Download GR Cosmology: The Robertson

Relativistic Cosmology
1. Introduction & history of ideas
1500-1700 Stars
Solar System, sun as a star:
Copernicus (1473-1543) Galileo (1564-1642), Kepler (1571-1630), Newton
(1642-1727) …
Steady State: prevention of gravitational collapse …..Newton's
gaffe (missing Gauss Theorem)
Isotropy and Homogeneity
assumed by Leibnitz (1646-1714) to apply to star
distribution, infinite universe
1700-1850
Our galaxy as a rotating disc
John Lambert (1728-1777).
Galaxies
Thomas Wright (1711-1786),
Immanuel Kant (1724-1804), Nebulae = Galaxies, homogeneity
& isotropy at Galaxy distribution level
William + John Herschel (1738-1822): …. no nebulae seen in
region of galactic disc ?!
…Richard Proctor (1837-1888) ... because dust obscures them.
Collapse prevented by angular momentum: Laplace (1749-1827)
THESE ARE NOT A COMPLETE SET OF LECTURE NOTES. USE THIS SPACE!
YOU WILL CERTAINLY NEED TO ADD
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DIAGRAMS AND OTHER ANNOTATIONS
EXPLANATIONS
NOTE OF ILLUSTRATIONS
COMMENTS AS TO DETAILS NOT EXAMINABLE
1850-1918
Relativity
Finite universe without boundaries: Riemann (1826-1866) curved
space
Ernst Mach (1838-1916) 1883: all motion relative
Einstein 1905 Special Relativity, 1915: General Relativity
with cosmological constant (quote Einstein…)
see:
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/General_relativity.html
1919-1939
Expanding Universe
better telescopes + benefit of spectroscopy:
Harold Shapley 1919 determined size of our galaxy and our
position in it from the distribution of globular star clusters
Edwin Hubble (1889-1953) 1925 first determined distances to
nebulae such as M31, proving they were separate galaxies.
Hubble 1929:
Hubble's Law: the universe is expanding
Shift of ideas from Steady State (but not without a fight - Fred
Hoyle) towards Big Bang
1945-1995
Big Bang Physics
1964: microwave technology (inc cryogenics):
CMBR 4K background radiation
- direct relic of Big Bang, redshifted x 104
Cosmology <-> Stellar Physics:
Abundance of Helium <-> Big Bang nucleosynthesis
Age of Universe approx 10 14 Gyr
(Stellar Physics gave way)
1995- Quantitative Cosmology
‘Standard Model’ 2003
Hubble Parameter H= 70 +/- 4 km/s Mpc-1
Age of Universe 13.70.3 G yr
Structure in CMBR
Spatial Curvature  0
4%
Regular Matter (well understood - baryons)
23%
Dark Matter (stuff we don’t understand)
73%
Dark Energy (aka Cosmological Constant – little idea)
2004: There remain dissenting voices! Galaxy clustering
claimed more consistent with Dark Energy replaced by Dark
Matter.
2. Size, Expansion and Age
Cosmological Distances
1 pc = 3.261 ly = 3 1016 m
1.3 pc Sun to alpha Centauri
0.3 kpc thickness of Milky Way
12 kpc radius of Milky Way
0.8 Mpc distance to M31
50 Mpc superclusters
100++ Mpc largest filaments, walls
4000 Mpc Hubble Radius
Age of Universe
Heavy isotope relative abundances
Especially 238U : 235U,
10-20 Gyr (including astrophysics)
White dwarf stars:
t0 > 12.7 Gyr (+/-0.7)
Globular Cluster ages:
CMBR :
11-16 Gyr
13.7 +/- 0.3 Gyr
Hubble Parameter (“Hubble Constant”)
v = H(t) r
H0 = H(now)
Hubble radius (where v  c ) rH  c H
H0 = 100 h km/s Mpc-1
Hubble Telescope 1995,
Cepheid plots:
h = 0.7 +/- 10%
H= 70 +/-7 km/s
Mpc-1
Red-shift measurement:
f emitted observed
1 v / c

 1 z 
f observer emitted
1 v2 / c2
Cosmology interpretation:
expansion of Universe between emission and observation
Linearity of Hubble’s Law
1. Experimental fact
2. Independence of Observer
The expansion law should look the same from all (co-expanding)
observers. This uniquely selects a linear law. (Blackboard…).
3. Relation to the Scale Factor
Cosmological expansion interpretation:
observed
S0
 1 z 
emitted
S (t emitted )
S(t) = Scale Factor of Universe at time t; S0 = value now
Expansion rate
H (t )  S (t ) S (t )
and
H 0  S0 S 0
Deceleration Parameter
S(t )
q(t )  
S (t ) H (t ) 2
and
q0  
S0
S0 H 0
3. The Metric of Space and Time
Euclidean space
Metric
d 2  dx 2  dy 2  dz 2
(dx, dy, dz )  coordinate separations between two nearby points
d = distance between them: its value is independent of
coordinate system used even if the expression looks different. For
example we can express the same metric in spherical polars as
d 2  dr 2  r 2 d 2  r 2 sin 2 d 2
where we have changed the coordinates but not the geometry they
describe.
Curved Space (constant curvature)
The two-sphere is the two dimensional spherical surface
of
the
three
(dimensional)-ball
x2  y2  z 2  S 2
2
2
2
2
x y z S .
Viewed as a two dimensional surface at constant radius in three
dimensions, its metric can be given as
d 2  S 2 (d 2  sin 2 d 2 )
where S is just the (fixed) radius.
However we can also ask what this looks like from the point of
view of someone living in the surface (without awareness of more
than two dimensions).
From the pole (   0 ), ‘radial’ distance away is given by
integrating d at constant  leading to ‘radius’
  S .
One can also trace out a circumference of points at that distance,
and the length of this is given by integrating d w.r.t.  at constant
 giving

C  2S sin   2S sin
S

which is not quite the same as 2 but agrees as  0 .
S
This mismatch fundamental to the space being curved. The
 2  C 
3

curvature is defined from the deficit as
K  lim 
  0  3 
giving K  1 S 2 in this case (and for higher dimensional spheres
also).
History and convenience have led to a slightly different notation
being widely favoured:
define a different ‘radius’ from the circumference by
C
 r  S ,
2
where   sin  and d  cos d   1   2 d .
Then you can express the metric of the 2-sphere as
d 2
2
2
d  S (
  2 d 2 )
2
1  k
where k  1 and  is just
circumference
.
2S
Now we have two ways to generalise.
possibilities for k .
The first is other
k  0 is flat space, that is Euclidean space, slightly disguised by
what amounts to plane polar coordinates. Restoring S  r and
setting k  0 gives d 2  dr 2  r 2 d 2 .
k  1 is a hyperbolic space, the surface x 2  y 2  z 2  S 2 , where
circumference grows faster than radial distance as you go out.
(picture)
The other generalisation is to three dimensional spaces (and higher
if you like!), which just amounts to adding more angular
coordinates.
As we look out from point in three dimensions the direction “in the
sky” is characterised by two angles and the corresponding metric is
given by
 d 2

d 2  S 2 
  2 d 2  sin 2  d 2 
2
 1  k

[The angle  is now just angle from North and NOT to be
confused with our usage above for sin 1  . That confusion is the
price we pay for having understood the two sphere from its
embedding in three flat dimensions.]


The 3-sphere, k  1 , is a very serious candidate for the geometry
of space in our universe, and it would mean that the further away
you look,  , the more:
a circumference drawn around at that distance would be
smaller than 2 and
the area at that distance would be less than 4 2 .
Ultimately when you reached our ‘antipode’ both shrink to zero,
and you have discovered there is only a finite volume to the
Universe (at fixed cosmological time – see later).
Locally it is hard to tell which value of k we have. In terms of
r  S the metric is given by
dr 2
d 2 
 r 2 d 2  sin 2  d 2
2
1  Kr
so it is the curvature K  k 2 which we need to detect, which is
S
2
hard to do for Kr  1 .
The furthest structure we can see today is that of the CMBR, and
its structure implies that
KrH 2  0.02 .
(2003)


Here rH  c
is the Hubble radius (next lecture): the result means
H
that looking so far out as to be seeing back to the beginning of
time, we can scarcely see any curvature of space. The Universe
looks flat.
Special Relativity: Minkowski space-time
Metric
ds 2  c 2 d 2  d 2
 c 2 dt 2  dx 2  dy 2  dz 2
(dt , dx, dy, dz )  time and space coordinate separations between
two events
ds = space-time interval between events: this is independent of
frame of reference & coordinate system.
Depending on the sign of ds 2 it is convenient to interpret it as
Time-like separations:
0  ds 2  c 2 d 2 :
d = proper time (interval) between
events: the time difference in a frame of reference where they
occur at the same place.
Space-like separations:
0  ds 2  d 2 :
d = proper distance between events:
the distance between them in a frame of reference where they
occur at the same time
ds  0 corresponds to null or lightlike separation. Two events
connected (or connectable) by a light signal have this separation.
GR Cosmology: The Robertson-Walker Metric
 d 2

ds 2  c 2 dt 2  S (t )2 
  2 d 2   2 sin 2 d 2 
2
 1  k

This metric (1934) can be constructed from the assumptions that:
the universe is isotropic (in space)
the universe is homogeneous (in space)
spacetime is locally Minkowski-like
and some ideas about time (to follow).
It is clearly a combination of Minkowski-like time signature with a
uniformly curved space, whose scale factor S (t ) varies with time
but not space. The spatial curvature k
varies with time, in
S (t ) 2
magnitude but not in sign.
The Cosmic Time coordinate used is constructed so that for a
"Fundamental Observer" at fixed  ,  ,  , ds 2  c 2 dt 2 , hence the
time coordinate is proper time for such an observer.
All the standard approaches to large-scale Relativistic Cosmology
approximate galaxies as points at fixed  ,  ,  . Then evidently the
spatial separation between them at equal cosmic time simply
inflates with the scale factor S (t ) , giving the Hubble flow (below).
From the point of view of large-scale cosmology, the remaining
problem is how S (t ) varies with time.
Our Particle Horizon
How much of the Universe today could have influenced us in the
past? This can be finite back to the beginning of time, even for
universes without positive curvature.
The useful way to express the answer is as a value of comoving
radius  p (away from us), and this is determined by the condition
that a light ray could have propagated from (time,
space)= (t min ,  p ) to (t 0 ,0) , remembering the convention that
t 0 means time now.
d 2
 0 all
1  k 2
along the ray path, which conveniently leads to the integral
For a light ray, we must have ds 2  c 2 dt 2  S (t ) 2
t0
c
dt

S (t )
p

d
.
1  k 2
We will see later that for simple cosmologies we expect
S (t )  t  where the exponent obeys  <1, then the time integral is
finite. This then leads to a finite particle horizon in the flat and
negatively curved cases.
t min
0
In the positively curved case the "radial" integral has a maximum
value sin 1 (1) =  / 2 , and it is possible that we might see light
which has traversed the Universe! Searches for this have been
made for quasars, the negative result setting
S 0  400h 1 Mpc. (only about 13% of the Hubble Radius)