Download ppt - 1 Astrophysical Cosmology Andy Taylor Institute for Astronomy, University of Edinburgh, Royal Observatory Edinburgh

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Astrophysical Cosmology

Andy TaylorInstitute for Astronomy,

University of Edinburgh,

Royal Observatory Edinburgh

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Lecture 1

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The large-scale distribution of galaxies

5

Temperature Variations in the Cosmic Microwave Background

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Properties of the Universe• Universe is expanding.• Components of the Universe are:

• Universe is 13.7 Billion years old.• Expansion is currently accelerating.

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1920’s: The Great Debate

Are these nearby clouds of gas?

Or distant stellar systems (galaxies)

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• In 1924 Edwin Hubble finds Cepheid Variable stars in M31.

• Cepheid intrinsic brightness correlate with variability (standard candle), so can measure their distance.

• Measured 3 million light years (1Mpc) to M31.

Eyenear far

brighter fainter

Edwin Hubble

1920’s: The Great Debate

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The Expanding Universe• Between 1912 and 1920 Vesto Slipher finds most

galaxy’s spectra are redshifted.

Slipher is first to suggest the Universe is expanding !

Vesto Slipher

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In 1929 Hubble also finds fainter galaxies are more redshifted. Infers that recession velocities increase with distance.

Hubble’s Law

HDV

DV

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The Expanding Universe

1/tH HD,

Time Velocity x Distance

V

1. Grenade Model:

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1/tH HD,


V

2. Scaling Model:

x(t) = R(t) x0

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1/tH HD,


V

D

t

Now Hubble Time:tH=1/H

H=70 km/s/MpctH=14Gyrs

tH

0

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In 1915 Albert Einstein showed that the geometry of spacetime is shaped by the mass-energy distribution.

General Theory of Relativityrequired to describe the evolution of spacetime.

Albert Einstein

Relativistic Cosmologies

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• Cosmological Coordinates (t , x):• How do we lay down a global coordinate system? • In general we cannot.

• Can we lay down a local coordinate system?• Yes, can use Special Relativity locally, if we can cancel gravity.

• We can cancel gravity by free-falling (equivalence principle).


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• Equivalence Principle:

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• Equivalence Principle:

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• In free-fall, a Fundamental Observer locally measures the spacetime of Special Relativity. • Special Relativity Minkowski-space line element:

• So all Fundamental Observers will measure time changing at the same rate, dt.

• Universal cosmological time coordinate, t.

22222222 dzdydxdtcdcds

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• How can we synchronize this Universal cosmological time coordinate, t, everywhere?

• With a Symmetry Principle:• On large-scales Universe seems isotropic (same in all directions, eg, Hubble expansion, galaxy distribution, CMB).

• Combine with Copernican Principle (we’re not in a special place).

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Relativistic Cosmologies• Isotropy + Copernican Principle = homogeneity

(same in all places)

A

0

B

12

So 2=1=0.

So uniform density everywhere

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Relativistic Cosmologies• Isotropy + homogeneity = Cosmological Principle

A

0

B

12

So 2=1=0.

So uniform density everywhere

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Relativistic Cosmologies• With the Cosmological Principle, we have uniform density everywhere.

• Density will decrease with expansion, so = (t).

• So can synchronize all Fundamental Observers clocks at pre-set density, 0, and time, t0:

000 )( : tt

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Relativistic Cosmologies• What is the line element (metric) of a relativistic cosmology?

• Locally Minkowski line element (Special Relativity):22222222 dzdydxdtcdcds

t

x

Worldline

Proper time: ddt – Universal time

dx/c

• Spacetime Diagram:

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Lecture 2

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Relativistic Cosmologies• A general line element (Pythagoras on curved surface):

• Minkowski metric tensor:

• We have Universal Cosmic Time of Special Relativity, t, so

),,,( 22 zyxctxdxdxgdc

1000

0100

0010

0001

g

metric ofpart Spatial23

23

2222

ddtcdc

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• What is spatial metric, 32?

• From Cosmological Principle (homogeneity + isotropy) spatial curvature must be constant everywhere.

• Only 3 possibilities:• Sphere – positive curvature• Saddle – negative curvature• Flat – zero curvature.


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• What is form of 32?

• Consider the metric on a 2-sphere of radius R, 2

2:


) sin( 222222 ddRd

d

d

d

R

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)sin( 222222 ddRd

dr

d

d

R)sin( 222222 rddrRd

• The metric on a 2-sphere of radius R:

• Now re-label as r and as :

where r = (0,) is a dimensionless distance.

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• Can generate other 2 models from the 2-sphere:


)(

)sinh(

)sin(

222222

222222

,

222222

drdrRd

rddrRd

rddrRd

Rr

iRRirr

k = +1

k = - 1

k = 0

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• General 3-metric for 3 curvatures:


k = +1

k = - 1

k = 0

1 ),sinh(

0 ,

1 ),sin(

)(

))(( 222222

kr

kr

kr

rS

drSdrRd

k

k

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• Different properties of triangles on curved surfaces:


k = 0

k = +1

r d

sin r d

r

r

d

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• Different properties of triangles on curved surfaces:


k = +1

k = - 1

k = 0

1 ),sinh(

0 ,

1 ),sin(

)(

))(( 222222

kr

kr

kr

rS

drSdrRd

k

k d

sin r d

sinh r d

r d

r

r

r

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• Finally add extra compact dimension:

• Promote a 2-sphere to a 3-sphere

• So metric of 3-sphere is


22222

2

222223

sin

),)((

dddd

drSdrRd k

2222 sin ddd

d

d

) sin)(()( 22222222 ddrSdrdrSdr kk

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• The Robertson-Walker metric generalizes the Minkowski line element for symmetric cosmologies:


k = +1

k = - 1

k = 0

))(( 22222222

2222222

drSdrRdtcdc

drdrdtcdc

k

- The Robertson-Walker Metric

Invariant proper time

UniversalCosmic

timeScale factor

Co-moving radial

distance

Co-moving angulardistance

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• Alternative form of the Robertson-Walker metric:


k = +1

k = - 1

k = 02

1

22

2222

22222222

1

1sin

sin

1

))(sin(

ydy

yd

ry

ydy

dyRdtc

drdrRdtcdc

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• Alternative form of the Robertson-Walker metric:


k = +1

k = - 1

k = 02

1

22

2222

22222222

1

1

)(

1

))((

kydy

ydS

rSy

ydky

dyRdtc

drSdrRdtcdc

k

k

k

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• The Robertson-Walker models.


k = +1

k = - 1

k = 0

))(( 22222222 drSdrRdtcdc k

• k = +1: positive curvature everywhere, spatially closed, finite volume, unbounded.

• k = - 1: negative curvature everywhere, spatially open, infinite volume, unbounded.

• k = 0: flat space, spatially open, infinite volume, unbounded.

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• The Robertson-Walker models.


• We have defined the comoving radial distance, r, to be dimensionless.

• The current comoving angular distance is: d = R0Sk(r) (Mpc).

• The proper physical angular distance is: d(t) = R(t)Sk(r) (Mpc).

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Lecture 3

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rtRtd )()(

• Superluminal expansion:The proper radial distance is

The proper recession velocity is:

?)()()( cdR

RrtRtdtv

What does this mean?Locally things are not moving (just Special Relativity).But distance (geometry) is changing. No superluminal information exchange.

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Light Propagation

• How does light propagate through the expanding Universe?

• Let a photon travel from the pole (r=0) along a line of constant longitude (d=0,d=0).

• The line element for a photon is a null geodesic (zero proper time):

0)( 222222 drtRdtcdc

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Light Propagation

• Equation of motion of a photon:

The comoving distance light travels.

t

tR

cdttr

drtRdtc

0

2222

)()(

)(

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Light Propagation

• Let’s assume R(t)=R0(t/t0):

1 ,1

'

''

)'(

')(

0

1

0

0

00

0

0

t

t

t

t

R

ct

dttR

ct

tR

cdttr

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Causal structure

• Lets assume > 1:

tt

tctrtRtl

ttR

cttr

ttRtR

1

2

10

20

200

)()()(

11 )(

:)/()(

t

lFor t >> t1 , r is constant. This is called an Event Horizon.As t1 tends to 0, l(t) diverges, everywhere is causally connected.

l

R

t1

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Causal structure

• Lets assume < 1:

cttrtRtl

tR

tctr

ttRtR

2)()()(

2)(

:)/()(

2/1

0

2/10

2/100

t

lAt early times all points are causally disconnected.The furthest that light can have travelled is called the Particle Horizon.

t=0

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Cosmological Redshifts• Consider the emission and observation of light:

0

00

Rrd

r

tt

111

)(1

1

)(

1

0

rtRd

rr

ttt

t tRdt

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Cosmological Redshifts• Consider the emission and observation of light:

0

00

Rrd

r

tt

111

)(1

1

)(

1

0

rtRd

rr

ttt

t tRdt

0

000

Rrd

r

ttt

1111

)(1

11

)(

11

00

rttRd

rr

ttttt

tt tRdt

A bit later:

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Cosmological Redshifts• But the comoving position of an observers is a constant:

)()( 0

0

1

1

)()()()()(1

00

0

11

1

1

0

11

00

1

0

tR

t

tR

t

rtt

t tRdt

tt

t tRdt

t

t tRdt

tt

tt tRdt

t

t tRdt

Say the wavelength of light is = ct:

1

1

0

0

RR

0

1

1

0)1(R

R

v

vz so

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Cosmological Redshifts• Can also understand as a series of small Doppler shifts:

R

Rt

R

RtH

c

V

v

v

1 Rv

d=ct

t=0 t=t

V=Hd=cHt

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Decay of particle momentum• Every particle has a de Broglie wavelength:• So momentum (seen by FO’s) is redshifted too:

• Why? (“Hubble drag”, “expansion of space”?)

Rp

vp

/1

d=Rr, V=Hd

t=0 t=t

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Lecture 4

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The Dynamics of the ExpansionIn 1922 Russian physicist Alexandre Friedmann

predicted the expansion of the Universe

Rr

Rr

GMmrRmE

EPEKEnergyTotal

TOT

2)(2

1

...

Newtonian Derivation:

m

M=4(Rr)3/3

Birkhoff’s TheoremrRV

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The Dynamics of the ExpansionIn 1922 Russian physicist Alexandre Friedmann

predicted the expansion of the Universe

2

22

2

3

4

R

kcG

R

RH

Friedmann Equation:Rr

m

M=4(Rr)3/3

Birkhoff’s TheoremV

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ctR

k

kcR

R

kcRG

R

1

0

3

8

22

2

222

• So a low-density model will evolve to an empty, flat expanding universe.

• There is a direct connection between density & geometry:

Geometry & Density

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03

8

3

8

2

2

22

k

GH

RR

cGH

• So with the right balance between H and , we have a flat model.

• There is a direct connection between density & geometry:

Geometry & Density

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• We can define a critical density for flat models and hence a density parameter which fixes the geometry.

Critical density & density parameter

k = +1 c >1

k = - 1 c

k = 0 c

2

2

3

88

3

H

GG

H

c

c

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• How does evolve with time?


)()(1)(

3

81

3

8

22

2

22

2

2

2

22

tHtR

kct

HR

kc

H

G

R

kcGH

t

1

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• What is present curvature length?


Mpch

kkH

cR 1

2/12/1

00

13000

1

Define a dimensionless Hubble parameter:

11100 Mpckms

Hh

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• What is present density?


1SUN

211

32260

)M(1078.2

)(1088.1

Mpch

mkgh

Or 1 small galaxy per cubic Mpc.Or 1 proton per cubic meter.

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The meaning of the expansion of space

)( 2222222 drdrdtcdc

22 11/' tcr

cV tttt

• Consider an expanding empty, spatially flat universe. c.f. a relativistic Grenade Model:

• Minkowski metric:

• Let v=Hr, H=1/t so v=r/t.

• Switch to comoving frame:

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The meaning of the expansion of space

22

2'

22222

)(1' dr

drdtcdc

ctr

22

'1

ct

r• Rewrite in terms of t’ (comoving time):

• Hence in the comoving frame:

• but this is a k=-1 open model with R=ct!

• So what is curvature?

• And is space expanding ?

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3/2

2

3

8

tR

GH

The matter dominated universe• Consider a universe with pressureless matter (dust, galaxies, or cold dark matter). • As Universe expands, density of matter decreases: =0(R/R0)-3.

• Consider a flat model: k=0, =1.R

t

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Lecture 5

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The matter dominated universe• The spatially flat, matter-dominated model is called the Einstein-de Sitter model.

R

t

Gyrs 3.93

2

3

2

00

00

3/2

Ht

tR

RH

tR

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The matter dominated universe• Consider an open or closed, matter-dominated universe.• Define a conformal time, dcdt/R(t).

R

t

)])(()[( 2222222 drSdrdtRdc k

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The matter dominated universe• Consider a closed, matter-dominated universe.• Define a conformal time, dcdt/R(t).

R

t

)sin()(

)cos1()(

2'

*

*

2

**

2

*

Rct

RR

R

R

R

R

R

R

2

300

* 3

4

c

RGR

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The matter dominated universe• Consider an open or closed, matter-dominated universe.• Define a conformal time, dcdt/R(t).

R

t

)()(

)1()(

2'

*

*

2

**

2

*

k

k

SkRct

CkRR

R

Rk

R

R

R

R

2

300

* 3

4

c

RGR

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The matter dominated universe

Expand forever

Eventual recollapse

“Big Bang’’ “Big Crunch’’

k = -1

k = +1

< 1

= 1

> 1

• So for matter-dominated models geometry/density=fate.

k = 0

69

The radiation dominated universe• As Universe expands, density of matter decreases:

m=0m(R/R0)-3.• Radiation energy density: r=0r(R/R0)-4.• At early enough times we have radiation-dominated Universe.

Log R

Log

m

rFor T(CMB)=2.73K, zeq=1000.

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The radiation dominated universe• At early enough times we also have a flat model: k=0

240,

30,

2

2240,

30,

2

)/()/(3

8

1 ,0

)/()/(3

8

RRRRRG

R

R

kcRRRRRG

R

oom

oom

2/1240,

2 ,3

8tRRR

GR o

So Particle Horizon!

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The radiation dominated universe• Timescales:

• Matter-dominated: R~t2/3

•Radiation dominated: R~t1/2

m

m

Gt

G

tH

6

1

3

8

3

2

Gt

G

tH

32

1

3

8

2

1

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The radiation dominated universe• Spatial flatness at early times:

Recall:

How close to 1 can this be? At Planck time (t=10-43s)?

02

2

1)(

1)(t

tk

RH

kct

60

0

1011)( t

tt pl

t

1

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Energy density and Pressure

• Thermodynamics and Special Relativity:

• So energy-density changes due to expansion.

0)/(3

)()(2

323

cpH

RpdcRd

pdVdE

2/33 cHpH

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Energy density and Pressure

• For pressureless matter (CDM, dust, galaxies):

• Radiation pressure:

•Cf. electromagnetism.

3 0 Rp

)/(3 2cpH

2

4

3

1

4

cp

HR

• Conservation of energy:

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Lecture 6

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Pressure and Acceleration

• Time derivative of Friedmann equation:

•Acceleration equation for R:

)2(3

82

3

8

2

222

RRRG

RR

kcRG

dt

dR

dt

d

)/(3 2cpH

RcpG

R 2/33

4

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Vacuum energy and acceleration

• Gravity responds to all energy. • What about energy of the vacuum?

• Two possibilities:1. Einstein’s cosmological constant.2. Zero-point energy of virtual particles.

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Einstein’s Cosmological Constant

Einstein introduced constant to make Universe static.

33

82

22

R

kcGH

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Einstein’s Cosmological Constant• Problem goes back to Newton (1670’s).

• Einstein’s 1917 solution:

rg rG4 4 22 G

0g G4

4

2 G

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Einstein’s Cosmological Constant

Einstein called this:“My greatest blunder.”

• But this is not stable to expansion/contraction.

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British physicist Paul Dirac predicted antiparticles. Werner Heisenberg’s Uncertainty Principle:

Vacuum is filled with virtual particles. Observable (Casmir Effect) for electromagnetism.

Zero-point vacuum energy

+ -+-

vnEn

2

1

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The Vacuum Energy Problem

• So Quantum Physics predicts vacuum energy.• But summation diverges.• If we cut summation at Planck energy it

predicts an energy 10120 times too big. Density of Universe = 10 atoms/m3

Density predicted = 1 million x mass of the Universe/m3

• Perhaps the most inaccurate prediction in science? Or is it right?

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Vacuum energy• Vacuum energy is a constant everywhere: V ~ R0

• Thermodynamics: Consider a piston:

2

33232

cp

RdpRdcRcd

VV

VVV

The equation of state of the vacuum.

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Vacuum energy and acceleration• Effect of negative pressure on acceleration:

• So vacuum energy leads to acceleration.

VVV cpR 2)/3( 2

Ht

V

eRR

HRR

H

0

2 const

R

t• Eddington: is the cause of the expansion.

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General equation of State• In general should include all contributions to energy-density.

20,0,

4,

3,

20

02

22

)1(

/)( ,3

8

aaaH

RtRaR

kcGH

mVoomV

Log R

Log

m

r

V

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General equation of State• In general must solve F.E. numerically.• Geometry is still governed by total density:

0 1 2 m

1V

0

-1

OPEN

CLOSEDFLAT

87

General equation of State• In general must solve F.E. numerically.• But in general no geometry-fate relation:

0 1 2 m

1V

0

-1

OPEN

CLOSEDFLAT

RECOLLAPSE

R t

EXPAND FOREVER

R t

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General equation of State

0 1 2 m

1V

0

-1

OPEN

CLOSEDFLAT

RECOLLAPSE

R t

EXPAND FOREVER

R t

R t

No singularity

89

Age and size of Universe• Evolution of redshift:

where

Age of the universe:

)()1(

)()1(

20

0

zHzR

RR

dt

dz

tR

Rz

20,0,

4,

3,

20

2 )1)(1()1()1()( zzzHzH mVoomV

0 )()1()(

zHz

dzzt z=0

z=infinity

90

Age and size of Universe• Usually evaluate t0 numerically, but approximately:

3.0

00 )1(3.07.0

3

2 VmHt

m

1

H0t0

2/3

0

v=0

v+m=1

91

Age and size of Universe• Comoving distance-redshift relation: dr=cdt/R=cdz/R0H(z).

m=1 m=0

1

00

2/1

00

)1()1(2

)1(12

zzH

crR

zH

crR

r(z)

z

w = 0

w = -1 v=1, R0r=cz/H0

92

Lecture 7

93

Age and size of Universe• Comoving distance-redshift relation: dr=cdt/R=cdz/R0H(z).

instein de Sitterm=1 de SitterV=1

zH

crR

zH

crR

00

2/1

00 )1(1

2

r(z)

z

m = 1

v=1

94

Observational Cosmology• Size and Volume:

• Start from line element:

•Angular sizes:

•Volumes:

22222222 )()( drSdrtRdtcdc k

k = 0

k = +1

r d

sin r d

r

r

d dzrSzRdl k )()(

drdzrSzRzdV k )]([)()( 23

95

Observational Cosmology• Angular diameter distance:

• Einstein-de Sitter universe de Sitter universe

)]([)()( zrSzRzD kA

2/1

0

)1(1)1(

2)(

z

zH

czDA

DA(z)

0 1 z

r=2c/H0

)1()(

0 zH

czzDA

dS

EdS

96

Observational Cosmology• Angular size:

• Einstein-de Sitter universe de Sitter universe

)()(

zD

dlzd

A

2/1

0

)1(1)1(

2)(

z

zH

czDA

d(z)

0 1 z

)1()(

0 zH

czzDA

1/zz EdS

dS

97

Observational Cosmology• Luminosity and flux density:

• Euclidean space:

• Curved, expanding space:•L=E/t~(1+z)-2

•Lv=dL/dv d/dv0=(1+z)d/dv•v=(1+z)v0

24

)(

r

vLS v

v

2220

220

00

)1(41

)1()(4

])1[()(

zSR

LS

z

dvS

zzrSR

vzLvS

k

TOTvTOT

k

vv

98

Observational Cosmology• Surface brightness, Iv:

dIS vv

33/3 )1(

])1[()/(

1

1

vz

vzBTvn

ev

I vkThv

v

d

So the high-redshift objects are heavily dimmed by expansion.

99

Observational Cosmology• Luminosity distance:

• Einstein-de Sitter:

• de Sitter:

)]([)1()( ,)(4 02

zrSRzzDzD

LS kL

L

TOTTOT

)1()(

)1(1)1(2

)(

0

2/1

0

zH

czzD

zH

zczD

L

L

DL(z)

0 1 z

dS

EdS

100

Observational Cosmology• Magnitude-redshift relation:

• The K-correction: redshift shifts frequency & passbands.

)(10

)( log5 zK

pc

zDMm L

][

])1[()1(log5.2)(

vL

vzLzzK

v

v

Lv

dv v(1+z)dv

101

Observational Cosmology• Galaxy Counts: Number of galaxies on sky as function of flux, N(>S).

•Euclidean Model: Consider n galaxies per Mpc3 with same luminosity, L, in a sphere of radius D.

2/3

2/33

2/1

)()(

SSnVSN

SDV

SD

102

Observational Cosmology• Olbers Paradox:

The Sky brightness:

which diverges as S goes to zero.

Too many sources as D increases, due to increase in volume.

02/1

0

2/3

0

)(1

SdSS

SdS

SdNdS

AI

103

Lecture 8

104

Observational Cosmology• Olbers Paradox: Why is the night sky dark?

which diverges as S goes to zero.

Too many sources as D increases, due to increase in volume.

02/1

0

2/3

0

)(1

SdSS

SdS

SdNdS

AI

D

105

Observational Cosmology• Relativistic Galaxy Count Model: =1, k=0 Einstein-de Sitter model:

Flux density: Lv~v

Number counts. z<<1 z>>1

30

2/1

)(3

))1(1(2

zrRA

V

zr

)1(22/112

0

)1())1(1()1()(4

zz

zrR

LS v

v

V(z)

0 1 z

EuclideanEdS

32

2/33

0)(

)(

HcVSN

SzVSN

106

Observational Cosmology• Relativistic Galaxy Count Model:

Log N(>S)

S

Euclidean

z>>1 sources

S-3/2

Counts converge due to finite volume/age /distance at high-z.So solves Olber’s paradox.

107

Distances and age of the Universe

• Cosmological Distances: c/H0 = 3000h-1Mpc.• Cosmological Time: 1/H0 = 14 Gyrs.

•Recall our solution for age of Universe:

So if we know m, V and H0, we can get t0. Or if we know H0 and t0 we can get m and V.

3.0

0

0

0

)1(3.07.03

2

)()1(

VmH

zHz

dzt

108


• Estimating the age of the Universe, t0:

• Nuclear Cosmo-chronology:Natural clock of radioactive decay, ~10Gyrs.

• Heavy elements ejected from supernova into ISM:Thorium (232Th) => Lead (208Pb) 20 GyrsUranium (235U) => Lead (207Pb) 1 GyrUranium (238U) => Lead (206Pb) 6.5 Gyrs

109


• Estimating the age of the Universe:

• No new nuclei produced after solar system forms, just nuclear decay:

• But don’t know D0, so how to measure D?

P0

D0

P D)1(

)1(/

/0

t

t

eP

eP

PD

110

Distances and age of the Universe• Estimating the age of the Universe:

• Take ratio with a stable isotope of D, S.

• Plot D/S versus P/S

D0/S

)1( /0 teS

P

S

D

S

D

D/S

P/S

Slope (et/-1)

Meterorites: SS = 4.57(+/-0.04) Gyrs+ Nuclear theory: MW = 9.5 Gyrs

111

Surface Temperature

Log

(L

umin

osit

y) Red GiantBranch

Main Sequence

Turnoff

•Age from Stellar evolution: t~M/L


GC = 13-17 Gyrs.

Recall for EdS t0=9.3Gyrs!

112

• Local distance:


• Use Cepheid Variables (cf Hubbles measurement to M31).• Mass M=3 – 9Msun

• Moving onto Red Giant Branch Luminosity~(Period)1.3

L~ 1/D2

D~1/L1/2

Need to know DLMC=51kpc +/- 6%.From parallax, or SN1987a.

Surface Temperature

Red GiantBranch

113

• Larger distances:


• Use supernova Hubble diagram.• SN Ia, Ib, II.• SNIa standard candles.

• Nuclear detonation of WD.

• HST Key programme:H0 = 72+/-8kms-1.

(error mainly distance to LMC)

White dwarf

Red Giant

114

• So now know H0 and t0 so now know H0t0=0.96.• What can we infer about m and V?

• This implies that if V=0, m=0….• Or V=2.3m! Implies vacuum domination…• And if flat (k=0) V=0.7, m=0.3.

m

1

H0t0

2/3

0

v=0

v+m=1


115

• Can measure m and V from luminosity distances to standard candles – the supernova Hubble diagram.

Cosmological Geometry

DL(z)

0 1 z

dS

EdS

z

VmL zzH

c

zH

dzzzD

0

2

0

4/)22()(

)1()(

116

• The supernova Type Ia are fainter than expected given their redshift velocity.

Type 1a supernova Hubble Law

Faintness= - log DL Decelerating

Universe

AcceleratingUniverse

Redshift


117

• Can measure m and V from luminosity distances to standard candles – the supernova Hubble diagram.


118

Lecture 9

119

The thermal history of the Universe• Recall that as Universe expands: m=0m(R/R0)-3 r=0r(R/R0)-4 = v0 (R/R0)0

• At early enough times we have radiation-dominated Universe.

Log R

Log

m

rFor T(CMB) = 2.73K today.

v

120

The thermal history of the Universe• Also expect a neutrino background, v=0.68 (see later).

Log R

Log

m

rFor T(CMB) = 2.73K today.

v

23

4

102.4

h

aT

r

r

vr

473.2

2

0,

0,

0,

0,

40,

30,

)(900,231

1)1(

)1(

KT

mr

m

r

meq

r

m

r

m

hz

z

z

121

The thermal history of the Universe• How far back do we think we can go to in time?• To the Quantum Gravity Limit:

• Quantum Mechanics: de Broglie:• General Relativity: Schwartzschild:

sc

Gt

mc

G

GeVG

cm

pl

pl

pl

455

353

19

10

10

10

2

2

22

c

Gmmcv

c

S

dB

Log dB

S

Log mmpl

Quantum Classical

122

Thermal backgrounds• If expansion rate < interaction rate we have thermal equilibrium.• Shall also assume a we have perfect gas.

• Occupation number for relativistic quantum states is:

2242 cpcm f(x)

x=(-)/kT

+ fermions– bosons 1

1)(

/])([ kTpe

pf

fermions

bosons

e-(-)/kT – Boltzmann

1/2

1

123

Thermal backgrounds

• The Chemical Potential, :

• A change of energy when change in number of particles.• As in equilibrium, expect total energy does not change:

dNpdVTdSdE

0

124

Thermal backgrounds

• The particle number density:

• N(p) is the density of discrete quantum states in a box of volume V with momentum p:

)()(1

pfpdNV

n

kp

g = degeneracy factor (eg spin states)

ky

k

kx

dk

kdV

gkdN 33)2(

)(

125

Thermal backgrounds

• The number density of relativistic quantum particles:


ky

k

kx

dk

0 /)(

2

3

3

3

3

1

4

)2(

)()2(

kTpe

dppg

pfpdg

n

126

Thermal backgrounds• The ultra-relativistic limit: p>>mc, kT>>mc2 (bosons)

• The non-relativistic limit: kT<<mc2 (Boltzmann)

3

0 2/

2

3

)3(

1

4

)2(

c

kTg

e

dppgn

kTpc

kTmckTcpmc emkT

gdpepg

n /2/3

20

/)2/(23

22

24

)2(

Log n

Log kT

Threshold mc2

T3

T3/2e-mc2/kT

Particle production

Particle annihilation

127

Thermal backgrounds

• Proton-antiproton production and annihilation:

• mp = 103 MeV so for T > 1013 K there is a thermal background of protons and antiprotons.

• But when T<1013K annihilation to photons.

• Should annihilate to zero, but in fact p/p=10-9! (or else we wouldn’t be here.)

• So there must have been a Matter-Antimatter Asymmetry!!

p

p

128

Thermal backgrounds

• The energy density of relativistic quantum particles (bosons):


ky

k

kx

dk 433

2

0 /)(

2

3

0 3

3

32

)(30

)(1

4

)2(

)()()2(

kTc

g

pe

dppg

ppfpdg

cu

kTp

129

Thermal backgrounds• The entropy, S, of relativistic quantum particles:

•The entropy is an extensive quantity (like E & V):

V

S

V

S

V

E

V

E

E1 V1 S1

E2

V2 S2

+

E=E1+E2

V=V1+V2 S=S1+S2

pV

TS

V

E

pdVdTT

STdV

V

STdT

T

EdV

V

E

pdVTVTdSTVdE

),(),(so

Hence

T

cp

V

Ss

3/

130

Thermal backgrounds• So in the ultra-relativistic case:

3

4

3

3

43

1

TT

s

up

Tu

Tn

• So

• But (entropy is a conserved quantity).

• Usual to quote ratios e.g. baryon density nB/s=10-9.

0s

ns

131

Lecture 10

132

Thermal backgrounds• Given these simple scalings with T for bosons, what is the scaling for n, u and s for fermions when kT >> mc2 ?

• Formally expand:

• So occupation numbers:

1

2

1

1

1

12

xxx eee

)2/(2)()( TfTfTf BBF

133


• For kT>>mc2:

• Number densities:

3

3

Tgn

Tgn

BB

FF

)(4

3

2

21)(

)2/(2)()(

3

Tng

g

Tng

g

TnTng

gTn

BB

F

BB

F

BBB

FF

134


• For kT>>mc2:

• Energy densities:

4

4

Tgu

Tgu

BB

FF

)(8

7

2

21)(

)2/(2)()(

4

Tug

g

Tug

g

TuTug

gTu

BB

F

BB

F

BBB

FF

135


• For kT>>mc2:

• Energy densities:

4

42

3/ , ,

Tgu

upTguTc

pus

BB

FF

)(8

7

2

21

3

4)(

)3/11)(2/(2)3/11)(()(

42

2

Tsg

g

Tc

Tu

g

g

TuTuTcg

gTs

BB

F

B

B

F

BBB

FF

136


• Define an effective number of relativistic particles:

• So energy of all relativistic particles is:

)(8

7)( ),(

8

7)( ),(

4

3)( Ts

g

gTsTu

g

gTuTn

g

gTn B

B

FFB

B

FFB

B

FF

Fermions

FBosons

B ggg8

7*

43

* )()(30

kTc

guTotal

137

Thermal backgrounds• The effective number of relativistic particles will change with time as kT<mc2 and particles become non-relativistic.

Fermions

FBosons

B ggg8

7*

g*

100

10

1 103 10-1 104 T (GeV)

• For high-T g*=100. If supersymmetric, g*=200.

138

Time and Temperature• At early times radiation and matter are strongly coupled and thermalized to temperature, T, of radiation.

• Recall: in a radiation-dominated universe,

and:

• Hence:

rGt

32

3

• Note also that T=2.73K(1+z), so z~T/(1 K).

seconds108.1

2

102/1

*

K

Tgt

22/1*

4* so , TgtTgr

139

A Thermal History of the Universe

• With time now related to temperature, and hence energy, we can map out the thermal history of the Universe.

140Inflation?

Dark Matter formed?

Proton-antiproton annihilation

e- e+ annihilation

• z ~ T/(1K)• E=kT ~ T/(3x103)eV

= 1010K

= 1013K

= 1015K

= 1028K

= 1032K

141

Freeze-out and Relic Particles• Electrons - Positrons annihilation:

•For first 3 seconds we have

• Then T drops and energy in becomes too low, so electrons and positrons annihilate.

• Stops when annihilation rate drops below expansion rate.

ee

e+

e-

142

Freeze-out and Relic Particles• Need the Boltzmann Equation to describe reactions:

2v3 nHnn

Rate of changeDilution by expansion

exp~1/H~R2

Loss due to annhilationint~1/(<v>n)~R3• Timescales:

log

log R

exp > int thermal equilibrium

exp < int Particle Freeze-out/

Decoupling

143

Freeze-out and Relic Particles• Creation, annihilation and freeze-out of particle relics:

kTmceT /2/3 2

Log n

log kT

T3

T3

Pair production

Pair annihilation

Freeze-out

144

Freeze-out and Relic Particles• Electrons-Positrons annihilation and neutrino decoupling:

• What happens to the energy released by ?

• At early times only have photons, neutrinos and e-e+ pairs in equilibrium.

• At T = 5x109K (3 seconds) e-e+ pairs annihilate.

ee

vv

ee

As weak force decoupled at T=1010K.

145

Freeze-out and Relic Particles• So radiation is boosted above neutrino temperature by neutrino decay.

• Before nv~n, after nv < n and Tv < T.

• But recall entropy is conserved

where

0s

3*Tgs

Bosons Fermions

FB ggg8

7*

146

Lecture 11

147

Freeze-out and Relic Particles• How much is photon temperature boosted ?

• Entropy: ge=2 g=2

• Neutrino Temperature:

)(4

11

)(1

)()(

87

vs

s

eess

before

beforeg

gg

beforeafter

ee

3*Tgs

KKTTv 95.1)73/2(11

43/1

148

Freeze-out and Relic Particles• How much is radiation energy boosted ?

• Neutrino energy:

• Enhances r by factor of 1.68 for 3 neutrino species:

68.1

)3( 2

cuu vr

uuuv 227.011

4

8

73/4

log

log R

v

Neutrinos annihilate

149

Relic massive neutrinos• Can we put cosmological constraints on the mass of neutrinos ?

• 1960’s: Particle physics models with mv = 0.

• 1970’s: Particle physics models with massive neutrinos.

• 1990’s: Non-zero mass detected (Superkamiokande, and Sudbury Neutrino Observatory (SNO) confirms Solar model).

150


• Number-density of cosmological neutrinos:

• Mass-density:

speciescmv

KTnvvn BF

// 113

)95.1|()(3

43

G

Hnm v

vvv

8

3 2

151


• Density-parameter of cosmological neutrinos:

• Re-arrange:

• Compare with lab: mve<15eV, mv<0.17MeV, mv<24MeV. m2=mi

2-mj2=[7x10-3 eV]2.

21

1 7.03.036.4

3

1

hN

eVmm vvN

ivv

v

i

v

i

N

ivv m

eVh 125.93

1

152

Big-Bang Nucleosynthesis (BBN)• As R tends to zero and T increases, eventually reach nuclear burning temperatures.• 1940’s: George Gamow suggests nuclear reactions in early

Universe led to Helium.

• Prediction of a radiation background (CMB)• Predicted 25% helium by mass, as found in stars.

• 1960’s: Details worked out by Hoyle, Burbidge and Fowler.

• First need free protons and neutrons to form.

153

Big-Bang Nucleosynthesis (BBN)• So first need proton & neutron freeze-out.• Recall:

• Below T<1013K (Mp~Mn ~ 103 MeV):

• Annihilation leaves a residual p/p~n/n~10-9.

• Protons and neutrons undergo Weak Interactions:

nn

pp

vpen

vnep

154

Big-Bang Nucleosynthesis (BBN)

• Assume equilibrium and low energy (kT<<mc2) limit:

• Ratio of neutrons to protons at temp T (m=mn-mp=1.3MeV):

kTmceTn /2/3 2

TKkTmc

p

n een

n /105.1/ 102

Log n

log kT

T3

Pair productionPair

annihilation

Freeze-out

155


• Annihilations stop when p & n freeze-out occurs:

Reaction expansion time > timeint~1/vn exp~1/H

• So ratio is frozen in at Tfreezeout

freezeoutfreezeout TKkTmc

p

n een

n /105.1/ 102

Log n

log kT

T3

Pair productionPair

annihilation

Freeze-out

156


• What is the observed neutron-proton ratio?

• Most He is in the form of 4He:

• He fraction by mass is:

• Observe Y=0.25 for stars.

• So np/nn=2/Y-1=7, or:

nppn

n

nnmnn

mnY

/1

2

)(

)2/4(

p

n

n

p

14.07

1

p

n

n

n

157


• At what time, then, does neutron freeze-out happen?

• Need to know weak interation rates: <v>weak

• This was calculated by Enrico Fermi in 1930’s.

• Find

• So expected neutron-proton ratio is:

34.0/105.1 10

freezeoutTK

p

n en

n

KnT freezeout10104.1)(

158


• Expect nn/np=0.34.

• But we said from observed stellar abundances.

• Close, but a bit big.

• But: 1. We have assumed kTfreezeout>>mec2

but really kTfreezeout~mec2

2. Neutrons decay. n=887 +/- 2 seconds for free neutrons. Need to be locked away

in a few seconds:25.0 14.034.0 Y

n

n

p

n

14.0p

n

n

n

159


• The onset of Nuclear Reactions.

• At the same time nuclear reactions become important.

•Neutrons get locked up in Deuteron via the strong interaction

• Happens at deuteron binding energy kT~2.2MeV

• Dominant when T(D formation)=8x108K, or at a time t=3 minutes.

Dpnn

p

D

n

pD

160

Big-Bang Nucleosynthesis (BBN)• The formation of Helium.

• 4He is preferred over H or D on thermodynamic grounds. Binding energies: E(He)= 7 MeV

E(D) = 1.1 MeV• After Deuteron forms:•

•Then

pTDD

nHeDD

3

nHeDT

pHeHeD

HepT

HeDD

4

43

4

4

T gets too low and reactionsstop at Li & Be.

BBN starts at 1010K, t=1s.Ends at 109K, t=3mins.

161

Lecture 12

162

Big-Bang Nucleosynthesis (BBN)• Summary of BBN:• T=1013K, t~0.1s: Neutron & proton annihilation (p/p~n/n~10-9).

• T=1010K, t=1s: Neutron freeze-out (nn/np=0.34.) Neutrons decay (nn/np=0.14.) Nuclear reactions start.

• T=109K, t=3mins: Formation of Helium. Peak of D formation. End of nuclear reactions H, D, 3He, 4He, 7Li, 7Be

Dpn

163

Big-Bang Nucleosynthesis (BBN)• The number of neutrino generations:• The ratio of neutrons to protons is:

• Depends weakly on B: higher baryon density means closer packed, so n locked up in nucleons (D) faster.

• Depends on Nv: More neutrinos, more r (g*), so Hubble rate increases, neutron freezeout happens sooner, so more n.

• Cosmological constraint that Nv<4.

• In 1990’s LEP at CERN sets Nv=3.

14.03

)(163.02.0

04.02/2

v

BkTmc

Decayp

n Nhef

n

nFreezeout

164

Big-Bang Nucleosynthesis (BBN)• Testing BBN:• The abundance of elements is sensitive to density of baryons:

• is number density of baryons per unit entropy.

• This gives us the matter-antimatter difference of p/p~10-9

Agreement between BBN theory and observation is a spectacular confirmation of the Big-Bang model !

10103

n

nn pn

10-11 10-10 10-9

165

Big-Bang Nucleosynthesis (BBN)• Using BBN to weigh the baryons:• The abundance of elements is sensitive to the density of baryons.

• The photon density scales as T-3, so:

• This yields

• But:

So most of the matter in the Universe cannot be made of Baryons !

103

28 10373.2

)(1074.2

K

Th

n

nnB

pn

2

7.02 04.0 002.002.0 h

BBh

Bm 3.0

166

Recombination of the Universe

• Energy-density of the Universe:

Log R

Log

m

r

Plasma Era:, p, e- in a plasma

Thomson scattering (g-e)

zeq~104

T=103Kzrec~103

Bound atoms (H)and free photons

Recombination (a misnomer)

167


• Ionization of a plasma:• Assume thermal equilibrium.

• Use Saha equation for ionization fraction, x:HH

Hx

kTen

mkT

x

x /3

2/32

)2(

)2(

1

=13.6 eV – H binding energy.

168


• Ionization of a plasma:• But equilibrium rapidly ceases to be valid. Interactions are too fast, and photons cannot escape.

• Escape bottleneck with 2-photon emission:

• This means the ionization fraction is higher than predicted by the Saha equation.

H H+E= >>

H+E= <

2s

1s

0

169


• The surface of last scattering:• In plasma photons random walk (Thomson scattering off electrons).

• After recombination photons travel freely and atoms form.• The last scattering surface forms a photosphere (like sun), The Cosmic Microwave Background.

IonizedPlasma

z

z=1100

z=80

170

The Cosmic Microwave Background• The CMB spectrum:• The CMB was discovered by accident in 1965 by Arno Penzias and Bob Wilson, two researchers at Bell Labs, New Jersey. • This confirmed the Big-Bang model, and ruled out the competing Steady-State model of Hoyle.

• They received the 1978 Nobel Prize for Physics.

171

The Cosmic Microwave Background• The CMB spectrum:• The Big Bang model predicts a thermal black-body spectrum (thermalized early on and adiabatic expansion).

• The observed CMB is an almost perfect BB spectrum:

KT 002.0725.2

• Accuracy limited by reference BB source.

• CMB contributes to 1% of TV noise.

172

The Cosmic Microwave Background

1/1/ 11)( 0

DkThvkThv eevn

• The CMB dipole:• The CMB dipole is due to our motion through universe.

• Doppler Shift: v=Dv0, D=1+v.r/rc - dipole.

• Same as temperature shift: T=(1+v.r/rc)T0

173


• The CMB dipole:• This gives us the absolute motion of the Earth (measured by George Smoot in 1977):

VEarth = 371+/- 1 km/s, (l,b) = (264o,48o)

assuming no intrinsic dipole.

• What is its origin?• Not due to rotation of sun around galaxy (wrong direction). v=300km/s, (l,b)=(90o,0o).• Motion of the Local Group? • Implies VLG=600km/s (l,b)=(270o,30o).

174

Lecture 13

175


• The CMB dipole:• What is its origin of the dipole?

• Motion of the Local Group.• VLG=600km/s (l,b)=(270o,30o).

• Motion due to gravitational attraction of large-scale structure:

• LG is falling into the Virgo Supercluster (~10Mpcs away)

• Which is being pulled by the Hercules Supercluster (the Great Attractor, ~150Mpc away).

176

Dark Matter• Recall from globular cluster ages, supernova and BBN:

• So we infer most of the matter in the Universe is non-baryonic.

• How secure is the density parameter measurement?

• If it’s wrong and lower, could all just be baryons.

04.03.0 Bm

177

Dark Matter

Lm L

M

• Mass-to-light ratio of galaxies:

• We can expect M/L=F(M)

Comets:

Low-Mass stars:

Galaxy stars: Sun

Sun

Sun

Sun

Sun

Sun

L

M

L

M

L

M

L

M

L

M

L

M

101

10

1012

M/L

M

M -3

luminosity densityfrom galaxy surveys

178

Dark Matter• In blue star-light:

• So we find:

This is way above the M/L=10 we see in stars.

• So not enough luminous baryons in stars.

• In fact not enough baryons in stars to makeB=0.04, so there must be baryonic dark matter too.

3211

38

1078.2

10)7.00.2(

MpcMh

MpchL

Sunmcrit

SunL

Sun

Sunm

Blue L

Mh

L

M

7.03.0100300

179

Dark Matter• Dark Matter in Galaxy Halos:

• In 1970’s Vera Rubin found galaxies rotate

like solid spheres, not Keplerian.

• For V=const, need M(<r)~r, so

• Density profile of Isothermal Sphere.• Yields dark matter = 5 x stellar mass

23

1

rr

M

r

rGMV

)(2

r

V

Keplerian 1/r1/2

180

• In 1933 Fritz Zwicky found the Doppler motion of galaxies in the Coma cluster were moving too fast to be gravitationally bound.

• First detection of dark matter.

• Assume hydrostatic equilibrium:

• So need 10 – 100 x stellar mass.

Dark Matter in Galaxy Clusters

Zwicky (1898-1974)

22

11)(Vr

Prr

rGMF

Coma

Velocity dispersion

181


rd

d

rd

Td

m

rkT

r

rGM gas

p ln

ln

ln

ln)()(

Coma

1.07.0

09.001.02/3

h

MMM

MM

M

M

DMstarsgas

starsgas

Tot

B

• X-ray emission from galaxy clusters. • Hot gas emits X-rays. • Assume hydrostatic equilibrium.• Equate gravitational and thermal potentials:

• Get both total mass, and baryonic (gas) mass.

• So MDM=10MB

182


)( EE M

• Gravitational lensing by clusters of galaxies.

• Use giant arcs around clusters

to measure projected mass.

• Strongest distortion at the

Einstein radius:

• Independent of state of cluster (equilibrium).

• Find again MTot =10 – 100 Mstars

Abell 2218

183

Dark Matter and m

3.0

1078.2

1010311

3314

m

Suncrit

Sunclusterclusterm

MpcM

MpcMnM

• So independent methods show in galaxy clusters:

MDM ~ 10 Mgas ~ 100 Mstars

• Can estimate mass-density of Universe from clusters:

184

Large-scale structure in the Universe

• The distribution of matter in the Universe is not uniform.

• There exists galaxies, stars, planets, complex life etc.

• Where does all this structure come from?

• Is there a fossil remnant from when it was formed?

• How do we reconcile this structure with the Cosmological Principle & Friedmann model ?

185


6 billion light years

The 2-degree Field Galaxy Redshift Survey (2dFGRS)

186


The 2-degree Field Galaxy Redshift Survey (2dFGRS)

187

Large-scale structure in the Universe• The matter density perturbation:

• Fourier decomposition:

)()(

rr

r

(r)

k

iker k.r )(

L

kx=n/L, n=1,2…

188

Large-scale structure in the Universe• The statistical properties:

– The Ergodic Theorem:

Volume averages are equal to ensemble averages.

• Moments of the density field:

• Define the power spectrum:

rdV

31

kk

k

ikerd

Vrrd

V

22

3232 1

)( 1

,0 k.r

)/2(ln2

)()(

)(

2

2

2

32

2

kRkd

dkPkk

kP k

189

Large-scale structure in the Universe• The statistical properties:• The correlation function:

• So correlation function is the Fourier transform of the power spectrum, P(k).

• For point processes, correlation function is the excess probability of finding a point at 2 given a point at 1: k

rikk

k

rxikk

k

xikk eeexd

Vrxxxd

Vrxxr .2

'

).('

.33 1

)()( 1

)()()(

212 ))(1()1|2( dVdVrndP

1

2r

190

Lecture 14

191

Large-scale structure in the Universe• The Matter Power Spectrum:• So 2-point statistics can be found from P(k).

• What is the form of P(k)?

• For simplicity let’s assume for now it’s a power-law:

where A is an amplitude and n is the spectral index.

2

32

2)(

)(

n

n

Akk

AkkP log 2(k)

log k

192

Large-scale structure in the Universe• The Potential Power Spectrum:• Can we put limits on spectral index, n?

• Consider the potential field, .

• So far we have assumed <<1 (so metric is Freidmann).

• Poisson equation:

• So

20

02

02

4

4

4

kG

Gk

G

kk

kk

12

42

)(

)(

n

nk

kk

kkP

193

Large-scale structure in the Universe• The Potential Power Spectrum:• Can we put limits on spectral index, n?

• To keep homogeneity, need n less than or equal to 1. • To avoid black holes, need n greater or equal to 1.

• So must have n=1, with 2=const~10-10 (from CMB).

• n=1 is scale invariant (fractal) in the potential field.

12

42

)(

)(

n

nk

kk

kkP

log 2(k)

log kn<1

n>1

n=1

194

Structure in the Universe

• Where did this structure come from ?

• In 1946 Russian physicist Evgenii Lifshitz suggested small variations in density in the Early Universe grow due to gravitational instability.

density

position

195

Dynamics of structure formation

r

t

r

)sin(

)cos1(

Bt

Ar

• Consider the gravitational collapse of a sphere:• Assume Einstein-de Sitter (m=1, pm=0).

• Behaves like a mini-universe, so

196

Dynamics of structure formation

• Linear theory growth:

• 0th order:

- expansion of universe.

• 1st order:

r

t

)1()sin(

)1()cos1(2

2013

61

21212

21

BBt

AAr

232

find we Since GMBAr

GMr

)(6

2

3/2

0 taB

tAr

3/23/26

20

11

6

2 B

t

B

tAr

r0

197

Dynamics of structure formation• Linear growth of over-densities:

• Density ~ 1/Vol:

• Linear growth in E-dS:

r

t)1()1(

1

31

03/1

0

3

00

rrr

r

r

)(6

20

33/2

taB

t

3/23/26

20

11

6

2 B

t

B

tArr

r0

position

198

Dynamics of structure formation• Nonlinear growth of over-densities:

• Linear growth:• Turnaround:• Collapse:• Virialization:

69.1 ,177 , ,r ,2

0,r ,2

55.5 , ,

23

max21

linrKV

Bt

a

r

virialization

r

t

r0

Linear turnaround collapse

199

Formation of a galaxy cluster

200

Dark Matter and the Power Spectrum• Since ~ a on all linear scales, the matter power spectrum preserves its shape in the linear regime.

• Linear regime, <<1, valid at early times and on large scales. • If n>-3 initial shape will be preserved on large scales.

• Power spectra is shaped by dark matter, so leaves imprint.

r

log 2(k)

log k

201

Dark Matter and the Power Spectrum• Have already seen we need non-baryonic dark matter• (B=0.04 < m=0.3, from BBN and clusters, SN, ages…).

• But what can the dark matter be?• Massive Neutrinos?

Now know to have mass, so possibly.

• Black holes? Also now know to exist at centre of all galaxies. But

if too large, disrupt galactic disk & lens stars in LMC and galactic bugle (MACHO and OGLE surveys). Too small and will over-produce Hawking radiation emission.

• A frozen-out particle relic from the early universe: Weakly Interacting Massive Particles (WIMPS).

202

• Must be weakly interacting to avoid detection so far.• A promising idea in particle physics is Supersymmetry:

• The lightest supersymmetric particle (the neutralino = gravitino+photino+zino) could be detected in 2007 at Europe’s CERN Large Hadron Collider (LHC).

What is Dark Matter ?

Matter Particles (fermions) Force Particles (boson)

electron photon

selectronphotino

203

Dark Matter and the Power Spectrum• It is convenient to divide dark matter candidates into 3 types: 1. Hot Dark Matter (HDM):

Relativistic at freeze-out (e.g. neutrinos), kT>>mc2.

2. Warm Dark Matter (WDM):Some momentum at freeze-out, kT~mc2.

3. Cold Dark Matter (CDM):No momentum at freeze-out, kT<<mc2.

eVmnn HDMHDM ,

keVmnn WDMWDM 101 ,

MeVmmemn /DMh2

m

2eV 10GeV

HDM CDM

204

Dark Matter and the Power Spectrum• The matter Transfer Functions: Dark matter affects the matter power spectrum of density perturbations.• HDM: Free-streaming and damping: HDM freezes-out relativistically.

log k2

log k

n=1

>>ctct

HDM escapes

HDM trapped

~a2

kH~1/Ldamp, Ldamp=Damping scale

205

Lecture 15

206

Dark Matter and the Power Spectrum• HDM: Free-streaming and damping:

• HDM freezes-out relativistically, v~c, so can free-stream out of density perturbations in matter-dominated regime.

• So if HDM, expect no structure (galaxies) on small-scales today!.• This rules out an HDM-dominated universe.

log k2

log k

~a2

MpchneutrinoL

hzzzDmckTctL

vdamp

meqeqmeqHdamp

12

22/12

)(16)(

)(900,23 ,)()(

207

Dark Matter and the Power Spectrum• Baryons + photons: Baryon Oscillations and Silk damping.

• t<trec: Baryon-photon plasma

log k2

log k

n=1

>>ctct

Photons & baryonstrapped in plasma

- No collapse

Photons & baryons trapped

~a2

kH~1/DH, DH=Horizon scale

208

Dark Matter and the Power Spectrum• Baryons + photons: Baryon Oscillations and Silk damping.

• t>trec: Baryon and photons free. Baryons oscillate.

log k2

log k

n=1

>>ctct

Photons free-streamcarrying baryons(Silk damping).

Baryons oscillate.

Photons & baryons trapped

~a2


209

Dark Matter and the Power Spectrum• CDM + photons: The Meszaros Effect.

• Recall at early times >>m

log k2

log k

No growth

n=1

>>ctct

Radiation escapes

Radiation trapped

~a4


210

Dark Matter and the Power Spectrum

• CDM + photons: The Meszaros Effect.• After matter-radiation equality, all scales grow the same.

• Produces a break in the matter power spectrum at comoving horizon scale at zeq=23,900mh2.

Predicts hierarchical sequence of structure formation (smallest first).

log k2

log k

n=-3~a2


~a2

MpchzrRzD meqHeqH12

0 )(16)()(

211

Dark Matter and the Power Spectrum• Transfer Functions:

• Can quantify all this with the transfer function, T(k):

log k2

log k

CDM

BaryonicHDM

)()0( 222 zTz kkk

212

Observations: 2dFGRS Power-Spectrum

• No large oscillations or damping.

• Rules out a pure baryonic or pure HDM universe.

• Smooth power – expected for CDM-dominated universe.

• Detection of baryon oscillations – trace baryons.

213

Cosmological Parameters from 2dFGRS

Likelihood contours from the shape of the power spectrum:

Break scale: Matter density: mh = 0.19 ± 0.02

Baryon oscillations: Baryon fraction

= 0.18 ± 0.06 (if n = 1)

So m=0.27 (h/0.7) -1

B=0.04 (h/0.7) -1

214

Observations: 2dFGRS Power-Spectrum • Information about the amplitude of the power spectrum is confused, as we are looking a galaxies, not matter.

• We usually assume a linear relation between matter and density perturbations:

mattergalaxies b b-bias parameter

galaxiesmatter

So amplitude of galaxy clustering mixes primordial power and process of galaxy formation.

215

Structure Formation in a CDM Universe

216

Cosmological Inflation

• Standard Model of Cosmology explains a lot (expansion, BBN, CMB, evolution of structure) but does not explain:

• Origin of the Expansion: Why is the Universe expanding

at t=0?

• Flatness Problem: Why is ~1?

R

t

t

1

217


• Standard Model of Cosmology explains a lot (expansion, BBN, CMB, evolution of structure) but does not explain:

• Horizon Problem: Why is the CMB so uniform over large angles, when the causal horizon is 1 degree?

• Structure Problem: What is the origin of the structure?

218

0/33

4 2 cpRG

R

t

lt

l

0R

• Lets tackle the horizon problem first.• Recall for R~t1/2 we have a particle horizon:

• But if R~t, >1, can causally connect universe:

• More generally

• This happens when:

which we get from Vacuum Energy, 2cp VV


219

We are here !

In 1980 Alan Guth proposed that the Early Universe had undergone acceleration, driven by vacuum energy.

He called this Cosmological Inflation.

Inflates a small, uniform causal patch to the size of observable Universe.

Explains why Universe (& CMB) looks so uniform.


220

• Expansion Problem:• Vacuum energy leads to acceleration of the Early Universe• This powers the expansion (recall Eddington).

Ht

V

eRR

HRR

H

0

2 const

R

t

• Need Inflation to end to start BB phase.


Inflation Big-Bang (radiation-dominated)

221

03

8

3

8

2

2

22

k

GH

RR

kcGH

V

V

• So Inflation predict =m+v=1 to high accuracy.• Compare with SN & galaxy clustering results.

• The Flatness Problem: Recall that if we expand a model with curvature, it looks locally flat:


222

• How much Inflation do we need?• Usually assume Inflation happens at GUT era, EGUT~1015GeV.• So how large is the current horizon at the GUT era?

• But causal horizon at GUT era is just dGUT=ctGUT=3x10-27m.

• So need to stretch GUT horizon by factor aInfl=1029 ~ e60

mMpcGUTd

GeV

GeV

E

Ez

ztodaydGUTd

Mpchtodayd

H

CMB

GUTGUT

GUTHH

H

224

2713

15

1

1010)(

10105.2

101

),1/()()(

6000)(


223

Lecture 16

Lecture Notes, PowerPoint notes,Tutorial Problems and Solutions are now available at:

http://www.roe.ac.uk/~ant/Teaching/Astro%20Cosmo/index.html

224

Dynamics of Inflation We need a dynamical process to switch off inflation. Simplest models are based on scalar fields (spin-0), , the inflaton

(e.g. -mesons, Higgs bosons). No idea what is…

Must obey energy equation: E2 = p2c2 + m2c4

Quantize to get Klein-Gordon equation:

Assume is uniform and add

expansion term, 3H:

Equation of a particle in a

potential V=(m2c4-2).

V

)( 24222 cmc

ddVV' ),('3 VH

tiEip ,

225

Dynamics of Inflation Evolution of a scalar field, , in a potential V().

Energy of scalar field:

Assume Slow Roll:

So like a vacuum energy with pressure

Drives acceleration of expansion.

)(2

1 2 V

V

.

2

constV

V

2cp

226

Dynamics of Inflation The End of Inflation. Eventually the inflaton will reach the bottom of the

potential and oscillate.

No longer slow-rolling.

The Inflaton can then decay into other particles and radiation, re-heating Universe for radiation-domination.

V

V2

227

Dynamics of Inflation

The Origin of Structure in Inflation. The evolution of the inflaton has quantum fluctuations

Fluctuations due to Hawking

Radiation (c.f. Black Holes).

So the universe expands at different rates, leading to density perturbations:

quantumclassical V

2

H

tHtH

R

R ,333

228

Dynamics of Inflation The Origin of Structure in Inflation. During Slow-Roll equation of motion simplifies:

The induced density field is:

The density power spectrum is

Exponential expansion generates a fractal in the potential field: so spectral index is n=1.

H

VVH

3

' ,'3 V

'V

'

V

'2

93

3/23

VV

HH

2

32

')(

V

VkP

229

• Structure created by freezing in quantum fluctuations during an inflationary epoch. Leaves imprint in structure.

Inflationary Epoch

Big Bang Epoch

t

r

Quantum fluctuationsof massless virtual particles

(inflaton, gravitational waves)excited by expansion.

Frozen-in structure & gravitational waves

Gravitational wave spectrum:PGW(k) ~ 2~ V

A Gravitational Wave Background from Inflation

230

The Multiverse in Inflation Inflation can happen lots of times, producing a Multiverse.

So somewhere life will appear.

231

Observerz =1000 z = infinity

Plasma

Recombination

T=2.73K


232


Plasma

Recombination

T=2.73K


233


Plasma

Recombination

T=2.73K


Causal horizon~1o

234

• Recall in 1930’s George Gamow predicts an afterglow from the Big Bang.

• Detected in 1969 by Arno Penzias & Robert Wilson (who won Nobel Prize).

• Formed at reionization at z = 1100.

• In 1960’s Sachs and Wolfe predicted there should be anisotropies due to potential perturbations.

Anisotropies in the CMB

235

• Anisotropies first detected in 1992 by the COBE satellite.

• On large angular scales T/T=10-5

• Detail images from the Wilkinson Microwave Anisotropy Probe (WMAP) in 2003.

• First year data.• Third year data expected any

day now.

Anisotropies in the CMB

236

Spherical Harmonic Analysis of the CMB

• Expand fluctuations in CMB temperature field in spherical harmonics on the celestial sphere:

m

mm YT

),( ),T(

237

• The squared harmonic modes of the temperature can be averaged to give a power spectrum:

where

TTm CT 2

)/2(2

)1(2

T

TCTT

Spherical Harmonic Analysis of the CMB

238

T on super-horizon scales (>1o)

• Sachs-Wolfe Effect: Gravitational redshift due to photons climbing out of potential wells,

3

1

T

T Newtonian potential

239

T on sub-horizon scales (<1o)

• Acoustic Oscillations: Baryons oscillate in and out of dark matter potential wells,

)cos(3

4

kAT

T

ct

240

T on sub-horizon scales (<1o)

• Photon Diffusion: Photons random walk out of potential wells,

)2/exp( 22sT

T

241

The CMB Temperature Power Spectrum

• Sachs-Wolfe Effect

• Acoustic Peaks

• Diffusion Damping

Wavenumber

02

)1(T

C

242

CMB Temperature Power Spectrum seen by WMAP (2003)

T/T)2

• Universe is spatially flat (m+v=1).• Potential amplitude: =10-5

• Spectral slope n~1

243

Geometry of the Universe from the CMB Temperature Power Spectrum

• The peak of the CMB power spectrum is given by the horizon size at recombination.• Hence it is a fixed “ruler”.

deg8.1

)(181)1(2

)(

2/1

2/122/1

00

mH

mmzH MpchzH

cdrRzD

Matter only universe:

244

Dark Matter and Vacuum Energy• The CMB acoustic peak

measure spatial curvature of Universe.

• Combine with supernova, galaxy clusters, galaxy clustering.

• Likelihood contours for m- V plane.

• So four independent methods converge on one model.

3.0,7.0 mV

245

Putting it all together

The “Standard Model” of Cosmology: • CMB acoustic peaks: m+V=1

• Supernova: 2V-m=1.1

• CMB + SN: V=0.7

• Galaxy clusters and galaxy clustering: m=0.3

• BBN and CMB: B=0.04• CMB Sachs-Wolfe effect: =10-5

• CMB + galaxy clustering: n~1• HST key programme and CMB: h=0.7• Age of Universe: 13.7+/- 0.2 Gyrs

246

Structure in the Universe with CDM

Amplitude of Structure,

247

Open Questions and Speculations

• Need to explain this strange Universe.• What is vacuum energy (dark energy)?• Why is V ~ (1 eV)4 ~ m??

• New particle physics, change gravity?• What is the Cold Dark Matter?

• CDM – neutralino in LHC?•Did inflation happen?

• Detect gravity wave background?• How did galaxies form?

•Watch them form at high-z?

248

A Multiverse? Inflation and superstring theory both predict a multiverse.

Find ~10120 universes…

Life will appear only in those Universes with small vacuum energy.

249

• Speculative ideas like the Ekpyrotic Universe try to unify Dark Energy and Inflation.

Parallel Universe Cosmologies

Time

Distance between galaxies

Our Universe Another Universe

250

The End