Cosmology ( 宇宙學 )

Chu Ming-chung 朱明中

Department of Physics

The Chinese University of Hong Kong

mcchu@phy.cuhk.edu.hk

http://www.phy.cuhk.edu.hk/gee/mctalks/mcpdp.html

Introduction to Cosmology

3.1 Cosmological Principle3.2 Expansion of the universe3.3 Newtonian theory3.4 General Relativistic theory*3.5 Big Bang*3.6 Fate of the universe*

*Extra materials for your reference only.

3.1 Cosmological Principle Homogeneity (均勻性 )

Large-scale distribution of matter in the universe is uniform.

Isotropy (各向同性 ) The universe looks the same in any direction,

on large scales.

They are not identical concepts.

Examples

A

The cube is homogeneous, but it is not isotropic as observed at A.

The sphere is not homogeneous because it is composed of different layers, but it is isotropic as observed at the centre.

Cosmological Principle ( 宇宙學原理 )

= Homogeneity + Isotropy

Any observer anywhere in the universe sees the

same general features of the universe, after

correcting for evolutionary changes

no special place in the universe; all locations are

typical

the universe has no edge, no center

4 years ago

8 years ago

Universality of physical laws ( 物理定律之普遍性 ): Physical laws apply

everywhere in the universe

3.2 Expansion of the universe

Red shifts of galaxies:

Observed spectra of

remote galaxies are red

shifted

Doppler effects: distant

galaxies are moving away

from us at high speeds

http://antwrp.gsfc.nasa.gov/diamond_jubilee/1996/hub_1929.html

http://skolor.nacka.se/samskolan/eaae/summerschools/Hubble.html

/ / , for .v c v c

from Hubble’s plates

Galaxy Velocity (km/s)

Distance (Mpc)

Virgo 16

Ursa Major 200

Corona 293

BoÖtes 520

Hydra

d Distance (100 Mpc)

Recession speed v (104 km/s)

2 4 6 8

2

4

6

0

Reproduce Hubble’s Law!

Slope = v/d ~ 70 km/s/Mpc

1 pc ~ 3.3 light years

Hubble’s Law

1929 1999

Hubble’s Law ( 哈勃定律 ): The speed of a receding galaxy is proportional to its distance v = Hd

the farther a galaxy is, the higher is its receding speed

the universe is expanding uniformly

every galaxy is moving away from all others, but there is no single center of expansion; every point is a center!

Uniform expansion

a(t)

From every point on the surface of a balloon, all other points recede away as the balloon is inflated. There is no single center of the expansion on the surface. Can detect the expansion locally by measuring the scale factor a(t). Uniform expansion → .v d

2D analogy: the entire universe exists only on the surface of the balloon. There is no single center or boundary and nothing outside of the universe.

3.3 Newtonian theory

A spherical universe of uniform density

Force on a unit mass on surface

343 RM

RR

GMF

2

RR

GMF

2 : expansion will slow down to stop, and then the universe collapses

= 0 : expands exactly at the escape velocity; expansion will slow down but continues forever

> 0 : expansion continues forever,

tR ,0

0R

2

2

2

1 1

21

2

RRR GM

R

dR dGM

dt dt RGM

RR

K.E. P.E. total

2

2

1 4

2 3

R G

R R

4

3

R G

R

Problems with Newtonian Cosmology

Unstable: forces not balanced anywhere except at the exact center, but unstable equilibrium even there.

Evolution depends on the ‘shape’ (boundary) of the universe.

Any small movement of any object exerts instant effects on all masses in the entire universe.

3.4* General relativistic theoryMass induces curvature on

spacetimeGeometry of spacetime

determined by distribution of matter

Fate of universe determined by geometry of spacetime

Consider the cosmic scale factor a(t) instead of R

a(t)

a(t’)

Note: Same as Newtonian, if P = 0!

2/ / 8 / 3 /v r H a a G k a

43

aG P

a

General Relativistic CosmologyFriedmann Equations

Note: Hubble’s ‘constant’ is not a constant!Note: Cannot make , the universe has to expand/contract!

0a a

Space-time has an intrinsic curvature k: 3 allowed types: positive, zero, negative

Gravity always decelerates the expansion.

2 2/ 8 / 3 /a a G k a

43

aG P

a

/ 3

/ 3

, tuned to make the universe static.Einstein introduced Cosmological constant 宇宙常數

Find the value of and k/a2 to make the universe static. Assume a matter dominated universe P = 0.

Ans.: 2 24 , / ( ).G k a H

Einstein and Hubble at Mount Wilson Observatory

‘(Adding the cosmological constant is) the greatest blunder in my life.’

3.5* The Big Bang ( 大爆炸 )

Tracing the expansion backward in time

all matters in the universe were confined in an extremely small volume a long time ago (say, at time t = 0)

matters were compressed in a primordial fireball ( 原始火球 )

For t < 0.0001s

The universe was filled with high-energy

photons (光子 ), temperature ~1012 K,

density ~1013 g cm-3

Radiation-dominated ( 輻射主導 ) universe:

The universe was dominated by high-energy

radiations (photons), which interact continuously

with matters

High energy photons energy is converted to matter according to E

= mc2, creating pairs of particle and anti-

particle; protons, neutrons, and electrons

were produced within the first few seconds

particles and anti-particles could annihilate

and became photons again

e-

e+

continuous interaction between

radiations and matter

universe is opaque (not transparent to

radiation)

e-

e+

Light changes to particles, can hardly go through the space

For some quantum mechanical reasons, there

were slightly more particles than anti-particles

our present universe is made up of matter

but very little anti-matter

The universe cooled during expansion

Human + Antihuman

By t = 3 minutes: proton + neutron

deuterium (heavy hydrogen nuclei)

formation of light nuclei began

By t = 30 minutes: nuclear reactions

stopped ~ 25% helium, ~ 75%

hydrogen and deuterium in the early universe

By t = 300,000 years: T ~ 3000K; nuclei combined

with electrons to form atoms radiation does not interact

much with neutral atoms the universe became

transparent to radiation, and was matter-dominated

Radiation in the early universe was red shifted as the universe expanded

→ wavelengths have increased by 1000 times by now

CMB: the best blackbody spectrum in Nature

becomes primordial microwave background radiation ( 微波背景輻射 ): has a wavelength of 1 mm (microwave), which looks as if it was emitted by a black body of temperature 2.735 K today.

The background radiation is approximately isotropic, except for a small blue shift in the direction of Leo due to the motion of our galaxy, and a red shift in the opposite direction.

Fluctuations ~ 10-5 K

CMB Anisotropies 不勻性

History of Cosmic Microwave Background Radiation detection

Detection of a uniform background T~2.7 K, l = 0

1965: Penzias and Wilson

Cobe Satellite 1992

First anisotropy map 7o resolution

WMAP Satellite 2002

First detailed anisotropy map l~2000Downloaded from http://map.gsfc.nasa.gov/m_uni/uni_101Flucts.html

From APS Newsletter

Wilkinson Microwave Anisotropy ProbeSatellite launched

1/4/02, in L2 pointHave been returning

data since 12/02Resolution <0.3o ;

sensitivity ~20K

http://map.gsfc.nasa.gov/Photos are from NASA/WMAP.

Estimated age of the universe ~10-20 billion years; determined by measuring the expansion rate of the universe

Newest measurement by Hubble Space Telescope ~12-14 billion years

http://map.gsfc.nasa.gov/m_uni/uni_101age.html By Cosmic Microwave Background: 13.7

billion years

Evidences for Big Bang: CMB, Big Bang Nucleosynthesis of light elements (eg. 4He), expansion of the universe, stellar evolution, …

3.6* The fate of the universe

Future expansion of the universe depends on the

competition between gravity (attractive) and

cosmological constant (attractive or repulsive)

Critical density (critical density) 4x10-30 g cm-3

- the boundary between open and closed universe

if

4 / 3.3

aG P

a

0. 2 2/ 8 / 3 /a a G k a

0

2oH

Hubble ‘constant’

Critical density

Accelerating universe1998:All wrong!

is positive and large →vacuum repulsive force.

Big crunch (大塌縮 ) oscillating universe (脈動宇宙 )?

a (t)S

cale

of

the

univ

erse

~14 billion years ago

Empty universe

ttoday

k > 0 closed universe

k = 0 flat universe

k < 0 open universe

D:\MCChu\stup\2006\Cosmology\accel.mov

SummaryCosmological Principle = Homogeneity + Isotropy →

the universe is similar everywhereExpansion of the universe: Hubble’s LawNewtonian Cosmology: unstableGeometry of the universe: close, flat, open depending

on the average density vs. critical density → different fates

General relativity: the universe has to expand/contract → cosmological constant

Big Bang Theory: primordial fireball, cosmic microwave background

Introduction to Cosmology

3.1 Cosmological Principle3.2 Expansion of the universe3.3 Newtonian theory3.4 General Relativistic theory*3.5 Big Bang*3.6 Fate of the universe*

*Extra materials for your reference only.

Animation courtesy STScI/NASA

Deep field survey

Deep field survey

Animation courtesy STScI/NASA