maverick280857
Jun23-08, 03:11 AM
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.
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.