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 that unvirialised protoclusters account for a significant number of dark lenses .
In our model , a protocluster consists of a small virialised region surrounded by in-falling 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 \rho \sim r ^ { - \alpha } 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 ( \sim 10 ^ { 13 } ~ { } \mathrm { M } _ { \odot } ) and large total mass ( \sim 10 ^ { 15 } ~ { } \mathrm { M } _ { \odot } ) , with a relatively flat density profile outside of the virial radius ( \alpha \sim 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 = 30 ~ { } \mathrm { arcmin ^ { -2 } } and intrinsic ellipticity dispersion \sigma _ { \epsilon } = 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 .