Is cosmological dimensional reduction possible?

Volume 138B, number 4

PHYSICS LETTERS

19 April 1984

IS COSMOLOGICAL DIMENSIONAL REDUCTION POSSIBLE? Kei-ichi MAEDA Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan Received 7 October 1983

Chodos and Detweiler have shown that "extra" dimensions in Kaluza-Klein theories may contract to an unobservable scale, by adopting an anisotropic Kasner-type vacuum solution of higher dimensions. However, if we consider a quantized matter field, the above anisotropic space-time may be isotropized by particle creation in the same manner with the process proposed by Zel'dovich in a conventional four-dimensional space-time. In this paper, the isotropization process in higher dimensions is computed by generalizing the method given by Hu and Parker for a conventional universe. It would be concluded that cosmological dimensional reduction does not occur.

The higher-dimensional theory o f Kaluza and Klein is one of the most interesting ways o f unifying gauge theories and gravitation. An important question in this approach is how the scale o f the " e x t r a " dimensions shrinks to be so small that one cannot observe it. Such a process must exist if we take the point o1" view that the " e x t r a " dimension is not simply a mathematical stratagem but rather a physical reality. F r o m the point of view o f the solutions o f the higher-dimensional Einstein equations, the methods o f this dimensional reduction may be classified into two types; one is the static solution and the other is the dynamical one. In the latter case, we will call it cosmological dimensional reduction, which has been proposed first by Chodos and Detweiler [1 ], and in a more realistic model, i.e. eleven-dimensional supergravity theory, by Freund and Rubin [2,3]. These cosmological dimensional reductions are very attractive because one can explain the smallness of the "extra" dimensions by the dynamical evolution of the universe. The vacuum Einstein equations have the Kasnertype solution, which describes an anisotropic and homogeneous space. Chodos and Detweiler have shown that the space o f the " e x t r a " dimension contracts to a very small scale by adopting the Kasnertype solution. For example, in the case o f five dimensions, the scale factor of a conventional three-space 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

expands as a(t) ~ t 1/2 and that o f the fifth dimensional space contracts as a5(t ) c~ t - l ~ 2" In the conventional four-dimensional Einstein theory, it is known that if we consider a quantum effect of matter field the particle creation mechanism isotropizes the anisotropic expansion. This isotropization process was first proposed b y Zel'dovich [4] and the quantitative analysis has been given by Zel'dovich and Starobinsky [5] and Hu and Parker [6]. The model given by Chodos and Detweiler is a Kasner-type anisotropic universe in higher dimensions. Then, if we take into consideration the matter lagrangian, which is requested in more realistic unified theories, the above isotropization mechanism due to particle creation may destroy the cosmological dimensional reduction process. In the model by Freund and Rubin, the eleven-dimensional universe is also anisotropic and the matter lagrangian of an antisymmetric tensor field Auv o has been included. Then, isotropization may occur also in this case. In this paper, we consider a quantum effect of a scalar field in five-dimensional s p a c e - t i m e as a concrete model and investigate how effectively particle creation works for the isotropization of the fivedimensional universe which begins with an anisotropic Kasner-type solution. In the case o f much higher dimensions, we can easily give a similar discussion and probably get a qualitatively same result. 269

Volume 138B, number 4

PHYSICS LETTERS

We write the five-dimensional Einstein equations as i ,

R v -- ~guvR = 8~rG(%u)reg,

(1)

where (Tupreg is a vacuum expectation value of the regularized e n e r g y - m o m e n t u m tensor of the matter field. We assume a generalized Bianchi type I universe in the five dimensions, which has the following form: 5 ds 2 = - d t 2 + ~ a 2 ( t ) ( d x A)2. A=I

The cosmological scale factors aA (t) are determined by eq. (1). It follows from eqs. (1) and (2) that the nonvanishing components o f (Tff) are

(ToO)--p,

The conservation equation (7) guarantees the validity of the constraint equation (4), if the constraint equation is satisfied at initial time. Therefore, we use eq. (7) instead of eq. (4). We assume the ordinary three-dimensional space to be isotropic as in the case of Chodos and Detweiler, i.e. a = a 1 = a 2 = a 3 . Then, the trace o f the e n e r g y momentum tensor is = -p + (3e

(2)

(TAB)--PA6AB,

(3)

19 April 1984

+%)~- r e ) ,

(8)

where P = P1 = P2 = P3. From eqs. (7) and (8), we can express the pressure P and P5 as

P=~@'-~')-l{p'+(lnV)'p+y[T(n)+o]),

(9)

e 5 = ( a ' - / 3 ' ) -1 {p'+ (In V)'p + a' [T(r/) + p] },

(10)

where p is an energy density and the PA are principal pressures. The Einstein equations ( I ) take the form:

where a = In a and/3 = c~5 = in a 5 . As the matter field we consider a quantized conformal massless scalar field. The lagrangian of the scalar field is

G00 = 1 V - 1 / 2 [ p ( O t A ) 2

£M = - - ½ X / ~ (gUVa.~ au~ + ~RO2),

_ ( A ~ A ),(

= - 8 r i G 5 p,

B~O~B); (4)

and

(11)

where ~ = ~ in the five-dimensional s p a c e - t i m e . The e n e r g y - m o m e n t u m tensor is given by %u = (V, ~b)(VvqS) - ½guu(Vp (p)(vpd~) + ~ 2 G u ,

RAA = V-1/2

+ 7~ctA

- ~ [Vu Vu(~ 2) - guvV ° Vo(~2)].

(12)

identity we get the conservation equation of the energy momentum tensor; (TUV);v = 0, that is,

Tuu vanishes on the classical level, but, in a quantum theory, a non-vanishing trace anomaly T(r/) may arise as a result o f the regularization. We write the initial epoch when the particle creation mechanism switches on as t o . Since in our calculations t O is greater than the Planck time tp, we have ignored the trace anomaly and put T(r/) = 0. So, if we can calculate the density p = -(T00)reg and p ' , we get from eqs. (9) and (10) the source term which arises as a result o f particle creation and we find the behavior o f a and a 5 by solving (5). Here, we shall estimate p and p' by the five-dimensional version of the method of Hu and Parker [6]. Let us put

V-I(pV)' + ~ a A P A - 0 .

0 = V-3/8X,

= a~o 5(eA - ' r),

(s)

since

R tile = 8,/TGs (%12 __1~guy T),

(1')

with T = Tu~. A prime denotes a derivative with respect to conformal time, which is defined by

t

-- f V -114 dt',

(6)

and ~A = lnaA and V = ala2a3a 5 . F r o m the Bianchi

A

(7)

+ 1 Greek letters take values from 0 to 3 and 5, capital Latins from 1 to 3 and 5, and small Latins from 1 to 3. Our signature is the same as that of Misner et al. [7]. 270

with

(13)

Volume 138B, number 4 X = (2~) 2 fd4k[Ak

PHYSICS LETTERS

Xk( 7?) exp (i k ' x )

+ A ; X~(r/) exp ( i k ' x ) ] . The function ,, 2

Xk(r/) satisfies

(14) the equation

Xk +(g2 +Q)xk=O,

(15)

19 April 1984

is valid, matter can be treated as classical incoherent matter. For low frequencies, i.e. in the domain q(t), we get the solution of eq. (16) using the approximation exp (2i f~o g2k dr/') ~ 1 as follows: 1 Ec~kexp( i i g 2 d r / , ) Xk _ x/5-fi

rio

where

+ 3k exp

i

f a dr/'

,

(20)

no

g2k = V 1 / 4 ( ~ k 2 1 / 2 \,4 a2 !

(16)

ak=Cl(g]l/2--ig2-1/2 f Qdr/')+c2f] -1/2, no

and Q=3

3k=Cl(g21]2+i~2-1/2 f Q d r / ' ) - c 2 ~ 2 - i / 2 ,

~ , , 2 64 A >B (aA -- aB) "

(17)

We can show that the operators A k and A?k obey the usual commutation relations of annihilation and creation operators, respectively, using the relations of canonical quantization provided that ,t

,

l

x k X k - X k X k =i.

(18)

Let 10A ) be a vacuum state defined by the annihilation operators A k. Adopting the adiabatic regularization proposed by Fulling et al. [8], we can write down the regularized energy density as follows :

rlo where c 1 and c 2 are integral constants [6]. We take the initial condition such that ak(r/0 ) = 1 and 3k(r/0) = 0, when the initial vacuum state l0A ) with respect to A k is regarded as a real vacuum because Xk becomes a positive-frequency solution. From this condition we find,

C1 - g g2k(r/0)-l/2, --

_

O)]Xkl2

32n 4

+

1

2

2

Q dr/

q(t)

+ 1 gZo

rio

•

- a - (1/2a)[l(a'/a)

¼(a'/g2)e'2(3)

½Qe2(2)]),

32rr 4

(j )2j

- f2 - (1/2f2) [l(fz'/~2) 2 - Ol

- ag2 (e2(2))

(22)

Then we can compute the quantum part of the energy density Pq(t) from eqs. (19)-(22) as

Pq(t)

1 V-5/4fd4k{lxkl2 + (f]2

+ (1/2a)[~(g2'/a)2¢2(2)

c2 = 'i ~k(r/0)l/2.

1

1

p = --(0 A I T00 [0A)reg

_

(21)

2 - Ol + (1/2a)[...1

}, (23)

(19)

where the definitions of e2(2) and e2(3) are the same as in eqs. (2.40) and (2.41) in ref. [8] and g2 = g2k. Following Hu and Parker, we decompose the energy density O into a quantum part Pq(t) and a classical part Pc(t), where q(t) is the quantum domain in kspace over which cok(-V-1/4g2k) G t -1 and c(t) is the classical one with cok > t 1. In the low frequency domain q(t) of the k-space, quantum effects such as particle creation are dominant and for the higher frequency region c(t) where the WKB approximation

where ao

= ~(r/0).

Since the quantum domain q(t) shrinks with time, some part Of pq(t) changes into Pc(t). Let k m = (kin(t), kSm(t)) be the maximum wave number defined by cok(t ) = t -1 with k = (Y,ik2) 1/2 . The part of the energy density which becomes classical in the time interval from r/to r/+ 2~r/is estimated as 6pq(r/) = pq(km(r/),r/)

-- pq(km(r/+

Ar/),r/).

(24)

We assume that the classical particles in the domain 271

Volume 138B, number 4

PHYSICS LETTERS

19 April 1984

c(t) behave as a collision-dominated relativistic fluid with "isotropic" pressure Pc(-~Pe), _1 and then Pc(t) is proportional to V -5/4. This assumption may be regarded as a phenomenological way of taking into account the effect of strong interactions occurring in the high-density fluid of newly created particles [6]. Then, the density at r/+ Ar/is given by

p(r/ + /Xrl) = [V(rl)/V(r/ + Ar/)] 5/4 [Pc(r/) + 8pq(r/)] + pq (km (r/+ Ar/), r/+ Ar/) = [V(r/)/V(r/+ Ar/)] 5/4/9(r/) I>

10 5

+ A~? [(O/3rl)pq(k m ,r/) + ~ ( V / V ) p q ( k m , 71)1. (25) Giving the initial energy density, we can compute p(r/) and p'(r/) from eqs. (23) and (25), and then P(~) and P5 (r/) follow from eqs. (9) and (10). We have taken our initial state to be vacuum. Then, through the particle creation process, the quantum energy density is created at first, and the particle whose frequency is larger than t - 1 begins to behave like a classical particle. By the vacuum polarization effect, #q(~) in eq. (23) does not vanish initially. The regularization terms in pq(r/), i.e. the last three terms of eq. (23) have been introduced in order to subtract the UV divergence by the adiabatic regularization. However, they will not play an important role for low frequency modes in the domain q(t). Therefore, we will neglect all these terms in this paper, and we take only first two terms in eq. (23) ,2. We choose the initial condition as follows: (i) P(to) = Pq(to) and p'(to) are given by eqs. (23) and (25). The initial energy density Pq(to) does not vanish because of the vacuum polarization. (ii) We cannot put c~'(r/0) = 2/3r/0 and/3'(r/0) = -2/37/0 as in the Kasner vacuum because this condition is inconsistent with the constraint equation (4) in the case of non-vanishing energy density. We choose c~'(r/0) = 2/3r/0 and/3'(r/0) is determined from eq. (4). /3'(~0) is nearly equal to - 2 / 3 7 0 , since Pq(to ) is not so large. We show the result by the numerical integration ,2 Even if we take into account the regularization terms adopting the other regularization which is valid in the low frequency region, our result will not change qualitatively as far as the created energy density is positive. 272

10 2

10 3

104

~p

¢

Fig. 1. A measure of anisotropy, zM1/H, is plotted with respect to t/tp for the initial time t o = tp, 2tp and 3tp. + and Q denote the characteristic isotropization time t F and the time when the fifth dimension turns from contraction to expansion, respectively. of eqs. (5) in fig. 1. The five-dimensional gravitational constant is chosen as G 5 = 27rG3/2 because G 5 ~ 27rGR 5 and R 5 ~ lp, where R 5 is the present size of the fifth dimension and lp is the Planck length. We define a measure of anisotropy by

In a spatially flat universe, we can freely scale the axes aA . For convenience, we set a = a 5 = 1 at t 0. F r o m fig. 1, we can see that the isotropization mechanism b y the particle creation is appreciably effective. We define the characteristic isotropization time t F as the time when AH/H becomes unity. If we take t o = tp, t F is about 5tp and the epoch when the fifth dimension turns from contraction to expansion is about 14tp. When t O = 2tp and t O = 3tp, t F ~ 200tp and t F ~ 1700tp respectively. If we take the value o f R 5 larger than lp, e.g.R 5 ~ 10/p, G 5 becomes greater than 27rG3/2 and then isotropization occurs more rapidly because the effect by the matter is enhanced as can be seen from eq. (5). Therefore, if we consider the quantum effect of a matter field in an anisotropic s p a c e - t i m e like a Kasner-type solution, s p a c e - t i m e is isotropized in five dimensions by the particle creation mechanism, and the cosmological dimensional reduction by Chodos and Detweiler may break down. If the initial time t O is before the Planck

Volume 138B, number 4

PHYSICS LETTERS

time, the above isotropization mechanism becomes more effective. In this case, however, since the trace anomaly term, T(rt), may become important, we cannot say anything within the framework o f this paper. Chodos and Detweiler consider that the five-dimensional s p a c e - t i m e is closed in order to compactify the fifth dimension, i.e., 0 ~< x A <~L. In the above calculation, we assume the s p a c e - t i m e to be not closed, l f L N l p , the low frequency region q(t) o f k-space contains many quantized waves because the critical wave number is about to1 ~ 1p 1 and the minimum wave number is about L -1 , and therefore we can turn E k into f d4k approximately. Therefore, even when s~ace time is closed, i f L >> lp, the above result is valid. When L ~ lp, the result may change quantitatively, but not qualitatively. As an other effect of the closed space, we can consider the Casimir effect. Recently, Appelquist and Chodos [9] show that the five-dimensional static K a l u z a - K l e i n vacuum (M 4 X S 1 ) is unstable for a contraction o f the fifth dimension when we consider a quantum effect of the gravitational field, calculating the effective potential. Therefore, we may be able to explain the smallness of the " e x t r a " dimensions by this mechanism. We can consider that this instability is due to the Casimir effect because the fifth dimension is closed. Does the Casimir effect work also sufficiently in time-dependent background s p a c e time like a universe? This effect may compete against the isotropization o f s p a c e - t i m e by particle creation. Which mechanism is more important and effective, particle creation o f matter fields or the Casimir effect o f matter fields and/or gravitons? It is also not known whether the creation of gravitons isotropizes space time, because gravity is not a conformally invariant theory. We must investigate the quantum effect and/or the topological effect o f the gravitational field and the matter field in anisotropic time-dependent s p a c e - t i m e . Freund and Rubin also give a model of cosmolog-

19 April 1984

ical dimensional reduction in a more realistic theory. This higher dimensional universe is also anisotropic. In this model, the matter field,Auvp, is included in the lagrangian. Is the anisotropic s p a c e - t i m e also isotropized by the creation o f this particle? The difference from the model by Chodos and Detweiler is that the existence of the anisotropic classical matter field is assumed to achieve compactification. Then, it is not so clear whether the mechanism of isotropization b y quantum particle creation is effective, because the anisotropic matter may prevent the isotropization of space time. However, this anisotropic classical matter give a non-zero cosmological constant and must vanish sooner or later. If the anisotropic matter vanishes near the Planck time and the universe is time-dependent, the particle creation mechanism may also isotropize the universe in eleven-dimensional s p a c e - t i m e . The author would like to thank Professor H. Sato for valuable discussions and continuous encouragement. He also thanks S. Midorikawa and U. Carow for useful discussions. References [1 ] A. Chodos and S. Detweiler, Phys. Rev. D21 (1980) 2167. [2] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233. [3] P.G.O. Freund, Nucl. Phys. B209 (1982) 146. [4] Ya.B. Zel'dovich, Sov. Phys. JETP Lett. 12 (1970) 307. [5 ] Ya.B. Zel'dovich and A.A. Starobinsky, Soy. Phys. JETP 34 (1971) 1159. [6] B.L. Hu and L. Parker, Phys. Rev. D17 (1978) 933. [7] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [8] S.A. Fulling, L. Parker and B.L. Hu, Phys. Rev. D10 (1974) 3905. [9] T. Appelquist and A. Chodos, Phys. Rev. Lett. 50 (1983) 141 ;Phys. Rev. D28 (1983) 772.

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