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Cosmological insights from an exponential $Om(z)$ function in $f(T,T_{G})$ gravity framework

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read An exponential Om(z) diagnostic in modified teleparallel gravity fits the combined CC, BAO, and Pantheon+ datasets and yields a Hubble constant in line with local distance-ladder measurements.

desk verdict New exponential Om(z) parameterization fit to standard datasets, but the best-fit parameters make H(z)^2 negative at z≳11, contradicting the paper's central 'well-behaved at all redshifts' claim. read the letter →

arxiv 2506.06399 v1 pith:ISU3QFV7 submitted 2025-06-06 gr-qc

Amit Samaddar , S. Surendra Singh This is my paper
classification gr-qc MSC 83F0583D05 PACS 98.80.-k98.80.Es04.50.Kd
keywords modifiedteleparallelgravityf(TT_G)Om(z)diagnosticexponentialparameterizationdarkenergylate-timecosmicaccelerationobservationalconstraintsHubbleconstanttension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that a two-parameter exponential form of the $Om(z)$ diagnostic, $Om(z)=\alpha e^{z/(1+z)}+\beta$, can describe the late-time acceleration of the universe when the gravity sector is the square-root teleparallel model $f(T,T_G)=T+\gamma\sqrt{T_G}+\delta\sqrt{T}$. A sympathetic reader would care because the model fits 31 cosmic-chronometer, 26 BAO, and 1701 Pantheon+ supernova points with just two extra parameters, and it returns a Hubble constant near the local distance-ladder value, offering one route toward the Hubble tension. The same fit produces a transition redshift and present deceleration and equation-of-state values close to those inferred in standard cosmology, so the model is a viable alternative to a cosmological constant. It also predicts a universe age of about 13.3–13.9 Gyr and satisfies the null and dominant energy conditions while violating the strong energy condition at late times.

What carries the argument

The central object is the exponential $Om(z)$ diagnostic, $Om(z)=\alpha e^{z/(1+z)}+\beta$, whose inversion through the definition $Om(z)=((H/H_0)^2-1)/((1+z)^3-1)$ produces the model Hubble function $H(z)=H_0\sqrt{(\alpha e^{z/(1+z)}+\beta)[(1+z)^3-1]+1}$. This Hubble function is fed into the modified Friedmann equations of $f(T,T_G)=T+\gamma\sqrt{T_G}+\delta\sqrt{T}$ (with $\gamma=0.05$ used for the derived plots), which turn it into predictions for the deceleration parameter, energy density and pressure, equation of state, energy conditions, statefinder pair, and cosmic age. The diagnostic carries the argument by converting a model-independent ratio of Hubble rates into a parameterized expansion history that can be fit directly to CC, BAO, and Pantheon+ data.

What would settle it

A reliable measurement of $H(z)$ at a redshift above about 10.6 would falsify the all-redshift claim, because the best-fit model makes $H(z)^2$ negative there. Even without new data, evaluating the reconstructed $H(z)$ at $z=10.6$ with $\alpha=-0.148$ and $\beta=0.369$ gives a negative square, contradicting the paper's statement that the model is well-behaved at all redshifts.

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Extended reading notes

Core claim

The paper's central claim is that the exponential diagnostic $Om(z)=\alpha e^{z/(1+z)}+\beta$, combined with the modified teleparallel gravity model $f(T,T_G)=T+\gamma\sqrt{T_G}+\delta\sqrt{T}$, accounts for the observed late-time expansion history. With the combined CC+BAO+Pantheon+ dataset the best-fit parameters $\alpha=-0.148$, $\beta=0.369$ yield $H_0\approx73.3$ km/s/Mpc with a 1$\sigma$ range of about 68.5–77.4, a deceleration-to-acceleration transition at $z_{tr}\approx0.48$–$0.54$, present $q_0\approx-0.32$ to $-0.34$ and $\omega_0\approx-0.33$, and a cosmic age of about 13.3–13.9 Gyr. The model satisfies the null and dominant energy conditions, violates the strong energy condition at late times, and its statefinder trajectory passes through the $\Lambda$CDM point $(r,s)=(1,0)$. The authors take these results to show that the exponential $Om(z)$ form is a viable, data-consistent description of dark energy evolution within this modified gravity framework.

Load-bearing premise

The load-bearing premise is that the exponential expansion formula used for $Om(z)$ is valid at every redshift, so the reconstructed expansion rate squared stays positive; at the paper's best-fit parameters the factor $\alpha e^{z/(1+z)}+\beta$ crosses zero near $z\approx10.6$, which would make the expansion rate imaginary there.

Editorial extensions

If this is right

  • The combined CC, BAO, and Pantheon+ data can be described by a two-parameter exponential $Om(z)$ diagnostic, so the present data do not require a cosmological constant if modified teleparallel gravity is allowed.
  • The best-fit $H_0$ of about 73.3 km/s/Mpc sits closer to local distance-ladder values than to the early-universe Planck value, giving a concrete path toward easing the Hubble tension.
  • The predicted transition redshift $z_{tr}\approx0.48$–$0.54$ and present deceleration $q_0\approx-0.32$ place the onset of acceleration at roughly the epoch inferred in $\Lambda$CDM.
  • The model's statefinder trajectory passes through the $\Lambda$CDM point $(r,s)=(1,0)$, so the modified-gravity model is observationally close to $\Lambda$CDM while allowing evolving dark energy.
  • The age estimate of 13.3–13.9 Gyr is consistent with globular-cluster and Planck age bounds, so the model does not require a revision of cosmic chronology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Read as a full-history model, the best-fit form fails at high redshift: $\alpha e^{z/(1+z)}+\beta$ changes sign near $z\approx10.6$, making the reconstructed expansion rate imaginary. That suggests the exponential diagnostic should be treated as a late-time parameterization rather than a complete description of the early universe.
  • Because $\gamma$ is set to 0.05 by hand for the energy-condition and equation-of-state plots, those conclusions are conditional on that choice; a joint fit of $\gamma$ with $\alpha$ and $\beta$ would test whether the NEC and DEC results survive.
  • Since the paper compares against $\Lambda$CDM but not against other two-parameter dark-energy forms, a model comparison on the same data would show whether the exponential shape itself is preferred or merely adequate.
  • The negative $\alpha$ produces an $Om(z)$ that grows with redshift, which the paper links to phantom-like dark energy; a derived check of whether this growth leads to a future singularity would extend the model's predictions beyond the current epoch.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes an exponential parameterization of the Om(z) diagnostic, Om(z) = α e^{z/(1+z)} + β, within the f(T,T_G) = T + γ√T_G + δ√T gravity model. The authors invert the standard Om diagnostic to reconstruct H(z), fit α, β, and H0 to 31 cosmic chronometer, 26 BAO, and 1701 Pantheon+ data points using MCMC, and then derive q(z), the effective equation of state, energy conditions, statefinder parameters, and the cosmic age. The reported best-fit results include H0 ≈ 73.3 km/s/Mpc, z_tr ≈ 0.48–0.54, q0 ≈ −0.34, ω0 ≈ −0.33, and an age of 13.28–13.87 Gyr, leading the authors to claim the model is observationally supported and well-behaved at all redshifts.

Significance. If the reconstructed expansion history and the derived cosmological parameters were reliable, the paper would add a useful example to the literature on Om(z) parameterizations in modified gravity. The use of 31 CC + 26 BAO + 1701 Pantheon+ points is a reasonable dataset combination, and the paper presents the field-equation algebra explicitly, which aids reproducibility of the derivation. However, the central claims are undermined by an internal inconsistency in the high-redshift regime, an error in the interpretation of the fitted parameters, and an ad hoc treatment of the modified-gravity parameters. The presently stated conclusions therefore do not provide a dependable test of the f(T,T_G) framework.

major comments (4)
  1. [§4, Eqs. (21), (25)] The claim that the exponential Om(z) form 'remains well-behaved at all redshifts from z=0 to high redshift z→∞' is contradicted by the paper's own best-fit values. With α = −0.148 and β = 0.369 (the CC+BAO+Pantheon+ case), Om(z) = α e^{z/(1+z)} + β vanishes at z ≈ 10.6 and becomes negative for larger z. Since H(z)^2/H0^2 = Om(z)[(1+z)^3 − 1] + 1, H(z)^2 becomes negative for z ≳ 11, so H(z) is imaginary in the early universe. Every quantity derived from H(z), including q(z) in Eq. (39), ρ and p in Eqs. (40)–(41), the energy conditions in Eqs. (43)–(45), the statefinder parameters, and the age integral in Eq. (49), is therefore not defined on the full redshift range used to support the claimed 'stable and physically realistic behavior across all epochs'.
  2. [§4, Eq. (21)] The phantom-like interpretation of the best fit is based on an incorrect derivative. The text states that a negative α produces an increasing Om(z), but dOm/dz = α e^{z/(1+z)}/(1+z)^2, which is negative for α < 0. Hence the best-fit parameter set gives a decreasing Om(z), not an increasing one. The claimed phantom behavior and the related discussion in Sections 5.2 and 6.4 need to be revised accordingly.
  3. [§2–3, 5] The observational analysis does not actually constrain the f(T,T_G) model. The reconstructed H(z) comes entirely from the Om(z) ansatz (Eqs. 21 and 25), and the MCMC fit in Section 5 involves only H0, α, and β; the gravity parameters γ and δ do not enter the likelihood. The field equations of f(T,T_G) are used only after the fit, with γ fixed to 0.05, to compute ρ, p, the EoS, and the energy conditions. Consequently, the claimed agreement with CC+BAO+Pantheon+ data is a statement about the kinematic Om(z) parameterization, not about f(T,T_G) gravity; the modified-gravity part of the model is not tested by the data.
  4. [§6.2–6.3, §7, Eqs. (40)–(45)] The energy-density, pressure, EoS, and energy-condition results are presented for γ = 0.05 with no explanation of how this value was chosen and no propagation of the MCMC uncertainties. Since γ is a free parameter in the action, the statement that the model 'satisfies NEC and DEC' is not a result of the observational fit; it is conditional on an arbitrary choice of γ. A joint fit over γ (or a demonstration of insensitivity to γ) is needed before these claims can be regarded as a test of the model.
minor comments (4)
  1. [§5.1.1, Eq. (34)] The BAO chi-square uses the notation σYi; this should read σ(Y_i) or σ_i. In addition, the manuscript does not itemize the 26 BAO data points or the exact covariance information, making the analysis difficult to reproduce.
  2. [Fig. 3 caption] The caption contains a typo: 'CC+ABO' should be 'CC+BAO'. The caption also refers to 'Pantheon+SHOES' while the text and abstract refer to the Pantheon+ dataset; the nomenclature should be unified.
  3. [§6.4 and Fig. 6] The text states that Om(z) 'starts near 0.05 at higher redshifts and approaches approximately 0.36 as z → −1', while the figure axis extends to z = −1, where the combination z/(1+z) is singular. The redshift range of the plot and the interpretation of the limiting behavior should be clarified.
  4. [§3 and §6.2, Eqs. (20), (40)–(41)] The parameter δ in f(T,T_G) = T + γ√T_G + δ√T does not appear in the reconstructed ρ and p or in any of the derived observables. If δ cancels in the FLRW background, this should be stated explicitly; if it does not, the expressions in Eqs. (40)–(41) are incomplete. As written, the paper advertises a two-parameter modification but effectively analyzes only the γ√T_G term.

Circularity Check

1 steps flagged · score 5.0 of 10

Fitted exponential Om(z) ansatz drives all derived 'predictions'; the MCMC fit itself is legitimate, and the main circularity is presenting algebraic consequences of the fit as model confirmations.

  1. fitted input called prediction [Abstract; Section 6.1, especially Eq. (39); Section 6.3]
    "Our model predicts a transition redshift z_tr ≈ (0.48−0.54), present q_0≈−0.34, and ω_0≈−0.33. ... By substituting the best-fit values of α and β from our observational analysis into the reconstructed Hubble parameter H(z), we examine the evolution of q(z)."

    The advertised 'predictions' are obtained by inserting the fitted α and β into expressions that are algebraic consequences of the assumed ansatz itself. Equation (25) is just the definition of Om(z) in Eq. (24), inverted after substituting Eq. (21). Every derived quantity is therefore a deterministic function of the fitted parameters. In particular, setting z=0 in Eq. (39) gives q_0 = −1 + 3(α+β), so the reported q_0 ≈ −0.34 is a re-labeling of the fitted combination α+β ≈ 0.22; z_tr, ω_0, the statefinder values, and the dimensionless age integral are likewise fixed by the same fitted α, β, and H_0. Presenting these algebraic outputs as 'predictions' and using them to claim validation adds no independent information beyond the fit itself.

full rationale

The parameter estimation against CC, BAO, and Pantheon+ is not circular: H_0, α, and β are genuinely fitted to external data through Eq. (25). The circularity is limited to the presentation of derived quantities. q_0, z_tr, ω_0, r_0, s_0, and the cosmic age are not independent outputs of the f(T,T_G) theory; they are obtained by substituting the best-fit parameters into formulas that follow immediately from the exponential Om(z) ansatz and the definition of Om(z). In particular, q_0 = −1 + 3(α+β) makes the abstract's 'our model predicts ... q_0≈−0.34' a restatement of the fitted combination α+β, not a test. The energy-condition results also depend on a hand-set γ=0.05 rather than a fitted or marginalized value, so they are limitations rather than additional validations. A separate, non-circular correctness problem is that the best-fit parameters make H²(z) negative for z ≳ 11, contradicting the paper's claim that the model is 'well-behaved at all redshifts'; this affects the high-redshift derived quantities and the age integral but is not a circularity. On balance, the fit has independent content, but the central claim's supporting 'predictions' reduce by construction to the fitted ansatz, warranting a partial circularity score of 5.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on an ad hoc functional form for Om(z) and an unconstrained choice of the gravity parameter gamma; no new physical entities are introduced.

free parameters (5)
  • alpha = -0.148 (CC+BAO+Pantheon+); range [-0.232, -0.068]
    Shape parameter in the exponential Om(z) ansatz; fitted to H(z) data.
  • beta = 0.369 (CC+BAO+Pantheon+); range [0.218, 0.560]
    Offset parameter of the Om(z) ansatz; fitted.
  • H0 = 73.309 km/s/Mpc (CC+BAO+Pantheon+)
    Hubble constant; fitted.
  • gamma = 0.05 (chosen, not fitted)
    Model parameter in f(T,T_G); set to 0.05 for all numerical results, without justification or uncertainty.
  • delta = unspecified
    Model parameter in f(T,T_G); no value given; term appears to cancel (or is silently set to zero) in rho and p.
assumptions (6)
  • domain assumption Spatially flat FLRW background with Weitzenbock gauge tetrad
    Used to set T=6H^2 and T_G=24H^2(Hdot+H^2), equations (15)-(17).
  • domain assumption Modified Friedmann equations (18)-(19) for f(T,T_G) gravity, adopted from the literature
    Taken as given from refs [45,46]; no independent derivation in this paper.
  • ad hoc to paper Specific model f(T,T_G)=T+gamma*sqrt(T_G)+delta*sqrt(T) is assumed ad hoc
    One of many possible f(T,T_G) forms; not derived or motivated beyond analogy with other sqrt modifications.
  • ad hoc to paper Exponential Om(z)=alpha*exp(z/(1+z))+beta is assumed ad hoc
    Chosen for smoothness; no physical derivation; the paper's results depend on this shape.
  • standard math Natural units kappa^2=1
    Adopted in Section 2 to simplify calculations.
  • domain assumption Data likelihoods (CC, BAO, Pantheon+) are correctly implemented
    The BAO compilation and Pantheon+ covariance handling are not fully specified.

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Cite this review

Pith. "Pith review of Cosmological insights from an exponential $Om(z)$ function in $f(T,T_{G})$ gravity framework." pith.science (2026). https://pith.science/paper/ISU3QFV7

@misc{pith2026250606399,
  author       = {Pith},
  title        = {Pith review of: Cosmological insights from an exponential $Om(z)$ function in $f(T,T_G)$ gravity framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISU3QFV7}},
  note         = {Machine review of arXiv:2506.06399}
}
abstract

We examine a modified teleparallel gravity model defined by $f(T,T_{G})=T+\gamma\sqrt{T_{G}}+\delta\sqrt{T}$ by introducing an exponential $Om(z)$ diagnostic of the form $Om(z)=\alpha e^{\frac{z}{1+z}}+\beta$. This novel form captures smooth redshift evolution and allows for a flexible, model-independent probe of dark energy dynamics. We derive a Hubble function from this expression and use MCMC analysis with $31$ CC, $26$ BAO and $1701$ Pantheon+ data points to constrain the model parameters. The best-fit results yield $H_{0} \in [68.46, 77.38]$km/s/Mpc for $\alpha \in [-0.232, -0.068]$ and $\beta \in [0.218, 0.560]$ which is consistent with local $H_{0}$ values. Our model predicts a transition redshift $z_{tr} \approx (0.48-0.54)$, present $q_{0}\approx -0.34$, and $\omega_{0}\approx-0.33$. It satisfies NEC and DEC, closely tracks $\Lambda$CDM in the statefinder plane and estimates a cosmic age of $(13.28-13.87)$ Gyr which confirms its strength in explaining late-time acceleration. Our findings demonstrate that the exponential $Om(z)$ parameterization provides a robust and insightful approach to trace dark energy evolution within modified gravity frameworks.

Figures

Figures reproduced from arXiv: 2506.06399 by the authors.

Figure 1
Figure 1. Joint confidence regions for (H0, α, β) parameters from various dataset combinations. The shaded regions denote the 1σ (68.27%), 2σ (95.45%) and 3σ (99.73%) confidence levels [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Error bar analysis depicting variations in parameter estima [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. q vs. z plot [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Energy density and pressure vs. redshift for the param [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: EoS parameter vs. redshift for the parameter [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Om(z) vs. redshift. The plot of Om(z) versus redshift shows a clear increasing trend, starting near 0.05 at higher redshifts and approaching approximately 0.36 as z → −1. This behavior is consistent with a dynamical dark energy scenario and reflects a deviation from th…
Figure 7
Figure 7. Figure 7: Redshift evolution of energy conditions for the paramete [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Statefinder trajectory in the {r, s} plane for the CC+BAO+Pantheon+ dataset. 9 Age of the Universe for f(T, TG) framework Accurately determining the age of the Universe serves as a crucial consistency test for any cosmological model. In our framework, the age is comput…
Figure 9
Figure 9. Figure 9: The variation of cosmic time with redshift for best fit values [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

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