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.
