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

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

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

CMB Interferometers: DASI, VSA DASI @ South Pole VSA @ Tenerife

CMB Interferometers: CBI CBI @ Chile

The Cosmic Background Imager

The Instrument 13 90-cm Cassegrain antennas 6-meter platform 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

3-Axis mount : rotatable platform

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

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

Site – Northern Chilean Andes

A Theoretical Digression

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

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

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

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

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

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

Images of the CMB WMAP Satellite BOOMERANG ACBAR

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

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

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

CMB Interferometry

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:

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

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

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:

Mosaicing in the uv plane offset & add phase gradients

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

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!

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

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

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

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

Covariance of Gridded Visibilities Or 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

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

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)

The Computational Problem

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)

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

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

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

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

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

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

Example – Mock deep field Noise removed Raw CMB Sources

CBI Results

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

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

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

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

CBI 2000 Mosaic Power Spectrum

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

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

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)

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

New: CBI 2000+2001 Results

CBI 2000+2001 Noise Power

CBI 2000+2001 and WMAP

CBI 2000+2001, WMAP, ACBAR

The CMB From NRAO HEMTs

Example: Post-WMAP parameters

CBI Polarization

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

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.

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)

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

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

CBI-Pol 2000 Cartwright thesis

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

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!

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

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

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)

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/0203048) P. Zhang, U. Pen, & B. Wang (astro-ph/0201375)

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

CBI SZE visibility function dominated by shortest baselines

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

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

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.