CosmoCoffee

This paper investigates the possible effect of decaying adiabatic scalar modes on the CMB power spectrum, accounting for the neutrino anisotropic stress. The interesting thing is that with neutrinos they only decay as [tex]1/x^{1/2}[/tex] (but with super-horizon oscillations), rather than 1/x before neutrino decoupling. This means that decaying modes with amplitudes only ~100 times larger than the growing mode at neutrino decoupling may be observable.

This is all very interesting. However since the decay before neutrino decoupling does go like 1/x, this still implies that very large amplitudes need to be generated in the decaying mode unless they are generated very close to neutrino decoupling (or after), which would be quite surprising.

I'm also not very clear on how linearity holds up in this sort of calculation. For very large wavenumbers [tex]x=k \tau [/tex] can be arbitrarily small at any time, and hence the decaying mode amplitude going like 1/x can be arbitrarily large. Is this a serious problem or not? Does it impose a physical constraint on the spectral index of the decaying mode power spectrum?
[similar things happen with vector and tensor modes, as I investigated recently]

(PS. I think there is a typo in the expression for [tex]\theta_\nu[/tex] in eq 8 - a numerical factor missing in the usual result. I haven't checked the full decaying mode result)

WRT your PS, there is a factor of 1/18 missing from the growing mode θν term in equation (8). The neutrino-less result for θr in equation (6) is wrong too - the decaying mode should be constant at -D*3k/8. Their result is out by a factor of τ.

The full decaying mode result does work; it solves the appropriate equations (e.g. (92) in Ma & Bertschinger 1995).

Dear Antony and Joe,
thanks a lot for your e-mails pointing out typos in relativistic velocities.

Linearity holds from nucleosynthesis at least, and therefore we think that the calculations using CMBFAST with the amplitude and spectra we used is valid. However, the point raised by Antony is interesting (in my opinion it applies to the known isocurvature modes -baryons, CDM, etc ..- as well).

Cheers,
Fabio