Which Model Wins the H₀ World Cup? Ranking 14 Hubble Tension Solutions Against ΛCDM

The 7.1σ Crisis and the World Cup Framework

1. The 7.1σ Discrepancy

   The escalating conflict over the Hubble constant is no longer a mere statistical fluctuation; it is a structural fracture in the standard cosmological model. Local universe measurements, primarily driven by the SH0ES collaboration using Cepheid-calibrated Type Ia supernovae, have consistently refined their precision, arriving at an expansion rate of 73.50 ± 0.81 km/s/Mpc. Conversely, the Planck satellite's mapping of the cosmic microwave background (CMB), when processed through the ΛCDM framework, demands a significantly lower value of 67.19 ± 0.38 km/s/Mpc. The resultant 7.1σ gap implies that the probability of this tension arising from random chance is infinitesimally small. Resolving this requires either an undiscovered, correlated systematic error across multiple independent local probes or, more intriguingly, new physics. The 2026 H₀ World Cup evaluates theoretical models designed to inject this new physics, assessing whether they can artificially raise the CMB-inferred Hubble constant without destroying the exquisite fit to the acoustic peaks of the early universe.

2. The Statistical Gauntlet

   To prevent theoretical overfitting, the H₀ World Cup deploys a merciless statistical framework. Evaluating 14 models requires more than simply checking if a model can achieve a higher H₀; it demands penalizing models for unwarranted complexity. The tournament relies on three primary scoring metrics: the Bayes factor (ln BF), the Akaike Information Criterion (ΔAIC), and the change in the maximum a posteriori fit (ΔDMAP). In our comparative evaluation matrix, a clear hierarchy emerges. Early Dark Energy and Early Modified Gravity both clear the strict thresholds of ln BF > 3 and −ΔAIC > 10, suffering only minor ΔDMAP penalties, proving their parameters are statistically justified. Varying Electron Mass sits in the middle, failing to break ln BF > 3 and taking heavy ΔDMAP penalties. Finally, Extra Radiation and Late-Time models score at the absolute bottom with negative Bayes factors and massive ΔAIC penalties. This matrix ensures that any model that dramatically improves the H₀ fit but degrades the fit to CMB polarization is severely penalized.

The Heavyweights: Early Dark Energy and Early Modified Gravity

1. Early Dark Energy

   Early Dark Energy (EDE) enters the World Cup as the undisputed crowd favorite. The premise of EDE is the introduction of a scalar field that acts as a temporary cosmological constant in the pre-recombination era, peaking in energy density near matter-radiation equality before rapidly diluting away. By injecting this extra energy, EDE decreases the physical size of the sound horizon, which acts as the fundamental standard ruler of the CMB. To keep the angular size of the sound horizon consistent with Planck observations, the model must compensate with a higher Hubble constant today. The 2026 analysis confirms that EDE successfully elevates H₀ to approximately 70 km/s/Mpc. While it does not fully bridge the gap to 73.50, it reduces the residual tension to a much more palatable 2.5–3.6σ. Crucially, EDE passes the stringent −ΔAIC > 10 threshold, proving that its physical mechanism is mathematically justified despite the addition of three new parameters.

2. Early Modified Gravity

   Closely tailing EDE in the rankings is Early Modified Gravity. Rather than introducing a transient scalar field, this class of models alters the fundamental laws of gravity in the early universe, typically by modifying the effective gravitational constant prior to recombination. The phenomenological outcome is remarkably similar to EDE: an altered expansion history that shrinks the sound horizon and subsequently forces a higher inferred H₀. In the World Cup scoring matrix, Early Modified Gravity performs exceptionally well, matching EDE's ability to reach H₀ ≈ 70 km/s/Mpc and leaving a similar 2.5–3.6σ residual tension. The Bayesian evidence heavily favors this model over baseline ΛCDM, though it faces slight penalties in the ΔDMAP metric due to subtle phase shifts in the CMB temperature anisotropy peaks. Nevertheless, alongside EDE, Early Modified Gravity stands out as one of the few theoretical frameworks capable of structurally surviving the full suite of cosmological data.

The Midfielders: Varying Electron Mass and Modified Recombination

1. Varying Electron Mass

   Occupying the middle of the pack is the varying electron mass model. This framework posits that the fundamental mass of the electron was slightly higher in the early universe. A heavier electron alters the binding energy of hydrogen, which in turn causes the epoch of recombination to occur earlier, at a higher redshift. An earlier recombination directly reduces the size of the sound horizon. Unlike EDE, which alters the expansion rate, varying the electron mass alters the atomic physics of the early universe. The World Cup results classify this model as an intermediate performer. While it successfully raises H₀ without requiring transient dark energy, it struggles with the high-ell polarization data from the CMB. A changing electron mass fundamentally alters the thickness of the last scattering surface, leading to enhanced damping that clashes with precision measurements, preventing it from achieving the top-tier Bayes factors seen in EDE.

2. Other Modified Recombination Schemes

   Beyond varying the electron mass, the World Cup evaluates a broader suite of modified recombination models, including primordial magnetic fields and non-standard recombination histories. These models attempt to inject energy or alter the ionization fraction to force an earlier decoupling of photons and baryons. However, the 2026 analysis reveals severe limitations within this tactical group. While they offer marginal improvements in H₀, their performance in the −ΔAIC > 10 metric is largely unimpressive. Primordial magnetic fields, for instance, induce small-scale clumping in the baryon fluid, which accelerates recombination but introduces spectral distortions that are tightly constrained by existing data. Ultimately, modified recombination schemes fail to cross the finish line; they are sufficiently penalized by the ΔDMAP evaluations that they cannot be considered complete solutions to the 7.1σ crisis, relegating them to the status of theoretically interesting but phenomenologically insufficient.

The Relegated: Extra Radiation and Late-Time Solutions

1. Extra Radiation Models

   For years, adding extra relativistic species—often parameterized as N_eff—was considered the most natural way to increase the pre-recombination expansion rate. Extra radiation, such as sterile neutrinos or thermal axions, increases the early Hubble rate, effectively shrinking the sound horizon. However, the H₀ World Cup decisively eliminates extra radiation from the viable pool of contenders. The fatal flaw of N_eff models is their profound impact on the CMB damping tail. Extra radiation increases the Silk damping scale relative to the sound horizon, a signature that is strictly forbidden by the pristine high-resolution data from the Planck satellite. In the 2026 scoring, extra radiation models fail universally; they yield negative Bayes factors and completely fail the ΔAIC criteria. The observational data clearly dictates that whatever new physics is introduced, it cannot behave like a simple free-streaming relativistic particle.

2. Late-Time Interventions

   If modifying the early universe is fraught with complications, why not alter the late universe? Late-time models attempt to solve the tension by modifying dark energy or gravity at low redshifts (typically z < 2), attempting to bend the expansion history to intersect the SH0ES value without touching the CMB sound horizon. The World Cup analysis demonstrates that late-time models are phenomenologically bankrupt. The primary executioner of late-time theories is the baryon acoustic oscillation (BAO) data, particularly from the recent DESI survey, combined with Type Ia supernovae catalogs. These low-redshift anchors rigorously map the expansion history from z = 0.1 to z = 2.0. Any late-time model that tries to violently accelerate the universe to reach H₀ = 73.50 violently clashes with the rigid BAO scaffolding. As a result, late-time models score at the absolute bottom of the 14-contender bracket.

The Ultimate Referees: Robustness and New Data

1. High-Resolution CMB Data

   A model's survival in the World Cup is not based solely on Planck data; it must withstand the relentless scrutiny of independent, ground-based, high-resolution CMB experiments. The Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT-3G) serve as the ultimate referees in this tournament. These observatories probe the CMB at smaller angular scales (higher multipoles) and with greater polarization sensitivity than Planck. The 2026 analysis by Schöneberg et al. shows that EDE and Early Modified Gravity remain remarkably robust when subjected to ACT and SPT-3G data. However, the inclusion of this high-resolution data severely punishes the intermediate models. Varying electron mass and primordial magnetic field models exhibit notable tension with the SPT-3G TE and EE polarization power spectra. This robustness testing is what ultimately separates the true champions from the statistical flukes, ensuring physical consistency across all observable scales.

2. DESI BAO and BBN Constraints

   The final hurdle for the World Cup contenders involves Big Bang Nucleosynthesis (BBN) and the latest Dark Energy Spectroscopic Instrument (DESI) BAO measurements. BBN provides a strict anchor on the baryon density and the expansion rate at T ≈ 1 MeV. Any model injecting new physics too early risks altering the primordial abundances of helium and deuterium. Early Dark Energy gracefully passes this test because its scalar field remains frozen and dynamically irrelevant during the BBN epoch. Meanwhile, DESI BAO data acts as an immovable wall for the late-time expansion history. DESI's sub-percent precision on the distance scale across multiple redshift bins ensures that the geometry of the late universe is strictly locked. The World Cup demonstrates that only models which modify the early universe sound horizon—while leaving the late-universe ΛCDM expansion history perfectly intact—can simultaneously satisfy DESI, BBN, and SH0ES.

Conclusion