How Density Environment Changes the Influence of the Dark Matter-Baryon Streaming Velocity on the Cosmological Structure Formation

We study the dynamical effect of relative velocities between dark matter and baryonic fluids, which remained supersonic after the epoch of recombination. The impact of this supersonic motion on the formation of cosmological structures was first formulated by Tseliakhovich & Hirata (2010), in terms of the linear theory of small-scale fluctuations coupled to large-scale, relative velocities in mean-density regions. In their formalism, they limited the large-scale density environment to be those of the global mean density. We improve on their formulation by allowing variation in the density environment as well as the relative velocities. This leads to a new type of coupling between large-scale and small-scale modes. We find that the small-scale fluctuation grows in a biased way: faster in the overdense environment and slower in the underdense environment. We also find that the net effect on the global power spectrum of the density fluctuation is to boost its overall amplitude from the prediction by Tseliakhovich & Hirata (2010). Correspondingly, the conditional mass function of cosmological halos and the halo bias parameter are both affected in a similar way. The discrepancy between our prediction and that by Tseliakhovich & Hirata (2010) is significant, and therefore the related cosmology and high-redshift astrophysics should be revisited. The mathematical formalism of this study can be used for generating cosmological initial conditions of small-scale perturbations in generic, overdense (underdense) background patches.