$\Lambda$CDM is alive and well

The $\Lambda$CDM model faces several tensions with recent cosmological data and their increased accuracy. The mismatch between the values of the Hubble constant $H_0$ obtained from direct distance ladder measurements and from the cosmic microwave background (CMB) is the most statistically significant, but the amplitude of the matter fluctuations is also regarded as a serious concern, leading to the investigation of a plethora of models. We first show that the combination of several recent measurements from local probes leads to a tight constraint on the present-day matter density $\Omega_M$ as well as on the amplitude of the matter fluctuations, both acceptably consistent with the values inferred from the CMB. Secondly, we show that the data on cosmic chronometers allow to derive an accurate value of the Hubble constant $H_0$ for $\Lambda$CDM models: $H_0 = 67.4 \pm 1.34$ km/s/Mpc. This implies that, within $\Lambda$CDM, some determinations of $H_0$ are biased. Considering a bias on the Hubble constant as a nuisance parameter within $\Lambda$CDM, we examine such a $\Lambda$CDM$+ H_0$ bias model on the same statistical grounds as alternative cosmological models. We show that the former statistically supersede most existing extended models proposed up to now. In a third step, we show that the value of $\Omega_M$ we obtained, combined with $H_0$ from SH0ES, leads to an accurate measurement of $\omega_M$, providing an additional low-redshift test for cosmological models. From this test, most extensions seem to be confronted with a new tension, whereas the $\Lambda$CDM with $H_0 \sim 67$ has none. We conclude that a standard $\Lambda$CDM model with an unknown bias in the Cepheids distance calibration represents a model that reaches a remarkable agreement, statistically better than previously proposed extensions with $H_0 \sim 73$ for which such a comparison can be performed. (abridged)