10/5/2004New Windows on the Universe Jan Kuijpers Part 1: Gravitation & relativityPart 1:...

10/5/2004 New Windows on the Universe

New Windows on the New Windows on the UniverseUniverseJan KuijpersJan Kuijpers

• Part 1: Gravitation & relativityPart 1: Gravitation & relativityJ.A. Peacock, Cosmological Physics, Chs. 1 & 2J.A. Peacock, Cosmological Physics, Chs. 1 & 2

• Part 2: Classical CosmologyPart 2: Classical CosmologyPeacock, Chs 3 & 4Peacock, Chs 3 & 4


Part 2: Classical cosmologyPart 2: Classical cosmology

• The isotropic universe (3)The isotropic universe (3)

• Gravitational lensing (4)Gravitational lensing (4)


The isotropic universeThe isotropic universe

• The RW metric (3.1)The RW metric (3.1)

• Dynamics of the expansion (3.2-3.3)Dynamics of the expansion (3.2-3.3)

• Observations (3.4)Observations (3.4)


Gravitational lensingGravitational lensing

• Lense equation; lensing potential (4.1)Lense equation; lensing potential (4.1)

• Simple lenses (4.2)Simple lenses (4.2)

• Fermat’s principle (4.3)Fermat’s principle (4.3)

• Observations (4.4-4.6)Observations (4.4-4.6)


The RW metric (3.1)The RW metric (3.1)

The isotropic universeThe isotropic universe

Define fundamental observers: at rest in local matter distributionDefine fundamental observers: at rest in local matter distributionGlobal time coordinate t can be defined as proper time measuredGlobal time coordinate t can be defined as proper time measured by these observersby these observers

22 2 2 2

2 2 2 2 2 2 2 2 2

2 2 2 2

SR:

Here:

c d c dt dr

c d c dt R (t )[f ( r )dr g ( r )d ]

d d sin d

Choose radial coordinate so that either Choose radial coordinate so that either f=1f=1 or or g=rg=r22


The RW metric (3.1)The RW metric (3.1)2 2 2 2 2 2 2 2

( 1)

( 1)

( 0)

k

k

c d c dt R (t )[dr S ( r )d ]

sin r k

S ( r ) sinhr k

r k

Different definition of comoving distance Different definition of comoving distance rr:: kS (r ) r2

2 2 2 2 2 2 221

drc d c dt R (t ) r d

kr

Or dimensionless scale factor:Or dimensionless scale factor:

Or isotropic form: Or isotropic form:

0

R(t )a(t )

R

2

2 2 2 2 2 2 22 21 4

R (t )c d c dt dr r d

( kr / )


The RW metric (3.1)The RW metric (3.1)

Or define conformal time:Or define conformal time:0

t cdt'

R(t ')

2 2 2 2 2 2 2c d R (t ) d dr r d


RedshiftRedshift

Proper (small) separation of two fundamental observers:Proper (small) separation of two fundamental observers:

d R(t )dr

d Rdv Rdr d

dt R

R(t )H(t )

R(t )

Hubble’s lawHubble’s law

Comoving distance between two fo’s is constant:Comoving distance between two fo’s is constant:

1

obs obs obs

em em em

t t dt

em obs

em obst t dt

em obs

obs em

cdt cdt dt dtr

R(t ) R(t ) R(t ) R(t )

R(t )z

R(t )


Dynamics of the expansion (3.2-3.3)Dynamics of the expansion (3.2-3.3)

GR required: - Birkhoff’s theoremGR required: - Birkhoff’s theorem - Integration constant- Integration constantFriedmann eqns: Use RW metric in field eqns (problem 3.1):Friedmann eqns: Use RW metric in field eqns (problem 3.1):

2 2 2 22

2

2

8

3 3

4 3

3 3

G R R cR kc

GR p RcR

c

Newton.: Newton.: 1. Energy eqn.1. Energy eqn.Take time derivative +Take time derivative +energy conservationenergy conservation

2 3 3 Eqn. 2d c R pd R

2

2 2 2

81

3 m r vc

G kc(a ) (a ) (a )

H H R


• Flatness problemFlatness problem• Matter radiation equality: Matter radiation equality:

• Recombination: Recombination: 1+zrec=1000• Matter dominated and flat:Matter dominated and flat:• Radiation dominated and flat:Radiation dominated and flat:

• Vacuum energy Vacuum energy (p=-c2 follows from energy conservation)follows from energy conservation):

1 21 1 23900m r eq/ ( z ) z h

2

2 2

2

8

3 3Empty De Sitter space:

8 where

3 3

vv

Ht v

G c

H H

G cR e H

2 3

1 2

/

/

R(t ) t

R(t ) t


Observations (3.4)Observations (3.4)

Luminosity distance:Luminosity distance: the apparent distance assuming the apparent distance assuming inverse square law for light intensity reductioninverse square law for light intensity reduction-Luminosity Luminosity L : power output/4 : power output/4-Radiation flux density Radiation flux density S: energy received per unit area per sec: energy received per unit area per sec

02 2 20

11 lum k

k

LS D ( z )R S ( r )

R S ( r )( z )

Redshift for photon energy and one for rateRedshift for photon energy and one for rateAngular-diameter distance:Angular-diameter distance: the apparent distance based on the apparent distance based onobserved diameter assuming euclidean universeobserved diameter assuming euclidean universe

0

1k

em k A

R S (r )R(t )S ( r ) D

z


Lensing equation; lensing potential (4.1)Lensing equation; lensing potential (4.1)

Gravitational lensingGravitational lensing

2 2

2 21 2

d y v d d

dt c dy dy

Relativistic particles in weak fields (eq. 2.24):Relativistic particles in weak fields (eq. 2.24):

Bend angle (use angular diameter distances):Bend angle (use angular diameter distances):

2

2a d

c Approximation: geometricallyApproximation: geometrically

thin lensesthin lenses


Gravitational lenses are flawed!!!Gravitational lenses are flawed!!!


Gravitational imaging


Lensing equationLensing equation

DLDLS

DS

where is

mapping between 2D object and image planes

SI S I SL I

LS

DD

D ��

Flux density from image is: Flux density from image is:

3

Amplification is ratio of areas

is invariantI

S

S I image area

I A

��

��


Lensing potentialLensing potential

2 22 2 2

2 2 2a d d d

c c c ��

LSI S IL I

S

DD

D

��Notation: Notation: - potential!- potential!

22

2

82 Poisson

surface density

critical sd 4

L LS

S c

Sc

L LS

D D G

D c

d

D c

D D G

21

c

( ) ( ') ln ' d '

��


2

4

where is closest distance

and is mass in projection l I

M bG

c b

b D

M b

Multiple imagesMultiple imagesSimple lenses (4.2)Simple lenses (4.2)

DLS

DL

Circularly symmetric surfaceCircularly symmetric surfacemass density:mass density:


Einstein ringEinstein ring

S OL

rE

SD

LD

1/ 2 1/ 22 2S LS S

ES L L

R D R

D D D


Typical numbersTypical numbers

Einstein Radius point mass:Einstein Radius point mass:

ER isothermal sphere:ER isothermal sphere:

Critical surface density:Critical surface density:

1 2 1 2

11

2

2c

arcsec10 M Gpc

arcsec186 km/s

Gpc3 5 kg/m

Gpc Gpc

/ /

L S LSE

v LSE

S

LS

L S

M D D / D

D

D

D /.

D / D /


DLDLS

DS

SI SL I

LS

DD

D ��

b

Time delaysTime delays

Time lags between multiple images because of:Time lags between multiple images because of:1.1. Path length difference:Path length difference:

2. Reduced coordinate speed of light (static weak fields):2. Reduced coordinate speed of light (static weak fields):

2

1 1 12 2 2

I S L I S L Sg L L L

LS

D D Dbc t z z z

D

22 2 2 22 2 2

2 2 21 1 1p Lc d c dt dr c t z d

c c c


Fermat’s principle (4.3)Fermat’s principle (4.3)Images form along paths where the time delay is stationaryImages form along paths where the time delay is stationary

21

1 2LS

I S I S I

L L S

D, c t

z D D

��

Note: differentiation wrt Note: differentiation wrt II recovers lens equation.recovers lens equation.

Example: from a to d: Example: from a to d: introduction of increasing mass introduction of increasing mass (increasing -(increasing -) leads to extra) leads to extraStationary points (minima,Stationary points (minima,Maxima, saddle points in Maxima, saddle points in ))


Caustics and catastrophe theoryCaustics and catastrophe theory


Lens model for flattened galaxy at two different relative distances.Lens model for flattened galaxy at two different relative distances.a: density contours c: caustics in image planea: density contours c: caustics in image planeb: time surface contoursb: time surface contours d: dual caustics in source plane d: dual caustics in source plane


Observations (4.4-4.6)Observations (4.4-4.6)Light deflection around the SunLight deflection around the Sun

The SunThe Sun 1.75”1.75”

1,75'' 2

24 SRGM

R c R


Newton/Soldner versus EinsteinNewton/Soldner versus Einstein



Total eclipseTotal eclipse21 september 192221 september 1922Western Australia,Western Australia,92 stars (dots are92 stars (dots arereference positions,reference positions,lines displacements,lines displacements,enlarged!)enlarged!)


Strong lensingStrong lensing



ModellingModelling






Robert J. NemiroffRobert J. Nemiroff1993:1993:Sky as seen pastSky as seen pasta compact star,a compact star,1/3 bigger than its1/3 bigger than itsSchwarzschildSchwarzschildradius, and at a radius, and at a distance of 10 distance of 10 Schwarzschild radii.Schwarzschild radii.The star has a The star has a terrestrial surfaceterrestrial surfacetopographytopography


OrionOrionOrion Orion

SiriusSiriusSirius Sirius


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