in the primordial universe (wave turbulence) · Øexplosiveinversecascade of wave action /...
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Turbulence of gravitational wavesin the primordial universe
(wave turbulence)Sébastien Galtier
& E. Buchlin, J. Laurie, S. Nazarenko, S. Thalabard
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2017
New detections [see arXiv:1811.12907v1]
LIGOVirgo
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tPlanck ≈ 10-43 s
t ≈ 10-36 s
TODAY
t ≈ 10-32 s
Quantum Gravity
INFLATION
t ≈ 10-10 sCERNE ≈ 102 Gev
E ≈ 1015 Gev
E ≈ 1019 Gev
t ≈ 4 105 yCosmic Microwave Background (CMB)
T ≈ 1032 K
T ≈ 1028 K
T ≈ 1015 K
T ≈ 3000 K
Electroweak transition
GUT
t ≈ 1010 yDark energy dominates
T ≈ 2.7 K
T ≈ 10 K
E ≈ 10-10 Gev
PLANCK TIME
GW production
GW production
Inflation is explained by introducing an hypothetical scalar field called inflaton
Quantum foam (space-time fluctuations)
Heinsenberg’suncertainty principle [Wheeler, PR, 1955]
lPlanck≈ 10-35m
History of the Universe
[Guth, 1981]
Causality disconnected
Causality connectedPhase transition
Phase transition
General Relativity
Nonlinearitiesare important
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The Cosmic Microwave Background (1964) is statistically uniformè horizon (causal) problem is solved with inflation
“So far, the details of inflation are unknown, and the whole idea of inflation remains a speculation, though one that is increasingly plausible.” Weinberg, Cosmology, 2008.
1989-1993 2001-2010 2009-2013
Homogeneousup to 1/10 000
2006
1978
T=2.72548 ± 0.00057 K
[Ijjas+, PLB, 2013; 2014]
We need a convincing physical mechanism for inflation
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Rµn: Ricci tensor Gµn : Christoffel symbol R : Ricci scalar Tµn : stress-energy gµn: metric tensor G = 6.67 10-11 m3kg-1s-2
L : cosmological constant c = 2.99 108 ms-1
Einstein equations
10 nonlinear partial differential equations
[Einstein, SPAW, 1915]
sccccx
a
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Gravitational waves
Exact linear solutions in an empty – flat – Universe:
Effect of a + gravitational wave on a ring of particles
(h = 0.5)
Poincaré-Minkowskispace-time metric
L=0
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Wave Turbulence
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Strong (eddy) turbulence
k
[Saddoughi et al., JFM, 1994]
Eu
Kolmogorov spectrum
Ocean current flows /eddies (Feb. 2005 to Jan. 2006)
[G. Shirah, NASA/GSFC]
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[Kiyani+, Phil. Trans. R. Soc. A, 2015] [David & SG, 2019, in press]
8 decades !!
Alfvén waves
Kinetic Alfvén wavesOblique whistler waves
Solar wind(~ 1 AU)
Strong & Weak Wave turbulence
-7/3 SWT
-8/3 WWT
intermittency
no intermittency
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Weak wave turbulence
Statistical theory of weakly nonlinear waves
2011
For magnetized plasmas see : SG, Nonlin. Proc. Geophys. (2009)
2013
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Weak wave turbulence theory
+ Natural asymptotical closure of the hierarchy of moment equations
+ The kinetic equations admits stationary finite flux solutions
- Finite flux spectra not valid over all k’s ® strong turbulence
- Experiments and dns show some limitations in the predictions
[Benney & Saffman, PRSLA, 1966; Benney & Newell, JMP, 1967]
[Zakharov & Filonenko, DAN, 1966; Kraichnan, PoF, 1967]
[Galtier+, JPP, 2000; Meyrand+, PRL, 2016]
[Morize+, PoF, 2005; Nazarenko, NJP, 2007]
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Capillary wave turbulence
water
F(x,y,z,t): velocity potentialh(x,y,t): free surface
incompressibleirrotational
with: s = g/rwater
g: surface tension (=0.07N/m)
[Zakharov & Filonenko, JAMTP, 1967]
⇒
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Capillary wave turbulence
Canonical variables:
⇒
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Capillary wave turbulence
[Zakharov & Filonenko, JAMTP, 1967]
⇒
We are ready for a statistical development⇒
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Capillary wave (homogeneous) turbulence
[Benney & Saffman, PRSLA, 1966; Benney & Newell, JMP, 1967]
Asymptotic closure:only resonance terms survive
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Capillary wave (homogeneous) turbulence
Kinetic equation
Resonant condition:
[Zakharov & Filonenko, JAMTP, 1967]
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Capillary wave (homogeneous) turbulence
Polarized energy spectrum:
Detailed energy conservation on the resonant manifold
⇒
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Zakharov transformation: [Zakharov & Filonenko, DAN, 1966; Kraichnan, PoF, 1967]
0 p/k
q/k
Two solutions for the energy spectrum
• zero-flux (thermodynamic)
• constant-flux (Kolmogorov-Zakharov)
[Kuznetsov, JETP, 1972]
Exact solutions of the kinetic equations
For three waves
E(k) = C P1/2 k-7/4
E(w) ~w-7/4 Eh(w) ~ w-17/6
1D isotropic (constant flux) spectra:
Direct cascade
[Zakharov & Filonenko, JAMTP, 1967]
k=p+q
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Capillary wave turbulence: experimental results[e.g. Brazhnikov+, EPL, 2002; Falcon+, PRL, 2007; Boyer+, PRL, 2008; Cobelli+, PRL, 2011; Berhanu+, PRE, 2013; Aubourg & Mordant, PRL, 2015; Michel+, PRL, 2017; Berhanu+, JFM, 2018] 3-wave
4-wave
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Back to cosmology
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Weakly nonlinear general relativity
Triadic interactions:
Three-wave interactions in GW turbulence do not contribute![SG & Nazarenko, PRL, 2017]
L=0
We found no contribution on the resonant manifold
+ …
Collinear wave vectors ⇒
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Einstein-Hilbert action:
Diagonal space-time metric:
Lagrangian density:
[Hadad & Zakharov, JGP, 2014]
⇒Give the linear contribution
Weakly nonlinear general relativity L=0
(Lamé coefficients)
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• Dynamical equations given by:
• Vacuum Einstein equations:
Hadad & Zakharov’s theorem (JGP, 2014)
4 equations
7 equations
It’s compatible
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Hamiltonian formalism
Normal variables:
Hamiltonian equation:
(Fourier space)
With R01=R02=R12=0 we find:
4 wave processes
8
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Kinetic equation of GW turbulence
Direct cascade Inverse cascade
Energy Wave actionConstant flux (isotropic) spectra:
finitecapacity
[SG & Nazarenko, PRL, 2017]
® Additional symmetry
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Local approximation: nonlinear diffusion model
Explosive inverse cascade
Initial spectrum (decay simulation)
[SG, Nazarenko, Buchlin & Thalabard, Physica D, 2019]
Constant negative flux
• Rigorous derivation is rare (in MHD it’s possible) [SG & Buchlin, ApJ, 2010]
• Here, it’s a phenomenological model[see also Dyachenko+, Physica D, 1992;
Passot & Sulem, JPP, 2019]
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Anomalous scaling
Self-similar solution of the second kind:
Wave action flux
[Thalabard+, JPAMT, 2015]
[see eg. Connaughton & Nazarenko, PRL, 2004;Nazarenko, JETPL, 2006; Boffetta+, JLTP, 2009]
[SG, Nazarenko, Buchlin & Thalabard, Physica D, 2019]
[SG+, JPP, 2000; Lacaze+, Physica D, 2001][Semikoz & Tkachev, PRL, 1995]
Old subject (weak & strong)
10�18 10�16 10�14 10�12
t⇤ � t
10�1
103
107
1011
1015
1019
1023
k fk f , t⇤ = 1.002167e � 11k f , t⇤ = 1.002168e � 11k f , t⇤ = 1.002169e � 11Slope 3.296
0.0 0.5 1.0t ⇥ 1011
107
1015
1023
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Extrapolation/phenomenology(beyond weak turbulence)
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Big-Bang scenario driven by space-time turbulence ?[SG, Laurie & Nazarenko, 2019]
Formation of a condensate
The growth of h0(t) is interpreted as an expansion of the Universe
critical balance
Inflation appears if the growth in time is fast enough
[Nazarenko & Onorato, 2006; 2007][During+, PRE, 2015] Speculative
scenario
/ Merger of PBH
t ≈ 10-36 s
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Comparison with observations
k
|h(k)|2
Expansion leads to dilution
fossil spectrumfor the CMB
Small fluctuations are treated in the Newtonian limit:
⇒Prediction compatible with the
Harrison-Zeldovich spectrum (ns=1)
k
[Dodelson+, science, 1996]
ns≈0.967Planck is compatible withthe fossil spectrum
|h(k)|2 ~ k-1.033
Latest data[Planck coll., A&A, 2015][Harrison, PRD, 1970; Zeldovich, MNRAS, 1972]
(dT/T ~ dr/r ~ 10-5)
k-1 Evolution from inflation
3
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Present/Future: direct numerical simulations
Pseudo-spectral code
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ConclusionØ Theory of weak GW – space-time – turbulence (4 waves)
Ø Explosive inverse cascade of wave action / anomalous scalingØ The Riemann (4th order) curvature tensor is non-trivial
v Phenomenological (CB) model of inflation (via a condensate)
è a standard model of inflation (‘no’ tuning parameter)è falsifiable predictions (with dns of general relativity)è close to elastic wave turbulence [Hassaini+, 2018]
v Fossil spectrum ~ compatible with CMB data
v Presence of an anomalous scaling in the Planck data (ns<1)
è necessary to explain the structures in the Universeè can this anomalous scaling be explained by turbulence ?
[Semikoz & Tkachev, PRL, 1995 ; Lacaze+, PD, 2001]