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

Posted: April 17 2015
Hello, I have a question about how $$\chi$$ is determined in CAMB. I know that it is set to $$-1$$, but see below.

\label{1}
\mathcal{R} = \pm (\Delta_{\mathcal{R}})^{1/2} = \pm \sqrt{A_s}

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

\label{2}
\mathcal{R} = \eta + \frac{\mathcal{H} v}{ k}

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,

\label{3}
\eta= 2C - \frac{5+4 R_{\nu}}{6(15+4R_{\nu})} C (k \tau)^2,

and

\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).

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,

\label{5}
C \approx \mp 2 \cdot 10^{-5}

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.

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

Posted: April 17 2015
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?

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

Posted: April 17 2015
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.

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

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_\ell.

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

Posted: April 17 2015
Oh, I see. Okay, thanks for the help!