averaging inhomogeneous cosmologies thomas buchert, cralyon xii th school of cosmology iesc cargèse...
TRANSCRIPT
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Averaging Inhomogeneous Cosmologies
Thomas Buchert, CRALyon
XIIth School of Cosmology IESC Cargèse September 2014
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Course 1 - Contents
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Newton-Galilei spacetime
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Simulations vs. Lagrangian Perturbation Theory
N-body 1st order LPT
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Volume Averaging
Spatial Average on a compact domain :
Restmass conservation on the domain D
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Commutation Rule
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Euler’s Equation
Euler’s Equation
Lagrangian derivative Kinematical decomposition :
Rate of expansion, shear and vorticity
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Raychaudhuri’s Equation
Raychaudhuri’s Equation
Newtonian field equations :
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Averaging Raychaudhuri’s Equation
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Volume acceleration despite local deceleration
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Averaged Cosmological Equations
Kinematical Backreaction :
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Physical Interpretation
Positive Backreaction : acts accelerating
Negative Backreaction :
acts decelerating
Large / Intermediate / Small Scales : Dominance of Expansion Variance / Shear / Vorticity
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How to treat boundary conditions ?
We can add to the gravitational potential an arbitrary harmonic potential A
i) Asymptotic fall-off conditions or ii) periodic functions
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global, scale-independent background is our choice structures do not interact with the backgroundstructures average out on the background
Integral Properties of Newtonian Models
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Properties of Backreaction in Euclidean Space
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Minkowski Functionals
Integral Properties of an averaging domain
Morphological interpretation of backreaction
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Statistical Analysis of galaxy catalogues
Germ-Grain Model
SDSS - DR7 – LRG Sample 700 Mpc/h :Wiegand, Buchert, Ostermann arXiv: 1311.3661
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Summary
Spatial Average on a mass preserving compact domain delivers scale-dependent acceleration and expansion laws
The extra inhomogeneity term is a consequence of the nonlocal nature of averaging and may lead to
volume acceleration despite local deceleration
Due to the Euclidean geometry of Newtonian spacethis extra term vanishes globally and describes
a scale-dependent morphological cosmic variance around an assumed background
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Course 2 - Contents
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t
1/3aD= VR
Einstein Spacetime
4g = - dt2 + gij dXi dXj
gijt
a(t)
Generalization of the standard cosmological model
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Averaging Einstein’s Equations
Spatial Average on a compact domain :
Restmass conservation on the domain D
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Non-commutativity
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Kullback-Leibler :
Relative Information Entropy
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Acceleration Law :
Expansion Law :
Conservation Law :
Integrability :
Cosmological Equations
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The Cosmic Quartet
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Effective Equations in Friedmannian form :
Effective Scalar Field :`Morphon´ Buchert, Larena, Alimi arXiv: gr-qc / 0606020
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Out-of-Equilibrium States
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G = T
m +
Pm + P
m +
Pm + P
=
=
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Average is friedmannian for :
Locally isotropic models (homogeneous) Special LTB models with homogeneous curvature
Integral Properties of Relativistic Models
Average is non-friedmannian :
generic scaling solutions : n = p relativistic perturbation theory : n = p = -1
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Effective Quintessence
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Unstable Sectors :
Q < 0 and <R> > 0
Q > 0 and <R> < 0
Averaged Cosmologies
Near FRW Cosmologies: Q small
Global Gravitational Instability
Roy, Buchert, Carloni, Obadia arXiv: 1103.1146
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Phase Space for = 0
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Unstable Sectors = 0
DE
DM
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Dark Energy Sector = 0
1/aD
Q > 0 and <R> < 0
1/aD
0
2
1
1/aD
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<R>D - 2 ≈ – 6 HD2
QD ≈ 0 <> ≈ 0 :
Volume-dominance of Voids
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Multi-scale Toy Models
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Volume Partitioning
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Volume Partitioning
D
M
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Volume Partitioning
D
M
D EM= υ
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Acceleration in the Multi-scale Model
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Acceleration in the Multi-scale Model
Wiegand, Buchert arXiv: 1002.3912
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Re-Interpretation of Observations
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Acceleration in the Standard Model
Λa(t) δij 2 t
local acceleration
global acceleration
apparent accelerationa(t) =V (t) ⅓
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Observational Strategies
C
log(1+z)
Template Metrics
Euclid
Larena, Alimi, Buchert, Kunz, Corasaniti arXiv: 0808.1161
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Structure formation and Dark Energy
Roukema, Ostrowski, Buchert arXiv: 1303.4444
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structure formation changes the geometry of the average cosmology
Dark Energy and Dark Matter exist in terms of “curvature energies“ qualitative understanding of the mechanism is completed and it works in the right direction
quantitative understanding in terms of non-perturbative models is in progress reinterpretation of observations !
Conclusions