Component Separation

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Component Separation in Multi-Frequency Sky Maps
Al Kogut Goddard Space Flight Center
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Simple observation Sky emission varies
dramatically with frequency
With non-trivial differences between polarized
and unpolarized emission
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Polarized vs Unpolarized Sky
Cross-variance, diagonal elements only
Unpolarized Emission Higher signal (better S/N
ratio) Component confusion important CMB,
synchrotron, free-free, thermal dust, spinning
dust,
Polarized Emission Fewer components, but lower
S/N Synchrotron, thermal dust, CMB No evidence
(yet) for more components
"Best" component separation algorithm depends on
what question you ask
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Sometimes, adding a few choices
Dust Emission
Synch
Free-Free
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Can make a simple situation complicated!
Haze
QU
I
Thermal Dust
Free-Free
Synchrotron
Anomalous Emission
HARD
SOFT
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A Menagerie of Methods
Cosmology
Astrophysics
Pixel-By-Pixel
Linear Combination
Template
Maximum Entropy
MCMC
Principal Component
Harmonic Filtering
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Template Methods

• Advantages
• Handful of fitted parameters
• Full use of template spatial structure
• Good for low S/N maps
• Retains simple noise properties
• Can fit more templates than channels

• Disadvantages
• No unique component identification
• Non-negligible parameter covariance
• Assumes spatially invariant spectra
• Need template for each component

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A Sampling of Template Maps
Free-Free
Synchrotron
Haslam 408 MHz
Finkbeiner H? Map
Thermal Dust
WMAP K-Ka (I Map)
FDS Model 8
WMAP K-band (Pol)
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Results Range From Surprisingly Efficient
Polarization cleaning good to 0.1 ?K rms 25
million pixels reduced to a handful of numbers!
Gold et al. 2008, arXiv0803.0715
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To Just Plain Surprising
Kogut et al. 1996, ApJ, 460, 1

• Microwave emission component correlated with
  far-IR dust emission
• but not with 408 MHz synchrotron emission
• Spectral index ? -2.2 in antenna temperature
• No definitive identification yet
• Spinning dust? Magnetic dust emission?
  Flat-spectrum synchrotron?

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Samuel Johnson and Cosmology

• Template fits hide real-world astrophysics
• Spatial variations in spectral index
• Additional or localized components
• Line-of-sight effects
• Exercise caution when interpreting results!

Spectral Index 408 -- 1420 MHz
But if all you want to do is remove
foregrounds, Theres no sign (yet) that weve hit
some limit
It is like a dog walking on its hind legs. It
is not that it is done well, but you are
surprised to find it done at all. -- Samuel
Johnson
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Linear Combinations
Describe sky map as
Noise foregrounds
True CMB
Beam
Generalize to set of k multi-frequency maps
so that
Minimal-variance solution is
where
Impractical ? N is (k Npix) x (k Npix) matrix
? We don't know foreground part
of N
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Approximations to Ideal Solution
I. WMAP Internal Linear Combination Map
Choose wi to minimize sky variance subject to
constraint
Hinshaw et al. 2007, ApJS, 170, 288 Gold et al.
2008, arXiv0803.0715
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Approximations to Ideal Solution
II. Harmonic Filtering
Replace pixel basis with spherical harmonics (or
other basis set)
Choose weights at each l to minimize sky
variance, with
with
computed from maps

• Advantages
• No smoothing (better resolution)
• Scale-dependent noise suppression
• Disadvantages
• Galactic plane contributes at all l
• Foreground leakage at smallest scales
• Iterative solution required

ILC
Tegmark et al. 2003, PRD, 68, 123523
Harmonic
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Principal Component Separation
Define k (dimensionless) maps
with moment matrix
Diagonalize to get
with eigenmaps
First few eigenmodes explain most of sky variance

• Advantages
• Broadly applicable "blind" technique
• Number of "important" components
• Spatial maps of each component
• Disadvantages
• Eigenmaps ? Physical foregrounds
• Not great for low S/N components

De Oliveira-Costa et al. 2008, arXiv0802.1525
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Eigen-maps of Radio Sky
First component "Total Stuff" 80 of total
variance
Inputs 11 maps 10 MHz to 94 GHz
Second component Synchrotron Fraction 19 of
total variance
99.7 of variance explained by only 3 components
Third component Thermal Dust Fraction 0.6 of
total variance
De Oliveira-Costa et al. 2008, arXiv0802.1525
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Pixel-By-Pixel Techniques
Explicitly model amplitude and frequency
dependence
Synchrotron ? ? -2.9 Free-Free ? ?
-2.14 Thermal Dust ? ? 2.0
Not an orthogonal frequency basis!
Worst-Case Parameter Count

• Advantages
• Explicit connection to astrophysics
• Allows imposition of constraints
• (spectral index or amplitude)

• Disadvantages
• Non-linear fits ? parameter runaway
• Non-trivial component covariance
• Requires more channels than components

"Some pixels are bigger than others"
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Fitting The Spectral Index

• Noise easily leads to unphysical values for ?
• Low S/N
• Small channel separation
• Multiple foreground components

Sim 7 Channels 25--38 GHz, Input Synch FF
Model Synch FF
Residual Error (?K)

• Possible Solutions To Problem
• Add More/Better Channels
• Use bigger pixels
• Impose Priors
• Fit "Effective" Index

Model One "Radio" Component
Instrument Noise (?K)
Brandt et al. 1994, ApJ, 424, 1
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Maximum Entropy Models
Solve Run-Away By Imposing Priors
Data
Minimize function H A ?B, where
Prior
Choose ? for smooth transition from prior (low
S/N regions) to data (high S/N)
WMAP Implementation
Amplitudes
Iterative Solution
Minimize H to get
Use residuals to update
Bennett et al. 2003, ApJS, 148, 97 Hinshaw et al.
2007, ApJS, 170, 288 Gold et al. 2008,
arXiv0803.0715
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WMAP Maximum Entropy vs Priors
MEM Map
Prior
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WMAP Maximum Entropy
3-component model fits sky data well
K
Q
W
3-Color Maps
Gold et al. 2008, arXiv0803.0715
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What About Polarization?

• Ratio of CMB/Foregrounds reversed for intensity
  vs B-mode polarization
• Intensity CMB 20x brighter
• Polarization CMB 20x fainter

Want component separation techniques that don't
require high signal-to-noise ratio
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Pixel-By-Pixel Fit to Polarization Maps
With free parameters
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Pixel-by-Pixel Polarized Model
Synchrotron Spectral Index
0.010 ? 0.004 in plane 0.036 ? 0.011 outside
P06 mask
Kogut et al. 2007, ApJ, 665, 355
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Fitting Intensity Polarization
Problem Polarization adds 6 more parameters
Tcmb, Ts, Tff, Td ?s, ?ff, ?d, Qcmb, Ucmb, Qs,
Us, Qd, Ud
"Brute-Force" techniques become inefficient

• Look to parameter-fitting algorithms to insert
  into pixel-by-pixel machinery
• Markov Chain Monte Carlo
• Gibbs Sampling

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Markov Chain Monte Carlo Techniques
WMAP 5-year IQU data Nside64 (0.9? pixels) 10
parameters per pixel Tcmb, Ts, Tff, Td, ?s, ?d,
Qs, Qd, Qcmb, Ucmb Fix ?ff -2.14 Fix synch
and dust polarization angle using K-band data
Mean
Best-Fit Point
Get parameter values, errors, and covariance for
each pixel
Gold et al. 2008, arXiv0803.0715
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WMAP MCMC Foreground Components
Gold et al. 2008, arXiv0803.0715
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WMAP MCMC Residuals

• "Base" power-law foreground model
• lt?dgt 1.8 ? 0.3
• lt?sgt -3 at high lat, -2.7 in plane
• High-lat residuals consistent with noise
• Galactic plane is more complicated
• Data brighter than model at Ka
• Data fainter than model at Q
• Spinning dust or synch curvature?

Gold et al. 2008, arXiv0803.0715
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Gibbs Sampling

• Instead of removing foreground templates at each
  channel, we parametrize the CMB, synchrotron, and
  dust in each low res pixel.
• Each pixel has 6 amplitudes. Sky has 30
  synchrotron indices in regions. Dust index is
  fixed.
• Gibbs sample the posterior distribution, split
  into amplitude slices and index slices
• Produces marginalized CMB map and errors, can be
  used for llt23 exact likelihood.

Q
U
Dust
Synchrotron
CMB
? 0.087? 0.017 (template cleaning) ? 0.100?
0.018 (Gibbs sampling) (WMAP5, Dunkley et al
2008) Good agreement but future investigation of
priors and sampling method will be useful
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Warning More Parameters Ahead!
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Frequency Dependence of Dust Polarization
Add parameters to include sub-mm bands

• Molecular Clouds Minimum near 350 GHz
• Different (local) environments
• Different dust species

Vaillancourt 2002, ApJ, 142, 53

• Diffuse Cirrus Monotonic change
• Same environment, different species

Hildebrand Kirby 2004, ASP Conf Series 309, 515
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Polarized Dust Polarization Angle
Heiles 2000, AJ, 119, 923 Page et al. 2007, ApJS,
170, 335, 327 Kogut et al. 2007, ApJ, 665, 355
Dust/Synchrotron Correlation
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Component Separation

• "Machinery" is under control
• Algorithms exist and work as advertised
• Multiple methods yield consistent results
• Demonstrated ability to "compress" data

• The problem is that darned data!
• Considerable uncertainty in fitted parameters
• Intensity data Component confusion
• Polarization data Signal-to-Noise ratio
• New parameters vs new components
• Biggest need is more data!

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Who knows what evil lurks in the heart of data?
The Shadow Knows