Adiabatic initial conditions for a particle gas in cosmology

In view of recent interest in the role of “dark” radiation in cosmology, such as cosmic gravitational waves, sterile neutrinos, and dark photons, we clarify the definition of adiabatic initial conditions in the kinetic theory of gases in an expanding universe. Without assuming any form for the phase space distribution function, we identify two possibilities: a and a adiabatic initial condition. The strong one corresponds to the standard adiabatic initial conditions, while the weak one is related to the strong via fluctuations. We show that both types of adiabatic initial conditions are consistent with the separate universe approach, although the latter requires initial internal isocurvature. In passing, we stress the importance of using the particle local momentum in the phase space to define the notion of adiabatic initial conditions. Doing so, we clarify that a gas of gravitons can have adiabatic initial conditions.