Late-transition vs smooth $H(z)$ deformation models for the resolution
  of the Hubble crisis
  Gravitational transitions at low redshifts ($z_t<0.1$) have been recently
proposed as a solution to the Hubble and growth tensions. Such transitions
would naturally lead to a transition in the absolute magnitude $M$ of type Ia
supernovae (SnIa) at $z_t$ (Late $M$ Transitions - $LMT$) and possibly in the
dark energy equation of state parameter $w$ (Late $w-M$ Transitions - $LwMT$).
Here, we compare the quality of fit to cosmological data of this class of
models, with the corresponding quality of fit of the cosmological constant
model ($\Lambda$CDM) and some of the best smooth $H(z)$ deformation models
($w$CDM, CPL, PEDE). We also perform model selection via the Akaike Information
Criterion and the Bayes factor. We use the full CMB temperature anisotropy
spectrum data, the baryon acoustic oscillations (BAO) data, the Pantheon SnIa
data, the SnIa absolute magnitude $M$ as determined by Cepheid calibrators and
the value of the Hubble constant $H_0$ as determined by local SnIa calibrated
using Cepheids. We find that smooth $H(z)$ deformation models perform worse
than transition models for the following reasons: 1) They have a worse fit to
low-$z$ geometric probes (BAO and SnIa data); 2) They favor values of the SnIa
absolute magnitude $M$ that are lower as compared to the value $M_c$ obtained
with local Cepheid calibrators at $z<0.01$; 3) They tend to worsen the
$\Omega_\mathrm{m,0}-\sigma_\mathrm{8,0}$ growth tension. We also find that the
$w-M$ transition model ($LwMT$) does not provide a better quality of fit to
cosmological data than a pure $M$ transition model ($LMT$) where $w$ is fixed
to the \lcdm value $w=-1$ at all redshifts. We conclude that the $LMT$ model
has significant statistical advantages over smooth late-time $H(z)$ deformation
models in addressing the Hubble crisis.
