Interferometric Imaging & Analysis of the CMB

1. Interferometric Imaging & Analysis of the CMB Steven T. Myers National Radio Astronomy Observatory Socorro, NM

2. Interferometers • Spatial coherence of radiation pattern contains information about source structure • Correlations along wavefronts • Equivalent to masking parts of a telescope aperture • Sparse arrays = unfilled aperture • Resolution at cost of surface brightness sensitivity • Correlate pairs of antennas • “visibility” = correlated fraction of total signal • Fourier transform relationship with sky brightness • Van Cittert – Zernicke theorem

3. CMB Interferometers • CMB issues: • Extremely low surface brightness fluctuations < 50 mK • Polarization less than 10% • Large monopole signal 3K, dipole 3 mK • No compact features, approximately Gaussian random field • Foregrounds both galactic & extragalactic • Traditional direct imaging • Differential horns or focal plane arrays • Interferometry • Inherent differencing (fringe pattern), filtered images • Works in spatial Fourier domain • Element gain effect spread in image plane • Limited by need to correlate pairs of elements • Sensitivity requires compact arrays

4. CMB Interferometers: DASI, VSA • DASI @ South Pole • VSA @ Tenerife

5. CMB Interferometers: CBI • CBI @ Chile

6. The Cosmic Background Imager

7. The Instrument • 13 90-cm Cassegrain antennas • 78 baselines • 6-meter platform • Baselines 1m – 5.51m • 10 1 GHz channels 26-36 GHz • HEMT amplifiers (NRAO) • Cryogenic 6K, Tsys 20 K • Single polarization (R or L) • Polarizers from U. Chicago • Analog correlators • 780 complex correlators • Field-of-view 44 arcmin • Image noise 4 mJy/bm 900s • Resolution 4.5 – 10 arcmin

8. 3-Axis mount : rotatable platform

9. CBI Instrumentation • Correlator • Multipliers 1 GHz bandwidth • 10 channels to cover total band 26-36 GHz (after filters and downconversion) • 78 baselines (13 antennas x 12/2) • Real and Imaginary (with phase shift) correlations • 1560 total multipliers

10. CBI Operations • Observing in Chile since Nov 1999 • NSF proposal 1994, funding in 1995 • Assembled and tested at Caltech in 1998 • Shipped to Chile in August 1999 • Continued NSF funding in 2002, to end of 2004 • Chile Operations 2004-2005 pending proposal • Telescope at high site in Andes • 16000 ft (~5000 m) • Located on Science Preserve, co-located with ALMA • Now also ATSE (Japan) and APEX (Germany), others • Controlled on-site, oxygenated quarters in containers • Data reduction and archiving at “low” site • San Pedro de Atacama • 1 ½ hour driving time to site

11. Site – Northern Chilean Andes

12. A Theoretical Digression

13. The Cosmic Microwave Background • Discovered 1965 (Penzias & Wilson) • 2.7 K blackbody • Isotropic • Relic of hot “big bang” • 3 mK dipole (Doppler) • COBE 1992 • Blackbody 2.725 K • Anisotropies 10-5

14. Thermal History of the Universe Courtesy Wayne Hu – http://background.uchicago.edu

15. CMB Anisotropies • Primary Anisotropies • Imprinted on photosphere of “last scattering” • “recombination” of hydrogen z~1100 • Primordial (power-law?) spectrum of potential fluctuations • Collapse of dark matter potential wells inside horizon • Photons coupled to baryons >> acoustic oscillations! • Electron scattering density & velocity • Velocity produces quadrupole >> polarization! • Transfer function maps P(k) >> Cl • Depends on cosmological parameters >> predictive! • Gaussian fluctuations + isotropy • Angular power spectrum contains all information • Secondary Anisotropies • Due to processes after recombination

16. Primary Anisotropies Courtesy Wayne Hu – http://background.uchicago.edu

17. Primary Anisotropies Courtesy Wayne Hu – http://background.uchicago.edu

18. Secondary Anisotropies Courtesy Wayne Hu – http://background.uchicago.edu

19. Images of the CMB WMAP Satellite BOOMERANG ACBAR

20. WMAP Power Spectrum Courtesy WMAP – http://map.gsfc.nasa.gov

21. CMB Polarization • Due to quadrupolar intensity field at scattering • E & B modes • E (gradient) from scalar density fluctuations predominant! • B (curl) from gravity wave tensor modes, or secondaries • Detected by DASI and WMAP • EE and TE seen so far, BB null • Next generation experiments needed for B modes • Science driver for Beyond Einstein mission • Lensing at sub-degree scales likely to detect • Tensor modes hard unless T/S~0.1 (high!) Hu & Dodelson ARAA 2002

22. CMB Imaging/Analysis Problems • Time Stream Processing (e.g. calibration) • Power Spectrum estimation for large datasets • MLM, approximate methods, efficient methods • Extraction of different components • From PS to parameters (e.g. MCMC) • Beyond the Power Spectrum • Non-Gaussianity • Bispectrum and beyond • Other • Optimal image construction • “object” identification • Topology • Comparison of overlapping datasets

23. CMB Interferometry

24. The Fourier Relationship • The aperture (antenna) size smears out the coherence function response • Lose ability to localize wavefront direction = field-of-view • Small apertures = wide field • An interferometer “visibility” in the sky and Fourier planes:

25. The uv plane and l space • The sky can be uniquely described by spherical harmonics • CMB power spectra are described by multipole l ( the angular scale in the spherical harmonic transform) • For small (sub-radian) scales the spherical harmonics can be approximated by Fourier modes • The conjugate variables are (u,v) as in radio interferometry • The uv radius is given by l / 2p • The projected length of the interferometer baseline gives the angular scale • Multipole l = 2pB / l • An interferometer naturally measures the transform of the sky intensity in l space

26. CBI Beam and uv coverage • 78 baselines and 10 frequency channels = 780 instantaneous visibilities • Frequency channels give radial spread in uv plane • Baselines locked to platform in pointing direction • Baselines always perpendicular to source direction • Delay lines not needed • Very low fringe rates (susceptible to cross-talk and ground) • Pointing platform rotatable to fill in uv coverage • Parallactic angle rotation gives azimuthal spread • Beam nearly circularly symmetric • CBI uv plane is well-sampled • few gaps • inner hole (1.1D), outer limit dominates PSF

27. CMB peaks smaller than this ! Field of View and Resolution • An interferometer “visibility” in the sky and Fourier planes: • The primary beam and aperture are related by: CBI:

28. Mosaicing in the uv plane offset & add phase gradients

29. Power Spectrum and Likelihood • Statistics of CMB (Gaussian) described by power spectrum: Construct covariance matrices and perform maximum Likelihood calculation: Break into bandpowers

30. Power Spectrum Estimation • Method described in CBI Paper 4 • Myers et al. 2003, ApJ, 591, 575 (astro-ph/0205385) • The problem - large datasets • > 105 visibilities in 6 x 7 field mosaic • ~ 104 distinct per mosaic pointing! • But only ~ 103 independent Fourier plane patches • More problems • Mosaic data must be processed together • Data also from 4 independent mosaics! • Polarization “data” x3 and covariances x6! • ML will be O(N3), need to reduce N!

31. Covariance of Visibilities • Write with operators • Covariance • But, need to consider conjugates v = P t + e < v v†> = P < t t † > P† + E E =<ee†> (~diagonal noise) < v v t> = P < t t t> P t = P < t t † > P t

32. Conjugate Covariances • On short baselines, a visibility can correlate with both another visibility and its conjugate

33. Gridded Visibilities • Solution - convolve with “matched filter” kernel • Kernel • Normalization • Returns true t for infinite continuous mosaic D = Q v + Q v* Deal with conjugate visibilities

34. Digression: Another Approach • Could also attempt reconstruction of Fourier plane • v = P t + e → v = M s + e • e.g. ML solution over e = v – Ms • x = H v = s + n H = (MtN-1M)-1MtN-1 n = H e • see Hobson & Maisinger 2002, MNRAS, 334, 569 • applied to VSA data

35. D = R t + n R = Q P + Q P n = Q e + Q e* M = < DD†> = R < t t † > R† + N N = <nn†> = QEQ†+ QEQ† M = < DD t> = R < t t t > Rt + N N = <nn t> = QEQt+ QEQt Covariance of Gridded Visibilities • Or • Covariances • Equivalent to linear (dirty) mosaic image

36. Complex to Real • pack real and imaginary parts into real vector • put into (real) likelihood equation

37. Gridded uv-plane “estimators” • Method practical & efficient • Convolution with aperture matched filter • Reduced to 103 to 104 grid cells • Not lossless, but information loss insignificant • Fast! (work spread between gridding & covariance) • Construct covariance matrices for gridded points • Complicates covariance calculation • Summary of Method • time series of calibrated visibilities V • grid onto D, accumulate R and N (scatter) • assemble covariances (gather) • pass to Likelihood or Imager • parallelizable! (gridding easy, ML harder)

38. The Computational Problem

39. Gridded “estimators” to Bandpowers • Output of gridder • estimators D on grid (ui,vi) • covariances N, CT, Csrc, Cres, Cscan • Maximum likelihood using BJK method • iterative approach to ML solution • Newton-Raphson • incorporates constraint matrices for projection • output bandpowers for parameter estimation • can also investigate Likelihood surface (MCMC?) • Wiener filtered images constructed from estimators • can IFFT D(u,v) to image T(x,y) • apply Wiener filters D‘=FD • tune filters for components (noise,CMB,srcs,SZ)

40. Maximum Likelihood • Method of Bond, Jaffe & Knox (1998)

41. Differencing & Combination • Differencing • 2000-2001 data taken in Lead-Trail mode • Independent mosaics • 4 separate equatorial mosaics 02h, 08h, 14h, 20h

42. Constraints & Projection • Fit for CMB power spectrum bandpowers • Terms for “known” effects • instrumental noise • residual source foreground • incorporate as “noise” matrices with known prefactors • Terms for “unknown effects” • e.g. foreground sources with known positions • known structure in C • incorporate as “noise” matrices with large prefactors • equivalent to downweighting contaminated modes in data projected noise fitted

43. Window Functions • Bandpowers as filtered integral over l • Minimum variance (quadratic) estimator • Window function:

44. Tests with mock data • The CBI pipeline has been extensively tested using mock data • Use real data files for template • Replace visibilties with simulated signal and noise • Run end-to-end through pipeline • Run many trials to build up statistics

45. Wiener filtered images • Covariance matrices can be applied as Wiener filter to gridded estimators • Estimators can be Fourier transformed back into filtered images • Filters CX can be tailored to pick out specific components • e.g. point sources, CMB, SZE • Just need to know the shape of the power spectrum

46. Example – Mock deep field Noise removed Raw CMB Sources

47. CBI Results

48. CBI 2000 Results • Observations • 3 Deep Fields (8h, 14h, 20h) • 3 Mosaics (14h, 20h, 02h) • Fields on celestial equator (Dec center –2d30’) • Published in series of 5 papers (ApJ July 2003) • Mason et al. (deep fields) • Pearson et al. (mosaics) • Myers et al. (power spectrum method) • Sievers et al. (cosmological parameters) • Bond et al. (high-l anomaly and SZ) pending

49. Calibration and Foreground Removal • Calibration scale ~5% • Jupiter from OVRO 1.5m (Mason et al. 1999) • Agrees with BIMA (Welch) and WMAP • Ground emission removal • Strong on short baselines, depends on orientation • Differencing between lead/trail field pairs (8m in RA=2deg) • Use scanning for 2002-2003 polarization observations • Foreground radio sources • Predominant on long baselines • Located in NVSS at 1.4 GHz, VLA 8.4 GHz • Measured at 30 GHz with OVRO 40m • Projected out in power spectrum analysis

50. CBI Deep Fields 2000 • Deep Field Observations: • 3 fields totaling 4 deg^2 • Fields at d~0 a=8h, 14h, 20h • ~115 nights of observing • Data redundancy  strong tests for systematics