Modified initial power spectrum and too big to fail problem

The galactic scale challenges of dark matter such as "missing satellite" problem and "too big to fail" problem are the main caveats of standard model of cosmology. These challenges could be solved either by implementing the complicated baryonic physics or it could be considered as an indication to a new physics beyond the standard model of cosmology. The modification of collisionless dark matter models or the standard initial conditions are two promising venues for study. In this work, we investigate the effects of the deviations from scale invariant initial curvature power spectrum on number density of dark matter halos. We develop the non-Markov extension of the excursion set theory to calculate the number density of dark matter substructures and dark matter halo progenitor mass distribution. We show that the plausible solution to "too big to fail" problem could be obtained by a Gaussian excess in initial power in the scales of $k_* \sim 3 \text{h/Mpc}$ that is related to the mass scale of $M_* \sim 10^{11} M_{\odot}$. We show that this deviation leads to the decrement of dark matter sub-halos in galactic scale, which is consistent with the current status of the non-linear power spectrum. Our proposal also has a prediction that the number density of Milky way type galaxies must be higher than the standard case.