Primordial Gravitational Waves from Inﬂation

(1)

Primordial Gravitational Waves from Inflation

Eiichiro Komatsu

[Max Planck Institute for Astrophysics]

University of Amsterdam

February 27, March 5, and 19, 2020

Lecture notes:

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

(2)

Plan

• Today: Vacuum Fluctuation

• March 5: Polarisation of the cosmic microwave background

• March 19: Sourced Contribution

⇤ h _ij = 0

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⇤ h _ij = 16⇡ GT ⇡ _ij _ij ^GW ^GW

(3)

GW = Area-conserving distortion

of distances between two points

(4)

Distance between two points in space

• Static (i.e., non-expanding) Euclidean space

• In Cartesian coordinates

(5)

Distance between two points in space

• Homogeneously expanding Euclidean space

• In Cartesian comoving coordinates

“scale factor”

(6)

Distance between two points in space

• Homogeneously expanding Euclidean space

• In Cartesian comoving coordinates

“scale factor” =1 for i=j

=0 otherwise

(7)

Distance between two points in space

• Inhomogeneous curved space

• In Cartesian comoving coordinates

“metric perturbation”

-> CURVED SPACE!

(8)

• Gravitational waves shall be:

• Transverse: the direction of the oscillation of space is perpendicular to the propagation direction

• This means

Four conditions

X 3

i=1

k ⁱ h _ij = 0

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~ k

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3 conditions for h ĳ

propagation direction of GW h ij ~ cos(kz)

~ k

(9)

• Area-conserving: the determinant of the distortion in space remains unchanged

• This means that the trace vanishes:

Four conditions

X 3

i=1

h _ii = 0

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1 condition for h ĳ

y

x

(10)

• Gravitational waves shall be:

• Transverse: the direction of the oscillation of space is perpendicular to the propagation direction

• This means

• Area-conserving: the determinant of the distortion in space remains unchanged

• This means that the trace vanishes:

Four conditions

X 3

i=1

k ⁱ h _ij = 0

~ k

X 3

i=1

h _ii = 0

6 components of h ĳ minus 4 conditions = 2 degrees of freedom

3 conditions for h ĳ

1 condition for h ĳ

(11)

• We should start with a space-time distance with a 4-by-4 metric tensor, g μν [μ,ν=0,1,2,3]

• It has 10 components:

• Coordinate condition eliminates 4 degree of freedom (DoF)

More precisely:

ds ² ₄ =

X 3

µ=0

X 3

⌫ =0

g _µ⌫ dx ^µ dx ^⌫

dx ^µ = (dt, dx ⁱ )

with

(12)

• We should start with a space-time distance with a 4-by-4 metric tensor, g μν [μ,ν=0,1,2,3]

• It has 10 components:

• Coordinate condition eliminates 4 degree of freedom (DoF)

• leaving 6 DoF: This is where we started; in this lecture we started with h 00 =0 and h 0i =0 (called “synchronous gauge”)

More precisely:

ds ² ₄ =

X 3

µ=0

X 3

⌫ =0

g _µ⌫ dx ^µ dx ^⌫

= ( 1 + h ₀₀ )dt ² + a(t)

X 3

i=1

h _0i dtdx ⁱ + a ² (t)

X 3

i=1

X 3

j =1

( _ij + h _ij )dx ⁱ dx ^j

dx ^µ = (dt, dx ⁱ )

with

(13)

• We should start with a space-time distance with a 4-by-4 metric tensor, g μν [μ,ν=0,1,2,3]

• It has 10 components:

• Coordinate condition eliminates 4 degree of freedom (DoF)

• leaving 6 DoF: This is where we started; in this lecture we started with h 00 =0 and h 0i =0 (called “synchronous gauge”)

More precisely:

ds ² ₄ =

X 3

µ=0

X 3

⌫ =0

g _µ⌫ dx ^µ dx ^⌫

= ( 1 + h ₀₀ )dt ² + a(t)

X 3

i=1

h _0i dtdx ⁱ + a ² (t)

X 3

i=1

X 3

j =1

( _ij + h _ij )dx ⁱ dx ^j

dx ^µ = (dt, dx ⁱ )

with

6 DoF = 2 scalar, 2 vector, 2 tensor DoF

This is GW, which can be extracted by

imposing transverse and traceless conditions

(14)

+ and x modes

• If the GW is propagating in the z (i=3) direction, we can write

h _ij =

0 @ h ₊ h _⇥ 0 h _⇥ h ₊ 0

0 0 0

1 A

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h

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₊ h

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_⇥

y

x

(15)

propagation direction of GW

h + =cos(kz)

h x =cos(kz)

~ k

z

(16)

Equation of motion (EoM)

• Writing Einstein’s gravitational field equation with

• and , ; We obtain

stress-energy source of GW

X 3

i=1

k ⁱ h _ij = 0

X 3

i=1

h _ii = 0

⇤ h _ij = 16⇡ GT _ij ^GW

a ²

(17)

EoM in a non-expanding space

with

stress-energy source of GW

⇤ h _ij = 16⇡ GT _ij ^GW

⇤ = @ ²

@ t ² + r ² =

X 3

µ=0

X 3

⌫ =0

⌘ ^µ⌫ @

@ x ^µ

@

@ x ^⌫

⌘ ⁰⁰ = 1, ⌘ ⁰ⁱ = 0, ⌘ ^ij = ^ij

a ²

(18)

EoM in an expanding Universe

⇤ ⌘ 1

p g

X 3

µ=0

X 3

⌫ =0

@

@ x ^µ

✓ p

gg ^µ⌫ @

@ x ^⌫

◆ ⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t + 1

a ² r ²

with

stress-energy source of GW

⇤ h _ij = 16⇡ GT _ij ^GW

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

g = a ³ (t)

a ²

(19)

⇤ ⌘ 1

p g

X 3

µ=0

X 3

⌫ =0

@

@ x ^µ

✓ p

gg ^µ⌫ @

@ x ^⌫

◆ ⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t

k ²

a ² In Fourier space

stress-energy source of GW

⇤ h _ij = 16⇡ GT _ij ^GW

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

g = a ³ (t)

EoM in an expanding Universe

with

r ² e ^ik ^· ^x = k ² e ^ik ^· ^x

a ²

(20)

⇤ ⌘ 1

p g

X 3

µ=0

X 3

⌫ =0

@

@ x ^µ

✓ p

gg ^µ⌫ @

@ x ^⌫

◆ ⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t

k ²

a ² In Fourier space

stress-energy source of GW

⇤ h _ij = 16⇡ GT _ij ^GW

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

g = a ³ (t)

EoM in an expanding Universe

with

r ² e ^ik ^· ^x = k ² e ^ik ^· ^x

k: comoving wavenumber

a ²

(21)

⇤ ⌘ 1

p g

X 3

µ=0

X 3

⌫ =0

@

@ x ^µ

✓ p

gg ^µ⌫ @

@ x ^⌫

◆ ⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t

k ²

a ² In Fourier space

stress-energy source of GW

⇤ h _ij = 16⇡ GT _ij ^GW

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

g = a ³ (t)

EoM in an expanding Universe

with

r ² e ^ik ^· ^x = k ² e ^ik ^· ^x

k/a: physical wavenumber

a ²

(22)

⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t

k ² a ²

In Fourier space

h ¨ _ij + 3 ˙ a

a h ˙ _ij + k ²

a ² h _ij = 16⇡ GT _ij ^GW

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stress-energy source of GW

⇤ h _ij = 16⇡ GT _ij ^GW

EoM in an expanding Universe

a ²

T _ij = a ² ⇡ _ij

⇡ _ij ^GW

We define:

(23)

⇤ = @ ²

@ t ² 3 a ˙ a

@

@ t

k ² a ²

In Fourier space

h ¨ _ij + 3 ˙ a

a h ˙ _ij + k ²

a ² h _ij = 16⇡ GT _ij ^GW

stress-energy source of GW

⇤ h _ij = 16⇡ GT _ij ^GW

expansion of the Universe aﬀects h ij

EoM in an expanding Universe

a ²

⇡ _ij ^GW

T _ij = a ² ⇡ _ij

⇡ _ij ^GW

We define:

(24)

Let’s solve EoM

• Two tricks:

h ¨ _ij + 3 ˙ a

a h ˙ _ij + k ²

a ² h _ij = 16⇡ GT _ij ^GW

⌘ =

Z dt a(t)

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(1) Define “conformal time”

and use this instead of time derivatives

a(t) @

@ t = @

@⌘

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expansion of the Universe aﬀects h ij

⇡ _ij ^GW

(25)

Let’s solve EoM

• Two tricks:

⌘ =

Z dt a(t)

(1) Define “conformal time”

and use this instead of time derivatives

a(t) @

@ t = @

@⌘

<latexit sha1_base64="tAD74QR0ehNcGl7YTxIV7iqnXJk=">AAACJnicdVDLSgMxFM34rPVVdelmsAh1U2aqoJtC0Y3LCvYBnVLupJk2NPMguSOUYb7Gjb/ixkVFxJ2fYtoOoq0eCBzOuTfJOW4kuELL+jBWVtfWNzZzW/ntnd29/cLBYVOFsaSsQUMRyrYLigkesAZyFKwdSQa+K1jLHd1M/dYDk4qHwT2OI9b1YRBwj1NALfUKVSjhmeNJoIkTgUQOIv1mJqbV/zyHIaS9QtEqWzOYy8TOSJFkqPcKE6cf0thnAVIBSnVsK8JuMr2SCpbmnVixCOgIBqyjaQA+U91kFjM1T7XSN71Q6hOgOVN/biTgKzX2XT3pAw7VojcV//I6MXpX3YQHUYwsoPOHvFjHD81pZ2afS0ZRjDUBKrn+q0mHoHtB3Wxel2AvRl4mzUrZPi9X7i6Kteusjhw5JiekRGxySWrkltRJg1DySJ7JhLwaT8aL8Wa8z0dXjGzniPyC8fkFVpunkg==</latexit>

h ⁰⁰ _ij + 2a ⁰

a h ⁰ _ij + k ² h _ij = 16⇡ Ga ² T _ij ^GW

<latexit sha1_base64="VjIb6+O9vdxPsPWea9csOUwlj1g=">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</latexit>

Primes =

conformal time derivatives

⇡ _ij ^GW

(26)

Let’s solve EoM

• Two tricks:

(2) Multiply h ij by the scale factor and define

h ⁰⁰ _ij + 2a ⁰

a h ⁰ _ij + k ² h _ij = 16⇡ Ga ² T _ij ^GW

<latexit sha1_base64="VjIb6+O9vdxPsPWea9csOUwlj1g=">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</latexit>

u _ij = ah _ij

<latexit sha1_base64="QIFDGr/9FJbMgr/WA90jezz/REw=">AAAB9XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6EYounFZwT6gHUsmzbSxSWZIMkoZ+h9uXCji1n9x59+YTmehrQcu93DOveTmBDFn2rjut7O0vLK6tl7YKG5ube/slvb2mzpKFKENEvFItQOsKWeSNgwznLZjRbEIOG0Fo+up33qkSrNI3plxTH2BB5KFjGBjpfukl7KHySUeZr1XKrsVNwNaJF5OypCj3it9dfsRSQSVhnCsdcdzY+OnWBlGOJ0Uu4mmMSYjPKAdSyUWVPtpdvUEHVulj8JI2ZIGZervjRQLrccisJMCm6Ge96bif14nMeGFnzIZJ4ZKMnsoTDgyEZpGgPpMUWL42BJMFLO3IjLEChNjgyraELz5Ly+SZrXinVaqt2fl2lUeRwEO4QhOwINzqMEN1KEBBBQ8wyu8OU/Oi/PufMxGl5x85wD+wPn8AQXEktk=</latexit>

⇡ _ij ^GW

(27)

Let’s solve EoM

• Two tricks:

(2) Multiply h ij by the scale factor and define

u _ij = ah _ij

<latexit sha1_base64="QIFDGr/9FJbMgr/WA90jezz/REw=">AAAB9XicbVDLSgMxFL3js9ZX1aWbYBFclZkq6EYounFZwT6gHUsmzbSxSWZIMkoZ+h9uXCji1n9x59+YTmehrQcu93DOveTmBDFn2rjut7O0vLK6tl7YKG5ube/slvb2mzpKFKENEvFItQOsKWeSNgwznLZjRbEIOG0Fo+up33qkSrNI3plxTH2BB5KFjGBjpfukl7KHySUeZr1XKrsVNwNaJF5OypCj3it9dfsRSQSVhnCsdcdzY+OnWBlGOJ0Uu4mmMSYjPKAdSyUWVPtpdvUEHVulj8JI2ZIGZervjRQLrccisJMCm6Ge96bif14nMeGFnzIZJ4ZKMnsoTDgyEZpGgPpMUWL42BJMFLO3IjLEChNjgyraELz5Ly+SZrXinVaqt2fl2lUeRwEO4QhOwINzqMEN1KEBBBQ8wyu8OU/Oi/PufMxGl5x85wD+wPn8AQXEktk=</latexit>

u ⁰⁰ _ij +

✓

k ² a ⁰⁰ a

◆ u _ij = 16⇡ Ga ³ T _ij ^GW

<latexit sha1_base64="eRbdQExFD0oauWoBlIGj4h2GAN0=">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</latexit>

⇡ _ij ^GW

(28)

Let’s solve EoM

u ⁰⁰ _ij +

✓

k ² a ⁰⁰ a

◆ u _ij = 16⇡ Ga ³ T _ij ^GW

u ⁰⁰ _ij + ⇥

k ² + m ² (⌘ ) ⇤

u _ij = 16⇡ Ga ³ T _ij ^GW

<latexit sha1_base64="PvPQpRBQaCypctU0Vah5KYcyV2c=">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</latexit>

m ² (⌘ ) = a ⁰⁰ a

<latexit sha1_base64="WiuhZ1iii0RnES1FMaD6l4ncXmU=">AAACAXicbVBNS8NAEN3Ur1q/ol4EL8EirQdLUgW9CEUvHivYD2hqmWw37dLdJOxuhBLixb/ixYMiXv0X3vw3btsctPpg4PHeDDPzvIhRqWz7y8gtLC4tr+RXC2vrG5tb5vZOU4axwKSBQxaKtgeSMBqQhqKKkXYkCHCPkZY3upr4rXsiJA2DWzWOSJfDIKA+xaC01DP3+F217BIFRxfHri8AJ1AqpQmkPbNoV+wprL/EyUgRZaj3zE+3H+KYk0BhBlJ2HDtS3QSEopiRtODGkkSARzAgHU0D4ER2k+kHqXWolb7lh0JXoKyp+nMiAS7lmHu6k4MaynlvIv7ndWLln3cTGkSxIgGeLfJjZqnQmsRh9akgWLGxJoAF1bdaeAg6B6VDK+gQnPmX/5JmteKcVKo3p8XaZRZHHu2jA1RGDjpDNXSN6qiBMHpAT+gFvRqPxrPxZrzPWnNGNrOLfsH4+AbiOZXl</latexit>

Defining

We obtain a harmonic oscillator with a time-dependent mass term!

eﬀect of the expansion of the Universe

⇡ _ij ^GW

(29)

Propagation of GW

in vacuum: Two regimes

u ⁰⁰ _ij + ⇥

k ² + m ² (⌘ ) ⇤

u _ij = 0

<latexit sha1_base64="XKsaOQjgVd+ZbhLgnaJzqpMvotI=">AAACFHicbVDLSsNAFJ34tr6iLt0Ei6gUShIF3QhFNy4VrBaaWCbTm3bs5MHMjVBCP8KNv+LGhSJuXbjzb5y2WWj1wMDhnHO5c0+QCq7Qtr+MqemZ2bn5hcXS0vLK6pq5vnGtkkwyqLNEJLIRUAWCx1BHjgIaqQQaBQJugt7Z0L+5B6l4El9hPwU/op2Yh5xR1FLLrGStnN8NdncrnoAQm71btxLdunseIN33JO900R9HTuyWWbar9gjWX+IUpEwKXLTMT6+dsCyCGJmgSjUdO0U/pxI5EzAoeZmClLIe7UBT05hGoPx8dNTA2tFK2woTqV+M1kj9OZHTSKl+FOhkRLGrJr2h+J/XzDA89nMepxlCzMaLwkxYmFjDhqw2l8BQ9DWhTHL9V4t1qaQMdY8lXYIzefJfcu1WnYOqe3lYrp0WdSyQLbJN9ohDjkiNnJMLUieMPJAn8kJejUfj2Xgz3sfRKaOY2SS/YHx8A1t7ncQ=</latexit>

• Two regimes:

1. Short wavelength (k >> |m|)

• u ij ~ exp(ikη) => h ij ~ a ^–1 exp(ikη) [decaying]

2. Long wavelength (k << |m|)

• u ij ~ a => h ij ~ constant

(30)

Meaning of m ²

• The inverse of the expansion rate, (aH) ^–1 , gives an estimate of the (comoving) size of the observable Universe, or “horizon”

• So, k << |m| is the “super-horizon” mode, and k >> |m| is the “sub-horizon” mode

m ² (⌘ ) = a ⁰⁰

a = a ² (2H ² + ˙ H )

<latexit sha1_base64="yV/jVbUevNVeMhccYfb1uR5/lcQ=">AAACFHicbVC7SgNBFJ31GeNr1dJmMUgiYthdBW0Cok1KBZMIeXF3MqtDZh/M3BXCsh9h46/YWChia2Hn3zh5FJp4YOBwzj3cuceLBVdo29/G3PzC4tJybiW/ura+sWlubddVlEjKajQSkbz1QDHBQ1ZDjoLdxpJB4AnW8PqXQ7/xwKTiUXiDg5i1A7gLuc8poJa65mHQcUsthnBQOWr5EmgKxWKWQlY5Au241Y572OpFmFazg65ZsMv2CNYscSakQCa46ppfOkqTgIVIBSjVdOwY2ylI5FSwLN9KFIuB9uGONTUNIWCqnY6Oyqx9rfQsP5L6hWiN1N+JFAKlBoGnJwPAezXtDcX/vGaC/lk75WGcIAvpeJGfCAsja9iQ1eOSURQDTYBKrv9q0XvQ1aDuMa9LcKZPniV1t+wcl93rk8L5xaSOHNkle6REHHJKzkmVXJEaoeSRPJNX8mY8GS/Gu/ExHp0zJpkd8gfG5w8wX5xn</latexit>

H = a ˙ a

<latexit sha1_base64="y+S02G8rLeCEGoO+lu1Gav7Z218=">AAAB/HicbVDLSsNAFJ34rPUV7dLNYBFclaQKuhGKbrqsYB/QhjKZTtqhk0mYuRFCiL/ixoUibv0Qd/6N0zYLbT1w4XDOPcyd48eCa3Ccb2ttfWNza7u0U97d2z84tI+OOzpKFGVtGolI9XyimeCStYGDYL1YMRL6gnX96d3M7z4ypXkkHyCNmReSseQBpwSMNLQrzZtBoAjNBqMIMpLnZoZ21ak5c+BV4hakigq0hvaXSdMkZBKoIFr3XScGLyMKOBUsLw8SzWJCp2TM+oZKEjLtZfPjc3xmlBEOImVGAp6rvxMZCbVOQ99shgQmetmbif95/QSCay/jMk6ASbp4KEgEhgjPmsAjrhgFkRpCqOLmVkwnxHQBpq+yKcFd/vIq6dRr7kWtfn9ZbdwWdZTQCTpF58hFV6iBmqiF2oiiFD2jV/RmPVkv1rv1sVhds4pMBf2B9fkDU1GVNw==</latexit>

Hubble’s expansion rate

(31)

Horizon Distance

• Horizon = the physical distance traveled by a photon

• The (unperturbed) photon path in the radial direction is given by

• Integrating it, we obtain the physical distance traveled by a photon, d horizon , as

ds ² ₄ = dt ² + a ² (t)dr ² = 0

d _horizon = a(t)r = a(t)

Z _t

0 dt ⁰ a(t ⁰ )

• Hubble length is given by H ^–1 , which is on the same order of magnitude as d horizon . Comoving Hubble length is (aH) ^–1 ,

which is on the same order of magnitude as d horizon /a(t)=r

(32)

GW “entering the horizon”

• This is a tricky concept, but it is important

• Suppose that GWs exist at all wavelengths

• ^{Let’s not} yet ask the origin of these “super-horizon GW”, but assume their existence

• As the Universe expands, the horizon size grows and we can see longer and longer wavelengths

• Fluctuations “entering the horizon”

(33)

10 Gpc today 1 Gpc today 100 Mpc today

10 Mpc today 1 Mpc today

“enter the horizon”

Radiation Era Matter Era

a k

<latexit sha1_base64="0WH/fC7ApXueVifNR9Qec6YYzK0=">AAAB8nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA0lM120y7dbMLuRCihP8OLB0W8+mu8+W/ctjlo64OBx3szzMwLUykMuu63s7a+sbm1Xdop7+7tHxxWjo7bJsk04y2WyER3Q2q4FIq3UKDk3VRzGoeSd8Lx3czvPHFtRKIecZLyIKZDJSLBKFrJ70WaspxO8/G0X6m6NXcOskq8glShQLNf+eoNEpbFXCGT1Bjfc1MMcqpRMMmn5V5meErZmA65b6miMTdBPj95Ss6tMiBRom0pJHP190ROY2MmcWg7Y4ojs+zNxP88P8PoJsiFSjPkii0WRZkkmJDZ/2QgNGcoJ5ZQpoW9lbARtSmgTalsQ/CWX14l7XrNu6zVH66qjdsijhKcwhlcgAfX0IB7aEILGCTwDK/w5qDz4rw7H4vWNaeYOYE/cD5/ANqrkaA=</latexit>

(34)

GW Evolution: Summary

• Super-horizon scales [k << aH]

• The amplitude of GW is conserved (i.e., h ij = constant)

• Sub-horizon scales [k >> aH]

• The amplitude of GW decays (i.e., h ij ~ 1/a)

Therefore, the long-wavelength

GW preserves the initial condition:

the beginning of the Universe!

(35)

Source of GW

in the early Universe?

• Was there any source of GW in the early Universe?

• Yes, in a sense that there are many papers on possible sources in the literature

• See a recent review article by C. Caprini and D. Figueroa, Classical and Quantum Gravity, 35, 163001 (2018),

arXiv:1801.04268

u ⁰⁰ _ij +

✓

k ² a ⁰⁰ a

◆ u _ij = 16⇡ Ga ³ T _ij ^GW

⇡ _ij ^GW

(36)

Quantum generation of GW in the early Universe!

• But, even if there was no source, GW can emerge quantum-mechanically!

• This is the subject of today’s lecture. We will talk about the right hand side on March 19

• To see this, we need to quantise the left hand side of the equation

u ⁰⁰ _ij +

✓

k ² a ⁰⁰ a

◆ u _ij = 16⇡ Ga ³ T _ij ^GW

Grishchuk (1974); Starobinsky (1979)

⇡ _ij ^GW

(37)

Cosmic Inflation

• Exponential expansion (inflation) stretches the wavelength of quantum fluctuations to very large scales

Starobinsky (1980); Sato (1981); Guth (1981); Linde (1982); Albrecht & Steinhardt (1982)

Quantum fluctuations on microscopic scales

Inflation!

(38)

Cosmic Inflation

• Inflation is the accelerated , quasi-exponential expansion. Thus, we must have

¨ a

a = ˙ H + H ² > 0 ^✏ ⌘ H ˙

H ² < 1

Actually, we rather need ε << 1, to have a sustained period of inflation. So H(t) is a

slowly-varying function of time

Starobinsky (1980); Sato (1981); Guth (1981); Linde (1982); Albrecht & Steinhardt (1982)

(39)

Cosmic Inflation

• Inflation is the accelerated , quasi-exponential expansion. Thus, we must have

¨ a

a = ˙ H + H ² > 0 ^✏ ⌘ H ˙

H ² < 1

Starobinsky (1980); Sato (1981); Guth (1981); Linde (1982); Albrecht & Steinhardt (1982)

˙ a

a = H ! a(t) = exp

Z t

t ₀

dt H (t ⁰ ) ⇡ exp[H (t t ₀ )]

During inflation, a(t) grows exponentially in time

Therefore,

(40)

GW from inflation

• During inflation, the scale factor grows exponentially in time,

• In conformal time, this means

u ⁰⁰ _ij + k ² 2a ² H ² u _ij = 0

<latexit sha1_base64="cRdLghSfCyX4p6F2KI5rlaxBi9A=">AAACEnicbVDLSsNAFJ3UV62vqEs3wSKtiCWJgm6EopsuK9gHNG2ZTCft2MmDmRuhhH6DG3/FjQtF3Lpy5984bbPQ1gMDh3PO5c49bsSZBNP81jJLyyura9n13Mbm1vaOvrtXl2EsCK2RkIei6WJJOQtoDRhw2owExb7LacMd3kz8xgMVkoXBHYwi2vZxP2AeIxiU1NWP427C7seFwonDqQfFYcc+tXHHrnRsR7D+ANLAldnV82bJnMJYJFZK8ihFtat/Ob2QxD4NgHAsZcsyI2gnWAAjnI5zTixphMkQ92lL0QD7VLaT6Ulj40gpPcMLhXoBGFP190SCfSlHvquSPoaBnPcm4n9eKwbvsp2wIIqBBmS2yIu5AaEx6cfoMUEJ8JEimAim/mqQARaYgGoxp0qw5k9eJHW7ZJ2V7NvzfPk6rSOLDtAhKiILXaAyqqAqqiGCHtEzekVv2pP2or1rH7NoRktn9tEfaJ8/7xycYg==</latexit>

a(t)

<latexit sha1_base64="ts/QEiAxzaPStsFASzho9PJIF3I=">AAAB/XicbVDLSsNAFJ34rPUVHzs3wSK0m5JUQZdFN11WsA9oQplMJ+3QyWSYuRFrKf6KGxeKuPU/3Pk3TtsstPXAhcM593LvPaHkTIPrflsrq2vrG5u5rfz2zu7evn1w2NRJqghtkIQnqh1iTTkTtAEMOG1LRXEcctoKhzdTv3VPlWaJuIORpEGM+4JFjGAwUtc+xkUo+VIlEhLHpw+yWINS1y64ZXcGZ5l4GSmgDPWu/eX3EpLGVADhWOuO50oIxlgBI5xO8n6qqcRkiPu0Y6jAMdXBeHb9xDkzSs+JEmVKgDNTf0+Mcaz1KA5NZ4xhoBe9qfif10khugrGTMgUqCDzRVHKHfPpNAqnxxQlwEeGYKKYudUhA6wwARNY3oTgLb68TJqVsndertxeFKrXWRw5dIJOURF56BJVUQ3VUQMR9Iie0St6s56sF+vd+pi3rljZzBH6A+vzB0J5lHM=</latexit>

/ exp(Ht)

a(⌘ ) = (H ⌘ ) ¹

<latexit sha1_base64="ZEyL8UrnGldIoCA+zBu8ZL49dHA=">AAACAHicbZC7SgNBFIZn4y3G26qFhc1gEGKRsBsFbYSgTcoI5gLJGmYnJ8mQ2Qszs0JYtvFVbCwUsfUx7HwbJ5stNPGHgY//nMOZ87shZ1JZ1reRW1ldW9/Ibxa2tnd298z9g5YMIkGhSQMeiI5LJHDmQ1MxxaETCiCey6HtTm5n9fYjCMkC/15NQ3A8MvLZkFGitNU3j0ipB4qc4WtcLtVTfIjLdtI3i1bFSoWXwc6giDI1+uZXbxDQyANfUU6k7NpWqJyYCMUoh6TQiySEhE7ICLoafeKBdOL0gASfameAh4HQz1c4dX9PxMSTcuq5utMjaiwXazPzv1o3UsMrJ2Z+GCnw6XzRMOJYBXiWBh4wAVTxqQZCBdN/xXRMBKFKZ1bQIdiLJy9Dq1qxzyvVu4ti7SaLI4+O0QkqIRtdohqqowZqIooS9Ixe0ZvxZLwY78bHvDVnZDOH6I+Mzx98Z5Rq</latexit>

for

<latexit sha1_base64="sfmiVUD15mwI1NCtwzmV4XCXGkk=">AAAB9XicbVBNS8NAEN3Ur1q/qh69LBbBiyWpgh56KHrxWMF+QBPLZrtpl242YXeihND/4cWDIl79L978N27bHLT1wcDjvRlm5vmx4Bps+9sqrKyurW8UN0tb2zu7e+X9g7aOEkVZi0YiUl2faCa4ZC3gIFg3VoyEvmAdf3wz9TuPTGkeyXtIY+aFZCh5wCkBIz2cuVwGkNZdBqRu98sVu2rPgJeJk5MKytHsl7/cQUSTkEmggmjdc+wYvIwo4FSwSclNNIsJHZMh6xkqSci0l82unuATowxwEClTEvBM/T2RkVDrNPRNZ0hgpBe9qfif10sguPIyLuMEmKTzRUEiMER4GgEecMUoiNQQQhU3t2I6IopQMEGVTAjO4svLpF2rOufV2t1FpXGdx1FER+gYnSIHXaIGukVN1EIUKfSMXtGb9WS9WO/Wx7y1YOUzh+gPrM8fsLGR+w==</latexit>

1 < ⌘ < 0

m ² (⌘) = a ⁰⁰

a = a ² (2H ² + ˙ H )

<latexit sha1_base64="yV/jVbUevNVeMhccYfb1uR5/lcQ=">AAACFHicbVC7SgNBFJ31GeNr1dJmMUgiYthdBW0Cok1KBZMIeXF3MqtDZh/M3BXCsh9h46/YWChia2Hn3zh5FJp4YOBwzj3cuceLBVdo29/G3PzC4tJybiW/ura+sWlubddVlEjKajQSkbz1QDHBQ1ZDjoLdxpJB4AnW8PqXQ7/xwKTiUXiDg5i1A7gLuc8poJa65mHQcUsthnBQOWr5EmgKxWKWQlY5Au241Y572OpFmFazg65ZsMv2CNYscSakQCa46ppfOkqTgIVIBSjVdOwY2ylI5FSwLN9KFIuB9uGONTUNIWCqnY6Oyqx9rfQsP5L6hWiN1N+JFAKlBoGnJwPAezXtDcX/vGaC/lk75WGcIAvpeJGfCAsja9iQ1eOSURQDTYBKrv9q0XvQ1aDuMa9LcKZPniV1t+wcl93rk8L5xaSOHNkle6REHHJKzkmVXJEaoeSRPJNX8mY8GS/Gu/ExHp0zJpkd8gfG5w8wX5xn</latexit>

⌘ =

Z dt a(t)

(41)

GW from inflation

• During inflation, the scale factor grows exponentially in time,

• In conformal time, this means

a(t)

<latexit sha1_base64="ts/QEiAxzaPStsFASzho9PJIF3I=">AAAB/XicbVDLSsNAFJ34rPUVHzs3wSK0m5JUQZdFN11WsA9oQplMJ+3QyWSYuRFrKf6KGxeKuPU/3Pk3TtsstPXAhcM593LvPaHkTIPrflsrq2vrG5u5rfz2zu7evn1w2NRJqghtkIQnqh1iTTkTtAEMOG1LRXEcctoKhzdTv3VPlWaJuIORpEGM+4JFjGAwUtc+xkUo+VIlEhLHpw+yWINS1y64ZXcGZ5l4GSmgDPWu/eX3EpLGVADhWOuO50oIxlgBI5xO8n6qqcRkiPu0Y6jAMdXBeHb9xDkzSs+JEmVKgDNTf0+Mcaz1KA5NZ4xhoBe9qfif10khugrGTMgUqCDzRVHKHfPpNAqnxxQlwEeGYKKYudUhA6wwARNY3oTgLb68TJqVsndertxeFKrXWRw5dIJOURF56BJVUQ3VUQMR9Iie0St6s56sF+vd+pi3rljZzBH6A+vzB0J5lHM=</latexit>

/ exp(Ht)

a(⌘ ) = (H ⌘ ) ¹

<latexit sha1_base64="ZEyL8UrnGldIoCA+zBu8ZL49dHA=">AAACAHicbZC7SgNBFIZn4y3G26qFhc1gEGKRsBsFbYSgTcoI5gLJGmYnJ8mQ2Qszs0JYtvFVbCwUsfUx7HwbJ5stNPGHgY//nMOZ87shZ1JZ1reRW1ldW9/Ibxa2tnd298z9g5YMIkGhSQMeiI5LJHDmQ1MxxaETCiCey6HtTm5n9fYjCMkC/15NQ3A8MvLZkFGitNU3j0ipB4qc4WtcLtVTfIjLdtI3i1bFSoWXwc6giDI1+uZXbxDQyANfUU6k7NpWqJyYCMUoh6TQiySEhE7ICLoafeKBdOL0gASfameAh4HQz1c4dX9PxMSTcuq5utMjaiwXazPzv1o3UsMrJ2Z+GCnw6XzRMOJYBXiWBh4wAVTxqQZCBdN/xXRMBKFKZ1bQIdiLJy9Dq1qxzyvVu4ti7SaLI4+O0QkqIRtdohqqowZqIooS9Ixe0ZvxZLwY78bHvDVnZDOH6I+Mzx98Z5Rq</latexit>

for

<latexit sha1_base64="sfmiVUD15mwI1NCtwzmV4XCXGkk=">AAAB9XicbVBNS8NAEN3Ur1q/qh69LBbBiyWpgh56KHrxWMF+QBPLZrtpl242YXeihND/4cWDIl79L978N27bHLT1wcDjvRlm5vmx4Bps+9sqrKyurW8UN0tb2zu7e+X9g7aOEkVZi0YiUl2faCa4ZC3gIFg3VoyEvmAdf3wz9TuPTGkeyXtIY+aFZCh5wCkBIz2cuVwGkNZdBqRu98sVu2rPgJeJk5MKytHsl7/cQUSTkEmggmjdc+wYvIwo4FSwSclNNIsJHZMh6xkqSci0l82unuATowxwEClTEvBM/T2RkVDrNPRNZ0hgpBe9qfif10sguPIyLuMEmKTzRUEiMER4GgEecMUoiNQQQhU3t2I6IopQMEGVTAjO4svLpF2rOufV2t1FpXGdx1FER+gYnSIHXaIGukVN1EIUKfSMXtGb9WS9WO/Wx7y1YOUzh+gPrM8fsLGR+w==</latexit>

1 < ⌘ < 0

u ⁰⁰ _ij +

✓

k ² 2

⌘ ²

◆ u _ij = 0

<latexit sha1_base64="fMKX4NtmSvNN7gURQs4Z+J3d5Mo=">AAACGXicbVDLSgNBEJz1GeMr6tHLYpBExLC7CnoRgl48RjCJkE3C7KQ3GTP7YKZXCEt+w4u/4sWDIh715N84eRw0WjBQVFXT0+XFgiu0rC9jbn5hcWk5s5JdXVvf2MxtbddUlEgGVRaJSN56VIHgIVSRo4DbWAINPAF1r3858uv3IBWPwhscxNAMaDfkPmcUtdTOWUk75XfDQuHQFeBjsd9yjlxfUuakLiBtOUNX8m4PDya5c6udy1slawzzL7GnJE+mqLRzH24nYkkAITJBlWrYVozNlErkTMAw6yYKYsr6tAsNTUMagGqm48uG5r5WOqYfSf1CNMfqz4mUBkoNAk8nA4o9NeuNxP+8RoL+WTPlYZwghGyyyE+EiZE5qsnscAkMxUATyiTXfzVZj+peUJeZ1SXYsyf/JTWnZB+XnOuTfPliWkeG7JI9UiQ2OSVlckUqpEoYeSBP5IW8Go/Gs/FmvE+ic8Z0Zof8gvH5DXC1n/U=</latexit>

(42)

GW from inflation

• The solution is

u ⁰⁰ _ij +

✓

k ² 2

⌘ ²

◆ u _ij = 0

u _ij = A _ij



cos(k ⌘) sin(k ⌘ )

k⌘ + B _ij

 cos(k ⌘ )

k ⌘ + sin(k ⌘)

<latexit sha1_base64="UozWQHnrK2zFgSG/1u7/lFEFm0E=">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</latexit>

• How do we fix the integration constants, A ij and B ij ? We need QM!

• ^{We find A} ^ij ^{and B} ^ij , such that the u ij coincides with the known

flat-space (Minkowski) results for the quantum fluctuation in

vacuum

(43)

Second-order Action

• The action that gives Einstein’s field equations is the so-

called “Einstein-Hilbert action”, given by the Ricci scalar R:

M _pl = (8⇡ G) ^1/2

<latexit sha1_base64="sb3kvG0b3zx6DLFWE8n+264RI2E=">AAACA3icbVDLSsNAFJ3UV62vqDvdDBahLqxJFexGKLrQjVDBPqCJYTKdtkMnyTAzEUoIuPFX3LhQxK0/4c6/cdpmodYDFw7n3Mu99/icUaks68vIzc0vLC7llwsrq2vrG+bmVlNGscCkgSMWibaPJGE0JA1FFSNtLggKfEZa/vBi7LfuiZA0Cm/ViBM3QP2Q9ihGSkueuXPtJY4IIGfpWanqcAovD+6SQ/uoknpm0SpbE8BZYmekCDLUPfPT6UY4DkioMENSdmyLKzdBQlHMSFpwYkk4wkPUJx1NQxQQ6SaTH1K4r5Uu7EVCV6jgRP05kaBAylHg684AqYH8643F/7xOrHpVN6EhjxUJ8XRRL2ZQRXAcCOxSQbBiI00QFlTfCvEACYSVjq2gQ7D/vjxLmpWyfVyu3JwUa+dZHHmwC/ZACdjgFNTAFaiDBsDgATyBF/BqPBrPxpvxPm3NGdnMNvgF4+MbU86WBQ==</latexit>

I _GR =

Z p

gd ⁴ x

✓ 1

2 M _pl ² R

◆

<latexit sha1_base64="lfxv+anUZPeBjI55rRtdURGiNfM=">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</latexit>

with

• Expanding this to second-order in h ij , we obtain the action that gives the equation of motion for h ij :

= Z

a ³ d ⁴ x 1

2 M _pl ² X

=+, ⇥

✓ 1

2 h ˙ ² ( r h ) ² 2a ²

◆

<latexit sha1_base64="LvUSE56AYQZkuBuLnPVniz6c/UY=">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</latexit>

p

<latexit sha1_base64="Lb76PZ1+iifm/4nfVHUao+y9C4M=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgxbCbCHoRgl48RjAPSNYwO5kkQ2YfmekNhCXf4cWDIl79GG/+jZNkD5pY0FBUddPd5UVSaLTtbyuztr6xuZXdzu3s7u0f5A+P6jqMFeM1FspQNT2quRQBr6FAyZuR4tT3JG94w7uZ3xhzpUUYPOIk4q5P+4HoCUbRSG5bjxQmF/3pDX0qd/IFu2jPQVaJk5ICpKh28l/tbshinwfIJNW65dgRuglVKJjk01w71jyibEj7vGVoQH2u3WR+9JScGaVLeqEyFSCZq78nEuprPfE90+lTHOhlbyb+57Vi7F27iQiiGHnAFot6sSQYklkCpCsUZygnhlCmhLmVsAFVlKHJKWdCcJZfXiX1UtEpF0sPl4XKbRpHFk7gFM7BgSuowD1UoQYMRvAMr/Bmja0X6936WLRmrHTmGP7A+vwBceyR4Q==</latexit>

g = a ³

I _GR ⁽²⁾ = Z

a ³ d ⁴ x 1

4 M _pl ²

✓ 1

2 h ˙ ² _ij ( r h _ij ) ² 2a ²

◆

<latexit sha1_base64="KQZvFqzAI+yMamne1do5rGVKL5o=">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</latexit>

h _ij = 0

@ h ₊ h _⇥ 0 h _⇥ h ₊ 0

0 0 0

1 A

with

(44)

Second-order Action

• The action that gives Einstein’s field equations is the so-

called “Einstein-Hilbert action”, given by the Ricci scalar R:

M _pl = (8⇡ G) ^1/2

<latexit sha1_base64="sb3kvG0b3zx6DLFWE8n+264RI2E=">AAACA3icbVDLSsNAFJ3UV62vqDvdDBahLqxJFexGKLrQjVDBPqCJYTKdtkMnyTAzEUoIuPFX3LhQxK0/4c6/cdpmodYDFw7n3Mu99/icUaks68vIzc0vLC7llwsrq2vrG+bmVlNGscCkgSMWibaPJGE0JA1FFSNtLggKfEZa/vBi7LfuiZA0Cm/ViBM3QP2Q9ihGSkueuXPtJY4IIGfpWanqcAovD+6SQ/uoknpm0SpbE8BZYmekCDLUPfPT6UY4DkioMENSdmyLKzdBQlHMSFpwYkk4wkPUJx1NQxQQ6SaTH1K4r5Uu7EVCV6jgRP05kaBAylHg684AqYH8643F/7xOrHpVN6EhjxUJ8XRRL2ZQRXAcCOxSQbBiI00QFlTfCvEACYSVjq2gQ7D/vjxLmpWyfVyu3JwUa+dZHHmwC/ZACdjgFNTAFaiDBsDgATyBF/BqPBrPxpvxPm3NGdnMNvgF4+MbU86WBQ==</latexit>

I _GR =

Z p

gd ⁴ x

✓ 1

2 M _pl ² R

◆

<latexit sha1_base64="lfxv+anUZPeBjI55rRtdURGiNfM=">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</latexit>

• Expanding this to second-order in h ij , we obtain the action that gives the equation of motion for h ij :

= Z

a ³ d ⁴ x 1

2 M _pl ² X

=+, ⇥

✓ 1

2 h ˙ ² ( r h ) ² 2a ²

◆ p

<latexit sha1_base64="Lb76PZ1+iifm/4nfVHUao+y9C4M=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgxbCbCHoRgl48RjAPSNYwO5kkQ2YfmekNhCXf4cWDIl79GG/+jZNkD5pY0FBUddPd5UVSaLTtbyuztr6xuZXdzu3s7u0f5A+P6jqMFeM1FspQNT2quRQBr6FAyZuR4tT3JG94w7uZ3xhzpUUYPOIk4q5P+4HoCUbRSG5bjxQmF/3pDX0qd/IFu2jPQVaJk5ICpKh28l/tbshinwfIJNW65dgRuglVKJjk01w71jyibEj7vGVoQH2u3WR+9JScGaVLeqEyFSCZq78nEuprPfE90+lTHOhlbyb+57Vi7F27iQiiGHnAFot6sSQYklkCpCsUZygnhlCmhLmVsAFVlKHJKWdCcJZfXiX1UtEpF0sPl4XKbRpHFk7gFM7BgSuowD1UoQYMRvAMr/Bmja0X6936WLRmrHTmGP7A+vwBceyR4Q==</latexit>

g = a ³

I _GR ⁽²⁾ = Z

a ³ d ⁴ x 1

4 M _pl ²

✓ 1

2 h ˙ ² _ij ( r h _ij ) ² 2a ²

◆ unwanted pre-factor

with

h _ij = 0

@ h ₊ h _⇥ 0 h _⇥ h ₊ 0

0 0 0

1 A

with

(45)

• Two tricks again:

• (1) Use the conformal time:

• (2) Define:

“Canonically-normalised”

mode function

= Z

a ³ d ⁴ x 1

2 M _pl ² X

=+, ⇥

✓ 1

2 h ˙ ² ( r h ) ² 2a ²

◆ I _GR ⁽²⁾ = Z

a ³ d ⁴ x 1

4 M _pl ²

✓ 1

2 h ˙ ² _ij ( r h _ij ) ² 2a ²

◆ unwanted pre-factor

a

<latexit sha1_base64="EP+DbbwIaXIFcY4RHL1jppARU80=">AAACAnicbVDLSgNBEJz1GeNr1ZN4GQyCp7CbBPQiBL14jGAekGxC7+xsMmT2wcysJCzBi7/ixYMiXv0Kb/6Nk2QPmljQUFR1093lxpxJZVnfxsrq2vrGZm4rv72zu7dvHhw2ZJQIQusk4pFouSApZyGtK6Y4bcWCQuBy2nSHN1O/+UCFZFF4r8YxdQLoh8xnBJSWeuYxdMvY61ZG+ApDt4K9DlWghfKoZxasojUDXiZ2RgooQ61nfnW8iCQBDRXhIGXbtmLlpCAUI5xO8p1E0hjIEPq0rWkIAZVOOnthgs+04mE/ErpChWfq74kUAinHgas7A1ADuehNxf+8dqL8SydlYZwoGpL5Ij/hWEV4mgf2mKBE8bEmQATTt2IyAAFE6dTyOgR78eVl0igV7XKxdFcpVK+zOHLoBJ2ic2SjC1RFt6iG6oigR/SMXtGb8WS8GO/Gx7x1xchmjtAfGJ8/+1CVQA==</latexit>

³ d ⁴ x = a ⁴ d⌘ d ³ x

u = M _pl

p 2 ah

<latexit sha1_base64="tu4YNXkaO8UpYsqDxBAKYORtl9w=">AAACGnicbVDLSgMxFM34rPU16tJNsAiuykwVdCMU3bgRKtgHdIYhk2ba0ExmTDJCCfMdbvwVNy4UcSdu/BvTdgRtPRA4nHMuN/eEKaNSOc6XtbC4tLyyWlorr29sbm3bO7stmWQCkyZOWCI6IZKEUU6aiipGOqkgKA4ZaYfDy7HfvidC0oTfqlFK/Bj1OY0oRspIge1mgcdMvIfgOfQigbC+DrQnYpiyPNeevBNK1/IcDX5ygV1xqs4EcJ64BamAAo3A/vB6Cc5iwhVmSMqu66TK10goihnJy14mSYrwEPVJ11COYiJ9PTkth4dG6cEoEeZxBSfq7wmNYilHcWiSMVIDOeuNxf+8bqaiM19TnmaKcDxdFGUMqgSOe4I9KghWbGQIwoKav0I8QKYfZdosmxLc2ZPnSatWdY+rtZuTSv2iqKME9sEBOAIuOAV1cAUaoAkweABP4AW8Wo/Ws/VmvU+jC1Yxswf+wPr8BjUAoZ8=</latexit>

(46)

• Two tricks again:

• (1) Use the conformal time:

• (2) Define:

“Canonically-normalised”

mode function

I _GR ⁽²⁾ = Z

a ³ d ⁴ x 1

4 M _pl ²

✓ 1

2 h ˙ ² _ij ( r h _ij ) ² 2a ²

◆ a

<latexit sha1_base64="EP+DbbwIaXIFcY4RHL1jppARU80=">AAACAnicbVDLSgNBEJz1GeNr1ZN4GQyCp7CbBPQiBL14jGAekGxC7+xsMmT2wcysJCzBi7/ixYMiXv0Kb/6Nk2QPmljQUFR1093lxpxJZVnfxsrq2vrGZm4rv72zu7dvHhw2ZJQIQusk4pFouSApZyGtK6Y4bcWCQuBy2nSHN1O/+UCFZFF4r8YxdQLoh8xnBJSWeuYxdMvY61ZG+ApDt4K9DlWghfKoZxasojUDXiZ2RgooQ61nfnW8iCQBDRXhIGXbtmLlpCAUI5xO8p1E0hjIEPq0rWkIAZVOOnthgs+04mE/ErpChWfq74kUAinHgas7A1ADuehNxf+8dqL8SydlYZwoGpL5Ij/hWEV4mgf2mKBE8bEmQATTt2IyAAFE6dTyOgR78eVl0igV7XKxdFcpVK+zOHLoBJ2ic2SjC1RFt6iG6oigR/SMXtGb8WS8GO/Gx7x1xchmjtAfGJ8/+1CVQA==</latexit>

³ d ⁴ x = a ⁴ d⌘ d ³ x

u = M _pl

p 2 ah

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= Z

d⌘d ³ x X

=+, ⇥

✓ 1

2 u ⁰ ² 1

2 ( r u ) ² + a ⁰⁰

a u ²

◆

<latexit sha1_base64="1ihUeZ2kHjR5aLyxpnvZLs2rWts=">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</latexit>

This is the correct (“canonical”) normalisation!

2

(47)

GW from inflation

• The solution is

u ⁰⁰ _ij +

✓

k ² 2

⌘ ²

◆ u _ij = 0

u _ij = A _ij



cos(k ⌘) sin(k ⌘ )

k⌘ + B _ij

 cos(k ⌘ )

k ⌘ + sin(k ⌘)

<latexit sha1_base64="UozWQHnrK2zFgSG/1u7/lFEFm0E=">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</latexit>

• How do we fix the integration constants, A ij and B ij ? We need QM!

• ^{We find A} ^ij ^{and B} ^ij , such that the u ij coincides with the known

flat-space (Minkowski) results for the quantum fluctuation in

vacuum

(48)

GW from inflation

• In the very short wavelength limit, kη->∞, we want to reproduce the quantum field theory result in the flat (Minkowski) space, which is

u ⁰⁰ +

✓

k ² 2

⌘ ²

◆ u = 0

<latexit sha1_base64="HmcyaSln2BD3vZjR4YdC5hsVZRs=">AAACH3icbVDLSsNAFJ3Ud31VXboJFqkiliSKuhFENy4rWBWaNkymN+3QyYOZG6GE/okbf8WNC0XEXf/GaQ2i1gMDh3PO5c49fiK4QssaGoWp6ZnZufmF4uLS8spqaW39RsWpZFBnsYjlnU8VCB5BHTkKuEsk0NAXcOv3Lkb+7T1IxePoGvsJNEPaiXjAGUUteaWj1HOFjrdppbLnCghwp9dy9t1AUuZkLiBtOQNX8k4Xd7+jp5ZXKltVawxzktg5KZMcNa/04bZjloYQIRNUqYZtJdjMqETOBAyKbqogoaxHO9DQNKIhqGY2vm9gbmulbQax1C9Cc6z+nMhoqFQ/9HUypNhVf72R+J/XSDE4aWY8SlKEiH0tClJhYmyOyjLbXAJD0deEMsn1X03Wpboa1JUWdQn235MnyY1TtQ+qztVh+ew8r2OebJItskNsckzOyCWpkTph5IE8kRfyajwaz8ab8f4VLRj5zAb5BWP4CZZtohU=</latexit>

u = A



cos(k⌘ ) sin(k⌘ )

k ⌘ + B

 cos(k⌘ )

k ⌘ + sin(k⌘ )

<latexit sha1_base64="D4rgBWCe2XA4a2Zy8T/YZtaUSGU=">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</latexit>

• The solution is

u ! exp( ik ⌘ ) p 2k

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Primordial Gravitational Waves from Inﬂation

Primordial Gravitational Waves from Inflation

Eiichiro Komatsu

[Max Planck Institute for Astrophysics]

University of Amsterdam

February 27, March 5, and 19, 2020

Lecture notes:

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

Plan

• Today: Vacuum Fluctuation

• March 5: Polarisation of the cosmic microwave background

• March 19: Sourced Contribution

⇤ h ij = 0

⇤ h ij = 16⇡ GT ⇡ ij ij GW GW

GW = Area-conserving distortion

of distances between two points

Distance between two points in space

• Static (i.e., non-expanding) Euclidean space

• In Cartesian coordinates

Distance between two points in space

• Homogeneously expanding Euclidean space

• In Cartesian comoving coordinates

“scale factor”

Distance between two points in space

• Homogeneously expanding Euclidean space

• In Cartesian comoving coordinates

“scale factor” =1 for i=j

=0 otherwise

Distance between two points in space

• Inhomogeneous curved space

• In Cartesian comoving coordinates

“metric perturbation”

-> CURVED SPACE!

• Gravitational waves shall be:

• Transverse: the direction of the oscillation of space is perpendicular to the propagation direction

• This means

Four conditions

X 3

i=1

k i h ij = 0

~ k

3 conditions for h ĳ

propagation direction of GW h ij ~ cos(kz)

~ k

• Area-conserving: the determinant of the distortion in space remains unchanged

• This means that the trace vanishes:

Four conditions

X 3

i=1

h ii = 0

1 condition for h ĳ

y

x

• Gravitational waves shall be:

• Transverse: the direction of the oscillation of space is perpendicular to the propagation direction

• This means

• Area-conserving: the determinant of the distortion in space remains unchanged

• This means that the trace vanishes:

Four conditions

X 3

i=1

k i h ij = 0

~ k

X 3

i=1

h ii = 0

6 components of h ĳ minus 4 conditions = 2 degrees of freedom

3 conditions for h ĳ

1 condition for h ĳ

• We should start with a space-time distance with a 4-by-4 metric tensor, g μν [μ,ν=0,1,2,3]

• It has 10 components:

• Coordinate condition eliminates 4 degree of freedom (DoF)

More precisely:

ds 2 4 =

X 3

µ=0

X 3

⌫ =0

g µ⌫ dx µ dx ⌫

dx µ = (dt, dx i )

⇤ h _ij = 0

⇤ h _ij = 16⇡ GT ⇡ _ij _ij ^GW ^GW

k ⁱ h _ij = 0

h _ii = 0

k ⁱ h _ij = 0

h _ii = 0

ds ² ₄ =

g _µ⌫ dx ^µ dx ^⌫

dx ^µ = (dt, dx ⁱ )

ds ² ₄ =

g _µ⌫ dx ^µ dx ^⌫

= ( 1 + h ₀₀ )dt ² + a(t)

h _0i dtdx ⁱ + a ² (t)

( _ij + h _ij )dx ⁱ dx ^j

dx ^µ = (dt, dx ⁱ )

ds ² ₄ =

g _µ⌫ dx ^µ dx ^⌫

= ( 1 + h ₀₀ )dt ² + a(t)

h _0i dtdx ⁱ + a ² (t)

( _ij + h _ij )dx ⁱ dx ^j

dx ^µ = (dt, dx ⁱ )

h _ij =

@ h ₊ h _⇥ 0 h _⇥ h ₊ 0

₊ h

_⇥