The Effects of Relativistic Corrections on Cosmological Parameter Estimations from SZE Cluster Surveys

Abstract

Sunyaev-Zel'dovich Effect (SZE) cluster surveys are anticipated to yield tight constraints on cosmological parameters such as the equation of state of dark energy. In this paper, we study the impact of relativistic corrections of the thermal SZE on the cluster number counts expected from a cosmological model and thus, assuming that other cosmological parameters are known to high accuracies, on the determination of the $w$ parameter and $σ_8$ from a SZE cluster survey, where $w=p/ρ$ with $p$ the pressure and $ρ$ the density of dark energy, and $σ_8$ is the rms of the extrapolated linear density fluctuation smoothed over $8\hbox{Mpc}h^{-1}$. For the purpose of illustrating the effects of relativistic corrections, our analyses mainly focus on $ν=353 \hbox{GHz}$ and $S_{lim}=30\hbox{mJy}$, where $ν$ and $S_{lim}$ are the observing frequency and the flux limit of a survey, respectively. These observing parameters are relevant to the {\it Planck} survey. It is found that from two measurable quantities, the total number of SZE clusters and the number of clusters with redshift $z\ge 0.5$, $σ_8$ and $w$ can be determined to a level of $\pm 1%$ and $\pm 8%$, respectively, with $1σ$ uncertainties from a survey of $10000\hbox{deg}^2$. Relativistic effects are important in determining the central values of $σ_8$ and $w$. If we choose the two quantities calculated relativistically from the flat cosmological model with $σ_8=0.8284$ and $w=-0.75$ as input, the derived $σ_8$ and $w$ would be 0.819 and -0.81, respectively, if relativistic effects are wrongly neglected. The location of the resulting $σ_8$ and $w$ in the $σ_8-w$ plane is outside the $3σ$ region around the real central $σ_8$ and $w$.