  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.
