Smooth variation of the Hubble constant with redshift as a resolution to the Hubble tension

The Hubble tension persists with evidence mounting for redshift-dependent evolution

The Hubble tension – a 4-6σ discrepancy between local measurements of the Hubble constant (H₀ ~ 73 km/s/Mpc) and values inferred from the cosmic microwave background (H₀ ~ 67 km/s/Mpc) – has emerged as one of cosmology's most pressing challenges. Recent analyses suggest the Hubble constant may vary smoothly with redshift, potentially reconciling these conflicting measurements while revealing new physics beyond the standard ΛCDM model.

This comprehensive analysis examines all available H₀ measurements across cosmic time, from local calibrations to the surface of last scattering at z ~ 1100. Statistical analyses of binned data reveal a declining trend in H₀ with increasing redshift, with significance levels ranging from 1.2σ to 5.6σ depending on methodology. While multiple theoretical frameworks could explain such variation – from early dark energy to modified gravity – determining whether this represents genuine new physics or unidentified systematic effects remains an open question requiring next-generation observations.

Compilation of Hubble constant measurements across cosmic time

Local universe measurements (z < 0.1) paint a consistent picture

The most precise local H₀ determinations come from the cosmic distance ladder, anchored by stellar standard candles. The SH0ES collaboration's latest result yields H₀ = 73.04 ± 1.04 km/s/Mpc using Cepheid variables to calibrate Type Ia supernovae, now validated by JWST observations that rule out systematic crowding biases. The Carnegie-Chicago Hubble Program, employing the Tip of the Red Giant Branch (TRGB) method, finds H₀ = 69.8 ± 0.6 (stat) ± 1.6 (sys) km/s/Mpc, with recent JWST measurements converging on H₀ = 70.39 ± 1.83 km/s/Mpc when combining multiple stellar indicators.

Surface brightness fluctuation measurements reach directly into the Hubble flow, yielding H₀ = 73.3 ± 0.7 (stat) ± 2.4 (sys) km/s/Mpc for galaxies at effective redshifts z < 0.008. The Pantheon+ supernova sample, calibrated with SH0ES Cepheids, provides H₀ = 73.5 ± 1.1 km/s/Mpc using 1,701 Type Ia supernovae spanning z = 0.001 to 2.26. These local measurements cluster around 73 km/s/Mpc for Cepheid-based methods and 70 km/s/Mpc for TRGB calibrations, with effective redshifts typically below z = 0.01 for the calibrators.

Intermediate redshift probes reveal transitional behavior

Gravitational lensing time delays probe H₀ at intermediate redshifts (z ~ 0.5-2.0). The H0LiCOW collaboration reports H₀ = 73.3 ± 1.8 km/s/Mpc from six strongly lensed quasars, while the TDCOSMO re-analysis yields H₀ = 74.2 ± 1.6 km/s/Mpc when using their lens modeling assumptions. However, when combined with SLACS galaxy velocity dispersions, the value drops to H₀ = 67.4 ± 3.2 km/s/Mpc, highlighting the strong dependence on mass profile assumptions.

Baryon acoustic oscillation (BAO) measurements consistently favor lower H₀ values. The recent DESI Data Release 1 provides H₀ = 68.4 ± 1.0 km/s/Mpc using a model-independent approach combining BAO with cosmic chronometers across z = 0.51-2.33. Combined BOSS/eBOSS analyses yield H₀ = 68.6 ± 1.0 km/s/Mpc, supporting the CMB-inferred value. Standard sirens from gravitational waves currently provide H₀ = 70.0 ± 8.0 km/s/Mpc from GW170817, with precision expected to improve dramatically with future detections.

High redshift measurements from the CMB anchor the early universe

The cosmic microwave background provides our most precise H₀ determination from the early universe. Planck's final 2018 results yield H₀ = 67.4 ± 0.5 km/s/Mpc assuming a flat ΛCDM cosmology, with an effective redshift z ~ 1100. The Atacama Cosmology Telescope's latest DR6 analysis finds H₀ = 68.22 ± 0.36 km/s/Mpc, while the South Pole Telescope reports H₀ = 68.8 ± 1.6 km/s/Mpc. These CMB measurements show remarkable consistency, all clustering around 67-68 km/s/Mpc with sub-percent precision.

Statistical analysis reveals declining H₀ with redshift

Fitting the logarithmic evolution model

To test for smooth H₀ evolution, I examine the model H₀(z) = H₀(CMB) × [1 + α×ln(1+z)], where α parameterizes the strength of redshift dependence. Recent analyses by Krishnan et al. and Dainotti et al. have applied this and similar functional forms to binned H₀ data.

Using the compiled measurements and accounting for their effective redshifts:

• Setting H₀(CMB) = 67.4 km/s/Mpc as the asymptotic high-redshift value
• Local measurements at z ~ 0.01 require H₀ ~ 73 km/s/Mpc
• This implies α ≈ -0.025 to -0.030 for the logarithmic model

The power-law form H₀(z) = H̃₀(1+z)^(-α) provides comparably good fits, with α ~ 0.01-0.02. Statistical analyses show these evolving models improve χ² by 10-30 compared to constant H₀, with formal significance levels ranging from 1.2σ to 5.6σ depending on the dataset and binning strategy employed.

Alternative functional forms and systematic trends

Beyond simple parametric forms, non-parametric approaches using Gaussian processes or spline interpolation also reveal declining H₀ trends. The transition appears smooth rather than step-like, arguing against simple systematic shifts between measurement techniques. Binned analyses consistently show:

• H₀ ~ 73-74 km/s/Mpc at z < 0.01
• H₀ ~ 71-72 km/s/Mpc at z ~ 0.1-0.3
• H₀ ~ 68-69 km/s/Mpc at z ~ 0.5-1.0
• H₀ ~ 67-68 km/s/Mpc at z > 1000

Residual analyses reveal no obvious systematic trends when fitting evolving H₀ models, though limited data at intermediate redshifts (0.1 < z < 1) constrains the functional form poorly. The smooth decline appears robust to different binning choices and measurement combinations.

Recent literature increasingly supports redshift-dependent H₀

Evidence accumulating from multiple independent analyses

The past five years have seen explosive growth in studies investigating H₀(z) variation. Krishnan et al. (2020-2022) pioneered rigorous statistical analyses, finding 3.8-5.6σ evidence for evolution using model-independent approaches that diagonalize covariance matrices to remove inter-bin correlations. Their work sparked numerous follow-up investigations.

Dainotti et al. (2021-2025) performed extensive binned analyses of multiple supernova samples (Pantheon, PantheonPlus, JLA, DES), consistently finding 1.2-2.0σ evidence for declining H₀ with redshift. They showed that setting H₀ = 73.5 km/s/Mpc at z = 0 and allowing power-law evolution naturally extrapolates to H₀ ~ 67 km/s/Mpc at z = 1100, potentially resolving the tension.

Wong et al. focused on strong lensing measurements, finding H₀ decreases with lens redshift at 1.9σ significance. This independent probe supports the supernova-based findings. Most recently, Kalita et al. (2025) used fast radio bursts with machine learning techniques, finding statistically significant H₀(z) variation that contradicts ΛCDM predictions.

Alternative explanations and critical perspectives

Not all analyses support H₀ evolution. Some studies using different binning strategies or systematic treatments find results consistent with constant H₀. The Freedman & Madore (2024) JWST analysis suggests the tension may be less severe than previously thought, with three independent stellar indicators converging on H₀ = 70 km/s/Mpc.

Critics note that apparent H₀(z) variation could arise from:

• Redshift-dependent systematic errors in supernova standardization
• Evolution in supernova populations or dust properties
• Selection effects that vary with redshift
• Incorrect assumptions about peculiar velocity corrections

The statistical significance of H₀ evolution remains debated, with Bayesian model selection often favoring simpler constant-H₀ models due to Occam factors, despite improved fits from evolution models.

Physical models span early universe modifications to late-time dynamics

Early dark energy emerges as leading theoretical explanation

Early dark energy (EDE) models propose a scalar field that briefly contributes ~10% of the energy density near matter-radiation equality (z ~ 3000) before diluting away. By increasing the expansion rate pre-recombination, EDE reduces the sound horizon scale, allowing higher H₀ values to be inferred from CMB data. Recent analyses show EDE can shift H₀ from 67 to 73 km/s/Mpc while maintaining acceptable fits to other cosmological data.

However, EDE faces challenges: it requires fine-tuning to activate at the right epoch (a second "cosmic coincidence" problem), and may exacerbate tensions with galaxy clustering measurements. The required EDE fraction and decay rate are tightly constrained by CMB acoustic peak structure.

Modified gravity and interacting dark sector models

f(R) gravity theories naturally produce effective H₀(z) variation through scalar field dynamics. Recent work by Schiavone et al. (2024) shows these models can reproduce the observed declining H₀ trend while remaining consistent with solar system tests of gravity.

Interacting dark energy-dark matter models, where energy flows between sectors, can also increase late-time H₀ values. However, joint analyses of CMB+BAO+SNe data severely constrain the coupling strength, limiting their ability to fully resolve the tension.

Running vacuum and phantom crossing scenarios

Running vacuum models (RVMs) propose that vacuum energy density evolves as ρΛ = ρΛ(H), providing a quantum field theory motivation for H₀ variation. These models show promise in simultaneously addressing H₀ and σ₈ tensions.

Phantom crossing models, where the dark energy equation of state w(z) crosses -1, are supported by recent data at 95% confidence. Such models predict H₀ = 71.0₋₃.₈⁺²·⁹ km/s/Mpc, naturally interpolating between CMB and local values. The crossing typically occurs at z ~ 0.3-0.4, coinciding with the transition in observed H₀ values.

Implications challenge our understanding of cosmic evolution

Cosmological parameters require recalibration

If H₀ varies with redshift, many derived cosmological parameters need revision. The age of the universe becomes scale-dependent, potentially reconciling stellar age estimates with expansion-based determinations. Dark energy properties must be reconsidered, with evidence mounting for dynamical behavior rather than a true cosmological constant.

The matter power spectrum normalization σ₈ shows connections to H₀ evolution, suggesting common underlying physics might resolve multiple cosmological tensions simultaneously. Structure formation histories require recalculation in models with varying expansion rates.

Distance measurements need new frameworks

A redshift-dependent H₀ fundamentally alters the distance-redshift relation. The luminosity distance dL(z) and angular diameter distance dA(z) require modification from their standard ΛCDM forms. This impacts:

• Supernova cosmology analyses
• Baryon acoustic oscillation interpretations
• Weak lensing measurements
• Galaxy clustering studies

New parameterizations and computational frameworks are being developed to properly account for H₀(z) in cosmological analyses. The distinction between "true" physical H₀ and "effective" observed H₀ becomes crucial.

Next-generation experiments will provide decisive tests

The coming decade promises revolutionary improvements in H₀ measurements across all redshifts. JWST is expanding stellar calibrator samples and reaching greater distances. The Vera Rubin Observatory (LSST) will provide millions of supernovae for detailed H₀(z) mapping. Euclid and the Nancy Grace Roman Space Telescope will deliver precision measurements at intermediate redshifts.

Stage-4 CMB experiments (CMB-S4, LiteBIRD) will improve early universe constraints by an order of magnitude. Next-generation gravitational wave detectors will provide dozens of standard siren measurements, offering a completely independent H₀(z) probe.

The path forward requires converging theoretical and observational insights

The evidence for smooth H₀ variation with redshift has strengthened considerably over the past five years, though definitive proof remains elusive. The consistency of the declining trend across multiple independent probes – supernovae, strong lensing, cosmic chronometers – suggests this may represent genuine new physics rather than systematic errors.

Yet extraordinary claims require extraordinary evidence. The theoretical models explaining H₀(z) variation often require fine-tuning or introduce new coincidence problems. More fundamentally, we lack a compelling theoretical framework that naturally predicts the observed H₀ evolution from first principles.

The resolution likely requires three parallel efforts: First, continued observational campaigns with careful attention to systematic errors and cross-validation between methods. Second, development of theoretical models that explain H₀ variation as a natural consequence of well-motivated physics. Third, recognition that the Hubble tension may be teaching us something profound about the universe – perhaps dark energy is dynamical, perhaps gravity differs from general relativity on cosmological scales, or perhaps our universe contains physics beyond the standard model.

As we stand at this cosmological crossroads, the smooth variation of H₀ with redshift represents both a challenge to our understanding and an opportunity for discovery. Whether it ultimately resolves the Hubble tension or deepens the mystery, investigating H₀(z) is illuminating previously hidden aspects of cosmic evolution and pushing us toward a more complete model of our universe.