  We investigate the radiative stability of Horndeski scalar-tensor theories
with luminally propagating gravitational waves (as extensively discussed in the
wake of GW170817) and show that in general there is a tension between obtaining
observable deviations from General Relativity (GR) in cosmology and the
requirement of radiative stability. Using this as a constraint, we discuss the
subsets of theories that are capable of evading this conclusion and yielding
observable, radiatively stable departures from GR. We find several classes of
theories that can do so, recovering known cases and identifying several
additional radiatively stable cases. Finally, we also extract the cosmological
signatures of two particularly well-motivated radiatively stable classes of
theories: shift-symmetric theories and theories with a conformal coupling
between the scalar and gravity. We find that cosmological parameter constraints
on dark energy and modified gravity parameters for both of these two classes,
which we explicitly compute using data from the Planck, SDSS/BOSS and 6dF
surveys, are significantly tightened with respect to generic Horndeski
theories.
