Testing the coincidence problem with strong gravitational lens, Type Ia supernovae and Hubble parameter observational data
In this paper, we use three different kinds of observational data, including 130 strong gravitational lensing (SGL) systems, type Ia supernovae (SNeIa: Pantheon and Union2.1) and 31 Hubble parameter data points ($H(z)$) from cosmic chronometers to constrain the phenomenological model ($ρ_x\varproptoρ_m a^ξ$). By combining these three kinds of data (Union2.1+SGL+$H(z)$), we get the parameter value at the confidence interval of $2σ$, $Ω_{X,0} = 0.69\pm0.34$, $ω_x = -1.24\pm0.61$, $ξ= 3.8\pm3.9$ and $H_0 = 70.22\pm0.86$ kms$^{-1}$Mpc$^{-1}$. According to our results, we find that the $Λ$CDM model is still the model which is in best agreement with the observational data at present, and the coincidence problem is not alleviated. In addition, the $Ω_X$ and $Ω_m$ have the same order of magnitude in $0 0.645$, it is necessary to introduce the dark energy interacting with dark matter.