Time-Delay Cosmography: Measuring the Hubble Constant and other cosmological parameters with strong gravitational lensing

Multiply lensed sources experience a relative time delay in the arrival of photons. This effect can be used to measure absolute distances and the Hubble constant ($H_0$) and is known as time-delay cosmography. The methodology is independent of the local distance ladder and early-universe physics and provides a precise and competitive measurement of $H_0$. With upcoming observatories, time-delay cosmography can provide a 1% precision measurement of $H_0$ and can decisively shed light on the current reported 'Hubble tension'. This paper presents the theoretical background and the current techniques applied for time-delay cosmographic studies and the measurement of the Hubble constant. The paper describes the challenges and systematics in the different components of the analysis and strategies to mitigate them. The current measurements are discussed in context and the opportunities with the anticipated data sets in the future are laid out.
26 pages, 11 figures, to be submitted to Space Science Reviews, Topical Collection 'Strong Gravitational Lensing', eds. J. Wambsganss et al, added references with minor changes