Black Hole Threshold in Gravitational Collapse

• Black Hole Threshold is the critical boundary in phase space that separates initial conditions leading to black hole formation from those that disperse.
• It is characterized by a control parameter (e.g., p) and exhibits universal dynamics such as scaling laws, self-similarity, and log-periodic fine structure near criticality.
• This concept applies across diverse settings—gravitational collapse, primordial black hole formation, and ultrarelativistic collisions—providing practical insights for numerical relativity and astrophysical modeling.

Searching arXiv for recent and foundational papers on black hole thresholds across gravitational collapse and primordial black hole formation. Search query 1: "black hole threshold gravitational collapse primordial black hole formation compaction function" A black hole threshold is a critical boundary separating initial data that form a black hole from initial data that do not. In general relativity this notion appears in several technically distinct settings: as a codimension-one critical surface in the phase space of gravitational collapse, as a critical amplitude for primordial overdensities in an expanding universe, as a critical boost in ultrarelativistic collisions, and as a threshold of immediate merger in compact-binary dynamics. Across these contexts, the threshold is characterized by a control parameter $p$ with a critical value $p_c$ such that subcritical data disperse or scatter, while supercritical data form a horizon or merge promptly. Near threshold, the dynamics often exhibit universality, self-similarity, and scaling laws, although the specific threshold variable and its degree of universality depend on the matter model, gauge choice, and physical setting (0711.4620).

1. Threshold as a critical surface in gravitational collapse

In the dynamical-systems description of gravitational collapse, the black hole threshold is the codimension-one boundary in the space of initial data separating evolutions that form a black hole from evolutions that disperse. For a one-parameter family of initial data $z(p)$, there is typically a unique critical value $p_*$ such that $p<p_*$ is subcritical and $p>p_*$ is supercritical. The critical surface is an infinite-dimensional hypersurface of codimension one, and the critical solution lying on it acts as an attractor within that surface with exactly one unstable mode (0711.4620).

This structure yields the standard language of critical phenomena. Near $p_*$, solutions approach a universal intermediate attractor, lose detailed memory of the initial data, and depart along the single unstable direction. For continuously self-similar critical solutions, the linearized expansion

$$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$

implies a black-hole mass law

$$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$

where $\lambda_0$ is the unique growing eigenvalue. For discretely self-similar solutions, the same power law acquires a log-periodic fine structure. In scalar collapse, the echoing period is $p_c$0, and the critical exponent is $p_c$1 (0711.4620).

This formulation makes “threshold” more than a phenomenological boundary. It is a geometric object in phase space, and its associated critical solution explains universality, scaling, and the appearance of arbitrarily large curvatures from smooth data. A plausible implication is that the term “black hole threshold” should be understood as a family of related codimension-one transition phenomena rather than a single numerical constant.

2. Critical phenomena and threshold observables

The threshold is commonly probed by one-parameter tuning. In Choptuik-style collapse, one identifies $p_c$2 by bracketing black-hole-forming and dispersing runs, and the lifetime of the near-critical phase grows logarithmically as the threshold is approached. In the generic dynamical-systems picture this follows from the single unstable mode: $p_c$3 This logarithmic law is the characteristic type-I signature when the critical solution is stationary or periodic, while type-II behavior is associated with self-similarity and black-hole mass scaling to zero (0711.4620).

A related threshold appears in the immediate-merger problem for binary black holes. In the extreme-mass-ratio limit, the threshold between prompt merger and scattering is controlled by unstable circular geodesics, and the number of whirl orbits obeys

$p_c$4

with $p_c$5 often taken to be the impact parameter. When radiation reaction is included through the self-force approximation, the critical solution becomes an adiabatically evolving sequence of unstable circular orbits. In that model, almost all the initial energy and angular momentum are radiated on the critical solution, and even for infinite initial energy this occurs over a finite number of orbits,

$p_c$6

with $p_c$7 the mass ratio (Gundlach et al., 2012).

The observable associated with the threshold is therefore context dependent. It may be a black-hole mass, a lifetime, a number of orbits, a compaction function, or a Lorentz factor. What is common is the existence of a sharply defined separator between outcomes and an amplification of universal dynamics near that separator.

3. Primordial black hole thresholds in cosmology

In primordial black hole formation, the threshold is the critical amplitude of a cosmological perturbation at horizon re-entry separating collapse from dispersal. For a one-parameter family labeled by $p_c$8, one writes $p_c$9 for subcritical evolution and $z(p)$0 for supercritical evolution. In the radiation era, the background equation of state is $z(p)$1 with $z(p)$2, and the perturbations are large-amplitude, spherically symmetric overdensities that re-enter the Hubble radius from super-horizon scales (Kehagias et al., 2024).

Several threshold variables are used. One is the density contrast at horizon crossing in uniform Hubble slicing,

$z(p)$3

with the corresponding comoving-gauge amplitude used in simulations

$z(p)$4

For radiation, $z(p)$5, these become

$z(p)$6

which improved substantially on the older estimate $z(p)$7 and $z(p)$8 (Harada et al., 2013).

Another widely used variable is the compaction function. In spherical symmetry, using the Misner–Sharp mass $z(p)$9 and areal radius $p_*$0,

$p_*$1

This is geometrical and gauge invariant. In the primordial case it measures the excess mass-energy inside a radius relative to that radius, and $p_*$2 signals approach to horizon formation. Thresholds written directly in terms of the local maximum $p_*$3 are profile dependent, but this dependence can be reduced by averaging procedures and by characterizing the curvature of the profile at its maximum (Kehagias et al., 2024).

A further refinement concerns the time at which the threshold is evaluated. Starting from the shape of the primordial power spectrum on superhorizon scales, one may compute a superhorizon threshold $p_*$4, but fully relativistic simulations show that the physically relevant threshold at non-linear horizon crossing is larger by about a factor two. In the radiation era, the paper gives

$p_*$5

for the superhorizon extrapolated threshold and

$p_*$6

for the threshold at non-linear horizon crossing, with

$p_*$7

over a broad range of shapes (Musco et al., 2020).

These formulations show that the primordial threshold is not a single number independent of conventions. It is a critical amplitude whose numerical value depends on the chosen variable, gauge, and stage of evolution, although some formulations are markedly more universal than others.

4. Universality, profile dependence, and compaction-based formulations

A major development in primordial-black-hole threshold theory is the identification of variables in which the threshold is approximately universal. The central result is that the threshold becomes profile independent when expressed in terms of the volume-averaged compaction function evaluated at the radius $p_*$8 where the local compaction reaches its maximum. In a radiation-dominated universe,

$p_*$9

and this value is independent of the detailed shape of the initial perturbation within numerical error (Kehagias et al., 2024).

This universality was first established numerically and then explained analytically. In one formulation, the averaged compaction is

$p<p_*$0

and for radiation domination the threshold is

$p<p_*$1

The corresponding local threshold $p<p_*$2 remains profile dependent, but that dependence is largely controlled by the normalized curvature at the compaction maximum,

$p<p_*$3

This leads to an analytic threshold formula $p<p_*$4 that reproduces full numerical results to within about $p<p_*$5 for a wide range of profile families (Escrivà et al., 2019).

The deeper explanation is self-similarity. At exact criticality, the collapsing core is self-similar all the way to the center. Because both the compaction function and its volume average are invariant under the associated dilatations, the threshold may be evaluated in the $p<p_*$6 limit, where regularity and self-similarity fix the answer. For a radiation fluid this gives the universal averaged threshold $p<p_*$7 (Kehagias et al., 2024).

This near-universality is not unlimited. For general equations of state $p<p_*$8, a semi-analytic formula for $p<p_*$9 is accurate for $p>p_*$0, but for $p>p_*$1 the full shape of the compaction function becomes important and a single curvature parameter $p>p_*$2 is not sufficient (Escrivà et al., 2020). Likewise, nonspherical collapse changes the threshold far less than once feared: in fully three-dimensional simulations of spheroidal super-horizon perturbations in radiation, ellipticities $p>p_*$3 shift the threshold amplitude in $p>p_*$4 by only $p>p_*$5, so the effect is negligibly small for large amplitudes relevant to PBH formation (Yoo et al., 2020).

More recent work qualifies the universal-threshold picture further by introducing the geometry of the inner core. In an asymptotically flat radiation-dominated FRW universe, initial data can be classified by an open, closed, or flat FRW core surrounded by a shell of higher three-dimensional curvature. In that classification, the threshold is not solely determined by the behavior of the compaction function at its maximum. Closed cores lower the threshold, open cores raise it, and flat cores lie in between; the distinction becomes especially important for sharp profiles and for the separation between Type-I and Type-II PBH formation (Germani et al., 2 Oct 2025). This suggests that compaction-based universality is highly effective but not exhaustive.

5. Thresholds beyond the standard radiation-fluid model

The threshold depends sensitively on the matter model and on the underlying theory of gravity. In anisotropic radiation fluids with $p>p_*$6, the threshold is still defined through the average mass excess within the radius of maximum compaction,

$p>p_*$7

but the anisotropy modifies the perturbation shape and therefore the effective threshold. Positive anisotropy, typically $p>p_*$8, increases the shape parameter $p>p_*$9 and raises $p_*$0; negative anisotropy lowers it. In the local anisotropic model $p_*$1, moderate anisotropy can vary $p_*$2 by up to $p_*$3 relative to the isotropic case (Musco et al., 2021).

Quantum-gravity modifications can also shift the threshold. In loop quantum cosmology, using the effective Friedmann equation

$p_*$4

the paper estimates semi-analytically that for low-mass PBHs forming close to the quantum bounce the threshold $p_*$5 can be reduced by up to $p_*$6 compared to the general relativistic regime. The reduction is tied to the modified horizon dynamics near the bounce and depends on the critical density $p_*$7, hence on the Barbero–Immirzi parameter $p_*$8 (Papanikolaou, 2023).

Conversely, pressure stiffening can increase the threshold. Using the saturation equation of state

$p_*$9

with density-dependent sound speed

$$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$0

spherically symmetric collapse simulations show that the PBH formation threshold increases relative to the radiation-fluid reference. The reported values are

$$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$1

with

$$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$2

At the same time, the critical mass-scaling exponent remains

$$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$3

consistent with the radiation-fluid value within numerical precision. The paper interprets this as evidence that the lattice equation of state is only a mild perturbation of the radiation case over the near-critical density range, not as evidence for a universal $$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$4 independent of matter content (Bakhti, 28 Jun 2026).

These examples show that “threshold” is a theory-dependent and microphysics-dependent object. A plausible synthesis is that the threshold is best viewed as a response functional of the effective sound speed, anisotropy, and background expansion rather than as a purely geometric number.

6. Thresholds in ultrarelativistic collisions, null-geodesic criteria, and extremal limits

In asymptotically flat strong-field dynamics, black hole thresholds take other forms. In ultrarelativistic head-on collisions of equal-mass self-gravitating fluid stars with compaction $$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$5 and $$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$6 equation of state, the control parameter is the Lorentz factor $$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$7. Full numerical relativity gives a critical boost

$$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$8

with $$Z(x,\tau) \simeq Z_*(x) + \sum_i C_i(p)e^{\lambda_i\tau}Z_i(x)$$9 subcritical and $$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$0 supercritical. This is lower than a naive hoop-conjecture estimate $$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$1, so

$$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$2

Near threshold, two distinct apparent horizons first form and later merge. The paper attributes the lowered threshold to gravitational focusing: each boosted star acts as a lens on the other, focusing matter into two dense regions after overlap. The gravitational-wave luminosity reaches

$$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$3

and for $$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$4 the emitted gravitational-wave energy is

$$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$5

(East et al., 2012).

A closely related geometric criterion arises in primordial collapse through null geodesics. In a spherically symmetric radiation background, the threshold compaction for PBH formation is well approximated by the threshold for the appearance of the first unstable circular null orbit. The condition for a circular null geodesic can be written as

$$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$6

and the first such orbit appears with vanishing Lyapunov exponent $$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$7. When converted to comoving gauge, the resulting critical compaction agrees well with the standard PBH threshold across realistic profile shapes. The same paper further argues that, once a self-similar stage develops, the PBH critical exponent is set by the inverse Lyapunov coefficient of the unstable null orbit, $$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$8 (Ianniccari et al., 2024). This suggests a geometric bridge between collapse thresholds and the stability theory of light rings.

Threshold behavior can also terminate at extremal black holes. In the Einstein–Maxwell–Vlasov system for spherically symmetric charged collisionless matter, the threshold between dispersal and black-hole formation is controlled by two regimes. For sufficiently large particle angular momentum, the threshold solution is a stationary horizonless shell with a type-I-like logarithmic lifetime scaling

$$M_{\rm BH} \propto (p-p_*)^\gamma,\qquad \gamma=\frac{1}{\lambda_0},$$9

As the charge-to-mass ratio approaches unity from below, the instability timescale diverges as

$\lambda_0$0

with $\lambda_0$1 the surface gravity of the associated near-extremal Reissner–Nordström geometry. Beyond the critical point, the threshold solution becomes an extremal black hole with $\lambda_0$2, and the dispersal time scales as

$\lambda_0$3

for $\lambda_0$4 (East, 25 Nov 2025). This provides a distinct threshold structure in which the critical solution is itself a zero-surface-gravity black hole.

A different, kinematic use of the phrase appears in compact-star thermodynamics. For nonrotating stars, the black-hole limit corresponds to compactness $\lambda_0$5, while isotropic stars satisfy the Buchdahl bound $\lambda_0$6. In anisotropic models pushed toward $\lambda_0$7, the rescaled thermodynamic entropy

$\lambda_0$8

tends to the black-hole value $\lambda_0$9, implying an entropy–area scaling $p_c$00 near the threshold. This is not a dynamical collapse threshold, but it defines a limiting pre-horizon regime in which black-hole-like thermodynamic behavior emerges continuously (Alexander et al., 2018).

Across these settings, the black hole threshold is always a separator between qualitatively distinct outcomes, but the associated control parameter may be an overdensity, a compaction average, a shape parameter, a Lorentz boost, an impact parameter, a charge-to-mass ratio, or a compactness limit. The unifying structure is the existence of a sharply defined critical boundary and, near it, the emergence of simplified universal dynamics.