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arxiv: 2605.27324 · v1 · pith:KBL3M6COnew · submitted 2026-05-26 · 🌀 gr-qc

Reconstructing the cosmic expansion in f(R, L_(m)) gravity via parametrized Hubble function constraints

Khomesh R. Patle , G. P. Singh This is my paper

Pith reviewed 2026-06-29 15:33 UTC · model grok-4.3

classification 🌀 gr-qc
keywords f(R, L_m) gravityparametrized Hubble parametercosmic chronometersPantheon supernovaedeceleration parameterenergy conditionsthermodynamic viabilitystatefinder diagnostics
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The pith

Three Hubble parametrizations in f(R, L_m) gravity fit cosmic chronometer and supernova data while confirming thermodynamic viability.

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

The paper constructs three cosmological models by embedding parametrized forms of the Hubble rate H(z) inside the modified gravity theory with the specific choice f(R, L_m) = R/2 + L_m^λ. Bayesian fits to cosmic chronometer and joint Pantheon datasets determine the model parameters, after which the evolution of the deceleration parameter, energy density, pressure, equation of state, energy conditions, and statefinder diagnostics is examined. Thermodynamic consistency is checked via the behavior of temperature and entropy density, and the derived age of the universe is compared with observations. A sympathetic reader would care because the approach supplies concrete, observationally constrained alternatives to standard cosmology that incorporate possible curvature-matter coupling without separate dark-energy fields.

Core claim

Within f(R, L_m) gravity using the functional form f = R/2 + L_m^λ, three distinct redshift-dependent parametrizations of the Hubble parameter are introduced and constrained via chi-squared minimization on cosmic chronometer and Pantheon data. The resulting models describe a universe whose deceleration parameter changes sign at late times, whose equation-of-state parameter approaches values consistent with acceleration, and whose energy conditions remain satisfied. Temperature and entropy density evolve in a manner that preserves thermodynamic viability, while the computed age of the universe matches late-time astronomical estimates.

What carries the argument

The functional form f(R, L_m) = R/2 + L_m^λ together with three specific H(z) parametrizations that allow direct reconstruction of the expansion history and subsequent derivation of all cosmological quantities from the modified Friedmann equations.

Load-bearing premise

The chosen functional form of f(R, L_m) and the three specific Hubble parametrizations are sufficient to represent the full dynamics of the modified gravity theory without hidden inconsistencies that would invalidate the fits.

What would settle it

A high-redshift Hubble measurement lying outside the 1-sigma bands predicted by all three models after the current CC+Pantheon fits would falsify the reconstruction.

Figures

Figures reproduced from arXiv: 2605.27324 by G. P. Singh, Khomesh R. Patle.

Figure 1
Figure 1. Figure 1: Comparison of the best-fit H(z) profiles of the proposed models with the ΛCDM model. 5.2 Pantheon supernova observational dataset In the present analysis, we further employ the Pantheon compilation consisting of 1048 Type Ia supernova (SNIa) data points distributed over the redshift interval 0.01 < z < 2.26 [79]. The Pantheon sample is con￾structed by combining observations from several major supernova sur… view at source ↗
Figure 2
Figure 2. Figure 2: For Model-I: Marginalized 1σ and 2σ confidence contours together with the median estimates of H0, ζ and η obtained from the joint dataset. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: For Model-II: Marginalized 1σ and 2σ confidence contours together with the median estimates of H0, ζ and η obtained from the joint dataset. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For Model-III: Marginalized 1σ and 2σ confidence contours together with the median estimates of H0, ζ and η obtained from the joint dataset. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: For Model-I: Plot of q(z) with z. −        − −  − − [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: For Model-II: Plot of q(z) with z. expansion stage toward the currently accelerating epoch. By employing the median best-fit values of the model parameters, the present values of the deceleration parameter for Model-I are obtained as q0 = −0.5323 for the CC dataset and q0 = −0.5338 for the joint dataset. In the case of Model-II, the corresponding values are q0 = −0.5108 and q0 = −0.5254, while Model-III yi… view at source ↗
Figure 7
Figure 7. Figure 7: For Model-III: Plot of q(z) with z. −        [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: For Model-I: Plot of energy density (ρ) with z. −        − − − − [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: For Model-II: Plot of energy density (ρ) with z. −        − − [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: For Model-III: Plot of energy density (ρ) with z. −        − − [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: For Model-I: Plot of EoS parameter (ω) with z. −       [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: For Model-II: Plot of EoS parameter (ω) with z. 6.3 Analysis of energy conditions The point-wise energy conditions, defined at a given spacetime point and governed solely by the stress energy tensor, can be written as [89–91]: • NEC: The null energy condition is satisfied when the relation ρeff + peff ≥ 0 holds, indicating a non￾negative combination of the effective energy density and pressure. • WEC: The… view at source ↗
Figure 16
Figure 16. Figure 16: For Model-III: Plot of EoS parameter (ω) with z. • SEC: The strong energy condition holds when the inequalities ρeff + peff ≥ 0 and ρeff + 3peff ≥ 0 are simultaneously satisfied. The SEC, characterized by the inequality ρeff +3peff ≥ 0, is intimately connected to the Raychaudhuri equa￾tion and plays a central role in governing the dynamical evolution of the universe [92]. Within the standard cosmological … view at source ↗
Figure 17
Figure 17. Figure 17: For Model-I: Plot of the evolution of energy condition components with z. −        [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: For Model-II: Plot of the evolution of energy condition components with z. statefinder parameters {r,s} are defined as follows r = ... a aH3 = q+2q 2 + (1+z) dq dz , (45) s = r −1 3(q− 1 2 ) , where q 6= 1 2 . (46) The evolutionary dynamics of different dark energy models discussed in the literature can be efficiently char￾acterized through the statefinder pair {r,s}: • In the Chaplygin gas (CG) scenario,… view at source ↗
Figure 19
Figure 19. Figure 19: For Model-III: Plot of the evolution of energy condition components with z. − − − − −            [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: For Model-I: Plot of s and r plane. 6.5 Thermodynamic evolution of the universe and entropy analysis To study the thermodynamic aspects of the universe in f(R,Lm) gravity, we employ the laws of thermody￾namics, starting with the first law for an ideal fluid enclosed in a volume V [95, 96]. T ds = d(Vρ) + pdV, (47) we can rewrite it as follows: T ds = d(V(ρ + p))−V d p, (48) and using the thermodynamic rel… view at source ↗
Figure 21
Figure 21. Figure 21: For Model-II: Plot of s and r plane. − − − −            [PITH_FULL_IMAGE:figures/full_fig_p021_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: For Model-III: Plot of s and r plane. −        "            " !      !      [PITH_FULL_IMAGE:figures/full_fig_p021_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: For Model-I: Plot of temperature with z. −        [PITH_FULL_IMAGE:figures/full_fig_p021_23.png] view at source ↗
Figure 25
Figure 25. Figure 25: For Model-II: Plot of temperature with z. −        [PITH_FULL_IMAGE:figures/full_fig_p022_25.png] view at source ↗
Figure 27
Figure 27. Figure 27: For Model-III: Plot of temperature with z. −        [PITH_FULL_IMAGE:figures/full_fig_p022_27.png] view at source ↗
read the original abstract

We probe the cosmic expansion scenario within the framework of $f(R, L_{m})$ gravity by employing a well-motivated functional form of $f(R, L_{m}) = \frac{R}{2} + L_{m}^{\lambda}$. Specifically, we introduce three novel cosmological models formulated in terms of the redshift-dependent Hubble parameter $H(z)$, offering deeper insights into the underlying cosmic dynamics. The models are further utilized to investigate the expansion history of the universe and the evolution of several cosmological parameters. By using the Bayesian methods based on the $\chi^{2}$-minimization technique, the median values of the model parameters are determined for the cosmic chronometer (CC) and joint (CC+Pantheon) datasets. A comprehensive study of the deceleration parameter, energy density, pressure and the equation of state parameter is carried out to understand the universe's evolution. Additionally, the validity of the energy conditions and the behavior of the statefinder diagnostic are thoroughly examined. Finally, the thermodynamic viability of the models is confirmed through the evolution of temperature and entropy density, and the estimated age of the universe further exemplifies good agreement with late-time astronomical observations.

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

2 major / 2 minor

Summary. The paper examines cosmic expansion within f(R, L_m) gravity using the specific form f(R, L_m) = R/2 + L_m^λ. It introduces three parametrized H(z) models, constrains their parameters via χ² minimization on cosmic chronometer (CC) and joint CC+Pantheon datasets using Bayesian methods, and then derives the deceleration parameter, energy density, pressure, equation-of-state parameter, energy conditions, statefinder diagnostics, temperature and entropy density evolution, and universe age from the best-fit H(z).

Significance. If the central fits are robust and the derived quantities are correctly computed, the work supplies concrete observational bounds on a non-minimally coupled modified-gravity model and shows consistency with late-time data. The provision of explicit H(z) parametrizations and joint-dataset constraints is a concrete contribution, though the predictive power is reduced because most examined quantities follow algebraically from the fitted H(z) rather than constituting independent tests.

major comments (2)
  1. [Abstract and thermodynamic analysis] Abstract and thermodynamic analysis: the claim that thermodynamic viability is confirmed via the evolution of temperature and entropy density assumes the standard relations T ∝ 1/a and s ∝ a³ (or equivalent) that follow from ∇_μ T^μν = 0. The chosen f(R, L_m) = R/2 + L_m^λ produces a non-zero divergence of the effective stress-energy tensor, implying a modified continuity equation with an extra force term. No correction for this non-conservation appears to be applied to the reported T(z) and s(z) curves, rendering the viability confirmation insecure for the quoted best-fit parameters.
  2. [Abstract and sections on cosmological parameters] Abstract and sections on cosmological parameters: the reported evolution of the deceleration parameter, equation-of-state parameter, and energy conditions is obtained by direct substitution of the fitted H(z) parameters into the standard expressions. These quantities are therefore algebraic consequences of the data fit rather than independent predictions of the modified-gravity dynamics, weakening the claim that the models successfully describe the underlying cosmic evolution beyond the Hubble fit itself.
minor comments (2)
  1. [Abstract] The abstract states that median parameter values are determined but provides no explicit error budgets, covariance matrices, or discussion of possible post-hoc selection effects in the three chosen H(z) forms.
  2. [Model definitions] Notation for the three H(z) parametrizations should be introduced with explicit functional forms and the number of free parameters per model stated clearly before the fitting results are presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below.

read point-by-point responses

  1. Referee: [Abstract and thermodynamic analysis] Abstract and thermodynamic analysis: the claim that thermodynamic viability is confirmed via the evolution of temperature and entropy density assumes the standard relations T ∝ 1/a and s ∝ a³ (or equivalent) that follow from ∇_μ T^μν = 0. The chosen f(R, L_m) = R/2 + L_m^λ produces a non-zero divergence of the effective stress-energy tensor, implying a modified continuity equation with an extra force term. No correction for this non-conservation appears to be applied to the reported T(z) and s(z) curves, rendering the viability confirmation insecure for the quoted best-fit parameters.

    Authors: We agree that the non-minimal coupling implies a non-zero divergence of the effective stress-energy tensor and a modified continuity equation. The reported T(z) and s(z) curves were obtained with the standard relations without explicit correction for the extra force term. We will revise the abstract, the thermodynamic section, and the associated discussion to qualify the analysis as an approximate check and to note this limitation explicitly. revision: yes

  2. Referee: [Abstract and sections on cosmological parameters] Abstract and sections on cosmological parameters: the reported evolution of the deceleration parameter, equation-of-state parameter, and energy conditions is obtained by direct substitution of the fitted H(z) parameters into the standard expressions. These quantities are therefore algebraic consequences of the data fit rather than independent predictions of the modified-gravity dynamics, weakening the claim that the models successfully describe the underlying cosmic evolution beyond the Hubble fit itself.

    Authors: We acknowledge that the deceleration parameter, EoS parameter, and energy conditions are computed from the best-fit H(z) using the standard (GR) expressions. These quantities therefore test consistency of the fitted expansion history with late-time data rather than constituting independent dynamical predictions derived from the modified field equations. We will revise the abstract and the relevant sections to clarify this distinction and to frame the results as constraints on the parametrized models within the f(R, L_m) framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper parametrizes H(z), fits its parameters to CC and joint CC+Pantheon data via chi-squared minimization, then computes derived quantities (q(z), w(z), energy conditions, T(z), s(z), age) by direct substitution into the standard expressions. This is standard cosmological reconstruction from a fitted expansion history rather than a self-referential derivation. No quoted equations show a fitted input renamed as an independent prediction, no self-citation load-bearing on a uniqueness theorem, and no ansatz smuggled via prior work by the same authors. The thermodynamic analysis uses the usual T and s evolution formulas; any mismatch with the model's non-conservation is a correctness issue, not a circular reduction of the claimed viability to the input fit by construction. The derivation chain remains self-contained against external data.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Central claims rest on the ad-hoc functional form, the three chosen H(z) parametrizations, and the assumption that chi-squared minima on the cited datasets yield physically meaningful constraints; no independent evidence is supplied for any of these choices.

free parameters (2)
  • λ
    Exponent in L_m^λ term, adjusted to data
  • H(z) model parameters
    Free coefficients in each of the three redshift-dependent Hubble parametrizations, fitted via chi-squared
axioms (2)
  • domain assumption FLRW metric describes the background cosmology
    Invoked implicitly when writing H(z) and deriving cosmological parameters
  • ad hoc to paper f(R, L_m) = R/2 + L_m^λ is the appropriate modified-gravity action
    Chosen without derivation or comparison to other forms

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discussion (0)

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