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 SPARC galaxies, we measure the energy conversion efficiency epsilon, defined as the fraction of supernova feedback energy that is used to change the central dark-matter potential. We find epsilon ~ 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 M_sun lie outside the forbidden region, whereas ultra-faint dwarf galaxies < 10^8 M_sun, galaxy groups and clusters > 10^11 M_sun 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.

Figures (4)

• Figure 1: Left: Radial profiles of density (upper-left) and circular velocity (lower-left) for a dark matter halo with mass $M_{200} = 10^9\,M_{\odot}$, illustrating the effect of the cusp--core transition. The orange curve represents the NFW profile. The solid and dashed green curves indicate Burkert profiles with mass loss fractions $f = 0$ and $f = f_\mathrm{b}$, respectively. The magenta dashed curve indicates the model where the virial radius is recalculated from the halo mass after mass loss, adopting a mass–loss function of $f = f_{\mathrm{b}}$. Upper-right: The dependence of $\eta$ on $M_{200}$. Solid and dashed curves correspond to mass loss fractions of $f = 0$ and $f = f_\mathrm{b}$, respectively. Lower-right: The dependence of $\psi_{\mathrm{NFW}}$ and $\psi_{\mathrm{BKT}}$ on $M_{200}$. The orange curve represents the NFW profile, while the solid and dashed green curves denote Burkert profiles with $f = 0$ and $f = f_\mathrm{b}$, respectively. The magenta dashed curve indicates a marginally smaller $\psi$ resulting from the redefinition of the virial radius.
• Figure 2: The left panel shows $M_{\ast,\mathrm{crit}}/M_{200}$ versus $M_{200}$; the right panel shows $M_{\ast,\mathrm{crit}}$ versus $M_{200}$. The orange solid, green solid, and green dashed lines correspond to $\varepsilon=1$, $0.1$, and $0.01$, respectively. The red solid and magenta dashed curves show the $z=0$ stellar-to-halo mass relations from Behroozi+2019 and Moster+2013, respectively. Regions with star formation efficiency (SFE) $>1$ are shaded gray. In the left panel, open white symbols indicate SPARC data, and filled blue symbols denote the sample of Gastaldello+2007. Objects in Gastaldello+2007 with stellar-mass uncertainties larger than their nominal values are shown in light blue.
• Figure 3: Upper-right: The full light orange plane represents the surface in the three-dimensional parameter space followed by NFW dark matter haloes determined by the $c$–$M$ relation. The dark orange region indicates the forbidden zone for the cusp-core transition, derived using our energy transport model explained in section \ref{['sec:theoreticalmodel']} assuming energy conversion efficiency $\varepsilon = 1$. The planes followed by Burkert-profile haloes, whose parameters derived by our cusp-core transition model explained in section \ref{['sec:theoreticalmodel']}, are shown in dark green and light green for cases the mass conservation model and mass loss model, respectively. Upper-left: The $V_{\mathrm{max}}$–$M_{\ast}$ projection, corresponding to a top-down view of the upper-right panel. The green solid and dashed lines show $M_{*, \mathrm{crit}}$ as a function of $V_{\rm{max}}$ using for mass conservation model and mass loss model, respectively. Here, we assume $\varepsilon=1$. The orange solid line also shows $M_{*, \mathrm{crit}}$ as a function of $V_{\rm{max}}$, but $V_{\rm{max}}$ is calculated for the cusp profile. Lower-left: The $V_{\mathrm{max}}$–$\bar{\Sigma}(<0.01\,r_{\mathrm{max}})$ projection, corresponding to the upper-right panel seen from lower-right. The orange solid curve shows $c$--$M$ relation converted on the $V_{\mathrm{max}}$–$\bar{\Sigma}(<0.01\,r_{\mathrm{max}})$ plane. The solid and dashed green curves show the $c$--$M$ relation for cored haloes, which is derived using our cusp-core transition model for the mass conservation case and the mass loss case, respectively. Here, we assume $\varepsilon=1$ again. Tick marks on green curves for $V_{\mathrm{max}}$ correspond to $M_{\ast,\mathrm{crit}}/M_{\odot} = 10^4,\ 10^6,\ 10^8$ and $\ 10^{10}$. Lower-right: The $M_{\ast}$–$\bar{\Sigma}(<0.01\,r_{\mathrm{max}})$ projection, corresponding to the upper-right panel seen from lower-left. In both the upper-left and lower-right panels, the cusp region shown in orange includes only the forbidden region from the upper-right panel.
• Figure 4: Relation between $V_{\max}$ and the energy conversion efficiency $\varepsilon$, derived from observed $V_{\max}$ and $M_{\ast}$. Background shading darkens with increasing stellar mass. The lower-right region, where stellar mass exceeds halo mass, is excluded. The solid line denotes $\varepsilon=1$ ($100\%$), above which cusp-core transitions are forbidden. SPARC data cluster around $\varepsilon\sim0.01$.