Short wave corrections

In the previous subsection we discussed the equations for the evolution of the first order perturbations. The non-linear structures we observe in the Universe today are the result of the growth of initial small perturbations. This suggests us the perturbative expansion we have considered will be broken at some time and some scale. However the cosmological observations propose that while there are highly dense regions at small scales but there are no strong gravitational fields objects at the scales of interest. Using the observational evidence we can pick the weak field approximation to study the structure formation in the Universe.

The weak-field limit requires that the metric perturbations about the back-ground FLRW metric and consequently the gravitational potentials remain small but the matter perturbations can become large and should not be treated per-turbatively. This scheme seems relevant for the cosmological studies since most of the objects of interest are within the weak field regime. In Fig. 1.2 we show the gravitational potentials and the densities of some structures in the Universe.

From the figure and the observational fact that by increasing the size of struc-tures their densities and gravitational potentials decrease, one can see that most of the structures important for cosmological studies are in the weak field regime.

To explain the objects in the strong field regime (neutron stars, black holes) we cannot use the weak field approximation anymore and we need to pick the appropriate metric according to the symmetries.

To this end, we assume that the potentials remain small, but their variation of fluctuations (like the curvature of the potential wells) at small scales can become large which leads to large density fluctuations, so δρ/ρ ∼ (k/H)²Φ can become very large in principle. To do the expansion consistently, considering a small parameter,we assume Φ,Ψ, B_i andh_ij are at most of orderO(). Also for non-relativistic components, time derivatives are usually of order Hubble thus do not change the order of a term in the expansion. But each spatial derivative decreases the order of a term by one half, for instance ∂_iΦ ∼ O(^1/2) and ∇²Φ ∼ O(⁰) (see Adamek et al. [2016a], Adamek et al. [2016b], Fidler et al. [2017], Hassani et al. [2019c] for detailed discussion about the weak field approximation). In Table 1.1 we show the order of quantities in the weak field framework. To summarize, in the equations we keep the terms up to linear order in the metric perturbations, but from the quadratic terms we only keep the ones that have exactly two or more spatial derivatives. For example at the first order in the weak field framework we keep Ψ∇²Ψ or δ^ijΦ_,iΦ_,j while we neglect the terms like ΦΦ⁰, δ^ijB_i∆B_j. The higher order terms in the standard perturbation theory contributing to the equations in the weak field framework are called “short wave corrections” as they become only relevant at small scales.

In the following part, we are going to include the short wave corrections into the Einstein field equations. To do so, we need to consider higher order terms

1.3. The cosmological perturbation theory

Figure 1.2: A Picture that shows the densities and gravitational potentials of some structures in the weak field and strong field regime, including the Sun, the Milky Way, a neutron star and a black hole. We have not depicted larger structures, however the structures with larger sizes (galaxies, clusters, filaments and super-clusters) have less density and less gravitational potential compared to the Sun and Milky way, so they also belong to the weak field regime. The numbers for the neutron star and black hole are computed for a typical kind.

in the perturbative expansion and keep track of the number of fields and spatial derivatives in order to use the weak field framework. To be consistent, since for the perturbations of order higher than one, the scalar, vector and tensor modes couple to each other, we need to write the metric in the full form, i.e., , considering the vector and tensor modes in addition to the scalar modes.

We thus start from the general metric in Eq. (1.48),

ds² =g_µνdx^µdx^ν =a²(τ)^h−(1 + 2Ψ)dτ²−2B_idxⁱdτ+ (1−2Φ)δ_ijdxⁱdx^j+h_ijdxⁱdx^ji . (1.87)

We use the Poisson gauge, which impliesδ^ijBi,j =δ^ijhij =δ^jkhij,k = 0 to remove the extra degrees of freedom. Solving the Einstein’s equations in the weak field limit G^µ_ν = 8πGT_ν^µ results in the following time-time component (for details see Adamek et al. [2016a, 2017b], Hassani et al. [2020b],

(1 + 4Φ)∆Φ−3HΦ⁰+ 3H²(χ−Φ) + 3

2δ^ijΦ_,iΦ_,j =−4πGa²T₀⁰ −T¯₀⁰ , (1.88) whereχ .

= Φ−Ψ. This equation in the first order limit and neglecting the rela-tivistic corrections reduce to the Newtonian Poisson equation ∆ψ_N = 4πGa²δρ

variable order Φ,Ψ,Φ⁰,Ψ⁰,Φ⁰⁰,Ψ⁰⁰

Φ_,i,Ψ_,i,Φ⁰_,i,Ψ⁰_,i ^1/2 Φ_,ij,Ψ_,ij 1 χ, χ⁰, χ⁰⁰, χ_,i, χ⁰_,i, χ_,ij B_i, B_i⁰, B_i⁰⁰, B_i,jB_i,j⁰ B_i,jk h_ij, h⁰_ij, h⁰⁰_ij, h_ij,k, h⁰_ij,k, h_ij,kl δT₀⁰/T¯₀⁰ 1 T_i⁰/T¯₀^{0 1/2} Π_ij/T¯₀⁰

vⁱ, q_i 1

Table 1.1: A table showing the order of perturbative quantities we consider to obtain the equations in the weak field regime. This table is similar to the table in Adamek et al. [2016a] which is the framework used in the gevolution N-body code.

and at the first order is similar to the Eq. (1.85). The other 5 degrees of free-dom i.e. χ (one scalar degree of freedom), B_i (two vector degree of freedom) and hij (two tensor degree of freedom) are determined from the traceless part of space-space Einstein’s equations which are fully relativistic without Newtonian analogue. In the weak-field expansion they read;

δ_kⁱδ^j_l − 1

3δ^ijδ_kl 1

2h⁰⁰_ij +Hh⁰_ij − 1

2∆h_ij +B_(i,j)⁰ + 2HB_(i,j)+χ_,ij

−2χΦ_,ij+ 2Φ_,iΦ_,j+ 4ΦΦ_,ij

= 8πGa²

δ_ikT_lⁱ−1 3δ_klT_iⁱ

.

= 8πGa²Π_kl.

(1.89)

The other four Einstein’s equations obtained from the time-space equations and the spatial trace, are not independent and can be obtained from the previous equations, however sometime it is useful some of those equations to do consis-tency checks of the solutions.