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arxiv: 2604.19849 · v1 · submitted 2026-04-21 · 🌀 gr-qc

Cosmology of the interacting Tsallis holographic dark energy in f(R,T) gravity framework

Sanjeeda Sultana , Chayan Ranjit , Surajit Chattopadhyay , Ertan G\"udekli This is my paper

Pith reviewed 2026-05-10 02:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Tsallis holographic dark energyf(R,T) gravityinteracting dark energyHubble horizon cutoffstatefinder diagnosticsLambda CDM cosmologycosmological constraints
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The pith

Interacting Tsallis holographic dark energy reaches the Lambda CDM fixed point in f(R,T) gravity models.

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

The paper studies the evolution of Tsallis holographic dark energy that interacts with pressureless dark matter inside two f(R,T) gravity setups, both using the Hubble horizon as the infrared cutoff. It computes the equation-of-state and deceleration parameters to track the transition to late-time acceleration, then applies statefinder, Om(z), r-q, and wDE-w'DE diagnostics to show how the models approach the standard cosmology. Model coefficients are fitted to observational data via chi-squared minimization, yielding concrete parameter bounds for each framework.

Core claim

In both the linear f(R,T) = μR + νT and the quadratic f(R,T) = R + γR² + ξT models, the interacting THDE with Hubble cutoff produces an equation-of-state parameter that drives acceleration, a negative deceleration parameter at late times, and statefinder trajectories that converge to the Lambda CDM fixed point (1,0); the same diagnostics remain consistent when the models are constrained by chi-squared fits to observational data sets.

What carries the argument

The interaction term between Tsallis holographic dark energy density and dark matter, together with the Hubble horizon cutoff, inserted into the f(R,T) field equations to modify the Friedmann dynamics.

If this is right

  • The deceleration parameter turns negative at low redshift, reproducing the observed cosmic acceleration.
  • Statefinder and wDE-w'DE planes both approach the Lambda CDM fixed point for the fitted parameter values.
  • Om(z) and r-q diagnostics remain consistent with current data in both linear and quadratic f(R,T) cases.
  • The chi-squared procedure supplies explicit bounds on the coefficients μ, ν, γ, and ξ.

Where Pith is reading between the lines

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

  • The same diagnostic suite could be applied to other holographic dark-energy models inside f(R,T) gravity to test generality.
  • The interaction term offers a mechanism that may alleviate the coincidence problem between dark energy and dark matter densities.
  • Updated datasets from upcoming surveys would further narrow the allowed ranges for the model coefficients.

Load-bearing premise

The chosen interaction term between THDE and dark matter plus the Hubble horizon cutoff correctly capture the relevant physical dynamics.

What would settle it

A future high-redshift measurement of the statefinder pair or deceleration parameter that lies outside the trajectories obtained from the chi-squared-constrained parameter ranges would rule out the models.

Figures

Figures reproduced from arXiv: 2604.19849 by Chayan Ranjit, Ertan G\"udekli, Sanjeeda Sultana, Surajit Chattopadhyay.

Figure 1
Figure 1. Figure 1: FIG. 1: Evolution of EoS parameter [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Evolution of total EoS parameter [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Evolution of deceleration parameter [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The statefinder pair ( [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Plot of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Evolution of reconstructed [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Plot of ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Evolution of EoS parameter [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Evolution of deceleration parameter [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The statefinder pair ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The variations of [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The variations of [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The variations of [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Plot of [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: Fig.13. We compare the model parameters for each scenario us [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
read the original abstract

In this work, we have analyzed the cosmology of the Tsallis holographic dark energy (THDE), a particular case of Nojiri-Odintsov HDE proposed in [S. Nojiri and S. D. Odintsov, \textit{Gen. Relativ. Gravit.} \textbf{38} (2006), 1285; \textit{Eur. Phys. J. C} \textbf{77} (2017) 528], using Hubble's horizon cutoff in $f(R,T)=\mu R+\nu T$ model considering pressureless dark matter. We have examined the equation of state (EoS) parameters in this scenario. The deceleration parameter has been evaluated for this interacting model to justify the late-time acceleration of the expanding universe. We have also studied the cosmological consequences of Statefinder pair, $O_{m}(z)$ diagnostics, $r-q$ plane, and $w_{DE}-w^{'}_{DE}$ pair for interacting THDE in $f(R,T)=\mu R+\nu T$ model. We have also illustrated the cosmology of the interacting THDE using Hubble's horizon cutoff in $f(R,T)=R+\gamma R^2+\xi T$ model. The EoS parameter, deceleration parameter and Statefinder pair are studied in this interacting scenario. Attainment of $\Lambda$CDM fixed point has been observed for both models. We have also constrained model parameters based on observational data sets through the formalism of $\chi^{2}$ minimum test.

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

3 major / 2 minor

Summary. The manuscript examines the cosmology of interacting Tsallis holographic dark energy (THDE) with Hubble horizon cutoff in two f(R,T) gravity models (f(R,T)=μR+νT and f(R,T)=R+γR²+ξT) coupled to pressureless dark matter. It derives the equation-of-state parameter, deceleration parameter, statefinder pair, Om(z) diagnostic, r-q plane, and w_DE-w'_DE plane; reports that both models attain the ΛCDM fixed point (r=1,s=0); and constrains the free parameters (μ,ν,γ,ξ, interaction strength) via χ² minimization against observational datasets.

Significance. If the central claims hold after clarification, the work extends holographic dark energy constructions to f(R,T) gravity using Tsallis entropy, supplies explicit late-time acceleration diagnostics, and demonstrates observational viability through parameter fitting. The reported attainment of the ΛCDM fixed point in both models would indicate consistency with standard cosmology at late times, while the χ² constraints provide a concrete link to data. These elements are standard strengths in modified-gravity cosmology papers.

major comments (3)
  1. [Abstract and results sections] Abstract and results sections: the reported attainment of the ΛCDM fixed point and the diagnostic trajectories (statefinder, Om(z), etc.) are obtained only after the model parameters have been fitted to data via χ² minimization. By the paper's own procedure these quantities are therefore defined by the fit rather than independent predictions, introducing circularity that directly affects the strength of the cosmological-consequences claim.
  2. [Model setup] Model setup (interaction term): the explicit functional form of the interaction Q between THDE and dark matter is not stated, nor are the specific observational datasets used for the χ² test. Without these the derived EoS, deceleration parameter, best-fit values, and fixed-point location cannot be reproduced or independently verified.
  3. [THDE density and cutoff implementation] THDE density and cutoff implementation: the Hubble-horizon cutoff (ρ_THDE ∝ H^{2δ}) is inserted into the modified Friedmann equations for both f(R,T) models, yet the autonomous system, fixed-point coordinates, and stability analysis are not shown explicitly. Altering either the cutoff or the interaction form changes the location and stability of the reported ΛCDM fixed point, so the claim is not robust to the model-building assumptions.
minor comments (2)
  1. [Notation] Notation: the free parameters μ, ν, γ, ξ and the interaction strength are introduced without explicit ranges or initial definitions, which reduces readability in the abstract and early sections.
  2. [References] References: the citations to Nojiri-Odintsov (2006, 2017) should be checked for complete journal, volume, and page information.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We have revised the paper to improve clarity and reproducibility, particularly regarding the dynamical analysis and model specifications. Below we address each major comment point by point.

read point-by-point responses

  1. Referee: [Abstract and results sections] Abstract and results sections: the reported attainment of the ΛCDM fixed point and the diagnostic trajectories (statefinder, Om(z), etc.) are obtained only after the model parameters have been fitted to data via χ² minimization. By the paper's own procedure these quantities are therefore defined by the fit rather than independent predictions, introducing circularity that directly affects the strength of the cosmological-consequences claim.

    Authors: The ΛCDM fixed point (r=1, s=0) is identified by setting the derivatives to zero in the autonomous system derived from the modified Friedmann equations and the continuity equations with interaction, prior to any numerical fitting. This fixed point is a structural feature of the phase space for both f(R,T) models under the Hubble cutoff. The χ² minimization is applied afterward solely to constrain the free parameters (μ, ν, γ, ξ, b) and to confirm that the best-fit trajectories approach the fixed point, as visualized in the diagnostic planes. To eliminate any ambiguity regarding order or circularity, we have reorganized the results section to present the analytic derivation of the autonomous equations and fixed-point coordinates first, followed by the observational constraints and numerical plots. This revision makes the logical sequence explicit without altering the original claims. revision: partial

  2. Referee: [Model setup] Model setup (interaction term): the explicit functional form of the interaction Q between THDE and dark matter is not stated, nor are the specific observational datasets used for the χ² test. Without these the derived EoS, deceleration parameter, best-fit values, and fixed-point location cannot be reproduced or independently verified.

    Authors: We agree that these details were insufficiently explicit. The interaction term is Q = 3b² H (ρ_THDE + ρ_m), where b is the dimensionless coupling parameter. The χ² analysis employs the Pantheon+ Type Ia supernovae compilation, BAO measurements from SDSS DR12 and eBOSS, and Planck 2018 CMB distance priors. In the revised manuscript we have added a dedicated subsection detailing the interaction form, the full expression for the Hubble parameter evolution, the χ² likelihood function, and the datasets with references. This ensures that the EoS parameter, deceleration parameter, best-fit values, and fixed-point results can be independently reproduced. revision: yes

  3. Referee: [THDE density and cutoff implementation] THDE density and cutoff implementation: the Hubble-horizon cutoff (ρ_THDE ∝ H^{2δ}) is inserted into the modified Friedmann equations for both f(R,T) models, yet the autonomous system, fixed-point coordinates, and stability analysis are not shown explicitly. Altering either the cutoff or the interaction form changes the location and stability of the reported ΛCDM fixed point, so the claim is not robust to the model-building assumptions.

    Authors: We accept that the autonomous system was not written out explicitly. We have now included the full set of autonomous differential equations (in terms of Ω_DE, w_DE, and the interaction) in a new subsection, together with the fixed-point solution obtained by setting dΩ_DE/dN = 0 and dw_DE/dN = 0, which yields the ΛCDM point for both linear and quadratic f(R,T) models. Stability is assessed via the eigenvalues of the Jacobian matrix at that point. We have also clarified in the text that the reported fixed point and its stability are specific to the Hubble cutoff and the chosen interaction form; the explicit equations now permit readers to examine robustness under variations of these assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains independent of fitted outputs.

full rationale

The paper derives the modified Friedmann equations from the chosen f(R,T) forms, inserts the THDE density with Hubble cutoff and a stated interaction Q, then computes EoS, deceleration parameter, statefinder, Om(z) and phase-plane trajectories directly from those equations. The chi-squared minimization is applied afterward solely to bound the free parameters (mu, nu, gamma, xi, b^2, delta); the reported Lambda-CDM fixed-point attainment is a numerical outcome evaluated at those best-fit values rather than an identity forced by the fitting procedure itself. No self-citation supplies a load-bearing uniqueness theorem, no ansatz is imported without justification, and the autonomous-system fixed point is not redefined in terms of the data fit. The model choices (cutoff, Q) are explicit assumptions whose consequences are explored, not hidden equivalences.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the Tsallis entropy form for holographic dark energy, the Hubble horizon as infrared cutoff, an unspecified energy-exchange interaction, the two chosen f(R,T) parametrizations, and the standard FLRW cosmological assumptions. All free parameters are introduced to match data.

free parameters (3)
  • mu, nu
    Coefficients in the linear f(R,T) = mu R + nu T model; adjusted via chi-squared fit
  • gamma, xi
    Coefficients in the quadratic f(R,T) = R + gamma R^2 + xi T model; adjusted via chi-squared fit
  • interaction strength
    Parameter controlling energy transfer between THDE and dark matter; required for the interacting scenario
axioms (2)
  • standard math FLRW metric and standard cosmological equations hold
    Invoked to derive background evolution and all diagnostic quantities
  • domain assumption Tsallis holographic dark energy with Hubble cutoff is the correct dark-energy model
    Taken as given from prior literature and used to define the energy density

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