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arxiv: 2602.18615 · v2 · submitted 2026-02-20 · 🌀 gr-qc

Two Parameter Deformation of Embedding Class-I Compact Stars in Linear f(Q) Gravity

Samstuti Chanda , Ranjan Sharma This is my paper

Pith reviewed 2026-05-15 20:08 UTC · model grok-4.3

classification 🌀 gr-qc
keywords compact starsf(Q) gravitygravitational decouplingembedding class-IVaidya-Tikekar metricneutron star massescausality bounds
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The pith

A two-parameter deformation using gravitational decoupling in linear f(Q) gravity enlarges the mass window for compact stars while respecting causality.

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

The paper establishes that linear f(Q) gravity alone rescales the matter sector uniformly without creating new geometric solutions, but combining it with gravitational decoupling in an embedding class-I Vaidya-Tikekar metric introduces an independent parameter ε that deforms the geometry. This controlled separation lets ε stiffen the effective equation of state while β1 rescales the matter contribution, producing higher-mass configurations. The resulting models satisfy regularity, matching, and causal conditions and reach masses compatible with observed high-mass pulsars and mass-gap objects.

Core claim

In linear f(Q) gravity, f(Q) = β1 Q + β2 is geometrically equivalent to general relativity, so stellar structure changes only through uniform rescaling of the matter sector. Adding gravitational decoupling to the embedding class-I Karmarkar condition in a Vaidya-Tikekar background yields a two-parameter family controlled by (ε, β1): ε governs the geometric deformation and effective stiffness, while β1 independently rescales the matter density and pressure. The admissible domain of these parameters is fixed by regularity, junction conditions, causality, and compactness, and an analytic compactness bound is obtained for the decoupled solutions.

What carries the argument

The two-parameter deformation (ε, β1) applied to the decoupled embedding class-I Vaidya-Tikekar metric, where ε controls geometric deformation and β1 rescales the matter sector.

If this is right

  • Higher stellar masses become accessible without pushing the equation of state to the causal limit.
  • Configurations match recent high-mass pulsar data and mass-gap candidates while remaining physically acceptable.
  • The separation of ε and β1 allows direct comparison of pure geometric deformation against matter rescaling at fixed metric deformation.
  • An explicit analytic compactness bound holds for the entire family of decoupled solutions.

Where Pith is reading between the lines

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

  • The same decoupling approach could be tested in other f(Q) or f(R) models to isolate whether mass enhancement is geometric or coupling-driven.
  • Gravitational-wave signals from mergers involving these deformed stars might carry distinguishable signatures from the two-parameter family.
  • Extending the construction to anisotropic or rotating configurations would test whether the mass-window enlargement survives rotation.

Load-bearing premise

Linear f(Q) gravity stays dynamically equivalent to general relativity at the geometric level, allowing β1 to rescale only the matter sector without new degrees of freedom or instabilities.

What would settle it

Observation of a compact star whose mass and radius violate the analytic compactness bound derived for admissible (ε, β1) while still satisfying the same regularity and matching conditions.

Figures

Figures reproduced from arXiv: 2602.18615 by Ranjan Sharma, Samstuti Chanda.

Figure 1
Figure 1. Figure 1: Structural distinction between GR-based decoupling and linear [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2-4
Figure 2-4. Figure 2-4 [PITH_FULL_IMAGE:figures/full_fig_p016_2-4.png] view at source ↗
Figure 2
Figure 2. Figure 2: Radial variation of the total energy den [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Radial variation of anisotropy ∆ for dif [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Radial variation of sound speeds ( [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Radial variation of the total radial pressure P tot r in GR and f(Q) gravity (with β1=0.8) in presence or absence of the decou￾pling parameter ϵ for assumed values of K = 2 and β2 = 0. PSR J0614−3329 f(Q) with decoupling; ϵ=0.2 Pure f(Q) Pure GR GR with decoupling; ϵ=0.2 f(Q) with decoupling; ϵ=1 GR with decoupling; ϵ=1 0 2 4 6 8 10 0 10 20 30 40 50 60 70 r (km) Pttot (MeV fm - 3 ) [PITH_FULL_IMAGE:figur… view at source ↗
Figure 11
Figure 11. Figure 11: Radial variation of the total tangen￾tial pressure P tot t in GR and f(Q) gravity (with β1=0.8) in presence or absence of the decou￾pling parameter ϵ for assumed values of K = 2 and β2 = 0. PSR J0614−3329 f(Q) with decoupling; ϵ=0.2 Pure f(Q) Pure GR GR with decoupling; ϵ=0.2 f(Q) with decoupling; ϵ=1 GR with decoupling; ϵ=1 0 2 4 6 8 10 0 5 10 15 20 r (km) Δ (MeV fm - 3 ) [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 13
Figure 13. Figure 13: Radial variation of the sound speeds [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Variation of the EOS in GR and f(Q) gravity (with β1=0.8) in presence or absence of the decoupling parameter ϵ for assumed values of K = 2 and β2 = 0 [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: M–R plots in GR and f(Q) gravity (with β1=0.8) in presence or absence of the decoupling parameter ϵ for assumed values of K = 2 and β2 = 0 (Set-I) [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Variation of compactness bound with ϵ for different values of K. The current study fits well in the context of the above developments. An in￾triguing feature of the present model is that for ϵ = 1, the maximum mass reaches ≈ 2.63 M⊙ for β1 = 0.9 and ≈ 2.8 M⊙ for β1 = 0.8, placing these configurations within the neutron star-black hole mass-gap region. In contrast, for the same geo￾metric deformation (i.e.… view at source ↗
read the original abstract

Recent multi-messenger observations, including gravitational wave detections of compact objects in the neutron star-black hole mass-gap region and precise measurements of high-mass pulsars, motivate mechanisms capable of enlarging the stellar mass window without arbitrarily stiffening the equation of state (EOS) toward the causal limit. In linear $f(Q)$ gravity of the form $f(Q)=\beta_1 Q+\beta_2$, the theory is dynamically equivalent to General Relativity at the geometric level and modifies stellar structure solely through a uniform rescaling of the matter sector governed by $\beta_1$. Consequently, linear $f(Q)$ alone does not introduce new geometric families of stellar solutions or alter classical compactness bounds. To overcome this structural limitation, we incorporate gravitational decoupling within an embedding class-I (Karmarkar) Vaidya-Tikekar configuration in linear $f(Q)$ gravity. While similar VT-based decoupling constructions exist in GR, the present framework introduces a controlled two-parameter deformation characterized by $(\epsilon,\beta_1)$: the decoupling parameter $\epsilon$ governs geometric deformation and EOS stiffness, whereas $\beta_1$ independently rescales the matter sector without altering the metric structure. This separation permits a direct comparison between GR and linear $f(Q)$ gravity at fixed geometric deformation, thereby isolating pure coupling-driven mass enhancement. We determine the admissible parameter domain from regularity, matching, causality and compactness requirements and derive an analytic compactness bound for the decoupled embedding class-I configuration. The combined action of $\epsilon$ and $\beta_1$ enlarges the accessible stellar mass window while preserving physical acceptability, allowing configurations compatible with recent high-mass pulsars and mass-gap candidates without exceeding causal limits.

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

1 major / 2 minor

Summary. The paper proposes a two-parameter deformation of embedding class-I Vaidya-Tikekar compact star solutions in linear f(Q) gravity, combining minimal geometric decoupling (controlled by ε, which deforms the metric and stiffens the effective EOS) with the coupling parameter β1 (which rescales the matter sector). It derives an analytic compactness bound and admissible domains for (ε, β1) from regularity, junction matching, causality (v_s^2 < 1), and compactness conditions, claiming that the combined action enlarges the accessible stellar mass window while preserving physical acceptability and allowing compatibility with high-mass pulsars and mass-gap candidates.

Significance. If the claimed separation of geometric and matter-sector effects holds, the construction supplies a controlled, falsifiable way to isolate coupling-driven mass enhancement from pure geometric deformation in modified gravity, with the analytic compactness bound providing a concrete, testable limit. This is a strength for applications to multi-messenger observations, though it rests on the persistence of linear f(Q)–GR equivalence under decoupling.

major comments (1)
  1. [Derivation of effective field equations and TOV structure] The central claim that β1 uniformly rescales the matter sector independently of the ε-driven geometric deformation assumes that linear f(Q) remains dynamically equivalent to GR once the extra anisotropic source θ_μν is introduced by minimal geometric decoupling. The non-metricity scalar Q for the deformed metric may acquire additional contributions not captured by a simple overall factor of β1, which could alter the effective TOV equation or violate uniform causality across the parameter domain. An explicit derivation of the decoupled field equations and verification that v_s^2 < 1 holds uniformly for all admissible (ε, β1) is required to support the separation.
minor comments (2)
  1. [Compactness bound derivation] Clarify in the text how the analytic compactness bound is obtained from the matching conditions and whether it reduces to the standard GR limit when ε = 0 and β1 = 1.
  2. [Numerical results and figures] Ensure that all plots of mass-radius relations explicitly state the fixed values of the remaining parameters and the range of the varying parameter.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The major comment raises a valid point about the need for explicit derivations to confirm the separation of effects under minimal geometric decoupling. We address this below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: The central claim that β1 uniformly rescales the matter sector independently of the ε-driven geometric deformation assumes that linear f(Q) remains dynamically equivalent to GR once the extra anisotropic source θ_μν is introduced by minimal geometric decoupling. The non-metricity scalar Q for the deformed metric may acquire additional contributions not captured by a simple overall factor of β1, which could alter the effective TOV equation or violate uniform causality across the parameter domain. An explicit derivation of the decoupled field equations and verification that v_s^2 < 1 holds uniformly for all admissible (ε, β1) is required to support the separation.

    Authors: We agree that an explicit derivation is necessary to rigorously support the claimed separation. In the linear f(Q) framework, the field equations for the deformed metric (with ε controlling the geometric deformation via the embedding class-I ansatz) reduce to β1 G_μν = 8π (T_μν + θ_μν), where the non-metricity scalar Q is computed directly from the full metric components and enters only linearly. Consequently, β1 provides a uniform rescaling of the total effective energy-momentum tensor without introducing ε-dependent cross terms that would alter the structure of the TOV equation. The sound-speed squared is obtained from the effective EOS and has been checked to satisfy v_s^2 < 1 uniformly over the admissible domain. In the revised manuscript we will add a dedicated appendix containing the full derivation of the decoupled field equations, the explicit form of the TOV equation, and the analytic/numeric verification of causality for representative (ε, β1) pairs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; parameters constrained independently by physical conditions

full rationale

The paper introduces ε as a geometric deformation parameter via minimal geometric decoupling on the Vaidya-Tikekar embedding class-I metric and β1 as a uniform rescaling factor from the linear f(Q) equivalence to GR. These are treated as free inputs and then restricted by explicit regularity, matching, causality (v_s^2 < 1), and compactness conditions derived from the field equations. No step reduces a prediction to a fit of the same inputs by construction, nor does any load-bearing claim rest on a self-citation chain that itself assumes the target result. The mass-window enlargement follows from solving the decoupled system and applying external bounds, remaining self-contained against observed pulsar masses and causal limits.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The model rests on standard assumptions of stellar structure in modified gravity and the validity of the chosen metric ansatz; no new entities are postulated.

free parameters (2)
  • ε
    Decoupling parameter controlling geometric deformation and effective EOS stiffness
  • β1
    Rescaling parameter in the linear f(Q) function that modifies the matter sector
axioms (3)
  • domain assumption Linear f(Q) gravity is dynamically equivalent to GR at the geometric level
    Invoked to justify that modifications occur only through uniform rescaling of the matter sector
  • domain assumption Karmarkar condition holds for the embedding class-I Vaidya-Tikekar metric
    Required to close the system for the interior solution
  • standard math Smooth matching to exterior Schwarzschild spacetime
    Standard junction conditions for stellar models

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Works this paper leans on

110 extracted references · 110 canonical work pages · 3 internal anchors

  1. [1]

    K. R. Karmarkar,Proc. Indian Acad. Sci. A27(1948) 56,https://doi.org/ 10.1007/BF03173443

    work page doi:10.1007/bf03173443 1948

  2. [2]

    P. Bhar, K. N. Singh and T. Manna,Int. J. Mod. Phys. D26(2017) 1750090, https://doi.org/10.1142/S0218271817500900

    work page doi:10.1142/s0218271817500900 2017

  3. [3]

    P. Bhar, K. N. Singh, F. Rahaman, N. Pant and S. Banerjee,Int. J. Mod. Phys. D26(2017) 1750078,https://doi.org/10.1142/S021827181750078X

    work page doi:10.1142/s021827181750078x 2017

  4. [4]

    P. Bhar, S. K. Maurya, Y. K. Gupta and T. Manna,Eur. Phys. J. A52(2016) 312,https://doi.org/10.1140/epja/i2016-16312-x

    work page doi:10.1140/epja/i2016-16312-x 2016

  5. [5]

    P. Bhar, K. N. Singh, N. Sarkar and F. Rahaman,Eur. Phys. J. C77(2017) 596,https://doi.org/10.1140/epjc/s10052-017-5149-2

    work page doi:10.1140/epjc/s10052-017-5149-2 2017

  6. [6]

    S. K. Maurya, Y. K. Gupta, S. Ray and B. Dayanandan,Eur. Phys. J. C75 (2015) 389,https://doi.org/10.1140/epjc/s10052-015-3615-2

    work page doi:10.1140/epjc/s10052-015-3615-2 2015

  7. [7]

    S. K. Maurya, B. S. Ratanpal and M. Govender,Ann. Phys.382(2017) 36–49, https://doi.org/10.1016/j.aop.2017.04.008

    work page doi:10.1016/j.aop.2017.04.008 2017

  8. [9]

    B. S. Ratanpal, V. O. Thomas and R. Patel,New Astron.100(2023) 101970, https://doi.org/10.1016/j.newast.2022.101970

    work page doi:10.1016/j.newast.2022.101970 2023

  9. [10]

    K. N. Singh, P. Bhar and N. Pant,Int. J. Mod. Phys. D25(2016) 1650099, https://doi.org/10.1142/S0218271816500991

    work page doi:10.1142/s0218271816500991 2016

  10. [11]

    K. N. Singh, S. K. Maurya, R. K. Bisht and N. Pant,Chin. Phys. C44(2020) 035101,https://doi.org/10.1088/1674-1137/44/3/035101

    work page doi:10.1088/1674-1137/44/3/035101 2020

  11. [12]

    P. D. Makalo, J. M. Sunzu and J. M. Mkenyeleye,New Astron.98(2023) 101935,https://doi.org/10.1016/j.newast.2022.101935

    work page doi:10.1016/j.newast.2022.101935 2023

  12. [13]

    Estevez-Delgado, J

    G. Estevez-Delgado, J. Estevez-Delgado, R. Soto-Espitia, A. Rend´ on Romero and J. M. Paulin-Fuentes,Commun. Theor. Phys.75(2023) 085403,https: //doi.org/10.1088/1572-9494/acded8

    work page doi:10.1088/1572-9494/acded8 2023

  13. [14]

    G´ omez-Leyton, H

    Y. G´ omez-Leyton, H. Javaid, L. S. Rocha and F. Tello-Ortiz,Phys. Scr.96 (2021) 025001,https://doi.org/10.1088/1402-4896/abcce3

    work page doi:10.1088/1402-4896/abcce3 2021

  14. [15]

    Sutar, K

    B. Sutar, K. L. Mahanta and R. R. Sahoo,Eur. Phys. J. Plus138(2023) 1115, https://doi.org/10.1140/epjp/s13360-023-04763-y

    work page doi:10.1140/epjp/s13360-023-04763-y 2023

  15. [16]

    Fayyaz and M

    I. Fayyaz and M. F. Shamir,Chin. J. Phys.66(2020) 287–296,https://doi. org/10.1016/j.cjph.2020.05.018

    work page doi:10.1016/j.cjph.2020.05.018 2020

  16. [17]

    Malik, A

    A. Malik, A. Hussain, M. Ahmad and M. F. Shamir,Eur. Phys. J. Plus139 (2024) 535,https://doi.org/10.1140/epjp/s13360-024-05277-x

    work page doi:10.1140/epjp/s13360-024-05277-x 2024

  17. [18]

    Sharif and M

    M. Sharif and M. Z. Gul,Gen. Relativ. Gravit.55(2023) 10,https://doi. org/10.1007/s10714-022-03062-8

    work page doi:10.1007/s10714-022-03062-8 2023

  18. [19]

    Zubair, S

    M. Zubair, S. Waheed, M. F. Jamal and G. Mustafa,Results Phys.24(2021) 104787,https://doi.org/10.1016/j.rinp.2021.104787

    work page doi:10.1016/j.rinp.2021.104787 2021

  19. [20]

    Chanda, R

    S. Chanda, R. Sharma and S. D. Maharaj,To appear in Grav. & Cosmol- ogy (2026); arXiv:2602.15486 [gr-qc] (2026).https://arxiv.org/abs/2602. 15486

    work page arXiv 2026

  20. [21]

    J. B. Jim´ enez, L. Heisenberg and T. S. Koivisto,Universe5(2019) 173,https: //doi.org/10.3390/universe5070173

    work page doi:10.3390/universe5070173 2019

  21. [22]

    Lazkoz, F

    R. Lazkoz, F. S. N. Lobo, M. Ortiz-Ba˜ nos and V. Salzano,Phys. Rev. D100 (2019) 104027,https://doi.org/10.1103/PhysRevD.100.104027

    work page doi:10.1103/physrevd.100.104027 2019

  22. [23]

    Ayuso, R

    I. Ayuso, R. Lazkoz and V. Salzano,Phys. Rev. D103(2021) 063505,https: //doi.org/10.1103/PhysRevD.103.063505

    work page doi:10.1103/physrevd.103.063505 2021

  23. [24]

    B. J. Barros, T. Barreiro, T. Koivisto and N. J. Nunes,Phys. Dark Univ.30 (2020) 100616,https://doi.org/10.1016/j.dark.2020.100616

    work page doi:10.1016/j.dark.2020.100616 2020

  24. [25]

    W. Wang, H. Chen and T. Katsuragawa,Phys. Rev. D105(2022) 024060, https://doi.org/10.1103/PhysRevD.105.024060

    work page doi:10.1103/physrevd.105.024060 2022

  25. [26]

    S. K. Maurya, A. Errehymy, Ksh. Newton Singh, O. Donmez, K. Sooppy Nisar and M. Mahmoud,Phys. Dark Univ.46(2024) 101619,https://doi.org/ 10.1016/j.dark.2024.101619

    work page doi:10.1016/j.dark.2024.101619 2024

  26. [27]

    Adeel, M

    M. Adeel, M. Zeeshan Gul, S. Rani and A. Jawad,Mod. Phys. Lett. A38 (2023) 2350152,https://doi.org/10.1142/S0217732323501523

    work page doi:10.1142/s0217732323501523 2023

  27. [28]

    S. K. Maurya, A. Errehymy, G.-E. Vˆ ılcu, H. I. Alrebdi, K. Sooppy Nisar and April 28, 2026 1:0 SR˙26˙IJGMMP˙R1 32REFERENCES A.-H. Abdel-Aty,Class. Quantum Grav.41(2024) 115009,https://doi.org/ 10.1088/1361-6382/ad3b5f

    work page doi:10.1088/1361-6382/ad3b5f 2026

  28. [29]

    S. K. Maurya, Ksh. Newton Singh, G. Mustafa, M. Govender, A. Errehymy and A. Aziz,JCAP09(2024) 048,https://doi.org/10.1088/1475-7516/ 2024/09/048

    work page doi:10.1088/1475-7516/ 2024

  29. [30]

    Errehymy, S

    A. Errehymy, S. K. Maurya, K. Boshkayev, A.-H. Abdel-Aty, H. I. Alrebdi and M. Mahmoud,Phys. Dark Univ.46(2024) 101622,https://doi.org/ 10.1016/j.dark.2024.101622

    work page doi:10.1016/j.dark.2024.101622 2024

  30. [31]

    S. Paul, J. Kumar, S. K. Maurya, S. Choudhary and S. Kiroriwala, arXiv:2409.16334 [gr-qc] (2024),https://arxiv.org/abs/2409.16334

    work page arXiv 2024

  31. [32]

    Kumar, S

    J. Kumar, S. Paul, S. K. Maurya, S. Chaudhary and S. Kiroriwal,Phys. Dark Univ.47(2025) 101764,https://doi.org/10.1016/j.dark.2024.101764

    work page doi:10.1016/j.dark.2024.101764 2025

  32. [33]

    Physical properties and the maximum compactness bound of a class of compact stars in $f(Q)$ gravity

    A. Ghosh, A. Paul, R. Sharma and S. Chanda, arXiv:2409.04487 [gr-qc] (2024),Mod. Phys. Lett. A(2026) 2650098 .https://doi.org/10.1142/ S0217732326500987

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  33. [34]

    Awais and M

    M. Awais and M. Azam, arXiv:2503.20792 (2025),https://doi.org/10. 48550/arXiv.2503.20792

    work page arXiv 2025

  34. [35]

    P. Bhar, A. Malik and A. Almas,Chin. J. Phys.88(2024) 839–856,https: //doi.org/10.1016/j.cjph.2024.02.016

    work page doi:10.1016/j.cjph.2024.02.016 2024

  35. [36]

    S. Rani, M. Adeel, M. Z. Gul and A. Jawad,Int. J. Geom. Methods Mod. Phys. 21(2024) 2450033,https://doi.org/10.1142/S0219887824500336

    work page doi:10.1142/s0219887824500336 2024

  36. [37]

    Bhar and J

    P. Bhar and J. M. Z. Pretel,Phys. Dark Univ.42(2023) 101322,https: //doi.org/10.1016/j.dark.2023.101322

    work page doi:10.1016/j.dark.2023.101322 2023

  37. [38]

    P. Bhar, A. Errehymy and S. Ray,Eur. Phys. J. C83(2023) 1151,https: //doi.org/10.1140/epjc/s10052-023-12340-x

    work page doi:10.1140/epjc/s10052-023-12340-x 2023

  38. [39]

    P. Bhar, S. Pradhan, A. Malik and A. Almas,Eur. Phys. J. C83(2023) 646, https://doi.org/10.1140/epjc/s10052-023-11745-y

    work page doi:10.1140/epjc/s10052-023-11745-y 2023

  39. [40]

    S. K. Maurya, M. K. Jasim, A. Errehymy, K. Sooppy Nisar, M. Mahmoud and R. Nag,Fortschr. Phys.72(2024) 202300229,https://doi.org/10.1002/ prop.202300229

    work page 2024

  40. [41]

    Kumar, S

    J. Kumar, S. K. Maurya, S. Chaudhary, A. Errehymy, K. Myrzakulov, and Z. Umbetova,Physics of the Dark Universe45(2024) 101593,https://doi. org/10.1016/j.dark.2024.101593

    work page doi:10.1016/j.dark.2024.101593 2024

  41. [42]

    S. K. Maurya, A. Ashraf, F. Al Khayari, A. Errehymy, and A.-H. Abdel- Aty,European Physical Journal C84(2024) 986,https://doi.org/10.1140/ epjc/s10052-024-13334-z

    work page 2024

  42. [43]

    S. Rani, M. Adeel, M. Z. Gul, A. Jawad, and S. Shaymatov,Physics of the Dark Universe47(2024) 101754,https://doi.org/10.1016/j.dark.2024. 101754

    work page doi:10.1016/j.dark.2024 2024

  43. [44]

    M. R. Shahzad, W. Habib, H. Nazar, S. Waheed, and M. A. R. Sakhi,Eu- ropean Physical Journal Plus140(2025) 469,https://doi.org/10.1140/ epjp/s13360-025-06308-x April 28, 2026 1:0 SR˙26˙IJGMMP˙R1 REFERENCES33

    work page 2025

  44. [45]

    De and T.-H

    A. De and T.-H. Loo,Class. Quantum Grav.40(2023) 115007.https:// iopscience.iop.org/article/10.1088/1361-6382/accef7

    work page doi:10.1088/1361-6382/accef7 2023

  45. [46]

    Heisenberg and C

    L. Heisenberg and C. Pastor-Marcos, arXiv:2512.03037 [gr-qc] (2025).https: //arxiv.org/abs/2512.03037

    work page arXiv 2025

  46. [47]

    Casadio, J

    R. Casadio, J. Ovalle and R. da Rocha,Class. Quantum Grav.32(2015) 215020,https://doi.org/10.1088/0264-9381/32/21/215020

    work page doi:10.1088/0264-9381/32/21/215020 2015

  47. [48]

    Ovalle,Phys

    J. Ovalle,Phys. Rev. D95(2017) 104019,https://doi.org/10.1103/ PhysRevD.95.104019

    work page 2017

  48. [49]

    Contreras and P

    E. Contreras and P. Bargue˜ no,Eur. Phys. J. C78(2018) 558,https://doi. org/10.1140/epjc/s10052-018-6048-x

    work page doi:10.1140/epjc/s10052-018-6048-x 2018

  49. [50]

    Ovalle, C

    J. Ovalle, C. Posada and Z. Stuchl´ ık,Class. Quantum Grav.36(2019) 205010

    work page 2019

  50. [51]

    V. A. Torres-S´ anchez and E. Contreras,Eur. Phys. J. C79(2019) 829,https: //doi.org/10.1140/epjc/s10052-019-7341-z

    work page doi:10.1140/epjc/s10052-019-7341-z 2019

  51. [52]

    Abell´ an,´A

    G. Abell´ an,´A. Rinc´ on, E. Fuenmayor and E. Contreras,Eur. Phys. J. Plus 135(2020) 606,https://doi.org/10.1140/epjp/s13360-020-00589-0

    work page doi:10.1140/epjp/s13360-020-00589-0 2020

  52. [53]

    Sultana,Symmetry13(2021) 1598, https://doi.org/10.3390/sym13091598

    J. Sultana,Symmetry13(2021) 1598, https://doi.org/10.3390/sym13091598

    work page doi:10.3390/sym13091598 2021

  53. [54]

    S. K. Maurya, K. N. Singh and B. Dayanandan,Eur. Phys. J. C80(2020) 448,https://doi.org/10.1140/epjc/s10052-020-8005-8

    work page doi:10.1140/epjc/s10052-020-8005-8 2020

  54. [55]

    S. K. Maurya, A. Errehymy, Ksh. Newton Singh, G. Mustafa and S. Ray, arXiv:2501.00735 [astro-ph.HE] (2025),https://doi.org/10.48550/arXiv. 2501.00735

    work page internal anchor Pith review doi:10.48550/arxiv 2025

  55. [57]

    S. K. Maurya and F. Tello-Ortiz,Phys. Dark Univ.29(2020) 100577,https: //doi.org/10.1016/j.dark.2020.100577

    work page doi:10.1016/j.dark.2020.100577 2020

  56. [60]

    S. K. Maurya, J. Kumar and S. Kiroriwal,J. High Energy Astrophys.44(2024) 194–209,https://doi.org/10.1016/j.jheap.2024.09.012

    work page doi:10.1016/j.jheap.2024.09.012 2024

  57. [61]

    Mushtaq, X

    F. Mushtaq, X. Tiecheng, A. Ditta, G. Mustafa and S. K. Maurya,Commun. Theor. Phys.77(2025) 025402,https://doi.org/10.1088/1572-9494/ ad7c36

    work page doi:10.1088/1572-9494/ 2025

  58. [62]

    Pradhan, S

    S. Pradhan, S. K. Maurya, P. K. Sahoo and G. Mustafa,Fortschr. Phys.72 (2024) 202400092,https://doi.org/10.1002/prop.202400092

    work page doi:10.1002/prop.202400092 2024

  59. [64]

    S. K. Maurya, A. Errehymy, K. N. Singh, G. Mustafa and S. Ray,Eur. Phys. J. C83(2023) 968,https://doi.org/10.1140/epjc/s10052-023-12127-0

    work page doi:10.1140/epjc/s10052-023-12127-0 2023

  60. [65]

    S. K. Maurya, Ksh. Newton Singh, M. Govender and S. Ray,Fortschr. Phys. 71(2023) 202300023,https://doi.org/10.1002/prop.202300023

    work page doi:10.1002/prop.202300023 2023

  61. [67]

    Al Busaidi, J

    A. Al Busaidi, J. Al Hosni, S. K. Maurya, A. Al Zarii, T. Al-Kasbi, M. Al Omairi, B. Al Zakwani and M. K. Jasim,Phys. Scr.98(2023) 075302,https: //doi.org/10.1088/1402-4896/acd442

    work page doi:10.1088/1402-4896/acd442 2023

  62. [68]

    S. K. Maurya, A. Errehymy, M. Govender, G. Mustafa and S. Ray,Eur. Phys. J. C83(2023) 348,https://doi.org/10.1140/epjc/s10052-023-11507-w

    work page doi:10.1140/epjc/s10052-023-11507-w 2023

  63. [69]

    M. K. Jasim, Ksh. Newton Singh, A. Errehymy, S. K. Maurya and M. V. Mandke,Universe9(2023) 208,https://doi.org/10.3390/ universe9050208

    work page 2023

  64. [70]

    Dayanandan, S

    B. Dayanandan, S. K. Maurya, S. T. T. Smitha and J. P. Maurya,Chinese J. Phys.82(2023) 155–170,https://doi.org/10.1016/j.cjph.2022.12.013

    work page doi:10.1016/j.cjph.2022.12.013 2023

  65. [71]

    M. A. Habsi, S. K. Maurya, S. A. Badri, S. Ray and B. Sahoo,Eur. Phys. J. C83(2023) 286,https://doi.org/10.1140/epjc/s10052-023-11420-2

    work page doi:10.1140/epjc/s10052-023-11420-2 2023

  66. [73]

    Al Hadhrami, S

    M. Al Hadhrami, S. K. Maurya, Z. Al Amri, M. Govender, A. Erre- hymy and A. Aziz,Pramana97(2023) 13,https://doi.org/10.1007/ s12043-022-02486-w

    work page 2023

  67. [75]

    S. K. Maurya, M. Govender, G. Mustafa, M. Shoaib and S. Ray,Eur. Phys. J. C82(2022) 1006,https://doi.org/10.1140/epjc/s10052-022-10935-4

    work page doi:10.1140/epjc/s10052-022-10935-4 2022

  68. [76]

    S. K. Maurya, B. Mishra, S. Ray and R. Nag,Chinese Phys. C46(2022) 105105,https://doi.org/10.1088/1674-1137/ac7d45

    work page doi:10.1088/1674-1137/ac7d45 2022

  69. [77]

    S. K. Maurya, A. Al Saadi, W. Al Amri, S. Al Hosni and R. Al Sharyani,Phys. Scr.97(2022) 105002,https://doi.org/10.1088/1402-4896/ac8d39

    work page doi:10.1088/1402-4896/ac8d39 2022

  70. [78]

    S. K. Maurya, A. Banerjee, A. Pradhan, M. Govender and A. Erre- hymy,Eur. Phys. J. C82(2022) 552,https://doi.org/10.1140/epjc/ s10052-022-10496-6

    work page doi:10.1140/epjc/ 2022

  71. [79]

    S. K. Maurya, A. Errehymy, R. Nag and M. Daoud,Fortschr. Phys.70(2022) 202200041,https://doi.org/10.1002/prop.202200041

    work page doi:10.1002/prop.202200041 2022

  72. [80]

    S. K. Maurya, A. Errehymy, M. K. Jasim, G. Mustafa and S. Ray,Eur. Phys. J. C83(2023) 317,https://doi.org/10.1140/epjc/s10052-023-11447-5

    work page doi:10.1140/epjc/s10052-023-11447-5 2023

  73. [81]

    S. K. Maurya, A. Aziz, K. N. Singh, G. Mustafa and S. Ray,Eur. Phys. J. C April 28, 2026 1:0 SR˙26˙IJGMMP˙R1 REFERENCES35 84(2024) 296,https://doi.org/10.1140/epjc/s10052-024-12626-8

    work page doi:10.1140/epjc/s10052-024-12626-8 2026

  74. [82]

    S. K. Maurya, A. Errehymy, Ksh. Newton Singh, A. Aziz, S. Hansraj and S. Ray,Astrophys. J.972(2024) 175,https://doi.org/10.3847/1538-4357/ ad5cf1

    work page doi:10.3847/1538-4357/ 2024

  75. [83]

    S. K. Maurya, Ksh. Newton Singh, M. Govender, G. Mustafa and S. Ray,Astro- phys. J. Suppl. Ser.269(2023) 35,https://doi.org/10.3847/1538-4365/ ad0154

    work page doi:10.3847/1538-4365/ 2023

  76. [84]

    S. K. Maurya, Ksh. Newton Singh, S. V. Lohakare and B. Mishra,Fortschr. Phys.70(2022) 202200061,https://doi.org/10.1002/prop.202200061

    work page doi:10.1002/prop.202200061 2022

  77. [85]

    Mustafa, A

    G. Mustafa, A. Ditta, S. Mumtaz, S. K. Maurya and D. Sofuo˘ glu,Chin. J. Phys.88(2024) 938–954,https://doi.org/10.1016/j.cjph.2024.02.022

    work page doi:10.1016/j.cjph.2024.02.022 2024

  78. [86]

    S. Paul, J. Kumar and S. K. Maurya,Int. J. Geom. Methods Mod. Phys.25 (2025) 50181,https://doi.org/10.1142/S0219887825501816

    work page doi:10.1142/s0219887825501816 2025

  79. [87]

    Newton Singh, P

    Ksh. Newton Singh, P. Bhar, M. Laishram and F. Rahaman,Heliyon5(2019) e01929,https://doi.org/10.1016/j.heliyon.2019.e01929

    work page doi:10.1016/j.heliyon.2019.e01929 2019

  80. [88]

    Schwarzschild,Sitz

    K. Schwarzschild,Sitz. Deut. Akad. Wiss. Math. Phys. Berlin24(1916) 424

    work page 1916

Showing first 80 references.