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Dark Energy from Backreaction
Thomas Buchert
LMU-ASC
Munich, Germany
& University of Bielefeld, Germany
Collaborations :
Mauro Carfora (Pavia, Italy):
Averaging Riemannian Geometry
Jürgen Ehlers (Golm, Germany):
Averaging Newtonian Cosmologies
George Ellis (Cape Town, South Africa):
Averaging Strategies in G.R.
Toshifumi Futamase (Sendai, Japan):
Averaging and Observations
Akio Hosoya (Tokyo, Japan):
Averaging and Information Theory
I. The Standard Model
II. Effective Einstein Equations
Buchert: GRG 32, 105 (2000) : `Dust’
Buchert: GRG 33, 1381 (2001) : `Perfect Fluids’
III. Dark Energy
from Backreaction
Räsänen:
astro-ph/ 0504005 (2005)
Kolb, Matarrese & Riotto: astro-ph/ 0506534 (2005)
Nambu & Tanimoto:
gr-qc/ 0507057 (2005)
Ishibashi & Wald:
gr-qc/ 0509108 (2005)
…
…
The
Triangle
The Cosmic
Standard
Model
Cosmological
Parameters
Bahcall et al. (1999)
The Concordance Model
0,3
0
0,7
Bahcall et al. (1999)
Simulations of Large Scale Structure
Euclidean
MPA Garching
Sloan Digital Sky Survey–Sample 12
Euclidean
Todai, Tokyo

150000 galaxies
II. Effective Einstein Equations
Averaging the scalar parts
Non-commutativity
The role of information entropy
The averaged equations
The cosmic equation of state
The Idea
Averaged Raychaudhuri Equation
Averaged Hamiltonian Constraint
Generic Domains
t
d2
s=-
dt2
+ gij
dXi
dXj
t aD=
1/3
VR
a(t)
Einstein
Spacetime
gij
Non-Commutativity
Relative Information Entropy
Kullback-Leibler :
S>0
t S > 0 :
Information in the Universe grows
in competition with its expansion
The Hamiltonian Constraint
The Hamiltonian constraint :
Averaged Hamiltonian Constraint :
R + K2 – Kij Kji = 16 G  + 2
< R > + < K2 – Kij Kji > = 16 G <  > + 2
Decompose extrinsic curvature :
Define : <  > = : 3 HD
-Ki J = 1/3  iJ + iJ
Define :
Q = 2/3 < ( - <  >)2 > - 2 < 2 >
The averaged Hamiltonian Constraint
Generalized Friedmann Equation
The Cosmic Quartet
The Cosmic Equation of State
Mean field description
Out-of-Equilibrium States
III. Dark Energy from Backreaction
Kolb et al. 2005 :
Estimates in Newtonian Cosmology
vanishes for periodic boundaries
vanishes for spherical motion
measures deviations from a sphere
is negligible on large scales
Global Integral Properties
of Newtonian Models
Boundary conditions are periodic !
Result : spatial scale 100 Mpc/h
Therefore …
A classical explanation of
Dark Energy through Backreaction
is only conceivable
in General Relativity !
Particular Exact Solutions I
Buchert 2000
However …
What happens,
if the averaged curvature
is coupled to backreaction ?
Particular Exact Solutions II
Buchert 2005 ; Kolb et al. 2005
Global Stationarity

Particular Exact Solutions III
Globally Static Cosmos without 
Buchert 2005
Particular Exact Solutions III
Globally Static Cosmos without 
Global Equation of State :
Particular Exact Solutions IV
Globally Stationary Cosmos without 
Buchert 2005
Particular Exact Solutions IV
Globally Stationary Cosmos without 
Global Equation of State :
Particular Exact Solutions V
Averaged Tolman-Bondi Solution
Nambu & Tanimoto 2005
Particular Exact Solutions VI
Scaling Solutions
Buchert, Larena, Alimi 2006
Cosmic Phase Diagram  = 0
Friedmann
=0
Phantom
quintessence
q
m
Evolution of Cosmological Parameters
today

Conclusions
`Near-Friedmannian’ : no coupling between Q and <R>
Standard Perturbation Theory : Q / V-2 <R> / a-2
`Hard Scenario’ : strong coupling between Q and <R>
Large backreaction out of `near-Friedmannian’ data
`Soft Scenario’ : regional fluctuations of a global
out-of-equilibrium state ( peff / -1/3 eff )
with strong initial expansion fluctuations