[PPT] - Inflationary Cosmology Drew Jamieson Stony Brook University March PowerPoint Presentation

SLIDE 1

Inflationary Cosmology

Drew Jamieson

Stony Brook University

March 23, 2016

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 1 / 28

SLIDE 2

Overview

1

Cosmic Expansion

2

Three Cosmological Problems

3

Mechanism for Inflationary Expansion

4

Exciting Quantum Oscillators According to Heisenberg

5

Generation of Cosmic Anisotropies

6

Comparison with CMB Data

7

Conclusions

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 2 / 28

SLIDE 3

The Metric

ds2 = −dt2 + a(t)2γijdxidxj γijdxidxj = 1 1 − Kr 2 dr 2 + r 2dΩ (spherical coordinates) ➔ K = +1: Spherical ➔ K = 0: Flat ➔ K = −1: Hyperbolic Einstein field equations: Rµν − 1

2gµνR = −8πGTµν

˙

a a

2

= 4πG 3 (ρ + Λ) − K a2 Scale factor evolves according to energy density, cosmological constant, and curvature

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 3 / 28

SLIDE 4

Matter

Smatter =

d4x√−gLmatter

Tµν = 1 √−g δSmatter δgµν T00 = ρ

1 3T i i = p

Continuity equation: ∇µT µ

ν = 0

˙ ρ = −3(ρ + p) ˙ a a For p = wρ: dρ da = −3(1 + w)ρ a ρ ∼ a−3(1+w)

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 4 / 28

SLIDE 5

Expansion Regimes

w a(t) Radiation:

1 3

∼ √t Nonrelativistic matter: ∼ t

2 3

Curvature: −1

3

∼ t Λ: −1 ∼ eαt

t a(t) Λ K NR R

Today: a = 1, ˙ a = H0 Percentages of total energy density: ΩR + ΩNR + ΩK + ΩΛ = 1 ˙ a a = H(a) = H0

ΩR

a4 + ΩNR a3 + ΩK a2 + ΩΛ ΩNR = 0.308 ± 0.012 ΩΛ = 0.692 ± 0.012 ΩR ≈ ΩK ≈ 0

0.01 0.05 0.10 0.50 1 5 10 0.005 0.010 0.050 0.100 0.500 1 Time (Billion years) a(t)

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 5 / 28

SLIDE 6

Three problems

Flatness: ΩK ≪ 1 = ⇒

ρK ρT ≪ 10−16 during e+e− annihilation

Horizons: CMB temperature fluctuations:

δT T ∼ 10−5

even though sky is mostly causally disconnected Monopoles: If you like GUTs, you get roughly 1 magnetic monopole per nucleon

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 6 / 28

SLIDE 7

Solution:

Inflation: Early on, the universe underwent a period

f temporary but rapid expansion

➔ a(t) ∼ eαt ➔ α → 0 eventually

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 7 / 28

SLIDE 8

The Inflaton and Slow Roll Inflation

SKG =

d4x√−g

1

2∂µφ∂µφ − V (φ)

ρ = 1

2 ˙ φ2 + V (φ) p = 1 2 ˙ φ2 − V (φ) w =

1 2 ˙

φ2 − V (φ)

1 2 ˙

φ2 + V (φ) If V (φ) ≫ ˙ φ2 w ≈ −1 Slow Roll Inflaton Potential

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 8 / 28

SLIDE 9

Intermission

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 9 / 28

SLIDE 10

Exciting an Oscillator

Two mechanisms: a) A driving force shakes things up: ¨ q(t) = −ω2q(t) + j(t) b) A time-dependent frequency: ¨ q(t) = −ω(t)2q(t) ➔ Both involve resonance if periodic ➔ Classically ω(t) won’t work in the ground state ➔ In quantum mechanics, even ground state has fluctuations

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 10 / 28

SLIDE 11

The Driven Oscillator

In the Heisenberg picture: [q(t), p(t)] = i1 q† = q, p† = p H = 1 2p(t)2 + ω2 2 q(t)2 − j(t)q(t) Heisenberg equation of motion: i ˙ f (t) = [f (t), H(t)] ˙ q(t) = p(t) ˙ p(t) = −ω2q(t) + j(t)

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 11 / 28

SLIDE 12

The Driven Oscillator

Usual trick: q(t) = 1 √ 2ω

a + a†

p(t) = −i

ω

2

a − a†

The Hamiltonian simplifies: H(t) = ω

a†(t)a(t) + 1

2

− j(t)

√ 2ω

a†(t) + a(t)
EOM for a(t):

i ˙ a(t) = ωa(t) − 1 √ 2ωj(t) Solution: a(t) = aine−iωt + 1 i √ 2ω

t

dt′j(t′)eiω(t′−t)

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 12 / 28

SLIDE 13

Finite Duration Driving Force

The force acts only for 0 < t < T Define j0 =

i √ 2ω

T

0 dt′j(t′)eiω(t′−T), then:

a(t) =

                            

aine−iωt for t < 0 aine−iωt + 1

i √ 2ω

t

0 dt′j(t′)eiω(t′−t) for 0 < t < T

(ain + j0) e−iωt for t > T

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 13 / 28

SLIDE 14

Finite Duration Driving Force

Before ain, a†

in

H = ω

a†

inain + 1

2

|nin =

1 √ n!

a†

in

n |Ωin

{|nin} form complete basis on the Hilbert space After aout, a†

ut

H = ω

a†
utaout + 1

2

|nout =

1 √ n!

a†
ut

n |Ωout

{|nout} form complete basis on the Hilbert space Both |nin and |nout have energy ω

n + 1

2

, but aout = ain + j0

|nin = |nout , Heisenberg picture |Ωin = |Ωout

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 14 / 28

SLIDE 15

Finite Duration Driving Force

In the ground state: |ψ = |Ωin: Before Fluctuations: Ωin| q(t) |Ωin = 0 Energy: Ωin| H |Ωin = ω 2 After Fluctuations: Ωin| q(t) |Ωin =

T

dt′ sin ((t − t′) ω) ω j(t′) Energy: Ωin| H |Ωin = ω

1

2 + |jo|2

What is |Ωin at late times?

aout |Ωin = (ain + j0) |Ωin = j0 |Ωin A coherent state!

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 15 / 28

SLIDE 16

Finite Duration Driving Force

Early vacuum state is superposition of late time excited states: |Ωin =

n

e− 1

2 |j0|2 j n

√ n! |nout , (Bogoliubov transformation) Particle production: P(n) = | nout|Ωin |2 = |j0|2n n! e−|j0|2

10 20 30 40 50 0.00 0.02 0.04 0.06 0.08 n P(n)

|j0|=5 Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 16 / 28

SLIDE 17

Quantum Fluctuations of the Inflaton

Consider the a Klein-Gordon field φI = ¯ φ(t) + φ(t, x) in flat FRW: SKG =

d4x√−g

1

2gµν∂µφ∂νφ − 1 2m2φ2

Let dt = a(η)dη:

ds2 = a(η)2(−dη2 + d x · d x) Let χ(η, x) = a(η)φ(η, x) SKG =

d3xdη
χ′2 −
∇χ

2 −

m2a2 − a′′

a

χ2
Drew Jamieson (SBU)

Inflationary Cosmology March 23, 2016 17 / 28

SLIDE 18

Quantum Fluctuations of the Inflaton

Equation of motion: χ′′ − ∇2χ +

m2a2 − a′′

a

χ = 0

Fourier Transform: χk(η) =

d3x

(2π)

3 2

χ(η, x)e−i

k· x

χ′′

k +

k2 + m2a2 − a′′

a

χk = 0

Each k-mode is a Harmonic oscillator with time dependent frequency: ωk(η) =

k2 + m2a(η)2 − a′′(η)

a(η)

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 18 / 28

SLIDE 19

Quantum Fluctuations of the Inflaton

ωk(η) =

k2 + m2a(η)2 − a′′(η)

a(η) For exponential expansion: a = eHt = − 1

Hη, assuming m ≪ H:

ωk(η) ≈

k2 − 2

η2 Becomes imaginary for: |η| < √ 2 k

r

1 H < a(η)2π k = ⇒ λ(η) ≈ H−1

3.5
3.0
2.5
2.0
1.5
1.0
1.0
0.5

0.0 0.5 1.0 1.5 2.0 η ω2 k=1 k= 2 

1

k= 2

A mode oscillates until its wavelength is stretched larger than the Hubble radius, when it crosses this horizon the mode grows This is an effective amplification of vacuum fluctuations!

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 19 / 28

SLIDE 20

Quantum Fluctuations of the Inflaton

Three cases: small wavelength Start small, never cross the horizon, (k >> η−1) these quantum fluctuations are not amplified medium wavelength Start small but eventually cross the horizon, (k ≈ η−1) freezing out, these are amplified by inflation large wavelength Are always larger than the horizon, (k << η−1) physically unclear what these are (Λ)?

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 20 / 28

SLIDE 21

Quantum Fluctuations of the Inflaton

χ′′

k + ωk(η)2χk = 0

Solve Hermiticity, commutation relations, and equation of motion with: χk(η) = 1 √ 2

v∗

k (η)ak + vk(η)a† k

where v∗

k (η)v′ k(η) − v′∗ k (η)vk(η) = −2i (Wronskian for ODE)

If mode functions vk(η) solve KG equation, so does χk(η) Solution: ω2

k(η) = k2 + m2 H2η2 − 2 η2

vk(η) =

π|η|

2

Jn (k|η|) − iYn (k|η|)
where n =
9

4 − m2 H2 ≈ 3 2

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 21 / 28

SLIDE 22

Quantum Fluctuations of the Inflaton

0.001 0.010 0.100 1 10 100 |η|

5

5 10 15 20 Im(vk) k=0.01 k=0.5 ⟵ Increasing time

Large k solution: v′′

k (η) + k2v′′ k (η) ≈ 0

vk(η) = 1 √ k eikη massless Klein-Gordon field At late times (η ≈ 0), for small k: v′′

k (η) +

m2

H2η2 − 2 η2

vk(η) ≈ 0

vk(η) = π 2k Γ(n) π 2n (k|η|)

1 2 −n ∼

1 k|η| growing mode

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 22 / 28

SLIDE 23

Quantum Fluctuations of the Inflaton

Fluctuations: δφ2

k = Ω| (φk − ✟✟

✟

φk)2 |Ω δφ2

k = 1 a(η)2 Ω| χ2 k |Ω

δφk = a(η)−1k

3 2 |vk(η)|

Large k: vk(η) = 1 √ k eikη δφk = k a(η) δφk ∼ 1 λ(η) Small k, end of inflation: vk(η) = 2nΓ(n) √ 2πk (k|η|)

1 2 −n

δφk = H2

3 2 Γ(3

2)

π δφk ∼ constant The large fluctuations stay constant even though space is expanding ➔ effective amplification

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 23 / 28

SLIDE 24

Quantum Fluctuations of the Inflaton

0.1 10 1000 105 107 109 0.01 0.10 1 10 100 λ δϕ Case 1: small modes δϕ ~ λ-1 slope = -1 Case 2: medium modes δϕ ~ H Case 3: large modes

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 24 / 28

SLIDE 25

Amount of Inflation

If inflation is the origin of cosmic structure, H determines the amplitude of fluctuations: δφ ∼ H Measurements of CMB indicate: H−1 ≈ 103ℓpl ∼ 10−30m The duration of inflation must be long enough to let modes expand to cosmological size: ∆t ≈ 10−32s Expansion is then very large: a(tf ) a(ti) = eH(tf −ti)c = e1030×10−32×3×108 = e3×105 ∼ 101.3×105 Only need 1030 to solve cosmology problems

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 25 / 28

SLIDE 26

Realistic Inflation

Must consider both inflaton and gravity perturbations (tree level quantum gravity): gµν = a(η)2ηµν + δgµν(η, x) φ = ¯ φ(η) + a(η)χ(η, x) Decompose δgµν into scalar (CMB temperature), vector (E-modes) and tensor parts (B-modes, not measured yet) Actual exponential expansion produces very large gravity perturbations = ⇒ inflation was not quite exponential, large power law: For V (φ) = V0Exp

−
12π

p φ mpl

=

⇒ a(t) ∼ tp = ⇒ a(η) ∼ − 1

Hη 1 1−ǫ

This potential comes from Starobinsky modified gravity: SG =

d4x√−g

m2

pl

2

R + R2

6M2

Drew Jamieson (SBU)

Inflationary Cosmology March 23, 2016 26 / 28

SLIDE 27

CMB observables

e-foldings: N = Log(af

ai )

CMB Features: ➔ Gaussian, adiabatic ➔ δT ∼ 10−5 ➔ Power law: P(k) ∼ kns−1 ➔ ns = 0.965 ± 0.006 ➔ tensor/scalar: r < 0.08 Power law inflation predicts: ➔ ns = 1 − 2

N

➔ r ≈ 12

N2

➔ From δT: N = 58 ± 4 ➔ ns = 0.966 ± 0.002 ➔ r ≈ 0.0036 ± 0.0005

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 27 / 28

SLIDE 28

Conclusions

➔ Inflation explains:

➔ Flatness ➔ Homogeneity ➔ Absence of monopoles (If you care about GUTs)

➔ Inflation agrees with CMB observations:

➔ Spectral tilt ns ➔ Smallness of tensor/scalar r ➔ Gaussianity/adiabticity

➔ Problems:

➔ Too many models ➔ Discriminating observables are hard to measure

➔ Spectral running dns/dlog(k) ➔ Tensor/scalar r ➔ Nongaussianity ➔ Nonadiabticity (isocurvature)

Drew Jamieson (SBU) Inflationary Cosmology March 23, 2016 28 / 28