AS 4022: Cosmology
Hongsheng Zhao
Online notes:
star-www.st-and.ac.uk/~hz4/cos/cos.html
take your own notes (including blackboard lectures
AS 4022
Cosmology
Look forward
Malcolm S. Longair’s “Galaxy Formation” 2nd edition [Library]
Chpt 1-2,5-8: expanding metrics, energy density,
curvature, distances
Chpt 4,11,15,20: DM, Structure growth, inflation
Chpt 9-10,13: Thermal History of Particle Reaction,
Neutrinos, WIMPs
Text (intro): Andrew Liddle: Intro to Modern Cosmology
(advanced): John Peacock: Cosmological Physics
Web Lecture Notes: John Peacock, Ned Wright
AS 4022
Cosmology
Why Study Cosmology?
• Fascinating questions:
– Birth, life, destiny of our Universe
– Hot Big Bang --> ( 75% H, 25% He ) observed in stars!
– Formation of structure ( galaxies … )
• Technology -> much recent progress:
– Precision cosmology: uncertainties of 50% --> 2%
• Deep mysteries remain:
– Dark Matter? Dark Energy? General Relativity wrong?
• Stretches your mind:
– Curved expanding spaces, looking back in time, …
AS 4022
Cosmology
Observable Space-Time and
Bands
See
What is out there? In all Energy bands
Pupil  Galileo’s Lens  8m telescopes  square km arrays
Radio, Infrared  optical  X-ray, Gamma-Ray (spectrum)
COBE satellites  Ground  Underground DM detector
Know
How were we created? XYZ & T ?
Us, CNO in Life, Sun, Milky Way, … further and further
 first galaxy  first star  first Helium  first quark
Now  Billion years ago  first second  quantum origin
AS 4022
Cosmology
The Visible Cosmos:
a hierarchy of structure and motion
“Cosmos in a computer”
AS 4022
Cosmology
Observe A Hierarchical Universe
Planets
moving around stars;
Stars grouped together,
moving in a slow dance around the center of galaxies.
AS 4022
Cosmology
Cosmic Village
The Milky Way and Andromeda galaxies,
along with about fifteen or sixteen smaller galaxies,
form what's known as the Local Group of galaxies.
The Local Group
sits near the outer edge of a supercluster, the Virgo cluster.
the Milky Way and Andromeda are moving toward each other,
the Local Group is falling into the middle of the Virgo cluster, and
the entire Virgo cluster itself,
is speeding toward a mass
known only as "The Great Attractor."
AS 4022
Cosmology
Hubble Deep Field:
At faint magnitudes,
we see thousands of
Galaxies for every
star !
~1010 galaxies in the
visible Universe
~1010 stars per
galaxy
~1020 stars in the
visible Universe
AS 4022
Cosmology
Galaxies themselves
some 100 billion of them in the observable universe—
form galaxy clusters bound by gravity as they journey through the
void.
But the largest structures of all are superclusters,
each containing thousands of galaxies
and stretching many hundreds of millions of light years.
are arranged in filament or sheet-like structures,
between which are gigantic voids of seemingly empty space.
AS 4022
Cosmology
Cosmology
Today
Accelerating/Decelerating Expansion
AS 4022
Cosmology
Introducing Gravity and DM
(Key players)
These structures and their movements
can't be explained purely by the expansion of the universe
must be guided by the gravitational pull of matter.
Visible matter is not enough
one more player into our hierarchical scenario:
dark matter.
AS 4022
Cosmology
Cosmologists hope to answer these questions:
How old is the universe? H0
Why was it so smooth? P(k), inflation
How did structures emerge from smooth? N-body
How did galaxies form? Hydro
Will the universe expand forever? Omega, Lamda
Or will it collapse upon itself like a bubble?
AS 4022
Cosmology
Looking Back in Time
• Blah
AS 4022
Cosmology
main concepts in cosmology
Expansion & Metric
Cosmological Redshift
Energy density
AS 4022
Cosmology
A few mins after New Year
Celebration at Trafalgar Square
He
He
Homogeneous
Isotropic Universe
Walking  Elevating  Earth Radius Stretching R(t )
AS 4022
Cosmology
Feb 14 t=45 days later
D2
D3
D1
C1
C2
C3
d
B1
A1
 R(t)
d
B2
A2
B3
dl 2 = [R(t )dc ] + [R(t ) sin cdf ]
2
A3
AS 4022
Cosmology
A1 - B2
2
1st concept Metric:
ant network on expanding sphere
3.
2
4
Angle
χ1
AS 4022
Cosmology
3
2
4
1
Angle
φ1
1
Expanding Radius
R(t)
Stretch of photon wavelength in
expanding space
Emitted with intrinsic wavelength λ0 from Galaxy A at
time t<tnow in smaller universe R(t) < Rnow
 Received at Galaxy B now (tnow ) with λ
λ / λ0 = Rnow /R(t) = 1+z(t) > 1
AS 4022
Cosmology
Redshift
• Expansion is a stretching of space.
• The more space there is between you and a galaxy,
the faster it appears to be moving away.
• Expansion stretches the wavelength of light,
causing a galaxy’s spectrum to be REDSHIFTED:
STATIONARY:
DOPPLER SHIFT:
REDSHIFT:
REDSHIFT IS NOT THE SAME AS DOPPLER SHIFT
2nd main concept: Cosmological
Redshift
The space/universe is expanding,
Galaxies (pegs on grid points) are receding from each other
As a photon travels through space, its wavelength
becomes stretched gradually with time.
Photons wave-packets are like links between grid points
This redshift is defined by:
l - lo
zº
lo
l
= 1+ z
lo
AS 4022
Cosmology
Galaxy
Redshift
Surveys
z = 0.3
Large Scale
Structure:
Empty voids
~50Mpc.
Galaxies are in
1. Walls between
voids.
2. Filaments where
walls intersect.
3. Clusters where
filaments intersect.
Like Soap
Bubbles !
AS 4022
Cosmology
2dF galaxy redshift
survey
z=0
E.g. Consider a quasar with redshift z=2. Since the time the
light left the quasar the universe has expanded by a factor of
1+z=3. At the epoch when the light left the quasar,
the distance between us and Virgo (presently 15Mpc) was 5Mpc.
the CMB temperature then (presently 3K) was 9K.
the quasar appears receding with Doppler speed V=2c.
The quasar appears at a look-back “distance” d = [t(0)-t(z)] c, where the
look-back time [t(0)-t(z)] ~ z/H0 at small z<<1, H0-1=Taylor expansion coeff.
lnow
1+ z =
(wavelength)
l (t )
Rnow
=
(expansion factor)
R(t )
T (t )
=
(Photon Blackbody T µ 1/ l , why ?)
Tnow
AS 4022
Cosmology
Universal Expansion
Hubble’s law appears to violate
The Copernican Principle.
Are we at a special location?
Milky Way
Is everything moving away from us?
Universal Expansion
Q : What is so special about our location ?
A : Nothing !
Me
You
According to Hubble’s Law:
v
I see:
You see:
3v
2v
v
v
We all see the same Hubble law expansion.
2v
3v
v
The Universal Expansion
• An observer in any galaxy sees all other galaxies moving
away, with the same Hubble law.
• Expansion (or contraction) produces a centre-less but
dynamic Universe.
AS 4022
Cosmology

Hubble Law  typical age
(at z=1)
V  H0 d
d
1
t0  
V H0
 1 Mpc 3 1019 km 1 yr 
 


7 
72 km/s  Mpc 3 10 s 
13 10 9 yr 13 Gyr.
d
Slope = c

Convert H as a
frequency Hertz,
find an integer
close to log10(H)?
Multiple choices
-30, -10, 0, 10, 30.
Look back t0-t
3rd concept: The changing rate of
expansion
Newtonian Analogy:
Consider a sphere of radius R(t) ,
 effective mass inside M = 4πρ R3/3
if energy density inside is e=ρ(t) c2.
On this expanding sphere, a test m
Kin.E. + Pot.E. = const Energy
 m (dR/dt)2/2 – G m M/R = (-k/2)m c2
(dR/dt)2 /(2c2) – (4πG/3)ρR2/c2 =-k/2
Unitless cst k <0, =0, >0  3 geometries
Newtonian expansion satisfies
H2 = (dR/dt/R)2 = (ρ + ρcur) (8πG/3)
the cst k absorbed in “density”
ρcur (t) = -k(cH0-1/R)2 (3H02/8πG)
~ -k R-2 * cst
Now H = H0
AS 4022
Cosmology
4th Concept: The Energy density of Universe
The Universe is made up of three things:
VACUUM
MATTER
PHOTONS (radiation fields)
The total energy density of the universe is made up
of the sum of the energy density of these three
components.
e (t ) = e vac + e matter + e rad
From t=0 to t=109 years the universe has expanded
by R(t).
AS 4022
Cosmology
Energy Density of expanding box
volume R 3
N particles
particle mass m
momentum p
2
p
energy

E  h  m 2c 4  p2c 2  m c 2 
 ...
2m
Radiation: ( m = 0 )
Hot neutrino: ( p >> mc > 0 )
Cold gas or neutrino ( p << mc)
  R (wavelengths stretch) :
E  m c 2  const
hc
1
E

h



R

N m c2
3
eM 
R
3
N h
R
e 
 R4
R
R3
COBE spectrum of CMB
A perfect Blackbody !
No spectral lines -- strong test of Big Bang.
Expansion preserves the blackbody spectrum.
T(z) = T0 (1+z)
AS 4022
Cosmology
T0 ~ 3000 K
z ~ 1100
Acronyms in Cosmology
• Cosmic Background Radiation (CBR)
– Or CMB (microwave because of present temperature 3K)
– Tutorial: Argue about 105 photons fit in a 10cmx10cmx10cm
microwave oven. [Hint: 3kT = h c / λ ]
• CDM/WIMPs: Cold Dark Matter, weakly-interact
massive particles
– If DM decoupled from photons at kT ~ 1014K ~ 0.04 mc2
– Then that dark particles were
– non-relativistic (v/c << 1), hence “cold”.
– And massive (m >> mproton =1 GeV)
AS 4022
Cosmology
Eq. of State for Expansion
& analogy of baking bread
Vacuum~air holes in bread
Matter ~nuts in bread
▲►
▼◄
▼▼
l
ll
Photons ~words painted
▲►
▼▼
▼◄
Verify expansion doesn’t
change Nhole, Nproton, Nphoton
No Change with rest energy of a
proton, changes energy of a
photon
AS 4022
Cosmology
ll
e (t ) = r eff (t )c 2
e (t )
= r eff (t )
2
c
VACUUM ENERGY:
r = constant
Þ Evac µ R3
MATTER:
r R3 = constant, Þ m » constant
RADIATION:number of photons Nph = constant
Þ n ph »
AS 4022
N ph
R
3
Cosmology
Wavelength stretches : ~ R
hc 1
Photons:E  h  ~
 R
hc 1
 e ph ~ n ph  ~ 4
 R
Total Energy Density rc2 e
e µ e vac + e matter + e ph
is given by:
µR
µR
0
µ R -3
loge
Radiation
Dominated
Matter
n=-4
Dominated Vacuum
Dominated
n=-3
n=0
R
AS 4022
Cosmology
-4
Tutorial: Typical scaling of
expansion
2/R2=8πG (ρ + ρ + ρ + ρ )/3
H2=(dR/dt)
cur
m
r
v
2
2
R 4p G r R
Assume -domination= cst
by a component ρ ~ R-n
2
3
Show Typical Solutions Are
r µ R µt
n = 2(curvature constant dominate)
n = 3(matter dominate)
n = 4(radiation dominate)
n ~ 0(vaccum dominate) : ln( R) ~ t
-n
-2
Tutorial: Eternal Static (R=cst)
and flat (k=0) Universe
Einstein introduced 
Vacuum
Dominated
to enable an eternal static universe.
8 G r    2
2
R  
R

k
c



3
Critical
Empty
SubCritical
Cycloid
2
2
3
k
c
R  0   
 8 G r
2
R
Einstein' s biggest blunder. (Or, maybe not.)
Static models unstable.
Fine tuning.
R
t
Density - Evolution - Geometry
H2 - (8πG/3) ρ = -k R-2, where H=(dR/dt/R)
r  rc
R(t)
t
r  rc
Flat
k=0
r  rc
Closed
k = +1
3 H0 2
rc 
8 G
AS 4022
Cosmology
Open
k = -1
11
1.4
10
Msun
26
3
 10 kg m 
(Mpc) 3
Isotropic Expansion
V (km/s)
Hubble law :
V  H0 d
Hubble "constant":
H0  500 km s Mpc
1
1
d (Mpc)
Why WRONG? Extinction by interstellar dust was not
then known, giving incorrect distances.
E.g.,: Empty Universe
without vacuum
Vacuum
Dominated


R2  8 G r    R 2  k c 2


3
Critical
Set r  0,   0. Then R  k c
 k  1 ( negative curvature )
R  c, R  c t
R
R 1
H 
R t
R
1
age : t0  0 
c
H0
2
Empty
2
SubCritical
t
Negative curvature drives
rapid expansion/flattening
AS 4022
Cosmology
Cycloid
Cosmic Distance Ladder
Hubble Law
Galaxies
Type Ia Supernovae
HST
Cepheids
Hipparcos
Parallax
Distance
H0 from the HST Key Project
H 0  72  3  7 km s Mpc
-1
Freedman, et al.
2001 ApJ 553, 47.
AS 4022
Cosmology
1
Re-collapse or Eternal Expansion ?
Inflation => expect FLAT
GEOMETRY
CRITICAL
DENSITY
Size of Universe R(t)
Empty
Open
Flat
1/H0
Closed
Now
Time
AS 4022
Cosmology
BIG CRUNCH !
Hubble Parameter Evolution -- H(z)
2
2




R
8

G

k
c
H 2    
r  2
3
3 R
R 
2
H2
k
c
4
3
2


x


x



x
R
M

H02
H02 R0 2
evaluate at x = 1
k c2
 1  0  2 2
H 0 R0
Dimensionless Friedmann Equation:
H2
4
3
2


x


x



(1


)
x
R
M

0
H0 2
x 1  z  R0 R
3 H02
rc 
8 G
rM
rR
M 
, R 
rc
rc
r

 

rc 3 H 0 2
 0   M  R   
Curvature Radius today:
c
R0 
H0
AS 4022
Cosmology
k  1

k
 k  0  
 0 1
k  1

0 1
0 1
0 1
Density
determines
Geometry
Possible
Universes
Vacuum
Dominated
km/s
H 0  70
Mpc
M ~ 0.3
 ~ 0.7
R ~ 8105
  1.0
AS 4022
Cosmology
Critical
Empty
Sub-Critical
Cycloid
Precision Cosmology
h  71  3
  1.02  0.02
 b  0.044  0.004
 M  0.27  0.04
   0.73  0.04
expanding
flat
baryons
Dark Matter
Dark Energy
t 0  13.7  0.2 10 9 yr
6
t  180 220
10
yr
80
t R  379  110 3 yr
now
z  20 10
5
z R  1090 1 recombination
( From the WMAP 1-year data analysis)
AS 4022
Cosmology
reionisation
Four Pillars of Hot Big Bang
Galaxies moving apart from each other
Redshift or receding from each other
Universe was smaller.
Helium production outside stars
Universe was hot, at least 3x109K to fuse 4H  He, to overcome a
potential barrier of 1MeV.
Nearly Uniform Radiation 3K Background (CMB)
Universe has cooled, hence expanded by at least a factor 109.
Photons (3K~10-4eV) are only 10-3 of baryon energy density, so
photon-to-proton number ratio ~ 10-3(GeV/10-4eV) ~ 108
Missing mass in galaxies and clusters (Cold DM)
Cluster potential well is deeper than the potential due to baryons.
CMB fluctuations: photons climb out of random potentials of DM.
If 1/10 of the matter density in 1GeV protons, 9/10 in dark particles
of e.g. 9GeV, then dark-to-proton number density ratio ~ 1
AS 4022
Cosmology
Cosmology Milestones
• 1925 Galaxy redshifts   0  1 z 
V =c z
– Isotropic expansion. ( Hubble law V = H0 d )
– Finite age. ( t0 =13 x 109 yr )

• 1965 Cosmic Microwave Background (CMB)
– Isotropic blackbody. T0 = 2.7 K
– Hot Big Bang T = T0 ( 1 + z )
• 1925 General Relativity Cosmology Models :
– Radiation era: R ~ t 1/2
– Matter era:
R ~ t 2/3
T ~ t -1/2
T ~ t -2/3
• 1975 Big Bang Nucleosynthesis (BBN)
– light elements ( 1H … 7Li ) t ~ 3 min T ~ 109 K
– primordial abundances (75% H, 25% He) as observed!
Tutorial: 3 Eras: radiation-matter-vacuum
radiation :
rR  R
matter :
vacuum:
rM  R
r = const
log r
4
3
r
4
a

rM ,0
3
a
rM

R
1
a

R0
1 z
r R,0
rR
r

  log R
 r
rR  rM at a ~ 104 t ~ 104 yr
e
log R
t
2/3
1/2
t

rM  r  at a ~ 0.7 t ~ 1010 yr
Presently vacuum is twice the density of matter.
log t
t
5th concept: Equation of State w
Equation of state :
r  R n
w
n  3(1  w)
pressure
p
n


1
2
energy density r c
3
Radiation : ( n  4, w  1/ 3)
d energy   work
1
rR c 2
3
Matter :
( n  3, w  0 )
d r c 2 R 3    p d R 3 
pR 
pM ~ rM c S  r M c 2
2
Vacuum :
( n  0, w  1)
p   r  c 2
Negative Pressure ! ?
r c 2  3 R 2 dR   R 3 c 2 dr   p  3 R 2 dR 
1
R dr
p
  2  w
3 r dR
rc
1 d ln r
w
1
3 d ln R 
w
AS 4022
Cosmology
n
1
3
Current Mysteries from
Observations
Dark Matter ?
Holds Galaxies together
Triggers Galaxy formation
Dark Energy ?
Drives Cosmic Acceleration
and negative w.
.
Modified Gravity ?
General Relativity wrong ?
AS 4022
Cosmology
Density Parameters
critical density :
density parameters (today) :
rR
rM
r

R 
M 
 

rc
rc
r c 3 H0 2
total density parameter today :
0  R   M  
density at a past/future epoch in units of today' s critical density :
r

   w x 3(1 w )   R x 4  M x 3   
x 1  z  R0 /R
rc w
in units of critical density at the past/future epoch :
3 H02
rc 
8 G
4
3
8 Gr H02

x


x
 
3(1 w )
R
M
(x) 


x

2
2 
3H
H w w
 R x 4   M x 3    (1  0 ) x 2
Note: radiation dominates at high z,
can be neglected at lower z.
AS 4022
Cosmology
Key Points
• Scaling Relation among
– Redshift: z,
– expansion factor: R
– Distance between galaxies
– Temperature of CMB: T
– Wavelength of CMB photons: lambda
• Metric of an expanding 2D+time universe
– Fundamental observers
– Galaxies on grid points with fixed angular coordinates
• Energy density in
– vacuum, matter, photon
– How they evolve with R or z
• If confused, recall the analogies of
– balloon, bread, a network on red giant star, microwave oven
AS 4022
Cosmology
Sample a wide range of topics
Theoretical and Observational
Universe of uniform density
Quest of H0 (obs.)
Metrics ds, Scale R(t) and Redshift
Applications of expansion models
EoS for mix of vacuum, photon,
matter, geometry, distances
Distances Ladders
Thermal history
Freeze-out of particles,
Neutrinos, CDM wimps
Cosmic Background
COBE/MAP/PLANCK etc.
Parameters of cosmos
Nucleo-synthesis He/D/H
Quest for Omega (obs.)
Structure formation
Inflation and origin of perturbations
Growth of linear perturbation
Relation to CMB peaks, sound horizon
Hongsheng Zhao
AS 4022
Cosmology
Galaxy and SNe surveys
Luminosity Functions
(thanks to slides from K. Horne)
6th concept:
Distances in Non-Euclidean Curved
Space
How Does Curvature affect
Distance Measurements ?
Is the universe very curved?
AS 4022
Cosmology
Geodesics
Gravity = curvature of space-time by matter/energy.
Freely-falling bodies follow geodesic trajectories.
Shortest possible path in curved space-time.
Local curvature replaces forces acting at distance.
Is our Universe Curved?
Closed
Curvature:
Flat
+
Open
0
--
= 180o
< 180o
Sum of angles of triangle:
> 180o
Circumference of circle:
<2 r
Parallel lines: converge
Size:
finite
Edge:
no
AS 4022
Cosmology
=2 r
remain parallel
infinite
no
>2r
diverge
infinite
no
Distance Methods
• Standard Rulers
==> Angular Size Distances
l

D
DA 
l
l

( for small angles << 1 radian )

• Standard Candles ==> Luminosity Distances

energy/time
L
F

area
4  D2
• Light Travel Time
t
AS 4022
distance 2 D

velocity
c
Cosmology
 L 1/ 2
DL  

4  F 
D


(e.g. within solar system)
Dt 
c
2t

Flat Space: Euclidean Geometry
Cartesian coordinates :
dl
dx
dz
dy
1D:
dl 2  dx 2
2D:
dl 2  dx 2  dy 2
3D:
dl 2  dx 2  dy 2  dz 2
4 D : dl 2  dw 2  dx 2  dy 2  dz 2
Metric tensor : coordinates - > distance
dl  ( dx dy dz ) 1 0 0dx 

 

0 1 0dy 

 
0
0
1

dz 
2
Summation convention :
dl 2  gij dx i dx j  
i
AS 4022
Cosmology

j
gij dx i dx j
Orthogonal coordinates
<--> diagonal metric
gx x  gy y  gz z 1
gx y  gx z  gy z  0
symmetric : g i j  g j i
Polar Coordinates
Radial coordinate r, angles  , ,  ,...
1 D : dl 2  dr 2
2 D : dl 2  dr 2  r 2 d 2
3 D : dl 2  dr 2  r 2  d 2  sin 2  d 2 

dl
r
d
dr

4 D : dl  dr  r  d  sin   d  sin  d  
2
2
dl 2  dr 2  r 2 d 2
2
2
2
2
2
2
generic angle : d 2  d 2  sin 2  d 2  ...
dl 2  ( dr d d ) 1 0

2
0
r


0 0
dr 
 
0 d 

2
2 
r sin  d 
0
gr r  ? gr  ?
g   ?
g   ?
g  ?
AS 4022
Cosmology
Embedded Spheres
R = radius of curvature
1 D : R 2  x 2
2  D : R2  x 2  y 2
3  D : R 2  x2  y 2  z 2
4  D : R2  x 2  y 2  z 2  w 2
0- D
1- D
2-D
3- D
2 points
circle
surface of 3 - sphere
surface of 4 - sphere
?
r

R
D

AS 4022
Cosmology
w
Reading: Non-Euclidean Metrics
k  1, 0,1 ( open, flat, closed )
dl 2 
dr
2
1 k  r /R 
open
2
2

r
d

2
dimensionless radial coordinates :
u  r /R  Sk  
2


du
2
2
2
2
dl  R 
 u d 
2
1 k u

 R 2  d 2  Sk   d 2 
2
S1(  )  sinh(  ) , S0 (  )   , S1(  )  sin(  )
AS 4022
Cosmology
closed
flat
Reading: Circumference
r
metric :
dl 
dr 2
2
1 k  r R 
 r d
2
2
2
radial distance ( for k  1 ) :
r
D

0
dr
1 k  r R 
2

R
D

 R sin 1 r R
circumference :
C
2
 r d  2  r
0
C
"circumferencial" distance : r 
 R S k (D/R)  R Sk (  )
2
If k = +1, coordinate r breaks down for r  R
AS 4022
Cosmology
w
Reading: Circumference
r
metric :

R
D
dl 2  R 2  d 2  Sk2   d 2 
radial distance :
D



g d   R d  R 
0
circumference :
C

g d 
 2 D
AS 4022
2
 R S ( ) d  2
k
R Sk (  )
0
Sk (  )
Cosmology

Same result for any choice of
coordinates.
w
Reading: Angular Diameter
r
metric :
R

D
dl 2  R 2  d 2  S k2    d 2 
radial distance :

D


w
g d   R d  R 
0
linear size : ( l  D )
l

g d  R S k   
angular size :
l

DA
AS 4022
Cosmology
DA
l

D  R   Radial Distance 
  Diameter Distance
DA  R Sk     Angular

Reading: Area of Spherical Shell
radial coordinate  , angles  ,  :
dl 2  R 2  d 2  Sk2   d 2  sin 2  d  2  
area of shell :
A

g d g d
 R 2 Sk2 (  )

 d
0
sin 
2
 d
0
 4  R 2 S k2 (  )
flux :
L
L
F 
A 4  DL2
AS 4022
Cosmology
D L  R Sk (  )  Luminosity Distance
Curved Space Summary
• The metric converts coordinate steps (grids) to physical lengths.

r
R
• Use the metric to compute lengths, areas, volumes, …
• Radial distance:

D
• “Circumferencial” distance
1/ 2
C  A 
1
r
   
4  
2 
2
2

D

grr dr  R 
g d   R Sk     R S k  D R
0
• “Observable” distances, defined in terms of local observables
(angles, fluxes), give r, not D.
DA 
l

r
Cosmology
open
 L 1/ 2
DL  
  r
4  F 
• r < D (positive curvature, S+1 (x)=sin x)
r>D (negative, S-1 (x)=sinh x) or r=D (flat, S0(x)=x)
AS 4022
w
or
closed
flat
Olber’s Paradox
Why is the sky dark at night ?
Flux from all stars in the sky :
F
n

F d Vol  
 max

0
 L 
n 
  A   R d 
A(  ) 
 n L R  max
  for flat space, R  .
A dark sky may imply :
(1) an edge (we don't observe one)
(2) a curved space (finite size)
(3) expansion ( R(t) => finite age, redshift )
AS 4022
Cosmology
Minkowski Spacetime Metric
ds 2  c 2dt 2  dl 2
2 



dl
1 dl
2
2
2
d  dt  2  dt 
1  2   


c
c

dt



2
Time-like intervals:
ds2 < 0, d2 > 0
Inside light cone.
Causally connected.
Proper time (moving clock):
Space-like intervals:
ds2 > 0 , d2 < 0
Outside light cone.
Causally disconnected.
World line
of massive
particle
at rest.
AS 4022
Cosmology
Null intervals
light cone:
v = c, ds2 = 0
Photons arrive
from our past
light cone.
Cosmological Principle (assumed)
+ Isotropy (observed)
=> Homogeneity
1
2
r1  r2
AS 4022
Cosmology
otherise not isotropic
for equidistan t fidos
7th Concept: Robertson-Walker metric
uniformly curved, evolving spacetime
ds2  c 2dt 2  R 2 (t)  d 2  Sk2    d 2 
 du 2

2
2
 c dt  R (t) 
 u d 
2
1 k u


2

dr
2
2

 c 2dt 2  a2 (t) 

r
d

1 k r R 2




0

2
2
 sin 

Sk (  )   

 sinh 
2
(k  1) closed
(k  0)
flat
(k  1) open
r

R
D

d 2  d 2  sin 2  d 2
a( t)  R(t) / R0
R0  R(t 0 )
radial distance  D(t)  R(t) 
circumference  2  r(t)
r(t)
  a(t) r  R(t) u  R(t)S k  
AS 4022
Cosmology
w

coordinate systems
Distance varies in
time:
t
D(t)
“Fiducial observers”
(Fidos)
D(t)  R(t) 
D
t
“Co-moving” coordinates
 or D0  R0 

Labels the Fidos

AS 4022
Cosmology

Distances-Redshift relation
• We observe the redshift : z 
  0 

1   observed,
0
0
0  emitted (rest)
• Hence we know the expansion factor:
 (t0 ) R(t0 ) R0
x 1  z 



R(t) R(t)
  0 (t)
t(z)  ?
• Need the time of light emitted
• Need coordinate of the source  (z)  ?
 them as functions of H0 M 
• 
Need
• Distances
D(t,  )  R(t) 
DA  r0 ( ) 1 z
 
r(t,  )  R(t) Sk (  )
DL  r0 ( ) 1 z 
• E.g. D_L is 4 x D_A for an object at z=1.
AS 4022
Cosmology
D
R
r


Tutorial: Time -- Redshift relation
R0
x 1  z 
Memorise this
R
derivation!
dx
R0 dR
 2
dt
R dt
R0 R

Hubble parameter :
R R
 x H( x)
dx
dz
 dt 

x H(x) 1 z H(z)
AS 4022
Cosmology
R
H
R
Tutorial:
Time
and
Distance
vs
Redshift


d
R
dx
x  1 z  0   dt 
dt 
R 
x H( x)
Look - back time :
t(z) 
t0

t
Age :
1 z
dx
dx
dt  
= 
1 z x H( x)
1 x H( x)
1
t 0  t(z  )
Distance : D  R 
 (z)   d 
t0

t
Horizon :
AS 4022
r


r  R Sk (  )
c dt c

R(t) R0
1 z

1
R0
dx
c
=
R(t) x H( x)
R0
1 z

1
 H   (z  )
Need to know R(t), or R0 and H(x).
Cosmology
D
R
dx
H( x)
Einstein’s General Relativity
• 1. Spacetime geometry tells matter how to move
– gravity = effect of curved spacetime
– free particles follow geodesic trajectories
– ds2 < 0 v < c time-like
massive particles
– ds2 = 0 v = c null
massless particles (photons)
– ds2 > 0 v > c space-like tachyons (not observed)
• 2. Matter (+energy) tells spacetime how to curve
– Einstein field equations
– nonlinear
– second-order derivatives of metric
with respect to space/time coordinates
AS 4022
Cosmology
8th concept: Einstein Field
Equations
1 
8 G
G  R  R  g  2 T   g
2
c
g  spacetime metric ( ds 2  g dx  dx  )
G  Einstein tensor (spacetime curvature)
R  Ricci curvature tensor
R  Ricci curvature scalar
G  Netwon's gravitational constant
T  energy - momentum tensor
  cosmological constant
AS 4022
Cosmology
Homogeneous perfect fluid
density r
Einstein field equations:
r c 2 0 0

p 0
8 G  0
G  2
c  0
0 p

0 0
 0
pressure p
c 2
0 


0 
0



 0
0


p
 0
0
1
0
0
0
0
1
0
0

0
0

1
---> Friedmann equations :
8 G r    2
R 2  

R  k c 2

3

   4 G  r c 2  3p R   R
R
3c 2
3

energy
momentum
Note: energy density and pressure decelerate,  accelerates.
AS 4022
Cosmology
Reading: Newtonian Analogy
m  2 G M m
4 3
E
R 
M
Rr
2
R
3
8 G
2E
2
2


R 
rR 
3
m
Friedmann equation:
m
R

8 G r    2
2
R  
R

k
c



3
2

2E
same equation if r  r 
,
 k c 2
8 G
m
AS 4022
Cosmology
r
Reading: Local Conservation of
Energy
d energy   work
d r c 2 R 3    p d R 3 
 c 2 R 3  r c 2 (3 R 2 R )  p (3 R 2 R )
r

p  R
r
  3 r  2 
 c  R
p  p( r)  equation of state
8 G

Friedmann 1 : R 2 
r R2  R2  k c 2
3
3
8 G

2










2
R
R

r
R

2
R
R
r





 2 R R 
3
3
2
r
 


8

G
R
R
 
   R r  R
3  2 R
 3


R
   4  G r  3 p R   R  Friedmann 2
3 
c 2 
3
AS 4022
Cosmology
Reading: Matterdominated Universes
Vacuum
Dominated
Critical
Empty
Curvature wins
R  t2/3
SubCritical
Cycloid
Matter wins
All evolve as
R ~ t2/3
at early times.

AS 4022
Cosmology
Critical Universe
(Einstein - de Sitter)
rM
M 
1
rc
R    0
Critical
 k  0 ( flat )
3
3 H0 2 R0 
r
 
8 G  R 
2
3
8

G
H
R
R 2 
r R2  0 0
3
R
dR R1/ 2  H 0 R0 3 / 2 dt
2 3/2
R  H 0 R0 3 / 2 t
3
2/3
R  t 
2 1
2 R0
   ,
age : t 0 

R0 t 0 
3 H0 3 c
AS 4022
Cosmology
Vacuum
Dominated
Empty
SubCritical
R  t2 / 3
Matter decelerates expansion.
Cycloid
Reading: Possible Universes

cosh(t)
exp(t)

sinh( t)


t
t2/3
 M    1
AS 4022
Cosmology
M
Reading: Rad. => Matter =>Vacu.
(z)   R x   M x       i x
4
3
R
3  wi 1
log r
i
w  p/r c2
x 1  z  R0 R  a1
R x 4  M x 3
M
 z  
 R

M



 x  M R

0.3
 3600
1 
5
8.4 10

log R
exp(t / t  )
M x 3    x 3   M
1
log R 
1
3
  3
0.7 
 z    1    1  0.33
0.3 
M 
t
2/3
1/2
t
log t
AS 4022
Cosmology
Tutorial: What observations justify
the “Concordance” Parameters?
km/s
km/s
H0  100 h
 70
Mpc
Mpc
h  0.7
 R  4.2 105 h2  8.4 105 (CMB photons  neutrinos)
 B ~ 0.02 h 2 ~ 0.04 (baryons)
 M ~ 0.3 (Dark Matter )
  ~ 0.7 (Dark Energy)
 0   R   M    1.0  Flat Geometry
AS 4022
Cosmology
9th Concept:“Concordance” Model
Three main constraints:
1. Supernova Hubble Diagram
2. Galaxy Counts  M/L ratios
M ~ 0.3
3. Flat Geometry
( inflation, CMB fluctuations)
0  M     1
2
1
3
concordance model
H0  72  M  0.3    0.7
AS 4022
Cosmology
Beyond H0
• Globular cluster ages:
t < t0 --> acceleration
• Radio jet lengths:
DA(z) --> deceleration
• Hi-Redshift Supernovae: DL(z) --> acceleration
• Dark Matter estimates --->
• Inflation ---> Flat Geometry
• CMB power spectra


• “Concordance Model”
AS 4022
Cosmology
M ~ 0.3
 0 1.0
 M ~ 0.3   ~ 0.7
 0   M    1.0
Deceleration parameter

R R
R
q    2  
R
R H2
R R 
 R 
q0    2   
2 
R
R
H
 0 
0
Friedmann momentum equation :
3 p 

R
   4 G 
r  2 R  R
3 
c 
3


R
4 G



r
1
3w



H02 R
3H 02
3H0 2
r, p  0 decelerate,   0 accelerates
Equation of state : p   wi ri 
c2
i
R
t - t0
q0  1/ 2
0

1/ 2

M
1
w M  0 w  1
 1
3
 R 
q0 > 0 =>
deceleration
1  3 wi 
M


q0  
  
 
 i   R 
2 
q0 = 0 => constant velocity
2
R H 0 i  2 
wR 
q0
< 0 => acceleration
AS 4022
Cosmology
Deceleration
Parameter
R
 R   M
q0    2  
 
2
 R 0
Matter decelerates
Vacuum (Dark) Energy
accelerates
q0 = -1
acceleration
0
deceleration
Measure q0 via :
1 . DA(z)
( e.g. radio jet lengths )
2. DL(z)
( curvature of Hubble Diagram )
Critical density matter-only --> q0=1/2.
AS 4022
Cosmology
+1
Tutorial: Observable Distances
angular diameter distance :

l
r0
c z  q0  3
=
DA 

z  ...
1
DA
(1 z ) H0 
2

luminosity distance :

L
c z  1 q0
F=
DL  r0 (1  z ) 
z  ...
1
2
H0 
2

4 DL
deceleration parameter :
M
q0 
 
2
Verify these low-z expansions.

AS 4022
Cosmology
r(z)
DA (z) 
1 z
DL (z)  1 z  r(z)


r(z)  R0 Sk  (z)

AS 4022
Cosmology
Reading: Hubble Diagram
 A  K(z)
m  apparent mag
M  absolute mag
apparent mag
DL (z) 
m  M  5 log
 25
 Mpc 
faint
slope = +5
vertical shift --> H0
curvature --> q0
acceleration
( q0 < 0 )
deceleration
( q0 > 0 )
A  extinction (dust in galaxies)
K(z)  K correction
bright
( accounts for redshift of spectra
log ( c z )
relative to observed bandpass )
Expand faster at large distance:
Expansion has slowed down.
Decelerating Universe
Constant Expansion: v = H0 d
AS 4022
Cosmology
Fainter
Brighter
Distance

c z  1  q0
DL (z) 
z  ... 
1
H 0 
2

Expand slower at large distance:
Expansion has speeded up.
Accelerating Universe
Recession Velocity ( v = c z )
Absolute magnitude M B ==>
Calibrating
“Standard
Bombs”
1. Brighter ones
decline more slowly.
2. Time runs slower
by factor (1+z).
AFTER correcting:
Constant peak brightness
MB = -19.7
Observed peak magnitude:
m = M + 5 log (d/Mpc) + 25
gives the distance!
AS 4022
Cosmology
Time ==>
SN Ia at z ~ 0.8
are ~25% fainter
than expected
Acceleration ( ! ? )
1. Bad Observations?
-- 2 independent teams agree
1. Dust ?
-- corrected using reddening
2. Stellar populations ?
-- earlier generation of stars
-- lower metalicity
3. Lensing?
-- some brighter, some fainter
-- effect small at z ~ 0.8
AS 4022
Cosmology
Reiss et al. 1998
Perlmutter et al. 1998
CMB Angular scale --> Geometry
AS 4022
Cosmology
10th Concept: Sound Horizon at z
= 1100
0
t0
LS
o
 
~ 0.8
DA
Standard Ruler :

tR
0
AS 4022
Cosmology

LS  c S tR ~ 100 kpc
zR  1100
cS  c


3
Tutorial: Sound Horizon at z=1100
distance travelled by a sound wave
recombination
at z = 1100
c S dt
expand each step by factor R(t R )/R(t) :
LS (t R )  R(t R )
tR

0
c S dt
R(t)
dt = - dx / x H( x )
R( t ) = R0 / x
R
x c S dx
 0 
1 z 1 z R0 x H( x)



1 z
cS
( 1 z ) H0
cS

( 1  z ) H0
AS 4022
dx
H( x)
1 z
dx
1 z
Cosmology
cS 
c
3
4
3

x R  x  M     1 0  x


 dx
dt 
x H(x)
H( x ) from Friedmann Eqn.


R0
R(t)
sound speed

cS

( 1 z )
x  1 z 
dx
x R  x  M
4
3
2
keep 2 largest terms.
Sound Horizon at z = 1100
cS
LS (t R ) 
(1 z )


1 z
dx
H(x)
cS

( 1 z ) H0 R



1 z
cS
( 1 z ) H0


1 z
dx
x3 ( x  x0 )



cS
2
x

1 0 

x 1 z
( 1 z ) H0 R  x 0


2c S
x0

1
 1 
1 z
( 1 z ) H0 M x 0 

dx
x4 R  x 3 M
x0 
 
M
 3500  M 
0.3 
R
cS 
c
3
2  4.6 1
c

H0 1100 3  0.3  3500
1/2




c
0.7
0.3
 3.4 105
110  
 kpc
 h  M 
H0
AS 4022
Cosmology
Expands by factor
1 + z = 1100
to ~120 Mpc today.
Reading: Angular scale  0
sound horizon :
angular diameter distance :
1  c S dx
LS (z) 

1  z 1 z H( x)
R0 SK   
DA (z) 
1 z

t0

t
c dt c 1 z dx


R(t) R0 1 H(x)
angular scale

LS (z)


DA (z)

1 z
c S dx
H( x)
 c 1 z dx 
R0 Sk 


R
H(
x)
 0 1

1.0o
Angular scale depends mainly
on the curvature.
Gives ~ 0.8o for flat geometry,
0.8o
0 = M + =1
0.6o
AS 4022
Cosmology
Evolution of Sound Speed
Rzc r (t )
Expand a box of fluid P (t )
Ryc
Rx
Sound Speed
c
¶R / ¶ (vol )
º
,
¶r / ¶( vol )
/ ¶R
= ¶R
¶r / ¶R
Cs2
Vol = R3 (t )× xc yc zc
µ R3 (t )
Brief History of Universe
• Inflation
– Quantum fluctuations of a tiny region
– Expanded exponentially
• Radiation cools with expansion T ~ 1/R ~t-2/n
– He and D are produced (lower energy than H)
– Ionized H turns neutral (recombination)
– Photon decouple (path no longer scattered by electrons)
• Dark Matter Era
– Slight overdensity in Matter can collapse/cool.
– Neutral transparent gas
• Lighthouses (Galaxies and Quasars) form
– UV photons re-ionize H
– Larger Scale (Clusters of galaxies) form
AS 4022
Cosmology
Coupled radiation-baryon
relativistic
fluid
Radiation
Matter
Where fluid density r(t )=
Fluid pressure R (t ) =
rr
rm × KT
m
m
c2 r
3 r
rr µ R -4
Note
rm
rm µ R
Matter number
density
-3
Neglect
Tutorial: Show C2s = c2/3 /(1+Q) , Q = (3 ρm) /(4 ρr) ,  Cs drops
from c/sqrt(3) at radiation-dominated era
to c/sqrt(5.25) at matter-radiation equality
Random motion energy
Non-Relativistic
IDEAL GAS
1
KTm << c 2
m
Reading: Sound Speed & Gas Temperature:
Cs2~T~Tph ~1500K*(z/500) before decouple.
Compton-scatter
r until redshift z
But After decoupling (z<500),
Cs ~ 6 (1+z) m/s because
d 3P d x3 invarient phase space volume
So: P µ x µ R
-1
dP
-1
2
3
´Te = mu µ R -2
2
2
Te ∞ Cs2 ∞ R-2
d
P
dX
æ 1+ z ö
Te ~ 1500 K ? ç
÷
è 500 ø
dX
2
T
Te
R
Except reionization z ~ 10 by stars quasars
What have we learned?
What determines the patterns of CMB at last
scattering
Analogy as patterns of fine sands on a drum at
last hit.
Heading to: origins of perturbations:
inflation brings patches in contact
Creates a flat universe naturally.
11th Concept: Inflation in Early Universe
Consider universe goes through a phase with
r (t ) ~ R(t )- n
R(t ) ~ t q where q=2/n
Problems with normal expansion theory (n=2,3,4):
What is the state of the universe at t0?
exotic scalar field?
Pure E&M field (radiation) or
Why is the initial universe so precisely flat?
What makes the universe homogeneous/similar in opposite directions of horizon?
Solutions: Inflation, i.e., n=0 or n<2
Maybe the horizon can be pushed to infinity?
Maybe there is no horizon?
Maybe everything was in Causal contact at early times?
Horizon
x
sun
x
Why are these two
galaxies so similar without
communicating yet?
2c
e K ( z ) e K (0) ´ R
n -2
=
~
R
~ 0 at t = 0
-n
e ( z ) e (0) ´ R
-2
Why is the curvature
term so small (universe
so flat) at early universe
if radiation dominates
n=4 >2?
Inflation broadens Horizon
Light signal travelling with speed c on an expanding sphere R(t), e.g., a
fake universe R(t)=1 lightyr ( t/1yr )q
Emitted from time ti
By time t=1yr will spread across (co-moving coordinate) angle c
Horizon in co-moving coordinates
(11- q - ti1-q )
cdt
cdt
xc = ò
=ò q =
R(t)
t
(1 - q)
ti
ti
1
1
1
is finite if q=2/n<1
(1 - q)
(e.g., n=3 matter-dominate or n=4 photon-dominate)
Normally xc <
(ti1- q - 1)
INFLATION phase xc =
can be very large for very small t i if q=2/n>1
(q - 1)
(e.g., t i = 0.01, q = 2, xc = 99 >> p , Inflation allows we see everywhere)
Tutorial: Inflation dilutes any
initial curvature of a quantum universe
e K ( R) e K ( Ri ) æ R ö
=
ç ÷
e ( R ) e ( Ri ) è Ri ø
n-2
~ 0 (for n<2) sometime after R>>R i
even if initially the universe is curvature-dominated
e K ( Ri )
=1
e ( Ri )
E.g.
If a toy universe starts with
e K ( Ri )
= 0.1 inflates from t i =10-40sec to t f =1sec with n=1,
e ( Ri )
and then expand normally with n=4 to t=1 year,
SHOW at this time the universe is far from curvature-dominated.
Exotic Pressure drives Inflation
d ( r c 2 R3 )
P=d ( R3 )
=>
r
P
d (r R2 ) n - 2
+ 2 ==
r if r ~ R - n
3 c
3RdR
3
=>
P/r c 2 =(n-3)/3
Inflation n < 2 requires exotic (negative) pressure,
define w=P/r c 2 , then w = (n-3)/3<0,
Verify negligble pressure for cosmic dust (matter),
Verify for radiation P=r c 2 / 3
Verify for vaccum P=-r c 2
Reading: 1980: Inflation (Alan Guth)
• Universe born from “nothing” ?
• A quantum fluctuation produces a tiny bubble of
“False Vacuum”.
• High vacuum energy drives exponential expansion,
also known as “inflation.”
• Universe expands by huge factor in tiny fraction of
second, as false vacuum returns to true vacuum.
• Expansion so fast that virtual particle-antiparticle
pairs get separated to become real particles and antiparticles.
• Stretches out all structures, giving a flat geometry and
uniform T and r, with tiny ripples.
• Inflation launches the Hot Big Bang!
AS 4022
Cosmology
Reading: Scalar Field Dynamics
V
During inflation:
Just after inflation:



V

0

V



Kinetic energy of the oscillations is damped.
Re-heats the Universe, creating all types of particleantiparticle pairs, launching the Hot Big Bang.

AS 4022
Cosmology


t

Today:
Reading: Scalar Field Equation of
State
Equation of State: w = p / e
c   
Uniform field:
1 1  2
2
2


c



  V ( )
3
2 hc
1 1  2
2
2
e 


c



  V ( )
3
2 h c
2
 2

p 
Required for inflation:

 2  2 h c 3V

1
w    2


e   2 hc 3V
3
p
+1
 2  hc 3V   0
e  3p  4 


2
w=0
-1/3
-1
 2  h c 3 V


Terminal velocity:
3

h
c
V
 
3H  
 2  h c 3 V

 V 2 9H 2
V
  
3
   hc

Need a long slow roll over a nearly flat potential.
AS 4022

Cosmology
Tutorial: ants on a sphere
Consider a micro-cosmos of N-ants inhabiting an expanding spherical surface of
radius R=R0 (t/t0)a, where presently we are at t=t0 =1min, R=R0 =1lightsecond.
Let a=1/2, N=100. What is the present rate of expansion dR/dt/R = in units of
1/min? How does the ant surface density change as function of cosmic time?
[due 6Mar]
Light emitted by ant-B travels a half circle and reaches ant-A now, what redshift
was the light emitted? What is the angular diameter distance to the emission
redshift? [due 10Apr]
Let each ant conserve its random angular momentum per unit mass
J=(1lightsecond)*(1m/min) with respect to the centre of the sphere. Estimate
the age of universe when the ants were moving relativistic. How far has ant B
travelled since the emission and since the beginning of the universe?
What have we learned?
Where are we heading?
Inflation as origin of perturbation, flatness, horizon.
Sound speed of gas before/after decoupling, and sound horizon.
Topics Next:
Growth of [bankruptcy of uniform universe]
Density Perturbations (how galaxies form by N-body simulations)
peculiar velocity (how galaxies move and merge)
CMB fluctuations (temperature variation in CMB)
Reading: Supercomputer
Simulations
AS 4022
Cosmology
Non-linear Collapse of an Overdense Sphere
An overdense sphere is a very useful non linear model as it behaves in
exactly the same way as a closed sub-universe.
Any spherically symmetric perturbation will clearly evolve at a given
radius in the same way as a uniform sphere containing the same amount
of mass.
rb + dr
rb
R,
R1
logρ
Rmax
t-2
Rmax/2
virialize
t
1
rb =
6p Gt 2
Background
density changes
this way
logt
Gradual Growth of perturbation
2
-4
ì
R
(mainly
radiation
r
µ
R
)
dr 3c 1
ï
®
=
µí
2
r 8p G r R ïî R (mainly matter r µ R -3 )
Perturbations Grow!
2
E.g., δ changes by a factor of 10 between z=10 and z=100 (matter era),
and a factor of 100 between z=105 and z=106 (Radiation Era)
Peculiar Motion
The motion of a galaxy has two parts:
d
v = [R(t )q (t )]
dt
= R(t ).q + R(t )q (t )
Uniform
expansion vo
Proper length vector
Peculiar motion v
Damping of peculiar motion
(in the absence of overdensity)
Generally peculiar velocity drops with expansion.
R q = R *( Rq ) = constant~"Angular Momentum"
2
Similar to the drop of (non-relativistic) sound speed with expansion
constant
dv = R(t ) xc =
R(t)
Reading: Equations governing Fluid Motion
Ñ f = 4p G r
2
(Poissons Equation)
1 dr d ln r
=
= -Ñ.v (Mass Conservation)
r dt
dt
dv
2
= -Ñf - cs Ñ ln r (Equation of motion)
dt
ÑP
since ¶P = cs2¶r
r
Reading: Decompose into unperturbed + perturbed
Let
r = ro + dr
v = vo + d v = R c c + R c c
f = fo + df
We define the Fractional Density Perturbation:
dr
d=
= d (t ) exp(-ik · x ),
ro
| k |= 2p / l , where l = R(t )lc
k · x = kc · xc
x(t ) = R(t ) c c
go (t ) + g1 (q , t )
Reading: Motion driven by gravity:
due to an overdensity:
r (t ) = ro (1 + d (q , t ))
Gravity and overdensity by Poisson’s equation:
-Ñ · g1 = 4p G rod
Continuity equation:
d
-Ñ · d v = (d (q , t ) )
dt
The over density will
rise if there is an
inflow of matter
Peculiar motion δv and peculiar gravity g1 both scale with δ and are in the
same direction.
12th Concept: THE equation for structure formation
In matter domination
Equation becomes
-c k
2 2
s
¶d
R ¶d
2 2
+2
= (4pGr o + cs Ñ )d
2
¶t
R ¶t
2
Gravity has the tendency to
make the density
perturbation grow
exponentially on large
scale.
Pressure makes it
oscillate on small
scale.
Each eq. is similar to a forced spring
F
m
Restoring
d 2x F
dx
2
= -w x - m
2
dt
m
dt
2
d x
dx
F (t )
2
+ m +w x =
2
dt
dt
m
Term due to friction
(Displacement for
Harmonic Oscillator)
x
t
e.g., Nearly Empty Pressure-less Universe
r ~0
¶ d 2 ¶d
R 1
+
= 0, H = =
(R µ t)
2
¶t t ¶t
R t
0
d µ t = constant
2
® no growth
Reading: Jeans Instability (no expansion)
Case 1- no expansion
- the density contrast  has a wave-like form
R=0
for the harmonic oscillator equation
d = d o exp(ik .r - iwt )
where we have the dispersion relation
¶ 2d
¶d
2
+
2
*
0
*
=
w
d
2
¶t
¶t
w 2 º cs2 k 2 - 4pGr
Pressure
support
gravity
Reading: At the (proper) JEANS LENGTH scale we switch from
Oscillations for shorter wavelength modes to
the exponential growth of perturbations for longer wavelength
p
lJ = cst , where timescale t =
Gr
<J, 2>0  oscillation of the perturbation.
J, 20exponential growth/decay
d µ exp(± Gt ) where G = - w 2
Jeans Instability
Case 2: on very large scale >>J = cs t of an Expanding universe
Neglect Pressure (restoring force) term
Grow as delta ~ R ~ t2/3 for long wavelength mode if Omega_m=1 universe.
cs2 k 2 << 4p G r = cs2 k J2
¶ 2d
¶d
+ 2H
= 4p G r md
2
¶t
¶t
2 /(3t )
2
2 /(3t )
Reading: E.g.,
Einstein de Sitter Universe
R 2
WM = 1, H = =
R 3t
2
3
1
Verify Growth Solution d µ R µ t µ
1
+
z
Ω =1
M
Generally
log
Log R/R0
Case III: Relativistic (photon) Fluid
equation governing the growth of perturbations being:
1/t
1/t2
dd
dd
æ 32pGr 2 2 ö
Þ 2 + 2H
= d .ç
- k cs ÷
dt
dt
è 3
ø
2
Oscillation solution happens on small scale 2π/k = λ<λJ
On larger scale, growth as
Þ d µ t µ R for length scale l >> lJ ~ cst
2
What have we learned?
Where are we heading?
Large scale OverDensity grows as
R (matter dominated) or R2 (radiation dominated)
Peculiar velocity points towards overdensity.
Small scale oscillation if baryon-radiation dominated.
Topics Next: CMB equations
T
~ 10 5
T
AS 4022
Cosmology
Recombination Epoch ( z~1100 )
ionised plasma
-->
neutral gas
• Redshift z > 1100
• Temp T > 3000 K
• H ionised
• z < 1100
• T < 3000 K
• H recombined
• electron -- photon
Thompson scattering
• almost no electrons
• neutral atoms
• photons set free
e - scattering optical depth
 z 13
 ( z)  

1080 
thin surface of last scattering
AS 4022
Cosmology
Cosmic Microwave Background
Almost isotropic
T = 2.728 K
Dipole anisotropy
V  T


10 3
c

T
Our velocity:
V  600 km/s

Milky Way sources

+ anisotropies T ~ 105
T
Theory of CMB Fluctuations
Linear theory of structure growth predicts that the perturbations:
dr D
d D in dark matter
rD
dr B
d B in baryons
rB
dr r
Or
d r in radiation
rr
dng
~ 3
dr = dr =
4
ng
will follow a set of coupled Harmonic Oscillator equations.
Acoustic Oscillations
   DA

Dark Matter potential wells - many sizes.
photon-electron-baryon fluid
fluid falls into DM wells
photon pressure pushes it out again
oscillations starting at t = 0 (post-inflation)
stopping at z = 1100 (recombination)
AS 4022
Cosmology
The solution of the Harmonic Oscillator
[within sound horizon] is:
d (t ) = A1 cos kcst + A2 sin kcst + A3
Amplitude is sinusoidal function of k cs t
if k=constant and oscillate with t [evolution]
or t=0.3Myr and oscillate with k (CMB peaks)
CMB Anisotropies
COBE
1994
T
~ 105
T

WMAP
2004
 ~ 1o

Snapshot of Universe at z = 1100
Seeds that later form galaxies.
Resonant Oscillations
z
size of potential well 
oscillation period P 

t  P1 /2
2  1 /2
AS 4022
Cosmology
1 
DA
1
 2 
2

angular size
2 cs t
1

nP n
~
2
2c s
1
 n 

DA
n
DA

T(t)  T(0) cos( 2 t /P )
n
z 0
1  2c s t

cs
c

sound speed c s 
3
temperature oscillations
max T at t 
z 1100
3  1 / 3

~ 0.8o
 3 

Smaller wells oscillate faster.

1
3
We don’t observe the baryon overdensity d B directly
-- what we actually observe is temperature fluctuations.
DT Dng
=
T
3ng
~
dB dR
=
=
3
3
ng ~ R -3 µ T 3
e g ~ ng kT µ T
4
The driving force is due to dark matter over densities.
The observed temperature is:
dB y
æ DT ö
ç
÷ = + 2
è T øobs 3 c
Effect due to having to climb out
of gravitational well
The observed temperature also depends on
how fast the Baryon Fluid is moving.
dd B
Velocity Field Ñv = dt
dB y v
æ DT ö
ç
÷ = + 2±
è T øobs 3 c c
Doppler Term
Tutorial: perturbation questions
The edge of the void are lined up by galaxies. What direction is their
peculiar gravity and peculiar motion? A patch of void is presently
cooler in CMB by 3 micro Kelvin than average. How much was it
cooler than average at the last scattering (z=1000)? Argue that a
void in universe now originates from an under-dense perturbation at
z=1010 with δ about 10-17.
List of keys
Scaling relations among
Redshift z, wavelength, temperature, cosmic time, energy density, number density,
sound speed
Definition formulae for pressure, sound speed, horizon
Metrics in simple 2D universe.
Describe in words the concepts of
Fundamental observers
thermal decoupling
Common temperature before,
Fixed number to photon ratio after
Hot and Cold DM.
gravitational growth.
Over-density,
direction of peculiar motion driven by over-density, but damped by expansion
pressure support vs. grav. collapse
Expectations
Remember basic concepts (or analogies)
See list
Can apply various scaling relations to do some of the short questions at the
lectures.
See list
*Relax*.
thermal history, structure formation, DM counting are advanced
subjects, just be able to recite the big picture.
Why Analogies in Cosmology
Help you memorizing
Cosmology calls for knowledge of many areas of physics.
Analogies help to you memorize how things move and
change in a mind-boggling expanding 4D metric.
*Help you reason*, avoid “more equations, more confusions”.
If unsure about equations, e.g. at exams, the analogies
*help you recall* the right scaling relations, and get the
big picture right.
*Years after the lectures*,
Analogies go a long way
Look back, Look forward?
Have done: Malcolm S. Longair’s “Galaxy Formation” 2 edition [Library]
Chpt 1-2,5-8: expanding metrics, energy
density, curvature, distances
Chpt 11,15,20: Structure growth, inflation
Heading to:
Chpt 4,9-10,13: Thermal History of Particle
Reaction, Neutrinos, WIMPs, DM budget
nd
Derive with concordance parameters
log T

Tutorial: Cooling History T(t)

 1 s 1 / 2
T
kT
Radiation era :   

 t 
2 1010 K 2 MeV
35,000 K
matter - radiation equality : r M  rR
 1010 yr 2/ 3
T
Matter era : 
 
2.7 K
 t


 11 s
~ 3 10
~ 10 4 yr
log t
Reading: A busy schedule for
the universe
Universe crystalizes with a sophisticated schedule, much more
confusing than simple expansion!
many bosonic/fermionic players changing (numbers conserved except
in phase transition!)
p + p- <->  (baryongenesis)
e + e+ <-> , v + e <-> v + e (neutrino decouple)
n < p + e- + v, p + n < D + (BBN)
H+ + e- < H + , + e <->  + e (recombination)
n  3H n    v  n 2  nT 2 

Reading: Significant Events
Event
T
Now
2.7 K
0.0002 eV
3.3
0
13 Gyr
First Galaxies
16 K
0.001 eV
3.3
5
1 Gyr
Recombination
3000 K
0.3 eV
3.3
1100
300,000 yr
Equal rM = rR
9500 K
0.8 eV
3.3
3500
50,000 yr
e+ e- pairs
109.7 K
0.5 MeV
11
109.5
3s
Nucleosynthesis 1010 K
1 MeV
11
1010
1s
Nucleon pairs
1013 K
1 GeV
70
1013
10-6.6 s
E-W unification
1015.5 K
250 GeV
100
1015
10-12 s
1019 GeV
100(?)
1032
10-43 s
Quantum gravity 1032 K
kT
geff
z
t
Here we will try to single out some rules of thumb.
We will caution where the formulae are not valid, exceptions.
You are not required to reproduce many details, but might be asked for general ideas.
Big Bang + 3 minutes
T ~ 109 K
First atomic nuclei forged.
Calculations predict:
75% H and 25% He
AS OBSERVED !
Helium abundance
Reading: Big Bang Nuclear Fusion
Oxygen abundance =>
+ traces of light elements
D, 3H, 3He, 7Be, 7Li
=> normal matter only 4% of
critical density.
AS 4022
Cosmology
13th Concept: Particle-Freeze-Out?
Freeze-out of equilibrium means NO LONGER
in thermal equilibrium.
Freeze-out temperature means a species of
particles have the SAME TEMPERATURE as
radiation up to this point, then they bifurcate.
Decouple = switch off the reaction chain
= insulation = Freeze-out
Reading: Thermal Schedule of Universe
At very early times, photons are typically energetic enough that they interact strongly with
matter so the whole universe sits at a temperature dictated by the radiation.
The energy state of matter changes as a function of its temperature and so a number of key
events in the history of the universe happen according to a schedule dictated by the
temperature-time relation.
Crudely
(1+z)~1/R ~ (T/3) ~109 (t/100s)(-2/n) ~ 1000 (t/0.3Myr)-2/n,
H~1/t, where n~4 during radiation domination
T(K)
Radiation
Matter
pp ~ 106 s
1010
e  e  ~ 1s
Recombination
He D ~100s
Neutrinos
decouple
103
1012 109
0.3My
10r 6 103
After this Barrier photons
free-stream in universe
1
1+z
A general history of a massive particle
Initially mass doesn’t matter in hot universe
relativistic, dense (comparable to photon
number density ~ R-3 ~ T3 ),
frequent collisions with other species to be in thermal
equilibrium and cools with photon bath.
Photon numbers (approximately) conserved, so is the number of
relativistic massive particles
Reading: Competition of two processes
Interactions keeps equilibrium:
E.g., a particle A might undergo the annihilation reaction:
A+ A ®g +g
depends on cross-section  and speed v. & most importantly
the number density n of photons ( falls as R-3 ~ t(-6/n) )
Decouples because of the increasing gap of space between particles due to
Hubble expansion H~ t-1.
e.g., equilibrium process dominates at small time; Hubble expansion
process dominates later because it falls slower.
Neutrino: smallest cross-section
and smallest mass
neutrinos (Hot DM) decouple from electrons (due to
very weak interaction) while still hot (relativistic
0.5 Mev ~ kT >mc2 ~ 0.02-2 eV)
Presently there are 3 x 113 neutrinos and 452 CMB photons per cm3 .
Details depend on
Neutrinos have 3 species of spin-1/2 fermions while photons are 1
species of spin-1 bosons
Neutrinos are a wee bit colder, 1.95K vs. 2.7K for photons [during
freeze-out of electron-positions, more photons created]
Reading: Rule of thumb
survival of the weakest particle
While in equilibrium, nA/nph ~ exp. q= mc2/kT (Heavier is rarer)
At decoupling, nA= Hdecouple /(A ) , nph ~ T3decoupl ,
Later on the abundance ratio is frozen at this value nA/nph ,
NA
N ph
Freeze out
mc2
kT
A LOW smallest
interaction, early freeze-out
while relativistic, Hot Matter
A HIGH later freeze-out
at lower T, reduced
abundance, Cold Matter
Question: why frozen while nA , nph both drop as T3 ~ R-3.
Energy density of species A ~ nph/(A) , if m ~ kTfreeze
number ratio of non-relativistic particles to photons are const
except for sudden reduction by certain transitions.
Reduction factor ~ exp(-y), ymc2/kT, which
drops sharply for heavier particles.
Non-relativistic particles (relic) become
*much rarer* by exp(-y) as universe cools
below mc2/y,
y~1025.
So rare that infrequent collisions can no longer
maintain coupled-equilibrium.
Reading: for example,
Antiprotons freeze-out t=10-6 sec,
earlier than positrons freeze-out t=1sec:
Because: anti-proton is ~1000 times heavier than
positron.
Hence factor of 1000 hotter in freeze-out
temperature
t goes as T2 in radiation-dominated regime
Dark Matter Evidences?
Galaxies: ( r ~ 20 Kpc )
Flat Rotation V ~ 200 km/s
Galaxy Clusters: ( r ~ 200 Kpc )
Galaxy velocities V ~ 1000 km/s
X-ray Gas T ~ 108 K
Giant Arcs
2
1
 M ~ 0.3
 b  0.04
AS 4022
Cosmology
3
Dark Matter in Galaxy Clusters
Probes gravity on 10x larger scales
200
Kpc
z  0.0767
cz
d
H0
 320 Mpc
X-ray
Optical
Chandra X-ray Image of Abell 2029

The galaxy cluster Abell 2029 is composed of thousands of galaxies enveloped
in a gigantic cloud of hot gas, and an amount of dark matter equivalent to
more than a hundred trillion Suns. At the center of this cluster is an
enormous, elliptically shaped galaxy that is thought to have been formed from
the mergers of many smaller galaxies.
AS 4022
Cosmology
Cluster Masses from X-ray Gas
hydrostatic equilibrium :
dP
G M ( r)
 r g  r
dr
r2
gas law :
rkT
P
 mH
X - ray emission from gas gives : T(r), n e (r)  r(r),P(r)
r 2 dP
M ( r)  
G r (r) dr
Coma Cluster:
M(gas)~M(stars)~3x1013 Msun
often M(gas) > M(stars)
AS 4022
Cosmology
M/L~100-200
Reading: Cluster Mass
from fitting X-ray Gas
T~108K
M ~ 1014 Msun
g ~ 3x10-8 cm s-2
total mass
stars
gas
AS 4022
Cosmology
Galaxy Rotation Curves
HI velocities
V2 = G M( r ) / r
Flat rotation curves
Dark Matter Halos
M( r ) = V2 r / G
Spirals, Ellipticals:
M/L~4-10
Some dwarf
galaxies:
M/L
~ 100.
AS 4022
Cosmology
M/L ~ 4
Reading: Masses from Gravitational
Lensing
1/2


R
4 G M DLS
 E  E   2

DL  c
DL DS 
2
M
DL DS /DLS   E 



11
10 M sun
Gpc arcsec 
Use redshifts, z L , zS ,
for the angular diameter distances.
General agreement with Virial Masses.
AS 4022
Cosmology
Dark Matter Candidates
• MACHOS = Massive Compact Halo Objects
– Black holes
– Brown Dwarfs
– Loose planets
– Ruled out by gravitational lensing experiments.
• WIMPS = Weakly Interacting Massive Particles
– Massive neutrinos
– Supersymmetry partners
– Might be found soon by Large Hadron Collider in CERN
AS 4022
Cosmology
Microlensing in the LMC
Massive Compace Halo
Objects (MACHOS) would
magnify LMC stars dozens
of times each year. Only a
few are seen.
Long events -> high mass
Short events -> low mass
AS 4022
Cosmology
Reading: LMC Microlensing
says NO to MACHOs
Afonso et al. (2003)
AS 4022
Cosmology
Mass Density by Direct Counting
• Add up the mass of all the galaxies per unit volume
– Volume calculation as in Tutorial problem.
• Need representative volume > 100 Mpc.
• Can’t see faintest galaxies at large distance.
Use local Luminosity Functions to include fainter ones.
• Mass/Light ratio depends on type of galaxy.
• Dark Matter needed to bind Galaxies and Galaxy Clusters
dominates the normal matter (baryons).
• Hot x-ray gas dominates the baryon mass of Galaxy Clusters.
AS 4022
Cosmology
Schechter Luminosity Function
3 Schechter parameters :

L 
luminosity of a typical big galaxy


dn
dx  L   x  1 e x dx
dx
0
0
add up the mass (need mass/light ratio)
rL 
rM 
L


0
AS 4022
M dn
M
L
dx 
r
L dx
L L
Cosmology
Exponential
cutoff above L*
Log( dn / dL)
L  1011 L sun
luminosity of any galaxy :
L
L  x L x  
L
number of galaxies per unit luminosity
dn
( x) 
  x  e x
dx
add up the luminosities
slope = alpha
log ( L / L* )
Measure Schechter parameters using:
galaxy clusters
galaxy redshift surveys
Measure M/L for :
Nearby galaxies, galaxy clusters
Reading: Galaxy Luminosity Function
Schechter
parameters
depend on
galaxy type.
AS 4022
Cosmology
14th Concept: Mass / Light ratios
galaxy luminosity distribution



 L 
dn
L

 ( L)      exp  
dL
L 
 L 
rL 
luminosity density
e.g. blue light
mass density
rM 


L (L) dL
 2  0.7 10 8 h Lsun Mpc 3
M 
 L (L) dL
 L 
 M rcrit = 2.8 1011 M h 2 M sun Mpc 3
Universe :
Sun :
M /L 1400 M h 2 ~ 200 ( M /0.3)(h /0.7) 2
M /L  1
(by definition)
main sequence stars : M /L  M 3
(since L  M 4 )
comets, planets : M /L ~ 10912
Is our Dark Matter halo filled with MACHOs ?
NO. Gravitational Lensing results rule them out.
AS 4022
Cosmology
Galaxy Rotation Curves
M/L >30
Small galaxies : V( r ) rises
Large galaxies : V( r ) flat
AS 4022
Cosmology
Quasars Lensed by Galaxies
AS 4022
Cosmology