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

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 $\sigma_8$ from a SZE cluster survey, where $w=p/\rho$ with $p$ the pressure and $\rho$ the density of dark energy, and $\sigma_8$ 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 $\nu=353 \hbox{GHz}$ and $S_{lim}=30\hbox{mJy}$, where $\nu$ 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 $z\ge 0.5$, $\sigma_8$ and $w$ can be determined to a level of $\pm 1$ and $\pm 8$, respectively, with $1\sigma$ uncertainties from a survey of $10000\hbox{deg}^2$. Relativistic effects are important in determining the central values of $\sigma_8$ and $w$. If we choose the two quantities calculated relativistically from the flat cosmological model with $\sigma_8=0.8284$ and $w=-0.75$ as input, the derived $\sigma_8$ and $w$ would be 0.819 and -0.81, respectively, if relativistic effects are wrongly neglected. The location of the resulting $\sigma_8$ and $w$ in the $\sigma_8-w$ plane is outside the $3\sigma$ region around the real central $\sigma_8$ and $w$.