On cosmological properties of black-hole hair in linearly coupled scalar-Gauss-Bonnet theory

Dar\'io Jaramillo-Garrido; Dra\v{z}en Glavan

arxiv: 2605.14132 · v2 · pith:QMBVFRGBnew · submitted 2026-05-13 · 🌀 gr-qc

On cosmological properties of black-hole hair in linearly coupled scalar-Gauss-Bonnet theory

Dra\v{z}en Glavan , Dar\'io Jaramillo-Garrido This is my paper

Pith reviewed 2026-05-15 01:43 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black hole hairscalar-Gauss-Bonnet theoryde Sitter spacetimesuperhorizon scalestest-field approximationmassless scalar fieldenergy fluxshift symmetry

0 comments

The pith

Scalar hair around black holes grows both temporally and spatially on superhorizon scales in de Sitter spacetime.

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

The paper examines the superhorizon behavior of scalar hair sourced by black holes in de Sitter spacetime within linearly coupled shift-symmetric scalar-Gauss-Bonnet theory. Working in the test-field regime where the scalar does not back-react on the metric, the hair shows both temporal and spatial growth beyond the horizon. This growth arises from the standard dynamics of a minimally coupled massless scalar field in expanding de Sitter spacetime, not from any special property of the black hole. The same growth appears around a point scalar charge, which shows that a scalarized black hole functions as a localized subhorizon source of scalar perturbations. The hair produces a steady outward energy flux that makes the test-field regime transient and accounts for the difficulty of finding static hairy solutions.

Core claim

We show that this hair exhibits both temporal and spatial growth on superhorizon scales. This growth is not a special consequence of the black hole, but instead follows from the dynamics of a minimally coupled massless scalar field in expanding de Sitter spacetime. Moreover, it is not even specific to black holes, but also arises for a point scalar charge in de Sitter, indicating that a scalarized black hole acts effectively as a localized subhorizon source of scalar perturbations. Backreaction, when important, first arises on subhorizon scales and does not by itself eliminate the superhorizon profile. The time-dependent scalar hair also carries a steady outward energy flux, which frames the

What carries the argument

Superhorizon evolution of massless scalar perturbations in de Sitter spacetime sourced by a localized object such as a black hole or point charge.

If this is right

The growth pattern is identical for any localized scalar source in de Sitter and is not tied to black-hole structure.
Back-reaction effects appear first on subhorizon scales and leave the superhorizon profile intact.
The outward energy flux renders the hair time-dependent, so the test-field regime is necessarily transient.
Difficulties in constructing self-consistent static hairy solutions follow directly from the presence of this flux.

Where Pith is reading between the lines

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

The transient character of the hair suggests that scalarized black holes in realistic cosmologies may contribute to scalar perturbations at early times before back-reaction becomes dominant.
The result implies that attempts to find static black-hole hair in de Sitter must fail once the cosmological expansion is properly included.
The same growth mechanism may appear in other shift-symmetric scalar-tensor theories with linear couplings, broadening the class of models where static hair is cosmologically disallowed.

Load-bearing premise

The test-field regime in which back-reaction of the scalar hair on the metric can be neglected remains valid long enough for the superhorizon growth to develop.

What would settle it

A calculation of the exact scalar field solution for a point charge in de Sitter spacetime that shows no temporal or spatial growth on superhorizon scales would falsify the claim that the growth is a generic feature of massless scalar dynamics.

read the original abstract

We investigate the superhorizon behavior of scalar hair sourced by black holes in de Sitter spacetime in the linearly coupled shift-symmetric scalar-Gauss-Bonnet theory. Working in the test-field regime, we show that this hair exhibits both temporal and spatial growth on superhorizon scales. This growth is not a special consequence of the black hole, but instead follows from the dynamics of a minimally coupled massless scalar field in expanding de Sitter spacetime. Moreover, it is not even specific to black holes, but also arises for a point scalar charge in de Sitter, indicating that a scalarized black hole acts effectively as a localized subhorizon source of scalar perturbations. Backreaction, when important, first arises on subhorizon scales and does not by itself eliminate the superhorizon profile. The time-dependent scalar hair also carries a steady outward energy flux, which frames the test-field regime as a transient, and helps explain the difficulties encountered in attempts to construct self-consistent static solutions.

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

Scalar hair growth on superhorizon scales here is just the standard massless scalar dynamics in de Sitter, with the black hole acting only as a subhorizon source, though the test-field regime's lifetime needs explicit checking.

read the letter

The main takeaway is that scalar hair around black holes in this linearly coupled shift-symmetric scalar-Gauss-Bonnet theory grows temporally and spatially on superhorizon scales in de Sitter, but the growth is not special to the black hole or the Gauss-Bonnet term. It follows from the usual wave equation for a massless minimally coupled scalar in expanding de Sitter spacetime. The same growth appears for a point scalar charge, so the black hole functions essentially as a localized subhorizon source of perturbations. Backreaction, when it matters, starts on subhorizon scales and leaves the superhorizon profile intact, while the outward energy flux carried by the time-dependent hair makes the test-field approximation transient. This framing also accounts for the difficulty in finding static solutions. The paper does a clean job of separating the generic de Sitter scalar behavior from the specific hair construction and of using the point-charge analogy to make the source role explicit. The logic from the test-field equations to the growth and flux conclusions is coherent on its own terms. The soft spot is the lack of a direct timescale comparison. The abstract notes that the flux renders the regime transient and that backreaction does not erase the superhorizon part, but without numbers showing that the growth rate outpaces the onset of order-one metric corrections over a Hubble time, it remains unclear whether the reported growth is actually realized inside the stated approximation. The assumption that linear coupling plus shift symmetry captures the leading cosmological behavior is therefore load-bearing. This paper is for people working on scalar-tensor theories and no-hair results in cosmological backgrounds. A reader who wants a clear separation between generic de Sitter effects and black-hole-specific hair will find it useful. I would send it to peer review because the central mapping is straightforward and the organizing insight is worth referee scrutiny even if the quantitative regime needs tightening.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates the superhorizon behavior of scalar hair sourced by black holes in de Sitter spacetime within the linearly coupled shift-symmetric scalar-Gauss-Bonnet theory, working in the test-field regime. It demonstrates that this hair exhibits temporal and spatial growth on superhorizon scales, a feature that arises from the dynamics of a minimally coupled massless scalar field in expanding de Sitter spacetime rather than being unique to black holes. The same growth occurs for a point scalar charge, positioning the scalarized black hole as a localized subhorizon source. Backreaction is argued to first appear on subhorizon scales without eliminating the superhorizon profile, while the time-dependent hair carries a steady outward energy flux, rendering the test-field regime transient and explaining challenges in static solutions.

Significance. If the central claims hold, the work is significant for clarifying the cosmological properties of scalar hair in this theory. By showing that the growth is a general consequence of scalar dynamics in de Sitter space and not specific to black holes, it provides a robust explanation for the absence of static scalarized solutions and underscores the transient nature of the test-field approximation due to energy flux. This contributes to understanding scalar-tensor theories in cosmological settings.

major comments (1)

The assertion that backreaction first arises on subhorizon scales and does not eliminate the superhorizon profile is load-bearing for the validity of the reported growth. However, the manuscript provides no explicit comparison of the superhorizon growth timescale (derived from the massless scalar wave equation in de Sitter) against the timescale on which the outward energy flux produces O(1) metric corrections. Without this quantitative check, it remains unclear whether the claimed growth can be observed before the test-field regime ceases to be valid.

minor comments (1)

The abstract and introduction would benefit from a brief explicit statement of the scalar wave equation in de Sitter coordinates used to derive the growth, to make the connection to the standard result more immediate for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the single major comment below and will revise the manuscript accordingly to strengthen the discussion of backreaction timescales.

read point-by-point responses

Referee: The assertion that backreaction first arises on subhorizon scales and does not eliminate the superhorizon profile is load-bearing for the validity of the reported growth. However, the manuscript provides no explicit comparison of the superhorizon growth timescale (derived from the massless scalar wave equation in de Sitter) against the timescale on which the outward energy flux produces O(1) metric corrections. Without this quantitative check, it remains unclear whether the claimed growth can be observed before the test-field regime ceases to be valid.

Authors: We agree that an explicit quantitative comparison of the superhorizon growth timescale against the backreaction timescale would strengthen the argument and clarify the regime of validity. In the revised manuscript we will add this comparison. The growth timescale follows directly from the exact solution of the massless scalar wave equation in de Sitter, which yields a linear-in-time growth on superhorizon scales. The backreaction timescale is set by the integrated energy flux carried by the time-dependent scalar profile; because this flux is diluted by the expanding volume on superhorizon scales, the metric corrections remain perturbatively small for a parametrically longer interval than the growth time. We will insert a short analytic estimate (and, if appropriate, a numerical check) demonstrating that O(1) backreaction first appears on subhorizon scales while the superhorizon profile continues to grow, thereby supporting the claim that the reported growth can be observed within the test-field regime. revision: yes

Circularity Check

0 steps flagged

Scalar hair growth follows from external de Sitter wave equation, no load-bearing self-reference

full rationale

The central claim traces superhorizon growth of the scalar hair directly to the standard massless minimally coupled scalar wave equation in expanding de Sitter spacetime, an independent external result. The abstract explicitly states this growth 'is not a special consequence of the black hole' and 'also arises for a point scalar charge,' confirming the derivation does not reduce to the hair construction or any fitted parameter. No self-citations, uniqueness theorems, or ansatzes are invoked as load-bearing steps in the provided text. The test-field regime is acknowledged as transient due to outward energy flux, but this does not create a circular reduction in the reported growth result. Overall minor self-citation risk at most, with the derivation remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from the stated theory and approximations; no new free parameters or invented entities are introduced in the summary.

axioms (2)

domain assumption The scalar field is minimally coupled and massless with linear shift-symmetric coupling to the Gauss-Bonnet invariant
This defines the theory under study and is invoked to derive the test-field equations.
domain assumption de Sitter spacetime provides the cosmological background
The expanding universe is modeled as de Sitter, standard for late-time cosmology.

pith-pipeline@v0.9.0 · 5474 in / 1535 out tokens · 33814 ms · 2026-05-15T01:43:35.668804+00:00 · methodology

discussion (0)

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IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

This growth is not a special consequence of the black hole, but instead follows from the dynamics of a minimally coupled massless scalar field in expanding de Sitter spacetime... also arises for a point scalar charge in de Sitter
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the scalar equation of motion is Φ=−αG... solution φ(η,x)=β/(a x)+β H ln(a/(1+a H x))

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