A convenient parametrisation of the SNeIa rate is [255, 256]
where [t.sub.min] is a suitably chosen minimum timescale for the occurrence of SNeIa (typically 30 Myr), [A.sub.Ia] is a normalisation constant, and D is the so-called delay time distribution (DTD), that is, the distribution of time intervals between the birth of the progenitor system (usually a binary system made of two intermediate-mass stars) and the SNIa explosion.
The procedures that the above four models commonly follow in the explanation of the SNeIa measurements include the following four steps: (1) Modifying the FE with an appropriate input of [LAMBDA], scalar field, perturbation, or external energy; (2) Determining the expansion rates (Hubble parameter) of the universe according to their modified FEs; (3) Submitting their expansion rates into the [D.sub.L] - Z relation; (4) Comparing the obtained redshift dependence of their luminosity distances with the SNeIa measurements.
This currently most accepted hypothesis for the standard cosmological model to quantitatively explain the measurements of distant type Ia supernovae strongly relies on the [D.sub.L] - Z relation that is used to bridge the measured SNeIa data and the theoretical model of cosmology.
In the upper panel of Figure 3, the distance modulus, which is defined by [mu] = 5 [log.sub.10] [D.sub.L] - 5 with [D.sub.L] in parsecs, is plotted as a function of redshift; while in the lower panel of Figure 3, the distance modulus difference between the measured SNeIa data and analytical results derived fromEq.