Cosmological perturbations in the (1+3+6)-dimensional space-times

Cosmological perturbations in the (1+3+6)-dimensional space-times including photon gas without viscous processes are studied on the basis of Abbott et al.'s formalism. Space-times consist of the outer space (the 3-dimensional expanding section) and the inner space (the 6-dimensional section). The inner space expands initially and contracts later. Abbott et al. derived only power-type solutions in the small wave-number limit which appear at the final stage of the space-times. In this paper, we derive not only small wave-number solutions, but also large wave-number solutions. It is found that the latter solutions depend on the two wave-numbers k_r and k_R (which are defined in the outer and inner spaces, respectively), and that the k_r-dependent and k_R-dependent parts dominate the total perturbations when (k_r/r(t))/(k_R/R(t))>>1 or<<1, respectively, where r(t) and R(t) are the scale-factors in the outer and inner spaces. By comparing the behaviors of these perturbations, moreover, changes in the spectrum of perturbations in the outer space with time are discussed.


Introduction
From the viewpoint of analyzing cosmological perturbations and discussing the evolution of their spectrum, we study the cosmological evolution of the (1 + 3 + 6)-dimensional spacetimes, in which it is assumed that our universe appears as an isotropic and homogeneous 10-dimensional space-time and evolves to the state consisting of the 3-dimensional inflating outer space and the 6-dimensional collapsing outer space. This scenario is supported by the present super-string theory (Kim et al. [1,2] in a matrix model).
In a previous paper [3], we discussed the entropy production at the stage when the above inflation and collapse coexist, and showed how viscous processes help the increase of cosmological entropy; we also discussed the possibility that we satisfy, at the same time, the condition that the entropy in the Guth level [4] is obtained and the condition that the inner space decouples from the outer space.
In this paper we study the evolution of cosmological perturbations in these space-times (in the case with no viscous processes), on the basis of Abbott et al.'s formalism [5]. They extended Bardeen's gauge-invariant formalism [6] in the 4-dimensional cosmological models to that in the multi-dimensional models. Abbott et al. derived only power-type solutions in the small wave-number limit which appear at the final stage of the space-times. In this paper, we derive not only small wave-number solutions, but also large wave-number solutions. It is found that the latter solutions depend on the two wave-numbers k r and k R which are defined in the outer and inner spaces, respectively, and that k r -dependent and k R -dependent parts dominate the total perturbations when (k r /r(t))/(k R /R(t)) ≫ 1 or ≪ 1, respectively, where r(t) and R(t) are the scale-factors in the outer and inner spaces. Using these solutions, we discuss the evolution of the k r -dependence (spectrum) of perturbations in the outer space.
In Sect. 2, we review our formalism based on that of Abbott et al., in which the perturbed quantities and Einstein equations are shown and they are classified into three modes, i.e., scalar, vector, and tensor modes. A new equation to be solved in the scalar mode is introduced. In Sect. 3, we derive solutions for the perturbed equations in the scalar mode, and approximate solutions in the cases of (k r /r(t))/(k R /R(t)) ≫ 1 or ≪ 1 are shown. In Sects. 4 and 5, we derive solutions in the vector and tensor modes, respectively. Similarly approximate solutions in the cases of (k r /r(t))/(k R /R(t)) ≫ 1 or ≪ 1are shown. In Sect. 6, changes in the spectrum of perturbations with time are discussed. In Sect. 7, concluding remarks are given. In Appendix A, we show the formulas of harmonics and gauge transformations in outer and inner spaces. In Appendices B and C, we show the derivations of approximate solutions in the scalar mode in the cases of x(≡ k r τ 4/3 ) ≪ 1 and y(≡ k R τ 2/3 ) ≪ 1, respectively, corresponding to the above cases, where τ ≡ t 0 − t and t 0 denotes the final time corresponding to r → ∞ and R = 0.

Formalism of the perturbation theory
The background space-time is expressed in the form of a product of two homogeneous spaces M d and M D as where d g ij and D g ab are the metrics of the outer space M d and the inner space M D with constant curvatures K r and K R , respectively. Here the dimensions of M d and M D are d = 3 and D = 6. The inner space M D expands initially and collapses after the maximum expansion with K R = 1, while the outer space M d continues to expand with K r = 0 or −1. As the collapse and expansion in these spaces proceed, however, the curvature terms of both spaces are negligible and the curvatures can be regarded approximately as K r = K R = 0. Then the background metric is g 00 = −1, g 01 = g 0a = g ia = 0, and the Ricci tensor is where i, j = 1, ..., d, a, b = d + 1, ..., d + D, and an overdot denotes d/dt. The background energy-momentum tensor is where u µ is the fluid velocity, ρ the energy density, and p the pressure. Here ρ and p are the common photon density and pressure in both spaces. The fluid is extremely hot and satisfies 2/34 the equation of state p = ρ/n of photon gas, where n = d + D = 9. Einstein equations are expressed as whereḠ is the (1 + d + D)-dimensional gravitational constant. In the following, we set 8πḠ = 1. The background equation of motion for the matter iṡ At the early stage, the expansion of the total universe is nearly isotropic (i.e. r ∝ R). At the later stage, the inner space collapses after the maximum expansion, and at the final stage we have an approximate solution and τ = t 0 − t, where t 0 is the final time corresponding to R = 0. For d = 3 and D = 6, we have For the solutions (7), Eqs. (4) and (5) lead to R 0 0 = 0 and T 0 0 − 1 2 T µ µ ∝ ρ, so that we have at the final stage.

Classification of perturbations
The simplest treatment of perturbations of geometrical and fluidal quantities is to expand them using harmonics, and to find the gauge-invariant quantities, as in Bardeen's theory for perturbations in the four-dimensional universe [6]. In the multi-dimensional universe consisting of the outer and inner homogeneous spaces M d and M D with different geometrical structures, we can have no harmonics in the (d + D)-dimensional space. Abbott et al. [5] considered In the classification adopted by Abbott et al., the six types of perturbations are divided into three groups: 1. scalar mode (SS), 2. vector mode (SV, VS, VV) , 3. tensor mode (ST, TS). In this paper we call these three groups as "modes", corresponding to Abbott et al's "problems". 3 where ab are scalar harmonics in M d and M D , respectively, and T , and G (0) are functions of t. The perturbations of fluid velocities and the energy-momentum tensor are expressed as and where we consider a perfect fluid, so that the anisotropic pressure terms vanish and we have The metric perturbations in Eq. (11) transform as shown in Appendix A for changes in the coordinates, and the following gauge-invariant quantities are defined: The gauge-invariant quantities Φ h and Φ

(r)
A in the outer space correspond to the gaugeinvariant perturbations defined by Bardeen[6] in the (1 + 3)-dimensional usual universes, 4/34 and Φ H and Φ

(R)
A in the inner space are similar to the above quantities. Φ h and Φ H represent the curvature perturbations in both spaces.
The gauge-invariant quantities for fluid velocity and energy density perturbations are given by and As a gauge-invariant quantity that has no counterpart in the usual universe, we have which was introduced by Abbott et al. [5].

2.2.2.
The vector mode. The metric perturbations are ij Q (0) ), g ab = R 2 ( D g ab + 2H (1) The perturbed fluid velocity is and the perturbed energy-momentum tensor is where we neglected anisotropic stresses.

5/34
For the VS part of the vector mode, we have the gauge-invariant metric perturbations defined by and fluidal perturbations are For the SV part of the vector mode, and fluidal perturbations are For the VV part, we have only one gauge-invariant quantity G (11) .

The tensor mode.
We have only metric perturbations given by and have no fluidal perturbations, where have we neglected anisotropic stresses. In this mode, h T and H T correspond to the TS and ST parts of curvature perturbations and they themselves are gauge-invariant.

Perturbed Einstein equations
The perturbed Einstein equations are 2.3.1. The scalar mode. First we take up the following three relations which hold in the perturbed Einstein equations Using the expressions of δR µ ν given in the Appendix of Ref. [5], we obtain three relations between the gauge-invariant quantities from the above relations: Φ G is a gauge-invariant quantity defined in Eq.(19), and This equation is rewritten in terms ofΦ G as where we have used the relation rR = const. Here Φ 6 is an auxiliary gauge-invariant quantity defined bẏ which satisfieṡ Next from another relation we obtain 7/34 As one of equations describing the time development of Φ h and Φ H , we have which is expressed using gauge-invariant quantities as where {l.c. ↔ u.c.} means the terms (in { }) given by the exchanges r ↔ R and d ↔ D.
As another equation describing the time development of Φ h and Φ H , we adopt (43) is expressed using the gauge-invariant quantities as where l.c. ↔ u.c. means the terms (in ) given by the exchanges r ↔ R and d ↔ D, and Φ 7 is another auxiliary gauge-invariant quantity defined by satisfying the relationṘ In Ref. [5], Eq. (43) was not adopted as the equation to be solved, but it is a fundamental equation to be solved to derive Φ h and Φ H in general situations. They paid attention only to 8/34 the case when ρ → 0 at the final stage r → ∞ and R → 0. In this case, Eq.(39) with ρǫ = 0 may be one of the conditions for constraining the behaviors of Φ h and Φ H , and they could derive the behavior of Φ h and Φ H using it in the limit of small wave-numbers. In present paper, however, we use Eqs. (43) and (44) to derive their behaviors in more general cases including the case of large wave-numbers. Then, equations to be solved are Eqs. (32), (37), (42), and (44) for the four quantitiesΦ G , Φ 6 , Φ h , and Φ H .

The vector mode.
In the VS case, we have the following three equations from δG µ ν = δT µ ν 1 2 In the SV case, we have similarly and In the VV case, we have for G (11) and in the ST case for H In these cases, we have h T and H T representing the tensor components, but no scalar and vector quantities.

Solutions in the scalar mode
Let us derive various approximate solutions for the equations of perturbations, at the final stage in the model with d = 3 and D = 6. In this model we have the relation rR = const (cf. Eqs. (7) and (9)), which is useful to simplify the derivation of solutions.

Basic equations
In the present model, Eq. (32) can be expressed as where τ = t 0 − t, ′ denotes ∂/∂τ , andΦ G is defined by Eq. (33). Next, eliminating Φ A , Eq.(42) can be expressed as 10/34 Equation (44) can similarly be expressed as or furthermore eliminating Φ 6 ′ by use of Eq. (58), we obtain Equations (59) and (61) and for the convenience of calculations. Then the above four equations are expressed using x as and where In the case of small τ and x ≪ 1, four equations Eqs.(65), (66), (67), and (68) are found to have a special set of solutions were solved using the Runge-Kutta method. An example of the numerical solutions is shown in Fig. 1, which gives Φ h and Φ H in the case of x i = 0.1 and µ = 0. It is found that, during the stage of x ≪ 1, Φ h and Φ H behave simply as ∝ x −2 and ∝ x −1 , respectively, but, as x (or τ ) increases, the behaviors change, and they oscillate when x ≫ 1. Now let us consider the case when x ≫ 1 and µ/x ≪ 1. In this case, the outer wavenumber (k (0) r /r) is ≫ 1 and much larger than the inner wave-number (k (0) R /R), as can be found from Eq. (72). Here we neglect the terms µ/x in four equations (65) -(68), and assume that all quantities have an oscillatory factor exp iωx, where ω is a constant frequency. Then Φ Gx , Φ 6x , Φ h , and Φ H are expressed as where Φ G0 (x), Φ 60 (x), Φ h0 (x), and Φ H0 (x) are slowly varying monotonic functions. The special case when the quantities cannot be expanded is separately treated.
The important character of this type of perturbation is that we have no curvature perturbations in the outer space (Φ h0 = 0).

Equations with respect to k
where Φ Gy ≡ ( 4 3 k and , y denotes d/dy. From Eqs. (89) and (96), we have where Φ hi , Φ Hi , and y i are the values of Φ h , Φ H , and y at the epoch τ = τ i . This is found to be identical with the special solution Eq. (73). The above four equations were solved numerically using Eq. (98) as the condition at the epoch y i in the direction of increasing y (or decreasing t). An example of it is shown in Fig. 2, which gives Φ h and Φ H in the case of y i = 0.2 and ν = 0. Now let us consider the case when y ≫ 1 and νy 2 (= µ/x) ≪ 1. In this case, the inner wave-number (k (0) R /R) is ≫ 1 and much larger than the outer wave-number (k (0) r /r), as can be found from Eq. (97). Here we neglect the terms νy 2 in the four equations, and assume that 16/34 all quantities have an oscillatory factor exp iωy, where ω is a frequency. Then Φ Gy , Φ 6y , Φ h , and Φ H are expressed as Φ Gy = Φ G0 (y) exp iωy, Φ 6y = Φ 60 (y) exp iωy, where Φ G0 (y), Φ 60 (y), Φ h0 (y), and Φ H0 (y) are slowly varying monotonic functions. The special case when the quantities cannot be expanded is treated separately. From Eq. (90), we obtain for the main terms with respect to 1/y (≪ 1) and, similarly, from Eq.(91) From the compatibility of Eqs. (100) and (101), we find that ω = 1 or Φ G0 = 0.
In this case, we have both non-zero perturbations in the inner and outer spaces, in different from Eq.(87).

Summary of approximate solutions in the scalar mode
In the case of small τ and x ≪ 1 (or R τ 2/3 . In the case of x ≫ 1 and µ/x ≪ 1 (or y ≫ 1 and νy 2 (= x/µ) ≪ 1), k r dependent perturbations (or k R dependent perturbations) dominate k R dependent ones (or k r dependent ones), respectively. Among the solutions in these cases, only the solutions in the cases of ω = 1, α ≃ 0.81 (of x ≫ 1, µ/x ≪ 1) and ω = (2/3) 1/2 , β ≃ 2.13 (of y ≫ 1, νy 2 ≪ 1) grow in the direction of increasing τ , and all the other perturbations grow in the direction of decreasing τ (or increasing t).
In Figs.1 and 2, we can see the behaviors of the solutions with ω = 1, α ≃ 0.81 in the region of x ≫ 1 and ω = (2/3) 1/2 , β ≃ 2.13 in the region of y ≫ 1. It is noted that only these perturbations appear as growing ones, when we solve in the direction of increasing x or y.

Solutions in the vector mode
We consider here the VS, SV and VV cases at the final stage with r = r 0 τ −1/3 and R = R 0 τ 1/3 , and ρ = p = 0.

The VS case
From Eqs. (48) and (49) , we obtain respectively. On the other hand, for ( k (1) where µ/x = (k so thatΨ for x ≫ 1 and µ/x ≪ 1, where J 0 (x) and N 0 (x) are the 0 th Bessel functions of first and second kinds, respectively. Here J 0 (x) and N 0 (x) for x ≫ 1 have asymptotic behaviors such as If we use y defined by Eq. (89), we obtain where νy 2 = (k so thatΨ for y ≫ 1 and νy 2 ≪ 1. It is found, therefore, that for (k (1) ψ r takes the wavy behaviors with respect to x or y. From the asymptotic behaviors of Bessel functions, the amplitudes of ψ r are found to be 2 πx x −1/2 and 2 πy y −1 for µ/x ≪ 1 and νy 2 (= x/µ) ≪ 1, respectively. In terms of x and y, solutions (116) are expressed as Ψ r ∝ 1, x −1 ( for x ≪ 1) or ∝ 1, y −2 (for y ≪ 1).

Solutions in the tensor mode
Here we consider the TS and ST cases at the final stage.

The TS case
The equation for h (2) T reduces to Here we use x, defined by k T ,x + (1 + µ/x)h (2) and h (2) T ,yy + 1 y h T ,y + (1 + νy 2 )h respectively, as the solutions which are regular at the epoch τ = 0, corresponding to x = 0 or y = 0. From the asymptotic behaviors of Bessel functions, the amplitudes of h T for x ≫ 1 (or y ≫ 1) are found to be 2 πx for µ/x ≪ 1 (or 2 πy for νy 2 (= x/µ) ≪ 1), respectively.
Here we use x and y which are defined by k T ,y + (1 + νy 2 )H (2) where µ/x and νy 2 (= 1/(µ/x)) are also given in the TS case. For µ/x ≪ 1 or νy 2 ≪ 1, we have respectively, as the solutions that are regular at the epoch τ = 0, corresponding to x = 0 or y = 0. From the asymptotic behaviors of Bessel functions, the amplitudes of H T for x ≫ 1 (or y ≫ 1) are found to be 2 πx for µ/x ≪ 1 (or 2 πy for νy 2 (= x/µ) ≪ 1), respectively.

Evolution of models, perturbations and their spectrum
We consider the evolution of the inner and outer spaces and the perturbations in their spaces at the final stage. Characteristic epochs in the evolution are as follows. Since the total universe starts from the isotropic state, we assume first for simplicity that, at the initial epoch t 1 of this final stage, both scale-factors are equal, i.e., R(τ 1 ) = r(τ 1 ), where τ 1 = t 0 − t 1 . As R(τ 1 ) = R 0 · (τ 1 ) 1/3 and r(τ 1 ) = r 0 · (τ 1 ) −1/3 , we have As for the k r and k R dependences of perturbations, moreover, we assume that initially both spaces have the same spectra (or k r and k R dependence) with equal mean valuesk R =k r . Then we have the relation (k R /R)/(k r /r) = 1 at epoch τ 1 .
Secondly, let us consider a characteristic epoch τ 2 , when the mean wavelength of k r dependent perturbations in the outer space is equal to the sound wavelength, that is, k r /r(τ 2 ) = 1/(c s τ 2 ) in all modes, where c s is the sound velocity of photon fluids (≈ 1). The mean value of x (≡ 3 4r0 k r τ 4/3 ) is ≈ 1 and ≫ 1 at τ = τ 2 and τ ≫ τ 2 , respectively. Thirdly, we consider another characteristic epoch τ 3 , when the mean wavelength of k R dependent perturbations in the outer space is equal to the sound wavelength, i.e.,k R /R = 1/(c s τ 3 ) in all modes. The mean values of y (≡ 3 2R0 k R τ 2/3 ) are ≈ 1 and ≫ 1 at τ = τ 3 and τ ≫ τ 3 , respectively. Then so that, using Eq. (146), we obtain the relation between the three epochs τ 1 , τ 2 and τ 3 .

22/34
Finally, we consider as the limiting epoch the epoch τ † when the cosmological entropy reaches the Guth level and the inner space decouples from the outer space. According to my previous work [3], we have R(τ † )/R(τ M ) ≃ 3.0 × 10 −13 , where τ M represents an epoch at the stage when the inner space has the maximum expansion. So, if we assume τ M ≈ τ 1 roughly, because of R ∝ τ 1/3 . Now we define the spectrum of perturbations in the outer space as the square of absolute values of curvature perturbations in the scalar mode (i.e., |Φ h | 2 ) which is expressed as their functions of k r . The form of the primeval spectrum appearing before the final stage is assumed to be a simple one with a peak in the mean value (k r ) of k r . At the final stage, the spectrum in the case of x ≫ 1 and µ/x ≪ 1 changes with time due to the evolution of the k r dependent perturbations, but the spectrum in the case of y ≫ 1 and νy 2 (= x/µ) ≪ 1 does not change with time, because k R dependent perturbations are dominant and do not depend on k r .
As we have seen, perturbations are functions of two wave-numbers k r and k R . Here let us consider the behaviors of perturbations with various values of k r . To simplify the treatment about them, we fix in the following the value of k R to the mean valuek R , when we compare the k r dependence with the k R dependence. (a) First we consider the perturbations with the mean wavelengths, i.e., k r =k r at epoch τ 1 . Then, at epochs τ 1 ≫ τ ≫ τ 3 , we have y ≫ 1 and because r increases and R decreases with time. In this case, from the results in Sect. 3, it is found that, in the outer space, the k R dependent perturbations dominate and the change in k r dependent perturbations is relatively small, because the k R dependent perturbations have no role to changing the k r dependence of the spectrum. At epochs τ † < τ < τ 3 , we have y < 1 and the wave-like behaviors disappear in both of k r and k R dependent perturbations, and so the change in perturbations is expressed in the form of power functions of τ . Accordingly the evolution of perturbations gives only small change in the k r dependence of the spectrum around k r =k r . (b) Next we consider the perturbations with k r <k r . At epochs τ 1 ≫ τ ≫ τ 3 , we have y ≫ 1 and similarly for the mean perturbations in the inner space (k R =k R ). So, the k R dependent perturbations dominate and the change in k r dependent perturbations are relatively small. That is, we have small changes in the spectrum for k r <k r .
(c) Thirdly we consider the perturbations with k r >k r . Then we have (k R /R)/(k r /r) < 1 or µ/x < 1 during the time interval of where 23/34 or τ k = (k r /k r ) 3/2 τ 1 .
In the last equation, we used the relationk R =k r . If τ 1 > τ k > τ 2 , we have x > 1 and the k r dependent wavy behavior in these perturbations dominates the k R dependent perturbations in the outer space for τ 1 > τ > τ k . If τ 2 > τ k > τ 3 , the k r dependent wavy behavior dominates the k R dependent perturbations for τ 1 > τ > τ 2 and the k r dependent power-like behavior dominates them for τ 2 > τ > τ k . If τ 3 > τ k > τ † , the k r dependent perturbations dominate the k R dependent perturbations for τ 1 > τ > τ 3 , and both of k r and k R dependent perturbations are expressed for τ 3 > τ > τ k as power functions of τ being independent of k r , so that there is no change in the spectrum. From the above analyses, it is found to be for k r >k r and τ k < τ 3 .
that, in the most effective way, the k r dependent perturbations dominate the k R dependent perturbations and can modify the spectrum of perturbations in the outer space. From Eqs.
(155) and (156), we obtain the condition where k rm is the critical values of k r with maximum spectral changes and This condition (157) means that, for larger values of τ 1 /τ 3 (≫ 1), we have spectral changes for larger k(> k rm ).

Concluding remarks
In this paper we studied the evolution of various perturbations (including the curvature perturbations) in three modes and found that both of k r and k R dependences appear in the perturbations. Sometimes the perturbations depend mainly on k r , and sometimes they depend mainly on k R . The k R dependent perturbations do not depend on k r , and so have no influence on the spectrum of k r dependent perturbations. It was found that, in the large interval of k r including the mean valuek r , the k R dependent perturbations dominate the k r dependent perturbations in the outer space, and so the spectrum of perturbations as functions of k r does not change with time, and that it is k r > k rm [≡k r (τ 1 /τ 3 ) 2/3 ] that k r dependent perturbations modify the spectrum by dominating the k R dependent perturbations, where τ 1 and τ 3 are the initial epoch at the final stage and the epoch when the mean wavelength of k R dependent perturbations is equal to the sound wavelength, respectively, and τ 1 /τ 3 ≫ 1. Thus, in our classical treatments of perturbations, the spectrum of perturbations in the outer space starts from the primeval one at the nearly isotropic stage (τ > τ 1 ), changes mainly for k r > k rm due to the evolution of perturbations at the final stage (τ 1 > τ > τ † ), and transfers to the observed spectrum of perturbations in the Friedmann stage (after epoch τ † ). During the long period from the primeval stage to the final anisotropic stage of the multi-dimensional universe, the perturbations may be created by the quantum fluctuations of scalar, vector and tensor fields. Their treatments may be very interesting, but are beyond the scope of the present work.
We have paid attention only to the spectrum in the scalar mode in Sect. 6. Similar discussions on the spectrum in the vector and tensor modes may be possible.
In this paper we have treated the case with no viscous processes. In the cases with viscous processes, the dynamical evolution of perturbations may be much more complicated, but the fundamental structure, that, in the outer space, perturbations depend on both k r and k R and the change in the spectrum depends on how the k r dependence dominates the k R dependence, may be invariant.

Acknowledgements
The numerical calculations in this work were carried out on SR16000 at YITP in Kyoto University.

A.2. Gauge transformations
Gauge transformations are shown in the following three modes.