Asymptotic solutions for the case of nearly symmetric gravitational lens systems

Abstract

Gravitational lensing provides a powerful tool to determine the Hubble parameter H0 from the measurement of the time delay Δt between two lensed images of a background variable source. Nevertheless, knowledge of the deflector mass distribution constitutes a hurdle. We propose in the present work interesting solutions for the case of nearly symmetric gravitational lens systems. For the case of a small misalignment between the source, the deflector and the observer, we first consider power-law (ɛ) axially symmetric models for which we derive an analytical relation between the amplification ratio and source position which is independent of the power-law slope ɛ. According to this relation, we deduce an expression for H0 also irrespective of the value ɛ. Secondly, we consider the power-law axially symmetric lens models with an external large-scale gravitational field, the shear γ, resulting in the so-called ɛ−γ models, for which we deduce simple first-order equations linking the model parameters and the lensed image positions, the latter being observable quantities. We also deduce simple relations between H0 and observables quantities only. From these equations, we may estimate the value of the Hubble parameter in a robust way. Nevertheless, comparison between the ɛ−γ and singular isothermal ellipsoid (SIE) models leads to the conclusion that these models remain most often distinct. Therefore, even for the case of a small misalignment, use of the first-order equations and precise astrometric measurements of the positions of the lensed images with respect to the centre of the deflector enables one to discriminate between these two families of models. Finally, we confront the models with numerical simulations to evaluate the intrinsic error of the first-order expressions used when deriving the model parameters under the assumption of a quasi-alignment between the source, the deflector and the observer. From these same simulations, we estimate for the case of the ɛ−γ family of models that the standard deviation affecting H0 is which merely reflects the adopted astrometric uncertainties on the relative image positions, typically arcsec. In conclusions, we stress the importance of getting very accurate measurements of the relative positions of the multiple lensed images and of the time delays for the case of nearly symmetric gravitational lens systems, in order to derive robust and precise values of the Hubble parameter.