Exact Third-Order Density Perturbation and One-Loop Power Spectrum in General Dark Energy Models

Under the standard perturbation theory (SPT), we obtain the fully consistent third-order density fluctuation and kernels for the general dark energy models without using the Einstein-de Sitter (EdS) universe assumption for the first time. We also show that even though the temporal and spatial components of the SPT solutions can not be separable, one can find the exact solutions to any order in general dark energy models. With these exact solutions, we obtain the less than \% error correction of one-loop matter power spectrum compared to that obtained from the EdS assumption for $k = 0.1 {\rm h\, Mpc}^{-1}$ mode at $z = 0$ (1, 1.5). Thus, the EdS assumption works very well at this scale. However, if one considers the correction for $P_{13}$, the error is about 6 (9, 11) \% for the same mode at $z = 0$ (1, 1.5). One absorbs $P_{13}$ into the linear power spectrum in the renormalized perturbation theory (RPT) and thus one should use the exact solution instead of the approximation one. The error on the resummed propagator $N$ of RPT is about 14 (8, 6) \% at $z =0$ (1, 1.5) for $k = 0.4 {\rm h\, Mpc}^{-1}$. For $k = 1 {\rm h\, Mpc}^{-1}$, the error correction of the total matter power spectrum is about 3.6 (4.6, 4.5) \% at $z = 0$ (1, 1.5). Upcoming observation is required to archive the sub-percent accuracy to provide the strong constraint on the dark energy and this consistent solution is prerequisite for the model comparison.

The equations of motion ofδ(τ, k) andθ(τ, k) in the Fourier space are given by where τ is the conformal time, k 12 ≡ k 1 + k 2 , δ D is the Dirac delta function, H ≡ 1 a ∂a ∂τ , Ω m is the matter energy density contrast, α( k 1 , k 2 ) ≡ k12· k1 k 2 1 , and β( k 1 , k 2 ) ≡ Due to the mode coupling of the nonlinear terms shown in the right hand side of Eqs. (1) - (2), one needs to make a perturbative expansion inδ andθ [1]. One can introduce the proper perturbative series of solutions for the fastest growing mode D n where one can define the each order solution aŝ To be consistent with the current observation, we consider the dark energy dominated flat universe as a background model. It has been known that the n-th order fastest growing mode solutions are proportional to the n-th power of the linear growth factor D 1 (i.e. D n ∝ D n 1 ) for the EdS universe. And this is not true for the general background models. There have been the investigations of the validity of these ansatz (3) and (4) by using the different growth rates forδ andθ [5,6]. However, the improper decomposition of fastest mode solutions and the incorrect initial conditions are used for the n-th order growth rate in both cases (see Appendix).
If one takes a derivatives of Eq. (1) and replace Eq. (2) into it, then one obtains From the Eqs.(1) and (11) , one obtains the expressions for the higher order solutions ofδ (2) ,θ (2) , andδ (3) aŝ where where explicit forms of F (s) 3i are given in the appendix. One can use the above equations to compute the power spectrum at any order in perturbation theory The one-loop power spectrum is defined as where P 22 and P 13 are obtained as where r = q k and x = q· k qk . The above equations (23) and (24) are identical to Eqs. (2.24) and (2.25) of [4] when one replace the coefficients of higher solutions c 2i and c 3i with those given in Eqs.(A-13) and (A-39). Thus, the terms with c 2i and c 3i represent the dark energy effect on the one-loop power spectrum. Now we obtain the one-loop power spectrum for ΛCDM model. We run the camb to obtain the linear power spectrum [12] using Ω b0 = 0.044, Ω m0 = 0.26, h = 0.72, n s = 0.96, and the numerical integration range for q in Eqs. (23) and (24) is 10 −6 ≤ q ≤ 10 2 . In Fig. 1, we show both the linear power spectra P L (thin lines) and the nonlinear power spectra P NL = P L + P 2 (thick lines) at the different redshift z = 0 (solid), 1.0 (dotted), and 1.5 (dotdashed), respectively. We demonstrate the ΛCDM model with Ω m0 = 0.26 in this figure. As one expects, the nonlinear power spectra are not simply enhanced by multiplying the differences of the square of the growth factor D 2 1 at the different redshifts. One also needs to emphasize that the exact kernels Eqs.(A-16), (A-18) and (A-33)-(A-38) also depend on time. The coefficient of each kernel changes at the different observational epoch. Now, we investigate the corrections in P 22 and P 13 compared to those using the EdS assumption. As one expects, the effect of the removing EdS assumption on P 22 and P 13 becomes larger as z increases. This is due to the fact that we use the Gaussianity initial conditions for the perturbed quantities. The coefficients c 21 -c 36 approach to those of EdS models as z decreases. This causes the fact that the kernels based on the EdS assumption deviate from the exact ones as z decreases. Thus, the exact P 22 and P 13 show the larger deviations from the EdS assumed P 22 and P 13 as z increases. This is shown in Fig. 2 The differences are about 5 (9, 11) % for k = 0.1h Mpc −1 mode at z = 0 (1.0, 1.5). In the right panel of Fig. 2, we show the errors in P 13 at the different redshift. We use the same notation as the left panel. The differences between the exact and EdS assumed P 13 are about 6 (9, 11) % for k = 0.1h Mpc −1 mode at z = 0 (1.0, 1.5).
We show the corrections on P NL ≡ P total and the resummed propagator N . The one loop correction is sum of the P 22 and P 13 . However, P 22 and P 13 have the different signs. Thus, if one considers the nonlinear power spectrum with the one loop correction, then the correction due to using the exact solution is very small compared to the nonlinear power spectrum based on EdS assumption. P NL = P L + P 2 where P 2 = P 22 + P 13 . As we show in the Fig. 2, each correction at each mode is about same at any epoch. Thus, the corrections on P 2 are canceled each other. This is shown in the left panel of Fig. 3. P ΛCDM total means the exact nonlinear matter power spectrum based on the ΛCDM models using the exact solution. While P EdS total means the nonlinear matter power spectrum based on the EdS assumption. The solid, dotted, and dotdashed lines correspond to errors of P total at z = 0, 1.0, and 1.5, respectively. The present nonlinear matter power spectrum is dominated by the one loop power spectrum at small scale k ≥ 0.1. The correction for the total matter power spectrum is about 2 % for k = 0.4 h/Mpc at any epoch. Thus, the EdS assumed nonlinear power spectrum is not a bad approximation. However, if one expands the SPT into RPT, then one needs to use the exact solution. This is shown in the right panel of Fig. 3 by using the resummed propagator N . For the same mode, the deviations of N from the exact values are about 14 (8, 6) % at z = 0 (1.0, 1.5). Thus, if one uses the EdS assumed nonlinear P 13 , then one is not able to avoid these amount of errors on the N .
The upcoming redshift surveys of galaxies such as BOSS, eBOSS, PFS, EUCLID, and MS-DESI will provide observational data of large scale structure of the universe in larger volume with higher density. The analysis of these observational data requires very accurate theoretical modeling down to the quasi-linear regime. In this Letter, we present an accurate perturbation theory without adopting the EdS assumption. The obtained results are general for any background universe model including time varying dark energy models, and will be useful for studies of future surveys.

Acknowledgments
This work were carried out using computing resources of KIAS Center for Advanced Computation. S.L would like to thank for the hospitality at APCTP during the program TRP.

APPENDIX
In this section, we show the spatial and temporal solutions of the each order by using Eqs. (2) and (11). The equations for the first order solution of δ (1) (τ, k) and θ (1) (τ, k) are given by From the above Eqs. (A-1) and (A-2), one obtains where we use If one uses the fact that the dark energy is dominated only at the late universe, then one can adopt the EdS conditions (i.e. Ω m = 1) for D 1 at early time (it i.e. a i ), Thus, one can obtain the exact solution for D 1 (a) for any dark energy model from Eqs.(A-1) and (A-7) except for the early dark energy one [13][14][15].
One can repeat the same process forδ (2) (τ, k) to get If we adopt the initial zero non-Gaussianity of the higher order solutions (δ (n) = 0), then one can obtain the equations for the fastest growing mode solutions with the initial Gaussianity and the EdS initial conditions where we use the fastest growing mode solutions for the EdS universe Often it is knows as the EdS coefficient as However, this is not the coefficients for the fastest growing mode solutions because of the existence of the second terms in Eq. (A-12). From Eq.(2), one can obtain equations forθ (2) by using other solutions From Eqs.(A-9)-(A-14) Now one can obtain the third order solutions from the previous solutions up to the second order. One can write the third order solution If one replaces Eq.(A-19) into Eq.(11), then one obtains In the above equation (A-32), we use the initial Gaussinity condition of δ (3) (i.e. D 3i (a i ) = 0) to obtain the coefficients for the last terms of D 3i .
One can find the third order kernels (F 3i ( k)) from the above Eqs. (A-20)-(A-25). For example, one obtain F (s) 31 as One can repeat the above process to obtain Thus, one can calculate P 22 (a, k) and P 13 (a, k) at any epoch.