Cosmological entropy production and viscous processes in the (1+3+6)-dimensional space-times

The cosmological entropy production is studied in the (1+3+6)-dimensional space-times consisting 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. First it is shown how the production of the 3-dimensional entropy S_3 within the horizon is strengthened by the dissipation due to viscous processes between the two spaces, in which we consider the viscosity caused by the gravitational-wave transport. Next it is shown under what conditions we can have the critical epoch when S_3 reaches the value 10^{88} in the Guth level and at the same time the outer space is decoupled from the inner space. Moreover, the total entropy S_9 in the 9-dimensional space at the primeval expanding stage is also shown corresponding to S_3.


Introduction
Our observed Universe consists of 4-dimensional space-time and its 3-dimensional spatial section is very isotropic, homogeneous and flat. According to the super-string theories, on the other hand, the space-time originally has 10 (= 1 + 9) dimensions and our Universe is considered as a partial section of the total space-time after its evolution. In order that the section may be our observed Universe, it must satisfy the famous cosmological condition that it has the vast entropy ∼ 10 88 within the horizon-size region. [1] Recently the evolution of the space-time was analyzed by Kim et al. [2,3] in a matrix model of super-string theory and it was shown that owing to the dimensional symmetrybreaking the total 9-dimensional space is separated into the outer space (the 3-dimensional expanding section) and the inner space (the 6-dimensional expanding section), and that the expansion rate of the inner space is smaller than that of the outer space. As suggested by them, this may show the beginning of the separation of our (1 + 3)-dimensional Universe from the other section. In order that our Universe may form in this direction, the evolution of the inner space must tend from the expanding phase to the contracting phase, collapse and finally decouple from the expanding outer space, while the outer space inflates and tends to the Friedman phase. A similar scenario of such a dynamic evolution of anisotropic multi-dimensional space-times was studied in the form of Kaluza-Klein models in 1980-1990 [4][5][6][7]. The cosmological entropy problem was also discussed in many papers [8][9][10][11][12][13][14] in the framework of classical relativity. Kolb et al. [12] paid attention to the freeze-out epoch and found it is impossible, if the dimension of the inner space is less than 16, that at that epoch we obtain the 3-dimensional entropy within the horizon in the Guth level and the outer space is decoupled from the inner space. Abbott et al. [9,10] showed that it is possible, if the dimension of the inner space is ∼ 40.
In this paper we consider the entropy production which is obtained at an epoch different from the freeze-out epoch, so as to avoid Kolb [14] for the dynamics of multi-dimensional space-times. As for the initial condition, the multi-dimensional universe is assumed to start from the state of nearly isotropic expansion, in the same way as Kolb et al.'s and Abbott et al.'s treatments, but in a different way from that in our previous one. [14] in which we treated only highly anisotropic cases. We include imperfect fluid with viscosity caused by the transport of gravitational waves, as well as the perfect fluid. In Sect. 3 we describe the difference between the freeze-out epoch (t * ), the decoupling (or stabilization) epoch and the epoch (t † ) when the 3-dimensional entropy S 3 within the horizon reaches the critical value 10 88 . In Sect. 4 we study first the behaviors of S 3 and the total entropy S, and compare them in two cases with non-viscous and viscous fluids. The role of viscosity is found to be so strong as to bring vast entropies at the final stage in the collapse of the compact inner space and the inflation of the outer space. Next, using approximate power solutions (at the final stage), the condition (A) for S 3 ∼ 10 88 is derived, together with the condition (B) that the outer space should be decoupled from the inner space, and the compatibility of the two conditions A and B at epoch t † (different from t * ) is shown.
In Sect. 5, moreover, we solve numerically the differential evolution equations for scale factors from the initial singular epoch to the final epoch, taking account of viscosity due to the gravitational-wave transport and derive the model parameters satisfying the above two conditions A and B, by comparing their solutions (at the later stage) with the asymptotic power solutions. In Sect. 6 we derive the (9-dimensional) primeval total entropy S 9 , and it is found that the primeval entropy in the viscous case is much smaller than that in the non-viscous case. In Sect. 7, the epoch which may cause the dimensional symmetry-breaking is discussed from the viewpoint of energy conservation, and the possible epoch of symmetrybreaking is estimated. In Sect. 8 concluding remarks are given. In Appendix A the treatment of imperfect fluids is shown, in Appendix B the Planck length in the outer space is derived in connection with the sizes of these two spaces, and in Appendix C the relations between S 3 and S 9 are derived.

Multi-dimensional space-times with viscous fluid
First we assume that the fluid consists of massless particles with the energy density ǫ and the pressure p expressed as ǫ = N a n T 4+n and p = ǫ/(3 + n), where T is the temperature, n(= 6) is the dimension of the inner space, a n is the (4 + n)-dimensional Stefan-Boltzmann constant defined by a n ≡ (3 + n) Γ((4 + n)/2) ζ(4 + n)/π (4+n)/2 , and N is the number of particle species. The units c = = k (the Boltzmann constant) = 1 are used.

2/22
Since the total entropy S within a comoving volume V is given from the second law of thermodynamics by S is expressed as S/V = [(4 + n)/(2 + n)]N a n (ǫ/N a n ) (3+n)/(4+n) .
As viscous quantities, we have the shear viscosity η and the bulk viscosity ζ, but ζ vanishes for the fluid of massless particles. In imperfect fluids, propagating gravitational waves are absorbed in multi-dimensional universes as well as in 4-dimensional universes. The corresponding η is expressed as where G = κ/8π is a (4 + n)-dimensional gravitational constant, and N r is the number of radiative particle species absorbing gravitational waves. We consider states so hot that there are many interactions between particles, and so we assume N r /N = 1 for simplicity. The definition of η and ζ and the derivation of Eq.(6) are shown in Appendix A.
The space-times are described by the line element where M, N = 0, ..., (3 + n), i, j = 1, 2, 3, and α, β = 4, ..., (3 + n). The spaces with metrics g ij and g αβ are the spaces with constant curvatures k f and k h (= 0 or ±1). The Einstein equations are where G = κ/8π is a (4 + n)-dimensional gravitational constant. The energy-momentum tensor T M N for fluids with energy density ǫ, pressure p and viscosity η is defined in Appendix A, and their components are expressed under the comoving condition (u L = δ L 0 ) as where κ i j = 2(ḟ /f )δ i j , κ α β = 2(ḣ/h)δ α β , (an overdot denoting ∂/∂t) and whereV /V = 3ḟ /f + nḣ/h. Then the above Einstein equations are expressed as where From the above equations we can geṫ For the change in the total entropy S, we obtain from Eqs. (4) and (14) ( Evidently S = const for the non-viscous case (η = ζ = 0). It is assumed that initially the universe expands isotropically, but the expansion of the n-dimensional inner space is slower than that of the 3-dimensional outer space. At the later anisotropic stage the inner space contracts, while the outer space continues to expand. These behaviors correspond to the solutions with k h = 1 and k f = 0 or −1. Their solutions can be derived solving Eq.(11) numerically, but, paying attention to their early isotropic stage and the later anisotropic stage (after the epoch of maximum expansion of the inner space), we can use approximate power solutions which are derived in the following, neglecting the curvature terms. They are expressed as with µ = ν at the isotropic stage, and with µ = ν at the anisotropic stage, where h I , f I , t I , h A , f A , and t A are constants. The times t I and t A represent the initial epoch of the isotropic stage and the final epoch of the anisotropic stage, respectively. From Eq. (11), we obtain the solutions expressed as where H ≡ 3ν 2 + 1 2 n(n − 1)µ 2 + 3nµν. In the isotropic case, the solutions of these equations and the energy density are In the anisotropic case, which we consider from now on, they are with and where and we must take account of higher terms in f and h due to the curvature terms, to derive ǫ. For η 0 = 0, on the other hand, S is constant and so from Eq.(4) where α ≡ (nµ + 3ν)(4 + n)/(3 + n) = 10/9 for n = 6. The temperature T (∝ ǫ 1/(n+4) ) is expressed as for η 0 = 0, = 0, respectively. Accordingly, the temperature change for η 0 = 0 is very rapid, compared with that for η 0 = 0. Around the epoch when the inner space takes maximum expansion, the above power solution cannot be used and curvature terms must be considered. Here we specify the epoch t M as the epoch which is comparatively near the epoch of maximum expansion but when the power solutions are approximately applicable. In Sect.4 we consider the anisotropic stage after t M for simplicity and use the approximate power solutions. In Sect. 5, 6 and 7, we treat the solutions applicable at all epochs and compare them with the approximate power solutions.

Epochs of freeze-out and decoupling of the inner space
While the inner space continues to collapse, the outer space inflates. In order that the expansion of the outer space may change to the Friedman-like slow one, the inner space must be separated from the outer space (see Fig. 1). This epoch is called the decoupling (or stabilization) epoch. At this epoch the size of the inner space is estimated to be comparable with or smaller than the Planck length in the outer space, so that the space-time may be completely quantized on that scale in the outer space.
Around this epoch, we may have an epoch when the physical state in the inner space may change. Kolb et al. [12] suggested that classical treatments of space-time and fluids in the inner space may be questionable around the epoch (t * ) when hT = 1, in which h represents the size of the inner space and T is related to ǫ by ǫ = T 4+n (n = 6). This is because h is comparable with the mean wavelength of massless particles. This epoch (t * ) is called the freeze-out epoch. Kolb et al. assumed that t * is near the decoupling epoch, and derived the 3-dimensional entropy S 3 within the horizon. But they found it is impossible for it to reach the expected value ∼ 10 88 in the Guth level, [1] if the dimension n of the inner space is less than 16. Abbott et al. [9,10] found that at t * it reaches the expected value, if n is ∼ 40.
In the next section we analyze the entropies at t † (which is different from t * ), and derive the condition (A) for S 3 = 10 88 using classical relativity. Moreover, we also consider the Planck length in the outer space (f pl ), corresponding to the Newtonian gravitational constant (G N ). It is defined in the connection with G (= κ/8π) by Using this relation, we derive the condition (B) that the size of the inner space h(t) is comparable with or smaller than f pl , and examine the compatibility between conditions A and B. The total entropy S in the multi-dimensional space-time is expressed at the stage of (t A − t) ≪ t A using Eqs. (4) and (17) as where β ≡ (3 + n)/(4 + n) − (nµ + 3ν) for η 0 = 0. For the non-viscous case, S is constant. The behavior of S was discussed in Ref. [14] using the inflation and collapse factors of the outer and inner spaces. The important quantity which is to be noticed directly from the viewpoint of cosmological observations, however, is not S, but the 3-dimensional entropy S 3 within the horizon l h of 3-dimensional outer space.
Here S 3 is defined by where the 3-dimensional entropy density s 3 is given by s 3 = ǫ 3 3/4 , and the 3-dimensional energy density ǫ 3 is In contrast to S, S 3 increases with time, not only in the viscous case but also in the nonviscous case, because the common temperature T increases with the decrease of the volume V of the total space in both cases. 6/22 Here and in the following we neglect the factors ≈ 1 such as a 0 . Then where the horizon-size l h is 4.1. Entropy in the freeze-out epoch t * where we assumed that this epoch t * is at the stage of (t A − t) ≪ t A .
In the non-viscous case, we have µ = −ν = 1/3, and where we used the notation t * v to discriminate t * in the viscous case from t * in the nonviscous case. Now let us compare quantities at t * and t * v . Using Eq.(24), we have for η 0 = 0, η 0 = 0, respectively. Since h(t * )T (t * ) = h(t * v )T (t * v ) = 1 and h(t M )T (t M ) is common, we obtain from these equations so that S 3 in the non-viscous case is expressed in terms of t * v as By comparing this with Eq.(33), therefore, it is found that (S 3 ) * in the viscous case is much larger than (S 3 ) * in the non-viscous case, since h(t * v )/h(t M ) ≪ 1.

Entropy at epoch t † (different from t * )
Now let us derive the conditions A and B suggested in Sect. 3. First we derive S 3 at epoch In this case we have (a) the viscous case (η 0 = 0.225) 7/22 Using Eq.(24) we obtain with µ = 0.345. For S 3 at t † , we obtain neglecting the factors of integrals and we assumed that the epoch t † is at the stage of (t A − t) ≪ t A . If we express (S 3 ) † in terms of λ, we have by use of Eq.(39) Moreover, we obtain from these relations Next we consider the relations of f and h to the Planck length f pl in the outer space (cf. Sect. 3), which are derived in Appendix B. In order that the space-time in the outer space can be treated classically at epoch t † , the condition f † /f pl ≫ 1 must be satisfied. This condition gives λ > 1.96. Moreover, h † /f pl is smaller than 1, if λ < 1.06 × 10 4 . It is found, therefore, that, if λ ∼ 1.06 × 10 4 , the outer space at epoch t † gets (S 3 ) † = 10 88 and is decoupled from the inner space at the same time, because it has the quantized space-time. Using Eq.(24) we obtain For S 3 at t † , we obtain neglecting the factors of integrals Moreover, we obtain from these relations Next we consider the relations of f and h to the Planck length f pl in the outer space, which are derived in Appendix B. In order that the space-time in the outer space can be treated classically at epoch t † , the condition f † /f pl ≫ 1 must be satisfied. This condition gives gives λ > 10 −5.88 . Moreover, h † /f pl is smaller than 1, if λ < 46.1. It is found, therefore, that, if λ ∼ 46.1, the outer space at epoch t † gets (S 3 ) † = 10 88 and is decoupled from the inner space at the same time, because it has the quantized space-time.

Numerical histories of multi-dimensional universes
In this section let us solve numerically equations of scale factors f (t) and h(t) in the outer and inner spaces and equations of the energy density ǫ(t) (given by Eqs. (11) and (13)) at the total stage and relate their behaviors at the final asymptotic stage (which were treated in the previous section) to the behaviors at the earliest stage. Here we assume that the inner space has positive curvature (k h = 1) and the outer space is flat or has negative curvature (k f = 0 or −1). Now let us transform variables (t, f, h, ǫ, p, η) to (t,f ,h,ǭ,p,η) as follows, for the convenience of numerical calculations: where ζ is a positive constant and η 0 , k f , k h are assumed to be invariant. For this transformation, the forms of Eqs. (11), (12), and (13) are invariant. In this section these equations for the new variables are solved, where ζ is determined so as to be consistent with the initial condition in the following.
To determine the initial condition at t = t i for solving these equations, we consider the approximate solutions f (t) and h(t) around the isotropic solution t 1/5 (in the limit of small t) as with constants h 0 and f 0 (= h 0 ), where we assume f 1 ≪ 1, h 1 ≪ 1. Here the isotropic solution is equal to Eqs. (16) and (19) with T I = 0, and t i is a small positive time. Then we obtain from Eq.(11)f where the last terms come from the positive curvature (k h = 1) in the inner space and the curvature k f (= 0 or 1) in the outer space, and from Eq.(13) Solving these equations, we obtain the following approximate solutions where 10/22 and f 10 − h 10 = 125 × 3 288 Next, using these solutions, we make 6 types of initial conditions f, h,ḟ , andḣ at t = t i which are discriminated with η 0 , h 0 and t i as a. (010)  (58) corresponding to these initial conditions, t is dimensionless, while ζ has a dimension of length. For these initial values, Eq.(11) was solved numerically using the Runge-Kutta method, and the result is shown by solid curves for t ≥ t i and by short-dashed curves for t < t i in Figs. 2, 3, 4, 5, 6, and 7 for k f = 0. The behavior of f and h for k f = −1 is similar to that for k f = 0.
In these solutions we estimated the singular epoch t = t A and derived the power solutions for η 0 = 0, 0.225, respectively, are shown as functions of t by solid curves in Figs. 8,9,10,11,12, and 13 for k f = 0. The temperature T [= (ǫ/N a n ) 1/10 ] is also shown by dashed curves in the same figures. These two quantities Φ 0 (t) and Φ 2 (t) are invariant for the transformation (58) and so they can be regarded as quantities on the ordinary scale of (t, f, h, ǫ, p, η). Moreover, λ, S 0 and the ratios of f, h, and T also, which are determined in the next section using these Φ 0 (t) and Φ 2 (t), can be treated as quantities on the ordinary scale.  1) and (4), the total entropy S is given by S = 5 4 N a n V T 9 for n = 6, so that S 9 is defined by   The values in cases a, ..., f on the ordinary scale are shown in Table 5 for k f = 0 and −1.
We assumed N a n = 1 in the above calculations. This factor N a n appears often in the calculations, as h i /f pl ∝ (N a n ) 0.1074 and T i /T pl ∝ (N a n ) −0.1215 for η 0 = 0, and h i /f pl ∝ (N a n ) 0.09118 and T i /T pl ∝ (N a n ) −0.1182 for η 0 = 0.225. But the product does not much depend on N a n as T i h i ∝ (N a n ) 0.014 , (N a n ) 0.027 , respectively.

Dimensional symmetry-breaking and the primeval state of multi-dimensional universes
Kim et al. [2,3] showed in a matrix model of super-string theory that, due to the symmetrybreaking, the (1 + 3 + 6)-dimensional universe with isotropic expansion changes to that with anisotropic expansion, in which the 3-dimensional space expands at the larger rate than the 16/22     6-dimensional space. In the present treatment due to classical relativity, such a symmetrybreaking cannot be studied accurately. From the viewpoint of energy balance, however, we may examine the behavior of the symmetry breaking in simplified situations.
1. Case of k f = 0 Let us assume that at epoch t br the symmetry-breaking occurred from the isotropic state with scale factorsf =h (kf = kh = 1) to the anisotropic state with f = h (k f = 0 and 17/22 k h = 1), without change inḟ /f,ḣ/h and ǫ. Then from Eq. (13), we obtain the following condition for consistency