This short review looks at one of cosmology’s persistent puzzles: different methods give different values for the Hubble constant, H0, which measures how fast the Universe is expanding today. Cosmic microwave background (CMB) measurements interpreted in the standard Λ cold dark matter (ΛCDM) model give a lower value than local, low-redshift methods that use astronomical distance ladders. The authors survey the theoretical ideas that try to reconcile those numbers.

The paper examines changes that could occur at different steps in the cosmological inference chain. That chain begins with the basic gravitational equations and runs through the pre-recombination era (when electrons and protons combined and the CMB was released) to the late-time relation between distance and redshift. The authors focus mostly on modified gravity theories — changes to how gravity works on large scales — and also review early-time proposals, like early dark energy (EDE), and late-time proposals that alter distances or the dark sector.

A central technical problem is the “acoustic ruler” set before recombination. The CMB fixes an angular scale, θs, which is the ratio of the sound horizon (the distance sound waves traveled in the early plasma) to the angular-diameter distance to the last-scattering surface. To raise H0 while keeping that observed angle requires coordinated shifts in the physical matter and baryon densities and the early expansion history. Planck 2018 data gives H0 = 67.36 ± 0.54 km s−1 Mpc−1 in ΛCDM, while the SH0ES distance-ladder program finds a local value near 73 km s−1 Mpc−1. Other experiments such as ACT (Atacama Cosmology Telescope), SPT-3G (South Pole Telescope third-generation), and BAO (baryon acoustic oscillation) measurements from surveys like DESI (Dark Energy Spectroscopic Instrument) provide independent checks and add more constraints.