Recognition: 2 theorem links
In the Realm of the Hubble tension - a Review of Solutions
Pith reviewed 2026-05-15 11:45 UTC · model grok-4.3
The pith
Some extensions to LambdaCDM restore agreement between Planck and local Hubble constant measurements within 1-2 sigma.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The review establishes that models with early or dynamical dark energy, neutrino interactions, interacting cosmologies, primordial magnetic fields, and modified gravity can bring the Planck 2018 CMB-derived Hubble constant into agreement with the SH0ES local measurement within 1-2 sigma when additional degrees of freedom are allowed, while preserving consistency with BAO and Pantheon supernova data.
What carries the argument
The Hubble tension, quantified as the statistically significant mismatch between early-time LambdaCDM predictions and late-time local distance-ladder determinations of the present-day expansion rate.
If this is right
- Extra parameters in successful models typically enlarge the uncertainty on H0 rather than simply shifting its central value.
- Additional independent probes of the expansion history will be required to discriminate among the remaining viable extensions.
- Solutions that alter early-universe physics may simultaneously affect other parameter tensions such as the clustering amplitude.
Where Pith is reading between the lines
- If any of these extensions holds, the assumption of a constant dark-energy density throughout cosmic time would need revision.
- A confirmed resolution via new physics would imply that the current data sets already contain hints of beyond-standard-model behavior at cosmological scales.
Load-bearing premise
The discrepancy is caused by missing physics rather than unrecognized systematic errors in the measurements.
What would settle it
A new, independent determination of the Hubble constant that falls between the early and late values while remaining inconsistent with all the extended models at high significance.
read the original abstract
The $\Lambda$CDM model provides a good fit to a large span of cosmological data but harbors areas of phenomenology. With the improvement of the number and the accuracy of observations, discrepancies among key cosmological parameters of the model have emerged. The most statistically significant tension is the $4-6\sigma$ disagreement between predictions of the Hubble constant $H_0$ by early time probes with $\Lambda$CDM model, and a number of late time, model-independent determinations of $H_0$ from local measurements of distances and redshifts. The high precision and consistency of the data at both ends present strong challenges to the possible solution space and demand a hypothesis with enough rigor to explain multiple observations--whether these invoke new physics, unexpected large-scale structures or multiple, unrelated errors. We present a thorough review of the problem, including a discussion of recent Hubble constant estimates and a summary of the proposed theoretical solutions. Some of the models presented are formally successful, improving the fit to the data in light of their additional degrees of freedom, restoring agreement within $1-2\sigma$ between {\it Planck} 2018, using CMB power spectra data, BAO, Pantheon SN data, and R20, the latest SH0ES Team measurement of the Hubble constant ($H_0 = 73.2 \pm 1.3{\rm\,km\,s^{-1}\,Mpc^{-1}}$ at 68\% confidence level). Reduced tension might not simply come from a change in $H_0$ but also from an increase in its uncertainty due to degeneracy with additional physics, pointing to the need for additional probes. While no specific proposal makes a strong case for being highly likely or far better than all others, solutions involving early or dynamical dark energy, neutrino interactions, interacting cosmologies, primordial magnetic fields, and modified gravity provide the best options until a better alternative comes along.[Abridged]
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript is a review of the Hubble tension, presenting the 4-6σ discrepancy between Planck 2018 CMB predictions under ΛCDM and late-time local H0 measurements (e.g., R20: H0 = 73.2 ± 1.3 km s^{-1} Mpc^{-1}). It summarizes classes of proposed solutions (early dark energy, neutrino interactions, interacting cosmologies, primordial magnetic fields, modified gravity) and states that some formally improve fits to Planck + BAO + Pantheon + R20 data, reducing tension to 1-2σ, while noting that extra parameters may increase H0 uncertainty via degeneracies and that no model is strongly preferred.
Significance. As a balanced review consolidating literature on a major cosmological discrepancy, the work is significant for mapping viable extensions to ΛCDM and underscoring the need for additional probes beyond current datasets. The explicit caveats on parameter freedom and the discussion of both physical and systematic interpretations strengthen its utility for the field.
major comments (2)
- [§4] §4 (early dark energy subsection): the statement that certain EDE models restore 1-2σ agreement should cite the precise Δχ² values and best-fit H0 shifts from the referenced fits to Planck 2018 + BAO + Pantheon + R20; without these numbers the claim that the improvement is 'formal' remains qualitative rather than quantitative.
- [§5.3] §5.3 (interacting cosmologies): the review asserts reduced tension but does not address whether the additional interaction parameters remain consistent with independent large-scale structure constraints (e.g., σ8 from weak lensing); this is load-bearing for the claim that these models are among the 'best options'.
minor comments (3)
- [Abstract] Abstract: define 'R20' on first use rather than assuming reader familiarity with the SH0ES notation.
- [Figures] Figure captions (throughout): expand to list the exact combination of datasets (Planck 2018, BAO, Pantheon, R20) used for each tension assessment shown.
- [§2] §2: the discussion of possible systematics would benefit from a short table comparing the magnitude of known calibration uncertainties in local distance ladders versus CMB foreground modeling.
Simulated Author's Rebuttal
We thank the referee for the positive evaluation of our review and for the constructive suggestions. We address the major comments point by point below.
read point-by-point responses
-
Referee: [§4] §4 (early dark energy subsection): the statement that certain EDE models restore 1-2σ agreement should cite the precise Δχ² values and best-fit H0 shifts from the referenced fits to Planck 2018 + BAO + Pantheon + R20; without these numbers the claim that the improvement is 'formal' remains qualitative rather than quantitative.
Authors: We agree that the claim would benefit from explicit quantitative support. In the revised manuscript we will add the relevant Δχ² improvements and best-fit H0 values (with uncertainties) drawn directly from the key EDE references already cited in §4 for the Planck 2018 + BAO + Pantheon + R20 combination. revision: yes
-
Referee: [§5.3] §5.3 (interacting cosmologies): the review asserts reduced tension but does not address whether the additional interaction parameters remain consistent with independent large-scale structure constraints (e.g., σ8 from weak lensing); this is load-bearing for the claim that these models are among the 'best options'.
Authors: This is a fair and important observation. In the revised version we will expand §5.3 with a concise discussion of the σ8 implications, referencing existing literature that examines consistency (or tension) with weak-lensing constraints, thereby clarifying why we still regard these models as among the stronger candidates. revision: yes
Circularity Check
No significant circularity: review of external literature
full rationale
This paper is a review article that summarizes existing literature on Hubble tension solutions without presenting any original derivation chain, new fits, or self-referential equations. All claims about model performance (e.g., restoring 1-2σ agreement) are explicitly attributed to cited external studies using Planck 2018, BAO, Pantheon, and R20 data. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central discussion remains a compilation of independent results with caveats on additional parameters and uncertainties. The paper is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
-
HierarchyForcingadditive_composition_is_minimal contradicts?
contradictsCONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.
Some of the models presented are formally successful, improving the fit to the data in light of their additional degrees of freedom, restoring agreement within 1-2σ between Planck 2018... and R20
-
DimensionForcingdimension_forced unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
solutions involving early or dynamical dark energy, neutrino interactions, interacting cosmologies, primordial magnetic fields, and modified gravity provide the best options
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Forward citations
Cited by 22 Pith papers
-
Dispersion Measure Distribution of Unlocalized Fast Radio Bursts as a Probe of the Hubble Constant
The DM distribution of unlocalized FRBs yields H0 = 73.8 +14.0/-12.3 km/s/Mpc with 18% uncertainty.
-
DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints
DESI DR2 BAO data exhibits 2.3 sigma tension with CMB in Lambda-CDM but prefers evolving dark energy (w0 > -1, wa < 0) at 3.1 sigma with CMB and 2.8-4.2 sigma when including supernovae.
-
DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations
First-year DESI BAO data are consistent with flat LambdaCDM and, when combined with CMB, show a 2.5-3.9 sigma preference for evolving dark energy (w0 > -1, wa < 0) that strengthens with certain supernova datasets.
-
Axion dark matter from extended misalignment with a constant-$\omega_\phi$ pre-oscillatory phase and dark radiation
Extended misalignment for axion-like particles with constant-ω_ϕ pre-oscillation and dark radiation coupling yields data-driven constraints favoring negative ω_ϕ and f_ϕ in [80, 1.5×10^10] TeV but does not ease cosmol...
-
Axion dark matter from extended misalignment with a constant-$\omega_\phi$ pre-oscillatory phase and dark radiation
The extended misalignment model with constant-ω_φ pre-oscillatory phase and dark radiation coupling is constrained by cosmological data to favor negative ω_φ and f_φ in [80, 1.5×10^10] TeV without resolving H0 or S8 tensions.
-
Geometric Constraints on the Pre-Recombination Expansion History from the Hubble Tension
Model-independent reconstruction shows that early-universe modifications resolving the Hubble tension exist at the background level, requiring a smooth ~15% pre-recombination expansion rate enhancement.
-
Double the axions, half the tension: multi-field early dark energy eases the Hubble tension
Two-field axion-like early dark energy reduces Hubble tension to 1.5 sigma residual and improves high-ell CMB fits over single-field models.
-
Bounding axion dark energy
An analytic bound on axion parameters in thawing quintessence is derived independently of initial conditions and used with cosmological observations plus quantum gravity constraints to exclude large regions of axion d...
-
Model-independent constraints on generalized FLRW consistency relations with bootstrap-based symbolic regression
Bootstrap-based symbolic regression on supernova and BAO data finds mild 2-4 sigma deviations from FLRW consistency relations, which if real would rule out most FLRW-based solutions to cosmological tensions.
-
Non-minimally coupled quintessence with sign-switching interaction
A new quintessence model with non-minimal coupling produces an effective sign-switching interaction that fits current data better than LambdaCDM or w0waCDM and accounts for late-time dark energy weakening without phan...
-
Model-independent test of the cosmic distance duality relation with recent observational data
Two model-independent methods applied to latest SN and BAO data find the cosmic distance duality relation consistent with observations within 1 sigma and no evidence of violation.
-
Cosmological intercept tension
Tensions in the supernova intercept a_B at z~0.01 in PantheonPlus and z~0.1 in DES-Y5 point to data systematics or inter-survey inconsistencies rather than new physics, aligning H0 measurements and reducing support fo...
-
Do equation of state parametrizations of dark energy faithfully capture the dynamics of the late universe?
Node-based reconstruction of cosmic expansion prefers stronger deceleration at z≈1.7 than smooth DE EoS parametrizations, isolating z~1.5-2 as a window where the latter may compress localized kinematic features permit...
-
Measuring neutrino mass in light of ACT DR6 and DESI DR2
New ACT and DESI data yield model-dependent upper limits on sum of neutrino masses, with holographic dark energy giving the tightest bounds and a consistent preference for degenerate hierarchy.
-
Precision Analysis for $\boldsymbol{H_0}$ Using Upcoming Multi-band Gravitational Wave Observations
Multi-band GW observations of PBHs can reduce H0 uncertainty to ≲2 km/s/Mpc (conservative) or O(0.1) km/s/Mpc (optimistic) via Fisher forecasts on M_PBH and f_PBH.
-
Testing $\Lambda$CDM with ANN-Reconstructed Expansion History from Cosmic Chronometers
The ANN-reconstructed Hubble parameter H(z) from cosmic chronometers aligns with Lambda CDM predictions within uncertainties.
-
Spontaneous Symmetry Breaking and the Vacuum Displacement Principle: From Galactic Scales to Cosmic Fine-Tuning
A vacuum scalar field with spontaneous symmetry breaking and matter coupling generates Yukawa-corrected gravity that accounts for flat galactic rotation curves and dynamically tracks the cosmological constant.
-
Buchdahl Limit and TOV Equations in Interacting Vacuum Scenarios
Interacting vacuum energy relaxes the pressure gradient inside stars, allowing finite central pressure and compactness beyond the Buchdahl bound for suitable coupling strengths.
-
Impact of the SNe Ia Magnitude Transition at 20 Mpc on Cosmological Parameter Estimation
A 0.19 mag step in supernova absolute magnitude at 20 Mpc improves data fit and increases the Hubble constant by 2% while leaving matter density and dark energy parameters stable.
-
Probing cosmic anisotropy with galaxy clusters and supernovae
Analysis of galaxy cluster and supernova data reveals a ~2σ directional variation in the Hubble constant, robust across calibration methods and aligned with the CMB dipole.
-
Cosmological constraints on the big bang quantum cosmology model
The JCDM model yields H0 of 66.95 plus or minus 0.51 km/s/Mpc and Omega_m of 0.3419 plus or minus 0.0065 in a flat universe, rising to H0 of 69.13 plus or minus 0.56 with slight positive curvature, fitting late-time d...
-
Off-diagonal solutions in Einsteingravity modeling f(R) gravity and dynamical darkenergy vs Lambda CDM cosmology
Off-diagonal Einstein solutions constructed via the anholonomic frame and connection deformation method can mimic f(R) gravity and dynamical dark energy effects inside GR and Lambda CDM cosmology.
Reference graph
Works this paper leans on
- [1]
- [2]
- [3]
- [4]
- [5]
- [6]
- [7]
- [8]
- [9]
- [10]
- [11]
- [12]
- [13]
- [14]
-
[15]
A gravitational-wave standard siren measurement of the Hubble constant
Abbott B P et al. (LIGO Scientific, Virgo, 1M2H, Dark Energy Camera GW-E, DES, DLT40, Las Cumbres Observatory, VINROUGE, MASTER) 2017 Nature 551 85–88 [arXiv:1710.05835]
work page internal anchor Pith review Pith/arXiv arXiv 2017
- [16]
- [17]
- [18]
-
[19]
Gayathri V, Healy J, Lange J, O’Brien B, Szczepanczyk M, Bartos I, Campanelli M, Klimenko S, Lousto C O and O’Shaughnessy R 2021 Astrophys. J. Lett. 908 L34
work page 2021
- [20]
- [21]
-
[22]
Llinares C, Mota D F and Winther H A 2014 Astron. Astrophys. 562 A78 [arXiv:1307.6748]
work page internal anchor Pith review Pith/arXiv arXiv 2014
- [23]
- [24]
-
[25]
Wu H Y and Huterer D 2017 Mon. Not. Roy. Astron. Soc. 471 4946–4955 [arXiv:1706.09723]
work page internal anchor Pith review Pith/arXiv arXiv 2017
-
[26]
Kenworthy W D, Scolnic D and Riess A 2019 Astrophys. J. 875 145 [arXiv:1901.08681]
work page internal anchor Pith review Pith/arXiv arXiv 2019
- [27]
- [28]
- [29]
-
[30]
Bengaly C A P, Bernui A and Alcaniz J S 2015 Astrophys. J. 808 39 [arXiv:1503.01413]
work page internal anchor Pith review Pith/arXiv arXiv 2015
-
[31]
Bengaly Jr C A P 2016 JCAP 04 036 [arXiv:1510.05545]
work page internal anchor Pith review Pith/arXiv arXiv 2016
-
[32]
Bolejko K, Nazer M A and Wiltshire D L 2016 JCAP 06 035 [arXiv:1512.07364]
work page internal anchor Pith review Pith/arXiv arXiv 2016
- [33]
-
[34]
Andrade U, Bengaly C A P, Santos B and Alcaniz J S 2018 Astrophys. J. 865 119 [arXiv:1806.06990]
work page internal anchor Pith review Pith/arXiv arXiv 2018
-
[35]
Clarkson C and Umeh O 2011 Class. Quant. Grav. 28 164010 [arXiv:1105.1886]
work page internal anchor Pith review Pith/arXiv arXiv 2011
- [36]
-
[37]
Clarkson C, Ellis G F R, Faltenbacher A, Maartens R, Umeh O and Uzan J P 2012 Mon. Not. Roy. Astron. Soc. 426 1121–1136 [arXiv:1109.2484]
work page internal anchor Pith review Pith/arXiv arXiv 2012
-
[38]
Adamek J, Clarkson C, Coates L, Durrer R and Kunz M 2019 Phys. Rev. D 100 021301 [arXiv:1812.04336]
work page internal anchor Pith review Pith/arXiv arXiv 2019
- [39]
-
[40]
Ellis G F R 1984 Fundam. Theor. Phys. 9 215–288
work page 1984
-
[41]
Ellis G F R and Stoeger W 1987 Class. Quant. Grav. 4 1697–1729
work page 1987
-
[42]
Green S R and Wald R M 2014 Class. Quant. Grav. 31 234003 [arXiv:1407.8084]
work page internal anchor Pith review Pith/arXiv arXiv 2014
-
[43]
Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?
Buchert T et al. 2015 Class. Quant. Grav. 32 215021 [arXiv:1505.07800]
work page internal anchor Pith review Pith/arXiv arXiv 2015
- [44]
-
[45]
Zalaletdinov R M 1992 Gen. Rel. Grav. 24 1015–1031
work page 1992
-
[46]
Zalaletdinov R M 1997 Bull. Astron. Soc. India 25 401–416 [arXiv:gr-qc/9703016]
work page internal anchor Pith review Pith/arXiv arXiv 1997
-
[47]
Gasperini M, Marozzi G and Veneziano G 2009 JCAP 03 011 [arXiv:0901.1303]
work page internal anchor Pith review Pith/arXiv arXiv 2009
-
[48]
Gasperini M, Marozzi G and Veneziano G 2010 JCAP 02 009 [arXiv:0912.3244]
work page internal anchor Pith review Pith/arXiv arXiv 2010
-
[49]
Korzynski M 2010 Class. Quant. Grav. 27 105015 [arXiv:0908.4593]
work page internal anchor Pith review Pith/arXiv arXiv 2010
-
[50]
Gasperini M, Marozzi G, Nugier F and Veneziano G 2011 JCAP 07 008 [arXiv:1104.1167]
work page internal anchor Pith review Pith/arXiv arXiv 2011
-
[51]
Buchert T 2000 Gen. Rel. Grav. 32 105–125 [arXiv:gr-qc/9906015]
work page internal anchor Pith review Pith/arXiv arXiv 2000
-
[52]
Buchert T 2001 Gen. Rel. Grav. 33 1381–1405 [arXiv:gr-qc/0102049]
work page internal anchor Pith review Pith/arXiv arXiv 2001
-
[53]
Buchert T and R¨ as¨ anen S 2012Ann. Rev. Nucl. Part. Sci. 62 57–79 [arXiv:1112.5335]
work page internal anchor Pith review Pith/arXiv arXiv
- [54]
- [55]
-
[56]
Bernal J L, Verde L and Riess A G 2016 JCAP 10 019 [arXiv:1607.05617]
work page internal anchor Pith review Pith/arXiv arXiv 2016
-
[57]
M¨ ortsell E and Dhawan S 2018JCAP 09 025 [arXiv:1801.07260] In the Realm of the Hubble tension − a Review of Solutions 114
work page internal anchor Pith review Pith/arXiv arXiv
-
[58]
Lin M X, Raveri M and Hu W 2019 Phys. Rev. D 99 043514 [arXiv:1810.02333]
work page internal anchor Pith review Pith/arXiv arXiv 2019
- [59]
-
[60]
Arendse N et al. 2020 Astron. Astrophys. 639 A57 [arXiv:1909.07986]
- [61]
- [62]
- [63]
- [64]
-
[65]
Evslin J, Sen A A and Ruchika 2018 Phys. Rev. D 97 103511 [arXiv:1711.01051]
work page internal anchor Pith review Pith/arXiv arXiv 2018
-
[66]
Aylor K, Joy M, Knox L, Millea M, Raghunathan S and Wu W L K 2019 Astrophys. J. 874 4 [arXiv:1811.00537]
work page internal anchor Pith review Pith/arXiv arXiv 2019
- [67]
- [68]
- [69]
-
[70]
Caldwell R R, Doran M, Mueller C M, Schafer G and Wetterich C 2003 Astrophys. J. Lett. 591 L75–L78 [arXiv:astro-ph/0302505]
work page internal anchor Pith review Pith/arXiv arXiv 2003
- [71]
-
[72]
Doran M, Karwan K and Wetterich C 2005 JCAP 11 007 [arXiv:astro-ph/0508132]
work page internal anchor Pith review Pith/arXiv arXiv 2005
-
[73]
Karwal T and Kamionkowski M 2016 Phys. Rev. D 94 103523 [arXiv:1608.01309]
work page internal anchor Pith review Pith/arXiv arXiv 2016
-
[74]
Pettorino V, Amendola L and Wetterich C 2013 Phys. Rev. D 87 083009 [arXiv:1301.5279]
work page internal anchor Pith review Pith/arXiv arXiv 2013
- [75]
- [76]
-
[77]
Fardon R, Nelson A E and Weiner N 2004 JCAP 10 005 [arXiv:astro-ph/0309800]
work page internal anchor Pith review Pith/arXiv arXiv 2004
-
[78]
Kaplan D B, Nelson A E and Weiner N 2004 Phys. Rev. Lett. 93 091801 [arXiv:hep-ph/0401099]
work page internal anchor Pith review Pith/arXiv arXiv 2004
-
[79]
Peccei R D 2005 Phys. Rev. D 71 023527 [arXiv:hep-ph/0411137]
work page internal anchor Pith review Pith/arXiv arXiv 2005
- [80]
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.