Karakci et al. (2013b)
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:
- It does not require rapid chopping and scanning.
- Interferometric beam patterns have lower sidelobes.[১]
- Insensitive to uniform brightness or fluctuations in the atmospheric emissions on scales larger than beam width.
- Measures Stokes parameters directly, thus inherently avoid leakage from total into polarized.
- Using redundant baselines systematic errors can be averaged out.
- 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:
- gain, <math>(g)</math>
- 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:
- beam width: each antenna have different beam widths (<math>\sigma</math>)
- beam shape: measure of ellipticity (determined by polar coordinates <math>\rho</math> and <math>\phi</math>)
- beam center: pointing error (determined by <math>\rho</math>, <math>\phi</math> and <math>\sigma</math>)
- cross-polarization of the antennae (determined by <math>\mu_p</math> term)
References
- ↑ Timbie et al. (2006)