Thomas Montandon, Vivian Poulin, Oliver Hahn, Jozef Bucko, Aurel Schneider

We present a semi-analytical framework for the halo mass function (HMF) in decaying dark matter (DDM) cosmologies, in which dark matter decays into a massive daughter particle inheriting a velocity kick $v_k$ and a massless dark radiation component. Building on the Press-Schechter formalism, we encode the DDM physics through a spherical collapse model that explicitly tracks the decay-induced mass loss, yielding a modified, mass-dependent critical collapse threshold $δ_c(M_0)$ and a mapping $M_{\rm coll}(M_0)$ between the initial Lagrangian mass and the collapsed halo mass. The critical threshold exhibits a characteristic transition between two analytically tractable plateaus: a large-mass limit, where all daughter particles are retained by the halo, and a small-mass limit, where all daughters escape and the collapse is equivalent to that of a dark matter species decaying entirely into dark radiation, making $δ_c$ independent of $M_0$ and $v_k$. We provide semi-analytical results and fits for both limits and a fitting formula for the transition, whose single free parameter $M_1 \propto v_k^3\,\tildeΓ^{-1/2} t_{\rm ta}$ has a transparent physical interpretation: it is the mass scale at which the kick velocity equals the halo orbital velocity. We validate our predictions against a suite of N-body simulations at $z=0$ and $z\approx 1$, finding good agreement across models spanning mild to strong HMF suppression relative to $Λ$CDM. Residual deviations for the largest kick velocities at $z=0$ are observed. Via a halo-by-halo comparison between simulations, we trace the discrepancy to the definition of the halo mass when daughter orbits extend beyond the halo boundary. The resulting fitting functions for $δ_c(M_0,Γ,v_k)$ and $M_{\rm coll}(M_0)$ provide an efficient and accurate route to DDM constraints from current and forthcoming probes of the halo mass function.