MA

SSAC

HU

SET

TS

INST

ITU

TE

OF

TE

CH

NO

LO

GY

Physics

Departm

entP

hysics8.286:

The

Early

Universe

September

18,2011

Prof.

Alan

Guth

PR

OB

LEM

SET

2

DU

ED

AT

E:Tuesday,

September

27,2011.

REA

DIN

GA

SSIG

NM

EN

T:

Barbara

Ryden,

Intro

ductio

nto

Cosm

olo

gy,

Chapters

1-3.

PLA

NN

ING

AH

EA

D:A

lthoughthis

problemset

isnot

dueuntilSeptem

ber27,

Irecom

mend

thatyou

finishit

bythis

coming

Friday,September

23.P

roblemSet

3w

illbe

relativelyshort,

butw

illbe

dueon

Thursday,

September

29,just

two

daysafter

thisset

isdue.

Ifyou

want

toread

ahead,the

readingassignm

entw

ithP

roblemSet

3w

illbe

Weinberg,

The

First

Three

Min

utes,

Chapter

3.P

roblemSets

1through

3,including

thereading

assignments,

will

beincluded

inthe

materialcovered

onQ

uiz1,

onT

hursday,O

ctober6.

SEP

TE

MB

ER

/OC

TO

BE

R

MO

NT

UE

SW

ED

TH

UR

SFR

I

19Septem

ber20

21StudentH

oliday

2223R

ecomm

endation:Finish

PS

2

2627P

S2

due28

29PS

3due

30

3O

ctober4

56Q

uiz1

–in

class

7

INT

RO

DU

CT

ION

TO

TH

EP

RO

BLEM

SET

Inthis

problemset

we

willconsider

auniverse

inw

hichthe

scalefactor

isgiven

bya(t)

=bt

2/3

,

where

bis

anarbitrary

constantof

proportionalityw

hichshould

notappear

inthe

answers

toany

ofthe

questionsbelow

.(W

ew

illsee

inLecture

Notes

3that

thisis

thebehavior

ofa

flatuniverse

with

am

assdensity

thatis

dominated

bynonrelativistic

matter.)

We

will

supposethat

adistant

galaxyis

observedw

itha

redshiftz.

As

aconcrete

example

we

will

considerthe

most

distantknow

nob

jectw

itha

well-determ

inedredshift,

thegalaxy

UD

Fy-38135539,

which

hasa

redshift

8.286P

RO

BLE

MSE

T2,

FA

LL

2011p.2

z=

8.55.

The

discoveryof

thisgalaxy

was

announcedin

September

2009by

threegroups

ofastronom

ers*,all

ofw

homdiscovered

itin

infraredim

agesin

theH

ubbleSpace

Telescope

Ultra

Deep

Field.

The

redshiftw

asinitially

estimated

“photometrically,”

which

means

thatbroad

featuresofthe

spectrumare

determined

bym

easuringthe

lightthat

comes

througha

rangeof

filters.T

heredshift

was

confirmed

spectrographicallyin

October

2010by

Lehnert

etal.†

The

rateat

which

thehighest

measured

redshifthas

beengrow

inghas

beendram

atic.In

1986the

highestm

easuredredshift

was

only3.78.

Itw

as4.01

in1988,

4.73in

1992,4.897

in1994,

and4.92

in1998,

5.34in

2000,6.28

in2002,

and6.58

in2003.

In2006

Iyeet

al.‡discovered

agalaxy

with

aredshift

of6.96.

In2006

Richard

McM

ahoncom

piledthe

graphon

theright,

which

was

publishedin

aN

ews

arti-cle

onp.128

ofthe

same

issueof

Nature

asthe

Iyeet

al.discovery.

The

searchfor

highred-

shiftob

jectscontinues

tobe

anexciting

areaof

research,as

astronomers

tryto

sortout

theconditions

inthe

universew

henthe

firstgalax-

iesbegan

toform

.

PR

OB

LEM

1:D

ISTA

NC

ET

OT

HE

GA

LA

XY

(5points)

Let

t0

denotethe

presenttim

e,andlet

te

denotethe

time

atw

hichthe

lightthat

we

arecurrently

receivingw

asem

ittedby

thegalaxy.

Interm

sof

thesequantities,

findthe

presentvalue

ofthe

physicaldistance

�p

between

thisdistant

galaxyand

us.

PR

OB

LEM

2:T

IME

OF

EM

ISSIO

N(5

points)

Express

theredshift

zin

termsof

t0

andte .

Find

theratio

te /

t0

forthe

z=

8.55

galaxy.

*R

.J.B

ouwens

etal.,

Astrophys.

J.Letters

709,L133-L

137(2010),

http://arxiv.org/abs/0912.4263;

Andrew

Bunker

etal.,

Monthly

Notices

ofthe

Royal

Astronom

icalSociety

409,855-866

(2010),http://arxiv.org/abs/1010.4312;

R.J.

McL

ureet

al.,M

onthlyN

oticesof

theR

oyalA

stronomical

Society403,

960-983(2010),http://arxiv.org/abs/0909.2437.

†M

.D.

Lehnert

etal.,

Nature

467,940–942

(2010),http://arxiv.org/abs/

1010.4312.‡

Iyeet

al.,“Agalaxy

ata

redshiftz

=6.96,”

Nature

vol.443,no.7108,pp.186–188

(14Septem

ber14

2006).

8.286P

RO

BLE

MSE

T2,

FA

LL

2011p.3

PR

OB

LEM

3:D

ISTA

NC

EIN

TER

MS

OF

RED

SH

IFT

z(5

points)

Express

thepresent

valueofthe

physicaldistancein

terms

ofthepresent

valueof

theH

ubbleexpansion

rateH

0and

theredshift

z.Taking

H0 ≈

72km

-sec −1-

Mpc −

1,how

faraw

ayis

thegalaxy?

Express

youransw

erboth

inlight-years

andin

Mpc.

PR

OB

LEM

4:SP

EED

OF

REC

ESSIO

N(5

points)

Find

thepresent

rateat

which

thephysical

distance�p

between

thedistant

galaxyand

usis

changing.E

xpressyour

answer

interm

sof

theredshift

zand

thespeed

oflight

c,and

evaluateit

numerically

forthe

casez

=8.55.

Express

youransw

eras

afraction

ofthespeed

oflight.[Ifyou

getit

right,this“fraction”

isgreater

thanone!

Our

expandinguniverse

violatesspecial

relativity,butis

consistentw

ithgeneral

relativity.]

PR

OB

LEM

5:A

PPA

REN

TA

NG

ULA

RSIZ

ES

(10points)

Now

supposefor

simplicity

thatthe

galaxyis

spherical,and

thatits

physicaldiam

eterw

asw

atthe

time

item

ittedthe

light.(T

heactual

galaxyis

seenas

anunresolved

pointsource,

sow

edon’t

knowit’s

actualsize

andshape.)

Find

theapparent

angularsize

θ(m

easuredfrom

oneedge

tothe

other)of

thegalaxy

asit

would

beobserved

fromE

arthtoday.

Express

youransw

erin

terms

ofw

,z,

H0 ,

andc.

You

may

assume

thatθ�

1.C

ompare

youransw

erto

theapparent

angularsize

ofa

circleof

diameter

win

astatic

Euclidean

space,at

adistance

equaltothe

presentvalue

ofthe

physicaldistance

tothe

galaxy,as

foundin

Problem

1.[H

int:draw

diagrams

which

tracethe

lightrays

inthe

com

ovin

gcoordinate

system.

Ifyou

haveit

right,youw

illfindthat

θhas

am

inimum

valuefor

z=

1.25,and

thatθ

increasesfor

largerz.

This

phenomenon

makes

senseifyou

thinkabout

thedistance

tothe

galaxyat

thetim

eof

emission.

Ifthe

galaxyis

very

faraw

aytoday,

thenthe

lightthat

we

nowsee

must

haveleft

theob

jectvery

early,w

henit

was

ratherclose

tous!]

8.286P

RO

BLE

MSE

T2,

FA

LL

2011p.4

PR

OB

LEM

6:R

EC

EIV

ED

RA

DIA

TIO

NFLU

X(10

points)

At

thetim

eof

emission,

thegalaxy

hada

power

outputP

(measured,

say,in

ergs/sec)w

hichw

asradiated

uniformly

inall

directions.T

hispow

erw

asem

ittedin

theform

ofphotons.W

hatis

theradiation

energyflux

Jfrom

thisgalaxy

atthe

earthtoday?

Energy

flux(w

hichm

ightbe

measured

inergs-cm

−2-sec −

1)is

definedas

theenergy

perunit

areaper

unittim

estriking

asurface

thatis

orthogonaltothe

directionof

energyflow

.T

heeasiest

way

tosolve

thisproblem

isto

considerthe

trajectoriesof

thephotons,as

viewed

incom

ovingcoordinates.

You

must

calculatethe

rateat

which

photonsarrive

atthe

detector,and

youm

ustalso

usethe

factthat

theenergy

ofeach

photonis

proportionalto

itsfrequency,

andis

thereforedecreased

bythe

redshift.Y

oum

ayfind

itusefulto

thinkofthe

detectoras

asm

allpart

ofa

spherethat

iscentered

onthe

source,as

shown

inthe

following

diagram: