Measurements of the Hubble constant, and more generally measurements of the expansion rate and distances over the interval 0 < z < 1, appear to be inconsistent with the predictions of the standard cosmological model (ΛCDM) given observations of cosmic microwave background temperature and polarization anisotropies. Here we consider a variety of types of departures from ΛCDM that could, in principle, restore concordance among these datasets, and we explain why we find almost all of them unlikely to be successful. We single out the set of solutions that increase the expansion rate in the decade of scale factor expansion just prior to recombination as the least unlikely. These solutions are themselves tightly constrained by their impact on photon diffusion and on the gravitational driving of acoustic oscillations of the modes that begin oscillating during this epoch -modes that project on to angular scales that are very well measured. We point out that a general feature of such solutions is a residual to fits to ΛCDM, like the one observed in Planck power spectra. This residual drives the modestly significant inferences of angular-scale dependence to the matter density and anomalously high lensing power, puzzling aspects of a data set that is otherwise extremely well fit by ΛCDM.
I. INTRODUCTIONEstimates of the Hubble constant from a distance ladder approach are generally higher than those derived from cosmic microwave background (CMB) data, assuming the standard "ΛCDM" cosmological model [2]. The SH 0 ES team calibrates a supernova sample with Cepheids and finds H 0 = 74.03 ± 1.42 km/s/Mpc [3, hereafter R19].Compared with the value inferred from Planck CMB temperature and polarization power spectra plus CMB lensing, assuming ΛCDM, H 0 = 67.27 ± 0.60 km/s/Mpc, there is a 4.4 σ discrepancy. The most recent result from strong-lensing time delays, from the H 0 LiCoW team [4,5], assuming the standard cosmological model and a prior of Ω m ∈ [0.05, 0.5], of H 0 = 76.8 ± 2.5 km/s/Mpc is consistent with R19 and discrepant with the ΛCDM Planck value at 3.1 σ. The Carnegie-Chicago Hubble Project have used their own Hubble flow set of supernovae that they have calibrated with the tip of the red giant branch method. They find H 0 = 69.8 ± 0.8(stat) ± 1.7(sys) km/s/Mpc [6], which at the 2 σ level is consistent with all of these results.Bernal et al. [7] showed that in addition to a discrepancy in the Hubble constant, there is a discrepancy in the comoving sound horizon at the end of the baryon drag epoch, r drag s . They used Cepheid-calibrated supernovae [8,9] to infer distances out to redshifts with precise measurements of the baryon acoustic oscillation feature [10][11][12]. With these distances, they could convert the BAO angles to inferences of r drag s . Using supernova data to control the shape of D(z) they obtained relatively model-independent inferences of this empirically- * with apologies to Gunion et al. [1]