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

TL;DR

This paper derives exact third-order density perturbation solutions for general dark energy models, improving the accuracy of one-loop power spectrum calculations beyond the Einstein-de Sitter approximation.

Contribution

It provides the first exact third-order solutions in general dark energy models without assuming EdS, enhancing precision in cosmological perturbation theory calculations.

Findings

Less than 1% error in one-loop matter power spectrum at certain scales using exact solutions.

Approximate EdS assumption works well at large scales but introduces errors at smaller scales.

Exact solutions are essential for sub-percent accuracy in future dark energy constraints.

Abstract

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.