Cusp-to-Core Transition in Dark Matter Halos

• Cusp-to-core transition is defined as the shift in the inner dark matter density from a steep NFW-like cusp (α ≈ -1) to a flatter core (α ≈ 0), foundational to the core–cusp problem.
• Observational analyses using rotation curves and high-resolution simulations (e.g., FIRE-2) reveal a mass-dependent pattern where baryonic feedback induces core formation primarily in galaxies with Vmax ~50–150 km/s.
• Energetic mappings and scaling relations (e.g., constant central surface density μ0D) underpin the transition, though uncertainties persist due to observational limits and alternative dark matter models.

The cusp-to-core transition denotes the transformation of the inner density structure of a dark-matter halo from a centrally divergent cusp, typically of Navarro–Frenk–White type, to a shallow or approximately constant-density core. It is the dynamical content behind the broader core–cusp problem: collisionless $\Lambda$CDM simulations predict inner profiles close to $\rho \propto r^{-1}$, whereas many observed dwarf, low-surface-brightness, and late-type galaxy rotation curves favor much flatter central mass distributions. Recent cross-scale analyses further indicate that the phenomenon is not universal in the sense of all halos becoming cored; rather, halo structure appears to depend on mass scale, with late-type galaxies often showing shallower-than-NFW inner profiles, while lower-mass dwarf spheroidals and higher-mass groups and clusters tend to remain cuspy (0910.3538, Hayashi et al., 29 Jul 2025).

1. Formal definition and parametrization

A cusp-to-core transition is usually expressed through the inner logarithmic density slope

$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$

A cuspy profile has $\alpha \approx -1$, as in the NFW form, whereas a truly cored profile has $\alpha \approx 0$. In the generalized Hernquist fits used by Hayashi et al., the free parameter $\gamma$ plays exactly this role, with $\gamma \simeq -\alpha$; thus $\gamma \to 1$ denotes a steep cusp and $\gamma \to 0$ a core (Hayashi et al., 29 Jul 2025).

The canonical cuspy reference model is the NFW profile,

$$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$

whose asymptotic inner behavior is $\rho \propto r^{-1}$0. Two widely used cored alternatives are the pseudo-isothermal profile,

$\rho \propto r^{-1}$1

and the Burkert profile,

$\rho \propto r^{-1}$2

both of which approach an approximately constant central density in the inner region (Popolo et al., 2022).

Two surface-density-like quantities are central in the literature. Ogiya et al. define the cored-halo central surface density as

$\rho \propto r^{-1}$3

and show that it can remain nearly mass-independent under simple cusp-to-core mappings (Ogiya et al., 2013). Hayashi et al. instead adopt a mass-scaled central quantity designed for cross-comparison from dwarfs to clusters,

$\rho \propto r^{-1}$4

with

$\rho \propto r^{-1}$5

The choice of $\rho \propto r^{-1}$6 is a compromise: it lies well inside the region that rotation curves typically resolve across galaxy types, while still scaling with halo size so as not to bias comparisons of dwarfs, spirals, and clusters (Hayashi et al., 29 Jul 2025).

2. Observational status across galaxy mass scales

The empirical basis for the cusp-to-core transition was established from high-resolution studies of low-surface-brightness and gas-rich dwarf galaxies. HI synthesis data with beam $\rho \propto r^{-1}$7–$\rho \propto r^{-1}$8 and linear resolution $\rho \propto r^{-1}$9–$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$0, together with H$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$1 long-slit observations with seeing $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$2–$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$3 and $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$4 resolution, showed slowly rising inner rotation curves and inferred inner density slopes $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$5 to $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$6 in the inner $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$7–$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$8. Two-dimensional velocity fields found non-circular motions of typical amplitude $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$9 and photometric–kinematic center offsets $\alpha \approx -1$0, both too small to hide a genuine cusp in all but contrived cases (0910.3538).

Hayashi et al. extended the problem to a uniform, mass-dependent comparison using 115 high-quality SPARC galaxies with inclination $\alpha \approx -1$1 and at least 9 data points. Their best-fit generalized-Hernquist inner slopes span $\alpha \approx -1$2 to $\alpha \approx -1$3. Roughly half the sample has $\alpha \approx -1$4, much shallower than NFW, and typical median values are $\alpha \approx -1$5–$\alpha \approx -1$6, with cores $\alpha \approx -1$7 in many Milky Way-mass disks. In the same framework, Milky Way dwarf spheroidals, including classical dSphs and ultra-faint dwarfs, generally lie near the cuspy NFW prediction in the $\alpha \approx -1$8–$\alpha \approx -1$9 plane and have $\alpha \approx 0$0 in the range $\alpha \approx 0$1–$\alpha \approx 0$2, while galaxy groups and clusters show $\alpha \approx 0$3–$\alpha \approx 0$4 at $\alpha \approx 0$5–$\alpha \approx 0$6, again matching a cuspy NFW baseline. Late-type SPARC galaxies with $\alpha \approx 0$7–$\alpha \approx 0$8 scatter systematically below the NFW line and track FIRE-2 core predictions, whereas ultra-faint dwarfs with $\alpha \approx 0$9 and clusters with $\gamma$0 revert to cuspy behavior. This yields a mass-dependent “cusp-to-core-to-cusp” pattern across $\gamma$1 (Hayashi et al., 29 Jul 2025).

That pattern is significant because it disfavors the idea of a single, mass-independent inner profile family. It also narrows the transition region to galaxy scales where baryonic processes are dynamically important but not overwhelmingly suppressed by either shallow star-formation budgets or deep cluster potentials.

3. Inference from rotation curves and mass-scaled halo diagnostics

Modern analyses infer cusp-to-core behavior from rotation-curve decomposition rather than from direct density measurements. In the SPARC analysis, each observed rotation curve is modeled as

$\gamma$2

The dark component is obtained by numerically integrating an axisymmetric generalized Hernquist density re-parameterized by $\gamma$3, where $\gamma$4 is the inner slope and $\gamma$5 controls the sharpness of the transition to outer slope $\gamma$6 (Hayashi et al., 29 Jul 2025).

The priors are deliberately broad. For the dark halo, flat priors are adopted over

$\gamma$7

$\gamma$8

The stellar mass-to-light ratios follow log-normal priors $\gamma$9 and $\gamma \simeq -\alpha$0, while distance $\gamma \simeq -\alpha$1 and inclination $\gamma \simeq -\alpha$2 are also varied, entering as $\gamma \simeq -\alpha$3 and $\gamma \simeq -\alpha$4, with Gaussian priors from SPARC catalog uncertainties. The posterior is sampled with emcee under

$\gamma \simeq -\alpha$5

This construction propagates uncertainties in $\gamma \simeq -\alpha$6, $\gamma \simeq -\alpha$7, and $\gamma \simeq -\alpha$8 into the posterior rather than treating them as fixed nuisance choices (Hayashi et al., 29 Jul 2025).

The importance of the mass-scaled surface density $\gamma \simeq -\alpha$9 lies in comparative inference. A fixed physical aperture would mix systems at different fractions of their characteristic halo radii; the chosen aperture instead normalizes to halo size and allows dwarfs, spirals, groups, and clusters to be placed on a common structural plane. Within that plane, departures below the NFW baseline can be interpreted as a compact empirical signature of core formation, while proximity to the baseline indicates retention of cuspy structure (Hayashi et al., 29 Jul 2025).

4. Baryonic routes to core formation

Within $\gamma \to 1$0CDM, the most developed physical explanation for the cusp-to-core transition is repeated baryonic feedback. In hydrodynamical simulations such as FIRE-2, repeated supernova outbursts generate rapid, oscillatory changes in the central potential. Dark-matter particles gain energy through resonant interactions, and the inner profile flattens toward $\gamma \to 1$1. Hayashi et al. emphasize that core-formation efficiency peaks at stellar-to-halo mass ratio $\gamma \to 1$2, corresponding to $\gamma \to 1$3–$\gamma \to 1$4, which coincides with the mass range where the SPARC galaxies most clearly depart from the cuspy baseline (Hayashi et al., 29 Jul 2025).

A controlled demonstration was provided by Teyssier et al. in an isolated adaptive-mesh-refinement experiment. They evolved an initially cuspy halo with $\gamma \to 1$5, $\gamma \to 1$6, and concentration $\gamma \to 1$7, using RAMSES with quasi-Lagrangian refinement to maximum level $\gamma \to 1$8, corresponding to $\gamma \to 1$9. Their feedback model introduces a non-thermal energy reservoir to represent unresolved turbulence, cosmic rays, and magnetic fields, with dissipation time $\gamma \to 0$0 and cooling suppressed where $\gamma \to 0$1. In the run with feedback, a clear core appears after $\gamma \to 0$2, and by $\gamma \to 0$3 the dark-matter profile is well fit by a pseudo-isothermal profile with core radius $\gamma \to 0$4 and central density $\gamma \to 0$5–$\gamma \to 0$6. Over the same interval the inner slope evolves from $\gamma \to 0$7 at $\gamma \to 0$8 to $\gamma \to 0$9, while the star-formation history oscillates by factors of $$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$0–$$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$1 with duty cycle $$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$2–$$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$3, and the final stellar component becomes thick and hot with $$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$4 (Teyssier et al., 2012).

Ogiya and Mori formulated the same basic process as a resonance problem. If the baryonic potential oscillates with period $$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$5, halo particles are efficiently heated at the radius where the local dynamical time satisfies

$$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$6

For an NFW halo, their analytic estimate gives

$$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$7

for $$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$8 and $$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$9. Their $\rho \propto r^{-1}$00-body tests show that runs with $\rho \propto r^{-1}$01 form a clear core at the radius predicted by $\rho \propto r^{-1}$02, and that the oscillation period fixes the eventual core radius, while baryon mass and oscillation amplitude mainly determine the rate of energy transfer and the number of cycles required (Ogiya et al., 2012).

Taken together, these results define the baryonic cusp-to-core transition not as a single explosive event but as a cumulative, non-adiabatic reorganization of collisionless orbits by repeated potential fluctuations.

5. Alternative dynamical and non-CDM channels

Baryonic supernova feedback is not the only proposed route. On cluster scales, Martizzi, Teyssier, and Moore showed in idealized simulations that repeated AGN-driven outflow/inflow cycles can flatten a cuspy NFW halo. In their setup, a halo with $\rho \propto r^{-1}$03 and $\rho \propto r^{-1}$04 hosts a central black hole of initial mass $\rho \propto r^{-1}$05 with radiative efficiency $\rho \propto r^{-1}$06 and coupling $\rho \propto r^{-1}$07. After $\rho \propto r^{-1}$08–$\rho \propto r^{-1}$09 AGN cycles over $\rho \propto r^{-1}$10–$\rho \propto r^{-1}$11, the dark mass within $\rho \propto r^{-1}$12 drops by $\rho \propto r^{-1}$13, the profile flattens over $\rho \propto r^{-1}$14–$\rho \propto r^{-1}$15, and the inner slope evolves from $\rho \propto r^{-1}$16 to $\rho \propto r^{-1}$17. However, the observational comparison by Hayashi et al. places groups and clusters close to the cuspy NFW baseline, so cluster-scale core formation remains system-dependent rather than an empirically established generic outcome (Martizzi et al., 2012, Hayashi et al., 29 Jul 2025).

In dwarf spheroidals, several mechanisms invoke dynamical heating by perturbers rather than gas feedback. Boldrini, Mohayaee, and Silk simulated Fornax with globular clusters embedded in dark-matter minihalos of mass $\rho \propto r^{-1}$18. In their recent-accretion scenario, five clusters on eccentric orbits pass through the central $\rho \propto r^{-1}$19 region 3–4 times within $\rho \propto r^{-1}$20, with a typical inner orbital period of $\rho \propto r^{-1}$21. The fitted inner slope evolves from $\rho \propto r^{-1}$22 at $\rho \propto r^{-1}$23 to $\rho \propto r^{-1}$24 at $\rho \propto r^{-1}$25, while the fitted core radius grows from $\rho \propto r^{-1}$26 to $\rho \propto r^{-1}$27 after four crossings. Between passages, phase mixing permits partial cusp regeneration, with $\rho \propto r^{-1}$28 recovering by $\rho \propto r^{-1}$29–$\rho \propto r^{-1}$30 and $\rho \propto r^{-1}$31 shrinking by $\rho \propto r^{-1}$32–$\rho \propto r^{-1}$33, yielding a sawtooth sequence of flattening and partial re-steepening (Boldrini et al., 2019). Related $\rho \propto r^{-1}$34-body calculations of globular-cluster evolution in static and time-varying halo potentials show that clusters in a static cusp remain compact, with $\rho \propto r^{-1}$35–$\rho \propto r^{-1}$36 by $\rho \propto r^{-1}$37, whereas those in a static core expand to $\rho \propto r^{-1}$38; time-varying cusp-to-core histories yield intermediate final sizes around $\rho \propto r^{-1}$39 but retain extra tidal debris from the earlier cusp phase (Orkney et al., 2019).

Such channels are especially relevant for gas-poor dwarf spheroidals, where the stellar-feedback mechanism appears negligible in the classical-dwarf regime and core formation must be sought either in perturber-driven reconfiguration of the potential or in non-standard dark-matter physics (Boldrini, 2022). The review literature groups the non-CDM alternatives into self-interacting dark matter, warm or wave-like dark matter, and other particle-physics mechanisms (Popolo et al., 2022). SIDM with cross section per unit mass $\rho \propto r^{-1}$40–$\rho \propto r^{-1}$41 produces cored profiles with core radii of a few kiloparsecs in halos of $\rho \propto r^{-1}$42–$\rho \propto r^{-1}$43, while larger $\rho \propto r^{-1}$44 is excluded by cluster lensing and ellipticity constraints. Warm dark matter with $\rho \propto r^{-1}$45–$\rho \propto r^{-1}$46 produces only $\rho \propto r^{-1}$47 cores in dwarfs and does not solve cusp–core and too-big-to-fail simultaneously (Popolo et al., 2022).

More specialized proposals include primordial-black-hole heating, oscillating asymmetric dark matter, and scalar-field dark matter. In two-component dwarf halos, PBHs of mass $\rho \propto r^{-1}$48–$\rho \propto r^{-1}$49 comprising at least $\rho \propto r^{-1}$50 of the dark matter can heat the cold component through dynamical friction and two-body relaxation, producing robust $\rho \propto r^{-1}$51–$\rho \propto r^{-1}$52 cores in $\rho \propto r^{-1}$53–$\rho \propto r^{-1}$54 (Boldrini et al., 2019). In late-time dark-matter oscillation models, a tiny dark-number-violating mass term $\rho \propto r^{-1}$55–$\rho \propto r^{-1}$56 reactivates annihilation in halo centers; simulations then produce dwarf-galaxy cores of order $\rho \propto r^{-1}$57–$\rho \propto r^{-1}$58 and cluster cores of order $\rho \propto r^{-1}$59–$\rho \propto r^{-1}$60 for $\rho \propto r^{-1}$61–$\rho \propto r^{-1}$62 (Cline et al., 2020). In an exponential-potential scalar-field model,

$\rho \propto r^{-1}$63

the self-pressure of the field yields a central slope $\rho \propto r^{-1}$64 and an outer $\rho \propto r^{-1}$65 envelope, so the halo interpolates analytically between a constant-density core and a flat-rotation-curve regime (Su et al., 2010).

6. Scaling relations, energetics, and unresolved issues

A major development in the cusp-to-core literature is the connection between inner-profile transformation and halo scaling relations. Ogiya et al. studied a mapping from an initial NFW halo to a final Burkert halo under two assumptions: conservation of total virial mass $\rho \propto r^{-1}$66 and preservation of the outer density profile beyond $\rho \propto r^{-1}$67. These imply

$\rho \propto r^{-1}$68

Under this mapping the central surface density

$\rho \propto r^{-1}$69

becomes nearly mass-independent, with the observed value

$\rho \propto r^{-1}$70

over more than 8 orders of magnitude in $\rho \propto r^{-1}$71. For dwarf-galaxy core radii $\rho \propto r^{-1}$72–$\rho \propto r^{-1}$73, the same relation yields

$\rho \propto r^{-1}$74

thereby recovering the Strigari relation as a consequence of constant $\rho \propto r^{-1}$75 in that regime. Because $\rho \propto r^{-1}$76 and $\rho \propto r^{-1}$77, the central density of a core also records halo formation redshift (Ogiya et al., 2013).

Kaneda, Mori, and Otaki recast this program in terms of the $\rho \propto r^{-1}$78CDM concentration–mass relation. They derive an analytic NFW-to-Burkert mapping by enforcing equality of density and enclosed mass at $\rho \propto r^{-1}$79, solving for $\rho \propto r^{-1}$80 and then recomputing $\rho \propto r^{-1}$81, $\rho \propto r^{-1}$82, and characteristic surface densities for the cored analogue. In that framework, the predicted cored and cuspy $\rho \propto r^{-1}$83 relations coincide in massive halos with $\rho \propto r^{-1}$84, and no core formation is expected above a “critical” mass $\rho \propto r^{-1}$85. At lower mass, dwarf galaxies should scatter between pure-cusp and full-core loci, and the density contrast between the two cases reaches $\rho \propto r^{-1}$86–$\rho \propto r^{-1}$87 dex only at radii $\rho \propto r^{-1}$88 with $\rho \propto r^{-1}$89 (Kaneda et al., 2024).

An energetics-based extension was developed by Shinozaki et al. and Kaneda et al., who compare the work needed to transform an NFW cusp into a Burkert core,

$\rho \propto r^{-1}$90

with the Type II supernova energy budget

$\rho \propto r^{-1}$91

where $\rho \propto r^{-1}$92 for a Chabrier IMF. Defining the energy-conversion efficiency

$\rho \propto r^{-1}$93

they infer from SPARC data that galaxies cluster around $\rho \propto r^{-1}$94, with a median $\rho \propto r^{-1}$95, mean $\rho \propto r^{-1}$96, and a simple fit $\rho \propto r^{-1}$97–$\rho \propto r^{-1}$98 with $\rho \propto r^{-1}$99 dex scatter. The same analysis defines a forbidden region in the $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$00 plane where $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$01 is insufficient for cusp removal: halos with masses $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$02 to $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$03 lie outside the forbidden region, while ultra-faint dwarfs below $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$04 and groups and clusters above $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$05 lie within it, consistent with inefficient core formation at the lowest and highest masses (Shinozaki et al., 20 Jan 2026).

The remaining uncertainties are observational and modeling as much as physical. Hayashi et al. note that beam smearing, non-circular motions, pressure support, and disequilibrium can artificially lower inferred $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$06, and that disk–halo degeneracy in $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$07 admits alternative low-$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$08 fits even in a Bayesian framework. The same study therefore concludes that observational limitations and modeling uncertainties still prevent a definitive conclusion (Hayashi et al., 29 Jul 2025). Review work reaches a similarly cautious position: no single mechanism yet provides a complete, guaranteed $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$09CDM solution across all scales (Popolo et al., 2022).

Future tests are correspondingly precise. Proposed observational advances include SKA rotation curves with $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$10 accuracy and sub-kpc resolution, high-precision stellar-kinematic surveys of dwarfs with Subaru-PFS and the Thirty-Meter Telescope, proper-motion constraints from Roman and Gaia in the $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$11–$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$12 regime, and sub-kpc or $$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$13–$$\alpha \equiv \left.\frac{d\ln\rho}{d\ln r}\right|_{r\to 0}.$$14 inner-profile measurements with ALMA, ngVLA, TMT, and Subaru-PFS where the predicted cusp–core differences are largest (Hayashi et al., 29 Jul 2025, Kaneda et al., 2024). In that sense, the cusp-to-core transition now functions not only as a small-scale structure problem, but also as a diagnostic of how baryons, halo assembly, and possibly dark-matter microphysics reshape the inner phase-space structure of halos.