On the Renormalization of the Effective Field Theory of Large Scale
  Structures
  Standard perturbation theory (SPT) for large-scale matter inhomogeneities is
unsatisfactory for at least three reasons: there is no clear expansion
parameter since the density contrast is not small on all scales; it does not
fully account for deviations at large scales from a perfect pressureless fluid
induced by short-scale non-linearities; for generic initial conditions, loop
corrections are UV-divergent, making predictions cutoff dependent and hence
unphysical. The Effective Field Theory of Large Scale Structures successfully
addresses all three issues. Here we focus on the third one and show explicitly
that the terms induced by integrating out short scales, neglected in SPT, have
exactly the right scale dependence to cancel all UV-divergences at one loop,
and this should hold at all loops. A particularly clear example is an Einstein
deSitter universe with no-scale initial conditions P_in=A k^n. After
renormalizing the theory, we use self-similarity to derive a very simple result
for the final power spectrum for any n, excluding two-loop corrections and
higher. We show how the relative importance of different corrections depend on
n. For n=-1.5, relevant for our universe, pressure and dissipative corrections
are more important than the two-loop corrections.
