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: Traditional direct imaging
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
South Pole
Tenerife

5. CMB Interferometers: CBI
Chile

6. The Cosmic Background Imager

7. The Instrument 13 90-cm Cassegrain antennas 6-meter platform
78 baselines
6-meter platform
Baselines 1m – 5.51m
10 1 GHz channels 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 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 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 K
Anisotropies 10-5

14. Thermal History of the Universe
Courtesy Wayne Hu –

15. CMB Anisotropies Primary Anisotropies Secondary 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. Courtesy Wayne Hu – http://background.uchicago.edu
Primary Anisotropies
Courtesy Wayne Hu –

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

18. Secondary Anisotropies
Courtesy Wayne Hu –

19. Images of the CMB
WMAP Satellite
BOOMERANG
ACBAR

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

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 = 2p B / 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. Field of View and Resolution
An interferometer “visibility” in the sky and Fourier planes:
The primary beam and aperture are related by:
CMB peaks smaller than this !
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/ )
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 =< e e† > (~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. Deal with conjugate visibilities
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 n = H e
see Hobson & Maisinger 2002, MNRAS, 334, 569
applied to VSA data

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

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
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
noise
fitted
projected

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 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

51. CBI 2000 Mosaic Power Spectrum
Mosaic Field Observations
3 fields totaling 40 deg^2
Fields at d~0 a=2h, 14h, 20h
~125 nights of observing
~ 600,000 uv points covariance matrix 5000 x 5000

52. CBI 2000 Mosaic Power Spectrum

53. Cosmological Parameters
wk-h: 0.45 < h < 0.9, t > 10 Gyr
HST-h: h = 0.71 ± 0.076
LSS: constraints on s8 and G from 2dF, SDSS, etc.
SN: constraints from Type 1a SNae

54. SZE Angular Power Spectrum
[Bond et al. 2002]
Smooth Particle Hydrodynamics (5123) [Wadsley et al. 2002]
Moving Mesh Hydrodynamics (5123) [Pen 1998]
143 Mpc 8=1.0
200 Mpc 8=1.0
200 Mpc 8=0.9
400 Mpc 8=0.9
Dawson et al. 2002
Review SZ effect – expected crossover
Use of simulations to predict power
Description of simulations
Parameters of simulations
Scaling of power with parameters confirms s8to the 7
Power for s8=1 and s8=0.9

55. Constraints on SZ “density”
Combine CBI & BIMA (Dawson et al.) 30 GHz with ACBAR 150 GHz (Goldstein et al.)
Non-Gaussian scatter for SZE
increased sample variance (factor ~3))
Uncertainty in primary spectrum
due to various parameters, marginalize
Explained in Goldstein et al. (astro-ph/ )
Use updated BIMA (Carlo Contaldi)
Courtesy Carlo Contaldi (CITA)

56. New : Calibration from WMAP Jupiter
Old uncertainty: 5%
2.7% high vs. WMAP Jupiter
New uncertainty: 1.3%
Ultimate goal: 0.5%

57. New: CBI Results

58. CBI Noise Power

59. CBI and WMAP

60. CBI , WMAP, ACBAR

61. The CMB From NRAO HEMTs

62. Example: Post-WMAP parameters

63. CBI Polarization

64. CBI Polarization CBI instrumentation 2000 Observations 2002 Upgrade
Use quarter-wave devices for linear to circular conversion
Single amplifier per receiver: either R or L only per element
2000 Observations
One antenna cross-polarized in 2000 (Cartwright thesis)
Only 12 cross-polarized baselines (cf. 66 parallel hand)
Original polarizers had 5%-15% leakage
Deep fields, upper limit ~8 mK
2002 Upgrade
Upgrade in 2002 using DASI polarizers (switchable)
Observing with 7R + 6L starting Sep 2002
Raster scans for mosaicing and efficiency
New TRW InP HEMTs from NRAO

65. Polarization Sensitivity
CBI is most sensitive at the peak of the polarization power spectrum
The compact configuration TE EE
Theoretical sensitivity (±1s) of CBI in 450 hours (90 nights) on each of 3 mosaic fields 5 deg sq (no differencing), close-packed configuration.

66. Stokes parameters
CBI receivers can observe either R or L circular polarization
CBI correlators can cross-correlate R or L from a given pair of antennas
Mapping of correlations (RR,LL,RL,LR) to Stokes parameters (I,Q,U,V)
Intensity I plus linear polarization Q,U important
CMB not circularly polarized, ignore V (RR = LL = I)

67. Polarization Interferometry
“Cross hands” sensitive to linear polarization (Stokes Q and U):
where the baseline parallactic angle is defined as:

68. E and B modes
A useful decomposition of the polarization signal is into gradient and curl modes – E and B:

69. CBI-Pol 2000 Cartwright thesis

70. Courtesy Wayne Hu – http://background.uchicago.edu
Pol 2003 – DASI & WMAP
Courtesy Wayne Hu –

71. Polarization Issues Low signal levels Instrumental polarization
High sensitivity and long integrations needed
Prone to systematics and foreground contamination
Use B modes a veto at E levels
Instrumental polarization
Well-calibrated system necessary
Somewhat easier to control in interferometry
Constraint matrix approach possible (e.g. DASI)
Stray radiation
Sky (atmosphere) ~unpolarized (good!)
Ground highly polarized (bad!)
Scan differencing or projection necessary
Computationally intensive!

72. CBI Current Polarization Data
Observing since Sep 2002
Four mosaics 02h, 08h, 14h, 20h
02h, 08h, 14h 6 x 6 fields, 45’ centers
20h deep strip 6 fields
Currently data to Mar 2003 processed
Preliminary data analysis available
Only 02h, 08h (partial), and 20h strip

73. CBI Polarization Projections
CBI funded for Chile ops until 2003 Dec 31
Projections using mock data available
NSF proposal pending for ops through 2005

74. Beyond Gaussianity Objects in CMB data The Sunyaev-Zeldovich Effect
our galaxy: diffuse, structure, different spectral components
see WMAP papers for example of template filtering
discrete source foregrounds
known sources catalogued, can project out or fit
faint sources merge into “Gaussian” foreground
scattering of CMB from clusters of galaxies (SZE)
The Sunyaev-Zeldovich Effect
Compton upscattering of CMB photons by keV electrons
decrement in I below CMB thermal peak (increment above)
negative extended sources (absorption against 3K CMB)
massive clusters mK, but shallow profile θ-1 → exp(-v)

75. 2ndary SZE Anisotropies
Spectral distortion of CMB
Dominated by massive halos (galaxy clusters)
Low-z clusters: ~ 10’-30’
z=1: ~1’  expected dominant signal in CMB on small angular scales
Amplitude highly sensitive to s8
pengjie map is 1.19 deg x 1.19 deg; color scale is dT/T=-2y
A. Cooray (astro-ph/ )
P. Zhang, U. Pen, & B. Wang (astro-ph/ )

76. SZE with CBI: z < 0.1 clusters
P. Udomprasert thesis (Caltech)

77. CBI SZE visibility function
dominated by shortest baselines

78. CL 0016+16, z = 0.55 (Carlstrom et al.)
X-Ray
SZE:  = 15 K, contours =2

79. CMB Interferometry Issues?
process issues
more clever compression (e.g. S/N eigen., MC)
uv-plane exploration (e.g. Hobson & Maisinger)
incorporation of time-series (e.g. calibration)
beyond ML (MCMC?), also projection and marginalization
application to general radio interferometry (e.g. mosaicing)
multi-components
spectral components (uv-coverage vs. frequency)
spatial components (CMB, SZE, point sources, diffuse fg)
non-Gaussianity (bispectrum etc., image-plane?)
SZE issues
modelfitting or multiscale imaging?
removal of CMB
substructure

80. The CBI Collaboration
Caltech Team: Tony Readhead (Principal Investigator), John Cartwright, Alison Farmer, Russ Keeney, Brian Mason, Steve Miller, Steve Padin (Project Scientist), Tim Pearson, Walter Schaal, Martin Shepherd, Jonathan Sievers, Pat Udomprasert, John Yamasaki.
Operations in Chile: Pablo Altamirano, Ricardo Bustos, Cristobal Achermann, Tomislav Vucina, Juan Pablo Jacob, José Cortes, Wilson Araya.
Collaborators: Dick Bond (CITA), Leonardo Bronfman (University of Chile), John Carlstrom (University of Chicago), Simon Casassus (University of Chile), Carlo Contaldi (CITA), Nils Halverson (University of California, Berkeley), Bill Holzapfel (University of California, Berkeley), Marshall Joy (NASA's Marshall Space Flight Center), John Kovac (University of Chicago), Erik Leitch (University of Chicago), Jorge May (University of Chile), Steven Myers (National Radio Astronomy Observatory), Angel Otarola (European Southern Observatory), Ue-Li Pen (CITA), Dmitry Pogosyan (University of Alberta), Simon Prunet (Institut d'Astrophysique de Paris), Clem Pryke (University of Chicago).
The CBI Project is a collaboration between the California Institute of Technology, the Canadian Institute for Theoretical Astrophysics, the National Radio Astronomy Observatory, the University of Chicago, and the Universidad de Chile. The project has been supported by funds from the National Science Foundation, the California Institute of Technology, Maxine and Ronald Linde, Cecil and Sally Drinkward, Barbara and Stanley Rawn Jr., the Kavli Institute,and the Canadian Institute for Advanced Research.