Interferometric Imaging

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

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CMB Interferometers

• CMB issues
• Extremely low surface brightness fluctuations lt
  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

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CMB Interferometers DASI, VSA

• DASI _at_ South Pole

• VSA _at_ Tenerife

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CMB Interferometers CBI

• CBI _at_ Chile

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The Cosmic Background Imager
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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

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3-Axis mount rotatable platform
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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

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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

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Site Northern Chilean Andes
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A Theoretical Digression
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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

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Thermal History of the Universe
Courtesy Wayne Hu http//background.uchicago.edu
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CMB Anisotropies

• Primary Anisotropies
• Imprinted on photosphere of last scattering
• recombination of hydrogen z1100
• Primordial (power-law?) spectrum of potential
  fluctuations
• Collapse of dark matter potential wells inside
  horizon
• Photons coupled to baryons gtgt acoustic
  oscillations!
• Electron scattering density velocity
• Velocity produces quadrupole gtgt polarization!
• Transfer function maps P(k) gtgt Cl
• Depends on cosmological parameters gtgt predictive!
• Gaussian fluctuations isotropy
• Angular power spectrum contains all information
• Secondary Anisotropies
• Due to processes after recombination

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Primary Anisotropies
Courtesy Wayne Hu http//background.uchicago.edu
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Primary Anisotropies
Courtesy Wayne Hu http//background.uchicago.edu
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Secondary Anisotropies
Courtesy Wayne Hu http//background.uchicago.edu
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Images of the CMB
WMAP Satellite
BOOMERANG
ACBAR
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WMAP Power Spectrum
Courtesy WMAP http//map.gsfc.nasa.gov
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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/S0.1 (high!)

Hu Dodelson ARAA 2002
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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

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CMB Interferometry
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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

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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

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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

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Field of View and Resolution

• An interferometer visibility in the sky and
  Fourier planes

• The primary beam and aperture are related by

CBI
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Mosaicing in the uv plane
offset add
phase gradients
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Power Spectrum and Likelihood

• Statistics of CMB (Gaussian) described by power
  spectrum

Construct covariance matrices and perform maximum
Likelihood calculation
Break into bandpowers
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Power Spectrum Estimation

• Method described in CBI Paper 4
• Myers et al. 2003, ApJ, 591, 575
  (astro-ph/0205385)
• The problem - large datasets
• gt 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!

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Covariance of Visibilities

• Write with operators
• Covariance
• But, need to consider conjugates

v P t e
lt v v gt P lt t t gt P E E lt e e gt
(diagonal noise)
lt v v t gt P lt t t tgt P t P lt t t gt P t
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Conjugate Covariances

• On short baselines, a visibility can correlate
  with both another visibility and its conjugate

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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
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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

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Covariance of Gridded Visibilities

• Or
• Covariances
• Equivalent to linear (dirty) mosaic image

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Complex to Real

• pack real and imaginary parts into real vector
• put into (real) likelihood equation

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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)

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The Computational Problem
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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 DFD
• tune filters for components (noise,CMB,srcs,SZ)

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Maximum Likelihood

• Method of Bond, Jaffe Knox (1998)

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Differencing Combination

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

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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
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Window Functions

• Bandpowers as filtered integral over l
• Minimum variance (quadratic) estimator
• Window function

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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

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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

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Example Mock deep field
Noise removed
Raw
CMB
Sources
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CBI Results
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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

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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 RA2deg)
• 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

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CBI Deep Fields 2000

• Deep Field Observations
• 3 fields totaling 4 deg2
• Fields at d0 a8h, 14h, 20h
• 115 nights of observing
• Data redundancy ? strong tests for systematics

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CBI 2000 Mosaic Power Spectrum

• Mosaic Field Observations
• 3 fields totaling 40 deg2
• Fields at d0 a2h, 14h, 20h
• 125 nights of observing
• 600,000 uv points ?covariance matrix 5000 x
  5000

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CBI 2000 Mosaic Power Spectrum
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Cosmological Parameters
wk-h 0.45 lt h lt 0.9, t gt 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
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SZE Angular Power Spectrum
Bond et al. 2002

• Smooth Particle Hydrodynamics (5123) Wadsley et
  al. 2002
• Moving Mesh Hydrodynamics (5123) Pen 1998

• 143 Mpc ??81.0
• 200 Mpc ??81.0
• 200 Mpc ??80.9
• 400 Mpc ??80.9

Dawson et al. 2002
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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/0212517)
• Use updated BIMA (Carlo Contaldi)

Courtesy Carlo Contaldi (CITA)
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New Calibration from WMAP Jupiter

• Old uncertainty 5
• 2.7 high vs. WMAP Jupiter
• New uncertainty 1.3
• Ultimate goal 0.5

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New CBI 20002001 Results
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CBI 20002001 Noise Power
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CBI 20002001 and WMAP
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CBI 20002001, WMAP, ACBAR
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The CMB From NRAO HEMTs
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Example Post-WMAP parameters
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CBI Polarization
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CBI Polarization

• CBI instrumentation
• 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

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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.
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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)

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Polarization Interferometry
Cross hands sensitive to linear polarization
(Stokes Q and U)
where the baseline parallactic angle is defined
as
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E and B modes

• A useful decomposition of the polarization signal
  is into gradient and curl modes E and B

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CBI-Pol 2000 Cartwright thesis
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Pol 2003 DASI WMAP
Courtesy Wayne Hu http//background.uchicago.edu
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Polarization Issues

• Low signal levels
• 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!

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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

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CBI Polarization Projections

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

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Beyond Gaussianity

• Objects in CMB data
• 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 ?-1 ?
  exp(-v)

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2ndary SZE Anisotropies

• Spectral distortion of CMB
• Dominated by massive halos (galaxy clusters)
• Low-z clusters 10-30
• z1 1 ? expected dominant signal in CMB on
  small angular scales
• Amplitude highly sensitive to s8

A. Cooray (astro-ph/0203048)
P. Zhang, U. Pen, B. Wang (astro-ph/0201375)
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SZE with CBI z lt 0.1 clusters
P. Udomprasert thesis (Caltech)
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CBI SZE visibility function

• dominated by shortest baselines

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CL 001616, z 0.55 (Carlstrom et al.)
X-Ray
SZE ? 15 ?K, contours 2?
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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

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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.
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