Constraining cosmological extra dimensions with gravitational wave
  standard sirens: from theory to current and future multi-messenger
  observations
  The propagation of gravitational waves (GWs) at cosmological distances offers
a new way to test the gravitational interaction at the largest scales. Many
modified theories of gravity, usually introduced to explain the observed
acceleration of the universe, can be probed in an alternative and complementary
manner with respect to standard electromagnetic (EM) observations. In this
paper we consider a homogeneous and isotropic cosmology with extra spatial
dimensions at large scales, which represents a simple phenomenological
prototype for extra-dimensional modified gravity cosmological models. By
assuming that gravity propagates through the higher-dimensional spacetime,
while photons are constrained to the usual four dimensions of general
relativity, we derive from first principles the relation between the luminosity
distance measured by GW detectors and the one inferred by EM observations. We
then use this relation to constrain the number of cosmological extra dimensions
with the binary neutron star event GW170817 and the binary black hole merger
GW190521. We further provide forecasts for the Laser Interferometer Space
Antenna (LISA) by simulating multi-messenger observations of massive black hole
binary (MBHB) mergers. This paper extends and updates previous analyses which
crucially neglected an additional redshift dependency in the GW-EM luminosity
distance relation which affects results obtained from multi-messenger GW events
at high redshift, in particular constraints expected from LISA MBHBs.
