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 a and running of running β of the spectral index, we exclude large positive α and β 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.