Constraint on the minimally extended varying speed of light using time dilations in type Ia supernovae

The Friedmann-Lema\^{i}tre-Robertson-Walker model establishes the correlation between redshifts and distances. It has a metric expansion of space. As a result, the wavelength of photons propagating through the expanding space is stretched, creating the cosmological redshift, $z$. It also relates the frequency of light detected by a local observer to that emitted from a distant source. In standard cosmology (\textit{i.e.}, a constant speed light model), this relation is given by a factor $1/(1+z)$. However, this ratio is modified in the minimally extended varying speed of light model (meVSL, $c = c_0 a^{b/4}$) as $1/(1+z)^{1-b/4}$. This time dilation effect is detected as the observed rate of the time variation in the intensity of emitted radiation. The spectra of type Ia supernovae (SNe Ia) provide a reliable way to measure the apparent aging rate of distant objects. We use data on 13 high-redshift ($0.28 \leq z \leq 0.62$) SNe Ia to obtain $b = 0.198 \pm 0.415$ at the $1$-$\sigma$ confidence interval. The current data is too sparse to give meaningful constrain on the meVSL and cannot distinguish the meVSL model from the standard model.