Study of Errors in Strong Gravitational Lensing

Abstract

We examine the accuracy of strong gravitational lensing determinations of the mass of galaxy clusters by comparing the conventional approach with the numerical integration of the fully relativistic null geodesic equations in the case of weak gravitational perturbations on Robertson-Walker metrics. In particular, we study spherically symmetric, three-dimensional singular isothermal sphere models and the three-dimensional matter distribution of Navarro and coworkers which are both commonly used in gravitational lensing studies. In both cases we study two different methods for mass-density truncation along the line of sight: hard truncation and conventional (no truncation). We find that the relative error introduced in the total mass by the thin-lens approximation alone is less than 0.3% in the singular isothermal sphere model and less than 2% in the model of Navarro and coworkers. The removal of hard truncation introduces an additional error of the same order of magnitude in the best case and up to an order of magnitude larger in the worst case studied. Our results ensure that the future generation of precision cosmology experiments based on lensing studies will not require the removal of the thin-lens assumption, but they may require a careful handling of truncation.

Accuracy of the thin-lens approximation in strong lensing by smoothly truncated dark matter haloes

“…In this paper we describe a study that provides a quantitative measure of the degree of accuracy of the method in a particular context. Together with those of a previous study of ours (Kling & Frittelli 2008), these results are among the first quantitative estimates of the accuracy of the thin‐lens approximation applied to gravitational lensing systems.…”

“…Hence, a truncation mechanism is needed for long‐range applications, such as gravitational lensing. Two truncation mechanisms for the NFW model have been used: hard truncation (Takada & Jain 2003; Kling & Frittelli 2008) and smooth truncation (Baltz, Marshall & Oguri 2009). The hard‐truncation mechanism consists of discontinuously terminating the model by assuming that the mass density is identically vanishing outside a given radius.…”

“…The Euler–Lagrange equations are equivalent to five first‐order ODEs, which we can write as These equations are accurate to first order in φ. The parameter b is a constant of integration and is related to the ‘observation angle’ at the observer, or the angle between the light ray and the optical (radial) axis connecting the observer to the lens as explained in Kling & Frittelli (2008): where the potential φ 0 , is evaluated at the observer position, r 0 .…”

“…The hard truncation model is a density profile ρ 0 with r s = 0.25 Mpc within the virial radius, and zero outside of the virial radius. The hard truncation model is discussed extensively in our previous work (Kling & Frittelli 2008), where the gravitational potential needed for the integration of the null geodesics is found (equation 34 in Kling & Frittelli 2008).…”

“…These equations are accurate to first order in ϕ. The parameter b is a constant of integration and is related to the 'observation angle' at the observer, or the angle between the light ray and the optical (radial) axis connecting the observer to the lens as explained in Kling & Frittelli (2008):…”