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Sunyaev-Zel'dovich scaling relations from a simple phenomenological model for galaxy clusters

We build a simple, {\it top-down} model for the gas density and temperature profiles for galaxy clusters. The gas is assumed to be in hydrostatic equilibrium along with a component of non-thermal pressure taken from simulations and the gas fraction approaches the cosmic mean value only at the virial radius or beyond. The free parameters of the model are the slope and normalisation of the concentration-mass relation, the gas polytropic index, and slope and normalisation of the mass-temperature relation. These parameters can be fixed from X-Ray and lensing observations. We compare our gas pressure profiles to the recently proposed `Universal' pressure profile by \cite{Arnaud09} and find very good agreement. We find that the Sunyaev-Zel'dovich Effect (SZE) scaling relations between the integrated SZE flux, $Y$, the cluster gas temperature, $T_{\rm sl}$, the cluster mass, $M_{\rm tot}$, and the gas mass, $M_{\rm gas}$ are in excellent agreement with the recently observed $r_{2500}$ SZE scaling relations by \cite{Bonamente08} and $r_{500}$ relation by \cite{Arnaud09}. The gas mass fraction increases with cluster mass and is given by $f_{\rm gas}(r_{500}) = 0.1324 + 0.0284 \,\rm{log}\, (\frac{M_{500}}{10^{15}h^{-1}M_\odot})$. This is within $10%$ of observed $f_{\rm gas}(r_{500})$. The consistency between the global properties of clusters detected in X-Rays and in SZE shows that we are looking at a common population of clusters as a whole, and there is no deficit of SZE flux relative to expectations from \xr scaling properties. Thus, it makes it easier to compare and cross-calibrate clusters from upcoming \xr and SZE surveys.