Flat to nonflat: Calculating nonlinear power spectra of biased tracers for a nonflat $\Lambda \mathrm{CDM}$ model

The growth of large-scale structures, together with the geometrical information of cosmic expansion history and cosmological distances, can be used to obtain constraints on the spatial curvature of the Universe that probes the early Universe physics, whereas modeling the nonlinear growth in a nonflat universe is still challenging due to the computational expense of simulations in a high-dimensional cosmological parameter space. In this paper, we develop an approximate method to compute the halo-matter and halo-autopower spectra for nonflat $\Lambda$ cold dark matter ($\Lambda \mathrm{CDM}$) model, from quantities representing the nonlinear evolution of the corresponding flat $\Lambda \mathrm{CDM}$ model, based on the separate universe method. By utilizing the fact that the growth response to long-wavelength fluctuations (equivalently, the curvature) $T_{{\delta }_b}\left(k\right)$ is approximated by the response to the Hubble parameter $T_h\left(k\right)$, our method allows one to estimate the nonlinear power spectra in a nonflat universe efficiently from the power spectra of the flat universe. We use $N$-body simulations to show that the estimator can provide the halo-matter (halo-auto) power spectrum at $\sim 1\%$ ($\sim 2\%$) accuracy up to $k\simeq 3\left(1\right)h{\mathrm{Mpc}}^{-1}$ even for a model with large curvature ${\Omega }_K=\pm 0.1$. Using the estimator, we can extend the prediction of the existing emulators such as dark emulator to nonflat models without degrading their accuracy. Since the response to long-wavelength fluctuations is also a key quantity for estimating the supersample covariance (SSC), we discuss that the approximate identity $T_{{\delta }_b}\left(k\right)\approx T_h\left(k\right)$ can be used to calculate the SSC terms analytically.