Karakci et al. (2013b)

ডিপথট থেকে

There are 2 polarization modes in the CMB signal: E and B. B is much weaker and expected to exhibit <math>T_b < 0.1 \ \mu K</math>. So systematic errors must be controlled very exquisitely to observe this signal. Interferometers are good because the authors say:

  1. It does not require rapid chopping and scanning.
  2. Interferometric beam patterns have lower sidelobes.[১]
  3. Insensitive to uniform brightness or fluctuations in the atmospheric emissions on scales larger than beam width.
  4. Measures Stokes parameters directly, thus inherently avoid leakage from total into polarized.
  5. Using redundant baselines systematic errors can be averaged out.
  6. Straightforward way to measure angular power spectra as visibilities are already in Fourier space.

Bunn (2007) studied systematic effects on CMB polarization observation. But he ignored: configuration of array, instrumental noise, sampling variance due to finite sky coverage and incomplete uv-coverage.

Zhang et al. (2012) presented a simulation pipeline to assess systematic errors, esp. pointing error. They analysed a mock data with maximum likelihood method.

This paper observed Stokes visibilities in the flat-sky approximation for a given input CMB angular power spectra. A beam width of <math>5^\circ</math> was used so that flat-sky approx. remains valid.

Systematics

The measurement equation in summation form-

<math>V_{pq} = J_p \left( \sum_s E_p^s H K_p B_{pq} K_q^H H^H E_q^s(H) \right) J_q^H</math>
<math>B_{pq} = \begin{pmatrix}

I+Q & U+iV \\ U-iV & I-Q \end{pmatrix}</math>

<math>H_{linear} = \begin{pmatrix}

1 & 0 \\ 0 & 1 \end{pmatrix}</math>

<math>H_{circular} = \frac{1}{\sqrt{2}} \begin{pmatrix}

1 & i \\ 1 & -i \end{pmatrix}</math>

<math>K_{p} = e^{-2\pi i(u_pl+v_pm+w_p(n-1))}</math>
<math>J_p = \begin{pmatrix}

1+g_1^p & \epsilon_1^p \\ \epsilon_2^p & 1+g_2^p \end{pmatrix}</math>

<math>E_p^s = E_p^0(\rho,\phi) \begin{pmatrix}

1+\frac{1}{2}\mu_p\frac{\rho^2}{\sigma^2}\cos 2\phi & \frac{1}{2}\mu_p\frac{\rho^2}{\sigma^2}\sin 2\phi \\ \frac{1}{2}\mu_p\frac{\rho^2}{\sigma^2}\sin 2\phi & 1-\frac{1}{2}\mu_p\frac{\rho^2}{\sigma^2}\cos 2\phi \end{pmatrix}</math>

The instrumental or direction independent effects (DIE) are:

  1. gain, <math>(g)</math>
  2. coupling due to mixing of 2 orthogonally polarized signals in the system, <math>(\epsilon)</math>

An elliptical Gaussian <math>(E_p^0(\rho,\phi))</math> has been used as the primary beam pattern. This beam is modified by the matrix that is used after it. But it is modified by a Jones matrix and the errors can be included in that matrix. The direction dependent (DDE) errors are:

  1. beam width: each antenna have different beam widths (<math>\sigma</math>)
  2. beam shape: measure of ellipticity (determined by polar coordinates <math>\rho</math> and <math>\phi</math>)
  3. beam center: pointing error (determined by <math>\rho</math>, <math>\phi</math> and <math>\sigma</math>)
  4. cross-polarization of the antennae (determined by <math>\mu_p</math> term)

References

  1. Timbie et al. (2006)