Cusp-Core Transition Forbidden Region

• The cusp-core transition forbidden region characterizes the parameter space where dark matter halos are prevented from shifting from a cuspy NFW profile to a flattened core by specific energetic or dynamical limits.
• It integrates diverse mechanisms—baryonic feedback, ultra-light axion and scalar dark matter, PBH heating, and resonance effects—with thresholds defined by halo mass, feedback efficiency, and temporal dynamics.
• This framework constrains dark matter properties and feedback efficiencies, ruling out formation channels that fail to exceed the required energy barrier for a cuspy-to-cored transition.

The cusp-core transition forbidden region delineates regimes in physical or parameter space where the transformation of a cuspy dark matter density profile into a cored profile cannot occur, given a specified microphysical or astrophysical mechanism. This concept arises in diverse theoretical and numerical studies addressing the “core-cusp problem,” which is the tension between cuspy predictions from ΛCDM dark-matter-only simulations and observed shallow central density cores in many galaxies. The forbidden region is rigorously defined by the failure of the relevant dynamical or energetic criteria for core formation and is highly dependent on the mechanism invoked—whether baryonic feedback, dark matter microphysics, or dynamical heating by a subdominant component.

1. Fundamental Definitions and Physical Motivation

The “cusp-core transition” refers to the flattening of the central density cusp (characterized by an inner slope $\rho \propto r^{-\gamma}$ with $\gamma\sim1$ as in NFW profiles) to a nearly constant-density core ($\gamma\to0$). The forbidden region, in this context, is the set of system parameters (e.g., halo mass, dark matter particle mass, coupling constant, baryon content, feedback energy, or PBH fraction) for which core formation is disallowed by physical constraints specific to the proposed transformation mechanism.

For each mechanism—such as SN feedback, axion dark matter, PBH heating, or resonant potential oscillations—the forbidden region is sharply defined by the energetic, relaxation, or resonance conditions that must be satisfied for a core to form within a physically or observationally relevant timescale (Hayashi et al., 29 Jul 2025, Boldrini et al., 2019, Shinozaki et al., 20 Jan 2026, Marsh et al., 2015, Ogiya et al., 2012, Cline et al., 2020, Deng et al., 2018, Kaneda et al., 2024).

2. Forbidden Regions in Baryonic Feedback and Halo Mass Space

Baryon-driven core formation is governed primarily by the interplay between the star-formation- and supernova-driven energy input and the gravitational binding energy of the dark matter halo. The forbidden region in this context can be mapped in the $(M_{200}, M_*)$ or $(M_{200}, V_{\max})$ plane:

• For halo masses $M_{200}\lesssim10^8$–$10^{10}\ M_\odot$ (or $V_{\max}\lesssim 30$ km/s), galaxies lack sufficient stellar feedback energy to unbind or reconfigure the central dark matter, and thus remain cuspy.
• For halo masses $M_{200}\gtrsim10^{11}$–$10^{12}\ M_\odot$ ($V_{\max}\gtrsim 200$ km/s), even substantial feedback is unable to counter the gravitational potential well, so the cusp is retained.
• Only in the intermediate mass range $10^{10}\lesssim M_{200}/M_\odot\lesssim 10^{12}$ is the energy input capable of effecting a transition, resulting in cores (Kaneda et al., 2024, Hayashi et al., 29 Jul 2025, Shinozaki et al., 20 Jan 2026).

The criterion is that the total available SN II energy (modulated by an efficiency parameter, $\varepsilon\sim0.01$), integrated over all bursts, must exceed the dynamical energy gap $\Delta E$ needed for the NFW $\rightarrow$ cored (Burkert) transition: $$E_{\rm avail} = \varepsilon e_{51} f_{\rm SNII} M_*$$

$$\Delta E = [\psi_{NFW}(c_{200}) - \psi_{BKT}(M_{200})]\, M_{200}\,\phi_{200}$$

A forbidden region comprises all $(M_{200}, M_*)$ values for which $E_{\rm avail}<\Delta E$ (Shinozaki et al., 20 Jan 2026).

3. Cusp-core Transition Forbidden Regions in Dark Matter Microphysics

3.1 Ultra-light Axion (Fuzzy) Dark Matter

For solitonic core formation via ultra-light axion (ULA) dark matter, the forbidden region appears in axion mass parameter space:

• For $m_a > 1.1\times10^{-22}\ \mathrm{eV}$, core sizes in dwarf spheroidals are too small: galaxies remain cuspy.
• For $m_a \lesssim 1\times10^{-22}\ \mathrm{eV}$, cosmological structure formation (e.g., galaxy counts at high redshift) is strongly suppressed and in tension with data.

Thus, there is no allowed window in $m_a$ satisfying both kpc-size core formation in dwarfs and cosmological structure constraints. The region between these two bounds constitutes the axion “Catch-22 forbidden region” (Marsh et al., 2015).

3.2 Self-interacting and Light Scalar Dark Matter

A general scaling analysis for minimally coupled complex scalars or general polytrope-like models shows that the empirical relation $\rho_c\propto 1/R_c$ ($\beta\simeq1$) observed in galactic cores is not accessible to any stable ground state or virialized configuration. Instead, three branches are found:

• $\beta=4$: quantum pressure vs. gravity
• $\beta=2$: quantum pressure vs. self-interaction, or strong gravity
• $\beta\to\infty$: repulsive self-interaction vs. gravity

The $\beta\sim1$ scaling observed is not reproducible; the $(\rho_c, R_c)$ plane around $\rho_c\propto 1/R_c$ is “forbidden” for these models (Deng et al., 2018).

3.3 Dynamical Heating by Primordial Black Holes

For halos containing a subdominant population of PBHs, forbidden regions are found in the $(m_{\rm PBH}, f_{\rm PBH})$ plane:

• Cores form within a Hubble time only for $m_{\rm PBH}\in[25, 100]\,M_\odot$ and $f_{\rm PBH}\gtrsim0.01$.
• For $m_{\rm PBH}<25\,M_\odot$, or $$f_{\rm PBH}<f_{\rm min}(m_{\rm PBH})\propto 1/m_{\rm PBH}$$, core formation time exceeds galaxy lifetimes and the cusp is preserved.
• The forbidden region is the set $(m_{\rm PBH}, f_{\rm PBH})$ where $t_{\rm core}>\,T_{\rm age}$, i.e., below the critical $f_{\min}(m_{\rm PBH})$ curve for physically motivated $T_{\rm age}$ (Boldrini et al., 2019).

3.4 Oscillation-Activated Dark Matter Annihilation

In scenarios with late-time oscillation-induced DM annihilation (e.g., asymmetric DM with small DM-number violation), the forbidden region is defined as:

• Coupling $g'$ below the threshold for effective annihilation heating:

$$g' < 2\pi^{1/2} [m_\chi^3 \sqrt{G\rho_s} / F(r_m)]^{1/4}$$

where $F(r_m)$ encodes the mediator-mass dependence.

• Mediator-to-DM mass ratios $m_\phi/m_\chi$ above process-specific thresholds (e.g., $>0.94$ for vectors), suppressing phase space.
• Additional relic density, CMB, and $N_{\rm eff}$ limits further restrict the allowed region.
• N-body results confirm that in the forbidden region no core is produced and the density slope remains $\alpha\simeq1$ (cuspy) (Cline et al., 2020).

4. Resonant Baryonic Forcing: Timescale-Forbidden Bands

For resonance-driven core formation via oscillating baryonic potentials (e.g., SNe-driven starburst cycles), the forbidden region emerges in the space of forcing period $T$ vs. local dynamical time $t_d(r)$:

• Resonance criterion: maximum core growth when $T\approx t_d(r)$ at some $r$ (“resonance band”).
• For $T\ll t_d(r)$ (“fast forcing”) or $T\gg t_d(r)$ (“adiabatic forcing”) everywhere in the halo, no core is generated; these are forbidden regimes for resonance-driven cusp-flattening.
• Only galaxies where the episodic feedback period overlaps the dynamical time of the DM-dominated region can develop cores (Ogiya et al., 2012).

5. Empirical Manifestations and Observational Signatures

Systematic empirical analyses of surface densities and inner slopes across galaxy samples, such as in the SPARC database, show that:

• Late-type galaxies and classical dwarfs with $M_{200} \sim 10^{10}-10^{12}M_\odot$ display a diversity of inner slopes, including many with cored profiles ($\gamma \lesssim 0.5$), indicating allowed core formation.
• Ultra-faint dwarfs ($V_{\max}\lesssim 30$ km/s, $M_{200}\lesssim 10^9M_\odot$) and massive groups/clusters ($V_{\max}\gtrsim 200$ km/s, $M_{200}\gtrsim 10^{12}M_\odot$) remain cuspy ($\gamma \gtrsim 1$), coinciding with the predicted forbidden regions for feedback-driven transformations (Hayashi et al., 29 Jul 2025).

A comparable forbidden region appears in the space of stellar-to-halo mass ratios ($M_*/M_{\rm halo}$), where core formation is maximal for $x \sim 10^{-3} - 10^{-2}$ and forbidden outside this window.

6. Comparative Summary Table

The following table summarizes the principal forbidden regions as reported in the literature:

Mechanism/Model	Forbidden Region (Key Parameters)	Reference
Baryonic (SN Feedback)	$M_{200} \lesssim 10^{10}M_\odot$, $M_{200} \gtrsim 10^{12}M_\odot$	(Kaneda et al., 2024, Shinozaki et al., 20 Jan 2026, Hayashi et al., 29 Jul 2025)
PBH Dynamical Heating	$m_{\rm PBH}<25\,M_\odot$ or $f_{\rm PBH}<f_{\min}(m_{\rm PBH})$	(Boldrini et al., 2019)
ULA Axion Dark Matter (“Fuzzy DM”)	$m_a > 1.1\times10^{-22}$ eV or $m_a\lesssim1\times10^{-22}$ eV	(Marsh et al., 2015)
Light Scalar DM and Polytropes	$\beta\approx1$ band in $(R_c, \rho_c)$ (“empirical core scaling”)	(Deng et al., 2018)
DM Annihilation via Oscillation	$g'<g'_{\rm crit}(m_\chi,m_\phi)$ and $m_\phi/m_\chi >$ threshold	(Cline et al., 2020)
Resonant Forcing	$T \ll t_d(r)$ or $T \gg t_d(r)$ for all $r$ (no resonance possible)	(Ogiya et al., 2012)

7. Physical Interpretation and Implications for Galaxy Formation

The emergence of forbidden regions for the cusp-core transition is a generic prediction of both analytic and simulation-based treatments across a wide array of DM models and feedback prescriptions. Their existence serves as a critical model-selection criterion: any viable solution to the cusp-core problem must reproduce the observed boundaries of core formation in physical parameter space.

The forbidden region’s location provides direct constraints on the energy-coupling efficiency of baryonic feedback ($\varepsilon\sim 0.01$), the allowable mass range of Fuzzy DM, or the required fraction and mass of PBHs, among others. Systems detected to host cores outside the allowed zone would rule out the corresponding mechanism for core formation.

Empirical confirmation of the boundaries—through measurements of surface density, inner slope, or via dynamical timescales—directly tests the predictions of simulation and theory, offering a powerful probe of both DM microphysics and the role of baryons in galaxy evolution.

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References:

(Boldrini et al., 2019, Shinozaki et al., 20 Jan 2026, Marsh et al., 2015, Deng et al., 2018, Cline et al., 2020, Kaneda et al., 2024, Hayashi et al., 29 Jul 2025, Ogiya et al., 2012)