  Gravitational wave (GW) constraints have recently been used to significantly
restrict models of dark energy and modified gravity. New bounds arising from GW
decay and GW-induced dark energy instabilities are particularly powerful in
this context, complementing bounds from the observed speed of GWs. We discuss
the associated linear cosmology for Horndeski gravity models surviving these
combined bounds and compute the corresponding cosmological parameter
constraints, using CMB, redshift space distortion, matter power spectrum and
BAO measurements from the Planck, SDSS/BOSS and 6dF surveys. The surviving
theories are strongly constrained, tightening previous bounds on cosmological
deviations from $\Lambda{}$CDM by over an order of magnitude. We also comment
on general cosmological stability constraints and the nature of screening for
the surviving theories, pointing out that a raised strong coupling scale can
ensure compatibility with gravitational wave constraints, while maintaining a
functional Vainshtein screening mechanism on solar system scales. Finally, we
discuss the quasi-static limit as well as (constraints on) related observables
for near-future surveys.
