Relativistic cosmology number densities in void-Lemaitre-Tolman-Bondi
  models
  The goal of this work is to compute the number density of far-IR selected
galaxies in the comoving frame and along the past lightcone of observationally
constrained LTB models and to compare those results with their standard model
counterparts.
  We derived integral number densities and differential number densities using
different cosmological distance definitions in the LTB models. Then, we
computed selection functions and consistency functions for the luminosity
functions in the combined fields of the Herschel/PACS evolutionary probe (PEP)
survey in both standard and void cosmologies, from which we derived the
observed values of the above-mentioned densities. We used the
Kolmogorov-Smirnov statistics to study both the evolution of the consistency
functions and its connection to the evolution of the comoving density of
sources. Finally, we fitted the power-law behaviour of the densities along the
observer's past lightcone.
  The analysis of the comoving number density shows that the increased
flexibility of the LTB models is not enough to fit the observed redshift
evolution of the number counts, if it is specialised to a recent best-fit giant
void parametrisation. The results for the power-law fits of the densities along
the observer's past lightcone show general agreement across both cosmological
models studied here around a slope of -2.5 $\pm$ 0.1 for the integral number
density on the luminosity-distance volumes. The differential number densities
show much bigger slope discrepancies.
  We conclude that the differential number densities on the observer's past
lightcone were still rendered dependent on the cosmological model by the flux
limits of the PEP survey. In addition, we show that an intrinsic evolution of
the sources must be assumed to fit the comoving number-density redshift
evolution in the giant void parametrisation for the LTB models used in this
work.
