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The enduring tension between local and distant measurements H0 remains unresolved. It was recently pointed out that cosmic microwave background (CMB) and large-scale structure (LSS) observables are invariant under a uniform rescaling of the gravitational free-fall rates of all species present and the Thomson scattering rate between photons and electrons. We show that a unique variation of the fine-structure constant α and the electron mass m_e can leverage this scaling transformation to reconcile the CMB and LSS data with a broad spectrum of Hubble constant values, encompassing those inferred from local measurements. Importantly, this study demonstrates that the constraints on the variation of fundamental constants imposed by the specific recombination history are not as stringent as previously assumed. Our work highlights the critical role of the Thomson scattering rate in the existing Hubble tension and offers a distinct avenue of exploration for particle model builders. 

Abstract The Hubble-Lemaître tension is currently one of the most important questions in cosmology. Most of the focus so far has been on reconciling the Hubble constant value inferred from detailed cosmic microwave background measurement with that from the local distance ladder. This emphasis on one number — namely H 0 — misses the fact that the tension fundamentally arises from disagreements of distance measurements. To be successful, a proposed cosmological model must accurately fit these distances rather than simply infer a given value of H 0 .Using the newly developed likelihood package ` distanceladder ', which integrates the local distance ladder into MontePython , we show that focusing on H 0 at the expense of distances can lead to the spurious detection of new physics in models which change late-time cosmology. As such, we encourage the observational cosmology community to make their actual distance measurements broadly available to model builders instead of simply quoting their derived Hubble constant values. 

We present here a lightning review of the status of the Hubble-Lemaître tension. Instead of discussing the broad array of proposed solutions found in the literature, we focus here on the assumptions made to measure the Hubble constant from cosmic microwave background and baryon acoustic oscillation data on the one hand, and from a cepheid-calibrated distance ladder on the other hand. From this discussion, we extract two important lessons that inform which kind of physics-based solutions could plausibly resolve this tension.