A new analysis has quantified a stringent limit, dubbed the "shape wall," on solutions to the Hubble tension that involve modifications to late-time cosmic expansion. The Hubble tension refers to the discrepancy between measurements of the universe's expansion rate (Hubble constant, H₀) obtained from the cosmic microwave background (CMB) and those derived from local observations, such as Type Ia supernovae. This study demonstrates that, even if new physics is allowed in the late universe, the shape of the expansion history, E(z) ≡ H(z)/H₀, severely restricts how much H₀ can increase.

Traditionally, solutions to the Hubble tension have faced a "normalization wall" due to Baryon Acoustic Oscillation (BAO) measurements, which constrain the product H₀r_d, where r_d is the sound horizon at baryon drag. However, this new work emphasizes that the shape of E(z) imposes an additional constraint. Researchers used a nonparametric approach, employing Gaussian Processes and the latest Type Ia supernovae (PantheonPlus SNeIa) and BAO (DESI DR2) data, to reconstruct E(z) and evaluate the impact of this "shape wall."

The results indicate that, if r_d and the acoustic angular scale θ_s are held fixed, a fractional increase in H₀ must be matched by a similar fractional increase in the dimensionless distance integral I ≡ ∫ dz/E(z). In their most conservative analysis, they found that the maximum fractional increase allowed in H₀ is less than 2%. This contrasts significantly with the approximately 8% increase required to fully resolve the Hubble tension. This shape wall persists even when considering early-time new physics, implying that late-time modifications to E(z) have very limited scope to alleviate the tension, either alone or in combination with early-time new physics.