  For a large class of dark energy (DE) models, for which the effective
gravitational constant is a constant and there is no direct exchange of energy
between DE and dark matter (DM), knowledge of the expansion history suffices to
reconstruct the growth factor of linearized density perturbations in the
non-relativistic matter component on scales much smaller than the Hubble
distance. In this paper we develop a non-parametric method for extracting
information about the perturbative growth factor from data pertaining to the
luminosity or angular size distances. A comparison of the reconstructed density
contrast with observations of large scale structure and gravitational lensing
can help distinguish DE models such as the cosmological constant and
quintessence from models based on modified gravity theories as well as models
in which DE and DM are either unified, or interact directly. We show that for
current SNe data, the linear growth factor at z = 0.3 can be constrained to 5%,
and the linear growth rate to 6%. With future SNe data, such as expected from
the JDEM mission, we may be able to constrain the growth factor to 2-3% and the
growth rate to 3-4% at z = 0.3 with this unbiased, model-independent
reconstruction method. For future BAO data which would deliver measurements of
both the angular diameter distance and Hubble parameter, it should be possible
to constrain the growth factor at z = 2.5 to 9%. These constraints grow tighter
with the errors on the datasets. With a large quantity of data expected in the
next few years, this method can emerge as a competitive tool for distinguishing
between different models of dark energy.
