Warm dark matter primordial spectra and the onset of structure formation at redshift $z$

overdensities and compare them to the cold dark matter (CDM) case. We consider the ratio of the WDM to CDM primordial spectrum and the ratio of the WDM to CDM overdensities: they turn to be self-similar functions of k/k1/2 and R/R1/2 respectively, k1/2 and R1/2 being the wavenumber and length where the WDM spectrum and overdensity are one-half of the respective CDM magnitudes. Both k1/2 and R1/2 show scaling as powers of the WDM particle mass m while the self-similar functions are independent of m. The WDM primordial spectrum sharply decreases around k1/2 with respect to the CDM spectrum, while the WDM overdensity slowly decreases around R1/2 for decreasing scales with respect to the CDM one. The nonlinear regions where WDM structure formation takes place are shown and compared to those in CDM: the WDM non-linear structures start to form later than in CDM, and as a general trend, decreasing the DM particle mass delays the onset of the non-linear regime. The non-linear regime starts earlier for smaller objects than for larger ones; smaller objects can form earlier both in WDM and CDM. We compute and analyze the differential mass function dN/dM for WDM at redshift z in the Press-Schechter approach. The WDM suppression effect of small scale structure increases with the redshift z. Our results for dN/dM are useful to be contrasted with observations, in particular for 4 . z . 12. We perfom all these studies for the most popular WDM particle physics models. Contrasting them to observations should point out the precise value of the WDM particle mass within the keV scale, and help to single out the best WDM particle physics model.