A Stochastic Approach to Reconstructing the Speed of Light in Cosmology

The Varying Speed of Light (VSL) model describes how the speed of light in a vacuum changes with cosmological redshift. Despite numerous models, there is little observational evidence for this variation. While the speed of light can be accurately measured by physical means, cosmological methods are rarely used. Previous studies quantified the speed of light at specific redshifts using Gaussian processes and reconstructed the redshift-dependent function $c(z)$. It is crucial to quantify the speed of light across varying redshifts. We use the latest data on angular diameter distances $D_A(z)$ and Hubble parameters $H(z)$ from baryon acoustic oscillation (BAO) and cosmic chronometer measurements in the redshift interval $z\in[0.07,1.965]$. The speed of light $c(z)$ is determined using Gaussian and deep Gaussian processes to reconstruct $H(z)$, $D_A(z)$, and $D^{\prime}_A(z)$. Furthermore, we conduct comparisons across three distinct models, encompassing two renowned VSL models. We get the result of the parameters constraints in the models (1) for the ``$c$-c" model, $c_0=29492.6 \pm^{6.2}_{5.3} \mathrm{~km} \mathrm{~s}^{-1}$. (2) For the ``$c$-cl" model, $c_0=29665.5 \pm^{11.2}_{11.4}\mathrm{~km} \mathrm{~s}^{-1}$ and $n=0.05535 \pm^{0.00008}_{0.00007}$. (3) For the ``$c$-CPL" model, $c_0=29555.7 \pm^{13.3}_{13.2} \mathrm{~km} \mathrm{~s}^{-1}$ and $n=-0.0607 \pm 0.0001$. Based on our findings, it may be inferred that Barrow's classical VSL model is not a suitable fit for our data. In contrast, the widely recognized Chevallier-Polarski-Linder (CPL) VSL model, under some circumstances, as well as the universal ``c is constant" model, demonstrate a satisfactory ability to account for our findings.