Separate universe approach to evaluate nonlinear matter power spectrum for non-flat $\Lambda$CDM model

The spatial curvature ($\Omega_K$) of the Universe is one of the most fundamental quantities that could give a link to the early universe physics. In this paper we develop an approximate method to compute the nonlinear matter power spectrum, $P(k)$, for "non-flat" $\Lambda$CDM models using the separate universe (SU) ansatz which states that the effect of the curvature on structure formation is equivalent to that of long-wavelength density fluctuation ($\delta_{\rm b}$) in a local volume in the "flat" $\Lambda$CDM model, via the specific mapping between the background cosmological parameters and redshifts in the non-flat and flat models. By utilizing the fact that the normalized response of $P(k)$ to $\delta_{\rm b}$ (equivalently $\Omega_K$), which describes how the non-zero $\Omega_K$ alters $P(k)$ as a function of $k$, is well approximated by the response to the Hubble parameter $h$ within the flat model, our method allows one to generalize the prediction of $P(k)$ for flat cosmologies via fitting formulae or emulators to that for non-flat cosmologies. We use $N$-body simulations for the non-flat $\Lambda$CDM models with $|\Omega_K|\leq 0.1$ to show that our method can predict $P(k)$ for non-flat models up to $k \simeq 6\,h{\rm Mpc}^{-1}$ in the redshift range $z\simeq [0,1.5]$, to the fractional accuracy within $\sim 1$% that roughly corresponds to requirements for weak lensing cosmology with upcoming surveys. We find that the emulators, those built for flat cosmologies such as EuclidEmulator, can predict the non-flat $P(k)$ with least degradation.