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Hi

For a summer project, I am required to read the paper "Measuring Primordial Non-Gaussianity in the Cosmic Microwave Background", Komatsu et al (http://arxiv.org/abs/astro-ph/0305189v2).

On page 3, the following arguments describe the method of reconstructing primordial fluctuations from temperature anisotropy:

(The point I am stuck at is given below and is boldfaced...you may want to scroll down skipping the background, which I have included to define the notation.)

The harmonic coefficients of the CMB anisotropy are given by

[tex]a_{lm} = \frac{1}{T}\int d^{2}\hat{n}\Delta T(\hat{n})Y_{lm}^{*}(\hat{n})[/tex]

They are related to the primordial fluctuations as

[tex]a_{lm} = b_{l}\int r^{2}dr \left[\Phi_{lm}(r)\alpha_{l}^{adi}(r) + S_{lm}(r)\alpha_{l}^{iso}(r)\right] + n_{lm}[/tex]

where [itex]\Phi_{lm}(r)[/itex] and [itex]S_{\lm}(r)[/itex] are the harmonic coefficients of the fluctuations at a given comoving distance [itex]r = |x|[/itex], [itex]b_{lm}[/itex] is the beam transfer function and [itex]n_{lm}[/itex] is the harmonic coefficient of noise.

Here,

[tex]\alpha_{l} \equiv \frac{2}{\pi}\int k^{2}dk g_{Tl}(k)j_{l}(kr)[/tex]

where [itex]g_{Tl}[/itex] is the radiation transfer function of either adiabatic (adi) or isocurvature (iso) perturbation; [itex]j_{l}(kr)[/itex] is the spherical Bessel function of order [itex]l[/itex].

This is where I'm stuck:

I can't see how the authors have obtained the solution for the filter from the partial differential equation. I would be grateful if someone could shed light on this step. Next, assumuming that [itex]\Phi(x)[/itex] dominates, we try to reconstruct [itex]\Phi(x)[/itex] from the observed [itex]\Delta T(\hat{n})[/itex]. A linear filter, [itex]O_{l}(r)[/itex], which reconstructs the underlying field, can be obtained by minimizing variance of difference between the filtered field [itex]O_{l}(r)a_{lm}[/itex] and the underlying field [itex]\Phi_{lm}(r)[/itex]. By evaluating

[tex]\frac{\partial}{\partial O_{l}(r)}\left\langle\left|O_{l}(r)a_{lm}-\Phi_{lm}(r)\right|^{2}\right\rangle = 0[/tex]

one obtains a solution for the filter as

[tex]O_{l}(r) = \frac{\beta_{l}(r)b_{l}}{C_{l}}[/tex]

where the function [itex]\beta_{l}(r)[/itex] is given by

[tex]\beta_{l}(r) \equiv \frac{2}{\pi} \int k^{2}dk P(k) g_{Tl}(k)j_{l}(kr)[/tex]

and [itex]P(k)[/itex] is the power spectrum of [itex]\Phi[/itex].

(Here [itex]C_{l} \equiv C_{l}^{th}b_{l}^{2} + \sigma_{0}^2[/itex] includes the effects of [itex]b_{l}[/itex] and noise, where [itex]C_{l}^{th}[/itex] is the theoretical power spectrum.)

Thanks in advance.

Vivek.

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# Reconstruction Primordial Fluctuations from Temperature Anisotropy

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