Black-Hole-Field Critical Mass Spectrum

• Black-hole-field critical mass spectrum is the set of discrete threshold values delineating transitions in stability, resonances, and bound-state behaviors near black holes.
• It is derived using analytic and numerical methods such as asymptotic matching, WKB, and quantum field theory techniques to solve master perturbation equations.
• Its implications are pivotal for astrophysical observations, primordial black hole formation, and constraining beyond-Standard-Model physics through spectroscopic analyses.

The black-hole-field critical mass spectrum is a set of discrete or threshold values in the mass parameter space of fundamental fields or field–black-hole systems that demarcate qualitative physical transitions in the behavior of linear or nonlinear field perturbations, stability, and resonance structures near or around black holes. This spectrum plays a central role in black-hole spectroscopy, the theory of superradiant instabilities, the phenomenology of primordial black holes, and the quantum-gravitational discretization of black-hole properties. Technically, these critical masses arise from exact or asymptotic solutions of the master equations governing field perturbations in black-hole backgrounds, matched to specific boundary conditions reflecting regularity at the horizon and bound-state behavior at spatial infinity or at singularities introduced by non-minimal couplings. The content below systematizes the theory, computational methodology, and phenomenological importance of the critical mass spectrum for black-hole-field systems across various classical and quantum frameworks.

1. Definitions and General Principles

A black-hole-field critical mass is a field or composite system parameter, typically of the form $\mu_n$ (field Compton mass) or a combination $M\mu_n$ (where $M$ is the black-hole mass), that marks a transition in the physical properties or spectral structure of the combined system. These transitions include:

• The threshold for stationary or marginally-stable bound states (“clouds”)
• The onset or cessation of instability (e.g., superradiant, monopole, or quasinormal mode instability)
• The point where quasi-bound or quasi-normal resonances switch from damped to infinitely long-lived
• Discrete spectra emerging from quantization or boundary conditions in quantum gravity

The spectrum can be discrete (indexed by overtone number $n$ or angular momentum $m,l$) or consist of critical thresholds (e.g., maximum/minimum allowed mass for resonance).

2. Analytic and Numerical Frameworks

Multiple analytic and computational approaches define and extract the critical mass spectrum:

Setting	Main Techniques	Key Eigenvalue Equation / Condition
Kerr/charged black holes + field	Asymptotic matching, WKB, gamma-function expansion	Discrete resonance condition from radial ODE matching
Schwarzschild + massive field	Frobenius/Hill-determinant, continued fraction, potential analysis	Quasinormal mode boundary, algebraic barrier/plateau condition
Non-minimal coupling/Proca mass	Interior-exterior split, Dirichlet at singularity, hypergeometric	Zeros of $_2F_1(-a,a;1;1)$, quantization of $\mu\,r_-$
PBH formation (early universe)	Critical collapse scaling, convolution with primordial spectrum	Power-law scaling $M_{\rm BH} = k M_H (\delta-\delta_c)^\gamma$
Quantum gravity (BTZ, 3D)	Quantum field theory in superposed geometry, detector response	Resonant condition $M_B/M_A = (m/n)^2$ for Bohr-Sommerfeld levels

The extraction of the spectrum involves either direct solution of transcendental eigenvalue equations or asymptotic expansions valid in extreme limits ($M\mu \gg 1$, ultra-relativistic or non-relativistic).

3. Stationary “Clouds” and Superradiant Critical Masses

The archetypal instance of the critical mass spectrum arises in massive scalar or vector fields minimally coupled to rotating (Kerr) or charged (Reissner–Nordström) black holes:

• In the large-mass and high-$m$ (azimuthal quantum number) limit, near-extremal Kerr holes support stationary bound-state solutions (“scalar clouds”) with analytic spectrum,

$$\mu_n = \sqrt{2}m\Omega_H \left[ 1 - \frac{\pi(\mathcal{R}+n)}{m|\ln\tau|} \right],$$

where $\Omega_H$ is the horizon angular velocity, $\tau$ is the dimensionless temperature, $\mathcal{R}$ a phase, and $n$ the radial overtone (Hod, 2016). This result is valid for $M\mu \gg 1$ and $1<\mu/(m\Omega_H)<\sqrt{2}$.

• In charged (Reissner–Nordström) spacetimes, the WKB quantization at large $l$ (eikonal regime) gives the critical mass as a root of coupled algebraic equations,

$$\bar\mu_{\rm crit}^2(\bar Q) = \frac{\bar r_{\rm crit}^2 - 3\bar r_{\rm crit} + 2\bar Q^2}{\bar r_{\rm crit}^3 - \bar Q^2\bar r_{\rm crit}^2}$$

where $\bar r_{\rm crit}$ satisfies a cubic in $\bar Q=Q/M$ (Hod, 13 Nov 2025).

• The spectrum is inherently discrete, and as $m\to\infty$ (rotational quantum number) the cloud masses saturate the interval $1 < \mu/(m\Omega_H) < \sqrt{2}$ with arbitrarily fine resolution.

4. Quasinormal Modes and Instability Thresholds

In Schwarzschild and other backgrounds, the spectrum of massive fields exhibits threshold masses at which qualitative changes in the spectrum occur. For massive scalar QNMs in Schwarzschild spacetime (Alves et al., 8 Sep 2025):

• $m_{\rm lim}$: Below this threshold the effective potential’s barrier peak lies above the asymptotic plateau; above, the roles reverse. Defined by setting the effective potential at the peak equal to $m^2$.
• $m_{\rm max}$: The potential barrier disappears as two critical points in the potential coalesce. This marks the end of the regime supporting proper QNMs, given by the vanishing of the discriminant of a cubic in the radial coordinate.
• $m_{zd}$: The “zero-damping mass” where the imaginary part of the lowest QNM frequency vanishes, associated with the emergence of infinitely long-lived (“quasi-BPS”-type) resonances. Remarkably, $m_{zd} > m_{\rm max}$; robust numerical treatments (Hill-determinant and Leaver’s continued fraction) confirm such long-lived modes persist even when the classical barrier is absent.

Critical thresholds control the regions in field parameter space relevant for black-hole spectroscopy and potential observational signatures.

5. Critical Collapse and Primordial Black Hole Mass Spectra

In the context of primordial black hole (PBH) formation, critical collapse phenomena lead to a continuous critical mass spectrum, modifying the classical notion of monochromatic, horizon-mass PBH formation.

• For Gaussian initial perturbations $\delta$ with threshold $\delta_c$, the PBH mass is

$M_{\rm BH}=k M_H (\delta-\delta_c)^\gamma,$

where $k$ is a dimensionless coefficient and $\gamma \simeq 0.36$ (radiation era) (Kuhnel et al., 2015, Gow et al., 2020).

• The resulting initial mass function (IMF) is broad, with a characteristic low-mass power-law tail $M^{1/\gamma-1}$ and a minimum width in $\ln M$, reflecting universality of critical scaling.
• Critical collapse leads to a systematic shift, broadening, and suppression of the PBH mass spectrum compared to the monochromatic approximation. This imposes a “minimum width” and removes the possibility of an arbitrarily localized mass spike.

Critical-mass considerations in this context are vital for accurate confrontation with observational limits on PBH abundance and for discriminating formation models.

6. Quantum and Non-Minimally-Coupled Theories: Discretization and Instabilities

In scenarios involving quantum gravitational effects or non-minimal couplings, black-hole-field critical mass spectra acquire different, often discrete, structures:

• In quantum gravity frameworks, such as BTZ black holes in superpositions of masses, operational measurements (Unruh–DeWitt detector response) reconstruct the Bekenstein–Hawking discrete area/mass spectrum:

$M_n = \left(\frac{n}{\ell}\right)^2,$

manifesting quantized resonance structure detectable via field-based probes whenever $\sqrt{M_B / M_A} = m/n \in \mathbb{Q}$ (Foo et al., 2021).

• For Schwarzschild black-holes coupled to non-minimally coupled Proca (vector) monopole fields, the marginally-stable resonance condition at the near-horizon pole yields a discrete spectrum:

$\mu_n r_- = \sqrt{3} n,$

where $r_-$ is the location of the potential pole set by the coupling parameter, and $\mu_c = \sqrt{3}/r_-$ is the lowest (critical) mass for instability onset (Hod, 24 Jun 2025).

• In critical extended gravities, the spectrum of spin-2 (graviton) perturbations demonstrates degenerate massless and logarithmic modes at the critical point $m^2 = 0$, and the stability is controlled by the allowed spectrum after truncating log modes (Liu et al., 2011).

7. Astrophysical Implications and Mass Exclusion Windows

The notion of a critical mass spectrum is tightly tied to astrophysical observations:

• Superradiant instabilities in rotating black holes exclude the existence of bosonic fields (scalars, vectors, tensors) in distinct mass windows, as measured from black-hole spin data across mass scales:

$$\text{For scalars: } 3.8 \times 10^{-14}\,\mathrm{eV} \leq \mu \leq 3.4 \times 10^{-11}\,\mathrm{eV},\;\dots$$

Similar bands exist for vector and tensor fields (Stott, 2020, Stott et al., 2018).

• Spin and mass distributions of astrophysical black holes inferred via microlensing require population-based critical mass estimates for distinguishing PBH from stellar-origin black holes, placing upper limits on allowed contributions to dark matter and constraining the location and width of mass “bumps” (Perkins et al., 2023).
• The observed spectrum serves as a powerful probe for phenomena such as the axiverse (multiple axion-like fields with characteristic mass distributions), fuzzy dark matter, and PBHs, as well as testing quantum and semiclassical conjectures regarding black hole mass quantization.

8. Significance and Open Directions

Understanding the black-hole-field critical mass spectrum underpins the following:

• Predictive models for the existence and observability of so-called “hairy” black holes, quasi-bound state resonances, and the onset of instabilities
• Reliable guidance for the exclusion of beyond-Standard-Model particles and fields via black-hole astrophysical observations
• The foundational structure of PBH cosmology, Bayesian inference of mass functions from survey data, and cross-disciplinary phenomenology
• Probing quantum gravity corrections and the emergence of discrete spectra in operationally accessible frameworks

Future directions include refining the correspondence between spectrum features and observable quantities (e.g., gravitational wave signals, BH spin distributions), extending analytic results to broader parameter regimes, and integrating quantum and semiclassical results for a complete account across physical scales.