3 Basics of Gravitational Lensing

The path, the size and the cross section of a light bundle propagating through spacetime in principle are affected by all the matter between the light source and the observer. For most practical purposes, we can assume that the lensing action is dominated by a single matter inhomogeneity at some location between source and observer. This is usually called the “thin lens approximation”: All the action of deflection is thought to take place at a single distance. This approach is valid only if the relative velocities of lens, source and observer are small compared to the velocity of light v ≪ c and if the Newtonian potential is small 2 |Φ| ≪ c. These two assumptions are justified in all astronomical cases of interest. The size of a galaxy, e.g., is of order 50 kpc, even a cluster of galaxies is not much larger than 1 Mpc. This “lens thickness” is small compared to the typical distances of order few Gpc between observer and lens or lens and background quasar/galaxy, respectively. We assume that the underlying spacetime is well described by a perturbed Friedmann–Robertson–Walker metric2:

( ) ( ) ds2 = 1 + 2Φ- c2dt2 − a2(t) 1 − 2Φ- dσ2. (3 ) c2 c2

 3.1 Lens equation
 3.2 Einstein radius
 3.3 Critical surface mass density
 3.4 Image positions and magnifications
 3.5 (Non-)Singular isothermal sphere
 3.6 Lens mapping
 3.7 Time delay and “Fermat’s” theorem

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