Day 2: Polarisation of the CMB

(1)

Day 2:

Polarisation of the CMB

Eiichiro Komatsu

[Max Planck Institute for Astrophysics]

University of Amsterdam March 5, 2020

Lecture notes:

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

(2)

How do we measure

gravitational waves?

(3)

Measuring GW

d`

²

= dx

²

= X

ij

dx

ⁱ

dx

^j

d`

²

= X

ij

(

_ij

+ h

_ij

)dx

ⁱ

dx

^j

•

GW changes distances between two points

(4)

Laser Interferometer

Mirror

detector _{No signal}

(5)

Laser Interferometer

Mirror

Signal!

detector

(6)

LIGO detected GW from a binary

blackholes, with a frequency of ~100 Hz

= the wavelength of thousands of kilometres

But, the primordial GW affecting the

CMB has a frequency of 10 –18 Hz = the wavelength of billions of light-years!!

How do we find it?

(7)

Detecting GW by CMB

Isotropic electro-magnetic fields

(8)

Detecting GW by CMB

GW propagating in isotropic electro-magnetic fields

h ⁺ _ij

<latexit sha1_base64="+b4698YGkh1ezVXsa0LwhPMRkAM=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBAEIeyKoMegF48RzAOTNcxOZpMxs7PLTK8Qlv0LLx4U8erfePNvnDwOmljQUFR1090VJFIYdN1vZ2l5ZXVtvbBR3Nza3tkt7e03TJxqxusslrFuBdRwKRSvo0DJW4nmNAokbwbD67HffOLaiFjd4SjhfkT7SoSCUbTS/aCbicf8ITvNu6WyW3EnIIvEm5EyzFDrlr46vZilEVfIJDWm7bkJ+hnVKJjkebGTGp5QNqR93rZU0YgbP5tcnJNjq/RIGGtbCslE/T2R0ciYURTYzojiwMx7Y/E/r51ieOlnQiUpcsWmi8JUEozJ+H3SE5ozlCNLKNPC3krYgGrK0IZUtCF48y8vksZZxXMr3u15uXo1i6MAh3AEJ+DBBVThBmpQBwYKnuEV3hzjvDjvzse0dcmZzRzAHzifP7yOkPE=</latexit><latexit sha1_base64="+b4698YGkh1ezVXsa0LwhPMRkAM=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBAEIeyKoMegF48RzAOTNcxOZpMxs7PLTK8Qlv0LLx4U8erfePNvnDwOmljQUFR1090VJFIYdN1vZ2l5ZXVtvbBR3Nza3tkt7e03TJxqxusslrFuBdRwKRSvo0DJW4nmNAokbwbD67HffOLaiFjd4SjhfkT7SoSCUbTS/aCbicf8ITvNu6WyW3EnIIvEm5EyzFDrlr46vZilEVfIJDWm7bkJ+hnVKJjkebGTGp5QNqR93rZU0YgbP5tcnJNjq/RIGGtbCslE/T2R0ciYURTYzojiwMx7Y/E/r51ieOlnQiUpcsWmi8JUEozJ+H3SE5ozlCNLKNPC3krYgGrK0IZUtCF48y8vksZZxXMr3u15uXo1i6MAh3AEJ+DBBVThBmpQBwYKnuEV3hzjvDjvzse0dcmZzRzAHzifP7yOkPE=</latexit><latexit sha1_base64="+b4698YGkh1ezVXsa0LwhPMRkAM=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBAEIeyKoMegF48RzAOTNcxOZpMxs7PLTK8Qlv0LLx4U8erfePNvnDwOmljQUFR1090VJFIYdN1vZ2l5ZXVtvbBR3Nza3tkt7e03TJxqxusslrFuBdRwKRSvo0DJW4nmNAokbwbD67HffOLaiFjd4SjhfkT7SoSCUbTS/aCbicf8ITvNu6WyW3EnIIvEm5EyzFDrlr46vZilEVfIJDWm7bkJ+hnVKJjkebGTGp5QNqR93rZU0YgbP5tcnJNjq/RIGGtbCslE/T2R0ciYURTYzojiwMx7Y/E/r51ieOlnQiUpcsWmi8JUEozJ+H3SE5ozlCNLKNPC3krYgGrK0IZUtCF48y8vksZZxXMr3u15uXo1i6MAh3AEJ+DBBVThBmpQBwYKnuEV3hzjvDjvzse0dcmZzRzAHzifP7yOkPE=</latexit><latexit sha1_base64="+b4698YGkh1ezVXsa0LwhPMRkAM=">AAAB8XicbVDLSgNBEOz1GeMr6tHLYBAEIeyKoMegF48RzAOTNcxOZpMxs7PLTK8Qlv0LLx4U8erfePNvnDwOmljQUFR1090VJFIYdN1vZ2l5ZXVtvbBR3Nza3tkt7e03TJxqxusslrFuBdRwKRSvo0DJW4nmNAokbwbD67HffOLaiFjd4SjhfkT7SoSCUbTS/aCbicf8ITvNu6WyW3EnIIvEm5EyzFDrlr46vZilEVfIJDWm7bkJ+hnVKJjkebGTGp5QNqR93rZU0YgbP5tcnJNjq/RIGGtbCslE/T2R0ciYURTYzojiwMx7Y/E/r51ieOlnQiUpcsWmi8JUEozJ+H3SE5ozlCNLKNPC3krYgGrK0IZUtCF48y8vksZZxXMr3u15uXo1i6MAh3AEJ+DBBVThBmpQBwYKnuEV3hzjvDjvzse0dcmZzRzAHzifP7yOkPE=</latexit>

h ^⇥ _ij

<latexit sha1_base64="NwsD1SgjkJfMeVFacsNc7+DBIRA=">AAAB+HicbVBNS8NAEJ34WetHox69BIvgqSQi6LHoxWMF+wFtDZvtpl272YTdiVBDfokXD4p49ad489+4bXPQ1gcDj/dmmJkXJIJrdN1va2V1bX1js7RV3t7Z3avY+wctHaeKsiaNRaw6AdFMcMmayFGwTqIYiQLB2sH4euq3H5nSPJZ3OElYPyJDyUNOCRrJtysjP+MP+X3WQx4xnft21a25MzjLxCtIFQo0fPurN4hpGjGJVBCtu56bYD8jCjkVLC/3Us0SQsdkyLqGSmK29LPZ4blzYpSBE8bKlERnpv6eyEik9SQKTGdEcKQXvan4n9dNMbzsZ1wmKTJJ54vCVDgYO9MUnAFXjKKYGEKo4uZWh46IIhRNVmUTgrf48jJpndU8t+bdnlfrV0UcJTiCYzgFDy6gDjfQgCZQSOEZXuHNerJerHfrY966YhUzh/AH1ucPjLyTpw==</latexit><latexit sha1_base64="NwsD1SgjkJfMeVFacsNc7+DBIRA=">AAAB+HicbVBNS8NAEJ34WetHox69BIvgqSQi6LHoxWMF+wFtDZvtpl272YTdiVBDfokXD4p49ad489+4bXPQ1gcDj/dmmJkXJIJrdN1va2V1bX1js7RV3t7Z3avY+wctHaeKsiaNRaw6AdFMcMmayFGwTqIYiQLB2sH4euq3H5nSPJZ3OElYPyJDyUNOCRrJtysjP+MP+X3WQx4xnft21a25MzjLxCtIFQo0fPurN4hpGjGJVBCtu56bYD8jCjkVLC/3Us0SQsdkyLqGSmK29LPZ4blzYpSBE8bKlERnpv6eyEik9SQKTGdEcKQXvan4n9dNMbzsZ1wmKTJJ54vCVDgYO9MUnAFXjKKYGEKo4uZWh46IIhRNVmUTgrf48jJpndU8t+bdnlfrV0UcJTiCYzgFDy6gDjfQgCZQSOEZXuHNerJerHfrY966YhUzh/AH1ucPjLyTpw==</latexit><latexit sha1_base64="NwsD1SgjkJfMeVFacsNc7+DBIRA=">AAAB+HicbVBNS8NAEJ34WetHox69BIvgqSQi6LHoxWMF+wFtDZvtpl272YTdiVBDfokXD4p49ad489+4bXPQ1gcDj/dmmJkXJIJrdN1va2V1bX1js7RV3t7Z3avY+wctHaeKsiaNRaw6AdFMcMmayFGwTqIYiQLB2sH4euq3H5nSPJZ3OElYPyJDyUNOCRrJtysjP+MP+X3WQx4xnft21a25MzjLxCtIFQo0fPurN4hpGjGJVBCtu56bYD8jCjkVLC/3Us0SQsdkyLqGSmK29LPZ4blzYpSBE8bKlERnpv6eyEik9SQKTGdEcKQXvan4n9dNMbzsZ1wmKTJJ54vCVDgYO9MUnAFXjKKYGEKo4uZWh46IIhRNVmUTgrf48jJpndU8t+bdnlfrV0UcJTiCYzgFDy6gDjfQgCZQSOEZXuHNerJerHfrY966YhUzh/AH1ucPjLyTpw==</latexit><latexit sha1_base64="NwsD1SgjkJfMeVFacsNc7+DBIRA=">AAAB+HicbVBNS8NAEJ34WetHox69BIvgqSQi6LHoxWMF+wFtDZvtpl272YTdiVBDfokXD4p49ad489+4bXPQ1gcDj/dmmJkXJIJrdN1va2V1bX1js7RV3t7Z3avY+wctHaeKsiaNRaw6AdFMcMmayFGwTqIYiQLB2sH4euq3H5nSPJZ3OElYPyJDyUNOCRrJtysjP+MP+X3WQx4xnft21a25MzjLxCtIFQo0fPurN4hpGjGJVBCtu56bYD8jCjkVLC/3Us0SQsdkyLqGSmK29LPZ4blzYpSBE8bKlERnpv6eyEik9SQKTGdEcKQXvan4n9dNMbzsZ1wmKTJJ54vCVDgYO9MUnAFXjKKYGEKo4uZWh46IIhRNVmUTgrf48jJpndU8t+bdnlfrV0UcJTiCYzgFDy6gDjfQgCZQSOEZXuHNerJerHfrY966YhUzh/AH1ucPjLyTpw==</latexit>

(9)

hot

cold

cold cold

hot hot

Detecting GW by CMB

Space is stretched => Wavelength of light is also stretched

h ⁺ _ij

h ^⇥ _ij

(10)

hot

cold

cold cold

hot hot

Detecting GW by CMB Polarisation

electron electron

Space is stretched => Wavelength of light is also stretched

h ⁺ _ij

h ^⇥ _ij

(11)

hot

cold

cold cold

hot hot

Detecting GW by CMB Polarisation

Space is stretched => Wavelength of light is also stretched

h ⁺ _ij

11

h ^⇥ _ij

(12)

horizontally polarised

Photo Credit: TALEX

(13)

Photo Credit: TALEX

(14)

Physics of CMB Polarisation

• Necessary and sufficient conditions for generating polarisation in CMB:

• Thomson scattering

• Quadrupolar temperature anisotropy around an electron

By Wayne Hu

(15)

(16)

Credit: ESA

(17)

Credit: ESA

(18)

E and B mode

• ^{E mode}

: Polarisation directions

parallel or perpendicular

to the wavevector

• ^{B mode}

: Polarisation directions

45 degree tilted

with respect to the wavevector

(19)

E&B decomposition:

A closer look

(20)

Polarisation

No polarisation

Polarised in x-direction

(21)

Stokes Parameters

[Flat Sky, Cartesian coordinates]

a

b

(22)

Stokes Parameters

change under coordinate rotation

x’

Under (x,y) -> (x’,y’):

y’

(23)

Compact Expression

•

Using an imaginary number, write

Then, under coordinate rotation we have

(24)

E and B decomposition

•

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

To achieve this, we need

to go to Fourier space

(25)

ˆ

n = (sin ✓ cos , sin ✓ sin , cos ✓ )

“Flat sky”,

if θ is small

(26)

Fourier-transforming Stokes Parameters?

•

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

•

^So…

where

(27)

Tweaking 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(*)

(*) Nevermind the overall minus sign. This is just for convention

(28)

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)

(29)

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

(30)

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

(31)

Geometric Meaning

• ^{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

(32)

Parity

• ^{E mode}

: Parity even

• ^{B mode}

: Parity odd

(33)

Parity

• ^{E mode}

: Parity even

• ^{B mode}

: Parity odd

(34)

Power Spectra

•

^However,

<EB> and <TB> vanish

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

(35)

Power Spectra

•

^However,

<EB> and <TB> vanish

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

<EB> and <TB> can be non-zero if physics that produced gravitational waves during information

broke parity! This will be the topic on March 19

(36)

How do gravitational waves generate E&B

polarisation?

(37)

Distance between two points in space

•

Inhomogeneous curved space

•

In Cartesian comoving coordinates

“metric perturbation”

-> CURVED SPACE!

(38)

Now I am going to change notation (sorry!): h ĳ -> D ĳ

•

Notation in today’s lecture follows that of the text book

“Cosmology” by Steven Weinberg

(39)

Space-time Distance

•

Einstein told us that a clock ticks slowly when gravity is strong…

•

Space-time distance, ds4, is modified by the presence of gravitational fields

: Newton’s gravitational potential

: Spatial scalar curvature perturbation

: Tensor metric perturbation [=gravitational waves]

(40)

Tensor perturbation D ĳ :

Area-conserving deformation

•

Determinant of a matrix

is given by

•

^{Thus, D}^ij must be trace-less

if it is area-conserving deformation of two points in space

(41)

Curvature Perturbation

•

Einstein told us that a clock ticks slowly when gravity is strong…

•

Space-time distance, ds4, is modified by the presence of gravitational fields

: Newton’s gravitational potential

: Spatial scalar curvature perturbation

is a perturbation to the determinant of spatial metric

(42)

Evolution of

photon’s coordinates

•

Photon’s path is determined such that the distance

traveled by a photon between two points is minimised.

This yields the equation of motion for photon’s coordinates

This equation is known as the “geodesic equation”.

The second term is needed to keep the form of the equation unchanged under general coordinate transformation => GRAVITATIONAL EFFECTS!

y

x

“u” labels photon’s path

(43)

Evolution of

photon’s momentum

•

It is more convenient to write down the geodesic equation in terms of the photon momentum:

y

x

“u” labels photon’s path then

Magnitude of the photon momentum is equal to the photon energy:

(44)

Some calculations…

With ( )

Scalar perturbation [valid to all orders] Tensor perturbation [valid to 1st order in D]

(45)

Recap

•

Requiring photons to travel between two points in

space-time with the minimum path length, we obtained the geodesic equation

•

The geodesic equation contains that is required to make the form of the equation unchanged under

general coordinate transformation

•

Expressing in terms of the metric perturbations, we obtain the desired result - the equation that describes the rate of change of the photon energy!

Math may be messy but the concept is transparent!

(46)

•

Let’s interpret this equation physically

The Result

γⁱ is a unit vector of the direction of photon’s momentum:

Sachs & Wolfe (1967)

(47)

•

Cosmological redshift

•

Photon’s wavelength is stretched in proportion to the scale factor, and thus the photon energy decreases as

The Result

γⁱ is a unit vector of the direction of photon’s momentum:

p / a ¹

(48)

•

Cosmological redshift - part II

•

The spatial metric is given by

•

Thus, locally we can define a new scale factor:

•

Then the photon momentum decreases as

The Result

ds

²

= a

²

(t) exp( 2 )dx

²

˜

a(t, x) = a(t) exp( )

p / a ˜ ¹

(49)

•

Gravitational blue/redshift (Scalar)

The Result

Potential well (φ < 0)

(50)

•

Gravitational blue/redshift (Tensor)

The Result

(51)

The Result

•

Gravitational blue/redshift (Tensor)

(52)

Formal Solution (Scalar)

or

Line-of-sight direction

Coming distance (r)

“L” for “Last scattering surface”

(53)

Formal Solution (Scalar)

Coming distance (r) Initial Condition

(54)

Formal Solution (Scalar)

Comoving distance (r) Gravitational Redshit

(55)

Formal Solution (Scalar)

Coming distance (r)

“integrated Sachs-Wolfe” (ISW) eﬀect

(56)

Formal Solution (Tensor)

negligible contribution before the last scattering

(57)

Formal Solution (Tensor)

negligible contribution before the last scattering Sachs & Wolfe (1967)

(58)

Formal Solution (Tensor)

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.

(59)

Formal Solution (Tensor)

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

(60)

E-mode!

(61)

E-mode!

Pol on the horizon is 1/2

of the zenith

(62)

B-mode!

Pol on the horizon vanishes

(63)

propagation direction of GW h

+

=cos(kx)

Polarisation directions perpendicular/parallel to the

wavenumber vector -> E mode polarisation

(64)

propagation direction of GW h

x

=cos(kx)

Polarisation directions 45 degrees tilted from to the

wavenumber vector -> B mode polarisation

(65)

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

Signature of gravitational

waves in the sky [?]

(66)

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

(67)

Propagation of cosmological gravitational waves

•

Tensor anisotropic stress can do two things:

•

It can generate gravitational waves

•

It can damp gravitational waves (neutrino anisotropic stress)

tensor

But we ignore the tensor anisotropic stress today

(68)

Super-horizon Solution

•

Super-horizon tensor perturbation is conserved

•

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

<latexit sha1_base64="a4Lh3bqarO5i6yyItf0Aivpni1I=">AAACFnicbVDLSgNBEJz1GeMr6tHLYBAEMeyKoBchqAePCkYDSQizk04yZmZ2mekVwrL+hBd/xYsHRbyKN//GyeOg0YKGoqqb7q4wlsKi7395U9Mzs3PzuYX84tLyymphbf3aRonhUOGRjEw1ZBak0FBBgRKqsQGmQgk3Ye904N/cgbEi0lfYj6GhWEeLtuAMndQs7J01U3Gb0WOa1o2iPNIWmcaM7o6EFnDWF7pzj2BU1iwU/ZI/BP1LgjEpkjEumoXPeiviiQKNXDJra4EfYyNlBgWXkOXriYWY8R7rQM1RzRTYRjp8K6PbTmnRdmRcaaRD9edEypS1fRW6TsWwaye9gfifV0uwfdRIhY4TBM1Hi9qJpBjRQUa0JQxwlH1HGDfC3Up5lxnGXQg270IIJl/+S673S4FfCi4PiuWTcRw5skm2yA4JyCEpk3NyQSqEkwfyRF7Iq/foPXtv3vuodcobz2yQX/A+vgEEVZ9N</latexit><latexit sha1_base64="a4Lh3bqarO5i6yyItf0Aivpni1I=">AAACFnicbVDLSgNBEJz1GeMr6tHLYBAEMeyKoBchqAePCkYDSQizk04yZmZ2mekVwrL+hBd/xYsHRbyKN//GyeOg0YKGoqqb7q4wlsKi7395U9Mzs3PzuYX84tLyymphbf3aRonhUOGRjEw1ZBak0FBBgRKqsQGmQgk3Ye904N/cgbEi0lfYj6GhWEeLtuAMndQs7J01U3Gb0WOa1o2iPNIWmcaM7o6EFnDWF7pzj2BU1iwU/ZI/BP1LgjEpkjEumoXPeiviiQKNXDJra4EfYyNlBgWXkOXriYWY8R7rQM1RzRTYRjp8K6PbTmnRdmRcaaRD9edEypS1fRW6TsWwaye9gfifV0uwfdRIhY4TBM1Hi9qJpBjRQUa0JQxwlH1HGDfC3Up5lxnGXQg270IIJl/+S673S4FfCi4PiuWTcRw5skm2yA4JyCEpk3NyQSqEkwfyRF7Iq/foPXtv3vuodcobz2yQX/A+vgEEVZ9N</latexit><latexit sha1_base64="a4Lh3bqarO5i6yyItf0Aivpni1I=">AAACFnicbVDLSgNBEJz1GeMr6tHLYBAEMeyKoBchqAePCkYDSQizk04yZmZ2mekVwrL+hBd/xYsHRbyKN//GyeOg0YKGoqqb7q4wlsKi7395U9Mzs3PzuYX84tLyymphbf3aRonhUOGRjEw1ZBak0FBBgRKqsQGmQgk3Ye904N/cgbEi0lfYj6GhWEeLtuAMndQs7J01U3Gb0WOa1o2iPNIWmcaM7o6EFnDWF7pzj2BU1iwU/ZI/BP1LgjEpkjEumoXPeiviiQKNXDJra4EfYyNlBgWXkOXriYWY8R7rQM1RzRTYRjp8K6PbTmnRdmRcaaRD9edEypS1fRW6TsWwaye9gfifV0uwfdRIhY4TBM1Hi9qJpBjRQUa0JQxwlH1HGDfC3Up5lxnGXQg270IIJl/+S673S4FfCi4PiuWTcRw5skm2yA4JyCEpk3NyQSqEkwfyRF7Iq/foPXtv3vuodcobz2yQX/A+vgEEVZ9N</latexit><latexit sha1_base64="a4Lh3bqarO5i6yyItf0Aivpni1I=">AAACFnicbVDLSgNBEJz1GeMr6tHLYBAEMeyKoBchqAePCkYDSQizk04yZmZ2mekVwrL+hBd/xYsHRbyKN//GyeOg0YKGoqqb7q4wlsKi7395U9Mzs3PzuYX84tLyymphbf3aRonhUOGRjEw1ZBak0FBBgRKqsQGmQgk3Ye904N/cgbEi0lfYj6GhWEeLtuAMndQs7J01U3Gb0WOa1o2iPNIWmcaM7o6EFnDWF7pzj2BU1iwU/ZI/BP1LgjEpkjEumoXPeiviiQKNXDJra4EfYyNlBgWXkOXriYWY8R7rQM1RzRTYRjp8K6PbTmnRdmRcaaRD9edEypS1fRW6TsWwaye9gfifV0uwfdRIhY4TBM1Hi9qJpBjRQUa0JQxwlH1HGDfC3Up5lxnGXQg270IIJl/+S673S4FfCi4PiuWTcRw5skm2yA4JyCEpk3NyQSqEkwfyRF7Iq/foPXtv3vuodcobz2yQX/A+vgEEVZ9N</latexit>

(69)

Matter-dominated Solution

•

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

•

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

/ 1 a(t)

<latexit sha1_base64="tt+hBjENX0OF7Ntykl1fzbjM13A=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2AR6qbMiKDLohuXFewDOkPJpJk2NJOEJCOUcRb+ihsXirj1N9z5N6btLLT1wIXDOfdy7z2RZFQbz/t2Siura+sb5c3K1vbO7p67f9DWIlWYtLBgQnUjpAmjnLQMNYx0pSIoiRjpROObqd95IEpTwe/NRJIwQUNOY4qRsVLfPQqkEtIIGMQK4czPM1QzZ3nfrXp1bwa4TPyCVEGBZt/9CgYCpwnhBjOkdc/3pAkzpAzFjOSVINVEIjxGQ9KzlKOE6DCb3Z/DU6sMYCyULW7gTP09kaFE60kS2c4EmZFe9Kbif14vNfFVmFEuU0M4ni+KUwbtv9Mw4IAqgg2bWIKwovZWiEfIBmFsZBUbgr/48jJpn9d9r+7fXVQb10UcZXAMTkAN+OASNMAtaIIWwOARPINX8OY8OS/Ou/Mxby05xcwh+APn8we83JXm</latexit><latexit sha1_base64="tt+hBjENX0OF7Ntykl1fzbjM13A=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2AR6qbMiKDLohuXFewDOkPJpJk2NJOEJCOUcRb+ihsXirj1N9z5N6btLLT1wIXDOfdy7z2RZFQbz/t2Siura+sb5c3K1vbO7p67f9DWIlWYtLBgQnUjpAmjnLQMNYx0pSIoiRjpROObqd95IEpTwe/NRJIwQUNOY4qRsVLfPQqkEtIIGMQK4czPM1QzZ3nfrXp1bwa4TPyCVEGBZt/9CgYCpwnhBjOkdc/3pAkzpAzFjOSVINVEIjxGQ9KzlKOE6DCb3Z/DU6sMYCyULW7gTP09kaFE60kS2c4EmZFe9Kbif14vNfFVmFEuU0M4ni+KUwbtv9Mw4IAqgg2bWIKwovZWiEfIBmFsZBUbgr/48jJpn9d9r+7fXVQb10UcZXAMTkAN+OASNMAtaIIWwOARPINX8OY8OS/Ou/Mxby05xcwh+APn8we83JXm</latexit><latexit sha1_base64="tt+hBjENX0OF7Ntykl1fzbjM13A=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2AR6qbMiKDLohuXFewDOkPJpJk2NJOEJCOUcRb+ihsXirj1N9z5N6btLLT1wIXDOfdy7z2RZFQbz/t2Siura+sb5c3K1vbO7p67f9DWIlWYtLBgQnUjpAmjnLQMNYx0pSIoiRjpROObqd95IEpTwe/NRJIwQUNOY4qRsVLfPQqkEtIIGMQK4czPM1QzZ3nfrXp1bwa4TPyCVEGBZt/9CgYCpwnhBjOkdc/3pAkzpAzFjOSVINVEIjxGQ9KzlKOE6DCb3Z/DU6sMYCyULW7gTP09kaFE60kS2c4EmZFe9Kbif14vNfFVmFEuU0M4ni+KUwbtv9Mw4IAqgg2bWIKwovZWiEfIBmFsZBUbgr/48jJpn9d9r+7fXVQb10UcZXAMTkAN+OASNMAtaIIWwOARPINX8OY8OS/Ou/Mxby05xcwh+APn8we83JXm</latexit><latexit sha1_base64="tt+hBjENX0OF7Ntykl1fzbjM13A=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2AR6qbMiKDLohuXFewDOkPJpJk2NJOEJCOUcRb+ihsXirj1N9z5N6btLLT1wIXDOfdy7z2RZFQbz/t2Siura+sb5c3K1vbO7p67f9DWIlWYtLBgQnUjpAmjnLQMNYx0pSIoiRjpROObqd95IEpTwe/NRJIwQUNOY4qRsVLfPQqkEtIIGMQK4czPM1QzZ3nfrXp1bwa4TPyCVEGBZt/9CgYCpwnhBjOkdc/3pAkzpAzFjOSVINVEIjxGQ9KzlKOE6DCb3Z/DU6sMYCyULW7gTP09kaFE60kS2c4EmZFe9Kbif14vNfFVmFEuU0M4ni+KUwbtv9Mw4IAqgg2bWIKwovZWiEfIBmFsZBUbgr/48jJpn9d9r+7fXVQb10UcZXAMTkAN+OASNMAtaIIWwOARPINX8OY8OS/Ou/Mxby05xcwh+APn8we83JXm</latexit>

/ 1

a ² (t)

<latexit sha1_base64="OECroyH6Bm5+Rqs2nC0WT7KE1do=">AAACAXicbVDLSgMxFM3UV62vUTeCm2AR6qbMFEGXRTcuK9gHdMaSSTNtaCYJSUYoQ934K25cKOLWv3Dn35i2s9DWA4HDOfdyc04kGdXG876dwsrq2vpGcbO0tb2zu+fuH7S0SBUmTSyYUJ0IacIoJ01DDSMdqQhKIkba0eh66rcfiNJU8DszliRM0IDTmGJkrNRzjwKphDQCBrFCOPMnGbqvVczZpOeWvao3A1wmfk7KIEej534FfYHThHCDGdK663vShBlShmJGJqUg1UQiPEID0rWUo4ToMJslmMBTq/RhLJR93MCZ+nsjQ4nW4ySykwkyQ73oTcX/vG5q4sswo1ymhnA8PxSnDNrE0zpgnyqCDRtbgrCi9q8QD5GtwtjSSrYEfzHyMmnVqr5X9W/Py/WrvI4iOAYnoAJ8cAHq4AY0QBNg8AiewSt4c56cF+fd+ZiPFpx85xD8gfP5A/Jfloo=</latexit><latexit sha1_base64="OECroyH6Bm5+Rqs2nC0WT7KE1do=">AAACAXicbVDLSgMxFM3UV62vUTeCm2AR6qbMFEGXRTcuK9gHdMaSSTNtaCYJSUYoQ934K25cKOLWv3Dn35i2s9DWA4HDOfdyc04kGdXG876dwsrq2vpGcbO0tb2zu+fuH7S0SBUmTSyYUJ0IacIoJ01DDSMdqQhKIkba0eh66rcfiNJU8DszliRM0IDTmGJkrNRzjwKphDQCBrFCOPMnGbqvVczZpOeWvao3A1wmfk7KIEej534FfYHThHCDGdK663vShBlShmJGJqUg1UQiPEID0rWUo4ToMJslmMBTq/RhLJR93MCZ+nsjQ4nW4ySykwkyQ73oTcX/vG5q4sswo1ymhnA8PxSnDNrE0zpgnyqCDRtbgrCi9q8QD5GtwtjSSrYEfzHyMmnVqr5X9W/Py/WrvI4iOAYnoAJ8cAHq4AY0QBNg8AiewSt4c56cF+fd+ZiPFpx85xD8gfP5A/Jfloo=</latexit><latexit sha1_base64="OECroyH6Bm5+Rqs2nC0WT7KE1do=">AAACAXicbVDLSgMxFM3UV62vUTeCm2AR6qbMFEGXRTcuK9gHdMaSSTNtaCYJSUYoQ934K25cKOLWv3Dn35i2s9DWA4HDOfdyc04kGdXG876dwsrq2vpGcbO0tb2zu+fuH7S0SBUmTSyYUJ0IacIoJ01DDSMdqQhKIkba0eh66rcfiNJU8DszliRM0IDTmGJkrNRzjwKphDQCBrFCOPMnGbqvVczZpOeWvao3A1wmfk7KIEej534FfYHThHCDGdK663vShBlShmJGJqUg1UQiPEID0rWUo4ToMJslmMBTq/RhLJR93MCZ+nsjQ4nW4ySykwkyQ73oTcX/vG5q4sswo1ymhnA8PxSnDNrE0zpgnyqCDRtbgrCi9q8QD5GtwtjSSrYEfzHyMmnVqr5X9W/Py/WrvI4iOAYnoAJ8cAHq4AY0QBNg8AiewSt4c56cF+fd+ZiPFpx85xD8gfP5A/Jfloo=</latexit><latexit sha1_base64="OECroyH6Bm5+Rqs2nC0WT7KE1do=">AAACAXicbVDLSgMxFM3UV62vUTeCm2AR6qbMFEGXRTcuK9gHdMaSSTNtaCYJSUYoQ934K25cKOLWv3Dn35i2s9DWA4HDOfdyc04kGdXG876dwsrq2vpGcbO0tb2zu+fuH7S0SBUmTSyYUJ0IacIoJ01DDSMdqQhKIkba0eh66rcfiNJU8DszliRM0IDTmGJkrNRzjwKphDQCBrFCOPMnGbqvVczZpOeWvao3A1wmfk7KIEej534FfYHThHCDGdK663vShBlShmJGJqUg1UQiPEID0rWUo4ToMJslmMBTq/RhLJR93MCZ+nsjQ4nW4ySykwkyQ73oTcX/vG5q4sswo1ymhnA8PxSnDNrE0zpgnyqCDRtbgrCi9q8QD5GtwtjSSrYEfzHyMmnVqr5X9W/Py/WrvI4iOAYnoAJ8cAHq4AY0QBNg8AiewSt4c56cF+fd+ZiPFpx85xD8gfP5A/Jfloo=</latexit>

η: “conformal time”, or the distance traveled by photons

(70)

Temperature C l from GW

Scale-invariant

(71)

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

(72)

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

(73)

•

E and B modes are produced nearly equally, but on small

scales B is smaller than E because B vanishes on the horizon

(74)

•

(75)

• This damping is actually due to the “Fuzziness”

damping from the finite extent of the

last-scattering

surface

(76)

No Fuzziness damping

Pritchard and Kamionkowski (2005)

(77)

With damping

Pritchard and Kamionkowski (2005)

(78)

Entered the horizon after the last scattering

Tensor ISW

Polarisation generated by scattering at the

last scattering

(79)

Polarisation generated by scattering at the

last scattering

TE correlation

(80)

B-mode from lensing E-mode

from sound waves

Temperature from sound waves

B-mode from GW

We understand this

(81)

Appendix:

Experimental Landscape

(82)

What comes next?

Advanced Atacama Cosmology Telescope

South Pole Telescope “3G”

CLASS

BICEP/Keck Array

(83)

Advanced Atacama

Cosmology Telescope

(84)

South Pole Telescope “3G”

CLASS BICEP/Keck Array

CMB-S4(?)

(85)

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!

(86)

CCAT-p Collaboration

(87)

Frank Bertoldi’s slide from the Florence meeting

(88)

Frank Bertoldi’s slide from the Florence meeting

Cornell U. + German consortium + Canadian consortium + …

(89)

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.

(90)

Simons Observatory (USA)

in collaboration

South Pole?

(91)

Simons Observatory (USA)

in collaboration

South Pole?

This could be

“CMB-S4”

(92)

To have even more

frequency coverage…

(93)

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–

(94)

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–

(95)

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)

(96)

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 200K100K30K

JAXA

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

4.5 m

Focal plane 0.1K

Slide courtesy Yutaro Sekimoto (ISAS/JAXA)

European Contribution

(97)

LiteBIRD Collaboration

(98)

Day 2: Polarisation of the CMB

Day 2:

Polarisation of the CMB

Eiichiro Komatsu

[Max Planck Institute for Astrophysics]

University of Amsterdam March 5, 2020

How do we measure

gravitational waves?

Measuring GW

d`

= dx

= X

dx

dx

d`

= X

(

+ h

)dx

dx

•

Laser Interferometer

Mirror

Mirror

detector No signal

Laser Interferometer

Mirror

Mirror

Signal!

detector

LIGO detected GW from a binary

blackholes, with a frequency of ~100 Hz

= the wavelength of thousands of kilometres

But, the primordial GW affecting the

CMB has a frequency of 10 –18 Hz = the wavelength of billions of light-years!!

How do we find it?

Detecting GW by CMB

Isotropic electro-magnetic fields

Detecting GW by CMB

GW propagating in isotropic electro-magnetic fields

h + ij

h ⇥ ij

hot

hot

cold

cold

cold cold

hot hot

Detecting GW by CMB

Space is stretched => Wavelength of light is also stretched

h + ij

h ⇥ ij

hot

hot

cold

cold

cold cold

hot hot

Detecting GW by CMB Polarisation

electron electron

Space is stretched => Wavelength of light is also stretched

h + ij

h ⇥ ij

hot

hot

cold

cold

cold cold

hot hot

Detecting GW by CMB Polarisation

Space is stretched => Wavelength of light is also stretched

h + ij

h ⇥ ij

horizontally polarised

Photo Credit: TALEX

Photo Credit: TALEX

Physics of CMB Polarisation

By Wayne Hu

E and B mode

• E mode

detector _{No signal}

h ⁺ _ij

h ^⇥ _ij

h ⁺ _ij

h ^⇥ _ij

h ⁺ _ij

h ^⇥ _ij

h ⁺ _ij

h ^⇥ _ij

• ^{E mode}

• ^{B mode}

• ^{E mode}

^Q

• ^{B mode}

^U

• ^{E mode}

• ^{B mode}

• ^{E mode}

• ^{B mode}