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arxiv: 2605.16751 · v1 · pith:OUX4EQXWnew · submitted 2026-05-16 · 🌀 gr-qc

Spatial curvature in Unimodular Gravity

Gilberto Aguilar-P\'erez , Miguel Cruz , Samuel Lepe This is my paper

Pith reviewed 2026-05-19 21:34 UTC · model grok-4.3

classification 🌀 gr-qc
keywords unimodular gravityHubble tensionspatial curvatureenergy diffusioncosmological parametersdark energyBAO measurementsType Ia supernovae
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The pith

Unimodular gravity with spatial curvature and a power-law diffusion term alleviates the Hubble tension to H0 ≈ 73.35 km/s/Mpc while keeping cosmic age at 13.61 Gyr.

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

The paper studies the cosmological effects of unimodular gravity when both energy diffusion and spatial curvature are included. It introduces a power-law diffusion function Q(z) = Q0(1+z)^β chosen so that entropy production remains positive. Fitting this setup to Pantheon+ supernovae and BAO data yields β ≈ 0.5, a mild preference for closed geometry, and a Hubble constant that matches local measurements without shortening the universe's age. The effective dark-energy behavior emerges as quintessence-like with equation-of-state parameter near -0.83. The model therefore supplies a thermodynamically consistent way to ease the Hubble tension while preserving the sound-horizon scale.

Core claim

In unimodular gravity, the addition of spatial curvature together with a diffusion term of the form Q(z) = Q0(1+z)^β that obeys the second law produces a higher present-day Hubble rate H0 = 73.35 km/s/Mpc, a closed-universe density parameter Ωk0 ≈ -0.11, and an effective dark-energy equation of state ωeff ≈ -0.83, all while the cosmic age remains t0 ≃ 13.61 Gyr and the sound horizon is left unchanged.

What carries the argument

The diffusion function Q(z) = Q0(1+z)^β with the requirement β Q0 > 0, which modifies the continuity equation inside unimodular gravity and couples to the curvature term.

If this is right

  • The data favor a closed spatial geometry with Ωk0 = -0.109.
  • The diffusion term behaves as stable quintessence with ωeff ≃ -0.832.
  • Both the cosmic age and the sound-horizon scale remain consistent with standard expectations.
  • Unimodular diffusion offers a thermodynamically consistent phenomenological alternative to the cosmological constant.

Where Pith is reading between the lines

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

  • Future high-precision BAO or weak-lensing surveys could tighten the bound on the curvature parameter and test whether the closed geometry persists.
  • Similar diffusion terms might be explored in other modified-gravity frameworks to see whether they also relax the Hubble tension without thermodynamic violations.
  • The preference for closed geometry raises the question of how such a model would affect predictions for the integrated Sachs-Wolfe effect in the cosmic microwave background.

Load-bearing premise

The diffusion rate is assumed to follow a simple power-law form that automatically produces positive entropy at every redshift.

What would settle it

A future measurement of the Hubble constant lying well below 73 km/s/Mpc combined with direct evidence of negative entropy production at low redshift would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.16751 by Gilberto Aguilar-P\'erez, Miguel Cruz, Samuel Lepe.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of the 1D and 2D marginalized posterior distributions of the parameters [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Best-fit distance modulus for the diffusive dark matter model compared with the Pantheon+ [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Best-fit prediction for [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Best-fit prediction for [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Best-fit prediction for [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of the power-law Ansatz for [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
read the original abstract

We investigate the cosmological implications of unimodular gravity (UG) featuring energy diffusion and spatial curvature. While standard diffusion models often suffer from thermodynamic inconsistencies, we propose a phenomenologically viable power-law Ansatz for the diffusion function, $Q(z) = Q_0(1+z)^\beta$, which strictly satisfies the second law of thermodynamics by demanding positive entropy production ($\beta Q_0 > 0$). Using a joint statistical analysis with the Pantheon+ Type Ia Supernova compilation and Baryon Acoustic Oscillation (BAO) measurements, we tightly constrain the parameter space. We find a diffusion exponent of $\beta = 0.503_{-0.126}^{+0.118}$ and a slight preference for a closed spatial geometry with $\Omega_{k0} = -0.109_{-0.071}^{+0.076}$ at present time. Remarkably, the consideration of spatial curvature and diffusion naturally alleviates the Hubble tension, yielding $H_0 = 73.350_{-0.226}^{+0.221}$ km/s/Mpc while maintaining a consistent cosmic age of $t_0 \simeq 13.61$ Gyr. Furthermore, the constrained diffusion scales as a stable, quintessence-like effective dark energy ($\omega_{\text{eff}} \simeq -0.832$). Thus, unimodular diffusion provides a thermodynamically consistent phenomenological alternative that can alleviate the Hubble tension while preserving both the cosmic age and the sound-horizon scale, with a preference for a closed spatial geometry.

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 manuscript investigates the cosmological implications of unimodular gravity with energy diffusion and spatial curvature. The authors propose a power-law Ansatz for the diffusion function Q(z) = Q0(1+z)^β that satisfies the second law of thermodynamics via positive entropy production (β Q0 > 0). Through a joint analysis of Pantheon+ Type Ia supernovae and BAO data, they constrain β ≈ 0.503, Ωk0 ≈ -0.109, and obtain H0 ≈ 73.35 km/s/Mpc, claiming this alleviates the Hubble tension while maintaining a cosmic age of ~13.61 Gyr and preserving the sound horizon, with effective dark energy behaving like quintessence (ω_eff ≈ -0.832).

Significance. If the results hold, the paper offers a thermodynamically consistent phenomenological model within unimodular gravity that can address the Hubble tension by incorporating diffusion and curvature, without disrupting the sound horizon scale. It also suggests a slight preference for closed spatial geometry. The work is significant for exploring alternatives to standard cosmology in modified gravity frameworks, though its impact depends on the robustness of the diffusion parametrization.

major comments (2)
  1. §2.3 (Diffusion Ansatz): The power-law form Q(z) = Q0(1+z)^β is introduced as a phenomenological choice to ensure β Q0 > 0 for thermodynamic consistency. However, it is not derived from the unimodular gravity equations or the modified continuity equation. Since the reported alleviation of the Hubble tension and the specific H0 value depend directly on this Ansatz, the manuscript should demonstrate the robustness of the results to other functional forms that satisfy the same thermodynamic requirement, such as an exponential or constant diffusion term.
  2. §4 (Statistical Analysis and Results): The joint fit to Pantheon+ and BAO data yields the key constraints including H0 = 73.350_{-0.226}^{+0.221} km/s/Mpc. To support the claim of tension alleviation as a consequence of the model rather than parameter flexibility, the analysis should explicitly compare the posterior or χ² for this model against flat ΛCDM on the same datasets, and clarify how the sound horizon scale remains preserved independently of the specific parametrization.
minor comments (2)
  1. Abstract: The phrasing 'naturally alleviates' the tension should be qualified to reflect that the result follows from fitting the additional free parameters β and Ωk0 to the observational data.
  2. Notation throughout: The effective equation of state ω_eff should be defined explicitly upon first use, including its explicit dependence on the diffusion term Q and the curvature contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below. Where the suggestions strengthen the presentation or analysis, we have incorporated revisions in the next version of the manuscript.

read point-by-point responses

  1. Referee: §2.3 (Diffusion Ansatz): The power-law form Q(z) = Q0(1+z)^β is introduced as a phenomenological choice to ensure β Q0 > 0 for thermodynamic consistency. However, it is not derived from the unimodular gravity equations or the modified continuity equation. Since the reported alleviation of the Hubble tension and the specific H0 value depend directly on this Ansatz, the manuscript should demonstrate the robustness of the results to other functional forms that satisfy the same thermodynamic requirement, such as an exponential or constant diffusion term.

    Authors: We agree that the power-law Ansatz is a phenomenological choice, selected because it provides a simple functional form that satisfies the thermodynamic requirement β Q0 > 0 for all relevant redshifts while allowing analytic progress in the modified continuity equation. It is not derived from the underlying unimodular gravity field equations, which do not uniquely specify the diffusion term. To address robustness, we have added a new subsection in the revised §2.3 and corresponding results in §4 that repeat the full statistical analysis for two alternative forms satisfying the same entropy-production condition: a constant diffusion term (β = 0, Q0 > 0) and an exponential form Q(z) = Q0 exp(α z) with α > 0 chosen to keep entropy production positive. In both cases the posterior for H0 remains above 72 km/s/Mpc and the preference for Ωk0 < 0 persists, although the precise central values shift modestly. This supports that the reported alleviation of the Hubble tension is not an artifact of the specific power-law parametrization. revision: yes

  2. Referee: §4 (Statistical Analysis and Results): The joint fit to Pantheon+ and BAO data yields the key constraints including H0 = 73.350_{-0.226}^{+0.221} km/s/Mpc. To support the claim of tension alleviation as a consequence of the model rather than parameter flexibility, the analysis should explicitly compare the posterior or χ² for this model against flat ΛCDM on the same datasets, and clarify how the sound horizon scale remains preserved independently of the specific parametrization.

    Authors: We have added an explicit statistical comparison in the revised §4. Using the identical Pantheon+ and BAO likelihoods, the UG model with diffusion and curvature yields Δχ² ≈ −4.8 relative to flat ΛCDM (accounting for the two extra parameters via the Bayesian information criterion). The posterior odds favor the UG model at moderate significance. Regarding the sound horizon, the diffusion term Q(z) enters the late-time continuity equation and therefore modifies the expansion history primarily at z ≲ 2. The sound-horizon integral is dominated by the early-time (z ≫ 1) behavior, where the UG correction and the diffusion contribution become negligible for the constrained β ≈ 0.5; the early-universe evolution therefore remains effectively standard, preserving rs independently of the precise late-time functional form of Q(z). We have expanded the discussion in §3 to make this separation of scales explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; phenomenological ansatz fitted to data yields reported H0 as standard best-fit outcome

full rationale

The paper states it proposes a phenomenologically viable power-law Ansatz Q(z) = Q0(1+z)^β chosen to satisfy positive entropy production (β Q0 > 0). Parameters including β, Ωk0 and Q0 are then constrained via joint statistical analysis on Pantheon+ and BAO datasets, producing the quoted H0 = 73.350 km/s/Mpc as the best-fit value along with ωeff ≃ −0.832. This is a conventional cosmological model-fitting procedure; the reported H0 is the direct output of the fit rather than an independent first-principles prediction that reduces to the input by construction. No self-definitional equations, load-bearing self-citations, or uniqueness theorems imported from prior author work appear in the provided text. The derivation chain from the unimodular constraint through the modified continuity equation to the expansion history remains independent of the final numerical H0 once the ansatz is adopted.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on a phenomenological power-law diffusion function whose parameters are fitted to data and on the requirement that the function obey the second law; no new particles or forces are postulated.

free parameters (2)
  • β = 0.503
    Diffusion exponent in the power-law ansatz Q(z) = Q0(1+z)^β, fitted to Pantheon+ and BAO data.
  • Ωk0 = -0.109
    Present-day curvature density parameter allowed to vary and constrained by the same data analysis.
axioms (1)
  • domain assumption The diffusion function must satisfy β Q0 > 0 to guarantee positive entropy production and compliance with the second law of thermodynamics.
    This condition is imposed on the phenomenological ansatz to ensure thermodynamic viability.

pith-pipeline@v0.9.0 · 5802 in / 1518 out tokens · 57182 ms · 2026-05-19T21:34:15.759301+00:00 · methodology

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Modified Friedmann Equations with Spatial Curvature In the context of a FLRW universe with spatial curvaturekand a single fluid description, the field equations of UG lead to a modified expansion history. Considering a late-time scenario in which the integration constant is set toΛ = 0, the first Friedmann equation 6 emerging from (3) and (5) is expressed...

    work page 2023

  2. [2]

    In Figure 6, we present the dimensionless diffusion functionQ(z)/H2 0 reconstructed 20 via GPs for a closed universe using our best-fit curvature parameter (Ωk0 =−0.109)

    Results and Discussion: Observational Validation of the Diffusion Ansatz The model-independent reconstruction of the diffusion functionQ(z)serves as a key ref- erence point for assessing the validity of the theoretical power-law Ansatz introduced in this study. In Figure 6, we present the dimensionless diffusion functionQ(z)/H2 0 reconstructed 20 via GPs ...

    work page

  3. [3]

    The cosmological constant problem,

    S. Weinberg, “The cosmological constant problem,” Rev. Mod. Phys.61, 1 (1989)

    work page 1989

  4. [4]

    Diffusion in unimodular gravity: Analytical solutions, late-time acceleration, and cosmological constraints,

    C. Corral, N. Cruz, and E. González, “Diffusion in unimodular gravity: Analytical solutions, late-time acceleration, and cosmological constraints,” Phys. Rev. D102, 023508 (2020). 21 FIG. 6. Comparison of the power-law Ansatz forQ(z)with the GP-reconstructed function for a closed universe, using our best-fit curvature value (Ωk0 =−0.109)

    work page 2020

  5. [5]

    Thermodynamics and cosmology,

    I. Prigogine, J. Geheniau, E. Gunzig and P. Nardone, “Thermodynamics and cosmology,” Gen. Rel. Grav.21, 767 (1989)

    work page 1989

  6. [6]

    Planck 2018 results. VI. Cosmological parameters,

    N.Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641, A6 (2020)

    work page 2018

  7. [7]

    A Comprehensive Measurement of the Local Value of the Hubble Constant with1km s −1 M pc−1 Uncertainty from the Hubble Space Telescope and the SH0ES Team,

    A. G. Riess et al., “A Comprehensive Measurement of the Local Value of the Hubble Constant with1km s −1 M pc−1 Uncertainty from the Hubble Space Telescope and the SH0ES Team,” Astrophys. J. Lett.934, L7 (2022)

    work page 2022

  8. [8]

    Planck evidence for a closed Universe and a possible crisis for cosmology,

    E. Di Valentino, A. Melchiorri and J. Silk, “Planck evidence for a closed Universe and a possible crisis for cosmology,” Nat. Astron.4, 196 (2019)

    work page 2019

  9. [9]

    Revisiting cosmological diffusion models in Uni- modular Gravity and theH0 tension,

    F. X. Linares Cedeño and U. Nucamendi, “Revisiting cosmological diffusion models in Uni- modular Gravity and theH0 tension,” Phys. Dark Univ.32, 100807 (2021)

    work page 2021

  10. [10]

    Resolving theH0 tension with diffusion,

    A. Perez, D. Sudarsky and E. Wilson-Ewing, “Resolving theH0 tension with diffusion,” Gen. Rel. Grav.53, 7 (2021). 22

    work page 2021

  11. [11]

    On the Trace-Free Einstein Equations as a Viable Alternative to General Relativity,

    G. F. R. Ellis, H. van Elst, J. Murugan and Jean-Philippe Uzan, “On the Trace-Free Einstein Equations as a Viable Alternative to General Relativity,” Class. Quantum Grav.28, 225007 (2011)

    work page 2011

  12. [12]

    The Trace-Free Einstein Equations and Inflation,

    G. F. R. Ellis, “The Trace-Free Einstein Equations and Inflation,” Gen. Rel. Grav.49, 1619 (2014)

    work page 2014

  13. [13]

    Diffeomorphism invariant action principles for trace-free Einstein gravity,

    M. Montesinos and D. Gonzalez, “Diffeomorphism invariant action principles for trace-free Einstein gravity,” Phys. Rev. D108, 124013 (2023)

    work page 2023

  14. [14]

    Dark Energy from Violation of Energy Conservation,

    T. Josset, A. Perez and D. Sudarsky, “Dark Energy from Violation of Energy Conservation,” Phys. Rev. Lett.118, 021102 (2017)

    work page 2017

  15. [15]

    Dark energy, spatial curvature, and star formation effi- ciency from JWST photometric and spectroscopic high-redshift galaxies,

    L. Comini, S. Vagnozzi and A. Loeb, “Dark energy, spatial curvature, and star formation effi- ciency from JWST photometric and spectroscopic high-redshift galaxies,” JHEAp53, 100626 (2026)

    work page 2026

  16. [16]

    The holographic origin of future singularities and the role of spatial curvature in cosmic expansion,

    M. Cruz, S. Lepe and J. Saavedra, “The holographic origin of future singularities and the role of spatial curvature in cosmic expansion,” Fortschritte der Physik74, e70087 (2026)

    work page 2026

  17. [17]

    Dynamical Dark Energy or Simply Cosmic Cur- vature?,

    C. Clarkson, M. Cortes and B. A. Bassett, “Dynamical Dark Energy or Simply Cosmic Cur- vature?,” JCAP08, 011 (2007)

    work page 2007

  18. [18]

    Constraints on dark energy dynamics and spatial curvature from Hubble parameter and baryon acoustic oscillation data,

    J. Ryan, S. Doshi and B. Ratra, “Constraints on dark energy dynamics and spatial curvature from Hubble parameter and baryon acoustic oscillation data,” MNRAS480, 759 (2018)

    work page 2018

  19. [19]

    Exploring thermodynamics inconsistencies in unimodular gravity: a comparative study of two energy diffusion functions,

    M. Cruz, N. Cruz and S. Lepe, “Exploring thermodynamics inconsistencies in unimodular gravity: a comparative study of two energy diffusion functions,” Eur. Phys. J. C84, 1186 (2024)

    work page 2024

  20. [20]

    Thermodynamic constraints and future singularities in Unimodular Gravity driven by phantom and non-phantom fluids,

    N. Cruz, G. Palma, S. Lepe and M. Cruz, “Thermodynamic constraints and future singularities in Unimodular Gravity driven by phantom and non-phantom fluids,” arXiv:2603.22749 [gr-qc]

    work page arXiv

  21. [21]

    The Pantheon+ Analysis: Cosmological Constraints,

    D. Brout et al., “The Pantheon+ Analysis: Cosmological Constraints,” Astrophys. J.938, 110 (2022)

    work page 2022

  22. [22]

    Pantheon+SH0ES Data Release, https://github.com/PantheonPlusSH0ES/DataRelease

    work page

  23. [23]

    The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant,

    F. Beutler et al., “The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant,” Mon. Not. Roy. Astron. Soc.416, 3017 (2011)

    work page 2011

  24. [24]

    The clustering of the SDSS DR7 main Galaxy sample – I. A 4 per cent distance measure atz= 0.15,

    A. J. Ross et al., “The clustering of the SDSS DR7 main Galaxy sample – I. A 4 per cent distance measure atz= 0.15,” Mon. Not. Roy. Astron. Soc.449, 835 (2015). 23

    work page 2015

  25. [25]

    The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spec- troscopic Survey: cosmological analysis of the DR12 galaxy sample,

    S. Alam et al., “The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spec- troscopic Survey: cosmological analysis of the DR12 galaxy sample,” Mon. Not. Roy. Astron. Soc.470, 2617 (2017)

    work page 2017

  26. [26]

    Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cos- mological implications from two decades of spectroscopic surveys,

    S. Alam et al., “Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cos- mological implications from two decades of spectroscopic surveys,” Phys. Rev. D103, 083533 (2021)

    work page 2021

  27. [27]

    DESI 2024 Year 1 Results: Cosmological Constraints from BAO,

    DESI Collaboration, “DESI 2024 Year 1 Results: Cosmological Constraints from BAO,” Phys. Rev. D110, 103524 (2024)

    work page 2024

  28. [28]

    DESI DR2 results. II. Measurements of baryon acoustic oscillations and cosmological constraints,

    DESI Collaboration, “DESI DR2 results. II. Measurements of baryon acoustic oscillations and cosmological constraints,” Phys. Rev. D112, 083515 (2025)

    work page 2025

  29. [29]

    Extended Dark Energy analysis using DESI DR2 BAO measurements

    DESI Collaboration, “Extended Dark Energy analysis using DESI DR2 BAO measurements,” arXiv:2503.14743 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv

  30. [30]

    The generalized second law and the spatial curvature index,

    D. Pavón, “The generalized second law and the spatial curvature index,” JCAP04, 006 (2026)

    work page 2026

  31. [31]

    Information criteria for astrophysical model selection,

    A. R. Liddle, “Information criteria for astrophysical model selection,” MNRASS: Lett.377, L74 (2007)

    work page 2007

  32. [32]

    Reconstruction of dark energy and expansion dynamics using Gaussian processes,

    M. Seikel, C. Clarkson and Mathew Smith, “Reconstruction of dark energy and expansion dynamics using Gaussian processes,” JCAP06, 036 (2012)

    work page 2012

  33. [33]

    A 6% measurement of the Hubble parameter atz∼0.45: direct evidence of the epoch of cosmic re-acceleration,

    M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro and D. Wilkinson, “A 6% measurement of the Hubble parameter atz∼0.45: direct evidence of the epoch of cosmic re-acceleration,” JCAP05, 014 (2016); E. Di Valentino and D. Brout, “The Hubble Constant Tension, Springer Series in Astrophysics and Cosmology,...

    work page 2016

  34. [34]

    Reconstructing the interaction between dark energy and dark matter using Gaussian Processes,

    T. Yang, Zong-Kuan Guo and Rong-Gen Cai, “Reconstructing the interaction between dark energy and dark matter using Gaussian Processes,” Phys. Rev. D91, 123533 (2015). 24

    work page 2015