Backreaction of the infrared modes of scalar fields on de Sitter - PowerPoint PPT Presentation

Backreaction of the infrared modes of scalar fields on de Sitter geometry Gabriel Moreau, Julien Serreau APC, AstroParticule et Cosmologie, Universit´ e Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cit´ e, 10, rue Alice Domon et L´ eonie Duquet, 75205 Paris Cedex 13, France. Based on arXiv:1809.03969, arXiv:1808.00338

Table of Contents Motivations Free scalar field in de Sitter Framework Flow in the infrared limit Results Conclusion

Quantum fields in curved spacetime We do a semi-classical treatment with • a classical background metric • quantum fields as a content → No graviton loops We study the effects of non trivial backgrounds on the dynamic of quantum fields. A well known result : Unruh-Hawking radiation for black hole physics Motivations Backreaction of the infrared modes of scalar fields on de Sitter geometry 2

De Sitter spacetime It is maximally symmetric x 0 We will consider the Expanding dS D Poincar´ e patch • d s 2 = − d t 2 + a 2 ( t ) d � X 2 , a ( t ) = e Ht with constant H. • Conformal time, d η = d t a ( t ) , d s 2 = a 2 ( η ) � − d η 2 + d � X 2 � x 2 → spatially homogeneous. x 1 • Lemaitre-Painlev´ e-Gullstrand d s 2 = − ( 1 − � x 2 ) d t 2 − 2 x 2 , � x · d � x d t + d � → stationary. Motivations Backreaction of the infrared modes of scalar fields on de Sitter geometry 3

Why de Sitter ? It is relevant for inflation History of the Universe Radius of the Visible Universe Cosmic Microwave Background Neutral Hydrogen Forms Nuclear Fusion Begins Nuclear Fusion Ends Modern Universe Protons Formed Inflation Big Bang 0 10 − 32 s 1 µs 0.01 s 3 min 380,000 yrs 13.8 Billion yrs Age of the Universe Inflation is a postulated phase in the history of the univers which answers several fine-tuning problems of the cosmological standard model • the horizon problem • the flatness problem Motivations Backreaction of the infrared modes of scalar fields on de Sitter geometry 4

Particle creation and backreaction Gravitational effects in de Sitter : particle creation Similar effects when you put a quantum field with a constant background field • Schwinger effect : pair creation from vacuum because of an electric field � E • Unruh-Hawking radiation : pair creation because of the horizon of a black hole In both cases : the creation of pairs draws energy from the background source Motivations Backreaction of the infrared modes of scalar fields on de Sitter geometry 5

Stability of de Sitter For scalar field in dS, • Large gravitational effects in the infrared (superhorizon scales) • Infrared modes are amplified • Interactions cannot be treated perturbatively A. A. Starobinsky, J. Yokoyama ’94 ; C. P. Burgess et al. ’10 ; N. C. Tsamis, R. P. Woodard ’05 It is interesting to study the backreaction of these infrared modes fluctuations to test whether de Sitter space is stable under their effects. A. M. Polyakov ’10, ’12 ; E. Mottola ’85 ; I. Antoniadis et al. ’86 ; R. H. Brandenberger et al. ’96 ; Unruh ’98 Motivations Backreaction of the infrared modes of scalar fields on de Sitter geometry 6

Table of Contents Motivations Free scalar field in de Sitter Framework Flow in the infrared limit Results Conclusion

Free scalar field With the action 2 ϕ � ϕ − m 2 d D x √− g � 1 � � 2 ϕ 2 S = � − � + m 2 � ϕ = 0 where We get the Klein Gordon equation � � 1 η + d − 1 ∂ η + � − ∂ 2 ∂ 2 � = X a ( η ) η It gives for the mode decomposition of ϕ d d k � � � � k � � k · � ϕ ( η ,� e i X H ν X ) ∼ a k + h . c . ( 2 π ) d a ( η ) � d 2 4 − m 2 H 2 ≈ d with ν = 2 Free scalar field in de Sitter Backreaction of the infrared modes of scalar fields on de Sitter geometry 7

Free scalar field 2 | H ν ( z ) | 2 1 z 3 1 z z 1 In the case of light scalar fields m ≪ H , amplified fluctuating modes m 0 H k p = a ( η ) Free scalar field in de Sitter Backreaction of the infrared modes of scalar fields on de Sitter geometry 8

Table of Contents Motivations Free scalar field in de Sitter Framework Flow in the infrared limit Results Conclusion

Non perturbative renormalization group A. Kaya ’13 ; J. Serreau ’13 ; M. Guilleux, J. Serreau ’15 � j ˆ � e i W [ j , g ] = ϕ e iS [ ˆ ϕ , g ]+ i ϕ , D ˆ Γ [ ϕ , g ] = W [ j , g ] − j · ϕ with g µν the background metric and S the action for an O ( N ) theory. Add a regulator : it defines a continuum of coarse grained theories � ϕ , g ] = i ϕ ( x ) ˆ ϕ ( y ) , Γ κ [ ϕ , g ] = W κ [ j , g ] − j · ϕ − ∆ S κ [ ϕ , g ] i ∆ S κ [ ˆ x , y R κ ( x , y ) ˆ R κ ( p ) κ 2 Γ κ → ∞ = S Γ κ → 0 = Γ p κ Framework Backreaction of the infrared modes of scalar fields on de Sitter geometry 9

Non perturbative renormalization group 2 We want to solve the flow of Γ κ : it obeys the Wetterich equation , which is IR and UV finite Γ κ = 1 R κ ( Γ ( 2 ) κ + R κ ) − 1 . ˙ 2 tr ˙ C. Wetterich ’93 The physical values for g and ϕ are simultaneously determined at each scale κ through δ Γ κ δ Γ κ G κ T κ � � δϕ = 0 , δ g µν = 0 or µν = µν We take constant values of ϕ and de Sitter spacetime : it gives the flow of the Hubble constant Framework Backreaction of the infrared modes of scalar fields on de Sitter geometry 10

Table of Contents Motivations Free scalar field in de Sitter Framework Flow in the infrared limit Results Conclusion

Zero dimensional theory Under the following assumptions, • Infrared regime ( → local potential approximation) : κ ≪ H κ • Small curvature : m 2 t / l , κ ≪ H 2 κ The theory flows towards a zero dimensional theory ϕ , h )+ κ 2 � � � ϕ e − H − D Ω D + 1 e H − D Ω D + 1 W κ ( j , h ) = ϕ 2 − j · ˆ d N ˆ V in ( ˆ 2 ˆ ϕ with the initial conditions V in that match the microscopic potential, • It coincides with the equilibrium probability distribution in the stochastic formalism A. A. Starobinsky, J. Yokoyama ’94 • It is the effective theory for the scalar field averaged over a Hubble patch at constant values of the field Flow in the infrared limit Backreaction of the infrared modes of scalar fields on de Sitter geometry 11

Flow of the physical quantities Taking as initial conditions � � α − β + λ 2 H 2 ϕ 2 a ) 2 . V in ( ˆ ϕ , h ) = N 8 N ( ˆ where α ∝ Λ and β H 2 ∝ R the cosmological constant and Einstein-Hilbert term in a de Sitter geometry. The minimization of the effective action gives  ϕ κ = � ˆ ϕ �   β + 2 κ 2 �� ˆ � ˆ κ = 4 α λ H 2 ϕ 2 � − ϕ 2 � ϕ 4 � +  N β 2 β N 2  The expectation values are to be computed in the zero dimensional theory. Flow in the infrared limit Backreaction of the infrared modes of scalar fields on de Sitter geometry 12

Table of Contents Motivations Free scalar field in de Sitter Framework Flow in the infrared limit Results Conclusion

Large N case In the large N regime, we can solve everything analytically while keeping the main effects. Define ρ = ϕ 2 a NU κ ( ρ , H ) = V κ ( φ a , H ) and 2 N We wish to solve � � U κ ( ρ , H ) � � ∂ ρ U κ ( ρ , H ) = 0 ∂ H = 0 and � � H 4 � � ρ κ , H κ ρ κ , H κ Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 13

Large N case 2 We get ρ κ = 0, � � � t , κ = − κ 2 β + H 4 κ 2 κ 4 4 + λ H 4 κ = 4 α κ m 2 H 2 2 + 2 Ω , 1 + m 2 β Ω t , κ + κ 2 We have finite asymptotic values which we can compute exactly � � � � 2 ∞ = β Ω 1 − 32 α ≈ 4 α β + 2 � 4 α H 2 1 − + ··· β 2 Ω 4 β Ω β � � � � 2 � 4 α 0 = β Ω 1 − 16 α ≈ 4 α β + 1 H 2 1 − + ··· β 2 Ω β Ω β 2 Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 14

κ Large N case 3 0.413 0.412 0.411 0.410 2 H κ N = 1 0.409 N = 5 0.408 N = ∞ 0.407 0.406 0.0 0.1 0.2 0.3 0.4 0.5 • The superhorizon modes of the massless scalar fields are greatly enhanced, drawing energy from the gravitational field • The dynamical generation of a mass screens this effect, leading to a finite renormalization of the Hubble constant Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 15

Perturbation theory Expansion parameter is λ H 4 κ 4 : perturbation theory breaks down when κ decreases ����� ����� ��� ���� � � κ ����� ��� ���� ������ ����� ����� ����� ��� ��� ��� ��� ��� κ Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 16

κ ρ κ κ Broken phase 2 Ω κ 2 − m 2 + ζ H 2 m 2 + ζ H 2 κ − 2 m 2 � � ρ κ = − H 4 + 2 H 4 κ κ κ κ 4 α − β H 2 , Ω = 0 λ λ 0.208 0.8 exact 0.206 approx. 0.6 2 0.204 H κ 0.4 0.202 0.2 0.200 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 • the symmetry is always restored • the Goldstone bosons do not renormalize h κ ! Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 17