Constraining the growth of perturbations with lensing of supernovae
  A recently proposed technique allows one to constrain both the background and
perturbation cosmological parameters through the distribution function of
supernova Ia apparent magnitudes. Here we extend this technique to alternative
cosmological scenarios, in which the growth of structure does not follow the
$\Lambda$CDM prescription. We apply the method first to the supernova data
provided by the JLA catalog combined with all the current independent redshift
distortion data and with low-redshift cluster data from Chandra and show that
although the supernovae alone are not very constraining, they help in reducing
the confidence regions. Then we apply our method to future data from LSST and
from a survey that approximates the Euclid satellite mission. In this case we
show that the combined data are nicely complementary and can constrain the
normalization $\sigma_8$ and the growth rate index $\gamma$ to within $0.6\%$
and $7\%$, respectively. In particular, the LSST supernova catalog is forecast
to give the constraint $\gamma (\sigma_8/0.83)^{6.7} = 0.55 \pm 0.1$. We also
report on constraints relative to a step-wise parametrization of the growth
rate of structures. These results show that supernova lensing serves as a good
cross-check on the measurement of perturbation parameters from more standard
techniques.
