  We exploit the recent determination of cosmic star formation rate (SFR)
density at redshifts $z\gtrsim 4$ to derive astroparticle constraints on three
common dark matter scenarios alternative to standard cold dark matter (CDM):
warm dark matter (WDM), fuzzy dark matter ($\psi$DM) and self-interacting dark
matter (SIDM). Our analysis relies on the UV luminosity functions measured by
the Hubble Space Telescope out to $z\lesssim 10$ and down to UV magnitudes
$M_{\rm UV}\lesssim -17$. We extrapolate these to fainter yet unexplored
magnitude ranges, and perform abundance matching with the halo mass functions
in a given DM scenario, so obtaining a relationship between the UV magnitude
and the halo mass. We then compute the cosmic SFR density by integrating the
extrapolated UV luminosity functions down to a faint magnitude limit $M_{\rm
UV}^{\rm lim}$, which is determined via the above abundance matching
relationship by two free parameters: the minimum threshold halo mass $M_{\rm
H}^{\rm GF}$ for galaxy formation, and the astroparticle quantity $X$
characterizing each DM scenario (namely, particle mass for WDM and $\psi$DM,
and kinetic temperature at decoupling $T_X$ for SIDM). We perform Bayesian
inference on such parameters via a MCMC technique by comparing the cosmic SFR
density from our approach to the current observational estimates at $z\gtrsim
4$, constraining the WDM particle mass to $m_X\approx
1.2^{+0.3\,(11.3)}_{-0.4\,(-0.5)}$ keV, the $\psi$DM particle mass to
$m_X\approx 3.7^{+1.8\,(+12.9.3)}_{-0.4\,(-0.5)}\times 10^{-22}$ eV, and the
SIDM temperature to $T_X\approx 0.21^{+0.04\,(+1.8)}_{-0.06\,(-0.07)}$ keV at
$68\%$ ($95\%$) confidence level. We then forecast how such constraints will be
strengthened by upcoming refined estimates of the cosmic SFR density, if the
early data on the UV luminosity function at $z\gtrsim 10$ from JWST will be
confirmed down to ultra-faint magnitudes.
