Ph - web.mit.eduweb.mit.edu/8.286/www/quiz18/q1rp-euf18-2up.pdf · 1 v=u (nonrelativistic, observ...

MASSACHUSETTSINSTITUTEOFTECHNOLOGY

PhysicsDepartment

Physics8.286:TheEarlyUniverse

September28,2018

Prof.AlanGuth

REVIEW

PROBLEMSFOR

QUIZ1

QUIZDATE:Wednesday,October3,2018,duringthenormalclasstime.

QUIZCOVERAGE:LectureNotes1,2,and3;ProblemSets1,2,and3;Weinberg,

Chapters1,2,and3;Ryden,Chapters1,2,and3.(WhileallofRyden'sChapter3

hasbeenassigned,questionsonthequizwillbelimitedtoSection3.1.Thematerial

inSections3.2and3.3willbediscussedinlecturelaterinthecourse,andyouwill

notberesponsibleforituntilthen.Section3.4(forthe�=0case)mayhelpyou

understandthecosmologicalDopplershift,alsodiscussedinLectureNotes2,but

therewillbenoquestionsspeci�callyfocusedonRyden'sdiscussion.)Oneofthe

problemsonthequizwillbetakenverbatim

(oratleastalmostverbatim)

from

eitherthehomeworkassignments,orfrom

thestarredproblems

from

thissetofReview

Problems.ThestarredproblemsaretheonesthatI

recommendthatyoureviewmostcarefully:Problems2,4,7,12,15,17,19,and22.

Thestarredproblemsdonotincludeanyreadingquestions,butpartsofthereading

questionsintheseReviewProblemsmayalsorecurontheupcomingquiz.Forthe

homeworkproblems,extracreditproblemsareeligibletobetheproblemusedon

thequiz.

PURPOSE:Thesereviewproblemsarenottobehandedin,butarebeingmadeavail-

abletohelpyoustudy.Theycomemainlyfromquizzesinpreviousyears.Except

forafewpartswhichareclearlymarked,theyareallproblemsthatIwouldconsider

fairforthecomingquiz.Insomecasesthenumberofpointsassignedtotheproblem

onthequizislisted|

inallsuchcasesitisbasedon100pointsforthefullquiz.

Inadditiontothissetofproblems,youwill�ndonthecoursewebpagethe

actualquizzesthatweregivenin1994,1996,1998,2000,2002,2004,2005,2007,

2009,2011,2013,and2016.Therelevantproblemsfromthosequizzeshavemostly

beenincorporatedintothesereviewproblems,butyoustillmaybeinterestedin

lookingattheoriginalquizzes,justtoseehowmuchmaterialhasbeenincludedin

eachquiz.Sincethescheduleandthenumberofquizzeshasvariedovertheyears,

thecoverageofthisquizwillnotnecessarilybethesameasQuiz1fromallprevious

years.Infact,however,the�rstquizthisyearcoversessentiallythesamematerial

asthe�rstquizineither2009,2011,2013,or2016.

REVIEW

SESSION:Tohelpyoustudyforthequiz,therewillbeareviewsessionled

byHonggeunKimonSunday,September30,at3:30pminRoom3-333.

FUTURE

QUIZZES:TheotherquizdatesthistermwillbeMonday,November5,

andWednesday,December5,2018.

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.2

INFORMATION

TO

BEGIVEN

ON

QUIZ:

Eachquizinthiscoursewillhaveasectionof\usefulinformation"atthebackof

thequiz.Forthe�rstquiz,thisusefulinformationwillbethefollowing:

DOPPLER

SHIFT(Formotionalongaline):

z=v=u

(nonrelativistic,sourcemoving)

z=

v=u

1�v=u

(nonrelativistic,observermoving)

z= s1+�

1��1

(specialrelativity,with�=v=c)

COSMOLOGICALREDSHIFT:

1+z��observed

�emitted

=a(tobserved )

a(temitted )

SPECIALRELATIVITY:

TimeDilationFactor:

�

1

p1��2

;

��v=c

Lorentz-FitzgeraldContractionFactor:

RelativityofSimultaneity:

Trailingclockreadslaterbyanamount�`0 =c.

KINEMATICSOFAHOMOGENEOUSLY

EXPANDING

UNI-

VERSE:

Hubble'sLaw:v=Hr,

wherev=

recessionvelocityofadistantobject,H

=

Hubble

expansionrate,andr=distancetothedistantobject.

PresentValueofHubbleExpansionRate(Planck2018):

H0=67:66�0:42km-s �1-Mpc �1

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.3

ScaleFactor:`p (t)=a(t)`c;

where`p (t)isthephysicaldistancebetweenanytwoobjects,a(t)

isthescalefactor,and`cisthecoordinatedistancebetweenthe

objects,alsocalledthecomovingdistance.

HubbleExpansionRate:H(t)=

1a(t)

da(t)

dt

.

LightRaysinComovingCoordinates:Lightraystravelinstraightlines

withspeeddxd

t=

ca(t).

EVOLUTION

OFA

MATTER-DOMINATED

UNIVERSE:H

2= �_aa �2

=8�3

G��kc2

a2

;

�a=�4�3

G�a;

�(t)=a3(t

i )

a3(t)�(ti )

��=�c;where�c=3H2

8�G

:

Flat(k=0):a(t)/t2=3;=1

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.4

PROBLEM

LIST

1.DidYouDotheReading(2000)?

...............5(Sol:25)

*2.TheSteady-StateUniverseTheory...............6(Sol:27)

3.DidYouDoTheReading(2007)?

...............7(Sol:29)

*4.AnExponentiallyExpandingUniverse

.............8(Sol:31)

5.DidYouDoTheReading(1986/1990composite)?

........9(Sol:32)

6.AFlatUniverseWithUnusualTimeEvolution

.........9(Sol:33)

*7.AnotherFlatUniverseWithAnUnusualTimeEvolution

.....10(Sol:34)


...............11(Sol:38)

9.AFlatUniverseWitha(t)/t3=5

...............12(Sol:39)


...............13(Sol:43)

11.AnotherFlatUniverseWitha(t)/t3=5

.............14(Sol:44)

*12.TheDecelerationParameter..................14(Sol:48)

13.ARadiation-DominatedFlatUniverse

.............15(Sol:48)


...............15(Sol:49)

*15.SpecialRelativityDopplerShift................16(Sol:50)


...............16(Sol:51)

*17.TracingALightPulseThroughARadiation-DominatedUniverse

.17(Sol:53)

18.TransverseDopplerShifts...................18(Sol:55)

*19.ATwo-LevelHigh-SpeedMerry-Go-Round

...........19(Sol:56)

20.SignalPropagationInAFlatMatter-DominatedUniverse

....20(Sol:59)


...............21(Sol:65)

*22.TheTrajectoryOfAPhotonOriginatingAtTheHorizon.....21(Sol:68)

23.DidYouDotheReading(2016)?

...............22(Sol:69)

24.ObservingaDistantGalaxyinaMatter-DominatedFlatUniverse.23(Sol:71)

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.5

PROBLEM

1:DID

YOU

DO

THEREADING

(2000)?(35points)

ThefollowingproblemwasProblem1,Quiz1,2000.Thepartswereeachworth5points.

a)TheDopplere�ectforbothsoundandlightwavesisnamedforJohannChristian

Doppler,aprofessorofmathematicsattheRealschuleinPrague.Hepredictedthe

e�ectforbothtypesofwavesinxx42.Whatarethetwodigitsxx?

b)Whentheskyisveryclear(asitalmostneverisinBoston),onecanseeaband

oflightacrossthenightskythathasbeenknownsinceancienttimesastheMilky

Way.Explaininasentenceortwohowthisbandoflightisrelatedtotheshapeof

thegalaxyinwhichwelive,whichisalsocalledtheMilkyWay.

c)Thestatementthatthedistantgalaxiesareonaveragerecedingfromuswithaspeed

proportionaltotheirdistancewas�rstpublishedbyEdwinHubblein1929,andhas

becomeknownasHubble'slaw.WasHubble'soriginalpaperbasedonthestudyof

2,18,180,or1,800galaxies?

d)Thefollowingdiagram,labeledHomogeneityandtheHubbleLaw,wasusedbyWein-

bergtoexplainhowHubble'slawisconsistentwiththehomogeneityoftheuniverse:

Thearrowsandlabelsfromthe\VelocitiesseenbyB"andthe\Velocitiesseenby

C"rowshavebeendeletedfromthiscopyofthe�gure,anditisyourjobtosketch

the�gureinyourexambookwiththesearrowsandlabelsincluded.(Actually,in

Weinberg'sdiagramthesearrowswerenotlabeled,butthelabelsarerequiredhere

sothatthegraderdoesnothavetojudgethepreciselengthofhand-drawnarrows.)

e)Thehorizonisthepresentdistanceofthemostdistantobjectsfromwhichlighthas

hadtimetoreachussincethebeginningoftheuniverse.Thehorizonchangeswith

time,butofcoursesodoesthesizeoftheuniverseasawhole.Duringatimeinterval

inwhichthelinearsizeoftheuniversegrowsby1%,doesthehorizondistance

(i)growbymorethan1%,or

(ii)growbylessthan1%,or

(iii)growbythesame1%?

f)Namethetwomenwhoin1964discoveredthecosmicbackgroundradiation.With

whatinstitutionweretheyaÆliated?

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.6

g)Atatemperatureof3000K,thenucleiandelectronsthat�lledtheuniversecom-

binedtoformneutralatoms,whichinteractveryweaklywiththephotonsofthe

backgroundradiation.Afterthisprocess,knownas\recombination,"thebackground

radiationexpandedfreely.Sincerecombination,howhaveeachofthefollowingquan-

titiesvariedasthesizeoftheuniversehaschanged?(Youranswersshouldresemble

statementssuchas\proportionaltothesizeoftheuniverse,"or\inverselypropor-

tionaltothesquareofthesizeoftheuniverse".Theword\size"willbeinterpreted

tomeanlinearsize,notvolume.)

(i)theaveragedistancebetweenphotons

(ii)thetypicalwavelengthoftheradiation

(iii)thenumberdensityofphotonsintheradiation

(iv)theenergydensityoftheradiation

(v)thetemperatureoftheradiation

�

PROBLEM

2:THESTEADY-STATEUNIVERSETHEORY

(25points)

ThefollowingproblemwasProblem2,Quiz1,2000.

Thesteady-statetheoryoftheuniversewasproposedinthelate1940sbyHermann

Bondi,ThomasGold,andFredHoyle,andwasconsideredaviablemodelfortheuniverse

untilthecosmicbackgroundradiationwasdiscoveredanditspropertieswerecon�rmed.

Asthenamesuggests,thistheoryisbasedonthehypothesisthatthelarge-scaleproperties

oftheuniversedonotchangewithtime.Theexpansionoftheuniversewasanestablished

factwhenthesteady-statetheorywasinvented,butthesteady-statetheoryreconcilesthe

expansionwithasteady-statedensityofmatterbyproposingthatnewmatteriscreated

astheuniverseexpands,sothatthematterdensitydoesnotfall.Liketheconventional

theory,thesteady-statetheorydescribesahomogeneous,isotropic,expandinguniverse,

sothesamecomovingcoordinateformulationcanbeused.

a)(10points)Thesteady-statetheoryproposesthattheHubbleconstant,likeother

cosmologicalparameters,doesnotchangewithtime,soH(t)=H0 .Findthemost

generalformforthescalefactorfunctiona(t)whichisconsistentwiththishypothesis.

b)(15points)Supposethatthemassdensityoftheuniverseis�0 ,whichofcoursedoes

notchangewithtime.Intermsofthegeneralformfora(t)thatyoufoundinpart

(a),calculatetherateatwhichnewmattermustbecreatedfor�0toremainconstant

astheuniverseexpands.Youranswershouldhavetheunitsofmassperunitvolume

perunittime.[Ifyoufailedtoanswerpart(a),youwillstillreceivefullcredithere

ifyoucorrectlyanswerthequestionforanarbitraryscalefactorfunctiona(t).]

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.7

PROBLEM

3:DID

YOU

DO

THEREADING

(2007)?(25points)

ThefollowingproblemwasProblem1onQuiz1,2007,whereeachofthe5questionswas

worth5points:

(a)Inthe1940's,threeastrophysicistsproposeda\steadystate"theoryofcosmology,

inwhichtheuniversehasalwayslookedaboutthesameasitdoesnow.Statethe

lastnameofatleastoneoftheseauthors.(Bonuspoints:youcanearn1pointeach

fornamingtheothertwoauthors,andhenceupto2additionalpoints,but1point

willbetakeno�foreachincorrectanswer.)

(b)In1917,aDutchastronomernamedWillemdeSitterdidwhichoneofthefollowing

accomplishments:

(i)measuredthesizeoftheMilkyWaygalaxy,�ndingittobeaboutonebillion

light-yearsindiameter.

(ii)resolvedCepheidvariablestarsinAndromedaandtherebyobtainedpersua-

siveevidencethatAndromedaisnotwithinourowngalaxy,butisapparently

anothergalaxylikeourown.

(iii)publishedacatalog,NebulaeandStarClusters,listing103objectsthatas-

tronomersshouldavoidwhenlookingforcomets.

(iv)publishedamodelfortheuniverse,basedongeneralrelativity,whichappeared

tobestaticbutwhichproducedaredshiftproportionaltothedistance.

(v)discoveredthattheorbitalperiodsoftheplanetsareproportionaltothe3/2

powerofthesemi-majoraxisoftheirellipticalorbits.

(c)In1964{65,ArnoA.PenziasandRobertW.Wilsonobserveda uxofmicrowave

radiationcomingfromalldirectionsinthesky,whichwasinterpretedbyagroupof

physicistsataneighboringinstitutionasthecosmicbackgroundradiationleftover

fromthebigbang.Circlethetwoitemsonthefollowinglistthatwerenotpartof

thestorybehindthisspectaculardiscovery:

(i)BellTelephoneLaboratory

(ii)MIT

(iii)PrincetonUniversity

(iv)pigeons

(v)groundhogs

(vi)Hubble'sconstant

(vii)liquidhelium

(viii)7.35cm

(Grading:3ptsfor1correctanswer,5for2correctanswers,and-2foreachincorrect

answer,buttheminimumscoreiszero.)

(d)ImportantpredictionsoftheCopernicantheorywerecon�rmedbythediscovery

oftheaberrationofstarlight(whichshowedthatthevelocityoftheEarthhasthe

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.8

time-dependenceexpectedforrotationabouttheSun)andbythebehaviorofthe

Foucaultpendulum(whichshowedthattheEarthrotates).Thesediscoverieswere

made

(i)duringCopernicus'lifetime.

(ii)approximatelytwoandthreedecadesafterCopernicus'death,respectively.

(iii)aboutonehundredyearsafterCopernicus'death.

(iv)approximatelytwoandthreecenturiesafterCopernicus'death,respectively.

(e)IfoneaveragesoversuÆcientlylargescales,theuniverseappearstobehomogeneous

andisotropic.Howlargemusttheaveragingscalebebeforethishomogeneityand

isotropysetin?

(i)1AU(1AU=1:496�1011m).

(ii)100kpc(1kpc=1000pc,1pc=3:086�1016m=3.262light-year).

(iii)1Mpc(1Mpc=106pc).

(iv)10Mpc.

(v)100Mpc.

(vi)1000Mpc.

�

PROBLEM

4:

AN

EXPONENTIALLY

EXPANDING

UNIVERSE

(20

points)

ThefollowingproblemwasProblem2,Quiz2,1994,andhadalsoappearedonthe1994

ReviewProblems.Asisthecasethisyear,itwasannouncedthatoneoftheproblems

onthequizwouldcomefromeitherthehomeworkortheReviewProblems.Theproblem

alsoappearedasProblem2onQuiz1,2007.

Considera at(i.e.,ak=0,oraEuclidean)universewithscalefactorgivenby

a(t)=a0 e�t;

wherea0and�areconstants.

(a)(5points)FindtheHubbleconstantH

atanarbitrarytimet.

(b)(5points)Let(x;y;z;t)bethecoordinatesofacomovingcoordinatesystem.Sup-

posethatatt=0agalaxylocatedattheoriginofthissystememitsalightpulse

alongthepositivex-axis.Findthetrajectoryx(t)whichthelightpulsefollows.

(c)(5points)Supposethatwearelivingonagalaxyalongthepositivex-axis,andthat

wereceivethislightpulseatsomelatertime.Weanalyzethespectrumofthepulse

anddeterminetheredshiftz.Expressthetimetratwhichwereceivethepulsein

termsofz,�,andanyrelevantphysicalconstants.

(d)(5points)Atthetimeofreception,whatisthephysicaldistancebetweenourgalaxy

andthegalaxywhichemittedthepulse?Expressyouranswerintermsofz,�,and

anyrelevantphysicalconstants.

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.9

PROBLEM

5:DID

YOU

DO

THEREADING

(1986/1990COMPOSITE)?

(a)Theassumptionsofhomogeneityandisotropygreatlysimplifythedescriptionofour

universe.We�ndthattherearethreepossibilitiesforahomogeneousandisotropic

universe:anopenuniverse,a atuniverse,andacloseduniverse.Whatquantityor

conditiondistinguishesbetweenthesethreecases:thetemperatureofthemicrowave

background,thevalueof=�=�c ,mattervs.radiationdomination,orredshift?

(b)Whatisthetemperature,inKelvin,ofthecosmicmicrowavebackgroundtoday?

(c)Whichofthefollowingsupportsthehypothesisthattheuniverseisisotropic:the

distancestonearbyclusters,observationsofthecosmicmicrowavebackground,clus-

teringofgalaxiesonlargescales,ortheageanddistributionofglobularclusters?

(d)IsthedistancetotheAndromedaNebula(roughly)10kpc,5billionlightyears,2

millionlightyears,or3lightyears?

(e)DidHubblediscoverthelawwhichbearshisnamein1862,1880,1906,1929,or

1948?

(f)WhenHubblemeasuredthevalueofhisconstant,hefoundH�1�100millionyears,

2billionyears,10billionyears,or20billionyears?

(g)Cepheidvariablesareimportanttocosmologybecausetheycanbeusedtoestimate

thedistancestothenearbygalaxies.WhatpropertyofCepheidvariablesmakes

themusefulforthispurpose,andhowaretheyused?

(h)Cepheidvariablestarscanbeusedasestimatorsofdistancefordistancesupto

about100light-years,104light-years,107lightyears,or1010light-years?[Notefor

2011:thisquestionwasbasedonthereadingfrom

JosephSilk'sTheBigBang,

andthereforewouldbenotbeafairquestionforthisyear.]

(i)Namethetwomenwhoin1964discoveredthecosmicbackgroundradiation.With

whatinstitutionweretheyaÆliated?

(j)Atthetimeofthediscoveryofthecosmicmicrowavebackground,anactivebut

independente�ortwastakingplaceelsewhere.P.J.E.Peebleshadestimatedthat

theuniversemustcontainbackgroundradiationwithatemperatureofatleast10 ÆK,

andRobertH.Dicke,P.G.Roll,andD.T.Wilkinsonhadmountedanexperimentto

lookforit.Atwhatinstitutionwerethesepeopleworking?

PROBLEM

6:AFLATUNIVERSEWITHUNUSUALTIMEEVOLUTION

ThefollowingproblemwasProblem3,Quiz2,1988:

Considera atuniverse�lledwithanewandpeculiarform

ofmatter,witha

Robertson{Walkerscalefactorthatbehavesas

a(t)=bt1=3:

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.10

Herebdenotesaconstant.

(a)Ifalightpulseisemittedattimete

andobservedattimeto ,�ndthephysical

separation`p (to )betweentheemitterandtheobserver,atthetimeofobservation.

(b)Againassumingthatteandtoaregiven,�ndtheobservedredshiftz.

(c)Findthephysicaldistance`p (to )whichseparatestheemitterandobserveratthe

timeofobservation,expressedintermsofc,to ,andz(i.e.,withoutteappearing).

(d)Atanarbitrarytimetintheintervalte<t<to ,�ndthephysicaldistance`p (t)

betweenthelightpulseandtheobserver.Expressyouranswerintermsofc,t,and

to .

�

PROBLEM

7:

ANOTHER

FLAT

UNIVERSE

WITH

AN

UNUSUAL

TIMEEVOLUTION

(40points)

ThefollowingproblemwasProblem3,Quiz1,2000.

Considera atuniversewhichis�lledwithsomepeculiarformofmatter,sothat

theRobertson{Walkerscalefactorbehavesas

a(t)=bt ;

whereband areconstants.[Thisuniversedi�ersfromthematter-dominateduniverse

describedinthelecturenotesinthat�isnotproportionalto1=a3(t).Suchbehavioris

possiblewhenpressuresarelarge,becauseagasexpandingunderpressurecanloseenergy

(andhencemass)duringtheexpansion.]Forthefollowingquestions,anyoftheanswers

maydependon ,whetheritismentionedexplicitlyornot.

a)(5points)Lett0denotethepresenttime,andlettedenotethetimeatwhichthe

lightthatwearecurrentlyreceivingwasemittedbyadistantobject.Intermsof

thesequantities,�ndthevalueoftheredshiftparameterzwithwhichthelightis

received.

b)(5points)Findthe\look-back"timeasafunctionofzandt0 .Thelook-backtime

isde�nedasthelengthoftheintervalincosmictimebetweentheemissionand

observationofthelight.

c)(10points)Expressthepresentvalueofthephysicaldistancetotheobjectasa

functionofH0 ,z,and .

d)(10points)Atthetimeofemission,thedistantobjecthadapoweroutputP(mea-

sured,say,inergs/sec)whichwasradiateduniformlyinalldirections,intheform

ofphotons.Whatistheradiationenergy uxJfromthisobjectattheearthto-

day?ExpressyouranswerintermsofP,H0 ,z,and .[Energy ux(whichmight

bemeasuredinerg-cm�2-sec �1)isde�nedastheenergyperunitareaperunittime

strikingasurfacethatisorthogonaltothedirectionofenergy ow.]

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.11

e)(10points)Supposethatthedistantobjectisagalaxy,movingwiththeHubble

expansion.Withinthegalaxyasupernovaexplosionhashurledajetofmaterial

directlytowardsEarthwithaspeedv,measuredrelativetothegalaxy,whichis

comparabletothespeedoflightc.Assume,however,thatthedistancethejethas

traveledfromthegalaxyissosmallthatitcanbeneglected.WithwhatredshiftzJ

wouldweobservethelightcomingfromthisjet?Expressyouranswerintermsof

allorsomeofthevariablesv,z(theredshiftofthegalaxy),t0 ,H0 ,and ,andthe

constantc.

PROBLEM

8:DID

YOU

DO

THEREADING

(1996)?(25points)


Thefollowingquestionsareworth5pointseach.

a)In1814-1815,theMunichopticianJosephFrauenhoferallowedlightfromthesun

topassthroughaslitandthenthroughaglassprism.Thelightwasspreadintoa

spectrumofcolors,showinglinesthatcouldbeidenti�edwithknownelements|

sodium,iron,magnesium,calcium,andchromium.Weretheselinesdark,orbright

(2points)?Why(3points)?

b)TheAndromedaNebulawasshownconclusivelytolieoutsideourowngalaxywhen

astronomersacquiredtelescopespowerfulenoughtoresolvetheindividualstarsof

Andromeda.WasthisfeataccomplishedbyGalileoin1609,byImmanuelKantin

1755,byHenriettaSwanLeavittin1912,byEdwinHubblein1923,orbyWalter

BaadeandAllanSandageinthe1950s?

c)Someoftheearliestmeasurementsofthecosmicbackgroundradiationweremade

indirectly,byobservinginterstellarcloudsofamoleculecalledcyanogen(CN).State

whethereachofthefollowingstatementsistrueorfalse(1pointeach):

(i)The�rstmeasurementsofthetemperatureoftheinterstellarcyanogenwere

madeovertwentyyearsbeforethecosmicbackgroundradiationwasdirectly

observed.

(ii)Cyanogenhelpstomeasurethecosmicbackgroundradiationbyre ectingit

towardtheearth,sothatitcanbereceivedwithmicrowavedetectors.

(iii)Onereasonwhythecyanogenobservationswereimportantwasthattheygave

the�rstmeasurementsoftheequivalenttemperatureofthecosmicbackground

radiationatwavelengthsshorterthanthepeakoftheblack-bodyspectrum.

(iv)Bymeasuringthespectrumofvisiblestarlightthatpassesthroughthecyanogen

clouds,astronomerscaninfertheintensityofthemicrowaveradiationthat

bathestheclouds.

(v)Byobservingchemicalreactionsinthecyanogenclouds,astronomerscaninfer

thetemperatureofthemicrowaveradiationinwhichtheyarebathed.

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.12

d)Inabout280B.C.,aGreekphilosopherproposedthattheEarthandtheother

planetsrevolvearoundthesun.Whatwasthenameofthisperson?[Notefor2011:

thisquestionwasbasedonreadingsfromJosephSilk'sTheBigBang,andtherefore

isnotappropriateforQuiz1ofthisyear.]

e)In1832HeinrichWilhelmOlberspresentedwhatwenowknowas\Olbers'Paradox,"

althoughasimilarargumenthadbeendiscussedasearlyas1610byJohannesKepler.

Olbersarguedthatiftheuniverseweretransparent,static,in�nitelyold,andwas

populatedbyauniformdensityofstarssimilartooursun,thenoneofthefollowing

consequenceswouldresult:

(i)Thebrightnessofthenightskywouldbein�nite.

(ii)Anypatchofthenightskywouldlookasbrightasthesurfaceofthesun.

(iii)Thetotalenergy uxfromthenightskywouldbeaboutequaltothetotal

energy uxfromthesun.

(iv)Anypatchofthenightskywouldlookasbrightasthesurfaceofthemoon.

WhichoneofthesestatementsisthecorrectstatementofOlbers'paradox?

PROBLEM

9:A

FLATUNIVERSEWITH

a(t)/

t3=5




a(t)=bt3=5;

wherebisaconstant.

a)(5points)FindtheHubbleconstantH

atanarbitrarytimet.

b)(5points)Whatisthephysicalhorizondistanceattimet?

c)(5points)SupposealightpulseleavesgalaxyAattimetA

andarrivesatgalaxyB

attimetB.Whatisthecoordinatedistancebetweenthesetwogalaxies?

d)(5points)WhatisthephysicalseparationbetweengalaxyAandgalaxyBattime

tA?AttimetB?

e)(5points)Atwhattimeisthelightpulseequidistantfromthetwogalaxies?

f)(5points)WhatisthespeedofBrelativetoAatthetimetA?(By\speed,"Imean

therateofchangeofthephysicaldistancewithrespecttocosmictime,d`p =dt.)

g)(5points)Forobservationsmadeattimet,whatisthepresentvalueofthephysical

distanceasafunctionoftheredshiftz(andthetimet)?Whatphysicaldistance

correspondstoz=1?Howdoesthiscomparewiththehorizondistance?(Note

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.13

thatthisquestiondoesnotrefertothegalaxiesAandBdiscussedintheearlier

parts.Inparticular,youshouldnotassumethatthelightpulseleftitssourceat

timetA.)

h)(5points)ReturningtothediscussionofthegalaxiesAandBwhichwereconsidered

inparts(c)-(f),supposetheradiationfromgalaxyAisemittedwithtotalpowerP.

WhatisthepowerperareareceivedatgalaxyB?

i)(5points)WhenthelightpulseisreceivedbygalaxyB,apulseisimmediatelysent

backtowardgalaxyA.AtwhattimedoesthissecondpulsearriveatgalaxyA?

PROBLEM

10:DID

YOU

DO

THEREADING

(1998)?(20points)

ThefollowingquestionsweretakenfromProblem1,Quiz1,1998:

Thefollowingquestionsareworth5pointseach.

a)In1917,Einsteinintroducedamodeloftheuniversewhichwasbasedonhisnewly

developedgeneralrelativity,butwhichcontainedanextraterm

intheequations

whichhecalledthe\cosmologicalterm."(ThecoeÆcientofthistermiscalledthe

\cosmologicalconstant.")WhatwasEinstein'smotivationforintroducingthisterm?

b)Whentheredshiftofdistantgalaxieswas�rstdiscovered,theearliestobservations

wereanalyzedaccordingtoacosmologicalmodelinventedbytheDutchastronomer

W.deSitterin1917.Atthetimeofitsdiscovery,wasthismodelthoughttobestatic

orexpanding?Fromthemodernperspective,isthemodelthoughttobestaticor

expanding?

c)Theearlyuniverseisbelievedtohavebeen�lledwiththermal,orblack-body,radi-

ation.Forsuchradiationthenumberdensityofphotonsandtheenergydensityare

eachproportionaltopowersoftheabsolutetemperatureT.Say

Numberdensity/Tn1

Energydensity/Tn2

Givethevaluesoftheexponentsn1andn2 .

d)Atabout3,000Kthematterintheuniverseunderwentacertainchemicalchange

initsform,achangethatwasnecessarytoallowthedi�erentiationofmatterinto

galaxiesandstars.Whatwasthenatureofthischange?

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.14

PROBLEM

11:ANOTHERFLATUNIVERSEWITHa(t)/

t3=5

(40points)

ThefollowingwasProblem3,Quiz1,1998:



a(t)=bt3=5;

wherebisaconstant.

a)(5points)FindtheHubbleconstantH

atanarbitrarytimet.

b)(10points)Supposeamessageistransmittedbyradiosignal(travelingatthespeed

oflightc)fromgalaxyAtogalaxyB.Themessageissentatcosmictimet1 ,whenthe

physicaldistancebetweenthegalaxiesis`0 .Atwhatcosmictimet2isthemessage

receivedatgalaxyB?(Expressyouranswerintermsof`0 ,t1 ,andc.)

c)(5points)Uponreceiptofthemessage,thecreaturesongalaxyBimmediatelysend

backanacknowledgement,byradiosignal,thatthemessagehasbeenreceived.At

whatcosmictimet3

istheacknowledgmentreceivedongalaxyA?(Expressyour

answerintermsof`0 ,t1 ,t2 ,andc.)

d)(10points)ThecreaturesongalaxyBspendsometimetryingtodecodethemessage,

�nallydecidingthatitisanadvertisementforKellogg'sCornFlakes(whateverthat

is).Atatime�tafterthereceiptofthemessage,asmeasuredontheirclocks,they

sendbackaresponse,requestingfurtherexplanation.Atwhatcosmictimet4isthe

responsereceivedongalaxyA?Inansweringthispart,youshouldnotassumethat

�tisnecessarilysmall.(Expressyouranswerintermsof`0 ,t1 ,t2 ,t3 ,�t,andc.)

e)(5points)WhentheresponseisreceivedbygalaxyA,theradiowaveswillbered-

shiftedbyafactor1+z.Giveanexpressionforz.(Expressyouranswerinterms

of`0 ,t1 ,t2 ,t3 ,t4 ,�t,andc.)

f)(5points;Nopartialcredit)Ifthetime�tintroducedinpart(d)issmall,thetime

di�erencet4 �t3canbeexpandedto�rstorderin�t.Calculatet4 �t3to�rstorder

accuracyin�t.(Expressyouranswerintermsof`0 ,t1 ,t2 ,t3 ,t4 ,�t,andc.)[Hint:

whilethispartcanbeansweredbyusingbruteforcetoexpandtheanswerinpart

(d),thereisaneasierway.]

�

PROBLEM

12:THEDECELERATION

PARAMETER

ThefollowingproblemwasProblem2,Quiz2,1992,whereitcounted10pointsoutof

100.Manystandardreferencesincosmologyde�neaquantitycalledthedeceleration

parameterq,whichisadirectmeasureoftheslowingdownofthecosmicexpansion.

Theparameterisde�nedby

q��a(t)a(t)

_a2(t):

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.15

Findtherelationshipbetweenqandforamatter-dominateduniverse.[Incaseyou

haveforgotten,isde�nedby

=�=�c;

where�isthemassdensityand�cisthecriticalmassdensity(i.e.,thatmassdensity

whichcorrespondstok=0).]

PROBLEM

13:A

RADIATION-DOMINATED

FLATUNIVERSE

Wehavelearnedthatamatter-dominatedhomogeneousandisotropicuniversecan

bedescribedbyascalefactora(t)obeyingtheequation

�_aa �2

=8�3

G��kc2

a2

:

Thisequationinfactappliestoanyformofmassdensity,sowecanapplyittoauniverse

inwhichthemassdensityisdominatedbytheenergyofphotons.Recallthatthemass

densityofnonrelativisticmatterfallso�as1=a3(t)astheuniverseexpands;themassof

eachparticleremainsconstant,andthedensityofparticlesfallso�as1=a3(t)because

thevolumeincreasesasa3(t).Forthephoton-dominateduniverse,thedensityofphotons

fallsofas1=a3(t),butinadditionthefrequency(andhencetheenergy)ofeachphoton

redshiftsinproportionto1=a(t).Sincemassandenergyareequivalent,themassdensity

ofthegasofphotonsfallso�as1=a4(t).

Fora at(i.e.,k=0)matter-dominateduniversewelearnedthatthescalefactor

a(t)isproportionaltot2=3.Howdoesa(t)behaveforaphoton-dominateduniverse?

PROBLEM

14:DID

YOU

DO

THEREADING?

ThefollowingproblemwastakenfromProblem1,Quiz1,2004,whereeachpartcounted

5points,foratotalof25points.Thereadingassignmentincludedthe�rstthreechapters

ofRyden,IntroductiontoCosmology,andthe�rstthreechaptersofWeinberg,The

FirstThreeMinutes.

(a)In1826,theastronomerHeinrichOlberwroteapaperonaparadoxregardingthe

nightsky.WhatisOlber'sparadox?Whatistheprimaryresolutionofit?

(b)WhatisthevalueoftheNewtoniangravitationalconstantGinPlanckunits?The

Plancklengthisoftheorderof10 �35m,10 �15m,1015m,or1035m?

(c)WhatistheCosmologicalPrinciple?IstheHubbleexpansionoftheuniversecon-

sistentwithit?(Forthelatterquestion,asimple\yes"or\no"willsuÆce.)

(d)Inthe\StandardModel"oftheuniverse,whentheuniversecooledtoabout3�10a

K,itbecametransparenttophotons,andtodayweobservetheseastheCosmic

MicrowaveBackground(CMB)atatemperatureofabout3�10bK.Whatarethe

integersaandb?

(e)Whatdidtheuniverseprimarilyconsistofatabout1/100thofasecondafterthe

BigBang?Includeanyconstituentthatisbelievedtohavemadeupmorethan1%

ofthemassdensityoftheuniverse.

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.16

�

PROBLEM

15:SPECIALRELATIVITY

DOPPLER

SHIFT

Thefollowingproblem

wastakenfrom

Problem

2,Quiz1,2004,whereitcounted20

points.

ConsidertheDopplershiftofradiowaves,foracaseinwhichboththesourceandthe

observeraremoving.Supposethesourceisaspaceshipmovingwithaspeedvsrelative

tothespacestationAlpha-7,whiletheobserverisonanotherspaceship,movinginthe

oppositedirectionfromAlpha-7withspeedvorelativetoAlpha-7.

(a)(10points)CalculatetheDopplershiftzoftheradiowaveasreceivedbytheobserver.

(Recallthatradiowavesareelectromagneticwaves,justlikelightexceptthatthe

wavelengthislonger.)

(b)(10points)Usetheresultsofpart(a)todeterminevtot ,thevelocityofthesource

spaceshipasitwouldbemeasuredbytheobserverspaceship.(8pointswillbegiven

forthebasicidea,whetherornotyouhavetherightanswerforpart(a),and2points

willbegivenforthealgebra.)

PROBLEM

16:DID

YOU

DO

THEREADING?

Thefollowingquestionwastakenfrom

Problem


points.

(a)(4points)Whatwasthe�rstexternalgalaxythatwasshowntobeatadistance

signi�cantlygreaterthanthemostdistantknownobjectsinourgalaxy?Howwas

thedistanceestimated?

(b)(5points)Whatisrecombination?Didgalaxiesbegintoformbeforeorafterrecom-

bination?Why?

(c)(4points)InChapterIVofhisbook,Weinbergdevelopsa\recipeforahotuniverse,"

inwhichthematteroftheuniverseisdescribedasagasinthermalequilbriumat

averyhightemperature,inthevicinityof109K(severalthousandmilliondegrees

Kelvin).Suchathermalequilibrium

gasiscompletelydescribedbyspecifyingits

temperatureandthedensityoftheconservedquantities.Whichofthefollowingis

onthislistofconservedquantities?Circleasmanyasapply.

(i)baryonnumber

(ii)energyperparticle

(iii)protonnumber

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.17

(iv)electriccharge

(v)pressure

(d)(4points)ThewavelengthcorrespondingtothemeanenergyofaCMB(cosmicmi-

crowavebackground)photontodayisapproximatelyequaltowhichofthefollowing

quantities?(Youmaywishtolookupthevaluesofvariousphysicalconstantsatthe

endofthequiz.)

(i)2fm(2�10 �15m)

(ii)2microns(2�10 �6m)

(iii)2mm(2�10 �3m)

(iv)2m.

(e)(4points)Whatistheequivalenceprinciple?

(f)(4points)WhyisitdiÆcultforEarth-basedexperimentstolookatthesmallwave-

lengthportionofthegraphofCMBenergydensityperwavelengthvs.wavelength?

�

PROBLEM

17:

TRACING

A

LIGHT

PULSE

THROUGH

A

RADIATION-DOMINATED

UNIVERSE

Thefollowingproblem

wastakenfrom

Problem


points.

Considera atuniversethatexpandswithascalefactor

a(t)=bt1=2;

wherebisaconstant.Wewilllearnlaterthatthisisthebehaviorofthescalefactorfor

aradiation-dominateduniverse.

(a)(5points)Atanarbitrarytimet=tf,whatisthephysicalhorizondistance?(By

\physical,"Imeanasusualthedistanceinphysicalunits,suchasmetersorcen-

timeters,asmeasuredbyasequenceofrulers,eachofwhichisatrestrelativetothe

comovingmatterinitsvicinity.)

(b)(3points)Supposethataphotonarrivesattheorigin,attimetf ,fromadistant

pieceofmatterthatispreciselyatthehorizondistanceattimetf.Whatisthetime

teatwhichthephotonwasemitted?

(c)(2points)Whatisthecoordinatedistancefromtheorigintothepointfromwhich

thephotonwasemitted?

(d)(10points)Foranarbitrarytimetintheintervalte �t�tf,whilethephotonis

traveling,whatisthephysicaldistance`p (t)fromtheorigintothelocationofthe

photon?

(e)(5points)Atwhattimetmaxisthephysicaldistanceofthephotonfromtheorigin

atitslargestvalue?

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.18

PROBLEM

18:TRANSVERSEDOPPLER

SHIFTS

Thefollowingproblem

wastakenfrom

Problem


points.

(a)(8points)SupposethespaceshipXanthu

isatrestatlocation(x=0;y=a;z=0)in

aCartesiancoordinatesystem.(Weas-

sumethatthespaceisEuclidean,and

thatthedistancescalesintheproblem

aresmallenoughsothattheexpansion

oftheuniversecanbeneglected.)

The

spaceshipEmmeracismovingatspeed

v0alongthex-axisinthepositivedirec-

tion,asshowninthediagram,wherev0

iscomparabletothespeedoflight.As

theEmmeraccrossestheorigin,itre-

ceivesaradiosignalthathadbeensent

sometimeearlierfrom

theXanthu.Is

theradiationreceivedredshiftedorblueshifted?Whatistheredshiftz(whereneg-

ativevaluesofzcanbeusedtodescribeblueshifts)?

(b)(7points)Now

supposethattheEm-

meracisatrestattheorigin,while

theXanthuismovinginthenegativex-

direction,aty=aandz=0,asshown

inthediagram.Thatis,thetrajectoryof

theXanthucanbetakenas

(x=�v0 t;y=a;z=0):

Att=0theXanthucrossesthey-axis,

andatthatinstantitemitsaradiosig-

nalalongthey-axis,directedattheori-

gin.Theradiationisreceivedsometime

laterbytheEmmerac.Inthiscase,is

theradiationreceivedredshiftedorblueshifted?Whatistheredshiftz(whereagain

negativevaluesofzcanbeusedtodescribeblueshifts)?

(c)(5points)Isthesequenceofeventsdescribedin(b)physicallydistinctfrom

the

sequencedescribedin(a),orisitreallythesamesequenceofeventsdescribedin

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.19

areferenceframethatismovingrelativetothereferenceframeusedinpart(a)?

Explainyourreasoninginasentenceortwo.(Hint:notethattherearethreeobjects

intheproblem:Xanthu,Emmerac,andthephotonsoftheradiosignal.)

�

PROBLEM

19:A

TWO-LEVELHIGH-SPEEDMERRY-GO-ROUND

(15

points)

ThisproblemwasProblem3onQuiz1,2007.

Considerahigh-speedmerry-go-roundwhichissimilartotheonediscussedinProb-

lem3ofProblemSet1,butwhichhastwolevels.Thatis,therearefourevenlyspaced

carswhichtravelaroundacentralhubatspeedvatadistanceRfromacentralhub,

andalsoanotherfourcarsthatareattachedtoextensionsofthefourradialarms,each

movingataspeed2vatadistance2Rfromthecenter.Inthisproblemwewillconsider

onlylightwaves,notsoundwaves,andwewillassumethatvisnotnegligiblecompared

toc,butthat2v<c.

WelearnedinProblemSet1thatthereisnoredshiftwhenlightfromonecaratradius

RisreceivedbyanobserveronanothercaratradiusR.

(a)(5points)Supposethatcars5{8areallemittinglightwavesinalldirections.Ifan

observerincar1receiveslightwavesfromeachofthesecars,whatredshiftzdoes

sheobserveforeachofthefoursignals?

(b)(10points)Supposethataspaceshipisrecedingtotherightatarelativisticspeed

ualongalinethroughthehub,asshowninthediagram.Supposethatanobserver

incar6receivesaradiosignalfromthespaceship,atthetimewhenthecarisinthe

positionshowninthediagram.Whatredshiftzisobserved?

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.20

PROBLEM

20:

SIGNAL

PROPAGATION

IN

A

FLAT

MATTER-

DOMINATED

UNIVERSE(55points)

ThefollowingproblemwasonQuiz1,2009.

Considera at,matter-dominateduniverse,withscalefactor

a(t)=bt2=3;

wherebisanarbitraryconstant.Forthefollowingquestions,theanswertoanypartmay

containsymbolsrepresentingtheanswerstopreviousparts,whetherornottheprevious

partwasansweredcorrectly.

(a)(10points)Attimet=t1 ,alightsignalissentfromgalaxyA.Let`p;sA(t)denote

thephysicaldistanceofthesignalfromAattimet.(Notethatt=0correspondsto

theoriginoftheuniverse,nottotheemissionofthesignal.)(i)Findthespeedof

separationofthelightsignalfromA,de�nedasd`p;sA=dt.Whatisthevalueofthis

speed(ii)atthetimeofemission,t1 ,and(iii)whatisitslimitingvalueatarbitrarily

latetimes?

(b)(5points)Supposethatthereisasecondgalaxy,galaxyB,thatislocatedata

physicaldistancecH�1fromAattimet1 ,whereH(t)denotestheHubbleexpansion

rateandcisthespeedoflight.(cH�1iscalledtheHubblelength.)Supposethatthe

lightsignaldescribedabove,whichisemittedfromgalaxyAattimet1 ,isdirected

towardgalaxyB.Atwhattimet2doesitarriveatgalaxyB?

(c)(10points)Let`p;sB(t)denotethephysicaldistanceofthelightsignalfromgalaxy

Battimet.(i)FindthespeedofapproachofthelightsignaltowardsB,de�nedas

�d`p;sB=dt.Whatisthevalueofthisspeed(ii)atthetimeofemission,t1 ,and(iii)

atthetimeofreception,t2 ?

(d)(10points)IfanastronomerongalaxyAobservesthelightarrivingfromgalaxyB

attimet1 ,whatisitsredshiftzBA?

(e)(10points)Supposethatthereisanothergalaxy,galaxyC,also

locatedataphysicaldistancecH�1fromAattimet1 ,butin

adirectionorthogonaltothatofB.IfgalaxyBisobserved

fromgalaxyCattimet1 ,whatistheobservedredshiftzBC?

Recallthatthisuniverseis at,soEuclideangeometryapplies.

(f)(10points)SupposethatgalaxyA,attimet1 ,emitselectromagneticradiationspher-

icallysymmetrically,withpoweroutputP.(Pmightbemeasured,forexample,in

watts,where1watt=1joule/second.)Whatistheradiationenergy uxJthatis

receivedbygalaxyBattimet2 ,whentheradiationreachesgalaxyB?(Jmightbe

measured,forexample,inwattspermeter2.Unitsarementionedhereonlytohelp

clarifythemeaningofthesequantities|

youranswershouldhavenoexplicitunits,

butshouldbeexpressedintermsofanyorallofthegivenquantitiest1 ,P,andc,

plusperhapssymbolsrepresentingtheanswerstopreviousparts.)

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.21

PROBLEM

21:DID

YOU

DO

THEREADING?(25points)

ThefollowingproblemappearedonQuiz1of2011.

(a)(10points)Hubble'slawrelatesthedistanceofgalaxiestotheirvelocity.

The

Dopplere�ectprovidesanaccuratetooltomeasurevelocity,whilethemeasure

ofcosmicdistancesismoreproblematic.Explainbrie ythemethodthatHubble

usedtoestimatethedistanceofgalaxiesinderivinghislaw.

(b)(5points)OneexpectsHubble'slawtoholdasaconsequenceoftheCosmological

Principle.WhatdoestheCosmologicalPrinciplestate?

(c)(10points)Giveabriefde�nitionforthewordshomogeneityandisotropy.Thensay

foreachofthefollowingtwostatementswhetheritistrueorfalse.Iftrueexplain

brie ywhy.Iffalsegiveacounter-example.YoushouldassumeEuclideangeometry

(whichWeinbergimplicitlyassumedinhisdiscussion).

(i)Iftheuniverseisisotropicaroundonepointthenithastobehomogeneous.

(ii)Iftheuniverseisisotropicaroundtwoormoredistinctpointsthenithastobe

homogeneous.

(d)Bonusquestion:(2pointsextracredit)Ifweallowcurved(i.e.,non-Euclidean)spaces,

isittruethatauniversewhichisisotropicaroundtwodistinctpointshastobe

homogeneous?Iftrueexplainbrie ywhy,andotherwisegiveacounter-example.

�

PROBLEM

22:THE

TRAJECTORY

OF

A

PHOTON

ORIGINATING

ATTHEHORIZON

(25points)

ThefollowingproblemappearedonQuiz1of2011.

Consideragaina atmatter-dominateduniverse,withascalefactorgivenby

a(t)=bt2=3;

wherebisaconstant.Lett0denotethecurrenttime.

(a)(5points)Whatisthecurrentvalueofthephysicalhorizondistance`p;horizon (t0 )?

Thatis,whatisthepresentdistanceofthemostdistantmatterthatcanbeseen,

limitedonlybythespeedoflight.

(b)(5points)Consideraphotonthatisarrivingnowfromanobjectthatisjustatthe

horizon.Ourgoalistotracethetrajectoryofthisobject.Supposethatwesetup

acoordinatesystemwithusattheorigin,andthesourceofthephotonalongthe

positivex-axis.Whatisthecoordinatex0ofthephotonatt=0?

(c)(5points)Asthephotontravelsfromthesourcetous,whatisitscoordinatex(t)

asafunctionoftime?

(d)(5points)Whatisthephysicaldistance`p (t)betweenthephotonandusasafunction

oftime?

(e)(5points)Whatisthemaximumphysicaldistance`p;max (t)betweenthephotonand

us,andatwhattimetmaxdoesitoccur?

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.22

PROBLEM

23:DID

YOU

DO

THEREADING

(2016)?

ThefollowingproblemwastakenfromQuiz1,2016,whereitcounted35points.

(a)(5points)TheMilkyWayhasbeenknownsinceancienttimesasabandoflight

stretchingacrossthesky.WenowrecognizetheMilkyWayasthegalaxyofstars

inwhichwelive,withalargecollectionofstars,includingoursun,arrangedina

giantdisk.Sincetheindividualstarsaremostlytoosmallforoureyestoresolve,we

observethecollectivelightfromthesestars,concentratedintheplaneofthedisk.

TheideathattheMilkyWayisactuallyadiskofstarswasproposedby

(i)ClaudiusPtolemy,inthe2ndcenturyAD.

(ii)JohannesKepler,in1610.

(iii)IsaacNewton,in1695.

(iv)ThomasWright,in1750.

(v)ImmanuelKant,in1755.

(vi)EdwinHubble,in1923.

(b)(5points)Onceitwasrecognizedthatweliveinagalaxy,itwasinitiallyassumed

thatourswastheonlygalaxy.Thesuggestionthatsomeofthepatchesoflight

knownasnebulaemightactuallybeothergalaxieslikeourownwasmadeby







(c)(5points)The�rst�rmevidencethatthereismorethanonegalaxystemmedfrom

theabilitytoobservetheAndromedaNebulawithhighenoughresolutiontodistin-

guishitsindividualstars.Inparticular,theobservationofCepheidvariablestarsin

AndromedaallowedadistanceestimatethatplaceitwelloutsidetheMilkyWay.

TheobservationofCepheidvariablestarsinAndromedawas�rstmadeby

(i)JohannesKepler,in1610.

(ii)IsaacNewton,in1695.

(iiiThomasWright,in1750.

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.23

(iv)ImmanuelKant,in1755.

(v)HenriettaSwanLeavittandHarlowShapleyin1915.


(d)(5points)The�rsthintthattheuniverseis�lledwithradiationwithane�ective

temperaturenear3K,althoughnotrecognizedatthetime,wasanobservationof

absorptionlinesincyanogen(CN)byAdamsandMcKellarin1941.Theyobserved

darkspectrallineswhichtheyinterpretedasabsorptionbythecyanogenoflight

comingfromthestarbehindthegascloud.Explaininafewsentenceshowtheseab-

sorptionlinescanbeusedtomakeinferencesaboutthecosmicbackgroundradiation

bathingthecyanogengascloud.

(e)(5points)Astheuniverseexpands,thetemperatureofthecosmicmicrowaveback-

ground

(i)goesupinproportiontothescalefactora(t).

(ii)staysconstant.

(iii)goesdowninproportionto1=a(t).

(iv)goesdowninproportionto1=a2(t).

(f)(5points)WhenHubblemeasuredthevalueofhisconstant,hefoundH�1�100

millionyears,2billionyears,10billionyears,or20billionyears?

(g)(5points)Explaininafewsentenceswhatismeantbytheequivalenceprinciple?

PROBLEM

24:

OBSERVING

A

DISTANT

GALAXY

IN

A

MATTER-

DOMINATED

FLATUNIVERSE

ThefollowingproblemwastakenfromQuiz1,2016,whereitcounted40points.

Supposethatwearelivinginamatter-dominated atuniverse,withascalefactor

givenby

a(t)=bt2=3;

wherebisaconstant.Thepresenttimeisdenotedbyt0 .

(a)(5points)Ifwemeasuretimeinseconds,distanceinmeters,andcoordinatedistances

innotches,whataretheunitsofb?

(b)(5points)Supposethatweobserveadistantgalaxywhichisonehalfofa\Hubble

length"away,whichmeansthatthephysicaldistancetodayis`p=12cH�1

0

,wherec

8.286QUIZ1REVIEW

PROBLEMS,FALL2018

p.24

isthespeedoflightandH0isthepresentvalueoftheHubbleexpansionrate.What

isthepropervelocityvp �d`p(t)

dt

ofthisgalaxyrelativetous?

(c)(5points)Whatisthecoordinatedistance`cbetweenusandthedistantgalaxy?

Ifyoudidnotanswerthepreviouspart,youmaystillcontinuewiththefollowingparts,

usingthesymbol`cforthecoordinatedistancetothegalaxy.

(d)(5points)Atwhattimetewasthelightthatwearenowreceivingfromthegalaxy

emitted?

(e)(5points)Whatistheredshiftzofthelightthatwearenowreceivingfromthe

distantgalaxy?

(f)(10points)Consideralightpulsethatleavesthedistantgalaxyattimete ,ascal-

culatedinpart(d),andarriveshereatthepresenttime,t0 .Calculatethephysical

distancerp (t)betweenthelightpulseandus.Findrp (t)asafunctionoftforallt

betweenteandt0 .

(g)(5points)Ifwesendaradiomessagenowtothedistantgalaxy,atwhattimetrwill

itbereceived?

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.25

SOLUTIONS

PROBLEM

1:DID

YOU

DO

THEREADING

(2000)?(35points)

a)DopplerpredictedtheDopplere�ectin1842.

b)Mostofthestarsofourgalaxy,includingoursun,lieina atdisk.Wetherefore

seemuchmorelightwhenwelookoutfromearthalongtheplaneofthediskthan

whenwelookinanyotherdirection.

c)Hubble'soriginalpaperontheexpansionoftheuniversewasbasedonastudyof

only

18galaxies.Well,atleastWeinberg'sbooksays18galaxies.Formyown

bookImadeacopyofHubble'soriginalgraph,whichseemstoshow24blackdots,

eachofwhichrepresentsagalaxy,asreproducedbelow.Theverticalaxisshows

therecessionvelocity,inkilometerspersecond.Thesolidlineshowsthebest�t

totheblackdots,eachofwhichrepresentsagalaxy.Eachopencirclerepresentsa

groupofthegalaxiesshownasblackdots,selectedbytheirproximityindirection

anddistance;thebrokenlineisthebest�ttothesepoints.Thecrossshowsa

statisticalanalysisof22galaxiesforwhichindividualdistancemeasurementswere

notavailable.IamnotsurewhyWeinbergrefersto18galaxies,butitispossible

thatthetextofHubble'sarticleindicatedthat18ofthesegalaxiesweremeasured

withmorereliabilitythantherest.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.26

d)e)

Duringatimeintervalinwhichthelinearsizeoftheuniversegrowsby1%,the

horizondistancegrowsbymorethan1%.Toseewhy,notethatthehorizondistance

isequaltothescalefactortimesthecomovinghorizondistance.Thescalefactor

growsby1%duringthistimeinterval,butthecomovinghorizondistancealsogrows,

sincelightfromthedistantgalaxieshashadmoretimetoreachus.

f)ArnoA.PenziasandRobertW.Wilson,BellTelephoneLaboratories.

g)

(i)the

averagedistancebetweenphotons:

proportionaltothesizeof

theuniverse

(Photonsareneithercreatednordestroyed,sotheonlye�ectisthattheaver-

agedistancebetweenthemisstretchedwiththeexpansion.Sincetheuniverse

expandsuniformly,alldistancesgrowbythesamefactor.)

(ii)the

typical

wavelength

of

the

radiation:

proportionaltothesizeof

theuniverse(SeeLectureNotes3.)

(iii)the

numberdensity

ofphotons

in

the

radiation:

inverselypropor-

tionaltothecubeofthesizeoftheuniverse(From

(i),theaveragedistance

betweenphotonsgrowsinproportiontothesizeoftheuniverse.Sincethevol-

umeofacubeisproportionaltothecubeofthelengthofaside,theaverage

volumeoccupiedbyaphotongrowsasthecubeofthesizeoftheuniverse.The

numberdensityistheinverseoftheaveragevolumeoccupiedbyaphoton.)

(iv)the

energy

density

of

the

radiation:

inverselyproportionaltothe

fourthpowerofthesizeoftheuniverse(Theenergyofeachphotonispropor-

tionaltoitsfrequency,andhenceinverselyproportionaltoitswavelength.So

from(ii)theenergyofeachphotonisinverselyproportionaltothesizeofthe

universe,andfrom(iii)thenumberdensityisinverselyproportionaltothecube

ofthesize.)

(v)the

temperature

of

the

radiation:

inverselyproportionaltothesize

oftheuniverse(Thetempera-

tureisdirectlyproportionaltotheaverageenergyofaphoton,whichaccording

to(iv)isinverselyproportionaltothesizeoftheuniverse.)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.27

PROBLEM

2:THESTEADY-STATEUNIVERSETHEORY

(25points)

a)(10points)AccordingtoEq.(3.7),H

(t)=

1a(t)

dad

t:

Sointhiscase

1a(t)

dad

t=H0;

whichcanberewrittenas

daa

=H0dt:

Integrating,

lna=H0t+c;

wherecisaconstantofintegration.Exponentiating,

a=beH0

t;

whereb=ecisanarbitraryconstant.

b)(15points)Consideracubeofside`cdrawnonthecomovingcoordinatesystem

diagram.Thephysicallengthofeachsideisthena(t)`c ,sothephysicalvolumeis

V(t)=a3(t)`3c:

Sincethemassdensityis�xedat�=�0 ,thetotalmassinsidethiscubeatanygiven

timeisgivenby

M(t)=a3(t)`3c�0:

Intheabsenceofmattercreationthetotalmasswithinacomovingvolumewouldnot

change,sotheincreaseinmassdescribedbytheaboveequationmustbeattributed

tomattercreation.Therateofmattercreationperunittimeperunitvolumeis

thengivenby

Rate=

1V(t)

dMd

t

=

1

a3(t)`3c3a2(t)dad

t`3c�0

=3adad

t�0

=

3H0�0:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.28

Youwerenotaskedtoinsertnumbers,butitisworthwhiletoconsiderthenumerical

valueaftertheexam,toseewhatthisansweristellingus.SupposewetakeH0=70

km-sec �1-Mpc �1,andtake�0tobethecriticaldensity,�c=3H20 =8�G.Then

Toputthisnumberintomoremeaningfulterms,notethatthemassofahydrogen

atom

is1:67�10 �27

kg,andthat1year=

3:156�107

s.Therateofmatter

productionrequiredforthesteady-stateuniversetheorycanthenbeexpressedas

roughlyonehydrogenatompercubicmeterperbillionyears!Needlesstosay,sucha

rateofmatterproductionistotallyundetectable,sothesteady-statetheorycannot

beruledoutbythefailuretodetectmatterproduction.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.29

PROBLEM

3:DID

YOU

DO

THEREADING

(2007)?(25points)

Thefollowing5questionsareeachworth5points:

(a)Inthe1940's,threeastrophysicistsproposeda\steadystate"theoryofcosmology,

inwhichtheuniversehasalwayslookedaboutthesameasitdoesnow.Statethe

lastnameofatleastoneoftheseauthors.(Bonuspoints:youcanearn1pointeach

fornamingtheothertwoauthors,andhenceupto2additionalpoints,but1point

willbetakeno�foreachincorrectanswer.)

Ans:(Weinberg,page8,orRyden,page16):HermannBondi,ThomasGold,and

FredHoyle.

(b)In1917,aDutchastronomernamedWillemdeSitterdidwhichoneofthefollowing

accomplishments:

(i)measuredthesizeoftheMilkyWaygalaxy,�ndingittobeaboutonebillion

light-yearsindiameter.

(ii)resolvedCepheidvariablestarsinAndromedaandtherebyobtainedpersua-

siveevidencethatAndromedaisnotwithinourowngalaxy,butisapparently

anothergalaxylikeourown.

(iii)publishedacatalog,NebulaeandStarClusters,listing103objectsthatas-

tronomersshouldavoidwhenlookingforcomets.

(iv)publishedamodelfortheuniverse,basedongeneralrelativity,whichappeared

tobestaticbutwhichproducedaredshiftproportionaltothedistance.

(v)discoveredthattheorbitalperiodsoftheplanetsareproportionaltothe3/2

powerofthesemi-majoraxisoftheirellipticalorbits.

Discussion:(i)isfalseinpartbecausedeSitterwasnotinvolvedinthemeasurement

ofthesizeoftheMilkyWay,butthemostobviouserrorisinthesizeoftheMilky

Way.ItsactualdiameterisreportedbyWeinberg(p.16)tobeabout100,000light-

years,althoughnowitisbelievedtobeabouttwicethatlarge.(ii)isanaccurate

descriptionofanobservationbyEdwinHubblein1923(Weinberg,pp.19-20).(iii)

describestheworkofCharlesMessierin1781(Weinberg,p.17).(v)isofcourseone

ofKepler'slawsofplanetarymotion.

(c)In1964{65,ArnoA.PenziasandRobertW.Wilsonobserveda uxofmicrowave

radiationcomingfromalldirectionsinthesky,whichwasinterpretedbyagroupof

physicistsataneighboringinstitutionasthecosmicbackgroundradiationleftover

fromthebigbang.Circlethetwoitemsonthefollowinglistthatwerenotpartof

thestorybehindthisspectaculardiscovery:

(i)BellTelephoneLaboratory

(ii)MIT

(iii)PrincetonUniversity

(iv)pigeons

(v)groundhogs

(vi)Hubble'sconstant

(vii)liquidhelium

(viii)7.35cm

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.30

(Grading:3ptsfor1correctanswer,5for2correctanswers,and-2foreachincorrect

answer,buttheminimumscoreiszero.)

Discussion:Thediscoveryofthecosmicbackgroundradiationwasdescribedinsome

detailbyWeinberginChapter3.TheobservationwasdoneatBellTelephone

Laboratories,inHolmdel,NewJersey.Thedetectorwascooledwithliquidhelium

tominimizeelectricalnoise,andthemeasurementsweremadeatawavelengthof

7.35cm.Duringthecourseoftheexperimenttheastronomershadtoejectapair

ofpigeonswhowereroostingintheantenna.PenziasandWilsonwerenotinitially

awarethattheradiationtheydiscoveredmighthavecomefromthebigbang,but

BernardBurkeofMITputthem

intouchwithagroupatPrincetonUniversity

(RobertDicke,JamesPeebles,P.G.Roll,andDavidWilkinson)whowereactively

workingonthishypothesis.

(d)ImportantpredictionsoftheCopernicantheorywerecon�rmedbythediscovery

oftheaberrationofstarlight(whichshowedthatthevelocityoftheEarthhasthe

time-dependenceexpectedforrotationabouttheSun)andbythebehaviorofthe

Foucaultpendulum(whichshowedthattheEarthrotates).Thesediscoverieswere

made

(i)duringCopernicus'lifetime.

(ii)approximatelytwoandthreedecadesafterCopernicus'death,respectively.

(iii)aboutonehundredyearsafterCopernicus'death.

(iv)approximatelytwoandthreecenturiesafterCopernicus'death,respectively.

Rydendiscussesthisonp.5.Theaberrationofstarlightwasdiscoveredin1728,

whiletheFoucaultpendulumwasinventedin1851.

(e)IfoneaveragesoversuÆcientlylargescales,theuniverseappearstobehomogeneous

andisotropic.Howlargemusttheaveragingscalebebeforethishomogeneityand

isotropysetin?

(i)1AU(1AU=1:496�1011m).

(ii)100kpc(1kpc=1000pc,1pc=3:086�1016m=3.262light-year).

(iii)1Mpc(1Mpc=106pc).

(iv)10Mpc.

(v)100Mpc.

(vi)1000Mpc.

ThisissueisdiscussedinRyden'sbookonp.11.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.31

PROBLEM

4:AN

EXPONENTIALLY

EXPANDING

UNIVERSE

(a)AccordingtoEq.(3.7),theHubbleconstantisrelatedtothescalefactorby

H=_a=a:

So

H=�a0 e�t

a0 e�t

=

�:

(b)AccordingtoEq.(3.8),thecoordinatevelocityoflightisgivenby

dxd

t=

ca(t)=

ca0e ��t:

Integrating,

x(t)=

ca0 Z

t0

e ��t0d

t 0

=

ca0 ��

1�e ��t0 �t0

=

c�a0 �1�e ��t �:

(c)FromEq.(3.11),orfromthefrontofthequiz,onehas

1+z=a(tr )

a(te ):

Herete=0,so

1+z=a0 e�tr

a0

=)

e�tr

=1+z

=)

tr=1�

ln(1+z):

(d)Thecoordinatedistanceisx(tr ),wherex(t)isthefunctionfoundinpart(b),and

tristhetimefoundinpart(c).So

e�tr

=1+z;

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.32

and

x(tr )=

c�a0 �1�e ��tr �

=

c�a0 �1�1

1+z �

=

cZ

�a0 (1+z):

Thephysicaldistanceatthetimeofreceptionisfoundbymultiplyingbythescale

factoratthetimeofreception,so

`p (tr )=a(tr )x(tr )=

cze�tr

�(1+z)=

cz�:

PROBLEM

5:DID

YOU

DO

THEREADING

(1986/1990composite)?

(a)Thedistinguishingquantityis��=�c .Theuniverseisopenif<1, atif=1,

orclosedif>1.

(b)Thetemperatureofthemicrowavebackgroundtodayisabout3Kelvin.(Thebest

determinationtodate*wasmadebytheCOBEsatellite,whichmeasuredthetem-

peratureas2:728�0:004Kelvin.Theerrorhereisquotedwitha95%con�dence

limit,whichmeansthattheexperimentersbelievethattheprobabilitythatthetrue

valueliesoutsidethisrangeisonly5%.)

(c)Thecosmicmicrowavebackgroundisobservedtobehighlyisotropic.

(d)ThedistancetotheAndromedanebulaisroughly2millionlightyears.

(e)1929.

(f)2billionyears.Hubble'svalueforHubble'sconstantwashighbymodernstandards,

byafactorof5to10.

(g)Theabsoluteluminosity(i.e.,thetotallightoutput)ofaCepheidvariablestar

appearstobehighlycorrelatedwiththeperiodofitspulsations.Thiscorrelation

canbeusedtoestimatethedistancetotheCepheid,bymeasuringtheperiodand

theapparentluminosity.Fromtheperiodonecanestimatetheabsoluteluminosity

ofthestar,andthenoneusestheapparentluminosityandthe1=r2

lawforthe

intensityofapointsourcetodeterminethedistancer.

(h)107light-years.

(i)ArnoA.PenziasandRobertW.Wilson,BellTelephoneLaboratories.

(j)PrincetonUniversity.

*AstrophysicalJournal,vol.473,p.576(1996):TheCosmicMicrowaveBackground

SpectrumfromtheFullCOBEFIRASDataSets,D.J.Fixsen,E.S.Cheng,J.M.Gales,

J.C.Mather,R.A.Shafer,andE.L.Wright.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.33

PROBLEM

6:AFLATUNIVERSEWITHUNUSUALTIMEEVOLUTION

Thekeytothisproblemistoworkincomovingcoordinates.

[Somestudentshaveaskedmewhyonecannotuse\physical"coordinates,forwhich

thecoordinatesreallymeasurethephysicaldistances.Inprincipleonecanuseany

coordinatesystemonlikes,butthecomovingcoordinatesarethesimplest.Inanyother

system

itisdiÆculttowritedownthetrajectoryofeitheraparticleoralight-beam.

Incomovingcoordinatesitiseasytowritethetrajectoryofeitheralightbeam,ora

particlewhichismovingwiththeexpansionoftheuniverse(andhencestandingstill

inthecomovingcoordinates).Note,bytheway,thatwhenonesaysthataparticle

isstandingstillincomovingcoordinates,onehasnotreallysaidverymuchaboutit's

trajectory.Onehassaidthatitismovingwiththematterwhich�llstheuniverse,but

onehasnotsaid,forexample,howthedistancebetweentheparticleandoriginvaries

withtime.Theanswertothislatterquestionisthendeterminedbytheevolutionofthe

scalefactor,a(t).]

(a)Thephysicalseparationattoisgivenbythescalefactortimesthecoordinatedis-

tance.Thecoordinatedistanceisfoundbyintegratingthecoordinatevelocity,so

`p (to )=a(to ) Z

to

te

cdt 0

a(t 0)=bt1=3

o Zto

te

cdt 0

bt 01=3

=32

ct1=3

o ht2=3

o

�t2=3

e i

=32

cto h1�(te =to )2=3 i:

(b)Fromthefrontoftheexam,

1+z=a(to )

a(te )= �to

te �

1=3

=)

z= �to

te �

1=3�

1:

(c)Bycombiningtheanswersto(a)and(b),onehas

`p (to )=32

cto �1�

1

(1+z)2 �:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.34

(d)Thephysicaldistanceofthelightpulseattimetisequaltoa(t)timesthecoordinate

distance.Thecoordinatedistanceattimetisequaltothestartingcoordinate

distance,`c (te ),minusthecoordinatedistancethatthelightpulsetravelsbetween

timeteandtimet.Thus,

`p (t)=a(t) �`c (te )� Z

tte

cdt 0

a(t 0) �

=a(t) �Z

to

te

cdt 0

a(t 0) � Z

tte

cdt 0

a(t 0) �

=a(t) Z

to

t

cdt 0

a(t 0)

=bt1=3 Zto

t

cdt 0

bt 01=3

=32

ct1=3 ht2=3

o

�t2=3 i

=

32ct "�tot �2=3�

1 #:

PROBLEM

7:ANOTHERFLATUNIVERSEWITHAN

UNUSUALTIME

EVOLUTION

(40points)

a)(5points)Thecosmologicalredshiftisgivenbytheusualform,

1+z=a(t0 )

a(te ):

Forlightemittedbyanobjectattimete ,theredshiftofthereceivedlightis

1+z=a(t0 )

a(te )= �t0

te �

:

So,

z= �t0

te �

�1:

b)(5points)Thecoordinatest0andtearecosmictimecoordinates.The\look-back"

timeasde�nedintheexamisthentheintervalt0 �te .Wecanwritethisas

t0 �te=t0 �1�te

t0 �:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.35

Wecanusetheresultofpart(a)toeliminatete =t0infavorofz.From(a),

te

t0

=(1+z) �1=

:

Therefore,

t0 �te=t0 h1�(1+z) �1= i:

c)(10points)Thepresentvalueofthephysicaldistancetotheobject,`p (t0 ),isfound

from

`p (t0 )=a(t0 ) Z

t0

te

ca(t)dt:

Calculatingthisintegralgives

`p (t0 )=

ct 0

1� "

1t �1

0

�1

t �1

e

#:

Factoringt �1

0

outoftheparenthesesgives

`p (t0 )=

ct0

1� "1� �t0

te �

�1 #

:

ThiscanberewrittenintermsofzandH0usingtheresultofpart(a)aswellas,

H0=

_a(t0 )

a(t0 )=

t0

:

Finallythen,

`p (t0 )=cH�1

0

1� h1�(1+z) �

1 i:

d)(10points)AnearlyidenticalproblemwasworkedthroughinProblem8ofProblem

Set1.

Theenergyoftheobservedphotonswillberedshiftedbyafactorof(1+z).In

additiontherateofarrivalofphotonswillberedshiftedrelativetotherateofphoton

emmission,reducingthe uxbyanotherfactorof(1+z).Consequently,theobserved

powerwillberedshiftedbytwofactorsof(1+z)toP=(1+z)2.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.36

Imagineahypotheticalsphereincomovingcoordinatesasdrawnabove,centeredon

theradiatingobject,withradiusequaltothecomovingdistance`c .Nowconsiderthe

photonspassingthroughapatchofthespherewithphysicalareaA.Incomoving

coordinatesthepresentareaofthepatchisA=a(t0 )2.Sincetheobjectradiates

uniformlyinalldirections,thepatchwillinterceptafraction(A=a(t0 )2)=(4�`2c )of

thephotonspassingthroughthesphere.ThusthepowerhittingtheareaAis

(A=a(t0 )2)

4�`2c

P

(1+z)2

:

Theradiationenergy uxJ,whichisthereceivedpowerperarea,reachingtheearth

isthengivenby

J=

1

4�`p (t0 )2

P

(1+z)2

whereweused`p (t0 )=a(t0 )`c .Usingtheresultofpart(c)towriteJintermsof

P;H0 ;z;and gives,

J=

H20

4�c2 �1�

�2

P

(1+z)2 h1�(1+z) �

1 i2

:

e)(10points)FollowingthesolutionofProblem1ofProblemSet1,wecanintroduce

a�ctitiousrelaystationthatisatrestrelativetothegalaxy,butlocatedjustnext

tothejet,betweenthejetandEarth.Asintheprevioussolution,therelaystation

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.37

simplyrebroadcaststhesignalitreceivesfromthesource,atexactlytheinstantthat

itreceivesit.Therelaystationthereforehasnoe�ectonthesignalreceivedbythe

observer,butallowsustodividetheproblemintotwosimpleparts.

Thedistancebetweenthejetandtherelaystationisveryshortcomparedtocos-

mologicalscales,sothee�ectoftheexpansionoftheuniverseisnegligible.Forthis

partoftheproblemwecanusespecialrelativity,whichsaysthattheperiodwith

whichtherelaystationmeasuresthereceivedradiationisgivenby

�trelaystation= s1�vc

1+vc

��tsource:

NotethatIhaveusedtheformulafromthefrontoftheexam,butIhavechanged

thesizeofv,sincethesourceinthiscaseismovingtowardtherelaystation,sothe

lightisblue-shifted.ToobserversonEarth,therelaystationisjustasourceatrest

inthecomovingcoordinatesystem,so

�tobserved=(1+z)�trelaystation:

Thus,

1+zJ ��tobserved

�tsource

=

�tobserved

�trelaystation

�trelaystation

�tsource

=(1+z)jcosmological �(1+z)jspecialrelativity

=(1+z) s1�vc

1+vc

:

Thus,

zJ=(1+z) s1�vc

1+vc

�1:

Noteadded:Inlookingoverthesolutionstothisproblem,Ifoundthatasubstan-

tialnumberofstudentswrotesolutionsbasedontheincorrectassumptionthatthe

Dopplershiftcouldbetreatedasifitwereentirelyduetomotion.Thesestudents

usedthespecialrelativityDopplershiftformulatoconverttheredshiftzofthe

galaxytoavelocityofrecession,thensubtractedfromthisthespeedvofthejet,

andthenagainusedthespecialrelativityDopplershiftformulato�ndtheDoppler

shiftcorrespondingtothiscompositevelocity.However,asdiscussedattheendof

LectureNotes3,thecosmologicalDopplershiftisgivenby

1+z��to

�te=a(to )

a(te );

(3.11)

andisnotpurelyane�ectcausedbymotion.Itisreallythecombinede�ectofthe

motionofthedistantgalaxiesandthegravitational�eldthatexistsbetweenthe

galaxies,sothespecialrelativityformularelatingztovdoesnotapply.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.38

PROBLEM

8:DID

YOU

DO

THEREADING

(1996)?

a)Thelinesweredark,causedbyabsorptionoftheradiationinthecooler,outerlayers

ofthesun.

b)IndividualstarsintheAndromedaNebulawereresolvedbyHubblein1923.

[Theothernamesanddatesarenotwithoutsigni�cance.In1609Galileobuilthis

�rsttelescope;during1609-10heresolvedtheindividualstarsoftheMilkyWay,and

alsodiscoveredthatthesurfaceofthemoonisirregular,thatJupiterhasmoons

ofitsown,thatSaturnhashandles(laterrecognizedasrings),thatthesunhas

spots,andthatVenushasphases.In1755ImmanuelKantpublishedhisUniversal

NaturalHistoryandTheoryoftheHeavens,inwhichhesuggestedthatatleast

someofthenebulaearegalaxieslikeourown.In1912HenriettaLeavittdiscovered

therelationshipbetweentheperiodandluminosityofCepheidvariablestars.In

the1950sWalterBaadeandAllanSandagerecalibratedtheextra-galacticdistance

scale,reducingtheacceptedvalueoftheHubbleconstantbyaboutafactorof10.]

c)

(i)True.[In1941,A.McKellardiscoveredthatcyanogencloudsbehaveasifthey

arebathedinmicrowaveradiationatatemperatureofabout2.3 ÆK,butno

connectionwasmadewithcosmology.]

(ii)False.[Anyradiationre ectedbythecloudsisfartooweaktobedetected.It

isthebrightstarlightshiningthroughthecloudthatisdetectable.]

(iii)True.[Electromagneticwavesatthesewavelengthsaremostlyblockedbythe

Earth'satmosphere,sotheycouldnotbedetecteddirectlyuntilhighaltitude

balloonsandrocketswereintroducedintocosmicbackgroundradiationresearch

inthe1970s.PrecisedatawasnotobtaineduntiltheCOBEsatellite,in1990.]

(iv)True.[ThemicrowaveradiationcanboosttheCNmoleculefrom

itsground

statetoalow-lyingexcitedstate,astateinwhichtheCandNatomsrotate

abouteachother.Thepopulationofthislow-lyingstateisthereforedetermined

bytheintensityofthemicrowaveradiation.Thispopulationismeasuredby

observingtheabsorptionofstarlightpassingthroughtheclouds,sincethere

areabsorptionlinesinthevisiblespectrumcausedbytransitionsbetweenthe

low-lyingstateandhigherenergyexcitedstates.]

(v)False.[Nochemicalreactionsareseen.]

d)Aristarchus.[TheheliocentricpicturewasneveracceptedbyotherGreekphiloso-

phers,however,andwasnotreviveduntilthepublicationofDeRevolutionibusOr-

bium

Coelestium

(OntheRevolutionsoftheCelestialSpheres)byCopernicusin

1543.]

e)(ii)Anypatchofthenightskywouldlookasbrightasthesurfaceofthesun.

[Explanation:Thecruxoftheargumentisthatthebrightnessofanobject,measured

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.39

forexamplebythepowerperarea(i.e., ux)hittingtheretinaofyoureye,doesnot

changeastheobjectismovedfurtheraway.Thepowerfallso�withthesquareof

thedistance,butsodoestheareaoftheimageonyourretina|

sothepowerper

areaisindependentofdistance.Undertheassumptionsstated,yourlineofsight

willeventuallyhitastarnomatterwhatdirectionyouarelooking.Theenergy ux

onyourretinawillthereforebethesameasintheimageofthesun,sotheentire

skywillappearasbrightasthesurfaceofthesun.]

PROBLEM

9:A

FLATUNIVERSEWITH

a(t)/

t3=5

a)Ingeneral,theHubbleconstantisgivenbyH

=_a=a,wheretheoverdotdenotesa

derivativewithrespecttocosmictimet.Inthiscase

H=

1bt3=5

35bt �2=5=

35t:

b)Ingeneral,the(physical)horizondistanceisgivenby

`p;horizon (t)=a(t) Z

t0

ca(t 0)dt 0:

Inthiscaseonehas

`p;horizon (t)=bt3=5 Zt

0

cbt 03=5dt 0=ct3=552 ht2=5�02=5 i=

52ct:

c)Thecoordinatespeedoflightisc=a(t),sothecoordinatedistancethatlighttravels

betweentA

andtB

isgivenby

`c= Z

tB

tA

ca(t 0)dt 0= Z

tB

tA

cbt 03=5dt 0=

5c

2b �t2=5

B

�t2=5

A �:

d)Thephysicalseparationisjustthescalefactortimesthecoordinateseparation,so

`p (tA)=a(tA)`c=

52ctA "�tBt

A �2=5�

1 #:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.40

`p (tB)=a(tB)`c=

52ctB "1� �tA

tB �

2=5 #

:

e)Letteqbethetimeatwhichthelightpulseisequidistantfromthetwogalaxies.At

thistimeitwillhavetraveledacoordinatedistance`c =2,where`cistheanswerto

part(c).Sincethecoordinatespeedisc=a(t),thetimeteqcanbefoundfrom:

Zteq

tA

ca(t 0)dt 0=12

`c

5c

2b �t2=5

eq

�t2=5

A �=5c

4b �t2=5

B

�t2=5

A �

Solvingforteq ,

teq= "t2=5

A

+t2=5

B

2

#5=2

:

f)AccordingtoHubble'slaw,thespeedisequaltoHubble'sconstanttimesthephysical

distance.Bycombiningtheanswerstoparts(a)and(d),onehas

v=H(tA)`p (tA)

=

35tA

52ctA "�tBt

A �2=5�

1 #=

32c "�tBt

A �2=5�

1 #:

g)Theredshiftforradiationobservedattimetcanbewrittenas

1+z=

a(t)

a(te );

whereteisthetimethattheradiationwasemitted.Solvingforte ,

te=

t

(1+z)5=3

:

Asfoundinpart(d),thephysicaldistancethatthelighttravelsbetweenteandt,

asmeasuredattimet,isgivenby

`p (t)=a(t) Z

tte

ca(t 0)dt 0=52

ct "1� �tet �2=5 #

:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.41

Substitutingtheexpressionforte ,onehas

`p (t)=52

ct �1�

1

(1+z)2=3 �:

Asz!1,thisexpressionapproaches

limz!1`p (t)=52

ct;

whichisexactlyequaltothehorizondistance.Itisageneralrulethatthehorizon

distancecorrespondstoin�niteredshiftz.

h)Againwewillviewtheproblemincomovingcoordinates.PutgalaxyBattheorigin,

andgalaxyAatacoordinatedistance`calongthex-axis.Drawasphereofradius`c ,

centeredgalaxyA.AlsodrawadetectorongalaxyB,withphysicalareaA(measured

atthepresenttime).

Theenergyfromthequasarwillradiateuniformlyonthesphere.Thedetectorhas

aphysicalareaA,sointhecomovingcoordinatepictureitsareainsquarenotches

wouldbeA=a(tB)2.Thedetectorthereforeoccupiesafractionofthespheregiven

by

[A=a(tB)2]

4�`2c

=

A

4�`p (tB)2

;

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.42

sothisfractionoftheemittedphotonswillstrikethedetector.

Nextconsidertherateofarrivalofthephotonsatthesphere.Inlecturewe�gured

outthatifaperiodicwaveisemittedattimetA

andobservedattimetB,thenthe

rateofarrivalofthewavecrestswillbeslowerthantherateofemissionbyaredshift

factor1+z=a(tB)=a(tA).Thesameargumentwillapplytotherateofarrivalof

photons,sotherateofphotonarrivalatthespherewillbeslowerthantherateof

emissionbythefactor1+z,reducingtheenergy uxbythisfactor.Inaddition,

eachphotonisredshiftedinfrequencyby1+z.Sincetheenergyofeachphotonis

proportionaltoitsfrequency,theenergy uxisreducedbyanadditionalfactorof

1+z.Thus,therateatwhichenergyreachesthedetectoris

Powerhittingdetector=

A

4�`p (tB)2

P

(1+z)2

:

TheredshiftzofthelightpulsereceivedatgalaxyBisgivenby

1+z=a(tB)

a(tA)= �tBt

A �3=5

:

Usingoncemoretheexpressionfor`P(tB)frompart(d),onehas

J=Powerhittingdetector

A

=

P(tA=tB)6=5

25�c2t2B �1� �tA

tB �

2=5 �2

:

TheproblemiswordedsothattA,andnotz,isthegivenvariablethatdetermines

howfargalaxyAisfromgalaxyB.Inpractice,however,itisusuallymoreusefulto

expresstheanswerintermsoftheredshiftzofthereceivedradiation.Onecando

thisbyusingtheaboveexpressionfor1+ztoeliminatetA

infavorofz,�nding

J=

P

25�c2t2B(1+z)2=3 �(1+z)2=3�1 �2

:

i)Lett 0AbethetimeatwhichthelightpulsearrivesbackatgalaxyA.Thepulsemust

thereforetravelacoordinatedistance`c(theanswertopart(c))betweentimetB

andt 0A,so

Zt0A

tB

ca(t 0)dt 0=`c:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.43

Usingtheanswerfrom(c)andintegratingtheleft-handside,

5c

2b �t 02=5

A

�t2=5

B �=5c

2b �t2=5

B

�t2=5

A �:

Solvingfort 0A;

t 0A= �2t2=5

B

�t2=5

A �5=2

:

PROBLEM

10:DID

YOU

DO

THEREADING

(1998)?

a)Einsteinbelievedthattheuniversewasstatic,andthecosmologicaltermwasneces-

sarytopreventastaticuniversefromcollapsingundertheattractiveforceofnormal

gravity.[Therepulsivee�ectofacosmologicalconstantgrowslinearlywithdistance,

soifthecoeÆcientissmallitisimportantonlywhentheseparationsareverylarge.

Suchatermcanbeimportantcosmologicallywhilestillbeingtoosmalltobede-

tectedbyobservationsofthesolarsystemoreventhegalaxy.Recentmeasurements

ofdistantsupernovas(z�1),whichyoumayhavereadaboutinthenewspapers,

makeitlooklikemaybethereisacosmologicalconstantafterall!Sincethecosmo-

logicalconstantisthehotissueincosmologythisseason,wewillwanttolookatit

morecarefully.ThebesttimewillbeafterLectureNotes7.]

b)Atthetimeofitsdiscovery,deSitter'smodelwasthoughttobestatic[althoughit

wasknownthatthemodelpredictedaredshiftwhich,atleastfornearbygalaxies,

wasproportionaltothedistance].Fromamodernperspectivethemodelisthought

tobeexpanding.

[Itseemsstrangethatphysicistsin1917couldnotcorrectlydetermineifthe

theorydescribedauniversethatwasstaticorexpanding,butthemathematical

formalismofgeneralrelativitycanberatherconfusing.Thebasicproblemisthat

whenspaceisnotEuclideanthereisnosimplewaytoassigncoordinatestoit.

Themathematicsofgeneralrelativityisdesignedtobevalidforanycoordinate

system,buttheunderlyingphysicscansometimesbeobscuredbyapeculiarchoice

ofcoordinates.Achangeofcoordinatescannotonlydistorttheapparentgeometryof

space,butitcanalsomixupspaceandtime.ThedeSittermodelwas�rstwritten

downincoordinatesthatmadeitlookstatic,soeveryonebelieveditwas.Later

ArthurEddingtonandHermannWeyl(independently)calculatedthetrajectoriesof

testparticles,discoveringthatthey ewapart.]

c)n1=3,andn2=4.

d)Above3,000Ktheuniversewassohotthattheatomswereionized,dissociatedinto

nucleiandfreeelectrons.Ataboutthistemperature,however,theuniversewascool

enoughsothatthenucleiandelectronscombinedtoformneutralatoms.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.44

[Thisprocessisusuallycalled\recombination,"althoughthepre�x\re-"is

totallyinaccurate,sinceinthebigbangtheorytheseconstituentshadneverbeen

previouslycombined.AsfarasIknowthewordwas�rstusedinthiscontextby

P.J.E.Peebles,soIonceaskedhimwhythepre�xwasused.Herepliedthatthisword

isstandardterminologyinplasmaphysics,andwascarriedoverintocosmology.]

[Regardlessofitsname,recombinationwascrucialfortheclumpingofmatter

intogalaxiesandstars,becausethepressureofthephotonsintheearlyuniversewas

enormous.Whenthematterwasionized,thefreeelectronsinteractedstronglywith

thephotons,sothepressureofthesephotonspreventedthematterfromclumping.

Afterrecombination,however,thematterbecameverytransparenttoradiation,and

thepressureoftheradiationbecameine�ective.]

[Incidentally,atroughlythesametimeasrecombination(withbiguncertain-

ties),themassdensityoftheuniversechangedfrombeingdominatedbyradiation

(photonsandneutrinos)tobeingdominatedbynonrelativisticmatter.Thereisno

knownunderlyingconnectionbetweenthesetwoevents,anditseemstobesome-

thingofacoincidencethattheyoccurredataboutthesametime.Thetransition

fromradiation-dominationtomatter-dominationalsohelpedtopromotetheclump-

ingofmatter,butthee�ectwasmuchweakerthanthee�ectofrecombination|

becauseoftheveryhighvelocityofphotonsandneutrinos,theirpressureremained

asigni�cantforceevenaftertheirmassdensitybecamemuchsmallerthanthatof

matter.]

PROBLEM

11:ANOTHERFLATUNIVERSEWITH

a(t)/

t3=5

a)AccordingtoEq.(3.7)oftheLectureNotes,

H(t)=

1a(t)

dad

t:

Forthespecialcaseofa(t)=bt3=5,thisgives

H(t)=

1bt3=535

bt �2=5=

35t:

b)AccordingtoEq.(3.8)oftheLectureNotes,thecoordinatevelocityoflight(in

comovingcoordinates)isgivenby

dxd

t=

ca(t):

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.45

SincegalaxiesAandBhavephysicalseparation`0

attimet1 ,theircoordinate

separationisgivenby

`c=

`0

bt3=5

1

:

Theradiosignalmustcoverthiscoordinatedistanceinthetimeintervalfromt1to

t2 ,whichimpliesthat

Zt2

t1

ca(t)dt=

`0

bt3=5

1

:

Usingtheexpressionfora(t)andintegrating,

5c

2b �t2=5

2

�t2=5

1 �=

`0

bt3=5

1

;

whichcanbesolvedfort2togive

t2= �1+

2`0

5ct1 �

5=2

t1:

c)Themethodisthesameasinpart(b).Thecoordinatedistancebetweenthetwo

galaxiesisunchanged,butthistimethedistancemustbetraversedinthetime

intervalfromt2tot3 .So,

Zt3

t2

ca(t)dt=

`0

bt3=5

1

;

whichleadsto

5c

2b �t2=5

3

�t2=5

2 �=

`0

bt3=5

1

:

Solvingfort3gives

t3= "�t2

t1 �

2=5

+

2`0

5ct1 #

5=2

t1:

Theaboveanswerisperfectlyacceptable,butonecouldalsoreplacet2byusingthe

answertopart(b),whichgivest

3= �1+

4`0

5ct1 �

5=2

t1:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.46

[Alternatively,onecouldhavebeguntheproblembyconsideringthefullround

tripoftheradiosignal,whichtravelsacoordinatedistance2`c

duringthetime

intervalfromt1tot3 .Theproblemthenbecomesidenticaltopart(b),exceptthat

thecoordinatedistance`cisreplacedby2`c ,andt2isreplacedbyt3 .Oneisled

immediatelytotheanswerintheformofthepreviousequation.]

d)Cosmictimeisde�nedbythereadingofsuitablysynchronizedclockswhichareeach

atrestwithrespecttothematteroftheuniverseatthesamelocation.(Forthis

problemwewillnotneedtothinkaboutthemethodofsynchronization.)Thus,the

cosmictimeintervalbetweenthereceiptofthemessageandtheresponseisthesame

aswhatismeasuredonthegalaxyBclocks,whichis�t.Theresponseistherefore

sentatcosmictimet2+�t.Thecoordinatedistancebetweenthegalaxiesisstill

`0 =a(t1 ),so

Zt4

t2+�t

ca(t)dt=

`0

bt3=5

1

:

Integrationgives

5c

2b ht

2=5

4

�(t2+�t)2=5 i=

`0

bt3=5

1

;

whichcanbesolvedfort4togive

t4= "�t2+�t

t1

�2=5

+

2`0

5ct1 #

5=2

t1:

e)Fromtheformulaatthefrontoftheexam,

1+z=a(tobserved )

a(temitted )=

a(t4 )

a(t2+�t)= �t4

t2+�t �

3=5

:

So,

z=a(tobserved )

a(temitted )=

a(t4 )

a(t2+�t)= �t4

t2+�t �

3=5�

1:

f)If�tissmallcomparedtothetimethatittakesa(t)tochangesigni�cantly,then

theintervalbetweenasignalsentatt3andasignalsentatt3+�twillbereceived

witharedshiftidenticaltothatobservedbetweentwosuccessivecrestsofawave.

Thus,theseparationbetweenthereceiptoftheacknowledgementandthereceiptof

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.47

theresponsewillbeafactor(1+z)timeslongerthanthetimeintervalbetweenthe

sendingofthetwosignals,andtherefore

t4 �t3=(1+z)�t+O(�t2)

= �t4

t2+�t �

3=5

�t+O(�t2):

Sincetheanswercontainsanexplicitfactorof�t,theotherfactorscanbeevaluated

tozerothorderin�t:

t4 �t3= �t4

t2 �

3=5

�t+O(�t2);

whereto�rstorderin�tthet4

inthenumeratorcouldequallywellhavebeen

replacedbyt3 .

Forthosewhopreferthebruteforceapproach,theanswertopart(d)canbe

Taylorexpandedinpowersof�t.To�rstorderonehas

t4=t3+

@t4

@�t ��t=0�t+O(�t2):

Evaluatingthenecessaryderivativegives

@t4

@�t= "�t2+�t

t1

�2=5

+

2`0

5ct1 #

3=2�

t2+�t

t1

��3=5

;

whichwhenspecializedto�t=0becomes

@t4

@�t ��t=0= "�t2

t1 �

2=5

+

2`0

5ct1 #

3=2�

t2

t1 �

�3=5

:

Usingthe�rstboxedanswertopart(c),thiscanbesimpli�edto

@t4

@�t ��t=0= �t3

t1 �

3=5 �t2

t1 �

�3=5

= �t3

t2 �

3=5

:

PuttingthisbackintotheTaylorseriesgives

t4 �t3= �t3

t2 �

3=5

�t+O(�t2);

inagreementwiththepreviousanswer.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.48

PROBLEM

12:THEDECELERATION

PARAMETER

Fromthefrontoftheexam,weareremindedthat

�a=�4�3

G�a

and

�_aa �2

=8�3

G��kc2

a2

;

whereadotdenotesaderivativewithrespecttotimet.Thecriticalmassdensity�cis

de�nedtobethemassdensitythatcorrespondstoa at(k=0)universe,sofromthe

equationaboveitfollowsthat

�_aa �2

=8�3

G�c:

Substitutingintothede�nitionofq,we�nd

q=��a(t)a(t)

_a2(t)=��aa �a_a �2

= �4�3

G� ��3

8�G�c �=12��

c=

12:

PROBLEM

13:A

RADIATION-DOMINATED

FLATUNIVERSE

The atnessofthemodeluniversemeansthatk=0,so

�_aa �2

=8�3

G�:

Since

�(t)/1

a4(t);

itfollowsthat

dad

t=const

a

:

Rewritingthisas

ada=constdt;

theinde�niteintegralbecomes

12a2=(const)t+c 0;

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.49

wherec 0isaconstantofintegration.Di�erentchoicesforc 0correspondtodi�erent

choicesforthede�nitionoft=0.Wewillfollowthestandardconventionofchoosing

c 0=0,whichsetst=0tobethetimewhena=0.Thustheaboveequationimplies

thata2/t,andtherefore

a(t)/t1=2

foraphoton-dominated atuniverse.

PROBLEM

14:DID

YOU

DO

THEREADING

(2004)?(25points)

(a)In1826,theastronomerHeinrichOlberwroteapaperonaparadoxregardingthe

nightsky.WhatisOlber'sparadox?Whatistheprimaryresolutionofit?

(Ryden,Chapter2,Pages6-8)

Ans:Olber'sparadoxisthatthenightskyappearstobedark,insteadofbeing

uniformlybright.Theprimaryresolutionisthattheuniversehasa�niteage,andso

thelightfromstarsbeyondthehorizondistancehasnotreachedusyet.(However,

eveninthesteady-statemodeloftheuniverse,theparadoxisresolvedbecausethe

lightfromdistantstarswillbered-shiftedbeyondthevisiblespectrum).

(b)WhatisthevalueoftheNewtoniangravitationalconstantGinPlanckunits?The

Plancklengthisoftheorderof10 �35m,10 �15m,1015m,or1035m?

(Ryden,Chapter1,Page3)

Ans:G=1inPlanckunits,byde�nition.

ThePlancklengthisoftheorderof10 �35

m.(Notethatthisanswercouldbe

obtainedbyaprocessofeliminationaslongasyourememberthatthePlancklength

ismuchsmallerthan10 �15m,whichisthetypicalsizeofanucleus).

(c)WhatistheCosmologicalPrinciple?IstheHubbleexpansionoftheuniversecon-

sistentwithit?

(Weinberg,Chapter2,Pages21-23;Ryden,Chapter2,Page11)

Ans:TheCosmologicalPrinciplestatesthatthereisnothingspecialaboutour

locationintheuniverse,i.e.theuniverseishomogeneousandisotropic.

Yes,theHubbleexpansionisconsistentwithit(sincethereisnocenterofexpansion).

(d)Inthe\StandardModel"oftheuniverse,whentheuniversecooledtoabout3�

10aK,itbecametransparenttophotons,andtodayweobservetheseastheCosmic

MicrowaveBackground(CMB)atatemperatureofabout3�10bK.Whatarethe

integersaandb?

(Weinberg,Chapter3;Ryden,Chapter2,Page22)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.50

a=3,b=0.

(e)Whatdidtheuniverseprimarilyconsistofatabout1/100thofasecondafterthe

BigBang?Includeanyconstituentthatisbelievedtohavemadeupmorethan1%

ofthemassdensityoftheuniverse.

(Weinberg,Chapter1,Page5)

Ans:Electrons,positrons,neutrinos,andphotons.

PROBLEM

15:SPECIALRELATIVITY

DOPPLER

SHIFT(20points)

(a)Theeasiestwaytosolvethisproblem

isbyadoubleapplicationofthestandard

special-relativityDopplershiftformula,whichwasgivenonthefrontoftheexam:

z= s1+�

1��1;

(18.1)

where�=v=c.Rememberingthatthewavelengthisstretchedbyafactor1+z,we

�ndimmediatelythatthewavelengthoftheradiowavereceivedatAlpha-7isgiven

by

�Alpha�7= s1+vs =c

1�vs =c�emitted:

(18.2)

ThephotonsthatarereceivedbytheobserverareinfactneverreceivedbyAlpha-

7,butthewavelengthfoundbytheobserverwillbethesameasifAlpha-7acted

asarelaystation,receivingthephotonsandretransmittingthem

atthereceived

wavelength.So,applyingEq.(18.1)again,thewavelengthseenbytheobservercan

bewrittenas

�observed= s1+vo =c

1�vo =c�Alpha�7:

(18.3)

CombiningEqs.(18.2)and(18.3),

�observed= s1+vo =c

1�vo =c s

1+vs =c

1�vs =c�emitted;

(18.4)

so�nally

z= s1+vo =c

1�vo =c s

1+vs =c

1�vs =c �1:

(18.5)

(b)AlthoughweusedthepresenceofAlpha-7indeterminingtheredshiftzofEq.(18.5),

theredshiftisnotactuallya�ectedbythespacestation.Sothespecial-relativity

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.51

Dopplershiftformula,Eq.(18.1),mustdirectlydescribetheredshiftresultingfrom

therelativemotionofthesourceandtheobserver.Thus

s1+vtot =c

1�vtot =c �1= s1+vo =c

1�vo =c s

1+vs =c

1�vs =c �1:

(18.6)

Theequationabovedeterminesvtotintermsofvoandvs ,sotherestisjustalgebra.

Tosimplifythenotation,let�tot �vtot =c,�o �vo =c,and�s �vs =c.Then

1+�tot=1+�o

1��o

1+�s

1��s(1��tot )

�tot �1+1+�o

1��o

1+�s

1��s �=1+�o

1��o

1+�s

1��s �1

�tot �(1��o ��s+�o �s )+(1+�o+�s+�o �s )

(1��o )(1��s )

�=

(1+�o+�s+�o �s )�(1��o ��s+�o �s )

(1��o )(1��s )

�tot [2(1+�o �s )]=2(�o+�s )

�tot=

�o+�s

1+�o �s

vtot=

vo+vs

1+vo vs

c2

:

(18.7)

The�nalformulaistherelativisticexpressionfortheadditionofvelocities.Note

thatitguaranteesthatjvtot j�caslongasjvo j�candjvs j�c.

PROBLEM

16:DID

YOU

DO

THEREADING

(2005)?(25points)

(a)(4points)Whatwasthe�rstexternalgalaxythatwasshowntobeatadistance

signi�cantlygreaterthanthemostdistantknownobjectsinourgalaxy?Howwas

thedistanceestimated?

Ans:(Weinberg,page20)The�rstgalaxyshowntobeatadistancebeyondthesize

ofourgalaxywasAndromeda,alsoknownbyitsMessiernumber,M31.Itisthe

nearestspiralgalaxytoourgalaxy.Thedistancewasdetermined(byHubble)using

Cepheidvariablestars,forwhichtheabsoluteluminosityisproportionaltothepe-

riod.AmeasurementofaparticularCepheid'sperioddeterminesthestar'sabsolute

luminosity,which,comparedtothemeasuredluminosity,determinesthedistance

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.52

tothestar.(Hubble'sinitialmeasurementofthedistancetoAndromedauseda

badly-calibratedversionofthisperiod-luminosityrelationshipandconsequentlyun-

derestimatedthedistancebymorethanafactoroftwo;nonetheless,theinitial

measurementstillshowedthattheAndromedaNebulawasanorderofmagnitude

moredistantthanthemostdistantknownobjectsinourowngalaxy.)

(b)(5points)Whatisrecombination?Didgalaxiesbegintoformbeforeorafterrecom-

bination?Why?

Ans:(Weinberg,pages64and73)Recombinationreferstotheformationofneutral

atomsoutofchargednucleiandelectrons.Galaxiesbegantoform

afterrecom-

bination.Priortorecombination,thestrongelectromagneticinteractionsbetween

photonsandmatterproducedahighpressurewhiche�ectivelycounteractedthe

gravitationalattractionbetweenparticles.Oncetheuniversebecametransparentto

radiation,thematternolongerinteractedsigni�cantlywiththephotonsandconse-

quentlybegantoundergogravitationalcollapseintolargeclumps.

(c)(4points)InChapterIVofhisbook,Weinbergdevelopsa\recipeforahotuniverse,"

inwhichthematteroftheuniverseisdescribedasagasinthermalequilbriumat

averyhightemperature,inthevicinityof109K(severalthousandmilliondegrees

Kelvin).Suchathermalequilibrium

gasiscompletelydescribedbyspecifyingits

temperatureandthedensityoftheconservedquantities.Whichofthefollowingis

onthislistofconservedquantities?Circleasmanyasapply.

(i)baryonnumber

(ii)energyperparticle

(iii)protonnumber

(iv)electriccharge

(v)pressure

Ans:(Weinberg,page91)Thecorrectanswersare(i)and(iv).Athirdconserved

quantity,leptonnumber,wasnotincludedinthemultiple-choiceoptions.

(d)(4points)ThewavelengthcorrespondingtothemeanenergyofaCMB(cosmicmi-

crowavebackground)photontodayisapproximatelyequaltowhichofthefollowing

quantities?(Youmaywishtolookupthevaluesofvariousphysicalconstantsatthe

endofthequiz.)

(i)2fm(2�10 �15m)

(ii)2microns(2�10 �6m)

(iii)2mm(2�10 �3m)

(iv)2m.

Ans:(Ryden,page23)Thecorrectansweris(iii).

Ifyoudidnotrememberthisnumber,youcouldestimatetheanswerbyremem-

beringthatthecharacteristictemperatureofthecosmicmicrowavebackgroundis

approximately3Kelvin.ThetypicalphotonenergyisthenontheorderofkT,from

whichwecan�ndthefrequencyasE=h�.Thewavelengthofthephotonisthen

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.53

�=�=c.Thisapproximationgives�=5:3mm,whichisnotequaltothecorrect

answer,butitismuchclosertothecorrectanswerthantoanyoftheotherchoices.

(e)(4points)Whatistheequivalenceprinciple?

Ans:(Ryden,page27)Initssimplestform,theequivalenceprinciplesaysthatthe

gravitationalmassofanobjectisidenticaltoitsinertialmass.Thisequalityimplies

theequivalentstatementthatitisimpossibletodistinguish(withoutadditional

information)betweenanobserverinareferenceframeacceleratingwithacceleration

~aandanobserverinaninertialreferenceframesubjecttoagravitationalforce

�mobs ~a.

(Actually,whattheequivalenceprinciplereallysaysisthattheratioofthegravi-

tationaltoinertialmassesmg =miisuniversal,thatis,independentofthematerial

propertiesoftheobjectinquestion.Theratiodoesnotnecessarilyneedtobe1.

However,onceweknowthatthetwotypesofmassesareproportional,wecansim-

plyde�nethegravitationalcouplingGtomakethemequal.Toseethis,considera

theoryofgravitywheremg =mi=q.Thenthegravitationalforcelawis

mi a=�GMmg

r2

;

or

a=�GqM

r2

:

Atthispoint,ifwede�neG0=Gq,wehaveagravitationaltheorywithgravitational

couplingG0andinertialmassequaltogravitationalmass.)

(f)(4points)WhyisitdiÆcultforEarth-basedexperimentstolookatthesmallwave-

lengthportionofthegraphofCMBenergydensityperwavelengthvs.wavelength?

Ans:(Weinberg,page67)TheEarth'satmosphereisincreasinglyopaqueforwave-

lengthshorterthan.3cm.Therefore,radiationatthesewavelengthswillbeabsorbed

andrescatteredbytheEarth'satmosphere;observationsofthecosmicmicrowave

backgroundatsmallwavelengthsmustbeperformedabovetheEarth'satmosphere.

PROBLEM

17:TRACING

A

LIGHTPULSETHROUGH

A

RADIATION-

DOMINATED

UNIVERSE

(a)Thephysicalhorizondistanceisgiveningeneralby

`p;horizon=a(t) Z

tf

0

ca(t)dt;

sointhiscase

`p;horizon=bt1=2 Ztf

0

cbt1=2dt=

2ctf:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.54

(b)Ifthesourceisatthehorizondistance,itmeansthataphotonleavingthesourceat

t=0wouldjustbereachingtheoriginattf.So,te=0.

(c)Thecoordinatedistancebetweenthesourceandtheoriginisthecoordinatehorizon

distance,givenby

`c;horizon= Z

tf

0

cbt1=2dt=

2ct1=2

fb

:

(d)Thephotonstartsatcoordinatedistance2c ptf=b,andbytimetitwillhavetraveled

acoordinatedistance

Zt

0

cbt 01=2dt 0=2c pt

b

towardtheorigin.Thusthephotonwillbeatcoordinatedistance

`c=2cb �p

tf �p

t �

fromtheorigin,andhenceaphysicaldistance

`p (t)=a(t)`c=

2c � pttf �t �:

(e)To�ndthemaximumof`p (t),wedi�erentiateitandsetthederivativetozero:

d`p

dt= rtft�2 !c;

sothemaximumoccurswhen

rtf

tmax

=2;

or

tmax=14

tf:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.55

PROBLEM

18:TRANSVERSEDOPPLER

SHIFTS

(a)Describingtheeventsinthecoordinatesystemshown,theXanthuisatrest,soits

clocksrunatthesamespeedasthecoordinatesystemtimevariable,t.Theemission

ofthewavecrestsoftheradiosignalarethereforeseparatedbyatimeintervalequal

tothetimeintervalasmeasuredbythesource,theXanthu:

�t=�ts:

SincetheEmmeracismovingperpendiculartothepathoftheradiowaves,atthe

momentofreceptionitsdistancefromtheXanthuisataminimum,andhenceits

rateofchangeiszero.Hencesuccessivewavecrestswilltravelthesamedistance,as

longasc�t�a.Sincethewavecreststravelthesamedistance,thetimeseparation

oftheirarrivalattheEmmeracis�t,thesameasthetimeseparationoftheir

emission.TheclocksontheEmmerac,however,andrunningslowlybyafactorof

=

1

q1�v2

c2

:

Thetimeintervalbetweenwavecrestsasmeasuredbythereceiver,ontheEmmerac,

isthereforesmallerbyafactorof ,�

tr=�ts

:

Thus,thereisablueshift.Theredshiftparameterzisde�nedby

�tr

�ts=1+z;

so

1 =1+z;

or

z=1�

:

Recallthat >1,sozisnegative.

(b)Describingthissituationinthecoordinatesystemshown,thistimethesourceonthe

Xanthuismoving,sotheclocksatthesourcearerunningslowly.Thetimebetween

wavecrests,measuredincoordinatetimet,isthereforelargerbyafactorof than

�ts ,thetimeasmeasuredbytheclockonthesource:

�t= �ts:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.56

SincetheradiosignalisemittedwhentheXanthuisatitsminimumseparationfrom

theEmmerac,therateofchangeoftheseparationiszero,soeachwavecresttravels

thesamedistance(againassumingthatc�t�a).SincetheEmmeracisatrest,its

clocksrunatthesamespeedasthecoordinatetimet,andhencethetimeinterval

betweencrests,asmeasuredbythereceiver,is

�tr=�t= �ts:

Thusthetimeintervalasmeasuredbythereceiverislongerthanthatmeasuredby

thesource,andhenceitisaredshift.Theredshiftparameterzisgivenby

1+z=�tr

�ts= ;

so

z= �1:

(c)Theeventsdescribedin(a)canbemadetolookalotliketheeventsdescribedin(b)

bytransformingtoaframeofreferencethatismovingtotherightatspeedv0|

i.e.,bytransformingtotherestframeoftheEmmerac.InthisframetheEmmerac

isofcourseatrest,andtheXanthuistravelingonthetrajectory

(x=�v0 t;y=a;z=0);

asinpart(b).However,justasthetransformationcausesthex-componentofthe

velocityoftheXanthutochangefromzerotoanegativevalue,sothex-component

ofthevelocityoftheradiosignalwillbetransformedfromzerotoanegativevalue.

Thusinthisframetheradiosignalwillnotbetravelingalongthey-axis,sothe

eventswillnotmatchthosedescribedin(b).Thesituationsdescribedin(a)and(b)

arethereforephysicallydistinct(whichtheymustbeiftheredshiftsaredi�erent,as

wecalculatedabove).

PROBLEM

19:A

TWO-LEVELHIGH-SPEED

MERRY-GO-ROUND

(15

points)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.57

(a)Sincetherelativepositionsofallthecarsremain�xedasthemerry-go-roundrotates,

eachsuccessivepulsefromanygivencartoanyothercartakesthesameamountof

timetocompleteitstrip.ThustherewillbenoDopplershiftcausedbypulsestaking

di�erentamountsoftime;theonlyDopplershiftwillcomefromtimedilation.

Wewilldescribetheeventsfrom

thepointofviewofaninertialreferenceframe

atrestrelativetothehubofthemerry-go-round,whichwewillcallthelaboratory

frame.Thisistheframeinwhichtheproblemisdescribed,inwhichtheinnercars

aremovingatspeedv,andtheoutercarsaremovingatspeed2v.Inthelaboratory

frame,thetimeintervalbetweenthewavecrestsemittedbythesource�tLab

S

willbe

exactlyequaltothetimeinterval�tLab

O

betweentwocrestsreachingtheobserver:

�tLab

O

=�tLab

S

:

Theclocksonthemerry-go-roundcarsaremovingrelativetothelaboratoryframe,

sotheywillappeartoberunningslowlybythefactor

1=

1

p1�v2=c2

fortheinnercars,andbythefactor

2=

1

p1�4v2=c2

fortheoutercars.Thus,ifwelet�tS

denotethetimebetweencrestsasmeasured

byaclockonthesource,and�tO

asthetimebetweencrestsasmeasuredbyaclock

movingwiththeobserver,thenthesequantitiesarerelatedtothelaboratoryframe

timesby

2 �tS=�tLab

S

and 1 �tO

=�tLab

O

:

Tomakesurethatthe -factorsareontherightsideoftheequation,youshouldkeep

inmindthatanytimeintervalshouldbemeasuredasshorteronthemovingclocks

thanonthelabclocks,sincetheseclocksappeartorunslowly.Puttingtogetherthe

equationsabove,onehasimmediatelythat

�tO

= 2

1�tS

:

Theredshiftzisde�nedby

�tO

�(1+z)�tS

;

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.58

so

z= 2

1 �1= s1�v2

c2

1�4v2

c2

�1:

(b)Forthispartoftheproblemisusefultoimaginearelaystationlocatedjusttothe

rightofcar6inthediagram,atrestinthelaboratoryframe.Therelaystation

rebroadcaststhewavesasitreceivesthem,andhencehasnoe�ectonthefrequency

receivedbytheobserver,butservesthepurposeofallowingustoclearlyseparate

theproblemintotwoparts.

The�rstpartofthediscussionconcernstheredshiftofthesignalasmeasuredbythe

relaystation.Thiscalculationwouldinvolveboththetimedilationandachange

inpathlengthsbetweensuccessivepulses,butwedonotneedtodoit.Itisthe

standardsituationofasourceandobservermovingdirectlyawayfromeachother,

asdiscussedattheendofLectureNotes1.TheDopplershiftisgivenbyEq.(1.33),

whichwasincludedintheformulasheet.Writingtheformulaforarecessionspeed

u,itbecomes

(1+z)jrelay= s1+uc

1�uc

:

Ifweagainusethesymbol�tS

forthetimebetweenwavecrestsasmeasuredbya

clockonthesource,thenthetimebetweenthereceiptofwavecrestsasmeasured

bytherelaystationis

�tR

= s1+uc

1�uc

�tS

:

Thesecondpartofthediscussionconcernsthetransmissionfromtherelaystation

tocar6.Thevelocityofcar6isperpendiculartothedirectionfromwhichthepulse

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.59

isbeingreceived,sothisisatransverseDopplershift.Anychangeinpathlength

betweensuccessivepulsesissecondorderin�t,soitcanbeignored.Theonly

e�ectisthereforethetimedilation.Asdescribedinthelaboratoryframe,thetime

separationbetweencrestsreachingtheobserveristhesameasthetimeseparation

measuredbytherelaystation:

�tLab

O

=�tR

:

Asinpart(a),thetimedilationimpliesthat

2 �tO

=�tLab

O

:

Combiningtheformulasabove,�

O

=

1 2 s

1+uc

1�uc

�tS

:

Again�tO

�(1+z)�tS,so

z=

1 2 s

1+uc

1�uc

�1= s�1�

4v2

c2 ��1+uc �

1�uc

�1:

PROBLEM

20:

SIGNAL

PROPAGATION

IN

A

FLAT

MATTER-

DOMINATED

UNIVERSE(55points)

(a)-(i)Ifwelet`c (t)denotethecoordinatedistanceofthelightsignalfromA,thenwecan

makeuseofEq.(3.8)fromthelecturenotesforthecoordinatevelocityoflight:

d`c

dt=

ca(t):

(20.1)

Integratingthevelocity,

`c (t)= Z

tt1

cdt 0

a(t 0)=cb Zt

t1

dt 0

t 02=3

=3cb ht1=3�t1=3

1 i:

(20.2)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.60

Thephysicaldistanceisthen

`p;sA(t)=a(t)`c (t)=bt2=33cb ht1=3�t1=3

1 i

=3c �t�t2=3t1=3

1 �

=3ct "1� �t1t �1=3 #

:

(20.3)

Wenowneedtodi�erentiate,whichisdonemosteasilywiththemiddlelineofthe

aboveequation:

d`p;sA

dt

=c "3�2 �t1t �1=3 #

:

(20.4)

(ii)Att=t1 ,thetimeofemission,theaboveformulagives

d`p;sA

dt

=c:

(20.5)

Thisiswhatshouldbeexpected,sincethespeedofseparationofthelightsignalat

thetimeofemissionisreallyjustalocalmeasurementofthespeedoflight,which

shouldalwaysgivethestandardvaluec.

(iii)Atarbitrarilylatetimes,thesecondterminbracketsinEq.(20.4)becomesnegligible,

so

d`p;sA

dt

!3c:

(20.6)

Althoughthisanswerislargerthanc,itdoesnotviolaterelativity.Oncethesignal

isfarfromitsoriginitiscarriedbytheexpansionoftheuniverse,andrelativity

placesnospeedlimitontheexpansionoftheuniverse.

(b)ThispartoftheprobleminvolvesH(t1 ),sowecanstartbyevaluatingit:

H(t)=

_a(t)

a(t)=

ddt (bt2=3)

bt2=3

=

23t:

(20.7)

Thus,thephysicaldistancefromAtoBattimet1is

`p;BA

=32

ct1:

(20.8)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.61

Thecoordinatedistanceisthephysicaldistancedividedbythescalefactor,so

`c;BA

=cH�1(t

1 )

a(t1 )

=

32ct1

bt2=3

1

=3c

2bt1=3

1

:

(20.9)

Sincelighttravelsatacoordinatespeedc=a(t),thelightsignalwillreachgalaxyB

attimet2if

`c;BA

= Zt2

t1

cbt 02=3dt 0

=3cb ht1=3

2

�t1=3

1 i:

(20.10)

Settingtheexpressions(20.9)and(20.10)for`c;BA

equaltoeachother,one�nds

12t1=3

1

=t1=3

2

�t1=3

1

=)

t1=3

2

=32

t1=3

1

=)

t2=278

t1:

(20.11)

(c)-(i)Physicaldistancesareadditive,soifoneaddsthedistancefrom

Aandthelight

signaltothedistancefromthelightsignaltoB,onegetsthedistancefromAtoB:

`p;sA+`p;sB

=`p;BA

:

(20.12)

But`p;BA(t)isjustthescalefactortimesthecoordinateseparation,a(t)`c;BA.Using

thepreviousrelations(20.3)and(20.9)for`p;sA(t)and`c;BA,we�nd

3ct "1� �t1t �1=3 #

+`p;sB(t)=32

ct1=3

1

t2=3;

(20.13)

so

`p;sB(t)=92

ct1=3

1

t2=3�3ct=3ct "32 �t1t �1=3�

1 #:

(20.14)

Asacheck,onecanverifythatthisexpressionvanishesfort=t2=(27=8)t1 ,and

thatitequals(3=2)ct1att=t1 .Butweareaskedto�ndthespeedofapproach,

thenegativeofthederivativeofEq.(20.14):

Speedofapproach=�d`p;sB

dt

=�3ct1=3

1

t �1=3+3c

=

3c "1� �t1t �1=3 #

:

(20.15)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.62

(ii)Atthetimeofemission,t=t1 ,Eq.(20.15)gives

Speedofapproach=0:

(20.16)

Thismakessense,sinceatt=t1

galaxyBisoneHubblelengthfromgalaxyA,

whichmeansthatitsrecessionvelocityisexactlyc.Therecessionvelocityofthe

lightsignalleavingAisalsoc,sotherateofchangeofthedistancefromthelight

signaltoBisinitiallyzero.

(iii)Atthetimeofreception,t=t2=(27=8)t1 ,Eq.(20.15)gives

Speedofapproach=c;

(20.17)

whichisexactlywhatisexpected.Asinpart(a)-(ii),thisisalocalmeasurementof

thespeedoflight.

(d)To�ndtheredshift,we�rst�ndthetimetBA

atwhichalightpulsemustbeemitted

fromgalaxyBsothatitarrivesatgalaxyAattimet1 .Usingthecoordinatedistance

givenbyEq.(20.9),thetimeofemissionmustsatisfy

3c

2bt1=3

1

= Zt1

tBA

cbt 02=3dt 0=3cb �

t1=3

1

�t1=3

BA �;

(20.18)

whichcanbesolvedtogive

tBA

=18

t1:

(20.19)

Theredshiftisgivenby1

+zBA

=

a(t1 )

a(tBA)= �t1

tBA �

2=3

=4:

(20.20)

Thus,

zBA

=3:

(20.21)

(e)ApplyingEuclideangeometrytothetriangleC-A-Bshowsthatthephysicaldistance

fromCtoB,attimet1 ,is p2cH�1.Thecoordinatedistanceisalsolargerthanthe

A-Bseparationbyafactorof p2.Thus,

`c;BC

=3 p2c

2b

t1=3

1

:

(20.22)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.63

IfwelettBC

bethetimeatwhichalightpulsemustbeemittedfromgalaxyBso

thatitarrivesatgalaxyCattimet1 ,we�nd

3 p2c

2b

t1=3

1

= Zt1

tBC

cbt 02=3dt 0=3cb �

t1=3

1

�t1=3

BC �;

(20.23)

whichcanbesolvedto�nd

tBC

= 1�p

22 !3

t1:

(20.24)

Then

1+zBC

=

a(t1 )

a(tBC)= �t1

tBC �

2=3

=

1

�1�p

22 �2

;

(20.25)

and

zBC

=

1

�1�p

22 �2 �1:

(20.26)

Fullcreditwillbegivenfortheanswerintheformabove,butitcanbesimpli�ed

byrationalizingthefraction:

zBC

=

1

�1�p

22 �2 �

1+p

22 �

2

�1+p

22 �

2 �1

=1+p

2+12

14

�1

=

5+4 p2:

(20.27)

Numerically,zBC

=10:657.

(f)FollowingthesolutiontoProblem6ofProblemSet2,wedrawadiagramincomoving

coordinates,puttingthesourceatthecenterofasphere:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.64

TheenergyfromgalaxyAwillradiateuniformlyoverthesphere.Ifthedetector

hasphysicalareaAD,theninthecomovingcoordinatepictureithascoordinate

areaAD=a2(t

2 ),sincethedetectionoccursattimet2Thefullcoordinateareaofthe

sphereis4�`2c

;BA,sothefractionofphotonsthathitthedetectoris

fraction= �A=a(t2 )2 �

4�`2c

;BA

:

(20.28)

AsinProblem6,thepowerhittingthedetectorisreducedbytwofactorsof(1+z):

onefactorbecausetheenergyofeachphotonisproportionaltothefrequency,and

henceisreducedbytheredshift,andonemorefactorbecausetherateofarrivalof

photonsisalsoreducedbytheredshiftfactor(1+z).Thus,

Powerhittingdetector=P �A=a(t2 )2 �

4�`2c

;BA

1

(1+z)2

=P �A=a(t2 )2 �

4�`2c

;BA

�a(t1 )

a(t2 ) �

2

=P

A

4�`2c

;BA

a2(t

1 )

a4(t

2 ):

(20.29)

Theenergy uxisgivenby

J=Powerhittingdetector

A

;

(20.30)

so

J=

P

4�`2c

;BA

a2(t

1 )

a4(t

2 ):

(20.31)

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.65

Fromhereitisjustalgebra,usingEqs.(20.9)and(20.11),anda(t)=bt2=3:

J=

P

4� h3c

2b t

1=3

1 i2b2t4=3

1

b4t8=3

2

=

P

4� h3c

2b t

1=3

1 i2

b2t4=3

1

�278 �8=3

b4t8=3

1

=

P

4� h3c2t1=3

1 i2

t4=3

1

�32 �8

t8=3

1

=

28

310�

Pc2t21

=

256

59;049�

Pc2t21

:

(20.32)

Itisdebatablewhichofthelasttwoexpressionsisthesimplest,soIhaveboxedbothof

them.Onecouldalsowrite

J=1:380�10 �3

Pc2t21

:

(20.33)

PROBLEM

21:DID

YOU

DO

THEREADING

(2011)?(25points) y

ySolutionwrittenbyDanieleBertolini.

(a)(10points)TodeterminethedistanceofthegalaxieshewasobservingHubbleused

socalledstandardcandles.Standardcandlesareastronomicalobjectswhoseintrin-

sicluminosityisknownandwhosedistanceisinferredbymeasuringtheirapparent

luminosity.First,heusedasstandardcandlesvariablestars,whoseintrinsiclumi-

nositycanberelatedtotheperiodofvariation.QuotingWeinberg'sTheFirstThree

Minutes,chapter2,pages19-20:

In1923EdwinHubblewasforthe�rsttimeabletoresolvetheAndromedaNeb-

ulaintoseparatestars.Hefoundthatitsspiralarmsincludedafewbrightvariable

stars,withthesamesortofperiodicvariationofluminosityaswasalreadyfamiliar

foraclassofstarsinourgalaxyknownasCepheidvariables.Thereasonthiswas

soimportantwasthatintheprecedingdecadetheworkofHenriettaSwanLeavitt

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.66

andHarlowShapleyoftheHarvardCollegeObservatoryhadprovidedatightrela-

tionbetweentheobservedperiodsofvariationoftheCepheidsandtheirabsolute

luminosities.(Absoluteluminosityisthetotalradiantpoweremittedbyanastro-

nomicalobjectinalldirections.Apparentluminosityistheradiantpowerreceived

byusineachsquarecentimeterofourtelescopemirror.Itistheapparentrather

thantheabsoluteluminositythatdeterminesthesubjectivedegreeofbrightnessof

astronomicalobjects.Ofcourse,theapparentluminositydependsnotonlyontheab-

soluteluminosity,butalsoonthedistance;thus,knowingboththeabsoluteandthe

apparentluminositiesofanastronomicalbody,wecaninferitsdistance.)Hubble,

observingtheapparentluminosityoftheCepheidsintheAndromedaNebula,and

estimatingtheirabsoluteluminosityfromtheirperiods,couldimmediatelycalculate

theirdistance,andhencethedistanceoftheAndromedaNebula,usingthesimple

rulethatapparentluminosityisproportionaltotheabsoluteluminosityandinversely

proportionaltothesquareofthedistance.

Healsousedparticularlybrightstarsasstandardcandles,aswededucefrompage

25:Returningnowto1929:Hubbleestimatedthedistanceto18galaxiesfromtheappar-

entluminosityoftheirbrigheststars,andcomparedthesedistanceswiththegalaxies'

respectivevelocities,determinedspectroscopicallyfromtheirDopplershifts.

Note:sincefromreadingjustthe�rstpartofWeinberg'sdiscussiononecouldbe

inducedtothinkthatHubbleusedjustCepheidsasstandardcandles,studentswho

mentionedonlyCepheidsgot9pointsoutof10.Infact,however,Hubblewasable

toidentifyCepheidvariablesinonlyafewgalaxies.TheCepheidswerecrucial,

becausetheyservedasacalibrationforthelargerdistances,buttheywerenotin

themselvessuÆcient.

(b)(5points)QuotingWeinberg'sTheFirstThreeMinutes,chapter2,page21:

Wewouldexpectintuitivelythatatanygiventimetheuniverseoughttolookthesame

toobserversinalltypicalgalaxies,andinwhateverdirectionstheylook.(Here,and

below,Iwillusethelabel\typical"toindicategalaxiesthatdonothaveanylarge

peculiarmotionoftheirown,butaresimplycarriedalongwiththegeneralcosmic

owofgalaxies.)Thishypothesisissonatural(atleastsinceCopernicus)thatithas

beencalledtheCosmologicalPrinciplebytheEnglishastrophysicistEdwardArthur

Milne.

SotheCosmologicalprinciplebasicallystatesthattheuniverseappearsashomo-

geneousandisotropic(onscalesofdistancelargeenough)toanytypicalobserver,

wheretypicalisreferredtoobserverswithsmalllocalmotioncomparedtotheex-

pansion ow.Rydengivesamoregeneralde�nitionofCosmologicalPrinciple,which

isvalidaswell.QuotingRyden'sIntroductiontoCosmology,chapter2,page11or

14(dependingonwhichversion):

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.67

However,moderncosmologistshaveadoptedthecosmologicalprinciple,which

states:Thereisnothingspecialaboutourlocationintheuniverse.Thecosmological

principleholdstrueonlyonlargescales(of100Mpcormore).

(c)(10points)QuotingagainRyden'sIntroductiontoCosmology,chapter2,page9or

11:

Sayingthattheuniverseisisotropicmeansthattherearenopreferreddirectionsin

theuniverse;itlooksthesamenomatterwhichwayyoupointyourtelescope.Saying

thattheuniverseishomogeneousmeansthattherearenopreferredlocationsin

theuniverse;itlooksthesamenomatterwhereyousetupyourtelescope.

(i)False.Iftheuniverseisisotropicaroundonepointitdoesnotneedtobe

homogeneous.Acounter-exampleisadistributionofmatterwithspherical

symmetry,thatis,withadensitywhichisonlyafunctionoftheradiusbutdoes

notdependonthedirection:�(r;�;�)��(r).Inthiscaseforanobserveratthe

centerofthedistributiontheuniverselooksisotropicbutitisnothomogeneous.

(ii)

True.ForthecaseofEuclideangeometryisotropyaroundtwoormoredistinct

pointsdoesimplyhomogeneity.Weinbergshowsthisinchapter2,page24.

Considertwoobservers,andtwoarbitrarypointsAandBwhichwewouldlike

toproveequivalent.ConsideracirclethroughpointA,centeredonobserver1,

andanothercirclethroughpointB,centeredonobserver2.IfCisapointon

theintersectionofthetwocircles,thenisotropyaboutthetwoobserversimplies

thatA=CandB=C,andhenceA=B.(Thisargumentwasgoodenough

forWeinbergandhencegoodenoughtodeservefullcredit,butitisactually

incomplete:onecan�ndpointsAandB

forwhichthetwocircleswillnot

intersect.Onyournextproblemsetyouwillhaveachancetoinventabetter

proof.)

(d)(2pointsextracredit)False.IfwerelaxthehypothesisofEuclideangeometry,

thenisotropyaroundtwopointsdoesnotnecessarilyimplyhomogeneity.Acounter-

examplewementionedinclassisatwo-dimensionaluniverseconsistingofthesurface

ofasphere.ThinkofthesphereinthreeEuclideandimensions,butthemodel\uni-

verse"consistsonlyofitstwo-dimensionalsurface.Imaginelatitudeandlongitude

linestogivecoordinatestothesurface,andimagineamatterdistributionthatde-

pendsonlyonlatitude.Thiswouldnotbehomogeneous,butitwouldlookisotropic

toobserversatboththenorthandsouthpoles.Whilethisexampledescribesatwo-

dimensionaluniverse,whichthereforecannotbeouruniverse,wewilllearnshortly

howtoconstructathree-dimensionalnon-Euclideanuniversewiththesesameprop-

erties.

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.68

PROBLEM

22:THETRAJECTORY

OFA

PHOTON

ORIGINATING

AT

THEHORIZON

(25points)

(a)Theykeyideaisthatthecoordinatespeedoflightisgivenby

dxd

t=

ca(t);

sothecoordinatedistance(innotches)thatlightcantravelbetweent=0andnow

(t=t0 )isgivenby

`c= Z

t0

0

cdt

a(t):

Thecorrespondingphysicaldistanceisthehorizondistance:

`p;horizon (t0 )=a(t0 ) Z

t0

0

cdt

a(t):

Evaluating,

`p;horizon (t0 )=bt2=3

0 Zt0

0

cdt

bt2=3

=t2=3

0 h3ct1=3

0 i=

3ct0:

(b)Asstatedinpart(a),thecoordinatedistancethatlightcantravelbetweent=0

andt=t0isgivenby

`c= Z

t0

0

cdt

a(t)=3ct1=3

0b

:

Thus,ifweareattheorigin,att=0thephotonmusthavebeenat

x0=3ct1=3

0b

:

(c)Thephotonstartsatx=x0att=0,andthentravelsinthenegativex-direction

atspeedc=a(t).Thus,it'spositionattimetisgivenby

x(t)=x0 � Z

t0

cdt 0

a(t 0)=3ct1=3

0b

�3ct1=3

b

=

3cb �

t1=3

0

�t1=3 �:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.69

(d)Sincethecoordinatedistancebetweenusandthephotonisx(t),measuredinnotches,

thephysicaldistance(in,forexample,meters)isjusta(t)timesx(t).Thus.

`p (t)=a(t)x(t)=

3ct2=3 �t1=3

0

�t1=3 �:

(e)To�ndthemaximumof`p (t),wesetthederivativeequaltozero:

d`p (t)

dt

=

ddt h3c �t2=3t1=3

0

�t �i=3c "23 �t0t �1=3�

1 #=0;

so

�t0

tmax �

1=3

=32

=)

tmax= �23 �3

t0=

827t0:

Themaximumdistanceisthen

`p;max=`p (tmax )=3c �23 �2

t2=3

0 �t1=3

0

� �23 �

t1=3

0 �=3c �23 �2 �13 �

t0

=

49ct0:

PROBLEM

23:DID

YOU

DO

THEREADING

(2016)?(35points)

(a)(5points)TheMilkyWayhasbeenknownsinceancienttimesasabandoflight

stretchingacrossthesky.WenowrecognizetheMilkyWayasthegalaxyofstars

inwhichwelive,withalargecollectionofstars,includingoursun,arrangedina

giantdisk.Sincetheindividualstarsaremostlytoosmallforoureyestoresolve,we

observethecollectivelightfromthesestars,concentratedintheplaneofthedisk.

TheideathattheMilkyWayisactuallyadiskofstarswasproposedby






8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.70


(b)(5points)Onceitwasrecognizedthatweliveinagalaxy,itwasinitiallyassumed

thatourswastheonlygalaxy.Thesuggestionthatsomeofthepatchesoflight

knownasnebulaemightactuallybeothergalaxieslikeourownwasmadeby







(c)(5points)The�rst�rmevidencethatthereismorethanonegalaxystemmedfrom

theabilitytoobservetheAndromedaNebulawithhighenoughresolutiontodistin-

guishitsindividualstars.Inparticular,theobservationofCepheidvariablestarsin

AndromedaallowedadistanceestimatethatplaceditwelloutsidetheMilkyWay.

TheobservationofCepheidvariablestarsinAndromedawas�rstmadeby

(i)JohannesKepler,in1610.

(ii)IsaacNewton,in1695.

(iiiThomasWright,in1750.

(iv)ImmanuelKant,in1755.

(v)HenriettaSwanLeavittandHarlowShapleyin1915.


(d)(5points)The�rsthintthattheuniverseis�lledwithradiationwithane�ective

temperaturenear3K,althoughnotrecognizedatthetime,wasanobservationof

absorptionlinesincyanogen(CN)byAdamsandMcKellarin1941.Theyobserved

darkspectrallineswhichtheyinterpretedasabsorptionbythecyanogenoflight

comingfromthestarbehindthegascloud.Explaininafewsentenceshowtheseab-

sorptionlinescanbeusedtomakeinferencesaboutthecosmicbackgroundradiation

bathingthecyanogengascloud.

Answer:

Whenanatomabsorbsaphoton,itisexcitedfromitsinitialstatetosome

�nalstate,andtheenergyofthephotonmustmatchtheenergydi�erence

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.71

betwenthetwostates.Oneoftheobservedcyanogenlineswasassociated

withatransitionstartinginthegroundstate,andtwootherobservedlines

wereassociatedwithtransitionsstartingfromanexcitedstate.Bycompar-

ingtheintensitiesoftheseabsorptionlines,theastronomerscouldinferthe

relativeabundanceofgroundstateandexcitedcyanogenmolecules,which

inturnallowedthemtoinferthetemperatureofthegascloud.Theyfound

atemperatureof2.2K.

(e)(5points)Astheuniverseexpands,thetemperatureofthecosmicmicrowaveback-

ground

(i)goesupinproportiontothescalefactora(t).

(ii)staysconstant.

(iii)goesdowninproportionto1=a(t).

(iv)goesdowninproportionto1=a2(t).

(f)(5points)WhenHubblemeasuredthevalueofhisconstant,hefoundH�1�100

millionyears,2billionyears,10billionyears,or20billionyears?

(g)(5points)Explaininafewsentenceswhatismeantbytheequivalenceprinciple?

Answer:

Rydenstatesthattheequivalenceprincipleisthefactthatthegravitational

massofanyobjectisequaltoitsinertialmass.Itwouldalsobecorrectto

saythatthegraviationalmassisproportionaltotheinertialmass.(Ifthey

areproportional,thereisalwaysavalueofGwhichmakesthemequal.)

Theequivalenceprinciplecanalsobedescribedmoregenerallybysaying

thatgravityisequivalenttoacceleration,sothatwithinasmallvolumethe

e�ectsofgravitycanberemovedbydescribingthesysteminanaccelerating

coordinatesystem.

PROBLEM

24:

OBSERVING

A

DISTANT

GALAXY

IN

A

MATTER-

DOMINATED

FLATUNIVERSE(40points)

Supposethatwearelivinginamatter-dominated atuniverse,withascalefactor

givenby

a(t)=bt2=3;

wherebisaconstant.Thepresenttimeisdenotedbyt0 .

(a)(5points)Ifwemeasuretimeinseconds,distanceinmeters,andcoordinatedistances

innotches,whataretheunitsofb?

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.72

Answer:

a(t)wouldbemeasuredinmeters/notch,andtwouldbemeasuredinsec-

onds.So

[b]=[a(t)]

[t] 2=3

=

m

notch-s2=3

:

(b)(5points)Supposethatweobserveadistantgalaxywhichisonehalfofa\Hubble

length"away,whichmeansthatthephysicaldistancetodayis`p=12cH�1

0

,wherec

isthespeedoflightandH0isthepresentvalueoftheHubbleexpansionrate.What

isthepropervelocityvp �d`p(t)

dt

ofthisgalaxyrelativetous?

Answer:

ByHubble'slaw,thevelocityofrecessionisequaltoH0timesthephysical

distance,so

vp=H0 �12

cH�1

0 �=

12c:

Acommonerrorinthispartwastouse

H0=

_aa=23bt �1=3

0bt2=3

0

=

23t0

towrite

`p=34

ct0;

andthentodi�erentiatethisexpressionwithrespecttot0 ,�ndingvp

=

3c=4.Theproblemwiththisapproachisthatitassumesthattherelation

`p

=

12cH�1

holdsforallt,sothatonecandi�erentiateitto�ndthe

velocity.Butanobjectthatisatdistance12cH�1

doesnotremainata

distance12cH�1astimeprogresses.Itisthecoordinatedistance`c ,andnot

thephysicaldistancemeasuredinHubblelengths,thatremainsconstant

astheuniverseexpands.

(c)(5points)Whatisthecoordinatedistance`cbetweenusandthedistantgalaxy?

Expressyouranswerintermsofb,t0 ,andc(butnotH0 ).

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.73

Answer:

Weknowthat`p (t)=a(t)`c ,so

`c=`p (t0 )

a(t0 )=

c

2bH0 t2=3

0

:

ToeliminateH0 ,whichisnotallowedintheanswer,wecanuse

H0=

1a(t0 )

da(t0 )

dt0

=

1bt2=3

0

�23

bt �1=3

0

�=

23t0

:

Insertingtheresultintothelineabove,

`c=34ct1=3

0b

:

Ifyoudidnotanswerthepreviouspart,youmaystillcontinuewiththefollowingparts,

usingthesymbol`cforthecoordinatedistancetothegalaxy.

(d)(5points)Atwhattimetewasthelightthatwearenowreceivingfromthegalaxy

emitted?

Answer:

Weknowthatthecoordinatevelocityoflightis

dxd

t=

ca(t)=

cbt2=3:

Wecan�ndtebytherequirementthatthecoordinatedistancethatlight

travelsbetweenteandt0mustbeequalto`cfoundinpart(c):

Zt0

te

cbt 02=3dt 0=34ct1=3

0b

:

Integrating,

3cb ht1=3

0

�t1=3

e i=34ct1=3

0b

:

Withalittlealgebrawesee

t1=3

e

=34

t1=3

0

=)

te=2764

t0:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.74

(e)(5points)Whatistheredshiftzofthelightthatwearenowreceivingfromthe

distantgalaxy?

Answer:

Theredshiftisrelatedtothescalefactorby

1+z=a(t0 )

a(te )= �t0

te �

2=3

= �642

7 �2=3

=169

;

so

z=79

:

(f)(10points)Consideralightpulsethatleavesthedistantgalaxyattimete ,ascal-

culatedinpart(d),andarriveshereatthepresenttime,t0 .Calculatethephysical

distancerp (t)betweenthelightpulseandus.Findrp (t)asafunctionoftforallt

betweenteandt0 .

Answer:

We�rstcalculatethecoordinateseparationrc (t)betweenthelightpulse

andus,asafunctionoft.Attimeteitisequaltothevalueof`cfoundin

part(c),andfromthattimeonwarditisreducedbythecoordinatedistance

thatlightcantravelbetweentimesteandt.Therefore,

rc (t)=34ct1=3

0b

� Zt

te

cbt 02=3dt 0

=34ct1=3

0b

�3cb ht1=3�t1=3

e i

=34ct1=3

0b

�3cb �

t1=3�34

t1=3

0 �

=3cb ht1=3

0

�t1=3 i:

Thephysicaldistanceisthen

rp (t)=bt2=3rc (t)=3c ht1=3

0

t2=3�t i=

3ct "�t0t �1=3�

1 #:

8.286QUIZ1REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.75

(g)(5points)Ifwesendaradiomessagenowtothedistantgalaxy,atwhattimetrwill

itbereceived?

Answer:

Wecalculatethetimetrbywhichalightray,startingatt0 ,cantravela

coordinatedistanceequaltothevaluewefoundinpart(c):

Ztr

t0

cbt 02=3dt 0=`c=34ct1=3

0b

:

Integrating,

3cb ht1=3

r

�t1=3

0 i=34ct1=3

0b

;

fromwhichwe�ndt

1=3

r

=54

t1=3

0

=)

tr=125

64t0:

Ph - web.mit.eduweb.mit.edu/8.286/www/quiz18/q1rp-euf18-2up.pdf · 1 v=u (nonrelativistic, observ...

Documents

Transcript of Ph - web.mit.eduweb.mit.edu/8.286/www/quiz18/q1rp-euf18-2up.pdf · 1 v=u (nonrelativistic, observ...