The Inflationary Universe Matt Johnson Perimeter Institute/York - - PowerPoint PPT Presentation

The Inflationary Universe

Matt Johnson Perimeter Institute/York University

Thursday, 4 July, 13

A Mystery

ds2 = a(⌘)2 ⇥ (1 + 2Ψ) d⌘2 + (1 + 2Φ) ijdxidxj⇤

P(k) = Akns1

Why?

Thursday, 4 July, 13

A Mystery

ds2 = a(⌘)2 ⇥ (1 + 2Ψ) d⌘2 + (1 + 2Φ) ijdxidxj⇤

P(k) = Akns1

Why?

• Horizon problem: Seemingly acausal correlations.
• Flatness problem: Incredibly finely tuned initial conditions.
• Source of density fluctuations.

Thursday, 4 July, 13

Horizon problem

time space Big Bang

Thursday, 4 July, 13

Horizon problem

time space Big Bang

Thursday, 4 July, 13

Horizon problem

time space Big Bang

particle horizon

∆x = Z dt a(t) = Z a0=1

a=0

dln(a) aH

comoving horizon

1 aH = η

(radiation)

Thursday, 4 July, 13

Horizon problem

time space Big Bang Never touched!

particle horizon

∆x = Z dt a(t) = Z a0=1

a=0

dln(a) aH

comoving horizon

1 aH = η

(radiation)

Thursday, 4 July, 13

Horizon problem

time space

How are these regions at nearly the same temperature?

Big Bang Never touched!

particle horizon

∆x = Z dt a(t) = Z a0=1

a=0

dln(a) aH

comoving horizon

1 aH = η

(radiation)

Thursday, 4 July, 13

Flatness Problem

curvature

✓ H H0 ◆2 = Ωm a3 + Ωr a4 + ΩΛ + Ωk a2

Thursday, 4 July, 13

Flatness Problem

• Curvature redshifts slower than matter or radiation.
• The Universe is very nearly flat today, so it must have

been extremely flat in the past! curvature

✓ H H0 ◆2 = Ωm a3 + Ωr a4 + ΩΛ + Ωk a2

Thursday, 4 July, 13

Flatness Problem

• Curvature redshifts slower than matter or radiation.
• The Universe is very nearly flat today, so it must have

been extremely flat in the past! curvature

Ωr a4

I

/Ωk a2

I

= 1 a2

I

Ωr Ωk ⇠ 1060 Ωr Ωk

If the radiation dominated phase began at the GUT scale:

✓ H H0 ◆2 = Ωm a3 + Ωr a4 + ΩΛ + Ωk a2

Thursday, 4 July, 13

Source of Density Fluctuations

• The primordial spectrum of density fluctuations

necessary to explain observations is:

• Gaussian
• Nearly scale invariant
• Small amplitude
• Superhorizon
• Adiabatic

Why this particular set of properties?

P(k) = Akns1

ns ' 1 A ⌧ 1

Thursday, 4 July, 13

A Solution: Inflation

1 aH ⇠ a(1+3w)/2 w < 1/3

shrinking comoving horizon

Thursday, 4 July, 13

A Solution: Inflation

1 aH ⇠ a(1+3w)/2 w < 1/3

shrinking comoving horizon

Thursday, 4 July, 13

A Solution: Inflation

1 aH ⇠ a(1+3w)/2 w < 1/3

∆x / a(1+3w)/2

f

a(1+3w)/2

i

shrinking comoving horizon

Thursday, 4 July, 13

A Solution: Inflation

1 aH ⇠ a(1+3w)/2 w < 1/3

∆x / a(1+3w)/2

f

a(1+3w)/2

i

shrinking comoving horizon Solves horizon problem!

Thursday, 4 July, 13

• Fluids with redshift slower than curvature.

A Solution: Inflation

w < 1/3

ρinf / a3(1+w)

Thursday, 4 July, 13

• Fluids with redshift slower than curvature.

A Solution: Inflation

w < 1/3

ρinf / a3(1+w)

Solves flatness problem!

Thursday, 4 July, 13

• Fluids with redshift slower than curvature.

A Solution: Inflation

w < 1/3

ρinf / a3(1+w)

Solves flatness problem!

• also implies:
• Negative pressure

pi = wiρi

d2a dt2 = 4πG 3 (1 + 3w)aρ

• Accelerated expansion:

w < 1/3

Thursday, 4 July, 13

A Solution: Inflation

T = H 2π

S = Area 4GN = π H2GN

• A de Sitter universe with w=-1 has thermodynamic

properties:

Thursday, 4 July, 13

A Solution: Inflation

T = H 2π

S = Area 4GN = π H2GN

• A de Sitter universe with w=-1 has thermodynamic

properties:

H−1 ' 10−28m

T ' 1012 GeV

S ' 1014

• The inflationary universe is a small system:

Thursday, 4 July, 13

A Solution: Inflation

T = H 2π

S = Area 4GN = π H2GN

T ρ1/4 ' 10−5

• A de Sitter universe with w=-1 has thermodynamic

properties:

H−1 ' 10−28m

T ' 1012 GeV

S ' 1014

• The inflationary universe is a small system:

Thursday, 4 July, 13

A Solution: Inflation

T = H 2π

S = Area 4GN = π H2GN

T ρ1/4 ' 10−5

• A de Sitter universe with w=-1 has thermodynamic

properties:

H−1 ' 10−28m

T ' 1012 GeV

S ' 1014

• The inflationary universe is a small system:

Simplistic picture, but gives flavour of solution.

Thursday, 4 July, 13

The Inflaton

• Scalar fields:

T µ

ν = ∂µφ∂νφ gµ ν

1 2gαβ∂αφ∂βφ + V (φ)

• Thursday, 4 July, 13

The Inflaton

• Scalar fields:

T µ

ν = ∂µφ∂νφ gµ ν

1 2gαβ∂αφ∂βφ + V (φ)

• Tµν =

X

i

(ρi + pi) uiµuiν + pigµν

VS

Thursday, 4 July, 13

The Inflaton

• Scalar fields:

T µ

ν = ∂µφ∂νφ gµ ν

1 2gαβ∂αφ∂βφ + V (φ)

• Tµν =

X

i

(ρi + pi) uiµuiν + pigµν

VS

∂iφ = 0

ρ = 1 2 ✓dφ dt ◆2 + V (φ)

p = 1 2 ✓dφ dt ◆2 V (φ)

kinetic potential

Thursday, 4 July, 13

The Inflaton

• Scalar fields:

T µ

ν = ∂µφ∂νφ gµ ν

1 2gαβ∂αφ∂βφ + V (φ)

• Tµν =

X

i

(ρi + pi) uiµuiν + pigµν

VS

V 1 2 ✓dφ dt ◆2

Accelerated expansion results when:

a ' eHt

∂iφ = 0

ρ = 1 2 ✓dφ dt ◆2 + V (φ)

p = 1 2 ✓dφ dt ◆2 V (φ)

kinetic potential

Thursday, 4 July, 13

The Inflaton

• Slow-roll inflation:

d2φ dt2 + 3H dφ dt = dV dφ

' V (φ)

φ

friction gradient

Thursday, 4 July, 13

The Inflaton

• Slow-roll inflation:

d2φ dt2 + 3H dφ dt = dV dφ

' V (φ)

φ

friction gradient

⌘ = M 2

p

24⇡ 1 V

• d2V

d2

• ✏ = M 2

p

48⇡ ✓ 1 V dV d ◆2

• Friction dominated motion: slow-roll parameters small

Thursday, 4 July, 13

The Inflaton

• Slow-roll inflation:

d2φ dt2 + 3H dφ dt = dV dφ

' V (φ)

φ

friction gradient

⌘ = M 2

p

24⇡ 1 V

• d2V

d2

• ✏ = M 2

p

48⇡ ✓ 1 V dV d ◆2

• Friction dominated motion: slow-roll parameters small
• Need a sufficiently high and/or flat potential.

Thursday, 4 July, 13

Inflaton is an Attractor

• Slow-roll inflation:

d2φ dt2 + 3H dφ dt = dV dφ

' V (φ)

φ

friction gradient

• Most initial conditions lead to indistinguishable evolution

after a very short time: thanks friction!

all same

Thursday, 4 July, 13

Inflaton is an Attractor

• Slow-roll inflation:

d2φ dt2 + 3H dφ dt = dV dφ

' V (φ)

φ

friction gradient

• Most initial conditions lead to indistinguishable evolution

after a very short time: thanks friction!

all same

Thursday, 4 July, 13

The Inflaton

• Many potentials can drive inflation.

' V (φ)

φ

large field

∆ > Mp

' V (φ)

φ

∆ < Mp

small field

high and steep low and flat

• Differ in the energy scale at which inflation occurs.

Thursday, 4 July, 13

The Inflaton

• Where do these potentials come from?
• Inflation is an effective theory: valid below some energy

scale.

e.g. Newtonian gravity and GR, Maxwell and electroweak, etc.

V = V0 ✓ 1 + c2 φ2 M 2

p

+ . . . ◆

Thursday, 4 July, 13

The Inflaton

• Where do these potentials come from?
• Inflation is an effective theory: valid below some energy

scale.

e.g. Newtonian gravity and GR, Maxwell and electroweak, etc.

V = V0 ✓ 1 + c2 φ2 M 2

p

+ . . . ◆

⌘ = M 2

p

24⇡ 1 V

• d2V

d2

• Inflation is an effective theory sensitive to the physics of

quantum gravity:

∆η ∼ c2

Why is this correction small?

Thursday, 4 July, 13

The Inflaton

• Connection between quantum gravity and inflation is a

blessing and a curse. Blessing: Curse:

Observational tests of quantum gravity We don’t have a complete theory of quantum gravity

Thursday, 4 July, 13

The Inflaton

• Connection between quantum gravity and inflation is a

blessing and a curse. Blessing: Curse:

Observational tests of quantum gravity We don’t have a complete theory of quantum gravity

• String theory is the current best candidate theory of

quantum gravity string inflation!

Thursday, 4 July, 13

String Theory

Nearly all of modern physics: point particles. This worked until........

Thursday, 4 July, 13

String Theory

(Graviton: supervillain from Marvel Comics)

Thursday, 4 July, 13

String Theory

(Graviton: particle associated with gravity)

Thursday, 4 July, 13

String Theory

(Graviton: particle associated with gravity)

The theory of gravitons does not work!

(not a good quantum theory of gravity)

∞ ∞

Thursday, 4 July, 13

String Theory

• String theory: A good theory of quantum gravity!
• Unifies all forces and fundamental particles!

Thursday, 4 July, 13

String Theory

• String theory: A good theory of quantum gravity!
• Unifies all forces and fundamental particles!
• This only works if there are 9 dimensions of space!

Thursday, 4 July, 13

String Theory

• String theory: A good theory of quantum gravity!
• Unifies all forces and fundamental particles!

The solution: make the extra dimensions small!

• This only works if there are 9 dimensions of space!

Thursday, 4 July, 13

String Theory

• To keep the extra dimensions small, need to add energy.

Thursday, 4 July, 13

String Theory

• To keep the extra dimensions small, need to add energy.
• The inflaton: some property of the compact extra dimesions.

e.g. vary the size as a function of position.

Thursday, 4 July, 13

String Theory

• To keep the extra dimensions small, need to add energy.
• The inflaton: some property of the compact extra dimesions.

e.g. vary the size as a function of position.

• Changing size changes potential energy: inflation can be driven

by the energy stored in the extra dimensions.

Thursday, 4 July, 13

String Inflaton

• A ``proof of principle’’ exists, but how predictive is this?

Thursday, 4 July, 13

String Inflaton

• A ``proof of principle’’ exists, but how predictive is this?

Many possible inflaton potentials!

• The extra dimensions can assume many configurations:

(Many possible values of the Cosmological Constant)

Thursday, 4 July, 13

String Inflaton

• A ``proof of principle’’ exists, but how predictive is this?

Many possible inflaton potentials!

• The extra dimensions can assume many configurations:

(Many possible values of the Cosmological Constant)

• To do list: how do we make predictions then?

Thursday, 4 July, 13

Reheating

• Inflation has to come to an end.

' V (φ)

φ

d2φ dt2 + 3H dφ dt = dV dφ

friction gradient

Thursday, 4 July, 13

Reheating

• Inflation has to come to an end.

' V (φ)

φ

• The inflaton oscillates around the minimum, fragments and

decays into standard model particles, dark matter, and perhaps other stuff.

d2φ dt2 + 3H dφ dt = dV dφ

friction gradient

Thursday, 4 July, 13

Reheating

• Inflation has to come to an end.

' V (φ)

φ

• The inflaton oscillates around the minimum, fragments and

decays into standard model particles, dark matter, and perhaps other stuff.

d2φ dt2 + 3H dφ dt = dV dφ

friction gradient

• The standard story of the hot big-bang follows.

Thursday, 4 July, 13

Number of e-folds

• Total expansion of the Universe during inflation:
• Ne = log aend

abegin ' Mp 4 p 3⇡ Z φbegin

φend

d p✏

Thursday, 4 July, 13

Number of e-folds

• Total expansion of the Universe during inflation:
• Ne = log aend

abegin ' Mp 4 p 3⇡ Z φbegin

φend

d p✏

• To solve the horizon problem, need our observable universe

to come from single primordial Hubble patch:

a0 abegin = a0 aeq aeq areh areh abegin = Hbegin H0 ' 1055

(GUT scale inflation)

Thursday, 4 July, 13

Number of e-folds

• Total expansion of the Universe during inflation:
• Ne = log aend

abegin ' Mp 4 p 3⇡ Z φbegin

φend

d p✏

3000 Treh Teq

• To solve the horizon problem, need our observable universe

to come from single primordial Hubble patch:

a0 abegin = a0 aeq aeq areh areh abegin = Hbegin H0 ' 1055

(GUT scale inflation)

Thursday, 4 July, 13

Number of e-folds

• Total expansion of the Universe during inflation:
• Ne = log aend

abegin ' Mp 4 p 3⇡ Z φbegin

φend

d p✏

3000 Treh Teq

• To solve the horizon problem, need our observable universe

to come from single primordial Hubble patch:

a0 abegin = a0 aeq aeq areh areh abegin = Hbegin H0 ' 1055

(GUT scale inflation)

eNe, Ne ⇠ 60

Thursday, 4 July, 13

Classical Fields

d2φ dt2 r2φ + m2φ = 0

S = Z d3xdt 1 2(∂tφ)2 1 2(∂iφ)2 V (φ)

• Scalar field in Minkowski space:

Thursday, 4 July, 13

Classical Fields

d2φ dt2 r2φ + m2φ = 0

S = Z d3xdt 1 2(∂tφ)2 1 2(∂iφ)2 V (φ)

• Scalar field in Minkowski space:

!2

~ k = k2 + m2

• Go to fourier space: free field theory is an infinite number
• f independent oscillators.

φ(t, x) = Z d3k (2π)3 φ~

k(t) ei~ k·~ x

d2φ~

k

dt2 + ω2

~ kφ~ k = 0

Thursday, 4 July, 13

Quantum Fields

φ ! ˆ φ

π ! ˆ π

[ˆ (t, ~ x), ˆ ⇡(t, ~ y)] = i(~ x − ~ y)

• Promote fields to operators:

Thursday, 4 July, 13

Quantum Fields

φ ! ˆ φ

π ! ˆ π

• In fourier space: quantize the infinite number of independent
• scillators:

ˆ φ = 1 p 2 Z d3k (2π)3 h a−

~ k v∗ kei~ k·~ x + a+ ~ k vke−i~ k·~ xi

[ˆ (t, ~ x), ˆ ⇡(t, ~ y)] = i(~ x − ~ y)

• Promote fields to operators:

Thursday, 4 July, 13

Quantum Fields

φ ! ˆ φ

π ! ˆ π

• In fourier space: quantize the infinite number of independent
• scillators:

ˆ φ = 1 p 2 Z d3k (2π)3 h a−

~ k v∗ kei~ k·~ x + a+ ~ k vke−i~ k·~ xi

[ˆ (t, ~ x), ˆ ⇡(t, ~ y)] = i(~ x − ~ y) vk = 1 pωk ei!kt [a

~ k , a+ ~ k0] = 3(~

k ~ k0)

mode function creation/annihilation operators

• Promote fields to operators:

Thursday, 4 July, 13

Quantum Fields

a

~ k |0i = 0

vacuum

Thursday, 4 July, 13

Quantum Fields

particle = positive frequency excitation

a+

~ k |0i = |1i~ k

Thursday, 4 July, 13

Quantum Fields

multi-particle states

a+

~ k a+ ~ k0|0i = |1i~ k|1i~ k0

Thursday, 4 July, 13

QFT in Curved Spacetime

• In curved space:

S = 1 2 Z d4xpg ⇥ g↵@↵@ + m2 ⇤

Thursday, 4 July, 13

QFT in Curved Spacetime

• In curved space:

S = 1 2 Z d4xpg ⇥ g↵@↵@ + m2 ⇤

= a d2~

k

d⌘2 + !2

~ k(⌘)~ k = 0

comoving k!

• Re-cast as canonical free scalar (for FRW):

Thursday, 4 July, 13

QFT in Curved Spacetime

• In curved space:

S = 1 2 Z d4xpg ⇥ g↵@↵@ + m2 ⇤

= a d2~

k

d⌘2 + !2

~ k(⌘)~ k = 0

comoving k!

• Re-cast as canonical free scalar (for FRW):
• In de Sitter, the scale factor is:

a = 1 Hη 1 < η  0

ω2

~ k(η) = k2 com + 1

η2 ✓m2 H2 − 2 ◆

ω2

~ k(η) < 0,

kη ⌧ 1, m ⌧ H

Thursday, 4 July, 13

• Classical solutions to the equation of motion:

χ~

k

< η

QFT in Curved Spacetime

Re Im

vk = r ⇡|⌘| 2 [Jn(k⌘) iYn(k⌘)]

n = r 9 4 m2 H2

Thursday, 4 July, 13

• Classical solutions to the equation of motion:

Initial conditions

χ~

k

< η

QFT in Curved Spacetime

Re Im

vk = r ⇡|⌘| 2 [Jn(k⌘) iYn(k⌘)]

n = r 9 4 m2 H2

Thursday, 4 July, 13

• Classical solutions to the equation of motion:

Initial conditions

χ~

k

< η

φ~

k

< η

QFT in Curved Spacetime

Re Im

vk = r ⇡|⌘| 2 [Jn(k⌘) iYn(k⌘)]

n = r 9 4 m2 H2

Thursday, 4 July, 13

• Classical solutions to the equation of motion:

Initial conditions

χ~

k

< η

φ~

k

< η

kη ⇠ 1

The field ``freezes in”

QFT in Curved Spacetime

Re Im

vk = r ⇡|⌘| 2 [Jn(k⌘) iYn(k⌘)]

n = r 9 4 m2 H2

Thursday, 4 July, 13

QFT in Curved Spacetime

• Quantize just as before, but.....

ω2

~ k(η) = k2 com + 1

η2 ✓m2 H2 − 2 ◆

Thursday, 4 July, 13

• The frequency is time-dependent, so no unambiguous

definition of positive frequency -- no unambiguous definition

• f the vacuum!

QFT in Curved Spacetime

• Quantize just as before, but.....

ω2

~ k(η) = k2 com + 1

η2 ✓m2 H2 − 2 ◆

Thursday, 4 July, 13

• The frequency is time-dependent, so no unambiguous

definition of positive frequency -- no unambiguous definition

• f the vacuum!

QFT in Curved Spacetime

• Quantize just as before, but.....

ω2

~ k(η) = k2 com + 1

η2 ✓m2 H2 − 2 ◆

• A prescription for the vacuum: Minkowski at small scales.

vk = 1 p!k ei!k⌘, k⌘ ! 1

Bunch-Davies

Thursday, 4 July, 13

• Classical solutions to the equation of motion:

Initial conditions

χ~

k

< η

QFT in Curved Spacetime

Thursday, 4 July, 13

• Find the correlation functions:

QFT in Curved Spacetime

vk = r ⇡|⌘| 2 [Jn(k⌘) iYn(k⌘)]

|vk|2 = 1 k + 1 ⌘2k3

(massless)

Thursday, 4 July, 13

• Find the correlation functions:

QFT in Curved Spacetime

h0|ˆ ~

k ˆ

⇤

~ k0|0i = (~

k ~ k0)|vk|2 2 h0|ˆ (x, t)ˆ ⇤(y, t)|0i = Z 1 dk (2⇡)2 k2|vk|2 sin(kL) kL

Fourier space: Real space:

vk = r ⇡|⌘| 2 [Jn(k⌘) iYn(k⌘)]

|vk|2 = 1 k + 1 ⌘2k3

(massless)

Thursday, 4 July, 13

• Find the correlation functions:

QFT in Curved Spacetime

h0|ˆ ~

k ˆ

⇤

~ k0|0i = (~

k ~ k0)|vk|2 2 h0|ˆ (x, t)ˆ ⇤(y, t)|0i = Z 1 dk (2⇡)2 k2|vk|2 sin(kL) kL

Fourier space: Real space:

vk = r ⇡|⌘| 2 [Jn(k⌘) iYn(k⌘)]

|vk|2 = 1 k + 1 ⌘2k3

(massless)

h0|ˆ (x, t)ˆ ⇤(x, t)|0i = Z 1 dk (2⇡)2 k2|vk|2 ' Z ˜

k

dk (2⇡)2 1 ⌘2k + Z 1

˜ k

dk (2⇡)2 k

Coincident limit: UV IR

Thursday, 4 July, 13

h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = Z 1 dk (2⇡)2 k2|vk|2 ' Z ˜

k

dk (2⇡)2 1 ⌘2k + Z 1

˜ k

dk (2⇡)2 k

QFT in Curved Spacetime

Thursday, 4 July, 13

h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = Z 1 dk (2⇡)2 k2|vk|2 ' Z ˜

k

dk (2⇡)2 1 ⌘2k + Z 1

˜ k

dk (2⇡)2 k

QFT in Curved Spacetime

Thursday, 4 July, 13

h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = Z 1 dk (2⇡)2 k2|vk|2 ' Z ˜

k

dk (2⇡)2 1 ⌘2k + Z 1

˜ k

dk (2⇡)2 k

QFT in Curved Spacetime

• Go back to original field:

h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = 1 a2 h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = ✓ H 2⇡ ◆2 Z ˜

k

dk k

Thursday, 4 July, 13

h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = Z 1 dk (2⇡)2 k2|vk|2 ' Z ˜

k

dk (2⇡)2 1 ⌘2k + Z 1

˜ k

dk (2⇡)2 k

QFT in Curved Spacetime

• Go back to original field:

h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = 1 a2 h0|ˆ (x, ⌘)ˆ ⇤(x, ⌘)|0i = ✓ H 2⇡ ◆2 Z ˜

k

dk k

• Assume inflation has a finite duration, count modes larger

than the comoving horizon, go back to proper time:

h0|ˆ (x, t)ˆ ⇤(x, t)|0i = H3 4⇡2 (t t0)

• Diverges with increasing time -- pile-up of superhorizon
• modes. Regulated for non-zero mass.

Thursday, 4 July, 13

P(k) = P a2 = H2 (2π)2

• Transform back to the original field:

(m ⌧ H)

QFT in Curved Spacetime

Fourier space:

h0|ˆ ~

k ˆ

⇤

~ k0|0i = (~

k ~ k0)(2⇡)2 k3 P(k)

Thursday, 4 July, 13

P(k) = P a2 = H2 (2π)2

• Transform back to the original field:

(m ⌧ H)

• The power spectrum of a free field in dS is:
• Nearly scale invariant
• Gaussian
• Small amplitude (compared to...)

uncoupled harmonic oscillators!

QFT in Curved Spacetime

Fourier space:

h0|ˆ ~

k ˆ

⇤

~ k0|0i = (~

k ~ k0)(2⇡)2 k3 P(k)

Thursday, 4 July, 13

P(k) = P a2 = H2 (2π)2

• Transform back to the original field:

(m ⌧ H)

• The power spectrum of a free field in dS is:
• Nearly scale invariant
• Gaussian
• Small amplitude (compared to...)

uncoupled harmonic oscillators!

QFT in Curved Spacetime

Fourier space:

h0|ˆ ~

k ˆ

⇤

~ k0|0i = (~

k ~ k0)(2⇡)2 k3 P(k)

!!Gravitational waves!!

Thursday, 4 July, 13

Quantum to Classical

h0|ˆ (x, ⌘)ˆ ⇤(y, ⌘)|0i ! hˆ (x, ⌘)ˆ ⇤(y, ⌘)i

Quantum expectation value Ensemble average Spatial average

???

(pure dS might not be best example...)

Thursday, 4 July, 13

Inflationary Fluctuations

• The field couples to the metric - can choose a convenient

coordinate system (gauge).

Thursday, 4 July, 13

Inflationary Fluctuations

• The field couples to the metric - can choose a convenient

coordinate system (gauge).

time space

φ = const.

Spatially varying field and metric.

Thursday, 4 July, 13

Inflationary Fluctuations

• The field couples to the metric - can choose a convenient

coordinate system (gauge).

time space

φ = const.

Spatially varying metric, spatially uniform field.

t = const.

Comoving curvature perturbation: R

Thursday, 4 July, 13

Inflationary Fluctuations

• The field couples to the metric - can choose a convenient

coordinate system (gauge).

time space

φ = const.

Spatially varying metric, spatially uniform field.

t = const.

Comoving curvature perturbation: R !!!Conserved on superhorizon scales!!!

Thursday, 4 July, 13

• An important scale: comoving horizon

1 aH

• Horizon crossing: k = aH

comoving scale conformal time η

a = 1 Hη

k|η| = 1

• scillating

frozen

QFT in Curved Spacetime

Thursday, 4 July, 13

Inflationary Fluctuations

S = Z d4x  R 16πG 1 2gµ⌫∂µφ∂⌫φ V (φ)

• The action:

gravity inflaton field

Thursday, 4 July, 13

Inflationary Fluctuations

S = Z d4x  R 16πG 1 2gµ⌫∂µφ∂⌫φ V (φ)

• Expand into background and fluctuations:

S ' S0 + S2

• The action:

gravity inflaton field

Thursday, 4 July, 13

Inflationary Fluctuations

S = Z d4x  R 16πG 1 2gµ⌫∂µφ∂⌫φ V (φ)

• Expand into background and fluctuations:

S ' S0 + S2

• The action:

gravity inflaton field

S2 = 1 2 Z d3xdη "✓dv dη ◆2 (rv)2 + d2z dη2 v2 z #

• With a few re-definitions, looks like a free field with time-

dependent mass:

v ⌘ zMpR

z ⌘ a2 H2 ✓dφ dt ◆2

Thursday, 4 July, 13

Inflationary Fluctuations

• Quantize v, choose the Bunch-Davies vacuum, and find the

correlation functions:

P(k) = Akns−1

A = V 3 12⇡2(@V )2M 6

P

ns 1 = 2⌘ 6✏

ns < 1

ns > 1

Red Blue

Thursday, 4 July, 13

Inflationary Fluctuations

• Quantize v, choose the Bunch-Davies vacuum, and find the

correlation functions:

P(k) = Akns−1

A = V 3 12⇡2(@V )2M 6

P

ns 1 = 2⌘ 6✏

ns < 1

ns > 1

Red Blue

• Single-field slow-roll inflation: small non-gaussianity (to have

slow-roll, interactions must be small).

Thursday, 4 July, 13

Inflationary Fluctuations

' V (φ)

φ

P(k) =

Thursday, 4 July, 13

Inflationary Fluctuations

• Tensor modes:

AT = ✓ H 2⇡ ◆2 r ⌘ AT A = 16✏

0.94 0.96 0.98 1.00 Primordial Tilt (ns) 0.00 0.05 0.10 0.15 0.20 0.25 Tensor-to-Scalar Ratio (r0.002) Convex Concave Planck+WP Planck+WP+highL Planck+WP+BAO Natural Inflation Power law inflation Low Scale SSB SUSY R2 Inflation V ∝ φ2/3 V ∝ φ V ∝ φ2 V ∝ φ3 N∗=50 N∗=60

Thursday, 4 July, 13

Initial Conditions for Inflation

• Inflation, once it gets off the ground, can predict everything

we observe about the linear universe.

• Under what conditions can inflation begin?

To do.... (singularity? vacuum? nothing?)

H−1

V (∂φ)2

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

' V (φ)

φ

' V (φ)

φ

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Black hole

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Black hole Classically, you cannot find a way to make an inflating region in the lab!

Thursday, 4 July, 13

Inflation in the Lab

• What happens if I try to make inflation happen in the lab?

H−1

Black hole Quantum mechanically, there is some probability that you succeed, but you will never know.

Thursday, 4 July, 13

hn(t)i = 0 hn(t)n(t0)i = δ(t t0)

H2 = 8πGN 3 V (φ)

dV dφ < H3

Eternal Inflation

• Stochastic eternal inflation:

' V (φ)

φ

• The jitters dominate the motion when:

˙ φ = 1 3H  −dV dφ + H3 2π n(t)

• Thursday, 4 July, 13

Eternal Inflation

• When this occurs, inflation becomes eternal.

t → ∞

t

' V (φ)

φ

SRI EI

Thursday, 4 July, 13