Gravitational lensing

Gravitational lensing has developed into one of the most powerful tools for the analysis of the dark universe. This review summarises the theory of gravitational lensing, its main current applications and representative results achieved so far. It has two parts. In the first, starting from the equation of geodesic deviation, the equations of thin and extended gravitational lensing are derived. In the second, gravitational lensing by stars and planets, galaxies, galaxy clusters and large-scale structures is discussed and summarised. 1. From general relativity to gravitational lensing 1.1. Preliminaries We assume throughout that a valid model for the geometry and the evolution of the universe is given by a Friedmann-Lemaı̂tre-Robertson-Walker model with cosmological parameters narrowly constrained by numerous cosmological observations. The observational evidence for this cosmological standard model has recently been reviewed elsewhere [16]. Here, we just summarise its main parameters in Tab. 1, adapted from [262]. Table 1. Main cosmological parameters, adapted from [262]. The Hubble constant h is dimension-less and defined by H0 = 100 h km s−1 Mpc−1. “CMB only ” means parameters obtained from the analysis of the 7-year WMAP cosmic microwave background data, “CMB, BAO and H0 ” takes additional constraints from baryonic acoustic oscillations and external measurements of the Hubble constant into account. See [262] for detail. Parameter CMB only CMB, BAO and H0 Hubble constant h 0.710 ± 0.025 0.704 + − 0.0130.014 baryon density parameter Ωb,0 0.0449 ± 0.0028 0.0456 ± 0.0016

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