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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

Posted: April 17 2015 


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
\endequation
at Planck's pivot scale , and In the synchronous gauge, using the (+   ) signature, the comoving curvature perturbation is
\beginequation
\label2
\endequation
where v≡θ / k using the notation of Ma and Bertschinger (\tt arXiv:astroph/9506072). For in the radiation epoch,
\beginequation
\label3
,
\endequation
and
\beginequation
\label4
.
\endequation
It follows from Eqs. (\ref1) and (\ref2) that, for values of τ early enough during radiation domination such that is superhorizon,
\beginequation
\label4
\endequation
for evaluated at . I used R_{ν} = ρ_{ν} / (ρ_{γ} + ρ_{ν}),
ρ_{ν} / ρ_{γ} = (7N_{ν} / 8)(4 / 11)^{4 / 3}, N_{ν} = 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 = 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 A_{s} 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: 1341 Affiliation: University of Sussex

Posted: April 17 2015 


The η of the CAMB notes, e.g. in Eq 43, is not the synchronous gauge quantity, which is η_{sync} = − 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

Posted: April 17 2015 


Sorry if my last post was a bit confusing. The η in my post is the η_{s} from the synchronous gauge. And I'm using Equation A6 from astroph/0212248 for my expression for the comoving curvature perturbation (or χ as CAMB uses), accounting for the relation between the η and η_{s}. (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 astroph/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 superhorizon modes as an initial condition? In principle, it seems to me that specifying the initial conditions from the relation (where A_{s} 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), χ = ±1 outputs the correct CMB angular spectrum, but does not. 

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Antony Lewis
Joined: 23 Sep 2004 Posts: 1341 Affiliation: University of Sussex

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_{}. 

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

Posted: April 17 2015 


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

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