The paper presents results from a large library of cosmological N-body simulations using power-law initial spectra. The nonlinear evolution of the matter power spectra is compared with analytic scaling formulae based on the work of Hamilton et al. The scaling approach assumes stable clustering, where highly nonlinear structures are frozen in proper coordinates. However, results show that the scale-free spectra define a nonlinear locus shallower than predicted by stable clustering. Small-scale nonlinear power increases with decreasing power spectrum index n and density parameter Ω, not well described by previous scaling formulae. This breakdown is due to dark-matter halo modifications by mergers, which are included in the halo model. The halo model is used to fit both scale-free results and previous CDM data, offering more accuracy than the Peacock–Dodds formula. The paper discusses the theoretical understanding of nonlinear evolution, including the stable clustering hypothesis, the HKLM scaling relations, and the halo model. It describes numerical simulations, power spectrum measurement methods, and compares results with existing fitting formulae. The halo model is proposed as a new method for predicting nonlinear spectra, applicable to more general power spectra. The paper is structured into sections discussing nonlinear evolution, numerical simulations, power spectrum measurement, results, and conclusions. The results show that the power spectra scale in a self-similar way for certain initial spectra, and that the halo model provides a more accurate description of nonlinear evolution than previous methods. The paper also discusses the challenges of simulating highly negative spectral indices and the effects of numerical errors on the power spectrum.The paper presents results from a large library of cosmological N-body simulations using power-law initial spectra. The nonlinear evolution of the matter power spectra is compared with analytic scaling formulae based on the work of Hamilton et al. The scaling approach assumes stable clustering, where highly nonlinear structures are frozen in proper coordinates. However, results show that the scale-free spectra define a nonlinear locus shallower than predicted by stable clustering. Small-scale nonlinear power increases with decreasing power spectrum index n and density parameter Ω, not well described by previous scaling formulae. This breakdown is due to dark-matter halo modifications by mergers, which are included in the halo model. The halo model is used to fit both scale-free results and previous CDM data, offering more accuracy than the Peacock–Dodds formula. The paper discusses the theoretical understanding of nonlinear evolution, including the stable clustering hypothesis, the HKLM scaling relations, and the halo model. It describes numerical simulations, power spectrum measurement methods, and compares results with existing fitting formulae. The halo model is proposed as a new method for predicting nonlinear spectra, applicable to more general power spectra. The paper is structured into sections discussing nonlinear evolution, numerical simulations, power spectrum measurement, results, and conclusions. The results show that the power spectra scale in a self-similar way for certain initial spectra, and that the halo model provides a more accurate description of nonlinear evolution than previous methods. The paper also discusses the challenges of simulating highly negative spectral indices and the effects of numerical errors on the power spectrum.