Spherical Collapse and Halo Formation in a Cosmology with Decaying Dark Matter and a Semi-Cosmographic Dark Energy

We investigate nonlinear structure formation in a cosmological model combining one-body decaying dark matter (DDM) with a semi-cosmographic reconstruction of dark energy. In this scenario, a nonrelativistic dark-matter component decays into relativistic dark radiation with decay rate = tau ddm-1, while the dark-energy sector is reconstructed directly from the expansion history rather than being fixed to a cosmological constant. Using DESI DR1 BAO and compressed ShapeFit measurements, we constrain the background evolution and propagate the resulting posterior into the nonlinear regime through spherical collapse and halo abundance calculations. This provides a unified framework connecting a reconstructed dark-energy sector and decaying dark matter (DDM) to the nonlinear formation of cosmic structures. We find that the reconstructed dark-energy equation of state can deviate from the CDM value, w=-1, while the critical density threshold for collapse remains close to its standard prediction. The most pronounced signatures emerge in the abundance of massive halos, reflecting modifications to the growth of structure driven by both dark-matter decay and dynamical dark energy. By combining DESI DR1 clustering constraints with halo mass function measurements from the DESI Legacy Imaging Surveys DR9, we obtain joint constraints on the DDM lifetime and dark-energy parameters, demonstrating that halo abundances provide a powerful complementary probe of non-standard dark-sector physics.