Emulating the nonlinear effects of modified gravity on the matter power spectrum for reconstruction

Including nonlinear information from modified gravity (MG) brings more constraining power to the model-independent reconstruction of MG functions. However, the calculation of the nonlinear matter power spectrum with MGCAMB+ReACT is expensive in repeated likelihood evaluations, which limits the exploration of the high-dimensional parameter space. In this work, we construct a neural-network emulator for the nonlinear correction $R_{\mathrm{MG}}(k,z)=P_{\mathrm{MG}}^{\mathrm{NL}}(k,z)/P_{\mathrm{MG}}^{\mathrm{L}}(k,z)$, which is trained with \texttt{CosmoPower} on approximately $9\times10^5$ samples and validated with representative spectra, an independent validation set, and MCMC tests with synthetic data. For $\Lambda$CDM and moderate MG nonlinear corrections, the emulated power spectra agree with the reference predictions to within $1.5\%$ over the full scale range considered. For the extreme nonlinear case, the same accuracy is retained for $k<0.8\,\mathrm{Mpc}^{-1}$. Over the independent validation set, the mean residual is close to zero, with the $2\sigma$ scatter below $1\%$ for $k<0.5\,\mathrm{Mpc}^{-1}$ and about $2\%$ on smaller scales. The synthetic data MCMC analyses recover the input $\Lambda$CDM cosmology and the GR limits of the reconstructed MG functions within the posterior uncertainties, showing the accuracy and reliability of the emulator for Stage-IV-like surveys. We also demonstrate three applications: using $R_{\mathrm{MG}}$ to distinguish models with nearly degenerate linear power spectra, using the emulator for theory predictions for current photometric-survey $3\times2$pt likelihoods, and forecasting DESI+CSST constraints with principal component analysis.