  We re-analyze the nonlocal gravity model of Deser and Woodard which was
proposed to account for the current phase of cosmic acceleration. We show that
the growth of perturbations predicted by this nonlocal gravity model when its
background evolution is fixed by some particular non-$\Lambda$CDM models can be
substantially lower than when its background is fixed by $\Lambda$CDM. This can
be seen when we consider the background expansion by a dark energy model with a
slightly less negative equation of state with respect to cosmological constant.
Our results hints towards a fact that the choice of the background expansion
can play a crucial role how this nonlocal gravity model can fit the growth
history data. While the growth data might show better consistency to GR models
(among the background models we studied so far), it seems the nonlocal gravity
model studied in this work is able to show comparable consistency to the growth
data as well. Showing this consistency can be considered as a significant
result since this model can stand as a distinguishable alternative to the
standard model of cosmology.
