The abundance of lensing protoclusters

Weak gravitational lensing provides a potentially powerful method for the detection of clusters. In addition to cluster candidates, a large number of objects with possibly no optical or X-ray component have been detected in shear-selected samples. Determining the nature of these so-called `dark' lenses is an important step towards understanding the reliability of shear-selection techniques. We develop an analytic model to investigate the claim of Weinberg & Kamionkowski that unvirialized protoclusters account for a significant number of dark lenses. In our model, a protocluster consists of a small virialized region surrounded by infalling matter. We use a simple model for the density profile that assumes the Navarro-Frenk-White form inside of the virial radius and a power law ρ ~ r^-α outside. We find that, in order for a protocluster to simultaneously escape X-ray detection and create a detectable weak lensing signal, it must have a small virial mass (~10¹³M_solar) and large total mass (~10¹⁵M_solar), with a relatively flat density profile outside of the virial radius (α ~ 0-1). Such objects would be characterized by rising tangential shear profiles well beyond the virial radius. We use a semi-analytic approach based on the excursion set formalism to estimate the abundance of lensing protoclusters with a low probability of X-ray detection. We find that they are extremely rare, accounting for less than 0.4 per cent of the total lenses in a survey with background galaxy density n = 30arcmin^-2 and intrinsic ellipticity dispersion σ_ɛ = 0.3. Their abundance decreases significantly if flat density profiles outside of the virial radius are not common. We conclude that lensing protoclusters with undetectable X-ray luminosities are too rare to account for a significant number of dark lenses.