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 $ρ_X\propto ρ_{m} a^ξ$ (Dalal et al.2001), we discuss the cosmic coincidence problem in light of current observational data. $ξ$ is a key parameter to denote the severity of the coincidence problem. In this scenario, $ξ=3$ and $ξ=0$ correspond to $Λ$CDM and the self-similar solution without coincidence problem, respectively. Hence, any solution with a scaling parameter $0<ξ<3$ makes the coincidence problem less severe. In addition, the standard cosmology without interaction between DE and DM is characterized by $ξ+3ω_X=0$, whereas the inequality $ξ+3ω_X\neq 0$ represents non-standard cosmology. We place observational constraints on the parameters $(Ω_{X,0},ω_{X},ξ)$ 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 $Ω_{X,0}=0.72\pm0.02$, $ω_{X}=-0.98\pm0.07$ and $ξ=3.06\pm0.35$ at 68.3% confidence level. The result shows that the $Λ$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.

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