Density contrast and Velocity dispersion

[Paul Anthony Fronteri]

Hey.

I have some trouble understanding the changed in the code for the density constrast and velocity dispersion.

First, CAMB is in synchronous gauge or conformal gauge?

Second the only places I can find the derivatives, in LambdCDM code and quintessence, or the density contrast and the velocity disperstion is under the subroutine derivs:

if (EV%MassiveNuApprox(nu_i)) then
!Now EV%iq0 = clx, EV%iq0+1 = clxp, EV%iq0+2 = G_1, EV%iq0+3=G_2=pinu
!see astro-ph/0203507
G11_t=EV%G11(nu_i)/a/a2
G30_t=EV%G30(nu_i)/a/a2
off_ix = EV%nu_ix(nu_i)
w=wnu_arr(nu_i)
ayprime(off_ix)=-k*z*(w+1) + 3*adotoa*(w*ay(off_ix) - ay(off_ix+1))-k*ay(off_ix+2)
ayprime(off_ix+1)=(3*w-2)*adotoa*ay(off_ix+1) - 5._dl/3*k*z*w - k/3*G11_t
ayprime(off_ix+2)=(3*w-1)*adotoa*ay(off_ix+2) - k*(2._dl/3*EV%Kf(1)*ay(off_ix+3)-ay(off_ix+1))
ayprime(off_ix+3)=(3*w-2)*adotoa*ay(off_ix+3) + 2*w*k*sigma - k/5*(3*EV%Kf(2)*G30_t-2*G11_t)
else

However, I am confused with some term that are missing from these equations.

First in the density contrast should it not be a -(1+w)Theta also here? And further are there w_dot/(1 +w)Theta term in the velocity dispersion that is missing. FOr lambdaCDM this should be zero, since w is constant, but for quintessence and dynamical w, should it not. Theta here is then the velocity dispersion or then here ay(EV%off_ix+2).

Second does there seem to lack a k in the last term of the derivative of the velocity dispersion, since it should be k^2 and not k. Also for the term with delta_P = ay(off_ix + 1), should also be divided by (1 +w).

So my question is, where are these term? or are there any approximation that are done which makes these term go away? In that case, could you refer me to the paper where this approximation are discussed, and the equations are written.

I can't find anything in the Camb notes about this.

Thanks in advanced.

[Antony Lewis]

The velocity weight hierarchy approximation for non-relativistic neutrinos is described in

http://arxiv.org/abs/astro-ph/0203507

(but that may not be what you want...)

[Paul Anthony Fronteri]

Hey again.

I'm sorry, but it's not what I need. I have already read this, and also the papers by anthony challinor andgebbie on the 1+3 covariant approach, which I understand CAMB is based on. And I can't find these equations anywhere, not why these terms are not there in the code. I have taken a look at an older code (about 2005), and these terms are not there, and then is not because of a recent approximation scheme.

My question remains, why are these terms not there, and where is the basis of using then this set of equations, compared to the whole thing?

[Paul Anthony Fronteri]

Hey again.

It these are the four-energy integrated scalar equation that you talk about in the appendix, and understanding the notation:

h^' = k*z/3
I_0^(0) = delta_nu
I_0^(1) = delta_p_nu
I_1^(0) = theta_nu
I_1^(1) = G11_t
I_2^(0) =sigma_nu
I_3^(0) = G30_t
beta_2 = EV%Kf(1)
beta_3 = EV%Kf(2)

are there some minor mistaked, either in the code or equations from the paper.

These are:

First equation change: 3*adotoa*(w*delta_nu - delta_p_nu) to adotoa*(3*w*delta_nu - delta_p_nu)

Second equation change: k/3*G11 to k*G11 and 5/3*k*z*w to 15/3*k*z*w

Third equation change: k*(2/3*beta_2*sigma_nu - delta_p_nu) to k/3*(2*beta_2*sigma_nu - delta_p_nu)

Then my question becomes, how does this approximation work together with the newest one, in http://arxiv.org/pdf/1201.3654v2.pdf

Or have I misunderstood something?

[Antony Lewis]

Factors of three are probably just the difference between pressure perturbations and the velocity weighted moments. These equations are only for the massive neutrino fluid, nothing to do with dark energy.