Revisiting the Hubble–Lemaître law with Pantheon+ via Gaussian process

In this paper, we reconstruct the observable [Formula: see text], which corresponds to 1 when the Hubble–Lemaître law (HLL) holds up, from the recently released Pantheon+ supernovae Ia samples via the Gaussian process. In order to test the validity of the HLL, or sort out the redshift range where the HLL holds up, we propose a Kullback–Leibler (KL) divergence like discrete tension [Formula: see text]. We divide the whole redshifts ranging in [Formula: see text] into 10 bins with an equal logarithmic space, and calculate the ratio between the number of data points with value [Formula: see text] and the total number of data points in each bins. If the consistence criterion is that above [Formula: see text] data points contained in a redshift range are consistent, one can conclude that the HLL holds up when the redshift is roughly about [Formula: see text].