from theoretical promise to observational reality ... · from theoretical promise to observational...
TRANSCRIPT
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From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography
Adrian Liu, MIT
100
0.02 0.04 0.06 0.08
101
10 mK
1 K 100 mK
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The Vision for 21cm Tomography
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Vision
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Vision
Hydrogen atom
CMB
CMB
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Vision
E.g. Spatial curvature:!WMAP+SDSS: !"tot= 0.01 Planck: !"tot= 0.003#21cm: !"tot=0.0002
Mao, Tegmark, McQuinn, Zahn, Zaldarriaga 2008
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Vision
Cl P(k) and much
more!
Image credit: WMAP team Image credit: Trac & Cen 2007
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Vision
Image credit: Pritchard & Loeb 2010
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The Problem
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Image credit: de Oliveira-Costa et. al. 2008
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Outline • Precision Calibration for Precision Cosmology
– What makes calibration a new problem in 21cm tomography?
– Why redundant calibration? What are some of its subtleties?
– How does redundant calibration relate to traditional algorithms?
• Precision Subtraction for Precision Cosmology – What are some “traditional” proposals for 21cm
foreground subtraction? – Can we do better?
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Precision Calibration for Precision Cosmology
A. Liu, M. Tegmark, S. Morrison, A. Lutomirski, M. Zaldarriaga, MNRAS 408, 1029, Oct. 2010
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21cm tomography requires interferometer arrays
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In principle, each baseline probes a Fourier mode of the sky, but…
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In principle, each baseline probes a Fourier mode of the sky, but…
Measured signal
Complex gain:
True sky signal
Instrumental noise
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21cm tomography requires compact, redundant interferometer arrays
• Traditional radio astronomy: – Imaging bright (SNR >> 1) localized sources in a dim
background
Image credit: ASKAP website
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21cm tomography requires compact, redundant interferometer arrays
• Traditional radio astronomy – Imaging bright (SNR >> 1) localized sources in a dim
background
• 21cm tomography: – Measuring dim fluctuations (SNR << 1) over a large area
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Unlike traditional interferometers, 21cm tomography experiments have
many redundant baselines
Image credit: Parsons et. al. 2011
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After redundant calibration, redundant baselines give
identical results
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Not (just) a theorist’s dream!
Noordam & de Bruyn 1982
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How does this compare to other calibration schemes?
• “Traditional” point source calibration – Assumes field of view contains a single point source.
• Self calibration – Construct a model of the sky, predict measurements, iterate.
• Redundant calibration – Requires a redundant array. – Independent of the sky.
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Can we do better? • Better characterization of calibration errors.
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Average errors in $ for an 18 by 18 square array
0.0278
0.0280
0.0282
0.0284
0.0286
0.0288
Antenna x-location
Ant
enna
y-lo
catio
n
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Average errors in $ as a function of array size
101 102 103 10-2
10-1
100
Number of antenna elements in square array
Ave
rage
err
or in
$#
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Can we do better? • Better characterization of calibration errors. • Old, logarithmic version of redundant calibration is
biased; new linear version is unbiased.
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Traditional methods exhibit a bias
Simulated input $#
Ens
embl
e av
erag
ed
outp
ut $#
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Our linearized methods remove the bias
Simulated input $#
Ens
embl
e av
erag
ed
outp
ut $#
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Can we do better? • Better characterization of calibration errors. • Old, logarithmic version of redundant calibration is
biased; new linear version is unbiased. • Correcting for deviations from perfect redundancy.
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Taylor expand the Fourier sky
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Taylor expand the Fourier sky
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0.4 10-4
10-3
10-1
Err
or in
reco
vere
d $#
10-2
0.8 1.2 1.6 2.0 2.4
Antenna position errors in units of %#
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Near-redundancy may be good enough
Image Credit: C. Williams
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Can we do better? • Better characterization of calibration errors. • Old, logarithmic version of redundant calibration is
biased; new linear version is unbiased. • Correcting for deviations from perfect redundancy. • Self calibration and redundant calibration are special
cases that complement each other.
Redundant calibration
Self calibration
• Sky independent • Baselines must be
perfectly redundant
• Solves for sky model
• Any baselines
More baseline corrections More Taylor expansion terms
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Precision Foreground Subtraction for Precision Cosmology
AL, Tegmark, arXiv:1103.0281, submitted to MNRAS AL, Tegmark, Phys. Rev. D 83, 103006 (2011) AL, Tegmark, Bowman, Hewitt, Zaldarriaga,
MNRAS 398, 401 (2009) AL, Tegmark, Zaldarriaga, MNRAS 394, 1575 (2009)
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Foreground Modeling
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Principal components of the sky
Image credit: de Oliveira-Costa et. al. 2008
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Principal components of the sky
1st principal component
2nd principal component
3rd principal component
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Can we do better? • Understand, using a simple theoretical toy model, why
the foregrounds are describable using so few components.
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Understanding “why so few” components
• Start with a simple but realistic model.
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Understanding “why so few” components
• Start with a simple but realistic model. • Write down covariance function.
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Understanding “why so few” components
• Start with a simple but realistic model. • Write down covariance function. • Non-dimensionalize to get correlation function.
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Understanding “why so few” components
• Start with a simple but realistic model. • Write down covariance function. • Non-dimensionalize to get correlation function. • To a good approximation, correlation function fits the
following form with coherence length &c=560 MHz!
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Understanding “why so few” components
• Start with a simple but realistic model. • Write down covariance function. • Non-dimensionalize to get correlation function. • To a good approximation, correlation function fits the
following form with coherence length &c=560 MHz! • Find principal components/eigenfunctions:
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Foreground Subtraction
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Method #1: Line-of-Sight Polynomial Subtraction
E.g. Wang et. al. (2006), Bowman et. al. (2009), AL et. al. (2009a,b), Jelic et. al. (2008), Harker et. al. (2009, 2010).
Foregrounds
z
%#
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AL, Tegmark, Zaldarriaga, MNRAS 394, 1575 (2009) AL, Tegmark, Bowman, Hewitt, Zaldarriaga, MNRAS 398, 401 (2009). See also: Wang et. al. (2006), Bowman et. al. (2009), Jelic et. al. (2008), Harker et. al. (2009,2010).
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Can we do better?
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Foreground Subtraction Wish-list
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Foreground Subtraction Wish-list • Lossless
Cleaned Data
Raw Data
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Foreground Subtraction Wish-list • Lossless • Small “vertical” error bars
P(k)
k
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Foreground Subtraction Wish-list • Lossless • Small “vertical” error bars • Small “horizontal” error bars/mode-mixing
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Foreground Subtraction Wish-list • Lossless • Small “vertical” error bars • Small “horizontal” error bars/mode-mixing Im
age credit: Tegmark, Z
aldarriaga ‘09
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Foreground Subtraction Wish-list • Lossless • Small “vertical” error bars • Small “horizontal” error bars/mode-mixing Im
age credit: SDSS C
ollaboration
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Foreground Subtraction Wish-list • Lossless • Small “vertical” error bars • Small “horizontal” error bars/mode-mixing
AL
, Tegmark, Phys. R
ev. D 83, 103006 (2011)
100
101
10-1 10-2 10-1 100
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Foreground Subtraction Wish-list • Lossless • Small “vertical” error bars • Small “horizontal” error bars/mode-mixing • No additive noise/foreground bias
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Foreground Subtraction Wish-list • Lossless • Small “vertical” error bars • Small “horizontal” error bars/mode-mixing • No additive noise/foreground bias
Foregrounds
Uncleaned Signal
AL, M
. Tegmark, M
. Zaldarriaga, M
NR
AS 394, 1575 (2009)
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Method #2: Fourier space filtering/Inverse variance weighting
For similar methods, see also N. Petrovic & S.P. Oh, MNRAS 413, 2103 (2011) G. Paciga et. al., MNRAS 413, 1174 (2011)
Filte
r
0.2
0.4
0.6
0.8
1.0
0.0 0.0 0.5 1.0 1.5 2.0
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Method #2: Fourier space filtering/Inverse variance weighting
For similar methods, see also N. Petrovic & S.P. Oh, MNRAS 413, 2103 (2011) G. Paciga et. al., MNRAS 413, 1174 (2011)
Filte
r
0.2
0.4
0.6
0.8
1.0
0.0 0.0 0.5 1.0 1.5 2.0
Rapidly fluctuating modes retained
Smooth modes suppressed
Hig
h fo
regr
ound
sce
nario
Foregroundless scenario
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Back to our wishlist…
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Lossless?
Line-of-Sight Polynomial Subtraction --- Lossy
Inverse Variance Subtraction --- Lossless
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Biased?
Line-of-Sight Polynomial Subtraction --- Biased in literature, fixable
Inverse Variance Subtraction --- Unbiased
![Page 64: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/64.jpg)
Biased?
Line-of-Sight Polynomial Subtraction --- Biased!
Inverse Variance Subtraction --- Unbiased!
But remember…
100
101
10-2 10-1 10-1
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
Polynomial Line-of-Sight Method
Log10 T (in mK)
![Page 65: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/65.jpg)
Biased?
Line-of-Sight Polynomial Subtraction --- Biased!
Inverse Variance Subtraction --- Unbiased!
But remember…
100
101
10-2 10-1 10-1
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
Inverse Variance Method
Log10 T (in mK)
![Page 66: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/66.jpg)
Mode Mixing?
Line-of-Sight Polynomial Subtraction --- Yes
Inverse Variance Subtraction --- Yes, but less.
![Page 67: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/67.jpg)
Mode Mixing?
Line-of-Sight Polynomial Subtraction --- Yes
Inverse Variance Subtraction --- Yes, but less. 100
101
10-2 10-1
100 10-1
1.0
0.6 0.5 0.4 0.3 0.2 0.1
0.7 0.8 0.9
![Page 68: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/68.jpg)
Mode Mixing?
Line-of-Sight Polynomial Subtraction --- Yes
Inverse Variance Subtraction --- Yes, but less. 100
101
10-2 10-1
100 10-1
1.0
0.6 0.5 0.4 0.3 0.2 0.1
0.7 0.8 0.9
![Page 69: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/69.jpg)
Mode Mixing?
Line-of-Sight Polynomial Subtraction --- Yes
Inverse Variance Subtraction --- Yes, but less. 100
101
10-2 10-1
100 10-1
1.0
0.6 0.5 0.4 0.3 0.2 0.1
0.7 0.8 0.9
![Page 70: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/70.jpg)
Measurement errors?
Line-of-Sight Polynomial Subtraction --- Larger
Inverse Variance Subtraction --- Smaller
Consider errors on the quantity
![Page 71: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/71.jpg)
Errors?
Line-of-Sight Polynomial Subtraction --- Larger
Inverse Variance Subtraction --- Smaller
100
0.02 0.04 0.06 0.08
101
<10 mK
3.0
2.5
2.0
1.5
1 Log10 T (in mK)
Errors with no foregrounds
20 mK
![Page 72: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/72.jpg)
Errors?
Line-of-Sight Polynomial Subtraction --- Larger
Inverse Variance Subtraction --- Smaller
100
0.02 0.04 0.06 0.08
101
10 mK
1 K 100 mK
3.0
2.5
2.0
1.5
1 Log10 T (in mK)
Errors using Line of Sight Method
![Page 73: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/73.jpg)
Errors?
Line-of-Sight Polynomial Subtraction --- Larger
Inverse Variance Subtraction --- Smaller
100
0.02 0.04 0.06 0.08
101
<10 mK
130 mK
3.0
2.5
2.0
1.5
1 Log10 T (in mK)
Errors using Inverse Variance Method
30 mK
![Page 74: From Theoretical Promise to Observational Reality ... · From Theoretical Promise to Observational Reality: Calibration and Foreground Subtraction in 21cm Tomography Adrian Liu, MIT](https://reader034.vdocuments.net/reader034/viewer/2022043012/5fabea4e40ec6a377b40e902/html5/thumbnails/74.jpg)
Precision Calibration for Precision Cosmology
• Foreground modeling – Know why foregrounds are describable by ~3 components
• Foreground subtraction and power spectrum estimation – Inverse variance foreground subtraction is lossless, unbiased,
has less mode-mixing, and gives smaller error bars.
• Redundant calibration – Better characterization of errors, removal of systematic
biases, correction for non-exact redundancy. – Complements self-calibration
Precision Subtraction for Precision Cosmology Foreground ^
And more!