Gondolo-Silk Spikes: Dark Matter Overdensities at BH Centers

• Gondolo-Silk spikes are dark matter overdensities formed when a black hole grows adiabatically at the center of a halo, steepening the original density profile.
• Modifications such as finite seed masses, stellar cusps, non-circular orbits, and relativistic effects can soften or deplete the canonical spike, altering its slope.
• These spikes influence indirect dark matter detection and gravitational-wave signals by enhancing or suppressing annihilation and dynamical observables.

Gondolo-Silk spikes are dark-matter overdensities predicted to form when a black hole grows adiabatically at the center of a pre-existing halo. In the canonical construction introduced by Gondolo and Silk, an initial cusp is compressed into a steeper inner density law, generating potentially large enhancements of annihilation, decay, scattering, and dynamical signatures. Subsequent work has shown that the canonical spike is a limiting case: finite seed masses, stellar cusps, non-circular orbits, fully relativistic capture, and long-term stellar heating all modify either the formation or the survival of the spike, often driving it toward shallower or even depleted configurations (Herrera et al., 1 May 2026).

1. Canonical adiabatic spike construction

The original paradigm assumes an initial halo density

$\rho_{\chi,i}(r)=\rho_0\,(r/r_0)^{-\gamma},$

with $\gamma \simeq 1$ for an NFW cusp, a black hole that grows adiabatically, purely circular orbits so that the adiabatic invariant is $r\,M(<r)=\mathrm{const}$, and a zero-mass seed so that the enclosed mass initially comes only from dark matter. Under these assumptions, each dark-matter shell at radius $r_i$ is carried inward to a final radius $r_f$ according to

$r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$

and the inner profile becomes

$$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$

which gives $\gamma_{\rm sp}=7/3$ for $\gamma=1$ (Herrera et al., 1 May 2026).

Gondolo and Silk also fixed the normalization and spike radius by continuity with the original halo at $R_{\rm sp}$ and by mapping the full phase-space distribution through adiabatic growth. In the notation used in later work, the resulting profile may be written

$\gamma \simeq 1$0

with

$\gamma \simeq 1$1

and $\gamma \simeq 1$2 used as a relativistic capture factor close to the hole. A related canonical parametrization writes the spike outer radius as $\gamma \simeq 1$3, where the sphere of influence is $\gamma \simeq 1$4 (Sharpe et al., 30 Mar 2026).

In this form, a Gondolo-Silk spike is a specific adiabatic-compression solution rather than a generic prediction of black-hole growth. Its steepness follows directly from the assumed initial cusp and orbital structure.

2. Relativistic formulation and the inner cutoff

A fully general-relativistic treatment replaces Newtonian orbits plus ad hoc capture factors with exact Schwarzschild dynamics. In that framework, one begins with an isotropic, spherically symmetric phase-space distribution $\gamma \simeq 1$5 and grows the black hole slowly enough that the action variables are conserved. For a Schwarzschild black hole, the three classical action variables reduce to the radial action

$\gamma \simeq 1$6

together with $\gamma \simeq 1$7 and $\gamma \simeq 1$8, which imply that $\gamma \simeq 1$9 and $r\,M(<r)=\mathrm{const}$0 remain unchanged. The equality

$r\,M(<r)=\mathrm{const}$1

determines the mapping from the initial to the final energy (Sadeghian et al., 2013).

The density is then obtained from the mass current in curved spacetime, with the local rest-mass density measured by a static observer given by

$r\,M(<r)=\mathrm{const}$2

The integration domain is set by bound orbits, real radial motion, and black-hole capture. For marginally bound particles with $r\,M(<r)=\mathrm{const}$3, the critical angular momentum is $r\,M(<r)=\mathrm{const}$4 and the unstable circular orbit lies at $r\,M(<r)=\mathrm{const}$5. As $r\,M(<r)=\mathrm{const}$6, the integration volume shrinks to zero, so

$r\,M(<r)=\mathrm{const}$7

whereas the Gondolo-Silk Newtonian estimate vanishes at $r\,M(<r)=\mathrm{const}$8. The relativistic treatment leaves the asymptotic adiabatic slope unchanged for $r\,M(<r)=\mathrm{const}$9, but it moves the true inner cutoff to $r_i$0 and yields significantly higher densities very close to the black hole (Sadeghian et al., 2013).

This corrected near-horizon structure matters primarily for observables sensitive to the innermost phase space. It does not alter the canonical outer spike argument, but it does remove the commonly repeated Newtonian inner boundary at $r_i$1.

3. Generalized spike formation beyond the zero-seed, circular-orbit limit

The canonical construction can be generalized by allowing a finite seed mass $r_i$2, a stellar cusp

$r_i$3

and non-circular orbits. In the circular-orbit generalization, one defines the total initial mass

$r_i$4

and conserves the adiabatic invariant as

$r_i$5

The final profile is reconstructed from

$r_i$6

Sampling over plausible priors,

$r_i$7

shows that finite seeds and stars systematically soften the spike relative to the canonical $r_i$8 (Herrera et al., 1 May 2026).

A more self-consistent adiabatic treatment conserves the full radial action

$r_i$9

so that

$r_f$0

In practice, initial $r_f$1 are Monte Carlo sampled in the composite seed potential, $r_f$2 are computed, $r_f$3 is evaluated numerically, and the final $r_f$4 is obtained by solving $r_f$5. The new orbit is then populated by time-averaging. This gives similar but slightly steeper inner slopes than the circular-orbit mapping, because particles on radial orbits can penetrate more deeply (Herrera et al., 1 May 2026).

This suggests that the canonical $r_f$6 result is best interpreted as a special-case benchmark. Once baryons, finite initial black-hole masses, and non-circular orbital structure are included, the formation problem no longer selects a unique spike slope.

4. Stellar heating, Fokker-Planck evolution, and redshift dependence

After formation, a spike evolves through gravitational encounters with stars. In a Keplerian potential $r_f$7, one defines the isotropic density of states $r_f$8, its cumulative $r_f$9, and coupled energy-space Fokker-Planck equations for dark matter and stars:

$r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$0

The energy fluxes are

$r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$1

with

$r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$2

and

$r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$3

Because the dark-matter drift term is suppressed by $r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$4, the dark-matter evolution is diffusion-dominated. In steady state, stars recover the Bahcall-Wolf cusp $r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$5 and $r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$6, while dark matter მიდ tends to $r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$7 and $r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$8. The relaxation rate is governed by the stellar heating timescale

$r_i\,M_{\chi,i}(r_i)=r_f\,M_{\rm BH},$9

and a redshift-dependent stellar bath can be modeled by an effective step $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$0 that tracks the Madau-Dickinson star-formation history (Herrera et al., 1 May 2026).

Numerical integration with plausible priors gives a consistent evolutionary picture. Spikes formed at $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$1 begin with $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$2; by $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$3, stellar heating drives the inner cusp toward $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$4, largely independent of the initial $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$5 or $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$6. Averaging $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$7 within $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$8 yields a median evolution from $$\rho_\chi(r)\propto r^{-\gamma_{\rm sp}}, \qquad \gamma_{\rm sp}=\frac{9-2\gamma}{4-\gamma},$$9 at $\gamma_{\rm sp}=7/3$0 to $\gamma_{\rm sp}=7/3$1 by $\gamma_{\rm sp}=7/3$2, with the profile remaining near $\gamma_{\rm sp}=7/3$3 thereafter. The corresponding overdensities are two to four orders of magnitude below canonical expectations, but still well above an NFW-like cusp (Herrera et al., 1 May 2026).

A related conclusion emerges from 1D orbit-averaged Fokker-Planck simulations of isotropic nuclear star clusters with a multi-mass stellar population. There, mass segregation increases the central $\gamma_{\rm sp}=7/3$4 and cuts the two-body relaxation time by $\gamma_{\rm sp}=7/3$5 relative to single-mass models. Starting from $\gamma_{\rm sp}=7/3$6, the dark-matter slope flattens to $\gamma_{\rm sp}=7/3$7 within $\gamma_{\rm sp}=7/3$8 Gyr down to $\gamma_{\rm sp}=7/3$9, whereas in a single-mass model the spike persists over $\gamma=1$0 Gyr (Sharpe et al., 30 Mar 2026).

A common misconception is that the canonical spike, once formed, should remain intact over cosmic time. The Fokker-Planck results indicate instead that a quasi-steady-state cusp with $\gamma=1$1 is the robust late-time attractor in stellar environments.

5. Dependence on the dark-matter phase-space distribution

The adiabatic-growth mechanism is also sensitive to the microscopic form of the initial dark-matter distribution. In a fully relativistic fermionic construction, the halo is modeled as a self-gravitating system of massive fermions at finite temperature, using the Ruffini-Argüelles-Rueda model. In Schwarzschild-like coordinates, the spacetime satisfies the Einstein-Tolman-Oppenheimer-Volkoff system

$\gamma=1$2

with a fermion distribution

$\gamma=1$3

subject to the Tolman and Klein conditions

$\gamma=1$4

A black hole then grows adiabatically so that

$\gamma=1$5

and the final density follows from the orbit-mapped distribution function $\gamma=1$6 (Crespi et al., 2024).

In the dilute Boltzmannian limit, where $\gamma=1$7 and the initial distribution reduces to a Maxwell-Boltzmann law,

$\gamma=1$8

the spike in the regime $\gamma=1$9 becomes

$R_{\rm sp}$0

recovering a universal $R_{\rm sp}$1 slope independent of the fermion mass. By contrast, for semi-degenerate fermions with a dense compact core surrounded by a diluted halo, the spike generally does not develop a simple power-law profile. Its morphology depends on the fermion mass and the core degeneracy parameter, and the black hole may deplete rather than enhance the surrounding density. For $R_{\rm sp}$2, even modest-mass black holes with $R_{\rm sp}$3–$R_{\rm sp}$4 can deplete the surrounding density rather than build up a spike. More generally, for each $R_{\rm sp}$5 there is a critical black-hole mass $R_{\rm sp}$6 above which $R_{\rm sp}$7 at all $R_{\rm sp}$8 (Crespi et al., 2024).

This indicates that “Gondolo-Silk spike” is not a universal morphology. A plausible implication is that adiabatic growth should be understood as a response of a specific phase-space distribution to a changing central potential, not as a guarantee of a steep inner power law.

6. Observational signatures, conservative benchmarks, and current points of dispute

For indirect searches, the relevant linear and quadratic observables are the column density and the $R_{\rm sp}$9-factor. Using

$\gamma \simeq 1$00

with $\gamma \simeq 1$01, a spike enhances $\gamma \simeq 1$02 by $\gamma \simeq 1$03–$\gamma \simeq 1$04 and $\gamma \simeq 1$05 by $\gamma \simeq 1$06–$\gamma \simeq 1$07 relative to an uncontracted NFW cusp at $\gamma \simeq 1$08. By $\gamma \simeq 1$09, however, the median ratios have fallen substantially: $\gamma \simeq 1$10 decreases from $\gamma \simeq 1$11 to $\gamma \simeq 1$12, and $\gamma \simeq 1$13 from $\gamma \simeq 1$14 to $\gamma \simeq 1$15. Relative to the canonical Gondolo-Silk spike, $\gamma \simeq 1$16 drops from $\gamma \simeq 1$17 at $\gamma \simeq 1$18 to $\gamma \simeq 1$19 at $\gamma \simeq 1$20, while $\gamma \simeq 1$21 falls from $\gamma \simeq 1$22 to $\gamma \simeq 1$23. Annihilation signals, scaling as $\gamma \simeq 1$24, are therefore the most strongly affected: canonical GS models overpredict $\gamma \simeq 1$25 by $\gamma \simeq 1$26–$\gamma \simeq 1$27 orders of magnitude at $\gamma \simeq 1$28, whereas decay- or scattering-driven signals, scaling as $\gamma \simeq 1$29, differ by $\gamma \simeq 1$30–$\gamma \simeq 1$31 orders of magnitude (Herrera et al., 1 May 2026).

The same issue appears in gravitational-wave applications. For an undepleted spike, typical EMRI dephasings are quoted as $\gamma \simeq 1$32–$\gamma \simeq 1$33 rad over $\gamma \simeq 1$34 yr, and the leading-order scaling is $\gamma \simeq 1$35. Post-Newtonian three-body simulations of MBH-sBH-DM encounters show that inspiraling EMRIs can eject dark-matter particles through slingshots, producing irreversible depletion because collisionless dark matter cannot efficiently refill the depleted phase space. With realistic EMRI rates of $\gamma \simeq 1$36–$\gamma \simeq 1$37, inner spike densities are reduced by $\gamma \simeq 1$38–$\gamma \simeq 1$39, often pushing the residual dephasing below the LISA detectability threshold of $\gamma \simeq 1$40 rad. The viable parameter space for detectable collisionless spikes is then substantially restricted, with the strongest depletion occurring for lower-mass massive black holes and higher EMRI rates (Sharpe et al., 30 Mar 2026).

For stellar dynamics near galactic centers, the fully relativistic spike calculation implies that the gravitational effects of the spike are significantly smaller than the relativistic effects of the black hole itself, including frame dragging and quadrupolar effects, for stars close enough to test black-hole no-hair theorems. For the S2 star, the quoted dark-matter no-annihilation pericenter advance is $\gamma \simeq 1$41, compared with the Schwarzschild contribution $\gamma \simeq 1$42; for a tighter “no-hair” target star, the dark-matter contribution is $\gamma \simeq 1$43, compared with $\gamma \simeq 1$44, frame dragging $\gamma \simeq 1$45, and quadrupole precession $\gamma \simeq 1$46. Only if the dark matter is non-annihilating might the spike become marginally detectable for outer stars like S2 at the level of a few $\gamma \simeq 1$47 (Sadeghian et al., 2013).

The main current dispute is therefore not whether adiabatic growth can create a spike under idealized conditions, but which benchmark should be regarded as physically relevant. The canonical $\gamma \simeq 1$48 profile remains a useful reference solution. For realistic galactic nuclei, however, the literature increasingly favors either a redshift-evolved cusp near $\gamma \simeq 1$49 or, in some particle-physics and dynamical regimes, a substantially depleted inner distribution.