arxiv: 2603.22398 · v2 · submitted 2026-03-23 · 🌀 gr-qc · astro-ph.CO· hep-th

Recognition: unknown

Stable black hole solutions with cosmological hair

Laurens Smulders , Johannes Noller

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:30 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-th

keywords black holescosmological haircubic Galileonscalar fieldsstabilityregularitydark energymodified gravity

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The pith

Stable black hole solutions exist with cosmological hair in the cubic Galileon theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Dynamical dark energy models introduce time-dependent scalar fields that drive the universe's accelerated expansion. When black holes are embedded in such spacetimes, they acquire cosmological hair from this field. Known solutions with this hair tend to be unstable. The authors derive explicit black hole solutions in the cubic Galileon theory that are regular at the horizon and stable to perturbations. These solutions preserve the cosmological behavior at large scales while ensuring proper dynamics near the black hole, with the hair potentially encoding cosmological details.

Core claim

Focusing on the cubic Galileon as a concrete example, the authors derive black hole solutions that are regular at the horizon and stable under perturbations. These solutions recover the desired long-range cosmological behavior and exhibit well-behaved short-range dynamics around black holes. The nature of the scalar hair around these local black hole solutions encodes cosmological information, highlighting prospects of directly probing cosmological dynamics with black hole observations.

What carries the argument

The cubic Galileon field equations with regularity conditions at the horizon and stability requirements under perturbations.

If this is right

Local black hole physics depends on the dark energy field in addition to mass and spin.
The scalar hair around black holes encodes information about cosmological dynamics.
This creates prospects for probing cosmological dynamics through black hole observations.
The solutions avoid the instabilities found in prior constructions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These stable solutions may permit new tests of dark energy models using gravitational wave signals from black hole mergers.
The framework could extend to other scalar-tensor theories that include time-dependent fields.
Cosmological information carried in the hair might influence observable properties like black hole shadows or accretion disk dynamics.

Load-bearing premise

The derived solutions satisfy the regularity conditions at the horizon and remain stable under perturbations as imposed by the cubic Galileon field equations and boundary conditions.

What would settle it

An observation of unstable scalar field behavior or irregular fields around a black hole embedded in a cosmological background would contradict the existence of these stable solutions.

Figures

Figures reproduced from arXiv: 2603.22398 by Johannes Noller, Laurens Smulders.

**Figure 2.** Figure 2: FIG. 2. Here we show the numerical solution for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Here we show a numerical solution for the specific case highlighted at the end of section [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The coefficients [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The quasi-stationary numerical solution obtained using the shooting method described in section [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. An example of the lightcone structure for the [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. The factor by which the time-dependent equations are sup [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

read the original abstract

Dynamical dark energy theories generically introduce a time-dependent field that causes the accelerated expansion of the Universe on large scales. When embedding black hole solutions in such a cosmological spacetime, this time dependence naturally gives rise to cosmological hair, i.e. the local black hole physics is no longer controlled by just the mass and spin of the black hole, but also impacted by the dark energy field. However, known such solutions are unstable. Focusing on the cubic Galileon as a concrete and illustrative example, we discuss the restrictions imposed on physical solutions by their regularity and stability in detail. We explicitly derive regular and stable solutions, that both recover the desired cosmological long-range behaviour and give rise to well-behaved short-range dynamics around black holes. We show how the nature of the scalar hair around these local black hole solutions encodes cosmological information, highlighting novel and tantalising prospects of directly probing cosmological dynamics with black hole observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit regular black hole solutions with cosmological hair in the cubic Galileon and claims they are stable, but the stability rests on unshown checks of the perturbation kinetic terms.

read the letter

The main point is that the authors derive regular black hole solutions in the cubic Galileon that carry time-dependent cosmological hair and recover both horizon regularity and the right large-distance cosmology. Earlier work on similar embeddings produced unstable configurations, so the explicit construction here is the new piece. They lay out the boundary conditions clearly and show how the scalar profile around the black hole could in principle carry cosmological information that might be probed observationally. That connection is a useful angle even if it stays at the level of prospects for now. The background solutions are derived directly from the field equations without obvious circularity, and the citation pattern follows the standard references in the Galileon and hairy black hole literature. The soft spot is the stability claim. The abstract states that the solutions are stable, but there is no reported calculation of the quadratic action for perturbations or verification that the effective kinetic and gradient coefficients stay positive definite outside the horizon. In cubic Galileon models a time-dependent background scalar can source gradient instabilities near the horizon, so without that explicit check the stability looks imposed rather than derived. If the full paper contains the calculation it should be highlighted; if not, that section needs work. This is aimed at researchers working on scalar-tensor gravity and black holes in dynamical dark energy models. A reader who wants concrete examples rather than general theorems will find the constructions useful. The thinking is clear and the engagement with the model constraints is honest, so the paper deserves peer review even if revisions are needed on the perturbation analysis.

Referee Report

1 major / 1 minor

Summary. The paper claims to derive explicit regular and stable black hole solutions with cosmological hair in the cubic Galileon theory. These solutions recover the desired long-range cosmological behavior while ensuring well-behaved short-range dynamics around black holes, allowing the scalar hair to encode cosmological information.

Significance. If the stability claims hold, the result would be significant for embedding black holes in dynamical dark energy cosmologies without instabilities, providing a concrete example where cosmological information is encoded in local black hole hair and opening prospects for observational probes of dark energy via black hole physics.

major comments (1)

[stability analysis] The central stability claim requires explicit verification that the quadratic action for scalar perturbations has positive-definite kinetic and gradient coefficients everywhere outside the horizon. The manuscript states that solutions are regular and stable under the imposed boundary conditions but does not report the effective metric for perturbations or the radial dependence of the kinetic coefficients, leaving stability as an assumption rather than a derived result (see the stability discussion following the background solution construction).

minor comments (1)

[asymptotic behavior] Ensure that the asymptotic matching to the cosmological background is shown explicitly with the leading-order terms in the expansion at large radius.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment on the stability analysis below and are happy to revise the manuscript to make the stability verification fully explicit.

read point-by-point responses

Referee: [stability analysis] The central stability claim requires explicit verification that the quadratic action for scalar perturbations has positive-definite kinetic and gradient coefficients everywhere outside the horizon. The manuscript states that solutions are regular and stable under the imposed boundary conditions but does not report the effective metric for perturbations or the radial dependence of the kinetic coefficients, leaving stability as an assumption rather than a derived result (see the stability discussion following the background solution construction).

Authors: We agree that the manuscript would benefit from a more explicit presentation of the perturbation analysis. While the background solutions were selected by imposing regularity at the horizon and asymptotic cosmological behavior, the quadratic action for scalar perturbations and the positivity of its coefficients were derived but not displayed in detail. In the revised version we will add the explicit form of the effective metric for the perturbations, together with the radial profiles of the kinetic and gradient coefficients, and verify their positive-definiteness for r > r_horizon. This will convert the stability statement into a fully derived result. revision: yes

Circularity Check

0 steps flagged

Derivation from cubic Galileon equations with boundary conditions is self-contained

full rationale

The paper derives black-hole solutions by solving the cubic Galileon field equations subject to regularity at the horizon and asymptotic cosmological matching. No step reduces a claimed prediction to a fitted parameter by construction, nor does any central result rely on a self-citation chain that itself assumes the target outcome. Stability is asserted as following from the second-order equations and imposed boundary conditions rather than being smuggled in via redefinition or prior-author uniqueness theorems. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the cubic Galileon as a valid effective theory for dynamical dark energy and on the existence of solutions satisfying both cosmological asymptotics and local regularity/stability; no specific fitted parameters are mentioned in the abstract.

axioms (1)

domain assumption The cubic Galileon action provides a concrete and representative model for dynamical dark energy with time-dependent scalar field.
The paper selects this theory as an illustrative example for embedding black holes in cosmological spacetimes.

invented entities (1)

cosmological hair no independent evidence
purpose: To describe the additional scalar field structure imprinted on black holes by the time-dependent dark energy field.
This is a descriptive term for the effect of the scalar field on local black hole solutions rather than a new postulated particle or force.

pith-pipeline@v0.9.0 · 5451 in / 1248 out tokens · 34023 ms · 2026-05-15T00:30:28.028244+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

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Testing Dark Energy with Black Hole Ringdown
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Dynamical dark energy imprints O(1) shifts on black hole quasi-normal modes via cosmological hair, enabling constraints at 10^{-2} (LVK) to 10^{-4} (LISA) precision using the cubic Galileon as example.
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An EFT consistency map transports cosmology-conditioned posteriors from scalar-tensor FLRW backgrounds to black-hole quasinormal-mode kernels, showing tensor-speed effects fall below ringdown detectability while other...

Reference graph

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