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Mass-Gap Neutron Stars from Vector \texorpdfstring{$f(R)$}{f(R)} Gravity Inflationary Deformations

T0 review · 3 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that vector f(R) gravity inflationary models, solved as scalar-tensor theories, support static neutron stars up to about 2.75 solar masses with the MPA1 equation of state, placing them in the mass-gap region below the…

desk verdict A standard scalar-tensor TOV scan for four vector-f(R) inflationary models whose central mass-gap claim is currently unreproducible because the coupling alpha is defined two incompatible ways. read the letter →

arxiv 2507.17384 v1 pith:7REI2RUX submitted 2025-07-23 gr-qc

V.K. Oikonomou This is my paper
classification gr-qc MSC 83C5583D0585A15 PACS 04.50.Kd95.36.+x98.80.-k98.80.Cq11.25.-w
keywords neutronstarsmass-gapregionvectorf(R)gravityscalar-tensorTolman-Oppenheimer-VolkoffequationsMPA1equationofstatemodifiedinflationarymodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Neutron stars are observed up to about 2.35 solar masses, while gravitational-wave observatories have detected compact objects in the 2.5-5 solar-mass gap without knowing whether they are heavy neutron stars or small black holes. This paper tries to show that modified gravity can populate that gap without breaking the general-relativistic ceiling of 3 solar masses for static stars. It solves the Tolman-Oppenheimer-Volkoff equations for four vector f(R) gravity inflationary models and nine piecewise-polytropic equations of state. The central result is that the MPA1 equation of state produces maximum masses near 2.749 solar masses for every model, inside the mass-gap region, and that even cosmologically non-viable inflationary models give acceptable neutron-star phenomenology. If this is right, the identity of mass-gap objects could be heavy static neutron stars, and the equation of state rather than the inflationary model would decide which scenarios survive.

What carries the argument

The load-bearing machinery is the conformally transformed scalar-tensor system together with the physical-mass conversion formula. After the transformation $\tilde{g}_{\mu\nu} = A^{-2}g_{\mu\nu}$ with $A(\phi) = e^{\phi/(2\sqrt{6+\beta^2})}$, each vector $f(R)$ model supplies a potential $V(\phi)$ that enters the TOV equations along with the coupling $\alpha(\phi) = \frac{1}{2}\sqrt{6+\beta^2}$. A double-shooting LSODA solver tunes the central values $\nu_c$ and $\phi_c$ so that the scalar field vanishes at numerical infinity, where the metric becomes Schwarzschild; Eq. (23) then converts the Einstein-frame ADM mass into the physical Jordan-frame mass, which receives contributions from outside the star because the scalar field does not vanish at the stellar surface. The four models share the same $A(\phi)$ and $\alpha(\phi)$ and differ mainly in $V(\phi)$, which is why their mass-radius curves nearly coincide.

What would settle it

Recompute the mass-radius curve for the near-general-relativity model ($\beta = 10^4$, coupling about $5\times10^{-5}$) with the MPA1 equation of state and compare its maximum mass with the general-relativistic result for the same equation of state; the two must agree if the mass conversion is correct, yet the paper reports about $2.749\,M_\odot$ for that model and gives no GR baseline.

Watch

Extended reading notes

Core claim

Vector $f(R)$ gravity, built by replacing the Ricci scalar $R$ with $R + A^\mu A_\mu + \beta\nabla_\mu A^\mu$ in the Lagrangian, becomes a scalar-tensor theory on shell, and its four inflationary models are solved numerically for static neutron stars. The paper reports that the four models produce almost indistinguishable Jordan-frame mass-radius curves. Confronted with the NICER bounds, a refined version of NICER, the PSR J0740+6620 constraints, and the CSI, CSII, and CSIII radius constraints, the MPA1 equation of state is the only one compatible with all of them for all four models, with maximum masses around 2.749 solar masses for each model, below the 3-solar-mass causal limit. The WFF1, MS1, and MS1b equations of state are excluded by the constraints used. The paper also asserts that a cosmologically non-viable inflationary model can still give viable neutron-star phenomenology, that this is a model-dependent feature, and closes by noting that the theoretical context complies with the general behavior of viable modified-gravity models rather than producing a new physics prediction.

Load-bearing premise

The paper's mass values rest on the assumption that the numerical integration and the formula converting the computed mass into the physical mass are correct, including the boundary condition that the scalar field vanishes at very large radius; the near-general-relativity model is never checked against the known GR maximum mass for the MPA1 equation of state, so a systematic offset in the reported masses cannot be ruled out.

Editorial extensions

If this is right

  • The MPA1 equation of state in these vector $f(R)$ models supports static neutron stars up to about $2.749\,M_\odot$, inside the mass-gap region and below the $3\,M_\odot$ causal limit.
  • The WFF1, MS1, and MS1b equations of state are ruled out, while MPA1 satisfies all the observational constraints used in the paper.
  • A single mass-gap neutron star observation would not distinguish the four inflationary models, because their mass-radius curves are nearly identical.
  • Cosmological viability of the inflationary model is not a prerequisite for viable neutron-star phenomenology in this theory class.
  • The paper's conclusion implies that mass-gap objects could be static, non-rotating neutron stars without requiring masses above the $3\,M_\odot$ limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The near-identical curves suggest the result is controlled by the shared conformal coupling functions rather than by the specific potential, so other scalar-tensor theories with the same $A(\phi)$ and $\alpha(\phi)$ would likely give the same neutron-star phenomenology.
  • Because no general-relativistic baseline for the MPA1 equation of state is reported, an independent integration should reproduce the known GR maximum mass before the $2.75\,M_\odot$ value is taken at face value.
  • Tidal deformability, moment of inertia, and oscillation spectra are more sensitive to the scalar field than mass-radius curves are, so those observables are the most promising way to break the degeneracy among the four models.
  • The MPA1 scenario predicts a specific radius near 11.3 km at maximum mass, so a future radius measurement of a heavy neutron star candidate could test the scenario independently of mass alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The manuscript studies static neutron stars in a class of vector f(R) gravity theories that are recast as scalar-tensor theories. The author integrates the Einstein-frame TOV equations (10)-(14) with a Python LSODA double-shooting solver for nine piecewise-polytropic equations of state, and converts the results to the Jordan frame using the conformal factor A(φ) and the ADM mass formula (23). Four inflationary models are considered: the Starobinsky R^2 model (Model I), a Starobinsky variant with β≈0.1 (Model II), a power-law f(R) model with β=1, n=1.8 (Model III), and a large-β limit with β=10^4, n=4 (Model IV), the last being cosmologically non-viable. The central claim is that the MPA1 equation of state produces maximum masses around 2.749 M⊙ for all four models, placing them in the mass-gap region while remaining below the 3 M⊙ causal limit, and that even the non-viable inflationary model produces viable neutron star phenomenology, with all models giving nearly indistinguishable mass-radius curves.

Significance. If the numerical results are correct, the paper would be a useful forward-prediction study: the inflationary parameters (β, n, M, m) are fixed before the neutron-star calculation, so the MPA1 maximum-mass prediction is not obtained by fitting theory parameters to neutron star data. The paper also confronts the results with multiple constraints (NICER I/II, PSR J0740+6620, CSI–CSIII) and honestly states that the model does not produce new physics beyond existing modified-gravity behavior. However, the central quantitative claim is currently unverifiable from the manuscript because of internal inconsistencies in the definitions of A(φ) and α(φ), and because no GR baseline is tabulated. The significance of the result therefore cannot be assessed until these issues are resolved.

major comments (3)
  1. [Sec. I.A, Eqs. (37)-(39)] Equations (37), (38), and (39) are mutually inconsistent. From Eq. (37), ϕ = exp(2φ/√(6+β²)), so with A = Ω^{-1/2} = ϕ^{-1/2} from Eq. (5) the correct conformal factor is A(φ) = exp(-φ/√(6+β²)) and α(φ) = d ln A/dφ = -1/√(6+β²). The manuscript instead gives A(φ) = exp(+φ/(2√(6+β²))) in Eq. (38) and α(φ) = √(6+β²)/2 in Eq. (39). The two α values differ by a factor of (6+β²), which for Model IV (β=10^4) is about 10^8. Since α enters the TOV equations (12) and (13), the computed mass-radius curves and maximum masses in Table II depend critically on which expression was used in the solver. The author must state explicitly which definitions were implemented, correct any typographical or conceptual errors, and rerun the analysis if the implemented expressions do not match the corrected ones.
  2. [Sec. I.B, Table II and Fig. 6] The near-GR limit of Model IV is not documented. For β=10^4, the potential in Eq. (43) is strongly suppressed and, if α is taken from the derivative of the (corrected) conformal factor, the model should reduce essentially to general relativity. In that case the reported MPA1 maximum mass of 2.749149 M⊙ should equal the GR maximum mass for the same piecewise-polytropic MPA1 equation of state. The paper never quotes the GR maximum mass (or GR radii at 1.4 M⊙ or 2 M⊙) for any of the nine EoSs, so the reader cannot tell whether Model IV is being computed in the weak-coupling near-GR regime or in a strongly coupled regime (if Eq. (39) with α ≈ 5×10³ was used). Without this baseline, the assertion that all four models—including the non-viable one—produce viable and nearly identical mass-radius curves is not testable. Please provide the GR M–R curve and tabulated GR maximum masses for all EoSs, and identify the actual α used in the code.
  3. [Sec. I, Eq. (23)] The Jordan-frame ADM mass formula (23) depends explicitly on A(φ(r_E)) and α(φ(r_E))(dφ/dr) evaluated at numerical infinity. Given the inconsistency of Eqs. (37)–(39), both the sign and the magnitude of the scalar-field correction term are ambiguous; for Model IV the correction could either vanish (small α) or dominate (large α from Eq. (39)). In addition, the manuscript does not provide convergence tests for the choice of the numerical infinity r_E, the shooting tolerance, or the residual of the scalar field at infinity. Please include such tests, together with the asymptotic values of A(φ) and α(φ) at r_E, to demonstrate that the extracted Jordan-frame masses are stable and that the quoted 2.749 M⊙ value is well defined.
minor comments (8)
  1. [Table VIII] Table VIII lists R_ENG = 1.437837 km for Model III, which is almost certainly a typo (likely 11.437837 km) and is inconsistent with the other entries for the ENG EoS; please correct it.
  2. [Table XI] Table XI lists M_max = 10.866 M⊙ for the SLy EoS in Model II, which is physically implausible and inconsistent with Model II's other rows and with the M–R figures; all tables should be carefully regenerated from the corrected numerical output.
  3. [Sec. I.A, after Eq. (42)] The formula for the parameter m is garbled: 'm = 5.1 × 10−4pc−1/2 n (2pN )−(p+2)/4' is not readable. Please write the expression with clear notation for the Planck mass, the e-folding number N, and the indices p and n.
  4. [Eq. (1)] The causal-limit formula (1) uses ρ_u and P_u without defining the transition density or the matching procedure; please clarify what ρ_u and P_u refer to in the causal EoS construction.
  5. [Table II] The maximum masses in Table II are quoted to as many as seven significant figures (e.g., 2.749149112 M⊙). This precision exceeds what is meaningful for piecewise-polytropic parametrizations and should be rounded consistently.
  6. [Fig. 1] Figure 1 is described as presenting the CSI, CSII, and CSIII constraints, but the figure appears to be an edited astronomical image without visible constraint bands; please overlay the actual mass-radius constraint regions or replace the figure with one that conveys the relevant information.
  7. [Abstract and Sec. I] The abstract says 'we solve the TOV equations', but the paper actually solves the Einstein-frame scalar-tensor TOV equations and then converts the results to the Jordan frame; this distinction should be stated explicitly in the abstract or in the opening of Section I.
  8. [Table XII] The header of Table XII contains a broken phrase ('the and the correspondent') and the table formatting of the EoS names is inconsistent; please correct the formatting.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: inflationary parameters are fixed before the TOV solve; the mass-gap result is a forward numerical prediction.

full rationale

The derivation is forward and self-contained with respect to the central mass-gap claim. The vector-f(R) potentials and parameters (beta, n, M, m) are fixed by the inflationary model (Eqs. 40-43) before any neutron-star calculation; none are fitted to the NS constraints or to the M-R curves. The TOV system Eqs. (10)-(14) and the Jordan-frame ADM mass formula Eq. (23) are derived from the action and conformal transformation, and the double-shooting condition phi -> 0 at numerical infinity is an external boundary condition, not an input that encodes the 2.75 M_sun result. The MPA1 preference is a post-hoc comparison across nine EoSs against external NICER/CSI/CSII/CSIII constraints, so it is model selection rather than a circular derivation. The self-citations ([11], [12]) motivating MPA1 and the 3 M_sun rule are contextual and not load-bearing, since the paper recomputes all cases. A genuine internal inconsistency exists between Eq. (38) and Eq. (39) for alpha(phi): the derivative of Eq. (38) is 1/(2 sqrt(6+beta^2)), not (1/2) sqrt(6+beta^2), which is a correctness/support caveat for the numerical implementation, but it does not make the prediction equivalent to its inputs. The paper even concedes it did not reveal new physics, consistent with a non-circular, independent benchmark calculation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard scalar-tensor TOV equations and on the choice of four specific models from the prior literature. No new particles or fields are introduced. The main numerical output depends on the boundary conditions and the Jordan-frame mass formula, which are not independently verified and are the most fragile elements. The free parameters are all fixed by inflationary phenomenology rather than by neutron star data.

free parameters (4)
  • beta (Starobinsky deformation parameter) = Model I: 0; Model II: 0.1; Model III: 1; Model IV: 10^4
    Chosen by hand: beta~0.1 gives a cosmologically viable Model II, beta=1 is used in the R^n model Model III, and beta=10^4 produces the cosmologically non-viable Model IV. These choices set the scalar coupling alpha(phi) = 1/(2 sqrt(6+beta^2)).
  • n (power-law exponent in f(R)=R+m^2(1-n) R^n) = n = 1.8 for Model III; n = 4 for Model IV
    Chosen for simplicity within the inflationary-viable range 1.75 < n < 2.39 for Model III; Model IV is a large-beta limiting case with n=4.
  • e-folding number N = N = 60
    Used in the Planck normalization for the Starobinsky mass scale M; affects the potential amplitude, though the potential is negligible at neutron star densities.
  • Mass scale M (Model I and II), m (Models III and IV) = M ~ 1.3e-5 sqrt(1+beta^2/6) (60/55)^{-1}; m = 5.1e-4 pc^{-1/2} n (2pN)^{-(p+2)/4}
    Fixed by the amplitude of scalar perturbations from inflation, not by neutron star data. Included because the TOV equations contain V(phi).
assumptions (5)
  • domain assumption The vector f(R) gravity action (25) is equivalent, after integrating out the auxiliary fields, to the scalar-tensor theory (28).
    Taken from Ozkan, Pang and Tsujikawa [20]. The neutron star analysis uses this equivalence to work in the Einstein frame.
  • standard math The Einstein-frame TOV equations (10)-(14) correctly describe static neutron stars in scalar-tensor gravity.
    They are the standard equations from Pani and Berti [47]. Their correctness is assumed, and the paper does not derive them.
  • domain assumption The Jordan-frame ADM mass formula (23) correctly relates the Einstein-frame mass to the physical mass.
    This formula is stated without derivation. The maximum mass values depend on it.
  • domain assumption The scalar field vanishes at numerical infinity and the metric becomes Schwarzschild.
    Used as the boundary condition for the double-shooting method. In a theory with a very light scalar, the asymptotic behavior may be nontrivial.
  • ad hoc to paper The 3 solar mass causal limit applies to these modified gravity theories.
    The paper excludes the MS1 and MS1b EoSs because their maximum masses exceed 3 solar masses, assuming the GR causal limit holds in modified gravity. This is a selection criterion imposed a priori.

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Cite this review

Pith. "Pith review of Mass-Gap Neutron Stars from Vector \texorpdfstring{$f(R)$}{f(R)} Gravity Inflationary Deformations." pith.science (2026). https://pith.science/paper/7REI2RUX

@misc{pith2026250717384,
  author       = {Pith},
  title        = {Pith review of: Mass-Gap Neutron Stars from Vector \texorpdfstring$f(R)$f(R) Gravity Inflationary Deformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7REI2RUX}},
  note         = {Machine review of arXiv:2507.17384}
}
abstract

The latest observations from the LIGO-Virgo indicated the existence of mass-gap region astrophysical objects. This is a rather sensational observation and there are two possibilities for the nature of these mass-gap region astrophysical objects, these are either small black holes that result from the mergers of ordinary mass neutron stars, or these are heavy neutron stars. In the line of research implied by the former possibility, in this work we shall examine the implied neutron star phenomenology from vector $f(R)$ gravity inflationary models. These theories are basically scalar-tensor deformations of the Starobinsky inflationary model. We shall present the essential features of cosmologically viable and non-viable deformations of the Starobinsky model, originating from vector $f(R)$ gravity inflationary theories, and we indicate which models and for which equations of state provide a viable neutron star phenomenology. We solve the Tolman-Oppenheimer-Volkov equations using a robust double shooting LSODA python based code, for the following piecewise polytropic equations of state the WFF1, the SLy, the APR, the MS1, the AP3, the AP4, the ENG, the MPA1 and the MS1b. We confront the resulting phenomenology with several well known neutron star constraints and we indicate which equation of state and model fits the phenomenological constraints. A remarkable feature, also known from other inflationary attractor models, is that the MPA1 is the equation of state which is most nicely fitted the constraints, for all the theoretical models used, and actually the maximum mass for this equation of state is well inside the mass-gap region. Another mentionable feature that stroked us with surprise is the fact that even cosmologically non-viable inflationary models produced a viable neutron star phenomenology, which most likely has to be a model-dependent feature.

Figures

Figures reproduced from arXiv: 2507.17384 by the authors.

Figure 1
Figure 1. FIG. 1: The constraints CSI [78] [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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