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setting constant of integration \chi for initial conditions
 
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Kevin J Ludwick



Joined: 17 Apr 2015
Posts: 4
Affiliation: University of Virginia

PostPosted: April 17 2015  Reply with quote

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≡θ / 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 Rν = ρν / (ργ + ρν),
ρν / ργ = (7Nν / 8)(4 / 11)4 / 3, Nν = 3.046, and ln(1010As) = 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 = 2C gives an angular power spectrum with amplitudes that are too small. And As is obtained from the CMB, so it makes sense
to me that χ should be constrained observationally.

Thank you for any help.
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Antony Lewis



Joined: 23 Sep 2004
Posts: 1273
Affiliation: University of Sussex

PostPosted: April 17 2015  Reply with quote

The η of the CAMB notes, e.g. in Eq 43, is not the synchronous gauge quantity, which is ηsync = − η / 2 (see Sec 1.A). Maybe that is the confusion?
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Kevin J Ludwick



Joined: 17 Apr 2015
Posts: 4
Affiliation: University of Virginia

PostPosted: April 17 2015  Reply with quote

Sorry if my last post was a bit confusing. The η in my post is the ηs 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 ηs. (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 As 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 = ±1 outputs the correct CMB angular spectrum, but \chi= \pm \sqrt{A_s} \approx \pm 10^{-5} does not.
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Antony Lewis



Joined: 23 Sep 2004
Posts: 1273
Affiliation: University of Sussex

PostPosted: April 17 2015  Reply with quote

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



Joined: 17 Apr 2015
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Affiliation: University of Virginia

PostPosted: April 17 2015  Reply with quote

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