Early models of an
expanding Universe
Paramita Barai
Astr 8900 : Astronomy Seminar
5th Nov, 2003
Contents

Introduction
 Discuss papers :
– 1922 : Friedmann
– 1927 : Lemaître
– 1932 : Einstein & De Sitter

Present cosmological picture
 Some results
– SN project, WMAP, SDSS
Cosmological foundations

Cosmological principle
– Universe is Homogeneous & Isotropic on large scales
(> 100Mpc)





Universe (space itself) expanding, dD/dt ~ D
(Hubble Law)
Universe expanded from a very dense, hot initial
state (Big Bang)
Expansion of universe – mass & energy content –
explained by laws of GTR  Dynamics of
universe
Structure formation in small scales (<10-100 Mpc)
by gravitational self organization
WHAT IS THE GEOMETRY OF OUR UNIVERSE,
& IT’S CONSEQUENCES ??
Cosmological parameters
R – Scale factor of Universe
 Critical density , C – density to make universe
flat (it just stops expanding)
 Density parameter,  =  / C
 H = Hubble constant = v / r
  = Cosmological Constant (still speculative!!)

– Dark Energy
– Repulsive force, opposing gravity
Curvature of space

Positive curvature –
Closed – contract in
future
–  > C
– >1

Zero curvature – Flat –
stop expansion in future
& stationary
–  = C
– =1

Negative curvature –
Open – expand forever
–  < C
– <1
Timeline
1905 – Einstein’s STR, 1915 – GTR
 1917 – Einstein & De Sitter static cosmological
models with 
 1922 – Friedmann

–
–
First non-static model
Universe contracts / expands (with )
1927 – Lemaître – expanding universe
 1930 – Hubble: expanding universe, Einstein drops
 (“biggest blunder”)
 1932 – Einstein & de Sitter

–
Expanding universe of zero curvature
Timeline – cont’d…
1948 – Particle theory (QED) predicts non zero
vacuum energy , but QED = 10120other
 1965 – CMBR
 Early 1980’s: LUM << C  Open universe
 1980’s –

–
–
Inflation theory  Flat universe (TOT = 1)
Dark matter
1990’s - LUM ~ 0.02-0.04, DARK ~ 0.2-0.4, REST = ?
 1998 – Accelerating universe
 Present model – universe very near to flat (with
matter and vacuum energy)

Two first models of universe:
De Sitter

Matter density = 0
 Advantage :
– Explains naturally
observed radial receding
velocities of extra
galactic objects
– From consequence of
gravitational field

Without assuming we are
at special position

Parameters
– c = velocity of light
–  = Cosmological constant
–  = Density of universe
3c 2
 2
R
 0
M 0
Einstein universe

Non zero matter density
 Relation between density
& radius of universe

Parameters
–  = Einstein constant
= 1.8710-27 (cgs)
 masses much greater
than known in universe at
that time
Can’t explain receding
motion of galaxies
 Advantage

– Explains existence of
matter
2
c
 2
R
M
2
 2
R
4 2

R
Curvature of space
Aleksandr Friedman
Zeitschrift fur Physik
10, 377-386, 1922
Summary

First non static model of universe
 Work immediately not noticed, but found
important later …
 R independent of t :
– Stationary worlds of Einstein & de Sitter

R depends on time only :
– Monotonically expanding world
– Periodically oscillating world
 depending on  chosen
Goal of the paper

Derive the worlds of Einstein & de Sitter
from more general considerations
Assumptions of 1st class

Same as Einstein & de Sitter
1.
Gravitational potentials obey Einstein
field equations with cosmological term
Matter is at relative rest
2.
Assumptions of 2nd class
1.
Space curvature is constant wrt 3 space
coordinates; but depends on time
2.
Metric coefficients: g14, g24, g34 = 0,
suitable choice of time coordinate
Solutions:
Einstein & de Sitter worlds as
special cases
Stationary world
R(x4) = 0

M = M0 = constant
–
–
Cylindrical world
Einstein’s results

M = (A0x4+B0) cos x1
– Transform x4 
– De Sitter spherical
world (M=cos x1)
Non stationary world
R(x4)  0
 M = M(x4)
 But – suitable x4 –
M=1

2
1  dR 
 
2 
c  dt 
 3
R
2
3c
R
A R 
1 R
x
t 
dx  B
c a A  x   x3
3c 2
M
A
6 2

3A
R 3
 > 4c /9A
2

R(>0)
– Increases with t
– Initial value, R = R0
2
R
1
x
t  t0  
dx
c R0 A  x   x 3
3c 2
(>0) at t = t0

R = 0, at t = t
 t = Time since creation
of world
 Monotonic world of
first kind
1
t 
c
R0

0
x
 3
A x  2 x
3c
dx
0 <  < 4c /9A
2

Time since creation of
world, t
t 
R0
1
x
dx

c x  A  x   x3
0
3c 2
2

R increases with t
 Initial R = x0
 x0 & x0 are roots of
equation:
A-x+(x3/3c2) = 0

Monotonic world of
second kind
- <  < 0





R – periodic function
of t
World Period = t
Periodic World
t  if  
Small ,
approximate

x
2 0
x
t  
dx

c 0 A x 
3
x
3c 2
A
M
t 

c
6c
Possible universes of
Friedmann

Monotonic worlds
–  > 4c2/9A2

First kind
– 0 <  < 4c2/9A2


Second kind
Periodic universe
– - <  < 0
Conclusions

Insufficient data to conclude which world our
universe is …
 Cosmological constant,  is undetermined …
21
 If  = 0, M = 5  10 M
– Then, world period = 10 billion yrs
– But this only illustrates calculation
A Homogeneous universe of Constant
Mass & Increasing Radius accounting for
the Radial Velocity of Extra – Galactic
Nebulae
Abbe Georges Lemaître


Annales de la Société
scientifique de Bruxelles,
A47, 49, 1927
English translation in
MNRAS, 91, 483-490,
1931
Summary

Dilemma between de Sitter & Einstein
world models
 Intermediate solution – advantages of both
 R = R(t)
– R(t)  as t 
– Similar differential equation of R(t) as
Friedmann
Summary cont’d.…

Accounted the following:
– Conservation of energy
– Matter density
– Radiation pressure


Role in early stages of expansion of universe
First idea:
– Recession velocities of galaxies are results of expansion
of universe
– Universe expanding from initial singularity, the
‘primeval atom’
Intermediate model

Solution intermediate to Einstein & De
Sitter worlds
– Both material content & explaining recession of
galaxies

Look for Einstein universe
– Radius varying with time arbitrarily
Assumptions of model


Universe ~ Sparsely
dense gas
 Molecules ~ galaxies
– Uniformly distributed
– p = (2/3) K.E.
– Negligible w.r.t energy of
matter

– Density – uniform in
Ignore local
condensation
Radiation pressure of
E.M. wave
– Weak
– Evenly distributed
space, time variable

Internal stresses ~
Pressure


Keep p in general eqn
For astronomical
applications, p = 0
Field equations : conservation
of energy

Einstein field equations
–  = Cosmological Constant (unknown)
–  = Einstein Constant

Total energy change + Work done by
radiation pressure in the expanding universe
=0
Equations: Universe of
constant mass






 = Total density
 = Matter density
 =  - 3p
Mass, M = V = constant
 = constant
 = integration constant
R 2  1


  2  3 4
2
3 R
R
3R
R
t
dR
R 2
3
1 

3R


R2
Existing solutions
 De
Sitter world  Einstein world
=0
=0
=0
R = constant
Lemaître solution

R0 = Initial radius of
universe (from which
expanding)
 R = Lemaître distance
scale at time t
 RE = Einstein distance
scale at t
 For  = 0 &  = 2R0 
 
2
3
RE
1
 2
R0
t  R0 3 
dR
R
R  R0 R  2 R0
R  RE R0
3
2
Solution
R
x 
R  2R0
2
t  R0
 3x  1 
1 x 
C
3 log 
  R0 log 

1

x
3
x

1




Cosmological Redshift

R1, R2 = Radius of Universe at times of
emission & observation of light
 Apparent Doppler effect
v R2

1
c R1

If nearby source,
r = distance of source
R v

R cr
Values Calculated

Einstein radius of
universe: by Hubble
from mean density
– RE = 2.7  1010 pc

If R0 from radial
velocities of galaxies
 R from
– R3 = RE2 R0

From data
– R/R = 0.6810-27 cm-1

R0/R = 0.0465
 R = 0.215RE
= 6  109 pc
 R0 = 2.7  108 pc
= 9  108 LY
Conclusions
Mass of universe –
constant
2. Radius of universe –
increases from
R0 (t = -)
3. Galaxies recede as
effect of expansion
of universe
2 2
1


M R0
1.
R0 

rc
v 3
Advantage of both
Einstein & de Sitter
solutions
Possible universe of Lemaître

Expanding space
Limitations & Further scopes

100 Mt. Wilson
telescope range:
– 5  107 pc = R / 200
– Doppler effect –
3000 km/s
– Visible spectrum
displaced to IR

Why universe expands?
 Radiation pressure does
work during expansion
 expansion set up by
radiation itself
On the relation between Expansion &
mean density of universe
Albert Einstein & Wilhelm de sitter
(Proceedings of the National Academy of Sciences 18, 213 – 214,
1932)
Summary

After Hubble discovered expansion of
universe: Einstein & de Sitter withdrew 
 Expanding universe – without space
curvature
 If matter = C = 3H2/(8G)
– Euclidean geometry
– Flat, infinite universe

Using H0 ~ 10  H0 today
– G (optically visible galaxies) ~ C  Flat space
Motivation

Observational data for curvature
– Mean density
– Expansion
 Universe – non static
Can’t find curvature sign or value
 If can explain observation without
curvature ??

Zero curvature

  to explain finite
mean density in static
universe
 Dynamic universe –
without 

– =0
Line element:
ds 2   R 2 (dx 2  dy 2  dz 2 )  c 2 dt 2

R = R(t)
 Neglect pressure (p)
 Field equation => 2
differential eqns
2
1  dR  1
  
2 
R  cdt 
3
Solutions

From observation
– H - coefficient of
expansion
–  - mean density
From
– H = 500 km sec-1 Mpc-1
or, RB = 2  1027 cm

Get
– RA = 1.63  1027 cm
1 dR
1
h
R cdt
RB
–  = 4  10-28 g cm-3
– Coincide exactly with
2

RA 2
1
2
h  
3

theoretical upper limit of
density for Flat space
RA
2
RB
2
2

3
Confidence limit of solution
H – depends on
measured redshifts
 Density – depends on
assumed masses of
galaxies & distance
scale
 Extragalactic distances

– Uncertain

H2 /  or RA2/RB2 ~
/M
–  = Side of a cube
containing 1 galaxy =
106 LY
– M = average galaxy
mass = 2  1011 M ~
close to Dr. Oort’s
estimate of milky way
mass
Conclusions


 - higher limit
– Correct magnitude
order
Possible to describe
universe without
curvature of 3-D space
 However,
– curvature is
determinable
– More precise data


Fix curvature sign
Get curvature value
Present status of cosmological
model

Search for cosmological parameters determining dynamics
of universe:
– Hubble constant, H0
– TOT = M +  + K
 M = M/C
– Matter (visible+dark)
2
  =  / 3H0
– Vacuum energy
2
2
 K = -k / R0 H0
– Curvature term
– If flat k = 0
Current values

H0
– Hubble key project
– WMAP


H0 = (71  3) km/s/Mpc
M
– Cluster velocity dispersion
– Weak gravitational lens
effect



visible ~ 0.02 – 0.04
dark ~ 0.25
M ~ 0.3


– Energy density of
vacuum
– Discrepancy of > 120
orders of magnitude with
theory
–  ~ 0.7



SN Type Ia
WMAP
Age of universe:
– t0 = 13.7 G yr
SN Type Ia

Giant star accreting onto
white dwarf
 Standard candle
– Compare observed
luminosity with predicted

Far off SN fainter than
expected
  Expansion of
Universe is accelerating
Hubble diagram for SN type Ia
Microwave background
fluctuations

Brightest microwave background
fluctuations (spots): 1 deg across
 Ground & balloon based experiments
– Flat – 15 % accuracy

WMAP
– Measures basic parameters of Big Bang theory
& geometry of universe
– Flat – 2 % accuracy
CMB fluctuation result of
balloon experiment

Result best
matches with
Flat Space
WMAP
Convergence region of -M
WMAP result summary

Light in WMAP picture from 379,000 years after
Big Bang
 First stars ignited 200 million years after Big Bang
 Contents of Universe :
– 4 % atoms, 23 % Cold Dark Matter, 73 % Dark Energy

Data places new constraints on nature of dark
energy (??)
 Fast moving neutrinos do not play any major role
in evolution of structure of universe. They would
have prevented the early clumping of gas in the
universe, delaying the emergence of the first stars,
in conflict with new WMAP data.
WMAP results
H0 = (71  3) km s-1 Mpc-1 (with a margin of error
of about 5%)
 New evidence for Inflation (in polarized signal)
 For the theory that fits WMAP data, the Universe
will expand forever. (The nature of the dark
energy is still a mystery. If it changes with time, or
if other unknown and unexpected things happen in
the universe, this conclusion could change.)

Canonical cosmological
parameters (from WMAP)
TOT = 1.02  0.02
  = 0.73  0.04
 M = 0.27  0.04
 Baryon = 0.044  0.004

– nb (baryon density) = (2.50.1)10-7 cm-3
tUniverse = 13.7  0.2 Gyr
 tdecoupling = (379  8) kyr
 treionization = 180 (+220 – 80) Myr (95% CL)
 H0 = 71 (+ 4 –3) km/s/Mpc

Possible kinds of universe
SDSS Result

Universe made of
– 5% atoms
– 25% dark matter
– 70% dark energy

Neutrinos couldn't be a major constituent of the
dark matter, putting the strongest constraints to
date on their mass
 Data consistent with the detailed predictions of the
inflation model
Galaxy map
Density fluctuations of universe
Fate of our Universe
Flat universe …
 Infinite volume …
 Will expand & stop
some day ...

Thank you all …