  Next generation surveys will observe the large-scale structure of the
Universe with unprecedented accuracy. This will enable us to test the
relationships between matter over-densities, the curvature perturbation and the
Newtonian potential. Any large-distance modification of gravity or exotic
nature of dark energy modifies these relationships as compared to those
predicted in the standard smooth dark energy model based on General Relativity.
In linear theory of structure growth such modifications are often parameterised
by virtue of two functions of space and time that enter the relation of the
curvature perturbation to, first, the matter over-density, and second, the
Newtonian potential. We investigate the predictions for these functions in
Brans-Dicke theory, clustering dark energy models and interacting dark energy
models. We find that each theory has a distinct path in the parameter space of
modified growth. Understanding these theoretical priors on the
parameterisations of modified growth is essential to reveal the nature of
cosmic acceleration with the help of upcoming observations of structure
formations.
