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arxiv: 2607.01937 · v1 · pith:3KGYNKCRnew · submitted 2026-07-02 · 🌀 gr-qc

Neutron stars with primary scalar hair

Hamza Boumaza , Christos Charmousis , David Langlois , Etienne Ligout This is my paper

Pith reviewed 2026-07-03 08:31 UTC · model grok-4.3

classification 🌀 gr-qc
keywords neutron starsscalar hairDHOST theoriesmodified gravityTolman-Oppenheimer-Volkoff equationsmass-radius relationcompact objects
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The pith

Neutron stars carrying primary scalar hair in a DHOST subfamily become more compact than in general relativity, developing singularities above a critical charge.

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

The paper constructs static spherically symmetric neutron star solutions with primary scalar hair by solving modified Tolman-Oppenheimer-Volkoff equations in a chosen DHOST subfamily. It compares polytropic and realistic equations of state to show how the scalar field alters the metric, scalar profiles, and mass-radius curves relative to general relativity. A sympathetic reader cares because these configurations provide concrete, observable signatures that could bound the strength of scalar hair and related beyond-GR parameters through astrophysical data. Positive scalar charges systematically increase compactness, and a threshold exists beyond which solutions cease to be regular.

Core claim

In the selected DHOST subfamily, equilibrium neutron star configurations exist that carry primary scalar hair. The modified TOV equations yield more compact stars for positive scalar charges than their GR counterparts; above a critical charge value the configurations develop singularities. The resulting mass-radius relations deviate measurably from GR predictions for both polytropic and realistic equations of state.

What carries the argument

Primary scalar hair implemented via the modified Tolman-Oppenheimer-Volkoff equations obtained from the DHOST action under static spherical symmetry.

If this is right

  • Positive scalar charges produce systematically smaller radii at fixed mass than in GR.
  • A critical scalar-charge threshold exists beyond which no regular stellar solution remains.
  • Mass-radius relations for both polytropic and realistic equations of state deviate from GR in a charge-dependent way.
  • Astrophysical measurements of compactness or maximum mass can therefore bound the DHOST parameters that control the hair.

Where Pith is reading between the lines

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

  • Tidal deformability measurements from gravitational-wave events could further constrain the allowed range of scalar charges.
  • The singularity threshold may indicate where the effective-field-theory description breaks down and higher-order operators become necessary.
  • If realistic equations of state already saturate the compactness limit in GR, the additional compaction from hair could exclude entire families of DHOST models.

Load-bearing premise

The modified Tolman-Oppenheimer-Volkoff equations derived from the chosen DHOST subfamily correctly describe static spherically symmetric equilibrium configurations that carry primary scalar hair.

What would settle it

An observed neutron-star mass-radius pair that lies outside the family of more-compact curves predicted for any allowed scalar charge, or the detection of a highly compact star without the expected singularity, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2607.01937 by Christos Charmousis, David Langlois, Etienne Ligout, Hamza Boumaza.

Figure 3.1
Figure 3.1. Figure 3.1: Upper bounds in the parameter space (ξ2, ρ), defined by the maximal central density ρ max c allowed for each ξ2, with λ = 2, 5, 10 and 50 km. The plot corresponds to the polytropic EoS defined in (4.1)-(4.2). 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Radial profiles of A, B and P for the polytropic EoS given by (4.1) and (4.2), assuming nc = 4 n0 and λ = 50 km. Here we focus on the effect of ξ2, plotting the profiles for ξ2 = 0 (GR theory, in black), ξ2 = 0.03, 0.04 and 0.05 (in shades of blue). As a further exploration, we now study how the length parameter λ affects the neutron star profile. We have plotted in [PITH_FULL_IMAGE:figures/full_fig_p01… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Radial profiles of A, B and P for the polytropic EoS, assuming nc = 8 n0 and ξ2 = 0.03. Here we focus on the effect of λ, plotting the profiles for λ = 2, 5, 10 and 50 km. These are to be compared with the GR case pictured in black. has the same typical radial evolution as in GR. Indeed, for λ = 5 km in the figure, the mass increases within the star and settles to its ADM mass value at the surface. For h… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Radial profiles of A, B and P for the central densities nc = 4 n0 (top) and nc = 8 n0 (bottom), using the polytropic EoS. Here, we study the combined effect of λ and ξ2, for λ = 5, 10 and 50 km; from left to right, ξ2 is taken to be 0.03, 0.04 and 0.046 (close to the critical value 0.047 where the singularity appears). 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Radial profiles of m, ρ, ρeff and X¯ for λ = 5, 10 and 50 km, with the polytropic EoS, nc = 8 n0 and ξ2 = 0.03. The left panel shows the profile of m, ρ and ρeff within the star, while the right one focuses on the behavior of m and X¯ at larger distances. Note that for λ = 50 km (blue curve), m continues to increase beyond the star’s surface until it reaches its ADM value far away. 4.2 Phenomenological e… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Polytropic, SLy, BSk21 and BSk21 EoS, plotted for [PITH_FULL_IMAGE:figures/full_fig_p017_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Radial profiles of A, B and P for ξ2 = 0.01, 0.03 and 0.05 (in shades of blue), with the SLy EoS, ρc = 4 ρ0 and λ = 5 km. As before, the GR profile is in black. As we stressed in Subsection 3.3, the differential system governing the radial profile of a relativistic star breaks down when the quantity τ˜, defined in (3.19), reaches the value 1. This condition can be fulfilled either if τ is large enough or… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Radial profiles of c 2 s and τ for two almost singular configurations. We are using the SLy EoS and λ = 10 km [PITH_FULL_IMAGE:figures/full_fig_p018_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Maximal central density ρ max c allowed for each ξ2 and for λ = 2, 5, 10 and 50 km, in the case of the SLy EoS. The sign of X outside the star is positive by construction, since the gradient of the scalar field is 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: Radial profile of A X¯ showing the possible change of sign of the scalar density X in the star interior. Here we consider the SLy equation of state, two central densities, ρc = 6.5ρ0 (left panel) and ρc = 8ρ0 (right panel), λ = 50 km and ξ2 = 0, 0.01, 0.02 and 0.025. 4.3 Mass-radius relation As noticed in the various star configurations shown earlier, the star’s radius in modified gravity differs from th… view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Mass-radius relation M(R) for compact stars with the polytropic EoS (with ρc ≥ 2ρ0) . The left panel focuses on the influence of the theory-fixed parameter λ, for λ = 5, 10, 50 and 100 km, with ξ2 = 0.04. The right panel focuses on the influence of the star-dependent parameter ξ2, showing the cases ξ2 = 0.01, 0.6, 0.12, for the theory parametrised by λ = 10 km. By varying the central density, we can com… view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: Mass-radius relation M(R) for the SLy EoS. In the top panels we consider λ = 5 and 10 km respectively for values ξ2 = 0.004, 0.016, 0.04 and 0.08. We notice that for small values of λ the mass range (in shades of red) is relatively independent of the value of ξ2. For the bottom panels however (considering the larger values λ = 50 and 100 km), the mass range (in shades of blue) differs significantly depe… view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Relation µ(R) for λ = 50 km, for all EoS studied. The left plot corresponds to the polytropic EoS, for ξ2 from 0.04 to 0.10 with an increment of 0.02. The right plot corresponds to the SLy, BSk21 and BSk22 EoS, for ξ2 from 0.008 to 0.032 with an increment of 0.008. 4.4 Case of negative scalar charge We now discuss briefly the cases where the coupling parameter ξp takes negative values, which appears to … view at source ↗
Figure 4.13
Figure 4.13. Figure 4.13: Radial profiles of m, ρ, ρeff and X¯ for the SLy EoS, for λ = 10, 50 and 100 km, ρc = 3ρ0 and and ξ2 = −0.02. The left plot gives the profile of m, ρ and ρeff within the star, while the right one focuses on the behavior of m and X¯ at larger distances. Note that for λ = 100 km (purple curve), m eventually becomes negative. At very high central densities, the star exhibits unusual behaviour: both its mas… view at source ↗
Figure 4.14
Figure 4.14. Figure 4.14: Mass-radius relation M(R) for the SLy EoS in the case of a negative scalar charge, for λ = 5, 10, 50 and 100 km and ξ2 varying from −0.002 to −0.020 with an increment of 0.002. 5 Conclusions We have constructed neutron star solutions within a subfamily of DHOST theories, exploiting their global shift and parity symmetries. These theories are parametrised by two coupling constants and a positive exponent… view at source ↗
read the original abstract

We investigate static and spherically symmetric neutron star solutions endowed with primary scalar hair in a subfamily of Degenerate-Higher-Order-Scalar-Tensor (DHOST) theories of gravity. By solving the modified Tolman-Oppenheimer-Volkoff (TOV) equations, we construct equilibrium configurations for polytropic and realistic equations of state and analyse the impact of the scalar hair on the stellar structure. We examine the resulting metric and scalar field profiles as well as the mass-radius relation, showing deviations from the predictions of General Relativity (GR). Positive scalar charges lead to more compact stars than in GR and, above a critical threshold, to singularities. Observations could therefore put stringent constraints on the parameters characterising the beyond-GR effects in these theories and their potential scalar hair.

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.

Referee Report

0 major / 3 minor

Summary. The paper constructs families of static, spherically symmetric neutron-star solutions carrying primary scalar hair in a chosen subfamily of DHOST theories. Modified Tolman-Oppenheimer-Volkoff equations are integrated numerically for both polytropic and realistic equations of state; the resulting metric and scalar profiles, mass-radius relations, and compactness are compared with GR. The central finding is that positive scalar charge increases compactness relative to GR and produces singularities above a critical threshold, with the implication that observations can constrain the theory parameters.

Significance. If the modified field equations and boundary conditions for primary hair are correctly implemented, the work supplies explicit, falsifiable examples of how primary scalar hair alters neutron-star structure in DHOST gravity. The use of both polytropic and realistic EOS, together with the reported singularity threshold, supplies a concrete mechanism for placing bounds on beyond-GR parameters from mass-radius data. The reduction to the GR limit when the scalar charge vanishes is a necessary consistency check that strengthens the result.

minor comments (3)
  1. [§2] §2: the precise functional form of the DHOST Lagrangian (the functions A_i, B_i, etc.) and the values of the free parameters retained in the subfamily should be written explicitly so that the modified TOV system can be reproduced without ambiguity.
  2. [§4.1] §4.1 and Fig. 2: the numerical integration scheme (shooting method, radial step size, convergence criterion) and the precise location of the reported singularities (coordinate vs. curvature) are not stated; adding this information would allow independent verification of the compactness increase and the critical-charge threshold.
  3. The mass-radius curves for varying scalar charge should be accompanied by a table of central densities, radii, and compactness values so that the quantitative deviation from GR can be assessed directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work on neutron-star solutions with primary scalar hair in DHOST theories. The report correctly identifies the key results, including the increased compactness for positive scalar charge, the singularity threshold, and the recovery of the GR limit. No major comments were raised, so we have no point-by-point rebuttals. We will incorporate any minor suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds by obtaining the modified TOV equations from the chosen DHOST subfamily action under static spherical symmetry, imposing boundary conditions that allow primary scalar hair (vanishing at infinity, nonzero central value), and numerically integrating for given EOS to obtain metric and scalar profiles. Mass-radius relations and compactness trends are direct outputs of these integrations; no parameter is fitted to a subset of the target data and then relabeled as a prediction, no self-citation supplies a uniqueness theorem that forces the result, and the GR limit is recovered by construction when the scalar charge vanishes without circular redefinition. The central claims therefore remain independent of the inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The abstract supplies no explicit list of free parameters or axioms; the ledger is therefore inferred from the stated approach and remains provisional.

free parameters (1)
  • scalar charge / theory parameters
    The abstract refers to 'parameters characterising the beyond-GR effects' that control the strength of the scalar hair; their concrete values are not given.
axioms (1)
  • domain assumption The chosen DHOST subfamily admits static spherically symmetric solutions with non-trivial primary scalar hair.
    This assumption is required to pose and solve the modified TOV system described in the abstract.

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