Reconstruction Primordial Fluctuations from Temperature Anisotropy

[maverick280857]

Reconstructing Primordial Fluctuations from Temperature Anisotropy

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

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

They are related to the primordial fluctuations as

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}

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

Here,

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

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

This is where I'm stuck:

Next, assumuming that \Phi(x) dominates, we try to reconstruct \Phi(x) from the observed \Delta T(\hat{n}). A linear filter, O_{l}(r), which reconstructs the underlying field, can be obtained by minimizing variance of difference between the filtered field O_{l}(r)a_{lm} and the underlying field \Phi_{lm}(r). By evaluating

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

one obtains a solution for the filter as

O_{l}(r) = \frac{\beta_{l}(r)b_{l}}{C_{l}}

where the function \beta_{l}(r) is given by

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

and P(k) is the power spectrum of \Phi.

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

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.

Thanks in advance.

Vivek.

 

[Wallace]

This is a very specific question. I would suggest that you try a few advanced textbooks (although you probably have done this already?). Other than that, review papers, in this case a review of CMB physics and measurement, often go into more detail about methods that other papers do not have the space for.

Hopefully someone else will be able to provide a more useful answer, but that is as much as I can suggest.

 

[matt.o]

My advice would be to email the author of the paper - they should be able to offer the best advice.

 

[maverick280857]

Thank you matt.o and Wallace.

I believe this is a construction of a Wiener filter, but I am not sure how the PDE has resulted in the given solution. I have mailed the author of the paper.

 

[maverick280857]

What is the expectation taken over?