Measuring gravity at cosmological scales
  This paper is a pedagogical introduction to models of gravity and how to
constrain them through cosmological observations. We focus on the Horndeski
scalar-tensor theory and on the quantities that can be measured with a minimum
of assumptions. Alternatives or extensions of General Relativity have been
proposed ever since its early years. Because of Lovelock theorem, modifying
gravity in four dimensions typically means adding new degrees of freedom. The
simplest way is to include a scalar field coupled to the curvature tensor
terms. The most general way of doing so without incurring in the Ostrogradski
instability is the Horndeski Lagrangian and its extensions. Testing gravity
means therefore, in its simplest term, testing the Horndeski Lagrangian. Since
local gravity experiments can always be evaded by assuming some screening
mechanism or that baryons are decoupled, or even that the effects of modified
gravity are visible only at early times, we need to test gravity with
cosmological observations in the late universe (large-scale structure) and in
the early universe (cosmic microwave background). In this work we review the
basic tools to test gravity at cosmological scales, focusing on
model-independent measurements.
