  Gravitational wave (GW) source counts have been recently shown to be able to
test how gravitational radiation propagates with the distance from the source.
Here, we extend this formalism to cosmological scales, i.e. the high redshift
regime, and we discuss the complications of applying this methodology to high
redshift sources. We also allow for models with compactified extra dimensions
like in the Kaluza-Klein model. Furthermore, we also consider the case of
intermediate redshifts, i.e. $0<z\lesssim1$, where we show it is possible to
find an analytical approximation for the source counts $\frac{dN}{d(S/N)}$.
This can be done in terms of cosmological parameters, such as the matter
density $\Omega_{m,0}$ of the cosmological constant model or the cosmographic
parameters for a general dark energy model. Our analysis is as general as
possible, but it depends on two important factors: a source model for the black
hole binary mergers and the GW source to galaxy bias. This methodology also
allows us to obtain the higher order corrections of the source counts in terms
of the signal-to-noise $S/N$. We then forecast the sensitivity of future
observations in constraining GW physics but also the underlying cosmology by
simulating sources distributed over a finite range of signal-to-noise with a
number of sources ranging from 10 to 500 sources as expected from future
detectors. We find that with 500 events it will be possible to provide
constraints on the matter density parameter at present $\Omega_{m,0}$ on the
order of a few percent and with the precision growing fast with the number of
events. In the case of extra dimensions we find that depending on the
degeneracies of the model, with 500 events it may be possible to provide
stringent limits on the existence of the extra dimensions if the aforementioned
degeneracies can be broken.
