Circular reasoning: Solving the Hubble tension with a non-$\pi$ value of $\pi$

Abstract

Recently, cosmology has seen a surge in alternative models that purport to solve the discrepancy between the values of the Hubble constant $H_0$ as measured by cosmological microwave background anisotropies and local supernovae, respectively. In particular, many of the most successful approaches have involved varying fundamental constants, such as an alternative value of the fine structure constant and time-varying values of the electron mass, the latter of which showed particular promise as the strongest candidate in several earlier studies. Inspired by these approaches, in this paper, we investigate a cosmological model where the value of the geometric constant $\pi$ is taken to be a free model parameter. Using the latest CMB data from Planck as well as baryon-acoustic oscillation data, we constrain the parameters of the model and find a strong correlation between $\pi$ and $H_0$, with the final constraint $H_0 = 71.3 \pm 1.1 \ \mathrm{ km/s/Mpc}$, equivalent to a mere $1.5\sigma$ discrepancy with the value measured by the SH0ES collaboration. Furthermore, our results show that $\pi = 3.206 \pm 0.038$ at $95 \%$ C.L., which is in good agreement with several external measurements discussed in the paper. Hence, we conclude that the $\pi \Lambda$CDM model presented in this paper, which has only a single extra parameter, currently stands as the perhaps strongest solution to the Hubble tension.