Cosmological renormalization of model parameters in the second-order perturbation theory

It is shown that the serious problem on the cosmological tension between the direct measurements of the Hubble constant at present and the constant derived from the Planck measurements of the CMB anisotropies can be solved by considering the renormalized model parameters. They are deduced by taking the spatial average of second-order perturbations in the flat Lambda-CDM model, which includes random adiabatic fluctuations.


Introduction
High precision cosmology has started with the measurements of fluctuations in the cosmic microwave background radiation (CMB) by WMAP [1] and Planck [2,3] collaboration. Their studies have been useful to determine the Hubble constant and the cosmological parameters. However, it has been found that there is a tension between the Hubble constant (H 0 ) due to the Planck measurements of the CMB anisotropies and that due to the direct measurements.
These data show that there is a large discrepancy of 9.7% ∼ 16.9%. To solve this problem, various mechanisms have been proposed, such as models of decaying dark matter [8], a local void model [9], and the dark radiation [4]. In this paper, it is shown that the tension can be solved by considering the renormalized model parameters which are deduced by taking the spatial average of second-order perturbations in the flat Λ-CDM model, which includes random adiabatic fluctuations. In Sect. 2, we show the general-relativistic second-order perturbation theory in the flat Λ-CDM model, which was derived by the present author [10]. In Sect. 3, we derive spatial averages of second-order density and metric perturbations, and in Sect. 4, we define the renormalized Hubble constant due to the average second-order metric perturbations, and show that it is consistent with the measured Hubble constants and their various observed values correspond to the different upper limits of wave-numbers of perturbations which can be included in the renormalized perturbations. Other renormalized model parameters are also derived due to the average of second-order density perturbations. In Sect. 5, the renormalization in the past is described and in Sect. 6, the concluding remarks are given. In Appendix A, we show the definition of various quantities included in the expressions for the second-order metric perturbations. In Appendix B, we show the model parameters corresponding to Hubble constants in Eq. (2).

Background and the perturbation theory
The background universe is expressed by a spatially flat model with the line-element where the Greek and Roman letters denote 0, 1, 2, 3 and 1, 2, 3, respectively. The conformal time η(= x 0 ) is related to the cosmic time t by dt = a(η)dη.
In the comoving coordinates, the velocity vector and energy-momentum tensor of pressureless matter are and where ρ is the matter density . From the Einstein equations, we obtain and where a prime denotes ∂/∂η, Λ is the cosmological constant, and ρ 0 is an integration constant, and we use the units 8πG = c = 1 for the gravitational constant G and the light velocity c.
The Hubble parameter H is defined as Eq. (6) gives which is also expressed as where H 0 is H at the present epoch t 0 and a 0 ≡ a(t 0 ) = 1, and In this paper we adopt the following background values : The significance of these values will be explained later. Next let us show the first-order density perturbations. The perturbations of metric, matter density and velocity are represented by δ 1 g µν (≡ h µν ), δ 1 ρ, and δ 1 u µ . When we adopt the 2/14 synchronous coordinates (useful in the pressureless case), the metric perturbations satisfy the condition h 00 = 0 and h 0i = 0.
The first-order perturbations are classified into the growing case and the decaying case. Both cases are found in the previous paper. [10] Here we show only those in the growing case, which are used in this paper : where F is an arbitrary function of spatial coordinates x 1 , x 2 and x 3 , ∆ ≡ ∇ 2 , h j i = g jl h li and P (η) satisfies Its solution is expressed as After a partial integration, we obtain The three-dimensional covariant derivatives |i are defined in the space with metric dl 2 = δ ij dx i dx j and their suffices are raised and lowered using δ ij , so that their derivatives are equal to partial derivatives, i.e. F |j |i = F ,ij , where F ,i ≡ ∂F/∂x i . The second-order perturbations were derived in the previous paper [10]. This is a simple extension of my paper [11] which derived the second-order perturbations in the case of zero Λ in the Lifshitz formalism with iterative second-order terms. The results in the latter paper were later derived independently by Russ et al. [12] and by Matarrese et al. [13] in the different formalisms, and the validity of this theory has been confirmed. Here let us show their components δ2 g µν (≡ ℓ µν ), δ2 ρ, and δ2 u µ in the case of nonzero Λ, where the total perturbations are Here assuming the synchronous gauge condition, we have ℓ 00 = 0 and ℓ 0i = 0. 3/14 The perturbations in the growing case are expressed as where N |j |i = δ jl N |li = N ,ij and Q(η) satisfies The expressions of other quantities L j i , M j i , N |j |i , and C j i are shown in Appendix A. The velocity and density perturbations are found to be and The gauge used here is not only synchronous, but also comoving (cf. Eqs. (14) and (22)). The above perturbations are therefore physical density perturbations which are measured by comoving observers.
It is to be noticed that the present general-relativistic gravitational equations are nonlinear and applicable also in the super-horizon case, in contrast to linear gravitational equations in the Newtonian treatment [14,15], and so the cosmological result in the following sections cannot be derived in the Newtonian treatment, because of their difference.
It is discussed in Sect. 6 what we should do, in order to obtain the consistency between the general-relativistic treatment and the Newtonian cosmology.

Average second-order perturbations
We consider random perturbations given by where α(k) is a random variable and the average of F expressed as F vanishes. Here we assume the average process with a power spectrum P F (k), given by Here we have for the first-order density perturbation. For the second-order perturbations, we have 4/14 so that we obtain Similarly, we have so that we obtain

Second-order density perturbations
It follows therefore from Eq. (23) that Here F is related to the curvature fluctuation R by F = 2 R, and so we have the relation where P R is expressed using the power spectrum [16,17] as and P R0 = 2.2 × 10 −9 according to the result of Planck measurements. [2,3] The transfer function T s (x) is expressed as a function of x = k/k eq , where k eq (≡ a eq H eq ) = 219 (Ω M h) H 0 = 32.4 H 0 .
Here H 0 (≡ 100h) is the present background Hubble constant, (a eq , H eq ) is (a, H) at the epoch of equal energy density, and (Ω M , h) = (0.22, 0.673) (given in Eq. (12)). Moreover, we assume n = 1 here and in the following. Then we obtain for arbitrary a whereρ ≡ ρ + Λ, and A and B are expressed as using the transfer function T s (x) for the interval (x max , x min ). Here we have and using Eq.(17), we obtain where

Second-order metric perturbations
The second-order perturbation of the scale factor δ 2 a can be derived as follows using the metric second-order perturbations l ij , which are given in Eqs. (20) and (21), and in Appendix A. The averaging of second-order metric perturbations leads to where we have Since we obtain Then we have using Eqs.(28) and (30) where A, B and Z(a) are given by Eqs. (36) and (37). The line-element can be expressed as where the renormalized scale-factor is defined by and the relative difference of scale-factors is given by The renormalized redshift z rem corresponding to an arbitrary time t is defined using the scale-factor a rem as where t 0 denotes the present epoch, the background redshift z is 1/a − 1, and ξ(t) is defined by Eq.(47).

6/14
The square of the background Hubble parameter H is (ȧ/a) 2 and its perturbation is given by so that From Eqs. (44) and (50), we obtain

Average perturbations of model parameters
In this paper we assume the simplest transfer function (BBKS) for cold matter, adiabatic fluctuations, given by [18][19][20] This function has the peak around x ≃ 1, so that the upper and lower limits (x max and x min ) in the integrals A and B in Eq. (35) should have the values such as x max ∼ 6(> 1) and x min ∼ 0.01(≪ 1). Here A and B depend sensitively on x max , but not on x min . In order to find the best value of x max , we derive a length L max corresponding to k max . Using Eq. and By the way, we derive (δ 1 ρ/ρ) 2 to estimate the dispersion of δ2 ρ/ρ : so that we obtain which is equal approximately to the measured Hubble constants. [6,7] It is found, therefore, that the renormalized Hubble constant H rem may be consistent with the directly measured Hubble constants. The model parameters Ω M and Ω Λ describe the evolution of the background universe. But since our real universe is described using the renormalized quantities H rem and ρ + δ2 ρ in the place of the background Hubble constant H and ρ, we may obtain the following new set of model parameters : and (Ω Λ ) rem = Ω Λ 1 1 + δ2 ρ/ρ .
As for the observations of baryon acoustic oscillations of CMB (Planck [2,3]) and largescale galactic correlations [22,23], we use the angular distance in the late time model, so that the derived model parameters are not (Ω M , Ω Λ ), but ((Ω M ) rem , (Ω Λ ) rem ), which are given by Eq.(65). On the other hand, the scale of acoustic oscillations is determined at the recombination epoch with a(≈ 10 −3 ) and so the Hubble constant is given by H 0 (≃ H rem at the recombination epoch), but not H rem at the present epoch. So the above renormalized model parameters are consistent with the cosmological observations. [2,3,22,23] The relative difference of scale-factors ξ (in Eq. (47)

Renormalization of model parameters in the past
In the previous section, we treated the quantities at the present epoch (a = 1). Here we consider the quantities at the epochs of a < 1. First we calculate δ2 ρ/ρ for a < 1 using Eq.(35) for x max = 5.7 and x min = 0.01. Its dependence on a is shown in Fig.1. It is found that δ2 ρ/ρ has a peak at around a ∼ 0.65, but δ2 ρ/ρ increases monotonically, and that δ2 ρ/ρ reduces to 0 in the limit of a → 0. Using Eqs. (51), (61), (63), and (64), moreover, we obtain H rem and ((Ω M ) rem , (Ω Λ ) rem ) in the past. They are shown in Fig. 2 and Fig. 3. It is found that H rem reduces to H 0 (in Eq. (12)), and ((Ω M ) rem , (Ω Λ ) rem ) reduces to (Ω M , Ω Λ ) in the limit of a → 0.
In Fig. 4, the relative difference of scale-factors ξ (in Eq. (47)) is shown and ξ is −0.097 ∼ −0.195 for a = 1 ∼ 0, respectively. It is found from Eq.(48) that z rem is larger than z.

Concluding remarks
It was found in this paper that the random adiabatic fluctuations bring a kind of energy density which has an influence upon the dynamics of the universe. For its derivation, the nonlinearity of general-relativistic perturbation theory was important. As k increases in the region of x(≡ k/k eq ) > 1, the amplitude of perturbations decreases rapidly, and the frequency of perturbed objects is so small that they cannot be renormalized as part of the background matter density. The upper limit of x for renormalized perturbations is x max (≈ 6). Because of their small frequency corresponding to large k, the value of x max has large fluctuations, and this may be the origin of the directional fluctuations of x max and the measured Hubble constant.
The background model parameters in Eq. (12) are rather different from the renormalized parameters in Eqs.(62) and (65). We should notice that the observed model parameters are the latter ones. The Hubble constant H 0 in the Planck measurements (Eq.(1)) is the 10/14 renormalized Hubble constant measured at the early stage (a ≪ 1), which is approximately equal to the background Hubble constant (H 0 ). In this paper we adopted tentatively the background model parameters (in Eq. (12)) and the value of x max . Their values should be selected so that they may reflect best the real observations of model parameters.
Large-scale perturbations such as x ≡ k/k eq = 0.01 ∼ 6, which were treated in this paper (cf. Eq.(53)) cross the horizon in the course of their evolution, so that taking the generalrelativistic effect into consideration is indispensable for their dynamical analyses, which are not only linear but also on second-order.
In the Newtonian theory, the terms representing the gravitational strength ξ (≡ GM/(c 2 R) ) are taken only linearly into account, assuming that it is exremely small, where M and R are characteristic mass and length of dynamical objects. In the cosmological 11/14 where H is the Hubble parameter, R ∼ c/H, and M ∼ ρR 3 . When |δρ/ρ| is not so small, we cannot neglect the nonzero mean of ξ 2 , which should be treated in the post-Newtonian approximation. In the general-relativistic treatment of second-order perturbations, this nonzero mean is automatically taken into account. In the linear level, we have ξ = 0, but in the second-order, ξ 2 ∼ (δρ/ρ) 2 = 0. Moreover, the necessity of considering nonlinear ξ comes also from the energy-momentum conservation law. In order that the Newtonian theory is compatible with the energy-momentum conservation in the same way as in the general-relativistic cosmology, we must add a nonzero term (δρ/ρ) 2 to the second-order density perturbation, so as to recover the post-Newtonian terms with ξ 2 . By this addition, the Newtonian theory may be consistent with the general-relativistic cosmology. The correspondence of general-relativistic approach and Newtonian approach has been studied by 12/14 Matarrese et al. [24] The discussions about it in detail are beyond the scope of the present paper.