  Analytic formulas reproducing the warm dark matter (WDM) primordial spectra
are obtained for WDM particles decoupling in and out of thermal equilibrium
which provide the initial data for WDM non-linear structure formation. We
compute and analyze the corresponding WDM overdensities and compare them to the
CDM case. We consider the ratio of the WDM to CDM primordial spectrum and the
WDM to CDM overdensities: they turn to be self-similar functions of k/k_{1/2}
and R/R_{1/2} respectively, k_{1/2} and R_{1/2} being the wavenumber and length
where the WDM spectrum and overdensity are 1/2 of the respective CDM
magnitudes. Both k_{1/2} and R_{1/2} 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 k_{1/2} with respect to the CDM
spectrum, while the WDM overdensity slowly decreases around R_{1/2}. 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 in the keV scale, and help to single out the best WDM particle
physics model (Abridged).
