Constraints on the time variation of $c$ and $G$ from Dark Energy Survey Snpernova Program

In the context of the dispersion relation $c = \lambda \nu$ and considering an expanding universe where the observed wavelength today is redshifted from the emitted wavelength by $\lambda_{0} = \lambda_{\text{emit}} (1+z)$, to keep $c$ constant, it must be that $\nu_{0} = \nu_{\text{emit}} /(1+z)$. However, although the theory for wavelength in the RW metric includes the cosmological redshift, the same is not simply deduced for frequency (the inverse of time). Instead, cosmological time dilation $T_{0} = T_{\text{emit}} (1+z)$ is an additional assumption made to uphold the hypothesis of constant speed of light rather than a relation directly derived from the RW metric. Therefore, verifying cosmological time dilation observationally is crucial. The most recent data employing supernovae for this purpose was released recently by the Dark Energy Survey. Results from the i-band specifically support variations in the speed of light within 1-$\sigma$. We used these observations to investigate variations in various physical quantities, including $c$ and $G$, using the minimally extended varying speed of light model. The speed of light was $0.4$\% to $2.2$\% slower, and Newton's constant may have decreased by $1.7$\% to $8.4$\% compared to their current values at redshift $2$. These findings, consistent with previous studies, hint at resolving tensions between different $\Lambda$CDM cosmological backgrounds but are not yet conclusive evidence of a varying speed of light, as the full-band data aligns with standard model cosmology.