The cosmological solution of Friedmann-Lemaitre

139

B represents the only spatial component (for the fundamental observers); it is therefore identified with the pressure that is thus seen to reduce to a unique number for a homogeneous and isotropic space. A = TjJ,vu jJ,u v == p is the energy density measured by the fundamental observers. In conclusion, the most general form of the energymomentum tensor is that of a perfect fluid (3.24) The energy density, p, and the pressure, P , only depend on time. This form is completely fixed by the cosmological principle and is not an independent choice. Thus, the only remaining freedom is the choice of the equation of state, that is the relation between the pressure and the energy density.

3.1.5.2 Friedm ann and conservation equations Using the expressions for the E instein t ensor and the form (3.24) of the energymomentum tensor, we can easily deduce that the two Einstein equations reduce to two independent equations

(3.25)

(3 .26 ) recalling [see (1.130)] that r;, == 87rG N • As for the matter conservation equation (\7 jJ,TjJ,v = 0) , it reduces to a single equation

I p + 3H (p + P) = o. I

(3.27)

We can convince ourselves that t he three equations (3.25) , (3.26) and (3 .27) are not independent [indeed, by differentiating (3.25) with respect to time and expressing (i / a using (3 .26) and (3 .25) to eliminate H2 , we find (3.27)]. This is a consequence of the Bianchi identities.

3.1.5.3

Covariant approach

In the covariant approach, a Robertson- Walker space is defined by the conditions V/-Lf = 0 for any scalar function f , which implies that ilP = 0, and CJ jJ,V = wjJ, = 0, so that \7 jJ,u v = 8 9jJ,v/3 . Space being isotropic , we can set S = a, and thus H = 8/3 and the Raychaudhuri equation (1.73) gives (i

1 r;, ujJ,u v = - -(p 2 jJ,V 2

3- = -- R a

+ 3P ) + A'

140

The homogeneous Universe

and the generalized Friedmann equation (1. 78) gives 4 ~ 6K 2 (3) R = - 2 = 2KP + 2A - _ 8 2 = 2KP + 2A - 6H2. a 3

We recover the Friedmann equations, but the regime of validity of the Raychaudh and generalized Friedmann equations is larger since they also apply to non-isotro Universes , such as Bianchi Universes (see below).

3.1 .5.4

Equations in con form al time

It will sometimes be convenient to work in t erms of the conformal time. Rememberi that dt = ad7] , we find that the derivative of any quantity X with respect to 7], X', related to that with respect to t by XI = aX. It will be convenient to remember th

X" - HX I = a 2 X.

XI=aX ,

al / a has b een introduced, which satisfies

The comoving Hubble constant H particular

a" aii = - - H 2

H = aH,

a

'

a2if = HI - H2.

We deduce that the Friedmann and conservation equations [(3.25) to (3.27)] take t form

(3.2

I

H =

K

2

-"6 a

(p

A

2

+ 3P) +"3a ,

I pI + 3H (p + P) = o. I

(3.2

(3.3

Notice also that if the scale factor is a power law in cosmic time , then

a(t) ex t n as long as n

3.1.5.5

i=

{==}

a(7]) ex 7] ''.'.n,

(3.3

1.

Eq uation of state

We hence have two independent equations for three unknowns (the scale factor , t energy density and the pressure). We therefore need some additional information solve this system. For that, the most convenient way is to give an equation of sta for the matter in the form P=wp. (3.3

For instance, pressureless matter will be described by w = 0 whereas for radiation , will have w = 1/ 3. For the cosmological constant, A, which corresponds to a consta

4Note t h at here (3) R refers to the Gaussian curvature of the hypersurfaces L;t with metric ~I This explains why (3) R = 6K Ia 2 .

The cosmological solution of Friedmann-Lemai'tre

141

energy density, (3 .27) implies P = - p and thus w = -l. The resolution of (3.27) implies that for any constant w, p ex

a- 3 (l+w).

(3.33)

Comparing this general result with equations (3.25) and (3.26), we notice that the curvature term can be assimilated with a fluid with equation of state w = -1/3. By differentiating w with respect to time and expressing pi with the conservation equation (3.30) we get Wi = -3H(1 + w)(c; - w), (3.34) where

Cs

is the speed of sound, defined by

pi

cS2 =pi For a barotropic fluid w

=

(3.35)

c;, so that w is a constant.

3.1.5.6 Reduced form It is useful to rewrite the Friedmann equations in a dimensionless form in terms of reduced quantities. For that, we introduce the energy density parameters

(3.36) respectively, for the matter , the cosmological constant and the curvature. The matter term can be decomposed as a sum of components with different equations of state as o = I:x Dx with

(3.37) and we will denote by D xo its value today. The first Friedmann equation (3.25) then takes the form of a constraint

(3.38)

Using the solutions of the conservation equation, it follows that

_

Dx - D xo for a constant equation of state 5 as 5 For

a

general

Oxo (Hal H)2 exp [- 3

time-dependent

(~) -3(1+WXO) ao

Wx

= Wxo, so that

equ ation

J:o (1 + wx)d In a].

(Ho)2 H '

of state,

E(a) == it

can

H/H

o

be

can be expressed

checked

that

Ox

142

The homogeneous Universe

(3.3

°

Quantities with an index are evaluat ed today. Thus, to will be the dynamical age the Universe and ao = aCto) . Notice t hat E only depends on x = a/ao = 1/( 1 + Depending on the case, it will be denoted by E(a), E(x) or E( z). For the matter , t he following different components can be distinguished Tot al matter

0

Different equations of state

Om ~

Db

Different species

Dc

where the total matter component is decomposed into matter (w = 0) and radiati (w = 1/3) and then, respectively, into baryons (b), cold dark matter (c) , photons h and neutrinos (v). Of course , other species can be added, whether as a species with new equation of state or as a new component.

3.1 .5.7

Units and conventions

Two conventions are often used for the choice of dimensions. The decomposition any physical length as the product of t he scale factor and a co moving length , C(phys) a(t) C(com) can indeed lead to an ambiguity since we can arbitrarily choose either t scale factor or t he co moving length to have dimension of length. We will use bo conventions according to the situation at hand.

- The scale factor can be chosen to have dimension of length , and X to be dime sionless. This implies that K can be normalized to K = 0, ± l. The value of can t hen be determined by evaluating the Friedmann equations today.

I [aJ ~ [, and [xl

LX

=x,

K = 0, ± 1,

1

ao = Ho

1

J 11- Dol

.

For a Universe such t hat Do = 1 (i. e. K = 0) , ao is arbitrary, which reflects t invariance of the Euclidean space under dilatation. - The scale factor can be chosen to be dimensionless. X will then have dimensi of length , and thus K the dimension of inverse length squared. It is t hen oft convenient to set ao = 1.

[aJ = 1 and [xJ

=

L: K = (0, ±1)/ R~,

We recover that for Do = 1 (i.e. K charact eristic curvature scale.

1

Rc = - aoHo 0) , Rc

= 00 .

1

Jl1 - Do l

.

A Euclidean space has

I 1, the conservation equation implies that p o( a the Friedmann equation (3.25) implies H2 x o-3(1+ru). It follows that a By integrating this relation, we obtain, As seen earlier, if uL

so o<

2

(3.40)

a(l) x ,3{1+u) From this expression, the conformal time can be computed by integrating

giving 4

o li#b if w I -1/3. o(r1)

that

a-Q+3u)/2.

dl :

dt I

a(t)

,

We cleduce that

21 x q' ts;

if

al

(3 41)

;.

and a(4) o( exp ? otherwise.

3.2.1.2

K:0andu:-\

If the matter density is dominated by a cosmological constant, we have H2 x p x so that a x a. By integrating this relation, we get a(t) x eHt

a0,

(3.42)

This space has an accelerated expansion and is called de Sitter space. In terms of the conformal time, the scale factor is of the form

aQi)x-h,

4<0.

This space can also be written in such a way that Chapter 8). ds2

3.2.1.3

:

-dt2

. *+f!

it

(3.43)

has spherical spatiai sections (see

(dx, +sin21d02)

.

(3 44)

K:IIandu+-Il3

It is more convenient to work in conformal time to obtain

{"0i,t(:d}.

Using p

:

Qr_ Jlr: , {)sl, (g) ll \osl

in units such that l1{l

n,(t) in the parametric form

po(alas)-3(1+u) and (3.25), we get

: 1 This equation

t'-t''

_*.

can be integrated to give

144

The homogeneous Universe

(aoa) = (11 -0 00

)1 / 20: {(Sinh o.'TJ)1 / 0: if 1 (sin o.'TJ) 1/ 0: if 0

K = - 1, K = + 1,

(3.4

with 20. = 1 +3w. This solution is valid for all constant w of. - 1/3. The case w = - 1 corresponds to a Universe with no matter since matter can then be absorbed into curvature term. 3.2.1.4

K = 0, w = 0 and A of. 0

Let us mention the solution for a spatially Euclidean model dominated by a pressurel fluid and a cosmological constant [11]

a(t)

=

(1 ) OAO -

1

1/ 3

sinh

2 3 /

(3Tt) '

(3.4

where we a = HOVOAO. This solution is relevant for describing the late time evolut of a flat ACDM model. 3.2.1.5

Other solutions

We present in Fig. 3.3 the profile of the scale-factor evolution in a large number of si ations, including for different kinds of matter, curvature and value of the cosmologi constant. We may notice the presence of periods of slower evolution (a 'hesitating' phase) various lengths depending on the values of the cosmological constant (see, e.g., ca F, L or K). During these phases, the Universe gets closer to an Einstein static Unive (i.e. a = 0) for which

Thus, this is a spherical Universe for which the gravitational attraction is just co pensated by the repulsive effect of the cosmological constant. The behaviour of th solutions depends mainly on the relative values of the cosmological constant and A Another interesting solution is the one obtained for an empty Universe with hyp bolic spatial sections (K = - 1) . This space, called the Milne space, has t he follow metric (3.4

It can be shown that this metric corresponds to a Minkowski metric by an appropri change of coordinates (T = t cosh X, R = t sinh X leads to the Minkowski metric spherical coordinates) , but this space only covers a quarter of the Minkowski spa time.

3.2.2

Dynamical evolution

In order to study the general dynamics of the solutions of t he Friedmann equatio it is interesting to rewrite the system (3.25) and (3.26) in the form of a dynami system for t he quantities 0 , OA and OK (see Refs . [12- 14]).

Dynamics of Friedmann-Lemaitre space-times 3.0

0

<

*·····

c\\\~..

\,>o\}~.. ····

2.0

145

LK

~............

H

1.0

OJ

..................... ...G

0.0

B

A

D

oj

- 1.0 0.0

1.0

2.0

3.0

QmO

3.0

3.0

a

a ao

3.0

2.0 a

ao 1.0

0.0 -3 -2 -1

0

1

2

1.0

0.0 - 3 -2 -1

3

Hot

2.0

0

1

2

0.0 -3 -2 -1

3

Hot

3.0

a

a

ao

A =0

2.0

2

3

a

\

......

1.0

1

1

2

3

1

2

3

3.0

ao 1.0

0

Hot

3.0 Hesi tating ~ (Spherical:

0

2.0

0.0 -3 -2 -1

Hot

0

1

Hot

2

2.0

Hyperbolic

ao 1.0

...:

3

0.0 - 3 -2 -1

0

Hot

Fig. 3.3 Depending on the values of the cosmological parameters (upper panel) , the scale factor can have very different evolutions.

3.2.2.1 Using

Dynamical system

if =

ii/a - H2 and expressing ii/a with (3.26) and H2 with (3.25) leads to H

H2 = -(1 + q) .

(3.48)

q is the deceleration parameter defined by

(3.49) with 'Y == w + 1. The system of Friedmann equations can thus be rewritten using the new time variable p == In( a/ ao). The derivative, XI, with respect to p of any quantity X is then given by XI = X/H.

146

The homogeneous Universe

The evolution equation for the Hubble parameter (3.48) then becomes

+ q)H.

H' = -(1

(3.5

n

If we differentiate n, n A and K with respect to p and use (3.50) to express H noticing that a' = a and that (3.30) can be used to obtain pi = - 3,p, we then obta the system

n ' = (2q + 2 - 3, )n , n~

(3.5

= 2(1 + q)nA '

n~ = 2qn K

(3.5

(3.5

.

It is useless to keep all of these equations since (i) H does not enter in (3.51)- (3.5 can be deduced algebraically using (3.38). and (ii)

n

The dynamics of Friedmann- Lemaitre space-times can thus be studied by considering the dynamical system

(3.54) where q is a function of

nA

and

nK

given by (3.49).

This system associates a unique tangent vector to each point so that only o integral curve passes through each point of the phase space where the tangent vect is defined. If two trajectories crossed each other, the tangent vector at this point wou not be unique. Points where the tangent vector is not defined are called fixed point 3.2.2.2 Determination of the fixed points The fixed points are defined by the conditions solutions of (1 + q)nA = 0,

n~

= 0 and

n~

o and

are thu

(3.5

They represent the equilibrium positions (stable or not). This equation has thr solutions (nK,n A) E {(O,O),(O, l) ,(l, O)} , (3.5 and each of these solutions represent a Universe with different characteristics.

- (n K , n A)

= (0, 0): the Einstein- de Si tter space (EdS). This is a space-time wi no curvature (Euclidean spatial sections) and no cosmological constant. - (nK' n A ) = (0, 1): the de Sitter space (dS). This is a space with no matter b with a cosmological constant and the spatial sections have a positive curvatur This Universe is in eternal exponential expansion. - (nK , n A) = (1 , 0): the Milne space (M). This is an empty space with no cosm logical constant but with hyperbolic spatial sections (K < 0).

Dynamics of Friedmann-Lemaitre space-times

147

3.2.2.3 Stability of the fixed points To determine if these fixed points are attractors (A), saddle points (S), or repellers (R), we should study the evolution of a small perturbation around each equilibrium position. For this we set

== 0,K +WK, == 0,1\ + W1\,

OK 01\

(3.57) (3.58)

with (0,1\, 0,K) the coordinates of a fixed point and (W1\' WK) a small deviation. Rewriting the system (3.54) in the form

(3.59) where FK and F1\ are two functions determined by the system (3 .54) that vanish at the fixed point , it can be expanded to linear order as

~j ' WK

_ p_ _

W

-

(!tA,!tK)

~WKj W

with

p(IT IT ) A,

K

==

(;~: ~~:) aF1\

(3.60)

aF1\

aO K a01\

(ITA ,IT K

)

The stability of a fixed point depends on the sign of the two eigenvalues (A1,2) of the matrix P(ITA,IT K ) ' If both eigenvalues are positive (resp., negative), the fixed point is unstable (resp ., stable). If the two eigenvalues have different sign, it is a saddle point. The eigenvector U.\ 1,2 then gives the stable and unstable directions . - EdS: we have PEdS

= (

3"1 - 2 0 ) 0 3"1 '

with eigenvalues 3"1 - 2 and 3"1- EdS is thus an attractor for "1 Ej- 00, 0[, a saddle point for "1 Ej O, 2/3[ and a repeller for "1 Ej2/3, +00[. The associated eigendirections are U(3,. - 2) = (1,0). - dS: we have PdS

=

(2 =~'Y -~'Y

) ,

with eigenvalues - 2 and - 3"1- If "1 Ej - 00, O[ then dS is a saddle point and for "1 EjO, +00[, it is an attractor. The two eigendirections are given by U(-2)

= (1 , -1).

- M: we have PM

_ (2 - 3"1 - 3"1 ) -

0

2

'

with eigenvalues 2 and 2 - 3"1- If "1 Ej - 00,2/3[ then M is a repeller and if "1 Ej2/3, +00[, it is a saddle point. The two associated eigenvectors are U (2)

= (1 , - 1).

148

The homogeneous Universe

These results are summarised in Table 3.1 and in Figs. 3.4 to 3.6. Table 3.1 Stability of the fixed points as a function of the value of the equation of state, , = w+ 1, (A: attractor, R: repeller, and S: saddle point). ] -

EdS dS M

00,

o[

°

A

N.A.

S S

A R

]0,2/3[

2/3

]2/3, +oo[

S A R

NA A R

R

A S

0.5

E

C

0.5

- 0.5

- 1~

- 0.5

__~~~__~__~__~~__- J 1.5 o

0 -0.5

0

0.5

1.5

QA

Fig. 3.4 Dynamical evolution in the plane (r:lK , r:l A) (left) and (r:lm , r:lA) (right) for, = (w = 0). From Ref. [14].

3.2.3

Expansion and contraction

When the total energy density vanishes, the expansion rate of the Universe can chan sign. For a Universe with non-flat spatial sections and with matter, radiation and cosmological constant , there are four possibilities (see Fig. 3.3) . The expansion lasts eternally starting from the Big Bang. This is the case if K or - 1 for a Universe without cosmological constant.

=

The expansion is followed by a contracting phase leading to a big crunch. This the case for a Universe with K = + 1 with no cosmological constant.

The expansion lasts eternally but was preceded by a contracting phase; the must therefore have been a bounce. This situation can occur when the Univer has spherical spatial sections (K = + 1). - The Universe undergoes a series of contraction and expansion phases; this is oscillating Universe .

149

Dynamics of Friedmann-Lemaitre space-times 1

1.4 Qm
0.8

1.2

1

0.6

" 0.8

cf°.4

cf

0.6 0.2 0.4 0 - 0. 2 - 0. 2

0.2 O ~

0

0.2

0 .4

0 .6

0.8

1

__

~~~~~-L_ _~-w~~

-0.4 -0.2

0

QA

0.2 0.4 QA

0. 6

0.8

Fig. 3.5 Dynamical evolution in the plane (O K , Ot>.) (left) and (Om , Ot>.) (right) for 'Y From Ref. [14].

1

=

1/3 .

E

c:

- 0.2

0

0.2 0.4 QA

0.6

0.8

Fig. 3.6 Dynamical evolution in the plane (OK , Ot>. ) (left) and (Om , Ot>. ) (right) for 'Y = - l. From Ref. [14].

The separation between a Universe with infinite expansion and a Universe with a big crunch is obtained when

2

nAO = 1 - nmo + :3 cos [arct an (S;;;~) - 7r] )

(3.61 )

with SmO = JI2n mo - 11. The separation between a Universe with infinite expansion and a bouncing Universe is obtained when

150

The homogeneous Universe

nmO = 0, or 1

_

o ::; nmo ::; 2'

nAO - 1 - nmO

3 [ + 2(n 1110 )1 /3 (1

- n mo

+ '::'~mO )1 /3

+ (1 - nmo - :=:mo) 1/3] , nAO = 1 - n mo

3.3

2

+ 3" cos [arctan (:=:;;;6) + 7r].

(3.6

Time and distances

The characteristic distance and time scales of any Friedmann- Lemaitre Universe a fixed by the value of the Hubble constant. We define t he Hubble time and distance 1

c

tH = H'

DH = H'

(3.6

The sphere with radius equal to the Hubble radius is called the Hubble sphere. T order of magnit ude of these quantities is obtained by expressing the current value t he Hubble parameter in the following way

(3.6 with h typically of the order of 0.7. We will come back later to the measurement of t he parameters introduced in t his chapter. We obtain t hat

DHo = 9.26 h- 1 x 10 25 m ~ 3000 h - 1 Mpc ,

tHo = 9.78 h - 1 x 10 9 years.

(3.6

All the quantities that will be introduced in this section (distance and time) will expressed in terms of E( z ) defined in (3.39). As can be seen in Fig. 3.7, this functi has some degeneracies that change with t he redshift. By combining observations different redshifts, one can lift these degeneracies and thus use them jointly to bett constrain the cosmological parameters.

3.3.1

Age of the Universe and look-back time

From the definition of the Hubble parameter, it follows that dt = da /a H . Thus, t expression (3 .39) for the Hubble constant gives

da dt = tHo aE(a)'

The age of the Universe is thus obtained by integrating this equality between a = and a = ao, or equivalently between z = 0 and z = 00,

r dx tHo J xE(x)

roo

1

to

=

o

=

tHo Jo

dz

(1

+ z)E(z)'

(3.6

This function is represented for various sets of cosmological parameters in Fig. 3.8.

Time and distances 0.9

0.')<=== == =-=---:--.

0.8

o.

0.7

151

o

0

a<

c:< 0.6 0.5 0.4 0.3 0.1

0.2

0.3

0.4

0.5

QmO

Fig. 3.7 Isocont ours of the funct ion E( z ) in terms of the cosmological parameter (Dmo , DAO) for a J\CDM modeL T he degeneracies at z = 1 (left), z = 3 (middle) and z = 10 (right) are not the same. The code of grey shade is arbitrary.

The analogous expression in conformal time is 1

'T}o

=

tHo

10 x2~x) ·

(3.67)

In a similar way, the age of the Universe at t he t ime when a photon with redshift

z. was emitted is defined by

roo

t( z.) = tHo Jz. (1

dz

+ z)E(z) ·

(3.68)

The look-back time is the difference between the age of the Universe and its age when the photon was emitted by the source. The previous relations lead to

llt( z*) = to - t( z*) = tHo

3.3.2

Jro ' (1 + dz z)E(z )·

(3 .69)

Comoving radial distance

The co moving radial distance X of an object of redshift z t hat is observed by an observer located at X = 0, is obtained by integrating along a radial null geodesic. The form of the metric (3.4) gives dX = dt /a, so that dz

H( z) . It follows that

aoX( z*)

=

DHo

Jr' o

dz

E( z )·

(3.70)

The dependance on ao simply reflects the arbitrary choice of the system of units. This function is represented for various sets of cosmological parameters in Fig. 3.9.

152

The homogeneous Universe

9

11

10

8 9

1.0

1.5

2.0

2.5

QmO

Fig. 3.8 Isocont ours of the functi on to as a fun ction of the cosmological parameters. curve indicat es t he value of to in units of 10 9 years. The grey zone represents the zo paramet ers for which there is no Big Bang.

For a fiat Universe with no cosmological const ant, we find, for instance,

(

For a Universe with a non-vanishing curvature, we can give the expression for radial dist ance in units of the curvature radius

( 3.3.3

3. 3.3.1

Angular distances

Comoving angular diam eter

The co moving angular diameter relates the comoving transverse size of an obj ect the solid angle under which it is observed . It is defined by the relation

between the solid angle and the apparent surface of an obj ect. Remembering t h comoving sphere centred a t X = 0 and with comoving radius X has a comoving su s (co m ) = 47f jk (X ), we deduce that

Rang (z) = jf( [X( z)] .

(

Time and distances

153

................__... -_.. ... .... .

..:.: :.: ..

-:":;'

3

5

4

Redshift z

Redshift z

Fig. 3.9 (left): Age and look-back time in units of the Hubble time, t( z )j tHo and 6t( z )j tHo as a function of z for three cosmological models defined by (Dmo, DAO ) = (1 , 0) , (0.05 , 0) and (0.2 , O.S), respectively, as plain , d otted and d ashed lines. (right) : Comoving radial distance in uni ts of t he Hubble length , aox(z ) j DHo, as a fun ction of z for the same cosmological models.

Notice that t his function is not necessarily increasing wit h z . Indeed , if K = +1 , then two objects with the same comoving size, located at X and 7r - X will have the same comoving angular diameter.

3.3.3.2 Angular dis tance The angular dist ance is the not ion that generalizes the concept of parallax. It is defined , for a geodesic bundle converging at t he point where the observer is, as the ratio between the transverse physical size of an object and the solid angle under which it is observed . The physical area of the source is related to the co moving one by dS p hys = a 2 dS com . With this defini t ion phy s com dSso 2 urce 2 dssource D A= 2 = a 2 dD o b s dD obs This quantity will also be denoted by T o . Using the definition of the co moving angular diameter , int roduced in the previous section , we obtain

aoRang(z ) l+z

(3.74)

3.3.3.3 R eciprocity theorem In the same way, we can define the angular distance from the source. It is defined using a geodesic bundle diverging from the source as the ratio between the physical transverse size of t he observer and the solid angle under which it would be observed from the source, phys _ .2 d n 2 dS o bs

-

rs

~ [,s o urc e'

154

The homogeneous Universe

It can be shown that if the tra jectories of the photons are null geodesics and if t equation for the geodesic deviation is valid, then there exists a relation between T s a To

(3.7

called the r eciprocity theorem (see Ref. [15] for a demonstration) . This relation cann be directly checked since Ts cannot be observed directly. 3 .3 .4

Luminosity distance

3.3.4.1

Defini tion

The luminosity distance relates the luminosity of a source, locat ed at a comoving rad dist ance X from the observer , to the observed flux as A.

_

'Po bs -

Lsource (X) 47rD[

(3.7

To relat e the luminosity of the source and the observed luminosity, notice that t luminosity of t he source is given by the emitted energy per unit of emission tim L source = !:::"Eem/ !:::"t em . Due to the expansion of the Universe, !:::" Eobs = !:::" Eem a/ and !:::"to b s = !:::"temaO /a (because !:::,.'rJobs = !:::"'rJem for two null geodesics) so that L obs L so u rce (l + z )-2 The flux received by the observer at X = 0 is given by

using

S (phys) = a6 s (com).

Identifying this with the definition (3.76), it follows t hat

I DL( z ) =

ao(1 + z )fJ( [X (z)].

I

(3.7

The luminosity of the source is indeed unknown. However , by comparing some sour to a reference source for which the redshift z. and the distance are known, we obta tha t

By extracting a family of astrophysical obj ects, called s tandard candles, wit h t he sa luminosity, the ratio of their fluxes can be used to measure the luminosity dist ance To finish , let us not e that deriving (3.77) allows us to extract the Hubble rat e a function of redshift as 1

H( z )

[

1 + DJ(oHg

(~) 2]-1/2~ (~) . 1+ z

dz

1+ z

Time and distances

,

" ... - - -

~ ~

155

15

... :..- . .......... .. ......... .

" ..... .

,''' " f

...

I,:

I :'

"

I:' I:'

2 3 Redshift z

4

5 Redshiftz

Fig. 3.10 (left): The angular distance D A (z) / DHo as a function of the redshift z for three cosmological models, defined by (n mo , n AO ) = (1,0), (0.05, 0) and (0.2,0.8), respectively, as plain, dotted and dashed lines. (right): The luminosity distance Ddz) / DHo as a function of the redshift z for the three same cosmological models.

3.3.4.2 Distances duality relation If the number of photons is conserved, which is usually the case if the Maxwell equations are valid, and if the absorption from the medium in which they propagate is small, then (for a proof, see Ref. [15]) DL

= rs(1 + z),

so that (3 .75) implies a duality relation between the angular and luminosity distances (3.78) Unlike (3 .75) , this relation can be tested experimentally [16]. 3.3.4.3

Distance modulus

The luminosity distance is usually expressed in terms of the distance modulus, m - M, defined such that it vanishes if the source is located conventionally 6 at 10 pc, m - M

= - 2.51og [¢(z)/¢(10pc)] .

(3.79)

Using (3.76) and noticing that at 10 pc, the luminosity distance is also 10 pc, we obtain m - M = 5 1og[DL(z)/1O pc]. Using the expression 10 pc = DHo/3 X 10 8 h , it can be deduced that m - M

= 25 + 5 log [3000 DL ] - 5 log h. DHo

(3.80)

6Note that the -2.5 factor is a rbitrary a nd was chosen to match the magnitudes of star s initia lly defined by Hipparcos .

156

The homogeneous Universe

m is the apparent magnitude and the absolute magnitude M is defined by comparis to the Solar luminosity M = - 2.5 log ( : 8 )

(3.8

+ 4.76

where L 8 ~ 3.8 x 10 26 W is the Sun luminosity and Mo = 4.76 its magnitude. Wi such a definition , the brighter an object is, the weaker is its magnitude.

3.3.4.4

Magnitudes

The luminosities considered previously were the luminosities integrated over the enti electromagnetic spectrum, defining the bolometric magnitude. It is difficult to measu these bolometric magnitudes and, in general, some filters are used, selecting a speci frequency band. The magnitude Mx in the band X is then defined by

Mx = -2.5 10g ( Lx ) L8X

+ Mo x.

(3 .8

An example of different frequency bands used in astronomy is given in the table belo Band

Central

Width (run)

MO

66

5.61

wavelength (nrn)

365

U (u ltraviolet)

B (blue)

445

94

5.48

V (visible)

551

88

4.64

R (red)

658

138

4.42

I (infra-red)

806

149

4.08

J

1200

213

3.64

K

2190

390

3.28

00

4.76

Bolometric

The flux detected in the frequency band 6.vx was emitted in the frequency ban (1 + z )6.vx . The flux received per frequency unit is thus A.

() - (1+ )L v [v(l+z)] z 47rDl '

'l'X v

(3.8

where Lv is the source luminosity per frequency band. It follows that

¢x (v)

=

(1 +D~) 47r L

r

Lv[v(l

+ z)]dv.

(3.8

} tlvx

The correction induced by the deformation of the spectrum does not appear when t source is at 10 pc so that the distance modulus in the band X defined by mx - Mx - 2.5 log[¢x/¢x(10 pc)], is given by

mx - Mx = 510g ( DL ) 10pc

+ K.

(3.8

Time and distances

157

The K-correction represents the distortion of the spectrum due to the cosmological expansion K

= -2.51og(1 + z) - 2.51og {

Lvx Lv[v(l

J

+ Z)]dV}

[]

D.vx Lv v dv

.

(3.86)

Finally, note that other effects of astrophysical origin, such as the absorption by the interstellar or intergalactic medium, can affect the apparent magnitude. We shall discuss them in the next chapter. 3.3.5

Volume and number counts

The geometric properties of space-time also playa role in the determination of the distribution of a family of objects (for instance, galaxies) in a given redshift band. Let us consider such a family with proper number density n(z) and proper radius r*. First, we can determine the probability for a line of sight to intersect such a galaxy with redshift lying between z and z+dz. This probability is proportional to the surface area of the galaxy, to the density, and to the distance light propagated in this redshift interval,

(3.87) The probability to intersect an object with redshift smaller than z , called the optical depth, is then given by

1 Z

T(Z)

=

2 7rr*DHo

o

n(z) (

dz) (). l+zEz

In particular, for galaxies diluted by the cosmic expansion, n(z) Universe with 0 = 1 and OA = 0 we find

(3.88)

= no(1 + z)3, in a

(3.89) If no rv 0.02h 3 Mpc~3 and r* rv 10h ~ 1 kpc, T rv 4% at z = 1 and 100% at z rv 20. The proper volume contained in a solid angle d0 2 and between the redshifts z and z + dz is the product of the area D~.d02 and the length D Ho dzj(l + z)E(z), so that

~ - D d0 2 dz -

Ho

(1

iI[x(z)] + z)3 E(z)'

(3.90)

The number of galaxies in a solid angle d0 2 and in a redshift band dz is then dN d02dz

iI[x(z)]

= DHon(z) (1 + z)3 E(z)'

Galaxies number counts are among the oldest cosmological tests (see Ref. [1]).

(3.91 )

158

The homogeneous Un iverse

3.4 3.4.1

Behaviour at small redshifts D ecelerat ion parameter

We can expand the scale factor around its value today as

a(t) with 0 < to - t

«

=

1 2 ao [ 1 + Ho(t - to) - 2qoHo(t - to) 2

+ ...] ,

(3.9

to. The factor qo is the deceleration parameter7 and is defined by qo

== -

a!2It=to '

(3 .9

so that the Universe accelerates if qo < 0 and decelerates otherwise. Using the Frie mann equations to express 0,0 / ao , we easily find t hat

for a ACDM model, since the contribution from radiation is negligible today. For a arbitrary matter composition, the deceleration parameter is given by

(3.9

Thus , in order to have qo < 0, there should be at least one cosmic fluid (dominatin today) with equation of state Wx < - 1/3 . We stress that t he expansion (3 .92) on relies on the hypothesis that the Universe is described by a Friedmann- Lemaitre spac time. In particular , it makes no assumptions on t he matter content of the Univers contrary to (3 .94). 3 .4.2

3.4.2.1

Expression of the time and distances

Small redshift expansions

At small redshifts, we can expand the function E(z) as

neglecting t he contribution of radiation. Thus, we conclude t hat the look-back t ime

6.t( z ) =

tHo

[1 - ~(qO + 2)Z] z + 0( z3 ).

(3.9

The radial distance takes the form

7The choice of sign is historical. At the time qO was introduced it was thought that the expansi of the Universe was d ecelerat ing so t hat qO > 0 was exp ected. C hapter 4 details why we now thi t h at qO < O.

Behaviour at small redshifts

159

the comoving angular distance,

Rang(Z)

=

DHo ao

[1 - ~(qO + I)Z] Z + O(z3), 2

(3.97)

the angular distance, (3.98) and the luminosity distance, (3.99) Locally (z « 1) all the distances reduce to DHoz and the deviations only depend on the value of the deceleration parameter. A distance-redshift diagram will thus be sensitive to Ho in its region z « 1 (slope of the diagram), to qo mainly in its intermediate region (z ~ 1) and to the full set of cosmological parameters for larger z. This illustrates the evolution of the degeneracy of the function E(z) represented in Fig. 3.7. 3.4.2.2

Time drift of the redshift

Suppose we observe the same galaxy at to and to redshift given by

+ Mo.

There will be a change in its

Now, null geodesics are straight lines in conformal coordinates so that br] = br]o Malao = btla from which we deduce that bzl(1 + z) = [Ho - H]br]o to first order in 61)0 and thus

bz = Ho [(1 uto

£

+ z)

- E(z)] ,

(3.100)

where E is defined by (3.39). This gives the time evolution of the redshift of any object at redshift z as a function of the cosmological parameters. At low redshift, it reduces to bzlbt = -qoHoz + O(z2). Indeed, this drift is extremely small and of the order of az I at ~ 10- 8 I century. However, it gives a direct information on E(z) and thus on the cosmological parameters (see Refs . [17,18]). Note that the redshifts are free of the evolutionary properties of the sources, and that a direct detection of their drift would represent a new (and yet never exploited) and direct approach in observational cosmology to constrain the evolution history of the Universe, alternative to those based on luminosity or diameter distances. It can be of importance in the study of the dark-energy problem (see Chapter 12).

160

3.5

The homogeneous Universe

Horizons

An horizon is a frontier t hat separates observable events from non-observable on T wo very different concepts of horizon have t o be distinguished in cosmology ( Refs. [10, 19]) . The different quantities defined in t his section are represented in Figs. and 3.12. 15

~

Geodesic of

>.

'"0

:::-

a particle _ entering into our particle horizon today _

10

~

0)

B

0. 8 0.6

(f'

:3

t)


"

. _ 0.4

.~

0)

0;

"

0

CIJ

U

- 0.2 0 0

10

20

40

30

50

Comovin g di stance, aoX, ( 109 light-years) 15

LO

~

>. 0

~

:::-

0. 8

IO

~

" E

~

"

.~

0.6

'" ..'": .9

~

"


5

0)

0;

"

CIJ

0

U

20

30

40

50

Physica l di stance, ax, (10 9 light-years)

Fig. 3.11 The particle horizon at a given time (3. 102) is defined at different moments the geod esic of the most distant observable comoving particles at this moment (plain verti lines). T his surface can be visualized as the intersection of the particle horizon (3.103) w a constant-time hypersurface . The equation of this surface is given by X = ¢(t) in comov coordinates (top) and by X = u(t) in physical coordinates . This diagram represents a U verse with t he following cosmological parameters (!lmo , !lAO) = (0. 3, 0.7) and h = 0.7. Fr Ref. [20].

3.5.1

Event horizon

For an observer 0 , the event horizon is the hypersurface that divides all events in two families: the ones that have been , are, or will be observable by and the on that are for ever outside the observational perimeter of 0. A necessary and sufficient condition fo r the existence of an event horizon is t h the integral OO dt

°

J

a(t)

Horizons

161

Physical distance (10 9 light-years)

Comoving distance, GoX (10 9 light-years) 60

~

50

;>,

,

lOpO

,,'

3.0

o~/

2.0

~~,

b-~~--~----~----:~r7~~-*~--~~____~__~~OT'~'______~1.0

40

0. 8
30

04

~

0

¥

'-'

oJ

.~

OJ

E = 0

<2

€


20

0.1

U

"5"

'"

10 LLL-_'------'L--'---_"----_~_-'.L-_

- 40

_"'____

__"__

__"__

_"__

- 20 20 o Como ving di stance, GoX ([0 9 light-years)

___L_"____L_

40

_i:>....:J

0.01 0.00 I

60

Fig. 3.12 Universe diagrams for a Friedmann- Lemaitre space with paramet ers OmO = 0.3, [lAO = 0.7 and h = 0.7 . The first two diagrams represent , respectively, the physical distance, a(t)x, and the comoving distance , X, in terms of the cosmic time, t (left scale) or in terms of the scale factor, normalized to 1 today (right scale). The last diagram represents the comoving distance in terms of the conformal time, 7). The dotted lines represent the worldlines of comoving observers and our world line is the central vertical line. Above each of these lines is indicated the redshift at which a galaxy on this world line becomes visible for the central observer. We represent the past light cone for the present central observer and the event horizon corresponding to a similar light cone originated from time-like infinity (plain bold line) . We also show t he Hubble sphere, defined by (3.63) (plain light line) and the particle horizon , defined by (3 .103) (clashed line) . From Ref. [20].

is convergent . If this integr a l converges , then at a ny time to , there exists a worldline

1

00

X = Xo =

to

dt

a(t) ,

such that the photon emitted at to in Xo towa rds the origin , reach es X

=

0 at t

= +00.

162

The homogeneous Universe

Any photon emitted at to in X > Xo never reaches the origin and any photon emitt at to in X < XO reaches the origin in a finite time. The de Sitter space, for which a = exp(Ht ), has such an event horizon, since

=

Xo

=

dt e- Hto exp H t = ~ <

1 to

00.

As a second example, let us consider Universes wit h scale-factor evolution of the for a(t ) = tn. The integral

J=

c ndt

converges if and only if n > l. Recalling that for a flat Universe (K = 0) n is relat to the parameter w of t he equation of st at e by 3n = 2/ (1 + w), we find that an eve horizon exist s if 1 (3 .10 w < - '3 {==} p + 3P < 0, i.e. if the strong energy condition is violated. 3.5.2

Particle horizon

For an observer 0 at to, t he particle horizon is t he surface of the hypersurface t = dividing all particles of the Universe into two non-empty families : the ones that ha already been observed by 0 at the time to and the ones that have not. For ea time to, it is thus t he intersection between the geodesics of the most dist ant comovi par ticles that can be observed, with the hypersurface t = to. Hence, it is a tw dimensional space-like surface, which reduces to a sphere of centre 0 for an isotrop and homogeneous Universe. A necessary and sufficient condition for the exist ence of a particle horizon is th the following integral be convergent:

· 1

dt

dt a(t) '

J_=

or

a a(t )

depending on whether or not a(t) continues for negative values of t . Indeed , at any time t o, any particle such that X > a - 1dt has not yet be observed by an observer 0 at the origin. The two-dimensional surface

J;o

X = ¢(to ) =

i

dt

to

a

-

a(t)

(3.10

defines t he particle horizon at a given time to and thus divides all particles into tw sub-families: the ones t hat have been observed at to or before to [X :s: ¢(to )] and th ones t hat have not yet been observed [X > ¢(to) ]. This surface is called the partic horizon of 0 a t to. It can be seen as the section at t = to of t he space-time surfa (J = ¢ (t ). We thus define the particle horizon as the hypersurface X

=

(J

= ¢(t ).

(3 .10

The hypersurface (3 .103) is the fu t ure light cone emitted from the position of t observer at t = O. Since a(t ) is a positive function , when ¢(t ) exist s, it is an increasi

Horizons

163

function of t so that as time elapses, more and more particles are visible from O. In conformal time, t he particle horizon is a cone and the part icle horizon at to is a sphere. If ¢(t ) is finite when t t ends t o infinity, then the Universe has an event horizon since only the particles with X ::; ¢ (oo) will be accessible to the observer O. This is represented in Fig. 3.11 where the vertical lines represent the particle horizons at various times, whereas the d ashed line represents the surface defined by X = ¢(t) . As an example, let us consider Universes wit h scale- factor evolution of the form a( t ) = tn. The integral

lC

n

dt

converges if and only if n < 1. Since for a flat Universe (K = 0) , 3n = 2/ (1 + w) , it follows that there exists a particle horizon if 1

w > -3

~ p

+ 3P > O.

(3.104)

From the conditions (3.101) and (3 .104), we thus conclude that a Universe containing a single fluid with constant equation of st ate can have either an event horizon or particle horizon, but not both at the same time. The two types of horizons are thus mutually exclusive in this particular case. The equation (3.102) gives the co moving radius of the particle horizon. Its physical diamet er at a time t 2 for a n event that occurred at tl < t 2 is thus (3.105) It can be checked that if t l and t 2 are two events from an era dominated by a fluid with const ant equation of st at e w , then _ 6(1 + w)

D p h (t 1 , t 2) -

[ 1 +3w t2 1 -

(h) t2

l

(1+3W / (3+3W l ]

.

(3.106)

If tl « t2 t hen t he horizon diameter is proportional to t he Hubble radius (3.63) at the time t 2

(3.107) The particle horizon is at t he origin of the 'horizon problem ' of the st andard Big-Bang model (see Section 4.5.2 of Chapter 4). 3.5.3

Global properties

To study the global structures, and in particular at infinity, of the cosmological spacetimes , we shall use the representation introduced by P enrose (see R ef. [10] for a general study). It is based on the idea of constructing, for any manifold M with metric g/J>v , another manifold ;\It with a boundary 3 and metric 9/J>v = W g/J>v such that M is conformal to the interior of ;\It , and so that the 'infinity' of M is represented by the 'finite' hypersurface 3. The last property implies that W vanishes on 3. All asymptotic properties of M can be investigated by studying 3 (see Ref. [21 ]) .

164

The homogeneous Universe

3.5.3.1

The Minkowski space-time as an example

The Minkowski metric (1.10) in spherical coordinates can be written in terms of t advanced and retarded null coordinates, respectively, defined by v = t + r and u = t as 1 ds 2 = - dvdu + 4(v - u) 2d0 2,

where v ?: u and v and u range from -00 to +00 . No terms of t he form dv 2 or d appear because the surfaces of constant v (or constant u) are null surfaces. We can make these coordinates vary in a finite interval by introducing tan V = v ,

tanU = u ,

so that - 71"/ 2 < U :S V < 71"/2. Then , introducing the coordinates T and R by

T = U + V,

R = V -U,

with - 71" < T + R < 71" ,

- 71" < T - R < 71" ,

(3. 10

the Minkowski metric turns out to be conformal to the metric 9 given by ds 2 = - dT 2 + dR 2 + sin 2 R d0 2,

with conformal factor W = {2 sin[(R + T) / 2] sin[(T - R) / 2]} - 2. The Minkowski spa time is thus conformal to the region (3.1 08) of the Einstein static space-time (a cyl der S3 x IR. that represents a st atic space-t ime with spherical spatial sect ions). T boundary of this region therefore represents the conformal structure of infinity of t Minkowski space-time. This boundary can be decomposed into - two 3-dimensional null hypersurfaces , .:7+ and .:7- defined by

(3 .10

or equivalently by T = ±( 71" - R) with R E [0,71"]. T he image of a null geode originates on .:7- and t erminates at .:7+, which represent future and past nu infinity. - two points i+ and i - defined by U = V =

±-=2'

(3 .11

or equivalently by R = 0 and T = ± 71". The image of a t ime-like geodesic origina at i - and terminates at i + . They represent future and past time-like infinity, th is, respectively, t he start- and end-point of all time-like geodesics. - one point iO defined by U = - V= --= (3.11 2' or equivalently by R = 71" and T = o. It is t he start- and end-point of all space-l geodesics so that it represents spatial infinity.

Horizons

165

The fact that i± and iO are single points follows from the fact that sin R = O. These are coordinate singularities of the same type as the one encountered at the origin of polar coordinates. The manifold Nt is regular at these points. J - is a future null cone with vertex i- and it refocuses to a point iO that is spatially diametrically opposite to i - . The future null cone of iO is J +, which refocuses at i+. Note that the boundary is determined by the space-time and is unique but that the conformal ext ension space-time (here the Einstein static space-time) is not fixed by the original metric and is not unique since another conformal transformation could have been chosen. For any spherically symmetric space-time, the region (3.108) can be represented in the (T, R) plane by ignoring the angular coordinates so that each point represents a sphere S2 In this representation the Minkowski space-time is a square lozenge (see Fig. 3.13) . Null geodesics are represented by straight lines at ±45 deg running from J - to J + .

i+

i+

J+

(1) =

+ on )

:

I

I I

I I I

I

I I

I

J -

(I) = 0)

J -

(I) = -

00.')

I

Fig. 3.13 Conformal diagrams of the Minkowski space-time (left) and Friedmann-Lemaitre space-times with Euclidean spatial sections with A = 0 and P > 0 (middle) and de Sitter space (right) .

3.5.3.2

Friedmann-Lemaltre space-times with K = 0

When written in terms of the conformal time, Friedmann- Lemaitre space-times with Euclidean spatial sections are conformal to Minkowski. It follows that they map on a part of region (3 .108) representing Minkowski space-time in the Einstein static Universe. The actual region is determined by the range of variation of 7]. For A = 0 and P > 0, 0 < 7] < 00 so that it is conformal to the upper half of the Minkowski diamond defined by T > 0 (see Fig. 3.13) with a singularity boundary, T = O.

166

The homogeneous Universe

3.5.3.3 de Sit ter space-time

The metric of t he de Sitter space-time is given by (3 .44) with t varying in R It spherical spatial sections and is conformal to the whole Einstein static space-time. study its infinity we define the coordinates

R =X,

T = 2 arctan

(eHt )

,

(3.1

with 0 < R < 7r and 0 < T < 7r. It is then conformal to a square region of the Eins static space with conformal factor W = H - 2 cosh2(Ht) (see Fig. 3.13). We see that this conformal diagram has a space-like infinity both for time-like null geodesics, in the future and in the past.

3. 5.3.4

Friedmann- Lem aitre space-times with K = ± 1

Friedmann- Lemaitre space-times wit h spherical spatial sections (K = + 1) are con mal to the Einstein static space-time (with 17 --7 T and X --7 R). They are thus map into the part of this space-time determined by the allowed values for 17· There are three general possibilities. When A = 0, 17 varies from 0 to 7r if P = 0 from 0 to a < 7r when P > O. The Friedmann-Lemaitre space-time is thus confor to a square region of the Einst ein st atic space-time so that both J + and J space-like and represent two singularities. When A i- 0, 17 can vary from 0 t o 00 the hesitating Universes (such as examples Nand M in Fig. 3.3) or from - 00 to for t he b ouncing Universes (such as examples G and H in Fig. 3.3). The Friedma Lem aitre space- time is thus conformal to either half of or the entire Einstein st space-time. As in the case of the de Sitter space, the conformal region will be a squ of the Einstein static space and J + and J - are also space-like but do not necess represent a singularity. In the case of Fr iedmann- Lemaitre space-times with hyperbolic spatial secti (K = -1 ), the metric can be shown to be conformal to part of t he region (3.108) means of the coordinat e transformation

T= R =

arctan [tanh ( 17 : arctan [tanh ( 17 :

X)] + arct an [t anh (17 ; X)] , X)] _arct an [t anh ( 17 ; X)] .

Again, the exact shape of this region depends on the matter content (equation of s and A). We see in these examples that some parts of the boundary correspond to the B Bang singularity a = O. When P > 0 and A = 0, the initial singularity is space-l which corresponds t o the existence of a particle horizon. When K = + 1 and A = the future boundary is space-like, which signals the exist ence of event horizons for fundamental observers. For most physically interesting space-times, J + and J - are either space-like null , which is closely related to the existence of one of the two types of horizon descri above.

Beyond the cosmological principle

167

If .]- is space-like, the worldlines of the fundamental observers do not all meet on .]- at the same point . For any particular observer and event on its worldline close to J -, the past light cone of this event will not intercept all the particles in the Universe before it reaches .]- and there will be a particle horizon. If .]- is null , it is expected that all the worldlines of the fundamental observers pass through the vertex i- so that the past light cone of a point P will intercept all the worldlines and there is no particle horizon. If .]+ is space-like, the worldline of any fundamental terminates on a point 0 of J+ and the past light cone of 0 divides the Universe into events that can be seen by the observer and events that he can never see; there is en event horizon. If .]+ is null , all the worldlines will pass through the vertex i+ , so that 0 = i+ and its past light cone is .]+ and there is no event horizon. In conclusion, .]- space-like <===>- existence of a particle horizon, .]+ space-like <===>- existence of an event horizon.

,....

J + {}I

0

=;: I

k

1

II

II I ;><

J+

...

( 1/ = + ,~ ,c- )

a

II ;><

1 1

1 ;>< 1 1 II

I

"1 1

..J.... I

J -

(II= 0 )

J+{ I/= +'X ·I I

I

....

1 1

..

(I I = 0 )

a

II

,,

, I

;><

1 1

;><

II "1

1 1

I I

J - (17 =

- cc· )

Fig. 3.14 Conformal diagrams of the Friedmann- Lemaitre space-times with spherical spatial sections with P > 0 for A = 0 (left), and A =F 0: hesitating case (middle) and bouncing case (right). T he dotted line represents a singularity. The three diagrams differ only by the nature of J±, even though they are all space-like surfaces .

3.6

Beyond the cosmological principle

The Friedmann- Lemaitre space-times have spatial sections with constant curvature, i.e. three-dimensional homogeneous and isotropic surfaces. These symmetries are characterised by six Killing vectors (Chapter 1) that completely fix the geometry of the spatial metric , up to a number, K. Reciprocally, we can construct some solutions by restraining the number of symmetries and by imposing the structure of their Lie algebra. Numerous solutions of Einst ein's equations are known in this case and have been classified [22- 24].

168

The homogeneous Universe

We give a general description of the classification of cosmological space-times cording to t heir symmetries and t hen we present two cosmological solutions in or to illustrate the richness that this provides. 3.6.1

Classifying space-times

Even t hough the previous discussion focused on a particular class of cosmological so tions with maximally symmetric spatial sections , most solutions of E instein's equati have no symmetry at all. This simplification was motivated by the cosmological pr ciple but it is important to have an idea of exact solutions with less symmetry particular to construct counterexamples to some conclusions or t est their robustne

3.6.1.1

Symmetries and Killing vectors

Symmetries of a given space are t ransformations of this space into itself leaving metric unchanged as well as all its geometrical properties . As explained in Chapte a continuous symmetry can be characterized by a Killing vector field , ~ , satisfying

The group of isometries of a Riemannian manifold forms a Lie group (that is a gro that is also a smooth m anifold with a group operation xy - l that is a smooth m into t he manifold itself; see Chapter 2). The set of Killing vector fields generators t hese isometries forms the associated Lie alge bra: it is a vector space, since t he lin combination of any two Killing vectors is also a Killing vector, of finite dimensio since the maximum number of Killing vectors for a space of dimension n is n(n + 1) reached for maximally symmetric spaces. The product operation is defined by commutator of two Killing vectors

(3 .1 of coordinates [~a, ~b]'" = (8f3~b) ~g [~a, [~b , ~cll

+

[~b, [~c,~all

+

[~c , [~a , ~b ll

-

(8f3~~n~f and it satisfies the J acobi ident

= O.

It is easily checked that if ~a and ~b are two Killing vectors then [~a, ~bl is als Killing vector (since £[l;a, l;blgj.LV = 8!L[~a , ~blv + 8v[~b,~al!L = 0). Considering a ba {~a}a = l..r of this algebra , any Killing vector can be decomposed on this basis a linear combination with constant coefficients. It follows t hat

(3 .1

C~b are t he structure constan ts of the Lie algebra. They are antisymmetr -Cba ' and must satisfy, from the J acobi ident ity C~[bC~d) = O. These relati are integrability conditions t hat ensure that t he Lie algebra is defined in a consist way.

where C~b =

3.6.1.2

Defini tions

T he action of a group of isometries defines orbits in the space where it acts and dimensionality of these orbits characterizes t he symmetries.

Beyond the cosmological principle

169

T he orbit of a point P is the set of all points into which P is transformed by the action of the isometries of the space. The orbits are thus invariant sets. An invariant variety is a set that is globally left invariant by the action of the group of isometries. It will be larger than all t he orbits it contains. T he orbits are t he smallest invariant subspaces (for all the isometries in the group) . An orbit that reduces to a single point is a fixed point. It is a point where all Killing vectors vanish. We distinguish general points where the dimension of Killing Lie algebra takes the value it has almost everywhere from special p oints, such as t he fixed points, where the dimension is lower. The group of isometries is transitive on a space S if it can move any point of S into any other point of S. Orbits are the largest spaces through each point on which the group is transitive and can be called surfaces of transitivity. Their dimension, s, has to be smaller than or equal to t he dimension of t he group of translation, s = dim(surfaceoftransitivity)

~

n.

The isotropy group, at a point P, is the subgroup of isometries letting that particular point be fixed. It is generat ed by the Killing vectors that vanish at P. Its dimension q is smaller t han the dimension of the group of rotations, n(n - 1) / 2, q

= dim(isotropygroup)

~

1 "2n(n -1).

The dimension of the group of isometries , r , is the sum of the dimensions of the isotropy group and of t he surface of transitivity. It follows t hat r has to be smaller t han the maximum dimension of the Killing Lie algebra, that is n( n + 1) / 2, that is for a space of constant curvature. r = dim(groupofisometries) = s

+ q ~ ~n(n + 1).

The group of isometries of a space is thus characterized by two of the t hree numbers (S, q,T) and by the structure constants of t he group (C~b)' 3.6.1.3

Classification of cosmological models

For an = 4 dimensiona l space-t imes, r ~ 10 and s can run from 0 to 4. The dimension of the group of isometries being a global property of t he space, r must not change over space. Indeed, the real Universe has no symmetry (r = 0). For cosmological spaces, we assume that p + P > 0, which implies that q E {3, I , O} because the congruence of t he worldline of the fundamental observers is invariant, so that the isotropy group at each point must be a subgroup acting orthogonally to uf.' and it can be shown t hat there is no subgroup of 0(3) of dimension 2. Furthermore, q can vary over space but not over an orbit and it can be greater at special points where s is smaller. The space will be said to be isotropic for q = 3 (this is the case of the Friedmann- Lemaitre spaces), locally invariant under rotation for q = 1 (there is a preferred spatial direction) and anisotropic for q = O. Table 3.2 summarizes all t he possibilities according to q and s. For more details, see Refs. [7,22- 24].

The homogeneous Universe

170

Table 3.2 Classification of cosmological models according to their isotropy and homogeneity properties . We recall that r = q + s :::; 10.

q= 6

8= 3

8= 2

spatially homogeneous

inhomogeneous

Minkowski , de Sitter anti-de Sitter

none

none

Einstein static

Friedma nn- Lemaitre

none

q= 1

Bianchi Kantowski- Sachs

Lemaitre- Tolman- Bondi

q= O

Bianchi

q= 3

3.6.2

8= 4 space-time homogeneous

Universe with non-homogeneous spatial sections

3.6.2.1 Lemaitre- Tolman-Bondi space-time

Among all the non-homogeneous spaces, the Lemaitre-Tolman- Bondi space-tim the simplest. It enjoys a spherical symmetry, so that s = 2, q = 1 and r = 3 (but n that the origin is a special point with q = 3 and s = 0). Its metric takes the gen form ds 2 = -dt 2 + S2(t , r)dr2 + R2(t , r)dS1 2. (3. 1 The Einstein equations for a fluid of dust (P = 0) reduce to 2

S (r, t)

= 1_

(R')2 K(r)r2

R? = 2M(r) _ K(r)r2

R M'(r) 47rG N P(r, t) = R2 R' '

(3.1

(3 .1

(3.1

where a prime and a dot represent the derivatives with respect to rand t , respectiv M(r) and K(r) are two integration functions. The Friedmann- Lemaitre space-ti correspond to the special case R = a(t)r and K(r) = K. These equations can integrated to give the solution

R(rJ , r)

=

M(r) d