[PPT] - Cosmological implications of polymer quantization Sanjeev Seahra PowerPoint Presentation

SLIDE 1

Polymer cosmology – 1 / 26

Cosmological implications of polymer quantization

Sanjeev Seahra (with G Hossain, V Husain and I Brown)

July 11, 2012

SLIDE 2

Polymer quantization

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 2 / 26

SLIDE 3

Approaches to quantum gravity

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 3 / 26

SLIDE 4

Approaches to quantum gravity

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 3 / 26

SLIDE 5

Approaches to quantum gravity

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 3 / 26

SLIDE 6

Approaches to quantum gravity

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 3 / 26

SLIDE 7

Approaches to quantum gravity

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 3 / 26

SLIDE 8

Approaches to quantum gravity

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 3 / 26

SLIDE 9

Cosmological applications

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 4 / 26

polymer quantization corrects Schr¨
dinger quantization at high

energies

SLIDE 10

Cosmological applications

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 4 / 26

polymer quantization corrects Schr¨
dinger quantization at high

energies

can look for effects in the (quantum cosmological) evolution
f the universe at high density

SLIDE 11

Cosmological applications

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 4 / 26

polymer quantization corrects Schr¨
dinger quantization at high

energies

can look for effects in the (quantum cosmological) evolution
f the universe at high density
polymer quantization also corrects Schr¨
dinger quantization on

small scales

SLIDE 12

Cosmological applications

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 4 / 26

polymer quantization corrects Schr¨
dinger quantization at high

energies

can look for effects in the (quantum cosmological) evolution
f the universe at high density
polymer quantization also corrects Schr¨
dinger quantization on

small scales

trans-Planckian problem: quantum primordial

perturbations that seed structure in the universe have physical scale ≪ lPl at beginning of inflation

SLIDE 13

Cosmological applications

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 4 / 26

polymer quantization corrects Schr¨
dinger quantization at high

energies

can look for effects in the (quantum cosmological) evolution
f the universe at high density
polymer quantization also corrects Schr¨
dinger quantization on

small scales

trans-Planckian problem: quantum primordial

perturbations that seed structure in the universe have physical scale ≪ lPl at beginning of inflation

should look for polymer quantization effects in the spectrum
f primordial perturbations and dynamics of the early

universe

SLIDE 14

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 15

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 16

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 17

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 18

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 19

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 20

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 21

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 22

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 23

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 24

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 25

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 26

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 27

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 28

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 29

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 30

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 31

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 32

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 33

Basic properties

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 5 / 26

SLIDE 34

Example: the simple harmonic oscillator

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 6 / 26

let’s find the energy eigenvalues of a polymer-quantized SHO of

mass m and frequency ω

SLIDE 35

Example: the simple harmonic oscillator

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 6 / 26

let’s find the energy eigenvalues of a polymer-quantized SHO of

mass m and frequency ω

conventional Hamiltonian: ˆ

H = 1 2m ˆ p2 + 1 2mω2ˆ x2

SLIDE 36

Example: the simple harmonic oscillator

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 6 / 26

let’s find the energy eigenvalues of a polymer-quantized SHO of

mass m and frequency ω

conventional Hamiltonian: ˆ

H = 1 2m ˆ p2 + 1 2mω2ˆ x2

polymer Hamiltonian:

ˆ H = 1 2m

i

ˆ Uλ − ˆ U †

λ

2λ 2 + 1 2mω2ˆ x2, M⋆ = 1 λ

SLIDE 37

Example: the simple harmonic oscillator

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 6 / 26

let’s find the energy eigenvalues of a polymer-quantized SHO of

mass m and frequency ω

conventional Hamiltonian: ˆ

H = 1 2m ˆ p2 + 1 2mω2ˆ x2

polymer Hamiltonian:

ˆ H = 1 2m

i

ˆ Uλ − ˆ U †

λ

2λ 2 + 1 2mω2ˆ x2, M⋆ = 1 λ

position eigenstate basis: |Ψ =

∞

j=−∞

cj|xj with xj = x0 + jλ

ˆ

x|xj = xj|xj

ˆ

Uλ|xj = |xj+1

xj|xj′ = δj,j′

SLIDE 38

Example: the simple harmonic oscillator

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 6 / 26

projection of energy eigenvalue equation ˆ

H|Ψ = E|Ψ onto |xj yields a difference equation for cj’s: 1 8mλ2(2cj − cj−2 − cj+2) + 1 2mω2xjcj = Ecj

what you would get from a simple finite differencing of the
rdinary Schr¨
dinger equation
could obtain energy eigenvalues numerically

SLIDE 39

Momentum representation of the SHO

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 7 / 26

easier to work in “momentum eigenstate” basis:

|p =

∞

j=−∞

e−ipxj|xj, p ∈

− π

2λ, π 2λ

SLIDE 40

Momentum representation of the SHO

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 7 / 26

easier to work in “momentum eigenstate” basis:

|p =

∞

j=−∞

e−ipxj|xj, p ∈

− π

2λ, π 2λ

wavefunction: Ψ(p) = p|Ψ

SLIDE 41

Momentum representation of the SHO

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 7 / 26

easier to work in “momentum eigenstate” basis:

|p =

∞

j=−∞

e−ipxj|xj, p ∈

− π

2λ, π 2λ

wavefunction: Ψ(p) = p|Ψ
perators: p| ˆ

Uλ|Ψ = eiλpΨ(p) and p|ˆ x|Ψ = i∂pΨ(p)

SLIDE 42

Momentum representation of the SHO

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 7 / 26

easier to work in “momentum eigenstate” basis:

|p =

∞

j=−∞

e−ipxj|xj, p ∈

− π

2λ, π 2λ

wavefunction: Ψ(p) = p|Ψ
perators: p| ˆ

Uλ|Ψ = eiλpΨ(p) and p|ˆ x|Ψ = i∂pΨ(p)

projecting eigenvalue equation ˆ

H|Ψ = E|Ψ onto |p: EΨ = ω 2

− ∂2

∂y2 + sin2(√gy) g

Ψ,

y = p √mω, g = mω M 2

⋆

SLIDE 43

Momentum representation of the SHO

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 7 / 26

easier to work in “momentum eigenstate” basis:

|p =

∞

j=−∞

e−ipxj|xj, p ∈

− π

2λ, π 2λ

wavefunction: Ψ(p) = p|Ψ
perators: p| ˆ

Uλ|Ψ = eiλpΨ(p) and p|ˆ x|Ψ = i∂pΨ(p)

projecting eigenvalue equation ˆ

H|Ψ = E|Ψ onto |p: EΨ = ω 2

− ∂2

∂y2 + sin2(√gy) g

Ψ,

y = p √mω, g = mω M 2

⋆

“low energy” quantum states with ∆y ≪ g−1/2 recover

standard eigenfunctions/energies

SLIDE 44

Momentum representation of the SHO

Polymer quantization Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions

Polymer cosmology – 7 / 26

the eigenvalue ODE is analytically solvable:
recover Schr¨
dinger quantization for g ≪ 1

SLIDE 45

Quantum cosmology

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 8 / 26

SLIDE 46

Quantum cosmologies

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 9 / 26

SLIDE 47

Quantum cosmologies

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 9 / 26

SLIDE 48

Quantum cosmologies

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 9 / 26

SLIDE 49

Quantum cosmologies

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 9 / 26

SLIDE 50

Quantum cosmologies

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 9 / 26

SLIDE 51

Quantum cosmologies

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 9 / 26

SLIDE 52

Avoiding the big bang

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 10 / 26

SLIDE 53

Avoiding the big bang

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 10 / 26

SLIDE 54

Avoiding the big bang

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 10 / 26

SLIDE 55

Avoiding the big bang

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 10 / 26

SLIDE 56

Avoiding the big bang

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 10 / 26

SLIDE 57

Avoiding the big bang

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 10 / 26

SLIDE 58

Semiclassical approximation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 11 / 26

SLIDE 59

Semiclassical approximation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 11 / 26

SLIDE 60

Semiclassical approximation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 11 / 26

SLIDE 61

Semiclassical approximation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 11 / 26

SLIDE 62

Semiclassical approximation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 11 / 26

SLIDE 63

Semiclassical approximation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 11 / 26

SLIDE 64

Effective Friedmann equation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 12 / 26

SLIDE 65

Effective Friedmann equation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 12 / 26

SLIDE 66

Effective Friedmann equation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 12 / 26

SLIDE 67

Effective Friedmann equation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 12 / 26

SLIDE 68

Effective Friedmann equation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 12 / 26

SLIDE 69

Effective Friedmann equation

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 12 / 26

SLIDE 70

Numerical results

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Primordial fluctuations Conclusions

Polymer cosmology – 13 / 26

SLIDE 71

Primordial fluctuations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 14 / 26

SLIDE 72

The problem

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 15 / 26

here we consider an inhomogeneous massless scalar in a de

Sitter background

ds2 =

−dt2 + a2dx2

a = exp(Ht) a2(−dη2 + dx2) a = −(Hη)−1 Hφ =

d3x a3

1 2a6π2 + 1 2a2(∇φ)2

SLIDE 73

The problem

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 15 / 26

here we consider an inhomogeneous massless scalar in a de

Sitter background

ds2 =

−dt2 + a2dx2

a = exp(Ht) a2(−dη2 + dx2) a = −(Hη)−1 Hφ =

d3x a3

1 2a6π2 + 1 2a2(∇φ)2

goal: power spectrum of fluctuations Pφ(k) produced during

inflation assuming polymer quantization

SLIDE 74

The problem

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 15 / 26

here we consider an inhomogeneous massless scalar in a de

Sitter background

ds2 =

−dt2 + a2dx2

a = exp(Ht) a2(−dη2 + dx2) a = −(Hη)−1 Hφ =

d3x a3

1 2a6π2 + 1 2a2(∇φ)2

goal: power spectrum of fluctuations Pφ(k) produced during

inflation assuming polymer quantization

problem: polymer QFT poorly understood

SLIDE 75

The problem

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 15 / 26

here we consider an inhomogeneous massless scalar in a de

Sitter background

ds2 =

−dt2 + a2dx2

a = exp(Ht) a2(−dη2 + dx2) a = −(Hη)−1 Hφ =

d3x a3

1 2a6π2 + 1 2a2(∇φ)2

goal: power spectrum of fluctuations Pφ(k) produced during

inflation assuming polymer quantization

problem: polymer QFT poorly understood
N.B.: no a priori relation between H and polymer energy scale

M⋆ assumed

SLIDE 76

Quantizing inflationary fluctuations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 16 / 26

scalar field Hamiltonian

Hφ = Hφ(φ(x), π(y))

SLIDE 77

Quantizing inflationary fluctuations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 16 / 26

scalar field Hamiltonian

Hφ = Hφ(φ(x), π(y))

textbook algorithm promote to operators

(φ(x), π(y)) → (ˆ φ(x), ˆ π(y))

Fourier transform

ˆ φ =

k fk(η)eik·xˆ

ak + h.c. ˆ φ = 0 ⇒ choose BCs to

recover flat QFT in ∞ past

Pφ(k) ∝ |fk|2

k≪Ha

SLIDE 78

Quantizing inflationary fluctuations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 16 / 26

scalar field Hamiltonian

Hφ = Hφ(φ(x), π(y))

textbook algorithm promote to operators

(φ(x), π(y)) → (ˆ φ(x), ˆ π(y))

Fourier transform

ˆ φ =

k fk(η)eik·xˆ

ak + h.c. ˆ φ = 0 ⇒ choose BCs to

recover flat QFT in ∞ past

Pφ(k) ∝ |fk|2

k≪Ha

unclear how to represent these

perators in polymer picture (see

attempts by Ashtekar et al 2003; Kreienbuehl and Husain 2010)

SLIDE 79

Quantizing inflationary fluctuations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 16 / 26

scalar field Hamiltonian

Hφ = Hφ(φ(x), π(y))

textbook algorithm promote to operators

(φ(x), π(y)) → (ˆ φ(x), ˆ π(y))

Fourier transform

ˆ φ =

k fk(η)eik·xˆ

ak + h.c. ˆ φ = 0 ⇒ choose BCs to

recover flat QFT in ∞ past

Pφ(k) ∝ |fk|2

k≪Ha

alternative algorithm Fourier transform

(φ(x), π(y)) → (φk, πk′) Hφ =

k Hk(φk, πk)

separately quantize each mode using QM: i ∂tψk = ˆ

Hkψk Pφ(k) ∝ ψk|ˆ φ2

k|ψk

k≪Ha

where ψk is “ground state”

SLIDE 80

Time dependent Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 17 / 26

“Fourier transform then quantize” algorithm leads to following Schr¨

dinger equations (recall a = exp Ht):
standard quantization:

i ∂ ∂tψ(t, πk) = 1 2a3π2

k − ak2

2 ∂2 ∂π2

k

ψ(t, πk)
polymer quantization:

i ∂ ∂tψ(t, πk) = 1 2λ sin2 λπk a3/2

− ak2

2 ∂2 ∂π2

k

ψ(t, πk)

SLIDE 81

Time dependent Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 17 / 26

“Fourier transform then quantize” algorithm leads to following Schr¨

dinger equations (recall a = exp Ht):
standard quantization:

i ∂ ∂tψ(t, πk) = 1 2a3π2

k − ak2

2 ∂2 ∂π2

k

ψ(t, πk)
polymer quantization:

i ∂ ∂tψ(t, πk) = 1 2λ sin2 λπk a3/2

− ak2

2 ∂2 ∂π2

k

ψ(t, πk)
following transformations and re-scaling make things simpler:

η = − 1 Ha, y = −kη

H2

k3 πk, ψ(t, πk) = H2 k3 1/4 −kηΨ(η, y) exp

−i y2

2kη

SLIDE 82

Effective Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 18 / 26

After transformations and re-scalings, PDEs governing power spectrum are:

standard quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + y2

Ψ

SLIDE 83

Effective Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 18 / 26

After transformations and re-scalings, PDEs governing power spectrum are:

standard quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + y2

Ψ
rdinary simple harmonic oscillator

SLIDE 84

Effective Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 18 / 26

After transformations and re-scalings, PDEs governing power spectrum are:

standard quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + y2

Ψ
rdinary simple harmonic oscillator
ground state unambiguous ⇒ gives Bunch-Davies vacuum

SLIDE 85

Effective Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 18 / 26

After transformations and re-scalings, PDEs governing power spectrum are:

standard quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + y2

Ψ
rdinary simple harmonic oscillator
ground state unambiguous ⇒ gives Bunch-Davies vacuum
polymer quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + sin2(√gy) g

Ψ

SLIDE 86

Effective Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 18 / 26

After transformations and re-scalings, PDEs governing power spectrum are:

standard quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + y2

Ψ
rdinary simple harmonic oscillator
ground state unambiguous ⇒ gives Bunch-Davies vacuum
polymer quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + sin2(√gy) g

Ψ
“polymer coupling”: g =

k M⋆a = physical wavenumber

polymer energy scale

SLIDE 87

Effective Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 18 / 26

After transformations and re-scalings, PDEs governing power spectrum are:

standard quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + y2

Ψ
rdinary simple harmonic oscillator
ground state unambiguous ⇒ gives Bunch-Davies vacuum
polymer quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + sin2(√gy) g

Ψ
“polymer coupling”: g =

k M⋆a = physical wavenumber

polymer energy scale

late time limit g → 0: recover standard wave equation

SLIDE 88

Effective Schr¨

dinger equations

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 18 / 26

After transformations and re-scalings, PDEs governing power spectrum are:

standard quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + y2

Ψ
rdinary simple harmonic oscillator
ground state unambiguous ⇒ gives Bunch-Davies vacuum
polymer quantization: i∂Ψ

∂η = k 2

− ∂2

∂y2 + sin2(√gy) g

Ψ
“polymer coupling”: g =

k M⋆a = physical wavenumber

polymer energy scale

late time limit g → 0: recover standard wave equation
time dependent potential makes ground state ambiguous

SLIDE 89

Formal solution of polymer Schr¨

dinger equation

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 19 / 26

wavefunction ansatz: Ψ(η, y) =

∞

n=0

cn(η)ei

ǫn(η)dηΨn(η, y)
Ψn are instantaneous energy eigenfunctions:

1 2k

−∂2

y + g−1 sin2(√gy)

Ψn(η, y) = ǫn(η)Ψn(η, y)

SLIDE 90

Formal solution of polymer Schr¨

dinger equation

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 19 / 26

wavefunction ansatz: Ψ(η, y) =

∞

n=0

cn(η)ei

ǫn(η)dηΨn(η, y)
Ψn are instantaneous energy eigenfunctions:

1 2k

−∂2

y + g−1 sin2(√gy)

Ψn(η, y) = ǫn(η)Ψn(η, y)
subbing ansatz into Schr¨
dinger equation gives:

d dgc = Ac, c =    c0 c1

. . .

   , A =    a00 a01 · · · a10 a11

. . . ...

  

where anm = anm(η) are matrix elements in the {Ψn} basis

SLIDE 91

Formal solution of polymer Schr¨

dinger equation

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 19 / 26

wavefunction ansatz: Ψ(η, y) =

∞

n=0

cn(η)ei

ǫn(η)dηΨn(η, y)
Ψn are instantaneous energy eigenfunctions:

1 2k

−∂2

y + g−1 sin2(√gy)

Ψn(η, y) = ǫn(η)Ψn(η, y)
subbing ansatz into Schr¨
dinger equation gives:

d dgc = Ac, c =    c0 c1

. . .

   , A =    a00 a01 · · · a10 a11

. . . ...

  

where anm = anm(η) are matrix elements in the {Ψn} basis

solve numerically: c(η = 0) gives final quantum state and

hence power spectrum Pφ(k)

SLIDE 92

Initial conditions

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 20 / 26

suppose we prepare a given k mode in the ground state at an

initial time

SLIDE 93

Initial conditions

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 20 / 26

suppose we prepare a given k mode in the ground state at an

initial time

standard quantization: it will stay in the ground state

SLIDE 94

Initial conditions

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 20 / 26

suppose we prepare a given k mode in the ground state at an

initial time

standard quantization: it will stay in the ground state
polymer quantization: it will not stay in the ground state

SLIDE 95

Initial conditions

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 20 / 26

suppose we prepare a given k mode in the ground state at an

initial time

standard quantization: it will stay in the ground state
polymer quantization: it will not stay in the ground state
we assume each mode is in ground state at the start of inflation

(c.f. Martin and Brandenberger 2001)

SLIDE 96

Initial conditions

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 20 / 26

suppose we prepare a given k mode in the ground state at an

initial time

standard quantization: it will stay in the ground state
polymer quantization: it will not stay in the ground state
we assume each mode is in ground state at the start of inflation

(c.f. Martin and Brandenberger 2001)

final quantum state determined by polymer coupling at start of

inflation g0 = k/k⋆

k⋆ = present day wavenumber of a mode with physical

wavelength M −1

⋆

at the start of inflation

SLIDE 97

Initial conditions

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 20 / 26

SLIDE 98

Results for the power spectrum

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 21 / 26

Solution for expansion coefficients with g0 = 20 and M⋆/H = 1:

SLIDE 99

Results for the power spectrum

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 21 / 26

recover standard result Pφ = P0 = (H/2π)2 for g0 ≪ 1
polymer effects vanish for M⋆/H → ∞

SLIDE 100

Semi-analytic results

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 22 / 26

perturbation theory/fitting functions to numeric results lead to

approximate power spectrum:

Pφ P0 ≈        1 + 1 2 k k⋆ , k ≪ k⋆ 1 + H 4M⋆ sin 2M⋆ H k2 k2

⋆

− 1

,

k ≫ k⋆ M⋆ ≫ H

k⋆ ∼ 3 × 10−6

Mpc M⋆ H Einf 1016 GeV e65 eN 100 G 1/12

N is the number of e-folds of inflation
Einf is the energy scale of inflation
G is the effective number of relativistic species at the end of

inflation

SLIDE 101

Observational consequences

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 23 / 26

1 2 3 4 5 6 l(l+1)CTTl / 1000 0.5 1 1.5 (l+1)CTEl HZ M*/H=1 M*/H=8 10 100 l 2.5 5 (l+1)CEEl / 100

polymer effects on CMB assuming k⋆ = 5 × 10−4 Mpc−1
unobservable due to cosmic variance

SLIDE 102

Observational consequences

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 23 / 26

10

5

10

4

10

3

10

2

10

1

10 k (Mpc

1)

10

2

10

3

10

4

P(k) (Mpc

2)

M*/H=1 M*/H=8 HZ

Present day matter power spectrum with k⋆ = 5 × 10−4 Mpc−1

SLIDE 103

Observational consequences

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨

dinger equations

Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions

Polymer cosmology – 23 / 26

0.1 0.2 0.3 k (Mpc

1)
0.05

0.05 log10[P(k)/P(k)smooth] M*/H=8 HZ

baryon acoustic oscillations with k⋆ = 5 × 10−4 Mpc−1
M⋆/H ∼ 1 already ruled out by current observations
future surveys (e.g. Euclid) will be able to rule out M⋆/H 10

SLIDE 104

Conclusions

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 24 / 26

SLIDE 105

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

polymer quantization (PQ) is an alternative to standard

quantization involving a notion of fundamental discreteness

SLIDE 106

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

polymer quantization (PQ) is an alternative to standard

quantization involving a notion of fundamental discreteness

modifies standard results for energies ≫ M⋆

SLIDE 107

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

polymer quantization (PQ) is an alternative to standard

quantization involving a notion of fundamental discreteness

modifies standard results for energies ≫ M⋆
M⋆ is a free parameter to be fixed by experiment

SLIDE 108

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

polymer quantization (PQ) is an alternative to standard

quantization involving a notion of fundamental discreteness

modifies standard results for energies ≫ M⋆
M⋆ is a free parameter to be fixed by experiment
PQ of matter in quantum cosmology results in early time de

Sitter inflation with H ∼ M 2

⋆/MPl

SLIDE 109

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

we found PQ corrections to scale invariant spectrum of

primordial perturbations in de Sitter universes

SLIDE 110

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

we found PQ corrections to scale invariant spectrum of

primordial perturbations in de Sitter universes

scillatory power spectrum for k k⋆

SLIDE 111

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

we found PQ corrections to scale invariant spectrum of

primordial perturbations in de Sitter universes

scillatory power spectrum for k k⋆
amplitude of oscillations ∝ H/M⋆

SLIDE 112

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

we found PQ corrections to scale invariant spectrum of

primordial perturbations in de Sitter universes

scillatory power spectrum for k k⋆
amplitude of oscillations ∝ H/M⋆
if polymer effects drive inflation H/M⋆ ∼ Einf/MPl

SLIDE 113

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

we found PQ corrections to scale invariant spectrum of

primordial perturbations in de Sitter universes

scillatory power spectrum for k k⋆
amplitude of oscillations ∝ H/M⋆
if polymer effects drive inflation H/M⋆ ∼ Einf/MPl
scillation amplitude ∼ 10−4 for GUT scale inflation

SLIDE 114

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

we found PQ corrections to scale invariant spectrum of

primordial perturbations in de Sitter universes

scillatory power spectrum for k k⋆
amplitude of oscillations ∝ H/M⋆
if polymer effects drive inflation H/M⋆ ∼ Einf/MPl
scillation amplitude ∼ 10−4 for GUT scale inflation
difficult to see in CMB power spectra

SLIDE 115

Summary

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 25 / 26

we found PQ corrections to scale invariant spectrum of

primordial perturbations in de Sitter universes

scillatory power spectrum for k k⋆
amplitude of oscillations ∝ H/M⋆
if polymer effects drive inflation H/M⋆ ∼ Einf/MPl
scillation amplitude ∼ 10−4 for GUT scale inflation
difficult to see in CMB power spectra
future observations of baryon acoustic oscillations could

constrain H/M⋆ 0.1

SLIDE 116

Future work

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 26 / 26

generalization to slow roll inflation

SLIDE 117

Future work

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 26 / 26

generalization to slow roll inflation
tensor-to-scalar ratio

SLIDE 118

Future work

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 26 / 26

generalization to slow roll inflation
tensor-to-scalar ratio
CMB bispectrum (non-gaussianities)

SLIDE 119

Future work

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 26 / 26

generalization to slow roll inflation
tensor-to-scalar ratio
CMB bispectrum (non-gaussianities)
“Fourier transform then quantize” approach to inflationary

fluctuations can be use to study effects of other alternative quantization schemes

SLIDE 120

Future work

Polymer quantization Quantum cosmology Primordial fluctuations Conclusions Summary Future work

Polymer cosmology – 26 / 26

generalization to slow roll inflation
tensor-to-scalar ratio
CMB bispectrum (non-gaussianities)
“Fourier transform then quantize” approach to inflationary

fluctuations can be use to study effects of other alternative quantization schemes

e.g. from modified uncertainty relations