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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 $\mathcal{R} = \pm (\Delta_{\mathcal{R}})^{1/2} = \pm \sqrt{A_s}$ \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.
Antony Lewis

Joined: 23 Sep 2004
Posts: 1332
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 ηsync = − η / 2 (see Sec 1.A). Maybe that is the confusion?
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 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.
Antony Lewis

Joined: 23 Sep 2004
Posts: 1332
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.
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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