Ph - web.mit.edu
Transcript of Ph - web.mit.edu
MASSACHUSETTSINSTITUTEOFTECHNOLOGY
PhysicsDepartment
Physics8.286:TheEarlyUniverse
November22,2020
Prof.AlanGuth
REVIEW
PROBLEMSFOR
QUIZ3
QUIZDATE:Wedneday,December2,2020,duringthenormalclasstime.
COVERAGE:LectureNotes6(pp.12{end)andLectureNotes7.ProblemSets7and
8;StevenWeinberg,TheFirstThreeMinutes,Chapter8andtheAfterword;Barbara
Ryden,IntroductiontoCosmology,Chapters8(TheCosmicMicrowaveBackground)
and10(In ationandtheVeryEarlyUniverse)[FirstEdition:Chapters9and11];
AlanGuth,In ationandtheNewEraofHigh-PrecisionCosmology,
http://web.mit.edu/physics/news/physicsatmit/physicsatmit_
02_
cosmology.pdf.
Oneoftheproblemsonthequizwillbetakenverbatim
(oratleastalm
ost
verbatim)from
eitherthehomeworkassignments,orfrom
thestarred
problemsfrom
thissetofReview
Problems.Thestarredproblemsarethe
onesthatIrecommendthatyoureviewmostcarefully:Problems6,7,11,15,17,
and19.
PURPOSE:Thesereviewproblemsarenottobehandedin,butarebeingmadeavail-
abletohelpyoustudy.Theycomemainlyfromquizzesinpreviousyears.Insome
casesthenumberofpointsassignedtotheproblemonthequizislisted|
inall
suchcasesitisbasedon100pointsforthefullquiz.
Inadditiontothissetofproblems,youwill�ndonthecoursewebpagetheactual
quizzesthatweregivenin1994,1996,1998,2000,2002,2004,2007,2009,2011,
2013,2016,and2018.Therelevantproblemsfromthosequizzeshavemostlybeen
incorporatedintothesereviewproblems,butyoustillmaybeinterestedinlooking
atthequizzes,justtoseehowmuchmaterialhasbeenincludedineachquiz.The
coverageoftheupcomingquizwillnotnecessarilymatchthecoverageofanyofthe
quizzesfrompreviousyears.Thecoverageforeachquizinrecentyearsisusually
describedatthestartofthereviewproblems,asIdidhere.
QUIZLOGISTICS:ThelogisticswillbeidenticaltoQuizzes1and2,exceptofcourse
forthedates.Thequizwillbeclosedbook,nocalculators,nointernet,and85
minuteslong.Iassumethatmostofyouwilltakeitduringourregularclasstimeon
December2,butyouwillhavetheoptionofstartingitanytimeduringa24-hour
windowfrom11:05amESTonDecember2to11:05amESTonThursday,December
3.Ifyouwanttostartlaterthan11:05am12/2/20,youshouldemailmeyourchoice
ofstartingtimeby11:59pmonthenightbeforethequiz(earlierisappreciated).
ThequizwillbecontainedinaPDF�le,whichIwilldistributebyemail.Youwill
eachbeexpectedtospendupto85minutesworkingonit,andthenyouwillupload
youranswerstoCanvasasaPDF�le.Iwon'tplaceanyprecisetimelimitonthe
uploading,becausethetimeneededforscanning,photographing,orwhateverkind
ofprocessingyouaredoingcanvary.Ifyouhavequestionsaboutthemeaningof
8.286QUIZ3REVIEW
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p.2
thequestions,IwillbeavailableonZoomduringtheDecember2classtime,and
wewillarrangeforeitherBrunoormetobeavailablebyemailasmuchaspossible
duringtheotherquiztimes.Ifyouhaveanyspecialcircumstancesthatmightmake
thisprocedurediÆcult,orifyouneedapostponementbeyondthe24-hourwindow,
pleaseletme([email protected])know.
PURPOSEOFTHEREVIEW
PROBLEMS:Thesereviewproblemsarenotto
behandedin,butarebeingmadeavailabletohelpyoustudy.Theycomemainly
fromquizzesinpreviousyears.Insomecasesthenumberofpointsassignedtothe
problemonthequizislisted|
inallsuchcasesitisbasedon100pointsforthefull
quiz.
REVIEW
SESSION
AND
OFFICEHOURS:AreviewsessionandspecialoÆce
hourswillbeheldtohelpyoustudyforthequiz.Detailswillfollow.
QUIZZESFROM
PREVIOUSYEARS:Inadditiontothissetofproblems,you
will�ndonthecoursewebpagetheactualquizzesthatweregivenin1994,1996,
1998,2000,2002,2004,2005,2007,2009,2011,2013,2016,and2018.Therelevant
problemsfromthosequizzeshavemostlybeenincorporatedintothesereviewprob-
lems,butyoustillmaybeinterestedinlookingatthequizzes,mainlytoseehow
muchmaterialhasbeenincludedineachquiz.Thecoverageoftheupcomingquiz
willnotnecessarilymatchexactlythecoveragefromallpreviousyears,butIbelieve
thatallthesereviewproblemswouldbefairproblemsfortheupcomingquiz.The
coverageforeachquizinrecentyearsisusuallydescribedatthestartofthereview
problems,asIdidhere.In2016we�nishedWeinberg'sbookbythetimeofQuiz2,
butotherwisethecoveragehasbeenthesamesince2016.
8.286QUIZ3REVIEW
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p.3
INFORMATION
TO
BEGIVEN
ON
QUIZ:
Forthethirdquiz,thefollowinginformationwillbemadeavailabletoyou:
DOPPLER
SHIFT(De�nition:)
1+z��tobserver
�tsource
=�observer
�source
;
where�tobserverand�tsourcearetheperiodofthewaveasmeasuredby
theobserverandbythesource,respectively,and�observerand�source
arethewavelengthofthewave,asmeasuredbytheobserverandby
thesource,respectively.
DOPPLER
SHIFT(Formotionalongaline):
Nonrelativistic,u=wavespeed,sourcemovingatspeedvawayfrom
observer:
z=v=u
Nonrelativistic,observermovingatspeedvawayfromsource:
z=
v=u
1�v=u
Dopplershiftforlight(specialrelativity),��v=c,wherecisthespeed
oflightandvisthevelocityofrecession,asmeasuredbyeitherthe
sourceortheobserver:
z= s1+�
1���1
COSMOLOGICALREDSHIFT:
1+z��observed
�emitted
=a(tobserved )
a(temitted )
SPECIALRELATIVITY:
TimeDilation.Aclockthatismovingatspeedvrelativetoaninertial
referenceframeappearstoberunningslowly,asmeasuredinthat
frame,byafactor :
�
1
p1��2
;
��v=c
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.4
Lorentz-FitzgeraldContraction.Arodthatismovingalongitslength,
relativetoaninertialframe,appearstobecontracted,asmeasuredin
thatframe,bythesamefactor:
�
1
p1��2
RelativityofSimultaneity.Iftwoclocksthataresynchronizedintheir
ownreferenceframe,andseparatedbyadistance`0intheirownframe,
aremovingtogether,inthedirectionofthelineseparatingthem,at
speedvrelativetoaninertialframe,thenmeasurementsintheinertial
framewillshowthetrailingclockreadinglaterbyanamount
�t=�`0
c
Energy-MomentumFour-Vector:
p�= �Ec
;~p �;~p= m0 ~v;E= m0 c2= q(m0 c2)2+j~pj 2c2;
p2�j~pj 2� �p0 �2
=j~pj 2�E2
c2
=�(m0 c)2
:
KINEMATICSOFAHOMOGENEOUSLY
EXPANDING
UNI-
VERSE:
Hubble'sLaw:v=Hr,
wherev=
recessionvelocityofadistantobject,H
=
Hubble
expansionrate,andr=distancetothedistantobject.
PresentValueofHubbleExpansionRate(Planck2018):
H0=67:66�0:42km-s�
1-Mpc�
1
ScaleFactor:`p (t)=a(t)`c;
where`p (t)isthephysicaldistancebetweenanytwoobjects,a(t)
isthescalefactor,and`cisthecoordinatedistancebetweenthe
objects,alsocalledthecomovingdistance.
HubbleExpansionRate:H(t)=
1a(t)
da(t)
dt
.
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.5
LightRaysinComovingCoordinates:Lightraystravelinstraight
lineswithphysicalspeedcrelativetoanyobserver.InCartesian
coordinates,coordinatespeeddxd
t=
ca(t).Ingeneral,ds2
=
g�� dx�dx�=0:
HorizonDistance:
`p;horizon (t)=a(t) Z
t0
ca(t0)dt0
= �3ct
( at,matter-dominated),
2ct
( at,radiation-dominated).
COSMOLOGICALEVOLUTION:
H2= �_aa �2
=8�3
G��kc2
a2
;
�a=�4�3
G ��+3pc
2 �a;
�m(t)=a3(t
i )
a3(t)�m(ti )(matter);
�r (t)=a4(t
i )
a4(t)�r (ti )(radiation):
_�=�3_aa �
�+
pc2 �;��=�c;where�c=3H2
8�G
:
EVOLUTION
OFA
MATTER-DOMINATED
UNIVERSE:
Flat(k=0):
a(t)/t2=3
=1:
Closed(k>0):
ct=�(��sin�);
apk=�(1�cos�);
=
2
1+cos�>1;
where��4�3G�
c2 �apk �3
:
Open(k<0):
ct=�(sinh���);
ap�=�(cosh��1);
=
2
1+cosh�<1;
where��4�3G�
c2 �ap� �3
;
���k>0:
8.286QUIZ3REVIEW
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p.6
MINKOWSKIMETRIC(SpecialRelativity):
ds2��c2d�2=�c2dt2+dx2+dy2+dz2:
ROBERTSON-WALKER
METRIC:
ds2��c2d�2=�c2dt2+a2(t) �dr2
1�kr2+r2 �d�2+sin2�d�2 � �:
Alternatively,fork>0,wecande�ner=sin p
k,andthen
ds2��c2d�2��c2dt2+~a2(t) �d 2+sin2 �d�2+sin2�d�2 �;
where~a(t)=a(t)= pk.Fork<0wecande�ner=sinh
p�k,andthen
ds2��c2d�2=�c2dt2+~a2(t) �d 2+sinh2 �d�2+sin2�d�2 �;
where~a(t)=a(t)= p�k.Notethat~acanbecalledaifthereisnoneed
torelateittothea(t)thatappearsinthe�rstequationabove.
SCHWARZSCHILD
METRIC:
ds2��c2d�2=� �1�2GM
rc2 �c2dt2+ �1�2GM
rc2 �
�
1dr2
+r2(d�2+sin2�d�2);
GEODESICEQUATION:
dds �gijdxj
ds �=12
(@i gk` )dxk
ds
dx`
ds
or:
dd� �g��dx�
d� �=12
(@�g��)dx�
d�
dx�
d�
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.7
BLACK-BODY
RADIATION:
WheneverkT�mc2foranyparticle,wherekistheBoltzmannconstant,
Tisthetemperature,andm
isthe(rest)massoftheparticle,in
thermalequilibriumtherewillbeablack-bodyradiation,inwhichthe
particlewillmakethefollowingcontributionstotheenergydensity,
massdensity,pressure,numberdensity,andenergydensity:
u=g�2
30
(kT)4
(�hc)3
(energydensity)
p=13
u
�=u=c2
(pressure,massdensity)
n=g�
�(3)
�2
(kT)3
(�hc)3
(numberdensity)
s=g2�2
45
k4T3
(�hc)3
;
(entropydensity)
whereg� (
1perspinstateforbosons(integerspin)
7/8perspinstateforfermions(half-integerspin)
g�� (1perspinstateforbosons
3/4perspinstateforfermions,
and
�(3)=
113+
123+
133+����1:202:
Thevaluesofgandg�
forphotons,neutrinos,andelectron-positronpairs
areasfollows:
g =g� =2;
g�=
78|{z}
Fermion
factor
�
3|{z}
3species
�e;��;�� �
2|{z}
Particle=
antiparticle �
1|{z}
Spinstates
=
214;
g��=
34|{z}
Fermion
factor
�
3|{z}
3species
�e;��;�� �
2|{z}
Particle=
antiparticle �
1|{z}
Spinstates
=
92;
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.8
ge+e�
=
78|{z}
Fermion
factor
�1
|{z}Species �
2|{z}
Particle=
antiparticle �
2|{z}
Spinstates
=
72;
g�e
+e�
=
34|{z}
Fermion
factor
�1
|{z}Species �
2|{z}
Particle=
antiparticle �
2|{z}
Spinstates
=
3:
SpectrumofBlack-BodyRadiation:
Theenergydensityforradiationinthefrequencyintervalbetween�
and�+d�isgivenby
�� (�)d�=8�2g�h�3
c3
1
e2��h�=kT�1d�:
EVOLUTION
OF
A
FLAT
RADIATION-DOMINATED
UNI-
VERSE:
�=
3
32�Gt2
kT= �45�h3c5
16�3gG �
1=4
1pt
Form�=106MeV�kT�me=0:511MeV,g=10:75andthen
kT=
0:860MeV
pt(insec) �
10:75
g �1=4
Afterthefreeze-outofelectron-positronpairs,
T�
T
= �41
1 �1=3
:
COSMOLOGICALCONSTANT:
uvac=�vac c2=
�c4
8�G
;
pvac=��vac c2=��c4
8�G
:
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.9
GENERALIZED
COSMOLOGICALEVOLUTION:
xdxd
t=H0 qm;0 x+rad;0+vac;0 x4+k;0 x2;
where
x�a(t)
a(t0 ) �
11+z;
k;0 ��
kc2
a2(t0 )H20
=1�m;0 �rad;0 �vac;0:
Ageofuniverse:
t0=
1H0 Z
10
xdx
pm;0 x+rad;0+vac;0 x4+k;0 x2
=
1H0 Z
10
dz
(1+z) pm;0 (1+z)3+rad;0 (1+z)4+vac;0+k;0 (1+z)2
:
Look-backtime:
tlook-back (z)=
1H0 Z
z0
dz0
(1+z0) pm;0 (1+z0)3+rad;0 (1+z0)4+vac;0+k;0 (1+z0)2
:
PHYSICALCONSTANTS:
G=6:674�10�
11m3�kg�
1�s�
2=6:674�10�
8cm3�g�
1�s�
2
k=Boltzmann'sconstant=1:381�10�
23joule=K
=1:381�10�
16erg=K
=8:617�10�
5eV=K
�h=
h2�=1:055�10�
34joule�s
=1:055�10�
27erg�s
=6:582�10�
16eV�s
c=2:998�108m/s
=2:998�1010cm/s
�hc=197:3MeV-fm;
1fm=10�
15m
1yr=3:156�107s
1eV=1:602�10�
19joule=1:602�10�
12erg
1GeV=109eV=1:783�10�
27
kg(wherec�1)
=1:783�10�
24g:
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.10
PlanckUnits:ThePlancklength`P,thePlancktimetP,thePlanckmass
mP,andthePlanckenergyEparegivenby
`P
= rG�h
c3
=1:616�10�
35m;
=1:616�10�
33cm;
tP
= r�hGc
5
=5:391�10�
44s;
mP
= r�hcG
=2:177�10�
8kg;
=2:177�10�
5g;
EP
= r�hc5
G
=1:221�1019GeV:
Wedonothaveacompletequantumtheoryofgravity,butweexpect
thePlanckscaletobethescaleatwhichthee�ectsofquantumgravity
becomesigni�cant.Thatis,weexpectthee�ectsofquantumgravity
tobeimportantforprocessesthatinvolvedistancesoforder`P
or
less,timesofordertP
orless,orparticleswithmassesofordermP
or
greater,orenergiesoforderEP
orgreater.
CHEMICALEQUILIBRIUM:
(ThistopicwillNOTbeincludedonQuiz3,buttheformulasare
nonethelessincludedhereforlogicalcompleteness.Theywillberele-
vanttoProblemSet9.)
GeneralIdealGas,RelativisticorNot,BosonsorFermions:
Thenumberdensityofparticlesoftypeiwithmomentawithina
boxofsized3pcenteredat~pisgivenby
ni;~p(~p)d3p=
�gi
(2��h)3
d3p
hexp �Ei (p)�
�i
kT
��1 i;
where�g
i=numberofspinstatesofparticle
Ei (p)= qm2i c4+p2c2=energyofparticlewithmomentump
mi=massofparticle
�i=chemicalpotential
�=+forfermions,and�forbosons.
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.11
Notethatunlikethequantitiesgandg�
de�nedinthesection
onblack-bodyradiation,�gisimplycountsspinstates,withno
correctionfactorassociatedwithfermions.
Chemicalpotentialsareassignedinitiallytoconservedquantities
(e.g.,electriccharge,baryonnumber,orleptonnumber),andare
awayofspecifyinghowmuchofthesequantitiesarepresent.The
chemicalpotentialofanytypeofparticleisthesumofthechemical
potentialsofitsconservedquantities.Forexample,aprotonhas
oneunitofbaryonnumberandoneunitofelectriccharge,so
�p=�baryon+�charge .
Thenumberdensityofparticleiisgivenby
ni=
�gi
(2��h)3 Z
10
4�p2dp
hexp �E(p)�
�i
kT
��1 i;
andtheenergydensityisgivenby
ui=
�gi
(2��h)3 Z
10
4�p2E(p)dp
hexp �E(p)�
�i
kT
��1 i:
IdealDiluteGasofNonrelativisticParticles:
Thenonrelativistic,dilutegaslimitoftheformulaaboveforthe
numberdensityniisgivenby
ni=�gi (2�mi kT)3=2
(2��h)3
e(�i�
mi c2)=kT
:
whereni=numberdensityofparticle
�gi=numberofspinstatesofparticle
mi=massofparticle
�i=chemicalpotential
Theformulaaboveassumesthatthegasisnonrelativistic(kT�
mi c2)anddilute(e(�i�
mi c2)=kT
�1).
Foranyreactionthatisconsistentwithallconservationlaws,the
sumofthe�iontheleft-handsideofthereactionequationmust
equalthesum
ofthe�iontheright-handside.Consequently,
theproductofthenumberdensitiesontheleft-handside,divided
bytheproductofthenumberdensitiesontheright-handside,
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.12
isalwaysindependentofallchemicalpotentials.Forexample,
sinceH+ !p+e�
(hydrogenatom+photon !proton+
electron)isapossiblereaction,�H
+� =�p+�e�
,andtherefore
nH
n
npne�
canbeevaluatedusingtheformulaabovefornumberdensities,
andallchemicalpotentialswillcancelout.(Photonshavenocon-
servedquantities,so� �0,sonH=(npne�
)isalsoindependent
ofanychemicalpotentials.)
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.13
PROBLEM
LIST
1.DidYouDotheReading(2018)?
...............14(Sol:33)
2.DidYouDotheReading(2016)?
...............15(Sol:35)
3.DidYouDotheReading(2013)?
...............17(Sol:37)
4.DidYouDotheReading(2009)?
...............19(Sol:39)
5.DidYouDotheReading(2007)?
...............20(Sol:41)
*6.TimeEvolutionofaUniverseIncludingaHypothetical
KindofMatter
......22(Sol:43)
*7.TheConsequencesofanAlt-Photon
..............22(Sol:46)
8.NumberDensitiesintheCosmicBackgroundRadiation......23(Sol:49)
9.PropertiesofBlack-BodyRadiation
..............23(Sol:50)
10.ANewSpeciesofLepton
...................23(Sol:52)
*11.ANewTheoryoftheWeakInteractions
............24(Sol:55)
12.DoublingofElectrons
....................25(Sol:61)
13.TimeScalesinCosmology
..................26(Sol:63)
14.EvolutionofFlatness.....................26(Sol:63)
*15.TheSloanDigitalSkySurveyz=5:82Quasar..........27(Sol:64)
16.SecondHubbleCrossing
...................28(Sol:70)
*17.TheEventHorizonforOurUniverse..............29(Sol:72)
18.TheE�ectofPressureonCosmologicalEvolution
........30(Sol:74)
*19.TheFreeze-outofaFictitiousParticleX
............31(Sol:76)
20.TheTimeofDecoupling
...................32(Sol:80)
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.14
PROBLEM
1:DID
YOU
DO
THEREADING
(2018)?(20points)
(a)(5points)WhichoneofthefollowingstatementsaboutCMBisNOTcorrect?
(i)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionofthe
observerrelativetoaframeofreferenceinwhichtheCMBisisotropic.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,themeantemperature
averagingovertheskyis<T>=2.725K.
(iii)AfterthedipoledistortionoftheCMBissubtractedaway,thetemperatureof
theCMBvariesby0.3microKelvinacrossthesky.
(iv)ThephotonsoftheCMBhavemostlybeentravelingonstraightlinessincethey
werelastscatteredatt�370;000yr,atalocationcalledthesurfaceoflast
scattering.
(b)(5points)Thenonuniformitiesinthecosmicmicrowavebackgroundallowusto
measuretheripplesinthemassdensityoftheuniverseatthetimewhentheplasma
combinedtoformneutralatoms,about300,000-400,000yearsafterthebigbang.
Theseripplesarecrucialforunderstandingwhathappenedlater,sincetheyarethe
seedswhichledtothecomplicatedtapestryofgalaxies,clustersofgalaxies,and
voids.Whichofthefollowingsentencesdescribeshowtheseripplesarecreatedin
thecontextofin ationarymodels:
(i)Magneticmonopolescanformrandomlyduringthegranduni�edtheoryphase
transition,resultinginnonuniformitiesinthemassdensity.
(ii)Cosmicstrings,whicharelineliketopologicaldefects,canformrandomlyduring
thegranduni�edtheoryphasetransition,resultinginnonuniformitiesinthe
massdensity.
(iii)Theyaregeneratedbyquantum uctuationsduringin ation.
(iv)Sincetheearlyuniversewasveryhot,therewerelargethermal uctuations
whichultimatelyevolvedintotheripplesinthemassdensity.
(c)(5points)InChapter8ofTheFirstThreeMinutes,StevenWeinbergdescribes
thefutureoftheuniverse(assuming,aswasthoughtthentobethecase,thatthe
cosmologicalconstantiszero).Onepossibilitythathediscussesisthatthecosmic
matterdensitycouldbegreaterthanthecriticaldensity.Assumingthatwelivein
suchauniverse,whichofthefollowingstatementsisNOTtrue?
(i)Theuniverseis�niteanditsexpansionwilleventuallycease,givingwaytoan
acceleratingcontraction.
(ii)Threeminutesafterthetemperaturereachesathousandmilliondegrees(109K),
thelawsofphysicsguaranteethattheuniversewillcrunch,andtimewillstop.
8.286QUIZ3REVIEW
PROBLEMS,FALL2020
p.15
(iii)Duringatleasttheearlypartofthecontractingphase,wewillbeabletoobserve
bothredshiftsandblueshifts.
(iv)Whentheuniversehasrecontractedtoone-hundredthitspresentsize,theradi-
ationbackgroundwillbegintodominatethesky,withatemperatureofabout
300K.
(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
redderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
bluerbecauseoftheDopplere�ect.
(iii)Photonstravelingtowardusfromthesurfaceoflastscatteringappearredder
becauseofabsorptionintheintergalacticmedium.
(iv)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappearbluer
becauseofabsorptionintheintergalacticmedium.
(v)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearredder
becausetheymustclimboutofthegravitationalpotentialwell.
(vi)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearbluer
becausetheymustclimboutofthegravitationalpotentialwell.
PROBLEM
2:DID
YOU
DO
THEREADING?(2016)(25points)
Exceptforpart(d),youshouldanswerthesequestionsbycirclingtheonestatementthat
iscorrect.
(a)(5points)IntheEpilogueofTheFirstThreeMinutes,SteveWeinbergwrote:\The
moretheuniverseseemscomprehensible,themoreitalsoseemspointless."
The
sentencewasquali�ed,however,byaclosingparagraphthatpointsoutthat
(i)thequestofthehumanracetocreateabetterlifeforallcanstillgivemeaning
toourlives.
(ii)iftheuniversecannotgivemeaningtoourlives,thenperhapsthereisanafterlife
thatwill.
(iii)thecomplexityandbeautyofthelawsofphysicsstronglysuggestthatthe
universemusthaveapurpose,evenifwearenotawareofwhatitis.
(iv)thee�orttounderstandtheuniversegiveshumanlifesomeofthegraceof
tragedy.
(b)(5points)IntheAfterwordofTheFirstThreeMinutes,Weinbergdiscussesthe
baryonnumberoftheuniverse.(Thebaryonnumberofanysystem
isthetotal
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numberofprotonsandneutrons(andcertainrelatedparticlesknownashyperons)
minusthenumberoftheirantiparticles(antiprotons,antineutrons,antihyperons)
thatarecontainedinthesystem.)Weinbergconcludedthat
(i)baryonnumberisexactlyconserved,sothetotalbaryonnumberoftheuniverse
mustbezero.Whilenucleiinourpartoftheuniversearecomposedofprotons
andneutrons,theuniversemustalsocontainantimatterregionsinwhichnuclei
arecomposedofantiprotonsandantineutrons.
(ii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe
partoftheuniversewecanobserve,andhenceapositivedensityofbaryon
number.Sincebaryonnumberisconserved,thiscanonlybeexplainedby
assumingthattheexcessbaryonswereputinatthebeginning.
(iii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe
partoftheuniversewecanobserve,andhenceapositivedensityofbaryonnum-
ber.Thiscanbetakenasapositivehintthatbaryonnumberisnotconserved,
whichcanhappenifthereexistasyetundetectedheavy\exotic"particles.
(iv)itispossiblethatbaryonnumberisnotexactlyconserved,butevenifthatisthe
case,itisnotpossiblethattheobservedexcessofmatteroverantimattercanbe
explainedbytheveryrareprocessesthatviolatebaryonnumberconservation.
(c)(5points)IndiscussingtheCOBEmeasurementsofthecosmicmicrowaveback-
ground,Rydendescribesadipolecomponentofthetemperaturepattern,forwhich
thetemperatureoftheradiationfromonedirectionisfoundtobehotterthanthe
temperatureoftheradiationdetectedfromtheoppositedirection.
(i)Thisdiscoveryisimportant,becauseitallowsustopinpointthedirectionofthe
pointinspacewherethebigbangoccurred.
(ii)ThisisthelargestcomponentoftheCMBanisotropies,amountingtoa10%
variationinthetemperatureoftheradiation.
(iii)Inadditiontothedipolecomponent,theanisotropiesalsoincludescontributions
fromaquadrupole,octupole,etc.,allofwhicharecomparableinmagnitude.
(iv)ThispatternisinterpretedasasimpleDopplershift,causedbythenetmotion
oftheCOBEsatelliterelativetoaframeofreferenceinwhichtheCMBis
almostisotropic.
(d)(5points)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBis
notcorrect:
(i)AfterthedipoledistortionoftheCMBissubtractedaway,themeantemperature
averagingovertheskyishTi=2:725K.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,therootmeansquare
temperature uctuationis D�ÆTT �2 E1=2
=1:1�10�
3.
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(iii)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionofthe
observerrelativetoaframeofreferenceinwhichtheCMBisisotropic.
(iv)Intheirgroundbreakingpaper,WilsonandPenziasreportedthemeasurement
ofanexcesstemperatureofabout3.5Kthatwasisotropic,unpolarized,and
freefromseasonalvariations.InacompanionpaperwrittenbyDicke,Peebles,
RollandWilkinson,theauthorsinterpretedtheradiationtobearelicofan
early,hot,dense,andopaquestateoftheuniverse.
(e)(5points)In ationisdrivenbya�eldthatisbyde�nitioncalledthein aton�eld.
Instandardin ationarymodels,the�eldhasthefollowingproperties:
(i)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitspotentialenergy.
(ii)Thein atonisavector�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitspotentialenergy.
(iii)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitskineticenergy.
(iv)Thein atonisavector�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitskineticenergy.
(v)Thein atonisatensor�eld,whichisresponsibleforonlyasmallfractionof
theenergydensityoftheuniverseduringin ation.
PROBLEM
3:DID
YOU
DO
THEREADING
(2013)?(35points)
ThiswasProblem1ofQuiz3,2013.
(a)(5points)RydensummarizestheresultsoftheCOBEsatelliteexperimentforthe
measurementsofthecosmicmicrowavebackground(CMB)intheform
ofthree
importantresults.The�rstwasthat,inanyparticulardirectionofthesky,the
spectrum
oftheCMBisveryclosetothatofanidealblackbody.TheFIRAS
instrumentontheCOBEsatellitecouldhavedetecteddeviationsfromtheblackbody
spectrumassmallas��=��10�
n,wherenisaninteger.Towithin�1,whatisn?
(b)(5points)ThesecondresultwasthemeasurementofadipoledistortionoftheCMB
spectrum;thatis,theradiationisslightlyblueshiftedtohighertemperaturesinone
direction,andslightlyredshiftedtolowertemperaturesintheoppositedirection.To
whatphysicale�ectwasthisdipoledistortionattributed?
(c)(5points)Thethirdresultconcernedthemeasurementoftemperature uctuations
afterthedipolefeaturementionedabovewassubtractedout.De�ning
ÆTT(�;�)�T(�;�)�hTi
hTi
;
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wherehTi=2:725K,theaveragevalueofT,theyfoundarootmeansquare uctu-
ation,
*�ÆTT �2 +1=2
;
equaltosomenumber.Towithinanorderofmagnitude,whatwasthatnumber?
(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
redderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
bluerbecauseoftheDopplere�ect.
(iii)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearredder
becausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearbluer
becausetheymustclimboutofthegravitationalpotentialwell.
(v)Photonstravelingtowardusfromthesurfaceoflastscatteringappearredder
becauseofabsorptionintheintergalacticmedium.
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappearbluer
becauseofabsorptionintheintergalacticmedium.
(e)(5points)The atnessproblemreferstotheextreme�ne-tuningthatisneededin
atearlytimes,inorderforittobeascloseto1todayasweobserve.Startingwith
theassumptionthattodayisequalto1withinabout1%,oneconcludesthatat
onesecondafterthebigbang,j
�1jt=1sec<10�
m
;
wheremisaninteger.Towithin�3,whatism?
(f)(5points)Thetotalenergydensityofthepresentuniverseconsistsmainlyofbaryonic
matter,darkmatter,anddarkenergy.Givethepercentagesofeach,accordingto
thebest�tobtainedfromthePlanck2013data.Youwillgetfullcreditifthe�rst
(baryonicmatter)isaccurateto�2%,andtheothertwoareaccuratetowithin�5%.
(g)(5points)Withintheconventionalhotbigbangcosmology(withoutin ation),it
isdiÆculttounderstandhowthetemperatureoftheCMBcanbecorrelatedat
angularseparationsthataresolargethatthepointsonthesurfaceoflastscattering
wasseparatedfrom
eachotherbymorethanahorizondistance.Approximately
whatangle,indegrees,correspondstoaseparationonthesurfacelastscatteringof
onehorizonlength?Youwillgetfullcreditifyouranswerisrighttowithinafactor
of2.
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PROBLEM
4:DID
YOU
DO
THEREADING
(2009)?(25points)
ThisproblemwasProblem1,Quiz3,2009.
(a)(10points)Thisquestionconcernssomenumbersrelatedtothecosmicmicrowave
background(CMB)thatoneshouldneverforget.Statethevaluesofthesenumbers,
towithinanorderofmagnitudeunlessotherwisestated.Inallcasesthequestion
referstothepresentvalueofthesequantities.
(i)TheaveragetemperatureToftheCMB(towithin10%).
(ii)ThespeedoftheLocalGroupwithrespecttotheCMB,expressedasafraction
v=cofthespeedoflight.(ThespeedoftheLocalGroupisfoundbymeasuring
thedipolepatternoftheCMBtemperaturetodeterminethevelocityofthe
spacecraftwithrespecttotheCMB,andthenremovingspacecraftmotion,the
orbitalmotionoftheEarthabouttheSun,theSunaboutthegalaxy,andthe
galaxyrelativetothecenterofmassoftheLocalGroup.)
(iii)Theintrinsicrelativetemperature uctuations�T=T,afterremovingthedipole
anisotropycorrespondingtothemotionoftheobserverrelativetotheCMB.
(iv)Theratioofbaryonnumberdensitytophotonnumberdensity,�=nbary =n .
(v)Theangularsize�H,indegrees,correspondingtowhatwastheHubbledistance
c=H
atthesurfaceoflastscattering.Thisanswermustbewithinafactorof3
tobecorrect.
(b)(3points)Becausephotonsoutnumberbaryonsbysomuch,theexponentialtailof
thephotonblackbodydistributionisimportantinionizinghydrogenwellafterkT
fallsbelowQH
=13:6eV.WhatistheratiokT =QH
whentheionizationfraction
oftheuniverseis1=2?
(i)1=5
(ii)1=50
(iii)10�
3
(iv)10�
4
(v)10�
5
(c)(2points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
redderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
bluerbecauseoftheDopplere�ect.
(iii)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearredder
becausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearbluer
becausetheymustclimboutofthegravitationalpotentialwell.
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(v)Photonstravelingtowardusfromthesurfaceoflastscatteringappearredder
becauseofabsorptionintheintergalacticmedium.
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappearbluer
becauseofabsorptionintheintergalacticmedium.
(d)(10points)Foreachofthefollowingstatements,saywhetheritistrueorfalse:
(i)Darkmatterinteractsthroughthegravitational,weak,andelectromagnetic
forces.
T
orF?
(ii)Thevirialtheoremcanbeappliedtoaclusterofgalaxiesto�nditstotalmass,
mostofwhichisdarkmatter.
T
orF?
(iii)Neutrinosarethoughttocompriseasigni�cantfractionoftheenergydensityof
darkmatter.
T
orF?
(iv)Magneticmonopolesarethoughttocompriseasigni�cantfractionoftheenergy
densityofdarkmatter.
T
orF?
(v)LensingobservationshaveshownthatMACHOscannotaccountforthedark
matteringalactichalos,butthatasmuchas20%ofthehalomasscouldbein
theformofMACHOs.
T
orF?
PROBLEM
5:DID
YOU
DO
THEREADING
(2007)?(25points)
ThefollowingproblemwasProblem1,Quiz3,in2007.Eachpartwasworth5points.
(a)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBisnotcorrect:
(i)AfterthedipoledistortionoftheCMBissubtractedaway,themeantemperature
averagingovertheskyishTi=2:725K.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,therootmeansquare
temperature uctuationis D�ÆTT �2 E1=2
=1:1�10�
3.
(iii)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionofthe
observerrelativetoaframeofreferenceinwhichtheCMBisisotropic.
(iv)Intheirgroundbreakingpaper,WilsonandPenziasreportedthemeasurement
ofanexcesstemperatureofabout3.5Kthatwasisotropic,unpolarized,and
freefromseasonalvariations.InacompanionpaperwrittenbyDicke,Peebles,
RollandWilkinson,theauthorsinterpretedtheradiationtobearelicofan
early,hot,dense,andopaquestateoftheuniverse.
(b)(CMBexperiments)ThecurrentmeanenergyperCMBphoton,about6�10�
4eV,
iscomparabletotheenergyofvibrationorrotationforasmallmoleculesuchasH2 O.
Thusmicrowaveswithwavelengthsshorterthan��3cmarestronglyabsorbedby
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watermoleculesintheatmosphere.TomeasuretheCMBat�<3cm,whichone
ofthefollowingmethodsisnotafeasiblesolutiontothisproblem?
(i)MeasureCMBfromhigh-altitudeballoons,e.g.MAXIMA.
(ii)MeasureCMBfromtheSouthPole,e.g.DASI.
(iii)MeasureCMBfromtheNorthPole,e.g.BOOMERANG.
(iv)MeasureCMBfromasatelliteabovetheatmosphereoftheEarth,e.g.COBE,
WMAPandPLANCK.
(c)(Temperature uctuations)Thecreationoftemperature uctuationsinCMBby
variationsinthegravitationalpotentialisknownastheSachs-Wolfee�ect.Which
oneofthefollowingstatementsisnotcorrectconcerningthise�ect?
(i)ACMBphotonisredshiftedwhenclimbingoutofagravitationalpotentialwell,
andisblueshiftedwhenfallingdownapotentialhill.
(ii)Atthetimeoflastscattering,thenonbaryonicdarkmatterdominatedtheen-
ergydensity,andhencethegravitationalpotential,oftheuniverse.
(iii)Thelarge-scale uctuationsinCMBtemperaturesarisefromthegravitational
e�ectofprimordialdensity uctuationsinthedistributionofnonbaryonicdark
matter.
(iv)Thepeaksintheplotoftemperature uctuation�T
vs.multipolelaredueto
variationsinthedensityofnonbaryonicdarkmatter,whilethecontributions
frombaryonsalonewouldnotshowsuchpeaks.
(d)(Darkmattercandidates)Whichoneofthefollowingisnotacandidateofnonbary-
onicdarkmatter?
(i)massiveneutrinos
(ii)axions
(iii)mattermadeoftopquarks(atypeofquarkswithheavymassofabout171
GeV).
(iv)WIMPs(WeaklyInteractingMassiveParticles)
(v)primordialblackholes
(e)(Signaturesofdarkmatter)Bywhatmethodscansignaturesofdarkmatterbe
detected?Listtwomethods.(Grading:3pointsforonecorrectanswer,5pointsfor
twocorrectanswers.Ifyougivemorethantwoanswers,yourscorewillbebased
onthenumberofrightanswersminusthenumberofwronganswers,withalower
boundofzero.)
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�
PROBLEM
6:TIME
EVOLUTION
OF
A
UNIVERSE
INCLUDING
A
HYPOTHETICALKIND
OFMATTER
(30points)
ThefollowingproblemwasProblem2,Quiz3,2018.
Supposethata atuniverseincludesnonrelativisticmatter,radiation,andalsomys-
ticium,wherethemassdensityofmysticiumbehavesas
�myst /1
a5(t)
astheuniverseexpands.Inthisproblemwewillde�ne
x(t)�a(t)
a(t0 );
wheret0
isthepresenttime.Forthefollowingquestions,youneednotevaluateany
oftheintegralsthatmightarise,buttheymustbeintegralsofexplicitfunctionswith
explicitlimitsofintegration;rememberthata(t)isnotgiven.Youmayexpressyour
answersintermsofthepresentvalueoftheHubbleexpansionrate,H0 ,andthevarious
contributionstothepresentvalueof:m;0 ,rad;0 ,andmyst;0 .
(a)(7points)WriteanexpressionfortheHubbleexpansionrateH(t).
(b)(7points)Writeanexpressionforthecurrentageoftheuniverse.
(c)(3points)Writeanexpressionforthetimet(x)intermsofthevalueofx.
(d)(3points)Writeanexpressionforthetotalmassdensity�(x)asafunctionofx.
(e)(10points)Writeanexpressionforthephysicalhorizondistance,`p;hor .
�
PROBLEM
7:THECONSEQUENCESOFAN
ALT-PHOTON
(25points)
Supposethat,inadditiontotheparticlesthatareknowntoexist,therealsoexisted
analt-photon,whichhasexactlythepropertiesofaphoton:itismassless,hastwospin
states(orpolarizationstates),andhasthesameinteractionswithotherparticlesthat
photonsdo.Likephotons,itisitsownantiparticle.
(a)(5points)InthermalequilibriumattemperatureT,whatisthetotalenergydensity
ofalt-photons?
(b)(5points)InthermalequilibriumattemperatureT,whatisthenumberdensityof
alt-photons?
(c)(10points)Inthissituation,whatwouldbethetemperatureratiosT�=T
and
T�=Talt today?
(d)(5points)Wouldtheexistenceofthisparticleincreaseordecreasetheabundanceof
helium,orwouldithavenoe�ect?
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PROBLEM
8:NUMBER
DENSITIESIN
THECOSMIC
BACKGROUND
RADIATION
Todaythetemperatureofthecosmicmicrowavebackgroundradiationis2:7ÆK.
Calculatethenumberdensityofphotonsinthisradiation.Whatisthenumberdensity
ofthermalneutrinosleftoverfromthebigbang?
PROBLEM
9:PROPERTIESOFBLACK-BODY
RADIATION
(25points)
ThefollowingproblemwasProblem4,Quiz3,1998.
Inansweringthefollowingquestions,rememberthatyoucanrefertotheformulas
atthefrontoftheexam.Sinceyouwerenotaskedtobringcalculators,youmayleave
youranswersintheformofalgebraicexpressions,suchas�32= p5�(3).
(a)(5points)Fortheblack-bodyradiation(alsocalledthermalradiation)ofphotonsat
temperatureT,whatistheaverageenergyperphoton?
(b)(5points)Forthesameradiation,whatistheaverageentropyperphoton?
(c)(5points)Nowconsidertheblack-bodyradiationofamasslessbosonwhichhasspin
zero,sothereisonlyonespinstate.Wouldtheaverageenergyperparticleand
entropyperparticlebedi�erentfromtheanswersyougaveinparts(a)and(b)?If
so,howwouldtheychange?
(d)(5points)Nowconsidertheblack-bodyradiationofelectronneutrinosattempera-
tureT.Theseparticlesarefermionswithspin1/2,andwewillassumethatthey
aremasslessandhaveonlyonepossiblespinstate.Whatistheaverageenergyper
particleforthiscase?
(e)(5points)Whatistheaverageentropyperparticlefortheblack-bodyradiationof
neutrinos,asdescribedinpart(d)?
PROBLEM
10:A
NEW
SPECIESOFLEPTON
ThefollowingproblemwasProblem2,Quiz3,1992,worth25points.
Supposethecalculationsdescribingtheearlyuniverseweremodi�edbyincludingan
additional,hypotheticallepton,calledan8.286ion.The8.286ionhasroughlythesame
propertiesasanelectron,exceptthatitsmassisgivenbymc2=0:750MeV.
Parts(a)-(c)ofthisquestionrequirenumericalanswers,butsinceyouwerenot
toldtobringcalculators,youneednotcarryoutthearithmetic.Youranswershould
beexpressed,however,in\calculator-ready"form|
thatis,itshouldbeanexpression
involvingpurenumbersonly(nounits),withanynecessaryconversionfactorsincluded.
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(Forexample,ifyouwereaskedhowmanymetersalightpulseinvacuumtravelsin5
minutes,youcouldexpresstheansweras2:998�108�5�60.)
a)(5points)Whatwouldbethenumberdensityof8.286ions,inparticlespercubic
meter,whenthetemperatureTwasgivenbykT=3MeV?
b)(5points)Assuming(asinthestandardpicture)thattheearlyuniverseisaccurately
describedbya at,radiation-dominatedmodel,whatwouldbethevalueofthemass
densityatt=:01sec?Youmayassumethat0:75MeV�kT�100MeV,sothe
particlescontributingsigni�cantlytotheblack-bodyradiationincludethephotons,
neutrinos,e+-e�
pairs,and8.286ion-anti8286ionpairs.Expressyouranswerinthe
unitsofg/cm3.
c)(5points)Underthesameassumptionsasin(b),whatwouldbethevalueofkT,in
MeV,att=:01sec?
d)(5points)Whennucleosynthesiscalculationsaremodi�edtoincludethee�ectofthe
8.286ion,istheproductionofheliumincreasedordecreased?Explainyouranswer
inafewsentences.
e)(5points)SupposetheneutrinosdecouplewhilekT�0:75MeV.Ifthe8.286ions
areincluded,whatdoesonepredictforthevalueofT�=T today?(HereT�denotes
thetemperatureoftheneutrinos,andT
denotesthetemperatureofthecosmic
backgroundradiationphotons.)
�
PROBLEM
11:ANEW
THEORYOFTHEWEAK
INTERACTIONS(40
points)
ThisproblemwasProblem3,Quiz3,2009.
SupposeaNewTheoryoftheWeakInteractions(NTWI)wasproposed,whichdif-
fersfrom
thestandardtheoryintwoways.First,theNTWIpredictsthattheweak
interactionsaresomewhatweakerthaninthestandardmodel.Inaddition,thetheory
impliestheexistenceofnewspin-12particles(fermions)calledtheR+
andR�
,witharest
energyof50MeV(where1MeV=106eV).Thisproblemwilldealwiththecosmological
consequencesofsuchatheory.
TheNTWIwillpredictthattheneutrinosintheearlyuniversewilldecoupleat
ahighertemperaturethaninthestandardmodel.Supposethatthisdecouplingtakes
placeatkT�200MeV.Thismeansthatwhentheneutrinosceasetobethermally
coupledtotherestofmatter,thehotsoupofparticleswouldcontainnotonlyphotons,
neutrinos,ande+-e�
pairs,butalso�+,��
,�+,��
,and�0particles,alongwiththe
R+-R�
pairs.(Themuonisaparticlewhichbehavesalmostidenticallytoanelectron,
exceptthatitsrestenergyis106MeV.Thepionsarethelightestofthemesons,with
zeroangularmomentumandrestenergiesof135MeVand140MeVfortheneutraland
chargedpions,respectively.The�+
and��
areantiparticlesofeachother,andthe�0
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isitsownantiparticle.Zeroangularmomentumimpliesasinglespinstate.)Youmay
assumethattheuniverseis at.
(a)(10points)Accordingtothestandardparticlephysicsmodel,whatisthemass
density�oftheuniversewhenkT�200MeV?Whatisthevalueof�atthis
temperature,accordingtoNTWI?Useeitherg/cm3orkg/m3.(Ifyouwish,youcan
savetimebynotcarryingoutthearithmetic.Ifyoudothis,however,youshould
givetheanswerin\calculator-ready"form,bywhichImeananexpressioninvolving
purenumbers(nounits),withanynecessaryconversionfactorsincluded,andwith
theunitsoftheanswerspeci�edattheend.Forexample,ifaskedhowfarlight
travelsin5minutes,youcouldanswer2:998�108�5�60m.)
(b)(10points)Accordingtothestandardmodel,thetemperaturetodayofthethermal
neutrinobackgroundshouldbe(4=11)1=3T
,whereT
isthetemperatureofthe
thermalphotonbackground.WhatdoestheNTWIpredictforthetemperatureof
thethermalneutrinobackground?
(c)(10points)Accordingtothestandardmodel,whatistheratiotodayofthenumber
densityofthermalneutrinostothenumberdensityofthermalphotons?Whatis
thisratioaccordingtoNTWI?
(d)(10points)Sincethereactionswhichinterchangeprotonsandneutronsinvolveneu-
trinos,thesereactions\freezeout"atroughlythesametimeastheneutrinosdecou-
ple.Atlatertimestheonlyreactionwhiche�ectivelyconvertsneutronstoprotons
isthefreedecayoftheneutron.Despitethefactthatneutrondecayisaweakinter-
action,wewillassumethatitoccurswiththeusual15minutemeanlifetime.Would
theheliumabundancepredictedbytheNTWIbehigherorlowerthantheprediction
ofthestandardmodel?Towithin5or10%,whatwouldtheNTWIpredictforthe
percentabundance(byweight)ofheliumintheuniverse?(Asinpart(a),youcan
eithercarryoutthearithmetic,orleavetheanswerincalculator-readyform.)
Usefulinformation:Theprotonandneutronrestenergiesaregivenbympc2
=
938:27MeVandmnc2=939:57MeV,with(mn �mp )c2=1.29MeV.Themean
lifetimefortheneutrondecay,n!p+e�
+��e,isgivenby�=886s.
PROBLEM
12:DOUBLING
OFELECTRONS(10points)
ThefollowingwasonQuiz3,2011(Problem4):
Supposethatinsteadofonespeciesofelectronsandtheirantiparticles,supposethere
wasalsoanotherspeciesofelectron-likeandpositron-likeparticles.Supposethatthenew
specieshasthesamemassandotherpropertiesastheelectronsandpositrons.Ifthis
werethecase,whatwouldbetheratioT� =T ofthetemperaturetodayoftheneutrinos
tothetemperatureoftheCMBphotons.
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PROBLEM
13:TIMESCALESIN
COSMOLOGY
Inthisproblemyouareaskedtogivetheapproximatetimesatwhichvariousim-
portanteventsinthehistoryoftheuniversearebelievedtohavetakenplace.Thetimes
aremeasuredfromtheinstantofthebigbang.Toavoidambiguities,youareaskedto
choosethebestanswerfromthefollowinglist:
10�
43sec.
10�
37sec.
10�
12sec.
10�
5sec.
1sec.
4mins.
10,000{1,000,000years.
2billionyears.
5billionyears.
10billionyears.
13billionyears.
20billionyears.
ForthisproblemitwillbesuÆcienttostateananswerfrommemory,withoutexplanation.
Theeventswhichmustbeplacedarethefollowing:
(a)thebeginningoftheprocessesinvolvedinbigbangnucleosynthesis;
(b)theendoftheprocessesinvolvedinbigbangnucleosynthesis;
(c)thetimeofthephasetransitionpredictedbygranduni�edtheories,whichtakes
placewhenkT�1016GeV;
(d)\recombination",thetimeatwhichthematterintheuniverseconvertedfrom
aplasmatoagasofneutralatoms;
(e)thephasetransitionatwhichthequarksbecamecon�ned,believedtooccur
whenkT�300MeV.
Sincecosmologyisfraughtwithuncertainty,insomecasesmorethanoneanswerwill
beacceptable.Youareasked,however,togiveONLY
ONEoftheacceptableanswers.
PROBLEM
14:EVOLUTION
OFFLATNESS(15points)
ThefollowingproblemwasProblem3,Quiz3,2004.
The\ atnessproblem"isrelatedtothefactthatduringtheevolutionofthestandard
cosmologicalmodel,isalwaysdrivenawayfrom1.
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(a)(9points)Duringaperiodinwhichtheuniverseismatter-dominated(meaningthat
theonlyrelevantcomponentisnonrelativisticmatter),thequantity
�1
growsasapoweroft,providedthatisnear1.Showthatthisistrue,andderive
thepower.(Statingtherightpowerwithoutaderivationwillbeworth3points.)
(b)(6points)Duringaperiodinwhichtheuniverseisradiation-dominated,thesame
quantitywillgrowlikeadi�erentpoweroft.Showthatthisistrue,andderivethe
power.(Statingtherightpowerwithoutaderivationwillagainbeworth3points.)
Ineachpart,youmayassumethattheuniversewasalwaysdominatedbythespeci�ed
formofmatter.
�
PROBLEM
15:THESLOANDIGITALSKYSURVEYz
=5:82QUASAR
(40points)
ThefollowingproblemwasProblem4,Quiz3,2004.
OnApril13,2000,theSloanDigitalSkySurveyannouncedthediscoveryofwhat
wasthenthemostdistantobjectknownintheuniverse:aquasaratz=5:82.Toexplain
tothepublichowthisobject�tsintotheuniverse,theSDSSpostedontheirwebsitean
articlebyMichaelTurnerandCraigWiegerttitled\HowCanAnObjectWeSeeToday
be27BillionLightYearsAwayIftheUniverseisonly14BillionYearsOld?"Usinga
modelwithH0=65km-s�
1-Mpc�
1,m
=0:35,and�
=0:65,theyclaimed
(a)thattheageoftheuniverseis13.9billionyears.
(b)thatthelightthatwenowseewasemittedwhentheuniversewas0.95billionyears
old.
(c)thatthedistancetothequasar,asitwouldbemeasuredbyarulertoday,is27
billionlight-years.
(d)thatthedistancetothequasar,atthetimethelightwasemitted,was4.0billion
light-years.
(e)thatthepresentspeedofthequasar,de�nedastherateatwhichthedistance
betweenusandthequasarisincreasing,is1.8timesthevelocityoflight.
Thegoalofthisproblemistocheckalloftheseconclusions,althoughyouareofcourse
notexpectedtoactuallyworkoutthenumbers.Youranswerscanbeexpressedinterms
ofH0 ,m,�,andz.De�niteintegralsneednotbeevaluated.
Notethatm
representsthepresentdensityofnonrelativisticmatter,expressedas
afractionofthecriticaldensity;and�
representsthepresentdensityofvacuumenergy,
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expressedasafractionofthecriticaldensity.Inansweringeachofthefollowingquestions,
youmayconsidertheanswertoanypreviouspart|
whetheryouanswereditornot|
asagivenpieceofinformation,whichcanbeusedinyouranswer.
(a)(15points)Writeanexpressionfortheaget0ofthismodeluniverse?
(b)(5points)Writeanexpressionforthetimeteatwhichthelightwhichwenowreceive
fromthedistantquasarwasemitted.
(c)(10points)Writeanexpressionforthepresentphysicaldistance`phys;0tothequasar.
(d)(5points)Writeanexpressionforthephysicaldistance`phys;ebetweenusandthe
quasaratthetimethatthelightwasemitted.
(e)(5points)Writeanexpressionforthepresentspeedofthequasar,de�nedasthe
rateatwhichthedistancebetweenusandthequasarisincreasing.
PROBLEM
16:SECOND
HUBBLECROSSING
(40points)
ThisproblemwasProblem3,Quiz3,2007.In2018wehavenotyettalkedaboutHubble
crossingsandtheevolutionofdensityperturbations,sothisproblemwouldnotbefairas
worded.Actually,however,youhavelearnedhowtodothesecalculations,sotheproblem
wouldbefairifitdescribedinmoredetailwhatneedstobecalculated.
InProblemSet9(2007)wecalculatedthetimetH1 (�)ofthe�rstHubblecrossing
foramodespeci�edbyits(physical)wavelength�atthepresenttime.Inthisproblem
wewillcalculatethetimetH2 (�)ofthesecondHubblecrossing,thetimeatwhichthe
growingHubblelengthcH�
1(t)catchesuptothephysicalwavelength,whichisalso
growing.AtthetimeofthesecondHubblecrossingforthewavelengthsofinterest,the
universecanbedescribedverysimply:itisaradiation-dominated atuniverse.However,
since�isde�nedasthepresentvalueofthewavelength,theevolutionoftheuniverse
betweentH2 (�)andthepresentwillalsoberelevanttotheproblem.Wewillneedto
usemethods,therefore,thatallowforboththematter-dominatederaandtheonsetof
thedark-energy-dominatedera.AsinProblemSet9(2007),themodeluniversethatwe
considerwillbedescribedbytheWMAP3-yearbest�tparameters:
Hubbleexpansionrate
H0
=
73:5km�s�
1�Mpc�
1
Nonrelativisticmassdensity
m
=
0.237
Vacuummassdensity
vac
=
0.763
CMBtemperature
T ;0
=
2.725K
Themassdensitiesarede�nedascontributionsto,andhencedescribethemassdensity
ofeachconstituentrelativetothecriticaldensity.Notethatthemodelisexactly at,
soyouneednotworryaboutspatialcurvature.Hereyouarenotexpectedtogivea
numericalanswer,sotheabovelistwillserveonlytode�nethesymbolsthatcanappear
inyouranswers,alongwith�andthephysicalconstantsG,�h,c,andk.
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(a)(5points)Foraradiation-dominated atuniverse,whatistheHubblelength`H(t)�
cH�
1(t)asafunctionoftimet?
(b)(10points)ThesecondHubblecrossingwilloccurduringtheinterval
30sec�t�50;000years,
whenthemassdensityoftheuniverseisdominatedbyphotonsandneutrinos.During
thiseratheneutrinosarealittlecolderthanthephotons,withT�=(4=11)1=3T
.
Thetotalenergydensityofthephotonsandneutrinostogethercanbewrittenas
utot=g1�2
30(kT )4
(�hc)3
:
Whatisthevalueofg1 ?(Forthefollowingpartsyoucantreatg1asagivenvariable
thatcanbeleftinyouranswers,whetherornotyoufoundit.)
(c)(10points)Fortimesintherangedescribedinpart(b),whatisthephotontemper-
atureT (t)asafunctionoft?
(d)(15points)Finally,wearereadyto�ndthetimetH2 (�)ofthesecondHubble
crossing,foragivenvalueofthephysicalwavelength�today.Makinguseofthe
previousresults,youshouldbeabletodeterminetH2 (�).Ifyouwerenotableto
answersomeofthepreviousparts,youmayleavethesymbols`H(t),g1 ,and/or
T (t)inyouranswer.
PROBLEM
17:THEEVENTHORIZON
FOROURUNIVERSE(25points)
ThefollowingproblemwasProblem3fromQuiz3,2013.
Wehavelearnedthattheexpansionhistoryofouruniversecanbedescribedinterms
ofasmallsetofnumbers:m;0 ,thepresentcontributiontofromnonrelativisticmatter;
rad;0 ,thepresentcontributiontofromradiation;vac ,thepresentcontributionto
fromvacuumenergy;andH0 ,thepresentvalueoftheHubbleexpansionrate.The
bestestimatesofthesenumbersareconsistentwitha atuniverse,sowecantakek=0,
m;0+rad;0+vac=1,andwecanusethe atRobertson-Walkermetric,
ds2=�c2dt2+a2(t) �dr2+r2 �d�2+sin2�d�2 ��:
(a)(5points)Supposethatweareattheoriginofthecoordinatesystem,andthatat
thepresenttimet0weemitasphericalpulseoflight.Itturnsoutthatthereisa
maximumcoordinateradiusr=rmaxthatthispulsewilleverreach,nomatterhow
longwewait.(Thepulsewillneveractuallyreachrmax ,butwillreachallrsuchthat
0<r<rmax .)rmax
isthecoordinateofwhatiscalledtheeventhorizon:events
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thathappennowatr�rmaxwillneverbevisibletous,assumingthatweremainat
theorigin.Assumingforthispartthatthefunctiona(t)isaknownfunction,write
anexpressionforrmax .Youranswershouldbeexpressedasanintegral,whichcan
involvea(t),t0 ,andanyoftheparametersde�nedinthepreamble.[Advice:Ifyou
cannotanswerthis,youshouldstilltrypart(c).]
(b)(10points)Sincea(t)isnotknownexplicitly,theanswertothepreviouspartis
diÆculttouse.Show,however,thatbychangingthevariableofintegration,youcan
rewritetheexpressionforrmaxasade�niteintegralinvolvingonlytheparameters
speci�edinthepreamble,withoutanyreferencetothefunctiona(t),exceptperhaps
toitspresentvaluea(t0 ).Youarenotexpectedtoevaluatethisintegral.[Hint:One
methodistouse
x=
a(t)
a(t0 )
asthevariableofintegration,justaswedidwhenwederivedthe�rstoftheexpres-
sionsfort0shownintheformulasheets.]
(c)(10points)Astronomersoftendescribedistancesintermsofredshifts,soitisuseful
to�ndtheredshiftoftheeventhorizon.Thatis,ifalightraythatoriginatedat
r=rmaxarrivedatEarthtoday,whatwouldbeitsredshiftzeh(eh=eventhorizon)?
Youarenotaskedto�ndanexplicitexpressionforzeh ,butinsteadanequationthat
couldbesolvednumericallytodeterminezeh .Forthispartyoucantreatrmax
asgiven,soitdoesnotmatterifyouhavedoneparts(a)and(b).Youwillget
halfcreditforacorrectanswerthatinvolvesthefunctiona(t),andfullcreditfora
correctanswerthatinvolvesonlyexplicitintegralsdependingonlyontheparameters
speci�edinthepreamble,andpossiblya(t0 ).
PROBLEM
18:THEEFFECTOFPRESSUREONCOSMOLOGICALEVO-
LUTION
(25points)
ThefollowingproblemwasProblem2ofQuiz3,2016.ItwasalsoProblem2ofProblem
Set7(2016),exceptthatsomenumericalconstantshavebeenchanged,sotheanswers
willnotbeidentical.
Aradiation-dominateduniversebehavesdi�erentlyfromamatter-dominateduni-
versebecausethepressureoftheradiationissigni�cant.Inthisproblemweexplorethe
roleofpressureforseveral�ctitiousformsofmatter.
(a)(8points)Forthe�rst�ctitiousformofmatter,themassdensity�decreasesasthe
scalefactora(t)grows,withtherelation
�(t)/1
a8(t):
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Whatisthepressureofthisformofmatter?[Hint:theanswerisproportionalto
themassdensity.]
(b)(9points)Findthebehaviorofthescalefactora(t)fora atuniversedominated
bytheformofmatterdescribedinpart(a).Youshouldbeabletodeterminethe
functiona(t)uptoaconstantfactor.
(c)(8points)Nowconsiderauniversedominatedbyadi�erentformof�ctitiousmatter,
withapressuregivenby
p=23
�c2:
Astheuniverseexpands,themassdensityofthisformofmatterbehavesas
�(t)/1
an(t):
Findthepowern.
�
PROBLEM
19:THE
FREEZE-OUT
OFA
FICTITIOUSPARTICLE
X
(25points)
ThefollowingproblemwasProblem3ofQuiz3,2016.
Supposethat,inadditiontotheparticlesthatareknowntoexist,therealsoexisteda
familyofthreespin-1particles,X+,X�
,andX0,allwithmasses0.511MeV/c2,exactly
thesameastheelectron.TheX�
istheantiparticleoftheX+,andtheX0isitsown
antiparticle.SincetheX'sarespin-1particleswithnonzeromass,eachparticlehasthree
spinstates.
TheX'sdonotinteractwithneutrinosanymorestronglythantheelectronsand
positronsdo,sowhentheX'sfreezeout,alloftheirenergyandentropyaregiventothe
photons,justliketheelectron-positronpairs.
(a)(5points)InthermalequilibriumwhenkT�0:511MeV/c2,whatisthetotalenergy
densityoftheX+,X�
,andX0particles?
(b)(5points)Inthermalequilibrium
whenkT
�0:511MeV/c2,whatisthetotal
numberdensityoftheX+,X�
,andX0particles?
(c)(10points)TheXparticlesandtheelectron-positronpairsfreezeoutofthethermal
equilibriumradiationatthesametime,askTdecreasesfromvalueslargecompared
to0.511MeV/c2tovaluesthataresmallcomparedtoit.IftheX's,electron-positron
pairs,photons,andneutrinoswereallinthermalequilibriumbeforethisfreeze-out,
whatwillbetheratioT�=T ,theratiooftheneutrinotemperaturetothephoton
temperature,afterthefreeze-out?
(d)(5points)IfthemassoftheX'swas,forexample,0.100MeV/c2,sothattheelectron-
positronpairsfrozeout�rst,andthentheX'sfrozeout,wouldthe�nalratioT� =T
behigher,lower,orthesameastheanswertopart(c)?Explainyouranswerina
sentenceortwo.
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PROBLEM
20:THETIMEtd
OFDECOUPLING
(25points)
ThefollowingproblemwasProblem4ofQuiz3,2016.
Theprocessbywhichthephotonsofthecosmicmicrowavebackgroundstopscatter-
ingandbegintotravelonstraightlinesiscalleddecoupling,andithappensataphoton
temperatureofaboutTd
�3;000K.InLectureNotes6weestimatedthetimetd
of
decoupling,workingintheapproximationthattheuniversehasbeenmatter-dominated
fromthattimetothepresent.Wefoundavalueof370,000years.Inthisproblemwe
willremovethisapproximation,althoughwewillnotcarryoutthenumericalevaluation
neededtocomparewiththepreviousanswer.
(a)(5points)Letusde�ne
x(t)�a(t)
a(t0 );
asontheformulasheets,wheret0isthepresenttime.Whatisthevalueofxd �
x(td )?Assumethattheentropyofphotonsisconservedfromtimetdtothepresent,
andletT0denotethepresentphotontemperature.
(b)(5points)Assumethattheuniverseis at,andthatm;0 ,rad;0 ,andvac;0denote
thepresentcontributionstofromnonrelativisticmatter,radiation,andvacuum
energy,respectively.LetH0denotethepresentvalueoftheHubbleexpansionrate.
Writeanexpressionintermsofthesequantitiesfordx=dt,thederivativeofxwith
respecttot.Hint:youmayuseformulasfromtheformulasheetwithoutderivation,
sothisproblemshouldrequireessentiallynowork.Toreceivefullcredit,youranswer
shouldincludeonlytermsthatmakeanonzerocontributiontotheanswer.
(c)(5points)Writeanexpressionfortd .Ifyouranswerinvolvesanintegral,youneed
nottrytoevaluateit,butyoushouldbesurethatthelimitsofintegrationareclearly
shown.
(d)(10points)Nowsupposethatinadditiontotheconstituentsdescribedinpart(b),
theuniversealsocontainssomeofthe�ctitiousmaterialfrompart(a)ofProblem
18(Quiz3ReviewProblems,2020),with
�(t)/1
a8(t):
Denotethepresentcontributiontofrom
this�ctitiousmaterialasf;0 .The
universeisstillassumedtobe at,sothenumericalvaluesofm;0 ,rad;0 ,and
vac;0
mustsum
toasmallervaluethaninparts(b)and(c).Withthisextra
contributiontothemassdensityoftheuniverse,whatisthenewexpressionfortd ?
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SOLUTIONS
PROBLEM
1:DID
YOU
DO
THEREADING
(2018)?(20points)
(a)(5points)WhichoneofthefollowingstatementsaboutCMBisNOTcorrect?
(i)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionofthe
observerrelativetoaframeofreferenceinwhichtheCMBisisotropic.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,themeantemperature
averagingovertheskyis<T>=2.725K.
(iii)AfterthedipoledistortionoftheCMBissubtractedaway,thetemperatureof
theCMBvariesby0.3microKelvinacrossthesky.
(iv)ThephotonsoftheCMBhavemostlybeentravelingonstraightlinessincethey
werelastscatteredatt�370;000yr,atalocationcalledthesurfaceoflast
scattering.
[Comment:Theactualvariationisabout30microKelvin,ormaybeafewtimesthat
much.RydenquotestheCOBErootmeansquarefractionalvariationoftheCMB
temperatureas
< �ÆTT �2
>1=2=1:1�10�
5
asEq.(8.8)(2ndEdition),whichgivesavalueofabout30microKelvin,giventhat
T�3K.InLectureNotes2wequotedavalueof4:14�10�
5computedfromPlanck
data.Therootmeansquare uctuationsincreasewithbetterangularresolution,
because uctuationswithsmallangularwavelengthsarenotseenunlesstheresolution
ishigh.
(b)(5points)Thenonuniformitiesinthecosmicmicrowavebackgroundallowusto
measuretheripplesinthemassdensityoftheuniverseatthetimewhentheplasma
combinedtoformneutralatoms,about300,000-400,000yearsafterthebigbang.
Theseripplesarecrucialforunderstandingwhathappenedlater,sincetheyarethe
seedswhichledtothecomplicatedtapestryofgalaxies,clustersofgalaxies,and
voids.Whichofthefollowingsentencesdescribeshowtheseripplesarecreatedin
thecontextofin ationarymodels:
(i)Magneticmonopolescanformrandomlyduringthegranduni�edtheoryphase
transition,resultinginnonuniformitiesinthemassdensity.
(ii)Cosmicstrings,whicharelineliketopologicaldefects,canformrandomlyduring
thegranduni�edtheoryphasetransition,resultinginnonuniformitiesinthe
massdensity.
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(iii)Theyaregeneratedbyquantum uctuationsduringin ation.
(iv)Sincetheearlyuniversewasveryhot,therewerelargethermal uctuations
whichultimatelyevolvedintotheripplesinthemassdensity.
(c)(5points)InChapter8ofTheFirstThreeMinutes,StevenWeinbergdescribes
thefutureoftheuniverse(assuming,aswasthoughtthentobethecase,thatthe
cosmologicalconstantiszero).Onepossibilitythathediscussesisthatthecosmic
matterdensitycouldbegreaterthanthecriticaldensity.Assumingthatwelivein
suchauniverse,whichofthefollowingstatementsisNOTtrue?
(i)Theuniverseis�niteanditsexpansionwilleventuallycease,givingwaytoan
acceleratingcontraction.
(ii)Threeminutesafterthetemperaturereachesathousandmilliondegrees(109K),
thelawsofphysicsguaranteethattheuniversewillcrunch,andtimewillstop.
(iii)Duringatleasttheearlypartofthecontractingphase,wewillbeabletoobserve
bothredshiftsandblueshifts.
(iv)Whentheuniversehasrecontractedtoone-hundredthitspresentsize,theradi-
ationbackgroundwillbegintodominatethesky,withatemperatureofabout
300K.
[Comment:Weinbergisveryclearnospeculationsabouttheendoftheuniverseare
guaranteedtobetrue:\Doestimereallyhavetostopsomethreeminutesafterthe
temperaturereachesathousandmilliondegrees?Obviously,wecannotbesure.All
theuncertaintiesthatwemetintheprecedingchapter,intryingtoexplorethe�rst
hundredthofasecond,willreturntoperplexusaswelookintothelasthundredthof
asecond."]
(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
redderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
bluerbecauseoftheDopplere�ect.
(iii)Photonstravelingtowardusfromthesurfaceoflastscatteringappearredder
becauseofabsorptionintheintergalacticmedium.
(iv)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappearbluer
becauseofabsorptionintheintergalacticmedium.
(v)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearredder
becausetheymustclimboutofthegravitationalpotentialwell.
(vi)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearbluer
becausetheymustclimboutofthegravitationalpotentialwell.
[Comment:RydendiscussestheSachs-Wolfee�ectonpp.161{162(2ndEdition).]
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PROBLEM
2:DID
YOU
DO
THEREADING
(2016)?(25points)
Exceptforpart(d),youshouldanswerthesequestionsbycirclingtheonestatementthat
iscorrect.
(a)(5points)IntheEpilogueofTheFirstThreeMinutes,SteveWeinbergwrote:\The
moretheuniverseseemscomprehensible,themoreitalsoseemspointless."
The
sentencewasquali�ed,however,byaclosingparagraphthatpointsoutthat
(i)thequestofthehumanracetocreateabetterlifeforallcanstillgivemeaning
toourlives.
(ii)iftheuniversecannotgivemeaningtoourlives,thenperhapsthereisanafterlife
thatwill.
(iii)thecomplexityandbeautyofthelawsofphysicsstronglysuggestthatthe
universemusthaveapurpose,evenifwearenotawareofwhatitis.
(iv)thee�orttounderstandtheuniversegiveshumanlifesomeofthegraceof
tragedy.
(b)(5points)IntheAfterwordofTheFirstThreeMinutes,Weinbergdiscussesthe
baryonnumberoftheuniverse.(Thebaryonnumberofanysystem
isthetotal
numberofprotonsandneutrons(andcertainrelatedparticlesknownashyperons)
minusthenumberoftheirantiparticles(antiprotons,antineutrons,antihyperons)
thatarecontainedinthesystem.)Weinbergconcludedthat
(i)baryonnumberisexactlyconserved,sothetotalbaryonnumberoftheuniverse
mustbezero.Whilenucleiinourpartoftheuniversearecomposedofprotons
andneutrons,theuniversemustalsocontainantimatterregionsinwhichnuclei
arecomposedofantiprotonsandantineutrons.
(ii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe
partoftheuniversewecanobserve,andhenceapositivedensityofbaryon
number.Sincebaryonnumberisconserved,thiscanonlybeexplainedby
assumingthattheexcessbaryonswereputinatthebeginning.
(iii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe
partoftheuniversewecanobserve,andhenceapositivedensityofbaryonnum-
ber.Thiscanbetakenasapositivehintthatbaryonnumberisnotconserved,
whichcanhappenifthereexistasyetundetectedheavy\exotic"particles.
(iv)itispossiblethatbaryonnumberisnotexactlyconserved,butevenifthatisthe
case,itisnotpossiblethattheobservedexcessofmatteroverantimattercanbe
explainedbytheveryrareprocessesthatviolatebaryonnumberconservation.
Explanation:Allstudentsweregivencreditforthispart,whethertheyansweredit
correctlyornot.IwasinSanFranciscowhenImadeupthisquiz,anddue
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topoorplanningIdidnothavemycopyofTheFirstThreeMinutes.SoI
foundaversiononline,butIcouldonly�ndtheBritishversion,publishedby
Flamingo/FontanaPaperbacks,ratherthantheUSversionpublishedbyBasic
Books.Iassumedthatthe\Afterword"inthetwoversionswouldbethesame,
butIwaswrong!Sothisquestionwasbasedonadi�erent\Afterword"than
theonethatyouread.55%ofyoustillgotitright,butobviouslythequestion
wasnotfair.Apologies.
(c)(5points)IndiscussingtheCOBEmeasurementsofthecosmicmicrowaveback-
ground,Rydendescribesadipolecomponentofthetemperaturepattern,forwhich
thetemperatureoftheradiationfromonedirectionisfoundtobehotterthanthe
temperatureoftheradiationdetectedfromtheoppositedirection.
(i)Thisdiscoveryisimportant,becauseitallowsustopinpointthedirectionofthe
pointinspacewherethebigbangoccurred.
(ii)ThisisthelargestcomponentoftheCMBanisotropies,amountingtoa10%
variationinthetemperatureoftheradiation.
(iii)Inadditiontothedipolecomponent,theanisotropiesalsoincludecontributions
fromaquadrupole,octupole,etc.,allofwhicharecomparableinmagnitude.
(iv)ThispatternisinterpretedasasimpleDopplershift,causedbythenetmotion
oftheCOBEsatelliterelativetoaframeofreferenceinwhichtheCMBis
almostisotropic.
Explanation:(i)isnonsense,sincetheconventionalbigbangtheorydescibesacom-
pletelyhomogeneousuniverse,whichhasnosinglepointatwhichthebigbang
occurred.(ii)iswrong,becausethevariationsinthetemperatureoftheCMB
aremuchsmallerthan10%.Thedipoletermhasamagnitudeofabout1/1000
ofthemeantemperature.(iii)iswrongbecausethedipoleisnotcomparableto
theotherterms,becausetheyhavemagnitudesofonlyabout1/100,000ofthe
mean.
(d)(5points)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBis
notcorrect:
(i)AfterthedipoledistortionoftheCMBissubtractedaway,themeantemperature
averagingovertheskyishTi=2:725K.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,therootmeansquare
temperature uctuationis D�ÆTT �2 E1=2
=1:1�10�
3.
(iii)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionofthe
observerrelativetoaframeofreferenceinwhichtheCMBisisotropic.
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(iv)Intheirgroundbreakingpaper,WilsonandPenziasreportedthemeasurement
ofanexcesstemperatureofabout3.5Kthatwasisotropic,unpolarized,and
freefromseasonalvariations.InacompanionpaperwrittenbyDicke,Peebles,
RollandWilkinson,theauthorsinterpretedtheradiationtobearelicofan
early,hot,dense,andopaquestateoftheuniverse.
Explanation:Therightvalueis*�
ÆTT �2 +1=2
=1:1�10�
5:
(e)(5points)In ationisdrivenbya�eldthatisbyde�nitioncalledthein aton�eld.
Instandardin ationarymodels,the�eldhasthefollowingproperties:
(i)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitspotentialenergy.
(ii)Thein atonisavector�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitspotentialenergy.
(iii)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitskineticenergy.
(iv)Thein atonisavector�eld,andduringin ationtheenergydensityofthe
universeisdominatedbyitskineticenergy.
(v)Thein atonisatensor�eld,whichisresponsibleforonlyasmallfractionof
theenergydensityoftheuniverseduringin ation.
Explanation:ThesefactswerementionedinbothSection11.5(ThePhysicsofIn-
ation)ofRyden'sbook,andalsointhearticlethatyouwereaskedtoread
calledIn ationandtheNewEraofHigh-PrecisionCosmology,writtenbyme
forthePhysicsDepartment2002newsletter.
PROBLEM
3:DID
YOU
DO
THEREADING
(2013)?(35points)
(a)(5points)RydensummarizestheresultsoftheCOBEsatelliteexperimentforthe
measurementsofthecosmicmicrowavebackground(CMB)intheform
ofthree
importantresults.The�rstwasthat,inanyparticulardirectionofthesky,the
spectrum
oftheCMBisveryclosetothatofanidealblackbody.TheFIRAS
instrumentontheCOBEsatellitecouldhavedetecteddeviationsfromtheblackbody
spectrumassmallas��=��10�
n,wherenisaninteger.Towithin�1,whatisn?
Answer:n=4
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.38
(b)(5points)ThesecondresultwasthemeasurementofadipoledistortionoftheCMB
spectrum;thatis,theradiationisslightlyblueshiftedtohighertemperaturesinone
direction,andslightlyredshiftedtolowertemperaturesintheoppositedirection.To
whatphysicale�ectwasthisdipoledistortionattributed?
Answer:ThelargedipoleintheCMBisattributedtothemotionofthesatellite
relativetotheframeinwhichtheCMBisverynearlyisotropic.(TheentireLocal
Groupismovingrelativetothisframeataspeedofabout0.002c.)
(c)(5points)Thethirdresultconcernedthemeasurementoftemperature uctuations
afterthedipolefeaturementionedabovewassubtractedout.De�ning
ÆTT(�;�)�T(�;�)�hTi
hTi
;
wherehTi=2:725K,theaveragevalueofT,theyfoundarootmeansquare uctu-
ation,
*�ÆTT �2 +1=2
;
equaltosomenumber.Towithinanorderofmagnitude,whatwasthatnumber?
Answer:
*�ÆTT �2 +1=2
=1:1�10�
5:
(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
redderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
bluerbecauseoftheDopplere�ect.
(iii)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearredder
becausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearbluer
becausetheymustclimboutofthegravitationalpotentialwell.
(v)Photonstravelingtowardusfromthesurfaceoflastscatteringappearredder
becauseofabsorptionintheintergalacticmedium.
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappearbluer
becauseofabsorptionintheintergalacticmedium.
(e)(5points)The atnessproblemreferstotheextreme�ne-tuningthatisneededin
atearlytimes,inorderforittobeascloseto1todayasweobserve.Startingwith
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.39
theassumptionthattodayisequalto1withinabout1%,oneconcludesthatat
onesecondafterthebigbang,j
�1jt=1sec<10�
m
;
wheremisaninteger.Towithin�3,whatism?
Answer:m=18.(SeethederivationinLectureNotes8.)
(f)(5points)Thetotalenergydensityofthepresentuniverseconsistsmainlyofbaryonic
matter,darkmatter,anddarkenergy.Givethepercentagesofeach,accordingto
thebest�tobtainedfromthePlanck2013data.Youwillgetfullcreditifthe�rst
(baryonicmatter)isaccurateto�2%,andtheothertwoareaccuratetowithin�5%.
Answer:Baryonicmatter:5%.Darkmatter:26.5%.Darkenergy:68.5%.The
Planck2013numbersweregiveninLectureNotes7.Totherequestedaccuracy,
however,numberssuchasRyden'sBenchmarkModelwouldalsobesatisfactory.
(g)(5points)Withintheconventionalhotbigbangcosmology(withoutin ation),it
isdiÆculttounderstandhowthetemperatureoftheCMBcanbecorrelatedat
angularseparationsthataresolargethatthepointsonthesurfaceoflastscattering
wasseparatedfrom
eachotherbymorethanahorizondistance.Approximately
whatangle,indegrees,correspondstoaseparationonthesurfacelastscatteringof
onehorizonlength?Youwillgetfullcreditifyouranswerisrighttowithinafactor
of2.
Answer:Rydengives1Æ
astheanglesubtendedbytheHubblelengthonthesurface
oflastscattering.Foramatter-dominateduniverse,whichwouldbeagoodmodel
forouruniverse,thehorizonlengthistwicetheHubblelength.Anynumberfrom
1Æ
to5Æ
wasconsideredacceptable.
PROBLEM
4:DID
YOU
DO
THEREADING
(2009)?(25points)
(a)(10points)Thisquestionconcernssomenumbersrelatedtothecosmicmicrowave
background(CMB)thatoneshouldneverforget.Statethevaluesofthesenumbers,
towithinanorderofmagnitudeunlessotherwisestated.Inallcasesthequestion
referstothepresentvalueofthesequantities.
(i)TheaveragetemperatureToftheCMB(towithin10%).2:725K
(ii)ThespeedoftheLocalGroupwithrespecttotheCMB,expressedasafraction
v=cofthespeedoflight.(ThespeedoftheLocalGroupisfoundbymeasuring
thedipolepatternoftheCMBtemperaturetodeterminethevelocityofthe
spacecraftwithrespecttotheCMB,andthenremovingspacecraftmotion,the
orbitalmotionoftheEarthabouttheSun,theSunaboutthegalaxy,andthe
galaxyrelativetothecenterofmassoftheLocalGroup.)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.40
Thedipoleanisotropycorrespondstoa\peculiarvelocity"(thatis,velocitywhich
isnotduetotheexpansionoftheuniverse)of630�20kms�
1,orintermsof
thespeedoflight,v=c�2�10�
3.
(iii)Theintrinsicrelativetemperature uctuations�T=T,afterremovingthedipole
anisotropycorrespondingtothemotionoftheobserverrelativetotheCMB.
1:1�10�
5
(iv)Theratioofbaryonnumberdensitytophotonnumberdensity,�=nbary =n .
TheWMAP5-yearvaluefor�=nb =n =
(6:225�0:170)�10�
10,whichto
closestorderofmagnitudeis10�
9.
(v)Theangularsize�H,indegrees,correspondingtowhatwastheHubbledistance
c=H
atthesurfaceoflastscattering.Thisanswermustbewithinafactorof3
tobecorrect.�1Æ
(b)(3points)Becausephotonsoutnumberbaryonsbysomuch,theexponentialtailof
thephotonblackbodydistributionisimportantinionizinghydrogenwellafterkT
fallsbelowQH
=13:6eV.WhatistheratiokT =QH
whentheionizationfraction
oftheuniverseis1=2?
(i)1=5
(ii)1=50
(iii)10�
3
(iv)10�
4
(v)10�
5
Thisisnotanumberonehastocommittomemoryifonecanrememberthe
temperatureof(re)combinationineV,orifonlyinKalongwiththeconversion
factor(k�10�
4eVK�
1).Onecanthencalculatethatnearrecombination,
kT =QH
�(10�
4eVK�
1)(3000K)=(13:6eV)�1=45.
(c)(2points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
redderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear
bluerbecauseoftheDopplere�ect.
(iii)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearredder
becausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearbluer
becausetheymustclimboutofthegravitationalpotentialwell.
(v)Photonstravelingtowardusfromthesurfaceoflastscatteringappearredder
becauseofabsorptionintheintergalacticmedium.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.41
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappearbluer
becauseofabsorptionintheintergalacticmedium.
Explanation:Denserregionshaveadeeper(morenegative)gravitationalpoten-
tial.Photonswhichtravelthroughaspatiallyvaryingpotentialacquirearedshift
orblueshiftdependingonwhethertheyaregoingupordownthepotential,re-
spectively.Photonsoriginatinginthedenserregionsstartatalowerpotential
andmustclimbout,sotheyendupbeingredshiftedrelativetotheiroriginal
energies.
(d)(10points)Foreachofthefollowingstatements,saywhetheritistrueorfalse:
(i)Darkmatterinteractsthroughthegravitational,weak,andelectromagnetic
forces.
T
orF?
(ii)Thevirialtheoremcanbeappliedtoaclusterofgalaxiesto�nditstotalmass,
mostofwhichisdarkmatter.
T
orF?
(iii)Neutrinosarethoughttocompriseasigni�cantfractionoftheenergydensityof
darkmatter.
T
orF?
(iv)Magneticmonopolesarethoughttocompriseasigni�cantfractionoftheenergy
densityofdarkmatter.
T
orF?
(v)LensingobservationshaveshownthatMACHOscannotaccountforthedark
matteringalactichalos,butthatasmuchas20%ofthehalomasscouldbein
theformofMACHOs.
T
orF?
PROBLEM
5:DID
YOU
DO
THEREADING?(2007)(25points)
Thefollowingpartsareeachworth5points.
(a)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBisnotcorrect:
(i)AfterthedipoledistortionoftheCMBissubtractedaway,themeantemperature
averagingovertheskyishTi=2:725K.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,therootmeansquare
temperature uctuationis D�ÆTT �2 E1=2
=1:1�10�
3.
(iii)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionofthe
observerrelativetoaframeofreferenceinwhichtheCMBisisotropic.
(iv)Intheirgroundbreakingpaper,WilsonandPenziasreportedthemeasurement
ofanexcesstemperatureofabout3.5Kthatwasisotropic,unpolarized,and
freefromseasonalvariations.InacompanionpaperwrittenbyDicke,Peebles,
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.42
RollandWilkinson,theauthorsinterpretedtheradiationtobearelicofan
early,hot,dense,andopaquestateoftheuniverse.
Explanation:Aftersubtractingthedipolecontribution,thetemperature uctua-
tionisabout1:1�10�
5.
(b)(CMBexperiments)ThecurrentmeanenergyperCMBphoton,about6�10�
4eV,
iscomparabletotheenergyofvibrationorrotationforasmallmoleculesuchasH2 O.
Thusmicrowaveswithwavelengthsshorterthan��3cmarestronglyabsorbedby
watermoleculesintheatmosphere.TomeasuretheCMBat�<3cm,whichone
ofthefollowingmethodsisnotafeasiblesolutiontothisproblem?
(i)MeasureCMBfromhigh-altitudeballoons,e.g.MAXIMA.
(ii)MeasureCMBfromtheSouthPole,e.g.DASI.
(iii)MeasureCMBfromtheNorthPole,e.g.BOOMERANG.
(iv)MeasureCMBfromasatelliteabovetheatmosphereoftheEarth,e.g.COBE,
WMAPandPLANCK.
Explanation:TheNorthPoleisatsealevel.Incontrast,theSouthPoleis
nearly3kilometersabovesealevel.BOOMERANGisaballoon-borneexperi-
mentlaunchedfromAntarctica.
(c)(Temperature uctuations)Thecreationoftemperature uctuationsinCMBby
variationsinthegravitationalpotentialisknownastheSachs-Wolfee�ect.Which
oneofthefollowingstatementsisnotcorrectconcerningthise�ect?
(i)ACMBphotonisredshiftedwhenclimbingoutofagravitationalpotentialwell,
andisblueshiftedwhenfallingdownapotentialhill.
(ii)Atthetimeoflastscattering,thenonbaryonicdarkmatterdominatedtheen-
ergydensity,andhencethegravitationalpotential,oftheuniverse.
(iii)Thelarge-scale uctuationsinCMBtemperaturesarisefromthegravitational
e�ectofprimordialdensity uctuationsinthedistributionofnonbaryonicdark
matter.
(iv)Thepeaksintheplotoftemperature uctuation�T
vs.multipolelaredueto
variationsinthedensityofnonbaryonicdarkmatter,whilethecontributions
frombaryonsalonewouldnotshowsuchpeaks.
Explanation:Thesepeaksareduetotheacousticoscillationsinthephoton-
baryon uid.
(d)(Darkmattercandidates)Whichoneofthefollowingisnotacandidateofnonbary-
onicdarkmatter?
(i)massiveneutrinos
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.43
(ii)axions
(iii)mattermadeoftopquarks(atypeofquarkswithheavymassofabout171
GeV).
(iv)WIMPs(WeaklyInteractingMassiveParticles)
(v)primordialblackholes
Explanation:Mattermadeoftopquarksissounstablethatitisseenonly eet-
inglyasaproductinhighenergyparticlecollisions.
(e)(Signaturesofdarkmatter)Bywhatmethodscansignaturesofdarkmatterbe
detected?Listtwomethods.(Grading:3pointsforonecorrectanswer,5pointsfor
twocorrectanswers.Ifyougivemorethantwoanswers,yourscorewillbebased
onthenumberofrightanswersminusthenumberofwronganswers,withalower
boundofzero.)
Answers:
(i)Galaxyrotationcurves.(I.e.,measurementsoftheorbitalspeedofstarsinspiral
galaxiesasafunctionofradiusRshowthatthesecurvesremain atatradii
farbeyondthevisiblestellardisk.Ifmostofthematterwerecontainedinthe
disk,thenthesevelocitiesshouldfallo�as1= pR.)
(ii)Usethevirialtheoremtoestimatethemassofagalaxycluster.(Forexample,
thevirialanalysisshowsthatonly2%ofthemassoftheComaclusterconsists
ofstars,andonly10%consistsofhotintraclustergas.
(iii)Gravitationallensing.(Forexample,themassofaclustercanbeestimatedfrom
thedistortionoftheshapesofthegalaxiesbehindthecluster.)
(iv)CMBtemperature uctuations.(I.e.,theanalysisoftheintensityofthe uc-
tuationsasafunctionofmultipolenumbershowsthattot �1,andthatdark
energycontributes�
�0:7,baryonicmattercontributesbary
�0:04,and
darkmattercontributesdarkmatter �0:26.)
Thereareotherpossibleanswersaswell,butthesearetheonesdiscussedbyRyden
inChapters8and9.
PROBLEM
6:TIMEEVOLUTION
OFAUNIVERSEINCLUDING
AHY-
POTHETICALKIND
OFMATTER
(30points)
Supposethata atuniverseincludesnonrelativisticmatter,radiation,andalsomys-
ticium,wherethemassdensityofmysticiumbehavesas
�myst /1
a5(t)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.44
astheuniverseexpands.Inthisproblemwewillde�ne
x(t)�a(t)
a(t0 );
wheret0
isthepresenttime.Forthefollowingquestions,youneednotevaluateany
oftheintegralsthatmightarise,buttheymustbeintegralsofexplicitfunctionswith
explicitlimitsofintegration;rememberthata(t)isnotgiven.Youmayexpressyour
answersintermsofthepresentvalueoftheHubbleexpansionrate,H0 ,andthevarious
contributionstothepresentvalueof:m;0 ,rad;0 ,andmyst;0 .
(a)(7points)WriteanexpressionfortheHubbleexpansionrateH(x).
(b)(7points)Writeanexpressionforthecurrentageoftheuniverse.
(c)(3points)Writeanexpressionforthetimet(x)intermsofthevalueofx.
(d)(3points)Writeanexpressionforthetotalmassdensity�(x)asafunctionofx.
(e)(10points)Writeanexpressionforpresentvalueofthephysicalhorizondistance,
`p;hor (t0 ).
Solution:
(a)Sincetheuniverseis at,the�rstFriedmannequationbecomes
H2=8�3
G�;
butthenwecanwrite�as
H2=8�3
G (�m;0 �a(t0 )
a(t) �
3+�rad;0 �a(t0 )
a(t) �
4+�myst;0 �a(t0 )
a(t) �
5 ):
Nowuse
�c;0=3H20
8�G
and���
c;
so
H2=
H20
�c;0 (�m;0 �a(t0 )
a(t) �
3+�rad;0 �a(t0 )
a(t) �
4+�myst;0 �a(t0 )
a(t) �
5 )
=H20 �m;0
x3
+rad;0
x4
+myst;0
x5 �:
Finally,
H(x)=H0
x2 rm;0 x+rad;0+myst;0
x
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.45
(b)To�ndthecurrentaget0 ,westartwith
H=
_aa=
_xx
=)
dxd
t=xH
=)
dt=
dx
xH
:
Sot0canbefoundbyintegratingovertherangeofx,from0to1:
t0= Z
10
dx
xH(x)
=
1H0 Z
10
xdx
qm;0 x+rad;0+myst;0
x
:
(c)To�ndthetimetcorrespondingtosomevalueofxotherthan1,onesimplyintegrates
dtfromx0
=0tox0
=x:
t(x)= Z
x0
dx0
x0H(x0)
=
1H0 Z
x0
x0dx0
qm;0 x0
+rad;0+myst;0
x0
:
(d)Fromthe�rstFriedmannequation,
H2=8�3
G�
=)
�=
38�GH2(x):
Giventheanswerinpart(a),thisbecomes
�(x)=
38�G
H20
x4 �m;0 x+rad;0+myst;0
x
�:
(e)Thegeneralformulaforthephysicalhorizondistanceisgivenontheformulasheet:
`p;hor (t)=a(t) Z
t0
ca(t0)dt0
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.46
Herewearenotgiventhefunctiona(t),butwecanchangethevariableofintegration
tointegrateoverx:
dt0
=dt0
dada=1_a
da=1aa_a
da=
da
aH(x):
So
`p;hor (t0 )=a(t0 ) Z
a(t0)
0
cda
a2H(a)
= Z1
0
cdx
x2H(x)
=
cH0 Z
10
dx
qm;0 x+rad;0+myst;0
x
:
PROBLEM
7:THECONSEQUENCESOFAN
ALT-PHOTON
(25points)
Supposethat,inadditiontotheparticlesthatareknowntoexist,therealsoexisted
analt-photon,whichhasexactlythepropertiesofaphoton:itismassless,hastwospin
states(orpolarizationstates),andhasthesameinteractionswithotherparticlesthat
photonsdo.Likephotons,itisitsownantiparticle.
(a)(5points)InthermalequilibriumattemperatureT,whatisthetotalenergydensity
ofalt-photons?
(b)(5points)InthermalequilibriumattemperatureT,whatisthenumberdensityof
alt-photons?
(c)(10points)Inthissituation,whatwouldbethetemperatureratiosT�=T
and
T�=Talt today?
(d)(5points)Wouldtheexistenceofthisparticleincreaseordecreasetheabundanceof
helium,orwouldithavenoe�ect?
Solution:
(a)Theenergydensitywillbethesameasforphotons,sincethereisnodi�erence.The
generalformulais
u=g�2
30(kT)4
(�hc)3
;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.47
asgivenontheformulasheets,andg=2foralt-photons(orphotons),sincethere
aretwopolarizationstates,andtheparticlesarebosons.So
ualt =�2
15(kT)4
(�hc)3
:
(4.1)
(b)Forthenumberdensity,thegeneralformulais
n=g�
�(3)
�2
(kT)3
(�hc)3
;
whereg�
=2sinceagainthealt-photonsarebosonswithtwopolarizationstates.So
nalt =2�(3)
�2
(kT)3
(�hc)3
:
(4.2)
(c)Asintheactualscenario,theeventthatcausesatemperaturedi�erenceisthe
disappearanceoftheelectron-positronpairsfromthethermalequilibriummix,which
occursaskTchangesfromvalueslargecomparedtome c2=0:511MeVtovalues
thataresmallcomparedtoit.Thekeypointisthatthisdisappearanceoccursafter
theneutrinoshavedecoupledfromtheotherparticles,soalloftheentropyfromthe
electron-positronpairsisgiventothephotons,andnoneisgiventotheneutrinos.
Inthiscasetheentropyisgiventoboththephotonsandthealt-photons.
Thegeneralformulaforentropydensityisontheformulasheet,anditcanbe
rewrittenas
s=AgT3;
(4.3)
where
A=2�2
45
k4
(�hc)3
:
(4.4)
ThevalueofAwillinfactnotbeneededforthisproblem.
Sincetheneutrinoshavedecoupledbythetimethee+e�
pairsdisappear,theentropy
ofneutrinosandtheentropyofeverythingelsewillbeseparatelyconserved.Entropy
conservationmeansthattheentropypercomovingvolumedoesnotchange.During
theperiodbeforee+e�
freeze-out,gisconstant,sotheconstancyofentropyper
comovingvolumeimpliesthat
S=sVphys=gT3AVphys=ga3T3AVcoord;
(4.5)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.48
soS=Vcoord=constimpliesthata3T3isconstant,andsoaTisconstant.HereT
isthecommontemperatureofphotons,alt-photons,electronsandpositrons,and
neutrinos,allofwhichwereinthermalequilibriumduringthisperiod.SinceaTis
constantduringthisperiod,wecangivetheconstantaname,
aT=[aT]before:
(4.6)
Fortheneutrinos,theformulasheettellsusthat
g�=
78|{z}
Fermion
factor
�
3|{z}
3species
�e;��;�� �
2|{z}
Particle=
antiparticle �
1|{z}
Spinstates
=
214;
(4.7)
while
ge+
e�
=
78|{z}
Fermion
factor
�1
| {z}Species �
2|{z}
Particle=
antiparticle �
2|{z}
Spinstates
=
72:
(4.8)
Thus
gelse=g +galt +ge+
e�
=2+2+72
=152
:
(4.9)
Thusbeforethee+e�
freezeout,thetwoconservedquantitieswere
S�
Vcoord
=Ag�[aT] 3before;
Selse
Vcoord
=Agelse [aT] 3before:
(4.10)
Aftere+e�
freezeout,thetemperatureoftheneutrinosT�
willnolongerbethe
sameasthetemperatureT
ofthephotonsandalt-photons,andofcoursee+e�
pairswillnolongerbepresent.ButT
andTalt willbeequaltoeachother,since
theyhavethesameinteractions;weknowthattheinteractionsofthephotonskeep
theminthermalequilibriumuntiltdecoupling �380;000years,soboththephotons
andthealt-photonswillremaininthermalequilibrium
untillongaftertheeraof
e+e�
freezeout,whichisoforder1{10seconds.Thusthetwoconservedquantities
willbe
S�
Vcoord
=Ag� [aT�] 3after;
Selse
Vcoord
=A(g +galt )[aT ] 3after:
(4.11)
ByequatingthevaluesofS� =Vcoordbeforeandafter,weseethat
[aT�]after=[aT]before;
(4.12)
andthenbyequatingthevaluesofSelse =Vcoordbeforeandafter,weseethat
[aT ]after= �gelse
g +galt �
1=3
[aT]before= �gelse
g +galt �
1=3
[aT� ]after;
(4.13)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.49
whereweusedEq.(4.12)inthelaststep.Itfollowsthat
�T�
T �after= �g +galt
gelse
�1=3
= �2+2
152 �
1=3
=
�81
5 �1=3
:
(4.14)
(d)Itwouldincreasetheabundanceofhelium.Themaine�ectofthealt-photonwould
betoincreasetheexpansionrateoftheuniverse,whichinturnwouldcausethe
neutrinostodecoupleearlierfromthethermalequilibriummix,whichinturnwould
meanthattherationn=np ,theratioofneutronstoprotons,wouldbecomefrozen
atalargervalue.Theincreasedexpansionratewouldalsomeanlesstimeavailable
forfreeneutrondecay,whichfurtherincreasesthenumberofneutronsthatremain
whenthetemperaturefallslowenoughforheliumformationtocomplete.Essentially
alltheneutronsbecomeboundintohelium,somoreneutronsimpliesmorehelium.
PROBLEM
8:NUMBER
DENSITIESIN
THECOSMIC
BACKGROUND
RADIATION
Ingeneral,thenumberdensityofaparticleintheblack-bodyradiationisgivenby
n=g�
�(3)�2 �kT�h
c �3
Forphotons,onehasg�
=2.Then
k=1:381�10�
16erg=ÆK
T=2:7ÆK
�h=1:055�10�
27erg-sec
c=2:998�1010cm/sec 9>>>>>=>>>>>;
=)
�kT�h
c �3
=1:638�103cm�
3:
Thenusing�(3)'1:202,one�ndsn
=399=cm3:
Fortheneutrinos,
g��=2�34
=32
perspecies.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.50
Thefactorof2istoaccountfor�and��,andthefactorof3/4arisesfromthePauli
exclusionprinciple.Soforthreespeciesofneutrinosonehas
g��=92
:
Usingtheresult
T3�
=
411T3
fromProblem8ofProblemSet3(2000),one�nds
n�= �g��
g� ��T�
T �
3n
= �94 ��411 �
399cm�
3
=)
n�=326=cm3(forallthreespeciescombined).
PROBLEM
9:PROPERTIESOFBLACK-BODY
RADIATION
(a)Theaverageenergyperphotonisfoundbydividingtheenergydensitybythenumber
density.Thephotonisabosonwithtwospinstates,sog=g�
=2.Usingthe
formulasonthefrontoftheexam,
E=
g�2
30(kT)4
(�hc)3
g�
�(3)
�2
(kT)3
(�hc)3
=
�4
30�(3)kT:
Youwerenotexpectedtoevaluatethisnumerically,butitisinterestingtoknowthat
E=2:701kT:
Notethattheaverageenergyperphotonissigni�cantlymorethankT,whichisoften
usedasaroughestimate.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.51
(b)Themethodisthesameasabove,exceptthistimeweusetheformulafortheentropy
density:
S=
g2�2
45
k4T3
(�hc)3
g�
�(3)
�2
(kT)3
(�hc)3
=
2�4
45�(3)k:
Numerically,thisgives3:602k,wherekistheBoltzmannconstant.
(c)Inthiscasewewouldhaveg=
g�
=
1.Theaverageenergyperparticleand
theaverageentropyparticledependsonlyontheratiog=g�,sotherewouldbe
nodi�erencefromtheanswersgiveninparts(a)and(b).
(d)Forafermion,gis7/8timesthenumberofspinstates,andg�
is3/4timesthe
numberofspinstates.Sotheaverageenergyperparticleis
E=
g�2
30(kT)4
(�hc)3
g�
�(3)
�2
(kT)3
(�hc)3
=
78�2
30(kT)4
(�hc)3
34�(3)
�2
(kT)3
(�hc)3
=
7�4
180�(3)kT:
Numerically,E=3:1514kT.
Warning:theMathematicianGeneralhasdetermined
thatthememorizationofthisnumbermayadversely
a�ectyourabilitytorememberthevalueof�.
Ifonetakesintoaccountbothneutrinosandantineutrinos,theaverageenergyper
particleisuna�ected|
theenergydensityandthetotalnumberdensityareboth
doubled,buttheirratioisunchanged.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.52
Notethattheenergyperparticleishigherforfermionsthanitisforbosons.This
resultcanbeunderstoodasanaturalconsequenceofthefactthatfermionsmust
obeytheexclusionprinciple,whilebosonsdonot.Largenumbersofbosonscan
thereforecollectinthelowestenergylevels.Infermionsystems,ontheotherhand,
thelow-lyinglevelscanaccommodateatmostoneparticle,andthenadditional
particlesareforcedtohigherenergylevels.
(e)Thevaluesofgandg�
areagain7/8and3/4respectively,so
S=
g2�2
45
k4T3
(�hc)3
g�
�(3)
�2
(kT)3
(�hc)3
=
782�2
45
k4T3
(�hc)3
34�(3)
�2
(kT)3
(�hc)3
=
7�4
135�(3)k:
Numerically,thisgivesS=4:202k.
PROBLEM
10:A
NEW
SPECIESOFLEPTON
a)Thenumberdensityisgivenbytheformulaatthestartoftheexam,
n=g�
�(3)
�2
(kT)3
(�hc)3
:
Sincethe8.286ionisliketheelectron,ithasg�
=3;thereare2spinstatesforthe
particlesand2fortheantiparticles,giving4,andthenafactorof3/4becausethe
particlesarefermions.So
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.53
Then
Answer=3�(3)
�2
� �
3�106�102
6:582�10�
16�2:998�1010 �
3
:
Youwerenotaskedtoevaluatethisexpression,buttheansweris1:29�1039.
b)Fora atcosmology�=0andoneoftheEinsteinequationsbecomes
�_aa �2
=8�3
G�:
Duringtheradiation-dominatederaa(t)/t1=2,asclaimedonthefrontcoverofthe
exam.So,
_aa=
12t:
Usingthisintheaboveequationgives
14t2
=8�3
G�:
Solvethisfor�,
�=
3
32�Gt2
:
Thequestionasksthevalueof�att=
0:01sec.
WithG
=
6:6732�
10�
8cm3sec�
2g�
1,then�
=
3
32��6:6732�10�
8�(0:01)2
inunitsofg=cm3.Youweren'taskedtoputthenumbersin,but,forreference,doing
sogives�=4:47�109g=cm3.
c)Themassdensity�=u=c2,whereuistheenergydensity.Theenergydensityfor
black-bodyradiationisgivenintheexam,
u=�c2=g�2
30(kT)4
(�hc)3
:
WecanusethisinformationtosolveforkTintermsof�(t)whichwefoundabove
inpart(b).Atatimeof0.01sec,ghasthefollowingcontributions:
Photons:
g=2
e+e�
:
g=4�78=312
�e ;��;�� :
g=6�78=514
8:286ion�anti8:286ion
g=4�78=312
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.54
gtot=1414
:
SolvingforkTintermsof�gives
kT= �30
�2
1gtot �h
3c5� �1=4
:
Usingtheresultfor�frompart(b)aswellasthelistoffundamentalconstantsfrom
thecoversheetoftheexamgives
kT= �90�(1:055�10�
27)3�(2:998�1010)5
14:24�32�3�6:6732�10�
8�(0:01)2 �
1=4
�
1
1:602�10�
6
wheretheanswerisgiveninunitsofMeV.PuttinginthenumbersyieldskT=8:02
MeV.
d)Theproductionofhelium
isincreased.Atanygiventemperature,theadditional
particleincreasestheenergydensity.SinceH/�1=2,theincreasedenergydensity
speedstheexpansionoftheuniverse|theHubbleconstantatanygiventemperature
ishigheriftheadditionalparticleexists,andthetemperaturefallsfaster.The
weakinteractionsthatinterconvertprotonsandneutrons\freezeout"whenthey
cannolongerkeepupwiththerateofevolutionoftheuniverse.Thereaction
ratesatagiventemperaturewillbeuna�ectedbytheadditionalparticle,butthe
highervalueofHwillmeanthatthetemperatureatwhichtheseratescannolonger
keeppacewiththeuniversewilloccursooner.Thefreeze-outwillthereforeoccur
atahighertemperature.Theequilibrium
valueoftheratioofneutrontoproton
densitiesislargerathighertemperatures:nn=np /exp(��mc2=kT),wherennand
nparethenumberdensitiesofneutronsandprotons,and�mistheneutron-proton
massdi�erence.Consequently,therearemoreneutronspresenttocombinewith
protonstobuildhelium
nuclei.Inaddition,thefasterevolutionrateimpliesthat
thetemperatureatwhichthedeuteriumbottleneckbreaksisreachedsooner.This
impliesthatfewerneutronswillhaveachancetodecay,furtherincreasingthehelium
production.
e)Aftertheneutrinosdecouple,theentropyintheneutrinobathisconservedseparately
fromtheentropyintherestoftheradiationbath.Justafterneutrinodecoupling,
alloftheparticlesinequilibriumaredescribedbythesametemperaturewhichcools
asT/1=a.Theentropyinthebathofparticlesstillinequilibriumjustafterthe
neutrinosdecoupleis
S/grest T3(t)a3(t)
wheregrest=gtot �g�=9.Bytoday,thee+�e�
pairsandthe8.286ion-anti8.286ion
pairshaveannihilated,thustransferringtheirentropytothephotonbath.Asaresult
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.55
thetemperatureofthephotonbathisincreasedrelativetothatoftheneutrinobath.
Fromconservationofentropywehavethattheentropyafterannihilationsisequal
totheentropybeforeannihilations
g T3 a3(t)=grest T3(t)a3(t):
So,
T
T(t)= �grest
g �
1=3
:
Sincetheneutrinotemperaturewasequaltothetemperaturebeforeannihilations,
wehavethat
T�
T
= �29 �1=3
:
PROBLEM
11:A
NEW
THEORY
OFTHEWEAK
INTERACTIONS(40
points)
(a)Inthestandardmodel,theblack-bodyradiationatkT�200MeVcontainsthe
followingcontributions:
Photons:
g=2
e+e�
:
g=4�78=312
�e ;��;�� :
g=6�78=514
�+��
:
g=4�78=312
�+��
�0
g=3
9>>>>>>>=>>>>>>>;gTOT
=1714
Themassdensityisthengivenby
�=
uc2=gTOT�2
30(kT)4
�h3c5
:
Inkg/m3,onecanevaluatethisexpressionby
�= �1714 ��2
30 �
200�106eV�1:602�10�
19J
eV
�4
(1:055�10�
34J-s)3(2:998�108m/s)5
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.56
Checkingtheunits,[�
]=
J4
J3-s3-m5-s�
5
=J-s2
m5
= �kg-m2-s�
2 �s2
m5
=kg/m3:
So,the�nalanswerwouldbe
�= �1714 ��2
30 �200�106�1:602�10�
19 �4
(1:055�10�
34)3(2:998�108)5kg
m3
:
Youwerenotexpectedtoevaluatethis,butwithacalculatoronewould�nd
�=2:10�1018kg/m3:
Ing/cm3,onewouldevaluatethisexpressionby
�= �1714 ��2
30
�200�106eV�1:602�10�
12erg
eV
�4
(1:055�10�
27erg-s)3(2:998�1010cm/s)5
:
Checkingtheunits,
[�]=
erg4
erg3-s3-cm5-s�
5=erg-s2
cm5
= �g-cm2-s�
2 �s2
cm5
=g/cm3:
So,inthiscasethe�nalanswerwouldbe
�= �1714 ��2
30 �200�
106�1:602�10�
12 �4
(1:055�10�
27)3(2:998�1010)5
gcm3
:
Noevaluationwasrequested,butwithacalculatoryouwould�nd
�=2:10�1015g/cm3;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.57
whichagreeswiththeanswerabove.
Note:Acommonmistakewastoleaveouttheconversionfactor1:602�10�
19J/eV
(or1:602�10�
12erg/eV),andinsteadtouse�h=6:582�10�
16eV-s.Butifone
worksouttheunitsofthisanswer,theyturnouttobeeV-sec2/m5(oreV-sec2/cm5),
whichisamostpeculiarsetofunitstomeasureamassdensity.
IntheNTWI,wehaveinadditionthecontributiontothemassdensityfromR+-R�
pairs,whichwouldactjustlikee+-e�
pairsor�+-��
pairs,withg=312.Thus
gTOT
=2034,so
�= �2034 ��2
30 �200�106�1:602�10�
19 �4
(1:055�10�
34)3(2:998�108)5kg
m3
or
�= �2034 ��2
30 �200�
106�1:602�10�
12 �4
(1:055�10�
27)3(2:998�1010)5
gcm3
:
Numerically,theanswerinthiscasewouldbe
�NTWI=2:53�1018kg/m3=2:53�1015g/cm3:
(b)Aslongastheuniverseisinthermalequilibrium,entropyisconserved.Theentropy
inagivenvolumeofthecomovingcoordinatesystemis
a3(t)sVcoord
;
wheresistheentropydensityanda3V
coordisthephysicalvolume.So
a3(t)s
isconserved.Aftertheneutrinosdecouple,
a3s
�
and
a3s
other
areseparatelyconserved,wheresotheristheentropyofeverythingexceptneutrinos.
Notethatscanbewrittenas
s=gAT3
;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.58
whereAisaconstant.Beforethedisappearanceofthee;�,R,and�particlesfrom
thethermalequilibriumradiation,s
�= �514 �
AT3
sother= �1512 �
AT3
:
So
s�
sother=
514
1512
:
Ifa3s
�anda3s
otherareconserved,thensoiss�=sother .Bytoday,theentropyprevi-
ouslysharedamongthevariousparticlesstillinequilibriumafterneutrinodecoupling
hasbeentransferedtothephotonssothat
sother=sphotons=2AT3
:
Theentropyinneutrinosisstill
s�= �514 �
AT3�
:
Sinces�=sotherisconstantweknowthat
�514 �T3�
2T3
=
s�
sother=
514
1512
=)
T�= �431 �1=3
T
:
(c)Onecanwrite
n=g�BT3
;
whereBisaconstant.Hereg� =2,andg��=6�34=412 .Inthestandardmodel,
onehastoday
n�
n
=g�� T3�
g� T3
= �412 �2
411=
911:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.59
IntheNTWI,
n�
n
= �412 �2
431=
931:
(d)AtkT=200MeV,thethermalequilibriumratioofneutronstoprotonsisgivenby
nn
np
=e�
1:29MeV=200MeV
�1:
Inthestandardtheorythisratiowoulddecreaserapidlyastheuniversecooledand
kTfellbelowthep-nmassdi�erenceof1.29MeV,butintheNTWItheratiofreezes
outatthehightemperaturecorrespondingtokT=200MeV,whentheratiois
about1.WhenkTfallsbelow200MeVintheNTWI,theneutrinointeractions
n+�e $p+e�
and
n+e+
$p+��e
thatmaintainthethermalequilibrium
balancebetweenprotonsandneutronsno
longeroccuratasigni�cantrate,sotheration= npisnolongercontrolledbyther-
malequilibrium.AfterkTfallsbelow200MeV,theonlyprocessthatcanconvert
neutronstoprotonsistheratherslowprocessoffreeneutrondecay,withadecay
time�dofabout890s.Thus,whenthedeuteriumbottleneckbreaksatabout200
s,thenumberdensityofneutronswillbeconsiderablyhigherthaninthestandard
model.SinceessentiallyalloftheseneutronswillbecomeboundintoHenuclei,the
higherneutronabundanceoftheNTWIimpliesa
higherpredictedHeabundance:
ToestimatetheHeabundance,notethatifwetemporarilyignorefreeneutrondecay,
thentheneutron-protonratiowouldbefrozenatabout1andwouldremain1until
thetimeofnucleosynthesis.Atthetimeofnucleosynthesisessentiallyallofthese
neutronswouldbeboundintoHenuclei(eachwith2protonsand2neutrons).For
aninitial1:1ratioofneutronstoprotons,alltheneutronsandprotonscanbebound
intoHenuclei,withnoprotonsleftoverintheformofhydrogen,soYwouldequal
1.However,thefreeneutrondecayprocesswillcausetherationn=nptofallbelow
1beforethestartofnucleosynthesis,sothepredictedvalueofYwouldbelessthan
1.Tocalculatehowmuchless,notethatRydenestimatesthestartofnucleosynthe-
sisatthetimewhenthetemperaturereachesTnuc ,whichisthetemperaturefor
whichathermalequilibriumcalculationgivesnD=nn=1.Thiscorrespondstowhat
Weinbergreferstoasthebreakingofthedeuteriumbottleneck.Thetemperature
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.60
Tnuciscalculatedintermsof�=nB=n andphysicalconstants,soitwouldnotbe
changedbytheNTWI.Thetimewhenthistemperatureisreached,however,would
bechangedslightlybythechangeintheratioT�=T .Sincethise�ectisrather
subtle,nopointswillbetakeno�ifyouomittedit.However,tobeasaccurate
aspossible,oneshouldrecognizethatnucleosynthesisoccursduringtheradiation-
dominatedera,butlongafterthee+-e�
pairshavedisappeared,sotheblack-body
radiationconsistsofphotonsattemperatureT andneutrinosatalowertemperature
T� .Theenergydensityisgivenby
u=�2
30(kT )4
(�hc)3 "2+ �214 ��T�
T �
4 #�ge��2
30(kT )4
(�hc)3
;
where
ge�
=2+ �214 ��T�
T �
4
:
Forthestandardmodel
gsm
e�
=2+ �214 ��411 �4=3
;
andfortheNTWI
gNTWI
e�
=2+ �214 ��43
1 �4=3
:
Therelationbetweentimeandtemperatureina atradiation-dominateduniverse
isgivenintheformulasheetsas
kT= �45�h3c5
16�3gG �
1=4
1pt:
Thus,
t/
1g1=2
e�
T2
:
InthestandardmodelRydenestimatesthetimeofnucleosynthesisastsmnu
c �200s,
sointheNTWIitwouldbelongerbythefactor
tNTWI
nuc
= sgsm
e�
gNTWI
e�
tsmn
uc:
Whileofcoureyouwerenotexpectedtoworkoutthenumerics,thisgives
tNTWI
nuc
=1:20tsmn
uc:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.61
NotethatRydengivestnuc �200s,whileWeinbergplacesitat334minutes�225s,
whichiscloseenough.
Tofollowthee�ectofthisfreedecay,itiseasiesttodoitbyconsideringtheratio
neutronstobaryonnumber,nn=nB,sincenB
doesnotchangeduringthisperiod.
Atfreeze-out,whenkT�200MeV,
nn
nB
�12
:
Justbeforenucleosynthesis,attimetnuc ,theratiowillbe
nn
nB
�12
e�
tnuc=�d
:
Iffreedecayisignored,wefoundY=1.Sinceallthesurvivingneutronsarebound
intoHe,thecorrectedvalueofYissimplydeceasedbymultiplyingbythefraction
ofneutronsthatdonotundergodecay.Thus,thepredictionofNTWIis
Y=e�
tnuc =�d
=exp 8<:� q
gsm
eff
gNTW
I
eff
200
890
9=;;
wheregsm
e�
andgNTWI
e�
aregivenabove.Whenevaluatednumerically,thiswouldgive
Y=PredictedHeabundancebyweight�0:76:
PROBLEM
12:DOUBLING
OFELECTRONS(10points)
Theentropydensityofblack-bodyradiationisgivenby
s=g �2�2
45
k4
(�hc)3 �T3
=gCT3;
whereC
isaconstant.Atthetimewhentheelectron-positronpairsdisappear,
theneutrinosaredecoupled,sotheirentropyisconserved.Alloftheentropyfrom
electron-positronpairsisgiventothephotons,andnonetotheneutrinos.Thesame
willbetruehere,forbothspeciesofelectron-positronpairs.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.62
TheconservedneutrinoentropycanbedescribedbyS� �a3s
� ,whichindicatesthe
entropypercubicnotch,i.e.,entropyperunitcomovingvolume.Weintroducethe
notationn�
andn+
forthenewelectron-likeandpositron-likeparticles,andalso
theconventionthat
Primedquantities:
valuesaftere+e�
n+n�
annihilation
Unprimedquantities:
valuesbeforee+e�
n+n�
annihilation.
Fortheneutrinos,
S0�=S�
=)
g� C(a0T0� )3=g�C(aT�)3
=)
a0T0�=aT�:
Forthephotons,beforee+e�
n+n�
annihilationwehave
T =Te+
e�
n+n�
=T�;
g =2;ge+
e�
=gn+n�
=7=2:
Whenthee+e�
andn+n�
pairsannihilate,theirentropyisaddedtothephotons:
S0 =Se+
e�
+Sn+
n�
+S
=)
2C �a0T0 �3
= �2+2�72 �
C(aT )3
=)
a0T0 = �92 �1=3
aT ;
soaT increasesbyafactorof(9=2)1=3.
Beforee+e�
annihilationtheneutrinoswereinthermalequilibriumwiththephotons,
soT =T� .Byconsideringthetwoboxedequationsabove,onehas
T0�= �29 �1=3
T0 :
Thisratiowouldremainunchangeduntilthepresentday.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.63
PROBLEM
13:TIMESCALESIN
COSMOLOGY
(a)1sec.[Thisisthetimeatwhichtheweakinteractionsbeginto\freezeout",sothat
freeneutrondecaybecomestheonlymechanismthatcaninterchangeprotonsand
neutrons.Fromthistimeonward,therelativenumberofprotonsandneutronsisno
longercontrolledbythermalequilibriumconsiderations.]
(b)4mins.[Bythistimetheuniversehasbecomesocoolthatnuclearreactionsareno
longerinitiated.]
(c)10�
37sec.[WelearnedinLectureNotes7thatkTwasabout1MeVatt=1sec.
Since1GeV=1000MeV,thevalueofkTthatwewantis1019timeshigher.Inthe
radiation-dominatederaT/a�
1/t�
1=2,soweget10�
38sec.]
(d)10,000{1,000,000years.[ThisnumberwasestimatedinLectureNotes7as200,000
years.]
(e)10�
5sec.[Asin(c),wecanuset/T�
2,withkT�1MeVatt=1sec.]
PROBLEM
14:EVOLUTION
OFFLATNESS(15points)
(a)WestartwiththeFriedmannequationfromtheformulasheetonthequiz:
H2= �_aa �2
=8�3
G��kc2
a2
:
Thecriticaldensityisthevalueof�correspondingtok=0,so
H2=8�3
G�c:
UsingthisexpressiontoreplaceH2ontheleft-handsideoftheFriedmannequation,
andthendividingby8�G=3,one�nds
�c=��3kc2
8�Ga2
:
Rearranging,
���c
�
=
3kc2
8�Ga2�:
Ontheleft-handsidewecandividethenumeratoranddenominatorby�c ,andthen
usethede�nition��=�ctoobtain
�1
=
3kc2
8�Ga2�:
(1)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.64
Foramatter-dominateduniverseweknowthat�/1=a3(t),andso
�1
/a(t):
Iftheuniverseisnearly atweknowthata(t)/t2=3,so
�1
/t2=3:
(b)Eq.(1)aboveisstilltrue,soouronlytaskistore-evaluatetheright-handside.For
aradiation-dominateduniverseweknowthat�/1=a4(t),so
�1
/a2(t):
Iftheuniverseisnearly atthena(t)/t1=2,so
�1
/t:
PROBLEM
15:THESLOAN
DIGITALSKY
SURVEY
z
=
5:82QUASAR
(40points)
(a)Sincem
+�
=0:35+0:65=1,theuniverseis at.Itthereforeobeysasimple
formoftheFriedmannequation,
H2= �_aa �2
=8�3
G(�m
+��);
wheretheoverdotindicatesaderivativewithrespecttot,andthetermproportional
tokhasbeendropped.Usingthefactthat�m
/1=a3(t)and��
=const,theenergy
densitiesontheright-handsidecanbeexpressedintermsoftheirpresentvalues
�m;0and��
���;0 .De�ning
x(t)�a(t)
a(t0 );
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.65
onehas
�_xx �2
=8�3
G ��m;0
x3
+�� �
=8�3
G�c;0 �m;0
x3
+�;0 �
=H20 �m;0
x3
+�;0 �:
Hereweusedthefactsthat
m;0 ��m;0
�c;0;
�;0 ���
�c;0
;
and
H20=8�3
G�c;0:
Theequationabovefor(_x=x)2impliesthat
_x=H0x rm;0
x3
+�;0;
whichinturnimpliesthat
dt=
1H0
dx
x qm
;0
x3
+�;0
:
Usingthefactthatxchangesfrom0to1overthelifeoftheuniverse,thisrelation
canbeintegratedtogive
t0= Z
t0
0
dt=
1H0 Z
10
dx
x qm
;0
x3
+�;0
:
Theanswercanalsobewrittenas
t0=
1H0 Z
10
xdx
pm;0 x+�;0 x4
or
t0=
1H0 Z
10
dz
(1+z) pm;0 (1+z)3+�;0
;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.66
whereinthelastanswerIchangedthevariableofintegrationusing
x=
11+z;
dx=�dz
(1+z)2
:
Notethattheminussignintheexpressionfordxiscanceledbytheinterchangeof
thelimitsofintegration:x=0correspondstoz=1,andx=1correspondsto
z=0.
Youranswershouldlooklikeoneoftheaboveboxedanswers.Youwerenotexpected
tocompletethenumericalcalculation,butforpedagogicalpurposesIwillcontinue.
Theintegralcanactuallybecarriedoutanalytically,giving
Z1
0
xdx
pm;0 x+�;0 x4=
2
3 p�;0ln pm
+�;0+ p�;0
pm
!:
Using
1H0
=9:778�109
h0
yr;
whereH0=100h0km-sec�
1-Mpc�
1,one�ndsforh0=0:65that
1H0=15:043�109yr:
Thenusingm
=0:35and�;0=0:65,one�nds
t0=13:88�109yr:
SotheSDSSpeoplewererightontarget.
(b)Havingdonepart(a),thispartisveryeasy.Thedynamicsoftheuniverseisof
coursethesame,andthequestionisonlyslightlydi�erent.Inpart(a)wefoundthe
amountoftimethatittookforxtochangefrom0to1.Thelightfromthequasar
thatwenowreceivewasemittedwhen
x=
11+z;
sincethecosmologicalredshiftisgivenby
1+z=a(tobserved )
a(temitted ):
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.67
Usingtheexpressionfordtfrompart(a),theamountoftimethatittooktheuniverse
toexpandfromx=0tox=1=(1+z)isgivenby
te= Z
te
0
dt=
1H0 Z
1=(1+z)
0
dx
x qm
;0
x3
+�;0
:
Againonecouldwritetheanswerotherways,including
t0=
1H0 Z
1z
dz0
(1+z0) pm;0 (1+z0)3+�;0
:
Againyouwereexpectedtostopwithanexpressionliketheoneabove.Continuing,
however,theintegralcanagainbedoneanalytically:
Zxmax
0
dx
x qm
;0
x3
+�;0
=
2
3 p�;0ln pm
+�;0 x3m
ax+ p�;0x3=2
max
pm
!:
Usingxmax=1=(1+5:82)=:1466andtheothervaluesasbefore,one�nds
te=0:06321
H0
=0:9509�109yr:
SoagaintheSDSSpeoplewereright.
(c)To�ndthephysicaldistancetothequasar,weneedto�gureouthowfarlightcan
travelfromz=5:82tothepresent.Sincewewantthepresentdistance,wemultiply
thecoordinatedistancebya(t0 ).Forthe atmetric
ds2=�c2d�2=�c2dt2+a2(t) �dr2+r2(d�2+sin2�d�2) ;
thecoordinatevelocityoflight(intheradialdirection)isfoundbysettingds2=0,
giving
dr
dt=
ca(t):
Sothetotalcoordinatedistancethatlightcantravelfromtetot0is
`c= Z
t0
te
ca(t)dt:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.68
Thisisnotthe�nalanswer,however,becausewedon'texplicitlyknowa(t).We
can,however,changevariablesofintegrationfromttox,using
dt=
dt
dxdx=dx_x
:
So
`c=
ca(t0 ) Z
1xe
dx
x_x;
wherexeisthevalueofxatthetimeofemission,soxe=1=(1+z).Usingthe
equationfor_xfrompart(a),thisintegralcanberewrittenas
`c=
c
H0 a(t0 ) Z
11=(1+z)
dx
x2 qm
;0
x3
+�;0
:
Finally,then
`phys;0=a(t0 )`c=
cH0 Z
11=(1+z)
dx
x2 qm
;0
x3
+�;0
:
Alternatively,thisresultcanbewrittenas
`phys;0=
cH0 Z
11=(1+z)
dx
pm;0x+�;0x4
;
orbychangingvariablesofintegrationtoobtain
`phys;0=
cH0 Z
z0
dz0
pm;0(1+z0)3+�;0
:
Continuingforpedagogicalpurposes,thistimetheintegralhasnoanalyticform,so
farasIknow.Integratingnumerically,
Z5:82
0
dz0
p0:35(1+z0)3+0:65=1:8099;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.69
andthenusingthevalueof1=H0frompart(a),
`phys;0=27:23light-yr:
Rightagain.
(d)`phys;e=a(te )`c ,so
`phys;e=a(te )
a(t0 )`phys;0=
`phys;0
1+z:
Numericallythisgives
`phys;e=3:992�109light-yr:
TheSDSSannouncementisstillokay.
(e)Thespeedde�nedinthiswayobeystheHubblelawexactly,so
v=H0`phys;0=c Z
z0
dz0
pm;0(1+z0)3+�;0
:
Then
vc= Z
z0
dz0
pm;0(1+z0)3+�;0
:
Numerically,wehavealreadyfoundthatthisintegralhasthevalue
vc=1:8099:
TheSDSSpeoplegetanA.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.70
PROBLEM
16:SECOND
HUBBLECROSSING
(40points)
(a)Fromtheformulasheets,weknowthatfora atradiation-dominateduniverse,
a(t)/t1=2:
Since
H=
_aa;
(whichisalsoontheformulasheets),H
=
12t:
Then
`H(t)�cH�
1(t)=
2ct:
(b)Wearetoldthattheenergydensityisdominatedbyphotonsandneutrinos,sowe
needtoaddtogetherthesetwocontributionstotheenergydensity.Forphotons,the
formulasheetremindsusthatg =2,so
u =2�2
30(kT )4
(�hc)3
:
Forneutrinostheformulasheetremindsusthat
g�=
78|{z}
Fermion
factor
�
3|{z}
3species
�e;��;�� �
2|{z}
Particle=
antiparticle �
1|{z}
Spinstates
=
214;
so
u�=214�2
30(kT�)4
(�hc)3
:
CombiningthesetwoexpressionsandusingT�=(4=11)1=3T ,onehas
u=u +u�= "2+214 �411 �4=3 #�2
30(kT )4
(�hc)3
;
so�nally
g1=2+214 �411 �4=3
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.71
(c)TheFriedmannequationtellsusthat,fora atuniverse,
H2=8�3
G�;
whereinthiscaseH=1=(2t)and
�=
uc2=g1�2
30
(kT )4
�h3c5
:
Thus
�12
t �2
=8�G3
g1�2
30
(kT )4
�h3c5
:
SolvingforT ,
T =1k �45�h3c5
16�3g1 G �
1=4
1pt:
(d)TheconditionforHubblecrossingis
�(t)=cH�
1(t);
andthe�rstHubblecrossingalwaysoccursduringthein ationaryera.Thusany
Hubblecrossingduringtheradiation-dominatederamustbethesecondHubble
crossing.
If�isthepresentphysicalwavelengthofthedensityperturbationsunderdiscussion,
thewavelengthattimetisscaledbythescalefactora(t):
�(t)=
a(t)
a(t0 )�:
BetweenthesecondHubblecrossingandnow,therehavebeennofreeze-outsof
particlespecies.Todaytheentropyoftheuniverseisstilldominatedbyphotonsand
neutrinos,sotheconservationofentropyimpliesthataT
hasremainedessentially
constantbetweenthenandnow.Thus,
�(t)=
T ;0
T (t)�:
UsingthepreviousresultsforcH�
1(t)andforT (t),thecondition�(t)=cH�
1(t)
canberewrittenas
kT ;0 �16�3g
1 G
45�h3c5 �
1=4p
t�=2ct:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.72
Solvingfort,thetimeofsecondHubblecrossingisfoundtobe
tH2 (�)=(kT ;0 �)2 ��3g
1 G
45�h3c9 �
1=2
:
Extension:Youwerenotaskedtoinsertnumbers,butitisofcourseinterestingto
knowwheretheaboveformulaleads.Ifwetake�=106lt-yr,itgives
tH2 (106lt-yr)=1:04�107s=0:330year:
For�=1Mpc,
tH2 (1Mpc)=1:11�108s=3:51year:
Taking�=2:5�106lt-yr,thedistancetoAndromeda,thenearestspiralgalaxy,
tH2 (2:5�106lt-yr)=6:50�107sec=2:06year:
PROBLEM
17:THEEVENTHORIZON
FOROURUNIVERSE(25points)
(a)Inasphericalpulseeachlightrayismovingradiallyoutward,sod�=d�=0.A
lightraytravelsalonganulltrajectory,meaningthatds2=0,sowehave
ds2=�c2dt2+a2(t)dr2=0:
(3.1)
fromwhichitfollowsthat
dr
dt=�c
a(t):
(3.2)
Weareinterestedinaradialpulsethatstartsatr=0attimet=t0 ,sothelimiting
valueofrisgivenby
rmax= Z
1t0
ca(t)dt:
(3.3)
(b)Changingvariablesofintegrationto
x=
a(t)
a(t0 );
(3.4)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.73
theintegralbecomes
rmax= Z
11
ca(t)
dt
dxdx=
ca(t0 ) Z
11
1xdt
dxdx;
(3.5)
whereweusedthefactthatt=t0correspondstox=a(t0 )=a(t0 )=1.Asgivento
usontheformulasheet,the�rst-orderFriedmannequationcanbewrittenas
xdxd
t=H0 qm;0 x+rad;0+vac;0 x4+k;0 x2:
(3.6)
Usingthissubstitution,
rmax=
c
a(t0 )H0 Z
11
dx
pm;0 x+rad;0+vac;0 x4
;
(3.7)
wherewehaveusedk;0=0,sincetheuniverseistakentobe at.
(c)To�ndthevalueoftheredshiftforthelightthatwearepresentlyreceivingfrom
coordinatedistancermax ,wecanbeginbynoticingthatthetimeofemissiontecan
bedeterminedbytheequationwhichimpliesthatthecoordinatedistancetraveled
byalightpulsebetweentimesteandt0mustequalrmax .UsingEq.(3.2)forthe
coordinatevelocityoflight,thisequationreads
Zt0
te
ca(t)dt=rmax:
(3.8)
The\half-credit"answertothequizproblem
wouldincludetheaboveequation,
followedbythestatementthattheredshiftzehcanbedeterminedfrom
z=a(t0 )
a(te ) �1:
(3.9)
The\full-credit"answerisobtainedbychangingthevariableofintegrationasin
part(b),soEq.(3.8)becomesr
max= Z
1xe
ca(t)
dt
dxdx
=
ca(t0 ) Z
1xe
1xdt
dxdx;
(3.10)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.74
wherexe
isthevalueofxcorrespondingtot=
te .ThenusingEq.(3.6)with
k;0=0,we�ndr
max=
c
a(t0 )H0 Z
1xe
dx
pm;0 x+rad;0+vac;0 x4
:
(3.11)
Tocompletetheanswerinthislanguage,weuse
z=
1xe �1:
(3.12)
Eqs.(3.11)and(3.12)constituteafullanswertothequestion,butonecouldgo
furtherandreplacermaxusingEq.(3.7),�nding
Z1
1
dx
pm;0 x+rad;0+vac;0 x4
= Z1
xe
dx
pm;0 x+rad;0+vac;0 x4
:
(3.13)
InthisformtheanswerdependsonlyonthevaluesofX;0 .
Youwereofcoursenotaskedtoevaluatethisformulanumerically,butyoumight
beinterestedinknowingthatthePlanck2013valuesm;0=0:315,vac;0=0:685,
andrad;0=9:2�10�
5leadtozeh=1:87.Thus,noeventthatishappeningnow
(i.e.,atthesamevalueofthecosmictime)inagalaxyatredshiftlargerthan1.87
willeverbevisibletousorourdescendants,eveninprinciple.
PROBLEM
18:THEEFFECTOFPRESSUREONCOSMOLOGICALEVO-
LUTION
(25points)
(a)(8points)Thisproblemisansweredmosteasilybystartingfromthecosmological
formulaforenergyconservation,whichIremembermosteasilyintheformmotivated
bydU=�pdV.Usingthefactthattheenergydensityuisequalto�c2,theenergy
conservationrelationcanbewritten
dUd
t=�pdVd
t
=)
ddt ��c2a3 �=�pdd
t �a3 �:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.75
Setting
�=
�a8
forsomeconstant�,theconservationofenergyformulabecomes
ddt �
�c2
a5 �=�pdd
t �a3 �;
whichimplies
�5�c2
a6
dad
t=�3pa2dad
t:
Thus
p=53�c2
a8
=
53�c2:
Alternatively,onemaystartfromtheequationforthetimederivativeof�,
_�=�3_aa �
�+
pc2 �:
Since�=
�a8 ,wetakethetimederivativeto�nd_�=�8(_a=a)�,andtherefore
�8_aa
�=�3_aa �
�+
pc2 �;
andtherefore
p=53
�c2:
(b)(9points)Fora atuniverse,theFriedmannequationreducesto
�_aa �2
=8�3
G�:
Using�/1=a8,thisimpliesthat
_a=
�a3
;
forsomeconstant�.Rewritingthisas
a3da=�dt;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.76
wecanintegratetheequationtogive
14a4=�t+const;
wheretheconstantofintegrationhasnoe�ectotherthantoshifttheoriginofthe
timevariablet.Usingthestandardbigbangconventionthata=0whent=0,the
constantofintegrationvanishes.Thus,
a/t1=4:
Thearbitraryconstantofproportionalityinthisanswerisconsistentwiththe
wordingoftheproblem,whichstatesthat\Youshouldbeabletodeterminethe
functiona(t)uptoaconstantfactor."
Notethatwecouldhaveexpressedthe
constantofproportionalityintermsoftheconstant�thatweusedinpart(a),
buttherewouldnotreallybeanypointindoingthat.Theconstant�wasnota
givenvariable.Ifthecomovingcoordinatesaremeasuredin\notches,"thenais
measuredinmeterspernotch,andtheconstantofproportionalityinouranswercan
bechangedbychangingthearbitraryde�nitionofthenotch.
(c)(8points)Westartfromtheconservationofenergyequationintheform
_�=�3_aa �
�+
pc2 �:
Substituting_�=�n(_a=a)�andp=(2=3)�c2,wehave
�nH�=�3H �53
� �
andtherefore
n=5:
PROBLEM
19:THEFREEZE-0UTOFA
FICTITIOUSPARTICLEX
(25
points)
(a)(5points)Theformulasheettellsusthattheenergydensityofblack-bodyradiation
is
u=g�2
30(kT)4
(�hc)3
;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.77
where
g� �1perspinstateforbosons(integerspin)
7/8perspinstateforfermions(half-integerspin).
SincetheX
isspin-1,and1isaninteger,theX
particlesarebosonsandg=1per
spinstate.Thereare3species,X+,X�
,andX0,andeachspecieswearetoldhas
threespinstates,sothereareatotalof9spinstates,sog=9.Thus,
u=9�2
30(kT)4
(�hc)3
:
Alternatively,onecouldcounttheX+
andX�
asonespecieswithadistinct
particleandantiparticle,sogX+X�
isgivenby
gX+X�
=
1|{z}
Fermion
factor
�1
|{z}Species �
2|{z}
Particle=
antiparticle �
3|{z}
Spinstates
=
6:
TheX0isitsownantiparticle,whichmeansthattheparticle/antiparticlefactoris
one,so
gX0
=
1|{z}
Fermion
factor
�1
|{z}Species �
1|{z}
Particle=
antiparticle �
3|{z}
Spinstates
=
3;
sothetotalgforX+,X�
,andX0isagainequalto9.
(b)(5points)Theformulasheettellsusthatthenumberdensityofparticlesinblack-
bodyradiationis
n=g�
�(3)
�2
(kT)3
(�hc)3
;
where
g�� �1perspinstateforbosons
3/4perspinstateforfermions.
Forbosonsg�
=g,sog�
fortheXparticlesis9.Then
nX
=9�(3)
�2
(kT)3
(�hc)3
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.78
(c)(10points)Wearetoldthat,whentheXparticlesfreezeout,alloftheirenergyand
entropyisgiventothephotons.Weuseentropyratherthanenergytodetermine
the�naltemperatureofthephotons,becausetheentropyinacomovingvolumeis
simplyconserved,whiletheenergydensityvariesas
_�=�3_aa �
�+
pc2 �:
Thus,totracktheenergy,weneedtoknowexactlyhowpbehaves,andthebehavior
ofpduringfreeze-outiscomplicated,andwehavenotcalculateditinthiscourse.
Theformulasheettellsusthattheentropydensityofaconstituentofblack-body
radiationisgivenby
s=g2�2
45
k4T3
(�hc)3
:
Ifweconsidersome�xedcoordinatevolumeVcoord ,thecorrespondingphysicalvol-
umeisVphys=
Vcoorda3(t),wherea(t)isthescalefactor.Thetotalentropyof
neutrinosinVcoordisthenS
�=g�2�2
45
k4T3� (t)
(�hc)3
Vcoorda3(t):
ThequantitiesT� (t)anda(t)dependontime,buttheexpressionontheright-hand-
sidedoesnot,sinceentropyisconserved.ForbrevityIwillwrite
S�=g� A(t)T3�(t);
(1)
where
A(t)�2�2
45
k4
(�hc)3Vcoorda3(t):
Thee+e�
pairsandtheX'scontributetotheblack-bodyradiationonlybefore
thefreeze-out,whenkT�0:511MeV/c2.Lettbdenoteanytimebeforethefreeze-
out.Beforethefreeze-out,thetotalentropyofphotons,e+e�
pairs,andXparticles
isgivenby
Sbefore; eX
=(g +ge+
e�
+gX)A(tb )T3 (tb ):
(2)
IcancallthetemperatureT ,becausethee+e�
pairsandtheX's(aswellasthe
neutrinos)areallinthermalequilibrium
atthispoint,sotheyallhavethesame
temperature.
Usingta
todenoteanarbitrarytimeafterthefreeze-out,theentropyofthe
photonsduringthistimeperiodcanbewritten
Safter; =g A(ta )T3 (ta ):
(3)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.79
Butsincethee+e�
pairsandX
particlesgivealltheirentropytothephotons,we
have
Safter; =Sbefore; eX
:
(4)
ThenusingEqs.(2)and(3)we�nd
g A(ta )T3 (t
a )=(g +ge+e�
+gX)A(tb )T3 (t
b ):
(5)
WecanrewritethelastfactorinEq.(5)byrememberingthatEq.(1)holdsatall
times,andthatT� (tb )=T (tb ).So,
A(tb )T3 (tb )=A(tb )T3�(tb )=S�
g�
=A(ta)T3�(ta ):
(6)
SubstitutingEq.(6)intoEq.(5),wehave
g A(ta )T3 (ta )=(g +ge+
e�
+gX)A(ta )T3�(ta );
fromwhichweseethat
T3 (ta )=g +ge+e�
+gX
g
T3�(ta );
andtherefore
T� (ta )
T (ta )= �g
g +ge+
e�
+gX �
1=3
= �2
2+72+9 �
1=3
=
�429 �1=3
:
(d)(5points)Theanswerwouldbethesame,sinceitwascompletelydeterminedbythe
conservationequation,Eq.(4)intheaboveanswer.Regardlessoftheorderinwhich
thefreeze-outsoccurred,thetotalentropyfromthee+e�
pairsandtheX'swould
ultimatelybegiventothephotons,sotheamountofheatingofthephotonswould
bethesame.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.80
PROBLEM
20:THETIMEtd
OFDECOUPLING
(25points)
(a)(5points)Iftheentropyofphotonsisconserved,thentheentropydensityfallsas
s/1
a3(t):
Sinces/T3,itfollowsthat
T/1
a(t):
Thus,theratioofthescalefactorsisequaltotheinverseoftheratiotemperatures:
xd=T0
Td
:
(b)(5points)Theformulasheetremindsusthat
xdxd
t=H0 qm;0 x+rad;0+vac;0 x4+k;0 x2;
where
k;0 ��
kc2
a2(t0 )H20
=1�m;0 �rad;0 �vac;0:
Sofora atuniversek;0=0,andwehave
dxd
t=H0
x qm;0 x+rad;0+vac;0 x4:
(c)(5points)Theanswertopart(b)canberewrittenas
dt=
xdx
H0 pm;0 x+rad;0+vac;0 x4
:
tdisthetimethatelapsesfromwhentheuniversehasx=0towhenithasx=xd ,
so
td=
1H0 Z
xd
0
xdx
pm;0 x+rad;0+vac;0 x4
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.81
Youwereofcoursenotaskedtoevaluatethisintegralnumerically,butwewill
dothatnow.WetakeT0
=2:7255KfromFixsenetal.(citedinLectureNotes
6)andthePlanck2015best�tvaluesofH0=67:7km-s�
1-Mpc�
1,m;0=0:309,
vac;0=0:691.Theenergydensityofradiation(photonsplusneutrinos)canthen
becalculatedtogiverad;0
=9:2�10�
5
(seeEq.(6.23)ofLectureNotes6and
thetextofthe2ndparagraphofp.12ofLectureNotes7).Tokeepourmodel
universeexactly at,Iammodifyingvac;0tosetitequalto0:691�rad;0 ,whichis
wellwithintheuncertainties.Numericalintegrationthengives366,000years,very
closetoouroriginalestimate.Ofcoursethisnumberisstillapproximate,sincewe
startedwithTd �3000K.Inanycase,thedecouplingofthephotonsintheCMBis
actuallyagradualprocess.In2003Imodi�edastandardprogramcalledCMBFast
tocalculatetheprobabilitydistributionofthetimeoflastscattering(publishedin
https://arxiv.org/abs/astro-ph/0306275),withthefollowingresults:
Theparametersusedwerevac;0=0:70,m;0=0:30,H0
=68km-s�
1-Mpc�
1.The
peakofthecurveisat367,000years,andthemedianisat388,000years.
(d)(10points)Thederivationstartswiththe�rst-orderFriedmannequation.Sincewe
aredescribinga atuniverse,wecanstartwiththeFriedmannequationfora at
universe,
H2=8�3
G�:
Nowweusethefactsthat�m
/1=a3,�rad /1=a4,�vac /1,and�f /1=a8towrite
H2=8�3
G h�m;0
x3
+�rad;0
x4
+�vac;0+�f;0
x8 i:
Thenweuse
�m;0=�c m;0=3H20
8�Gm;0 ;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2020
p.82
withsimilarrelationsfortheothercomponentsofthemassdensity,torewritethe
Friedmannequationas
H2=H20 �m;0
x3
+rad;0
x4
+vac;0+f;0
x8 �:
NextwerewriteH2as
H2= �_aa �2
= �_xx �2
;
so
�_xx �2
=H20 �m;0
x3
+rad;0
x4
+vac;0+f;0
x8 �;
whichcanberewrittenas
xdxd
t=H0 rm;0 x+rad;0+vac;0 x4+f;0
x4
:
Fromherethederivationisidenticaltothatinpart(c),leadingto
td=
1H0 Z
xd
0
xdx
qm;0 x+rad;0+vac;0 x4+f;0
x4
;
whichcanalsobewrittenmoreneatlyas
td=
1H0 Z
xd
0
x3dx
pm;0 x5+rad;0 x4+vac;0 x8+f;0
: