Late-time quantum backreaction in cosmology Dra zen Glavan - - PowerPoint PPT Presentation

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Late-time quantum backreaction in cosmology Dra zen Glavan - - PowerPoint PPT Presentation

Aug 18, 2022 296 likes •598 views

Late-time quantum backreaction in cosmology Dra zen Glavan Institute for Theoretical Physics and Spinoza Institute, Center for Extreme Matter and Emergent Phenomena EMME , Science Faculty, Utrecht University MITP, Mainz, 22.06.2015. DG,

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Late-time quantum backreaction in cosmology

Draˇ zen Glavan

Institute for Theoretical Physics and Spinoza Institute, Center for Extreme Matter and Emergent Phenomena EMMEΦ, 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

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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

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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

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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: Gµν = 8πG

N

  • T cl

µν + ˆ

Tµν

  • (1)

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

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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

  • Gµν = 8πG

N

  • T cl

µν

  • +
  • ˆ

Tµν

  • (2)

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

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FLRW space-time

Line element (c = = 1): ds2 = −dt2 + a2(t)dx2 = a2(η)

  • − dη2 + dx2

, dt = adη (3) Hubble rate H = ˙

a a and conformal Hubble rate H = a′ a = aH

Friedmann equations: H a 2 = 8πG 3

  • i

ρi , H′ − H2 a2 = −4πG

  • i

(ρi + pi) (4) Ideal fluids: pi = wiρi, ρi ∼ a−3(1+wi) Dominance of one fluid ⇒ constant ǫ parameter ǫ = − ˙ H H2 = 1 − H′ H2 = 3 2(1 + w) (5)

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 6 / 29

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Evolution of the Universe: history

ǫ = − ˙ H H2 = 1 − H′ H2 (6)

ΕR ΕM ΕI ΕR Η

3 2

2 ΕΗ

Transition between periods fast τ ≪ H.

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 7 / 29

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Evolution of the Universe: hierarchy of scales

Η0 Η1 Η2 0, 2 1 Η Η

Hierarchy of scales H0, H ≪ H2 ≪ H1 For minimal inflation H0 ∼ H, but H0 ≪ H not disallowed

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 8 / 29

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Non-minimally coupled massive scalar: quantization

Action S =

  • dDx L(x) =
  • dDx√−g
  • −1

2gµν∂µφ∂νφ − m2 2 φ2 − 1 2ξRφ2

  • (7)

Canonically conjugate momentum π(x) = ∂L ∂φ′(x) = aD−2φ′(x) (8) Hamiltonian H(η) =

  • dD−

1x aD

2

  • a2−2Dπ2 + a−2(∇φ)2 + m2φ2 + ξRφ2

(9) Canonical commutation relations ˆ φ(η, x), ˆ π(η, y)

  • = iδD−

1(x − 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

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Non-minimally coupled massive scalar: quantization

Heisenberg equations of motion ˆ φ′′+(D−2)H′ ˆ φ′−∇2 ˆ φ+m2 ˆ φ+ξ(D−1)

  • 2H′+(D−2)H2

ˆ φ = 0 (11) Expand in Fourier modes ˆ φ(η, x) = a

2−D 2

  • dD−1k

(2π)

D−1 2

  • eik·xU(k, η)ˆ

b(k) + e−ik·xU ∗(k, η)ˆ b†(k)

  • (12)

Commutation relations ˆ b(k),ˆ b†(q)

  • = δD−

1(k − q)

ˆ b(k),ˆ b(q)

  • = 0 =

ˆ b†(k),ˆ b†(q)

  • (13)
  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 10 / 29

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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 U ′′(k, η) +

  • k2 + M2(η)
  • U(k, η) = 0

(15) M2(η) = m2a2 − 1 4

  • D−2−4ξ(D−1)
  • 2H′+H2

(16) 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 U(k, η) k→∞ − − − → e−ikη √ 2k

  • 1 + O(k−1)
  • (17)
  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 11 / 29

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Non-minimally coupled massive scalar: Energy-momentum tensor

Energy-momentum tensor operator ˆ Tµν = −2 √−g δS δgµν

  • ˆ

φ

= ∂µ ˆ φ ∂ν ˆ φ − 1 2gµνgαβ∂α ˆ φ ∂β ˆ φ + gµνm2 ˆ φ2 + ξ

  • Gµν − ∇µ∇ν + gµν
  • ˆ

φ2 (18) Expectation value in state |Ω diagonal ρQ = a−D (4π)

D − 1 2 Γ( D−

1 2 ) ∞

  • dk kD−

2

  • 2k2|U|2 − 1

2

  • D−2 −4ξ(D−1)
  • H′|U|2

+ 2m2a2|U|2 − 1 2

  • D−2 −4ξ(D−1)
  • H2 ∂

∂η|U|2 + 1 2 ∂2 ∂η2 |U|2

  • (19)
  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 12 / 29

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Non-minimally coupled massive scalar: Energy-momentum tensor

pQ = a2−D (4π)

D − 1 2 Γ( D−

1 2 ) ∞

  • dk kD−

2

  • 2k2

D−1|U|2 − 1 2

  • D−2 −4ξ(D−1)
  • H′|U|2

− 1 2

  • D−2 −4ξ(D−1)
  • H2 ∂

∂η|U|2 + 1 2(1−4ξ) ∂2 ∂η2 |U|2

  • (20)

Goal of perturbative calculation: find if and when ρQ/ρB ∼ 1 ρQ and pQ 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 (R2, (Rµν)2, (Rµναβ)2)

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 13 / 29

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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? (ρQ/ρB ∼ 1?) Check the tendency of the backreaction – to accelerate or decelerate the expansion (ρQ > 1?, wQ?) 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

⇒

  • therwise ’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

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Perturbative computation: choice of initial state

Natural choice for initial state in inflation: adiabatic vacuum - analytic extension of positive-frequency UV expansion U(k, η) = π 4H H(1)

ν

k H

  • (21)

ν =

  • 1

4 + 2(1−6ξ) − m2 H2

I

> 3 2 (22) Problem: IR divergent state! |U|2 ∼ k−2ν IR regulator: comoving IR cutoff k0. To be identified with the Hubble rate at the beginning of inflation H0 (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

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Perturbative computation: evolution of the mode function

U ′′ +

  • k2 + M2

U = 0 (23) UV modes evolve adiabatically, IR highly non-adiabatically. BD mode functions known for constant ǫ-periods (in massive case their expansion in small mass known) Represent the full mode function as expansion in terms of BD ones U(k, η) = α(k)u(k, η) + β(k)u∗(k, η) (24) Bogolyubov coefficients suppressed in UV, α → 1 and β → 0 faster than any power of k for k → ∞ For fast transitions well approximated by sudden transition approximation (limit τ = 0) – calculated from continuity of U and U ′ at the point of transition

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 16 / 29

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Perturbative computation: performing the integrals

Integrals cannot be performed exactly Different approximations available on different intervals (in small k/Hi or Hi/k) Identify intervals where dominant contribution comes from Example: consider one transition from accelerating to decelerating era at some H0 Approximating the integrand in the end amounts to approximating in small ratios of physical scales (e.g. H ≪ H0)

k 1Τ

  • Μ0

Μ

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 17 / 29

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Perturbative computation: Results

What (probably) does not work: massless, minimally coupled scalar

DG, Prokopec, Prymidis, PRD 89 (2014) 024024

massless, non-minimally coupled scalar (ξ <0)

DG, Prokopec, van der Woude, PRD 89 (2014) 024024

What works: Very light massive (m∼H), minimally coupled scalar

Ringeval, Suyama, Takahashi, Yamaguchi, Yokoyama, PRL 105 (2010) 121301

NI ∼1060 for HI ∼104GeV inflation

Aoki, Iso, arXiv:1411.5129 [gr-qc]

NI ∼1012 for HI ∼1013GeV inflation

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 18 / 29

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Perturbative computation: Results

Negative nonminimal coupling leads to growth of backreaction during inflation Constraint on (ξ−NI) parameter space: ρQ/ρB < 1 at the end of inflation

0.5 0.4 0.3 0.2 0.1 20 40 60 80 100 0.057 65.47 Ξ NI

During radiation period backreaction scales away as radiation

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 19 / 29

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Perturbative computation: Results

Backreaction in late-time matter era ρQ = 3H2

I H2

32π2 × e8|ξ|NI |ξ| ×

  • −|ξ| + 1

6 m H

  • 2

(25) Inequality 6|ξ| < (m/H)2 for the CC term to dominate Backreaction becomes large in late-time matter era and has the behavior of a CC! (ρQ/ρB ∼ 1)

0.01 0.02 0.03 0.04 0.05 200 400 600 800 1000 Ξ NI

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 20 / 29

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Self-consistent computation

Backreaction becomes large at late-time matter era – how exactly does it influence the dynamics of the expansion? Full self-consistent solution of the Friedmann equations needed – numerical problem Solve full integro-differential equations: mode function EOM, Friedmann equations with a sum over modes as a source

Suen, Anderson, PRD 35 (1987) 2904

Approximate: dominant contribution to ρQ comes from super-Hubble (deep IR) modes Gradients are negligible for super-Hubble modes and they are populated because of the (long enough) inflationary period – stochastic approximation

Starobinsky, Lect. Notes Phys. 246 (1986) 107

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 21 / 29

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Self-consistent: Stochastic formalism

Split the fields in super-Hubble part + the rest ˆ Φ(t, x) = ˆ φ(t, x) + ˆ φs(t, x) (26) ˆ φ(t, x) =

  • d3k

(2π)3/2 θ(µaH−k)

  • eik·xϕ(k, t)ˆ

b(k) + h.c.

  • (27)

ˆ φs(t, x) =

  • d3k

(2π)3/2 θ(k−µaH)

  • eik·xϕ(k, t)ˆ

b(k) + h.c.

  • (28)

µ ≪ 1. Analogously for conjugate momentum operator ˆ Π(t, x) Field operators satisfy Langevin-type equations

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 22 / 29

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Self-consistent: Stochastic formalism

Easy to close the equations for correlators in linear theory (neglect gradients) E(t) = ˆ φ2(t, x) (29) F(t) = a−3{ˆ φ(t, x)ˆ π(t, x)} (30) G(t) = a−6ˆ π2(t, x) , (31) Equations of motion (M2 = m2 + 6ξ(2−ǫ)H2) ˙ E − F = nE (32) ˙ F + 3HF − 2G + 2M2E = nF (33) ˙ G + 6HG + M2F = nG (34) Field-theoretic computations reproduced for backreaction in perturbative regime Noise sources mostly subdominant

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 23 / 29

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Self-consistent: Results

Energy density and pressure ρQ = 1 2G + 3ξHF + m2 2 E + 3ξH2E (35) pQ = (1−4ξ) 2 G + ξHF − m2 2 (1−4ξ)E + 12ξ2(2−ǫ)H2E − ξ(3−2ǫ)H2E (36) Include this as a source in Friedmann equations and solve numerically Impose initial condition at z = 10 corresponding to ones obtained by considering the Universe today with ΩΛ ≈ 0.68, ΩM ≈ 0.32

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 24 / 29

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Self-consistent: Results

m/H = 0.5, ξ = 0

2 2 4 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 N Ε

Slow roll parameter

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 25 / 29

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Self-consistent: Results

m/H = 0.1, ξ = 0

20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 N Ε

Slow roll parameter

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 26 / 29

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Conclusion & outlook

Quantum backreaction can indeed grow large given enough time and influence the expansion dynamics Particular model exhibiting effects of late-time Universe acceleration – Dark Energy model Tiny mass parameter necessary – can it be generated dynamically? Clustering properties of backreaction ˆ ρ(t, x)ˆ ρ(t, x′)? Different (interacting) models? (screening of the cosmological constant?)

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 27 / 29

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Bonus frame 1: stochastic forces

nE = 1 2π2 µ3a3H4(1−ǫ)|ϕ(k, t)|2

  • k=µaH

(37) nF = 1 2π2 µ3a3H4(1−ǫ) ∂ ∂t|ϕ(k, t)|2

  • k=µaH

(38) nG = 1 2π2 µ3a3H4(1−ǫ)| ˙ ϕ(k, t)|2

  • k=µaH

(39)

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 28 / 29

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Bonus frame 2: correlators

2 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 N eN

Φ2

2 2 4 6 0.15 0.10 0.05 0.00 N fN

Π,Φ

2 2 4 6 0.000 0.002 0.004 0.006 0.008 N gN

Π2

  • D. Glavan (ITF Utrecht)

Late-time quantum backreaction... Mainz, 22.05.2015. 29 / 29