Diagnosing the cosmic coincidence problem and its evolution with recent observations

In the framework of a phenomenological cosmological model with the assumption of $\rho_{X} \propto \rho_{m} a^{\xi}$ ($\rho_{X}$ and $\rho_{m} $ are the energy densities of dark energy and matter, respectively.), we intend to diagnose the cosmic coincidence problem by using the recent samples of Type Ia supernovae (SNe Ia), baryon acoustic oscillation (BAO) and cosmic microwave background (CMB). $\xi$ is a key parameter to characterize the severity of the coincidence problem, wherein $\xi=3$ and $0$ correspond to the $\Lambda$CDM scenario and the self-similar solution without the coincidence problem, respectively. The case of $\xi = Constant$ has been investigated in the previous studies, while we further consider the case of $\xi(z) = \xi_{0} + \xi_{z}*\frac{z}{1+z}$ to explore the possible evolution. A joint analysis of the Pantheon SNe Ia sample with the recent BAO and CMB data figures out that $\xi=3.28\pm0.15$ in the case of $\xi = Constant$ at $68\%$ confidence level (CL), in addition, $\xi_{0} = 2.78_{-1.01}^{+0.28}$ and $\xi_{z} = 0.93_{-0.91}^{+1.56}$ in the case of $\xi(z)$ at $68\%$ CL. It implies that the temporal evolution of the scaling parameter $\xi$ is accepted by the joint sample at $68\%$ CL; however, the joint sample also cannot distinguish whether the scaling parameter $\xi$ is variable or not at 95\% CL. Moreover, the $\Lambda$CDM scenario is accepted by the joint sample at $95\%$ CL in both cases, and the coincidence problem still exists. In addition, we apply the Bayesian evidence to compare the models with the analysis of the joint sample, it turns out that the $\Lambda$CDM scenario is most supported by the joint sample; furthermore, the joint sample prefers the scenario with a constant $\xi$ to the one with a variable $\xi(z)$.