Cosmological tests of modified gravity: constraints on $F(R)$ theories
  from the galaxy clustering ratio
  The clustering ratio $\eta$, a large-scale structure observable originally
designed to constrain the shape of the power spectrum of matter density
fluctuations, is shown to provide a sensitive probe of the nature of gravity in
the cosmological regime. We apply this analysis to $F(R)$ theories of gravity
using the luminous red galaxy (LRG) sample extracted from the spectroscopic
Sloan Digital Sky Survey (SDSS) data release 7 and 10 catalogues. We find that
General Relativity (GR), complemented with a Friedmann-Robertson-Walker (FRW)
cosmological model with parameters fixed by the Planck satellite, describes
extremely well the clustering of galaxies up to $z\sim 0.6$. On large cosmic
scales, the absolute amplitude of deviations from GR, $|f_{R_0 }|$, is
constrained to be smaller than $4.6 \times 10^{-5}$ at the $95\%$ confidence
level. This bound makes cosmological probes of gravity almost competitive with
the sensitivity of Solar System tests, although still one order of magnitude
less effective than astrophysical tests. We also extrapolate our results to
future large surveys like Euclid and show that the astrophysical bound will
certainly remain out of reach for such a class of modified-gravity models that
only differ from $\Lambda$CDM at low redshifts.
