The trouble with $H_0$

We perform a comprehensive cosmological study of the $H_0$ tension between the direct local measurement and the model-dependent value inferred from the Cosmic Microwave Background. With the recent measurement of $H_0$ this tension has raised to more than $3\sigma$. We consider changes in the early time physics without modifying the late time cosmology. We also reconstruct the late time expansion history in a model independent way with minimal assumptions using distances measures from Baryon Acoustic Oscillations and Type Ia Supernovae, finding that at $z<0.6$ the recovered shape of the expansion history is less than 5 % different than that of a standard LCDM model. These probes also provide a model insensitive constraint on the low-redshift standard ruler, measuring directly the combination $r_s h$ where $H_0=h \times 100$ km/s/Mpc and $r_s$ is the sound horizon at radiation drag (the standard ruler), traditionally constrained by CMB observations. Thus $r_s$ and $H_0$ provide absolute scales for distance measurements (anchors) at opposite ends of the observable Universe. We calibrate the cosmic distance ladder and obtain a model-independent determination of the standard ruler for acoustic scale, $r_s$. The tension in $H_0$ reflects a mismatch between our determination of $r_s$ and its standard, CMB-inferred value. Without including high-l Planck CMB polarization data (i.e., only considering the "recommended baseline" low-l polarisation and temperature and the high l temperature data), a modification of the early-time physics to include a component of dark radiation with an effective number of species around 0.4 would reconcile the CMB-inferred constraints, and the local $H_0$ and standard ruler determinations. The inclusion of the "preliminary" high-l Planck CMB polarisation data disfavours this solution.