CosmoCoffee

Kevin J Ludwick · Posted: April 17 2015

Hello, I have a question about how

χ

is determined in CAMB. I know that it is set to

- 1

, but see below.

\beginequation
\label1
$\mathcal{R} = \pm (\Delta_{\mathcal{R}})^{1/2} = \pm \sqrt{A_s}$
\endequation
at Planck's pivot scale $k_{\star} = 0.05 ~\mathrm{Mpc}^{-1}$ , and In the synchronous gauge, using the (+ - - -) signature, the comoving curvature perturbation is
\beginequation
\label2
$\mathcal{R} = \eta + \frac{\mathcal{H} v}{ k}$
\endequation
where

v \equivθ / k

using the notation of Ma and Bertschinger (\tt arXiv:astro-ph/9506072). For $k<<\mathcal{H}$ in the radiation epoch,
\beginequation
\label3
$\eta= 2C - \frac{5+4 R_{\nu}}{6(15+4R_{\nu})} C (k \tau)^2$ ,
\endequation
and
\beginequation
\label4
$v_{rad} \equiv (1-R_{\nu}) v_{\gamma}+ R_{\nu} v_{\nu} = - \frac{C}{18} (k \tau)^3 \biggl(1-R_{\nu}+R_{\nu} \frac{23+4R_{\nu}}{15+4R_{\nu}}\biggr)$ .
\endequation
It follows from Eqs. (\ref1) and (\ref2) that, for values of

τ

early enough during radiation domination such that $k=k_{\star}$ is super-horizon,
\beginequation
\label4
$C \approx \mp 2 \cdot 10^{-5}$
\endequation
for $\pm \sqrt{A_s}$ evaluated at $k=k_{\star}$ . I used

R ν = ρ ν / (ρ γ + ρ ν)

,

ρ ν / ρ γ = (7 N ν / 8)(4 / 11) 4 / 3

,

N ν = 3.046

, and

ln(10 10 A s) = 3.064

, from Planck 2015.
Comparing equations for initial conditions in CAMB notes, we see that

C = χ / 2

.
However, in CAMB,

χ

is set to

- 1

.

Am I doing something wrong here? Why this discrepancy? I know that using

χ = - 1

in CAMB
gives a CMB angular power spectrum that agrees with
Planck's 2015 results, and using

χ = 2 C

gives an angular power spectrum with amplitudes that are too small. And

A s

is obtained from the CMB, so it makes sense
to me that

χ

should be constrained observationally.

Thank you for any help.

Antony Lewis · Posted: April 17 2015

The

η

of the CAMB notes, e.g. in Eq 43, is not the synchronous gauge quantity, which is

η s y n c = - η / 2

(see Sec 1.A). Maybe that is the confusion?

Kevin J Ludwick · Posted: April 17 2015

Sorry if my last post was a bit confusing. The

η

in my post is the

η s

from the synchronous gauge. And I'm using Equation A6 from astro-ph/0212248 for my expression for the comoving curvature perturbation $\mathcal{R}$ (or

χ

as CAMB uses), accounting for the relation between the

η

and

η s

. (Sorry, my comment about

C = χ / 2

was wrong. What CAMB does is set

C = - 1 / 2

, or

χ = - 1

, for flat space. Bertschinger and Ma in astro-ph/9506072 set

C = - 1 / 6

for their plots.)

I guess my question is more of a conceptual one:
Why is the comoving curvature parameter

χ = - 1

for super-horizon modes as an initial condition? In principle, it seems to me that specifying the initial conditions from the relation $\mathcal{\chi}= \pm \sqrt{A_s}$ (where

A s

is the primordial scalar power spectrum amplitude) when the pivot scale is super-horizon should be correct and consistent with initial conditions that lead to the correct angular power spectrum for the CMB. But according to CAMB (I've tested this),

χ = \pm1

outputs the correct CMB angular spectrum, but $\chi= \pm \sqrt{A_s} \approx \pm 10^{-5}$ does not.

Antony Lewis · Posted: April 17 2015

CAMB evolves transfer functions, which are nicely normalized to fixed unit amplitude. The actual power spectrum goes in later when calculating the C

\text{[math]}

.

Kevin J Ludwick · Posted: April 17 2015

Oh, I see. Okay, thanks for the help!