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/rho 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 Omega(0). If we fix Omega(0) the number count of weak lenses for w = -2/3 is a factor of similar to2 smaller than for the Lambda cold dark matter (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 per cent for -1 less than or equal to w < -0.4. We show that a more promising method for constraining 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 haloes. For aperture sizes of similar to 15 arcmin, the predicted ratio of the non-virialized to virialized lenses is greater than 40 per cent and varies by similar to 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 similar to 40 deg(2) 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 similar to 50 deg(2) of sky coverage in order to differentiate a Lambda CDM model from a w = -0.6 model to 3 sigma.