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We provide predictions on small-scale cosmological density power spectrumfrom supernova lensing dispersion. Parameterizing the primordial power spectrumwith running $\alpha$ and running of running $\beta$ of the spectral index, weexclude large positive $\alpha$ and $\beta$ parameters which induce too largelensing dispersions over current observational upper bound. We ran cosmologicalN-body simulations of collisionless dark matter particles to investigatenon-linear evolution of the primordial power spectrum with positive runningparameters. The initial small-scale enhancement of the power spectrum islargely erased when entering into the non-linear regime. For example, even ifthe linear power spectrum at $k>10h {\rm Mpc}^{-1}$ is enhanced by $1-2$ ordersof 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 matterfluctuations weakly constrains the running parameters. When includingbaryon-cooling effects (which strongly enhance the small-scale clustering), theconstraint is comparable or tighter than the PLANCK constraint, depending onthe UV cut-off. Further investigations of the non-linear matter spectrum withbaryonic processes is needed to reach a firm constraint.

Constraints on small-scale cosmological fluctuations from SNe lensing dispersion