Lecture 9: CMB Polarisation from the Sound Waves

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Lecture 9: CMB Polarisation from the Sound Waves

1

The lecture slides are available at

https://wwwmpa.mpa-garching.mpg.de/~komatsu/

lectures--reviews.html

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2

Credit: ESA

CMB is weakly polarised!

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Stokes Parameters

change under coordinate rotation

x’

y’

Under (x,y) -> (x’,y’):

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Compact Expression

•

Using an imaginary number, write

Then, under the coordinate rotation we have

4

C.f.

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Part I: E- and B-mode Polarisation

5

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ˆ

n = (sin ✓ cos , sin ✓ sin , cos ✓ )

“Flat sky”, if θ is small

6

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Fourier transform the Stokes Parameters?

•

As Q+iU changes under rotation, the Fourier coeﬃcients change as well

•

^So…

where

7

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Tweaking the Fourier Transform

•

Under rotation, the azimuthal angle of a Fourier wavevector, φl, changes as

•

^This

cancels

the factor in the left hand side:

where we write the coeﬃcients as(*)

(*) Never mind the overall minus sign. This is just for convention.

8

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Tweaking Fourier Transform

•

We thus write

•

And, defining

By construction El and Bl do not pick up a factor of exp(2iφ) under coordinate rotation.

That’s

great!

What kind of polarisation patterns do these quantities represent?

Seljak (1997); Zaldarriaga & Seljak (1997); Kamionkowski, Kosowky, Stebbins (1997)

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Pure E, B Modes

•

Q and U produced by E and B modes are given by

•

Let’s consider Q and U that are produced by a single Fourier mode

•

Taking the x-axis to be the direction of a wavevector, we obtain

10

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Pure E, B Modes

•

Q and U produced by E and B modes are given by

•

Let’s consider Q and U that are produced by a single Fourier mode

•

Taking the x-axis to be the direction of a wavevector, we obtain

11

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12

Credit: ESA

CMB is weakly polarised!

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Geometric Meaning (1)

• ^{E mode}

: Polarisation directions

parallel or perpendicular

^to

the wavevector

• ^{B mode}

: Polarisation directions

45 degree tilted

with respect to the wavevector

13

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Geometric Meaning (2)

• ^{E mode}

^{: Stokes}

^Q

, defined with respect to as the x-axis

• ^{B mode}

^{: Stokes}

^U

, defined with respect to as the y-axis IMPORTANT: These are all coordinate-independent statements

14

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Parity

• ^{E mode}

: Parity even

• ^{B mode}

: Parity odd

15

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Power Spectra

•

^However,

<EB> and <TB> vanish

for parity- preserving fluctuations because <EB> and <TB> change sign under parity flip.

16

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17

https://www.mpa-garching.mpg.de/896049/news20201123

MPA Press Release (November 23, 2020)

A hint of <EB> correlation, pointing to new physics that

violates parity?

Minami & Komatsu, PRL, 125, 221301 (2020)

For explanation, see the YouTube video of “Cosmology Talks”,

hosted by Dr. Shaun Hotchkiss at the Univ. of Auckland.

https://youtu.be/9W9rDlEHg3c

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B-mode from lensing E-mode

from sound waves

Temperature from sound waves

B-mode from GW

We understand this now!

18

Next Topic

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Part II: E-mode Polarisation from the Sound Waves

19

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The Single Most Important

Thing You Need to Remember

• Polarisation

is generated by

scattering

^{of the}

local

quadrupole temperature

anisotropy

, which is proportional to

viscosity.

20

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(l,m)=(2,0) (l,m)=(2,1)

(l,m)=(2,2)

21

Local quadrupole

temperature anisotropy

seen from an electron

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(l,m)=(2,0) (l,m)=(2,1)

(l,m)=(2,2)

22

Let’

s symbolise (l,m)=(2,0) as

Hot

Cold Cold

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(l,m)=(2,0) (l,m)=(2,1)

(l,m)=(2,2)

Let’

s symbolise (l,m)=(2,0) as

Polarisation pattern you will see

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Polarisation pattern in the sky

generated by a single Fourier mode

rL

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Polarisation pattern in the sky

generated by a single Fourier mode

rL

E-mode!

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E-mode Power Spectrum

•

Viscosity at the last-scattering surface is given by the spatial gradient of the velocity:

•

Velocity potential is

Sin(qr

L

)

, whereas the temperature power spectrum is predominantly

Cos(qr

L

)

= 32 45

¯

⇢

T n ¯ _e @ _i @ _j u

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γ

•

Using the energy conservation,

26

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WMAP 9-year Power Spectrum

Bennett et al. (2013)

27

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Planck 29-mo Power Spectrum

Planck Collaboration (2016)

28

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SPTPol Power Spectrum South Pole Telescope Collaboration (2018)

29

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[1] Troughs in T -> Peaks in E

[2] T damps -> E rises

because ClTT ~ cos²(qrs) whereas ClEE ~ sin²(qrs)

because

T damps by viscosity, whereas

E is created by viscosity.

[3] Peaks in E are sharper

because ClTT is the sum of cos²(qrL) and the Doppler shift’s sin²(qrL), whereas ClEE is just sin²(qrL)

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[1] Troughs in T -> Peaks in E

[2] T damps -> E rises

because ClTT ~ cos²(qrs) whereas ClEE ~ sin²(qrs)

because

T damps by viscosity, whereas

E is created by viscosity.

[3] Peaks in E are sharper

because ClTT is the sum of cos²(qrL) and the Doppler shift’s sin²(qrL), whereas ClEE is just sin²(qrL)

31

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Polarisation from Re-ionisation

32

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Polarisation from Re-ionisation

ClEE ~

33

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Cross-correlation between T and E

•

Velocity potential is

Sin(qr

L

)

, whereas the temperature power spectrum is predominantly

Cos(qr

L

)

•

Thus, the TE correlation is

Sin(qr

L

)Cos(qr

L

)

^which

can change sign

34

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WMAP 9-year Power Spectrum

Bennett et al. (2013)

35

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Planck 29-mo Power Spectrum

Planck Collaboration (2016)

36

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SPTPol Power Spectrum South Pole Telescope Collaboration (2018)

37

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TE correlation is useful for understanding physics

•

T roughly traces gravitational potential, while E traces velocity

•

With TE, we witness how plasma falls into gravitational potential wells!

38

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Example:

Gravitational Effects

Gravitational Potential, Φ

Plasma motion

Coulson et al. (1994)

39

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B-mode from GW

40

We understand this now!

Next Topic

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Part III: B-mode from Gravitational Lensing

41

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Gravitational lensing effect on the CMB

What does it do to CMB?

•

The important fact: the gravitational lensing eﬀect does not change the surface brightness.

•

This means that the value of CMB temperature does not change by lensing;

only the directions change.

•

Only the anisotropy (and polarisation) is aﬀected:

42

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T _lensed (ˆ n) = T _unlensed (ˆ n + d)

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Basak, Prunet & Benumbed (2008)

<latexit sha1_base64="C0uxUZ6GXmgVBv0pQ3eEvYFLOLc=">AAACCHicbVA9SwNBEN3zM8avqKWFi0GITbgTRcugFpYR8gW5EPY2k2TJ3t6xOyeEI6WNf8XGQhFbf4Kd/8bNR6GJDwYe780wMy+IpTDout/O0vLK6tp6ZiO7ubW9s5vb26+ZKNEcqjySkW4EzIAUCqooUEIj1sDCQEI9GNyM/foDaCMiVcFhDK2Q9ZToCs7QSu3ckX8LEhmttFNfhzRREpSBzqjg9xlSddrO5d2iOwFdJN6M5MkM5Xbuy+9EPAlBIZfMmKbnxthKmUbBJYyyfmIgZnzAetC0VLEQTCudPDKiJ1bp0G6kbSmkE/X3RMpCY4ZhYDtDhn0z743F/7xmgt2rVipUnCAoPl3UTSTFiI5ToR2hgaMcWsK4FvZWyvtMM442u6wNwZt/eZHUzoreRdG9P8+XrmdxZMghOSYF4pFLUiJ3pEyqhJNH8kxeyZvz5Lw4787HtHXJmc0ckD9wPn8AvkKZJw==</latexit>

T _unlensed (ˆ n)

⁴³

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Basak, Prunet & Benumbed (2008)

T _lensed (ˆ n) = T _unlensed (ˆ n + d)

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Gravitational lensing effect on the CMB

Deflection angle and the “lens potential”

•

The vector “d” is called the deflection angle. For the scalar perturbation, we can write d as a gradient of a scalar potential (like the electric field):

with

45

T _lensed (ˆ n) = T _unlensed (ˆ n + d)

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d = @

@ n ˆ

r

L

: the comoving distance from the observer to the last scattering surface

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(ˆ n) =

Z _r

_L

0 dr r _L r

r _L r ( + )(r, nr ˆ )

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E->B conversion due to lensing

“Mode Coupling”: Mixing of diﬀerent Fourier wavenumbers

•

In the flat-sky approximation, lensing aﬀects the Stokes parameters as

•

Fourier-transforming both sides, we find

Taylor-expand this

46

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E->B conversion due to lensing

“Mode Coupling”: Mixing of diﬀerent Fourier wavenumbers

•

In the flat-sky approximation, lensing aﬀects the Stokes parameters as

•

Fourier-transforming both sides, we find

Mode mixing!

47

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E->B conversion due to lensing

“Mode Coupling”: Mixing of diﬀerent Fourier wavenumbers

•

The final results:

48

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E->B conversion due to lensing

“Mode Coupling”: Mixing of diﬀerent Fourier wavenumbers

•

Even if there was no intrinsic B-mode polarisation at the last scattering surface:

•

Important: There is no monopole (l=0) for the lensing potential. This means that E and B cannot be correlated at the same multipole: lensing does not violate parity globally.

•

The lensing eﬀect is the convolution. As the lensing potential is “smooth” and does not contain the acoustic oscillation, convolution will smear out the

acoustic oscillation in the E-mode polarisation.

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B-mode from GW

50

We understand this now!

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Part IV: EB Correlation:

The Cosmic Birefringence

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How does the electromagnetic wave of the CMB reach us?

Now shown: The cosmological redshift due to the expansion of the Universe

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How does the electromagnetic wave of the CMB reach us?

Note: rotation of the polarisation plane is massively exaggerated! ?

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Cosmic Birefringence

The Universe filled with a “birefringent material”

•

If the Universe is filled with a pseudo-scalar field (e.g., an axion field) coupled to the electromagnetic tensor via a Chern-Simons coupling:

Carroll, Field & Jackiw (1990); Harari & Sikivie (1992); Carroll (1998)

Turner & Widrow (1988)

Chern-Simons term

F˜^µ⌫ = X

↵

✏^µ⌫↵

2p

gF_↵

X

µ⌫

F_µ⌫ F ^µ⌫ = 2(B · B E · E)

Parity Even Parity Odd

X

µ⌫

F_µ⌫F˜^µ⌫ = 4B · E

•

The axion field, θ, is a “pseudo scalar”, which is parity odd;

thus, the last term in Eq.3.7 is parity even as a whole.

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Cosmic Birefringence

The Universe filled with a “birefringent material”

• Carroll, Field & Jackiw (1990); Harari & Sikivie (1992); Carroll (1998)

F˜^µ⌫ = X

↵

✏^µ⌫↵

2p

gF_↵

The “Cosmic Birefringence” (Carroll 1998)

This term makes the phase velocities of right- and left-handed polarisation states

of photons diﬀerent, leading to rotation of the linear polarisation direction.

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Cosmic Birefringence

The eﬀect accumulates over the distance

• Carroll, Field & Jackiw (1990); Harari & Sikivie (1992); Carroll (1998)

F˜^µ⌫ = X

↵

✏^µ⌫↵

2p

gF_↵

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= 2g

_a

Z

_t_observed

t_emission

dt ✓ ˙

The larger the distance the photon travels, the larger the effect becomes.

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Motivation

Why study the cosmic birefringence?

•

The Universe’s energy budget is dominated by two dark components:

•

Dark Matter

•

Dark Energy

•

Either or both of these can be an axion-like field!

•

See Marsh (2016) and Ferreira (2020) for reviews.

•

Thus, detection of parity-violating physics in polarisation of the cosmic microwave background can transform our understanding of Dark Matter/

Energy.

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(Simpler) Motivation

Why study the cosmic birefringence?

•

We know that the weak interaction violates parity (Lee & Yang 1956; Wu et al.

1957).

•

Why should the laws of physics governing the Universe conserve parity?

•

Let’s look!

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EB correlation from the cosmic birefringence

E <-> B conversion by rotation of the linear polarisation plane

•

The intrinsic EE, BB, and EB power spectra 13.8 billion years ago would yield the observed EB as

Lue, Wang & Kamionkowski (1999); Feng et al. (2005, 2006); Liu, Lee & Ng (2006)

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C

_`^EB,obs

= 1

2 (C

_`^EE

C

_`^BB

) sin(4 )

<latexit sha1_base64="IJuJCTE1N6kUusDX9hjw6FJU+kI=">AAACAXicbVBNS8NAEN3Ur1q/ol4EL8EiVISSSEWPpUXwWMF+QBPDZjttl242YXcjlFAv/hUvHhTx6r/w5r9x2+ag1QcDj/dmmJkXxIxKZdtfRm5peWV1Lb9e2Njc2t4xd/daMkoEgSaJWCQ6AZbAKIemoopBJxaAw4BBOxjVp377HoSkEb9V4xi8EA847VOClZZ88+C07rvA2F16VZu4JJKlihuAwie+WbTL9gzWX+JkpIgyNHzz0+1FJAmBK8KwlF3HjpWXYqEoYTApuImEGJMRHkBXU45DkF46+2BiHWulZ/UjoYsra6b+nEhxKOU4DHRniNVQLnpT8T+vm6j+pZdSHicKOJkv6ifMUpE1jcPqUQFEsbEmmAiqb7XIEAtMlA6toENwFl/+S1pnZee8bN9UitVaFkceHaIjVEIOukBVdI0aqIkIekBP6AW9Go/Gs/FmvM9bc0Y2s49+wfj4BhFLlgA=</latexit>

+C

_`^EB

cos(4 )

• Traditionally, one would find β by fitting C

lEE,CMB

– C

lBB,CMB

to the observed C

lEB,obs

using the best-fitting CMB model, and assuming the intrinsic EB to vanish, C

lEB

=0.

•

How do we infer β from the observational data?

59

(60)

Searching for the birefringence

Improvement #1 (Zhao et al. 2015)

•

If we look at how EE and BB spectra are also modified,

•

^{We find}

•

^Thus,<latexit sha1_base64="910OSkLqtnmy1k8Fk2NJltHkYrg=">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</latexit>

C

_`^EB,obs

= 1

2 (C

_`^EE

C

_`^BB

) sin(4 )

<latexit sha1_base64="hpnvjz53moKGSW47Mmmx7UGiDtA=">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</latexit>

= 1

2 (C

_`^EE,obs

C

_`^EE,obs

) tan(4 ) + C

_`^EB

cos(4 )

No need to assume a model

60

<latexit sha1_base64="HmjuqqPWXFnD0VGS/AXBkQIICXY=">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</latexit>

C_`ÊE,obs = C_`ÊE cos²(2 ) + C_`^BB sin²(2 ) C_`^BB,obs = C_`ÊE sin²(2 ) + C_`^BB cos²(2 )

<latexit sha1_base64="IJuJCTE1N6kUusDX9hjw6FJU+kI=">AAACAXicbVBNS8NAEN3Ur1q/ol4EL8EiVISSSEWPpUXwWMF+QBPDZjttl242YXcjlFAv/hUvHhTx6r/w5r9x2+ag1QcDj/dmmJkXxIxKZdtfRm5peWV1Lb9e2Njc2t4xd/daMkoEgSaJWCQ6AZbAKIemoopBJxaAw4BBOxjVp377HoSkEb9V4xi8EA847VOClZZ88+C07rvA2F16VZu4JJKlihuAwie+WbTL9gzWX+JkpIgyNHzz0+1FJAmBK8KwlF3HjpWXYqEoYTApuImEGJMRHkBXU45DkF46+2BiHWulZ/UjoYsra6b+nEhxKOU4DHRniNVQLnpT8T+vm6j+pZdSHicKOJkv6ifMUpE1jcPqUQFEsbEmmAiqb7XIEAtMlA6toENwFl/+S1pnZee8bN9UitVaFkceHaIjVEIOukBVdI0aqIkIekBP6AW9Go/Gs/FmvM9bc0Y2s49+wfj4BhFLlgA=</latexit>

+C

_`^EB

cos(4 )

<latexit sha1_base64="hjpgDlCA1j3OvcMR57Y1OklL/pA=">AAACAXicbVBNS8NAEN3Ur1q/ol4EL8Ei1IMlkYoeS4vgsYL9gCaGzXbaLt1swu5GKKFe/CtePCji1X/hzX/jts1Bqw8GHu/NMDMviBmVyra/jNzS8srqWn69sLG5tb1j7u61ZJQIAk0SsUh0AiyBUQ5NRRWDTiwAhwGDdjCqT/32PQhJI36rxjF4IR5w2qcEKy355sFp3XeBsbv0qjZxJeWlihuAwie+WbTL9gzWX+JkpIgyNHzz0+1FJAmBK8KwlF3HjpWXYqEoYTApuImEGJMRHkBXU45DkF46+2BiHWulZ/UjoYsra6b+nEhxKOU4DHRniNVQLnpT8T+vm6j+pZdSHicKOJkv6ifMUpE1jcPqUQFEsbEmmAiqb7XIEAtMlA6toENwFl/+S1pnZee8bN9UitVaFkceHaIjVEIOukBVdI0aqIkIekBP6AW9Go/Gs/FmvM9bc0Y2s49+wfj4BhxXlgc=</latexit>

C_`^EB sin(4 )

<latexit sha1_base64="Vpo7VpGoPmGR0Y3QiCM+efALLgU=">AAACAXicbVBNS8NAEN3Ur1q/ol4EL8EiVISSSEWPpUXwWMF+QBPDZjttl242YXcjlFAv/hUvHhTx6r/w5r9x2+ag1QcDj/dmmJkXxIxKZdtfRm5peWV1Lb9e2Njc2t4xd/daMkoEgSaJWCQ6AZbAKIemoopBJxaAw4BBOxjVp377HoSkEb9V4xi8EA847VOClZZ88+C07rvA2F16VZu4kvJSxQ1A4RPfLNplewbrL3EyUkQZGr756fYikoTAFWFYyq5jx8pLsVCUMJgU3ERCjMkID6CrKcchSC+dfTCxjrXSs/qR0MWVNVN/TqQ4lHIcBrozxGooF72p+J/XTVT/0kspjxMFnMwX9RNmqciaxmH1qACi2FgTTATVt1pkiAUmSodW0CE4iy//Ja2zsnNetm8qxWotiyOPDtERKiEHXaAqukYN1EQEPaAn9IJejUfj2Xgz3uetOSOb2Ue/YHx8AxkhlgU=</latexit>

+C_`^EB sin(4 )

<latexit sha1_base64="xFQX4OYFonzzXQhVgERGeHlHvOo=">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</latexit>

C_`ÊE,obs C_`^BB,obs = (C_`ÊE C_`^BB ) cos(4 ) 2C_`ÊB sin(4 )

<latexit sha1_base64="rH9/JqgfhSPyB1gLDTe50dN0BBU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WOpF49V7Ae0oWy2k3bpZhN2N0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0UK/3yxW36s5BVomXkwrkaPTLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5pdOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzXhjZ9xmaQGJVssClNBTExmb5MBV8iMmFhCmeL2VsJGVFFmbDglG4K3/PIqaV1Uvauqe39ZqdXzOIpwAqdwDh5cQw3uoAFNYBDCM7zCmzN2Xpx352PRWnDymWP4A+fzBx7wjRY=</latexit>

BB

(61)

The Biggest Problem:

Miscalibration of detectors

61

(62)

Impact of miscalibration of polarisation angles

•

Is the plane of linear polarisation rotated by the genuine cosmic birefringence eﬀect, or

simply because the polarisation-sensitive directions of detectors are rotated with respect to the sky coordinates (and we did not know it)?

•

If the detectors are rotated by α, it seems that we can measure only the

sum α+β

^.

62

OR

Polarisation-sensitive detectors on the focal plane

rotated by an angle “α”

(but we do not know it)

α

Wu et al. (2009); Komatsu et al. (2011); Keating, Shimon & Yadav (2012)

Cosmic or Instrumental?

(63)

The past measurements

The quoted uncertainties are all statistical only (68%CL)

•

α+β = –6.0 ± 4.0 deg (Feng et al. 2006)

•

α+β = –1.1 ± 1.4 deg (WMAP Collaboration, Komatsu et al. 2009; 2011)

•

α+β = 0.55 ± 0.82 deg (QUaD Collaboration, Wu et al. 2009)

•

^…

•

α+β = 0.31 ± 0.05 deg (Planck Collaboration 2016)

•

α+β = –0.61 ± 0.22 deg (POLARBEAR Collaboration 2020)

•

α+β = 0.63 ± 0.04 deg (SPT Collaboration, Bianchini et al. 2020)

•

α+β = 0.12 ± 0.06 deg (ACT Collaboration, Namikawa et al. 2020)

•

α+β = 0.09 ± 0.09 deg (ACT Collaboration, Choi et al. 2020)

63

first measurement

} Why not yet

discovered?

(64)

The past measurements

Now including the estimated systematic errors on α

•

β = –6.0 ± 4.0 ± ?? deg (Feng et al. 2006)

•

β = –1.1 ± 1.4 ± 1.5 deg (WMAP Collaboration, Komatsu et al. 2009; 2011)

•

β = 0.55 ± 0.82 ± 0.5 deg (QUaD Collaboration, Wu et al. 2009)

•

^…

•

β = 0.31 ± 0.05 ± 0.28 deg (Planck Collaboration 2016)

•

β = –0.61 ± 0.22 ± ?? deg (POLARBEAR Collaboration 2020)

•

β = 0.63 ± 0.04 ± ?? deg (SPT Collaboration, Bianchini et al. 2020)

•

β = 0.12 ± 0.06 ± ?? deg (ACT Collaboration, Namikawa et al. 2020)

•

β = 0.09 ± 0.09 ± ?? deg (ACT Collaboration, Choi et al. 2020)

64

Uncertainty in the calibration

of α has been the major

limitation

(65)

The Key Idea: The polarised Galactic foreground emission as a calibrator

65

(66)

Credit: ESA

Directions of the magnetic field inferred from polarisation of the thermal dust emission in the Milky Way

Emitted “right there” - it would not be affected by the cosmic

birefringence.

Polarised dust emission within our Milky Way!

ESA’s Planck

(67)

Searching for the birefringence

Improvement #2 (Minami et al. 2019)

•

Idea: Miscalibration of the polarization angle α rotates both the foreground and CMB, but β aﬀects only the CMB.

•

^Thus,

measured known accurately

Key: No explicit modelling of the foreground EE and BB is necessary

67

Emitted 13.8 billions years ago

But the source of foreground is much closer!

noise

(68)

Assumption for the baseline result

What about the intrinsic EB correlation of the foreground emission?

•

For the baseline result, we ignore the intrinsic EB correlations of the foreground and the CMB .

•

The latter is justifiable but the former is not. We will revisit this important issue at the end.

68

(69)

Likelihood for the simplest case

Single-frequency case, full sky data

Minami et al. (2019)

69

•

We determine α and β simultaneously from this likelihood.

•

We first validate the algorithm using simulated data.

•

For analysing the Planck data, we use the multi-frequency likelihood developed in Minami and Komatsu (2020a).

(70)

How does it work?

Simulation of future CMB data (LiteBIRD)

•

When the data are dominated by CMB, the sum of two

angles, α+β, is determined precisely.

•

This is the diagonal line.

•

The foreground determines α with some uncertainty,

breaking the degeneracy.

Then σ(β) ~ σ(α) because σ(α+β) << σ(α).

•

When the data are dominated by the foreground, it can

determine α but not β due to the lack of sensitivity to the CMB.

Minami et al. (2019)

70

(CMB-dominated)

(Dust-dominated)

(71)

Main Results

β > 0 at 2.4σ

•

^{All α}^ν’s are consistent with zero either

statistically, or within the ground calibration error of 0.28 deg.

•

Removing 100 GHz did not change β

•

β=0.35 deg also agrees well with the Planck determination assuming αν=0:

•

^β(α^ν=0) = 0.29 ± 0.05 (stat. from EB) ± 0.28 (syst.) [Planck Int. XLIX]

71

αν=0

0.289 ± 0.048

Minami & Komatsu (2020b)

(72)

72

<latexit sha1_base64="U2HTxgnW56tVqpANuPNA8D56uZo=">AAACFXicbZDLSgMxFIYz9VbrbdSlm2ARKmiZKYouS0tBcFPBXqAzLZk0bUOTmSHJCGUYH8KNr+LGhSJuBXe+jelF0NYDIR//fw7J+b2QUaks68tILS2vrK6l1zMbm1vbO+buXl0GkcCkhgMWiKaHJGHUJzVFFSPNUBDEPUYa3rA89ht3REga+LdqFBKXo75PexQjpaWOeeIQxnLlzvhqx5VKcvrDpVJyfN9yeBQ7gsPrpF1wO2bWyluTgotgzyALZlXtmJ9ON8ARJ77CDEnZsq1QuTESimJGkowTSRIiPER90tLoI06kG0+2SuCRVrqwFwh9fAUn6u+JGHEpR9zTnRypgZz3xuJ/XitSvUs3pn4YKeLj6UO9iEEVwHFEsEsFwYqNNCAsqP4rxAMkEFY6yIwOwZ5feRHqhbx9nrduzrLF0iyONDgAhyAHbHABiuAKVEENYPAAnsALeDUejWfjzXiftqaM2cw++FPGxzda+J5Q</latexit>

` ( C EE ` C BB ` )[ µ K 2 ]

•

Can we see β = 0.35

± 0.14 deg by eyes?

• First, take a look at the observed EE–BB spectra.

• ^{Red: Total}

• Blue: The best-fitting CMB model

• The diﬀerence is due to the FG (and potentially systematics)

rotated only

by α rotated by α+β

(73)

73

Minami & Komatsu (2020b)

<latexit sha1_base64="i5vVyDDvKmvMRLDcX4JKq8FVr5w=">AAACCHicbVDLSgMxFM34rPU16tKFwSK4KjNF0WVpEQQ3FewDOtOSSTNtaJIZkoxQhnHnxl9x40IRt36CO//G9LHQ1gOXezjnXpJ7gphRpR3n21paXlldW89t5De3tnd27b39hooSiUkdRyySrQApwqggdU01I61YEsQDRprBsDr2m/dEKhqJOz2Kic9RX9CQYqSN1LWPPMIYrHbHrZNeVbKHtseT1JMc3mSdkt+1C07RmQAuEndGCmCGWtf+8noRTjgRGjOkVNt1Yu2nSGqKGcnyXqJIjPAQ9UnbUIE4UX46OSSDJ0bpwTCSpoSGE/X3Roq4UiMemEmO9EDNe2PxP6+d6PDST6mIE00Enj4UJgzqCI5TgT0qCdZsZAjCkpq/QjxAEmFtssubENz5kxdJo1R0z4vO7VmhXJnFkQOH4BicAhdcgDK4BjVQBxg8gmfwCt6sJ+vFerc+pqNL1mznAPyB9fkDCT2ZWA==</latexit>

` C

EB `

[ µ K

2

]

<latexit sha1_base64="i5vVyDDvKmvMRLDcX4JKq8FVr5w=">AAACCHicbVDLSgMxFM34rPU16tKFwSK4KjNF0WVpEQQ3FewDOtOSSTNtaJIZkoxQhnHnxl9x40IRt36CO//G9LHQ1gOXezjnXpJ7gphRpR3n21paXlldW89t5De3tnd27b39hooSiUkdRyySrQApwqggdU01I61YEsQDRprBsDr2m/dEKhqJOz2Kic9RX9CQYqSN1LWPPMIYrHbHrZNeVbKHtseT1JMc3mSdkt+1C07RmQAuEndGCmCGWtf+8noRTjgRGjOkVNt1Yu2nSGqKGcnyXqJIjPAQ9UnbUIE4UX46OSSDJ0bpwTCSpoSGE/X3Roq4UiMemEmO9EDNe2PxP6+d6PDST6mIE00Enj4UJgzqCI5TgT0qCdZsZAjCkpq/QjxAEmFtssubENz5kxdJo1R0z4vO7VmhXJnFkQOH4BicAhdcgDK4BjVQBxg8gmfwCt6sJ+vFerc+pqNL1mznAPyB9fkDCT2ZWA==</latexit>

` C

EB `

[ µ K

2

]

•

Can we see β = 0.35

± 0.14 deg by eyes?

•

Red: The signal attributed to the miscalibration angle, αν

•

Blue: The signal attributed to the cosmic birefringence, β

•

Red + Blue is the best-fitting model for explaining the data points

(74)

How about the foreground EB?

•

If the intrinsic foreground EB power spectrum exists, our method interprets it as a miscalibration angle α.

•

Thus, α -> α+γ, where γ is the contribution from the intrinsic EB.

•

The sign of γ is the same as the sign of the foreground EB.

•

From FG: α+γ. From CMB: α+β.

•

Thus, our method yields β–γ = 0.35 ± 0.14 deg.

•

There is evidence for the dust-induced TEdust > 0 and TBdust > 0. Then, we’d

expect EBdust > 0 (Huﬀenberger et al. 2020), i.e., γ>0. If so, β increases further…

74

Minami et al. (2019); Minami & Komatsu (2020b)

(75)

Implications

What does it mean for your models of dark matter and energy?

•

When the Lagrangian density includes a Chern-Simons coupling between a pseudo scalar field and the electromagnetic tensor given by

75

•

The birefringence angle is

•

Our measurement yields

φ

LSS

φ

obs

+δφ

obs

Minami & Komatsu (2020b)

(76)

Summary of the Result

β = 0.35 ± 0.14 (68%CL)

•

We perfectly understand what 2.4σ means!

•

Higher statistical significance is need to confirm this signal.

•

Our new method finally allowed us to make this “impossible” measurement, which may point to new physics.

•

Our method can be applied to any of the existing and future CMB experiments.

•

The confirmation (or otherwise) of the signal should be possible immediately.

•

If confirmed, it would have important implications for dark matter/energy.

76

β = 0.35 ± 0.14

Lecture 9: CMB Polarisation from the Sound Waves