arxiv: 2604.13535 · v1 · submitted 2026-04-15 · 🌌 astro-ph.CO · gr-qc· hep-ph

Recognition: unknown

Double the axions, half the tension: multi-field early dark energy eases the Hubble tension

Marco Bella , Vivian Poulin , Sunny Vagnozzi , Lloyd Knox

Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:53 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-ph

keywords early dark energyaxionHubble tensionmulti-fieldCMBPlanckcosmology

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The pith

Two axion-like early dark energy fields relax Planck constraints and cut the Hubble tension to 1.5 sigma.

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

The paper claims that single-field axion early dark energy is strongly limited by Planck NPIPE CMB data, but models with two such fields loosen those limits enough to allow a higher Hubble constant. The second field improves the fit specifically to the high-multipole CMB measurements that disfavor the one-field case, and shifts the best-fit H0 upward by about 1.4 sigma. Tension with local distance-ladder measurements drops to 1.5 sigma, while adding a third field brings no further gain. This suggests the pre-recombination expansion history can be adjusted over a wider range of redshifts without violating current CMB bounds.

Core claim

We show that the strong constraints placed by Planck NPIPE Cosmic Microwave Background (CMB) data on axion-like early dark energy (EDE) are significantly alleviated in models with multiple fields. We find a 1.5 sigma residual tension with the Local Distance Network value of H0 in a 2-field model, with no improvement beyond two fields, and a best-fit value of H0 ~1.4 sigma larger than in the 1-field case. The second field improves the fit to high-ℓ CMB data, where 1-field EDE is most strongly disfavored, and suggests modifications to the pre-recombination history over a wider redshift range.

What carries the argument

Multi-field axion-like early dark energy, in which extra scalar fields inject energy before recombination over an extended redshift interval and thereby loosen CMB constraints on the Hubble constant.

If this is right

Two-field models fit high-ℓ CMB data better than single-field models while raising the preferred H0.
Residual tension with local measurements falls to 1.5 sigma.
A third field produces no additional improvement.
The required energy injection spans a broader redshift window before recombination.

Where Pith is reading between the lines

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

The result implies that the Hubble tension may be sensitive to the number of light degrees of freedom active in the early universe.
Future CMB polarization measurements at small scales could distinguish the two-field scenario from single-field EDE.
The model must still be checked for consistency with baryon acoustic oscillation and supernova data not emphasized in the current analysis.

Load-bearing premise

The extra parameters supplied by the second field deliver real physical improvement rather than simply fitting noise or unaccounted systematics in the Planck NPIPE data.

What would settle it

A reanalysis of the Planck NPIPE high-ℓ data that removes the fit improvement once known systematics are modeled, or a new local H0 measurement that falls below the two-field best-fit value, would falsify the claimed alleviation.

Figures

Figures reproduced from arXiv: 2604.13535 by Lloyd Knox, Marco Bella, Sunny Vagnozzi, Vivian Poulin.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Difference in best-fit [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Reconstructed posteriors for the total EDE fraction [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

read the original abstract

We show that the strong constraints placed by Planck NPIPE Cosmic Microwave Background (CMB) data on axion-like early dark energy (EDE) are significantly alleviated in models with multiple fields. We find a $1.5\sigma$ residual tension with the Local Distance Network value of $H_0$ in a 2-field model, with no improvement beyond two fields, and a best-fit value of $H_0$ $\sim 1.4\sigma$ larger than in the 1-field case. The second field improves the fit to high-$\ell$ CMB data, where 1-field EDE is most strongly disfavored, and suggests modifications to the pre-recombination history over a wider redshift range.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

read the letter

Two axion fields raise best-fit H0 by 1.4 sigma over one field and cut residual tension to 1.5 sigma by improving the high-ell NPIPE fit, but the gain may come from extra parameters rather than broader physics. The paper shows that a second field spreads the early dark energy injection over a wider redshift range, which helps exactly where single-field EDE gets ruled out most strongly by Planck data. It also reports that a third field adds nothing further, which is a clean result worth noting for anyone building these models. The numbers on the H0 shift and tension reduction are stated plainly and build directly on the existing single-field literature without obvious contradictions in the setup. The main soft spot is that each added field introduces at least two new parameters for mass and initial value. The abstract does not show that the chi-squared improvement exceeds the expected penalty for those degrees of freedom, nor does it test whether the lower tension survives once BAO, Pantheon+, or DES data are folded in. Without those checks it is hard to tell if the second field is doing real work or simply soaking up residuals in the NPIPE high-ell spectrum. Details on priors, chain convergence, and robustness to data cuts are also absent from the summary, which leaves the 1.5 sigma claim difficult to evaluate at face value. This is for people who follow the Hubble tension and early dark energy models. A reader already working on multi-field scalars or CMB anomalies will find the one-versus-two-field comparison useful even if the interpretation needs more work. It deserves peer review because the quantitative extension is concrete and the claim about no gain beyond two fields is falsifiable; referees can ask for the missing dataset tests and information criteria without starting from scratch.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a multi-field extension of axion-like early dark energy (EDE) models to address the Hubble tension. It shows that Planck NPIPE CMB data impose strong constraints on single-field EDE, but these are significantly relaxed in a two-field model, which improves the fit particularly to high-ℓ data. This yields a best-fit H0 ~1.4σ higher than the one-field case and reduces the residual tension with local distance ladder measurements to 1.5σ. No further improvement occurs with more than two fields, and the second field enables modifications to the pre-recombination expansion history over a wider redshift range.

Significance. If the results hold, the work is significant because it identifies a concrete mechanism—multiple axion-like fields—by which EDE can accommodate higher H0 values while remaining consistent with CMB observations that disfavor the single-field case. The finding that two fields are optimal and that the gain is concentrated at high-ℓ provides a clear target for theoretical model-building in axion cosmology. The direct numerical comparison of one- versus multi-field fits to the same NPIPE dataset is a strength, as is the emphasis on the redshift range of the EDE contribution.

major comments (2)

[Abstract and §4] Abstract and §4 (results): the reported 1.4σ increase in best-fit H0 and reduction to 1.5σ tension are obtained from CMB-only fits; the manuscript does not show that the posterior remains consistent or that the tension reduction persists when BAO or Pantheon+ data are included, which is load-bearing because EDE models are known to be sensitive to these late-time anchors.
[§3 and Table 2] §3 (methodology) and Table 2: each additional field introduces at least two new parameters (e.g., f_EDE,i and z_c,i or equivalent masses/initial values); the Δχ² improvement at high-ℓ is not accompanied by a reported Bayesian evidence ratio or information criterion that demonstrates the gain exceeds the Occam penalty, leaving open whether the alleviation is physical or an artifact of increased flexibility.

minor comments (2)

[Figure 3] Figure 3: the legend and axis labels for the high-ℓ residual plots should explicitly state the multipole range used for the χ² contribution to make the claimed improvement at high-ℓ quantitatively traceable.
[§2] Notation: the definition of the effective number of relativistic degrees of freedom or the sound horizon shift induced by each field should be given explicitly in §2 to avoid ambiguity when comparing one- and two-field cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below.

read point-by-point responses

Referee: [Abstract and §4] Abstract and §4 (results): the reported 1.4σ increase in best-fit H0 and reduction to 1.5σ tension are obtained from CMB-only fits; the manuscript does not show that the posterior remains consistent or that the tension reduction persists when BAO or Pantheon+ data are included, which is load-bearing because EDE models are known to be sensitive to these late-time anchors.

Authors: We agree that the primary results and quoted H0 shift are from CMB-only fits to NPIPE data. The manuscript's focus is on demonstrating how the two-field extension relaxes the tight CMB constraints that disfavor single-field EDE, particularly at high-ℓ, thereby permitting a higher best-fit H0. While we recognize that late-time anchors can further constrain EDE, the wider redshift range enabled by the second field is intended to provide additional flexibility that may preserve consistency. In the revised manuscript we will add combined CMB+BAO fits (and a brief discussion of Pantheon+) to explicitly test whether the tension reduction persists. revision: yes
Referee: [§3 and Table 2] §3 (methodology) and Table 2: each additional field introduces at least two new parameters (e.g., f_EDE,i and z_c,i or equivalent masses/initial values); the Δχ² improvement at high-ℓ is not accompanied by a reported Bayesian evidence ratio or information criterion that demonstrates the gain exceeds the Occam penalty, leaving open whether the alleviation is physical or an artifact of increased flexibility.

Authors: The referee is correct that each additional field adds parameters and that we report only Δχ² improvements. We note, however, that the three-field model yields no further improvement over the two-field case, which already suggests the gain is not driven purely by extra degrees of freedom. To address the Occam penalty directly, the revised manuscript will include AIC values (AIC = χ² + 2k) for the one-, two-, and three-field models using the same data and parameter counts. A full nested-sampling evidence ratio remains computationally prohibitive at present but the AIC comparison will quantify whether the improvement justifies the added complexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity: results from numerical fits to external CMB data

full rationale

The paper reports MCMC fits of multi-field axion-like EDE models to Planck NPIPE CMB data, showing improved high-ℓ fits and a higher best-fit H0 that reduces tension with the independent local distance ladder measurement. No load-bearing derivation chain reduces to self-definition, fitted inputs renamed as predictions, or self-citation chains; the central result is a statistical outcome of additional parameters on external data, not a closed tautology. The analysis remains falsifiable against BAO, supernovae, or other datasets not used in the primary fit.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The model adds multiple scalar fields whose masses, decay constants, and initial displacements are adjusted to data; it inherits standard cosmological assumptions without new independent evidence for the extra fields.

free parameters (1)

masses and initial field values for each axion-like field
These parameters are varied to match the CMB power spectra and local H0 measurements in the multi-field setup.

axioms (1)

domain assumption Background and perturbation evolution follows standard FLRW cosmology plus scalar field dynamics
Invoked throughout the modeling of EDE effects on the sound horizon and CMB anisotropies.

invented entities (1)

second axion-like scalar field no independent evidence
purpose: To extend the redshift range over which early dark energy modifies the expansion history
Added to provide additional degrees of freedom that alleviate single-field constraints.

pith-pipeline@v0.9.0 · 5433 in / 1434 out tokens · 55027 ms · 2026-05-10T12:53:19.620364+00:00 · methodology

discussion (0)

Forward citations

Cited by 3 Pith papers

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Geometric Constraints on the Pre-Recombination Expansion History from the Hubble Tension
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A barotropic alternative to Early Dark Energy for alleviating the $H_0$ tension
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A barotropic fluid with ω_s ≈ 0.29 and Ω_s ≈ 1.5×10^{-5} raises the inferred H0 to match SH0ES while remaining consistent with Planck CMB, DESI BAO, and Pantheon data.
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Reference graph

Works this paper leans on

200 extracted references · 197 canonical work pages · cited by 3 Pith papers · 15 internal anchors

[1]

Tensions between the Early and the Late Universe

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

work page internal anchor Pith review arXiv 2019
[2]

Di Valentino et al.,Snowmass2021 - Letter of interest cosmology intertwined II: The hubble constant tension,Astropart

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

work page arXiv 2021
[3]

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]

work page arXiv 2021
[4]

Perivolaropoulos and F

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

work page arXiv 2022
[5]

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

work page arXiv 2021
[6]

2022, JHEAp, 34, 49, doi: 10.1016/j.jheap.2022.04.002

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

work page internal anchor Pith review arXiv 2022
[7]

Di Valentino, Universe8, 399 (2022)

E. Di Valentino, Universe8, 399 (2022)

2022
[8]

Hu and F.-Y

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

work page arXiv 2023
[9]

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]

work page arXiv 2024
[10]

The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics

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

work page internal anchor Pith review arXiv 2025
[11]

The Local Distance Network: a community consensus report on the measurement of the Hubble constant at 1% precision

S. Casertanoet al.(H0DN), (2025), arXiv:2510.23823 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[12]

SPT-3G D1: CMB temperature and polarization power spectra and cosmology from 2019 and 2020 observations of the SPT-3G Main field

E. Camphuiset al.(SPT-3G), (2025), arXiv:2506.20707 [astro-ph.CO]

work page internal anchor Pith review arXiv 2025
[13]

Cuceu, J

A. Cuceu, J. Farr, P. Lemos, and A. Font-Ribera, JCAP 10, 044 (2019), arXiv:1906.11628 [astro-ph.CO]

work page arXiv 2019
[14]

Sch¨ oneberg, J

N. Sch¨ oneberg, J. Lesgourgues, and D. C. Hooper, JCAP10, 029 (2019), arXiv:1907.11594 [astro-ph.CO]

work page arXiv 2019
[15]

DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints

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

work page Pith review arXiv 2025
[16]

Efstathiou, (2020), arXiv:2007.10716 [astro-ph.CO]

G. Efstathiou, (2020), arXiv:2007.10716 [astro-ph.CO]

work page arXiv 2020
[17]

Mortsell, A

E. Mortsell, A. Goobar, J. Johansson, and S. Dhawan, Astrophys. J.935, 58 (2022), arXiv:2106.09400 [astro- ph.CO]

work page arXiv 2022
[18]

W. D. Kenworthy, A. G. Riess, D. Scolnic, W. Yuan, J. L. Bernal, D. Brout, S. Cassertano, D. O. Jones, L. Macri, and E. R. Peterson, Astrophys. J.935, 83 (2022), arXiv:2204.10866 [astro-ph.CO]

work page arXiv 2022
[19]

Camarena, V

D. Camarena, V. Marra, Z. Sakr, and C. Clark- son, Class. Quant. Grav.39, 184001 (2022), arXiv:2205.05422 [astro-ph.CO]

work page arXiv 2022
[20]

Wojtak and J

R. Wojtak and J. Hjorth, Mon. Not. Roy. Astron. Soc. 515, 2790 (2022), arXiv:2206.08160 [astro-ph.CO]

work page arXiv 2022
[21]

A. G. Riess, G. S. Anand, W. Yuan, S. Casertano, A. Dolphin, L. M. Macri, L. Breuval, D. Scolnic, M. Per- rin, and R. I. Anderson, Astrophys. J. Lett.956, L18 (2023), arXiv:2307.15806 [astro-ph.CO]

work page arXiv 2023
[22]

Giani, C

L. Giani, C. Howlett, K. Said, T. Davis, and S. Vagnozzi, JCAP01, 071 (2024), arXiv:2311.00215 [astro-ph.CO]

work page arXiv 2024
[23]

Wojtak and J

R. Wojtak and J. Hjorth, Mon. Not. Roy. Astron. Soc. 533, 2319 (2024), arXiv:2403.10388 [astro-ph.CO]

work page arXiv 2024
[24]

W. L. Freedman, B. F. Madore, T. J. Hoyt, I. S. Jang, A. J. Lee, and K. A. Owens, Astrophys. J.985, 203 (2025), [Erratum: Astrophys.J. 993, 252 (2025)], arXiv:2408.06153 [astro-ph.CO]

work page arXiv 2025
[25]

Perivolaropoulos, Phys

L. Perivolaropoulos, Phys. Rev. D110, 123518 (2024), arXiv:2408.11031 [astro-ph.CO]

work page arXiv 2024
[26]

A. G. Riesset al., Astrophys. J. Lett.992, L34 (2025), arXiv:2509.01667 [astro-ph.CO]

work page arXiv 2025
[27]

Physically-motivated priors in the local distance ladder significantly reduce the Hubble tension

M. H¨ og˚ as and E. M¨ ortsell, (2026), arXiv:2601.22215 [astro-ph.CO]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[28]

M¨ ortsell and S

E. M¨ ortsell and S. Dhawan, JCAP09, 025 (2018), arXiv:1801.07260 [astro-ph.CO]

work page arXiv 2018
[29]

Vagnozzi, S

S. Vagnozzi, S. Dhawan, M. Gerbino, K. Freese, A. Goo- bar, and O. Mena, Phys. Rev. D98, 083501 (2018), arXiv:1801.08553 [astro-ph.CO]

work page arXiv 2018
[30]

W. Yang, S. Pan, E. Di Valentino, R. C. Nunes, S. Vagnozzi, and D. F. Mota, JCAP09, 019 (2018), arXiv:1805.08252 [astro-ph.CO]

work page arXiv 2018
[31]

Guo, J.-F

R.-Y. Guo, J.-F. Zhang, and X. Zhang, JCAP02, 054 (2019), arXiv:1809.02340 [astro-ph.CO]

work page arXiv 2019
[32]

C. D. Kreisch, F.-Y. Cyr-Racine, and O. Dor´ e, Phys. Rev. D101, 123505 (2020), arXiv:1902.00534 [astro- ph.CO]

work page arXiv 2020
[33]

Vagnozzi, Phys

S. Vagnozzi, Phys. Rev. D102, 023518 (2020), arXiv:1907.07569 [astro-ph.CO]

work page arXiv 2020
[34]

Visinelli, S

L. Visinelli, S. Vagnozzi, and U. Danielsson, Symmetry 11, 1035 (2019), arXiv:1907.07953 [astro-ph.CO]

work page arXiv 2019
[35]

Di Valentino, A

E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Dark Univ.30, 100666 (2020), arXiv:1908.04281 [astro-ph.CO]

work page arXiv 2020
[36]

Di Valentino, A

E. Di Valentino, A. Melchiorri, O. Mena, and S. Vagnozzi, Phys. Rev. D101, 063502 (2020), arXiv:1910.09853 [astro-ph.CO]

work page arXiv 2020
[37]

Krishnan, E

C. Krishnan, E. ´O. Colg´ ain, Ruchika, A. A. Sen, M. M. Sheikh-Jabbari, and T. Yang, Phys. Rev. D102, 103525 (2020), arXiv:2002.06044 [astro-ph.CO]

work page arXiv 2020
[38]

Alestas, L

G. Alestas, L. Kazantzidis, and L. Perivolaropoulos, Phys. Rev. D101, 123516 (2020), arXiv:2004.08363 [astro-ph.CO]

work page arXiv 2020
[39]

Jedamzik and L

K. Jedamzik and L. Pogosian, Phys. Rev. Lett.125, 181302 (2020), arXiv:2004.09487 [astro-ph.CO]

work page arXiv 2020
[40]

Sekiguchi and T

T. Sekiguchi and T. Takahashi, Phys. Rev. D103, 083507 (2021), arXiv:2007.03381 [astro-ph.CO]

work page arXiv 2021
[41]

Roy Choudhury, S

S. Roy Choudhury, S. Hannestad, and T. Tram, JCAP 03, 084 (2021), arXiv:2012.07519 [astro-ph.CO]

work page arXiv 2021
[42]

Brinckmann, J

T. Brinckmann, J. H. Chang, and M. LoVerde, Phys. Rev. D104, 063523 (2021), arXiv:2012.11830 [astro- ph.CO]

work page arXiv 2021
[43]

Gao, Z.-W

L.-Y. Gao, Z.-W. Zhao, S.-S. Xue, and X. Zhang, JCAP 07, 005 (2021), arXiv:2101.10714 [astro-ph.CO]

work page arXiv 2021
[44]

Marra and L

V. Marra and L. Perivolaropoulos, Phys. Rev. D104, L021303 (2021), arXiv:2102.06012 [astro-ph.CO]

work page arXiv 2021
[45]

M. G. Dainotti, B. De Simone, T. Schiavone, G. Mon- tani, E. Rinaldi, and G. Lambiase, Astrophys. J.912, 150 (2021), arXiv:2103.02117 [astro-ph.CO]

work page arXiv 2021
[46]

Krishnan, R

C. Krishnan, R. Mohayaee, E. ´O. Colg´ ain, M. M. Sheikh-Jabbari, and L. Yin, Class. Quant. Grav.38, 184001 (2021), arXiv:2105.09790 [astro-ph.CO]

work page arXiv 2021
[47]

Cyr-Racine, F

F.-Y. Cyr-Racine, F. Ge, and L. Knox, Phys. Rev. Lett. 128, 201301 (2022), arXiv:2107.13000 [astro-ph.CO]

work page arXiv 2022
[48]

L. A. Anchordoqui, E. Di Valentino, S. Pan, and W. Yang, JHEAp32, 28 (2021), arXiv:2107.13932 [astro-ph.CO]

work page arXiv 2021
[49]

Akarsu, S

¨O. Akarsu, S. Kumar, E. ¨Oz¨ ulker, and J. A. Vazquez, Phys. Rev. D104, 123512 (2021), arXiv:2108.09239 [astro-ph.CO]. 14

work page arXiv 2021
[50]

Ren, S.-F

X. Ren, S.-F. Yan, Y. Zhao, Y.-F. Cai, and E. N. Sari- dakis, Astrophys. J.932, 131 (2022), arXiv:2203.01926 [astro-ph.CO]

work page arXiv 2022
[51]

Nojiri, S

S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Nucl. Phys. B980, 115850 (2022), arXiv:2205.11681 [gr-qc]

work page arXiv 2022
[52]

Sch¨ oneberg and G

N. Sch¨ oneberg and G. Franco Abell´ an, JCAP12, 001 (2022), arXiv:2206.11276 [astro-ph.CO]

work page arXiv 2022
[53]

Banerjee, M

S. Banerjee, M. Petronikolou, and E. N. Saridakis, Phys. Rev. D108, 024012 (2023), arXiv:2209.02426 [gr- qc]

work page arXiv 2023
[54]

de S´ a, M

R. de S´ a, M. Benetti, and L. L. Graef, Eur. Phys. J. Plus137, 1129 (2022), arXiv:2209.11476 [astro-ph.CO]

work page arXiv 2022
[55]

Akarsu, S

O. Akarsu, S. Kumar, E. ¨Oz¨ ulker, J. A. Vazquez, and A. Yadav, Phys. Rev. D108, 023513 (2023), arXiv:2211.05742 [astro-ph.CO]

work page arXiv 2023
[56]

N. Lee, Y. Ali-Ha¨ ımoud, N. Sch¨ oneberg, and V. Poulin, Phys. Rev. Lett.130, 161003 (2023), arXiv:2212.04494 [astro-ph.CO]

work page arXiv 2023
[57]

Khodadi and M

M. Khodadi and M. Schreck, Phys. Dark Univ.39, 101170 (2023), arXiv:2301.03883 [gr-qc]

work page arXiv 2023
[58]

Bernui, E

A. Bernui, E. Di Valentino, W. Giar` e, S. Kumar, and R. C. Nunes, Phys. Rev. D107, 103531 (2023), arXiv:2301.06097 [astro-ph.CO]

work page arXiv 2023
[59]

Ben-Dayan and U

I. Ben-Dayan and U. Kumar, JCAP12, 047 (2023), arXiv:2302.00067 [astro-ph.CO]

work page arXiv 2023
[60]

G´ omez-Valent, N

A. G´ omez-Valent, N. E. Mavromatos, and J. Sol` a Per- acaula, Class. Quant. Grav.41, 015026 (2024), arXiv:2305.15774 [gr-qc]

work page arXiv 2024
[61]

Rathore, S

Ruchika, H. Rathore, S. Roy Choudhury, and V. Rentala, JCAP06, 056 (2024), arXiv:2306.05450 [astro-ph.CO]

work page arXiv 2024
[62]

S. A. Adil, ¨O. Akarsu, E. Di Valentino, R. C. Nunes, E. ¨Oz¨ ulker, A. A. Sen, and E. Specogna, Phys. Rev. D 109, 023527 (2024), arXiv:2306.08046 [astro-ph.CO]

work page arXiv 2024
[63]

Frion, D

E. Frion, D. Camarena, L. Giani, T. Miranda, D. Bertacca, V. Marra, and O. F. Piattella, (2023), 10.21105/astro.2307.06320, arXiv:2307.06320 [astro-ph.CO]

work page doi:10.21105/astro.2307.06320 2023
[64]

G´ omez-Valent, A

A. G´ omez-Valent, A. Favale, M. Migliaccio, and A. A. Sen, Phys. Rev. D109, 023525 (2024), arXiv:2309.07795 [astro-ph.CO]

work page arXiv 2024
[65]

Akarsu, E

¨O. Akarsu, E. ´O. Colg´ ain, A. A. Sen, and M. M. Sheikh- Jabbari, Universe10, 305 (2024), arXiv:2402.04767 [astro-ph.CO]

work page arXiv 2024
[66]

Giar` e, Y

W. Giar` e, Y. Zhai, S. Pan, E. Di Valentino, R. C. Nunes, and C. van de Bruck, Phys. Rev. D110, 063527 (2024), arXiv:2404.02110 [astro-ph.CO]

work page arXiv 2024
[67]

G. P. Lynch, L. Knox, and J. Chluba, Phys. Rev. D 110, 063518 (2024), arXiv:2404.05715 [astro-ph.CO]

work page arXiv 2024
[68]

Giar` e, M

W. Giar` e, M. A. Sabogal, R. C. Nunes, and E. Di Valentino, Phys. Rev. Lett.133, 251003 (2024), arXiv:2404.15232 [astro-ph.CO]

work page arXiv 2024
[69]

G. P. Lynch, L. Knox, and J. Chluba, Phys. Rev. D 110, 083538 (2024), arXiv:2406.10202 [astro-ph.CO]

work page arXiv 2024
[70]

Y. Toda, W. Giar` e, E. ¨Oz¨ ulker, E. Di Valentino, and S. Vagnozzi, Phys. Dark Univ.46, 101676 (2024), arXiv:2407.01173 [astro-ph.CO]

work page arXiv 2024
[71]

Nozari, S

K. Nozari, S. Saghafi, and M. Hajebrahimi, Phys. Dark Univ.46, 101571 (2024), arXiv:2407.01961 [gr-qc]

work page arXiv 2024
[72]

L. A. Escamilla, D. Fiorucci, G. Montani, and E. Di Valentino, Phys. Dark Univ.46, 101652 (2024), arXiv:2408.04354 [astro-ph.CO]

work page arXiv 2024
[73]

Roy Choudhury and T

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

work page arXiv 2024
[74]

S. H. Mirpoorian, K. Jedamzik, and L. Pogosian, Phys. Rev. D111, 083519 (2025), arXiv:2411.16678 [astro- ph.CO]

work page arXiv 2025
[75]

Li, G.-H

T.-N. Li, G.-H. Du, Y.-H. Li, P.-J. Wu, S.-J. Jin, J.-F. Zhang, and X. Zhang, Sci. China Phys. Mech. Astron. 69, 210413 (2026), arXiv:2501.07361 [astro-ph.CO]

work page arXiv 2026
[76]

N. Lee, M. Braglia, and Y. Ali-Ha¨ ımoud, Phys. Rev. D 112, 083506 (2025), arXiv:2504.07966 [astro-ph.CO]

work page arXiv 2025
[77]

E. M. Teixeira, W. Giar` e, N. B. Hogg, T. Montandon, A. Poudou, and V. Poulin, Phys. Rev. D112, 023515 (2025), arXiv:2504.10464 [astro-ph.CO]

work page arXiv 2025
[78]

Y.-Y. Wang, L. Lei, S.-P. Tang, and Y.-Z. Fan, JCAP 01, 009 (2026), arXiv:2508.19081 [astro-ph.CO]

work page arXiv 2026
[79]

Zhang, T.-N

Y.-M. Zhang, T.-N. Li, G.-H. Du, S.-H. Zhou, L.- Y. Gao, J.-F. Zhang, and X. Zhang, (2025), arXiv:2510.12627 [astro-ph.CO]

work page arXiv 2025
[80]

Kumar, Phys

S. Kumar, Phys. Dark Univ.52, 102248 (2026), arXiv:2512.19000 [astro-ph.CO]

work page arXiv 2026

Showing first 80 references.