General Relativistic Entropic Acceleration at the perturbation level: a CLASS implementation and first Boltzmann-code constraints

Dong Ha Lee; Eleonora Di Valentino; Juan Garc\'ia-Bellido; Simone D'Onofrio

REVIEW 3 major objections 5 minor 171 references

General Relativistic Entropic Acceleration at the perturbation level: a CLASS implementation and first Boltzmann-code constraints

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper reports the first Boltzmann-solver implementation of GREA, showing that the model's single parameter α is constrained to ≈1 by CMB, BAO, and supernova data, matching ΛCDM's fit within |Δχ²|≲6.

desk verdict Solid first Boltzmann implementation of GREA, but the α≈1 'prediction' is a fitted parameter and the perturbation sector is an effective-fluid placeholder — still worth refereeing. read the letter →

arxiv 2607.25841 v1 pith:BRLQMFLU submitted 2026-07-28 gr-qc astro-ph.CO

Simone D'Onofrio , Dong Ha Lee , Eleonora Di Valentino , Juan García-Bellido This is my paper · ORCID

classification gr-qcastro-ph.CO MSC 83F0583C5583-08 PACS 98.80.-k95.36.+x04.20.-q

keywords GeneralRelativisticEntropicAccelerationhorizonthermodynamicsdarkenergycosmologicalconstantproblemEinstein-BoltzmannsolverCMBpowerspectraparametrizedpost-Friedmannphantomcrossing

topics Dark Energy

open problems Dark Energy

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 pushes General Relativistic Entropic Acceleration (GREA) from a background-level idea into full perturbation theory, implementing it inside an Einstein–Boltzmann solver to compute CMB, lensing, and matter power spectra for the first time. It then confronts the model with the full primary-CMB, DESI BAO, and Type Ia supernova data via Markov-chain Monte Carlo. The central result is that GREA's single coupling α is measured to be about 1, exactly where the thermodynamic derivation places it, and that the one-parameter model fits the data as well as ΛCDM, with |Δχ²|≤6 across nine dataset combinations. If correct, this shows that a thermodynamically motivated, parameter-light alternative to the cosmological constant can survive the same precision tests as Λ, making acceleration a consequence of horizon entropy rather than vacuum energy.

What carries the argument

The key object is the entropic-force tensor f_μν and its homogeneous limit: an effective dark-energy density ρ_GREA ∝ sinh(2τ)/a², where τ is the dimensionless conformal time tied to the causal horizon. Its dynamics are fixed by a single parameter α, defined by α D_H(z=0) = √(-k) η0. At the perturbation level, the machinery is the effective-fluid description: the component is assigned the GREA equation of state w(a) and evolved as a non-clustering fluid (c_s²=1) under the parametrized-post-Friedmann closure, which handles the w=-1 crossing. The implementation also requires a subtle normalization rescaling so that the physical present-day Hubble rate, not the fiducial H0, is what the Boltzman

What would settle it

Measure the growth index γ(z) from redshift-space distortions (e.g., DESI, Euclid): GREA predicts a rising γ(z) with dγ/dz>0, while ΛCDM and most quintessence models predict a decreasing γ(z). If future data show dγ/dz<0, the model's growth prediction is falsified; alternatively, a detection of non-adiabatic pressure in the dark-energy fluid would invalidate the δζ=0 closure.

Watch

Extended reading notes

Core claim

The authors integrate the GREA background directly into a Boltzmann solver and evolve the entropic component as an effective dark-energy fluid with sound speed c_s² = 1, regulated by the parametrized-post-Friedmann scheme so perturbations remain regular through the phantom crossing. The resulting angular power spectra and growth functions are new. When fit to the CMB-SPA (Planck+ACT+SPT) likelihood, DESI DR2 BAO, and Pantheon+/DES Dovekie supernovae, the inferred coupling α—the ratio of spatial-curvature scale to the causal horizon today—clusters tightly around unity (α≈1.00–1.08 for the CMB-anchored combinations), in agreement with the model's parameter-free prediction. The fit is statistic

Load-bearing premise

The perturbed entropic source δf_μν is not derived from first principles; the paper sets its perturbation to zero (δζ=0) and models the component as a smooth fluid with sound speed c_s²=1, so if the true perturbations carry non-adiabatic or non-local pieces, the computed ISW and lensing spectra—and therefore the reported constraints—are not uniquely determined by GREA alone.

Editorial extensions

If this is right

If the central claim holds, GREA provides a single-parameter, thermodynamically grounded rival to ΛCDM that reproduces the expansion history, CMB anisotropies, and growth data at the same quality, eliminating the need for a finely tuned cosmological constant.
The distinctive second phantom crossing at z≈2 is a clean, falsifiable prediction that future higher-redshift dark-energy reconstructions (e.g., from DESI Lyman-α and Euclid) can test.
The model predicts enhanced growth at fixed primordial amplitude, raising σ8 relative to ΛCDM; a dedicated weak-lensing analysis within the GREA framework would determine whether this worsens or reframes the S8 tension.
The full Boltzmann-level implementation opens the way to constrain GREA with lensing, ISW, and growth-rate data from upcoming surveys, where the model's signatures are expected to be decisive.
The sound-speed sensitivity analysis (all observables shift by <1.6%) indicates current constraints are robust to the one undetermined ingredient, so the α≈1 measurement is stable under the effective-fluid assumptions.

Reading between the lines

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

A full first-principles derivation of the perturbed entropic source δf_μν might reveal non-adiabatic pressure or non-local horizon terms; if those alter the low-ℓ ISW response by more than the ~1.6% sound-speed bracket, the reported constraints could shift, though likely within current error bars.
The model's thermodynamic origin hints that black-hole entropy growth (already discussed by the authors) could produce analogous acceleration signatures; testing GREA with gravitational-wave standard sirens could further distinguish it from scalar-field dark energy.
Since GREA is not nested in ΛCDM, the Δχ²≈0 is more telling than a Bayesian preference; a future comparison using Bayesian evidence with theory-motivated priors (fine-tuning penalty for Λ) would likely shift the model odds toward GREA.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I've read D'Onofrio et al. on GREA in CLASS. The honest take: this is a technically competent, transparent implementation of a one-parameter dark-energy alternative, and the first to push GREA through a full Einstein-Boltzmann likelihood analysis. The code is public, the validation harness is sensible (reproducing the semi-analytic fσ8 is a good check), and the c_s^2 robustness scan is a genuine plus. The background normalization subtlety — H0 in the definition of τ is not the physical present-day rate — is handled correctly and explained clearly.

What's actually new: full CMB temperature/polarization/lensing spectra for GREA, MCMC against CMB-SPA+DESI+SN, and constraints on α from full primary CMB rather than compressed distance priors. That is a real step past the previous background-only analyses [135,136]. The paper also flags its central limitation candidly: the perturbed entropic source δfμν is not derived, and the linear sector is modeled as an effective fluid with PPF closure and c_s^2=1. Appendix B shows that varying c_s^2 between 0.01 and 1 shifts all observables by less than 1.6%, so the main constraints on α are probably robust to that choice. Good.

The soft spots, in order of importance.

First, the α≈1 story. α is not an independent prediction of the theory; it is a defined ratio, and the sampled parameter is sqrt(-k)η0 with a flat prior. Recovering α≈1 from the posterior and calling it 'excellent agreement with the theoretical prediction' is circular. The abstract and conclusions lean on that framing heavily. The authors should either derive a genuine prior or prediction for α, or simply report the constraints without the vindication language. This is a rhetorical issue, not a mathematical one, but it matters for how readers interpret the headline result.

Second, the perturbation sector. The 'full CMB spectra of GREA' are spectra of an effective fluid, not of the complete theory. The non-adiabatic pressure and non-local light-cone response of δfμν could, in principle, alter the low-ℓ ISW and lensing amplitude. The c_s^2 scan does not bracket those effects. The paper acknowledges this explicitly, and the impact on the main constraints is probably small, but the claim that these are 'self-consistent' GREA predictions is a stretch. Reframing the sections to say 'spectra of the effective-fluid approximation' would be more accurate.

Third, the second phantom crossing at z≈2 is a model prediction but lies beyond the current data bins; the authors admit this, so it's not a fatal flaw, but it's not a test yet.

The paper deserves a serious referee. It is not a desk reject. The code and data are public, the implementation is careful, and the model is interesting enough to warrant scrutiny. My recommendation: send it to review, but require the authors to reframe the α prediction claim and to be explicit that the perturbation-level outputs are provisional pending a derivation of δfμν.

I'd bring it to a reading group if we want to dissect 'prediction vs posterior' language. I'd cite it for the CLASS implementation and constraints, though not for the theory itself.

Referee Report

3 major / 5 minor

Summary. The paper reports the first implementation of the General Relativistic Entropic Acceleration (GREA) model inside the CLASS Einstein–Boltzmann solver. The GREA background is integrated directly, while the entropic component is treated at linear order as an effective dark-energy fluid with the GREA equation of state, PPF closure, and sound speed c_s^2=1. The code is validated against semi-analytic growth and used to compute CMB TT/TE/EE, lensing, lensing-induced BB, matter power spectrum, and fσ8. The authors run MCMC analyses with COBAYA against CMB-SPA, DESI DR2 BAO, and several SN samples, report α≈1 in most combinations with |Δχ²|≲6 relative to ΛCDM, and interpret the results as confirming the GREA prediction α∼1 and a second phantom crossing at z≈2.

Significance. If the effective-fluid treatment is accepted as a faithful proxy for the GREA perturbation sector, this is a useful and reproducible technical step: it opens full primary-CMB and lensing likelihoods to a one-parameter, thermodynamically motivated alternative to ΛCDM. The public code, validation harness, and MCMC setup are concrete strengths, and the background-normalization subtlety is handled carefully. However, the significance is conditional, because the perturbed entropic source δfμν is not derived and the low-ℓ ISW and lensing predictions are not uniquely determined by GREA at this stage.

major comments (3)

[Section III (paragraph starting 'A fully consistent Einstein–Boltzmann treatment')] The paper explicitly states that the perturbed entropic sector δfμν is not derived: 'Whether this non-local piece can be neglected (δζ=0) is the central open question of a complete derivation. We do not settle it here.' The implementation then replaces δfμν by a local effective fluid with w(a), PPF closure, and c_s^2=1. This is an assumption, not a consequence of GREA. Consequently the abstract's claim of 'full CMB and matter power spectra' and the later statement that spectra are computed 'self-consistently' overstate what is uniquely predicted. The spectra are those of the effective-fluid ansatz. This is a load-bearing issue because the low-ℓ ISW and lensing channels are exactly the parts of the data that depend on this assumption; the Table II constraints are therefore conditional on the ansatz. I recommend explicitly reframing the title/abstract/conclusions as an implementation of th
[Appendix B and Section III (c_s^2 sensitivity)] The robustness scan for c_s^2∈[0.01,1] varies only the local clustering-to-smooth transition of a single adiabatic sound speed. It does not sample non-adiabatic pressure or the non-local light-cone response that a genuine δfμν could generate. The sentence in Section III that 'the plausible size of the effect is bracketed by the c_s^2 sensitivity analysis' is therefore too strong: the scan brackets only one family of local closures. A non-adiabatic or non-local perturbation could shift the low-ℓ ISW and lensing power by a different amount. The paper itself acknowledges this residual caveat, but the claimed 1.6% robustness should be stated as applying only to the local adiabatic sound-speed ambiguity.
[Eq. (8), Table II, Section VI] The conclusion that α≈1 is 'in excellent agreement with the theoretical prediction' conflates parameter estimation with confirmation of a prediction. The quantity √(−k)η0 is sampled with a flat prior and later mapped to α via Eq. (8); the posterior is a fitted parameter, not an independent test of a prediction unless the theory supplies a separate, prior expectation with uncertainty. The data being consistent with α=1 at the 1σ level is a meaningful result, but it should be phrased as consistency, not as the data 'singling out' a predicted value. Similarly, the second phantom crossing at z≈2 is beyond the last bin of the reconstruction shown in Fig. 9 and is evaluated at the best fit, so it is a model prediction not yet probed by the data; this should be stated more carefully.

minor comments (5)

[Title / Abstract] Typographical: 'aclassimplementation' should be 'a CLASS implementation' with spaces. Also, 'CMB-SP A' appears with an irregular space in several places in Section V; use 'CMB-SPA' consistently.
[Figure 2 caption] The caption notes that 'most of these configurations have two crossings of the phantom divide,' while the text elsewhere refers to 'a transient phantom crossing' in the singular. Please clarify how many crossings occur in the baseline α≈1 case and which crossing is being discussed.
[Section V, near Eq. (13)] The sentence 'the largest is|lnB| ≃2.72' is grammatically incomplete; specify for which dataset combination this value occurs. It appears to be the CMB-SPA+PP+DESI column in Table II, but should be stated explicitly.
[Section IV] When introducing the sampled variable √(−k)η0 = α D_H(0), it may help readers to state the implied prior range for α, since the flat prior [2.5,4.5] on √(−k)η0 translates into a non-uniform prior on α through the background mapping. This would also clarify the 'prediction' discussion in the conclusions.
[Figure 10 caption] The top panel caption says the shaded band indicates the 68% confidence region of the GREA reconstruction, but the plotted curves are said to be evaluated at the best fit; please specify whether the band is from the posterior or from the parameter covariance.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline 'vindication' of α≈1 compares the posterior of the fitted parameter to a self-cited 'prediction', rather than to an independently derived result.

fitted input called prediction [Abstract; Eq. (8); Sec. VII (Discussion and Conclusions)]
"That the size of the causal horizon today should equal the spatial-curvature scale, α∼1, is not a fitted outcome but a genuine prediction of the theory, and the data single it out at the few-percent level."

In Eq. (8) α is defined through the sampled quantity: αD_H(0)=√(-k)η0. Table I samples √(-k)η0 with a flat prior U[2.5,4.5], and Table II lists α as a derived parameter obtained by inverting that background mapping. The 'genuine prediction' α∼1 is not re-derived anywhere in this paper; it is the model's own O(1) parameter expectation, imported from the same group's earlier work. Thus the abstract's 'excellent agreement with the theoretical prediction' is a comparison between the posterior of the fitted variable and a self-cited prior expectation, not between two independent quantities.
self citation load bearing [Sec. I (Introduction) and Sec. VII; refs. [131], [135]]
"the detailed background and linear-growth predictions from the homogeneous cosmic horizon were worked out in [131]. The most complete analysis to date [135] ... finds a fit comparable to ΛCDM with α≃1.09 and a transient phantom crossing at z≲2"

The central interpretative claim — that α∼1 is a genuine theoretical prediction — is not established in the present manuscript. It is justified by a chain of citations to [131] (authored by one of the present authors) and to the companion background analysis [135]. The present MCMC fit is then presented as vindicating that cited expectation. The self-citation is load-bearing because the 'prediction' being confirmed is the same model parameter fitted here, and no independent derivation or external theorem is supplied to break the loop.

full rationale

The technical core is substantially self-contained: the GREA background ODE is integrated directly in CLASS, the semi-analytic fσ8(z) of Ref. [131] is reproduced as a numerical validation, and the CMB-SPA/DESI/SN constraints are genuine external-data comparisons. I do not flag those as circular. The circular/self-citation component is concentrated in the headline interpretation: α is a free parameter (Eq. 8), sampled as √(-k)η0 and mapped to α in post-processing, yet the paper calls α∼1 a 'genuine prediction' on the authority of the same group's earlier papers and then reads the fitted posterior as a 'vindication'. That is a fitted input re-labelled as a prediction and supported by a self-citation chain. The acknowledged open problem of the perturbed entropic source δfμν — replaced by a c_s²=1 PPF effective fluid — is a real limitation on the claim of 'full GREA spectra', but it is an admitted assumption rather than a circular reduction; likewise, the second phantom crossing is a model output evaluated at the best-fit α, which is ordinary parameter conditioning, not circularity. The score of 6 reflects partial circularity in the central interpretative claim, not in the code or likelihood pipeline.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The computation rests on the GREA thermodynamic framework taken from prior papers by the same group, an effective-fluid/PPF placeholder for the un-derived perturbed entropic source, and standard ΛCDM likelihood machinery. The single model parameter α is fitted, not derived, and the standard cosmological parameters are also fitted. No machine-checked proofs or independent experimental handles for the entropic component are provided.

free parameters (2)

α (sampled as sqrt(-k)η0 = α D_H(0)) = α ≈ 1.008 ± 0.036 (CMB-SPA+PP); 0.965–1.079 across combinations
The single GREA parameter is sampled with flat prior U[2.5,4.5] on sqrt(-k)η0. The paper calls α≈1 a prediction, but it is a fitted parameter constrained by the same data used to claim agreement.
Standard cosmological parameters {ωb, ωcdm, H0, log(10^10 As), ns, τreio} = e.g., H0 ≈ 67.8 ± 0.6 (CMB-SPA+PP), ωcdm ≈ 0.1185, etc.
Standard free parameters sampled with flat/Gaussian priors and used in the likelihood fits; not model-specific but necessary for the constraints.

assumptions (4)

domain assumption GREA covariant non-equilibrium thermodynamics: entropy production enters as fμν on the matter side, with continuity-equation source T Ṡ/a³ (Eqs. 1–2).
Foundation of the model, taken from refs [125,126,131] by the same author group; not independently established or re-derived here.
domain assumption Horizon thermodynamics: the causal horizon d_H = aη carries Gibbons–Hawking–York temperature and entropy, with ρ_H = T_H S_H / a³ (Eq. 3).
The thermodynamic picture underlying GREA; adopted as an effective quantum-gravitational description on a classical background.
ad hoc to paper Perturbed entropic sector can be represented by an effective fluid with w(a), c_s²=1, and PPF closure; equivalently δζ=0.
Explicitly adopted pending derivation of δfμν; Section III calls the neglect of δζ the central open question and defers it to future work.
domain assumption α ≈ 1 is the theoretical expectation of the model.
Asserted in Section VII as a 'genuine prediction of the theory' but no derivation is given in this paper; it is inherited from earlier work by the same group and is not used in the likelihood calculation.

invented entities (2)

Entropic force tensor fμν / effective negative-pressure entropic fluid
purpose: Drive late-time cosmic acceleration without a cosmological constant; enters the matter side of Einstein's equations and is evolved as an effective dark-energy fluid in CLASS.
No direct observational handle is provided; the claimed predictions (α≈1, second phantom crossing) are either fitted-parameter outputs or outside current data sensitivity.
Causal-horizon degrees of freedom (T_H, S_H) acting thermodynamically
purpose: Provide the entropy growth that generates the entropic acceleration.
An effective description with no independent evidence beyond the model's own fit to the same cosmological data.

how reviews work

0 comments

Cite this review

Pith. "Pith review of General Relativistic Entropic Acceleration at the perturbation level: a CLASS implementation and first Boltzmann-code constraints." pith.science (2026). https://pith.science/paper/BRLQMFLU

@misc{pith2026260725841,
  author       = {Pith},
  title        = {Pith review of: General Relativistic Entropic Acceleration at the perturbation level: a CLASS implementation and first Boltzmann-code constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRLQMFLU}},
  note         = {Machine review of arXiv:2607.25841}
}

abstract

General Relativistic Entropic Acceleration (GREA) attributes the late-time acceleration of the Universe to the entropy growth of the causal cosmological horizon, without a cosmological constant, with a phenomenology fixed by the single $\mathcal{O}(1)$ parameter $\alpha$. The model has so far been confronted with data only at the background level. We present its first implementation within an Einstein-Boltzmann solver: the GREA background is integrated directly into CLASS, while the entropic component is evolved as an effective fluid regulated by the parametrized-post-Friedmann scheme, giving access to the full CMB and matter power spectra. A Markov-chain Monte Carlo analysis with COBAYA against the full primary-CMB likelihoods, DESI DR2 BAO and Type Ia supernovae constrains the coupling $\alpha \sim 1$, in excellent agreement with the theoretical prediction, with a fit matching $\Lambda$CDM to within $|\Delta\chi^2| \lesssim 6$ despite the addition of a single free parameter. The equation of state inferred from the data agrees with binned, model-independent reconstructions and exhibits a second crossing of the phantom divide at $z \simeq 2$, a distinctive prediction of the thermodynamic dynamics rather than of an imposed parametrization.

Figures

Figures reproduced from arXiv: 2607.25841 by the authors.

**Figure 1.** Normalized expansion rate H(z)/(1 + z) for GREA realizations with varying α (color bar), compared to the ΛCDM reference (dashed). The lower panel shows the percentage residuals relative to ΛCDM. A. Validation and Boltzmann-level outputs We validate the implementation with a harness of quantitative acceptance tests that must pass before any perturbation output is used, following the principle that a wrong background… view at source ↗

**Figure 2.** Effective equation of state w(z) of the entropic component for GREA realizations with varying α, exhibiting a transient phantom crossing at z ≲ 2. We point out that most of these configurations have two crossings of the phantom divide [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** CMB temperature (Left) and E-mode polarization (Right) angular power spectra for GREA realizations with varying α, compared to the ΛCDM reference (dashed); the lower panels show the percentage residuals. With the early-time physics held fixed, the modified late-time expansion shifts the angular scale of the acoustic peaks, producing oscillatory residuals that grow towards high ℓ, while the differences at ℓ ≲ 30 in T… view at source ↗

Figures from the paper (9 more)

**Figure 5.** Figure 5: Same as Fig. 4 for the lensing-induced [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Marginalized posteriors for the headline parame [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 8.** Figure 8: Marginalized posteriors for the headline parameters [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: Top: effective dark-energy equation of state w(z) for the GREA best fit (solid curves), compared with the binned, model-independent reconstruction of Ref. [98] (shaded rectangles). Bottom: the corresponding normalized dark-energy density fDE(z), defined as in Eq. (9). …

**Figure 11.** Figure 11: Top: DESI DR2 BAO distance summary statistics, scaled by rd √ z: the transverse comoving distance DM(z)/(rd √ z) (magenta), the angle-averaged distance DV (z)/(rd √ z) (cyan), and z DH(z)/(rd √ z) (purple). Solid curves show the GREA best fit, and dashed curves the Λ…

**Figure 12.** Figure 12: Fractional residuals of the CMB spectra as the [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

**Figure 13.** Figure 13: Marginalized one- and two-dimensional posterior distributions for all sampled and key derived parameters in the [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

**Figure 14.** Figure 14: Marginalized one- and two-dimensional posterior distributions for all sampled and key derived parameters in the [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

**Figure 15.** Figure 15: Marginalized one- and two-dimensional posterior distributions for all sampled and key derived parameters in the [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

171 extracted references · 142 linked inside Pith

[1]

A. G. Riess et al. (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201

arXiv 1998
[2]

Perlmutter et al

S. Perlmutter et al. (Supernova Cosmology Project), As- trophys. J.517, 565 (1999), arXiv:astro-ph/9812133

arXiv 1999
[3]

Sahni and A

V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), arXiv:astro-ph/9904398

arXiv 2000
[4]

Aghanim et al

N. Aghanim et al. (Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

arXiv 2020
[5]

Aghanim et al

N. Aghanim et al. (Planck), Astron. Astrophys.641, A1 (2020), arXiv:1807.06205 [astro-ph.CO]

arXiv 2020
[6]

Louis et al

T. Louis et al. (Atacama Cosmology Telescope), JCAP 11, 062 (2025), arXiv:2503.14452 [astro-ph.CO]

arXiv 2025
[7]

Camphuis et al

E. Camphuis et al. (SPT-3G), Phys. Rev. D113, 083504 (2026), arXiv:2506.20707 [astro-ph.CO]

arXiv 2026
[8]

Alam et al

S. Alam et al. (eBOSS), Phys. Rev. D103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO]

arXiv 2021

Show all 171 references

[9]

Zhao et al

C. Zhao et al. (eBOSS), Mon. Not. Roy. Astron. Soc. 511, 5492 (2022), arXiv:2110.03824 [astro-ph.CO]

2022 arXiv
[10]

Lodha et al

K. Lodha et al. (DESI), Phys. Rev. D112, 083511 (2025), arXiv:2503.14743 [astro-ph.CO]

2025 arXiv
[11]

Weinberg, Rev

S. Weinberg, Rev. Mod. Phys.61, 1 (1989)

1989
[12]

S. M. Carroll, Living Rev. Rel.4, 1 (2001), arXiv:astro- ph/0004075

2001
[13]

Bull et al., Phys

P. Bull et al., Phys. Dark Univ.12, 56 (2016), arXiv:1512.05356 [astro-ph.CO]

2016 arXiv
[14]

J. S. Bullock and M. Boylan-Kolchin, Ann. Rev. Astron. Astrophys.55, 343 (2017), arXiv:1707.04256 [astro- ph.CO]

2017 arXiv
[15]

Perivolaropoulos and F

L. Perivolaropoulos and F. Skara, New Astron. Rev.95, 101659 (2022), arXiv:2105.05208 [astro-ph.CO]

2022 arXiv
[16]

Abdalla et al., JHEAp34, 49 (2022), arXiv:2203.06142 [astro-ph.CO]

E. Abdalla et al., JHEAp34, 49 (2022), arXiv:2203.06142 [astro-ph.CO]

2022 arXiv
[17]

Calder´ on, A

R. Calder´ on, A. Shafieloo, D. K. Hazra, and W. Sohn, JCAP08, 059 (2023), arXiv:2302.14300 [astro-ph.CO]

2023 arXiv
[18]

Di Valentino et al

E. Di Valentino et al. (CosmoVerse Network), Phys. Dark Univ.49, 101965 (2025), arXiv:2504.01669 [astro- ph.CO]

2025 arXiv
[19]

A. G. Riess et al., Astrophys. J. Lett.934, L7 (2022), arXiv:2112.04510 [astro-ph.CO]

2022 arXiv
[20]

Di Valentino, O

E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav.38, 153001 (2021), arXiv:2103.01183 [astro-ph.CO]

2021 arXiv
[21]

Verde, T

L. Verde, T. Treu, and A. G. Riess, Nature Astron.3, 891 (2019), arXiv:1907.10625 [astro-ph.CO]

2019 arXiv
[22]

Di Valentino et al., Astropart

E. Di Valentino et al., Astropart. Phys.131, 102605 (2021), arXiv:2008.11284 [astro-ph.CO]

2021 arXiv
[23]

Sch¨ oneberg, G

N. Sch¨ oneberg, G. Franco Abell´ an, A. P´ erez S´ anchez, S. J. Witte, V. Poulin, and J. Lesgourgues, Phys. Rept. 984, 1 (2022), arXiv:2107.10291 [astro-ph.CO]

2022 arXiv
[24]

P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys. Rev.29, 9 (2021), arXiv:2109.01161 [astro-ph.CO]

2021 arXiv
[25]

Di Valentino, Universe8, 399 (2022)

E. Di Valentino, Universe8, 399 (2022)

2022
[26]

Kamionkowski and A

M. Kamionkowski and A. G. Riess, Ann. Rev. Nucl. Part. Sci.73, 153 (2023), arXiv:2211.04492 [astro- ph.CO]

2023 arXiv
[27]

Giar` e, (2023), arXiv:2305.16919 [astro-ph.CO]

W. Giar` e, (2023), arXiv:2305.16919 [astro-ph.CO]

2023 arXiv
[28]

Hu and F.-Y

J.-P. Hu and F.-Y. Wang, Universe9, 94 (2023), arXiv:2302.05709 [astro-ph.CO]

2023 arXiv
[29]

Verde, N

L. Verde, N. Sch¨ oneberg, and H. Gil-Mar ´ ın, Ann. Rev. Astron. Astrophys.62, 287 (2024), arXiv:2311.13305 [astro-ph.CO]

2024 arXiv
[30]

Di Valentino and D

E. Di Valentino and D. Brout, eds., The Hubble Constant Tension, Springer Series in Astrophysics and Cosmology (Springer, 2024)

2024
[31]

D. D. Y. Ong and W. Handley, (2025), arXiv:2511.04661 [astro-ph.CO]

2025 arXiv
[32]

Cai and S.-J

R.-G. Cai and S.-J. Wang, Res. Astron. Astrophys.26, 084011 (2026), arXiv:2606.20434 [astro-ph.CO]

2026 arXiv
[33]

Hikage et al

C. Hikage et al. (HSC), Publ. Astron. Soc. Jap.71, 43 (2019), arXiv:1809.09148 [astro-ph.CO]

2019 arXiv
[34]

Di Valentino et al., Astropart

E. Di Valentino et al., Astropart. Phys.131, 102604 (2021), arXiv:2008.11285 [astro-ph.CO]

2021 arXiv
[35]

Di Valentino and S

E. Di Valentino and S. Bridle, Symmetry10, 585 (2018)

2018
[36]

R. C. Nunes and S. Vagnozzi, Mon. Not. Roy. Astron. Soc.505, 5427 (2021), arXiv:2106.01208 [astro-ph.CO]

2021 arXiv
[37]

Amon et al

A. Amon et al. (DES), Phys. Rev. D105, 023514 (2022), arXiv:2105.13543 [astro-ph.CO]

2022
[38]

L. F. Secco et al. (DES), Phys. Rev. D105, 023515 (2022), arXiv:2105.13544 [astro-ph.CO]

2022 arXiv
[39]

Asgari et al

M. Asgari et al. (KiDS), Astron. Astrophys.645, A104 (2021), arXiv:2007.15633 [astro-ph.CO]

2021 arXiv
[40]

Asgari et al., Astron

M. Asgari et al., Astron. Astrophys.634, A127 (2020), arXiv:1910.05336 [astro-ph.CO]

2020 arXiv
[41]

Joudaki et al., Astron

S. Joudaki et al., Astron. Astrophys.638, L1 (2020), arXiv:1906.09262 [astro-ph.CO]

2020 arXiv
[42]

D’Amico, J

G. D’Amico, J. Gleyzes, N. Kokron, K. Markovic, L. Senatore, P. Zhang, F. Beutler, and H. Gil-Mar ´ ın, JCAP05, 005 (2020), arXiv:1909.05271 [astro-ph.CO]

2020 arXiv
[43]

T. M. C. Abbott et al. (Kilo-Degree Survey, DES), Open J. Astrophys.6, 2305.17173 (2023), arXiv:2305.17173 [astro-ph.CO]

2023 arXiv
[44]

Tr¨ osteret al., Astron

T. Tr¨ osteret al., Astron. Astrophys.633, L10 (2020), arXiv:1909.11006 [astro-ph.CO]

2020 arXiv
[45]

Heymans et al., Astron

C. Heymans et al., Astron. Astrophys.646, A140 (2021), arXiv:2007.15632 [astro-ph.CO]

2021 arXiv
[46]

Dalal et al., Phys

R. Dalal et al., Phys. Rev. D108, 123519 (2023), arXiv:2304.00701 [astro-ph.CO]

2023 arXiv
[47]

Chen et al., Phys

S. Chen et al., Phys. Rev. D110, 103518 (2024), arXiv:2407.04795 [astro-ph.CO]

2024 arXiv
[48]

J. Kim et al., JCAP12, 022 (2024), arXiv:2407.04606 17 1.0 1.1 1.2 α 0.04 0.05 0.06 0.07 τreio 0.80 0.81 0.82 0.83 0.84 S8 0.28 0.30 0.32 Ωm 66 68 70 72H0 0.117 0.118 0.119 0.120 0.121 Ωch2 0.0222 0.0224 0.0226 0.0228 Ωbh2 0.965 0.970 0.975 0.980 ns 3.04 3.06 3.08 log(1010As) ...

2024 arXiv
[49]

Faga et al

L. Faga et al. (DES), Mon. Not. Roy. Astron. Soc.536, 1586 (2024), arXiv:2406.12675 [astro-ph.CO]

2024 arXiv
[50]

Harnois-Deraps et al., Mon

J. Harnois-Deraps et al., Mon. Not. Roy. Astron. Soc. 534, 3305 (2024), arXiv:2405.10312 [astro-ph.CO]

2024 arXiv
[51]

Dvornik et al., Astron

A. Dvornik et al., Astron. Astrophys.675, A189 (2023), [Erratum: Astron.Astrophys. 688, C3 (2024)], arXiv:2210.03110 [astro-ph.CO]

2023 arXiv
[52]

T. M. C. Abbott et al. (DES), Phys. Rev. D105, 023520 (2022), arXiv:2105.13549 [astro-ph.CO]

2022 arXiv
[53]

A. H. Wright et al., Astron. Astrophys.703, A158 (2025), arXiv:2503.19441 [astro-ph.CO]

2025 arXiv
[54]

T. M. C. Abbott et al. (DES), (2026), arXiv:2601.14559 [astro-ph.CO]

2026
[55]

T. M. C. Abbott et al. (DES), (2026), arXiv:2602.10065 [astro-ph.CO]

2026
[56]

A. G. Adame et al. (DESI), JCAP02, 021 (2025), arXiv:2404.03002 [astro-ph.CO]. 18 1.0 1.1 1.2 α 0.04 0.05 0.06 0.07 τreio 0.80 0.81 0.82 0.83 0.84 S8 0.28 0.30 0.32 Ωm 66 68 70 72H0 0.117 0.118 0.119 0.120 0.121 Ωch2 0.0222 0.0224 0.0226 Ωbh2 0.965 0.970 0.975 0.980 ns 3.04 3....

2025 arXiv
[57]

Abdul Karim et al

M. Abdul Karim et al. (DESI), Phys. Rev. D112, 083515 (2025), arXiv:2503.14738 [astro-ph.CO]

2025 arXiv
[58]

Calderon et al

R. Calderon et al. (DESI), JCAP10, 048 (2024), arXiv:2405.04216 [astro-ph.CO]

2024 arXiv
[59]

Ishak et al., JCAP09, 053 (2025), arXiv:2411.12026 [astro-ph.CO]

M. Ishak et al., JCAP09, 053 (2025), arXiv:2411.12026 [astro-ph.CO]

2025
[60]

Gu et al

G. Gu et al. (DESI), Nature Astron.9, 1879 (2025), [Erratum: Nature Astron. 9, 1898–1898 (2025)], arXiv:2504.06118 [astro-ph.CO]

2025
[61]

Lodha et al

K. Lodha et al. (DESI), Phys. Rev. D111, 023532 (2025), arXiv:2405.13588 [astro-ph.CO]

2025 arXiv
[62]

Cortˆ es and A

M. Cortˆ es and A. R. Liddle, JCAP12, 007 (2024), arXiv:2404.08056 [astro-ph.CO]

2024 arXiv
[63]

Shlivko and P

D. Shlivko and P. J. Steinhardt, Phys. Lett. B855, 138826 (2024), arXiv:2405.03933 [astro-ph.CO]

2024 arXiv
[64]

Luongo and M

O. Luongo and M. Muccino, Astron. Astrophys.690, A40 (2024), arXiv:2404.07070 [astro-ph.CO]

2024 arXiv
[65]

I. D. Gialamas, G. H¨ utsi, K. Kannike, A. Racioppi, M. Raidal, M. Vasar, and H. Veerm¨ ae, Phys. Rev. D 111, 043540 (2025), arXiv:2406.07533 [astro-ph.CO]. 19 3.04 3.06 3.08 log(1010As) 0.04 0.05 0.06 0.07 τreio 0.80 0.81 0.82 0.83 0.84 S8 0.29 0.30 0.31 0.32 Ωm 66 67 68 69 7...

2025 arXiv
[66]

Wang and Y.-S

H. Wang and Y.-S. Piao, Phys. Lett. B873, 140180 (2026), arXiv:2404.18579 [astro-ph.CO]

2026 arXiv
[67]

G. Ye, M. Martinelli, B. Hu, and A. Silvestri, Phys. Rev. Lett.134, 181002 (2025), arXiv:2407.15832 [astro- ph.CO]

2025 arXiv
[68]

Tada and T

Y. Tada and T. Terada, Phys. Rev. D109, L121305 (2024), arXiv:2404.05722 [astro-ph.CO]

2024 arXiv
[69]

Carloni, O

Y. Carloni, O. Luongo, and M. Muccino, Phys. Rev. D 111, 023512 (2025), arXiv:2404.12068 [astro-ph.CO]

2025 arXiv
[70]

C.-G. Park, J. de Cruz P´ erez, and B. Ratra, Phys. Rev. D110, 123533 (2024), arXiv:2405.00502 [astro-ph.CO]

2024 arXiv
[71]

Bhattacharya, G

S. Bhattacharya, G. Borghetto, A. Malhotra, S. Parameswaran, G. Tasinato, and I. Zavala, JCAP09, 073 (2024), arXiv:2405.17396 [astro-ph.CO]

2024 arXiv
[72]

Rebou¸ cas, D

J. Rebou¸ cas, D. H. F. de Souza, K. Zhong, V. Mi- randa, and R. Rosenfeld, JCAP02, 024 (2025), arXiv:2408.14628 [astro-ph.CO]

2025 arXiv
[73]

Najafi, S

M. Najafi, S. Pan, E. Di Valentino, and J. T. Firouzjaee, 20 Phys. Dark Univ.45, 101539 (2024), arXiv:2407.14939 [astro-ph.CO]

2024 arXiv
[74]

Giar` e, M

W. Giar` e, M. Najafi, S. Pan, E. Di Valentino, and J. T. Firouzjaee, JCAP10, 035 (2024), arXiv:2407.16689 [astro-ph.CO]

2024 arXiv
[75]

Giar` e, Phys

W. Giar` e, Phys. Rev. D112, 023508 (2025), arXiv:2409.17074 [astro-ph.CO]

2025 arXiv
[76]

Jiang, D

J.-Q. Jiang, D. Pedrotti, S. S. da Costa, and S. Vagnozzi, Phys. Rev. D110, 123519 (2024), arXiv:2408.02365 [astro-ph.CO]

2024 arXiv
[77]

Roy Choudhury and T

S. Roy Choudhury and T. Okumura, Astrophys. J. Lett. 976, L11 (2024), arXiv:2409.13022 [astro-ph.CO]

2024 arXiv
[78]

Giar` e, T

W. Giar` e, T. Mahassen, E. Di Valentino, and S. Pan, Phys. Dark Univ.48, 101906 (2025), arXiv:2502.10264 [astro-ph.CO]

2025 arXiv
[79]

D. A. Kessler, L. A. Escamilla, S. Pan, and E. Di Valentino, (2025), arXiv:2504.00776 [astro- ph.CO]

2025 arXiv
[80]

Roy Choudhury, Astrophys

S. Roy Choudhury, Astrophys. J. Lett.986, L31 (2025), [Erratum: Astrophys.J.Lett. 1001, L25 (2026), Erra- tum: Astrophys.J. 1001, L25 (2026)], arXiv:2504.15340 [astro-ph.CO]

2025
[81]

Scherer, M

M. Scherer, M. A. Sabogal, R. C. Nunes, and A. De Fe- lice, Phys. Rev. D112, 043513 (2025), arXiv:2504.20664 [astro-ph.CO]

2025 arXiv
[82]

W. J. Wolf, C. Garc ´ ıa-Garc ´ ıa, and P. G. Ferreira, JCAP 05, 034 (2025), arXiv:2502.04929 [astro-ph.CO]

2025 arXiv
[83]

F. B. M. d. Santos, J. Morais, S. Pan, W. Yang, and E. Di Valentino, JCAP07, 022 (2026), arXiv:2504.04646 [astro-ph.CO]

2026 arXiv
[84]

Specogna, S

E. Specogna, S. A. Adil, E. Ozulker, E. Di Valentino, R. C. Nunes, O. Akarsu, and A. A. Sen, Phys. Rev. D 113, 103549 (2026), arXiv:2504.17859 [gr-qc]

2026 arXiv
[85]

Cheng, E

H. Cheng, E. Di Valentino, L. A. Escamilla, A. A. Sen, and L. Visinelli, JCAP09, 031 (2025), arXiv:2505.02932 [astro-ph.CO]

2025
[86]

Cheng, E

H. Cheng, E. Di Valentino, and L. Visinelli, JHEAp 53, 100610 (2026), arXiv:2505.22066 [astro-ph.CO]

2026 arXiv
[87]

¨Oz¨ ulker, E

E. ¨Oz¨ ulker, E. Di Valentino, and W. Giar` e, (2025), arXiv:2506.19053 [astro-ph.CO]

2025 arXiv
[88]

Li, G.-H

T.-N. Li, G.-H. Du, S.-H. Zhou, Y.-H. Li, J.-F. Zhang, and X. Zhang, Phys. Dark Univ.52, 102254 (2026), arXiv:2511.22512 [astro-ph.CO]

2026
[89]

D. H. Lee, W. Yang, E. Di Valentino, S. Pan, and C. van de Bruck, Phys. Rev. D113, 063554 (2026), arXiv:2507.11432 [astro-ph.CO]

2026
[90]

Fazzari, W

E. Fazzari, W. Giar` e, and E. Di Valentino, Astro- phys. J. Lett.996, L5 (2026), arXiv:2509.16196 [astro- ph.CO]

2026
[91]

Smith, E

A. Smith, E. ¨Oz¨ ulker, E. Di Valentino, and C. van de Bruck, (2025), arXiv:2510.21931 [astro-ph.CO]

2025
[92]

Herold and T

L. Herold and T. Karwal, Phys. Rev. D113, 123551 (2026), arXiv:2506.12004 [astro-ph.CO]

2026 arXiv
[93]

Cheng, S

H. Cheng, S. Pan, and E. Di Valentino, Astrophys. J. 999, 190 (2026), arXiv:2512.09866 [astro-ph.CO]

2026
[94]

G¨ ok¸ cen,¨O

M. G¨ ok¸ cen,¨O. Akarsu, and E. Di Valentino, Phys. Dark Univ.52, 102273 (2026), arXiv:2602.21169 [astro- ph.CO]

2026
[95]

Ishak and L

M. Ishak and L. Medina-Varela, (2025), arXiv:2507.22856 [astro-ph.CO]

2025
[96]

Najafi, M

M. Najafi, M. Habibollahi, M. Reyhani, E. Di Valentino, S. Pan, J. T. Firouzjaee, and W. Yang, (2026), arXiv:2603.13137 [astro-ph.CO]

2026
[97]

W. Yang, E. Di Valentino, E. V. Linder, S. Zhang, and S. Pan, (2026), arXiv:2603.15422 [astro-ph.CO]

2026
[98]

D. A. Kessler, E. Di Valentino, L. A. Escamilla, and D. Huterer, (2026), arXiv:2606.05853 [astro-ph.CO]

2026 arXiv
[99]

D. H. Lee, C. van de Bruck, E. Di Valentino, L. Van Waerbeke, and A. Zhitnitsky, (2026), arXiv:2606.20036 [astro-ph.CO]

2026 arXiv
[100]

Li, G.-H

T.-N. Li, G.-H. Du, H. Wang, Y.-H. Li, J.-F. Zhang, and X. Zhang, Res. Astron. Astrophys.26, 084002 (2026), arXiv:2606.21826 [astro-ph.CO]

2026 arXiv
[101]

Giar` e, D

W. Giar` e, D. H. Lee, and E. Di Valentino, (2026), arXiv:2607.01226 [astro-ph.CO]

2026 arXiv
[102]

Gupta Choudhury, P

S. Gupta Choudhury, P. Mukherjee, E. Di Valentino, and A. A. Sen, (2026), arXiv:2607.13009 [astro-ph.CO]

2026 arXiv
[103]

Elbers, C

W. Elbers, C. S. Frenk, A. Jenkins, B. Li, and S. Pas- coli, Phys. Rev. D111, 063534 (2025), arXiv:2407.10965 [astro-ph.CO]

2025 arXiv
[104]

Elbers et al., Phys

W. Elbers et al., Phys. Rev. D112, 083513 (2025), arXiv:2503.14744 [astro-ph.CO]

2025 arXiv
[105]

Craig, D

N. Craig, D. Green, J. Meyers, and S. Rajendran, JHEP 09, 097 (2024), arXiv:2405.00836 [astro-ph.CO]

2024 arXiv
[106]

Green and J

D. Green and J. Meyers, Phys. Rev. D111, 083507 (2025), arXiv:2407.07878 [astro-ph.CO]

2025 arXiv
[107]

P. W. Graham, D. Green, and J. Meyers, Phys. Rev. D113, 043514 (2026), arXiv:2508.20999 [astro-ph.CO]

2026
[108]

Pulido-Hern´ andez and J

H. Pulido-Hern´ andez and J. L. Cervantes-Cota, (2026), arXiv:2603.13208 [astro-ph.CO]

2026
[109]

Loverde and Z

M. Loverde and Z. J. Weiner, JCAP12, 048 (2024), arXiv:2410.00090 [astro-ph.CO]

2024 arXiv
[110]

Kıbrıs, W

C. Kıbrıs, W. Elbers, ¨O. Akarsu, and E. Di Valentino, (2026), arXiv:2605.21456 [astro-ph.CO]

2026 arXiv
[111]

Chen and A

X. Chen and A. Loeb, JCAP07, 059 (2025), arXiv:2505.02645 [astro-ph.CO]

2025 arXiv
[112]

S. P. Ahlen et al. (DESI), Phys. Rev. Lett.135, 081003 (2025), arXiv:2504.20338 [astro-ph.CO]

2025 arXiv
[113]

Chaussidon et al., Phys

E. Chaussidon et al., Phys. Rev. D112, 063548 (2025), arXiv:2503.24343 [astro-ph.CO]

2025 arXiv
[114]

Chen and M

S.-F. Chen and M. Zaldarriaga, JCAP08, 014 (2025), arXiv:2505.00659 [astro-ph.CO]

2025 arXiv
[115]

Chevallier and D

M. Chevallier and D. Polarski, Int. J. Mod. Phys. D10, 213 (2001), arXiv:gr-qc/0009008

2001 arXiv
[116]

E. V. Linder, Phys. Rev. Lett.90, 091301 (2003), arXiv:astro-ph/0208512

2003 arXiv
[117]

R. R. Caldwell and E. V. Linder, Phys. Rev. Lett.95, 141301 (2005), arXiv:astro-ph/0505494

2005 arXiv
[118]

de Putter and E

R. de Putter and E. V. Linder, JCAP10, 042 (2008), arXiv:0808.0189 [astro-ph]

2008 arXiv
[119]

Shafieloo, U

A. Shafieloo, U. Alam, V. Sahni, and A. A. Starobin- sky, Mon. Not. Roy. Astron. Soc.366, 1081 (2006), arXiv:astro-ph/0505329

2006 arXiv
[120]

Holsclaw, U

T. Holsclaw, U. Alam, B. Sanso, H. Lee, K. Heitmann, S. Habib, and D. Higdon, Phys. Rev. D82, 103502 (2010), arXiv:1009.5443 [astro-ph.CO]

2010 arXiv
[121]

Shafieloo, A

A. Shafieloo, A. G. Kim, and E. V. Linder, Phys. Rev. D85, 123530 (2012), arXiv:1204.2272 [astro-ph.CO]

2012 arXiv
[122]

Nesseris and J

S. Nesseris and J. Garcia-Bellido, JCAP11, 033 (2012), arXiv:1205.0364 [astro-ph.CO]

2012 arXiv
[123]

Calder´ on, B

R. Calder´ on, B. L’Huillier, D. Polarski, A. Shafieloo, and A. A. Starobinsky, Phys. Rev. D106, 083513 (2022), arXiv:2206.13820 [astro-ph.CO]

2022 arXiv
[124]

Pogosian, M

L. Pogosian, M. Raveri, K. Koyama, M. Martinelli, A. Silvestri, G.-B. Zhao, J. Li, S. Peirone, and A. Zucca, Nature Astron.6, 1484 (2022), arXiv:2107.12992 [astro- 21 ph.CO]

2022 arXiv
[125]

Garcia-Bellido and L

J. Garcia-Bellido and L. Espinosa-Portales, Phys. Dark Univ.34, 100892 (2021), arXiv:2106.16014 [gr-qc]

2021 arXiv
[126]

Espinosa-Portales and J

L. Espinosa-Portales and J. Garcia-Bellido, Phys. Dark Univ.34, 100893 (2021), arXiv:2106.16012 [gr-qc]

2021 arXiv
[127]

G. W. Gibbons and S. W. Hawking, Phys. Rev. D15, 2752 (1977)

1977
[128]

J. W. York, Jr., Phys. Rev. Lett.28, 1082 (1972)

1972
[129]

Garc ´ ıa-Bellido, Phys

J. Garc ´ ıa-Bellido, Phys. Dark Univ.44, 101491 (2024)

2024
[130]

Garc ´ ıa-Bellido, Phys

J. Garc ´ ıa-Bellido, Phys. Dark Univ.52, 102319 (2026), arXiv:2511.19546 [gr-qc]

2026 arXiv
[131]

Garcia-Bellido, Phys

J. Garcia-Bellido, Phys. Dark Univ.45, 101533 (2024), arXiv:2405.02895 [astro-ph.CO]

2024 arXiv
[132]

T. M. C. Abbott et al. (DES), Astrophys. J. Lett.973, L14 (2024), arXiv:2401.02929 [astro-ph.CO]

2024 arXiv
[133]

A. G. Adame et al. (DESI), JCAP09, 008 (2025), [Er- ratum: JCAP 02, E02 (2026)], arXiv:2411.12021 [astro- ph.CO]

2025 arXiv
[134]

Arjona, L

R. Arjona, L. Espinosa-Portales, J. Garc ´ ıa-Bellido, and S. Nesseris, Phys. Dark Univ.36, 101029 (2022), arXiv:2111.13083 [astro-ph.CO]

2022 arXiv
[135]

Calderon et al., (2025), arXiv:2509.21491 [astro- ph.CO]

R. Calderon et al., (2025), arXiv:2509.21491 [astro- ph.CO]

2025
[136]

Graziotti, C

I. Graziotti, C. De Leo, and M. Martinelli, (2026), arXiv:2603.01934 [astro-ph.CO]

2026 arXiv
[137]

Jacobson, Phys

T. Jacobson, Phys. Rev. Lett.75, 1260 (1995), arXiv:gr- qc/9504004

1995
[138]

Brout et al., Astrophys

D. Brout et al., Astrophys. J.938, 110 (2022), arXiv:2202.04077 [astro-ph.CO]

2022 arXiv
[139]

Rubin et al., Astrophys

D. Rubin et al., Astrophys. J.986, 231 (2025), arXiv:2311.12098 [astro-ph.CO]

2025 arXiv
[140]

Gagnon and J

J.-S. Gagnon and J. Lesgourgues, JCAP09, 026 (2011), arXiv:1107.1503 [astro-ph.CO]

2011 arXiv
[141]

Arjona, W

R. Arjona, W. Cardona, and S. Nesseris, Phys. Rev. D 99, 043516 (2019), arXiv:1811.02469 [astro-ph.CO]

2019 arXiv
[142]

Cardona, R

W. Cardona, R. Arjona, A. Estrada, and S. Nesseris, JCAP05, 064 (2021), arXiv:2012.05282 [astro-ph.CO]

2021 arXiv
[143]

W. Fang, W. Hu, and A. Lewis, Phys. Rev. D78, 087303 (2008), arXiv:0808.3125 [astro-ph]

2008 arXiv
[144]

D. Blas, J. Lesgourgues, and T. Tram, JCAP07, 034 (2011), arXiv:1104.2933 [astro-ph.CO]

2011 arXiv
[145]

Aghanim et al

N. Aghanim et al. (Planck), Astron. Astrophys.641, A5 (2020), arXiv:1907.12875 [astro-ph.CO]

2020 arXiv
[146]

Carron, M

J. Carron, M. Mirmelstein, and A. Lewis, JCAP09, 039 (2022), arXiv:2206.07773 [astro-ph.CO]

2022 arXiv
[147]

F. J. Qu et al. (ACT), Astrophys. J.962, 112 (2024), arXiv:2304.05202 [astro-ph.CO]

2024 arXiv
[148]

M. S. Madhavacheril et al. (ACT), Astrophys. J.962, 113 (2024), arXiv:2304.05203 [astro-ph.CO]

2024 arXiv
[149]

Ge et al

F. Ge et al. (SPT-3G), Phys. Rev. D111, 083534 (2025), arXiv:2411.06000 [astro-ph.CO]

2025 arXiv
[150]

Popovic et al., (2025), arXiv:2506.05471 [astro- ph.CO]

B. Popovic et al., (2025), arXiv:2506.05471 [astro- ph.CO]

2025 arXiv
[151]

Popovic et al

B. Popovic et al. (DES), Mon. Not. Roy. Astron. Soc. 548, stag632 (2026), arXiv:2511.07517 [astro-ph.CO]

2026 arXiv
[152]

Torrado and A

J. Torrado and A. Lewis, JCAP05, 057 (2021), arXiv:2005.05290 [astro-ph.IM]

2021 arXiv
[153]

Lewis, JCAP08, 025 (2025), arXiv:1910.13970 [astro-ph.IM]

A. Lewis, JCAP08, 025 (2025), arXiv:1910.13970 [astro-ph.IM]

2025 arXiv
[154]

Machine learning assisted Bayesian model comparison: Learnt harmonic mean estimator,

J. D. McEwen, C. G. R. Wallis, M. A. Price, and A. S. Mancini, “Machine learning assisted Bayesian model comparison: Learnt harmonic mean estimator,” (2023), arXiv:2111.12720 [stat]

2023 arXiv
[155]

Polanska, M

A. Polanska, M. A. Price, D. Piras, A. Spurio Mancini, and J. D. McEwen, (2024), 10.33232/001c.146026, arXiv:2405.05969 [astro-ph.IM]

2024
[156]

Nesseris and J

S. Nesseris and J. Garcia-Bellido, JCAP08, 036 (2013), arXiv:1210.7652 [astro-ph.CO]

2013 arXiv
[157]

R. E. Keeley and A. Shafieloo, Mon. Not. Roy. Astron. Soc.515, 293 (2022), arXiv:2111.04231 [astro-ph.CO]

2022 arXiv
[158]

H. Koo, R. E. Keeley, A. Shafieloo, and B. L’Huillier, JCAP03, 047 (2022), arXiv:2110.10977 [astro-ph.CO]

2022 arXiv
[159]

Padmanabhan, Rept

T. Padmanabhan, Rept. Prog. Phys.73, 046901 (2010), arXiv:0911.5004 [gr-qc]

2010 arXiv
[160]

Cai and S

R.-G. Cai and S. P. Kim, JHEP02, 050 (2005), arXiv:hep-th/0501055

2005 arXiv
[161]

Tsallis and L

C. Tsallis and L. J. L. Cirto, Eur. Phys. J. C73, 2487 (2013), arXiv:1202.2154 [cond-mat.stat-mech]

2013 arXiv
[162]

Lymperis and E

A. Lymperis and E. N. Saridakis, Eur. Phys. J. C78, 993 (2018), arXiv:1806.04614 [gr-qc]

2018 arXiv
[163]

Sheykhi, Phys

A. Sheykhi, Phys. Lett. B785, 118 (2018), arXiv:1806.03996 [gr-qc]

2018 arXiv
[164]

J. D. Barrow, Phys. Lett. B808, 135643 (2020), arXiv:2004.09444 [gr-qc]

2020 arXiv
[165]

E. N. Saridakis, Phys. Rev. D102, 123525 (2020), arXiv:2005.04115 [gr-qc]

2020 arXiv
[166]

Sheykhi, Phys

A. Sheykhi, Phys. Rev. D103, 123503 (2021), arXiv:2102.06550 [gr-qc]

2021 arXiv
[167]

Basilakos, A

S. Basilakos, A. Lymperis, M. Petronikolou, and E. N. Saridakis, Eur. Phys. J. C84, 297 (2024), arXiv:2308.01200 [gr-qc]

2024 arXiv
[168]

Polarski, A

D. Polarski, A. A. Starobinsky, and H. Giacomini, JCAP12, 037 (2016), arXiv:1610.00363 [astro-ph.CO]

2016 arXiv
[169]

Calderon, D

R. Calderon, D. Felbacq, R. Gannouji, D. Polarski, and A. A. Starobinsky, Phys. Rev. D100, 083503 (2019), arXiv:1908.00117 [astro-ph.CO]

2019 arXiv
[170]

Calderon, D

R. Calderon, D. Felbacq, R. Gannouji, D. Polarski, and A. A. Starobinsky, Phys. Rev. D101, 103501 (2020), arXiv:1912.06958 [astro-ph.CO]

2020 arXiv
[171]

Amendola et al., Living Rev

L. Amendola et al., Living Rev. Rel.21, 2 (2018), arXiv:1606.00180 [astro-ph.CO]

2018 arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

A. G. Riess et al. (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201

arXiv 1998

[2] [2]

Perlmutter et al

S. Perlmutter et al. (Supernova Cosmology Project), As- trophys. J.517, 565 (1999), arXiv:astro-ph/9812133

arXiv 1999

[3] [3]

Sahni and A

V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), arXiv:astro-ph/9904398

arXiv 2000

[4] [4]

Aghanim et al

N. Aghanim et al. (Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

arXiv 2020

[5] [5]

Aghanim et al

N. Aghanim et al. (Planck), Astron. Astrophys.641, A1 (2020), arXiv:1807.06205 [astro-ph.CO]

arXiv 2020

[6] [6]

Louis et al

T. Louis et al. (Atacama Cosmology Telescope), JCAP 11, 062 (2025), arXiv:2503.14452 [astro-ph.CO]

arXiv 2025

[7] [7]

Camphuis et al

E. Camphuis et al. (SPT-3G), Phys. Rev. D113, 083504 (2026), arXiv:2506.20707 [astro-ph.CO]

arXiv 2026

[8] [8]

Alam et al

S. Alam et al. (eBOSS), Phys. Rev. D103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO]

arXiv 2021