Constraints on small-scale cosmological fluctuations from SNe lensing dispersion

We provide predictions on small-scale cosmological density power spectrum from supernova lensing dispersion. Parametrizing 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 > 10 h Mpc(-1) is enhanced by 1-2 orders of magnitude, the enhancement much decreases to a factor of 2-3 at late time (z <= 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 to 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.