Late-time quantum backreaction in cosmology Draˇ zen Glavan Institute for Theoretical Physics and Spinoza Institute, Center for Extreme Matter and Emergent Phenomena EMME Φ , Science Faculty, Utrecht University MITP, Mainz, 22.06.2015. DG, Tomislav Prokopec, Tomo Takahashi in preparation DG, Tomislav Prokopec, Aleksei A. Starobinsky in preparation D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 1 / 29
Outline Physical problem & motivation Theoretical setting – what is quantum backreaction? Model & definition of quantities to calculate Perturbative computation - Calculation scheme - Approximations - Results Self-consistent computation - Stochastic approximation - Results Conclusions & outlook D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 2 / 29
Physical problem – Dark Energy Universe today expanding in an accelerating fashion – unknown physical origin Could be a cosmological constant (CC) (but we think we can calculate it?) New matter content? (70% of total energy density) Modifications of General Relativity on cosmological scales Backreaction from non-linear structures Other effects – quantum backreaction D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 3 / 29
Quantum fluctuations All matter is quantum and exhibits quantum fluctuations Quantum fluctuations carry energy All energy is the source for Einstein’s equation Semiclassical gravity: � � T cl µν + � ˆ G µν = 8 πG T µν � (1) N Quantum correction to the equations of motion descending from the effective action D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 4 / 29
Perturbative vs self-consistent computations Are there any interesting phenomena in cosmology arising from quantum corrections? – solve the full equation self consistently Dark Energy: looking for an effect that becomes important only at late times – it must be small during most of the history of the expansion Solve simpler problem first – backreaction on a fixed FLRW background � � � �� � � ˆ T cl G µν = 8 πG + T µν � (2) µν N Backreaction initially small ⇒ there is a regime where it can safely be treated perturbatively Perturbative computation will determine if there is any effect, in what regimes, and for which ranges of parameters After establishing that backreaction grows to be large – tackle the self-consistent problem – start numerical evolution at late matter era D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 5 / 29
FLRW space-time Line element ( c = � = 1 ): ds 2 = − dt 2 + a 2 ( t ) d x 2 = a 2 ( η ) � − dη 2 + d x 2 � , dt = adη (3) a and conformal Hubble rate H = a ′ Hubble rate H = ˙ a a = aH Friedmann equations: H ′ − H 2 � 2 � H = 8 πG � � ρ i , = − 4 πG ( ρ i + p i ) (4) a 2 a 3 i i Ideal fluids: p i = w i ρ i , ρ i ∼ a − 3(1+ w i ) Dominance of one fluid ⇒ constant ǫ parameter ˙ H 2 = 1 − H ′ H H 2 = 3 ǫ = − 2(1 + w ) (5) D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 6 / 29
Evolution of the Universe: history ˙ H 2 = 1 − H ′ H ǫ = − (6) H 2 Ε � Η � Ε R Ε R 2 Ε M 3 2 Ε I 0 Η Transition between periods fast τ ≪ H . D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 7 / 29
Evolution of the Universe: hierarchy of scales Η 0 Η 1 Η 2 � 1 � � Η � � 2 � 0 , � Η Hierarchy of scales H 0 , H ≪ H 2 ≪ H 1 For minimal inflation H 0 ∼ H , but H 0 ≪ H not disallowed D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 8 / 29
Non-minimally coupled massive scalar: quantization Action 2 g µν ∂ µ φ∂ ν φ − m 2 d D x √− g � − 1 2 φ 2 − 1 � � � d D x L ( x ) = 2 ξRφ 2 S = (7) Canonically conjugate momentum ∂ L ∂φ ′ ( x ) = a D − 2 φ ′ ( x ) π ( x ) = (8) Hamiltonian 1 x a D � � a 2 − 2 D π 2 + a − 2 ( ∇ φ ) 2 + m 2 φ 2 + ξRφ 2 � d D − H ( η ) = (9) 2 Canonical commutation relations � ˆ = iδ D − 1 ( x − y ) , � φ ( η, x ) , ˆ π ( η, y ) � ˆ φ ( η, x ) , ˆ � � � φ ( η, y ) = 0 = ˆ π ( η, x ) , ˆ π ( η, y ) (10) D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 9 / 29
Non-minimally coupled massive scalar: quantization Heisenberg equations of motion φ ′′ +( D − 2) H ′ ˆ φ ′ −∇ 2 ˆ φ + m 2 ˆ � 2 H ′ +( D − 2) H 2 � ˆ ˆ φ + ξ ( D − 1) φ = 0 (11) Expand in Fourier modes d D − 1 k � � � 2 − D ˆ e i k · x U ( k, η )ˆ b ( k ) + e − i k · x U ∗ ( k, η )ˆ b † ( k ) φ ( η, x ) = a 2 D − 1 (2 π ) 2 (12) Commutation relations � ˆ b ( k ) , ˆ b † ( q ) = δ D − 1 ( k − q ) � � ˆ b ( k ) , ˆ � ˆ b † ( k ) , ˆ b † ( q ) � � b ( q ) = 0 = (13) D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 10 / 29
Non-minimally coupled scalar: mode function Wronskian normalization for mode function U ( k, η ) U ′∗ ( k, η ) − U ′ ( k, η ) U ∗ ( k, η ) = i (14) Equation of motion for modes – HO with time-dependent frequency � k 2 + M 2 ( η ) � U ′′ ( k, η ) + U ( k, η ) = 0 (15) M 2 ( η ) = m 2 a 2 − 1 � �� 2 H ′ + H 2 � D − 2 − 4 ξ ( D − 1) (16) 4 Construction of Fock space: ˆ b ( k ) | Ω � = 0 , and creation operators generate the rest; mode function determines the properties of | Ω � State with no condensate: � Ω | ˆ φ | Ω � = 0 Choice of U ( k, η ) not unique! Basic requirements: IR finiteness and reduces to positive-frequency Bunch-Davies (adiabatic) in the UV → e − ikη U ( k, η ) k →∞ � � 1 + O ( k − 1 ) − − − √ (17) 2 k D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 11 / 29
Non-minimally coupled massive scalar: Energy-momentum tensor Energy-momentum tensor operator � − 2 δS φ − 1 ˆ = ∂ µ ˆ φ ∂ ν ˆ 2 g µν g αβ ∂ α ˆ φ ∂ β ˆ � T µν = √− g φ � δg µν � ˆ φ + g µν m 2 ˆ φ 2 + ξ � � φ 2 ˆ G µν − ∇ µ ∇ ν + g µν � (18) Expectation value in state | Ω � diagonal ∞ � a − D � 2 k 2 | U | 2 − 1 � � dk k D − 2 H ′ | U | 2 ρ Q = D − 2 − 4 ξ ( D − 1) D 1 − 2 2 Γ( D − 1 (4 π ) 2 ) 0 � ∂ 2 + 2 m 2 a 2 | U | 2 − 1 H 2 ∂ ∂η | U | 2 + 1 � � ∂η 2 | U | 2 D − 2 − 4 ξ ( D − 1) 2 2 (19) D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 12 / 29
Non-minimally coupled massive scalar: Energy-momentum tensor ∞ � a 2 − D 2 k 2 � D − 1 | U | 2 − 1 � � dk k D − 2 H ′ | U | 2 p Q = D − 2 − 4 ξ ( D − 1) D 1 − 2 2 Γ( D − 1 (4 π ) 2 ) 0 � 2(1 − 4 ξ ) ∂ 2 − 1 H 2 ∂ ∂η | U | 2 + 1 � � ∂η 2 | U | 2 D − 2 − 4 ξ ( D − 1) (20) 2 Goal of perturbative calculation: find if and when ρ Q /ρ B ∼ 1 ρ Q and p Q exhibit standard quartic, quadratic and logarithmic divergences. Dimensional regularization automatically subtracts power-law divergences and logarithmic one has to be absorbed into CC and mass counterterms, and higher-derivative counterterms ( R 2 , ( R µν ) 2 , ( R µναβ ) 2 ) D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 13 / 29
Perturbative computation Compute the backreaction while it is small – neglect its influence on the background dynamics. Background is FLRW consisting of inflationary, radiation, and matter eras Check whether backreaction ever becomes important – when and for which parameters? ( ρ Q /ρ B ∼ 1 ?) Check the tendency of the backreaction – to accelerate or decelerate the expansion ( ρ Q > 1 ?, w Q ? ) Interesting range of parameters: - ξ < 0 ⇒ IR instability for modes (inflation and matter period) , - | ξ | ≪ 1 ⇒ backreaction still perturbative during inflation - ( m/H ) = ( ma/ H ) ≪ 1 ⇒ otherwise ’particle production’ stops and backreaction behaves as a non-relativistic matter fluid D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 14 / 29
Perturbative computation: choice of initial state Natural choice for initial state in inflation: adiabatic vacuum - analytic extension of positive-frequency UV expansion � π � k � 4 H H (1) U ( k, η ) = (21) ν H � 4 + 2(1 − 6 ξ ) − m 2 1 > 3 ν = (22) H 2 2 I Problem: IR divergent state! | U | 2 ∼ k − 2 ν IR regulator: comoving IR cutoff k 0 . To be identified with the Hubble rate at the beginning of inflation H 0 (comparison with explicit matching to pre-inflationary radiation era DG, Prokopec, van der Woude, PRD 89 (2014) 024024 ). D. Glavan (ITF Utrecht) Late-time quantum backreaction... Mainz, 22.05.2015. 15 / 29
Recommend
More recommend
