setting constant of integration \chi for initial conditions

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

setting constant of integration \\chi for initial conditions

Post by Kevin J Ludwick » April 17 2015

Hello, I have a question about how [tex]\chi[/tex] is determined in CAMB. I know that it is set to [tex]-1[/tex], 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 [tex]k_{\star} = 0.05 ~\mathrm{Mpc}^{-1}[/tex], 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 [tex]v \equiv \theta/k[/tex] using the notation of Ma and Bertschinger ({\tt arXiv:astro-ph/9506072}). For [tex]k<<\mathcal{H}[/tex] 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 [tex]\tau[/tex] early enough during radiation domination such that [tex]k=k_{\star}[/tex] is super-horizon,
\begin{equation}
\label{5}
C \approx \mp 2 \cdot 10^{-5}
\end{equation}
for [tex]\pm \sqrt{A_s}[/tex] evaluated at [tex]k=k_{\star}[/tex]. I used [tex]R_{\nu}=\rho_{\nu}/(\rho_{\gamma}+\rho_{\nu})[/tex],
[tex]\rho_{\nu}/\rho_{\gamma}=(7 N_{\nu}/8)(4/11)^{4/3}[/tex], [tex]N_{\nu}=3.046[/tex], and [tex]\ln(10^{10} A_s)= 3.064[/tex], from Planck 2015.
Comparing equations for initial conditions in CAMB notes, we see that [tex]C = \chi/2[/tex].
However, in CAMB, [tex]\chi[/tex] is set to [tex]-1[/tex].

Am I doing something wrong here? Why this discrepancy? I know that using [tex]\chi=-1[/tex] in CAMB
gives a CMB angular power spectrum that agrees with
Planck's 2015 results, and using [tex]\chi=2C[/tex] gives an angular power spectrum with amplitudes that are too small. And [tex]A_s[/tex] is obtained from the CMB, so it makes sense
to me that [tex]\chi[/tex] should be constrained observationally.

Thank you for any help.

Antony Lewis
Posts: 1353
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

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

Post by Antony Lewis » April 17 2015

The [tex]\eta[/tex] of the CAMB notes, e.g. in Eq 43, is not the synchronous gauge quantity, which is [tex]\eta_{sync} = -\eta/2[/tex] (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

Post by Kevin J Ludwick » April 17 2015

Sorry if my last post was a bit confusing. The [tex]\eta[/tex] in my post is the [tex]\eta_s[/tex] from the synchronous gauge. And I'm using Equation A6 from astro-ph/0212248 for my expression for the comoving curvature perturbation [tex]\mathcal{R}[/tex] (or [tex]\chi[/tex] as CAMB uses), accounting for the relation between the [tex]\eta[/tex] and [tex]\eta_s[/tex]. (Sorry, my comment about [tex]C=\chi/2[/tex] was wrong. What CAMB does is set [tex]C=-1/2[/tex], or [tex]\chi=-1[/tex], for flat space. Bertschinger and Ma in astro-ph/9506072 set [tex]C=-1/6[/tex] for their plots.)

I guess my question is more of a conceptual one:
Why is the comoving curvature parameter [tex]\chi=-1[/tex] for super-horizon modes as an initial condition? In principle, it seems to me that specifying the initial conditions from the relation [tex]\mathcal{\chi}= \pm \sqrt{A_s}[/tex] (where [tex]A_s[/tex] 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), [tex]\chi=\pm 1[/tex] outputs the correct CMB angular spectrum, but [tex]\chi= \pm \sqrt{A_s} \approx \pm 10^{-5}[/tex] does not.

Antony Lewis
Posts: 1353
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

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

Post by Antony Lewis » 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.

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

setting constant of integration \\chi for initial conditions

Post by Kevin J Ludwick » April 17 2015

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

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