Curvature perturbation spectrum from false vacuum inflation - PowerPoint PPT Presentation

Curvature perturbation spectrum from false vacuum inflation Jinn-Ouk Gong University of Wisconsin-Madison 1150 University Avenue, Madison WI 53706-1390 USA Cosmo 08 University of Wisconsin-Madison, USA 26th August, 2008 Based on JG and M. Sasaki, arXiv:0804.4488[astro-ph]

Introduction Two-point Correlation functions Power spectra Conclusions Outline Introduction 1 Motivation Physical picture Two-point Correlation functions 2 Inflaton field 2-point correlation function Energy density 2-point correlation function Power spectra 3 P Φ P R Conclusions 4 Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions Predictions of slow-roll inflation Scale invariant spectrum P R One of the greatest triumphs of inflation 1 Confirmed by recent observations e.g. WMAP5: n R ≈ 0.96 2 Naturally generated 3 Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

� Introduction Two-point Correlation functions Power spectra Conclusions Predictions of slow-roll inflation Scale invariant spectrum P R One of the greatest triumphs of inflation 1 Confirmed by recent observations e.g. WMAP5: n R ≈ 0.96 2 Naturally generated under the slow-roll approximation 3 true � inflation slow-roll false Inflation = nearly scale invariant P R : NOT necessarily true e.g. false vacuum inflation Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions R c during false vacuum inflation A drunken sailor cannot move in a deep, narrow hole: ˙ φ = 0 R c ∼ H δφ ♥ ˙ φ No preferred rest frame in pure dS space: Meaningless quantity! Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions R c during false vacuum inflation A drunken sailor cannot move in a deep, narrow hole: ˙ φ = 0 R c ∼ H δφ ♥ ˙ φ No preferred rest frame in pure dS space: Meaningless quantity! We DO have a preferred frame φ + µ 2 m 2 eff = m 2 a 2 = m eff ( t ) Breaking the perfect dS phase Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions R c during false vacuum inflation A drunken sailor cannot move in a deep, narrow hole: ˙ φ = 0 R c ∼ H δφ ♥ ˙ φ No preferred rest frame in pure dS space: Meaningless quantity! We DO have a preferred frame φ + µ 2 m 2 eff = m 2 a 2 = m eff ( t ) Breaking the perfect dS phase He can go home for more rum Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions How can we proceed? Situation is OK, but the method is inadequate Full quantum computation goes as: Calculate the inflaton 2-point correlation function 1 G ( x , x ′ ) = 〈 φ ( x ) φ ( x ′ ) 〉 Calculate the energy density 2-point correlation function 2 D ( x , x ′ ) ∼ 〈 δρ ( x ) δρ ( x ′ ) 〉 / ρ 2 ∼ D [ G ( x , x ′ )] Calculate P Φ using ∇ 2 Φ ∼ δρ / ρ 3 � Calculate P R using Φ ∼ R c d η 4 Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions Inflaton field 2-point correlation function Given φ + µ 2 � � V = V 0 + 1 m 2 φ 2 a 2 2 G ( x , x ′ ) =〈 φ ( x ) φ ( x ′ ) 〉 � � H � 2 � ∞ � ds cosh( ν s )1 + p 2cosh s − 2(1 − u ) [2cosh s − 2(1 − u )] 3/2 e − p 2cosh s − 2(1 − u ) = 2 π 0 m 2 , u = r 2 − ( η − η ′ ) 2 , r 2 = | x − x ′ | 2 , ν 2 = 9 � φ µ 2 ηη ′ p = 4 − 2 ηη ′ H 2 Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions Inflaton field 2-point correlation function Given φ + µ 2 � � V = V 0 + 1 m 2 φ 2 a 2 2 Expansion near s ≈ 0 G ( x , x ′ ) =〈 φ ( x ) φ ( x ′ ) 〉 ✏✏✏✏✏✏✏✏ ✶ � � H � 2 � ∞ � ds cosh( ν s )1 + p 2cosh s − 2(1 − u ) ❦ [2cosh s − 2(1 − u )] 3/2 e − p 2cosh s − 2(1 − u ) = 2 π 0 m 2 µ 2 ηη ′ ≫ 1, u = r 2 − ( η − η ′ ) 2 ≫ 1 , r 2 = | x − x ′ | 2 , ν 2 = 9 � φ p = 4 − 2 ηη ′ H 2 Early times Super-horizon separation � H � µηη ′ � 2 � π 2 � 1/2 � r 2 − ( η − η ′ ) η − η ′ � 2 � 3/4 e − µ G ( x , x ′ ) ≈ 2 π 2 � r 2 − � Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions Energy density 2-point correlation function (1/2) Density perturbation on the comoving hypersurface ∇ 2 ( ρ ∆ ) = ∇ 2 � − T 0 + 3 H ∂ i � − T 0 � � 0 i 2-point correlation function of ρ ∆ : D ( x , x ′ ) = ∇ 2 ∇ 2 ρ ∆ ( x ′ ) � � � � �� ρ ∆ ( x ) x x ′ ( t ) f σ ′ α ′ β ′ = f ρµν ( t ) ∂ i ∂ j ′ �� ∂ ρ ∂ α ′ ∂ β ′ G ( x , x ′ ) ∂ σ ′ ∂ µ ∂ ν G ( x , x ′ ) �� � j ′ i ∂ ρ ∂ σ ′ G ( x , x ′ ) ∂ µ ∂ ν ∂ α ′ ∂ β ′ G ( x , x ′ ) � �� �� + with the time dependent coefficients = 1 f 00 j = f 0 j 0 j 2 δ i i i f j 00 j =− δ i i � l δ jk �� j δ kl + 1 � f jkl k δ jl + δ i = a − 2 δ i δ i i 2 Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions Energy density 2-point correlation function (2/2) Useful properties of G ( x , x ′ ): Function of r = | x − x ′ | : G = G ( r ) 1 Anti-symmetric w.r.t. spatial derivative: ∂ x ′ = − ∂ x 2 Symmetric w.r.t. time: G ( r ; t , t ′ ) = G ( r ; t ′ , t ) 3 ··· After some calculations, we find that to leading order � H � 4 16 π ( H η ) 4 ( µη ) 4 ( µ r ) 3 µ 8 e − 2 µ r D ( r , η ) ≈ 2 π Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions 2-point correlation function of Φ in configuration space Φ : gauge invariant curvature perturbation in the Newtonian gauge Poisson equation: ∇ 2 a 2 Φ = − ρ ∆ 2 m 2 Pl Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions 2-point correlation function of Φ in configuration space Φ : gauge invariant curvature perturbation in the Newtonian gauge Poisson equation: ∇ 2 a 2 Φ = − ρ ∆ 2 m 2 Pl Super-horizon separation: ∇ 2 → (2 µ ) 2 D ( x , x ′ ) = 4 m 4 Pl ( H η ) 4 � ∇ 2 ∇ 2 ∇ 2 ∇ 2 x ′ Φ ( x ′ ) � � � �� x Φ ( x ) x x ′ ≈ 4 m 4 Pl ( H η ) 4 (2 µ ) 8 � Φ ( x ) Φ ( x ′ ) � Thus the 2-point correlation function of Φ in configuration space ξ Φ ( x ) ≡〈 Φ ( x ) Φ ( x + r ) 〉 � 4 ( µη ) 4 = π � H ( µ r ) 3 e − 2 µ r 64 2 π m Pl Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions Power spectrum of Φ By inverse Fourier transform P Φ = k 3 � d 3 r ξ Φ ( r ) e − i k · r 2 π 2 We have to integrate dr e − 2 µ r � ∞ j 0 ( kr ): blows up to infinity at r = 0 r 0 Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions Power spectrum of Φ By inverse Fourier transform P Φ = k 3 � d 3 r ξ Φ ( r ) e − i k · r 2 π 2 We have to integrate dr e − 2 µ r � ∞ j 0 ( kr ): blows up to infinity at r = 0 r 0 We are interested in the correlations of 2 points with r ≫ | η |··· The singularity at r = 0 should NOT matter Cutoff scale 1/ µ dr e − 2 µ r dr e − 2 µ r � ∞ � ∞ j 0 ( kr ) → j 0 ( kr ) ≈ − Ei( − 2) r r 0 1/ µ � k � 4 � 3 P Φ ( k ; η ) ≈ − Ei( − 2) � H ( µη ) 4 32 2 π m Pl µ Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions R c and Φ On super-horizon scales the general solution for Φ � η Φ = 3 H (1 + w ) a 2 ( η ′ ) d η ′ 2 C 1 a 2 η i Given Φ , R c is expressed as R c = 2 Φ ′ + (5 + 3 w ) H Φ = C 1 3(1 + w ) H Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong

Introduction Two-point Correlation functions Power spectra Conclusions R c and Φ On super-horizon scales the general solution for Φ � η Φ = 3 H (1 + w ) a 2 ( η ′ ) d η ′ 2 C 1 a 2 η i Given Φ , R c is expressed as R c = 2 Φ ′ + (5 + 3 w ) H Φ = C 1 3(1 + w ) H We need the information of a and 1 + w a : perfect dS expansion a = − 1/( H η ) 1 1 + w : we need to evaluate 〈 ρ + p 〉 = ??? 2 Curvature perturbation spectrumfrom false vacuum inflation Jinn-Ouk Gong