Download g=r 2

New Windows on the
Universe
Jan Kuijpers
• Part 1: Gravitation & relativity
J.A. Peacock, Cosmological Physics, Chs. 1 & 2
• Part 2: Classical Cosmology
Peacock, Chs 3 & 4
10/5/2004
New Windows on the Universe
Part 2: Classical cosmology
• The isotropic universe (3)
• Gravitational lensing (4)
10/5/2004
New Windows on the Universe
The isotropic universe
• The RW metric (3.1)
• Dynamics of the expansion (3.2-3.3)
• Observations (3.4)
10/5/2004
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Gravitational lensing
•
•
•
•
Lense equation; lensing potential (4.1)
Simple lenses (4.2)
Fermat’s principle (4.3)
Observations (4.4-4.6)
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The isotropic universe
The RW metric (3.1)
Define fundamental observers: at rest in local matter distribution
Global time coordinate t can be defined as proper time measured
by these observers
SR:
c d  c dt  dr
2
2
2
2
2
Here: c 2d 2  c 2dt 2  R 2 (t )[f 2 ( r )dr 2  g 2 ( r )d 2 ]
d 2  d 2  sin 2  d 2
Choose radial coordinate so that either f=1 or g=r2
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New Windows on the Universe
The RW metric (3.1)
c 2d 2  c 2dt 2  R 2 ( t )[dr 2  Sk2 ( r )d 2 ]
(k  1)
 sin r

Sk ( r )   sinh r (k  1)
r
(k  0)

Different definition of comoving distance r:
Sk ( r )  r
2

dr
2
2
2
2
2
2
2
c d  c dt  R ( t ) 
 r d 
2
 1  kr

Or dimensionless scale factor:
a( t ) 
R( t )
R0
Or isotropic form:
2
R
(t )
2
2
2
2
2
2
2
c d  c dt 
dr

r
d


2
2 
( 1  kr / 4 )
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The RW metric (3.1)
t
Or define conformal time:
cdt'

R( t')
0
c d  R ( t )  d  dr  r d
2
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2
2
2
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2
2
2

Redshift
Proper (small) separation of two fundamental observers:
d  R( t )dr
dv 
d
R
 R dr  d
dt
R
H( t ) 
Hubble’s law
R( t )
R( t )
Comoving distance between two fo’s is constant:
r
tobs

tem
cdt

R( t )
t obs dt obs

tem dtem
cdt
dt em
dt obs



R( t )
R( t em ) R( t obs )
 em
R( t obs )
 1 z 
 obs
R( t em )
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Dynamics of the expansion (3.2-3.3)
GR required: - Birkhoff’s theorem
- Integration constant
Friedmann eqns: Use RW metric in field eqns (problem 3.1):
2 2
8 G  R 2

R
c
2
R 
 kc 
3
3
4 GR 
3 p  Rc 2
R
  2 

3 
c 
3
Newton.:
1. Energy eqn.
Take time derivative +
energy conservation
2
d   c 2R 3    pd  R 3   Eqn. 2
 8 G 
kc 2


  m ( a )   r ( a )  v ( a )  1  2 2
2
c
3H
H R
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• Flatness problem
• Matter radiation equality:
m / r  ( 1  z )1  1  zeq
23900h2
• Recombination:
1+zrec=1000
• Matter dominated and flat: R(t )  t 2 / 3
• Radiation dominated and flat:
1/ 2
R(t )  t
• Vacuum energy (p=-c2 follows from energy conservation):
8 G v c 2
v 

2
3H
3H 2
Empty De Sitter space:
R  e Ht where H 
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8 G v

3
c 2
3
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Observations (3.4)
Luminosity distance: the apparent distance assuming
inverse square law for light intensity reduction
-Luminosity L : power output/4
-Radiation flux density S: energy received per unit area per sec
L
S 2 2
 Dlum  ( 1  z )R0Sk ( r )
2
R0 Sk ( r )( 1  z )
Redshift for photon energy and one for rate
Angular-diameter distance: the apparent distance based on
observed diameter assuming euclidean universe
 R( t em )Sk ( r )  DA 
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

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R0Sk ( r )
1 z
Gravitational lensing
Lensing equation; lensing potential (4.1)
Relativistic particles in weak fields (eq. 2.24):

d 2y
v 2  d
d
  1 2 
 2
2
dt
c  dy
dy

Bend angle (use angular diameter distances):
2
  2  ad
c
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Approximation: geometrically
thin lenses
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Gravitational lenses are flawed!!!
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Gravitational imaging
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Lensing equation
DLS
DL
DS

DS
  DL I  
 I S
DLS

 
where  I  S is
mapping between 2D object and image planes
Flux density from image is:
 Amplification is ratio of areas
S  I   image area  

I
3
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is invariant


 I

A



 S


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Lensing potential
2
2
2
  2  a d  2   2  d  2  2   d
c
c
c
 
DLS
I S 
  DL I     I
DS
DLDLS 8 G

 
2
Poisson
2
DS
c
c
2
surface density     d
DS
c2
critical sd c 
DLDLS 4 G
  ( ) 
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1
2

(

')ln



'
d
'

 c
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Notation:
- potential!
Simple lenses (4.2)
Multiple images
DLS
Circularly symmetric surface
mass density:
4G M   b 
 2
c
b
where b  Dl I is closest distance
and M   b  is mass in projection
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New Windows on the Universe
DL
Einstein ring
S
r
L

E
DL
DS
1/ 2
1/ 2
 2 RS DLS 
 2 RS 
E  
 

 DL 
 DS DL 
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O
Typical numbers
Einstein Radius point mass:
 M

 E   11

10
M


1/ 2
 DLDS / DLS 


Gpc


1 / 2
arcsec
ER isothermal sphere:
v

 DLS
E  
arcsec

 186 km/s  DS
2
Critical surface density:
DLS / Gpc 

 c 3.5
kg/m2
 DL / Gpc  DS / Gpc 
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  DL I  
Time delays

DS
 I S
DLS

b
DLS
DL
DS
Time lags between multiple images because of:
1. Path length difference:
2
c t g 
b
2
1  zL  
  I   S  DL
2
1  zL  
 I   S 
DLDS
2DLS
1  zL 
2. Reduced coordinate speed of light (static weak fields):
2 
2  2
2


c 2d 2   1  2  c 2dt 2   1  2  dr  c t p    1  zL  2 d
c 
c 
c


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Fermat’s principle (4.3)
Images form along paths where the time delay is stationary

  I , S

 
2
DLS
1

c t   I   S    I
2
1  zL  DLDS
Note: differentiation wrt I
recovers lens equation.
Example: from a to d:
introduction of increasing mass
(increasing -) leads to extra
Stationary points (minima,
Maxima, saddle points in )
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Caustics and catastrophe theory
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Lens model for flattened galaxy at two different relative distances.
a: density contours
c: caustics in image plane
b: time surface contours d: dual caustics in source plane
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Observations (4.4-4.6)
Light deflection around the Sun
1.75”
The Sun
1,75''
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2 RS
4GM
 

2
R c
R
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Newton/Soldner versus Einstein
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Total eclipse
21 september 1922
Western Australia,
92 stars (dots are
reference positions,
lines displacements,
enlarged!)
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Strong lensing
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Modelling
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Robert J. Nemiroff
1993:
Sky as seen past
a compact star,
1/3 bigger than its
Schwarzschild
radius, and at a
distance of 10
Schwarzschild radii.
The star has a
terrestrial surface
topography
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Sirius
Orion
Orion
Sirius
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