CAMB - Factors of a^2 kappa

[Paul Anthony Fronteri]

Hey Antony.

I have some unclear issues about some factors, and something that I did not find at all in the Camb notes unfortunately. Could you please help me and clarify?

I understand that in CAMB, does it propagate values with the factors of a^2 kappa like:
!grho = a^2 kappa rho
!dgrho = a^2 kappa \delta\rho

But there is also an cancellation of the scale factor as well like

grhob_t=grhob/a
grhoc_t=grhoc/a
grhor_t=grhornomass/a2
grhog_t=grhog/a2

leading to: grho=grhob_t+grhoc_t+grhor_t+grhog_t+grhov_t

and dgrho=grhob_t*clxb+grhoc_t*clxc.

Does grho and dgrho still contain the factor a^2 kappa, or have they been removed?

In other word, why are you dividing by terms of a and a^2? Is this the background and need to account for the scale factor? If so, are there in clxb and clxc factors of a^2 kappa, and does the end variables contain only the factor a^2 kappa?

Most confusing is for the neutrinos:

grhormass_t=grhormass(nu_i)/a**2

removing of a factor a^2, again background, or?

call Nu_Integrate_L012(EV, y, a, nu_i, clxnu,qnu)

!clxnu_here = rhonu*clxnu, qnu_here = qnu*rhonu
qnu=qnu/rhonu
clxnu = clxnu/rhonu

Here you remove a factor of rhonu, which I thaugh was the current density of neutrinos, is this correct? So because of the integrating process is it proportional to the density?

clxnu is delta_nu or density contrast?

grhonu_t=grhormass_t*rhonu

Here you multiply the changed density (at radiation?) ghrormass with the current density rhonu, or how are these related to each other? The difference between these, and what then grhonu_t becomes could I not find in anywhere in the notes.

grho = grho + grhonu_t
dgrho= dgrho + grhonu_t*clxnu

Are these values as written grho = a^2 kappa rho and dgrho = a^2 kappa \delta\rho?

Thank you very much for clarifying some issues with these factors. Thank you in advance.

Sincerely,

Paul

[Antony Lewis]

Hey Antony.

I understand that in CAMB, does it propagate values with the factors of a^2 kappa like:
!grho = a^2 kappa rho
!dgrho = a^2 kappa \delta\rho

Correct. grhob, etc are the value today, so the value at time t at scale factor [tex]a[/tex] is grhob/a^3 and hence grhob_t\propto grhob/a (where grhob_t = a^2 \kappa \rho_b )

Neutrinos are mostly done in in terms relative to the massless case. This is more confusing, but if you track through the code it should work out.

clxxx quantities are fractional perturbations.

[Paul Anthony Fronteri]

Thank you for a fast responce.

So this means that delta = (grhoc*y(3)+grhob*y(4))/(grhob+grhoc) will give me the contrast at a particular time t (scale factor a), and be of the correct dimension since the factors will cancel each other out? same for neutrinos drgho/grho?

If I would like to for example plot the perturbations instead as a function of scale factor, meaning delta rho_c (b og massive neutrino) then I need to remove the a^2 kappa terms which are propagated in Camb?

Meaning I have to take grhoc*y(3)/a/kappa, same for baryons, but then for neutrinos I will have to take dgrho_neutrino= grhonu_t*clxnu/kappa, or do I miss /a**2 on both?

Paul

[Antony Lewis]

yes, you can see sample code that is basically exactly that in the transfer function output routine outtransf in equations.f90.

[Paul Anthony Fronteri]

Hey again.

Sorry, but to make it perfectly clear. in outtranst in equations.f90 one have

dgrho = (clxc*grhoc + clxb*grhob)/a
grho = (grhoc+grhob)/a

for baryons and cdm, while

dgrho or dgrho= grhonu_t*clxnu for neutrinos. These quantity still have factors of a^2*kappa in them, such that I have to set

dgrho = drgho/a**2/kappa for it to be the proper delta rho(a)? And not the "numerical" value in Camb?

However, when I do this, does the pertubation decline. Therefore does it seem they have not the factor a^2, but without dividing by kappa, does the numerical value seem to small to be the realistic value. Am I doing something wrong here? Thank you very much for helping.

Paul

[Antony Lewis]

Mixing up \delta\rho with \delta\rho/\rho?

a^2\delta\rho will decline once dark energy kicks in because \delta\rho/\rho then grows slower than [tex]a[/tex].