  We provide predictions on small-scale cosmological density power spectrum
from supernova lensing dispersion. Parameterizing the primordial power spectrum
with running $\alpha$ and running of running $\beta$ of the spectral index, we
exclude large positive $\alpha$ and $\beta$ parameters which induce too large
lensing dispersions over current observational upper bound. We ran cosmological
N-body simulations of collisionless dark matter particles to investigate
non-linear evolution of the primordial power spectrum with positive running
parameters. The initial small-scale enhancement of the power spectrum is
largely erased when entering into the non-linear regime. For example, even if
the linear power spectrum at $k>10h {\rm Mpc}^{-1}$ is enhanced by $1-2$ orders
of magnitude, the enhancement much decreases to a factor of $2-3$ at late time
($z \leq 1.5$). Therefore, the lensing dispersion induced by the dark matter
fluctuations weakly constrains the running parameters. When including
baryon-cooling effects (which strongly enhance the small-scale clustering), the
constraint is comparable or tighter than the PLANCK constraint, depending on
the UV cut-off. Further investigations of the non-linear matter spectrum with
baryonic processes is needed to reach a firm constraint.
