Cosmological implications of polymer quantization Sanjeev Seahra - PowerPoint PPT Presentation

Example: the simple harmonic oscillator Polymer quantization let’s find the energy eigenvalues of a polymer-quantized SHO of � Quantum gravity mass m and frequency ω Cosmological applications 1 p 2 + 1 conventional Hamiltonian: ˆ 2 mω 2 ˆ x 2 Basic properties H = 2 m ˆ � Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions Polymer cosmology – 6 / 26

Example: the simple harmonic oscillator Polymer quantization let’s find the energy eigenvalues of a polymer-quantized SHO of � Quantum gravity mass m and frequency ω Cosmological applications 1 p 2 + 1 conventional Hamiltonian: ˆ 2 mω 2 ˆ x 2 Basic properties H = 2 m ˆ � Example: SHO Momentum polymer Hamiltonian: representation � � ˆ �� 2 � U λ − ˆ Quantum cosmology U † 1 + 1 M ⋆ = 1 ˆ 2 mω 2 ˆ x 2 , λ H = i Primordial fluctuations 2 m 2 λ λ Conclusions Polymer cosmology – 6 / 26

Example: the simple harmonic oscillator Polymer quantization let’s find the energy eigenvalues of a polymer-quantized SHO of � Quantum gravity mass m and frequency ω Cosmological applications 1 p 2 + 1 conventional Hamiltonian: ˆ 2 mω 2 ˆ x 2 Basic properties H = 2 m ˆ � Example: SHO Momentum polymer Hamiltonian: representation � � ˆ �� 2 � U λ − ˆ Quantum cosmology U † 1 + 1 M ⋆ = 1 ˆ 2 mω 2 ˆ x 2 , λ H = i Primordial fluctuations 2 m 2 λ λ Conclusions ∞ � position eigenstate basis: | Ψ � = c j | x j � with x j = x 0 + jλ � j = −∞ x | x j � = x j | x j � ˆ � ˆ U λ | x j � = | x j +1 � � � x j | x j ′ � = δ j,j ′ � Polymer cosmology – 6 / 26

Example: the simple harmonic oscillator projection of energy eigenvalue equation ˆ Polymer quantization H | Ψ � = E | Ψ � onto � Quantum gravity | x j � yields a difference equation for c j ’s: Cosmological applications Basic properties 8 mλ 2 (2 c j − c j − 2 − c j +2 ) + 1 1 Example: SHO 2 mω 2 x j c j = Ec j Momentum representation Quantum cosmology what you would get from a simple finite differencing of the � Primordial fluctuations ordinary Schr¨ odinger equation Conclusions � could obtain energy eigenvalues numerically Polymer cosmology – 6 / 26

Momentum representation of the SHO Polymer quantization � easier to work in “momentum eigenstate” basis: Quantum gravity Cosmological ∞ − π 2 λ, π applications � � � e − ipx j | x j � , | p � = p ∈ Basic properties 2 λ Example: SHO j = −∞ Momentum representation Quantum cosmology Primordial fluctuations Conclusions Polymer cosmology – 7 / 26

Momentum representation of the SHO Polymer quantization � easier to work in “momentum eigenstate” basis: Quantum gravity Cosmological ∞ − π 2 λ, π applications � � � e − ipx j | x j � , | p � = p ∈ Basic properties 2 λ Example: SHO j = −∞ Momentum representation wavefunction: Ψ( p ) = � p | Ψ � � Quantum cosmology Primordial fluctuations Conclusions Polymer cosmology – 7 / 26

Momentum representation of the SHO Polymer quantization � easier to work in “momentum eigenstate” basis: Quantum gravity Cosmological ∞ − π 2 λ, π applications � � � e − ipx j | x j � , | p � = p ∈ Basic properties 2 λ Example: SHO j = −∞ Momentum representation wavefunction: Ψ( p ) = � p | Ψ � � Quantum cosmology operators: � p | ˆ U λ | Ψ � = e iλp Ψ( p ) and � p | ˆ Primordial fluctuations x | Ψ � = i � ∂ p Ψ( p ) � Conclusions Polymer cosmology – 7 / 26

Momentum representation of the SHO Polymer quantization � easier to work in “momentum eigenstate” basis: Quantum gravity Cosmological ∞ − π 2 λ, π applications � � � e − ipx j | x j � , | p � = p ∈ Basic properties 2 λ Example: SHO j = −∞ Momentum representation wavefunction: Ψ( p ) = � p | Ψ � � Quantum cosmology operators: � p | ˆ U λ | Ψ � = e iλp Ψ( p ) and � p | ˆ Primordial fluctuations x | Ψ � = i � ∂ p Ψ( p ) � Conclusions projecting eigenvalue equation ˆ H | Ψ � = E | Ψ � onto | p � : � ∂y 2 + sin 2 ( √ gy ) − ∂ 2 E Ψ = ω � � p g = mω Ψ , y = √ mω, M 2 2 g ⋆ Polymer cosmology – 7 / 26

Momentum representation of the SHO Polymer quantization � easier to work in “momentum eigenstate” basis: Quantum gravity Cosmological ∞ − π 2 λ, π applications � � � e − ipx j | x j � , | p � = p ∈ Basic properties 2 λ Example: SHO j = −∞ Momentum representation wavefunction: Ψ( p ) = � p | Ψ � � Quantum cosmology operators: � p | ˆ U λ | Ψ � = e iλp Ψ( p ) and � p | ˆ Primordial fluctuations x | Ψ � = i � ∂ p Ψ( p ) � Conclusions projecting eigenvalue equation ˆ H | Ψ � = E | Ψ � onto | p � : � ∂y 2 + sin 2 ( √ gy ) − ∂ 2 E Ψ = ω � � p g = mω Ψ , y = √ mω, M 2 2 g ⋆ “low energy” quantum states with ∆ y ≪ g − 1 / 2 recover � standard eigenfunctions/energies Polymer cosmology – 7 / 26

Momentum representation of the SHO Polymer quantization the eigenvalue ODE is analytically solvable: � Quantum gravity Cosmological applications Basic properties Example: SHO Momentum representation Quantum cosmology Primordial fluctuations Conclusions odinger quantization for g ≪ 1 recover Schr¨ � Polymer cosmology – 7 / 26

Polymer quantization Quantum cosmology Quantum cosmologies Avoiding the big bang Semiclassical approx Friedmann equation Numerical results Quantum cosmology Primordial fluctuations Conclusions Polymer cosmology – 8 / 26

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨ odinger equations Primordial fluctuations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 14 / 26

The problem Polymer quantization � here we consider an inhomogeneous massless scalar in a de Quantum cosmology Sitter background Primordial fluctuations − dt 2 + a 2 d x 2 � The problem a = exp( Ht ) ds 2 = Quantization algorithms a 2 ( − dη 2 + d x 2 ) a = − ( Hη ) − 1 Schr¨ odinger equations � 1 Effective equations � � 2 a 6 π 2 + 1 d 3 x a 3 2 a 2 ( ∇ φ ) 2 Formal solution H φ = Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 15 / 26

The problem Polymer quantization � here we consider an inhomogeneous massless scalar in a de Quantum cosmology Sitter background Primordial fluctuations − dt 2 + a 2 d x 2 � The problem a = exp( Ht ) ds 2 = Quantization algorithms a 2 ( − dη 2 + d x 2 ) a = − ( Hη ) − 1 Schr¨ odinger equations � 1 Effective equations � � 2 a 6 π 2 + 1 d 3 x a 3 2 a 2 ( ∇ φ ) 2 Formal solution H φ = Initial conditions Power spectrum Semi-analytic results Observational goal: power spectrum of fluctuations P φ ( k ) produced during � consequences inflation assuming polymer quantization Conclusions Polymer cosmology – 15 / 26

The problem Polymer quantization � here we consider an inhomogeneous massless scalar in a de Quantum cosmology Sitter background Primordial fluctuations − dt 2 + a 2 d x 2 � The problem a = exp( Ht ) ds 2 = Quantization algorithms a 2 ( − dη 2 + d x 2 ) a = − ( Hη ) − 1 Schr¨ odinger equations � 1 Effective equations � � 2 a 6 π 2 + 1 d 3 x a 3 2 a 2 ( ∇ φ ) 2 Formal solution H φ = Initial conditions Power spectrum Semi-analytic results Observational goal: power spectrum of fluctuations P φ ( k ) produced during � consequences inflation assuming polymer quantization Conclusions � problem: polymer QFT poorly understood Polymer cosmology – 15 / 26

The problem Polymer quantization � here we consider an inhomogeneous massless scalar in a de Quantum cosmology Sitter background Primordial fluctuations − dt 2 + a 2 d x 2 � The problem a = exp( Ht ) ds 2 = Quantization algorithms a 2 ( − dη 2 + d x 2 ) a = − ( Hη ) − 1 Schr¨ odinger equations � 1 Effective equations � � 2 a 6 π 2 + 1 d 3 x a 3 2 a 2 ( ∇ φ ) 2 Formal solution H φ = Initial conditions Power spectrum Semi-analytic results Observational goal: power spectrum of fluctuations P φ ( k ) produced during � consequences inflation assuming polymer quantization Conclusions � problem: polymer QFT poorly understood N.B.: no a priori relation between H and polymer energy scale � M ⋆ assumed Polymer cosmology – 15 / 26

Quantizing inflationary fluctuations Polymer quantization Quantum cosmology Primordial fluctuations scalar field Hamiltonian The problem H φ = H φ ( φ ( x ) , π ( y )) Quantization algorithms Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 16 / 26

Quantizing inflationary fluctuations Polymer quantization textbook algorithm Quantum cosmology Primordial fluctuations scalar field Hamiltonian The problem H φ = H φ ( φ ( x ) , π ( y )) Quantization algorithms Schr¨ odinger equations promote to operators Effective equations ( φ ( x ) , π ( y )) → (ˆ φ ( x ) , ˆ π ( y )) Formal solution Initial conditions Power spectrum Fourier transform Semi-analytic results ˆ k f k ( η ) e i k · x ˆ φ = � Observational a k + h . c . consequences Conclusions � ˆ φ = 0 ⇒ choose BCs to recover flat QFT in ∞ past P φ ( k ) ∝ | f k | 2 � � � k ≪ Ha Polymer cosmology – 16 / 26

Quantizing inflationary fluctuations Polymer quantization textbook algorithm Quantum cosmology Primordial fluctuations scalar field Hamiltonian The problem H φ = H φ ( φ ( x ) , π ( y )) Quantization algorithms Schr¨ odinger equations promote to operators Effective equations unclear how to represent these ( φ ( x ) , π ( y )) → (ˆ φ ( x ) , ˆ π ( y )) Formal solution operators in polymer picture (see Initial conditions attempts by Ashtekar et al 2003; Power spectrum Kreienbuehl and Husain 2010) Fourier transform Semi-analytic results ˆ k f k ( η ) e i k · x ˆ φ = � Observational a k + h . c . consequences Conclusions � ˆ φ = 0 ⇒ choose BCs to recover flat QFT in ∞ past P φ ( k ) ∝ | f k | 2 � � � k ≪ Ha Polymer cosmology – 16 / 26

Quantizing inflationary fluctuations Polymer quantization textbook algorithm alternative algorithm Quantum cosmology Primordial fluctuations scalar field Hamiltonian The problem H φ = H φ ( φ ( x ) , π ( y )) Quantization algorithms Schr¨ odinger equations promote to operators Fourier transform Effective equations ( φ ( x ) , π ( y )) → (ˆ φ ( x ) , ˆ π ( y )) Formal solution ( φ ( x ) , π ( y )) → ( φ k , π k ′ ) Initial conditions Power spectrum Fourier transform H φ = � k H k ( φ k , π k ) Semi-analytic results ˆ k f k ( η ) e i k · x ˆ φ = � Observational a k + h . c . consequences Conclusions separately quantize each mode � ˆ using QM: i ∂ t ψ k = ˆ φ = 0 ⇒ choose BCs to H k ψ k recover flat QFT in ∞ past � P φ ( k ) ∝ � ψ k | ˆ φ 2 k | ψ k � � P φ ( k ) ∝ | f k | 2 � � k ≪ Ha � where ψ k is “ground state” � k ≪ Ha Polymer cosmology – 16 / 26

Time dependent Schr¨ odinger equations Polymer quantization “Fourier transform then quantize” algorithm leads to following odinger equations (recall a = exp Ht ): Quantum cosmology Schr¨ Primordial fluctuations � standard quantization: The problem � 1 Quantization k − ak 2 ∂ 2 i ∂ � algorithms 2 a 3 π 2 ∂tψ ( t, π k ) = ψ ( t, π k ) ∂π 2 Schr¨ odinger equations 2 k Effective equations polymer quantization: Formal solution � � 1 − ak 2 ∂ 2 Initial conditions i ∂ � λπ k � � 2 λ sin 2 ∂tψ ( t, π k ) = ψ ( t, π k ) Power spectrum ∂π 2 a 3 / 2 2 Semi-analytic results k Observational consequences Conclusions Polymer cosmology – 17 / 26

Time dependent Schr¨ odinger equations Polymer quantization “Fourier transform then quantize” algorithm leads to following odinger equations (recall a = exp Ht ): Quantum cosmology Schr¨ Primordial fluctuations � standard quantization: The problem � 1 Quantization k − ak 2 ∂ 2 i ∂ � algorithms 2 a 3 π 2 ∂tψ ( t, π k ) = ψ ( t, π k ) ∂π 2 Schr¨ odinger equations 2 k Effective equations polymer quantization: Formal solution � � 1 − ak 2 ∂ 2 Initial conditions i ∂ � λπ k � � 2 λ sin 2 ∂tψ ( t, π k ) = ψ ( t, π k ) Power spectrum ∂π 2 a 3 / 2 2 Semi-analytic results k Observational following transformations and re-scaling make things simpler: consequences � Conclusions � H 2 η = − 1 Ha, y = − kη k 3 π k , � 1 / 4 � � H 2 − i y 2 � � ψ ( t, π k ) = − kη Ψ( η, y ) exp k 3 2 kη Polymer cosmology – 17 / 26

Effective Schr¨ odinger equations Polymer quantization After transformations and re-scalings, PDEs governing power Quantum cosmology spectrum are: Primordial fluctuations − ∂ 2 standard quantization: i∂ Ψ ∂η = k � � The problem ∂y 2 + y 2 Ψ � Quantization 2 algorithms Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 18 / 26

Effective Schr¨ odinger equations Polymer quantization After transformations and re-scalings, PDEs governing power Quantum cosmology spectrum are: Primordial fluctuations − ∂ 2 standard quantization: i∂ Ψ ∂η = k � � The problem ∂y 2 + y 2 Ψ � Quantization 2 algorithms Schr¨ odinger equations � ordinary simple harmonic oscillator Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 18 / 26

Effective Schr¨ odinger equations Polymer quantization After transformations and re-scalings, PDEs governing power Quantum cosmology spectrum are: Primordial fluctuations − ∂ 2 standard quantization: i∂ Ψ ∂η = k � � The problem ∂y 2 + y 2 Ψ � Quantization 2 algorithms Schr¨ odinger equations � ordinary simple harmonic oscillator Effective equations Formal solution ground state unambiguous ⇒ gives Bunch-Davies vacuum � Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 18 / 26

Effective Schr¨ odinger equations Polymer quantization After transformations and re-scalings, PDEs governing power Quantum cosmology spectrum are: Primordial fluctuations − ∂ 2 standard quantization: i∂ Ψ ∂η = k � � The problem ∂y 2 + y 2 Ψ � Quantization 2 algorithms Schr¨ odinger equations � ordinary simple harmonic oscillator Effective equations Formal solution ground state unambiguous ⇒ gives Bunch-Davies vacuum � Initial conditions Power spectrum ∂y 2 + sin 2 ( √ gy ) Semi-analytic results − ∂ 2 � � polymer quantization: i∂ Ψ ∂η = k Observational Ψ � consequences 2 g Conclusions Polymer cosmology – 18 / 26

Effective Schr¨ odinger equations Polymer quantization After transformations and re-scalings, PDEs governing power Quantum cosmology spectrum are: Primordial fluctuations − ∂ 2 standard quantization: i∂ Ψ ∂η = k � � The problem ∂y 2 + y 2 Ψ � Quantization 2 algorithms Schr¨ odinger equations � ordinary simple harmonic oscillator Effective equations Formal solution ground state unambiguous ⇒ gives Bunch-Davies vacuum � Initial conditions Power spectrum ∂y 2 + sin 2 ( √ gy ) Semi-analytic results − ∂ 2 � � polymer quantization: i∂ Ψ ∂η = k Observational Ψ � consequences 2 g Conclusions k M ⋆ a = physical wavenumber “polymer coupling”: g = � polymer energy scale Polymer cosmology – 18 / 26

Effective Schr¨ odinger equations Polymer quantization After transformations and re-scalings, PDEs governing power Quantum cosmology spectrum are: Primordial fluctuations − ∂ 2 standard quantization: i∂ Ψ ∂η = k � � The problem ∂y 2 + y 2 Ψ � Quantization 2 algorithms Schr¨ odinger equations � ordinary simple harmonic oscillator Effective equations Formal solution ground state unambiguous ⇒ gives Bunch-Davies vacuum � Initial conditions Power spectrum ∂y 2 + sin 2 ( √ gy ) Semi-analytic results − ∂ 2 � � polymer quantization: i∂ Ψ ∂η = k Observational Ψ � consequences 2 g Conclusions k M ⋆ a = physical wavenumber “polymer coupling”: g = � polymer energy scale late time limit g → 0 : recover standard wave equation � Polymer cosmology – 18 / 26

Effective Schr¨ odinger equations Polymer quantization After transformations and re-scalings, PDEs governing power Quantum cosmology spectrum are: Primordial fluctuations − ∂ 2 standard quantization: i∂ Ψ ∂η = k � � The problem ∂y 2 + y 2 Ψ � Quantization 2 algorithms Schr¨ odinger equations � ordinary simple harmonic oscillator Effective equations Formal solution ground state unambiguous ⇒ gives Bunch-Davies vacuum � Initial conditions Power spectrum ∂y 2 + sin 2 ( √ gy ) Semi-analytic results − ∂ 2 � � polymer quantization: i∂ Ψ ∂η = k Observational Ψ � consequences 2 g Conclusions k M ⋆ a = physical wavenumber “polymer coupling”: g = � polymer energy scale late time limit g → 0 : recover standard wave equation � � time dependent potential makes ground state ambiguous Polymer cosmology – 18 / 26

Formal solution of polymer Schr¨ odinger equation ∞ � Polymer quantization � c n ( η ) e i ǫ n ( η ) dη Ψ n ( η, y ) wavefunction ansatz: Ψ( η, y ) = � Quantum cosmology n =0 Primordial fluctuations Ψ n are instantaneous energy eigenfunctions: � The problem Quantization y + g − 1 sin 2 ( √ gy ) algorithms 1 − ∂ 2 � � 2 k Ψ n ( η, y ) = ǫ n ( η )Ψ n ( η, y ) Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 19 / 26

Formal solution of polymer Schr¨ odinger equation ∞ � Polymer quantization � c n ( η ) e i ǫ n ( η ) dη Ψ n ( η, y ) wavefunction ansatz: Ψ( η, y ) = � Quantum cosmology n =0 Primordial fluctuations Ψ n are instantaneous energy eigenfunctions: � The problem Quantization y + g − 1 sin 2 ( √ gy ) algorithms 1 − ∂ 2 � � 2 k Ψ n ( η, y ) = ǫ n ( η )Ψ n ( η, y ) Schr¨ odinger equations Effective equations Formal solution subbing ansatz into Schr¨ odinger equation gives: � Initial conditions Power spectrum     c 0 a 00 a 01 · · · Semi-analytic results d Observational c 1 a 10 a 11 consequences dg c = Ac , c =  , A =        . . ... Conclusions . . . . where a nm = a nm ( η ) are matrix elements in the { Ψ n } basis Polymer cosmology – 19 / 26

Formal solution of polymer Schr¨ odinger equation ∞ � Polymer quantization � c n ( η ) e i ǫ n ( η ) dη Ψ n ( η, y ) wavefunction ansatz: Ψ( η, y ) = � Quantum cosmology n =0 Primordial fluctuations Ψ n are instantaneous energy eigenfunctions: � The problem Quantization y + g − 1 sin 2 ( √ gy ) algorithms 1 − ∂ 2 � � 2 k Ψ n ( η, y ) = ǫ n ( η )Ψ n ( η, y ) Schr¨ odinger equations Effective equations Formal solution subbing ansatz into Schr¨ odinger equation gives: � Initial conditions Power spectrum     c 0 a 00 a 01 · · · Semi-analytic results d Observational c 1 a 10 a 11 consequences dg c = Ac , c =  , A =        . . ... Conclusions . . . . where a nm = a nm ( η ) are matrix elements in the { Ψ n } basis solve numerically: c ( η = 0) gives final quantum state and � hence power spectrum P φ ( k ) Polymer cosmology – 19 / 26

Initial conditions Polymer quantization suppose we prepare a given k mode in the ground state at an � Quantum cosmology initial time Primordial fluctuations The problem Quantization algorithms Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 20 / 26

Initial conditions Polymer quantization suppose we prepare a given k mode in the ground state at an � Quantum cosmology initial time Primordial fluctuations The problem � standard quantization: it will stay in the ground state Quantization algorithms Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 20 / 26

Initial conditions Polymer quantization suppose we prepare a given k mode in the ground state at an � Quantum cosmology initial time Primordial fluctuations The problem � standard quantization: it will stay in the ground state Quantization algorithms polymer quantization: it will not stay in the ground state � Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 20 / 26

Initial conditions Polymer quantization suppose we prepare a given k mode in the ground state at an � Quantum cosmology initial time Primordial fluctuations The problem � standard quantization: it will stay in the ground state Quantization algorithms polymer quantization: it will not stay in the ground state � Schr¨ odinger equations Effective equations Formal solution we assume each mode is in ground state at the start of inflation � Initial conditions ( c.f. Martin and Brandenberger 2001) Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 20 / 26

Initial conditions Polymer quantization suppose we prepare a given k mode in the ground state at an � Quantum cosmology initial time Primordial fluctuations The problem � standard quantization: it will stay in the ground state Quantization algorithms polymer quantization: it will not stay in the ground state � Schr¨ odinger equations Effective equations Formal solution we assume each mode is in ground state at the start of inflation � Initial conditions ( c.f. Martin and Brandenberger 2001) Power spectrum Semi-analytic results � final quantum state determined by polymer coupling at start of Observational consequences inflation g 0 = k/k ⋆ Conclusions k ⋆ = present day wavenumber of a mode with physical � wavelength M − 1 at the start of inflation ⋆ Polymer cosmology – 20 / 26

Initial conditions Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 20 / 26

Results for the power spectrum Solution for expansion coefficients with g 0 = 20 and M ⋆ /H = 1 : Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions Polymer cosmology – 21 / 26

Results for the power spectrum Polymer quantization Quantum cosmology Primordial fluctuations The problem Quantization algorithms Schr¨ odinger equations Effective equations Formal solution Initial conditions Power spectrum Semi-analytic results Observational consequences Conclusions recover standard result P φ = P 0 = ( H/ 2 π ) 2 for g 0 ≪ 1 � polymer effects vanish for M ⋆ /H → ∞ � Polymer cosmology – 21 / 26

Semi-analytic results Polymer quantization perturbation theory/fitting functions to numeric results lead to � Quantum cosmology approximate power spectrum: Primordial fluctuations 1 + 1 k The problem  , k ≪ k ⋆ Quantization   2 k ⋆ algorithms P φ  ≈ Schr¨ odinger equations � k 2 � 2 M ⋆ �� k ≫ k ⋆ 1 + H P 0 Effective equations sin − 1 ,   k 2 4 M ⋆ H Formal solution M ⋆ ≫ H  ⋆ Initial conditions Power spectrum � 1 / 12 k ⋆ ∼ 3 × 10 − 6 � � e 65 � M ⋆ � � E inf � � 100 Semi-analytic results � Observational 10 16 GeV e N Mpc H G consequences Conclusions N is the number of e -folds of inflation � E inf is the energy scale of inflation � G is the effective number of relativistic species at the end of � inflation Polymer cosmology – 22 / 26