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Reconstruction Primordial Fluctuations from Temperature Anisotropy

  1. Jun 23, 2008 #1
    Reconstructing Primordial Fluctuations from Temperature Anisotropy


    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.


    [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.

    Thanks in advance.

  2. jcsd
  3. Jun 23, 2008 #2


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    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.
  4. Jun 23, 2008 #3
    My advice would be to email the author of the paper - they should be able to offer the best advice.
  5. Jun 23, 2008 #4
    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.
    Last edited: Jun 23, 2008
  6. Jun 25, 2008 #5
    What is the expectation taken over?
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