ETSU Astrophysics 3415:
“The Concordance Model in Cosmology:
Should We Believe It?…”
Martin Hendry
Nov 2005
AIM:
To review the current status of
cosmological models and currently
accepted values of cosmological
parameters
What do we mean by the “Concordance Model” anyway?
Since the late 1990s a remarkably (spookily?) consistent picture
has emerged from all sorts of different observations.
(This picture is not particularly simple or elegant!)
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The Universe began with a Big Bang, about 15 billion years ago, and has been
expanding ever since
It has a flat geometry (prediction of inflation)
Energy budget – 1: 30% gravitating matter (a few percent is baryonic, the
rest known as CDM – but we don’t know what that is)
Energy budget – 2: 70% ‘dark energy’ – we really don’t know what that is
but it is now causing the expansion of the Universe to accelerate, and probably
has something to do with the energy of the vacuum.
Energy budget – 3: a few percent probably also comes from massive
neutrinos (but those can’t be the CDM)
Large Scale Structure in the Universe grew from tiny quantum fluctuations
(probably generated during inflation), first seen in the CMBR, under the
influence of gravity.
What do we mean by the “Concordance Model” anyway?
 CDM
 CDM
From Lineweaver (1998)
What’s the problem with the “Concordance Model”?
 Lack of elegance
 Lack of observational evidence
 A theoretician’s headache
 It gives us nothing to argue about!
 The lessons of history…
Modelling the Universe:Background cosmological model described by the
Robertson-Walker metric
2

dr
2
2
2
2
2
ds  dt  R(t ) 
 r d 
2
1  kr

R(t )  cosmic scale factor
R(t )
1

R0
1 z
obs  emit
z
 redshift
emit
Modelling the Universe:Background cosmological model described by the
Robertson-Walker metric
2

dr
2
2
2
2
2
ds  dt  R(t ) 
 r d 
2
1  kr

Metric describes the geometry of the Universe
  1, open

k  curvature constant   0, flat
  1, closed

Closed
Open
  1, open

k  curvature constant   0, flat
  1, closed

Flat
General Relativity:Geometry
matter / energy
“Spacetime tells matter how to move and
matter tells spacetime how to curve”
Einstein’s Field Equations
G
Einstein tensor

R

Ricci tensor
1 
 g R  8 G T 
2
Metric tensor
Curvature scalar
Energy-momentum tensor
of gravitating mass-energy
General Relativity:Geometry
matter / energy
“Spacetime tells matter how to move and
matter tells spacetime how to curve”
Einstein’s Field Equations
Given
g 
R  and R ;

by T
can compute
These are generated
Treat Universe as a perfect fluid
T   (   P)u  u  Pg 
Density
Pressure
Four-velocity
Solve to give Friedmann’s Equations
2

R
8 G k
2
H    
 2
3
R
R

R
4 G
   3P 

R
3
N.B. c  1
Einstein originally sought static solution i.e. :-
R  0 for all t
But if
, P  0
can’t have
  0
R
2

R
8 G k
2
H    
 2
3
R
R

R
4 G
   3P 

R
3
Einstein originally sought static solution i.e. :-
R  0 for all t
But if
, P  0
can’t have
  0
R
However, GR actually gives
G

;
T
Can add a constant times
G

R

g
Covariant derivative


;
0
to
G 
Einstein’s cosmological
constant
1 
 g R  g  
2
Friedmann’s Equations now give:2

R
8 G  k
2
H    
  2
3
3 R
R

R
4 G

   3P  

R
3
3
Can tune
 to give R  0 for all t but unstable
(and Hubble expansion made idea redundant)
Einstein’s
greatest blunder?
Friedmann’s Equations now give:2

R
8 G  k
2
H    
  2
3
3 R
R

R
4 G

   3P  

R
3
3
Can tune
 to give R  0 for all t but unstable
(and Hubble expansion made idea redundant)
But Lambda term could still be non-zero anyway !
Can instead think of Lambda term as added to energymomentum tensor:-
T

But what is
T


g 
 ?…
Particle physics motivates  as energy density of the
vacuum but scaling arguments suggest:-
  (obs)
  (theory)
 10120
So historically it was easier to believe
0
Re-expressing Friedmann’s Equations:For
0
8 G k
H 
 2
3
R
2
1

 8 G 
k 0  
  crit
2 
 3H 
Define
8 G
m  

 crit 3H 2

 
3H 2
It follows that, at any time
m    k  1
k
k   2 2
R H
Re-expressing Friedmann’s Equations:Consider pressureless fluid (dust); assume mass conservation
R   0 R  constant    0
3
and
3
0
R03
 0 1  z 
3
R
3
8 G 0 (1  z )3 H 02
H 02

3
m 



(
1

z
)
m0
crit
3H 2
H 02
H2
More generally:-
ni
1/ 2

ni 
H  H 0   i 0 (1  z ) 
 i

Matter
Radiation
Curvature
Vacuum
Expansion rate dominated by different
terms at different redshifts
3
4
2
0
“Concordance model” predicts:-
k 0  0
 m 0  0.3
But at redshift,
k 2  0
At redshift,
k 6  0
 rad,0  0
  0  0.7
 rad, 2  0
 2  0.08
 rad,6  0
 6  0.007
 rad  0
  0.95
z2
 m 2  0.92
z 6
 m6  0.993
And in about another 15 billion years
k  0
 m  0.05
Value of

1.2

1
0.8
0.6
m
0.4
0.2
0
0
1
Present-day
2
3
4
R / R0
If the Concordance Model is right, we
live at a special epoch. Why?…
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