How to do 3x2pt analysis of SZ and galaxies Eiichiro Komatsu - - PowerPoint PPT Presentation

How to do 3x2pt analysis of SZ and galaxies

Eiichiro Komatsu (Max-Planck-Institut für Astrophysik) “Methods for Statistical Inference”, Institut Henri Poincaré October 25, 2018

• Ando, Benoit-Lévy, EK (2018)
• Bolliet, Comis, EK, Macias-Pérez (2018)
• Makiya, Ando, EK (2018)

2MASS Auto SZ Auto SZ-2MASS Cross

How to do 3x2pt analysis of SZ and galaxies

How to remove CMB foregrounds with spatially varying spectra

If I had some time left towards the end, I would also talk about:

• Ichiki et al., to be submitted on November 9

This paper was completed during this trimester program: Merci beaucoup for hospitality!

Auto 2-point Correlation

TCMB(1) x TCMB(2) ngal(1) x ngal(2) CMB LSS Cosmology Cosmology

“Joint Constraints”

1 2 1 2

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Hubble const. H0 [km/s/Mpc]

Dark Matter Density, Ωch2

CMB+LSS CMB +Supernova CMB Only WMAP, final result

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TCMB(1) x TCMB(2) ngal(1) x ngal(2) CMB LSS TCMB(1) x ngal(2) ngal(1) x TCMB(2) Why cross-correlation? 1 2 1 2

Cross 2-point Correlation

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Joint Analysis

• Joint analysis including all cross-correlations

between, e.g., CMB, spectroscopic LSS, and imaging LSS

• let us write the posterior of cosmological parameters,

given the data, as P(parameters | data)

• Usually done: P(parameters|data) = P1(parameters|CMB)

x P2(parameters|specLSS) x P3(parameters|imagingLSS)

• What needs to be done: P(parameters | data)

= P(parameters | CMB, specLSS, imagingLSS)

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What creates cross-correlations?

CMB Lensed CMB ISW Thermal SZ Kinetic SZ SpecLSS 3D galaxy map Velocity fields ImagingLSS Matter density

P(param.|data) = P(param. | CMB, specLSS, imagingLSS)

P(param.|data) = P1(param.|CMB) x P2(param.|specLSS) x P3(param.|imagingLSS)

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3x2pt

• The term popularised by the Dark Energy Survey

(DES) collaboration (of which I am not a part)

• Auto correlation of weak lensing
• Auto correlation of galaxies
• Cross correlation of galaxies-lensing
• If you have two tracers of the same underlying

matter density field, you should do all three!

Prat, Sanchez, et al. (2018) Troxel, et al. (2018) Elvin-Poole, et al. (2018) Krause, Eifler, et al. (2018)

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First step toward the goal:

DES Collaboration Lens auto Galaxies auto + Gal-lens Cross

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Why Cross-correlation? Two Signal-to-Noise Regimes

• Consider that we correlate tracers X and Y, both

probing the same underlying matter distribution

• 3x2pt: <XX>, <YY>, and <XY>
• 1. When X has a high signal-to-noise, but Y

has a low signal-to-noise

• Then <XY> is always more powerful than <YY>

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X: CMB Temperature Y: CMB Polarisation

<XX>: Temperature Auto

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X: CMB Temperature Y: CMB Polarisation

<YY>: Polarisation Auto

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X: CMB Temperature Y: CMB Polarisation

<XY>: Cross!

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Proof

Variance of the temperature(T)-polarisation(E) correlation: When T is signal dominated but E is noise dominated: Thus, (Signal-to-noise)2 of <TE> vs <EE>:

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Proof

Variance of the temperature(T)-polarisation(E) correlation: When T is signal dominated but E is noise dominated: Thus, (Signal-to-noise)2 of <TE> vs <EE>:

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Proof

Variance of the temperature(T)-polarisation(E) correlation: When T is signal dominated but E is noise dominated: Thus, (Signal-to-noise)2 of <TE> vs <EE>:

cross-correlation coefficient

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Proof

Variance of the temperature(T)-polarisation(E) correlation: When T is signal dominated but E is noise dominated: Thus, (Signal-to-noise)2 of <TE> vs <EE>:

cross-correlation coefficient

>> 1!

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Successful Examples

• Noisy CMB polarisation data buried in noise, cross-

correlated with high S/N temperature data

• Noisy Integrated Sachs-Wolfe (ISW) effect buried in

the primary CMB, cross-correlated with high S/N galaxy maps

• Noisy 21cm intensity mapping buried in noise and

junk, cross-correlated with high S/N galaxy maps

• Etc. If you have noisy data (e.g., stochastic

gravitational waves!), cross-correlating is the way to go until you have higher S/N!

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Why Cross-correlation? Two Signal-to-Noise Regimes

• Consider that we correlate tracers X and Y, both

probing the same underlying matter distribution

• 3x2pt: <XX>, <YY>, and <XY>
• 2. When both X and Y have high signal-to-noise
• Then the statistical constraining power of <XY> is

usually lower than that of <XX> and <YY>, but <XY> is often useful for breaking degeneracy with nuisance parameters affecting X or Y alone “Constraining known unknowns” (Licia Verde)

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Why Cross-correlation? Redshift Tomography!

• Consider that a map Y contains astrophysical

signals integrated over all redshifts

• And we have a number of other maps, Xi, which

contain objects within a known redshift range zmin,i < z < zmax,i

• Then cross-correlating them <XiY> allows to

measure the signals in Y as a function redshift: Redshift Tomography

In this talk, I present the way to do this for Y = SZ map

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Why Cross-correlation? Mass Tomography!

• Consider that a map Y contains astrophysical

signals integrated over all masses

• And we have a number of other maps, Xi, which

contain objects within a known mass range Mmin,i < M < Mmax,i

• Then cross-correlating them <XiY> allows to

measure the signals in Y as a function mass: Mass Tomography

And you can do this for any other quantities you wish, as long as you have appropriate tracers

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An Intuitive Example

• You have N galaxies in your galaxy catalog with

known redshifts (or masses or anything else)

• 3-dimensional positions of N galaxies
• You have an SZ map (“Y”). Then you stack signals
• f Y at the locations of galaxies

remove the mean

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An Intuitive Example

• You have N galaxies in your galaxy catalog with

known redshifts (or masses or anything else)

• 3-dimensional positions of N galaxies
• You have an SZ map (“Y”). Then you stack signals
• f Y at the locations of galaxies

remove the mean

A fancier way of writing it…

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An Intuitive Example

• You have N galaxies in your galaxy catalog with

known redshifts (or masses or anything else)

• 3-dimensional positions of N galaxies
• You have an SZ map (“Y”). Then you stack signals
• f Y at the locations of galaxies

A fancier way of writing it… continuous limit

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An Intuitive Example

• You have N galaxies in your galaxy catalog with

known redshifts (or masses or anything else)

• 3-dimensional positions of N galaxies
• You have an SZ map (“Y”). Then you stack signals
• f Y at the locations of galaxies

A fancier way of writing it… continuous limit

Ω

cross corr. function

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An Intuitive Example

• You have N galaxies in your galaxy catalog with

known redshifts (or masses or anything else)

• 3-dimensional positions of N galaxies
• You have an SZ map (“Y”). Then you stack signals
• f Y at the locations of galaxies

Ω

cross corr. function cross power spectrum

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Where is a galaxy cluster?

Subaru image of RXJ1347-1145 (Medezinski et al. 2010) http://wise-obs.tau.ac.il/~elinor/clusters

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Where is a galaxy cluster?

Subaru image of RXJ1347-1145 (Medezinski et al. 2010) http://wise-obs.tau.ac.il/~elinor/clusters

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Subaru image of RXJ1347-1145 (Medezinski et al. 2010) http://wise-obs.tau.ac.il/~elinor/clusters

Subaru

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Hubble image of RXJ1347-1145 (Bradac et al. 2008)

Hubble

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Chandra X-ray image of RXJ1347-1145 (Johnson et al. 2012)

Chandra

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Chandra X-ray image of RXJ1347-1145 (Johnson et al. 2012) ALMA Band-3 Image of the Sunyaev-Zel’dovich effect at 92 GHz (Kitayama et al. 2016)

ALMA!

5” resolution

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1σ=17 μJy/beam =120 μKCMB

• T. Kitayama

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Multi-wavelength Data

Optical:

• 102–3 galaxies
• velocity dispersion
• gravitational lensing

X-ray:

• hot gas (107–8 K)
• spectroscopic TX
• Intensity ~ ne2L

IX = Z dl n2

eΛ(TX)

SZ [microwave]:

• hot gas (107-8 K)
• electron pressure
• Intensity ~ neTeL

ISZ = gν σT kB mec2 Z dl neTe projected thermal e– pressure

Full-sky Thermal Pressure Map

North Galactic Pole South Galactic Pole Planck Collaboration

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• SZ map = Detected sources + undetected sources

+ diffuse emission

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• You can count objects; but you can also do

intensity mapping! [see Eric Switzer’s talk]

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• You can count objects; but you can also do

intensity mapping!

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But no redshift information from SZ alone

2MASS Redshift Survey

• ~40K galaxies with the median redshift of 0.02

Huchra et al. (2012)

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2MASS Redshift Survey

• ~40K galaxies with the median redshift of 0.02

Huchra et al. (2012)

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Cross-correlation extracts SZ signals at z<0.1

Cross-power!

Makiya, Ando & EK (2018)

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• R. Makiya

(Kavli IPMU)

Cross-power!

Makiya, Ando & EK (2018)

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• R. Makiya

(Kavli IPMU)

But, what do we learn from this? We need auto power spectra. We need 3x2pt!

2MRS Auto Power

Ando, Benoit-Lévy & EK (2018)

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• S. Ando

(GRAPPA, U. Amsterdam)

2MRS Auto Power

• ~40,000 galaxies over full sky, but a lot of power on

all scales, indicating extremely strong clustering

• Far from Gaussian. We need to include non-

Gaussian error bars [connected trispectrum]

• Many “nuisance” parameters [nuisance for

cosmologists but not necessarily for others]

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Nuisance parameters, or: How galaxies populate halos?

• Halo model (Seljak 2000)

Projecting 3-d galaxy power spectrum onto 2-d:

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• Halo model (Seljak 2000)

5 nuisance parameters (just for galaxies)

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Nasty degeneracy among nuisance parameters… But who cares, we just marginalise

Dominated by 1-halo term in most

• f the angular scales => Good for

cross-correlation with SZ clusters

Ando, Benoit-Lévy & EK (2018)

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Why should we believe this?

• Nuisance parameters - too phenomenological? And

horrible posterior… How do we know that these numbers make any sense?

• Uniqueness of a low-z survey like 2MASS: we

actually see these parameters in the sky, because we resolve all galaxy clusters/groups!

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Satellite galaxy radial profiles

Ando, Benoit-Lévy & EK (2018)

Number of satellites per halo

Ando, Benoit-Lévy & EK (2018)

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SZ Auto Power

• Far from Gaussian.

We need to include non- Gaussian error bars [connected trispectrum]

• When fitting, the Planck team

used Gaussian covariance ignoring the non-Gaussian term

• We also have a bunch of

nuisance parameters

Bolliet, Comis, EK, Macias-Pérez (2018) with non-Gaussian error without

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• B. Bolliet

Planck Collaboration (2016)

Foregrounds = Nuisance Parameters

Interpreting Planck’s SZ Power Spectrum

• Planck Collaboration (2015)
• Ignored trispectrum; Nuisance parameters

marginalised over

• Horowitz & Seljak (2017); Salvati et al. (2018)
• Included trispectrum; Nuisance parameters not

marginalised over

• Hurier & Lacasa (2017)
• Included trispectrum; Nuisance parameters not

marginalised over but performed cleaning in Cl space

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Interpreting Planck’s SZ Power Spectrum

• Planck Collaboration (2015)
• Ignored trispectrum; Nuisance parameters

marginalised over

• Horowitz & Seljak (2017); Salvati et al. (2018)
• Included trispectrum; Nuisance parameters not

marginalised over

• Hurier & Lacasa (2017)
• Included trispectrum; Nuisance parameters not

marginalised over but performed cleaning in Cl space

We vary/marginalise over everything:

• SZ model parameters
• All relevant cosmological parameters
• Nuisance parameters

with the trispectrum that depends also

• n the SZ+cosmological parameters

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SZ power is lower than Planck

Bolliet, Comis, EK, Macias-Pérez (2018) with trispectrum without

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Simple Interpretation

• Randomly-distributed point sources

= Poisson spectrum = ∑i(fluxi)2 / 4π multipole Cl [not “l2Cl”]

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EK, Kitayama (1999); EK, Seljak (2002)

Simple Interpretation

• Extended sources = the power

spectrum reflects intensity profiles multipole Cl [not “l2Cl”]

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EK, Kitayama (1999); EK, Seljak (2002)

Multipole l(l+1)Cl/2π [μK2]

>2x1015 Msun >1015 Msun >5x1014 Msun >5x1013 Msun Adding smaller clusters

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Planck Mass Bias

• The key ingredient of the power spectrum is a

profile of thermal pressure of electrons

C` = Z dz dV dz Z dM dn dM |y`(M, z)|2

˜ M500c = M500c,true/B

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Inferred SZ Amplitude

Bolliet, Comis, EK, Macias-Pérez (2018)

2.6% measurement! Essentially cosmological model-independent

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Inferred SZ Amplitude

Bolliet, Comis, EK, Macias-Pérez (2018)

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˜ M500c = M500c,true/B

Mass Bias [B=(1–b)–1]

Joint Analysis

2MASS Auto

Makiya, Ando & EK (2018)

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Joint Analysis

SZ Auto

Makiya, Ando & EK (2018)

Joint Analysis

Cross!

Makiya, Ando & EK (2018)

Full covariance including the trispectrum term

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Marignalise over EVERYTHING!

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[for Planck TT+lowP+lensing]

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Makiya, Ando & EK (2018) SZ Auto SZ-2MASS Cross + 2MASS Auto

No obvious systematics for B from the SZ auto power spectrum alone

Mass-bias Consistency

• B = 1.54 ± 0.098

[for Planck TT+lowP+lensing]

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Makiya, Ando & EK (2018) SZ Auto SZ-2MASS Cross + 2MASS Auto

Mass-bias Consistency

• B = 1.54 ± 0.098
• 1–b =B–1= 0.649 ± 0.041

Similar value was inferred from the SZ cluster number counts: additional consistency

Mass Dependence of SZ Auto

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Makiya, Ando & EK (2018)

Mass Tomography

Mass Dependence of Cross

Cross is sensitive to less massive halos: We can use this to explore the mass bias as a function of mass!

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Makiya, Ando & EK (2018)

Mass Tomography

Mass Tomography

Makiya, Ando & EK (2018)

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Pressure ~ (M/B)2/3+αp

Summary, Part I

• If you have two data sets, do 3x2pt
• Not just auto alone, or cross alone. Many people do just autos or cross
• 3x2pt analysis requires good knowledge of both data sets. Challenging

but rewarding! Can your machine learn to do this too?

• Novelty in our 3x2pt of SZ and galaxies
• Parameter-dependent trispectrum in the full covariance
• Marginalised over everything; deeper understanding of 2MASS auto

and SZ auto spectra

• Joint analysis revealed mass-(in)dependent of mass bias; and

consistency of mass bias inferred from auto and cross

• Useful (and rewarding)!

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• To be submitted on November 9

Do I have 5 more minutes?

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“Internal Template” Cleaning

• CMB = [ Map(140GHz) – α x Map(other freq) ] / [1– α]

Jean-François said this method does not assume anything about foregrounds, but it does: It assumes that the spectral property of foreground does not depend on sky directions

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Example: Synchrotron

• Power-law index: δTsynch ~ νβ

from WMAP 23 GHz and Haslam 408 MHz

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“Tensor-to-scalar Ratio” Parameter, r

Bias of r~2x10–3 if β is assumed constant Katayama, EK (2011)

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“Tensor-to-scalar Ratio” Parameter, r

Bias of r~2x10–3 if β is assumed constant Katayama, EK (2011)

My first reaction: Ah, not bad at al!

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ESA

2025– [proposed]

JAXA

LiteBIRD

2027– [proposed]

Target: δr<0.001 (68%CL)

+ possible participations

from USA, Canada, Europe

ESA

2025– [proposed]

JAXA

LiteBIRD

2027– [proposed]

Target: δr<0.001 (68%CL)

+ possible participations

from USA, Canada, Europe

Gosh, we need to do better

Delta-map Method

• Foreground spectrum varies spatially due to spatially varying

spectral parameters p (e.g., β)

• Inference on p(n)? Time consuming/non-linear…
• Spectral variations are actually not so large. (Bias in the tensor-to-

scalar ratio parameter is only of order r~O(10–3))

• Let’s Taylor-expand!

Ichiki, Kanai, Katayama, EK (to be submitted)

• bserved

polarisation foreground emission at frequency ν* foreground spectrum

[pI: Ith parameter]

• Key: the fluctuation part simply acts as an additional foreground

component with spatially-uniform coefficients, Dν,I

• We can then form a linear combination of frequencies to remove

both [Qf,Uf] and δp[Qf,Uf]. Easy!

• Long story short: this eliminates bias in r to

negligible level. But…

Ichiki, Kanai, Katayama, EK (to be submitted) foreground contribution mean spectrum fluctuation

Delta-map Method

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What am I doing?

• This sounds a bit ad hoc
• Sure, maybe it is unbiased, but is it minimum-

variance?

• What is the Bayesian foundation for this method?
• I can answer that

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CMB Solutions

• Simplest: Uniform spectral index. We need at least

two frequencies to remove it. The CMB solution is This is nothing but the simplest template cleaning Bottom-up

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CMB Solutions

• Spatially-varying power law. We need at least

three frequencies to remove it. The CMB solution is Bottom-up

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Now, likelihood

• m: Data (maps)
• D: “mixing matrix” (frequency dependence of stuff

in the sky) This depends on foreground parameters

• s: Signal (CMB and foregrounds)
• N: Noise covariance of data

(we expand this to first order in perturbation)

Top-down

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Maximum Likelihood

When we have a uniform spectral index and two frequencies, this gives the internal template solution: Top-down

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Expanding D to 1st order

Top-down

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Expanding D to 1st order

Top-down When we have a spatially-varying power law and three frequencies, this gives what we saw already:

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Posterior

• OK, these cleaned maps are ML solutions when the

number of frequencies is equal to the number of

• components. Sweet.
• What about the posterior?
• Let’s derive it in a heuristic way

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Posterior [Simplest]

Bottom-up

• Square and average to get covariance and plug it

into a Gaussian…

–2ln(posterior[r,β|m]) =

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Likelihood: Prior: [Flat on the mixing matrix] Top-down. We do it better now. Marginalise over sCMB because all we care is its signal covariance SCMB

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Top-down

Now what?

What you do from now on will decide which foreground-removal method you are using. (Almost?) all foreground removal methods in the literature can be categorised by the way you go from here. (Flavien Vansyngel’s thesis, 2014)

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Top-down

Now what?

We take the ML estimate of sf:

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Top-down

Now what?

We take the ML estimate of sf: And plug it into the top:

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Top-down We take the ML estimate of sf: And plug it into the top:

Looks terrible. However… actually… when we have, e.g., a uniform spectral index and two frequencies…

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Top-down

Looks terrible. However… actually… when we have, e.g., a uniform spectral index and two frequencies…

where

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That’s it! Summary:

• We can account for spatially-varying foreground

spectra and estimate the tensor-to-scalar ratio by

• I talked only about a power-law synchrotron, but you can use

this for any number of other components. All you need is the derivative of emission law as a function of spectral parameters.

• For example…

Ichiki, Kanai, Katayama, EK (to be submitted)

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• And, we figured out how to account for polarised

anomalous microwave emission (AME; spinning dust) and frequency decorrelation due to a superposition of varying spectra. If you are interested, wait for November!

Back-up Slides

Ando, Benoit-Lévy & EK (2018)

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Redshift Dependence

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Makiya, Ando & EK (2018)

Redshift Dependence

High-ell data of Compton-Y auto is the key. But… foreground contamination

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Makiya, Ando & EK (2018)

Z-dependence Poorly Constrained

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Makiya, Ando & EK (2018)

1-point PDF fits!! Dolag, EK, Sunyaev (2016)

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