The Projected Halo Model.

Abstract

I present a semi-analytical halo model to study line-of-sight projection effects in the galaxy cluster population. The model produces realizations of sight-lines using halo space densities and two-halo correlations, and employs a log-normal covariance model to relate halo mass to multivariate observables. The model sums idealized red-sequence galaxy counts (optical richness), thermal Sunyaev-Zel'dovich (SZ) effect, and X-ray flux within an angular annulus set by a given target halo of size $rtwo$ at redshift $z_rt$ after convolution with top-hat, matched, and background-subtracted filters.

The observed signal bias, variance, and covariance for a fiducial target of mass $halo{5}{14}$ at $z_rt = 0.3$ in a WMAP7 cosmology have been measured. Optical richness is most susceptible to projection, with a top-hat filtered signal bias of $16%$, compared to $6%$ and $4%$ in SZ and X-ray, respectively, for the fiducial target. Projection in the optical is also more localized in redshift and therefore more dependent on nearby, correlated haloes. Background subtraction removes the bias but increases the variance in observed signals. I examine the sensitivity of these effects to target mass and redshift, finding at $z_rt = 0.3$ that the top-hat projected SZ signal bias exceeds that in the optical at halo masses below $sim ! 10^{13.9} hms$. The underlying mass distribution at fixed observed signal is shown to be non-Gaussian with its scatter increased by projection and a skewed towards lower mass.

Covariance in observed signals is studied via extending the model to include intrinsic correlations. The projected contributions are found to be positively correlated, whose approximate scatter of $7%$ mostly come from cosmic variance, uncertainties in the spatial distributions of projected haloes. A $-20%$ intrinsic correlation between the optical and SZ / X-ray signals will be reduced to $-3%$ and $-10%$, respectively, when observed, due to projections for the fiducial target. The degree of such distortions to the intrinsic correlations are reduced at $z_rt = 0.7$ as fractional projection amplitude decreases.