arxiv: 2605.04113 · v1 · submitted 2026-05-05 · 🌀 gr-qc

Recognition: 3 theorem links

· Lean Theorem

Comparative Study of f(T) Gravity Models with Observational Constraints from textit{OHD} and textit{Pantheon+ datasets}

Anirudh Pradhan , S. A. Salve , V. B. Raut , S. H. Shekh

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:52 UTC · model grok-4.3

classification 🌀 gr-qc

keywords f(T) gravitylate-time accelerationenergy conditionsdeceleration parameterMCMC constraintsHubble parameterPantheon+ datasetOm(z) diagnostic

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The pith

Three f(T) gravity models all approach the ΛCDM cosmology at late times while sharing a violation of the strong energy condition.

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

This paper compares three different f(T) gravity models using recent observational data to see how well they explain the universe's accelerated expansion without invoking dark energy. The models are a power-law form, an exponential form, and a logarithmic form. By parameterizing the deceleration parameter and fitting to Hubble parameter and Pantheon+ supernova data via MCMC, the analysis shows that all three models converge to the standard ΛCDM behavior in the late universe. They differ however in their stability and how they satisfy or violate various energy conditions. The common violation of the strong energy condition supports the observed acceleration across all models.

Core claim

The comparative study reveals that the power-law, exponential, and logarithmic f(T) models, when constrained by OHD and Pantheon+ datasets using q(z) parameterization, asymptotically approach ΛCDM behavior at late times. They exhibit distinct stability properties and energy condition behaviors, with the violation of the strong energy condition emerging as a common feature consistent with current accelerated expansion.

What carries the argument

The deceleration parameter parameterization q(z) leading to an expression for H(z), applied to three specific f(T) functional forms to derive effective energy density, equation of state, sound speed, Om(z), and energy conditions (NEC, DEC, SEC).

If this is right

All three models remain consistent with current Hubble and supernova observations of late-time acceleration.
Violation of the strong energy condition appears in every case and aligns with the repulsive effect needed for expansion.
Stability properties differ across the models, with squared sound speed providing a way to distinguish viable forms.
The Om(z) diagnostic can separate these models from pure ΛCDM in upcoming data.

Where Pith is reading between the lines

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

The parameterization approach trades full dynamical freedom for direct observational constraints, so extensions to phase-space analysis could reveal hidden instabilities.
If the common SEC violation holds, it points to a geometric requirement shared by torsional modifications rather than a model-specific feature.
These forms could be further tested against CMB or BAO data at higher redshifts to tighten bounds on the free parameters.

Load-bearing premise

The chosen functional forms of f(T) and the parameterization of q(z) are sufficient to describe the relevant cosmological dynamics without requiring additional terms or a full phase-space analysis.

What would settle it

A future high-precision measurement of the squared sound speed at intermediate redshifts or the Om(z) diagnostic that deviates from the predictions of all three models while remaining consistent with ΛCDM evolution.

Figures

Figures reproduced from arXiv: 2605.04113 by Anirudh Pradhan, S. A. Salve, S. H. Shekh, V. B. Raut.

**Figure 2.** Figure 2: Behavior of the energy density (Left panel), isotropic pressure (Middle panel) and the equation view at source ↗

**Figure 3.** Figure 3: Behavior of the null energy condition (Left panel), dominant energy condition (Middle panel) view at source ↗

**Figure 4.** Figure 4: Behavior of the energy density (Left panel), isotropic pressure (Middle panel) and the equation view at source ↗

**Figure 5.** Figure 5: Behavior of the null energy condition (Left panel), dominant energy condition (Middle panel) view at source ↗

**Figure 6.** Figure 6: Behavior of the energy density (Left panel), isotropic pressure (Middle panel) and the equation view at source ↗

**Figure 7.** Figure 7: Behavior of the null energy condition (Left panel), dominant energy condition (Middle panel) view at source ↗

**Figure 8.** Figure 8: Behavior of the Stability of the model for the constraint value of constant parameters towords view at source ↗

read the original abstract

The late-time acceleration of the universe remains one of the most significant open problems in modern cosmology. Modified gravity frameworks such as $f(T)$ gravity provide a geometric alternative to dark energy by attributing cosmic acceleration to torsional effects. In this study, we present a comparative analysis of three different forms of $f(T)$ models: (i) a simple power-law form $f(T) = \eta (-T)^{n}$, (ii) the exponential form $f(T) = \beta T_{0}\left(1-e^{-p \sqrt{T/T_{0}}}\right)$ and (iii) a logarithmic form $f(T) = \gamma T \ln\!\left(\frac{T}{T_{0}}\right)$. Using parameterization of the deceleration parameter $q(z)$ and the corresponding $H(z)$ expression, we constrain the model parameters with the recent Hubble parameter and BAO data through a Markov Chain Monte Carlo (MCMC) approach. The physical behavior of the effective energy density, equation of state parameter, squared sound speed, cosmological $Om(z)$ diagnostics, and energy conditions (NEC, DEC, SEC) were investigated for all three models. Our comparative analysis shows that all models asymptotically approach the $\Lambda$CDM behavior at late times, while they differ in stability properties and energy condition behaviors. In particular, the violation of the strong energy condition (SEC) has emerged as a common feature consistent with current accelerated expansion. This study highlights how different $f(T)$ functional forms can yield distinct cosmological dynamics while maintaining consistency with observational data.

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

This paper fits three standard f(T) models to OHD and Pantheon+ data via a q(z) parameterization but leaves the dynamical consistency with the modified Friedmann equations unaddressed.

read the letter

Hey, the core of this paper is a side-by-side comparison of three f(T) forms—power-law, exponential, and logarithmic—constrained by recent Hubble and supernova data. They parameterize q(z) to get H(z), run MCMC fits, and then plot effective density, equation of state, sound speed squared, Om(z), and the energy conditions. All three models end up close to LambdaCDM at late times and violate the strong energy condition, which matches the observed acceleration. The work is straightforward and the plots are clear enough for a direct comparison across the functional choices. They also use updated data sets, which is a small plus. The soft spot is the parameterization itself. f(T) gravity supplies its own second-order equations for H(z) once f(T) is fixed, yet here q(z) is imposed from outside and H(z) is derived kinematically before the parameters are fitted. Nothing in the abstract shows they verified that the resulting H(z) actually solves the field equations for each chosen f(T). Without that check the reported differences in stability and the shared SEC violation could simply trace back to the ansatz rather than to the gravity models. Details on exact data cuts and error-bar sizes are also missing. This is the sort of incremental comparison that specialists in teleparallel gravity might skim for fresh bounds on these particular forms, but it does not open new theoretical ground. I would send it for peer review only if the authors add an explicit consistency test between the fitted H(z) and the f(T) equations; otherwise it stays too close to a kinematic exercise to be worth referee time.

Referee Report

2 major / 3 minor

Summary. The manuscript compares three f(T) gravity models—power-law f(T)=η(-T)^n, exponential f(T)=β T0(1−e^{−p√(T/T0)}), and logarithmic f(T)=γ T ln(T/T0)—by imposing a q(z) parameterization to generate H(z), then using MCMC to fit the free parameters to Hubble parameter and BAO data. It computes effective energy density, equation-of-state, squared sound speed, Om(z), and energy conditions (NEC/DEC/SEC), concluding that all three models approach ΛCDM asymptotically at late times while differing in stability and sharing SEC violation consistent with acceleration.

Significance. If the kinematic parameterization is shown to be dynamically consistent with the f(T) field equations, the work supplies a useful side-by-side comparison of distinct functional forms under the same data, identifying shared late-time behavior and the common SEC violation as a geometric signature of acceleration. It adds to the f(T) literature by examining model-specific stability via c_s² and Om(z) diagnostics.

major comments (2)

[§3 (Methodology)] §3 (Methodology): H(z) is generated from an externally imposed q(z) parameterization and then used to constrain the f(T) parameters via MCMC. In f(T) gravity the modified Friedmann equations are second-order differential equations for H(z); the effective ρ_eff, p_eff, c_s² and energy conditions must be evaluated along solutions of those equations. An independent kinematic ansatz therefore risks that the fitted trajectories do not satisfy the second Friedmann equation for the chosen f(T) forms, rendering the reported differences in stability and the universality of SEC violation potentially artifacts of the ansatz rather than intrinsic model properties.
[§4 (Results)] §4 (Results): The claims that all models asymptotically approach ΛCDM and that SEC violation is a common feature are obtained directly from the same parameters fitted to the data through the q(z)-derived H(z). These quantities therefore reduce to consequences of the data fit itself and do not constitute independent predictions or tests of the f(T) dynamics.

minor comments (3)

[Abstract] Abstract and title: The title cites OHD and Pantheon+ datasets while the abstract states 'Hubble parameter and BAO data'; reconcile the exact datasets employed and state the selection criteria explicitly.
[Tables] Parameter tables: Best-fit values and 1σ uncertainties for η, n, β, p and γ should be summarized in a table with the corresponding χ² or information criteria to allow direct comparison of model performance.
[Figures] Notation: Define T0 consistently (present-day torsion scalar) and ensure all figures label the three models distinctly with error bands where applicable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important issues regarding methodological consistency and the interpretation of our results, which we address point by point below. We are prepared to revise the paper accordingly to strengthen its rigor.

read point-by-point responses

Referee: [§3 (Methodology)] H(z) is generated from an externally imposed q(z) parameterization and then used to constrain the f(T) parameters via MCMC. In f(T) gravity the modified Friedmann equations are second-order differential equations for H(z); the effective ρ_eff, p_eff, c_s² and energy conditions must be evaluated along solutions of those equations. An independent kinematic ansatz therefore risks that the fitted trajectories do not satisfy the second Friedmann equation for the chosen f(T) forms, rendering the reported differences in stability and the universality of SEC violation potentially artifacts of the ansatz rather than intrinsic model properties.

Authors: We acknowledge the validity of this concern. Our approach employs the q(z) parameterization to derive an observationally constrained H(z) from OHD and Pantheon+ data, after which the f(T) parameters are determined via MCMC by matching the first modified Friedmann equation to the matter density evolution. This ensures consistency with the algebraic relation involving f(T) and H. However, to fully address potential inconsistencies with the second (differential) Friedmann equation, we will revise §3 to include an explicit post-fit verification: for the best-fit parameters of each model, we will substitute the kinematic H(z) back into the second equation and quantify any residuals. Should significant violations appear, we will either adjust the parameterization or note the limitation. This addition will demonstrate whether the reported stability differences and SEC behavior are intrinsic or ansatz-dependent. revision: yes
Referee: [§4 (Results)] The claims that all models asymptotically approach ΛCDM and that SEC violation is a common feature are obtained directly from the same parameters fitted to the data through the q(z)-derived H(z). These quantities therefore reduce to consequences of the data fit itself and do not constitute independent predictions or tests of the f(T) dynamics.

Authors: We agree that the late-time approach to ΛCDM and the SEC violation are evaluated along the data-constrained H(z) rather than emerging from an ab initio solution of the f(T) field equations. These features therefore reflect the observational constraints on the expansion history, which all three models are required to reproduce. The model-to-model differences in c_s² and Om(z) nevertheless provide a comparative test of how each functional form responds to the same trajectory. We will revise the discussion in §4 to explicitly state that these are properties of the models when fitted to the same data, rather than pure dynamical predictions, thereby clarifying the scope of the claims without overstating their independence from the kinematic ansatz. revision: partial

Circularity Check

1 steps flagged

Kinematic q(z) ansatz supplies H(z) for all MCMC fits and diagnostics, so reported f(T) behaviors reduce to the imposed parameterization

specific steps

fitted input called prediction [Abstract (and Methodology section)]
"Using parameterization of the deceleration parameter q(z) and the corresponding H(z) expression, we constrain the model parameters with the recent Hubble parameter and BAO data through a Markov Chain Monte Carlo (MCMC) approach. The physical behavior of the effective energy density, equation of state parameter, squared sound speed, cosmological Om(z) diagnostics, and energy conditions (NEC, DEC, SEC) were investigated for all three models."

The quoted procedure first imposes q(z) to generate H(z), fits the three f(T) parameter sets to data on that H(z), and then evaluates all listed diagnostics on the identical fitted H(z). The resulting behaviors (asymptotic ΛCDM limit, SEC violation, stability via c_s²) are therefore direct algebraic consequences of the chosen q(z) form and the data fit, not independent predictions obtained by integrating the second-order f(T) Friedmann equations for each model.

full rationale

The paper derives H(z) from an externally chosen q(z) parameterization, fits the f(T) model parameters (η,n; β,p; γ) to OHD+BAO via MCMC on that H(z), then computes Om(z), w_eff, c_s² and energy conditions directly from the same fitted H(z). Because the modified Friedmann equations of f(T) gravity are not solved to obtain the expansion history, the claimed asymptotic ΛCDM approach, stability differences, and universal SEC violation are outputs of the kinematic fit rather than independent consequences of the torsional field equations. This matches the fitted-input-called-prediction pattern and produces partial circularity (score 6) while still allowing some model-specific parameter constraints.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The study rests on standard FLRW cosmology plus the assumption that f(T) gravity is a viable framework; it introduces no new postulated entities but carries multiple free parameters per model that are fitted to data.

free parameters (4)

eta, n (power-law model)
Amplitude and exponent in f(T) = eta (-T)^n, adjusted via MCMC to match observations.
beta, p (exponential model)
Parameters in f(T) = beta T0 (1 - exp(-p sqrt(T/T0))), fitted to data.
gamma (logarithmic model)
Coefficient in f(T) = gamma T ln(T/T0), constrained by the same data.
q(z) parameterization coefficients
Free parameters in the chosen deceleration-parameter form that determine H(z) and are fitted jointly with the f(T) parameters.

axioms (2)

standard math FLRW metric and standard Friedmann equations hold in f(T) gravity
Used to derive H(z) from the parameterized q(z).
domain assumption f(T) gravity provides a consistent geometric description of late-time acceleration
The entire comparative analysis presupposes this framework is physically viable.

pith-pipeline@v0.9.0 · 5614 in / 1600 out tokens · 69478 ms · 2026-05-08T18:52:16.619629+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.

Foundation/AlphaDerivationExplicit.lean (RS derives constants parameter-free; this paper fits parameters) alphaProvenanceCert unclear

?

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

Using parameterization of the deceleration parameter q(z) and the corresponding H(z) expression, we constrain the model parameters with the recent Hubble parameter and BAO data through a Markov Chain Monte Carlo (MCMC) approach.
Cost/FunctionalEquation.lean (J is the unique calibrated reciprocal cost; no analogous uniqueness drives the f(T) choice here) washburn_uniqueness_aczel unclear

?

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

f(T) = η(−T)^n, f(T) = βT₀(1−e^{−p√(T/T₀)}), f(T) = γT ln(T/T₀)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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