  Dimensional flow, the scale dependence of the dimensionality of spacetime, is
a feature shared by many theories of quantum gravity (QG). We present the first
study of the consequences of QG dimensional flow for the luminosity distance
scaling of gravitational waves in the frequency ranges of LIGO and LISA. We
find generic modifications with respect to the standard general-relativistic
scaling, largely independent of specific QG proposals. We constrain these
effects using two examples of multimessenger standard sirens, the binary
neutron-star merger GW170817 and a simulated supermassive black-hole merger
event detectable with LISA. We apply these constraints to various QG
candidates, finding that the quantum geometries of group field theory, spin
foams and loop quantum gravity can give rise to observable signals in the
gravitational-wave spin-2 sector. Our results complement and improve GW
propagation-speed bounds on modified dispersion relations. Under more
model-dependent assumptions, we also show that bounds on quantum geometry can
be strengthened by solar-system tests.
