Lecture 10: Gravitational Waves

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Lecture 10: Gravitational Waves

1 The lecture slides are available at

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

lectures--reviews.html

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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!

2 We understand this now!

We understand this now!

Today’s topic

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Part I: Basics of the Gravitational Waves

3

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Gravitational waves are coming towards you!

To visualise the waves, watch motion of test particles.

4

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Gravitational waves are coming towards you!

To visualise the waves, watch motion of test particles.

5 y

x

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Distance between two points

y

x

Scale Factor

6 • In Cartesian coordinates, the distance between two points in Euclidean space is

• To include the isotropic expansion of space,

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Distortion in space

x ²

x ¹

δ ij = 1 for i=j;

δ ij = 0 otherwise

Distortion in space!

7 • Compact notation using Kronecker’s delta symbol:

• To include distortion in space,

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D _ij

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• The gravitational wave shall be transverse.

• The direction of distortion is perpendicular to the propagation direction

Four conditions for gravitational waves

3 conditions for D ĳ

Thus,

8

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~ q

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X 3

i=1

q ⁱ D _ij = 0

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Four conditions for gravitational waves

• The gravitational wave shall not change the area

• The determinant of δ ij +D ij is 1

Thus,

X 3

i=1

h _ii = 0

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x ²

x ¹

9 1 condition for D ĳ

D _ij

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D _ii

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• The symmetric matrix D ij has 6 components, but there are 4 conditions. Thus, we have two degrees of freedom.

• If the GW propagates in the x ³ =z axis, non-vanishing components of D ij are

6 – 4 = 2 degrees of freedom for GW

We call them “plus” and “cross” modes

h _ij =

0 @ h ₊ h _⇥ 0 h _⇥ h ₊ 0

0 0 0

1 A

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x ²

x ¹

h

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₊ h

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_⇥

10 D _ij

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11 h + =cos(kz)

Propagation direction of GW

h + =cos(qz)

h x =cos(qz)

z

~ q

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Tensor-to-scalar Ratio, the “r”

Everyone is after this!

• The current upper bound is r < 0.044 (95%CL) [Tristram et al., arXiv:2010.01139]

• We want to find this in the B-mode polarisation of the CMB.

12

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r =

P

ij h D _ij D _ij i

h ⇣ ² i

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How to detect GW?

Laser interferometer technique, used by LIGO and VIRGO

13 Mirror

Detector _{No Signal}

Mirror

Signal!

Detector Mirror

Beam splitter Beam splitter

The wavelength of GW detectable by this method is the size of Earth

(a few thousand km).

How do we detect GW with

billions of light-years’s wavelength?

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Detecting GW by CMB

14 Isotropic radiation field (CMB) Isotropic radiation field (CMB)

h

₊ h

_⇥

h

₊ h

_⇥

C H C

H C

C H H

Quadrupole temperature anisotropy generated by red- and blue-shifting of photons

Sachs & Wolfe (1967)

From Lecture 2:

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Detecting GW by CMB

15 Isotropic radiation field (CMB) Isotropic radiation field (CMB)

h

₊ h

_⇥

h

₊ h

_⇥

Electron

C H C

H C

C H H

Quadrupole temperature anisotropy generated by red- and blue-shifting of photons

Sachs & Wolfe (1967)

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Detecting GW by CMB Polarisation

Quadrupole temperature anisotropy scattered by an electron

16 Isotropic radiation field (CMB) Isotropic radiation field (CMB)

h

₊ h

_⇥

h

₊ h

_⇥

冷冷

熱

熱冷

冷熱熱

Electron

C H C

H C

C H H

Polnarev (1985)

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Generation and erasure

of tensor quadrupole (viscosity)

• Gravitational waves create quadrupole temperature anisotropy (i.e., tensor viscosity of a photon-baryon fluid) gravitationally, without a velocity potential.

• Still, tight-coupling between photons and baryons erases the tensor viscosity exponentially before the last scattering.

negligible contribution before the last scattering

17

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Part II: Propagation of

Gravitational Waves in an Expanding Universe

18

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Equation of Motion of the Gravitational Wave

Wave equation in a non-expanding Universe

• Einstein’s equation gives a wave equation for the GW:

19

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a ² ⇤ D _ij = 16⇡ GT _ij ^GW The stress-energy

source of GW

⇤ = @ ²

@ t ² + r ² =

X 3

µ=0

X 3

⌫ =0

⌘ ^µ⌫ @

@ x ^µ

@

@ x ^⌫

⌘ ⁰⁰ = 1, ⌘ ⁰ⁱ = 0, ⌘ ^ij = ^ij

where

with

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ds ² ₄ = dt ² + dx ² = X

µ⌫

⌘ _µ⌫ dx ^µ dx ^⌫

( )

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Equation of Motion of the Gravitational Wave

Wave equation in an expanding Universe

• Einstein’s equation gives a wave equation for the GW:

20 a ² ⇤ D _ij = 16⇡ GT _ij ^GW The stress-energy

source of GW

where

with

⇤ ⌘ 1

p g

X 3

µ=0

X 3

⌫ =0

@

@ x ^µ

✓ p

gg ^µ⌫ @

@ x ^⌫

◆ g ⁰⁰ = 1, g ⁰ⁱ = 0, g ^ij = a ² (t) ^ij , p

g = a ³ (t) (

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)

ds ² ₄ = dt ² + a ² (t)dx ²

= X

µ⌫

g _µ⌫ dx ^µ dx ^⌫

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Equation of Motion of the Gravitational Wave

Wave equation in an expanding Universe

• Einstein’s equation gives a wave equation for the GW:

21 a ² ⇤ D _ij = 16⇡ GT _ij ^GW The stress-energy

source of GW

where

⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t + 1

a ² r ²

Eﬀect of the expansion

of the Universe!

(22)

Equation of Motion of the Gravitational Wave

Wave equation in an expanding Universe

• Einstein’s equation gives a wave equation for the GW:

22 a ² ⇤ D _ij = 16⇡ GT _ij ^GW The stress-energy

source of GW

where

⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t + 1

a ² r ²

Eﬀect of the expansion of the Universe!

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q ² a ²

In Fourier space

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r ² exp(iq · x) = q ² exp(iq · x)

(23)

Equation of Motion of the Gravitational Wave

Wave equation in an expanding Universe

• Einstein’s equation gives a wave equation for the GW:

23 a ² ⇤ D _ij = 16⇡ GT _ij ^GW The stress-energy

source of GW

where

⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t + 1

a ² r ²

Eﬀect of the expansion of the Universe!

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q ² a ²

In Fourier space

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r ² exp(iq · x) = q ² exp(iq · x)

Physical wavenumber, q/a

(24)

Super-horizon Solution (q << aH)

• Super-horizon tensor perturbation is conserved!

• Similar to the conserved scalar perturbation, ζ.

• Thus, no ISW temperature anisotropy on super-horizon scales

• It does not look like “gravitational waves”, but it will start

oscillating and behaving like waves once it enters the horizon

D _ij = constant + decaying term

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(25)

Matter-dominated Solution

• ^∂D ^ij /∂t gives the ISW. It peaks at the horizon crossing, qr~2.

• The energy density is given by (∂D ij /∂t) ² , which indeed decays like radiation, a ^–4 .

/ 1

a(t)

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/ 1

a ² (t)

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25

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r

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r (t) = c

Z _t

0 dt ⁰ a(t ⁰ )

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j ₁ (qr ) = sin(qr ) (qr ) ²

cos(qr ) qr

Cf: Sound wave

The oscillation of the GW solution is given by

the photon horizon, not the sound horizon,

as the GW propagates at the speed of light.

(26)

Energy density of the gravitational waves

• The equation of motion yields

26 h ˙ _ij / a ² (t) ⇢ _GW (t) / a ⁴ (t)

The GW density redshifts

as relativistic particles (radiation)!

⇢ _GW (t) = 1

4 M _pl ² X

ij

h h ˙ _ij (t, x) ˙ h _ij (t, x) i

= 1

2 M _pl ² X

=+, ⇥

h h ˙ ² (t, x) i

M _pl = (8⇡ G) ^1/2

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D _ij

h _ij =

0 @ h ₊ h _⇥ 0 h _⇥ h ₊ 0

0 0 0

1 A

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D _ij

(27)

Part III: Temperature Anisotropy from Gravitational Waves

27

(28)

Formal Solution (Tensor)

Sachs & Wolfe (1967)

negligible contribution before the last scattering

28

(29)

Formal Solution (Tensor)

negligible contribution before the last scattering

Sachs & Wolfe (1967)

29

(30)

Formal Solution (Tensor)

negligible contribution before the last scattering

When a plane wave gravitational wave propagates in the z direction, no temperature anisotropy is

seen towards the poles (θ=0, π). The anisotropy is maximised on the horizon (θ=π/2) with cos(2φ) &

sin(2φ) modulation in the azimuthal directions.

Sachs & Wolfe (1967)

(31)

Formal Solution (Tensor)

negligible contribution before the last scattering

Spherical harmonics Y lm (θ,φ) with (l,m)=(2,2)

Sachs & Wolfe (1967)

31

(32)

Temperature C l from GW

Scale-invariant

32 r = 1

(33)

Entered the horizon after the last scattering

Tensor mode damped by

redshifts between the horizon re-

entry and the decoupling

Tensor ISW

Temperature C l from GW

Scale-invariant

33

(34)

Temperature C l from GW

Scale-invariant

This is NOT a Silk- like damping!

It’s not

exponential, but a power-law due

simply to redshifts

34

(35)

Part IV: E- and B-mode

Polarisation from Gravitational Waves

35

(36)

propagation direction of GW h + =cos(qz)

Polarisation directions perpendicular/parallel to the wavenumber vector -> E mode polarisation

36 ~ q

(37)

propagation direction of GW h x =cos(qz)

Polarisation directions 45 degrees tilted from to the wavenumber vector -> B mode polarisation

37 ~ q

(38)

CAUTION: we are NOT seeing a single plane wave propagating perpendicular to our line of sight

Signature of gravitational waves in the sky [?]

38

(39)

CAUTION: we are NOT seeing a single plane wave propagating perpendicular to our line of sight

Signature of gravitational waves in the sky [?]

if you wish, you could associate one pattern with one plane wave…

BUT

39

(40)

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

(l,m)=(2,2)

Let’s symbolise (l,m)=(2,2) as

Cold Hot

40

(41)

E-mode!

Pol on the horizon is 1/2

of the zenith

(42)

B-mode!

Pol on the

horizon

vanishes

(43)

• E and B modes are produced nearly equally, but on small scales B is smaller than E because B vanishes on the horizon.

43 r = 1

(44)

• E and B modes are produced nearly equally, but on small scales B is smaller than E because B vanishes on the horizon.

44 r = 1

(45)

• E and B modes are produced nearly equally, but on small scales B is smaller than E because B vanishes on the horizon

This damping is

actually due to the

“Fuzziness”

damping from the finite extent of the

last-scattering surface

45 r = 1

(46)

No Fuzziness damping

Pritchard and Kamionkowski (2005) ⁴⁶

(47)

With damping

Pritchard and Kamionkowski (2005) ⁴⁷

(48)

Entered the horizon after the last scattering

Tensor ISW

Polarisation generated by

tensor viscosity at the last scattering

48 r = 1

(49)

Polarisation generated by

tensor viscosity at the last scattering

TE correlation

49

(50)

B-mode from lensing E-mode

from sound waves

Temperature from sound waves

B-mode from GW

We understand this now!

50 We understand this now!

We understand this now!

Enjoy starting at these power spectra, and

being able to explain all

the features in words!

(51)

Appendix:

Experimental Landscape

(52)

What comes next?

Advanced Atacama Cosmology Telescope

South Pole Telescope “3G”

CLASS

BICEP/Keck Array

(53)

Advanced Atacama

Cosmology Telescope

(54)

South Pole Telescope “3G”

CLASS BICEP/Keck Array

CMB-S4(?)

(55)

The Biggest Enemy:

Polarised Dust Emission

• The upcoming data will NOT be limited by statistics, but by systematic eﬀects such as the Galactic contamination

• Solution: Observe the sky at multiple frequencies, especially at high frequencies (>300 GHz)

• This is challenging, unless we have a superb, high-altitude site with low water vapour

• ^CCAT-p!

(56)

CCAT-p Collaboration

(57)

Frank Bertoldi’s slide from the Florence meeting

(58)

Frank Bertoldi’s slide from the Florence meeting

Cornell U. + German consortium + Canadian consortium + …

(59)

A Game Changer

• ^CCAT-p : 6-m, Cross-dragone design, on Cerro Chajnantor (5600 m)

• Germany makes great telescopes!

• Design study completed, and the contract has been signed by

“VERTEX Antennentechnik GmbH”

• CCAT-p is a great opportunity for Germany to make

significant contributions towards the CMB S-4 landscape

(both US and Europe) by providing telescope designs and

the “lessons learned” with prototypes.

(60)

Simons Observatory (USA)

in collaboration

South Pole?

(61)

Simons Observatory (USA)

in collaboration

South Pole?

This could be

“CMB-S4”

(62)

To have even more

frequency coverage…

(63)

ESA

2025– [proposed]

JAXA

LiteBIRD

+ participations from

USA, Canada, Europe

Polarisation satellite dedicated to measure CMB

polarisation from primordial GW, with a few thousand TES bolometers in space

2028–

(64)

ESA

2025– [proposed]

JAXA

LiteBIRD

May 21, 2019: JAXA has chosen LiteBIRD as the strategic large-class mission.

We will go to L2!

+ participations from

USA, Canada, Europe

Selected!

2028–

(65)

• Polarized foregrounds

• Synchrotron radiation and thermal emission from inter-galactic dust

• Characterize and remove foregrounds

• 15 frequency bands between 40 GHz - 400 GHz

• Split between Low Frequency Telescope (LFT) and High Frequency Telescope (HFT)

• LFT: 40 GHz – 235 GHz

• HFT: 280 GHz – 400 GHz

Foreground Removal

7 Polarized galactic emission (Planck X) LiteBIRD: 15 frequency bands

Slide courtesy Toki Suzuki (Berkeley)

(66)

LiteBIRD

LiteBIRD Spacecraft

LiteBIRD for B-mode from Space

2018/7/21 11

LFT (5K)

HG-antenna HFT (5K)

V-groove

radiators SVM/BUS

PLM 200K 100K 30K

JAXA

H3 LFT (Low frequency telescope) 34 – 161 GHz : Synchrotron + CMB HFT (high frequency telescope) 89 – 448 GHz : CMB + Dust

Lecture 10: Gravitational Waves