  The standard model of cosmology with nearly Gaussian, isotropic, scale
invariant and adiabatic initial conditions describes the cosmological
observations well. However, the study of any deviation from the mentioned
conditions will open up a new horizon to the physics of early universe. In this
work, we study the effect of the oscillatory and step-like features in
potentials of inflationary models in late time large scale structure
observations. Mainly we study the matter power spectrum, number density of the
structures, dark matter halo bias and specifically CMB lensing. We show that
the oscillatory models can introduce some degeneracy with late time effects on
BAO scale. We also conclude that high frequency oscillatory models which are
favored by Planck data do not have significant effect on the non linear
structure formation. Finally we show that inflationary models with step
functions which deviates from the standard model in small scales $l \leq 1 Mpc$
can be constrained by future experiments via CMB lensing. We propose the idea
that CMB lensing is a bias independent observation which can be used as a small
scale physics probe due to distribution of the lenses in low redshifts.
Meantime this model can alter the prediction of the cosmological model for the
number density of small structures and can be used as a probable explanation
for galactic scale crisis of $\Lambda$CDM.
