The characteristic mass and energy conversion efficiency in the cusp–core transition of dark matter haloes: implications for scaling relations and supernova feedbacks

ABSTRACT Galaxies in the nearby Universe, particularly dwarf systems, exhibit inner mass profiles of dark matter haloes that systematically depart from canonical cold dark matter expectations, signalling an interplay between baryonic feedback and the collisionless halo. We update an analytical cusp–core transition model by incorporating the effect of supernova-driven mass-loss. Adapting this model to Spitzer Photometry & Accurate Rotation Curves (SPARC) galaxies (F. Lelli et al. 2016), we measure the energy conversion efficiency $\varepsilon$, defined as the fraction of supernova feedback energy that is used to change the central dark-matter potential. We find $\varepsilon \approx 0.01$ for nearby SPARC galaxies. Building on these measurements, we compare the dynamical energy required for a cusp–core transformation with the feedback energy available over burst cycles and identify a cusp–core transition forbidden region on the halo-stellar mass plane where transformation cannot occur. Galaxies with halo masses from $10^{8}$ to $10^{11}\, \mathrm{M}_\odot$ lie outside the forbidden region, whereas ultrafaint dwarf galaxies ${\lt} 10^{8}\, \mathrm{M}_\odot$, galaxy groups, and clusters ${\gt} 10^{11}\, \mathrm{M}_\odot$ fall within it, consistent with their high-central densities and the inefficiency of core formation at very low and very high masses. This approach also explains the observed diversity of inner density profiles in low-mass systems, showing that both the star formation rate and the energy conversion efficiency govern them, with the latter emerging as a key parameter setting the strength of the cusp–core transition. Beyond the cusp–core problem, our observationally inferred energy conversion efficiency provides a model independent benchmark that strongly constrains galaxy formation models.