A de Sitter limit analysis for dark energy and modified gravity models
  The effective field theory of dark energy and modified gravity is supposed to
well describe, at low energies, the behaviour of the gravity modifications due
to one extra scalar degree of freedom. The usual curvature perturbation is very
useful when studying the conditions for the avoidance of ghost instabilities as
well as the positivity of the squared speeds of propagation for both the scalar
and tensor modes, or the St\"uckelberg field performs perfectly when
investigating the evolution of linear perturbations. We show that the viable
parameters space identified by requiring no-ghost instabilities and positive
squared speeds of propagation does not change by performing a field
redefinition, while the requirement of the avoidance of tachyonic instability
might instead be different. Therefore, we find interesting to associate to the
general modified gravity theory described in the effective field theory
framework, a perturbation field which will inherit the whole properties of the
theory. In the present paper we address the following questions: 1) how can we
define such a field? and 2) what is the mass of such a field as the background
approaches a final de Sitter state? We define a gauge invariant quantity which
identifies the density of the dark energy perturbation field valid for any
background. We derive the mass associated to the gauge invariant dark energy
field on a de Sitter background, which we retain to be still a good
approximation also at very low redshift ($z\simeq 0$). On this background we
also investigate the value of the speed of propagation and we find that there
exist classes of theories which admit a non-vanishing speed of propagation,
even among the Horndeski model, for which in literature it has previously been
found a zero speed. We finally apply our results to specific well known models.
