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

The growth of large-scale structure, 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 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-auto power spectra for nonflat $\Lambda$CDM model, from quantities representing the nonlinear evolution of the corresponding flat $\Lambda$CDM model, based on the separate universe (SU) method. By utilizing the fact that the growth response to long-wavelength fluctuations (equivalently the curvature), $T_{\delta_{\rm b}}(k)$, is approximated by the response to the Hubble parameter, $T_h(k)$, 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 $k \simeq 3 (1) \, h {\rm Mpc}^{-1}$ 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 super sample covariance (SSC), we discuss that the approximate identity $T_{\delta_{\rm b}}(k) \approx T_h(k)$ can be used to calculate the SSC terms analytically.