Black Hole Formation Threshold

• Black hole formation threshold is defined by a critical density contrast or compaction function required for gravitational collapse amidst opposing forces like pressure or rotation.
• It encompasses analytical criteria—such as the Jeans and sound-crossing conditions—and is sensitive to perturbation shape, non-sphericity, and dynamical variables in both cosmological and ultrarelativistic contexts.
• Accurate threshold determination is crucial as small parameter shifts can exponentially affect primordial black hole production rates and serve as probes for modified gravity theories.

The black hole formation threshold is the quantitative criterion delineating the conditions under which a self-gravitating system or cosmological perturbation will undergo gravitational collapse to form a black hole, rather than dispersing or stabilizing under opposing forces such as pressure gradients or velocity dispersion. This concept is central to theoretical and numerical studies of gravitational collapse in both idealized vacuum collisions and cosmological settings—particularly in the context of primordial black hole (PBH) formation. Its precise formulation and the mechanisms that set its value are highly sensitive to the physical context (ultrarelativistic collision, cosmological background, fluid properties), the nature of the perturbation (sphericity, rotation, anisotropy, inhomogeneity), and the underlying gravitational theory.

1. Fundamental Principles and Physical Context

The threshold for black hole formation is often expressed as a critical value of a dimensionless parameter (typically a density contrast, compaction function, or curvature variable) associated with the initial state of the collapsing region.

• Ultrarelativistic Collisions: In head-on collisions of boosted, self-gravitating objects, the threshold is set by a combination of the total energy and the effective spatial extent after gravitational focusing. For instance, in numerical studies of head-on collisions of equal-mass fluid spheres at boosts with Lorentz factor $\gamma$ such that 88–92% of the spacetime energy is kinetic, black holes form when $\gamma > \gamma_c$, with $\gamma_c \approx 8.5 \pm 0.5$ (East et al., 2012). This is significantly lower than the naive "hoop conjecture" estimate, which simply compares the combined energy within the Schwarzschild radius, reflecting the profound impact of dynamical focusing.
• Cosmological Perturbations and PBH Formation: For primordial overdensities, the threshold relates to the amplitude of a superhorizon-scale density (or curvature) perturbation upon horizon reentry. Analytic and numerical studies show that this threshold typically lies between $0.4$ and $0.6$ for a radiation equation of state ($w=1/3$), but the precise value and its gauge—comoving, uniform Hubble, or averaged compaction—affect the conversion to observable PBH abundance (Harada et al., 2013, Harada, 2016, Escrivà et al., 2019, Kehagias et al., 8 May 2024).

2. Analytical Formulations and Physical Mechanisms

The physical basis for the threshold is the competition between self-gravity (favoring collapse) and opposing effects such as pressure gradients, rotation, or velocity dispersion.

• Jeans/Sound-Crossing Criterion: For a perfect fluid with $p = w\rho c^2$, gravitational collapse requires that the collapsing region's free-fall time is less than the sound-crossing time. The critical density excess in the uniform Hubble slice at horizon entry is given by (Harada et al., 2013, Harada, 2016):

$$\delta^{UH}_{Hc} = \sin^2\left( \frac{\pi \sqrt{w}}{1+3w} \right)$$

For radiation ($w=1/3$), this yields $\delta^{UH}_{Hc} \approx 0.62$ and $\tilde\delta_c \approx 0.41$ in the comoving (simulation) gauge.

• Compaction Function/Curvature Approach: The compaction function $\mathcal{C}(r)$ encodes the excess Misner–Sharp mass per areal radius above background. For spherically symmetric super-horizon fluctuations,

$$\mathcal{C}(r) \equiv \frac{2 [M(r) - M_b(r)]}{R(r)}$$

The maximum value of the volume-averaged compaction function within the shell is a universal diagnostic: collapse occurs when $\overline{\mathcal{C}}(r_m) \geq 0.4$ for radiation (Escrivà et al., 2019, Kehagias et al., 8 May 2024).

• Effect of Perturbation Shape and Profile: In the case of shell-like or highly inhomogeneous perturbations, both the value at the compaction peak and the profile's second derivative at that point can enter analytic threshold estimates (Escrivà et al., 2020, Escrivà et al., 2019). Sharper transitions (steeper boundaries) increase the threshold, while broader profiles reduce it.

3. Effects of Inhomogeneity, Non-Sphericity, Rotation, and Anisotropy

Realistic scenarios often depart from perfect sphericity, homogeneity, or zero angular momentum, all of which can impact the threshold for collapse.

• Inhomogeneity and Velocity Dispersion: In matter-dominated universes, the random motions generated by small-scale substructure effectively limit collapse by generating velocity dispersion. The critical density perturbation at horizon entry must satisfy $\tilde\delta_{th}\propto \sigma_0^{2/5}$ for small $\sigma_0$ (where $\sigma_0$ is the rms amplitude on the PBH scale), reflecting the heightened role of velocity dispersion in halting collapse (Harada et al., 2022). When the distribution over perturbation shapes is accounted for, the effective threshold for cosmological PBH abundance is further increased, $\zeta_{th} \sim \zeta_{rms}^{1/10}$ (Ebrahimian et al., 24 Jul 2025).
• Non-Spherical Perturbations: For deviations parameterized by ellipticity $e$ and prolateness $p$ (as in peak theory), the threshold amplitude for collapse increases as $(e, p)$ depart from zero. For amplitudes extremely close to the spherical threshold, even small nonsphericity can prevent collapse, following a power law $e_c \propto (\mu - \mu_{c,sp})^\gamma$ (Escrivà et al., 4 Oct 2024). However, for high peaks (large $\nu$), the vast majority of configurations remain effectively spherical, and the overall PBH production rate is only mildly sensitive to non-sphericity (Yoo et al., 2020, Escrivà et al., 4 Oct 2024).
• Rotation: For initial overdensities with nonzero angular momentum, the threshold density contrast for collapse increases with the square of the dimensionless spin parameter $a_K = J/(GM^2)$. In the radiation era, $\delta_{Hc}^{UH}\simeq 0.62 + 0.015 a_K^2$ (He et al., 2019). This indicates that rotation hinders collapse due to centrifugal support, a correction that is robust but requires numerical relativity for precise calibration in the nonlinear regime.
• Anisotropic Fluids: When the initial conditions are set by an anisotropic equation of state (non-equal radial and tangential pressure), the effective profile shape is modified, and the threshold can vary up to $\sim$25% depending on the anisotropy parameter (Musco et al., 2021).

4. Universality, Profile Independence, and the Role of Critical Phenomena

A salient result of recent research is the universality of the collapse threshold when expressed in terms of an invariant, profile-averaged compaction function, a property ultimately rooted in the emergence of critical (self-similar) behavior at the threshold.

• Volume-Averaged Threshold: For a vast class of spherically symmetric perturbations, the critical value for collapse is $\overline{\mathcal{C}}_c = 2/5$ in the radiation era, regardless of the detailed initial profile (Escrivà et al., 2019, Kehagias et al., 8 May 2024). This arises because, at criticality, variables such as the compaction function become functions only of the self-similar variable $z = r/(-t)$, and the dynamics are scale-invariant.
• Null Geodesic Correspondence: The appearance of an unstable circular null geodesic at the center of the collapsing region is tightly correlated with the threshold: the threshold amplitude matches the condition for the formation of the first such orbit, with mass scaling law exponent $\gamma = 1/\lambda$, where $\lambda$ is the Lyapunov exponent of the unstable orbit (Ianniccari et al., 3 Apr 2024).
• Type-I and Type-II Thresholds: The three-dimensional curvature at the perturbation "core" can distinguish classes of initial conditions: closed (type-C), flat (type-F), or open (type-O) FRW cores. The lowest collapse threshold arises for type-C (closed core) profiles, whereas open or flat cases require higher amplitudes (Germani et al., 2 Oct 2025). The structure of the power spectrum affects the statistical prevalence of each type.

5. Time-Dependent Equations of State and Modified Gravity

The threshold depends not only on the initial configuration but also on the governing dynamical framework.

• Time-Dependent Equation of State: During cosmological transitions (e.g., QCD epoch), the equation-of-state parameter $w$ and the sound speed $c_s^2$ can vary rapidly. This temporal variation must be incorporated in the threshold calculation: the effective sound speed modifies the sound-crossing time and can lower the threshold during epochs of "softening," enhancing PBH formation (Papanikolaou, 2022).
• Modified Gravity: If the cosmological background is governed by theories such as Eddington-inspired Born–Infeld gravity (EiBI), the threshold depends explicitly on background energy scale and the sign of the new coupling: a positive Born–Infeld parameter raises the threshold (suppressing PBH formation), while a negative value lowers it (Chen, 2019). This suggests that observational constraints on PBH abundance could provide probes of strong-field gravity beyond general relativity.

6. Observational and Cosmological Implications

Precise knowledge of the collapse threshold is crucial for translating primordial fluctuation statistics into PBH abundances, connecting theoretical predictions to constraints from gravitational wave observations, lensing, and cosmic background radiation.

• Exponentially Sensitive Abundance: PBH abundance typically scales as $\exp[-\delta_c^2/2\sigma^2]$, so even $\mathcal{O}(10\%)$ changes in $\delta_c$ result in orders-of-magnitude differences in expected PBH production (Harada et al., 2013, Escrivà et al., 2020). Profile-dependent, non-linear, and non-Gaussian corrections are therefore essential for accurate modeling.
• Spin and Kinematic Properties: PBHs formed in matter-dominated eras acquire slightly higher but still small spins, with typical $a_{rms} \sim \zeta_{rms}^{7/4} \ll 1$ (Ebrahimian et al., 24 Jul 2025). In radiation domination, the scaling is different due to shorter tidal torque timescales.
• Impact of Power Spectrum Shape: The width and sharpness of the primordial power spectrum strongly influence the typology of initial conditions (core-dominated versus shell-dominated), the likelihood of non-sphericity, and the mass spectrum of PBHs, with broad spectra tending to favor closed-core Type-I black holes and potentially explaining features in low-frequency gravitational wave backgrounds (Germani et al., 2 Oct 2025).

7. Summary Table: Collapse Conditions and Threshold Formulas

Physical Scenario	Threshold Criterion	Reference(s)
Ultrarelativistic fluid collision	$\gamma_c \approx 8.5$ ($\gamma_h \sim 20$)	(East et al., 2012)
Radiation-dominated PBH (uniform Hubble slice)	$$\delta^{UH}_{Hc} = \sin^2\left( \frac{\pi \sqrt{w}}{1+3w} \right)$$	(Harada et al., 2013)
Radiation-dominated PBH (comoving gauge)	$$\tilde\delta_c = \frac{3(1+w)}{5+3w} \sin^2\left( \frac{\pi \sqrt{w}}{1+3w} \right)$$	(Harada et al., 2013, Harada, 2016)
Universal compaction function average	$\overline{\mathcal{C}}_c = 0.4$ (radiation)	(Escrivà et al., 2019, Kehagias et al., 8 May 2024)
Rotating PBHs	$\delta_{Hc}^{UH} \simeq 0.62 + 0.015\, a_K^2$	(He et al., 2019)
Inhomogeneous, matter era	$\tilde\delta_{th}\propto \sigma_0^{2/5}$; Shape-averaged $\zeta_{th}\sim\zeta_{rms}^{1/10}$	(Harada et al., 2022, Ebrahimian et al., 24 Jul 2025)
Anisotropic fluid	$\delta_c(\lambda)$ varies up/down by up to $\sim25\%$	(Musco et al., 2021)

8. Outlook and Future Directions

Ongoing progress is aimed at clarifying the influence of higher-order non-linearities, non-Gaussian initial statistics, the impact of full 3D (non-spherical) collapse, and the effects of additional matter fields or modified gravity on the threshold. The interplay between analytic results—grounded in self-similarity and critical phenomena—and high-resolution numerical simulations continues to refine the predictive power of PBH abundance calculations and ashes out phenomenological signatures in both gravitational wave astronomy and early-universe cosmology.

In conclusion, the black hole formation threshold is a multi-faceted, context-dependent concept whose precise determination is central to gravitational collapse theory and PBH phenomenology. The threshold value encapsulates the complex interplay of gravitational focusing, pressure support, profile structure, non-linearities, and critical phenomena, with profound implications for both fundamental theory and observational cosmology.