Astroparticle Physics: Proceedings of the First Ncts Workshop, Kenting, Taiwan, 6-8 December 2001

143

shrinking in an accelerating way, or " implosively". Using again the complex scalar field as our language, we then have "imploding solitons". In what follows, we attempt to solve the problem of an exploding soliton, assuming that the values of both the potential parameters /J? and A are fairly stable during the period of the soliton expansion. The scalar field must satisfy:

7^r^)-W = v{(t')-

(12)

The radius of the soliton is R(t) while the thickness of the wall is A: (j> = o, = 0,

A r < Ro+vt—, A r> Ro + vt+—,

for

for

(13)

with R(t) = Ro + vt and v the radial expansion velocity of the soliton. We may write = f(r + vt);

w=(l-v2)r,

(14)

so that the field equation becomes

We will be looking for a solution of / across the wall so that it connects smoothly the true-vacuum solution inside and the false vacuum solution outside. Introducing g = wf(w), we find

^'^(l-^-^^l^-l2-^},

(16)

w an equation which we may solve in exactly the same manner as the collidingwall problem to be elucidated in the next section. 3.2

Colliding Walls

When bubbles of true vacua grow explosively, the nearby pair of bubbles will soon squeeze or collide with each other, resulting in merging of the two bubbles while producing cosmological objects that have specific coupling to the system. The situation is again extremely complicated. We try to disentangle the complexities by looking at between the two bubble walls that are almost ready to touch and for the initial attempt neglecting the coupling of the vacuum dynamics to the matter content. Between the two bubble walls, especially

144

between the centers of the two bubbles, it looks like a problem of plane walls in collision - and this is where we try to solve the problem to begin with. Consider the problem of two walls approaching each other. The wall, each of thickness A, separates the true vacuum on one side from the false vacuum on the other side of the wall. For the sake of simplicity, both walls are assumed parallel to the (xy)— plane and are infinite in both the x and y directions. In addition, the wall at z = R is moving to the left with the velocity v while that at z = —R moving to the right with +v. For z > R + y and all x and y, the complex scalar field <j> assumes <j>0, a value of the true vacuum (the ground state). On the other hand, for z < —R — y and all x and y, the complex scalar field 4> assumes 4>'0, another value of the true vacuum which differs in general from <J>Q. In between the two walls, i.e., —R + | < z < R — y , we have 0 = 0, the false vacuum. As indicated earlier, the field 4> must satisfy the field equation everywhere in spacetime: d2

^

vw\

nn

We may write the wall on the right hand side but moving toward the left with the velocity v. 4>= f(z-

vt),

for z-vt>0,

t< R/v.

(18)

so that

(1-«Y = A/(|/|2-A

0.

(19)

For a real scalar field / , this equation can be integrated by multiplying both sides by / ' : ( l -

2

) ^ = | /

4

- ^ /

2

+ Co,

(20)

with CQ the integration constant. This in turn gives rise to

I

df y 7 4 - 2
±a£ +

Cl,

(21)

*f

with £ = = z — vt — R+-2, a= J2(i-v2) an< ^ C l a n ° t h e r integration constant. What remains to be done is to choose CQ and c\ so that / connects smoothly to the solutions on both sides of the wall. The function in Eq. (19) may also be complex: / = ueie,

(22)

145

so that, with A = A/(l - v2), 2

u "-u(0')

= \u(u2-a2),

2u'e' + u0" = 0.

(23) (24)

Integrating the second equation, we find u29' = K,

(25)

with K an integation constant. The second equation can be integrated out in the same way as in the real case. We record the result as follows:

f =-

f

2 Jo

dV

(26)

J-K + ay-2py2+(3y^

with dy 2 3 2 Jo ^~K J^K + ay-2f3y +py

(27)

Here j3 = ^
146

5. Cosmological Implications of the MAXIMA-I High Resolution Cosmic Microwave Background Anisotropy Measurement, R. Stompor et al. Ap. J. Lett. 561, L7 (2001); A High Spatial Resolution Analysis of the MAXIMA-1 Cosmic Microwave Background Anisotropy Data, A.T. Lee et al., Ap. J. Lett. 561, LI (2001); Tests for Gaussianity of the MAXIMA-1 CMB Map, J.H.P. Wu et al., Phys. Rev. Lett. 87, 251303 (2001). 6. Complex Visibilities of Cosmic Microwave Background Anisotropics, K.W. Ng, Phys. Rev. D63, 123001 (2001). 7. S. Perhnutter et al., Astrophys. J. 517, 565 (1999) [arXiv:astroph/9812133]. 8. Je-An Gu and W-Y. P. Hwang, Phys. Lett. B517, 1 (2001).

N O N C O M M U T A T I V E EARLY U N I V E R S E PEI-MING HO Department

of Physics.

National Taiwan University. Taipei 106. Taiwan. pmho @phys.ntu. edu. tw

R. O. G.

We study t h e implication of the stringy spacetime uncertainty relation on the spectrum of metric perturbations. Assuming that the metric perturbations originate from quantum fluctuations of a scalar field, we find t h a t t h e initial state is uniquely determined by the uncertainty relation, and the spectrum is modified in the infrared.

1

Introduction

Cosmology needs string theory because classical gravity and the standard model of particle physics are expected to break down at very early stages of the universe. In this paper I would like to discuss possible implications of the stringy spacetime uncertainty relation (SSUR) on cosmology. SSUR was first proposed by Yoneya 2 to be a universal property of string theory. It says that the physical time and space coordinates satisfy AtpAxp

> l2s,

(1)

where ls is the string length scale. In comparison, another well known stringy uncertainty relation 2 AxpAp

> 1 + l2sAp2,

(2)

which implies a minimal length scale Axp > ls

(3)

can be violated by D-branes. In a previous work 3 , we suggested that SSUR can be used to solve the flatness problem and the horizon problem of cosmology, even without inflation. Inflation 4 is successful in solving many cosmological problems, such as the flatness problem, the horizon problem, the monopole problem, etc., and in explaining the observed mass perturbations and cosmic microwave background radiation. However, inflationary cosmology also suffers several conceptual problems 5 . Assume that the universe is created at the time t = 0. Then the temporal uncertainty is bounded by At < t.

147

(4)

148

Combined with SSUR, this implies that I2 Ax > - i .

(5)

Suppose that the earliest moment when the time coordinate makes sense is at t = £*, then the radius of spatial curvature, or any other physical length scale, has to be greater than l2/t* according to (5). For a small £* and a relatively larger ls, the radius of curvature at the Planck time can be much greater than the Planck scale and solve the flatness problem. Similarly, the horizon problem is solved because the correlation length of any physical degree of freedom is bounded from below by (5). Explicit calculation shows that if we choose i* = lp ~ (10 1 9 GeV) _ 1 and ls ~ (TeV) - 1 , then both problems can be solved without inflation. Now we would like to study the effect of SSUR on primordial metric perturbations, which are the origin of the observed large-scale structure and of cosmic microwave background anisotropics. As a first step, we consider a simple model with a single scalar field in a homogeneous, isotropic background. Quantum fluctuations of the scalar field are responsible for creating the metric perturbations. As such the cosmological perturbations are automatically Gaussian. The assumed flat, homogeneous isotropic background, can be described by the Friedmann-Robertson-Walker (FRW) metric ds2 = dt2

-a2(t)dx2

= a2{ri){dr}2 - dx2) = a-2(r)dT2 - a2(T)dx2,

(6)

where dt = a(r])dr] = a~1(r)dT. The coordinate r is useful because it can be used to write SSUR in a time-independent way AT

Ax > l2s.

(7)

If we use T], for instance, naively SSUR should look like a2(?7)A?7A:r > I2.

(8)

However, more precisely the factor a2 should be smeared over a length of time depending on A77, and this smearing needs to be further specified. Since the scale factor has no spatial dependence, SSUR imposes no restriction on the Hubble constant H = a/a, or H, etc. On the other hand, had we imposed the constraint (3), we would have H < Ms, and H < M2, etc., where Ms = Z"1 is the string energy scale.

149

The evolution of the scale factor should in principle be determined by string theory. However, since we do not know how to construct a gravitational theory incorporating SSUR, we will first try to understand the problem for a generic a, and then consider some examples. 2

Power Spectrum

In this section we would like to study the power spectrum of metric perturbations in a slightly more general setting than the focus of this work. We will consider a generic field theory for a free scalar field whose Fourier modes are decoupled. The most general form of the action is S~V

[

dTd3k\ (f3+(T)dT4>-kdrk)

.

(9)

Z

JkSKa

To simply our notation, we have denoted 0g as k, a n d \k\ as k. For later consideration of SSUR, we also allow the possibility that 0^ evolves nontrivially with time according to this action only for k within a range K0. Similar restrictions also appeared in field theories obeying (2) 6 . By a change of the time coordinate T —* fj, this action can be further simplified as S~V

[

dr,d?k\zl{fj) z

JkeKo

{'k ~ k24>-kk),

(10)

where / ' is the derivative of / with respect to fj. The new time coordinate fj is defined by 1/2

dfj

(11)

and

** = (tt + ) 1 / 4 -

(12)

For many models, the problem of finding metric perturbations can be reduced to solving the equation of motion for an action of the form (10). For an ordinary inflationary model with a scalar field, fj = r\ is the conformal time, and 00 (13) Vk = z

n

1

= -a /a, for scalar metric perturbations. where 00 is the zero mode of and Ji We have Vk — a

(14)

150

for tensor metric perturbations. Detailed review of the calculation of metric perturbations can be found in 7 . Assume that we are dealing with metric perturbations which are described by the action (10). By a change of variable k = ~,

(15)

Zk

the equation of motion is simplified to be

v'i + (k2 ~f-)vk

= 0.

(16)

From the action (10), the momentum canonically conjugate to v is

nk = v'_k-£.v-k,

(17)

Zk

and this leads to the Hamiltonian H

dk

I'

7

t

i' Zzk

I

t

t

\

ka\ak + - — (akaLk - OfcO-fe)

(18)

^ Zk

where a'k(fj) and ak(fj) are the creation and annihilation operators at time fj. They are related to vk and 11^ by Vk = -^=(ak+aik),

(19)

nk = -i\j^(a-k-al).

(20)

In the Heisenberg picture, the creation and annihilation operators evolve with time, and the state does not. Assume that the Universe is in the state |0) defined by a,(77°)|0) = 0 ,

(21)

which is the vacuum for mode k at some intial time rfc. It is in general not the vacuum at later times. The Bogoliubov transformation relates the creation and annihilation operators at fjk with the corresponding operators at the time 77: ak(fj) = ak{r,)ak{r,l) + (3k{ri)alk(f&),

(22)

4(V) = Pk(vh-k(f}°k) + ak(f,)al(f,°k),

(23)

where ak and j3k satisfy akak -/3kpk

= 1,

Vfc.

(24)

151

The Hamilton equations [H, ak] = iak and its Hermitian conjugate are

tf+(fc2-^)&=0,

(26)

in terms of the new variables Cfc = < * * - & ,

£k = ak+pk.

(27)

2

The larger the value of \Pk\ (fj), the larger the number of particles at time fj created out of the vacuum state |0). Let us denote the factors in (25), (26) by z" (z-1)" 2 M =k -^, (28) N = k ~^~. z k (zk*) For small fj (scales smaller than the Hubble radius), both M and N are positive and approximately equal to k2. Thus, £k and £t oscillate. If the initial state is taken to be the local vacuum state, 2

ak(fj°k) = l,

/?*($) = 0 ,

(29)

then the magnitudes of £ and f are of order 1 until the size of the fluctuation grows outside the Hubble radius. This represents the oscillation of quantum vacuum fluctuations. On scales larger than the Hubble radius (at later times), M and N are dominated by the second (negative) term. In this period, C a n d £ correspond to frozen fluctuations which are undergoing quantum squeezing and which scale like zk and z^1, respectively. If zk increases with time, by (27) we can approximate vk by Ck/V2k for sufficiently late times. When k2 is much smaller than both z'£/zk and {zk1)"/(zk~1), and if zk is an increasing function, the dominant solutions are simply Cfc ^ Ckzk,

& ~ -J—

(30)

<^kZk

for real uk, vk. Note that the condition (24) is satisfied for this solution, but not for other solutions of the second order differential equations. Suppose that for sufficiently late times, zk ]» 1 (zk -C 1) then vk ~ ^Qk/y/2k (vk ~ l£k/\/2k~). Since the initial condition is Ck(fjo) = €k(fJo) = 1, it is equivalent to say that for zk ~S> 1, for sufficiently large fj, vk(fj) is given by the solution to the differential equation (25) with the initial condition v(fjo) = l/v2fc. For zk
152

The power spectrum for the state |0) is defined by mp

zk

where Mp is the Planck mass. The normal ordering is defined for the operators a(rjl), a t ( ^ ) at 7$, so that V(k) = 0 at fj%. (Certainly we must have V{k) = 0 for 77 < 77°, before the mode k is allowed to appear.) For fj > 77°, we have

V

^^W^fM

P

(32)

z

k

According to the analysis above, for expanding zk, the spectrum is 1 k2 T{k) ~ a a Mpz2k(rik)

(33)

where fjk is the time when the fluctuation mode k crosses the Hubble radius (M = 0). In this paper we also consider the case of fluctuations outside the Hubble radius starting at the vacuum. For them 77* should be taken to be the time when they start at the vacuum. In general, fjk is the time when the mode k first appears outside the Hubble radius. 3

S S U R and Spectrum

Eq. (33) for the power spectrum is very convenient. In the usual expanding cosmological models, we have zk oc a for both scalar and tensor metric perturbations for scale factors of the form a(t) = a0tn = OOT^ ,

(34)

where a0 = ((n + l ) n a 0 ) "+ 1 . This occurs for constant equation of state w — p/p, and

n

= WT^y

(35)

For classical cases, we replacing zk by a and fj by 77 in the previous section. For n > 1 we have 77^ oc A;-1, and thus Zk{rfk) ex k1^. Therefore, from (33) we can easily find the spectrum for metric perturbations Pk~ck^,

(36)

where c = ((2n 2 - n) n a;j) " _ 1 /MP. For the spectrum to be exactly scale invariant, we need n = 00, that is, exponential inflation, for which the spectrum

153

H2 For the more general case when the scalar field is described by the action (10), the major difference from the classical case is that we have an effective scale factor Zk for each mode k, replacing a. The reason why SSUR will have such an effect is easy to understand at least intuitively. To describe a mode k, the spatial uncertainty can not be bigger than its wavelength. SSUR thus implies a lower bound on the uncertainty in time. Therefore, when this mode interacts with the scale factor a, or any other physical quantity, the quantity is smeared over some range of time. Suppose that the replacement of a by Zk everywhere is the only modification due to SSUR. The physical momentum for the mode k would be k/z}. instead of k/a. Being a massless scalar field, its energy is also k/zk- Therefore, the mode fc has Axp < zk/k,

Atp < zk/k,

(38)

and SSUR implies that Ms > - ^ - =

Ek(fj),

(39)

which is the energy for a particle in the mode k. For an expanding universe, o is monotonically increasing, and so is its smeared version z^. This bound implies that the mode A; can not exist before a certain time fj° when

*s4=al*)-

(40)

The mode k first appears at time 77°. The range of k whose dynamics is described by the action (10) is thus K0(fj)

= {k:fj>

fj°k}.

(41)

Remarkably, this constraint also uniquely determines the initial state of the universe. Before fjl, the fluctuation mode k does not exist. For a smooth transition to its existence after fj%, the only possibility is for it to start at the vacuum at 77°. This is precisely what we assumed to derive (33) for the power spectrum of metric perturbations. Interestingly, the power spectrum (33) can also be written as

Hk) * ^

-

(42)

154

Hence for all modes k which first appears outside the Hubble radius, their spectrum is scale invariant and is given by M2

"«=4-

(43)

This is independent of the precise form of z^\ Note that the modes whose wavelengths are larger than the Hubble radius are in the infra-red. For modes which first appears deep inside the Hubble radius, that is, the ultra-violet modes, the effective scale factor zj. is roughly the same as a when it crosses the Hubble radius at rjk = fjk — l/^> a n d their spectrum is given by the old'result (36). The whole spectrum, therefore, interpolates between a scale invariant one in the IR and the classical result in the UV. The fact that SSUR can change the power spectrum significantly seems to contradict which claims that the effect of new physics is hard to be observed 8 . Although our effective action is of the same kind considered there, the major difference is that our action only describes those modes k below a cutoff, and this cutoff determines our initial state, which is very different from the usual choice. Acknowledgments The author thanks Tzihong Chiueh, Je-An Gu, Hsien-chung Kao, Feng-Li Lin, Sanjaye Ramgoolam, Hyun-Seok Yang for helpful discussions. This work is supported in part by the National Science Council, the CosPA project of the Ministry of Education, the National Center for Theoretical Sciences, Taiwan, R.O.C. and the Center for Theoretical Physics at National Taiwan University. References 1. T. Yoneya, in "Wandering in the Fields", eds. K. Kawarabayashi, A. Ukawa (World Scientific, 1987), p. 419; M. Li and T. Yoneya, "Dparticle dynamics and the space-time uncertainty relation," Phys. Rev. Lett. 78, 1219 (1997) [arXiv:hep-th/9611072]; T. Yoneya, "String theory and space-time uncertainty principle," Prog. Theor. Phys. 103, 1081 (2000) [arXiv:hep-th/0004074]. 2. G. Veneziano, Europhys. Lett. 2, 199 (1986); D. J. Gross and P. F. Mende, Nucl. Phys. B 303, 407 (1988); D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216, 41 (1989); R. Guida, K. Konishi and P. Provero, Mod. Phys. Lett. A 6, 1487 (1991). 3. J.-A. Gu, P.-M. Ho and S. Ramgoolam, [hep-th/0101058].

155

4. A. Guth, "The Inflationary Universe: A Possible Solution To The Horizon And Flatness Problems," Phys. Rev. D 23, 347 (1981). A. D. Linde,. "Chaotic Inflation," Phys. Lett. B 129, 177 (1983). 5. R. H. Brandenberger, "Inflationary cosmology: Progress and problems," [hep-ph/9910410]. 6. A. Kempf and J. C. Niemeyer, "Perturbation spectrum in inflation with cutoff," Phys. Rev. D 64, 103501 (2001) [astro-ph/0103225]. 7. V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, "Theory of Cosmological Perturbations." Phys. Rept. 215, 203 (1992). A. Liddle and D. Lyth, 'Cosmological inflation and large-scale structure' (Cambridge Univ. Press, Cambridge, 2000). 8. N. Kaloper, M. Kleban, A. Lawrence and S. Shenker, "Signatures of short distance physics in the cosmic microwave background," arXiv:hepth/0201158.

COSMOLOGICAL C O N S T A N T , Q U I N T E S S E N C E A N D MINI-UNIVERSES XIAO-GANG HE * Department

of Physics.

National

Taiwan

University.

Taipei,

Taiwan

10764,

R-O.C.

In this talk, I report my recent work with Tzihong Chiueh (astro-ph/0107453) on bound object formation in a background universe accelerated by cosmological constant and by quintessence. If the acceleration lasts forever, due to t h e existence of event horizon, one would have naively expected the universe to approach a state of cold death. However, we find that many local regions in the universe can in fact be protected by their own gravity to form mini-universes, if their present matter densities exceed certain critical value. Within these mini-universes, there can be no event horizon although all of them will eventually fall onto each other's horizon and become isolated island universes. In the case with cosmological constant t h e condition of forming mini-universe is that the ratio of present density parameters &i / ^ A ought t o be larger than a critical value 3.63. In the case with quintessence, t h e condition is that the final ratio Cli/Clq of a mini-universe must be larger t h a n Wq — Wq, where w<j is the equation of state parameter.

1

Introduction

Recently direct evidences from the studies of Hubble diagram for Type la supernovae 1 indicate that our universe is expanding with an increasing rate, i.e., accelerating expansion. Accelerating expansion implies that the deceleration parameter q — (fi m + (3wq + l)Clq)/2 to be negative, where wq is given by the cosmic equation of state, pq = wqpq, of the quintessence Q. As long as wq < —1/3, the corresponding dark energy provides a negative contribution to q. The cosmological constant A has wq — — 1. The acceleration may stop in the future, i.e., a transient phenomenon with a time-dependent wq, or may last forever, depending on the nature of the dark energy 2 . A positive cosmological constant, consistent with the supernovae data at z = 1.7 3 , leads to a forever acceleration. Quintessence scenario with a constant wq, not ruled out by data, can also lead to forever acceleration. There are profound implications for a forever accelerating universe 4 , s . The universe will exhibit an event horizon; that is, there exist regions of the universe inaccessible to light probes. It has been argued that such a universe presents a challenge for string theories or any of its alternatives 4 . *B-MAIL: [email protected]

156

157

Naively speaking, in a forever accelerating universe, two points at different spatial locations will, as measured at any one of the two points, eventually be pulled apart at the speed of light to approach the future horizon. Hence existing structures will be eternally frozen and the universe will approach a state of cold death. Here we show that a local region with strong enough slef-gravity of matter can in fact protect the region from being torn up. A large number of mini-universes, including our own Local Group, will become self-bound and survive the destruction of cosmic repulsive force due to the quintessence. They can naturally arise despite that these mini-universes will still be falling into each other's horizon at a sufficiently late time and become isolated island universes 6 . 2

Cosmological Constant and Mini-Universe

We now study the conditions for bound object formation in the presence of cosmological constant. We adopt a toy evolutionary model for the sutdy of bound object formation which assumes the universe to consist of the background and an isolated cold dark matter fluctuation that grew from a small-amplitude since the early epoch. The localized perturbation possesses spherical symmetry and consists of two concentric solid spheres of different densities. The inner sphere has a radius r,- with a uniform dark-matter density pt, which is greater than the background matter density pm. The outer sphere has a radius rout and a uniform compensating under-density, such that the averaged matter density within rout equals pm. Any particle outside rout feels no extra gravity resulting from the over-density and expands as it would have been in a homogeneous universe. The inner sphere can be regarded as a closed Friedmann universe (the mini-universe) which has a Hubble parameter, Hi, given by

*-(*)'-¥<«+<*>-*•

(1)

where p\ = (l/8irG)A is a constant and Kj is the curvature of the miniuniverse. A test particle at Ti feels an effective potential V(ri), V(ri) =

GMi T%

ATTG

2

5-PA^i •

(2)

o

Conservation of matter yields a constant mass Mi within Ti with Mi = 47rpi(rj)r?/3. The potential V(r) has a maximum at rmax determined by V'(rmax) = 0 with V{rmax) = -AitGpKr2max and rmax = (p^/2pA))1/3r0,

158

where the density at present has been denoted as p° = pi(ro) with ro being the present radius of the over-dense sphere. When the test particle reaches r-max with a zero velocity, it will be marginally bound. The density pi corresponding to this situation is called the critical binding density, denoted as p\c at present, and any local region with density larger than p°c will eventually turn around and collapse. The curvature of the mini-universe can be conveniently obtained at the turn-around, after which gravitational collapse ensues, by setting r j 2 = 0 and equating —Kj/2 to the potential at the turn-around. That is,

Ki =

^(^n[ •*

(3)

+ 1)rL

PA

"ta

where rta is the turn-around radius. In particular for a critically bound miniuniverse, we have rta = rmax and Kic = Ki(rmax) = d>-KGpfj-^nax. The time span needed for the mini-universe to evolve to the present epoch is given by t

fr° _dri_= fro dri * ~ Jo uHi ~ 7 0 riy/{SnG/3)(pi + p A ) - (/Ci/r2) =

1

d

t

=

JL

(4)

When U is known, Eq.(4) can be inverted to solve for the present local density p°. Thus one needs to obtain information about tj. To this end we note that the mini-universe should have started to grow from a small amplitude perturbation at the beginning of matter domination around z = 3 x 10 3 . Therefore to a good approximation, the evolution time ij is equal to the background evolution time tB. The time ijg for the background Friedmann universe to reach the present ratio 0,^/Q,^ is tH =

r00 da

I

1

f1

V^Gp-ZJo

dy V^/"A)+2/

2

Jo

= _^=Binh-1(«/^).

(5)

Setting ti —tB, one can determine the present density ratio p°/pA for a given We consider three cases of interest for the discussions of the density ratio Pi/pA- i) the critical mini-universe, where the test particle reaches rmax with zero velocity and is therefore marginally bound; ii) the case where the overdensity is larger than that of case i) and the test particle has a zero velocity

159 and turns around at present; iii) the over-density is even larger so that the turn around occurred at the time when the background matter density p%% equaled p\, a particular epoch prior to which the matter in-fall had been vigorous. For case i), we have Ki = K,C = SirGpAr^ax- Given the background density parameters determined from various experimental data 7 £1^ = 0.35 and £lA = 0.65, Eq.(4) along with Eq.(5) yield the present critical density ratio P°JPA = 3.63 (Q.ic(r0) — 2.36) and r0 = rmax/1.22. We note that over-dense regions with the present density parameter larger than the critical value fiiC(ro) are abundant in the universe. In fact nearly all future bound objects will never reach rmax with a turn-around radius rta smaller than rmax. Inserting nt of Eq.(3) into Eq.(4), one obtains their present densities p«(ro) for a given ratio rta/ro. For case ii) the mini-universe that is presently turning around, we have rta — ro, and it is found that p°/pA = 5.90 with Q.ito = 3.83. Any over-dense region with a density greater than this value has already undergone collapse. For case iii), it is found that p^q / pA ~ 8.55 with £li<eq = 5.52. Although the toy evolutionary model discussed may seem somewhat simplistic, the model nevertheless illustrates the essential physics 8 . From the above discussions, we clearly see that local bound objects can form against the cosmic repulsive force, as long as the local density parameter fij(ro) at present is greater than 2.36. This value is at least a factor of several less than the mean density parameter of the Local Group, which includes the Milky Way, the Andromeda galaxy and more than a dozen of smaller galaxies such as the Magellanic clouds. It suggests that the Local Group has defied the cosmic repulsive force and gravitational collapse is underway. All the bound objects will virialize to form more compact objects after recollapse. The virialized kinetic energy of a test particle is given by, < T >= — < F • r > / 2 . For the simplified model under consideration, < T > is given by < T > = (GMi/2rv) - {AnrGp\rl/'S), where rv is the virialized radius. This leads to

- - l = < T > + < ^ > = - ( — i + —-p A rl).

(6)

2 Zrv 6 From both energy and mass conservation together with the expression offcjat the turn-around, eq.(3), with the subscripf'O" replaced by "ia", one finds 9 2

£t*£(2-!^) Ph

r

v

+

2 ( l - 2 - ^ - ) = 0.

(7)

r

ta

Regions with a space curvature Kj infinitesimally greater than Kic of case i) will finally collapse in the infinite future, and we shall also regard this

160

situation as case i) in a broad sense. It is straightforward to obtain that Tv/vta = 1/2.73 for case i), since pitta/pA = 2 and the corresponding rv/r0 is 1/2.24. The ratio of the virialized density to the A energy density is then found to be pv/p\ — 41. For cases ii), we find rv/r0 = 1/2.20 and pv/pA = 62. For case iii), rv/rta = 1/2.13 and pv/pA = 83 6 . 3

Quintessence and Mini-Universe

In this section we study the condition for bound object formation under the repulsive force of the quintessence with a constant wq. At present experimental data allow for the possibility that the accelerating cosmic expansion be due to quintessence with a relatively wide range of constant wq(> —1) and energy densities 10 . In order to have an accelerating universe at present, the equation-of-state parameter wq should satisfy

»,<-(! + §)•

W

Unfortunately, the value of 0,^/0,° is still rather uncertain, and its plausible value deduced from data correlates with wq, with a larger f2^/fi° for a smaller wq in a flat universe. For example, when £2^ = 35% and 0 ° = 65% to make up the energy budget for a flat universe, eq.(8) implies wq < —0.5, but the existing data pushes the plausible wq to be near —1 10 . Nevertheless, to have an idea as to how the results vary with wq, we shall consider three representative values, wq = —0.5, —0.75 and —1, in our later discussions. The quintessence energy density varies with the scaling factor as pq = p(q,(a/a0)~3(1+wi\ The reference background cosmological time t& in this case is given by

ts = -rL=

f

,

dy

""

ITTGO? JO ' ' ' + y-^ yjteGfi Jo y^VQO

(9)

For the study of bound mini-universes, the situation becomes more complicated than that with a cosmological constant because the space curvature Ki within the collapse region is time-dependent, and eq.(4) is no longer valid. (The analysis presented in Ref. 11 , which assumes a constant Ki, is hence invalid.) This problem should be analyzed using the momentum component of the Einstein equation, which describes the force balance of inertia with gravity and quintessence force. It can be understood as follows. Since the self-gravity of quintessence is negligible, the repulsive force of quintessence at a radius r

161

varies with time as (a0/a(t))3(1+w"\ This is a time-dependent force, thereby leading to a time-dependent space curvature. The momentum component of the Einstein equation is a second-order differential equation: r 4nG - = — g - [ f t + (l + 3«;,)p,]-

(10)

Defining x = (a/ao), y = (r/ro), one has ±=

J-^-p0^(n°m/n°q)(i/x) +

x-^

a = -^f-p°*yl§y~3 + a + a*;,)*- 3 * 1 ^].

(ii)

Combining these two equations one arrives at 2y"x(l + x-3w")

= y'(l + (1 +

3wq)x~!iw-)

- ( ^ + (l + 3wq)^x-3m"), (12) x y where x = x(n°m/^Y/3a^ y = y(£l0m/n0qY/3w<(Sl0m/n?) and y' = dy/dx. We may solve eq.(12) to obtain the conditions for bound object formation at any background cosmic time ts, c.f., eq.(9). However, due to the wide range of possible parameters for the quintessence cosmology, we shall instead concentrate on the critically bound mini-universe. The asymptotic state of the critical mini-universe can be found by letting both y —> oo and x —> oo in the infinite future. The existence of such a state also requires that the background expansion be accelerating, i.e., wq < —1/3. It then follows that eq.(12) has a scaling solution, y = (£licc/£lqooY^3x^1+Wq\ where -—

=wq-

wq,

(13)

Mqoo

by matching the coefficients in eq.(12). The subscript "oo" refers to evaluation of a quantity at a reference epoch in the infinite future. The space curvature Kj for this mini-universe defined as minus of twice the sum of kinetic energy and potential energy (of the force used in eq.(ll)) is given, in the asymptotic limit, by «i = - ( — ) - ( 1 + ^ ( l + 3 ^ ) 4 7 ^ 0 0 7 ^ , ,

(14)

where eq.(13) has been used. This confirms the statement made earlier that with quintessence the curvature is time dependent.

162

The space curvature decreases with time as a~^1+w''' but its magnitude cannot be determined from the scaling solution; it can be fixed only by integrating eq.(12) to obtain the entire solution. In this asymptotic state, the absolute values of kinetic, potential and total energies all have the same time dependence with comparable magnitudes. For wq = —1, we recover the cosmological constant case with pioo/pqoo = 2 and Kj given in eq.(3). The above critical mini-universe is very different from that of the case with a cosmological constant, in that its size grows indefinitely since r grows as a^1+w,'\ However, within it there is no event horizon, a fact that can be seen from the expanding velocity f/ro which approaches zero as (1 + wq)(a/ao)~(l+wi\ Thus, the radius of the mini-universe can never grow faster than light travels. One may further extrapolate the ratio pioo/pqoo back to the present by integrating eq.(12) numerically. The results are shown in Fig. 1. In Fig. 1 we show log(fij/Q q ) as a function of log(O g /fi m ) for wq — —0.5, —0.75 and —1. From Fig.l, one can read off the densities of the critically bound miniuniverse on the vertical axis for various background densities on the horizontal axis. (The horizontal axis can be regarded as another way for expressing the background cosmic time, c.f., eq.(9).) Regions whose present local densities exceed by a finite amount above a given curve shown in Fig. 1 will turn around within a finite time and become virialized in a manner similar to that discussed in the previous section, though the details differ 12 . However, it is interesting to note that there are differences for the virialized objects between the general quintessence cosmology and the ACDM cosmology. For the case with quintessence, a local region with an energy infinitesimally smaller than the critically-bound condition (local matter density larger than the critical density by an infinitesimal amount), the region will undergo collapse in the infinite future. The bound object has a nearly zero binding energy, as can be seen from eq. (14), and hence after virialization its virial radius rv discussed in Sec.(2) will be infinitely large, a great contrast to the wq = —1 case where the just bound objects have finite sizes. As has been mentioned earlier in this section, the precise value of £1^/0° is yet to be determined and the plausible value of fi^/fi° in fact anti-correlates with the value of wq. Though for wq = - 1 the most plausible $1^/0,° — 0.35/0.65, for wq = —0.75 and —0.5 the most plausible £2^/f}° decreases to approximately 0.1/0.9 and (RS)0/1, respectively 10 . Precise knowledge for the content of energy forms, which can be acquired from more accurate calibration of Type la supernovae and measurements CMB radiation as well as other observations 10>13, are crucial for making good use of Fig.l.

163

{

\ \

\

\\\ Y\\

\v*. \ \ '-_

\ ' s --. v

W--0.5 w=.-0.75 w—1

-

4

-

2

0 iog(iyn,.>

2

4

6

Figure 1. T h e barely bound mini-universe critical density as a function of time. The vertical axis is log(Qi/Clq) and the horizontal axis is log(Clq/Clm).

4

Conclusions

In this work, we have quantitatively shown that mini-universes can form against the repulsive force of cosmological constant or quintessence with a constant equation-of-state parameter wq, as long as they have sufficiently high densities at present. In the case with cosmological constant the condition of forming mini-universe is that the ratio of present density parameters 0°/fiA ought to be larger than a critical value 3.63. In the case with quintessence, the final ratio £li/Slq of a mini-universe is found to be always larger than Wq~wq. The corresponding densities at present time for a given wq and given values of fim,g can be read off from Fig. 1. The detailed properties of the mini-universes, depend on whether the accelerating expansion is due to the cosmological constant or quintessence. New experimental data from various observations in the coming decades are expected to reveal more information. By then we will have a better under-

164

standing of the properties of mini-universes. If the acceleration of our universe at present is indeed due to cosmological constant or quintessence with constant wq, our universe will enter a new era, the era where all island universes are falling onto each other's horizon and appear to fade away. Within each of which most physics remains the same as is today 14 . I thank Professer Tzihong Chiueh for collaboration on the work presented here. This work is supported in part by National Science Council under grants NSC 89-2112-M-002-058 and NSC 89-2112-M-002-065, and in part by the Ministry of Education Academic Excellence Project 89-N-FA01-1-4-3. References 1. S. Perlmutter et al., Nature 391, 51(1998); Astrophys. J. 517, 565(1999); A.G. Riess et al., Astron. J. 116, 1009(1998). 2. J. Barrow, R. Bean and J. Magueijo, Mon. Not. Astron. Soc. 316, L414(2000); A. Albrecht and C. Skordis, Phys. Rev. Lett, bf 84, 2076(2000); S.M. Carroll, Phys. Rev. Lett. 8 1 , 3067(1998); E. Halyo e-print hepph/0105216; J. Cline, e-print hep-ph/0105251. 3. A.G. Riess et al., e-print astro-ph/0104455. 4. S. Hellerman, N. Kaloper and L. Susskind, e-print hep-th/0104180; W. Fischler et al., e-print hep-th/0104181. 5. X.-G. He, e-print astro-ph/0105005; J. Moffat, e-print hep-th/0105017; J.-A. Gu and W.-Y, Hwang, e-print astro-ph/0106387. 6. Tzihong Chiueh and Xiao-Gang He, e-print astro-ph/0107453. 7. P. de Bernardis et al., Boomerang Coll., Nature 404, 955(2000); S. Hanany et al., Maxima Coll., Astrophys. J. 545, L1-L4 (2000). 8. J.A. Peacock, "Cosmological Physics", Cambridge Univ. Press (1999); A.R. Liddle and D. Lyth, "Cosmological Inflation and Large-Scale Structures", Cambridge Univ. Press (2000). 9. O. Lahav, P.B. Lilje, J.R. Primack and M.J. Rees, Mon. Not. Roy. Astro. Soc. 251, 128 (1991). 10. See for example M. Signore and D. Puy, e-print astro-ph/0108515; P. Binetray, e-print hep-ph/0005037. 11. E. Lokas and Y. Hoffman, e-print astro-ph/0108283. 12. L. Wang and P. Steinhardt, e-print astro-ph/9804015, Astrophys. J. 508, 483(1998). 13. P. Nugent, SNAP collaboration, in San Juan 2000, Particle Physics and Cosmology, pp263 (2000). 14. F. Adams and G. Laughlin, Rev. Mod. Phys. 69, 337(1997).

STABILITY OF T H E A N I S O T R O P I C B R A N E COSMOLOGY W. F. KAO Institute

of Physics, Chiao Tung University, Hsin email :wfgore@cc. nctu. edu. tui

Chu,

Taiwan

The stability of the Bianchi type I anisotropic brane cosmology will be analyzed in this talk. Analysis is presented for t h e model with a perfect fluid. It is shown t h a t t h e anisotropic expansion is smeared out dynamically in the large time limit. It is shown that t h e initial state is highly isotropic for the brane universe except for a very particular case. Moreover, it is also shown t h a t the Bianchi type I anisotropic cosmology is stable against any anisotropic perturbation for the brane model in t h e large time limit. This talk is based on a paper in collaboration with Chiang-Mei Chen l

Introduction The observation of the cosmic microwave background (CMB) radiation 2 ' 3 indicates that our Universe is globally isotropic to a very high degree of precision. Therefore, our Universe is usually assumed to be described by the Friedmann-Robertson-Walker (FRW) metric in most of the literatures. The origin of the isotropic universe has also become an interesting research topic ever since. On the other hand, it is known that there is a small large-angle CMB anisotropics, AT/T ~ 10~ 5 , under the CMB background 2 ' 3 . The isotropy of our Universe may have to do with the choice of initial conditions and the stability of the evolutionary solutions. Therefore, the present isotropic phase could be a dynamic result of the evolution of our Universe, no matter what the initial state started out. In this talk, we will address on this issue in both the standard Einstein's theory and the newly developed brane world scenario 4>5. In particular, we will consider the evolution of an anisotropic cosmology described by the Bianchi type I models 6 - 1 3 for with a perfect fluid as the matter source. Our result shows that the anisotropic model considered evolves dynamically from the anisotropic Bianchi type I universe into the isotropic FRW space in the large time limit. This property reveals that the isotropy of the cosmological principle may be justified and made consistent with our current observational data. For the conventional Einstein's theory (CET), the anisotropic expansion tends to be large in the very early stage. In another words, the universe has to begin from a highly anisotropic initial expansion and then decays to zero as the time increases. On the other hand, in the brane world scenario 1 4 - 1 7 ! due to the quadratic correction which significantly changes the early time behavior of the Universe. As a result, any initially non-vanishing anisotropy parameter

165

166

tends to vanish in the very early period. There is a characteristic time, tc, that divides the evolution of mean anisotropy parameter into two different stages. The mean anisotropy parameter is increasing when t < tc and reaches its maximal value at t = tc. After that, mean anisotropy parameter starts to decay. This result remains true for both the model with a perfect fluid and the model with a scalar field. And this appears to be a general feature independent of the types of matter considered. We will also consider the stability of this solution 18 - 27 . The Brane World The brane world scenario assumes that our Universe is a four-dimensional space-time, a 3-brane, embedded in the 5D bulk space-time. All the matter fields and the gauge fields except the graviton are confined on the the 3-brane as a prior requirement in order to avoid any violations with the empirical results. Moreover, inspired by the string theory/M-theory, the ^-symmetry with respect to the brane is imposed 28 . A formal realization of the brane world scenario, which recovers the Newton gravity in the linear theory, is the Randall-Sundrum model 4 , s in which the 4D flat brane(s) is embedded in the 5D anti-de Sitter (AdS) space-time. Later on, a covariant formulation of the effective gravitational field equations on the 3-brane has been obtained via a geometric approach by Shiromizu, Maeda and Sasaki 14>16. It is shown that the effective four-dimensional gravitational field equations on the brane take the the following form

where G^ and T^„ are the Einstein and energy-momentum tensors. quadratic contribution of T^ defined as S^v = ~^2TTv-v ~ 4 T M aT "a + -^9iu> (3T a/3 T Q/3 - T 2 ) .

«.?pi/ i s a

(2)

The effective 4D parameters, e.g., the cosmological constant A and gravitational coupling &4, are determined by the 5D bulk cosmological constant A5, the 5D gravitational coupling £5 and the tension of the brane A via the following relations

Here g^ is the metric tensor on the brane. In addition, the quantity E^ is a pure bulk effect defined by the bulk Weyl tensor 14 . iFrom the Eq. (1), it is easy to realize that the brane world scenario is different from the CET by two parts: (a) the matter fields contribute local "quadratic" energy-momentum correction via the tensor <SM„, and (b) there are "nonlocal" effects from the free field in the bulk, transmitted via the

167

projection of the bulk Weyl tensor Eav. Therefore, the CET can be treated as a limit of the brane theory by taking Eav = 0 and k^ —» 0 with properly adjusted values of the constants k± and A. In addition to the generalized Einstein equations (1), the energymomentum tensor also satisfies the conservation law V /2 T' il/ = 0. Therefore, there is a constraint on the tensor E^, V^-E^" = /c|V M 5 M,/ , due to the Bianchi identity on the brane. Here, the operator V is the covariant derivative with respect to the metric gal/ on the brane. One should point out here that the field equations on the brane, namely the generalized Einstein equations, the conservation of energy-momentum and the constraint on E^ are, in general, not a closed system in the 4D brane since the quantity E^ is five-dimensional. It can only be evaluated by solving the field equations in the bulk. In this paper, we will, however, only consider the quadratic effect on the brane world in the anisotropic background. Therefore, we will set E^ = 0 which is equivalent to embedding the 3-brane in the pure AdS bulk space-time. Bianchi type I metric The line element of the Bianchi type I space, an anisotropic generalization of the fiat FRW geometry, can be written as ds2 = -dt2 + a((t)dx2 + a?,(t)dy2 + al(t)dz2,

(4)

with a,i(t), i = 1,2,3 the expansion factors on each different spatial directions. For later convenience, we will introduce the following variables V = Yli=i ai> as the volume scale factor; H = ^-, for i — 1,2, 3 as the directional Hubble factors; and H = | J2i=i Hi = W> a s t n e m e a n Hubble factor. In addition, we will also introduce two basic physical observational quantities in cosmology: A = X)i=i 3^2 as the mean anisotropy parameter, and q = ^{H"1) — 1 as the deceleration parameter. Note that A ~ 0 for an isotropic expansion. Moreover, the sign of the deceleration parameter indicates how the Universe expands. Indeed, a positive sign corresponds to "standard" decelerating models whereas a negative sign indicates an accelerating expansion. In this section, we will consider the case that the matter energymomentum tensor, T^, is a perfect fluid whose components are given by T° 0 = - p ,

T\=T22=T33=p.

(5)

Here the energy density p and the pressure p of the cosmological fluid obey a linear barotropic equation of state of the form p = (7 — \)p with 7 a constant in the range 1 < 7 < 2. The gravitational field equations and the conservation law on the brane

168

take the form, 3

37-2,2 37-l ZH + £ f l ? = A - ^L-t klP - ± L ^ kip', ±±{VHi)

(6)

= A-1-^klp-1-^-kip\

i = l,2,3,

p + 3~/Hp = 0.

(7) (8)

The general solution of the above system was obtained in the exact parametric form in 15 . We are able to present all variables, including time, in terms of volume scale factor, V, with V > 0. For instance, the time variable can be expressed as

t-to = JFivy^dV,

(9)

where F(V) is denned as F(V) = 3AF 2 + 3klPoV2->

+ ifclpgV 2 " 2 7 + C,

(10)

with po and C the constants of integration. The other variables are

Oi —

a0iV1/3exp

Ki

fv^Fivy^'dv

A = 3K2F(V)~\ 3 7 (6fc|poV 2 -^ + kjplV2-^) 4F(V)

1 = 1,2,3,

(12) (13)

+ 12C

- 1,

(14)

where aoi, i^j, i = 1, 2, 3 are constants of integration and K2 = ^ i = i ^t 2 - I n addition, the arbitrary integration constants Ki and C must satisfy the same consistency condition K2 = 2C/3. One can immediately show that the effect of the energy density quadratic term becomes significant at the high energy epoch, or in another words, at the early stages of the Universe by looking at the Eq. (10). Indeed, F(V) oc V2~2ry at the limit t —• 0 when V is extremely small. As a result, the F diverges as t —• 0. Therefore, the mean anisotropy parameter A —* 0 at the early universe. On the other hand, in the large time limit, the properties of the Universe should be more or less the same as the case for the CET by looking at the same Eq. (10). Hence the early evolution of the anisotropic Bianchi type I brane Universe is dramatically changed due to the presence of the brane

169

correction terms proportional to the square of the energy density. The time variation of the mean anisotropy parameter of the Bianchi type I space-time is presented, for different values of 7, in Fig.l. From the Fig.l, it is clear that at high energy density the evolution of the brane Universe always starts out from an isotropic state with A —> 0. The mean anisotropy parameter increases and reaches a maximum value after a finite time interval tc. One can show that, when t > tc, the mean anisotropy parameter is a monotonically decreasing function approaching zero in the large time limit. This behavior is in sharp contrast to the usual evolution in the CET, as shown in Fig.2, in which the Universe has to kick off from a state of maximum anisotropy due to the constraint from the field equation. In addition, the early time evolution of the brane universe is normally not in an inflationary phase. On the other hand, the brane Universe always ends up in an inflationary phase in the large time limit in the presence of a nonvanishing cosmological constant. These are generic features of the brane Universe due to the constraint of the field equations on the brane cosmology. Indeed, a more detailed information can be extracted from the Eq. (9) in the limit t —» 0, or equivalently, the case with a small V. For simplicity, one will take C = 0 again. Indeed, we can show that V oc i 1 ' 7 as t —* 0. Hence, the expansion of the early universe is of the form of power law expansion. In addition, in the early stages of evolution of the brane Universe the mean anisotropy parameter varies as A oc i 2 - 2 / 7 approaching zero as t ~ 0. Moreover, the deceleration parameter is given by q = 37 — 1 which is always positive for all possible values of 7 for the case C •= 0. Stability Analysis The general perturbations for the FRW background with perfect fluid can be found in Ref. 17 . The same consideration is, however, more complicate for the anisotropic background. For the primary effect, we will only consider the scalar mode and neglect the vector and tensor modes 19,20 -p n e m e t r i c perturbation considered here is O-i —• O-Bi + ?>ai =

a

B i ( l + $h),

(15)

and the perturbations with respect to the perfect fluid considered in this paper is p -> PB + Sp,

p->PB + (l~ l)fy>-

(16)

Here the variables with subscript B are the exact solutions presented in the previous sections. For technical convenience, we will use variables Sbi instead of 6ai in our analysis. The perturbation equations for various quantities, can be obtained by substituting the perturbations (15, 16) into the field equations (6, 7, 8). Leading

170

Figure 1. Mean anisotropy parameter of the Bianchi type I brane Universe with a perfect fluid: 7 = 2 (solid curve), 7 = 1.5 (dotted curve) and 7 = 1 (dashed curve). The normalization of t h e parameters is chosen as 3A = 1,3fc|po = 2, k5p0 = 4, and C = 1.

Figure 2. Mean anisotropy parameter of the Bianchi type I conventional Einstein's theory with a perfect fluid: 7 = 2 (solid curve), 7 = 1.5 (dotted curve) and 7 = 1 (dashed curve). The normalization of t h e parameters is chosen as 3A = 1,3fc|/3o = 2, and C = 1.

terms will reproduce the field equations. Therefore, one has the following perturbation equations from the first order terms 0(5bi, Sp),

J2Sbi + 21£HBiSbi

=- ^

2 2

'-kl6p-?l—LktpB5p,

(17)

171

+ HBi J2 Sbj = -1—^L

Sh + ^8bi

k\ 6p -

1

—^ k%PB 6p,

Vi, (18)

3

6p + 3jHB 6p + j^26bipB

= 0.

(19)

t=i

In order to solve the above system of differential equations, we need an inspiration from the process of constructing the exact solutions. Their dynamics on the perturbation variables follow the constraints 3

6b

(2°)

6p=-7PBJ2 i> 3

3 B

j=l

j=l

By summing the Eqs. (18) and then using the result (20), we end up with a second order differential equation for a variable of the combination X)«=i ^&» 3

3

J2 6k + 6HB ^

3

1

Sbt - -1PB

[3( 7 - 2)k2A + ( 7 - 1)A*pB] £

Sbt = 0.

(22)

The task is to solve ^ <56j from the above equation and then to construct Sbi and Sp from the Eqs. (21) and (20). For the purpose of the stability analysis in the final stage, it is sufficient to consider the large time limit behaviors of the perturbation variables Sbi and 5p. From the discussion on the asymptotic behavior of the exact solutions, qualitative outcome in both the CET and the brane theory, we should divide our analysis into two different cases in the presence of absence of the cosmological constant. For the case with a positive cosmological constant, one can extract the asymptotic forms of the background variables from the exact solutions which give HB -» v/A/3,

VB OC exp(V3At),

pB ex exp(-V3A-yt).

(23)

The asymptotic expression of pB indicates that the third term in Eq. (22) can be neglected in the large time limit. As a result, we have the following equation for the asymptotic 5Z i = 1 Sbi which remains valid in both the CET and the brane world 3

3

6

52 h + vi2A J2 Sbi = °i=l

4=1

(24)

172

This in term leads to the final result 3

Y, Sbi oc exp(-VT2At).

(25)

i=l

Therefore, from Eqs. (20, 21) we can obtain the asymptotic expressions of the following perturbation variables 6bi oc exp(-v / 12Ai),

8p oc exp[->/3A(7 + 2)t].

(26)

This indicates that that the background solutions are stable. Similarly, for the case with A = 0, we can also show that this model is stable against the metric perturbations. Indeed, the asymptotic behavior of perturbation variables Sbi and Sp can be shown to be 6bi(xta,

6p<xta-2,

(27)

which shows that the background solutions are always stable even when the cosmological constant is vanishing. Conclusion We have discussed the anisotropic property of cosmological models in the brane theory. Comparison between the models with and without brane corrections indicates that the initial expansions behave very different. A realistic model, being consistent with the current observations, should produce a small value of anisotropic parameter at the later stage of the evolution of our Universe near the last scattering surface. By assuming the Bianchi type I space-time for the evolution of our Universe, we found that the final state of the evolving Universe always approaches the phase of isotropic expansion in both theories. These two different theories give completely different initial anisotropy at the very early stage of evolution. Indeed, for the CET, the anisotropy tends to be large in the very early stage. In another words, the universe tends to begin from a highly anisotropic initial state. The mean anisotropy parameter A will then decay to zero as the time increases. On the other hand, the early time behavior of the Universe in the brane world scenario changes significantly due to the quadratic correction on the brane. As a result, any non-vanishing mean anisotropy parameter, A(t), tends to vanish in the very early period. There is a characteristic time, tc, that divides the evolution of A(t) into two different stages. The mean anisotropy parameter is increasing when t < tc and reaches its maximal value at t = tc. After that, A(t) starts to decay. This kind of behavior is clearly shown in the Fig.l and Fig.2. This result remains true for both the model with a perfect fluid and the model with a scalar field. And this appears to be a general feature independent of the types of matter considered.

173

It is worth noting that the only exception is the model with p = 0 (i.e. 7 = 1) of the perfect fluid model. The mean anisotropy of this model behaves similar to the models in the CET where mean anisotropy parameter is large in the very early time. Moreover, we also analyzed the stability problem for those exactly solved anisotropic models shown in this paper. The result indicates that all of the solution known to us are stable in the large time limit. Therefore, the evolution of the Universe in the CET starts with highly initial anisotropic expansion. The dynamic of the system will take the Universe to the phase of isotropic expansion in the large time limit. We also show that the final isotropic expansion will remain stable in the large time limit. In addition, the mean anisotropy parameter will keep decreasing as time increases. The model provided here is a useful and explicit model that is capable of providing us with a Universe that has a tiny anisotropy left over near the last scattering surface. Acknowledgments This work is supported in part by the National Science Council under the grant numbers NSC90-2112-M009-021. References 1. Chiang-Mei Chen, W.F. Kao, hep-th/0201188. 2. S. Gulkis, P. M. Lubin, S. S. Meyer and R. F. Silverberg, Sci. Am. 262(1) 122-129 (1990). 3. D . J . Fixsen, et al, Astrophys. J. 473 (1996) 576-587. 4. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370-3373 (1999); hep-ph/9905221. 5. L. Randall and R. Sundrum, Phys. Rev. Lett 83, 4690-4693 (1999); hep-th/9906064. 6. C. W. Misner, Phys. Rev. Lett. 22 (1969) 1071-1074. 7. E. M. Lifshitz and I. M. Khalatnikov, Sov. Phys. Uspekhi 6 (1963) 495522. I. M. Khalatnikov and E. M. Lifshitz, Phys. Rev. Lett. 24 (1970) 76-79. 8. J. D. Barrow, J. Stein-Schabes, Phys. Rev. D32 (1985) 1595-1596. J. D. Barrow, K. E. Kunze, Phys. Rev. D56 (1997) 741-752; hep-th/9701085. 9. N. J. Cornish and J. J. Levin Phys. Rev. D 5 5 (1996) 7489-7510, gr-qc/9612066. Phys. Rev. Lett. 78 (1997) 998-1001, gr-qc/9605029. 10. M. Weaver, J. Isenberg and B. K. Berger, Phys. Rev. Lett. 80 (1998) 2984-2987; gr-qc/9712055. 11. J. D. Barrow and A. C. Ottewill, J. Phys. A: Math. Gen. 16 (1983) 2757-2776. 12. M. S. Madsen and J. D. Barrow, Nucl. Phys. B323 (1989) 242-252.

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D A R K E N E R G Y , P R I M O R D I A L M A G N E T I C FIELDS, A N D TIME-VARYING F I N E - S T R U C T U R E C O N S T A N T

KIN-WANG NG Institute of Physics, Academia Sinica, Taipei, Taiwan E-mail:

nkwQphys.sinica.edu.tw

Evidences indicate that the dark energy constitutes about two thirds of the critical density of the universe. We discuss the cosmological implications for the dark energy being an evolving scalar field that couples to electromagnetism.

1

Introduction

Recent astrophysical and cosmological observations such as dynamical mass, Type la supernovae, gravitational lensing, and cosmic microwave background (CMB) anisotropics, concordantly prevail a spatially flat universe containing a mixture of matter and a dominant smooth component with effective negative pressure l . The simplest possibility for this component is a cosmological constant. A dynamical variation calls for the existence of dark energy whose equation of state approaches that of the cosmological constant at recent epochs. Many possibilities have been proposed to explain for the dark energy. Most of the dark energy models involve the dynamical evolution of classical scalar fields or quintessence (Q). So far, many different kinds of scalar field potentials have been proposed. They include pseudo Nambu-Goldstone boson, inverse power law, exponential, tracking oscillating, and others 2 . Although the physical state of the dark energy can be probed through its gravitational effects on the cosmological evolution, it is important in fundamental physics to understand whether the quintessence is a nearly massless, slowing rolling scalar field. As long as an ultralight <j>fieldcouples to photon where for a slow-roll condition, the mass m^ is comparable to the Hubble constant H0, it is conceivable to have very long-wavelength electromagnetic fields generated via spinodal instabilities from the dynamics of 0 as a possible source of seed magnetic fields for the galactic dynamo 3 . Furthermore, the coupling may lead to an effective time-varying fine structure constant. Here we will investigate the implication of the electromagnetic coupling of the evolving 4> field to the origin of the primordial magnetic field (PMF) and the time-varying fine structure constant.

175

176

2

Primordial Magnetic Fields

As we know, the issue of the origin of the observed cluster and galactic magnetic fields of about a few p,G 4 remains a puzzle 5 . These magnetic fields could have been resulted from the amplification of a seed field of Bseed ~ 10 _ 2 3 G on a comoving scale larger than Mpc via the so-called galactic dynamo effect. A number of scenarios have been proposed for generating seed fields in the early universe, mainly relying on non-equilibrium conditions such as inflation, the electroweak and the QCD phase transitions. After the phase transitions, the universe became a highly conducting plasma so that the magnetic flux which existed is frozen in, and the ratio of the magnetic energy density and the thermal background, PB/PJ, remains constant thereafter. The required Bseea translates into ps — 10 _34 /9 7 . However, it turns out that the generated fields in these models are too small, except in somewhat contrived cases, to be of cosmological interest. In an equilibrium condition, a large damping term induced by the high plasma conductivity suppresses significantly any electromagnetic field fluctuations6'7.

3

Time-varying Fine Structure Constant

The idea of time-changing fundamental physical constants such as gauge coupling constants, the gravitational constant, and the speed of light has a long history 8 . Recently, it was claimed that the results of a search for time variability of the fine structure constant, a, using absorption systems in the spectra of distant quasars yield a smaller a in the past and the optical sample shows a 4-CT deviation: Aa/a = -0.72±0.18 x 10~ 5 over the redshift range 0.5 < z < 3.5 9 . An analysis of the isotropic abundances in the Oklo natural uranium fission reactor, active about 1.8 x 109 years ago (corresponding to a redshift of z ~ 0.1), suggests Aa/a = - 0 . 4 ± 1.4 x 1 0 - 8 10 . The most recent CMB data are consistent with a being smaller by a few percent at the time of recombination (z ~ 1000) in a ACDM model 1 1 , but the bounds imposed on the variation of a can be significantly ralaxed in models with a change in the equation of state of the dark energy 12 . Many models have been constructed to accommodate a changing a. Bekenstein formulated a dynamical model which consists of a massless scalar field coupled to electromagnetism 13 . This model was recently generalized by allowing an additional coupling of the scalar field to the supersymmetric dark matter 1 4 . Interestingly, it was proposed that this scalar field can be just the quintessence 1 5 , i e , but so far this idea has not been further pursuited.

177

4

Q-photon Coupling in Expanding Universe

Since the scalar potential V{<j>) of the Q field is scarcely known, it is convenient to discuss the evolution of through the equation of state, p

w^ > —1, where the latter equality holds for a pure vacuum state. Here we consider the ^-photon coupling, L^

= -f e^F^F^

+ -f g^g^F^F^,

(1)

where FM„ = d^A,, — dvA^, / is a mass scale, and c and c are coupling constants which we treat as free parameters. For the present consideration, we pick / equal to the reduced Planck mass Mp = (87rG) - 1 / 2 . The most stringent limit on the 0-photon coupling c comes from the cooling via the Primakoff conversion of horizontal branch (HB) stars in globular clusters 17 , c/f < 1.5xlO _ 1 1 GeV _ 1 , which gives c < 3.7 x 10 7 . The fifth-force experiments limit c < 10~ 3 (see e.g., Olive and Pospelov14). We are thus led to study the cosmological evolution of the 0-photon system in a flat universe. The effective action is S=

SM + jd^Xy/g

-J^-lg^g^F^F^

-lgf"/dlidv-V(4>) + ~L
,

(2)

where the signature is (—h + + ) and SM is the classical action for matter. We assume that the universe today has matter f i ^ = 0.3 and quintessence fl^ = 0.7, and define an ^-weighted average1 (w0)= /

%{ii)w4>{'n)dri x ( /

n^(ri)dr])

,

(3)

where rjo and TJU are respectively the conformal time today and at the last scattering, defined byr] = Hof dto _ 1 (t) with the scale factor a and the Hubble constant HQ. Assuming a spatially homogeneous cj>field,the evolution of the cosmic background is governed by

^

= -3aMl + «v)p«,

dh 3 ,o 1 - = --ah --awm,

(4) /c\ (5)

178

p^ = ptj>/(MpHo)2,

where h = H/H0,

and we have used 4>2 — (1 + w}p<j> a n d

V{<j>) = (1 — w^,)p/2. We have numerically solved the background equations by proposing a simple square-wave form for w^, as shown in Fig. 1. In order to satisfy the existing observational constraints on w/Mv. 5

G e n e r a t i o n of P M F

From the action (2) with c = 0, the comoving magnetic field in the comoving coordinates (r, x) with r = 7}/HQ satisfies ^ - V x ( v x B )

+ 4c?VxB,

(6)

CLT

where V = d/dx, a is the plasma conductivity, and v = dx/dr is the peculiar plasma velocity. After hydrogen recombination, the residual ionization keeps the conductivity high, a/H ~ 10 2 2 (T/eV) - 3 / 2 > 1. As a result, the a term under the assumption that v = 0 damps out any growth of the B field on scales above ~ A.U. 6 . But this assumption may not hold when the universe has entered the non-linear regime. It could be understood from the recent study by Kulsrud et al.18 where the battery mechanism has been proposed as a source for generating a small initial magnetic field during the large-scale structure (LSS) formation. There they showed that magnetic fields are built up in regions about to collapse into galaxies where velocity flows grow rapidly after extreme nonlinearities develop in the cosmic fluid. As such, the a term would be significantly reduced and the high plasma conductivity is no longer a hindrance to the growth of large-scale magnetic fields. Here we will show that the temporal variation of can also be a generating source for large-scale magnetic fields. Here we will simply omit the a term and solve for the photon equation self-consistently. Now we write B = V x A T and decompose the transverse field A T ( T , X) into Fourier modes, d3k •

/

3 %/2(27r) fc

eik-x$>Ak^Ak(T)eAk+h.c. A=±

(7)

where 6±k are destruction operators, and e±k are circular polarization unit vectors. Then, defining q = k/H0, the mode equations are d

dr)2

V±q+(q2T4cqf-)V±q V

dr]

= 0,

(8)

179

with initial conditions at early epoch given by

r*-1- i s } 1 " *

(9)

Hence, the comoving energy density of the magnetic field is given by pB = (B 2 )/8TT = Jdq(dpB/dq) with dpB

T0H$

2

^

2

where To is the current temperature of the CMB whose energy density is Pl

= TT 2 T 0 4 /15.

We have numerically solved the mode equations (8) using c = 130 and the background solution as shown in Fig. 1, and plotted the ratio (dpB/dq)/p-, in Fig. 2. Although photons are being produced as the scalar field starts rolling at z ~ 60, we have counted the photons produced only after z = 10 when the universe has presumably entered the non-linear regime. The result shows that a sufficiently large seed magnetic field of 10 Mpc scale has been produced before z ~ 4. Moreover, we notice that when c ~ 130 the spinoidal instability 19 gives rise to magnetic seed fields of the right magnitude and length scale. The self-consistency condition to reduce significantly this high conductivity effect can be justified in the sense that for the exponentially growing modes that we are interested in, dB/dr ~ amaxHGB ~ (10 - 1 6 s~ 1 )B with the coupling c ~ 100 and dB/dr] ~ 1, which is of the same order of the twisting term |V x (v x B)| ~ BV• v during the LSS formation obtained from the numerical simulations18. Although the value of c = 130 is well below the HB limit, undoubtedly it is much larger than the theoretical expectation. However, as suggested by Carroll 15 , an unsuppressed (^photon coupling may arise in higher dimensional theories. One possible way to reduce the value of the coupling and at the same time to produce the sufficiently large PMF on large correlation length scales during the LSS formation is to combine the mechanism we proposed here with the battery source proposed by Kulsrud et al.18. Also, note that the growth rate is controlled by the exponent a ~ 2c(dd/drj). As such, it may be possible to reduce the value of c by having a large d9/drj. In the case of scalar quintessence, we have seen in Fig. 1 that dO/dr) < 0.8 for z < 10. We cannot further increase dB/dr] since w^ has already reached the maximum value. Perhaps, in the fcessence models 2 0 , in which the kinetic term is modified to K(<j))<j)2/2 and we have ft — (1 + w^p^/Kicf)), we may make dd/dr] much larger than one by tuning the prefactor K(<j>). This possibility is under investigation.

180

Figure 1: The quantities dO/dq, w^, Hot, and fi^ as a function of redshift. Note that dO/dr) is drawn 4 times smaller.

Figure 2: Ratios of the spectral magnetic energy density to the present CMB energy density at various redshifts. The present wavelength of the magnetic field is given by 2-w/(qHo).

181

6

Variation of Fine Structure Constant

From the action (2) with c = 0, we have readily the change of a from the present time to a time tfi, mo

Aa

=

-107TC

a We have used the dO/dt] in Fig. 1 and c = 1 0 - 3 to calculate the change of a at the recombination time, Aa/a ~ —10~3. Given the equation of state w^ in Fig. 1, A a / a ~ - 1 0 - 3 at the time of the big bang nucleosynthesis. Although it is fine-tuned, we found that it is easy to modify the equation of state w^ at low redshifts to make the change of a consistent with the observations of quasar absorption spectra and the Oklo limits. Note that we have assumed no interaction between the quintessence and the dark matter. It is interesting to combine the present consideration with the supersymmetric generalization of the Bekenstein model 14 . 7

Conclusions

In conclusion, we have made an effort to link the dark-energy problem to a solution to another important problem in cosmology, namely, the generation of primordial magnetic fields. The high conductivity effect due to residual ionization after hydrogen recombination can be argued to be significantly reduced as a result of the existence of the cosmic flow with nonlinear, twisting peculiar velocity to avoid a hindrance to the growth of the magnetic fields. We have shown that the nonlinear instability that drives the rapid growth of magnetic fields is of spinodal instability where the long-wavelength modes about the Mpc scales evolve as being the inverted harmonic oscillators and the amplitude fluctuations begin to grow up to a time at which the scalar field velocity approaches zero at redshift z ~ 4. We have also given a simple quintessencephoton model that does not stick to a particular quintessential potential but satisfies all observational constraints to explain the observational data on the time-variation of the fine structure constant. Acknowledgments This work was in collaboration with Da-Shin Lee and Wolung Lee, and was supported in part by the National Science Council of the R.O.C. under the grant number NSC90-2112-M-001-028.

182

References 1. See, e.g., L. Wang, R. R. Caldwell, J. P. Ostriker, and P. J. Steinhardt, Astrophys. J. 530, 17 (2000). 2. I. Wagaand J. A. Frieman, Phys. Rev. D 62, 043521 (2000); S. Dodelson, M. Kaplinghat, and E. Stewart, Phys. Rev. Lett. 85, 5276 (2000); L. A. Boyle, R. R. Caldwell, M. Kamionkowski, astro-ph/0105318; T. Chiba, astro-ph/0106550, and references therein. 3. D.-S. Lee, W.-L. Lee, and K.-W. Ng, astro-ph/0109184. 4. P. P. Kronberg, Rep. Prog. Phys. 57, 325 (1994). 5. For reviews, see A. V. 01into,"in Proceedings of the 3rd RESCEU Symposium, Tokyo, Japan, 1997, edited by K. Sato, T. Yanagida, and T. Shiromizu; D. Grasso and H. R. Rubinstein, astro-ph/0009061. 6. M. S. Turner and L. M. Widrow, Phys. Rev. D 37, 2743 (1988). 7. M. Giovannini and M. Shaposhnikov, Phys. Rev. D 62, 103512 (2000); astro-ph/0011105. 8. P. A. M. Dirac, Nature, 139, 323 (1937). 9. J. K. Webb et al, Phys. Rev. Lett. 87, 091301 (2001). 10. A. I. Shylakhter, Nature, 264, 340 (1976); T. Damour and F. J. Dyson, Nucl. Phys. B , 480, 37 (1996); Y. Fujii et al, Nucl. Phys. B , 573, 377 (2000). 11. C. J. A. P. Martins et al, astro-ph/0203149. 12. G. Huey, S. Alexander, and L. Pogosian, astro-ph/0110562. 13. J. D. Bekenstein, Phys. Rev. D 25, 1527 (1982). 14. K. A. Olive and M. Pospelov, astro-ph/0110377. 15. S. M. Carroll, Phys. Rev. Lett. 8 1 , 3067 (1998). 16. G. Dvali and M. Zaldarriaga, astro-ph/0108217; T. Chiba and K. Kohri, astro-ph/0111086. 17. G. G. RafFelt, Stars as Laboratories for Fundamental Physics (University of Chicago Press, Chicago, 1996). 18. R. M. Kulsrud, R. Cen, J. P. Ostriker, and D. Ryu, Astrophys. J. 480, 481 (1997). 19. D. Boyanovsky, D.-S. Lee, and A. Singh, Phys. Rev. D 48, 800 (1993); D.-S. Lee and K.-W. Ng, Phys. Rev. D 61, 085003 (2000). 20. C. Armendariz-Picon, V. Mukhanov, and P. J. Steinhardt, Phys. Rev. D 63, 103510 (2001).

List of Participants Athar Husain Bor-Luen Huang Chiang-Mei Chen Chih-Yuan Liu Chi-Wei Wang Chun-Yi Lee Chung-Hsien Chou Chung-Ting Li Feng-Yin Chang Francis Halzen George W. S. Hou Guey-Lin Lin Hsueh-wei Huang Hsungrow Chan Henry Wong I-chin Wang Je An Gu Jie-Jun Tseng Jin Li

NCTS DeptofPhys,NTHU DeptofPhys,NTU Inst. Of Astronomy, NTHU DeptofPhys,NTU Electro-optical Eng, NCTU InstofPhys,AS Electro-optical Eng, NCTU Inst of Phys, NCTU Univ of Wisconsin DeptofPhys,NTU Inst of Phys, NCTU Dept of Phys, NTHU Ping-Tung Normal College Inst of Phys, AS Inst of Phys, NCKU Dept of Phys, NTU Inst of Phys, NCTU Inst of Phys, AS Dept of Phys, NTU NCTS Inst of Phys, AS Univ of Minnesota TMU, Japan Inst of Phys, AS Dept of Phys, NTU Inst of Astronomy, NCU Inst of Phys, AS

Kwang-Chang Lai Kingman Cheung K.-W. Ng K. Olive M. Chiba Ming-Huey Huang Pei-Ming Ho Ping-Hung Kuo Qian Yue

183

184

Ru-feng Liu Shiau-jing Wu Shih-Hung Chen Shih-Yuin Lin Sine-Nan Chiue Shu-Wei Lai Tzu-Chen Peng Venktesh Singh W. F. Kao W.-Y. Pauchy Hwang Wei-Chien Lai Wen-long Lin Wolung Lee Xiao-Gang He Xianwen Wang Y.-H. Chang Ya-Wen Tang Yu-chieh Chung Yu-Chung Chen

DeptofPhys,NCKU DeptofPhys,NCKU DeptofPhys,NTHU InstofPhys,AS Electro-optical Eng, NCTU Inst of Astronomy, NTHU DeptofPhys,NTHU InstofPhys,AS DeptofPhys,NCTU InstofPhys,NTU DeptofPhys,NCKU DeptofPhys,NTNU InstofPhys,AS Dept of Phys, NTU InstofPhys,AS Inst of Phys, NCU Inst of Astronomy, NTHU Dept of Phys, NCKU Dept of Phys, NTU

Astroparticle Physics This volume contains detailed articles by theorists and experimentalists in the newly developing field of astroparticle physics. A large variety of topics are covered. These include the role of neutrinos in astroparticle physics, big bang nucleosynthesis, string theory and cosmology.

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