On the perturbation theory in spatially closed background

In this article,we investigate some features of the perturbation theory in spatially closed universe. We will show that the perturbative field equations in a spatially closed universe always have two independent adiabatic solutions provided that the wavelengths of perturbation modes are very longer than the Hubble horizon. It will be revealed that these adiabatic solutions do not depend on the curvature directly. We also propound a new interpretation for the curvature perturbation in terms of the unperturbed geometry.


Introduction
The theory of the linear perturbations is an important part of the modern cosmology which explains CMB anisotropies and structure formation origin. This theory has been investigated for a spatially flat universe exceedingly [1,2,3,4,5,6,7,8,9,10]. However, observational data points out a universe with Ω Λ ∼ = .68 [11]. Existence of a positive cosmological constant necessitates a de Sitter spacetime for the vacuum background. From the different forms of the de Sitter spacetime with K = 0, ±1, merely K = 1 case, namely, Lorentzian de Sitter spacetime is maximally symmetric, maximally extended and also geodesically complete [12]. So in the following we assume Λ > 0 and K = 1 for the vacuum background. Furthermore, it seems hard to believe that the total density of the universe has been tuned in ρ crit 0 exactly, because despite the fact that the observational data indicate Ω K = 0 [11], this fine-tuning seems somehow unlikely. Moreover, if Ω tot equals to +1 exactly, this cannot last forever because of the instability [13]. On the other hand, there are some reasons that the universe may have positive spatially curvature with non-trivial topology. In other words, some positive curvature models with non-trivial topology can solve the problem of the CMB quadrupole and octopole suppression and also mystery of the missing fluctuations which appear in the concordance model of cosmology [14,15,16,17,18]. So these reasons augment the probability of spatially closed case and it seems necessary to investigate the theory of small fluctuations in spatially closed universes. The outline of this article is as follows. In Section 2 we derive the equations governing the linear perturbations in FLRW universe without fixing K. In Section 3 we study the spectral and stochastic properties of these perturbations for the case K = 1 and in Section 4 the gauge problem will be discussed. Finally, in the last section we derive two independent adiabatic solutions for the obtained equations with K = 1 while the perturbations scales go outside of the Hubble horizon. It will be seen that one of these solutions is decaying, so it has not cosmological significance. We also deduce a new geometrical interpretation for the curvature perturbation as the conformal factor of the spatial section of the background spacetime. Furthermore, we will show that for the supper-Hubble scales, curvature has no direct effect on the universe evolution.

The perturbed spacetime
We assume during most of the time the departures from homogeneity and isotropy have been very small, so that they can be treated as the first order perturbations. The total perturbed metric is whereḡ µν and h µν are the unperturbed metric and the first order perturbation respectively. Note thatḡ µν is the FLRW metric which in the comoving quasi-Cartesian coordinates can be written as [2] g 00 = −1, g 0i = g i0 = 0, Bar over any quantity denotes its unperturbed value. Perturbing the metric leads to perturbing the connection and Ricci tensor as [2] and The perturbative form of the Einstein's field equations may be written as where On the other hand, the perturbation of the energy-momentum conservation law gives Setting ν equal to 0 and i gives equations of energy and momentum conservation respectively. The explicit form of these equations is too lengthy and complicated, so we avoid expressing them here. Fortunately there is a mathematical technique, which simplifies these equations remarkably [3,4,5]. According to this technique we can decompose h µν into four scalars, two divergenceless, spatial vector and a symmetric, traceless, divergenceless spatial tensor as follows where ∇ i is the covariant derivative respect to the spatial unperturbed metricḡ ij (= a 2g ij ) and H ij = ∇ i ∇ j is the covariant Hessian operator . All the perturbations A, B, E, F, C i , G i and D ij are functions of t and x which satisfy Eq.(8) is generalization of the Helmholtz's decomposition theorem from R 3 to the Riemannian manifolds. Eq. (9) is also a theorem in Riemannian geometry [19,20]. According to this theorem, every rank 2 symmetric tensor on a compact Riemannian manifold can be uniquely represented in the Eq.(9) form. It is possible to carry out a similar decomposition of the energymomentum tensor. One can show that [2] δT 00 = −ρh 00 + δρ, We can decompose velocity perturbation δu i into the gradient of a scalar (velocity potential) δu and a transverse vector We may consider imperfectness of the cosmic fluid by adding a term Π ij to δT ij . Π ij is known as anisotropic inertia tensor field of the fluid and may be decomposed just like as h ij where Π V i and Π T ij satisfy conditions analogous to the Eqs. (10) and (11), which satisfied by C i and D ij in return. In Eq. (13) there is no term proportional tog ij , because δT ij itself contains such term. Finally, we have Now let's define Laplace-Beltrami operator Thus, for scalar field S we have Also for vector field V i and tensor field T ij we can write Substituting the Eqs.(7), (8), (9), (12), (14), (15) and (16) in the field and conservation equations namely Eqs.(4) and (6) and also separating the terms containingg ij ,∇ i and H ij , accompanied by using Eqs. (17), (18) and (19) results in three idependent sets of coupled equations

Vector mode equations
.

Tensor mode equation
As previously mentioned, in the linear perturbation theory, the scalar, vector and tensor modes evolve independently. The vector and tensor modes are not important for structure formation because they produce no density perturbation, albeit they affect on the CMB anisotropy.

Fourier decomposition and random fields
In this section, we study the spectral and stochastic properties of the perturbations for the case K = +1. Albeit the equations have been derived in Section 2 describe the time evolution of the perturbative quantities, viewed as functions of position (at fixed time) they are considered as random fields on S 3 (a), because they are defined on a homogeneous and isotropic space [7,21]. Now we investigate the stochastic properties of perturbations for every mode separately.

Scalar perturbations and scalar random fields
An important class of random fields are described by their Fourier transformations. There are many different Fourier transform conventions, however here our intention is the expansion of each mode of the perturbation fields in terms of the corresponding eigenfunctions of the Laplace-Beltrami operator. Thus, we have to find the eigen functions of ∇ 2 on S 3 (a). For scalar mode we have where ∇ 2 =ḡ ij H ij . In pseudo-spherical coordinates with the line element the Eq. (30) gives Solving Eq.(32) gets to the following eigenvalues and eigenfunctions [22,23,24,25] are known as Fock harmonics [22,25]. Also Y lm and C λ n are scalar spherical harmonics on S 2 and Gegenbauer (ultraspherical) polynomials respectively. It can be shown that where dµ = a 3 sin 2 χ sin θdχdθdϕ is the invariant volume element on S 3 (a). Scalar harmonics on S 3 (a) also can be expressed in terms of Jacobi polynomials or associated Legendre functions [26,27]. Furthermore Y nlm s constitute a complete orthonormal set for expansion of any scalar field on S 3 (a). Thus, for scalar perturbative quantity A(t, x) at some instant (which thereafter will be denoted byA(x)) we can write A nlm just like A (x) is a scalar random field. Apart from distribution function of A nlm , its simplest statistics are mean value and two-point covariance function, the latter is defined by A nlm A * n ′ l ′ m ′ . Here means ensemble average which equals to spatial average according to the ergodic theorem [7]. The homogeneity of S 3 (a) implies for any pair of scalar random field A and B A It means A nlm and A n ′ l ′ m ′ are uncorrelated random variables for different indices (indeed it results from the homogeneity of the spatial section of the backgrond spacetime). The homogeneity also implies that the coefficient of proportionality in Eq.(39) is just function of n i.e.
P 0 A (n) is power spectrum or spectral density of A (the superscript "0" over P states corresponding spin of the random field) which depends on distribution function governing on A. Moreover we have which P 0 A,B (n) is joint power spectrum of A and B [28,29]. One may define the correlation coefficient between A andB: −1 ≤ ∆ A,B (n) ≤ 1 and two extreme values ∆ A,B (n) = +1 and ∆ A,B (n) = −1 correspond respectively to full correlation and full anti-correlation [29]. Finally, let's define spectral index of random field A as Now we prove that the homogeneity of the universe yields Eq. (41). At first, On the other hand, according to the addition formula of Gegenbauer polynomials (Fock harmonics) [30] we have where cos γ = cos χ cos χ ′ + sin χ sin χ ′ (x.x ′ ). Consequently which is invariant under following transformations obviously Moreover one can show This shows for χ = χ ′ , cos γ is a function of |x − x ′ |, thus we conclude that Eq.(41) depends merely on |x − x ′ |. Now let's turn to the vector mode.

Vector perturbations and vector random fields
In order to investigate vector perturbation we should find vector spherical harmonics on S 3 (a) at first. They are solutions of the following equation The transversality condition is added as a constraint, because every vector perturbation in cosmology (C i , G i , Π V i ) is divergenceless. It can be shown that the vector spectrum of S 3 (a) is [22,23,24,25] and there are two independent eigenfunctions which in pseudo-spherical coordinates are and the other One can show that These vector harmonics constitute a complete orthonormal set for the expansion of any transverse vector field on S 3 (a). Thus, for vector perturbation where A o nlm and A e nlm are two random fields and like scalar perturbations we have

It yields
On the other hand, A i (x)A * j (x) must not change under parity transformation, because probability distribution function is invariant under spatial inversion so, P oe A (n) = 0. Furthermore, Because the power spectrum just depends on the probability distribution function and it cannot be function of parity. Thus, The last relation means A o nlm and A e nlm are statistically uncorrelated random fields, however, they have the same spectrum.

Tensor perturbations and tensor random fields
Every symmetric, traceless and transverse covariant tensor of rank 2 on S 3 (a) can be expanded in terms of t − t tensor spherical harmonics [22]. These harmonics can be classified into two groups: Odd parity a 2 2 (n 2 − 1) l (l − 1) (l + 1) (l + 2) × sin χ (l + 2) cos χΠ nl (χ) − n 2 − (l + 1) 2 sin χΠ nl+1 (χ) sin θW lm (θ, ϕ) , where Even parity (T e 11 ) nlm = a 2 n l (l − 1) (l + 1) (l + 2) 2 (n 2 − 1) where G nl (χ) = (l + 2) cos 2 χΠ nl (χ) − n 2 − 1 sin 2 χΠ nl (χ) It is also possible to express the tensor harmonics on S 3 (a) in terms of the Chebyshev polynomials of the first kind [31] which constitute the tensor Cartesian harmonics. It can be shown and also The set {(T o ij ) nlm , (T e ij ) nlm } constitutes a complete orthonormal basis for the expansion of any symmetric traceless-divergence-free covariant tensor field of rank 2 on S 3 (a). On the other hand, the tensor mode is completely characterized by two traceless-transverse symmetric tensors D ij (t, x) and Π T ij (t, x). We can expand them in terms of t − t tensor spherical harmonics on S 3 (a) : There is a similar expansion for Π T ij (t, x) (Note that we drop t here, because all quantities are considered at a fixed instant). D o nlm and D e nlm just like D ij (t, x) are two random fields, so where P +2 D (n) is the power spectrum of the gravitational wave D ij [32]. The probability distribution is independent of parity, so we cannot expect D o nlm D o * n ′ l ′ m ′ and D e nlm D e * n ′ l ′ m ′ having different values. In addition, because scalar, vector and tensor modes are independent, their joint power spectrums are vanished.

The gauge problem
In this section, we investigate the behavior of the perturbations under the gauge transformations. The equations derived in Section 2 may have physically equivalent solutions. This problem is called gauge freedom. Similar to the Einstein's field equations this gauge freedom may be fixed by choosing a coordinate system. For this purpose, let's consider a spacetime coordinate transformation with small ǫ µ (x) in the same sence that h µν and other perturbations are small. In cosmology, we call Eq.(70) a gauge transformation, if it affects only the field perturbations and preserves unperturbed metric [2,33]. Under such gauge transformation, the metric of spacetime changes as equivalently It yields After simplification we have Thus where ∇ µ is the covariant derivative corresponding toḡ µν . Consequently where ∇ i is the covariant derivative respect toḡ ij . Similarly we can derive the effect of gauge transformation Eq.(70) on the energy-momentum tensor or in more detail In order to derive the gauge transformations of the scalar, vector and tensor parts of h µν and T µν , it is necessary to decompose the spatial part of ǫ µ as follows Obviously Π S , Π V i , Π T ij , D ij and δu V i are gauge invariant quantities. Besides, one can construct more gauge invariant quantities by combination of the perturbative quantities, e.g. ζ = A 2 − H δρ . ρ (H =ȧ a ) which is known as thecurvature perturbation on the uniform density slices [34,35]. Note in particular that ζ is a pivotal quantity in cosmology which is related to the fluctuations of inflaton as well as, CMB angular power spectrum [32,36] and consequently connects the primordial perturbations to the present observational data. All of the tensor quantities are gauge invariant and in result gauge-fixing is not required. On the other hand, for the vector mode, we can fix a gauge by choosing ǫ V i so that either C i or G i vanishes. For the scalar perturbations, fixing a gauge means choosing ǫ 0 and ǫ S , so there are several ways to fix a gauge [5], but here we concentrate on a special gauge which was introduced by Mukhanov et al. [37] and is known as Newtonian gauge. In this gauge we choose ǫ 0 and ǫ S by setting B = F = 0. It is convenient to write E and A in this gauge as Φ and Ψ are known as Bardeen's potentials [34]. This gauge eliminates the gauge freedom completely in contrast to the synchronous gauge [2,38] which was introduced first time by Lifshitz [3]. In the Newtonian gauge the line element of the universe takes the form and the gravitational field and conservation equations become 3 (ρ +p)Ψ = ∂δρ ∂t + 3H (δρ + δp) + ∇ 2 (ρ +p) δu + a 2 HΠ S (91) In the next section we shall show that this system of equations has two independent adiabatic solutions.

Adiabatic modes in a spatially closed universe
In this section, we want to generalize the Weinberg's theorem [2,39] which has been proved for a spatially flat universe to the spatially closed case. According to this theorem whatever the contents of the universe, the perturbative field equations have two independent adiabatic solutions in the time intervals when the perturbation scales are often very longer than the Hubble horizon of the universe. These two solutions in Newtonian gauge and which ζ(x) is the curvature perturbation on the uniform density slices when the perturbations are outside of the Hubble horizon or equivalently conformal factor of S 3 and χ(x) is an arbitrary function of position.
In order to prove, initially we put Π S = 0, because the cosmic fluid is approximately perfect; thus, from Eq.(88) we have Now suppose the gauge transformation which converts the present Newtonian gauge to another Newtonian gauge. Consequently Eq.(98) results in which t 0 and η i (x) respectively are arbitrary time and arbitrary 3-vector field on S 3 . Substituting Eq.(100) in Eq.(99) yields sequently, we have a set of solutions of the Newtonian gauge field equations: Furthermore, It can be concluded from Eq.(109) that ζ is conserved i.e. it doesn't depend on the time, so that above solutions are appropriate to a period when the perturbations are outside of the Hubble horizon. In order to see conservation of ζ in the super-Hubbles scales, it is sufficient to write the Fourier transformation of Eq. (24) ∂δρ n ∂t + 1 − n 2 a 2 −a (ρ +p) F n + (ρ +p) δu n + aȧΠ S n + 3 2 (ρ +p)Ȧ n + 1 2 (ρ +p) 1 − n 2 Ḃ n + 3ȧ a (δρ n + δp n ) = 0.
On the other hand, we have By substituting Eq.(112) in Eq.(111) and using conservation law of energy in unperturbed universe we can writė Thus for adiabatic perturbations for which δρn where χ(x) is an arbitrary function on the S 3 (a). Note that in this case η i is a Killing vector of S 3 . By substituting Eq.(121) in Eqs.(104)-(108) we derived the second set of solutions as follows Unlike the first solution, this solution is a decaying mode, so it can be neglected at late times and its existence is significant just for counting of adiabatic solutions. In both solutions δρ(t,x) . ρ = δp(t,x) . p which means they are adiabatic solutions. In general, S 3 has four independent gradient conformal Killing vectors and six independent Killing vectors, however, we have totally two independent solutions for perturbations equations in super-Hubble scales. It can be shown that whatever would happen during the inflation, if the universe subsequently spends sufficient time in a state of local thermal equilibrium with conserved quantities, then the perturbations become adiabatic and they remain adiabatic, even when the conditions of local thermal equilibrium are no longer satisfied [44].

Conclusion and summary
The de Sitter background is maximally extended and also maximally symmetric if only if K = 1 i.e. its spatial section is closed. For this purpose, we obtained the required linear perturbation field equations and then proved the existence of two independent adiabatic solutions for these equations in the time interval when perturbations scales go outside of the Hubble horizon. We showed the curvature perturbation on the uniform density slices in a spatially closed universe is proportional to the divergence of the conformal Killing vector of S 3 . This indicates some perturbative cosmological potentials in the time intervals when the scales of the majority of perturbative modes become longer than the Hubble horizon, reduce to the geometrical properties of the background. In comparison with the adiabatic solutions in the spatially flat background, it seems the curvature has no direct role when aH ≫ 1 , but dependence of ζ(x) to the background geometry manifests even outside the horizon the curvature is significant. We also investigate stochastic properties of the perturbation fields in a spatially closed background and show that the spectrums of them are discrete due to the compactness of S 3 (a).