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MASSACHUSETTSINSTITUTEOFTECHNOLOGY
PhysicsDepartment
Physics8.286:TheEarlyUniverse
September28,2016
Prof.AlanGuthR
EVIEW
PROBLEMSFOR
QUIZ1
QUIZDATE:Wednesday,October5,2016,duringthenormalclasstime.
QUIZCOVERAGE:LectureNotes1,2,and3;ProblemSets1,2,and3;Wein-
berg,Chapters1,2,and3;Ryden,Chapters1,2,and3.(WhileallofRyden's
Chapter3hasbeenassigned,questionsonthequizwillbelimitedtoSection
3.1.ThematerialinSections3.2and3.3willbediscussedinlecturelaterin
thecourse,andyouwillnotberesponsibleforituntilthen.Section3.4(for
the�=0case)mayhelpyouunderstandthecosmologicalDopplershift,also
discussedinLectureNotes2,buttherewillbenoquestionsspeci�callyfocused
onRyden'sdiscussion.)Oneoftheproblemsonthequizwillbetaken
verbatim
(oratleastalmostverbatim)from
eitherthehomework
assignments,orfrom
thestarredproblemsfrom
thissetofReview
Problems.ThestarredproblemsaretheonesthatIrecommendthatyou
reviewmostcarefully:Problems2,4,7,12,15,17,19,and22.Thestarred
problemsdonotincludeanyreadingquestions,butpartsofthereadingques-
tionsintheseReviewProblemsmayalsorecurontheupcomingquiz.
PURPOSE:Thesereviewproblemsarenottobehandedin,butarebeingmade
availabletohelpyoustudy.Theycomemainlyfromquizzesinpreviousyears.
Exceptforafewpartswhichareclearlymarked,theyareallproblemsthatI
wouldconsiderfairforthecomingquiz.Insomecasesthenumberofpoints
assignedtotheproblemonthequizislisted|
inallsuchcasesitisbasedon
100pointsforthefullquiz.
Inadditiontothissetofproblems,youwill�ndonthecoursewebpage
theactualquizzesthatweregivenin1994,1996,1998,2000,2002,2004,2005,
2007,2009,2011,and2013.Therelevantproblemsfromthosequizzeshave
mostlybeenincorporatedintothesereviewproblems,butyoustillmaybe
interestedinlookingattheoriginalquizzes,justtoseehowmuchmaterialhas
beenincludedineachquiz.Sincethescheduleandthenumberofquizzeshas
variedovertheyears,thecoverageofthisquizwillnotnecessarilybethesame
asQuiz1from
allpreviousyears.Infact,however,the�rstquizthisyear
coversessentiallythesamematerialasthe�rstquizineither2009,2011,or
2013.
REVIEW
SESSION:Tohelpyoustudyforthequiz,therewillbeareview
session.Detailstofollow.
FUTUREQUIZZES:TheotherquizdatesthistermwillbeWedneday,Novem-
ber9,andWednesday,December7,2016.
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.2
INFORMATION
TO
BEGIVEN
ON
QUIZ:
Eachquizinthiscoursewillhaveasectionof\usefulinformation"attheback
ofthequiz.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.
KINEMATICSOF
A
HOMOGENEOUSLY
EXPANDING
UNIVERSE:
Hubble'sLaw:v=Hr,
wherev=recessionvelocityofadistantobject,H=Hubble
expansionrate,andr=distancetothedistantobject.
PresentValueofHubbleExpansionRate(Planck2015):
H0=67:7�0:5km-s �1-Mpc �1
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.3
ScaleFactor:`p (t)=a(t)`c;
where`p (t)isthephysicaldistancebetweenanytwoobjects,
a(t)isthescalefactor,and`c
isthecoordinatedistance
betweentheobjects,alsocalledthecomovingdistance.
HubbleExpansionRate:H(t)=
1a(t)
da(t)
dt
.
LightRaysinComovingCoordinates:
Lightraystravelin
straightlineswithspeeddxd
t=
ca(t).
EVOLUTION
OFA
MATTER-DOMINATED
UNIVERSE:
H2= �_aa �2
=8�3
G��kc2
a2
;
�a=�4�3
G�a;
�(t)=a3(t
i )
a3(t)�(ti )
��=�c;where�c=3H2
8�G
:
Flat(k=0):a(t)/t2=3;=1
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.4
PROBLEM
LIST
1.DidYouDotheReading(2007)?
............
5(Sol:23)
*2.TheSteady-StateUniverseTheory............
6(Sol:25)
3.DidYouDoTheReading?
...............
7(Sol:27)
*4.AnExponentiallyExpandingUniverse
..........
8(Sol:29)
5.DidYouDoTheReading?
...............
9(Sol:30)
6.AFlatUniverseWithUnusualTimeEvolution
......
10(Sol:31)
*7.AnotherFlatUniverseWithAnUnusualTimeEvolution
..
11(Sol:32)
8.DidYouDoTheReading?
...............
12(Sol:36)
9.AFlatUniverseWitha(t)/t3=5
............
13(Sol:37)
10.DidYouDoTheReading?
...............
14(Sol:41)
11.AnotherFlatUniverseWitha(t)/t3=5
..........
14(Sol:43)
*12.TheDecelerationParameter...............
15(Sol:47)
13.ARadiation-DominatedFlatUniverse
..........
16(Sol:47)
14.DidYouDoTheReading?
...............
16(Sol:48)
*15.SpecialRelativityDopplerShift.............
17(Sol:49)
16.DidYouDoTheReading?
...............
17(Sol:51)
*17.TracingALightPulseThroughARadiation-DominatedUniverse18(Sol:53)
18.TransverseDopplerShifts................
19(Sol:54)
*19.ATwo-LevelHigh-SpeedMerry-Go-Round
........
20(Sol:56)
20.SignalPropagationInAFlatMatter-DominatedUniverse
.
21(Sol:59)
21.DidYouDoTheReading?
...............
22(Sol:65)
*22.TheTrajectoryOfAPhotonOriginatingAtTheHorizon..
22(Sol:67)
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.5
PROBLEM
1:DID
YOU
DO
THEREADING
(2007)?(35points)
ThefollowingproblemwasProblem1,Quiz1,2000.Thepartswereeachworth5
points.
a)TheDopplere�ectforbothsoundandlightwavesisnamedforJohannChris-
tianDoppler,aprofessorofmathematicsattheRealschuleinPrague.He
predictedthee�ectforbothtypesofwavesinxx42.Whatarethetwodigits
xx?
b)Whentheskyisveryclear(asitalmostneverisinBoston),onecanseeaband
oflightacrossthenightskythathasbeenknownsinceancienttimesasthe
MilkyWay.Explaininasentenceortwohowthisbandoflightisrelatedto
theshapeofthegalaxyinwhichwelive,whichisalsocalledtheMilkyWay.
c)Thestatementthatthedistantgalaxiesareonaveragerecedingfromuswith
aspeedproportionaltotheirdistancewas�rstpublishedbyEdwinHubblein
1929,andhasbecomeknownasHubble'slaw.WasHubble'soriginalpaper
basedonthestudyof2,18,180,or1,800galaxies?
d)Thefollowingdiagram,labeledHomogeneityandtheHubbleLaw,wasusedby
WeinbergtoexplainhowHubble'slawisconsistentwiththehomogeneityof
theuniverse:
Thearrowsandlabelsfromthe\VelocitiesseenbyB"andthe\Velocitiesseen
byC"rowshavebeendeletedfromthiscopyofthe�gure,anditisyourjob
tosketchthe�gureinyourexambookwiththesearrowsandlabelsincluded.
(Actually,inWeinberg'sdiagramthesearrowswerenotlabeled,butthelabels
arerequiredheresothatthegraderdoesnothavetojudgethepreciselength
ofhand-drawnarrows.)
e)Thehorizonisthepresentdistanceofthemostdistantobjectsfrom
which
lighthashadtimetoreachussincethebeginningoftheuniverse.Thehorizon
changeswithtime,butofcoursesodoesthesizeoftheuniverseasawhole.
Duringatimeintervalinwhichthelinearsizeoftheuniversegrowsby1%,
doesthehorizondistance
(i)growbymorethan1%,or
8.286QUIZ1REVIEW
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p.6
(ii)growbylessthan1%,or
(iii)growbythesame1%?
f)Namethetwomenwhoin1964discoveredthecosmicbackgroundradiation.
WithwhatinstitutionweretheyaÆliated?
g)Atatemperatureof3000K,thenucleiandelectronsthat�lledtheuniverse
combinedtoformneutralatoms,whichinteractveryweaklywiththephotons
ofthebackgroundradiation.Afterthisprocess,knownas\recombination,"the
backgroundradiationexpandedfreely.Sincerecombination,howhaveeachof
thefollowingquantitiesvariedasthesizeoftheuniversehaschanged?(Your
answersshouldresemblestatementssuchas\proportionaltothesizeofthe
universe,"or\inverselyproportionaltothesquareofthesizeoftheuniverse".
Theword\size"willbeinterpretedtomeanlinearsize,notvolume.)
(i)theaveragedistancebetweenphotons
(ii)thetypicalwavelengthoftheradiation
(iii)thenumberdensityofphotonsintheradiation
(iv)theenergydensityoftheradiation
(v)thetemperatureoftheradiation
�
PROBLEM
2:
THE
STEADY-STATE
UNIVERSE
THEORY
(25
points)
ThefollowingproblemwasProblem2,Quiz1,2000.
Thesteady-statetheoryoftheuniversewasproposedinthelate1940sbyHer-
mannBondi,ThomasGold,andFredHoyle,andwasconsideredaviablemodelfor
theuniverseuntilthecosmicbackgroundradiationwasdiscoveredanditsproperties
werecon�rmed.Asthenamesuggests,thistheoryisbasedonthehypothesisthat
thelarge-scalepropertiesoftheuniversedonotchangewithtime.Theexpansion
oftheuniversewasanestablishedfactwhenthesteady-statetheorywasinvented,
butthesteady-statetheoryreconcilestheexpansionwithasteady-statedensityof
matterbyproposingthatnewmatteriscreatedastheuniverseexpands,sothat
thematterdensitydoesnotfall.Liketheconventionaltheory,thesteady-statethe-
orydescribesahomogeneous,isotropic,expandinguniverse,sothesamecomoving
coordinateformulationcanbeused.
a)(10points)Thesteady-statetheoryproposesthattheHubbleconstant,like
othercosmologicalparameters,doesnotchangewithtime,soH(t)=H0 .Find
themostgeneralformforthescalefactorfunctiona(t)whichisconsistentwith
thishypothesis.
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.7
b)(15points)Supposethatthemassdensityoftheuniverseis�0 ,whichofcourse
doesnotchangewithtime.Intermsofthegeneralformfora(t)thatyoufound
inpart(a),calculatetherateatwhichnewmattermustbecreatedfor�0
to
remainconstantastheuniverseexpands.Youranswershouldhavetheunitsof
massperunitvolumeperunittime.[Ifyoufailedtoanswerpart(a),youwill
stillreceivefullcredithereifyoucorrectlyanswerthequestionforanarbitrary
scalefactorfunctiona(t).]
PROBLEM
3:DID
YOU
DO
THEREADING?(25points)
ThefollowingproblemwasProblem1onQuiz1,2007,whereeachofthe5questions
wasworth5points:
(a)Inthe1940's,threeastrophysicistsproposeda\steadystate"theoryofcos-
mology,inwhichtheuniversehasalwayslookedaboutthesameasitdoes
now.Statethelastnameofatleastoneoftheseauthors.(Bonuspoints:you
canearn1pointeachfornamingtheothertwoauthors,andhenceupto2
additionalpoints,but1pointwillbetakeno�foreachincorrectanswer.)
(b)In1917,aDutchastronomernamedWillem
deSitterdidwhichoneofthe
followingaccomplishments:
(i)measuredthesizeoftheMilkyWaygalaxy,�ndingittobeaboutone
billionlight-yearsindiameter.
(ii)resolvedCepheidvariablestarsinAndromedaandtherebyobtainedper-
suasiveevidencethatAndromedaisnotwithinourowngalaxy,butis
apparentlyanothergalaxylikeourown.
(iii)publishedacatalog,NebulaeandStarClusters,listing103objectsthat
astronomersshouldavoidwhenlookingforcomets.
(iv)publishedamodelfortheuniverse,basedongeneralrelativity,which
appearedtobestaticbutwhichproducedaredshiftproportionaltothe
distance.
(v)discoveredthattheorbitalperiodsoftheplanetsareproportionaltothe
3/2powerofthesemi-majoraxisoftheirellipticalorbits.
(c)In1964{65,ArnoA.PenziasandRobertW.Wilsonobserveda uxofmi-
crowaveradiationcomingfromalldirectionsinthesky,whichwasinterpreted
byagroupofphysicistsataneighboringinstitutionasthecosmicbackground
radiationleftoverfromthebigbang.Circlethetwoitemsonthefollowinglist
thatwerenotpartofthestorybehindthisspectaculardiscovery:
8.286QUIZ1REVIEW
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p.8
(i)BellTelephoneLaboratory
(ii)MIT
(iii)PrincetonUniversity
(iv)pigeons
(v)groundhogs
(vi)Hubble'sconstant
(vii)liquidhelium
(viii)7.35cm
(Grading:3ptsfor1correctanswer,5for2correctanswers,and-2foreach
incorrectanswer,buttheminimumscoreiszero.)
(d)ImportantpredictionsoftheCopernicantheorywerecon�rmedbythediscov-
eryoftheaberrationofstarlight(whichshowedthatthevelocityoftheEarth
hasthetime-dependenceexpectedforrotationabouttheSun)andbythebe-
havioroftheFoucaultpendulum(whichshowedthattheEarthrotates).These
discoveriesweremade
(i)duringCopernicus'lifetime.
(ii)approximatelytwoandthreedecadesafterCopernicus'death,respectively.
(iii)aboutonehundredyearsafterCopernicus'death.
(iv)approximatelytwoandthreecenturiesafterCopernicus'death,respec-
tively.
(e)IfoneaveragesoversuÆcientlylargescales,theuniverseappearstobeho-
mogeneousandisotropic.Howlargemusttheaveragingscalebebeforethis
homogeneityandisotropysetin?
(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:ANEXPONENTIALLYEXPANDINGUNIVERSE(20
points)
ThefollowingproblemwasProblem2,Quiz2,1994,andhadalsoappearedonthe
1994ReviewProblems.Asisthecasethisyear,itwasannouncedthatoneofthe
problemsonthequizwouldcomefromeitherthehomeworkortheReviewProblems.
TheproblemalsoappearedasProblem2onQuiz1,2007.
Considera at(i.e.,ak=0,oraEuclidean)universewithscalefactorgiven
by
a(t)=a0 e�t;
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.9
wherea0and�areconstants.
(a)(5points)FindtheHubbleconstantH
atanarbitrarytimet.
(b)(5points)Let(x;y;z;t)bethecoordinatesofacomovingcoordinatesystem.
Supposethatatt=0agalaxylocatedattheoriginofthissystememitsalight
pulsealongthepositivex-axis.Findthetrajectoryx(t)whichthelightpulse
follows.
(c)(5points)Supposethatwearelivingonagalaxyalongthepositivex-axis,and
thatwereceivethislightpulseatsomelatertime.Weanalyzethespectrumof
thepulseanddeterminetheredshiftz.Expressthetimetratwhichwereceive
thepulseintermsofz,�,andanyrelevantphysicalconstants.
(d)(5points)Atthetimeofreception,whatisthephysicaldistancebetweenour
galaxyandthegalaxywhichemittedthepulse?Expressyouranswerinterms
ofz,�,andanyrelevantphysicalconstants.
PROBLEM
5:DID
YOU
DO
THEREADING?
(a)Theassumptionsofhomogeneityandisotropygreatlysimplifythedescription
ofouruniverse.We�ndthattherearethreepossibilitiesforahomogeneous
andisotropicuniverse:anopenuniverse,a atuniverse,andacloseduni-
verse.Whatquantityorconditiondistinguishesbetweenthesethreecases:the
temperatureofthemicrowavebackground,thevalueof=�=�c ,mattervs.
radiationdomination,orredshift?
(b)Whatisthetemperature,inKelvin,ofthecosmicmicrowavebackgroundto-
day?
(c)Whichofthefollowingsupportsthehypothesisthattheuniverseisisotropic:
thedistancestonearbyclusters,observationsofthecosmicmicrowaveback-
ground,clusteringofgalaxiesonlargescales,ortheageanddistributionof
globularclusters?
(d)IsthedistancetotheAndromedaNebula(roughly)10kpc,5billionlightyears,
2millionlightyears,or3lightyears?
(e)DidHubblediscoverthelawwhichbearshisnamein1862,1880,1906,1929,
or1948?
(f)WhenHubblemeasuredthevalueofhisconstant,hefoundH�1�100million
years,2billionyears,10billionyears,or20billionyears?
(g)Cepheidvariablesareimportanttocosmologybecausetheycanbeusedtoesti-
matethedistancestothenearbygalaxies.WhatpropertyofCepheidvariables
makesthemusefulforthispurpose,andhowaretheyused?
8.286QUIZ1REVIEW
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p.10
(h)Cepheidvariablestarscanbeusedasestimatorsofdistancefordistancesup
toabout100light-years,104
light-years,107
lightyears,or1010
light-years?
[Notefor2011:thisquestionwasbasedonthereadingfromJosephSilk'sThe
BigBang,andthereforewouldbenotbeafairquestionforthisyear.]
(i)Namethetwomenwhoin1964discoveredthecosmicbackgroundradiation.
WithwhatinstitutionweretheyaÆliated?
(j)Atthetimeofthediscoveryofthecosmicmicrowavebackground,anactivebut
independente�ortwastakingplaceelsewhere.P.J.E.Peebleshadestimated
thattheuniversemustcontainbackgroundradiationwithatemperatureofat
least10 ÆK,andRobertH.Dicke,P.G.Roll,andD.T.Wilkinsonhadmounted
anexperimenttolookforit.Atwhatinstitutionwerethesepeopleworking?
PROBLEM
6:AFLATUNIVERSEWITH
UNUSUALTIMEEVOLU-
TION
ThefollowingproblemwasProblem3,Quiz2,1988:
Considera atuniverse�lledwithanewandpeculiarformofmatter,witha
Robertson{Walkerscalefactorthatbehavesas
a(t)=bt1=3:
Herebdenotesaconstant.
(a)Ifalightpulseisemittedattimeteandobservedattimeto ,�ndthephys-
icalseparation`p (to )betweentheemitterandtheobserver,atthetimeof
observation.
(b)Againassumingthatteandtoaregiven,�ndtheobservedredshiftz.
(c)Findthephysicaldistance`p (to )whichseparatestheemitterandobserverat
thetimeofobservation,expressedintermsofc,to ,andz(i.e.,withoutte
appearing).
(d)Atanarbitrarytimetintheintervalte<t<to ,�ndthephysicaldistance
`p (t)betweenthelightpulseandtheobserver.Expressyouranswerinterms
ofc,t,andto .
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.11
�
PROBLEM
7:ANOTHERFLATUNIVERSEWITHAN
UNUSUAL
TIMEEVOLUTION
(40points)
ThefollowingproblemwasProblem3,Quiz1,2000.
Considera atuniversewhichis�lledwithsomepeculiarformofmatter,so
thattheRobertson{Walkerscalefactorbehavesas
a(t)=bt ;
whereband areconstants.[Thisuniversedi�ersfrom
thematter-dominated
universedescribedinthelecturenotesinthat�isnotproportionalto1=a3(t).Such
behaviorispossiblewhenpressuresarelarge,becauseagasexpandingunderpressure
canloseenergy(andhencemass)duringtheexpansion.]Forthefollowingquestions,
anyoftheanswersmaydependon ,whetheritismentionedexplicitlyornot.
a)(5points)Lett0denotethepresenttime,andlettedenotethetimeatwhich
thelightthatwearecurrentlyreceivingwasemittedbyadistantobject.In
termsofthesequantities,�ndthevalueoftheredshiftparameterzwithwhich
thelightisreceived.
b)(5points)Findthe\look-back"timeasafunctionofzandt0 .Thelook-back
timeisde�nedasthelengthoftheintervalincosmictimebetweentheemission
andobservationofthelight.
c)(10points)Expressthepresentvalueofthephysicaldistancetotheobjectas
afunctionofH0 ,z,and .
d)(10points)Atthetimeofemission,thedistantobjecthadapoweroutputP
(measured,say,inergs/sec)whichwasradiateduniformlyinalldirections,in
theformofphotons.Whatistheradiationenergy uxJfromthisobjectat
theearthtoday?ExpressyouranswerintermsofP,H0 ,z,and .[Energy
ux(whichmightbemeasuredinerg-cm�2-sec �1)isde�nedastheenergyper
unitareaperunittimestrikingasurfacethatisorthogonaltothedirectionof
energy ow.]
e)(10points)Supposethatthedistantobjectisagalaxy,movingwiththeHubble
expansion.Withinthegalaxyasupernovaexplosionhashurledajetofmaterial
directlytowardsEarthwithaspeedv,measuredrelativetothegalaxy,which
iscomparabletothespeedoflightc.Assume,however,thatthedistancethe
jethastraveledfrom
thegalaxyissosmallthatitcanbeneglected.With
whatredshiftzJ
wouldweobservethelightcomingfrom
thisjet?Express
youranswerintermsofallorsomeofthevariablesv,z(theredshiftofthe
galaxy),t0 ,H0 ,and ,andtheconstantc.
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.12
PROBLEM
8:DID
YOU
DO
THEREADING?(25points)
ThefollowingproblemwasProblem1,Quiz1,1996:
Thefollowingquestionsareworth5pointseach.
a)In1814-1815,theMunichopticianJosephFrauenhoferallowedlightfromthe
suntopassthroughaslitandthenthroughaglassprism.Thelightwasspread
intoaspectrumofcolors,showinglinesthatcouldbeidenti�edwithknown
elements|
sodium,iron,magnesium,calcium,andchromium.Werethese
linesdark,orbright(2points)?Why(3points)?
b)TheAndromedaNebulawasshownconclusivelytolieoutsideourowngalaxy
whenastronomersacquiredtelescopespowerfulenoughtoresolvetheindivid-
ualstarsofAndromeda.WasthisfeataccomplishedbyGalileoin1609,by
ImmanuelKantin1755,byHenriettaSwanLeavittin1912,byEdwinHubble
in1923,orbyWalterBaadeandAllanSandageinthe1950s?
c)Someoftheearliestmeasurementsofthecosmicbackgroundradiationwere
madeindirectly,byobservinginterstellarcloudsofamoleculecalledcyanogen
(CN).Statewhethereachofthefollowingstatementsistrueorfalse(1point
each):
(i)The�rstmeasurementsofthetemperatureoftheinterstellarcyanogen
weremadeovertwentyyearsbeforethecosmicbackgroundradiationwas
directlyobserved.
(ii)Cyanogenhelpstomeasurethecosmicbackgroundradiationbyre ecting
ittowardtheearth,sothatitcanbereceivedwithmicrowavedetectors.
(iii)Onereasonwhythecyanogenobservationswereimportantwasthatthey
gavethe�rstmeasurementsoftheequivalenttemperatureofthecosmic
backgroundradiationatwavelengthsshorterthanthepeakoftheblack-
bodyspectrum.
(iv)Bymeasuringthespectrum
ofvisiblestarlightthatpassesthroughthe
cyanogenclouds,astronomerscaninfertheintensityofthemicrowave
radiationthatbathestheclouds.
(v)Byobservingchemicalreactionsinthecyanogenclouds,astronomerscan
inferthetemperatureofthemicrowaveradiationinwhichtheyarebathed.
d)Inabout280B.C.,aGreekphilosopherproposedthattheEarthandtheother
planetsrevolvearoundthesun.Whatwasthenameofthisperson?[Notefor
2011:thisquestionwasbasedonreadingsfromJosephSilk'sTheBigBang,
andthereforeisnotappropriateforQuiz1ofthisyear.]
e)In1832HeinrichWilhelm
Olberspresentedwhatwenowknowas\Olbers'
Paradox,"althoughasimilarargumenthadbeendiscussedasearlyas1610by
8.286QUIZ1REVIEW
PROBLEMS,FALL2016
p.13
JohannesKepler.Olbersarguedthatiftheuniverseweretransparent,static,
in�nitelyold,andwaspopulatedbyauniformdensityofstarssimilartoour
sun,thenoneofthefollowingconsequenceswouldresult:
(i)Thebrightnessofthenightskywouldbein�nite.
(ii)Anypatchofthenightskywouldlookasbrightasthesurfaceofthe
sun.
(iii)Thetotalenergy uxfromthenightskywouldbeaboutequaltothe
totalenergy uxfromthesun.
(iv)Anypatchofthenightskywouldlookasbrightasthesurfaceofthe
moon.
WhichoneofthesestatementsisthecorrectstatementofOlbers'paradox?
PROBLEM
9:A
FLATUNIVERSEWITH
a(t)/
t3=5
ThefollowingproblemwasProblem3,Quiz1,1996:
Considera atuniversewhichis�lledwithsomepeculiarformofmatter,so
thattheRobertson{Walkerscalefactorbehavesas
a(t)=bt3=5;
wherebisaconstant.
a)(5points)FindtheHubbleconstantH
atanarbitrarytimet.
b)(5points)Whatisthephysicalhorizondistanceattimet?
c)(5points)SupposealightpulseleavesgalaxyAattimetA
andarrivesatgalaxy
BattimetB.Whatisthecoordinatedistancebetweenthesetwogalaxies?
d)(5points)WhatisthephysicalseparationbetweengalaxyAandgalaxyBat
timetA?AttimetB?
e)(5points)Atwhattimeisthelightpulseequidistantfromthetwogalaxies?
f)(5points)WhatisthespeedofBrelativetoAatthetimetA?(By\speed,"I
meantherateofchangeofthephysicaldistancewithrespecttocosmictime,
d`p =dt.)
g)(5points)Forobservationsmadeattimet,whatisthepresentvalueofthe
physicaldistanceasafunctionoftheredshiftz(andthetimet)?Whatphysical
distancecorrespondstoz=
1?
Howdoesthiscomparewiththehorizon
distance?
(NotethatthisquestiondoesnotrefertothegalaxiesAandB
discussedintheearlierparts.Inparticular,youshouldnotassumethatthe
lightpulseleftitssourceattimetA.)
h)(5points)ReturningtothediscussionofthegalaxiesAandBwhichwere
consideredinparts(c)-(f),supposetheradiationfrom
galaxyAisemitted
withtotalpowerP.WhatisthepowerperareareceivedatgalaxyB?
i)(5points)WhenthelightpulseisreceivedbygalaxyB,apulseisimmediately
sentbacktowardgalaxyA.Atwhattimedoesthissecondpulsearriveatgalaxy
A?
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PROBLEM
10:DID
YOU
DO
THEREADING?(20points)
ThefollowingquestionsweretakenfromProblem1,Quiz1,1998:
Thefollowingquestionsareworth5pointseach.
a)In1917,Einsteinintroducedamodeloftheuniversewhichwasbasedonhis
newlydevelopedgeneralrelativity,butwhichcontainedanextraterminthe
equationswhichhecalledthe\cosmologicalterm."
(ThecoeÆcientofthis
termiscalledthe\cosmologicalconstant.")WhatwasEinstein'smotivation
forintroducingthisterm?
b)Whentheredshiftofdistantgalaxieswas�rstdiscovered,theearliestobserva-
tionswereanalyzedaccordingtoacosmologicalmodelinventedbytheDutch
astronomerW.deSitterin1917.Atthetimeofitsdiscovery,wasthismodel
thoughttobestaticorexpanding?Fromthemodernperspective,isthemodel
thoughttobestaticorexpanding?
c)Theearlyuniverseisbelievedtohavebeen�lledwiththermal,orblack-body,
radiation.Forsuchradiationthenumberdensityofphotonsandtheenergy
densityareeachproportionaltopowersoftheabsolutetemperatureT.Say
Numberdensity/Tn1
Energydensity/Tn2
Givethevaluesoftheexponentsn1andn2 .
d)Atabout3,000Kthematterintheuniverseunderwentacertainchemical
changeinitsform,achangethatwasnecessarytoallowthedi�erentiationof
matterintogalaxiesandstars.Whatwasthenatureofthischange?
PROBLEM
11:ANOTHER
FLAT
UNIVERSEWITH
a(t)/
t3=5
(40
points)
ThefollowingwasProblem3,Quiz1,1998:
Considera atuniversewhichis�lledwithsomepeculiarformofmatter,so
thattheRobertson{Walkerscalefactorbehavesas
a(t)=bt3=5;
wherebisaconstant.
a)(5points)FindtheHubbleconstantH
atanarbitrarytimet.
b)(10points)Supposeamessageistransmittedbyradiosignal(travelingatthe
speedoflightc)fromgalaxyAtogalaxyB.Themessageissentatcosmic
timet1 ,whenthephysicaldistancebetweenthegalaxiesis`0 .Atwhatcosmic
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timet2isthemessagereceivedatgalaxyB?(Expressyouranswerintermsof
`0 ,t1 ,andc.)
c)(5points)Uponreceiptofthemessage,thecreaturesongalaxyBimmediately
sendbackanacknowledgement,byradiosignal,thatthemessagehasbeen
received.Atwhatcosmictimet3istheacknowledgmentreceivedongalaxyA?
(Expressyouranswerintermsof`0 ,t1 ,t2 ,andc.)
d)(10points)ThecreaturesongalaxyBspendsometimetryingtodecodethe
message,�nallydecidingthatitisanadvertisementforKellogg'sCornFlakes
(whateverthatis).Atatime�tafterthereceiptofthemessage,asmeasured
ontheirclocks,theysendbackaresponse,requestingfurtherexplanation.At
whatcosmictimet4
istheresponsereceivedongalaxyA?Inansweringthis
part,youshouldnotassumethat�tisnecessarilysmall.(Expressyouranswer
intermsof`0 ,t1 ,t2 ,t3 ,�t,andc.)
e)(5points)WhentheresponseisreceivedbygalaxyA,theradiowaveswillbe
redshiftedbyafactor1+z.Giveanexpressionforz.(Expressyouranswerin
termsof`0 ,t1 ,t2 ,t3 ,t4 ,�t,andc.)
f)(5points;Nopartialcredit)Ifthetime�tintroducedinpart(d)issmall,the
timedi�erencet4 �t3canbeexpandedto�rstorderin�t.Calculatet4 �t3
to�rstorderaccuracyin�t.(Expressyouranswerintermsof`0 ,t1 ,t2 ,t3 ,
t4 ,�t,andc.)[Hint:whilethispartcanbeansweredbyusingbruteforceto
expandtheanswerinpart(d),thereisaneasierway.]
�
PROBLEM
12:THEDECELERATION
PARAMETER
ThefollowingproblemwasProblem2,Quiz2,1992,whereitcounted10pointsout
of100.
Manystandardreferencesincosmologyde�neaquantitycalledthedeceler-
ationparameterq,whichisadirectmeasureoftheslowingdownofthecosmic
expansion.Theparameterisde�nedby
q���a(t)a(t)
_a2(t):
Findtherelationshipbetweenqandforamatter-dominateduniverse.[Incase
youhaveforgotten,isde�nedby
=�=�c;
where�isthemassdensityand�c
isthecriticalmassdensity(i.e.,thatmass
densitywhichcorrespondstok=0).]
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PROBLEM
13:A
RADIATION-DOMINATED
FLATUNIVERSE
Wehavelearnedthatamatter-dominatedhomogeneousandisotropicuniverse
canbedescribedbyascalefactora(t)obeyingtheequation
�_aa �2
=8�3
G��kc2
a2
:
Thisequationinfactappliestoanyformofmassdensity,sowecanapplyittoa
universeinwhichthemassdensityisdominatedbytheenergyofphotons.Recall
thatthemassdensityofnonrelativisticmatterfallso�as1=a3(t)astheuniverse
expands;themassofeachparticleremainsconstant,andthedensityofparticles
fallso�as1=a3(t)becausethevolumeincreasesasa3(t).Forthephoton-dominated
universe,thedensityofphotonsfallsofas1=a3(t),butinadditionthefrequency
(andhencetheenergy)ofeachphotonredshiftsinproportionto1=a(t).Sincemass
andenergyareequivalent,themassdensityofthegasofphotonsfallso�as1=a4(t).
Fora at(i.e.,k=0)matter-dominateduniversewelearnedthatthescale
factora(t)isproportionaltot2=3.Howdoesa(t)behaveforaphoton-dominated
universe?
PROBLEM
14:DID
YOU
DO
THEREADING?
Thefollowingproblem
wastakenfrom
Problem1,Quiz1,2004,whereeachpart
counted5points,foratotalof25points.Thereadingassignmentincludedthe�rst
threechaptersofRyden,IntroductiontoCosmology,andthe�rstthreechapters
ofWeinberg,TheFirstThreeMinutes.
(a)In1826,theastronomerHeinrichOlberwroteapaperonaparadoxregarding
thenightsky.WhatisOlber'sparadox?Whatistheprimaryresolutionofit?
(b)WhatisthevalueoftheNewtoniangravitationalconstantGinPlanckunits?
ThePlancklengthisoftheorderof10 �35m,10 �15m,1015m,or1035m?
(c)WhatistheCosmologicalPrinciple?IstheHubbleexpansionoftheuniverse
consistentwithit?(Forthelatterquestion,asimple\yes"or\no"willsuÆce.)
(d)Inthe\StandardModel"oftheuniverse,whentheuniversecooledtoabout
3�10aK,itbecametransparenttophotons,andtodayweobservetheseasthe
CosmicMicrowaveBackground(CMB)atatemperatureofabout3�10bK.
Whataretheintegersaandb?
(e)Whatdidtheuniverseprimarilyconsistofatabout1/100thofasecondafter
theBigBang?Includeanyconstituentthatisbelievedtohavemadeupmore
than1%ofthemassdensityoftheuniverse.
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�
PROBLEM
15:SPECIALRELATIVITY
DOPPLER
SHIFT
Thefollowingproblemwastakenfrom
Problem2,Quiz1,2004,whereitcounted
20points.
ConsidertheDopplershiftofradiowaves,foracaseinwhichboththesource
andtheobserveraremoving.Supposethesourceisaspaceshipmovingwithaspeed
vsrelativetothespacestationAlpha-7,whiletheobserverisonanotherspaceship,
movingintheoppositedirectionfromAlpha-7withspeedvorelativetoAlpha-7.
(a)(10points)CalculatetheDopplershiftzoftheradiowaveasreceivedbythe
observer.(Recallthatradiowavesareelectromagneticwaves,justlikelight
exceptthatthewavelengthislonger.)
(b)(10points)Usetheresultsofpart(a)todeterminevtot ,thevelocityofthe
sourcespaceshipasitwouldbemeasuredbytheobserverspaceship.(8points
willbegivenforthebasicidea,whetherornotyouhavetherightanswerfor
part(a),and2pointswillbegivenforthealgebra.)
PROBLEM
16:DID
YOU
DO
THEREADING?
ThefollowingquestionwastakenfromProblem1,Quiz1,2005,whereitcounted
25points.
(a)(4points)Whatwasthe�rstexternalgalaxythatwasshowntobeatadistance
signi�cantlygreaterthanthemostdistantknownobjectsinourgalaxy?How
wasthedistanceestimated?
(b)(5points)Whatisrecombination?Didgalaxiesbegintoformbeforeorafter
recombination?Why?
(c)(4points)InChapterIVofhisbook,Weinbergdevelopsa\recipeforahot
universe,"inwhichthematteroftheuniverseisdescribedasagasinthermal
equilbriumataveryhightemperature,inthevicinityof109K(severalthou-
sandmilliondegreesKelvin).Suchathermalequilibrium
gasiscompletely
describedbyspecifyingitstemperatureandthedensityoftheconservedquan-
tities.Whichofthefollowingisonthislistofconservedquantities?Circleas
manyasapply.
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(i)baryonnumber
(ii)energyperparticle
(iii)protonnumber
(iv)electriccharge
(v)pressure
(d)(4points)ThewavelengthcorrespondingtothemeanenergyofaCMB(cosmic
microwavebackground)photontodayisapproximatelyequaltowhichofthe
followingquantities?(Youmaywishtolookupthevaluesofvariousphysical
constantsattheendofthequiz.)
(i)2fm(2�10 �15m)
(ii)2microns(2�10 �6m)
(iii)2mm(2�10 �3m)
(iv)2m.
(e)(4points)Whatistheequivalenceprinciple?
(f)(4points)WhyisitdiÆcultforEarth-basedexperimentstolookatthesmall
wavelengthportionofthegraphofCMBenergydensityperwavelengthvs.
wavelength?
�
PROBLEM
17:
TRACING
A
LIGHT
PULSE
THROUGH
A
RADIATION-DOMINATED
UNIVERSE
Thefollowingproblemwastakenfrom
Problem3,Quiz1,2005,whereitcounted
25points.
Considera atuniversethatexpandswithascalefactor
a(t)=bt1=2;
wherebisaconstant.Wewilllearnlaterthatthisisthebehaviorofthescalefactor
foraradiation-dominateduniverse.
(a)(5points)Atanarbitrarytimet=tf,whatisthephysicalhorizondistance?
(By\physical,"Imeanasusualthedistanceinphysicalunits,suchasmeters
orcentimeters,asmeasuredbyasequenceofrulers,eachofwhichisatrest
relativetothecomovingmatterinitsvicinity.)
(b)(3points)Supposethataphotonarrivesattheorigin,attimetf,fromadistant
pieceofmatterthatispreciselyatthehorizondistanceattimetf.Whatis
thetimeteatwhichthephotonwasemitted?
(c)(2points)Whatisthecoordinatedistancefromtheorigintothepointfrom
whichthephotonwasemitted?
(d)(10points)Foranarbitrarytimetintheintervalte �t�tf,whilethephoton
istraveling,whatisthephysicaldistance`p (t)fromtheorigintothelocation
ofthephoton?
(e)(5points)Atwhattimetmax
isthephysicaldistanceofthephotonfromthe
originatitslargestvalue?
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PROBLEM
18:TRANSVERSEDOPPLER
SHIFTS
Thefollowingproblemwastakenfrom
Problem4,Quiz1,2005,whereitcounted
20points.
(a)(8points)Supposethespaceship
Xanthu
is
at
rest
at
location
(x=0;y=a;z=0)inaCartesianco-
ordinatesystem.(Weassumethat
thespaceisEuclidean,andthatthe
distancescalesintheproblem
are
smallenoughsothattheexpansion
oftheuniversecanbeneglected.)
ThespaceshipEmmeracismoving
atspeedv0
alongthex-axisinthe
positivedirection,asshowninthe
diagram,wherev0iscomparableto
thespeedoflight.AstheEmmerac
crossestheorigin,itreceivesara-
diosignalthathadbeensentsome
timeearlierfrom
theXanthu.
Is
theradiationreceivedredshiftedorblueshifted?Whatistheredshiftz(where
negativevaluesofzcanbeusedtodescribeblueshifts)?
(b)(7points)Now
supposethatthe
Emmeracisatrestattheorigin,
whiletheXanthuismovinginthe
negativex-direction,aty=aand
z=
0,asshowninthediagram.
Thatis,thetrajectoryoftheXan-
thucanbetakenas
(x=�v0 t;y=a;z=0):
Att=0theXanthucrossesthey-
axis,andatthatinstantitemits
aradiosignalalongthey-axis,di-
rectedattheorigin.
Theradi-
ationisreceivedsometimelater
bytheEmmerac.Inthiscase,is
theradiationreceivedredshiftedorblueshifted?Whatistheredshiftz(where
againnegativevaluesofzcanbeusedtodescribeblueshifts)?
(c)(5points)Isthesequenceofeventsdescribedin(b)physicallydistinctfromthe
sequencedescribedin(a),orisitreallythesamesequenceofeventsdescribed
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inareferenceframethatismovingrelativetothereferenceframeusedinpart
(a)?Explainyourreasoninginasentenceortwo.(Hint:notethatthereare
threeobjectsintheproblem:Xanthu,Emmerac,andthephotonsoftheradio
signal.)
�
PROBLEM
19:
A
TWO-LEVEL
HIGH-SPEED
MERRY-GO-
ROUND
(15points)
ThisproblemwasProblem3onQuiz1,2007.
Considerahigh-speedmerry-go-roundwhichissimilartotheonediscussedin
Problem3ofProblemSet1,butwhichhastwolevels.Thatis,therearefourevenly
spacedcarswhichtravelaroundacentralhubatspeedvatadistanceRfroma
centralhub,andalsoanotherfourcarsthatareattachedtoextensionsofthefour
radialarms,eachmovingataspeed2vatadistance2Rfromthecenter.Inthis
problemwewillconsideronlylightwaves,notsoundwaves,andwewillassume
thatvisnotnegligiblecomparedtoc,butthat2v<c.
WelearnedinProblemSet1thatthereisnoredshiftwhenlightfromonecarat
radiusRisreceivedbyanobserveronanothercaratradiusR.
(a)(5points)Supposethatcars5{8areallemittinglightwavesinalldirections.If
anobserverincar1receiveslightwavesfromeachofthesecars,whatredshift
zdoessheobserveforeachofthefoursignals?
(b)(10points)Supposethataspaceshipisrecedingtotherightatarelativistic
speedualongalinethroughthehub,asshowninthediagram.Supposethat
anobserverincar6receivesaradiosignalfromthespaceship,atthetimewhen
thecarisinthepositionshowninthediagram.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,theanswertoany
partmaycontainsymbolsrepresentingtheanswerstopreviousparts,whetheror
notthepreviouspartwasansweredcorrectly.
(a)(10points)Attimet=t1 ,alightsignalissentfromgalaxyA.Let`p;sA(t)
denotethephysicaldistanceofthesignalfromAattimet.(Notethatt=0
correspondstotheoriginoftheuniverse,nottotheemissionofthesignal.)
(i)FindthespeedofseparationofthelightsignalfromA,de�nedasd`p;sA=dt.
Whatisthevalueofthisspeed(ii)atthetimeofemission,t1 ,and(iii)what
isitslimitingvalueatarbitrarilylatetimes?
(b)(5points)Supposethatthereisasecondgalaxy,galaxyB,thatislocatedat
aphysicaldistancecH�1
fromAattimet1 ,whereH(t)denotestheHubble
expansionrateandcisthespeedoflight.(cH�1iscalledtheHubblelength.)
Supposethatthelightsignaldescribedabove,whichisemittedfromgalaxy
Aattimet1 ,isdirectedtowardgalaxyB.Atwhattimet2
doesitarriveat
galaxyB?
(c)(10points)Let`p;sB(t)denotethephysicaldistanceofthelightsignalfrom
galaxyBattimet.(i)Findthespeedofapproachofthelightsignaltowards
B,de�nedas�d`p;sB=dt.Whatisthevalueofthisspeed(ii)atthetimeof
emission,t1 ,and(iii)atthetimeofreception,t2 ?
(d)(10points)IfanastronomerongalaxyAobservesthelightarrivingfromgalaxy
Battimet1 ,whatisitsredshiftzBA?
(e)(10points)Supposethatthereisanothergalaxy,galaxy
C,alsolocatedataphysicaldistancecH�1
from
Aat
timet1 ,butinadirectionorthogonaltothatofB.If
galaxyBisobservedfromgalaxyCattimet1 ,whatis
theobservedredshiftzBC?Recallthatthisuniverseis
at,soEuclideangeometryapplies.
(f)(10points)SupposethatgalaxyA,attimet1 ,emitselectromagneticradiation
sphericallysymmetrically,withpoweroutputP.(Pmightbemeasured,for
example,inwatts,where1watt=1joule/second.)Whatistheradiation
energy uxJthatisreceivedbygalaxyB
attimet2 ,whentheradiation
reachesgalaxyB?(Jmightbemeasured,forexample,inwattspermeter2.
Unitsarementionedhereonlytohelpclarifythemeaningofthesequantities|
youranswershouldhavenoexplicitunits,butshouldbeexpressedintermsof
anyorallofthegivenquantitiest1 ,P,andc,plusperhapssymbolsrepresenting
theanswerstopreviousparts.)
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PROBLEM
21:DID
YOU
DO
THEREADING?(25points)
ThefollowingproblemappearedonQuiz1of2011.
(a)(10points)Hubble'slawrelatesthedistanceofgalaxiestotheirvelocity.The
Dopplere�ectprovidesanaccuratetooltomeasurevelocity,whilethemea-
sureofcosmicdistancesismoreproblematic.Explainbrie ythemethodthat
Hubbleusedtoestimatethedistanceofgalaxiesinderivinghislaw.
(b)(5points)OneexpectsHubble'slawtoholdasaconsequenceoftheCosmo-
logicalPrinciple.WhatdoestheCosmologicalPrinciplestate?
(c)(10points)Giveabriefde�nitionforthewordshomogeneityandisotropy.
Thensayforeachofthefollowingtwostatementswhetheritistrueorfalse.
Iftrueexplainbrie ywhy.Iffalsegiveacounter-example.Youshouldassume
Euclideangeometry(whichWeinbergimplicitlyassumedinhisdiscussion).
(i)Iftheuniverseisisotropicaroundonepointthenithastobehomogeneous.
(ii)Iftheuniverseisisotropicaroundtwoormoredistinctpointsthenithas
tobehomogeneous.
(d)Bonusquestion:(2pointsextracredit)Ifweallowcurved(i.e.,non-Euclidean)
spaces,isittruethatauniversewhichisisotropicaroundtwodistinctpoints
hastobehomogeneous?
Iftrueexplainbrie ywhy,andotherwisegivea
counter-example.
�
PROBLEM
22:THETRAJECTORY
OFA
PHOTON
ORIGINAT-
ING
ATTHEHORIZON
(25points)
ThefollowingproblemappearedonQuiz1of2011.
Consideragaina atmatter-dominateduniverse,withascalefactorgivenby
a(t)=bt2=3;
wherebisaconstant.Lett0denotethecurrenttime.
(a)(5points)Whatisthecurrentvalueofthephysicalhorizon
distance
`p;horizon (t0 )?Thatis,whatisthepresentdistanceofthemostdistantmatter
thatcanbeseen,limitedonlybythespeedoflight.
(b)(5points)Consideraphotonthatisarrivingnowfromanobjectthatisjustat
thehorizon.Ourgoalistotracethetrajectoryofthisobject.Supposethatwe
setupacoordinatesystemwithusattheorigin,andthesourceofthephoton
alongthepositivex-axis.Whatisthecoordinatex0ofthephotonatt=0?
(c)(5points)Asthephotontravelsfromthesourcetous,whatisitscoordinate
x(t)asafunctionoftime?
(d)(5points)Whatisthephysicaldistance`p (t)betweenthephotonandusasa
functionoftime?
(e)(5points)Whatisthemaximumphysicaldistance`p;max (t)betweenthephoton
andus,andatwhattimetmaxdoesitoccur?
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SOLUTIONS
PROBLEM
1:DID
YOU
DO
THEREADING?(35points)
a)DopplerpredictedtheDopplere�ectin1842.
b)Mostofthestarsofourgalaxy,includingoursun,lieina atdisk.Wetherefore
seemuchmorelightwhenwelookoutfromearthalongtheplaneofthedisk
thanwhenwelookinanyotherdirection.
c)Hubble'soriginalpaperontheexpansionoftheuniversewasbasedonastudy
ofonly18galaxies.Well,atleastWeinberg'sbooksays18galaxies.Formy
ownbookImadeacopyofHubble'soriginalgraph,whichseemstoshow
24blackdots,eachofwhichrepresentsagalaxy,asreproducedbelow.The
verticalaxisshowstherecessionvelocity,inkilometerspersecond.Thesolid
lineshowsthebest�ttotheblackdots,eachofwhichrepresentsagalaxy.Each
opencirclerepresentsagroupofthegalaxiesshownasblackdots,selectedby
theirproximityindirectionanddistance;thebrokenlineisthebest�ttothese
points.Thecrossshowsastatisticalanalysisof22galaxiesforwhichindividual
distancemeasurementswerenotavailable.IamnotsurewhyWeinbergrefers
to18galaxies,butitispossiblethatthetextofHubble'sarticleindicatedthat
18ofthesegalaxiesweremeasuredwithmorereliabilitythantherest.
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d)e)
Duringatimeintervalinwhichthelinearsizeoftheuniversegrowsby1%,the
horizondistancegrowsbymorethan1%.Toseewhy,notethatthehorizon
distanceisequaltothescalefactortimesthecomovinghorizondistance.The
scalefactorgrowsby1%duringthistimeinterval,butthecomovinghorizon
distancealsogrows,sincelightfromthedistantgalaxieshashadmoretimeto
reachus.
f)ArnoA.PenziasandRobertW.Wilson,BellTelephoneLaboratories.
g)
(i)theaveragedistance
between
photons:
proportionaltothesizeof
theuniverse
(Photonsareneithercreatednordestroyed,sotheonly
e�ectisthattheaveragedistancebetweenthem
isstretchedwiththe
expansion.Sincetheuniverseexpandsuniformly,alldistancesgrowby
thesamefactor.)
(ii)thetypicalwavelengthoftheradiation:
proportionaltothesizeof
theuniverse(SeeLectureNotes3.)
(iii)thenumberdensityofphotonsintheradiation:
inverselypropor-
tionaltothecubeofthesizeoftheuniverse(From(i),theaveragedis-
tancebetweenphotonsgrowsinproportiontothesizeoftheuniverse.
Sincethevolumeofacubeisproportionaltothecubeofthelengthof
aside,theaveragevolumeoccupiedbyaphotongrowsasthecubeof
thesizeoftheuniverse.Thenumberdensityistheinverseoftheaverage
volumeoccupiedbyaphoton.)
(iv)theenergy
density
oftheradiation:
inverselyproportionaltothe
fourthpowerofthesizeoftheuniverse
(Theenergyofeachphotonis
proportionaltoitsfrequency,andhenceinverselyproportionaltoitswave-
length.Sofrom(ii)theenergyofeachphotonisinverselyproportional
tothesizeoftheuniverse,andfrom(iii)thenumberdensityisinversely
proportionaltothecubeofthesize.)
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(v)thetemperatureoftheradiation:
inverselyproportionaltothesize
oftheuniverse(Thetemperatureisdirectlyproportionaltotheaver-
ageenergyofaphoton,whichaccordingto(iv)isinverselyproportional
tothesizeoftheuniverse.)
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 ,sothephysicalvolume
is
V(t)=a3(t)`3c:
Sincethemassdensityis�xedat�=�0 ,thetotalmassinsidethiscubeatany
giventimeisgivenby
M(t)=a3(t)`3c�0:
Intheabsenceofmattercreationthetotalmasswithinacomovingvolume
wouldnotchange,sotheincreaseinmassdescribedbytheaboveequation
8.286QUIZ1REVIEW
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mustbeattributedtomattercreation.Therateofmattercreationperunit
timeperunitvolumeisthengivenby
Rate=
1V(t)
dMd
t
=
1
a3(t)`3c3a2(t)dad
t`3c�0
=3adad
t�0
=
3H0�0:
Youwerenotaskedtoinsertnumbers,butitisworthwhiletoconsiderthe
numericalvalueaftertheexam,toseewhatthisansweristellingus.Suppose
wetakeH0
=
70km-sec �1-Mpc �1,andtake�0
tobethecriticaldensity,
�c=3H20 =8�G.Then
Toputthisnumberintomoremeaningfulterms,notethatthemassofahy-
drogenatom
is1:67�10 �27
kg,andthat1year=3:156�107
s.Therate
ofmatterproductionrequiredforthesteady-stateuniversetheorycanthen
beexpressedasroughlyonehydrogenatompercubicmeterperbillionyears!
Needlesstosay,sucharateofmatterproductionistotallyundetectable,so
thesteady-statetheorycannotberuledoutbythefailuretodetectmatter
production.
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PROBLEM
3:DID
YOU
DO
THEREADING?(25points)
Thefollowing5questionsareeachworth5points:
(a)Inthe1940's,threeastrophysicistsproposeda\steadystate"theoryofcos-
mology,inwhichtheuniversehasalwayslookedaboutthesameasitdoes
now.Statethelastnameofatleastoneoftheseauthors.(Bonuspoints:you
canearn1pointeachfornamingtheothertwoauthors,andhenceupto2
additionalpoints,but1pointwillbetakeno�foreachincorrectanswer.)
Ans:(Weinberg,page8,orRyden,page16):HermannBondi,ThomasGold,
andFredHoyle.
(b)In1917,aDutchastronomernamedWillem
deSitterdidwhichoneofthe
followingaccomplishments:
(i)measuredthesizeoftheMilkyWaygalaxy,�ndingittobeaboutone
billionlight-yearsindiameter.
(ii)resolvedCepheidvariablestarsinAndromedaandtherebyobtainedper-
suasiveevidencethatAndromedaisnotwithinourowngalaxy,butis
apparentlyanothergalaxylikeourown.
(iii)publishedacatalog,NebulaeandStarClusters,listing103objectsthat
astronomersshouldavoidwhenlookingforcomets.
(iv)publishedamodelfortheuniverse,basedongeneralrelativity,which
appearedtobestaticbutwhichproducedaredshiftproportionaltothe
distance.
(v)discoveredthattheorbitalperiodsoftheplanetsareproportionaltothe
3/2powerofthesemi-majoraxisoftheirellipticalorbits.
Discussion:(i)isfalseinpartbecausedeSitterwasnotinvolvedinthemea-
surementofthesizeoftheMilkyWay,butthemostobviouserrorisinthesize
oftheMilkyWay.ItsactualdiameterisreportedbyWeinberg(p.16)tobe
about100,000light-years,althoughnowitisbelievedtobeabouttwicethat
large.(ii)isanaccuratedescriptionofanobservationbyEdwinHubblein
1923(Weinberg,pp.19-20).(iii)describestheworkofCharlesMessierin1781
(Weinberg,p.17).(v)isofcourseoneofKepler'slawsofplanetarymotion.
(c)In1964{65,ArnoA.PenziasandRobertW.Wilsonobserveda uxofmi-
crowaveradiationcomingfromalldirectionsinthesky,whichwasinterpreted
byagroupofphysicistsataneighboringinstitutionasthecosmicbackground
radiationleftoverfromthebigbang.Circlethetwoitemsonthefollowinglist
thatwerenotpartofthestorybehindthisspectaculardiscovery:
8.286QUIZ1REVIEW
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(i)BellTelephoneLaboratory
(ii)MIT
(iii)PrincetonUniversity
(iv)pigeons
(v)groundhogs
(vi)Hubble'sconstant
(vii)liquidhelium
(viii)7.35cm
(Grading:3ptsfor1correctanswer,5for2correctanswers,and-2foreach
incorrectanswer,buttheminimumscoreiszero.)
Discussion:Thediscoveryofthecosmicbackgroundradiationwasdescribed
insomedetailbyWeinberginChapter3.TheobservationwasdoneatBell
TelephoneLaboratories,inHolmdel,NewJersey.Thedetectorwascooledwith
liquidheliumtominimizeelectricalnoise,andthemeasurementsweremadeat
awavelengthof7.35cm.Duringthecourseoftheexperimenttheastronomers
hadtoejectapairofpigeonswhowereroostingintheantenna.Penziasand
Wilsonwerenotinitiallyawarethattheradiationtheydiscoveredmighthave
comefromthebigbang,butBernardBurkeofMITputthemintouchwith
agroupatPrincetonUniversity(RobertDicke,JamesPeebles,P.G.Roll,and
DavidWilkinson)whowereactivelyworkingonthishypothesis.
(d)ImportantpredictionsoftheCopernicantheorywerecon�rmedbythediscov-
eryoftheaberrationofstarlight(whichshowedthatthevelocityoftheEarth
hasthetime-dependenceexpectedforrotationabouttheSun)andbythebe-
havioroftheFoucaultpendulum(whichshowedthattheEarthrotates).These
discoveriesweremade
(i)duringCopernicus'lifetime.
(ii)approximatelytwoandthreedecadesafterCopernicus'death,respectively.
(iii)aboutonehundredyearsafterCopernicus'death.
(iv)approximatelytwoandthreecenturiesafterCopernicus'death,respec-
tively.
Rydendiscussesthisonp.5.Theaberrationofstarlightwasdiscoveredin
1728,whiletheFoucaultpendulumwasinventedin1851.
(e)IfoneaveragesoversuÆcientlylargescales,theuniverseappearstobeho-
mogeneousandisotropic.Howlargemusttheaveragingscalebebeforethis
homogeneityandisotropysetin?
(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),
andtristhetimefoundinpart(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):
Thephysicaldistanceatthetimeofreceptionisfoundbymultiplyingbythe
scalefactoratthetimeofreception,so
`p (tr )=a(tr )x(tr )=
cze�tr
�(1+z)=
cz�:
PROBLEM
5:\DID
YOU
DO
THEREADING?"
(a)Thedistinguishingquantityis��=�c .Theuniverseisopenif<1, atif
=1,orclosedif>1.
(b)Thetemperatureofthemicrowavebackgroundtodayisabout3Kelvin.(The
bestdeterminationtodate*wasmadebytheCOBEsatellite,whichmeasured
thetemperatureas2:728�0:004Kelvin.Theerrorhereisquotedwitha
95%
con�dencelimit,whichmeansthattheexperimentersbelievethatthe
probabilitythatthetruevalueliesoutsidethisrangeisonly5%.)
(c)Thecosmicmicrowavebackgroundisobservedtobehighlyisotropic.
(d)ThedistancetotheAndromedanebulaisroughly2millionlightyears.
(e)1929.
(f)2billionyears.Hubble'svalueforHubble'sconstantwashighbymodern
standards,byafactorof5to10.
(g)Theabsoluteluminosity(i.e.,thetotallightoutput)ofaCepheidvariable
starappearstobehighlycorrelatedwiththeperiodofitspulsations.This
correlationcanbeusedtoestimatethedistancetotheCepheid,bymeasuring
theperiodandtheapparentluminosity.Fromtheperiodonecanestimatethe
absoluteluminosityofthestar,andthenoneusestheapparentluminosityand
the1=r2
lawfortheintensityofapointsourcetodeterminethedistancer.
(h)107light-years.
(i)ArnoA.PenziasandRobertW.Wilson,BellTelephoneLaboratories.
(j)PrincetonUniversity.
*AstrophysicalJournal,vol.473,p.576(1996):TheCosmicMicrowaveBack-
groundSpectrumfromtheFullCOBEFIRASDataSets,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:AFLATUNIVERSEWITH
UNUSUALTIMEEVOLU-
TION
Thekeytothisproblemistoworkincomovingcoordinates.
[Somestudentshaveaskedmewhyonecannotuse\physical"coordinates,for
whichthecoordinatesreallymeasurethephysicaldistances.Inprincipleonecan
useanycoordinatesystemonlikes,butthecomovingcoordinatesarethesimplest.
InanyothersystemitisdiÆculttowritedownthetrajectoryofeitheraparticle
oralight-beam.Incomovingcoordinatesitiseasytowritethetrajectoryofeither
alightbeam,oraparticlewhichismovingwiththeexpansionoftheuniverse(and
hencestandingstillinthecomovingcoordinates).Note,bytheway,thatwhenone
saysthataparticleisstandingstillincomovingcoordinates,onehasnotreallysaid
verymuchaboutit'strajectory.Onehassaidthatitismovingwiththematter
which�llstheuniverse,butonehasnotsaid,forexample,howthedistancebetween
theparticleandoriginvarieswithtime.Theanswertothislatterquestionisthen
determinedbytheevolutionofthescalefactor,a(t).]
(a)Thephysicalseparationatto
isgivenbythescalefactortimesthecoordi-
natedistance.Thecoordinatedistanceisfoundbyintegratingthecoordinate
velocity,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)timesthe
coordinatedistance.Thecoordinatedistanceattimetisequaltothestarting
coordinatedistance,`c (te ),minusthecoordinatedistancethatthelightpulse
travelsbetweentimeteandtimet.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:ANOTHER
FLAT
UNIVERSEWITH
AN
UNUSUAL
TIMEEVOLUTION
(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)Thecoordinatest0
andtearecosmictimecoordinates.The\look-
back"timeasde�nedintheexamisthentheintervalt0 �te .Wecanwrite
thisas
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 ),is
foundfrom
`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)aswell
as,
H0=
_a(t0 )
a(t0 )=
t0
:
Finallythen,
`p (t0 )=cH�1
0
1� h1�(1+z) �
1 i:
d)(10points)Anearlyidenticalproblem
wasworkedthroughinProblem8of
ProblemSet1.
Theenergyoftheobservedphotonswillberedshiftedbyafactorof(1+z).In
additiontherateofarrivalofphotonswillberedshiftedrelativetotherateof
photonemmission,reducingthe uxbyanotherfactorof(1+z).Consequently,
theobservedpowerwillberedshiftedbytwofactorsof(1+z)toP=(1+z)2.
8.286QUIZ1REVIEW
PROBLEM
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Imagineahypotheticalsphereincomovingcoordinatesasdrawnabove,cen-
teredontheradiatingobject,withradiusequaltothecomovingdistance`c .
Nowconsiderthephotonspassingthroughapatchofthespherewithphysical
areaA.IncomovingcoordinatesthepresentareaofthepatchisA=a(t0 )2.
Sincetheobjectradiatesuniformlyinalldirections,thepatchwillintercept
afraction(A=a(t0 )2)=(4�`2c )ofthephotonspassingthroughthesphere.Thus
thepowerhittingtheareaAis
(A=a(t0 )2)
4�`2c
P
(1+z)2
:
Theradiationenergy uxJ,whichisthereceivedpowerperarea,reachingthe
earthisthengivenby
J=
1
4�`p (t0 )2
P
(1+z)2
whereweused`p (t0 )=a(t0 )`c .Usingtheresultofpart(c)towriteJinterms
ofP;H0 ;z;and gives,
J=
H20
4�c2 �1�
�2
P
(1+z)2 h1�(1+z) �
1 i2
:
e)(10points)FollowingthesolutionofProblem
1ofProblem
Set1,wecan
introducea�ctitiousrelaystationthatisatrestrelativetothegalaxy,but
8.286QUIZ1REVIEW
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locatedjustnexttothejet,betweenthejetandEarth.Asintheprevious
solution,therelaystationsimplyrebroadcaststhesignalitreceivesfromthe
source,atexactlytheinstantthatitreceivesit.Therelaystationtherefore
hasnoe�ectonthesignalreceivedbytheobserver,butallowsustodividethe
problemintotwosimpleparts.
Thedistancebetweenthejetandtherelaystationisveryshortcomparedto
cosmologicalscales,sothee�ectoftheexpansionoftheuniverseisnegligible.
Forthispartoftheproblemwecanusespecialrelativity,whichsaysthatthe
periodwithwhichtherelaystationmeasuresthereceivedradiationisgivenby
�trelaystation= s1�vc
1+vc
��tsource:
NotethatIhaveusedtheformulafrom
thefrontoftheexam,butIhave
changedthesizeofv,sincethesourceinthiscaseismovingtowardtherelay
station,sothelightisblue-shifted.ToobserversonEarth,therelaystationis
justasourceatrestinthecomovingcoordinatesystem,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,Ifoundthatasub-
stantialnumberofstudentswrotesolutionsbasedontheincorrectassumption
thattheDopplershiftcouldbetreatedasifitwereentirelyduetomotion.
ThesestudentsusedthespecialrelativityDopplershiftformulatoconvert
theredshiftzofthegalaxytoavelocityofrecession,thensubtractedfrom
thisthespeedvofthejet,andthenagainusedthespecialrelativityDoppler
shiftformulato�ndtheDopplershiftcorrespondingtothiscompositevelocity.
8.286QUIZ1REVIEW
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However,asdiscussedattheendofLectureNotes3,thecosmologicalDoppler
shiftisgivenby
1+z��to
�te=a(to )
a(te );
(3.11)
andisnotpurelyane�ectcausedbymotion.Itisreallythecombinede�ect
ofthemotionofthedistantgalaxiesandthegravitational�eldthatexists
betweenthegalaxies,sothespecialrelativityformularelatingztovdoesnot
apply.
PROBLEM
8:DID
YOU
DO
THEREADING?
a)Thelinesweredark,causedbyabsorptionoftheradiationinthecooler,outer
layersofthesun.
b)IndividualstarsintheAndromedaNebulawereresolvedbyHubblein1923.
[Theothernamesanddatesarenotwithoutsigni�cance.In1609Galileo
builthis�rsttelescope;during1609-10heresolvedtheindividualstarsofthe
MilkyWay,andalsodiscoveredthatthesurfaceofthemoonisirregular,that
Jupiterhasmoonsofitsown,thatSaturnhashandles(laterrecognizedas
rings),thatthesunhasspots,andthatVenushasphases.In1755Immanuel
KantpublishedhisUniversalNaturalHistoryandTheoryoftheHeavens,in
whichhesuggestedthatatleastsomeofthenebulaearegalaxieslikeourown.
In1912HenriettaLeavittdiscoveredtherelationshipbetweentheperiodand
luminosityofCepheidvariablestars.Inthe1950sWalterBaadeandAllan
Sandagerecalibratedtheextra-galacticdistancescale,reducingtheaccepted
valueoftheHubbleconstantbyaboutafactorof10.]
c)
(i)True.[In1941,A.McKellardiscoveredthatcyanogencloudsbehaveasif
theyarebathedinmicrowaveradiationatatemperatureofabout2.3 ÆK,
butnoconnectionwasmadewithcosmology.]
(ii)False.[Anyradiationre ectedbythecloudsisfartooweaktobedetected.
Itisthebrightstarlightshiningthroughthecloudthatisdetectable.]
(iii)True.[Electromagneticwavesatthesewavelengthsaremostlyblocked
bytheEarth'satmosphere,sotheycouldnotbedetecteddirectlyuntil
highaltitudeballoonsandrocketswereintroducedintocosmicbackground
radiationresearchinthe1970s.Precisedatawasnotobtaineduntilthe
COBEsatellite,in1990.]
(iv)True.[ThemicrowaveradiationcanboosttheCNmoleculefromitsground
statetoalow-lyingexcitedstate,astateinwhichtheCandNatoms
rotateabouteachother.Thepopulationofthislow-lyingstateistherefore
8.286QUIZ1REVIEW
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determinedbytheintensityofthemicrowaveradiation.Thispopulation
ismeasuredbyobservingtheabsorptionofstarlightpassingthroughthe
clouds,sincethereareabsorptionlinesinthevisiblespectrumcausedby
transitionsbetweenthelow-lyingstateandhigherenergyexcitedstates.]
(v)False.[Nochemicalreactionsareseen.]
d)Aristarchus.[TheheliocentricpicturewasneveracceptedbyotherGreek
philosophers,however,andwasnotreviveduntilthepublicationofDeRevo-
lutionibusOrbiumCoelestium(OntheRevolutionsoftheCelestialSpheres)by
Copernicusin1543.]
e)(ii)Anypatchofthenightskywouldlookasbrightasthesurfaceofthesun.
[Explanation:Thecruxoftheargumentisthatthebrightnessofanobject,
measuredforexamplebythepowerperarea(i.e., ux)hittingtheretinaof
youreye,doesnotchangeastheobjectismovedfurtheraway.Thepower
fallso�withthesquareofthedistance,butsodoestheareaoftheimageon
yourretina|
sothepowerperareaisindependentofdistance.Underthe
assumptionsstated,yourlineofsightwilleventuallyhitastarnomatterwhat
directionyouarelooking.Theenergy uxonyourretinawillthereforebethe
sameasintheimageofthesun,sotheentireskywillappearasbrightasthe
surfaceofthesun.]
PROBLEM
9:A
FLATUNIVERSEWITH
a(t)/
t3=5
a)Ingeneral,theHubbleconstantisgivenbyH=_a=a,wheretheoverdotdenotes
aderivativewithrespecttocosmictimet.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:
8.286QUIZ1REVIEW
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c)Thecoordinatespeedoflightisc=a(t),sothecoordinatedistancethatlight
travelsbetweentA
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 #:
`p (tB)=a(tB)`c=
52ctB "1� �tA
tB �
2=5 #
:
e)Letteqbethetimeatwhichthelightpulseisequidistantfromthetwogalaxies.
Atthistimeitwillhavetraveledacoordinatedistance`c =2,where`cisthe
answertopart(c).Sincethecoordinatespeedisc=a(t),thetimeteq
canbe
foundfrom:
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'sconstanttimesthe
physicaldistance.Bycombiningtheanswerstoparts(a)and(d),onehas
v=H(tA)`p (tA)
=
35tA
52ctA "�tBt
A �2=5�
1 #=
32c "�tBt
A �2=5�
1 #:
8.286QUIZ1REVIEW
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p.39
g)Theredshiftforradiationobservedattimetcanbewrittenas
1+z=
a(t)
a(te );
whereteisthetimethattheradiationwasemitted.Solvingforte ,
te=
t
(1+z)5=3
:
Asfoundinpart(d),thephysicaldistancethatthelighttravelsbetweente
andt,asmeasuredattimet,isgivenby
`p (t)=a(t) Z
tte
ca(t 0)dt 0=52
ct "1� �tet �2=5 #
:
Substitutingtheexpressionforte ,onehas
`p (t)=52
ct �1�
1
(1+z)2=3 �:
Asz!1,thisexpressionapproaches
limz!1`p (t)=52
ct;
whichisexactlyequaltothehorizondistance.Itisageneralrulethatthe
horizondistancecorrespondstoin�niteredshiftz.
h)Againwewillviewtheproblem
incomovingcoordinates.PutgalaxyBat
theorigin,andgalaxyAatacoordinatedistance`calongthex-axis.Drawa
sphereofradius`c ,centeredgalaxyA.AlsodrawadetectorongalaxyB,with
8.286QUIZ1REVIEW
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physicalareaA(measuredatthepresenttime).
Theenergyfromthequasarwillradiateuniformlyonthesphere.Thedetector
hasaphysicalareaA,sointhecomovingcoordinatepictureitsareainsquare
notcheswouldbeA=a(tB)2.Thedetectorthereforeoccupiesafractionofthe
spheregivenby
[A=a(tB)2]
4�`2c
=
A
4�`p (tB)2
;
sothisfractionoftheemittedphotonswillstrikethedetector.
Nextconsidertherateofarrivalofthephotonsatthesphere.Inlecturewe
�guredoutthatifaperiodicwaveisemittedattimetA
andobservedattime
tB,thentherateofarrivalofthewavecrestswillbeslowerthantherateof
emissionbyaredshiftfactor1+z=a(tB)=a(tA).Thesameargumentwill
applytotherateofarrivalofphotons,sotherateofphotonarrivalatthe
spherewillbeslowerthantherateofemissionbythefactor1+z,reducingthe
energy uxbythisfactor.Inaddition,eachphotonisredshiftedinfrequency
by1+z.Sincetheenergyofeachphotonisproportionaltoitsfrequency,the
energy uxisreducedbyanadditionalfactorof1+z.Thus,therateatwhich
energyreachesthedetectoris
Powerhittingdetector=
A
4�`p (tB)2
P
(1+z)2
:
TheredshiftzofthelightpulsereceivedatgalaxyBisgivenby
1+z=a(tB)
a(tA)= �tBt
A �3=5
:
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
p.41
Usingoncemoretheexpressionfor`P(tB)frompart(d),onehas
J=Powerhittingdetector
A
=
P(tA=tB)6=5
25�c2t2B �1� �tA
tB �
2=5 �2
:
TheproblemiswordedsothattA,andnotz,isthegivenvariablethatdeter-
mineshowfargalaxyAisfromgalaxyB.Inpractice,however,itisusually
moreusefultoexpresstheanswerintermsoftheredshiftzofthereceived
radiation.Onecandothisbyusingtheaboveexpressionfor1+ztoeliminate
tA
infavorofz,�nding
J=
P
25�c2t2B(1+z)2=3 �(1+z)2=3�1 �2
:
i)Lett 0AbethetimeatwhichthelightpulsearrivesbackatgalaxyA.Thepulse
mustthereforetravelacoordinatedistance`c(theanswertopart(c))between
timetB
andt 0A,so
Zt0A
tB
ca(t 0)dt 0=`c:
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?
a)Einsteinbelievedthattheuniversewasstatic,andthecosmologicaltermwas
necessarytopreventastaticuniversefromcollapsingundertheattractiveforce
ofnormalgravity.[Therepulsivee�ectofacosmologicalconstantgrowslin-
earlywithdistance,soifthecoeÆcientissmallitisimportantonlywhenthe
separationsareverylarge.Suchatermcanbeimportantcosmologicallywhile
stillbeingtoosmalltobedetectedbyobservationsofthesolarsystemoreven
thegalaxy.Recentmeasurementsofdistantsupernovas(z�1),whichyou
8.286QUIZ1REVIEW
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mayhavereadaboutinthenewspapers,makeitlooklikemaybethereisa
cosmologicalconstantafterall!Sincethecosmologicalconstantisthehotissue
incosmologythisseason,wewillwanttolookatitmorecarefully.Thebest
timewillbeafterLectureNotes7.]
b)Atthetimeofitsdiscovery,deSitter'smodelwasthoughttobestatic[although
itwasknownthatthemodelpredictedaredshiftwhich,atleastfornearby
galaxies,wasproportionaltothedistance].Fromamodernperspectivethe
modelisthoughttobeexpanding.
[Itseemsstrangethatphysicistsin1917couldnotcorrectlydetermineif
thetheorydescribedauniversethatwasstaticorexpanding,butthemath-
ematicalformalism
ofgeneralrelativitycanberatherconfusing.Thebasic
problemisthatwhenspaceisnotEuclideanthereisnosimplewaytoassign
coordinatestoit.Themathematicsofgeneralrelativityisdesignedtobevalid
foranycoordinatesystem,buttheunderlyingphysicscansometimesbeob-
scuredbyapeculiarchoiceofcoordinates.Achangeofcoordinatescannot
onlydistorttheapparentgeometryofspace,butitcanalsomixupspaceand
time.ThedeSittermodelwas�rstwrittendownincoordinatesthatmadeit
lookstatic,soeveryonebelieveditwas.LaterArthurEddingtonandHermann
Weyl(independently)calculatedthetrajectoriesoftestparticles,discovering
thatthey ewapart.]
c)n1=3,andn2=4.
d)Above3,000Ktheuniversewassohotthattheatomswereionized,dissociated
intonucleiandfreeelectrons.Ataboutthistemperature,however,theuniverse
wascoolenoughsothatthenucleiandelectronscombinedtoform
neutral
atoms.
[Thisprocessisusuallycalled\recombination,"althoughthepre�x\re-
"istotallyinaccurate,sinceinthebigbangtheorytheseconstituentshad
neverbeenpreviouslycombined.AsfarasIknowthewordwas�rstusedin
thiscontextbyP.J.E.Peebles,soIonceaskedhimwhythepre�xwasused.
Herepliedthatthiswordisstandardterminologyinplasmaphysics,andwas
carriedoverintocosmology.]
[Regardlessofitsname,recombinationwascrucialfortheclumpingof
matterintogalaxiesandstars,becausethepressureofthephotonsintheearly
universewasenormous.Whenthematterwasionized,thefreeelectronsinter-
actedstronglywiththephotons,sothepressureofthesephotonspreventedthe
matterfromclumping.Afterrecombination,however,thematterbecamevery
transparenttoradiation,andthepressureoftheradiationbecameine�ective.]
[Incidentally,atroughlythesametimeasrecombination(withbiguncer-
tainties),themassdensityoftheuniversechangedfrombeingdominatedby
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radiation(photonsandneutrinos)tobeingdominatedbynonrelativisticmat-
ter.Thereisnoknownunderlyingconnectionbetweenthesetwoevents,andit
seemstobesomethingofacoincidencethattheyoccurredataboutthesame
time.Thetransitionfrom
radiation-dominationtomatter-dominationalso
helpedtopromotetheclumpingofmatter,butthee�ectwasmuchweaker
thanthee�ectofrecombination|
becauseoftheveryhighvelocityofphotons
andneutrinos,theirpressureremainedasigni�cantforceevenaftertheirmass
densitybecamemuchsmallerthanthatofmatter.]
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)isgivenbyd
xdt=
ca(t):
SincegalaxiesAandBhavephysicalseparation`0attimet1 ,theircoordinate
separationisgivenby
`c=
`0
bt3=5
1
:
Theradiosignalmustcoverthiscoordinatedistanceinthetimeintervalfrom
t1tot2 ,whichimpliesthatZ
t2
t1
ca(t)dt=
`0
bt3=5
1
:
Usingtheexpressionfora(t)andintegrating,
5c
2b �t2=5
2
�t2=5
1 �=
`0
bt3=5
1
;
8.286QUIZ1REVIEW
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whichcanbesolvedfort2togive
t2= �1+
2`0
5ct1 �
5=2
t1:
c)Themethodisthesameasinpart(b).Thecoordinatedistancebetweenthe
twogalaxiesisunchanged,butthistimethedistancemustbetraversedinthe
timeintervalfromt2tot3 .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,butonecouldalsoreplacet2byusing
theanswertopart(b),whichgives
t3= �1+
4`0
5ct1 �
5=2
t1:
[Alternatively,onecouldhavebeguntheproblembyconsideringthefull
roundtripoftheradiosignal,whichtravelsacoordinatedistance2`cduring
thetimeintervalfromt1tot3 .Theproblemthenbecomesidenticaltopart(b),
exceptthatthecoordinatedistance`cisreplacedby2`c ,andt2isreplacedby
t3 .Oneisledimmediatelytotheanswerintheformofthepreviousequation.]
d)Cosmictimeisde�nedbythereadingofsuitablysynchronizedclockswhichare
eachatrestwithrespecttothematteroftheuniverseatthesamelocation.(For
thisproblemwewillnotneedtothinkaboutthemethodofsynchronization.)
Thus,thecosmictimeintervalbetweenthereceiptofthemessageandthe
8.286QUIZ1REVIEW
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responseisthesameaswhatismeasuredonthegalaxyBclocks,whichis�t.
Theresponseisthereforesentatcosmictimet2+�t.Thecoordinatedistance
betweenthegalaxiesisstill`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,
thentheintervalbetweenasignalsentatt3andasignalsentatt3+�twillbe
receivedwitharedshiftidenticaltothatobservedbetweentwosuccessivecrests
ofawave.Thus,theseparationbetweenthereceiptoftheacknowledgement
andthereceiptoftheresponsewillbeafactor(1+z)timeslongerthanthe
timeintervalbetweenthesendingofthetwosignals,andtherefore
t4 �t3=(1+z)�t+O(�t2)
= �t4
t2+�t �
3=5
�t+O(�t2):
8.286QUIZ1REVIEW
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Sincetheanswercontainsanexplicitfactorof�t,theotherfactorscanbe
evaluatedtozerothorderin�t:
t4 �t3= �t4
t2 �
3=5
�t+O(�t2);
whereto�rstorderin�tthet4inthenumeratorcouldequallywellhavebeen
replacedbyt3 .
Forthosewhopreferthebruteforceapproach,theanswertopart(d)can
beTaylorexpandedinpowersof�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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PROBLEM
12:THEDECELERATION
PARAMETER
Fromthefrontoftheexam,weareremindedthat
�a=�4�3
G�a
and
�_aa �2
=8�3
G��kc2
a2
;
whereadotdenotesaderivativewithrespecttotimet.Thecriticalmassdensity
�cisde�nedtobethemassdensitythatcorrespondstoa at(k=0)universe,so
fromtheequationaboveitfollowsthat
�_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;
8.286QUIZ1REVIEW
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p.48
theinde�niteintegralbecomes
12a2=(const)t+c 0;
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?(25points)
(a)In1826,theastronomerHeinrichOlberwroteapaperonaparadoxregarding
thenightsky.WhatisOlber'sparadox?Whatistheprimaryresolutionofit?
(Ryden,Chapter2,Pages6-8)
Ans:Olber'sparadoxisthatthenightskyappearstobedark,insteadofbeing
uniformlybright.Theprimaryresolutionisthattheuniversehasa�niteage,
andsothelightfrom
starsbeyondthehorizondistancehasnotreachedus
yet.(However,eveninthesteady-statemodeloftheuniverse,theparadox
isresolvedbecausethelightfromdistantstarswillbered-shiftedbeyondthe
visiblespectrum).
(b)WhatisthevalueoftheNewtoniangravitationalconstantGinPlanckunits?
ThePlancklengthisoftheorderof10 �35m,10 �15m,1015m,or1035m?
(Ryden,Chapter1,Page3)
Ans:G=1inPlanckunits,byde�nition.
ThePlancklengthisoftheorderof10 �35m.(Notethatthisanswercouldbe
obtainedbyaprocessofeliminationaslongasyourememberthatthePlanck
lengthismuchsmallerthan10 �15m,whichisthetypicalsizeofanucleus).
(c)WhatistheCosmologicalPrinciple?IstheHubbleexpansionoftheuniverse
consistentwithit?
(Weinberg,Chapter2,Pages21-23;Ryden,Chapter2,Page11)
Ans:TheCosmologicalPrinciplestatesthatthereisnothingspecialaboutour
locationintheuniverse,i.e.theuniverseishomogeneousandisotropic.
Yes,theHubbleexpansionisconsistentwithit(sincethereisnocenterof
expansion).
8.286QUIZ1REVIEW
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(d)Inthe\StandardModel"oftheuniverse,whentheuniversecooledtoabout
3�10aK,itbecametransparenttophotons,andtodayweobservetheseasthe
CosmicMicrowaveBackground(CMB)atatemperatureofabout3�10bK.
Whataretheintegersaandb?
(Weinberg,Chapter3;Ryden,Chapter2,Page22)
a=3,b=0.
(e)Whatdidtheuniverseprimarilyconsistofatabout1/100thofasecondafter
theBigBang?Includeanyconstituentthatisbelievedtohavemadeupmore
than1%ofthemassdensityoftheuniverse.
(Weinberg,Chapter1,Page5)
Ans:Electrons,positrons,neutrinos,andphotons.
PROBLEM
15:SPECIALRELATIVITYDOPPLER
SHIFT(20points)
(a)Theeasiestwaytosolvethisproblemisbyadoubleapplicationofthestandard
special-relativityDopplershiftformula,whichwasgivenonthefrontofthe
exam:
z= s1+�
1���1;
(18.1)
where�=v=c.Rememberingthatthewavelengthisstretchedbyafactor
1+z,we�ndimmediatelythatthewavelengthoftheradiowavereceivedat
Alpha-7isgivenby
�Alpha�7= s1+vs =c
1�vs =c�emitted
:
(18.2)
Thephotonsthatarereceivedbytheobserverareinfactneverreceivedby
Alpha-7,butthewavelengthfoundbytheobserverwillbethesameasif
Alpha-7actedasarelaystation,receivingthephotonsandretransmittingthem
atthereceivedwavelength.So,applyingEq.(18.1)again,thewavelengthseen
bytheobservercanbewrittenas
�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)
8.286QUIZ1REVIEW
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so�nally
z= s1+vo =c
1�vo =c s
1+vs =c
1�vs =c �1:
(18.5)
(b)AlthoughweusedthepresenceofAlpha-7indeterminingtheredshiftzof
Eq.(18.5),theredshiftisnotactuallya�ectedbythespacestation.Sothe
special-relativityDopplershiftformula,Eq.(18.1),mustdirectlydescribethe
redshiftresultingfromtherelativemotionofthesourceandtheobserver.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 ,sotherestisjust
algebra.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.
Notethatitguaranteesthatjvtot j�caslongasjvo j�candjvs j�c.
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PROBLEM
16:DID
YOU
DO
THEREADING?(25points)
(a)(4points)Whatwasthe�rstexternalgalaxythatwasshowntobeatadistance
signi�cantlygreaterthanthemostdistantknownobjectsinourgalaxy?How
wasthedistanceestimated?
Ans:(Weinberg,page20)The�rstgalaxyshowntobeatadistancebeyondthe
sizeofourgalaxywasAndromeda,alsoknownbyitsMessiernumber,M31.
Itisthenearestspiralgalaxytoourgalaxy.Thedistancewasdetermined
(byHubble)usingCepheidvariablestars,forwhichtheabsoluteluminosityis
proportionaltotheperiod.AmeasurementofaparticularCepheid'speriod
determinesthestar'sabsoluteluminosity,which,comparedtothemeasured
luminosity,determinesthedistancetothestar.(Hubble'sinitialmeasurement
ofthedistancetoAndromedausedabadly-calibratedversionofthisperiod-
luminosityrelationshipandconsequentlyunderestimatedthedistancebymore
thanafactoroftwo;nonetheless,theinitialmeasurementstillshowedthat
theAndromedaNebulawasanorderofmagnitudemoredistantthanthemost
distantknownobjectsinourowngalaxy.)
(b)(5points)Whatisrecombination?Didgalaxiesbegintoformbeforeorafter
recombination?Why?
Ans:(Weinberg,pages64and73)Recombinationreferstotheformationof
neutralatomsoutofchargednucleiandelectrons.Galaxiesbegantoform
afterrecombination.Priortorecombination,thestrongelectromagneticinter-
actionsbetweenphotonsandmatterproducedahighpressurewhiche�ectively
counteractedthegravitationalattractionbetweenparticles.Oncetheuniverse
becametransparenttoradiation,thematternolongerinteractedsigni�cantly
withthephotonsandconsequentlybegantoundergogravitationalcollapseinto
largeclumps.
(c)(4points)InChapterIVofhisbook,Weinbergdevelopsa\recipeforahot
universe,"inwhichthematteroftheuniverseisdescribedasagasinthermal
equilbriumataveryhightemperature,inthevicinityof109K(severalthou-
sandmilliondegreesKelvin).Suchathermalequilibrium
gasiscompletely
describedbyspecifyingitstemperatureandthedensityoftheconservedquan-
tities.Whichofthefollowingisonthislistofconservedquantities?Circleas
manyasapply.
(i)baryonnumber
(ii)energyperparticle
(iii)protonnumber
(iv)electriccharge
(v)pressure
Ans:(Weinberg,page91)Thecorrectanswersare(i)and(iv).Athirdcon-
servedquantity,leptonnumber,wasnotincludedinthemultiple-choiceoptions.
(d)(4points)ThewavelengthcorrespondingtothemeanenergyofaCMB(cosmic
microwavebackground)photontodayisapproximatelyequaltowhichofthe
8.286QUIZ1REVIEW
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followingquantities?(Youmaywishtolookupthevaluesofvariousphysical
constantsattheendofthequiz.)
(i)2fm(2�10 �15m)
(ii)2microns(2�10 �6m)
(iii)2mm(2�10 �3m)
(iv)2m.
Ans:(Ryden,page23)Thecorrectansweris(iii).
Ifyoudidnotrememberthisnumber,youcouldestimatetheanswerbyremem-
beringthatthecharacteristictemperatureofthecosmicmicrowavebackground
isapproximately3Kelvin.Thetypicalphotonenergyisthenontheorderof
kT,fromwhichwecan�ndthefrequencyasE=h�.Thewavelengthofthe
photonisthen�=�=c.Thisapproximationgives�=5:3mm,whichisnot
equaltothecorrectanswer,butitismuchclosertothecorrectanswerthanto
anyoftheotherchoices.
(e)(4points)Whatistheequivalenceprinciple?
Ans:(Ryden,page27)Initssimplestform,theequivalenceprinciplesaysthat
thegravitationalmassofanobjectisidenticaltoitsinertialmass.Thisequality
impliestheequivalentstatementthatitisimpossibletodistinguish(without
additionalinformation)betweenanobserverinareferenceframeaccelerating
withacceleration~aandanobserverinaninertialreferenceframesubjecttoa
gravitationalforce�mobs ~a.
(Actually,whattheequivalenceprinciplereallysaysisthattheratioofthe
gravitationaltoinertialmassesmg =miisuniversal,thatis,independentofthe
materialpropertiesoftheobjectinquestion.Theratiodoesnotnecessarily
needtobe1.However,onceweknowthatthetwotypesofmassesarepro-
portional,wecansimplyde�nethegravitationalcouplingG
tomakethem
equal.Toseethis,consideratheoryofgravitywheremg =mi=q.Thenthe
gravitationalforcelawis
mi a=�GMmg
r2
;
or
a=�GqM
r2
:
Atthispoint,ifwede�neG0=Gq,wehaveagravitationaltheorywithgravi-
tationalcouplingG0andinertialmassequaltogravitationalmass.)
(f)(4points)WhyisitdiÆcultforEarth-basedexperimentstolookatthesmall
wavelengthportionofthegraphofCMBenergydensityperwavelengthvs.
wavelength?
8.286QUIZ1REVIEW
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Ans:(Weinberg,page67)TheEarth'satmosphereisincreasinglyopaquefor
wavelengthshorterthan.3cm.Therefore,radiationatthesewavelengthswill
beabsorbedandrescatteredbytheEarth'satmosphere;observationsofthe
cosmicmicrowavebackgroundatsmallwavelengthsmustbeperformedabove
theEarth'satmosphere.
PROBLEM
17:
TRACING
A
LIGHT
PULSE
THROUGH
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:
(b)Ifthesourceisatthehorizondistance,itmeansthataphotonleavingthe
sourceatt=0wouldjustbereachingtheoriginattf .So,te=0.
(c)Thecoordinatedistancebetweenthesourceandtheoriginisthecoordinate
horizondistance,givenby
`c;horizon= Z
tf
0
cbt1=2dt=
2ct1=2
fb
:
(d)Thephotonstartsatcoordinatedistance2c ptf=b,andbytimetitwillhave
traveledacoordinatedistanceZ
t0
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 �:
8.286QUIZ1REVIEW
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(e)To�ndthemaximumof`p (t),wedi�erentiateitandsetthederivativetozero:
d`p
dt= rtft�2 !c;
sothemaximumoccurswhen
rtf
tmax
=2;
or
tmax=14
tf:
PROBLEM
18:TRANSVERSEDOPPLER
SHIFTS
(a)Describingtheeventsinthecoordinatesystemshown,theXanthuisatrest,
soitsclocksrunatthesamespeedasthecoordinatesystemtimevariable,t.
Theemissionofthewavecrestsoftheradiosignalarethereforeseparatedbya
timeintervalequaltothetimeintervalasmeasuredbythesource,theXanthu:
�t=�ts:
SincetheEmmeracismovingperpendiculartothepathoftheradiowaves,
atthemomentofreceptionitsdistancefrom
theXanthuisataminimum,
andhenceitsrateofchangeiszero.Hencesuccessivewavecrestswilltravel
thesamedistance,aslongasc�t�a.Sincethewavecreststravelthesame
distance,thetimeseparationoftheirarrivalattheEmmeracis�t,thesame
asthetimeseparationoftheiremission.TheclocksontheEmmerac,however,
andrunningslowlybyafactorof
=
1
q1�v2
c2
:
Thetimeintervalbetweenwavecrestsasmeasuredbythereceiver,onthe
Emmerac,isthereforesmallerbyafactorof ,
�tr=�ts
:
Thus,thereisablueshift.Theredshiftparameterzisde�nedby
�tr
�ts=1+z;
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
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so
1 =1+z;
or
z=1�
:
Recallthat >1,sozisnegative.
(b)Describingthissituationinthecoordinatesystemshown,thistimethesource
ontheXanthuismoving,sotheclocksatthesourcearerunningslowly.The
timebetweenwavecrests,measuredincoordinatetimet,isthereforelargerby
afactorof than�ts ,thetimeasmeasuredbytheclockonthesource:
�t= �ts:
SincetheradiosignalisemittedwhentheXanthuisatitsminimumsepara-
tionfromtheEmmerac,therateofchangeoftheseparationiszero,soeach
wavecresttravelsthesamedistance(againassumingthatc�t�a).Sincethe
Emmeracisatrest,itsclocksrunatthesamespeedasthecoordinatetimet,
andhencethetimeintervalbetweencrests,asmeasuredbythereceiver,is
�tr=�t= �ts:
Thusthetimeintervalasmeasuredbythereceiverislongerthanthatmeasured
bythesource,andhenceitisaredshift.Theredshiftparameterzisgivenby
1+z=�tr
�ts= ;
so
z= �1:
(c)Theeventsdescribedin(a)canbemadetolookalotliketheeventsdescribed
in(b)bytransformingtoaframeofreferencethatismovingtotherightat
speedv0
|
i.e.,bytransformingtotherestframeoftheEmmerac.Inthis
frametheEmmeracisofcourseatrest,andtheXanthuistravelingonthe
trajectory
(x=�v0 t;y=a;z=0);
asinpart(b).However,justasthetransformationcausesthex-component
ofthevelocityoftheXanthutochangefromzerotoanegativevalue,sothe
x-componentofthevelocityoftheradiosignalwillbetransformedfromzeroto
anegativevalue.Thusinthisframetheradiosignalwillnotbetravelingalong
they-axis,sotheeventswillnotmatchthosedescribedin(b).Thesituations
describedin(a)and(b)arethereforephysicallydistinct(whichtheymustbe
iftheredshiftsaredi�erent,aswecalculatedabove).
8.286QUIZ1REVIEW
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PROBLEM
19:A
TWO-LEVELHIGH-SPEED
MERRY-GO-ROUND
(15points)
(a)Sincetherelativepositionsofallthecarsremain�xedasthemerry-go-round
rotates,eachsuccessivepulsefrom
anygivencartoanyothercartakesthe
sameamountoftimetocompleteitstrip.ThustherewillbenoDopplershift
causedbypulsestakingdi�erentamountsoftime;theonlyDopplershiftwill
comefromtimedilation.
Wewilldescribetheeventsfrom
thepointofviewofaninertialreference
frameatrestrelativetothehubofthemerry-go-round,whichwewillcallthe
laboratoryframe.Thisistheframeinwhichtheproblemisdescribed,inwhich
theinnercarsaremovingatspeedv,andtheoutercarsaremovingatspeed
2v.Inthelaboratoryframe,thetimeintervalbetweenthewavecrestsemitted
bythesource�tLab
S
willbeexactlyequaltothetimeinterval�tLab
O
between
twocrestsreachingtheobserver:
�tLab
O
=�tLab
S
:
Theclocksonthemerry-go-roundcarsaremovingrelativetothelaboratory
frame,sotheywillappeartoberunningslowlybythefactor
1=
1
p1�v2=c2
fortheinnercars,andbythefactor
2=
1
p1�4v2=c2
fortheoutercars.Thus,ifwelet�tS
denotethetimebetweencrestsas
measuredbyaclockonthesource,and�tO
asthetimebetweencrestsas
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
p.57
measuredbyaclockmovingwiththeobserver,thenthesequantitiesarerelated
tothelaboratoryframetimesby
2 �tS=�tLab
S
and 1 �tO
=�tLab
O
:
Tomakesurethatthe -factorsareontherightsideoftheequation,you
shouldkeepinmindthatanytimeintervalshouldbemeasuredasshorteron
themovingclocksthanonthelabclocks,sincetheseclocksappeartorun
slowly.Puttingtogethertheequationsabove,onehasimmediatelythat
�tO
= 2
1�tS
:
Theredshiftzisde�nedby
�tO
�(1+z)�tS
;
so
z= 2
1 �1= s1�v2
c2
1�4v2
c2
�1:
(b)Forthispartoftheproblemisusefultoimaginearelaystationlocatedjustto
therightofcar6inthediagram,atrestinthelaboratoryframe.Therelay
stationrebroadcaststhewavesasitreceivesthem,andhencehasnoe�ecton
thefrequencyreceivedbytheobserver,butservesthepurposeofallowingus
toclearlyseparatetheproblemintotwoparts.
The�rstpartofthediscussionconcernstheredshiftofthesignalasmeasured
bytherelaystation.Thiscalculationwouldinvolveboththetimedilationand
8.286QUIZ1REVIEW
PROBLEM
SOLUTIONS,FALL2011
p.58
achangeinpathlengthsbetweensuccessivepulses,butwedonotneedtodo
it.Itisthestandardsituationofasourceandobservermovingdirectlyaway
fromeachother,asdiscussedattheendofLectureNotes1.TheDopplershift
isgivenbyEq.(1.33),whichwasincludedintheformulasheet.Writingthe
formulaforarecessionspeedu,itbecomes
(1+z)jrelay= s1+uc
1�uc
:
Ifweagainusethesymbol�tS
forthetimebetweenwavecrestsasmeasured
byaclockonthesource,thenthetimebetweenthereceiptofwavecrestsas
measuredbytherelaystationis
�tR
= s1+uc
1�uc
�tS
:
Thesecondpartofthediscussionconcernsthetransmissionfrom
therelay
stationtocar6.Thevelocityofcar6isperpendiculartothedirectionfrom
whichthepulseisbeingreceived,sothisisatransverseDopplershift.Any
changeinpathlengthbetweensuccessivepulsesissecondorderin�t,soitcan
beignored.Theonlye�ectisthereforethetimedilation.Asdescribedinthe
laboratoryframe,thetimeseparationbetweencrestsreachingtheobserveris
thesameasthetimeseparationmeasuredbytherelaystation:
�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:
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
p.59
PROBLEM
20:
SIGNAL
PROPAGATION
IN
A
FLAT
MATTER-
DOMINATED
UNIVERSE(55points)
(a)-(i)Ifwelet`c (t)denotethecoordinatedistanceofthelightsignalfromA,then
wecanmakeuseofEq.(3.8)fromthelecturenotesforthecoordinatevelocity
oflight:
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)
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,whichisdonemosteasilywiththemiddleline
oftheaboveequation:
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,sincethespeedofseparationofthelight
signalatthetimeofemissionisreallyjustalocalmeasurementofthespeed
oflight,whichshouldalwaysgivethestandardvaluec.
8.286QUIZ1REVIEW
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(iii)Atarbitrarilylatetimes,thesecondterminbracketsinEq.(20.4)becomes
negligible,so
d`p;sA
dt
!3c:
(20.6)
Althoughthisanswerislargerthanc,itdoesnotviolaterelativity.Oncethe
signalisfarfromitsoriginitiscarriedbytheexpansionoftheuniverse,and
relativityplacesnospeedlimitontheexpansionoftheuniverse.
(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)
Thecoordinatedistanceisthephysicaldistancedividedbythescalefactor,so
`c;BA
=cH�1(t
1 )
a(t1 )
=
32ct1
bt2=3
1
=3c
2bt1=3
1
:
(20.9)
Sincelighttravelsatacoordinatespeedc=a(t),thelightsignalwillreachgalaxy
Battimet2if
`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
�nds12
t1=3
1
=t1=3
2
�t1=3
1
=)
t1=3
2
=32
t1=3
1
=)
t2=278
t1:
(20.11)
(c)-(i)Physicaldistancesareadditive,soifoneaddsthedistancefromAandthelight
signaltothedistancefromthelightsignaltoB,onegetsthedistancefromA
toB:
`p;sA+`p;sB
=`p;BA
:
(20.12)
8.286QUIZ1REVIEW
PROBLEM
SOLUTIONS,FALL2011
p.61
But`p;BA(t)isjustthescalefactortimesthecoordinateseparation,a(t)`c;BA.
Usingthepreviousrelations(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 ,
andthatitequals(3=2)ct1
att=t1 .Butweareaskedto�ndthespeedof
approach,thenegativeofthederivativeofEq.(20.14):
Speedofapproach=�d`p;sB
dt
=�3ct1=3
1
t �1=3+3c
=
3c "1� �t1t �1=3 #
:
(20.15)
(ii)Atthetimeofemission,t=t1 ,Eq.(20.15)gives
Speedofapproach=0:
(20.16)
Thismakessense,sinceatt=t1
galaxyBisoneHubblelengthfromgalaxy
A,whichmeansthatitsrecessionvelocityisexactlyc.Therecessionvelocity
ofthelightsignalleavingAisalsoc,sotherateofchangeofthedistancefrom
thelightsignaltoBisinitiallyzero.
(iii)Atthetimeofreception,t=t2=(27=8)t1 ,Eq.(20.15)gives
Speedofapproach=c;
(20.17)
whichisexactlywhatisexpected.Asinpart(a)-(ii),thisisalocalmeasure-
mentofthespeedoflight.
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
p.62
(d)To�ndtheredshift,we�rst�ndthetimetBA
atwhichalightpulsemustbe
emittedfromgalaxyBsothatitarrivesatgalaxyAattimet1 .Usingthe
coordinatedistancegivenbyEq.(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)
Theredshiftisgivenby
1+zBA
=
a(t1 )
a(tBA)= �t1
tBA �
2=3
=4:
(20.20)
Thus,
zBA
=3:
(20.21)
(e)ApplyingEuclideangeometrytothetriangleC-A-Bshowsthatthephysical
distancefromCtoB,attimet1 ,is p2cH�1.Thecoordinatedistanceisalso
largerthantheA-Bseparationbyafactorof p2.Thus,
`c;BC
=3 p2c
2b
t1=3
1
:
(20.22)
IfwelettBC
bethetimeatwhichalightpulsemustbeemittedfromgalaxy
BsothatitarrivesatgalaxyCattimet1 ,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)
8.286QUIZ1REVIEW
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p.63
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,wedrawadiagramin
comovingcoordinates,puttingthesourceatthecenterofasphere:
TheenergyfromgalaxyAwillradiateuniformlyoverthesphere.Ifthedetector
hasphysicalareaAD,theninthecomovingcoordinatepictureithascoordinate
areaAD=a2(t
2 ),sincethedetectionoccursattimet2Thefullcoordinatearea
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
p.64
ofthesphereis4�`2c
;BA,sothefractionofphotonsthathitthedetectoris
fraction= �A=a(t2 )2 �
4�`2c
;BA
:
(20.28)
AsinProblem6,thepowerhittingthedetectorisreducedbytwofactorsof
(1+z):onefactorbecausetheenergyofeachphotonisproportionaltothe
frequency,andhenceisreducedbytheredshift,andonemorefactorbecause
therateofarrivalofphotonsisalsoreducedbytheredshiftfactor(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(t2 ):
(20.31)
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)
8.286QUIZ1REVIEW
PROBLEM
SOLUTIONS,FALL2011
p.65
Itisdebatablewhichofthelasttwoexpressionsisthesimplest,soIhaveboxed
bothofthem.Onecouldalsowrite
J=1:380�10 �3
Pc2t21
:
(20.33)
PROBLEM
21:DID
YOU
DO
THEREADING?(25points) y
(a)(10points)TodeterminethedistanceofthegalaxieshewasobservingHubble
usedsocalledstandardcandles.Standardcandlesareastronomicalobjects
whoseintrinsicluminosityisknownandwhosedistanceisinferredbymeasuring
theirapparentluminosity.First,heusedasstandardcandlesvariablestars,
whoseintrinsicluminositycanberelatedtotheperiodofvariation.Quoting
Weinberg'sTheFirstThreeMinutes,chapter2,pages19-20:
In1923EdwinHubblewasforthe�rsttimeabletoresolvetheAndromeda
Nebulaintoseparatestars.Hefoundthatitsspiralarmsincludedafewbright
variablestars,withthesamesortofperiodicvariationofluminosityaswas
alreadyfamiliarforaclassofstarsinourgalaxyknownasCepheidvariables.
Thereasonthiswassoimportantwasthatintheprecedingdecadetheworkof
HenriettaSwanLeavittandHarlowShapleyoftheHarvardCollegeObserva-
toryhadprovidedatightrelationbetweentheobservedperiodsofvariationof
theCepheidsandtheirabsoluteluminosities.(Absoluteluminosityisthetotal
radiantpoweremittedbyanastronomicalobjectinalldirections.Apparent
luminosityistheradiantpowerreceivedbyusineachsquarecentimeterofour
telescopemirror.Itistheapparentratherthantheabsoluteluminositythatde-
terminesthesubjectivedegreeofbrightnessofastronomicalobjects.Ofcourse,
theapparentluminositydependsnotonlyontheabsoluteluminosity,butalso
onthedistance;thus,knowingboththeabsoluteandtheapparentluminosities
ofanastronomicalbody,wecaninferitsdistance.)Hubble,observingtheap-
parentluminosityoftheCepheidsintheAndromedaNebula,andestimating
theirabsoluteluminosityfromtheirperiods,couldimmediatelycalculatetheir
distance,andhencethedistanceoftheAndromedaNebula,usingthesimple
rulethatapparentluminosityisproportionaltotheabsoluteluminosityand
inverselyproportionaltothesquareofthedistance.
Healsousedparticularlybrightstarsasstandardcandles,aswededucefrom
page25:
Returningnowto1929:Hubbleestimatedthedistanceto18galaxiesfrom
theapparentluminosityoftheirbrigheststars,andcomparedthesedistances
withthegalaxies'respectivevelocities,determinedspectroscopicallyfromtheir
Dopplershifts.
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
p.66
Note:sincefromreadingjustthe�rstpartofWeinberg'sdiscussiononecould
beinducedtothinkthatHubbleusedjustCepheidsasstandardcandles,stu-
dentswhomentionedonlyCepheidsgot9pointsoutof10.Infact,however,
HubblewasabletoidentifyCepheidvariablesinonlyafewgalaxies.The
Cepheidswerecrucial,becausetheyservedasacalibrationforthelargerdis-
tances,buttheywerenotinthemselvessuÆcient.
(b)(5points)QuotingWeinberg'sTheFirstThreeMinutes,chapter2,page21:
Wewouldexpectintuitivelythatatanygiventimetheuniverseoughttolook
thesametoobserversinalltypicalgalaxies,andinwhateverdirectionsthey
look.(Here,andbelow,Iwillusethelabel\typical"toindicategalaxiesthatdo
nothaveanylargepeculiarmotionoftheirown,butaresimplycarriedalong
withthegeneralcosmic owofgalaxies.)
Thishypothesisissonatural(at
leastsinceCopernicus)thatithasbeencalledtheCosmologicalPrincipleby
theEnglishastrophysicistEdwardArthurMilne.
SotheCosmologicalprinciplebasicallystatesthattheuniverseappearsasho-
mogeneousandisotropic(onscalesofdistancelargeenough)toanytypicalob-
server,wheretypicalisreferredtoobserverswithsmalllocalmotioncompared
totheexpansion ow.Rydengivesamoregeneralde�nitionofCosmological
Principle,whichisvalidaswell.QuotingRyden'sIntroductiontoCosmology,
chapter2,page11or14(dependingonwhichversion):
However,moderncosmologistshaveadoptedthecosmologicalprinciple,
whichstates:Thereisnothingspecialaboutourlocationintheuniverse.The
cosmologicalprincipleholdstrueonlyonlargescales(of100Mpcormore).
(c)(10points)QuotingagainRyden'sIntroductiontoCosmology,chapter2,page
9or11:
Sayingthattheuniverseisisotropicmeansthattherearenopreferreddirec-
tionsintheuniverse;itlooksthesamenomatterwhichwayyoupointyour
telescope.Sayingthattheuniverseishomogeneousmeansthatthereareno
preferredlocationsintheuniverse;itlooksthesamenomatterwhereyouset
upyourtelescope.
(i)False.Iftheuniverseisisotropicaroundonepointitdoesnotneedtobe
homogeneous.Acounter-exampleisadistributionofmatterwithspherical
symmetry,thatis,withadensitywhichisonlyafunctionoftheradius
butdoesnotdependonthedirection:�(r;�;�)��(r).Inthiscaseforan
observeratthecenterofthedistributiontheuniverselooksisotropicbut
itisnothomogeneous.
(ii)
True.ForthecaseofEuclideangeometryisotropyaroundtwoormore
distinctpointsdoesimplyhomogeneity.Weinbergshowsthisinchapter
2,page24.Considertwoobservers,andtwoarbitrarypointsAandB
8.286QUIZ1REVIEW
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SOLUTIONS,FALL2011
p.67
whichwewouldliketoproveequivalent.Consideracirclethroughpoint
A,centeredonobserver1,andanothercirclethroughpointB,centered
onobserver2.IfCisapointontheintersectionofthetwocircles,then
isotropyaboutthetwoobserversimpliesthatA=CandB
=C,and
henceA=B.(ThisargumentwasgoodenoughforWeinbergandhence
goodenoughtodeservefullcredit,butitisactuallyincomplete:onecan
�ndpointsAandBforwhichthetwocircleswillnotintersect.Onyour
nextproblemsetyouwillhaveachancetoinventabetterproof.)
(d)(2pointsextracredit)False.IfwerelaxthehypothesisofEuclideangeome-
try,thenisotropyaroundtwopointsdoesnotnecessarilyimplyhomogeneity.
Acounter-examplewementionedinclassisatwo-dimensionaluniversecon-
sistingofthesurfaceofasphere.ThinkofthesphereinthreeEuclidean
dimensions,butthemodel\universe"consistsonlyofitstwo-dimensionalsur-
face.Imaginelatitudeandlongitudelinestogivecoordinatestothesurface,
andimagineamatterdistributionthatdependsonlyonlatitude.Thiswould
notbehomogeneous,butitwouldlookisotropictoobserversatboththenorth
andsouthpoles.Whilethisexampledescribesatwo-dimensionaluniverse,
whichthereforecannotbeouruniverse,wewilllearnshortlyhowtoconstruct
athree-dimensionalnon-Euclideanuniversewiththesesameproperties.
ySolutionwrittenbyDanieleBertolini.
PROBLEM
22:THETRAJECTORYOFAPHOTON
ORIGINATING
ATTHEHORIZON
(25points)
(a)Theykeyideaisthatthecoordinatespeedoflightisgivenby
dxd
t=
ca(t);
sothecoordinatedistance(innotches)thatlightcantravelbetweent=0and
now(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:
8.286QUIZ1REVIEW
PROBLEM
SOLUTIONS,FALL2011
p.68
(b)Asstatedinpart(a),thecoordinatedistancethatlightcantravelbetween
t=0andt=t0isgivenby`
c= Z
t0
0
cdt
a(t)=3ct1=3
0b
:
Thus,ifweareattheorigin,att=0thephotonmusthavebeenat
x0=3ct1=3
0b
:
(c)Thephotonstartsatx=x0
att=0,andthentravelsinthenegativex-
directionatspeedc=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 �:
(d)Sincethecoordinatedistancebetweenusandthephotonisx(t),measuredin
notches,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: