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?