## setting constant of integration \\chi for initial conditions

Kevin J Ludwick
Posts: 4
Joined: April 17 2015
Affiliation: University of Virginia

### setting constant of integration \\chi for initial conditions

Hello, I have a question about how $\chi$ is determined in CAMB. I know that it is set to $-1$, but see below.
\begin{equation}
\label{1}
\mathcal{R} = \pm (\Delta_{\mathcal{R}})^{1/2} = \pm \sqrt{A_s}
\end{equation}
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
\begin{equation}
\label{2}
\mathcal{R} = \eta + \frac{\mathcal{H} v}{ k}
\end{equation}
where $v \equiv \theta/k$ using the notation of Ma and Bertschinger ({\tt arXiv:astro-ph/9506072}). For $k<<\mathcal{H}$ in the radiation epoch,
\begin{equation}
\label{3}
\eta= 2C - \frac{5+4 R_{\nu}}{6(15+4R_{\nu})} C (k \tau)^2,
\end{equation}
and
\begin{equation}
\label{4}
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).
\end{equation}
It follows from Eqs. (\ref{1}) and (\ref{2}) that, for values of $\tau$ early enough during radiation domination such that $k=k_{\star}$ is super-horizon,
\begin{equation}
\label{5}
C \approx \mp 2 \cdot 10^{-5}
\end{equation}
for $\pm \sqrt{A_s}$ evaluated at $k=k_{\star}$. I used $R_{\nu}=\rho_{\nu}/(\rho_{\gamma}+\rho_{\nu})$,
$\rho_{\nu}/\rho_{\gamma}=(7 N_{\nu}/8)(4/11)^{4/3}$, $N_{\nu}=3.046$, and $\ln(10^{10} A_s)= 3.064$, from Planck 2015.
Comparing equations for initial conditions in CAMB notes, we see that $C = \chi/2$.
However, in CAMB, $\chi$ is set to $-1$.

Am I doing something wrong here? Why this discrepancy? I know that using $\chi=-1$ in CAMB
gives a CMB angular power spectrum that agrees with
Planck's 2015 results, and using $\chi=2C$ 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 $\chi$ should be constrained observationally.

Thank you for any help.

Antony Lewis
Posts: 1470
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

### Re: setting constant of integration \\\\chi for initial cond

The $\eta$ of the CAMB notes, e.g. in Eq 43, is not the synchronous gauge quantity, which is $\eta_{sync} = -\eta/2$ (see Sec 1.A). Maybe that is the confusion?

Kevin J Ludwick
Posts: 4
Joined: April 17 2015
Affiliation: University of Virginia

### setting constant of integration \\chi for initial conditions

Sorry if my last post was a bit confusing. The $\eta$ in my post is the $\eta_s$ 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 $\chi$ as CAMB uses), accounting for the relation between the $\eta$ and $\eta_s$. (Sorry, my comment about $C=\chi/2$ was wrong. What CAMB does is set $C=-1/2$, or $\chi=-1$, 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 $\chi=-1$ 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), $\chi=\pm 1$ outputs the correct CMB angular spectrum, but $\chi= \pm \sqrt{A_s} \approx \pm 10^{-5}$ does not.

Antony Lewis
Posts: 1470
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

### Re: setting constant of integration \\\\chi for initial cond

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

Kevin J Ludwick
Posts: 4
Joined: April 17 2015
Affiliation: University of Virginia

### setting constant of integration \\chi for initial conditions

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