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 $\xi=3.75_{-0.21}^{+0.13}$ 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 $\xi(z) = \xi_{0} + \xi_{z}*\frac{z}{1+z}$ at $68\%$ CL . It implies that the temporal evolution of the scaling parameter $\xi$ is supported by the joint sample at $68\%$ CL; moreover, the $\Lambda$CDM model is excluded by the joint sample at $68\%$ CL in both cases, and the coincidence problem still exists. In addition, according to the model selection techniques, the $\Lambda$CDM model is the favorite one in terms of the AIC and BIC techniques, however, the scenario of $\xi(z)$ is most supported in term of the DIC technique.