setting constant of integration \\chi for initial conditions

 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 [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:astroph/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 (1R_{\nu}) v_{\gamma}+ R_{\nu} v_{\nu} =  \frac{C}{18} (k \tau)^3 \biggl(1R_{\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 superhorizon,
\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.
\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:astroph/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 (1R_{\nu}) v_{\gamma}+ R_{\nu} v_{\nu} =  \frac{C}{18} (k \tau)^3 \biggl(1R_{\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 superhorizon,
\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.

 Posts: 1413
 Joined: September 23 2004
 Affiliation: University of Sussex
 Contact:
Re: setting constant of integration \\\\chi for initial cond
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?

 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 [tex]\eta[/tex] in my post is the [tex]\eta_s[/tex] from the synchronous gauge. And I'm using Equation A6 from astroph/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 astroph/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 superhorizon 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 superhorizon 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.
I guess my question is more of a conceptual one:
Why is the comoving curvature parameter [tex]\chi=1[/tex] for superhorizon 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 superhorizon 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.

 Posts: 1413
 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.

 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!