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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
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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
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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
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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)
8.DidYouDoTheReading(1996)?
...............11(Sol:38)
9.AFlatUniverseWitha(t)/t3=5
...............12(Sol:39)
10.DidYouDoTheReading(1998)?
...............13(Sol:43)
11.AnotherFlatUniverseWitha(t)/t3=5
.............14(Sol:44)
*12.TheDecelerationParameter..................14(Sol:48)
13.ARadiation-DominatedFlatUniverse
.............15(Sol:48)
14.DidYouDoTheReading(2004)?
...............15(Sol:49)
*15.SpecialRelativityDopplerShift................16(Sol:50)
16.DidYouDoTheReading(2005)?
...............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.DidYouDoTheReading(2011)?
...............21(Sol:65)
*22.TheTrajectoryOfAPhotonOriginatingAtTheHorizon.....21(Sol:68)
23.DidYouDotheReading(2016)?
...............22(Sol:69)
24.ObservingaDistantGalaxyinaMatter-DominatedFlatUniverse.23(Sol:71)
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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?
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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).]
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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
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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.
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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:
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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.]
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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)
ThefollowingproblemwasProblem1,Quiz1,1996:
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.
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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
ThefollowingproblemwasProblem3,Quiz1,1996:
Considera atuniversewhichis�lledwithsomepeculiarformofmatter,sothat
theRobertson{Walkerscalefactorbehavesas
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
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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?
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PROBLEM
11:ANOTHERFLATUNIVERSEWITHa(t)/
t3=5
(40points)
ThefollowingwasProblem3,Quiz1,1998:
Considera atuniversewhichis�lledwithsomepeculiarformofmatter,sothat
theRobertson{Walkerscalefactorbehavesas
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):
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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.
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�
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
1,Quiz1,2005,whereitcounted25
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
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(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
3,Quiz1,2005,whereitcounted25
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?
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PROBLEM
18:TRANSVERSEDOPPLER
SHIFTS
Thefollowingproblem
wastakenfrom
Problem
4,Quiz1,2005,whereitcounted20
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
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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?
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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.)
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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?
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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
(i)ClaudiusPtolemy,inthe2ndcenturyAD.
(ii)JohannesKepler,in1610.
(iii)IsaacNewton,in1695.
(iv)ThomasWright,in1750.
(v)ImmanuelKant,in1755.
(vi)EdwinHubble,in1923.
(c)(5points)The�rst�rmevidencethatthereismorethanonegalaxystemmedfrom
theabilitytoobservetheAndromedaNebulawithhighenoughresolutiontodistin-
guishitsindividualstars.Inparticular,theobservationofCepheidvariablestarsin
AndromedaallowedadistanceestimatethatplaceitwelloutsidetheMilkyWay.
TheobservationofCepheidvariablestarsinAndromedawas�rstmadeby
(i)JohannesKepler,in1610.
(ii)IsaacNewton,in1695.
(iiiThomasWright,in1750.
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(iv)ImmanuelKant,in1755.
(v)HenriettaSwanLeavittandHarlowShapleyin1915.
(vi)EdwinHubble,in1923.
(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
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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?
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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.
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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.)
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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
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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
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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
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(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
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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
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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
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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
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(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
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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
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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
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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
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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
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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 #:
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`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
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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
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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
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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
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[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
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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
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[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
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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
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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
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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
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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
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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
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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
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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
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SOLUTIONS,FALL2018
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(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
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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
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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
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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
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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
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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
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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
(i)ClaudiusPtolemy,inthe2ndcenturyAD.
(ii)JohannesKepler,in1610.
(iii)IsaacNewton,in1695.
(iv)ThomasWright,in1750.
(v)ImmanuelKant,in1755.
8.286QUIZ1REVIEW
PROBLEM
SOLUTIONS,FALL2018
p.70
(vi)EdwinHubble,in1923.
(b)(5points)Onceitwasrecognizedthatweliveinagalaxy,itwasinitiallyassumed
thatourswastheonlygalaxy.Thesuggestionthatsomeofthepatchesoflight
knownasnebulaemightactuallybeothergalaxieslikeourownwasmadeby
(i)ClaudiusPtolemy,inthe2ndcenturyAD.
(ii)JohannesKepler,in1610.
(iii)IsaacNewton,in1695.
(iv)ThomasWright,in1750.
(v)ImmanuelKant,in1755.
(vi)EdwinHubble,in1923.
(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.
(vi)EdwinHubble,in1923.
(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:
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