arxiv: 2507.01070 · v2 · submitted 2025-07-01 · 🌀 gr-qc · hep-th

Barrow holographic dark energy interacting model in the presence of radiation and matter

Gopinath Guin , Souvik Paul , Sunandan Gangopadhyay This is my paper

Pith reviewed 2026-05-19 06:59 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords Barrow holographic dark energyinteracting modelsdark energy equation of stateHubble tensionnon-flat universecosmological constraintsradiation and matter

0 comments p. Extension

The pith

In interacting Barrow holographic dark energy models with radiation, higher values of the Barrow exponent cause the dark energy equation of state to transition from quintessence to phantom at early times.

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

The paper investigates the dynamics of an interacting Barrow holographic dark energy model that includes radiation and matter in both open and closed universes. It derives evolution equations for the energy density parameters under four different linear interaction forms and solves them numerically to track their behavior across redshift. The results show that for larger Barrow exponents, the dark energy equation of state parameter moves into the phantom region from the quintessence region during early epochs. Observational constraints from cosmic chronometer, BAO, and Pantheon+ data indicate a higher Hubble constant than in the standard Lambda CDM model, pointing to a potential way to address the Hubble tension.

Core claim

For all four interaction models in the presence of radiation and matter, the dark energy equation of state shows a transition into the phantom region from the quintessence region at early times when the Barrow exponent takes higher values, and the derived constraints on the Hubble parameter are higher than those of the Lambda CDM model.

What carries the argument

The Barrow holographic dark energy density with exponent Delta, together with four chosen linear phenomenological interaction terms Q that couple the dark energy, dark matter, and radiation sectors in a non-flat FLRW universe, allowing numerical integration of the density parameter evolution equations.

If this is right

For all four interaction models, the dark energy equation of state parameter transitions from quintessence to phantom for higher Barrow exponent values at early times.
Various epochs of dark energy-dark matter, dark energy-radiation, and dark matter-radiation crossings are identified and match the thermal history of the universe.
The constrained Hubble parameter values are higher compared to the Lambda CDM model, suggesting a possible resolution to the Hubble tension problem.

Where Pith is reading between the lines

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

The transition to phantom behavior at early times for large Barrow exponent may be testable with future high-redshift observations of dark energy properties.
Since the model is applied to both open and closed universes, it could be compared to curvature constraints from CMB data to see if non-flat geometries are preferred.
Extending the interaction forms beyond the four linear ones chosen might reveal whether the phantom crossing is a generic feature or specific to these choices.

Load-bearing premise

The four selected linear phenomenological interaction forms between dark energy, dark matter, and radiation are the appropriate ones to model the energy exchanges in this cosmological setup.

What would settle it

An observation that the dark energy equation of state remains in the quintessence region even at high redshifts for large Barrow exponent values, or a precise Hubble constant measurement that does not exceed the Lambda CDM value in this framework.

read the original abstract

We have studied the effect of dynamical radiation in the interacting barrow holographic dark energy model for a non-flat universe. For both open and closed universes, we have obtained the evolution equation for the energy density parameters for dark energy, dark matter and radiation for four different kinds of interaction among the seven possible linear phenomenological interactions. We have then numerically solved those coupled differential equations to show their behaviour with the redshift parameter. Also, the dynamics of the dark energy equation of state parameter with redshift for different interaction models are shown. For all four interaction models, it is also found that for higher values of the Barrow exponent, the dark energy equation of state parameter shows a transition into the phantom region from the quintessence region in the early time, that is, for lower redshift values. We have also found different epochs corresponding to dark energy-dark matter, dark energy-radiation and dark matter-radiation crossings. These crossing points are also consistent with the thermal history of the universe. We have also obtained various observational constraints for different cosmological parameters for our interacting Barrow holographic dark matter model using the Cosmic chronometer, Baryon Acoustic Oscillator and Pantheon+ data sets. The constraint values of the Hubble parameter in our cosmological shows higher values compared to the $\Lambda$CDM model, therefore indicating towards a possible resolution to the Hubble tension problem.

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.

Desk Editor's Note private letter to a colleague

The paper adds radiation and non-flat geometry to Barrow holographic DE with four chosen interactions, produces numerical EoS plots and data fits with higher H0, but leaves the ODE integrations untested and the interaction selection unexplained.

read the letter

The main takeaway is that this work extends earlier Barrow holographic dark energy papers by including radiation, allowing curvature, and running four linear interaction terms through the coupled density equations. They integrate numerically, plot the evolution of the density parameters and the dark energy equation of state, and then fit the model to cosmic chronometer, BAO, and Pantheon+ data. The reported outcomes are phantom crossings at higher Barrow exponent and best-fit H0 values above the usual LambdaCDM number.

Referee Report

3 major / 2 minor

Summary. The manuscript examines an interacting Barrow holographic dark energy model in a non-flat FLRW universe that includes radiation and matter. It derives evolution equations for the density parameters Ω_DE, Ω_m and Ω_r under four selected linear interaction terms Q (chosen from seven possible phenomenological forms), numerically integrates the coupled system to display redshift evolution and epoch crossings, reports that w_DE(z) transitions from quintessence to phantom for larger Barrow exponent Δ, and performs MCMC fits to Cosmic Chronometers, BAO and Pantheon+ data that yield H0 values higher than those of ΛCDM, interpreted as a possible resolution of the Hubble tension.

Significance. If the numerical integrations are shown to be stable and the interaction selection is physically motivated, the work adds a concrete example of how Barrow entropy modifications combined with energy exchange can produce phantom crossings and elevated H0 constraints. The explicit inclusion of radiation and non-flat geometry broadens the model’s applicability to early-universe epochs, but the overall significance remains modest until the numerical fidelity and fitting procedure are verified.

major comments (3)

[Numerical solutions] Numerical integration section: the evolution equations for the density parameters are integrated numerically to obtain the reported w_DE(z) phantom crossings and crossing redshifts, yet no convergence tests, step-size independence, integrator choice, or initial-condition sensitivity at high z are provided. Small truncation or stiffness errors near radiation-matter-DE equality could shift the reported transition redshifts and bias the subsequent MCMC posteriors.
[Interaction terms] Interaction models: four linear phenomenological forms are retained out of seven possible without stated selection criteria or comparison to the three omitted cases. Because the phantom-transition and crossing results are asserted to hold “for all four interaction models,” the lack of justification makes the generality of the central claim difficult to assess.
[Observational analysis] Observational constraints: the Hubble parameter is fitted simultaneously with Δ and the interaction couplings to the same CC+BAO+Pantheon+ data sets used to constrain the model. The reported “higher H0” is therefore a fitted outcome rather than an independent prediction, weakening the claim that the model offers a resolution to the Hubble tension.

minor comments (2)

[Abstract] The final sentence of the abstract refers to a “Barrow holographic dark matter model”; this appears to be a typographical error for “dark energy.”
[Evolution equations] Explicit presentation of the coupled ODE system (including the precise definitions of the four retained Q terms) would allow readers to reproduce the numerical setup without ambiguity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating where revisions will be made to improve clarity and rigor.

read point-by-point responses

Referee: [Numerical solutions] Numerical integration section: the evolution equations for the density parameters are integrated numerically to obtain the reported w_DE(z) phantom crossings and crossing redshifts, yet no convergence tests, step-size independence, integrator choice, or initial-condition sensitivity at high z are provided. Small truncation or stiffness errors near radiation-matter-DE equality could shift the reported transition redshifts and bias the subsequent MCMC posteriors.

Authors: We appreciate the referee highlighting this omission. The integrations were carried out with scipy.integrate.solve_ivp using the RK45 method and adaptive step-size control, with initial conditions imposed at z ≈ 1000 under radiation domination. We agree that explicit documentation of numerical fidelity is necessary. In the revised manuscript we will add a dedicated paragraph (or short subsection) reporting the integrator choice, tolerance settings, step-size independence checks via repeated runs with tightened tolerances, and sensitivity tests to small variations in high-z initial conditions. These additions will confirm that the reported phantom-crossing redshifts and epoch crossings are robust. revision: yes
Referee: [Interaction terms] Interaction models: four linear phenomenological forms are retained out of seven possible without stated selection criteria or comparison to the three omitted cases. Because the phantom-transition and crossing results are asserted to hold “for all four interaction models,” the lack of justification makes the generality of the central claim difficult to assess.

Authors: The four interaction terms were chosen because they are among the most commonly adopted linear forms in the interacting dark-energy literature and because they produce stable, physically acceptable solutions (non-negative densities, no early-time divergences) across the full redshift range we consider. The three omitted forms were discarded after preliminary tests revealed unphysical behavior such as negative energy densities or numerical instabilities. We will insert a short explanatory paragraph in the revised text stating these selection criteria and clarifying that the phantom-transition and crossing results are demonstrated for the four representative, stable cases. A complete comparison with all seven forms lies beyond the present scope but could be addressed in follow-up work. revision: yes
Referee: [Observational analysis] Observational constraints: the Hubble parameter is fitted simultaneously with Δ and the interaction couplings to the same CC+BAO+Pantheon+ data sets used to constrain the model. The reported “higher H0” is therefore a fitted outcome rather than an independent prediction, weakening the claim that the model offers a resolution to the Hubble tension.

Authors: We agree that the higher H0 values emerge from the joint MCMC fit rather than constituting an a-priori prediction independent of the data. In the revised discussion we will rephrase the relevant statements to make this explicit: the model, once constrained by the combined Cosmic Chronometers, BAO and Pantheon+ datasets, yields H0 posteriors that are higher than the corresponding ΛCDM value and lie closer to local measurements. This is the standard manner in which extended models are assessed for their potential to alleviate the Hubble tension; we will avoid any implication of an independent forecast. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains independent of fitted outputs

full rationale

The paper derives the evolution equations for the density parameters Ω_DE, Ω_m and Ω_r directly from the Friedmann equations and the chosen linear interaction terms Q in the Barrow holographic dark energy model. These coupled ODEs are numerically integrated to obtain the redshift dependence of w_DE and the crossing epochs. Model parameters including the Barrow exponent Δ and interaction coefficients are then constrained via MCMC using external datasets (CC, BAO, Pantheon+). The reported higher H0 values and phantom transitions for larger Δ are outputs of this fitting procedure applied to the solved dynamics, not reductions of the central claims to the inputs by construction. No self-definitional steps, fitted quantities relabeled as predictions, or load-bearing self-citations appear in the derivation chain. The analysis is self-contained against the external observational benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard FLRW cosmology plus a phenomenological holographic cutoff and four ad-hoc linear interaction terms whose coefficients are fitted rather than derived.

free parameters (2)

Barrow exponent Delta
Varied to control the strength of the entropy modification and to produce the reported EoS transition.
Interaction coupling constants
Four separate linear coefficients chosen from seven possible forms and adjusted to data.

axioms (2)

standard math FLRW metric with curvature parameter k = +/-1
Invoked for open and closed universes in the evolution equations.
domain assumption Holographic principle with Barrow entropy
Basis for the dark energy density expression.

pith-pipeline@v0.9.0 · 5770 in / 1417 out tokens · 37857 ms · 2026-05-19T06:59:15.958130+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We have then numerically solved those coupled differential equations to show their behaviour with the redshift parameter... for higher values of the Barrow exponent, the dark energy equation of state parameter shows a transition into the phantom region
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

ρ_de = C L^{Δ-2}

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