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Modified matter power spectra with approximately Gaussian bump on sub-Mpcscales can be a result of a complex inflation. We consider five spectra withdifferent Gaussian amplitudes $A$ and locations $k_0$ and run N-bodysimulations in a cube $(5 Mpc/h)^3$ at $z>8$ to reveal the halo mass functionsand their evolution with redshift. We have found that the Sheth-Tormen formulaprovides a good approximation to a such kind of halo mass functions. In theconsidered models the dark matter halo formation starts much more earlier thanin $\Lambda$CDM, which in turn can result in an earlier star formation and anuclear activity in galaxies and can be detected and tested by, e.g., JWST. At$z=0$ the halo mass functions are hardly distinguishable from the standard$\Lambda$CDM, therefore the models with the bumpy spectra can be identified inobservations by their excess in number of bright sources at high redshift only.

Excess of high-$z$ galaxies as a test for bumpy power spectrum of density perturbations