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Rotating Fermion-Boson Stars in $R$-squared Gravity

T0 review · 3 major / 6 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read R-squared gravity plus a dark bosonic component raises the maximum mass of rotating mixed stars and widens the allowed mass-radius band relative to general relativity, while remaining compatible with current NICER and gravitational-wave bou

desk verdict First rotating fermion-boson stars in R^{2} gravity; solid numerical catalog with a real but well-flagged RPV limitation. read the letter →

arxiv 2607.04744 v1 pith:YGJYXBTN submitted 2026-07-06 gr-qc

Saeed Fakhry , Jorge Castelo Mourelle , Nicolas Sanchis-Gual , Daniela Doneva , Stoytcho Yazadjiev , José A. Font This is my paper · ORCID
classification gr-qc
keywords fermion-bosonstarsR-squaredgravityrotatingcompactscalardegreeoffreedomdarkmatteradmixedneutronmass-radiusrelationGW190814
topics Dark Matter
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

The paper constructs static and uniformly rotating equilibrium stars that contain both ordinary nuclear matter and a self-interacting complex bosonic field, all inside the simplest quadratic modification of Einstein gravity, f(R)=R+aR^{2}. The extra scalar degree of freedom that this theory carries redistributes both the bosonic amplitude and the fermionic pressure, systematically enlarges the space of allowed solutions, and lifts the maximum supported mass (static and mass-shedding) above the general-relativity values. For the largest a considered, the static maximum reaches 2.62 solar masses and the Keplerian maximum 3.29 solar masses. Because these sequences still intersect the NICER mass-radius ellipses and the secondary-mass window of GW190814, the models offer a concrete way to accommodate compact objects near the low-mass black-hole gap without forcing an unrealistically stiff nuclear equation of state.

What carries the argument

The RPV high-coupling approximation that replaces the complex bosonic field by an effective perfect-fluid source of compact support, solved together with the Einstein-frame scalar-tensor equations of R-squared gravity inside a modified RNS self-consistent-field code.

What would settle it

A numerical-relativity evolution of a high-mass, low-frequency model that retains the full bosonic tail would show whether the configuration remains long-lived or disperses; if it disperses while the truncated-tail equilibrium predicts stability, the maximum-mass claims fail.

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Extended reading notes

Core claim

In R-squared gravity the scalar degree of freedom modifies the spatial profiles of both the bosonic field and the fermionic energy density, expands the domain of admissible mixed-star equilibria, and raises both the static and Keplerian maximum masses relative to pure general relativity, all while the resulting sequences remain compatible with current astrophysical and gravitational-wave constraints.

Load-bearing premise

The calculation discards the exponentially decaying outer tail of the bosonic field and treats the bosonic sector as a perfect fluid of compact support; if that tail or the strong-self-interaction assumption is not valid, the reported mass-radius bands are unreliable.

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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 / 6 minor

Summary. The paper constructs static and uniformly rotating equilibrium configurations of mixed fermion-boson stars in R-squared gravity, f(R)=R+aR², formulated in the Einstein frame as a scalar-tensor theory. The fermionic sector uses the tabulated AkmalPR EOS; the bosonic sector is a self-interacting complex field treated in the RPV high-coupling approximation, so that its energy-momentum tensor is replaced by an effective perfect fluid of compact support. Equilibrium models are obtained with a modified RNS/KEH code for GR and two representative values a=10 and a=10^4. The main claims are that the scalaron modifies the spatial distribution of both components, enlarges the domain of admissible equilibria, and raises static and Keplerian maximum masses relative to GR (e.g. static 2.44→2.62 M⊙ and Keplerian 2.89→3.29 M⊙ for a=10^4), while the sequences remain compatible with NICER, GW170817, and GW190814 constraints for bosonic mass fractions ≲10%.

Significance. This is, to my knowledge, the first self-consistent construction that combines a mixed fermion-boson matter sector, uniform rotation, and a viable f(R) model with a massive scalaron. Prior work treated rotating fermion-boson stars in GR or static mixed stars in f(R), but not the three ingredients together. The explicit Einstein-frame field equations (Appendices A–B), the transparent RPV implementation, and the mass–radius maps with external multimessenger bounds make the contribution useful for the strong-field modified-gravity and dark-matter-admixed compact-star communities. If the reported mass gains and enlarged solution space survive beyond the RPV truncation, the models offer a concrete, observationally testable channel for objects near the low-mass black-hole gap without requiring an extremely stiff nuclear EOS.

major comments (3)
  1. [Section III A, Eqs. (29)–(33); Fig. 4; Sec. V C] Section III A, Eqs. (29)–(33) and the mass/angular-momentum integrals (42)–(43): the headline mass–radius bands and maximum-mass gains in Fig. 4 and Sec. V C rest on the RPV truncation of the bosonic tail and the algebraic replacement of T^b_μν by a compact-support perfect fluid. The paper itself notes (Sec. VI) that the neglected tail can become non-negligible for halo-dominated configurations (¯w ≳ 0.08–0.09), which populate a substantial fraction of the colored regions in Fig. 4 and the high-¯w columns of Figs. 1–3. Without a quantitative estimate of the truncated ADM mass/angular momentum for the adopted (μ_b, λ, ¯w, η) values—or an explicit restriction of the observational claims to the compact/core-like regime where RPV is controlled—the quoted static/Keplerian maxima and the GW190814 compatibility statement are not yet fully reliable. A short validation (e.g. order-of-magnitude ta
  2. [Section V C; Fig. 4] Section V C and the GW190814 discussion: the claim that a=10^4 allows a near-static mixed star with M_b/M_T ≲ 10% to sit inside the GW A band relies on the RPV sequences and on the single nuclear EOS AkmalPR. The paper correctly notes that stiff EOSs can reach ~2.6 M⊙ in GR, but the quantitative advantage attributed to R-squared gravity is not separated from (i) the RPV truncation and (ii) the fixed 10% bosonic-fraction ceiling. At minimum, the text should state how sensitive the static maximum (2.62 M⊙) is to modest changes in the bosonic fraction ceiling and to restoring a non-zero tail, and should avoid presenting the GW190814 resolution as robust until that sensitivity is shown.
  3. [Section V; comparison with Refs. [11, 16]] Units and physical scale of a: results are reported for a=10 and a=10^4 with no explicit statement of the unit system for a (length² in geometrized units). Prior R-squared NS literature typically quotes a in km² or relative to a gravitational radius. Without this, the claimed “near-saturation / Brans–Dicke-like” regime for a=10^4 cannot be mapped to a scalaron mass or to Solar-System/strong-field bounds. Please specify the units of a consistently with the dimensionless code variables and with the literature values used for comparison.
minor comments (6)
  1. [Abstract; Sec. I] Abstract and Introduction: “enlarges the domain of admissible equilibrium solutions” is used without a precise definition (parameter volume, mass–radius area, or existence of new topologies). A one-sentence operational definition would help.
  2. [Section III B] Eq. (36) and surrounding text: the compactified coordinate uses s, and μ=cos θ is introduced with a footnote warning against confusion with μ_b; still, several source-term symbols (S^x_T) reuse subscripts that collide with total quantities. A short notation table would reduce ambiguity.
  3. [Figs. 1–3] Fig. 1–3: relative-percentage panels are useful, but the vertical scales differ across columns; stating the peak percentage in the caption (as done for δφ²_max in Fig. 2) for the fermionic panels would aid comparison.
  4. [Appendix A; Sec. V A] Appendix A: the static metric is Schwarzschild-like while the rotating metric is quasi-isotropic; the text already warns that the static equations are not a direct coordinate limit of Eq. (24). Cross-referencing that warning again when static profiles are extracted as J_T→0 of the rotating code would avoid reader confusion.
  5. [Sec. V B; Introduction] Typographical: “dimentionless” (Sec. V B); “taht” in the Introduction’s appendix description; “inequivalent definitions” list is fine but “axion couplings [25]” could be checked for consistency with the reference list numbering after any revision.
  6. [Section III B] Bosonic microphysical parameters (μ_b ≃ 10^{-16} MeV, λ=100) are stated once in Sec. III B; a brief remark on why this ultralight scale is appropriate for the dark-matter interpretation (or that it is chosen purely to sit in the RPV regime) would help non-specialist readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: mass–radius gains and profile changes are numerical outputs of the field equations under stated approximations, not forced by definition or by fitting the target observables.

full rationale

The paper’s central claims—that the scalaron in f(R)=R+aR² redistributes fermionic and bosonic matter, enlarges the equilibrium domain, and raises static/Keplerian maximum masses relative to GR—are obtained by solving the Einstein-frame field equations (Eqs. 14–15, B1–B6) with the AkmalPR EOS and the RPV algebraic bosonic source (Eqs. 29–33) for chosen a, w̄, JT, MT. Those maxima (e.g. 2.44→2.62 M⊙ static and 2.89→3.29 M⊙ Keplerian for a=10^4) are turning points of the computed sequences in Fig. 4, not quantities already encoded in the inputs. Observational bands (NICER, GW170817, GW190814, RMW/NMR) are external benchmarks overlaid after the fact; a is not fitted to them. Self-citations to prior RNS/RPV and R-squared NS work supply numerical infrastructure and context, not a uniqueness theorem that forces the headline result. The RPV tail truncation is a load-bearing modeling assumption (correctness risk), not a circular reduction of prediction to input. Derivation chain is self-contained numerical construction under explicit approximations.

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

The central claim rests on the R-squared action, minimal gravitational coupling of two matter sectors, the RPV truncation of the bosonic field, a single tabulated nuclear EOS, and hand-chosen bosonic and curvature parameters. None of these is derived from the data; they are modeling choices that fix the solution space. No new fundamental particle is postulated beyond the standard scalaron of f(R) and a complex scalar dark-matter field already used in the cited literature.

free parameters (4)
  • a (R-squared coupling) = 10 and 10^4 (representative)
    Positive constant controlling the scalaron mass scale; representative values a=10 and a=10^4 are chosen by hand to span moderate and near-Brans–Dicke regimes, not fitted to data.
  • bosonic mass scale and couplings (μ_b, λ, η, w̄) = μ̄_b=0.1, η̄=0.01, λ̄=1; w̄ in [0.07,0.09] and broader scans
    Code inputs μ̄_b=0.1, η̄=0.01, λ̄=1 (corresponding to μ_b≃10^{-16} MeV, η=1, λ=100) place the model in the strong-self-interaction RPV regime; w̄ is scanned as a control parameter.
  • bosonic mass-fraction ceiling = ≤10%
    Mass–radius maps restrict M_b/M_T ≲ 10% by hand, motivated by accretion/capture scenarios in the cited literature rather than derived from the field equations.
  • nuclear EOS choice (AkmalPR) = AkmalPR
    Single tabulated zero-temperature EOS fixes the fermionic absolute mass scale; results for maximum mass and radius are EOS-dependent.
assumptions (5)
  • domain assumption f(R)=R+aR² with a≥0 is a viable strong-field modification equivalent to a massive scalar-tensor theory with κ(φ)=-1/√3.
    Invoked throughout Section II A; Solar-System viability and ghost-freedom conditions are assumed from the f(R) literature.
  • domain assumption Fermionic and bosonic sectors interact only gravitationally (no direct non-gravitational coupling).
    Stated in Section II B and used to write T*_μν = T*^f_μν + T*^b_μν.
  • ad hoc to paper RPV high-coupling approximation: bosonic tail may be truncated and the bosonic EMT replaced by an effective perfect fluid with compact support (Eqs. 29–33).
    Central numerical assumption of Section III A; enables reuse of the RNS/KEH fluid infrastructure but is not derived from the full Klein–Gordon problem for these parameters.
  • domain assumption Uniform (rigid) rotation of the fermionic fluid; no enforced co-rotation with the bosonic sector.
    Section III A; standard for KEH models but leaves differential-rotation and synchronization effects for future work.
  • domain assumption Zero-temperature perfect-fluid description with tabulated AkmalPR EOS is adequate for the fermionic sector.
    Section II B; finite-temperature and composition effects are neglected.

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Pith. "Pith review of Rotating Fermion-Boson Stars in $R$-squared Gravity." pith.science (2026). https://pith.science/paper/YGJYXBTN

@misc{pith2026260704744,
  author       = {Pith},
  title        = {Pith review of: Rotating Fermion-Boson Stars in $R$-squared Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGJYXBTN}},
  note         = {Machine review of arXiv:2607.04744}
}
abstract

Fermion-boson stars are compact equilibrium configurations composed of ordinary fermionic matter and a bosonic dark component interacting only through gravity. Such systems provide a natural framework for exploring deviations from standard neutron-star models, including the possible accumulation of dark matter inside neutron stars, and may be relevant for compact objects near the low-mass black-hole gap. We construct static and uniformly rotating fermion-boson stars within the framework of $R$-squared $f(R)$ gravity, characterized by the functional form $f(R)=R+aR^{2}$, where $a$ is a positive parameter governing the effective mass scale from the scalar degree of freedom. The fermionic sector is modeled as a perfect fluid described by a tabulated equation of state at zero temperature, while the bosonic component is represented by a self-interacting complex bosonic field. Our results show that the scalar degree of freedom modifies the spatial distribution of both the bosonic field and the fermionic pressure, enlarges the domain of admissible equilibrium solutions, and increases the maximum supported masses relative to general relativity. Our models remain compatible with current astrophysical and gravitational-wave constraints, suggesting that fermion-boson stars in $R$-squared gravity offer a promising framework to investigate the combined effects of dark bosonic matter, rotation, and strong-field modifications of gravity in compact objects.

Figures

Figures reproduced from arXiv: 2607.04744 by the authors.

Figure 1
Figure 1. FIG. 1: Radial profiles of the bosonic field ( [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Radial profiles of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Mass-radius relations for mixed star configurations in the framework of GR and two distinct models of [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Similar to Fig [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The mass-radius relation for AkmalPR EOS in the case of static stars and stars rotating at the Kepler limit. Different [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Fermionic core mass of the mixed star configurations, [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Total mass as a function of the bosonic field frequency within the framework of GR and two distinct [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]

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