Visibility based angular power spectrum estimation in low frequency radio interferometric observations

Abhik GhoshKapteyn Astronomical Institute

NenuFar30 March, 2015

Somnath Bharadwaj, Samir Choudhuri, Sk. Saiyad Ali

Motivation:

Characterization and power law modeling of diffuse foregrounds (Synchrotron) at low frequencies – both in image and uv domian. Short baselines of NenuFar will be extremelyuseful.

Diffuse Galactic turbulence: determination of outer scale of fluctuations, average emissivityand order to random magnetic field strength constraint.

Physical understanding of the origin of Synchrotron anisotropy: variation of magnetic fieldor fluctuations in cosmic ray electron density? Will the power law form of the angular powerspectrum hold down to small scales, relevant for 21 cm observations? Also physical modelingwill be useful to gain knowledge about the Galactic distribution of cosmic ray electrons and the magnetic fields.

Quantifying Foreground/HI Signal:

Angular power spectrum estimators :

1. Bare estimator : Uses individual visibilities to estimate the angular power spectrum.

And the expectation value is given by,

The weight is chosen such that It is zero if same visibilities are correlated.

Hence, Eb(a) gives an unbiased estimate of the angular power spectrum in bin a.

2014MNRAS.445.4351C

Estimated PS for GMRT with analytical and simulated error bars from different realizations:

Analytical error bars

Simulated error bars

Bare Estimator:

Estimated PS for LOFAR :Bare Estimator:

THE TAPERED GRIDDED ESTIMATOR:

The telescope primary beam is usually not very well quantified at large angles where we have the frequency dependent pattern of nulls and sidelobes. Point sources located nearthe nulls and the sidelobes are a problem for estimating the angular power spectrum of the diffuse background radiation.

Point sources located far away from the pointing center, particularly those located near thenulls, introduce ripples along the frequency direction in the angular power spectrum. This poses a severe problem for separating the foregrounds from the cosmological 21-cm signal.

We introduce a Frequency Independent window function that falls before the first null of the primary beam to taper array’s sky response.

Similar, to the Bare estimator we define the regularized gridded estimator such that the self-noise bias at the grid points can be avoided.

Expectation value of Tapered estimator:

This shows this gives an unbiased estimate of the binned power spectrum.

Haslam Map:

408 MHz 150 MHz

2012MNRAS.426.3295G

Cℓ ~ ℓ-2

Studying Galactic interstellar turbulence:

Outer scale of turbulence: The critical scale where the power spectrum transits from kolmogorov to a shallower slope is ℓ

cr ~ πL

max/L

out (Cho & Lazarian, 2002). For a characteristic

scale height of Synchrotron emission Lmax

can be determined depending on the Galactic

latitude. This gives an idea of the average emissivity and the energy injection scale (L

out) to the ISM.

Bo/B

r from L

out: Eilek 1989 has shown the source function variance

(σ2

SI)1/2/S

I ~ (√L

max/L

out)(σ

I/<I>). Variation of S

I reflects variation in random magnetic field

and for sub-sonic turbulence the fluctuation in Synchrotron emission is mainly caused by magnetic field fluctuation and (σ2

SI)1/2/S

I ~ B

r

2/(Br

2 + Bo

2). For transonic turbulence the

electron density fluctuations is also important and (σ2

SI)1/2/S

I ~ B

r

4/(Br

2 + Bo

2)2. For a turbulent

outer scale size of Lout

~ 20 pc, Iacobelli et. al., 2013 found Bo/B

r > 0.3.

Physical understanding of Synchrotron anisotropy:

For a power law spectrum P ≈ kn, Cℓ ≈ ℓn. If we assume that the spectrum is driven by

magnetic field fluctuations and it follows a kolmogorov spectrum then PB(k) ~ k-11/3 and C

ℓ ~

ℓ-3.7. Too steep compared to observations.

Variation of cosmic ray electron density: Supernovae remnants. The injection-diffusion model suggests C

ℓ ≈ ℓ-4, again the spectrum is too steep compared to observation. Further

investigations are needed to identify and understand the small scale anisotropy power.

Chen, 2013

Keypoints/conclusions:

It is possible to estimate the angular power spectrum of the sky signal from the synthesized radio image (e.g. Bernardi et al. 2009, 2010; Iacobelli et al. 2013). The noise properties ofthe visibilities are better understood than those of the image pixels. The noise in the differentvisibilities is uncorrelated, whereas the noise in the image pixels may be correlated depending

on the baseline uv coverage. The visibility based power spectrum estimators have the added advantage that they avoid possible imaging artifacts due to the dirty beam (Trott et al. 2011).

The two estimators considered here both avoid the positive noise bias which arises due tothe system noise contribution in the visibilities. This is achieved by not including the contribution from the correlation of a visibility with itself. As an alternative one could consideran estimator which straight away squared the measured or the gridded visibilities. In this

situation it is necessary to separately identify the noise bias contribution and subtract it out. We find the errors in the amplitude of the calibrated gains affect the noise bias. Frequency and baseline dependent gain errors would manifest themselves as the frequency and ℓ dependence of the noise bias. This is a major obstacle which is bypassed by our estimators.