Using a phenomenological model to test the coincidence problem of dark
  energy
  By assuming a phenomenological form for the ratio of the DE and DM densities
$\rho_X\propto \rho_{m} a^{\xi}$ (Dalal et al.2001), we discuss the cosmic
coincidence problem in light of current observational data. $\xi$ is a key
parameter to denote the severity of the coincidence problem. In this scenario,
$\xi=3$ and $\xi=0$ correspond to $\Lambda$CDM and the self-similar solution
without coincidence problem, respectively. Hence, any solution with a scaling
parameter $0<\xi<3$ makes the coincidence problem less severe. In addition, the
standard cosmology without interaction between DE and DM is characterized by
$\xi+3\omega_X=0$, whereas the inequality $\xi+3\omega_X\neq 0$ represents
non-standard cosmology. We place observational constraints on the parameters
$(\Omega_{X,0},\omega_{X},\xi)$ of this model, by using the Constitution Set,
the CMB shift parameter from the 5-year WMAP and the SDSS baryon acoustic peak.
Combining the three samples, we get $\Omega_{X,0}=0.72\pm0.02$,
$\omega_{X}=-0.98\pm0.07$ and $\xi=3.06\pm0.35$ at 68.3% confidence level. The
result shows that the $\Lambda$CDM model still remains a good fit to the recent
observational data, as well as, the coincidence problem indeed exists and is
quite severe, in the framework of this simple phenomenological model. We
further constrain the model with the transition redshift (deceleration /
acceleration). It shows that if the transition from deceleration to
acceleration happens at the redshit $z > 0.73$, within the framework of this
model, we can conclude that the interaction between DE and DM is necessary.
