Constraining dark energy from the abundance of weak gravitational lenses

We examine the prospect of using the observed abundance of weak gravitational lenses to constrain the equation-of-state parameter w = p/ of 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 formulae 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 0 . If we fix 0 the number count of weak lenses for w = -2/3 is a factor of 2 smaller than for the cold dark matter (CDM) model w = -1. However, if we allow 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 per cent for -1 w < -0.4. We show that a more promising method for constraining dark energy -one that is largely unaffected by the 0 -w degeneracy as well as uncertainties in observational noiseis to compare the relative abundance of virialized X-ray lensing clusters with the abundance of non-virialized, X-ray underluminous, lensing haloes. For aperture sizes of 15 arcmin, the predicted ratio of the non-virialized to virialized lenses is greater than 40 per cent and varies by 20 per cent between w = -1 and -0.6. Overall, we find that, if all other weak-lensing parameters are fixed, a survey must cover at least 40 deg 2 in order for the weak-lens number count to differentiate a CDM cosmology from a dark-energy model with w = -0.9 at the 3 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 50 deg 2 of sky coverage in order to differentiate a CDM model from a w = -0.6 model to 3 .