Spectrum of the Temperature Anisotropy and

Lecture 8: Understanding the Power

Spectrum of the Temperature Anisotropy and

Introduction to CMB Polarisation

1

The lecture slides are available at

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

lectures--reviews.html

Part I: Cosmological Parameter

Dependence of the Temperature Power Spectrum

2

Before starting: Let’s recap the Big Picture

In words!

•

How does the power spectrum constrain the baryon density?

•

Via the speed of sound, the increased inertia of a photon-baryon fluid, and Silk damping.

•

They all depend on R (the baryon-photon energy density ratio), which is proportional to ΩBh². Note h²! It is not just ΩB.

3

q q

q

q

EQ

= a

EQ

H

EQ

~ 0.01 (Ω

M

h

²

/0.14) Mpc

^–1

Before starting: Let’s recap the Big Picture

In words!

•

How does the power spectrum constrain the total matter density?

•

Via the boost of the amplitude of sound waves and the ISW due to a decaying gravitational after the horizon re-entry.

•

They all depend on qEQ (the wavenumber of the matter-radiation equality), which is proportional to ΩMh². Note h²! It is not just ΩM.

q q q

Before starting: Things you haven’t learned yet

•

How does the power spectrum constrain the Hubble constant?

•

Via the (comoving) distance to the last scattering surface, rL. Particularly interesting now because of the “Hubble constant tension”.

•

How does the power spectrum constrain the dark energy?

•

The same: via rL.

•

How does the power spectrum constrain the epoch of reionization?

•

The temperature power spectrum cannot constrain this.

5

How does r L depend on H 0 and dark energy?

•

We must know ΩMh² in advance. The CMB peak heights tell us ΩMh², which then enables us to determine Η0 if we assume a flat Universe. Otherwise, we cannot really determine H0 or ΩΛ! (Unless we use gravitational lensing of the CMB.)

6

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a ₀ r _L = ca ₀

Z _t

₀

t

_L

dt

a(t) = ca ₀

Z _a

₀

a

_L

da

a a ˙ = ca ₀

Z _a

₀

a

_L

da

a ² H (a)

Friedmann’s equation

<latexit sha1_base64="ijez57ORDXAE+vUBx+qcNBcbc/A=">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</latexit>

H

²

(a) = H

₀²

⌦

_M

a

³

+ ⌦

_k

a

²

+ ⌦

_⇤

! H

₀²

⌦

_M

a

³

+ 1 ⌦

_M

Flat Universe Ωk=0

<latexit sha1_base64="DUsrl46UZlYf7xAdF0txgiZC9V8=">AAACH3icbZDJSgNBEIZ7XGPcoh69DAYhIoYZ92PQixcxgjGBTBJ6OjWZxp6F7hohDHkTL76KFw+KiLe8jZ3lEBN/aPj4q4rq+t1YcIWW1Tfm5hcWl5YzK9nVtfWNzdzW9qOKEsmgwiIRyZpLFQgeQgU5CqjFEmjgCqi6T9eDevUZpOJR+IDdGBoB7YTc44yitlq5cyeWUYyR6QjwsODcBdChrVu/eUyb6dFJ71DT0YTrSN7x8aCVy1tFayhzFuwx5MlY5Vbux2lHLAkgRCaoUnXbirGRUomcCehlnURBTNkT7UBdY0gDUI10eF/P3NdO2/QiqV+I5tCdnEhpoFQ3cHVnQNFX07WB+V+tnqB32Uh5GCcIIRst8hJh6jwGYZltLoGh6GqgTHL9V5P5VFKGOtKsDsGePnkWHo+L9lnRuj/Nl67GcWTILtkjBWKTC1IiN6RMKoSRF/JGPsin8Wq8G1/G96h1zhjP7JA/Mvq/cj+h9g==</latexit>

/ ⌦

_M

h

²

a

³

+ h

²

⌦

_M

h

²

ΩM + Ωk + ΩΛ = 1

7

The sound horizon, rs, changes when the baryon density changes, resulting in a shift in the peak positions.

Adjusting it makes the physical effect at the last scattering manifest

8

Zero-point shift of the oscillations

9

Zero-point shift effect

compensated by (1+R)^–1/4 and Silk damping

10

Less tight coupling:

Enhanced Silk damping for low baryon density

11

Total Matter Density

12

Total Matter Density

13

Total Matter Density

First Peak:

More ISW and boost due to the decay of Φ

14

Total Matter Density

2nd, 3rd, 4th Peaks:

Boosts due to the decay of Φ

Less and less effects at larger multipoles

15

Two Other Effects

• Spatial curvature

•

We have been assuming a spatially-flat Universe with zero curvature (i.e., Euclidean space). What if it is curved?

• Optical depth to Thomson

scattering in a low-redshift Universe

•

We have been assuming that the Universe is transparent to photons since the last scattering at z=1090. What if there is an extra scattering in a low-redshift Universe?

16

Spatial Curvature

•

It changes the

angular diameter distance, d

A, to the last scattering surface; namely,

•

^{rL -> dA = R sin(rL/R) = rL(1}

–

^rL2/6R2) + … for positively- curved space

•

^{rL -> dA = R sinh(rL/R) = rL(1}

+

^rL2/6R2) + … for negatively- curved space

Smaller angles (larger multipoles) for a negatively curved Universe

17

18

19

late-time ISW

20

Optical Depth

•

Extra scattering by electrons in a low-redshift Universe damps temperature anisotropy

• ^C

^l

^{-> C}

^l

^exp(–2τ)

at l >~ 10

•

where τ is the optical depth

re-ionisation

21

22

23

•

Since the power spectrum is uniformly suppressed by exp(–2τ) at l>~10, we cannot determine the amplitude of the power spectrum of the gravitational potential, Pφ(q), independently of τ.

•

Namely, what we constrain is the combination: exp(–2τ)Pφ(q)

Important consequence of the optical depth

•

Breaking this degeneracy requires an independent determination of the optical depth. This requires

POLARISATION

of the CMB.

/ exp( 2⌧ )A _s

24

+CMB Lensing Planck

[100 Myr]

Cosmological Parameters Derived from the Power Spectrum

The Hubble Constant Tension

The role of the CMB

•

Can we explain, in words, how this measurement was obtained?

Wong et al. (2020)

•

The CMB peak positions are controlled by cos(qrs).

•

We measure q in the angular wavenumber, l ~ qrL.

•

Thus, the CMB power spectrum gives a direct measurement of the distance ratio: rs/rL.

•

You already know how to obtain a0rs = 145 Mpc (see Lecture 5).

•

Today we saw how rL depends on cosmology (in a flat Universe):

27

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a

₀

r

_L

= ca

₀

Z

_a0

a_L

da

a

²

H (a) /

Z

_a0

a_L

da a

²

p

⌦

_M

h

²

a

³

+ h

²

⌦

_M

h

²

CMB -> Distance Ratio -> (Physics) -> H 0

Physical Assumption: the sound horizon r

s

CMB -> Distance Ratio -> (Physics) -> H 0

Physical Assumption: the sound horizon r

s

•

The CMB peak positions are controlled by cos(qrs).

•

We measure q in the angular wavenumber, l ~ qrL.

•

Thus, the CMB power spectrum gives a direct measurement of the distance ratio: rs/rL.

•

You already know how to obtain a0rs = 145 Mpc (see Lecture 5).

•

Today we saw how rL depends on cosmology (in a flat Universe):

28

<latexit sha1_base64="tvvcBXmPIDrpOwDLma0bwz/eZtw=">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</latexit>

a

₀

r

_L

= ca

₀

Z

_a0

a_L

da

a

²

H (a) /

Z

_a0

a_L

da a

²

p

⌦

_M

h

²

a

³

+ h

²

⌦

_M

h

²

If we used an incorrect value of r

s

, we would infer r

L

incorrectly, hence

an incorrect value of H

0

!

So…?

Bernal, Verde and Riess (2016); Poulin et al. (2019)

•

The CMB-inferred value of H0 is too low, by 10 percent.

•

This means that the inferred value of rL is too high, by 10 percent.

•

This may mean that the value of rs we calculated using the standard understanding of physics was too high by 10 percent.

•

If we managed to reduce the calculated value of rs by 10 percent, we could resolve the Hubble constant tension.

•

Is that possible? Not really, but one way to achieve this would be to increase H(a) by 10 percent in the radiation era. => Early Dark Energy?

29

Part II: Basics of the CMB Polarisation

30

31

Credit: ESA

32

Credit: ESA

CMB is weakly polarised!

Polarisation

No polarisation

Polarised in x-direction

33

Photo Credit: TALEX

34

horizontally polarised Photo Credit: TALEX

35

Photo Credit: TALEX

36

Generation of polarisation

The necessary and sufficient conditions

•

To generate polarisation, we must satisfy the following two conditions

•

Scattering

•

Anisotropic incident light

•

However, the Universe does not have a preferred direction. How do we generate anisotropic incident light?

37

Physics of CMB Polarisation

Necessary and sufficient condition: Scattering and Local Quadrupole Anisotropy

Credit：Wayne Hu 38

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

(l,m)=(2,2)

39

Quadrupole

temperature anisotropy

seen by an electron

Generation of temperature quadrupole

The punch line

•

When the Thomson scattering is efficient (i.e., tight coupling between photons and baryons via electrons), the distribution of photons from the rest frame of

baryons appears isotropic.

• Only when tight coupling weakens

^, a local quadrupole

temperature anisotropy in the rest frame of a photon-baryon fluid can be generated.

•

In fact, “a local temperature anisotropy in the rest frame of a photon-baryon fluid” is equal to

viscosity.

40

Part III: Stokes Parameters

41

Stokes Parameters

[Flat Sky, Cartesian coordinates]

a b

42

Stokes Parameters

change under coordinate rotation

x’

Under (x,y) -> (x’,y’): y’

Compact Expression

•

Using an imaginary number, write

Then, under the coordinate rotation we have

44

Alternative Expression

•

With the polarisation amplitude, P, and angle, , defined by

• Then, under coordinate rotation we have We write

and P is invariant under rotation.

45

Help!

•

That Q and U depend on coordinates is not very convenient…

•

Someone said, “I measured Q!” but then someone else may say, “No, it’s U!”. They flight to death, only to realise that their coordinates are 45

degrees rotated from one another…

•

The best way to avoid this unfortunate fight is to define a coordinate-

independent quantity for the distribution of polarisation patterns in the sky

46

To achieve this, we need

to go to Fourier space

Part IV: E- and B-mode Polarisation

47

ˆ

n = (sin ✓ cos , sin ✓ sin , cos ✓ )

“Flat sky”, if θ is small

48

Fourier transform the Stokes Parameters?

•

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

•

^So…

where

49

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.

50

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)

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

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

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

54

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

55

Parity

• ^{E mode}

: Parity even

• ^{B mode}

: Parity odd

56

Parity

• ^{E mode}

: Parity even

• ^{B mode}

: Parity odd

57

Power Spectra

•

^However,

<EB> and <TB> vanish

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

58

59

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

B-mode from lensing E-mode

from sound waves

Temperature from sound waves

B-mode from GW

We understand this now!

60

Next Target

The Single Most Important

Thing You Need to Remember

• Polarisation

is generated by the local

quadrupole temperature anisotropy

^,

which is proportional to

viscosity.

61

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

(l,m)=(2,2)

62

Local quadrupole

temperature anisotropy

seen from an electron

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

(l,m)=(2,2)

63

Let’

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

Hot

Hot

Cold Cold

(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

Polarisation pattern in the sky

generated by a single Fourier mode

rL

Polarisation pattern in the sky

generated by a single Fourier mode

rL

E-mode!

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

γ

•

Using the energy conservation,

67

WMAP 9-year Power Spectrum

Bennett et al. (2013)

68

Planck 29-mo Power Spectrum

Planck Collaboration (2016)

69

SPTPol Power Spectrum South Pole Telescope Collaboration (2018)

70

[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] Eeaks 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] Eeaks 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)

72

Polarisation from Re-ionisation

73

Polarisation from Re-ionisation

ClEE ~

74

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

75

WMAP 9-year Power Spectrum

Bennett et al. (2013)

76

Planck 29-mo Power Spectrum

Planck Collaboration (2016)

77

SPTPol Power Spectrum South Pole Telescope Collaboration (2018)

78

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!

79

Example:

Gravitational Effects

Gravitational Potential, Φ

Plasma motion

Coulson et al. (1994)

80

B-mode from lensing E-mode

from sound waves

Temperature from sound waves

B-mode from GW

We understand this now!

81

We understand this now!

Next Lecture

Appendix: Effects of Neutrinos on the Temperature Power Spectrum

82

The Effects of Relativistic Neutrinos

•

To see the effects of relativistic neutrinos, we

artificially increase the number of neutrino species from 3 to 7

•

Great energy density in neutrinos, i.e., greater energy density in radiation

•

Longer radiation domination -> More ISW and boosts due to potential decay

(1)

83

84

After correcting for more ISW and boosts due to

potential decay

85

(2): Viscosity Effect on the Amplitude of Sound Waves

The solution is

where

Hu & Sugiyama (1996)

Bashinsky & Seljak (2004) Phase shift!

86

After correcting for the viscosity effect on the

amplitude

87

(3): Change in the Silk Damping

•

Greater neutrino energy density implies greater Hubble expansion rate,

Η

²

=8πG∑ρ

α

/3

•

^This

reduces

the sound horizon in proportion to H^–1, as rs ~ csH^–1

•

This also reduces the diffusion length, but in proportional to H^–1/2, as a/qsilk ~ (σTneH)^–1/2

•

As a result,

l

silk

decreases relative to the first peak position

, enhancing the Silk damping

Consequence of the random walk!

Bashinsky & Seljak (2004)

88

After correcting for the diffusion length

89

Zoom in!

90

91

(4): Viscosity Effect on the Phase of Sound Waves

The solution is

where

Hu & Sugiyama (1996)

Bashinsky & Seljak (2004) Phase shift!

92

After correcting for the phase shift

Now we understand everything quantitatively!!

93