Nonlinear Matter Power Spectrum from relativistic $N$-body Simulations: $\Lambda_{\rm s}$CDM versus $\Lambda$CDM

We present relativistic $N$-body simulations of a $\Lambda_{\rm s}$CDM - sign-switching cosmological constant (CC) - scenario under general relativity and compare its nonlinear matter power spectrum to $\Lambda$CDM at ${z = 15,\,2,\,1,\,0}$, using best-fit parameters from Planck-only and a combined ''full'' dataset. During the AdS-like CC ($\Lambda_{\rm s}<0$) phase, prior to the transition redshift $z_\dagger$, reduced Hubble friction dynamically enhances the growth of perturbations; after the switch, with dS-like CC ($\Lambda_{\rm s}>0$), the larger late-time expansion rate partly suppresses, but does not erase, the earlier amplification. Consequently, the ratio $P_{\Lambda_{\rm s}\rm CDM}/P_{\Lambda\rm CDM}$ exhibits a pronounced, redshift-dependent shape feature: a crest peaking at ${\sim 20-25\%}$ around ${k \simeq 1-3\,h\,\mathrm{Mpc}^{-1}}$ near the transition, which then migrates to larger physical scales and persists to ${z = 0}$ as a robust ${\sim 15-20\%}$ uplift at ${k \simeq 0.6-1.0\,h\,\mathrm{Mpc}^{-1}}$. These wavenumbers correspond to group or poor-cluster environments and lie within the sensitivity range of weak lensing, galaxy-galaxy lensing, cluster counts, and tSZ power, providing a concrete, falsifiable target that cannot be mimicked by a scale-independent change in $\sigma_8$ or $S_8$. The timing (earlier for Planck-only, later for the full dataset) and the amplitude of the crest align with the ''cosmic noon'' epoch (${z \simeq 1-2}$), offering a gravitational prior for the observed peak in the cosmic star-formation rate.