Fluctuations of the cosmic background radiation appearing in the 10-dimensional cosmological model

We consider a cosmological model starting from (1) 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) and reaching (2) the Friedmann model after the decoupling between the outer space and the inner space, and derive fluctuations of the background radiation appearing in the above 10-dimensional space-times. For this purpose we first derive the fluid-dynamical perturbations in the above 10-dimensional space-times, corresponding to two kinds of curvature perturbations (in the scalar mode) in the non-viscous case, and next study the quantum fluctuations in the scalar and tensor modes, appearing at the stage when the perturbations are within the horizon of the inflating outer space. Lastly we derive the wave-number dependence of fluctuations (the power spectrum) in the two modes, which formed at the above decoupling epoch and are observed in the Friedmann stage. It is found that it can be consistent with the observed spectra of the cosmic microwave background radiation.


Introduction
In order to derive the observed fluctuations of cosmic microwave background radiation, we study the cosmological evolution of the (1 + 3 + 6)-dimensional space-times, in which it is assumed that our universe was born as an isotropic and homogeneous 10-dimensional spacetimes and evolved to the state consisting of the 3-dimensional inflating outer space and the 6-dimensional collapsing inner space. Our 4-dimensional Friedmann universe appeared after decoupling of the outer space from the inner 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 the subsequent paper [5] , we studied the evolution of cosmological perturbations in the non-viscous case, solving the equations for geometrical perturbations.
In this paper we treat the fluidal perturbations corresponding to the geometrical perturbations, the quantum fluctuations in the scalar and tensor modes, and the consistency with the observations of cosmic microwave background radiation (CMB) in the non-viscous case. In Sect. 2, we first review the background model and the perturbed quantities. In Sect. 3, we derive the perturbed fluid-dynamical equations, corresponding to the geometrical perturbations in the 10-dimensional space-times, and solve them. In Sect. 4, we consider the quantum fluctuations in the scalar and tensor modes at the stage when they were within the horizon of the outer space with the inflationary expansion, and derive the initial conditions for their perturbations appearing after this stage. In Sect. 5, we derive the spectra of perturbations in these two modes, and compare them with the observed ones. In Sect. 6, concluding remarks are given. In Appendix A, we derive the higher-order terms in the two curvature perturbations with respect to small wave-numbers in the outer space.
2.1.1. Background 10-dimensional model before the decoupling. The background 10dimensional 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. 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. At the singular stage when R is near 0, the curvature terms with K r /r 2 and K R /R 2 are negligible, compared with the main terms, and the curvatures can be treated approximately as K r = K R = 0. 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 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 The Einstein equations for r and R were solved numerically in the previous paper [3] and their behaviors were shown in Figs. 1 -7 of [3]. 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 3/23 Fig. 2: Ratios of physical sizes of perturbations with the wave-number k to the Hubble length 1/H. The ratio r(τ )H/k in the outer space of the 10-dimensional space-times (τ ≡ t − t A ) is shown as the solid line on the left-hand side, and the ratio a(t f )H/k in the Friedmann model is shown as the solid line on the right-hand side. the maximum expansion, and at the final stage we have an approximate solution with and where t A 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. The curvature tensor is singular, on the other hand, in the limit τ → 0, like that in the 4-dimensional Kasner space-time. [6] 2.1.2. Decoupling of the two spaces and the Friedmann stage. As τ decreases, the inner space contracts and finally the size reaches the Planck length at the decoupling epoch. We 4/23 discussed the decoupling condition at this quantum-gravitational epoch and the entropy production to this epoch in the previous paper [3]. At present we cannot analyze the process of decoupling accurately, because no quantum theory of gravitation has been established yet. However it is expected that the inner space is so homogeneous and quietly evolves without violent phenomena. This is because in both the inner and outer spaces the perturbations are assumed to be caused by quantum fluctuations before the decoupling, grow gravitationally, and remain very small and at the linear stage, before the decoupling. Note here that gravitational instability in the outer and inner spaces was treated in the previous paper [5]. Thus the inner space separates quietly from the outer space and disappears, while in the outer space the Friedmann model appears after the decoupling After this decoupling epoch it is assumed here that the outer space is separated from the inner space and described using the Friedmann model with the metric with the cosmic time t f and σ(χ) = sin χ, χ, sinh χ for the curvature +, 0, − in the space, and the scale-factor a(t f ) ∝ t f 1/2 at the radiation-dominated hot stage. At the decoupling epoch t dec and (t f ) dec , the entropy is assumed to be conserved in the 10-dimensional space-time and the Friedmann model. The behavior of scale-factors is shown in Fig. 1.
After the decoupling epoch we have the ratio at the Friedmann stage, which decreases with time. The two ratios are nearly equal at the decoupling epoch. These ratios are shown schematically in Fig. 2. They can take the value 1 in both sides before and after the decoupling epoch, that is, we can have the horizon crossing in both sides. Quantum fluctuations are created at the epoch of r(τ )H/k < 1 in the outer space of the 10-dimensional space-time, and the fluctuations are observed as the fluctuations of the background radiation at the epochs of a(t f )H/k < 1 at the Friedmann stage.

Perturbed quantities
The simplest treatment of perturbations of geometrical and fluidal quantities is to expand them using harmonics, and to find the gauge-invariant quantities. For the four-dimensional universe (in the Friedmann model) it was shown in Bardeen's theory on perturbations [7]. 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. [8]  In this paper only scalar and tensor modes are considered in the 10-dimensional spacetimes. So quantities in these modes are shown here.
2.2.1. The scalar mode. The metric perturbations are expressed as where q (0) , q and where we consider a perfect fluid, so that the anisotropic pressure terms vanish and we have For the metric perturbations in Eq. (14) the following gauge-invariant quantities are defined: 6/23 The gauge-invariant quantities Φ h and Φ

(r)
A in the outer space correspond to the gaugeinvariant perturbations defined by Bardeen[7] in the (1 + 3)-dimensional usual universes, 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 It should be noticed that v and ǫ m do not vanish, though ρ = 0 at the final stage as in Eq. (10). As a gauge-invariant quantity that has no counterpart in the usual universe, we have Moreover the auxiliary quantities (Φ 6 and Φ 7 ) andΦ G are defined bẏ 2.2.2. 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.
More details about perturbations can be seen in the previous paper [5].

Evolution of fluidal perturbations in the scalar mode
In the previous paper [5], we derived the equations for geometrical perturbations Φ h , Φ H ,Φ G and Φ 6 in the 10-dimensional space-times, and found their behavior by solving them. In this section we derive the equations for gauge-invariant variables representing fluidal perturbations ǫ m , v First we obtain the following equation forǫ m from δT A 0;A = 0 R are the wave-numbers in the outer and inner spaces, respectively, and r = r 0 τ −1/3 , R = R 0 τ 1/3 , τ = t 0 − t, and t 0 denotes the epoch of r → ∞ and R = 0.
Equations forv s andV s are obtained from δT A i;A = 0 and δT A a;A = 0, respectively, aṡ From the latter two equations we obtain and integrating this above equation, we have From Eqs. (27) and (31), we obtain where Next, differentiating Eq. (32) with respect to t and eliminating v s andv s using Eqs. (28) and (32), we obtain the following equations for ǫ m and v s and whereC

Outside the horizons
At epoch t dec when the outer space and the inner space decouple, τ is assumed to be so small that Under these conditions the perturbations with wave-numbers k R are outside the horizons in the outer and inner spaces, and we have the relations where ∆Φ h and ∆Φ H consist of higher-order terms O(x 2 ) and O(y 2 ) with respect to x and y, respectively, which are shown in Appendix A, and τ i is an arbitrary epoch at the stage when Eqs. (44) and (45) are satisfied. Now let us derive ǫ m , v s and V s corresponding to the above curvature perturbations, neglecting higher-order terms, such as ∆Φ h , ∆Φ H ,Φ G and Φ 6 . Here we must pay attention to Φ 7 .

Inside the horizons
At earlier epochs of τ ≫ τ dec , x and y are comparable with 1 or larger than 1. It was shown in Sect. 3 of [5] that in the case of x ≫ 1 and y ≫ 1 under the condition the perturbations show wavy behaviors, depending on the wave-number k r /r(t)] 2 ≪ 1. In this case, the perturbations with large k in the inner space are negligible. In the following we study their wavy behaviors proportional to exp(iωx). This is because such perturbations will survive and may be connected with the present observational information through the CMB radiation, after the decoupling of the outer space from the inner space. The other perturbations including the components (∝ exp iωy) in the inner space will be disturbed or erased when the inner space is decoupled and disapear.
In Sect. 3 of [5], it was found that the approximate wavy solutions of equations for curvature perturbations are given only for ω = 1 and 1/3, and in these cases the solutions have the following relations and Φ H0 = Φ h0 for ω = 1/3.

Quantum fluctuations
In Sect. 2 and the previous paper [5], the perturbations were classified into three modes. In this paper we treat only their scalar and tensor modes and consider the perturbations created by the quantum effect in the comparably later stage of the 10-dimensional universe which is associated with the inflating outer space and collapsing inner space. Here Weinberg's procedure is used for the quantization [9]. 12/23

The scalar mode
At the stage of x ≫ 1 andȳ ≫ 1, the length of perturbations in the outer space can be smaller than the horizon size and they may be caused by the quantum effect, while at the later stage of x < 1 the length of perturbations is larger than the horizon size and they are frozen. So we should first consider the quantum fluctuations at earlier epochs of x ≫ 1 and y ≫ 1. Additionally, moreover, we assume thatμ/x ≪ 1, corresponding to the perturbations in the inner space with the average value (k (0) R ), which was described in Sect. 3. Then these perturbations appear mainly in the outer space, and hence we can treat the perturbations, as if they are in the 4-dimensional space-time (consisting of the time t and the outer space).
The energy density perturbation ǫ m is expressed by Eq. (37) in connection with gravitational perturbations. This equation is also derived from the action principle as where L ǫ and L g are the fluidal and gravitational parts of the total Lagrangian, and x is the coordinate in the outer space. Here L ǫ can be derived from Eq. (37) as follows. At the stage of x ≫ 1,ȳ ≫ 1 andμ/x ≪ 1, we have D 0 and D 1 in Eq. (56), and then the fluidal part in the equation of motion is derived using the following Lagrangian where r = r 0 (t 0 − t) −1/3 , and ∂ǫ m /∂x = ik . On the other hand, ǫ m can be expanded as and Φ h and Φ H also can be written as where the reality of these fields requires to take the above forms. The interaction of the photon field with the gravitational field makes the commutation relation of α(k (0) r ) and α * (k (0) r ) complicated, but they become simple at very early times. [9] In many cases when quantum fluctuations have so far been treated in a system of a scalar (inflaton) field and the gravitational field, the quantization of the scalar field is first tried. [9] In the present case also when we consider a system of a photon scalar field and the gravitational field, we try first the quantization of the photon scalar field in the following.
The canonical conjugate to ǫ m (x, t) is then The commutator of ǫ m and π m is 13/23 These commutation relations imply that α(k) and α ⋆ (k) behave as conventionally normalized annihilation and creation operators where t * is arbitrary and ω is a constant (= 1 or 1/3). This expression of ǫ m (k (0) r , t) is used as the initial condition for created fields of energy density ǫ m . Here we choose the quantum state during the inflation of the outer space under the simple assumption that the state of the universe is the vacuum state |0 , defined so that α(k)|0 = 0 and 0|0 = 1. (76) This corresponds to the Bunch-Davies vacuum [10] in the outer space within the 10dimensional universe. As described in Sect. 3, ǫ m and the curvature perturbations as quantum fluctuations are proportional each other. So the behavior of ǫ m in Eq. (75) is common to that of Φ h and Φ H , and, using Eqs. (64) -(67), we obtain for ω = 1, and for ω = 1/3.

The tensor mode
In the tensor mode, there are two types (ST) and (TS), as described in Sect. 2 and [5]. (ST) has the 3-dimensional scalar and the 6-dimensional tensor, while (TS) has the 3-dimensional tensor and the 6-dimensional scalar. Here we take up (TS) with the amplitude h T , which may not be connected with the observation in the 3-dimensional outer space, after the decoupling of the inner space.
In [5], we studied the behavior of tensor perturbations h T . They satisfÿ Here we consider the case of (k where k  14/23 where we used the relation R ∝ 1/r ∝ τ 1/3 and d = D/2 = 3. This equation can be also derived from the action principle as where for ∂h (2) T ∝ exp ik (2) r x. On the other hand, the amplitude ∂h (2) T takes the form and the canonical conjugate to h (2) T (x, t) is then The commutator of h (2) T and π T is T (x, t), ∂h These commutation relations imply that α(k) and α * (k) behave as conventionally normalized annihilation and creation operators, in the same way as Eq. (73), when h This expression of h T (k (2) r , t) is used as the initial condition of created fields in the tensor mode h (2) T . Here we choose the quantum state during the inflation of the outer space, so that the state of the universe may satisfy the relation in Eq. (76).

The scalar mode
The information about the perturbations which are created by the quantum fluctuations inside the horizon can be used to make an initial condition for the evolution of perturbations which re-enter the horizon after the long inflation. For this purpose, we use the quantities which are conserved outside the horizon. In the 4-dimensional universe with 3-dimensional space-section, we have a gauge-invariant curvature perturbation, represented as which is a conserved quantity. [7] In the 10-dimensional universe, on the other hand, we have the following two independent similar quantities as the candidates where τ dec represents the epoch when the inner space decouples from the outer space. For r τ 4/3 ) < 1 and y(≡ (3/2R 0 )k (0) R τ 2/3 ) < 1, R h and R H are nearly constant, and so these can be regarded as quantities conserved outside the horizon. 15/23 As other candidates for conserved quantities, we may consider but, for x ≪ 1, they are not independent of R h and R H , with respect to the main terms. For x ≫ 1 and y ≫ 1, moreover, we find that v s and V s are comparable with Φ h and Φ H , respectively, and R v and R V are ∼ v s /x and ∼ V s /y, respectively, which are small, compared with Φ h and Φ H . This means that the roles of R v and R V are small, compared with those of R h and R H , respectively. In this paper, therefore, we adopt R h and R H as the conserved quantities in the 10-dimensional universe. Neither of them, however, is necessarily a conserved quantity which is directly connected at epoch τ dec with R 4 in the 4-dimensional universe.
Here we construct the 10-dimensional gauge-invariant conserved quantity R 10 using R h and R H , by imposing the following two conditions : (1) R 10 = R 4 at epoch (τ dec ) of the decoupling of the outer space from the inner space, and (2) R 10 is consistent with the spectral constraint given by the CMB observation.
As the first candidate of R 10 , we consider a linear combination of Φ h and Φ H as where constants λ 0 and λ 1 are determined so as to satisfy the above two conditions (1) where we used r ∝ τ −1/3 and α is an arbitrary constant. Inserting Eq.(94) into Eq.(89), we obtain As x decreases and becomes smaller than 1, the x dependence of R h and R H changes from the wavy behavior (∝ exp(ix) and exp(ix/3)) to the constant ones. At epoch τ eq with x = 1, 16/23 we have therefore where where r τ dec 4/3 , and ζ h , ζ ′ h , ζ H and ζ ′ H are constants. The exact values of these constants are determined by solving dynamical equations for Φ h and Φ H given in [5], but they are estimated to be ≈ 1, because R h and R H are nearly constant for x < 1 and y < 1. Now we assume that the CMB spectrum is determined at epoch τ eq when x = 1 (indicating the horizon exit), and consider the k (0) r dependence of R 10 at this epoch. Here R 10 at epoch τ eq is expressed as around the observed wave-number (k where Re means the real part, and λ 0 and λ 1 are coefficients in Eq.(93). The CMB observation shows that the k  [11]. The condition that Eq.(100) and Eq.(103) should be consistent in the neighborhood of z = 1 is where . From the continuity of this equation and its first derivative at z = 1, it is found that That is, the observational spectrum (103) can be reproduced when R H is main and R h is about 10% of the total R 10 . The above definition of R 10 satisfies the condition of continuity of Eq.(104) in the first derivative, but not in the second derivative. In order to satisfy also the condition of continuity 17/23 in the second derivative, we consider the second candidate of R 10 at τ eq expressed as where constants λ 0 , λ 1 and λ 2 are determined so as to satisfy the above two conditions (1) and (2). Here R 10 at epoch τ eq is rewritten as where R 0 and R 1 are defined by Eqs. (101) and (102), and Then from the condition that Eqs. (107) and (103) should be consistent in the neighborhood of z = 1, we have where δ 1 ≡ R 1 /R 0 and δ 2 ≡ R 2 /R 0 . From the continuity of this equation and its first and second derivatives at z = 1, it is found that δ 1 = −0.260 and δ 2 = 0.0696.
Now let us define the power spectrum of curvature perturbations as [11,12] Then for R 10 , δ 1 and δ 2 in Eqs. (106) and (110), On the other hand, we have where x dec = (τ dec /τ eq ) 4/3 = (r eq /r dec ) 4 ≪ 1, and the factor Re(− 1 It is concluded that R 10 is consistent with the observed spectra of CMB radiation under the condition of (114) and (115).
Thus we could derive the condition that the parameters λ 0 , λ 1 and λ 2 in R 10 should satisfy for the consistency with the CMB observation. From their ratios the role of the curvature perturbation in the inner space is found to be larger than that in the outer space. This 18/23 condition and its consequeces are concerned with the condition at the earlier stage and the initial condition of the universe, which should be expressed as a perturbation model with the theoretical model parameters, and the above three parameters should be related to the latter parameters. They may be influenced through R 10 by the process of decoupling, which has not been discussed here, because its quantum-gravitational process cannot be treated at present. This situation in the observational aspect is compared with the situations in other inflation models, later in the subsection 5.3

The tensor mode
In the limit of x (≡ (3/4r 0 )k (2) r τ 4/3 ) → 0, the gauge-invariant perturbation h (2) T tends to a + b ln τ , as seen from the analyses in Abbott et al. [8] and the previous paper [5], where a and b are constants. So, as the quantity R t which is conserved outside the horizon, we adopt so that R t leads to a constant in the limit of x → 0. At the epoch τ eq of x = 1, we have the relation so that r 2 [k r ] −1/2 = const, and from Eq. (87) where λ t is a constant. Then it is found from Eq. (116) that As x decreases and becomes < 1, the x dependence of R t changes from the wavy behavior to the stationary constant one. But the k (2) r dependence does not change, so that the spectrum in the tensor mode have the form of The corresponding power spectrum is The amplitude of R t (τ eq ) should be determined, corresponding to the observation, which has not been given yet. At present, we have the condition r ≡ P t /P s < 0.24 for k

Comparison with the spectral analyses in other inflation models
In the 4-dimensional universe due to the Einstein theory, the quantity conserved outside the horizon is uniquely defined using one of curvature perturbations. [7] But in hypothetical inflation models with inflaton scalar fields (including the non-minimal coupling with the Ricci scalar), the values of parameters such as slow-roll parameters (ǫ, η) and the number N of inflationary e-folds, [9,13] and the coupling parameter ξ in the scalar field equation [14][15][16][17] are not unique. The observed spectral index n s (≈ 0.97) is, therefore, obtained by adjusting the above parameters ǫ, η, N and ξ.
In the R + R 2 modified gravitational theory, we have an inflation model associated with the de Sitter type solution which was derived first by Nariai and Tomita [18] and rederived later by Starobinsky. [19] Mukhanov and Chibisov [20] derived the quantum fluctuations generated at the de Sitter stage, and it was found that the spectral index n s of these fluctuations can be expressed as where k obs is the observed wave-number, N is the inflationary e-fold, and a and H are the scale factor and the Hubble constant at the epoch when the de Sitter expansion ends. This number N is determined to be 70, so that we may have n s ≃ 0.97 (the observed value).
In the present case of a photon scalar field in the 10-dimensional universe, the inflation of the outer space is unique, because the scale factor r of the outer space is ∝ τ −1/3 (in the non-viscous case). On the other hand, the conserved quantity is not unique, because there are two independent curvature perturbations Φ h and Φ H before the decoupling of the outer and inner spaces. It is, therefore, a key point to determine how to combine them in this case, to derive R 10 (connecting the two epochs outside the horizon). To obtain the observed spectral index n s , we made the examples of the combination of Φ h and Φ H as the conserved quantity, so that the theoretical spectral index n s may be consistent with the observed one.

Concluding remarks
In this paper I showed the possibility of deriving the observed fluctuation of CMB radiation from the quantum fluctuations which appeared at the inflating stage of the outer space in the 10-dimensional universe. In contrast to the rapid inflation in the inflaton scalar field, our inflation is a power type, but we have two independent curvature perturbations which make possible the consistency with the observed spectra.
For simplicity, on the other hand, I neglected the viscosity which may play important roles in dynamics between the outer space and the inner space. If we take the viscosity into account, not only much entropy is produced (as shown in the previous paper [3]), but also the severe condition such as λ 2 /λ 0 ≃ (λ 1 /λ 0 ) 2 ≪ 1 for producing the observed CMB fluctuations in the scalar mode may be softened. The next step is to study the perturbations and quantum fluctuations to derive the condition, in the case with viscous processes due to the transport of 10-dimensional gravitational waves. [3,21]