Can the homogeneity scale be used as a standard ruler?
  The Universe on comoving scales larger than 100 Mpc/h is assumed to be
statistically homogeneous, with the transition scale where we have $1\%$
deviation from homogeneity being known as the homogeneity scale $R_H$. The
latter was recently proposed in Ntelis et al. (2018) [arXiv:1810.09362] to be a
standard ruler. In this paper we perform a comparison between the baryon
acoustic oscillations and the homogeneity scales $R_H$, assuming a spatially
flat universe, linear cosmological perturbation theory and the cosmological
constant cold dark matter $\Lambda$CDM model. By direct theoretical
calculations, we demonstrate that the answer to the question of whether the
homogeneity scale can be used as a standard ruler, is clearly negative due to
the non-monotonicity of the homogeneity scale $R_H$ with the matter density
parameter $\Omega_{m0}$ of the $\Lambda$CDM model. Furthermore, we also
consider the effect of redshift space distortions and we find that they do not
break the degeneracies but instead only change the value of $R_H$ by a few
percent. Finally, we also confirm our findings with the help of an N-Body
simulation.
