Cosmological Tests of Modified Gravity
  We review recent progress in the construction of modified gravity models as
alternatives to dark energy as well as the development of cosmological tests of
gravity. Einstein's theory of General Relativity (GR) has been tested
accurately within the local universe i.e. the Solar System, but this leaves the
possibility open that it is not a good description of gravity at the largest
scales in the Universe. This being said, the standard model of cosmology
assumes GR on all scales. In 1998, astronomers made the surprising discovery
that the expansion of the Universe is accelerating, not slowing down. This
late-time acceleration of the Universe has become the most challenging problem
in theoretical physics. Within the framework of GR, the acceleration would
originate from an unknown dark energy. Alternatively, it could be that there is
no dark energy and GR itself is in error on cosmological scales. In this
review, we first give an overview of recent developments in modified gravity
theories including $f(R)$ gravity, braneworld gravity, Horndeski theory and
massive/bigravity theory. We then focus on common properties these models
share, such as screening mechanisms they use to evade the stringent Solar
System tests. Once armed with a theoretical knowledge of modified gravity
models, we move on to discuss how we can test modifications of gravity on
cosmological scales. We present tests of gravity using linear cosmological
perturbations and review the latest constraints on deviations from the standard
$\Lambda$CDM model. Since screening mechanisms leave distinct signatures in the
non-linear structure formation, we also review novel astrophysical tests of
gravity using clusters, dwarf galaxies and stars.
