Low-energy modified gravity signatures on the large-scale structures
  A large number of dark energy and modified gravity models lead to the same
expansion history of the Universe, hence, making it difficult to distinguish
them from observations. To make the calculations transparent, we consider
$f(R)$ gravity with a pressureless matter without making any assumption about
the form of $f(R)$. Using the late-time expansion history realizations
constructed by Shafieloo et al~\cite{2018-Shafieloo.etal-PRD}, we explicitly
show for any $f(R)$ model that the Bardeen potentials $\Psi$ and $\Phi$ evolve
differently. For an arbitrary $f(R)$ model that leads to late-time accelerated
expansion, we explicitly show that $|\Psi + \Phi|$ and its time-derivative
evolves differently than the $\Lambda$CDM model at lower redshifts. We show
that the $\Psi/\Phi$ has a significant deviation from unity for larger
wave-numbers. We discuss the implications of the results for the cosmological
observations.
