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

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