We examine the prospect of using the observed abundance of weak gravitational lenses to constrain the equation-of-state parameter w = p / \rho of the dark energy .
Dark energy modifies the distance-redshift relation , the amplitude of the matter power spectrum , and the rate of structure growth .
As a result , it affects the efficiency with which dark-matter concentrations produce detectable weak-lensing signals .
Here we solve the spherical-collapse model with dark energy , clarifying some ambiguities found in the literature .
We also provide fitting formulas for the non-linear overdensity at virialization and the linear-theory overdensity at collapse .
We then compute the variation in the predicted weak-lens abundance with w .
We find that the predicted redshift distribution and number count of weak lenses are highly degenerate in w and the present matter density \Omega _ { 0 } .
If we fix \Omega _ { 0 } the number count of weak lenses for w = -2 / 3 is a factor of \sim 2 smaller than for the \Lambda CDM model w = -1 .
However , if we allow \Omega _ { 0 } to vary with w such that the amplitude of the matter power spectrum as measured by the Cosmic Background Explorer ( COBE ) matches that obtained from the X-ray cluster abundance , the decrease in the predicted lens abundance is less than 25 % for -1 \leq w < -0.4 .
We show that a more promising method for constraining the dark energy—one that is largely unaffected by the \Omega _ { 0 } - w degeneracy as well as uncertainties in observational noise—is to compare the relative abundance of virialized X-ray lensing clusters with the abundance of non-virialized , X-ray underluminous , lensing halos .
For aperture sizes of \sim 15 arcmin , the predicted ratio of the non-virialized to virialized lenses is greater than 40 % and varies by \sim 20 % between w = -1 and w = -0.6 .
Overall , we find that if all other weak lensing parameters are fixed , a survey must cover at least \sim 40 square degrees in order for the weak lens number count to differentiate a \Lambda CDM cosmology from a dark-energy model with w = -0.9 at the 3 \sigma level .
If , on the other hand , we take into account uncertainties in the lensing parameters , then the non-virialized lens fraction provides the most robust constraint on w , requiring \sim 50 square degrees of sky coverage in order to differentiate a \Lambda CDM model from a w = -0.6 model to 3 \sigma .