On the perturbation theory in spatially closed background

In this article, we investigate some features of the perturbation theory in a 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 much longer than the Hubble horizon. It will be revealed that these adiabatic solutions do not depend on the curvature directly. We also propose a new interpretation for the curvature perturbation in terms of the unperturbed background geometry.


Introduction
The theory of the linear perturbations is an important part of the modern cosmology which explains CMB anisotropies and the origin of structure formation. This theory has been investigated for a spatially flat universe to great extent [1][2][3][4][5][6][7][8][9][10]. However, observational data point out a universe with ∼ = .68 [11]. The 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, a Lorentzian de Sitter spacetime, is maximally symmetric, maximally extended, and also geodesically complete [12]. So in the following we assume > 0 and K = 1 for the vacuum background. Furthermore, it seems hard to believe that the total density of the universe has exactly been tuned in ρ crit0 , because despite the fact that the observational data indicate K = 0 [11], this fine-tuning seems somehow unlikely. Moreover, if tot equals +1 exactly, this cannot last forever because of the instability [13]. On the other hand, there are some reasons why the universe may have positive spatially a e-mail: aliakbar.asgari@modares.ac.ir b e-mail: ahabbasi@modares.ac.ir c e-mail: j.gholizadeh@modares.ac.ir 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 the mystery of the missing fluctuations which appears in the concordance model of cosmology [14][15][16][17][18]. So these reasons augment the probability of a 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 Sect. 2 we derive the equations governing the linear perturbations in a FLRW universe without fixing K . In Sect. 3 we study the spectral and stochastic properties of these perturbations for the case K = 1 and in Sect. 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 no 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 super-Hubble scales, curvature has no direct effect on the universe's evolution.

The perturbed spacetime
We assume that during most of the time the departures from homogeneity and isotropy have been very small, so that they can be treated as 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, A 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 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 the 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 with respect to the spatial unperturbed metricḡ i j (= a 2g i j ) and H i j = ∇ i ∇ j is the covariant Hessian operator. All the perturbations A, B, E, F, C i , G i and D i j are functions of t and x which satisfy Equation (8) is generalization of the Helmholtz decomposition theorem from R 3 to Riemannian manifolds. Equation (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 form of Eq. (9). It is possible to carry out a similar decomposition of the energy-momentum tensor. One can show that [2] δT 00 = −ρh 00 + δρ, We can decompose the velocity perturbation δu i into the gradient of a scalar (velocity potential) δu and a transverse vector δu V i , We may account for the imperfectness of the cosmic fluid by adding a term i j to δT i j . i j is known as the anisotropic inertia tensor field of the fluid and may be decomposed just like as h i j , where V i and T i j satisfy conditions analogous to Eqs. (10) and (11), which are satisfied by C i and D i j in return. In Eq. (13) there is no term proportional tog i j , because δT i j itself contains such a term. Finally, we have Now let us define the Laplace-Beltrami operator, Thus, for scalar field S we have Also for the vector field V i and the tensor field T i j we can write Substituting 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 i j , ∇ i , and H i j , accompanied by using Eqs. (17), (18), and (19) results in three independent sets of coupled equations.

Scalar mode equations
These equations involve just scalars:

Vector mode equations
We have .p

Tensor mode equation
We have As previously mentioned, in 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 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 Sect. 2 to 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 eigenfunctions of ∇ 2 on S 3 (a). For the scalar mode we have where ∇ 2 =ḡ i j H i j . In pseudo-spherical coordinates with the line element Eq. (30) gives Solving Eq. (32) one gets 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 θ dχ dθ dϕ 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 the Y nlm constitute a complete orthonormal set for the expansion of any scalar field on S 3 (a). Thus, for the scalar perturbative quantity A(t, x) at some instant (which thereafter will be denoted by A(x)) we can write A nlm just like A (x) is a scalar random field. Apart from the distribution function of A nlm , its simplest statistics are described by the mean value and two-point covariance function, and the latter is defined by A nlm A * n l m . Here means the ensemble average which equals the spatial average according to the ergodic theorem [7]. The homogeneity of S 3 (a) implies for any pair of scalar random fields A and B (R is an arbitrary 3-vector in R 3 .) Thus A(x)B * (x ) must be just a function of x − x . This implies that It means that A nlm and A n l m are uncorrelated random fields for different indices (indeed it results from the homogeneity of the universe). The homogeneity also implies that the coefficient of proportionality in Eq. (39) is just a function of n i.e.
P 0 A (n) is a power spectrum or spectral density of A (the superscript "0" over P states corresponding to the spin of the random field) which depends on the distribution function governing A. Moreover, we have in which P 0 A,B (n) is a joint power (cross-correlation) spectrum of A and B [28,29]. One may define the correlation coefficient between A and B: −1 ≤ A,B (n) ≤ 1 and the two extreme values A,B (n) = +1 and A,B (n) = −1 correspond, respectively, to full correlation and full anti-correlation [29].
Finally, let us define the spectral index of the random field A as Now we prove that the homogeneity of the universe yields Eq. (41). First, let us calculate 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 obviously invariant under the following transformations: Moreover, one can show that This shows for χ = χ , that the cos γ is a function of |x − x |, and thus we conclude that Eq. (41) depends merely on |x − x |. Now let us turn to the vector mode.

Vector perturbations and vector random fields
In order to investigate the vector perturbation we should find the vector spherical harmonics on S 3 (a) first. They are solutions of the following equation: The transversality condition is added as a constraint, because every vector perturbation in cosmology 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 and These vector harmonics constitute a complete orthonormal set for the expansion of any transverse vector field on S 3 (a).

Thus, for the vector perturbation A i (x) we can write
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 a parity transformation, because the probability distribution function is invariant under a spatial inversion, and we have P oe A (n) = 0. Furthermore, Because the power spectrum just depends on the probability distribution function it cannot be a function of parity. Thus, The last relation means that A o nlm and A e nlm are statistically independent 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
× sin θ X lm (θ, ϕ) , where Even parity ∂Y lm ∂θ , where G nl (χ ) = (l + 2) cos 2 χ nl (χ ) − (n 2 − 1) sin 2 χ nl (χ ) It is also possible to express the tensor spherical harmonics in terms of the Chebyshev polynomials of the first kind [31]. It can be shown that and also dμḡ ikḡ jl (T e i j ) nlm (T e kl ) * n l m = δ nn δ ll δ mm . (67) The set {(T o i j ) nlm , (T e i j ) nlm } constitutes a complete orthonormal basis for the expansion of any symmetric tracelessdivergence-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 i j (t, x) and T i j (t, x). We can expand them in terms of t-t tensor spherical harmonics on S 3 (a): There is a similar expansion for T i j (t, x).
where P +2 D (n) is the power spectrum of the gravitational wave D i j [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 to have different values. In addition, because the scalar, vector, and tensor modes are independent, their joint power spectra vanish.

The gauge problem
In this section, we investigate the behavior of the perturbations under the gauge transformations. The equations derived in Sect. 2 may have physically equivalent solutions. This problem is called gauge freedom. Similar to the Einstein field equations this gauge freedom may be fixed by choosing a coordinate system. For this purpose, let us consider a spacetime coordinate transformation, with small μ (x) in the same sense as that h μν and the other perturbations are small. In cosmology, we call Eq. (70) a gauge transformation if it affects only the field perturbations and preserves the unperturbed metric [2,33]. Under such a gauge transformation, the metric of the 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 i j . 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 μ as follows: Obviously S , V i , T i j , D i j and δu V i are gauge invariant quantities. Besides, one can construct more gauge invariant quantities by combination of the perturbative quantities, e.g.
, which is known as the curvature 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 inflation as well as the 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 as a result gauge fixing is not required. On the other hand, for the vector mode, we can fix a gauge by choosing V i so that either C i or G i vanishes. For the scalar perturbations, fixing a gauge means choosing 0 and 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 the Newtonian gauge. In this gauge we choose 0 and S by setting B = F = 0. It is convenient to write E and A in this gauge as and 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 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)˙ = ∂δρ ∂t + 3H (δρ + δp) 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 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 the Newtonian gauge are and in which ζ(x) is the curvature perturbation on the uniform density slices when the perturbations are outside of the Hubble horizon or equivalently the conformal factor of S 3 and χ(x) is an arbitrary function of position. In order to prove this, initially we put S = 0, because the cosmic fluid is approximately perfect; thus, from Eq. (88) we have Now take the gauge transformation which converts the present Newtonian gauge to another Newtonian gauge. Consequently in which t 0 and η i (x), respectively, are arbitrary time and an arbitrary 3-vector field on S 3 . Substituting Eq. (100) into Eq. (99) yields Now suppose that η i (x) is a conformal Killing vector of S 3 where γ (x) = 1 3 ∇ i η i is a function on S 3 , the so-called conformal factor of S 3 [40]. Note that S 3 has no homothetic Killing vector [40,41], but due to its conformal symmetry, it has a conformal Killing vector. Indeed in [42] it has been proved that S 3 has a four-gradient conformal Killing vector. For instance, η i = δ m i (m = 1, 2, 3) is a conformal Killing vector of S 3 with conformal factor −x m : On the other hand, on the super-Hubble scales we can ignore the first term on the left side of Eq. (101), because 2 a 2 (τ ) dτ is of the order of 2 t t 0 ∇ 2 0 (τ, x) dτ , so its Fourier transform has same order of 2 t t 0 1−n 2 a 2 (τ ) 0 nlm (τ ) dτ , which is negligible for super-Hubble scales. Thus Eq. (101), on the time intervals when the perturbation scales are very longer than the Hubble horizon, turns to Besides, in the Newtonian gauge both and + are solutions, so that it results from the linearity of equations that is another solution of the Newtonian field equations too. It is also true for other perturbations. Consequently, 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 does not depend on the time, so that the above solutions are appropriate for a period when the perturbations are outside of the Hubble horizon. In order to see the conservation of ζ on 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 + aȧ S n ] + 3 2 (ρ +p)Ȧ n + 1 2 (ρ +p) (1 − n 2 )Ḃ n + 3ȧ a (δρ n + δp n ) = 0; for simplicity we drop l and m indices. On the super-Hubble scales (n << a H) we can approximate this equation as follows: ∂δρ n ∂t + 3 2 (ρ +p)Ȧ n + 3ȧ a (δρ n + δp n ) = 0.
On the other hand, we have By substituting Eq. (112) in Eq. (111) and using the conservation law of energy in an unperturbed universe we can writė Thus for adiabatic perturbations for which δρ n .ρ = δp n .p , we havė ζ n = 0.
Consequently, if the perturbations are adiabatic 1 , ζ is conserved of course in the epoch when the wavelength of most perturbations are very much longer than the Hubble radius. Indeed, the conservation of ζ is a general theorem in cosmology which has been proved even for a nonlinear generalization of ζ [43]. Note that ignoring the first term of the left hand side of Eq. (101) causes ζ to be independent of time, which is equivalent to going outside of the Hubble horizon.