arxiv: 2605.01904 · v2 · submitted 2026-05-03 · 🌌 astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

Alleviating the Hubble Tension Using ΛsCDM Model: A Coupled Dark Energy - Dark Matter Interaction

Amare Abebe Gidelew, Shambel Sahlu, Solomon Belay Tessema, Yismaw Wassie Ambelu

Pith reviewed 2026-05-11 01:42 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords Hubble tensioninteracting dark energydark mattercosmological parameterslate-time expansionstructure growthBAOCMB

0 comments

The pith

Coupled dark matter-dark energy interaction raises H0 to 71.8 km/s/Mpc and cuts tension to 1.2 sigma.

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

The paper tests whether a specific form of energy transfer between dark matter and dark energy can reconcile early-universe and late-universe measurements of the expansion rate. It introduces the ΛsCDM model with a gauge-invariant coupling that transfers energy from dark matter to dark energy only after recombination, which speeds up the late-time expansion without altering the sound horizon by more than one percent. When the model is fit to Planck CMB, DESI BAO, Pantheon+ supernovae, Hubble parameter data, and redshift-space distortions, it returns H0 = 71.8 with uncertainties of 0.3 to 0.4, moving the discrepancy with local measurements from roughly five sigma down to 1.2 sigma. The same late-time transfer also reduces the amplitude of matter fluctuations to σ8 = 0.744, which moves toward values favored by weak-lensing surveys. The resolution therefore arises from post-recombination dynamics rather than early-universe changes.

Core claim

The ΛsCDM model, defined by the interaction term Q = ξ H ρ_de together with a dynamically induced effective pressure in the dark matter fluid, yields H0 = 71.8_{-0.3}^{+0.4} km s^{-1} Mpc^{-1} when constrained by Planck 2018 CMB, DESI DR2 BAO, Pantheon+ supernovae, Hubble data, and RSD measurements. This value reduces the tension with the SH0ES local Hubble measurement from about 5σ in ΛCDM to 1.2σ while leaving the pre-recombination sound horizon within 1% of its ΛCDM value and producing σ8 = 0.744 ± 0.0185.

What carries the argument

The gauge-invariant coupling Q = ξ H ρ_de that transfers energy from dark matter to dark energy at late times and induces effective pressure in the dark matter component.

If this is right

Late-time energy transfer from dark matter to dark energy raises the present-day expansion rate while leaving early cosmology nearly unchanged.
Structure growth is suppressed relative to ΛCDM, producing a lower σ8 that moves closer to weak-lensing preferences.
The model remains consistent with all precision datasets employed in the MCMC analysis.
The interaction is automatically suppressed at high redshift, preserving the sound horizon to within 1%.

Where Pith is reading between the lines

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

This approach shows that purely late-time modifications can address the Hubble tension without requiring changes before recombination.
Future measurements of the growth rate or cluster abundances could test whether the lowered σ8 is realized in the data.
The specific functional form of the coupling may be generalized to other interaction kernels while retaining the late-time activation.

Load-bearing premise

The chosen coupling form and the induced dark-matter pressure correctly describe the interaction without introducing instabilities or violating other constraints.

What would settle it

A precise local H0 measurement lying well outside the 71.8 ± 0.4 range, or a future dataset showing early-universe deviations in the sound horizon larger than one percent.

read the original abstract

The considerable difference between early and late universe measurements of the Hubble constant, called the Hubble tension, poses a potential challenge to the standard $\Lambda$CDM cosmological model. We examine an interacting dark matter-dark energy model, $\Lambda_s$CDM, characterized by a gauge-invariant coupling $Q = \xi H\rho_{\mathrm{de}}$ and an effective pressure dynamically induced within the dark matter fluid. Using the CLASS Boltzmann code modified in this work, we analyze both the background and perturbation observables and compute an extensive Markov Chain Monte Carlo analysis with the latest cosmological datasets, including observational Hubble parameter data, Planck 2018 CMB compressed likelihood, BAO (from DESI DR2), Pantheon+ Type Ia supernovae, and redshift-space distortion measurements. The model predicts $H_0 = 71.8_{-0.3}^{+0.4}\mathrm{kms^{-1}Mpc^{-1}}$, reducing the tension with the SH0ES local measurement from about $5\sigma$ in $\Lambda$CDM to $1.2\sigma$ in $\Lambda_s$CDM. In contrast to the early dark energy model, the resolution emerges from late-time modification of the expansion history induced by the energy transfer from dark matter to dark energy. Moreover, the model suppresses late-time structure growth, providing $\sigma_8 = 0.744 \pm 0.0185$, lying below the $\Lambda$CDM value and moves in the direction preferred by weak lensing surveys. Since the interaction term is suppressed at high redshift, the pre-recombination sound horizon departs by less than $1\%$ from its $\Lambda$CDM value, suggesting that the alleviation of the tension dominantly originates from the late-time expansion rather than early-universe effects. We conclude that $\Lambda_s$CDM constitutes a phenomenologically viable interacting dark sector framework that addresses key cosmological tensions while remaining consistent with current precision data. }

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

This late-time interacting DE-DM model fits recent data with H0 near 72 and reduced tension, but the induced DM pressure term needs explicit stability checks before the posteriors can be trusted.

read the letter

The paper's core result is a gauge-invariant coupling Q = ξ H ρ_de that transfers energy from dark matter to dark energy at late times, producing an effective pressure in the DM fluid and yielding H0 = 71.8 km/s/Mpc with tension down to 1.2σ while leaving the sound horizon almost unchanged. It also returns a lower σ8 that moves toward weak-lensing preferences. That combination is the main thing worth noting: a concrete MCMC run on Planck compressed likelihood, DESI BAO, Pantheon+, Hubble data, and RSD that keeps early-universe physics close to ΛCDM and modifies only the expansion and growth after recombination.

Referee Report

1 major / 2 minor

Summary. The paper introduces the ΛsCDM model, an interacting DE-DM scenario with gauge-invariant coupling Q = ξ H ρ_de that dynamically induces an effective pressure in the dark matter fluid. Using a modified CLASS Boltzmann code, the authors perform MCMC analyses combining Planck 2018 compressed likelihood, DESI DR2 BAO, Pantheon+ supernovae, Hubble parameter data, and RSD measurements. They report H0 = 71.8_{-0.3}^{+0.4} km s^{-1} Mpc^{-1} (reducing tension with SH0ES to 1.2σ) and σ8 = 0.744 ± 0.0185, attributing the resolution primarily to late-time expansion history modifications while keeping the sound horizon within 1% of ΛCDM.

Significance. If the linear perturbation implementation remains stable, the model offers a viable late-time mechanism for alleviating the Hubble tension through energy transfer from DM to DE, while simultaneously suppressing structure growth in a direction favored by weak-lensing surveys. The extensive use of current datasets (including DESI DR2) and the explicit modification of CLASS for both background and perturbations constitute clear strengths.

major comments (1)

[linear perturbations / modified CLASS] Perturbation implementation (described in the section on linear perturbations and the modified CLASS code): the manuscript provides no explicit verification that the effective sound speed squared remains positive and subluminal for the posterior value of ξ across z ≈ 0–10. Because the reported H0 and σ8 posteriors are obtained from the MCMC that relies on these perturbation equations, absence of this stability check renders the central claim load-bearing on an unverified assumption.

minor comments (2)

[abstract] The abstract states that the interaction is 'suppressed at high redshift' but does not quantify the redshift at which |Q| drops below a given fraction of Hρ_de; a brief plot or statement would improve clarity.
[results] Table or figure presenting the full posterior constraints on ξ and the derived H0, σ8 values is referenced but its caption could explicitly note the tension metric calculation (e.g., how the 1.2σ figure is obtained).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript on the ΛsCDM model. We address the single major comment below regarding verification of the linear perturbation implementation.

read point-by-point responses

Referee: [linear perturbations / modified CLASS] Perturbation implementation (described in the section on linear perturbations and the modified CLASS code): the manuscript provides no explicit verification that the effective sound speed squared remains positive and subluminal for the posterior value of ξ across z ≈ 0–10. Because the reported H0 and σ8 posteriors are obtained from the MCMC that relies on these perturbation equations, absence of this stability check renders the central claim load-bearing on an unverified assumption.

Authors: We acknowledge that the original manuscript did not include an explicit numerical verification that the effective sound speed squared remains positive and subluminal (0 < c_s² < 1) for the posterior values of ξ over z ≈ 0–10. The perturbation equations in the modified CLASS implementation were derived from the gauge-invariant coupling and the induced effective pressure in the dark matter fluid, and we had verified internal consistency during code development, but a dedicated post-MCMC stability check across the relevant redshift range was omitted. We have now performed this check using the best-fit and 1σ posterior samples of ξ from our chains; the effective sound speed squared remains positive and well below unity throughout z = 0–10, with no sign of instabilities or superluminal propagation. We will add this verification (as a short paragraph and/or supplementary figure) to the revised manuscript to make the robustness of the perturbation sector fully explicit. revision: yes

Circularity Check

1 steps flagged

Fitted posterior H0 presented as independent model prediction creates partial circularity in tension-alleviation claim

specific steps

fitted input called prediction [Abstract]
"The model predicts H_0 = 71.8_{-0.3}^{+0.4} kms^{-1}Mpc^{-1}, reducing the tension with the SH0ES local measurement from about 5σ in ΛCDM to 1.2σ in ΛsCDM."

H0 and the quoted tension reduction are obtained by fitting the coupling strength ξ together with standard cosmological parameters to the listed datasets via MCMC; labeling the posterior mean a 'prediction' makes the alleviation a direct statistical consequence of the fit rather than an independent test of the interaction model.

full rationale

The paper defines a phenomenological interacting DE-DM model with gauge-invariant coupling Q = ξ H ρ_de plus induced DM effective pressure, modifies CLASS, and runs MCMC on Planck compressed, BAO, Pantheon+, OHD and RSD data to obtain parameter posteriors. It then labels the resulting H0 posterior mean a 'prediction' that reduces the SH0ES tension. This matches the fitted-input-called-prediction pattern: the reported H0 and tension reduction are direct outputs of the same fit that determines ξ, not an a-priori derivation from the model equations alone. No self-citation load-bearing steps, uniqueness theorems, or ansatz smuggling appear in the provided text, so the circularity is limited to the presentation of the fit result as a prediction. The underlying numerical pipeline itself is not circular.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on a chosen interaction form and an effective pressure term introduced to capture energy transfer; these are not derived from first principles but postulated for phenomenological viability.

free parameters (1)

ξ
Dimensionless coupling strength between dark energy and dark matter; its value is determined by MCMC fitting to data.

axioms (2)

standard math Standard FLRW background and linear perturbation theory remain valid under the interaction
Invoked to compute background evolution and observables with the modified Boltzmann code.
ad hoc to paper The interaction term Q = ξ H ρ_de is gauge-invariant and does not introduce instabilities
Chosen form of the coupling; stated as a defining feature of the ΛsCDM model.

invented entities (1)

Dynamically induced effective pressure in the dark matter fluid no independent evidence
purpose: To model the back-reaction of the energy transfer on dark matter dynamics
Introduced as a consequence of the coupling; no independent evidence outside the model equations is provided.

pith-pipeline@v0.9.0 · 5688 in / 1601 out tokens · 43556 ms · 2026-05-11T01:42:22.547341+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
gauge-invariant coupling Q = ξ H ρ_de and an effective pressure dynamically induced within the dark matter fluid... modified CLASS... MCMC analysis
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
p_thermo_dm = ξ/3 ρ_de ... g(a) = g0(1-a) ... stability condition ξ(1+ws)>0

Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages · 5 internal anchors

[1]

Riess et al.,Observational evidence from supernovae for an accelerating universe and a cosmological constant,Astron

A.G. Riess et al.,Observational evidence from supernovae for an accelerating universe and a cosmological constant,Astron. J.116(1998) 1009

work page 1998
[2]

and Aldering, G

S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent, P.G. Castro et al., Measurements ofωandλfrom 42 high-redshift supernovae,Astrophys. J.517(1999) 565 [astro-ph/9812133]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[3]

Verde, T

L. Verde, T. Treu and A.G. Riess,Tensions between the early and late universe,Nat. Astron.3 (2019) 891

work page 2019
[4]

Valentino et al.,Snowmass2021: Cosmology intertwined i: Perspectives for the next decade,Class

E.D. Valentino et al.,Snowmass2021: Cosmology intertwined i: Perspectives for the next decade,Class. Quantum Grav.38(2021) 153001

work page 2021
[5]

A.G. Riess et al.,A comprehensive measurement of the local value of the hubble constant with 1.3 km/s/mpc uncertainty from the hubble space telescope and the sh0es team,The Astrophysical Journal Letters934(2022) L7 [2112.04510]

work page internal anchor Pith review arXiv 2022
[6]

Abdalla et al.,Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies,J

E. Abdalla et al.,Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies,J. High Energy Astrophys. 34(2022) 49

work page 2022
[7]

The Hubble Tension and Early Dark Energy,

M. Kamionkowski and A.G. Riess,The hubble tension and early dark energy,Ann. Rev. Nucl. Part. Sci.73(2023) 153 [2211.04492]

work page arXiv 2023
[8]

Amendola,Coupled quintessence,Phys

L. Amendola,Coupled quintessence,Phys. Rev. D62(2000) 043511

work page 2000
[9]

Sahlu, U

S. Sahlu, U. Mukhopadhyay, R.R. Mekuria and A. Abebe,Observational constraints of diffusive dark-fluid cosmology,Monthly Notices of the Royal Astronomical Society546(2026) staf1921

work page 2026
[10]

Saturation Mechanisms in the Interacting Dark Sector

A. Paliathanasis and K.J. Duffy,Saturation Mechanisms in the Interacting Dark Sector, 2604.21671

work page internal anchor Pith review Pith/arXiv arXiv
[11]

Paliathanasis, Phys

A. Paliathanasis,Indications of a late-time transition to a strongly interacting dark sector, Phys. Dark Univ.52(2026) 102306 [2601.02789]

work page arXiv 2026
[12]

van der Westhuizen, D

M. van der Westhuizen, D. Figueruelo, R. Thubisi, S. Sahlu, A. Abebe and A. Paliathanasis, Compartmentalization in the dark sector of the universe after desi dr2 bao data,Physics of the Dark Universe(2025) 102107. – 26 –

work page 2025
[13]

Sahlu, A

S. Sahlu, A. Paliathanasis, G. Leon and A. Abebe,Testing the coexistence of dark energy and dark matter with late-time observational data,Physics of the Dark Universe(2026) 102280

work page 2026
[14]

Bolotin, A

Y.L. Bolotin, A. Kostenko, O.A. Lemets and D.A. Yerokhin,The cosmological constant problem and quintessence,Int. J. Mod. Phys. D24(2015) 1530007

work page 2015
[15]

W. Yang, S. Pan, E.D. Valentino, R.C. Nunes and S. Vagnozzi,Dynamical dark energy after planck cmb final release andh0 tension,J. Cosmol. Astropart. Phys.2018(2018) 016

work page 2018
[16]

Amendola, V

L. Amendola, V. Pettorino, C. Quercellini and A. Vollmer,Testing coupled dark energy with next-generation large-scale observations,Phys. Rev. D85(2012) 103008

work page 2012
[17]

Di Valentinoet al.(CosmoVerse Network), The Cos- moVerseWhitePaper: Addressingobservationaltensions in cosmology with systematics and fundamental physics, Phys

T.C. Network, E.D. Valentino, J.L. Said, A.G. Riess and over 400 additional authors,The cosmoverse white paper: Addressing observational tensions in cosmology with systematics and fundamental physics,Phys. Dark Universe49(2025) 101965 [2504.01669]

work page arXiv 2025
[18]

Non-minimally coupled quintessence with sign-switching interaction

J.-Q. Wang, R.-G. Cai, Z.-K. Guo, Y.-H. Li, S.-J. Wang and X. Zhang,Non-minimally coupled quintessence with sign-switching interaction,arXiv preprint(2026) [2604.02204v2]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[19]

Hoeming, R.G

G.A. Hoeming, R.G. Landim, L.O. Ponte, R.P. Rolim, F.B. Abdalla and E. Abdalla, Constraints on interacting dark energy revisited: Implications for the hubble tension,Phys. Rev. D112(2025) 023523

work page 2025
[20]

Chevallier and D

M. Chevallier and D. Polarski,Accelerating universes with scaling dark matter,Int. J. Mod. Phys. D10(2001) 213

work page 2001
[21]

Linder,Exploring the expansion history of the universe,Phys

E.V. Linder,Exploring the expansion history of the universe,Phys. Rev. Lett.90(2003) 091301

work page 2003
[22]

Akarsu et al.,Testing bulk viscous models with pantheon+ type ia supernovae and observational hubble data,Mon

Ö. Akarsu et al.,Testing bulk viscous models with pantheon+ type ia supernovae and observational hubble data,Mon. Not. R. Astron. Soc.527(2023) 2070

work page 2023
[23]

Efstathiou,To h0 or not to h0?,Mon

G. Efstathiou,To h0 or not to h0?,Mon. Not. R. Astron. Soc.505(2021) 3866

work page 2021
[24]

Gavela, D

M.B. Gavela, D. Hernandez, L.L. Honorez, O. Mena and S. Rigolin,Dark coupling,JCAP 0907(2009) 034

work page 2009
[25]

Valiviita, E

J. Valiviita, E. Majerotto and R. Maartens,Instability in interacting dark energy and dark matter fluids,J. Cosmol. Astropart. Phys.2008(2008) 020

work page 2008
[26]

Majerotto, J

E. Majerotto, J. Valiviita and R. Maartens,Perturbations in a coupled scalar field cosmology, Mon. Not. R. Astron. Soc.402(2010) 2344

work page 2010
[27]

Akarsu, S

Ö. Akarsu, S. Kumar, E. Özülker and J.A. Vazquez,Relaxing cosmological tensions with a sign switching cosmological constant,Phys. Rev. D104(2021) 123512

work page 2021
[28]

Harko and F.S.N

T. Harko and F.S.N. Lobo,f(r, lm)gravity,Phys. Rev. D87(2013) 044018

work page 2013
[29]

Hu,Structure formation with generalized dark matter,Astrophys

W. Hu,Structure formation with generalized dark matter,Astrophys. J.506(1998) 485

work page 1998
[30]

Harko, K

T. Harko, K. Asadi, H. Moshafi and H. Sheikhahmadi,Observational constraints on the interacting dark energy - dark matter (idm) cosmological models,Physics of the Dark Universe 38(2022) 101131 [2203.08907]

work page arXiv 2022
[31]

Kodama and M

H. Kodama and M. Sasaki,Cosmological perturbation theory,Prog. Theor. Phys. Suppl.78 (1984) 1

work page 1984
[32]

Ma and E

C.-P. Ma and E. Bertschinger,Cosmological perturbation theory in the synchronous and conformal newtonian gauges,Astrophys. J.455(1995) 7

work page 1995
[33]

Di Valentino, A

E.D. Valentino, A. Melchiorri, O. Mena and S. Vagnozzi,Interacting dark energy in the early 2020s: A promising solution to theh0 and cosmic shear tensions,Phys. Dark Universe30 (2020) 100666 [1908.04281]. [34]DES Collaborationcollaboration,Dark energy survey year 3 results: Cosmological constraints from galaxy clustering and weak lensing,Phys. Rev. D105(2...

work page arXiv 2020
[34]

Aboubrahim and P

A. Aboubrahim and P. Nath,A study of dark matter-dark energy interaction under the desi dr2 data constraint,Journal of Cosmology and Astroparticle Physics10(2025) 081

work page 2025
[35]

Cosmic Growth History and Expansion History,

E.V. Linder,Cosmic growth history and expansion history,Physical Review D72(2005) 043529 [astro-ph/0507263]

work page arXiv 2005
[36]

Kazantzidis and L

L. Kazantzidis and L. Perivolaropoulos,Evolution of thef σ8 tension with the planck15/λcdm determination and implications for modified gravity theories,Phys. Rev. D97(2018) 103503

work page 2018
[37]

D. Blas, J. Lesgourgues and T. Tram,The cosmic linear anisotropy solving system (class) ii: Approximation schemes,JCAP07(2011) 034 [1104.2933]

work page internal anchor Pith review arXiv 2011
[38]

Planck Collaboration,Planck 2018 results. vi. cosmological parameters,Astron. Astrophys.641 (2020) A6

work page 2018
[39]

J.(2025)

DESI Collaboration,The early data release of the dark energy spectroscopic instrument, Astron. J.(2025)

work page 2025
[40]

Brout et al.,The pantheon+ analysis: Cosmological constraints,Astrophys

D. Brout et al.,The pantheon+ analysis: Cosmological constraints,Astrophys. J.938(2022) 110

work page 2022
[41]

Moresco et al.,A 6% measurement of the hubble parameter atz∼0.45: Direct evidence of the epoch of cosmic re-acceleration,J

M. Moresco et al.,A 6% measurement of the hubble parameter atz∼0.45: Direct evidence of the epoch of cosmic re-acceleration,J. Cosmol. Astropart. Phys.2016(2016) 014

work page 2016
[42]

Moresco et al.,Unveiling the universe with emerging cosmological probes,Living Rev

M. Moresco et al.,Unveiling the universe with emerging cosmological probes,Living Rev. Relativ.25(2022) 6. [44]WiggleZ Collaborationcollaboration,The wigglez dark energy survey: Joint measurements of the expansion and growth history at z < 1,Mon. Not. R. Astron. Soc.425(2012) 405

work page 2022
[43]

Blake et al.,The wigglez dark energy survey: Mapping the distance-redshift relation with baryon acoustic oscillations,Mon

C. Blake et al.,The wigglez dark energy survey: Mapping the distance-redshift relation with baryon acoustic oscillations,Mon. Not. R. Astron. Soc.415(2011) 2892. [46]BOSS Collaborationcollaboration,The clustering of galaxies in the completed sdss-iii baryon oscillation spectroscopic survey: Cosmological analysis of the dr12 galaxy sample,Mon. Not. R. Astr...

work page 2011
[44]

H. Gil-Marín et al.,The completed sdss-iv extended baryon oscillation spectroscopic survey: Measurement of the bao and growth rate of structure of the luminous red galaxy sample from the anisotropic power spectrum between redshifts 0.6 and 1.0,Mon. Not. R. Astron. Soc.498 (2020) 2492

work page 2020
[45]

Baker et al.,The cosmology dependence of the galaxy correlation function,Rev

T. Baker et al.,The cosmology dependence of the galaxy correlation function,Rev. Mod. Phys. 93(2021) 015003

work page 2021
[46]

Nesseris and J

S. Nesseris and J. Garcia-Bellido,Is the jeffreys’ scale a reliable tool for bayesian model comparison in cosmology?,J. Cosmol. Astropart. Phys.2013(2013) 036

work page 2013
[47]

Liddle,Information criteria for astrophysical model selection,Mon

A.R. Liddle,Information criteria for astrophysical model selection,Mon. Not. R. Astron. Soc. 377(2007) L74

work page 2007
[48]

Foreman-Mackey, D.W

D. Foreman-Mackey, D.W. Hogg, D. Lang and J. Goodman,emcee: The mcmc hammer,Publ. Astron. Soc. Pac.125(2013) 306

work page 2013
[49]

Gelman and D.B

A. Gelman and D.B. Rubin,Inference from iterative simulation using multiple sequences,Stat. Sci.7(1992) 457

work page 1992
[50]

Akaike,A new look at the statistical model identification,IEEE Trans

H. Akaike,A new look at the statistical model identification,IEEE Trans. Autom. Control19 (1974) 716

work page 1974
[51]

Schwarz,Estimating the dimension of a model,Ann

G. Schwarz,Estimating the dimension of a model,Ann. Stat.6(1978) 461

work page 1978
[52]

Burnham and D.R

K.P. Burnham and D.R. Anderson,Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Springer, New York, 2nd ed. (2002). – 28 –

work page 2002
[53]

Carroll,Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, San Francisco (2004)

S.M. Carroll,Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, San Francisco (2004)

work page 2004
[54]

Poulin, T.L

V. Poulin, T.L. Smith, T. Karwal and M. Kamionkowski,Early dark energy can resolve the hubble tension,Phys. Rev. Lett.122(2019) 221301

work page 2019
[55]

Riess et al.,A comprehensive measurement of the local value of the hubble constant with 1 km/s/mpc uncertainty from the hubble space telescope and the sh0es team,Astrophys

A.G. Riess et al.,A comprehensive measurement of the local value of the hubble constant with 1 km/s/mpc uncertainty from the hubble space telescope and the sh0es team,Astrophys. J. Lett. 908(2021) L6

work page 2021
[56]

Lyons,Discovering the significance of 5σ,arXiv preprint(2013) [1310.1284]

L. Lyons,Discovering the significance of 5σ,arXiv preprint(2013) [1310.1284]

work page arXiv 2013
[57]

Ganesan, A

V. Ganesan, A. Chakraborty, T. Ray, S. Das and A. Banerjee,On the stability of coupled dark energy with dark matter,J. Cosmol. Astropart. Phys.2024(2024) 035

work page 2024
[58]

Giarè, M.A

W. Giarè, M.A. Sabogal, R.C. Nunes and E.D. Valentino,Quantifying the global bayesian evidence for interaction in the dark sector,Phys. Rev. D109(2024) 123512

work page 2024
[59]

Paraskevas, A

E.A. Paraskevas, A. Cam, L. Perivolaropoulos and Ö. Akarsu,Testing the self-interacting dark matter scenario with strong gravitational lensing,Phys. Rev. D109(2024) 103522

work page 2024
[60]

Kass and A.E

R.E. Kass and A.E. Raftery,Bayes factors,J. Am. Stat. Assoc.90(1995) 773

work page 1995
[61]

the euclid wide survey,Astron

Euclid Collaboration,Euclid preparation: I. the euclid wide survey,Astron. Astrophys.662 (2022) A112. – 29 –

work page 2022