Towards a general parametrization of the warm dark matter halo mass function

Studies of flux anomalies statistics and perturbations in stellar streams have the potential to constrain models of warm dark matter (WDM), including sterile neutrinos. Producing these constraints requires a parametrization of the WDM mass function relative to that of the cold dark matter (CDM) equivalent. We use five WDM models with half-mode masses, $M_\mathrm{hm}=[1.3,35]\times10^{8}$~$M_{\odot}$, spread across simulations of the Local Group, lensing ellipticals and the $z=2$ universe, to generate such a parametrization: we fit parameters to a functional form for the WDM-to-CDM halo mass function ratio, $n_\mathrm{WDM}(M_{X})/n_\mathrm{CDM}(M_{X})$, of ($1+(\alpha M_\mathrm{hm}/M_{X})^{\beta})^{\gamma}$. For $M_{X}\equiv$ virial mass of central halos we obtain $\alpha=2.3$, $\beta=0.8$, and $\gamma=-1.0$, and this fit is steeper than the extended Press-Schechter formalism predicts. For $M_{X}\equiv$ mass of subhalos we instead obtain $\alpha=4.2$, $\beta=2.5$ and $\gamma=-0.2$; in both mass definitions the scatter is $\sim20$~per~cent. The second fit typically underestimates the relative abundance of $z=2$ WDM subhaloes at the tens of per cent level. We caution that robust constraints will require bespoke simulations and a careful definition of halo mass, particularly for subhalos of mass $<10^{8}M_{\odot}$.