Linear and non-linear Modified Gravity forecasts with future surveys
  Modified Gravity theories generally affect the Poisson equation and the
gravitational slip (effective anisotropic stress) in an observable way, that
can be parameterized by two generic functions ($\eta$ and $\mu$) of time and
space. We bin the time dependence of these functions in redshift and present
forecasts on each bin for future surveys like Euclid. We consider both Galaxy
Clustering and Weak Lensing surveys, showing the impact of the non-linear
regime, treated with two different semi-analytical approximations. In addition
to these future observables, we use a prior covariance matrix derived from the
Planck observations of the Cosmic Microwave Background. Our results show that
$\eta$ and $\mu$ in different redshift bins are significantly correlated, but
including non-linear scales reduces or even eliminates the correlation,
breaking the degeneracy between Modified Gravity parameters and the overall
amplitude of the matter power spectrum. We further decorrelate parameters with
a Zero-phase Component Analysis and identify which combinations of the Modified
Gravity parameter amplitudes, in different redshift bins, are best constrained
by future surveys. We also extend the analysis to two particular
parameterizations of the time evolution of $\mu$ and $\eta$ and consider, in
addition to Euclid, also SKA1, SKA2, DESI: we find in this case that future
surveys will be able to constrain the current values of $\eta$ and $\mu$ at the
$2-5\%$ level when using only linear scales (wavevector k < 0.15 h/Mpc),
depending on the specific time parameterization; sensitivity improves to about
$1\%$ when non-linearities are included.
