Third-Order Density Perturbation and One-Loop Power Spectrum in Dark-Energy-Dominated Universe

TL;DR

The paper extends cosmological perturbation theory to third order in dark-energy models with a general time-varying equation of state $w(a)$, deriving the full second- and third-order density perturbations and the corresponding one-loop power spectrum. It demonstrates that the dark-energy cosmology induces only small corrections relative to the Einstein–de Sitter (EdS) baseline, with the EdS-based one-loop expression accurately reproducing the true power spectrum to better than about 1% for $k<0.4\,h\,{ m Mpc}^{-1}$ at $z=0$ (and even less at higher redshifts). The authors provide explicit analytical forms and fitting expressions for the growth factors and one-loop terms, along with exact kernel-based integrals for $P_{22}$ and $P_{13}$. The results have practical implications for precise BAO modeling and for probing nonlinear evolution in dark-energy–dominated universes on mildly nonlinear scales, while highlighting the limits of perturbation theory in deeply nonlinear regimes.

Abstract

We investigate the third-order density perturbation and the one-loop correction to the linear power spectrum in the dark-energy cosmological model. Our main interest is to understand the dark-energy effect on baryon acoustic oscillations in a quasi-nonlinear regime ($k \approx 0.1h$/Mpc). Analytical solutions and simple fitting formulae are presented for the dark-energy model with the general time-varying equation of state $w(a)$. It turns out that the power spectrum coincides with the approximate result based on the EdS (Einstein de-Sitter) model within 1% for $k<0.4h/$Mpc at $z=0$ in the WMAP (Wilkinson Microwave Anisotropy Probe) 5yr best-fitting cosmological model, which suggests that the cosmological dependence is very small.

Figures (4)

• Figure 1: Top panels: the contour lines show the relative differences between $D_{2 A,B}$ and $D_1^2$ at present ($a=1$) in the $\Omega_M-w$ plane. The flat cosmological model and the constant equation of state are assumed. The top left (right) panel is the result for $D_{2 A}$ ($D_{2 B}$). Middle and bottom panels: same as top panels, but for the relative differences between $D_{3}$ and $D_1^3$.
• Figure 2: Same as Fig. \ref{['fig_f2h3']}, but for the time-varying equation of state, $w(a)=w_0+w_a a \left( 1-a \right)$. The results are shown in the $w_0 - w_a$ plane in the flat cosmological model with $\Omega_M=0.28$.
• Figure 3: Relative differences of $P_{22}(k)$ (top left), $P_{13}(k)$ (top right), $P_{22}(k)+P_{13}(k)$ (bottom left) and $P(k)$ (bottom right) between the correct results and the approximate results denoted by $[...]_{\rm EdS}$. The cosmological model is consistent with the WMAP 5yr result.
• Figure 4: Same as top panels in Fig.\ref{['fig_dp']}, but at various redshifts of $z=0,1,3$.