Isocurvature initial conditions for second order Boltzmann solvers
  We study how to set the initial evolution of general cosmological
fluctuations at second order, after neutrino decoupling. We compute approximate
initial solutions for the transfer functions of all the relevant cosmological
variables sourced by quadratic combinations of adiabatic and isocurvature
modes. We perform these calculations in synchronous gauge, assuming a Universe
described by the $\Lambda$CDM model and composed of neutrinos, photons, baryons
and dark matter. We highlight the importance of mixed modes, which are sourced
by two different isocurvature or adiabatic modes and do not exist at the linear
level. In particular, we investigate the so-called compensated isocurvature
mode and find non-trivial initial evolution when it is mixed with the adiabatic
mode, in contrast to the result at linear order and even at second order for
the unmixed mode. Non-trivial evolution also arises when this compensated
isocurvature is mixed with the neutrino density isocurvature mode. Regarding
the neutrino velocity isocurvature mode, we show it unavoidably generates
non-regular (decaying) modes at second order. Our results can be applied to
second order Boltzmann solvers to calculate the effects of isocurvatures on
non-linear observables.
