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 $\xi(z)$.
