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

Backreaction of the infrared modes of scalar fields

• n 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 We will consider the Expanding Poincar´ e patch

• ds2 = −dt2 +a2(t)d

X2, a(t) = eHt with constant H.

• Conformal time, dη =

dt a(t),

ds2 = a2(η)

• −dη2 +d

X2 → spatially homogeneous.

• Lemaitre-Painlev´

e-Gullstrand ds2 = −(1− x2)dt2 −2

• x·d
• xdt +d
• x2,

→ stationary.

dSD x0 x1 x2

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

Age of the Universe Radius of the Visible Universe Inflation Protons Formed Nuclear Fusion Begins Nuclear Fusion Ends Cosmic Microwave Background Neutral Hydrogen Forms Modern Universe Big Bang 10 −32s 1 µs 0.01 s 3 min 380,000 yrs 13.8 Billion yrs

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
• f 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 S =

• dDx√−g

1 2ϕϕ − m2 2 ϕ2

• We get the Klein Gordon equation
• −+m2

ϕ = 0 where = 1 a(η)

• −∂ 2

η + d −1

η ∂η + ∂ 2

X

• It gives for the mode decomposition of ϕ

ϕ(η, X) ∼

• ddk

(2π)d

• ei
• k·

XHν

• k

a(η)

• ak +h.c.
• with ν =
• d2

4 − m2 H2 ≈ d 2

Free scalar field in de Sitter Backreaction of the infrared modes of scalar fields on de Sitter geometry 7

Free scalar field 2

1

1 z3 1 z

z |Hν(z)|2

In the case of light scalar fields m ≪ H,

m H p =

k a(η)

amplified fluctuating modes

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

eiW [j,g] =

• D ˆ

ϕeiS[ ˆ

ϕ,g]+i

j ˆ

ϕ,

Γ[ϕ,g] = W [j,g]−j·ϕ with gµν the background metric and S the action for an O(N) theory. Add a regulator : it defines a continuum of coarse grained theories i∆Sκ[ ˆ ϕ,g] = i

• x,y Rκ(x,y) ˆ

ϕ(x) ˆ ϕ(y), Γκ[ϕ,g] = Wκ[j,g]−j·ϕ −∆Sκ[ϕ,g]

p Rκ(p) κ κ2

Γκ→∞ = S Γκ→0 = Γ

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 Γκ : it obeys the Wetterich equation, which is IR and UV finite ˙ Γκ = 1 2 tr ˙ Rκ(Γ(2)

κ +Rκ)−1.

• C. Wetterich ’93

The physical values for g and ϕ are simultaneously determined at each scale κ through δΓκ δϕ = 0, δΓκ δgµν = 0

• r

Gκ

µν =

• Tκ

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 ≪ Hκ
• Small curvature : m2

t/l,κ ≪ H2 κ

The theory flows towards a zero dimensional theory eH−DΩD+1Wκ(j,h) =

• dN ˆ

ϕ e−H−DΩD+1

• Vin( ˆ

ϕ,h)+ κ2

2 ˆ

ϕ2−j· ˆ ϕ

• with the initial conditions Vin 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 Vin( ˆ ϕ,h) = N

• α − β

2 H2

• + λ

8N ( ˆ ϕ2

a)2.

where α ∝ Λ and βH2 ∝ R the cosmological constant and Einstein-Hilbert term in a de Sitter geometry. The minimization of the effective action gives      ϕκ = ˆ ϕ H2

κ = 4α

β + 2κ2 Nβ ˆ ϕ2 −ϕ2 + λ 2βN2 ˆ ϕ4 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 = ϕ2

a

2N and NUκ(ρ,H) = Vκ(φa,H) We wish to solve ∂ρUκ(ρ,H)

• ρκ,Hκ

= 0 and ∂H Uκ(ρ,H) H4

• ρκ,Hκ

= 0

Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 13

Large N case 2

We get ρκ = 0, m2

t,κ = −κ2

2 +

• κ4

4 + λH4 2Ω , H2

κ = 4α

β + H4

κ

βΩ

• 1+

κ2 m2

t,κ +κ2

• We have finite asymptotic values which we can compute exactly

H2

∞ = βΩ

4

• 1−
• 1− 32α

β 2Ω

• ≈ 4α

β + 2 βΩ 4α β 2 +··· H2

0 = βΩ

2

• 1−
• 1− 16α

β 2Ω

• ≈ 4α

β + 1 βΩ 4α β 2 +···

Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 14

Large N case 3

N=1 N=5 N=∞

0.0 0.1 0.2 0.3 0.4 0.5 0.406 0.407 0.408 0.409 0.410 0.411 0.412 0.413

κ

Hκ

2

• 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 λH4

κ4 : perturbation theory breaks down when

κ decreases

• κ

κ

• Results

Backreaction of the infrared modes of scalar fields on de Sitter geometry 16

Broken phase

ρκ = − H4

κ

2Ωκ2 − m2 +ζH2

κ

λ , 4α −βH2

κ − 2m2

m2 +ζH2

κ

• λ

+ 2H4

κ

Ω = 0

0.00 0.02 0.04 0.06 0.08 0.10 0.0 0.2 0.4 0.6 0.8 κ ρκ

exact approx.

0.00 0.02 0.04 0.06 0.08 0.10 0.200 0.202 0.204 0.206 0.208

κ

Hκ

2

• the symmetry is always restored
• the Goldstone bosons do not renormalize hκ!

Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 17

Broken phase 2

The backreaction from the longitudinal mode is more and more important for decreasing N N=1 N=5 N=∞

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.398 0.400 0.402 0.404 0.406 0.408 0.410

κ

Hκ

2 Results Backreaction of the infrared modes of scalar fields on de Sitter geometry 18

Table of Contents

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

Conclusion

The backreaction we studied is influenced by several phenomena :

• The mass generation screens the renormalization of the Hubble

parameter

• Non minimal coupling between the scalar fields and gravitational

field has a non trivial effect on the flow

• Goldstone modes do not contribute

A full non perturbative treatment is needed as the perturbative approach breaks down Perspectives :

• Work in a more general FLRW spacetime

Conclusion Backreaction of the infrared modes of scalar fields on de Sitter geometry 19