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Ongoing and future wide-field galaxy surveys can be used to locate a numberof clusters of galaxies with cosmic shear measurement alone. We studyconstraints on cosmological models using statistics of weak lensing selectedgalaxy clusters. We extend our previous theoretical framework to model thestatistical properties of clusters in variants of cosmological models as wellas in the standard $\Lambda$CDM model. Weak lensing selection of clusters doesnot rely on the conventional assumption such as the relation between luminosityand mass and/or hydrostatic equilibrium, but a number of observational effectscompromise robust identification. We use a large set of realistic mockweak-lensing catalogs as well as analytic models to perform a Fisher analysisand make forecast for constraining two competing cosmological models, $w$CDMmodel and $f(R)$ model proposed by Hu & Sawicki, with our lensing statistics.We show that weak lensing selected clusters are excellent probe of cosmologywhen combined with cosmic shear power spectrum even in presence of galaxy shapenoise and masked regions. With the information of weak lensing selectedclusters, the precision of cosmological parameter estimate can be improved by afactor of $\sim1.6$ and $\sim8$ for $w$CDM model and $f(R)$ model,respectively. Hyper Suprime-Cam survey with sky coverage of $1250$ squareddegrees can constrain the equation of state of dark energy $w_{0}$ with a levelof $\Delta w_0 \sim0.1$. It can also constrain the additional scalar degree offreedom in $f(R)$ model with a level of $|f_{R0}| \sim5\times10^{-6}$, whenconstraints from cosmic microwave background measurements are incorporated.Future weak lensing surveys with sky coverage of $20,000$ squared degrees willplace tighter constraints on $w_{0}$ and $|f_{R0}|$ even without cosmicmicrowave background measurements.

Probing cosmology with weak lensing selected clusters II: Dark energy and f(R) gravity models