Propagator NR

[veintemillas]

When the axion field is quite non-relativistic, its main evolution is a fast oscillation with frequency the axion mass which can be absorbed in the definition of a new field. With this definition, one can do a WKB approximation by time averaging the interactions and convert the real 2nd order PDE into a 1st order complex PDE.

Since it is first order one needs saving only the FIELD (not its first derivative), but it is complex so no savings with respect to theta (good to have kept the full complex field m allocated). The propagator becomes simpler

The evolution of MCs could be carried much further if we had functions:

• converting theta_conformal into the new complex, say psi
• a new propagator (most of the code can be adapted, but derivative doesn't have to be adapted, and since the sin(theta) is time averaged one gets a BesselFunction which is non-periodic so no need of finding the smallest laplacian)

[AlexVaq]

Taking it easy, since it's not urgent right now.