How to do 3x2pt analysis of SZ and galaxies

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

3

H u b b le c o n s t. H

[k m /s /Mp c ]

Dark Matter Density, Ω c h ²

CMB+LSS CMB

+Supernova

CMB Only WMAP, final result

4

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

5

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)

6

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)

7

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)

8

First step toward the goal:

DES Collaboration

Lens auto

Galaxies auto + Gal-lens Cross

9

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

has a low signal-to-noise

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

10

X: CMB Temperature Y: CMB Polarisation

<XX>:

Temperature Auto

11

X: CMB Temperature Y: CMB Polarisation

<YY>:

Polarisation Auto

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X: CMB Temperature Y: CMB Polarisation

<XY>:

Cross!

13

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>:

14

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>:

15

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

16

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!

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!

18

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) 19

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

20

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

21

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 of Y at the locations of galaxies

remove the mean

22

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 of Y at the locations of galaxies

remove the mean

A fancier way of writing it…

23

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 of Y at the locations of galaxies

A fancier way of writing it… continuous limit

24

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 of Y at the locations of galaxies

A fancier way of writing it… continuous limit

Ω

cross corr. function

25

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 of Y at the locations of galaxies

Ω

cross corr. function

cross power spectrum

26

Where is a galaxy cluster?

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

27

Where is a galaxy cluster?

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

28

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

Subaru

29

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

Hubble

30

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

Chandra

31

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

32

1σ=17 μJy/beam

=120 μKCMB

T. Kitayama

33

Multi-wavelength Data

Optical:

•10^2–3 galaxies

•velocity dispersion

•gravitational lensing

X-ray:

•hot gas (10^7–8 K)

•spectroscopic TX

•Intensity ~ ne2L

I_X = Z

dl n²_e⇤(T_X)

SZ [microwave]:

•hot gas (10^7-8 K)

•electron pressure

•Intensity ~ neTeL

I_SZ = g_⌫ ^T k_B m_ec²

Z

dl n_eT_e projected thermal e^– pressure

Full-sky Thermal Pressure Map

North Galactic Pole South Galactic Pole

Planck Collaboration ³⁵

• SZ map = Detected sources + undetected sources

+ diffuse emission 36

• You can count objects; but you can also do

intensity mapping! [see Eric Switzer’s talk] 37

• You can count objects; but you can also do

intensity mapping! 38

But no redshift

information from SZ alone

2MASS Redshift Survey

• ~40K galaxies with the median redshift of 0.02

Huchra et al. (2012)

39

2MASS Redshift Survey

• ~40K galaxies with the median redshift of 0.02

Huchra et al. (2012)

40

Cross-correlation extracts SZ signals at z<0.1

Cross-power!

41

R. Makiya (Kavli IPMU)

Cross-power!

42

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)

43

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

• Many “nuisance” parameters [nuisance for cosmologists but not necessarily for others]

44

Nuisance parameters, or:

How galaxies populate halos?

• Halo model (Seljak 2000)

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

45

• Halo model (Seljak 2000)

5 nuisance parameters (just for galaxies)

46

47

Nasty degeneracy among nuisance parameters…

But who cares,

we just marginalise

Dominated by 1-halo term in most of the angular scales => Good for cross-correlation with SZ clusters

Ando, Benoit-Lévy & EK (2018)

48

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

because we resolve all galaxy clusters/groups!

49

Satellite galaxy radial pr ofiles

Number of satellites per halo

Ando, Benoit-Lévy & EK (2018)

51

SZ Auto Power

• Far

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

52

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

53

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 on the SZ+cosmological parameters

54

SZ power is lower than Planck

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

with

trispectrum without

55

Simple Interpretation

• Randomly-distributed point sources

= Poisson spectrum = ∑i(fluxi)² / 4π

multipole Cl [not “l2 Cl”]

56

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

Simple Interpretation

• Extended sources = the power

spectrum reflects intensity profiles

multipole Cl [not “l2 Cl”]

57

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

Multipole l(l+1)C l /2 π [ μ K 2 ]

>2x10¹⁵ Msun

>10¹⁵ Msun

>5x10¹⁴ Msun

>5x10¹³ Msun

Adding smaller clusters

58

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

M ˜ _500c = M _500c,true /B

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Inferred SZ Amplitude

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

2.6% measurement!

Essentially cosmological model-independent

60

Inferred SZ Amplitude

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

61

M ˜ _500c = M _500c,true /B

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Mass Bias [B=(1–b) ^–1 ]

Joint Analysis

2MASS Auto

Makiya, Ando & EK (2018)

62

Joint Analysis

SZ Auto

Makiya, Ando & EK (2018)

Joint Analysis

Cross!

Makiya, Ando & EK (2018)

Full covariance including the trispectrum term

65

Marignalise over EVERYTHING!

66

[for Planck TT+lowP+lensing]

67

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]

68

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

69

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!

70

Makiya, Ando & EK (2018)

Mass Tomography

Mass Tomography

Makiya, Ando & EK (2018)

71

Pressure ~ (M/B)

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)! ⁷²

• To be submitted on November 9 Do I have 5 more minutes?

73

“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 ⁷⁴

Example: Synchrotron

• Power-law index: δTsynch ~ ν^β

from WMAP 23 GHz and Haslam 408 MHz

75

“T ensor -to-scalar Ratio” Parameter , r

Bias of r~2x10^–3

if β is assumed constant

Katayama, EK (2011)

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“T ensor -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!

77

ESA

2025– [proposed]

JAXA

LiteBIRD

2027– [proposed]

Target: δr<0.001 (68%CL) +

from USA, Canada, Europe

ESA

2025– [proposed]

JAXA

LiteBIRD

2027– [proposed]

Target: δr<0.001 (68%CL) +

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)

observed polarisation

foreground emission at frequency ν*

foreground spectrum

[p^I: I^th parameter]

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

spatially-uniform

• We can then form a linear combination of frequencies to remove both [Q^f,U^f] and δp[Q^f,U^f]. 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

81

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

82

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

83

CMB Solutions

• Spatially-varying power law. We need at least

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

84

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

85

Maximum Likelihood

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

Top-down

86

Expanding D to 1st order

Top-down

87

Expanding D to 1st order

Top-down

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

88

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

89

Posterior [Simplest]

Bottom-up

• Square and average to get covariance and plug it into a Gaussian…

–2ln(posterior[r,β|m]) =

90

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 S^CMB

91

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) ⁹²

Top-down

Now what?

We take the ML estimate of sf:

93

Top-down

Now what?

We take the ML estimate of sf:

And plug it into the top: ₉₄

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…

95

Top-down

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

where

96

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)

97

• 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)

101

Redshift Dependence

103

Makiya, Ando & EK (2018)

Redshift Dependence

High-ell data of Compton-Y auto is the key.

But…

foreground contamination

104

Makiya, Ando & EK (2018)

Z-dependence Poorly Constrained

105

Makiya, Ando & EK (2018)

1-point PDF fits!!

Dolag, EK, Sunyaev (2016)

106