New probe of gravity: strongly lensed gravitational wave multi-messenger
  approach
  Strong gravitational lensing by galaxies provides us with a unique
opportunity to understand the nature of gravity on galactic and extra-galactic
scales. In this paper, we propose a new multimessenger approach using data from
both gravitational wave (GW) and the corresponding electromagnetic (EM)
counterpart to infer the constraint of the modified gravity (MG) theory denoted
by the scale dependent phenomenological parameter. To demonstrate the
robustness of this approach, we calculate the time-delay predictions by
choosing various values of the phenomenological parameters and then compare
them with that from the general relativity (GR). For the third generation
ground-based GW observatory, with one typical strongly lensed GW+EM event, and
assuming that the dominated error from the stellar velocity dispersions is 5\%,
the GW time-delay data can distinguish an 18\% MG effect on a scale of tens of
kiloparsecs with a $68\%$ confidence level. Assuming GR and a Singular
Isothermal Sphere mass model, there exists a simplified consistency
relationship between time-delay and imaging data. This relationship does not
require for the velocity dispersion measurement, and hence can avoid major
uncertainties. By using this relationship, the multimessenger approach is able
to distinguish an $8\%$ MG effect. Our results show that the GW multimessenger
approach can play an important role in revealing the nature of gravity on the
galactic and extra-galactic scales.
