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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j
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2

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ul

vm
V (u, v)   I (l , m)  e
dl.dm
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CMB Interferometers
• CMB issues:
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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
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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 @ South Pole
• VSA @ Tenerife
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CMB Interferometers: CBI
• CBI @ 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
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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
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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)
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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 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
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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/S~0.1 (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
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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 / 2
• The projected length of the interferometer baseline
gives the angular scale
– Multipole l = 2 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:
CMB
peaks
smaller
than this !
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:
Break into bandpowers
Construct covariance
matrices and perform
maximum Likelihood
calculation:
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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
– > 105 visibilities in 6 x 7 field mosaic
– ~ 104 distinct per mosaic pointing!
– But only ~ 103 independent Fourier plane patches
• More problems
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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
v=Pt+e
• Covariance
< v v† > = P < t t † > P† + E
E =< e e† > (~diagonal noise)
• But, need to consider conjugates
< v v t > = P < t t t> P t = P < t t † > 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
D = Q v + Q v*
• Kernel
Deal with
conjugate
visibilities
• Normalization
– Returns true t for infinite continuous mosaic
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Digression: Another Approach
• Could also attempt reconstruction of Fourier plane
– v=Pt+e → v=Ms+e
• e.g. ML solution over e = v – Ms
– x=Hv=s+n
H = (MtN-1M)-1MtN-1
n=He
• see Hobson & Maisinger 2002, MNRAS, 334, 569
– applied to VSA data
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Covariance of Gridded Visibilities
• Or
D=Rt+n
R=QP+QP
n = Q e + Q e*
• Covariances
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
• 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
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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
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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
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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)
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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”
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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
projected
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
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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
Raw
Noise
removed
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)
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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 RA=2deg)
– Use scanning for 2002-2003 polarization observations
• Foreground radio sources
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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 deg^2
•Fields at d~0 a=8h, 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 deg^2
• Fields at d~0 a=2h, 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 < 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
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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 s8=1.0
Dawson et al. 2002
•200 Mpc s8=1.0
•200 Mpc s8=0.9
•400 Mpc s8=0.9
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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
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Old uncertainty: 5%
2.7% high vs. WMAP Jupiter
New uncertainty: 1.3%
Ultimate goal: 0.5%
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New: CBI 2000+2001 Results
Future plans
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CBI 2000+2001 Noise Power
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CBI 2000+2001 and WMAP
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CBI 2000+2001, 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
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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
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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
TE
The compact configuration
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
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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)
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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
A. Cooray (astro-ph/0203048)
P. Zhang, U. Pen, & B. Wang (astro-ph/0201375)
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SZE with CBI: z < 0.1 clusters
P. Udomprasert thesis (Caltech)
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CBI SZE visibility function
• dominated by shortest
baselines
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CL 0016+16, z = 0.55 (Carlstrom et al.)
X-Ray
SZE: s = 15 mK, contours =2s
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CMB Interferometry Issues?
• process issues
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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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