Transcript Backreaction as an explanation for Dark Energy? with some

Backreaction as an explanation for
Dark Energy?
with some remarks on
cosmological perturbation theory
James M. Bardeen
University of Washington
The Very Early Universe 25 Years On
Cambridge, December 17, 2007
3+1 Approach to Cosmological Perturbations
Ref: J. M. Bardeen in Cosmology and Particle Physics, ed. Li-Zhi Fang
and A. Zee (Gordon and Breach 1988)
Background homogeneity and isotropy:
 


Metric: ds 2  N 0 2 dt 2  a 2 dr 2 / 1 Cr 2  r 2 d2 ,
a&
Expansion rate: H 
,
aN 0
Perturbations completely in terms of spatially gauge invariant variables.
First-order perturbed geometry for scalar perturbations:
Lapse N  N 0 1  ,
Spatial curvature 3 Rij 
2C i 
4C  i
i








D
D j ,


j
j
2
2


a
a
Extrinsic curvature
K  K ii  3H   ,
1
1


K ij   ij K    D i D j   ij    ,


3
3
  3&/ N 0  H    .
Total energy-momentum tensor:
Energy density E  E0   ,
Momentum density J i  Di ,
1


Stress tensor S ij  P0    ij   D i D j   ij    .


3
Time gauge transformations:
g
    N 0T  / N 0 , %   H N 0T ,
&  N 0 T ,
%   N 0T , %   3HT
%   E&0T    3H E0  P0 N 0T , %   E0  P0 N 0T ,
%   P&0T   
dP0
3H E0  P0 N 0T , %  .
dE0
Gauge choices:
  0 (synchronous), % 0 (uniform curvature), % 0 (zero shear),
% 0 (uniform expansion), % 0 (uniform energy density), % 0 (comoving).
The gauge transformation to a particular gauge is singular if and when
the coefficient of T in the gauge condition vanishes. If so, the gauge is
badly behaved there.
Gauge-invariant variables:
Combine two or more variables to make a gauge-invariant combination,
e.g.,
 A    &/ N 0 ,  H    H  , Vzs 

E0  P0
 ,
H
 ,  zs    3H E0  P0  ,
E0  P0

dP
  
,     0  , etc.
3E0  P0 
dE0
 c    3H ,  c   
In most cases the physical meaning of a gauge-invariant variable is
gauge-dependent.
Background evolution equations:
H2 
8
C
C
GE0  2 , H& / N 0  4 G E0  P0  2 , E&0  3E0  P0 H.
3
a
a
First-order perturbation equations:
spatial metric evolution &/ N 0  H       / 3;
Einstein equations
3C 




   H  4 G ,
a2 
&/ N 0  2H    H& / N 0 
3C 




     12 G ,
a2 
 4 G   3 ,
1 i 
 i
D
D

 j   &/ N 0  H       8 G   0;

j

3
local energy-momentum conservation
&/ N 0  3H      E0  P0 3H        0,
2
3
&/ N 0  3H  E0  P0       

3C 
   0;
a2 
supplement with other matter and/or field evolution equations as appropriate.
Example of a single scalar field:

dV
,
d
background
   0  ;
&0  g
&0 dV
1 

 3H

 0;
N 0  N 0 
N 0 d
first-order
g
&0  g &
&0
&0
1  & 
&
d 2V
2  

 3H
  



;
N 0  N 0 
N0
d 2
N 0  N 0 
N02
N0
energy-momentum tensor
&0  2
1
E0  
 V,
2  N 0 
&0  2
1
P0  
 V,
2  N 0 
&0&  
&0  2

dV
  2    
,
N0  N0 
d
2
&
&
&




dV
  02   0   
,
N0  N0 
d
&0  2

E0  P0  
;

N


0
&0 

     ,
N 
0
  0.
Solution Strategies
•
There is no particular virtue in using gauge-invariant variables to carry
out a calculation. They do facilitate transforming results from one
gauge to another.
•
Be carefult to avoid gauge singularities, such as arise in the comoving
gauge if and when E0  P0  0. The synchronous gauge is good in this
respect and greatly simplifies the matter/field dynamical equations.
•
Be careful in choosing a gauge and in choosing which of the redundant
Einstein equations and matter evolution equations to use, to ensure
that the the problem is well posed numerically, without near
cancellations between large terms in the equations or in extracting the
physics. This is mainly an issue when k / aH  1.
•
While certainly not the only “conserved quantity” when k / aH  1, the
gauge-invariant variable  introduced in BST is perhaps best suited as
a measure of the overall amplitude of the perturbation. In the long
wavelength limit any change in  is of order the average over an efolding of expansion of the non-adiabatic stress perturbation ( and/or
(k/a)2) divided by E0+P0.
Backreaction and Dark Energy
The Claim (Buchert, Celerier, Rasanen, Kolb et al, Wiltshire, etc.):
The average expansion in a locally inhomogeneous universe
behaves differently than expected from the Friedmann equation based
on the large scale average energy density. Due to the non-linearity of
the Einstein equations spatial averaging and solving the Einstein
equations do not commute (Ellis).
Observations of the CMB radiation indicate that the primordial
amplitude of perturbations, the amplitude of curvature potential
fluctuations, which in a matter-dominated universe correspond to timeindependent fluctuations in the Newtonian potential on scales small
compared to the Hubble radius, is very small, about 10-5. However,
density perturbations grow and become non-linear, first on smaller
scales, and at present on scales the order of 100 Mpc, leading to
formation of structure in the universe. Can non-linearity in the density
cause the average expansion to deviate enough from background
Einstein-deSitter model to convert the Einstein-deSitter deceleration
into the effective acceleration inferred from the high-Z Type Ia
supernovae magnitude-redshift relation?
Counter-arguments (Ishibashi and Wald, Flanagan):
The local dynamics of a matter-dominated universe should be
Newtonian to a good approximation, as long as potential perturbations
and peculiar velocities are non-relativistic, which they are both from
direct observation and as inferred from the CMB anisotropy. Since
Newtonian gravity is linear, averaging and evolution do commute in the
Newtonian limit and should commute to a good approximation in
general relativity. Any relativistic corrections should be much to small
to turn Einstein-deSitter deceleration into an effective acceleration.
In very local regions, where black holes are forming, etc.,
deviations from Newton gravity may be large, but by Birkhoff’s theorem
in GR longer range gravitational interactions should be independent of
the internal structure of compact objects.
Simulations based on local Newtonian dynamics and a global
zero-curvature CDM model with acceleration seem to give a very
good account of all observations of large scale structure as well as the
supernovae data.
The Buchert equations (see Buchert gr-qc/0707.2153):
Exact GR equations constraining the evolution of averaged
quantities assuming a zero-pressure dust energy-momentum tensor.
Averaging is weighted by proper volume on hypersurfaces orthogonal
to the dust worldlines. Define:
aD t   VD t  ,
1/3

D

 t1 
aD
3


MD
2
, QD 
 
VD
3

2
D
2 2
D
Equations:
2
 a&D 
8 G


 a 
3
D
1
1

Q

D
D
6
6


3
R
a&&D
4 G


aD
3
,
D

1 d
1 d
6
2
a
Q

a
D
D
D
aD 6 dt
aD 2 dt
3
R
D
1

QD ,
D
3
 0.
The equations are indeterminate. They say nothing about the
time dependence of QD and whether QD can become large enough to
make the average expansion accelerate. Also, these equations
become invalid once the dust evolves to form caustics, which
generically happens as the density perturbations become large.
.
Lemaitre-Tolman-Bondi (LTB) Models
Spherically symmetric (zero pressure) dust,
ds 2  dt 2  b t, r  dr 2  R t, r  d 2 ,
2
metric
2m r 
R& r 2 k r  
,
R
b
2
R
,
 r 
  1 r 2 k r .
Choose comoving radius coordinate r such that m r  
With u 
3
2
2 3
r .
9
k R / r,
u 1 u 2  sinh 1 u k  0  2 
9 3/2
1 2

3
k t  t 0 r   

u
1

u K  u  1,


1
2

4
5
k  0  3 
sin u  u 1 u
and setting t 0 r   0,
1  R rk   r 9
R 
b  
t
 k   .
  r
k  R 4
r  
2k  rk   r 
 r
R  2k    2
  .
 R
b
R
2
The scalar curvature is
3
Initial Conditions
In cosmological perturbation theory with an Einstein-deSitter background the
primordial amplitude of the curvature potential perturbation in a comoving gauge
is the same as the gauge-invariant amplitude . If the background scale factor is
S(t) = t2/3 consistent with S = R/r as t  0 in the LTB solution,
 
 
4
   2 r 3 k   k  2  .
2
r


r 2S 2
r 2S 2
r
Consider the class of models with
3
R
  a 1 cr
2
1 r  ,
2 2



k r   4a 2  c  3cr 2 1 r 2 ,
0  r  1,
and  r   k r   0, r  1.
With k 0   0 the matter expands more rapidly and becomes underdense near
the center. If c  1 there is an outer region which expands more slowly than
Einstein-deSitter, part of which becomes overdense. If a void develops near the
center, a caustic must eventually develop away from the center sooner or later.
Where the caustic forms, the density becomes infinite and the dust solution
breaks down. Any discontinuity in k r  would imply a caustic or a separation
is present right from t  0, which is why we force continuity at r  1.
Dust Shell Evolution
Once a caustic forms, assume all matter flowing into it stays in an
infinitesmally thin shell.
The shell is characterized by its circumferential radius Rsh as a function of
its own proper time  , its internal "rest mass" msh , and its position at a
given  in the interior and exterior LTB spacetimes, t   , r  . Note


2 3
r  r 3  msh . The Israel junction
9
conditions give the equations
that R  Rsh , but m  m 
2
 m  m 
m  m  msh 
 dRsh 


1



 ,
 m


d 
R
2R
sh
2
2
dmsh 2  r 2 dt  dr r 2 dt  dr 
 

,

d
3    d d   d d 
2
 m  m msh 
2
m



 m
2R 
dr
sh
b

,
d


dR
m  m msh
  sh  R&  
m 
d
2R 
 msh
2
dt 
 dr 
 1  b   .
 d 
d
Application to the Nambu-Tanimoto model (gr-qc/0507057), which is still cited as
evidence for getting acceleration out of backreaction:
Two uniform curvature LTB regions are combined, an inner region (0  r  r0 )
with k(r)  k1  0 and an outer region with k(r)  k2  0. These are embedded
in an EdS model for r  1. As the outer region starts to collapse, they find a
volume average accelerated expansion in the Buchert sense.
Problems:
• Shell crossing starts immediately at t = 0, so the full LTB solution is never valid.
• Assuming a surface layer shell forms at the interface, the outer LTB region is
completely swallowed up by the shell before it starts to recollapse.
• Volume averaging over the LTB regions makes no sense, since most of the
mass ends up in the shell, and a completely empty region opens up between the
outer LTB region and the EdS region. Averaging over the LTB regions has
nothing to do with an average cosmological expansion.
• The shell does start to expand significantly faster than the EdS region once the
outer LTB region is swallowed, but this is a smaller deceleration, not an
acceleration.
• All of the dynamics is Newtonian to a very good approximation once t >> 1. The
LTB regions deviate from EdS expansion only at t >> 1, if |k|r2 << 1.
• Genuinely relativistic back-reaction effects are completely negligible.
Conclusions
•
•
•
•
Exact GR calculations indicate that non-linear backreaction modifying
average expansion rates is completely insignificant in our universe.
Newtonian gravity is a perfectly adequate description of dynamics on
sub-horizon scales (but clearly evident only in a Newtonian gauge).
A close to horizon-scale perturbation close to spherically symmetric
about our location could modify the supernova magnitude-redshift
relation to mimic dark energy, but the primordial perturbation amplitude
would have to be ~ a thousand times larger than than what is seen in
CMB anisotropy (e.g. Biswas, et al 2006, Vanderveld, et al 2006).
Effects of inhomogeneities on light propagation (weak lensing) would in
principle dim distant sources on average, but estimates by Bonvin, et al
(2006) and Vanderveld, et al (2007) indicate that the effect is much too
small to mimic apparent acceleration.
Various modifications of GR remain on the table, but are they any less
contrived than a cosmological constant?