Constraining dark energy from the abundance of weak gravitational lenses

Abstract

We examine the prospect of using the observed abundance of weak gravitational lenses to constrain the equation-of-state parameter w = p/ρ 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 Ω₀. If we fix Ω₀ the number count of weak lenses for w = −2/3 is a factor of ∼2 smaller than for the Λ cold dark matter (CDM) model w = −1. However, if we allow Ω₀ 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 Ω₀−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 ∼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 deg² in order for the weak-lens number count to differentiate a ΛCDM cosmology from a dark-energy model with w = −0.9 at the 3σ 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 deg² of sky coverage in order to differentiate a ΛCDM model from a w = −0.6 model to 3σ.