GR

AV

ITA

TIO

NA

LA

ST

RO

PH

YS

ICS

AN

DC

OS

MO

LO

GY

LE

CT

UR

E2

GR

AV

ITAT

ION

—T

his

isfir

stex

tend

edto

pic

(6le

ctur

es)

—W

hatd

ow

ew

antt

obe

able

todo

afte

r6

lect

ures

?

Aim

s—

wan

tto

unde

rsta

nd:

•ne

edfo

rG

ener

alR

elat

ivity

theo

ry•

Prin

cipl

eof

Equ

ival

ence

•m

etric

san

dge

odes

ics

•be

ndin

gof

light

ina

grav

itatio

nalfi

eld

•or

bits

ofpl

anet

s(p

rece

ssio

nof

peri-

helio

n)

•N

on-r

otat

ing

blac

kho

les

—w

hata

reth

ey?

—w

hath

appe

nsw

hen

thin

gsfa

llin

toth

em?

•so

me

aspe

cts

ofro

tatin

gbl

ack

hole

s•

ener

gies

ofor

bits

and

liber

atio

nof

en-

ergy

from

accr

etio

ndi

sks

arou

ndbl

ack

hole

s

1

The

nin

latte

rpa

rtof

cour

se

CO

SM

OLO

GY

—th

est

udy

ofth

eun

iver

seon

the

larg

ests

cale

s—

ques

tions

conc

ern

univ

erse

asa

who

le

Big

Que

stio

ns

1.H

owla

rge

isth

eun

iver

se?

2.H

owol

dis

the

univ

erse

?3.

Wha

tis

the

univ

erse

mad

eof

?4.

How

did

the

stru

ctur

ew

ese

ein

the

univ

erse

gett

here

?5.

Wha

tis

the

futu

reof

the

univ

erse

?—

will

itco

ntin

ueex

pand

ing

fore

ver?

—w

illit

even

tual

lyre

colla

pse?

The

stud

yof

grav

ityw

ein

itiat

ehe

reis

fund

amen

talt

oan

swer

ing

thes

equ

estio

ns.

Usi

ngth

ete

chni

ques

from

the

next

6le

ctur

es,w

ew

illbe

able

tofin

dth

em

etric

ofth

eun

iver

se(F

riedm

ann-

Rob

erts

on-W

alke

r)an

dge

tful

ldyn

amic

aleq

uatio

nsfo

rth

eev

olut

ion

ofth

eun

iver

se.

2

•In

this

first

lect

ure,

we

will

star

tby

cons

ider

ing

som

esi

mpl

eas

pect

sof

cosm

olog

y,w

ithth

eai

mof

show

ing

that

New

toni

angr

avity

can

geta

surp

risin

gly

long

way

inth

est

udy

ofth

eun

iver

seas

aw

hole

.

•H

owev

er,w

eru

nad

riftw

hen

itco

mes

toco

nsid

erin

gw

hatt

heun

iver

sew

aslik

eat

early

times

,or

how

light

prop

agat

esin

the

univ

erse

atan

ytim

e.

•T

his

will

show

usw

hyw

ene

eda

new

theo

ryof

grav

itatio

nin

orde

rto

stud

yth

eun

iver

se.

•T

hen,

we

will

mov

eon

toco

nsid

erw

hyG

ener

alre

lativ

ity(G

R)

isre

quire

d,an

dno

tfor

exam

ple,

just

asi

mpl

eex

tens

ion

toS

peci

alR

elat

ivity

whi

chin

corp

orat

esgr

avity

.

3

MA

INO

BS

ER

VA

TIO

NA

LF

EA

TU

RE

SO

FT

HE

UN

IVE

RS

E

Tost

art,

wha

tare

the

mai

nfe

atur

esof

the

univ

erse

onth

ela

rges

tsca

les,

and

how

does

itch

ange

with

time?

Beg

inw

ithH

ubbl

e’s

Law

(Hub

ble,

1929

—pr

edic

ted

byW

eyl,

1923

).T

hela

wis

v=

H0d

i.e.

gala

xies

are

syst

emat

ical

lym

ovin

gaw

ayfr

omus

with

spee

dpr

opor

tiona

lto

dist

ance

.

Infe

rth

isfr

omlo

okin

gat

spec

tra:

4

Iden

tifiab

lepa

ttern

ofem

issi

on(o

rab

sorp

tion)

lines

(H,M

g,C

a,N

a,et

c.al

lgiv

eop

tical

lines

).

Exp

ress

this

quan

titat

ivel

yvi

ath

ere

dshi

ftz

defin

edby

z=

λobs−

λL

λL

.

Not

eth

atot

her

defin

ition

sof

reds

hift

wou

ldbe

poss

ible

(e.g

.di

vidi

ngby

λobs)

—th

eon

egi

ven

isth

eon

eus

edin

cosm

olog

y.

Ifw

ech

oose

toin

terp

rett

his

asa

(non

-rel

ativ

istic

)D

oppl

ersh

ift,t

hen

v≈

zc

(sin

ceνobs=

νL(1−

v/c)

).

So

the

Hub

ble

effe

ct,i

sth

atfo

rca

ses

whe

rew

ekn

owth

edi

stan

ced

toth

eob

ject

(e.g

.as

prov

ided

byC

ephe

ids)

then

we

find

v≈

zc=

H0d.

H0

isH

ubbl

e’s

Con

stan

t.E

xact

valu

eis

cont

rove

rsia

l,bu

tpre

ttysu

reH

0=

50−

80km

s−1M

pc−1.

Tran

slat

edin

toS

Iuni

tsth

isis

1.6−

2.6×

10−18s−

1.

Thi

sha

sbi

gim

plic

atio

nsfo

rth

etim

eev

olut

ion

ofth

eun

iver

se.

Con

side

ra

gala

xya

dist

ance

dfr

omus

,e.g

.M

101

whi

chis≈

4M

pc

away

.

Pro

ject

ing

back

intim

e,w

hen

wou

ldit

have

been

onto

pof

us?

v=

4×

H0km

s−1

=⇒

t=

d v=

4M

pc

4M

pc×

H0

=H−1

0s

I.e.t c

ollid

eis

appr

oxim

atel

y:

15bi

llion

year

s(b

yr)

forH

0in

the

mid

dle

ofth

ekn

own

rang

e;

12.5

byr

ifH

0=

80km

s−1M

pc−1;

and

20by

rif

H0

=50km

s−1M

pc−1.

Cle

arly

this

time

isin

depe

nden

tofw

hich

gala

xyw

ear

eco

nsid

erin

g,an

dth

usth

ew

hole

ofth

eun

iver

sem

usth

ave

been

ina

very

smal

lvol

ume

this

time

ago.

But

the

age

ofth

eE

arth

iskn

own

tobe∼

4.5

byr

and

the

olde

stst

ars

inth

eG

alax

yar

e∼

11

byr

old.

Thi

spo

ses

grea

tpro

blem

sfo

rhi

gher

valu

esof

H0,s

ince

the

univ

erse

isth

enin

dang

erof

bein

gyo

unge

rth

anso

me

ofits

cons

titue

nts!

(Ifi

nclu

dedy

nam

ics,

find

t collid

eis

mod

ified

som

ewha

t,an

dis

mos

tlik

ely

even

shor

ter

than

we

have

just

dedu

ced!

)

Inan

yca

se,i

tis

clea

rth

atth

eun

iver

sem

usth

ave

com

efr

oman

earli

erst

age

atw

hich

itw

asm

uch

smal

ler

and

dens

er.

5

Rou

ghly

,con

serv

atio

nof

mat

ter

tells

usth

atρ∝

R−3,w

here

ρis

the

mat

ter

dens

ityan

dR

som

ech

arac

teris

ticsi

ze—

the

scal

efa

ctor

—as

soci

ated

with

the

univ

erse

—w

ew

illse

ela

ter

how

tode

fine

this

phys

ical

ly.

6

CO

SM

ICM

ICR

OW

AV

EB

AC

KG

RO

UN

D

The

CM

Bw

asfir

stdi

scov

ered

1965

byP

enzi

asan

dW

ilson

.

(Act

ually

the

effe

cts

wer

efir

stno

ticed

1939

inex

cita

tion

ofC

yano

gen

mol

ecul

esin

the

inte

rste

llar

med

ium

—op

tical

spec

tral

abso

rptio

nw

ork

tow

ards

star

sby

Ada

ms

and

McK

ella

r.)

Any

whe

rein

empt

ysp

ace

atth

em

o-m

ent

ther

eis

radi

atio

npr

esen

tco

rre-

spon

ding

tow

hat

abl

ackb

ody

wou

ldem

itat

ate

mpe

ratu

reof∼

2.7

4K

.

Bes

tmea

sure

men

toft

hesp

ectr

umof

this

radi

atio

nso

far

has

com

efr

omth

eC

OB

Esa

telli

te(C

osm

icB

ackg

roun

dE

xplo

rer

sate

llite

.)

7

CO

BE

show

sth

esp

ectr

umis

accu

rate

lybl

ackb

ody

toap

prox

.1

part

in104.

Idea

isth

atth

isra

diat

ion

was

emitt

edin

the

early

univ

erse

(hot

,den

seco

nditi

ons)

,w

hen

mat

ter

and

radi

atio

nw

ere

inth

erm

aleq

uilib

rium

.

—ho

tmea

nsm

atte

rw

asio

nize

d—

ther

efor

eph

oton

ssc

atte

red

freq

uent

lyof

fthe

free

elec

tron

s

Thi

sis

extr

emel

yim

port

ant.

Rad

iatio

nm

ustc

ome

from

are

gion

whe

reth

em

atte

ran

dra

diat

ion

are

inst

rictt

herm

odyn

amic

equi

libriu

m(T

E).

Thu

sun

iver

sew

asno

tjus

tsm

alla

ndde

nse

atea

rlytim

es,b

utin

TE

.Thi

sis

very

diffe

rent

from

toda

y.

Sta

rssh

ine

outa

t∼6000K

into

empt

ysp

ace

at2.7

K.

Ear

lyun

iver

sew

asno

tonl

yin

TE

,but

very

hot,

sinc

eth

era

diat

ion

cool

sas

the

univ

erse

expa

nds.

Thi

nkof

this

eith

eras

stre

tchi

ngof

the

phot

ons’

wav

elen

gth:

oras

the

wor

kdo

neby

the

phot

onga

sin

expa

nsio

n.

Eith

erw

ayfin

d:

Tem

pera

ture

ofra

diat

ion

cool

sas

T∝

1/R

whe

re(R

isth

esc

alef

acto

rof

the

univ

erse

).

Thu

sat

times∼

t collid

ebe

fore

the

pres

ent,

univ

erse

was

fille

dw

ithbl

ack

body

radi

atio

nat

ave

ryhi

ghte

mpe

ratu

re:

Run

ning

the

film

back

war

ds,s

eeth

atun

iver

seco

ols

away

from

infin

itete

mpe

ratu

reat

som

eso

rtof

sing

ular

ity—

the

Big

Ban

g.

As

the

univ

erse

expa

nded

,itc

oole

ddo

wn

and

mat

ter

beca

me

neut

ral:

—T

his

happ

ens

atte

mp.

T∼

4000K

—ph

oton

sno

long

ersc

atte

red

—tim

eab

out 0

.3−

0.5

mill

ion

year

saf

ter

the

big

bang

—re

dshi

ft≈

1400

Kno

wn

asth

eep

coh

ofre

com

bina

tion

—co

rres

pond

sto

last

scat

terin

gfo

rth

eco

smic

mic

row

ave

back

grou

nd(C

MB

)ph

oton

sw

ese

eto

day.

So

the

CM

Bpr

ovid

esus

with

api

ctur

eof

the

univ

erse

asit

was

atth

ese

early

times

:

FL

UC

TU

AT

ION

SIN

TH

EC

MB

The

radi

atio

nde

coup

les

from

the

mat

ter

—pr

opag

ates

unhi

nder

edto

war

dsus

toda

ygi

ving

usa

pict

ure

ofth

eun

iver

seat

atim

eap

prox

.300,0

00

year

saf

ter

the

big

bang

.

Firs

tevi

denc

efo

rst

ruct

ure

inth

ispi

ctur

e(fl

uctu

atio

ns)

cam

efr

omth

eC

OB

Esa

telli

te.

—le

velf

ound

was

atju

st1

part

in105

atan

angu

lar

scal

eof

10degre

es

Thi

sun

iform

ityis

asto

nish

ing

onS

tand

ard

Hot

Big

Ban

gpi

ctur

e.

—ca

nsh

owth

atre

gion

son

sky

mor

eth

anap

prox

.1.5

degre

es

apar

t,ca

nnot

have

been

inca

usal

cont

acti

nth

eir

past

atth

etim

eth

era

diat

ion

was

emitt

ed.

How

did

they

know

tosy

nchr

oniz

eth

eir

tem

pera

ture

sto

the

leve

lof1

part

in105?

Leav

eth

isqu

estio

nfo

ra

mom

ent.

Inst

ead

conc

entr

ate

onth

eid

eaof

CM

Bst

ruct

ure

asgi

ving

api

ctur

eof

the

seed

sne

eded

tofo

rmga

laxi

eset

c.

The

1pa

rtin

105

unifo

rmity

ofth

eC

MB

can

beco

ntra

sted

with

the

non-

unifo

rmity

ofth

epr

esen

tday

mat

ter

dist

ribut

ion:

•tr

eatin

gth

est

ars

asa

cont

inuo

usflu

idth

em

ean

dens

ityof

aga

laxy

is∼

6×

10−21kgm−3

•tr

eatin

gth

ega

laxi

esth

emse

lves

asa

cont

inuo

usflu

id,t

hem

ean

dens

ityof

univ

erse

asa

who

leis∼

5×

10−28kgm−3

Dis

cuss

how

tofin

dth

ese

num

bers

late

r.

Ass

umin

gth

atth

ean

isot

ropy

ofth

eC

MB

prov

ides

anes

timat

eof

the

frac

tiona

lpe

rtur

batio

nsin

the

mat

ter

dist

ribut

ion

atre

com

bina

tion,

we

see

that

the

gala

xies

,lik

est

ars,

repr

esen

tstr

ong

cond

ensa

tions

(δρ/ρ∼

107)

outo

fan

orig

inal

lyve

ryho

mog

eneo

usflu

id( δ

ρ/ρ∼

10−5).

How

did

this

com

eab

out?

ST

RU

CT

UR

EF

OR

MA

TIO

N

Bas

icco

smol

ogic

alqu

estio

nis

the

orig

inof

stru

ctur

e

—ho

wdi

dga

laxi

esan

dcl

uste

rsof

gala

xies

gett

here

?

Bas

ican

swer

toth

isis

grav

itatio

nali

nsta

bilit

y(e

.g.

Ben

tley/

New

ton

corr

espo

nden

ce).

—un

iform

grav

itatin

gm

ediu

mw

ithsl

ight

pert

urba

tion

δρ/ρ

—w

illco

llaps

efu

rthe

r

=⇒

mor

eδρ/ρ=⇒

mor

eco

llaps

e.

Jean

san

alys

is=⇒

gete

xpon

entia

lgro

wth

,eve

nin

am

ediu

mw

ithpr

essu

re,i

fmas

sof

pert

urba

tion

isab

ove

ace

rtai

nsi

ze.

Fin

d

δρ ρ∝

exp(A×

t)

(Ais

apo

sitiv

eco

nsta

nt)

The

refo

rem

ight

expe

cton

lyne

edve

rysm

alls

eeds

ofst

ruct

ure

tobe

pres

enti

nth

eea

rlyun

iver

se.

BU

T:U

nive

rse

isex

pand

ing

—co

unte

ract

sco

llaps

ete

nden

cy

Infa

ctge

tδρ ρ∝

t2/3

only

(Lifs

hitz

,194

6)

Now

disc

uss

how

muc

hco

smol

ogy

we

can

unde

rsta

ndus

ing

New

toni

anph

ysic

son

ly