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



Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
 Japan
ketomita@ybb.ne.jp


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 Λ-CDM model, which includes
 random  adiabatic  fluctuations.




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

First version:     January 17, 2017This version:     February 29, 2020
========================================================================================






§ INTRODUCTION


High precision cosmology has started with the measurements of  fluctuations
in the cosmic microwave background radiation (CMB) by WMAP<cit.> and
 Planck<cit.> 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.

Due to the Planck measurements<cit.>, we have the Hubble constant


    H_0 = 67.3  km  s^-1 Mpc^-1.


On the other hand, there are many direct measurements<cit.>, and the three
 examples of direct measurements give us<cit.>
 

    H_0 = 73.8,  74.3,  and 78.7   km  s^-1 Mpc^-1.


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<cit.>, a local void model<cit.>, and 
the dark radiation<cit.>. 
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<cit.>.
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


    ds^2 = g_μν dx^μ dy^ν = a^2 (η) [-dη^2 + δ_ij dx^i dx^j ],


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 


    u^0 = 1/a,     u^i = 0


and


    T^0_0 = -ρ,    T^0_i = 0,    T^i_j = 0,


where ρ is the matter density . 

From the Einstein equations, we obtain


    ρ a^2 = 3(a'/a)^2 - Λ a^2,


and


    ρ a^3 = ρ_0,


where a prime denotes ∂/∂η,  Λ is the
cosmological constant, and ρ_0 is an integration constant, and we use
the units 8π G = c = 1 for the gravitational constant G and the light velocity c. 
The Hubble parameter  H is defined as 


    H  =  ȧ/a = a'/a^2.


Eq.(<ref>) gives 


    H^2 = 1/3  (ρ + Λ),


which is also expressed as
 

    H^2 = H_0^2    (Ω_M a^-3 + Ω_Λ ),


where H_0 is H at the present epoch t_0 and a_0 ≡ a(t_0) = 1, and


    Ω_M  = 8π G ρ_0/3H_0^2 = 1/3ρ_0/H_0^2  and  Ω_Λ = Λ c^2/3H_0^2 = 1/3Λ/H_0^2.


In this paper we adopt the following background values : 


    H_0 = 67.3   km  s^-1 Mpc^-1 and(Ω_M,  Ω_Λ )  = (0.22, 0.78).


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_1ρ, and δ_1 u^μ.   When we adopt the 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.<cit.> 
Here we show only those in the growing case, which are used in this paper :


    h^j_i    = δ^j_i F + P(η) F^|j_|i, 
    δ_1 u^0    = 0,    δ_1 u^i = 0, 
    δ_1 ρ/ρ   = 1/ρ a^2(a'/a P'
    -1) Δ F,


where F is an arbitrary function of spatial coordinates x^1, x^2
and x^3,  Δ≡∇^2,  h^j_i = g^jl h_li and
P(η) satisfies  


    P” + 2a'/a P' - 1 =0.


Its solution is expressed as


    P = ∫^η_0 dη' a^-2 (η') ∫^η'_0 dη” a^2(η”).


After a partial integration, we obtain 


    (H_0)^2  P    = -2/3Ω_M a^-3/2√(Ω_M +Ω_Λ 
    a^3)∫^a_0
    db  b^3/2/√(Ω_M +Ω_Λb^3) + 2/3Ω_M a,
    
    H_0  η   = ∫^a_0
    db  b^-1/2/√(Ω_M +Ω_Λb^3).


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<cit.>.  This is a simple extension of my paper<cit.> 
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.<cit.>
and by Matarrese et al.<cit.>
in the different formalisms, and the validity of this theory has been confirmed.
Here let us show their components 

δ_2 g_μν (≡ℓ_μν), δ_2
ρ,  and δ_2 u^μ in the case of
 nonzero Λ, where the total perturbations are 



    δ g_μν   = h_μν + ℓ_μν, 
    δ u^μ   = δ_1 u^μ +δ_2 u^μ, 
    δρ/ρ   = δ_1 ρ/ρ + δ_2 ρ/ρ.


Here assuming the synchronous gauge condition, we have


    ℓ_00 = 0    and  ℓ_0i = 0.


The perturbations in the growing case are expressed as


    ℓ^j_i = P(η) L^j_i + P^2 (η) M^j_i + Q(η) N^|j_|i + C^j_i,
 

where N^|j_|i = δ^jl N_|li = N_,ij and Q(η) satisfies


    Q” +2a'/a Q' = P - 5/2 (P')^2.
 

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


    δ_2 u^0 =0,     δ_2 u^i =0
 
and


    δ_2 ρ/ρ   = 1  2ρ a^2{1 
    2(1 - a'/aP') (3F_,lF_,l + 8FΔ F) +1/2P
    [(Δ F)^2 + F_,klF_,kl] 
       + 1/4[(P')^2 - 2/7a'/aQ'] [(Δ F)^2 -
    F_,klF_,kl] - 1  7a'/aPP' [4F_,klF_,kl +
    3(Δ F)^2]  }.
 


The gauge used here is not only synchronous, but also comoving  (cf. Eqs.
(<ref>) and (<ref>)).  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 <cit.>,
 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


    F( k)  = ∫ d k α ( k)  e^i kx,


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


    ⟨α ( k) α ( k') ⟩ = (2π)^-2 P_F ( k) δ ( k +  k').


Here we have 


    ⟨δ_1 ρ/ ρ⟩ = 0


for the first-order density perturbation. For the second-order  perturbations,
we have


    F_,i  F_,i   = - ∫∫ d k  d k'⟨α ( k) α ( k') ⟩ k k' e^i( k+
     k')  x, 
    
     F Δ F    = - ∫∫ d k  d k'⟨α ( k) α ( k') ⟩ k^2 e^i( k+
     k')  x,


so that we obtain


    ⟨ F_,i  F_,i⟩  =  ⟨ F Δ F ⟩ = (2π)^-2∫ d kk^2  P_F ( k).


Similarly, we have


    F_,ij  F_,ij   =  ∫∫ d k  d k'⟨α ( k) α ( k') ⟩ ( k k')^2 e^i( k+
     k')  x, 
    
     (Δ F)^2    =  ∫∫ d k  d k'⟨α ( k) α ( k') ⟩ k^2 (k')^2 e^i( k+
     k')  x,


so that we obtain


    ⟨ F_,ij  F_,ij⟩  =  ⟨ (Δ F)^2 ⟩ = (2π)^-2∫ d kk^4  P_F ( k).




 §.§ Second-order density perturbations

It follows therefore from Eq. (<ref>) that


    ⟨δ_2 ρ/ρ⟩ = 
    1 - a'/aP'/2ρ a^2 (2π)^-2[11/2∫ d k k^2 
      P_F ( k) + P ∫  d  k k^4  P_F ( k)].


Here F is related to the curvature fluctuation R by F = 2  R,
and so we have the relation 


    P_F  ( k) = 4  P_ R  ( k),


where P_ R is  expressed using the power spectrum<cit.>
  as


    P_ R = 2 π^2   P_ R0 k^-3 (k/k_eq)^n-1 T_s^2
      (k/k_eq)


and P_ R0 = 2.2× 10^-9 according to the result of Planck
measurements.<cit.>
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. (<ref>)).

Moreover, we assume n = 1 here and in the following. 
Then we obtain for arbitrary a 


    ⟨δ_2 ρ/ρ̃⟩ = 4π/3 32.4^4  P_ R0 [1 - Y(a)]/(Ω_M/a + 
     Ω_Λ a^2)[11/2 32.4^-2 A + Z(a) B],


where ρ̃≡ρ + Λ,  and A and B are expressed as 


    A ≡∫^x_max_x_min dx  x  T_s^2 (x),       B ≡∫^x_max_x_min
     dx  x^3  T_s^2 (x)


using the transfer function T_s (x) for the interval (x_max, x_min).  
Here we have


    Y(a)  ≡a'/aP',    Z(a)  ≡ (H_0)^2  P,


and using Eq.(<ref>), we obtain 


    Y(a) = a^-5/2 (Ω_M + Ω_Λ a^3)^1/2  I(a),    
    Z(a) = 2/3Ω_M a [1 - Y(a)],


where


    I(a) ≡∫^a_0  db  [b^3/(Ω_M + Ω_Λ b^3)]^1/2.





 §.§ 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.(<ref>) and (<ref>), and in  Appendix A. 
The averaging of second-order metric perturbations leads to 


    δ_2 (a^2) = 1/3⟨ l^m_m  ⟩ ,
 

where we have


    ⟨ l_ij⟩ = P(η) ⟨ L_ij⟩ + P^2 (η) ⟨ M_ij⟩
    + Q(η) ⟨ N_,ij⟩ + ⟨ C_ij⟩.
 

Since L^i_j = L_ij,  M^i_j = M_ij and 


    L^i_i    = - 1/2 [2 FΔ F + 3/2 F_,l F_,l], 
    
    M^i_i    = - 1/28 [10 (F_,ll)^2 - 3 (Δ F)^2 ],  
    Δ N    = - 1/28 [ (Δ F)^2 -  F_,kl F_,kl], 
     C^i_i    = 0,
 

we obtain


    ⟨ L^i_i  ⟩   = - 7/4 ⟨ F Δ F  ⟩, 
    ⟨ M^i_i  ⟩   = - 1/4 ⟨ (Δ F)^2  ⟩, 
    ⟨Δ N  ⟩   = ⟨ C^i_i  ⟩  = 0.
 

Then we have using Eqs.(<ref>) and (<ref>)


    ⟨ l_ii⟩ =  ⟨ l^i_i  ⟩  =  - 2π 32.4^4   P_R0  Z(a) [7 × 32.4^-2 A + Z(a) B],
 

where A, B and Z(a) are given by Eqs.(<ref>) and (<ref>).

The line-element can be expressed as


    ds^2 = -dt^2 + a_rem (t)^2  [(dx^1)^2 + (dx^2)^2 + (dx^3)^2],
 

where the renormalized scale-factor is defined by


    a_rem (t)^2 ≡  a^2 + δ_2 (a^2) = a^2 + 1/3⟨ l^i_i  ⟩,
 

and the relative difference of scale-factors is given by 


    ξ≡  a_rem (t)/a(t) - 1 = [1 + 1/3⟨ l^i_i 
        ⟩/a^2]^1/2 - 1.
 


The renormalized redshift z_rem corresponding to an arbitrary time t is defined using the 
scale-factor a_rem as


    1 +  z_rem≡ a_rem (t_0) / a_rem (t) = 1+ξ (t_0)/1+ξ (t) (1+z),
 

 where t_0 denotes the present epoch,  the background redshift z is 1/a -1, 
 and ξ (t) is defined by Eq.(<ref>). 

The square of the background Hubble parameter H is (ȧ/a)^2 and its
perturbation is given by


    δ_2 (H^2) = δ_2 (ȧ^2)/a^2 - (ȧ)^2 δ_2 (a^2)/a^4,
 

so that


    δ_2 (H^2)/H^2 = 2/3[⟨ l^m_m  ⟩'/2aa' - 
    ⟨ l^m_m  ⟩/a^2].
 

From Eqs.(<ref>) and (<ref>), we obtain


    δ_2 (H^2)/H^2 = 4π/3 32.4^4   P_R01/a^2[7 (32.4)^-2(Z - Y(a) a/2(Ω_M + Ω_Λ
     a^3))
    A + Z (Z - Y(a) a/Ω_M + Ω_Λ a^3) B],
 

where Y(a) and Z(a) are given by Eq.(<ref>).




 §.§ Average perturbations of model parameters

In this paper we assume the simplest transfer function (BBKS) for cold matter,
adiabatic fluctuations,  given by<cit.>


    T_s (x) = ln (1+0.171 x)/0.171x [1+0.284 x +(1.18 x)^2 + (0.399 x)^3 +
    (0.490 x)^4]^-1/4.


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. (<ref>) 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.(<ref>), we have     


    L_max≡ 2π/k_max = 102/h   Mpc


for x_max (≡ k_max/k_eq) = 5.7.  This L_max corresponds to
the cosmological distance, over which the smooth  observations on
cosmological scales may be possible.  


So we adopt x_max = 5.7 and x_min = 0.01. Then we obtain


    A = 2.22,     B = 20.95,


and 


    ⟨δ_2 ρ/ρ̃⟩ = 0.121   and  δ_2 (H^2)/H^2 = 0.210


at the present epoch (a = 1).

  In similar cases with B ≈ 10 A,
 the terms with A in Eqs. (<ref>) and (<ref>) are negligible, and then we have 


    ⟨δ_2 ρ/ρ̃⟩≃8π/9 32.4^4  P_ R0 [1-Y(a)]^2 a^2/Ω_M/Ω_M + Ω_Λ 
     a^3 B ,


and


    δ_2 (H^2)/H^2≃4π/3 32.4^4   P_ R01/a^2  Z(a) [Z(a) - Y(a) a/Ω_M + Ω_Λ a^3] B.



By the way, we derive (δ_1 ρ/ρ̃)^2 to estimate the dispersion of 
⟨δ_2 ρ/ρ̃⟩:


    ⟨  (δ_1 ρ/ρ)^2 ⟩ = 
    [1 - Y(a)/ρ a^2]^2 (2π)^-2∫  d  k k^4  P_F ( k),


so that we obtain


    ⟨  (δ_1 ρ/ρ̃)^2 ⟩ = 8π/9× 32.4^4  P_R0 [1-Y(a)]^2 a^2/(Ω_M + Ω_Λ a^3)^2 
     B ≃⟨δ_2 ρ/ρ̃⟩Ω_M/Ω_M +Ω_Λ a^3.









§ RENORMALIZATION OF MODEL PARAMETERS


Now let us consider the renormalization of the background density and the Hubble
constant. 
Since ⟨δ_2 ρ/ρ⟩ is spatially constant, 
we may 
assume that it is a part of the background density. Here we regard 


    ρ_rem≡ρ + ⟨δ_2 ρ⟩


as the renormalized background density.

Similarly ⟨δ_2 H^2 ⟩ is spatially constant, and so we can regard 


    H_rem≡ [H^2 +  ⟨δ_2 (H^2) ⟩]^1/2


as the renormalized background Hubble parameter. 

For  ⟨δ_2  H^2  ⟩/H^2 in Eq. (<ref>) and 
H = H_0 in Eq. (<ref>), we obtain at present epoch


    H_rem = 74.0  km  s^-1 Mpc^-1 ,


which is equal approximately 
 to the measured Hubble constants.<cit.>  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 + ⟨δ_2 ρ⟩ in the place of the
   background Hubble constant H and ρ, we may obtain the
   following new set of model parameters :


    (Ω_M)_rem = Ω_M 1 +  ⟨δ_2
     ρ/ρ⟩/1 + ⟨δ_2 ρ/ρ̃⟩


and


    (Ω_Λ)_rem  = Ω_Λ1/1 + ⟨δ_2
     ρ/ρ̃⟩.


Using the background model parameters Eq.(<ref>) and ⟨δ_2 ρ/ρ̃⟩ in Eq. (<ref>), 
we obtain at the present epoch


    (Ω_M)_rem = 0.305    and   (Ω_Λ)_rem = 0.695.



The recent observations of the redshift-magnitude relation<cit.> 
include many supernova
 with redshifts z = 0.1 - 1.0, so that the present Hubble constant (H_rem)
is used, and  ((Ω_M)_rem,  (Ω_Λ)_rem)  (which are
 consistent with Eq.(<ref>)) are obtained.

As for  the observations of baryon acoustic oscillations of CMB
 (Planck<cit.>) and large-scale galactic correlations<cit.>,
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.(<ref>). 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. <cit.>

The relative difference of scale-factors ξ (in Eq.(<ref>)) is -0.097 at
 present epoch. Moreover, the present values of ⟨δ_2
  ρ/ρ̃⟩ and 
 ((Ω_M)_rem,   (Ω_Λ)_rem)
in the case when H_rem is 73.8, 74.3 or 78.7 are shown in Appendix B.




§ 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 
 ⟨δ_2 ρ/ρ̃⟩ for a < 1 
 using Eq.(<ref>) for
  x_max = 5.7 and x_min = 0.01. Its dependence on a is shown in Fig.1.
  It is found that ⟨δ_2 ρ/ρ̃⟩ has a peak at 
 around a ∼ 0.65, but ⟨δ_2 ρ/ρ⟩ increases
  monotonically,  and that 
⟨δ_2 ρ/ρ̃⟩ reduces to 0 in the limit of a 
→ 0. 

  Using Eqs. (<ref>),  (<ref>), (<ref>), and (<ref>), 
  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. (<ref>)), 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.(<ref>)) is
 shown and ξ is  -0.097 ∼ -0.195 for a = 1 ∼ 0, respectively. 
 It is found from Eq.(<ref>) 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.(<ref>) are rather different from the
renormalized parameters in Eqs.(<ref>) and (<ref>). 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 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. (<ref>)) 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.(<ref>)) cross the 
horizon in the course of their evolution, so that taking the 
general-relativistic 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 circumstances, however, we have  


    ξ∼ G ρ /H^2  · (δρ/ ρ)  ∼ δρ/ ρ,
 

where H is the Hubble parameter, R ∼ c/H, and M  ∼ ρ R^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 Matarrese et al.<cit.> 
 The discussions about it in detail are beyond the scope of the present 
 paper. 
      
            



§ ACKNOWLEDGEMENTS

The author thanks Masumi Kasai for helpful discussions. 





§ QUANTITIES IN THE SECOND-ORDER METRIC PERTURBATION

 
The quantity Q  in the second-order metric perturbation is expressed as


    Q = ∫^η_0 dη' a^-2(η') ∫^η'_0 dη”
    a^2(η”) [P(η”) -5  2(P'(η”))^2].
 

In the case Λ = 0, we have  P - (5/2)(P')^2 = 0 because of a 
∝η^2 and P = η^2/10, so that Q vanishes.  
The functions L^j_i and M^j_i are defined by


    L^j_i    = 1  2[-3 F_,i F_,j -2 F · F_,ij + 1  2δ_ij F_,l F_,l], 
    
    M^j_i    = 1  28{19F_,il F_,jl - 12 F_,ijΔ F -
    3δ_ij[F_,kl F_,kl -(Δ F)^2 ]}
 

and N is defined by


    Δ N = 1  28[(Δ F)^2 - F_,klF_,kl].
 


The last term C^j_i satisfies the wave equation


    C^j_i = 3  14(P/a)^2 G^j_i + 1  7[P - 5  2
    (P')^2 ] G̃^j_i,
 

where the operator  is defined by

    ϕ≡ g^μνϕ_;μν = -a^-2(∂^2/∂η^2 + 2a'  a∂/∂η -
    Δ) ϕ
 

for an arbitrary function ϕ by use of the four-dimensional covariant
derivative ;, and G^j_i and G̃^j_i are expressed as 


    G^j_i    ≡Δ(F_,ijΔ F - F_,il F_,jl) + (F_,ij F_,kl
    - F_,ik F_,jl)_,kl - 1  2δ_ijΔ [(Δ F)^2 -
    F_,klF_,kl], 
    G̃^j_i    ≡ F_,ijΔ F - F_,il F_,jl - 1 
    4δ_ij[(Δ F)^2 - F_,klF_,kl] - 7 N_,ij.

These functions satisfy the traceless and transverse relations


    G^l_l = 0,    G^l_i,l = 0, 
    G̃^l_l = 0,   G̃^l_i,l = 0,


so that C^j_i also satisfies


    C^l_l = 0,    C^l_i,l = 0.
 

This means that C^j_i represents the second-order gravitational
radiation emitted by first-order density perturbations.
The solution of the above inhomogeneous wave equation
(Eq.(<ref>)) can be represented in an explicit form using 
the retarded Green function for the operator .






§ CASE WHEN H_REM IS 73.8, 74.3, OR 78.7

If we assume that the three values of  Hubble constant due to direct measurements
 (= 73.8, 74.3, 78.7) are H_rem, we obtain


    ⟨δ_2 ρ/ρ̃⟩ = 0.117,  0.126,  0.213,


respectively, and


    ⟨  (δ_1 ρ/ρ̃ )^2 ⟩^1/2 = 0.160,  0.166,
     0.216.


Moreover,  corresponding to H_rem = 73.8, 74.3, 78.7 and H = H_0 in Eq.
 (<ref>), it is found 
for a fixed x_min = 0.01 using Eq. (<ref>) that 


    x_max = 5.6,  5.8,  7.4,



    A = 2.20,  2.25,  2.59,     B = 20.14,  21.78,  36.65,


respectively.
The values of A and B depend rather sharply on x_max, but not on
 x_min. It is found therefore that the directly observed Hubble constants 
 appear, corresponding to various values of x_max (≈ 6). 
 These values of x_max may depend on the direction in which we measure
 the Hubble constant.

For  H_rem = 73.8, 74.3, 78.7, moreover, we obtain 



    (Ω_M)_rem = (0.302,  0.307,  0.357)   and   
     (Ω_Λ)_rem = (0.698,  0.693,  0.643),


respectively.






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