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MASSACHUSETTSINSTITUTEOFTECHNOLOGY
PhysicsDepartment
Physics8.286:TheEarlyUniverse
December2,2016
Prof.AlanGuthR
EVIEW
PROBLEMSFOR
QUIZ3
QUIZDATE:Wedneday,December7,2016,duringthenormalclasstime.
COVERAGE:LectureNotes6(pp.12{end),LectureNotes7and8.Problem
Sets7{9;StevenWeinberg,TheFirstThreeMinutes,Chapter8andtheAf-
terword;BarbaraRyden,IntroductiontoCosmology,Chapters9(TheCosmic
MicrowaveBackground)and11(In ationandtheVeryEarlyUniverse);Alan
Guth,In ationandtheNewEraofHigh-PrecisionCosmology,
http://web.mit.edu/physics/news/physicsatmit/physicsatmit_
02_
cosmology.pdf.
Oneoftheproblemsonthequizwillbetakenverbatim
(oratleast
alm
ostverbatim)fromeitherthehomeworkassignments,orfromthe
starredproblemsfrom
thissetofReviewProblems.Thestarredprob-
lemsaretheonesthatIrecommendthatyoureviewmostcarefully:Problems
2,4,5,7,8,10,11,and14.
PURPOSE:Thesereviewproblemsarenottobehandedin,butarebeingmade
availabletohelpyoustudy.Theycomemainlyfromquizzesinpreviousyears.
Insomecasesthenumberofpointsassignedtotheproblemonthequizislisted
|
inallsuchcasesitisbasedon100pointsforthefullquiz.
Inadditiontothissetofproblems,youwill�ndonthecoursewebpagethe
actualquizzesthatweregivenin1994,1996,1998,2000,2002,2004,2007,
2009,2011,and2013.Therelevantproblemsfromthosequizzeshavemostly
beenincorporatedintothesereviewproblems,butyoustillmaybeinterested
inlookingatthequizzes,justtoseehowmuchmaterialhasbeenincludedin
eachquiz.Thecoverageoftheupcomingquizwillnotnecessarilymatchthe
coverageofanyofthequizzesfrompreviousyears.Thecoverageforeachquiz
inrecentyearsisusuallydescribedatthestartofthereviewproblems,asIdid
here.
REVIEW
SESSIONANDOFFICEHOURS:Tohelpyoustudyforthequiz,
VictorLiwillholdareviewsessiononSunday,December4,at7:30pm,in
Room
4-237.Inaddition,VictorLiwillbeholdingspecialoÆcehourson
Monday,December5,at4:00pminRoom4-163(ourregularclassroom),and
alsoonTuesday,December6,at5:00pminthesameroom.I(AlanGuth)
willunfortunatelybeoutoftown(inSanFranciscofortheBreakthroughPrizes
AwardCeremonyandSymposium)untilTuesdaynight,butIwilltrytoanswer
emailstotheextentthattimeallows.
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.2
INFORMATION
TO
BEGIVEN
ON
QUIZ:
Forthethirdquiz,thefollowinginformationwillbemadeavailabletoyou:
DOPPLER
SHIFT(Formotionalongaline):
z=v=u
(nonrelativistic,sourcemoving)
z=
v=u
1�v=u
(nonrelativistic,observermoving)
z= s1+�
1���1
(specialrelativity,with�=v=c)
COSMOLOGICALREDSHIFT:
1+z��observed
�emitted
=a(tobserved )
a(temitted )
SPECIALRELATIVITY:
TimeDilationFactor:
�
1
p1��2
;
��v=c
Lorentz-FitzgeraldContractionFactor:
RelativityofSimultaneity:
Trailingclockreadslaterbyanamount�`0 =c.
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
:
KINEMATICSOF
A
HOMOGENEOUSLY
EXPANDING
UNIVERSE:
Hubble'sLaw:v=Hr,
wherev=recessionvelocityofadistantobject,H=Hubble
expansionrate,andr=distancetothedistantobject.
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.3
PresentValueofHubbleExpansionRate(Planck2015):
H0=67:7�0:5km-s�
1-Mpc�
1
ScaleFactor:`p (t)=a(t)`c;
where`p (t)isthephysicaldistancebetweenanytwoobjects,
a(t)isthescalefactor,and`c
isthecoordinatedistance
betweentheobjects,alsocalledthecomovingdistance.
HubbleExpansionRate:H(t)=
1a(t)
da(t)
dt
.
LightRaysinComovingCoordinates:
Lightraystravelin
straightlineswithspeeddxd
t=
ca(t).
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
:
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.4
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:
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~acanbecalledaifthereis
noneedtorelateittothea(t)thatappearsinthe�rstequation
above.
SCHWARZSCHILD
METRIC:
ds2=�c2d�2=� �1�2GM
rc2 �c2dt2+ �1�2GM
rc2 �
�
1dr2
+r2d�2+r2sin2�d�2;
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.5
GEODESICEQUATION:
dds �gijdxj
ds �=12
(@i gk` )dxk
ds
dx`
ds
or:
dd� �g��dx�
d� �=12
(@�g��)dx�
d�
dx�
d�
BLACK-BODY
RADIATION:
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)
where
g� (1perspinstateforbosons(integerspin)
7/8perspinstateforfermions(half-integerspin)
g�� (1perspinstateforbosons
3/4perspinstateforfermions,
and
�(3)=
113+
123+
133+����1:202:
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.6
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;
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:
EVOLUTION
OF
A
FLAT
RADIATION-DOMINATED
UNIVERSE:
�=
3
32�Gt2
kT= �45�h3c5
16�3gG �
1=4
1pt
Form�=106MeV�kT�me=0:511MeV,g=10:75and
then
kT=
0:860MeV
pt(insec) �
10:75
g �1=4
Afterthefreeze-outofelectron-positronpairs,
T�
T
= �411 �1=3
:
COSMOLOGICALCONSTANT:
uvac=�vac c2=
�c4
8�G
;
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.7
pvac=��vac c2=��c4
8�G
:
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
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.8
=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:
PlanckUnits:ThePlancklength`P,thePlancktimetP,thePlanck
massmP,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:
CHEMICALEQUILIBRIUM:
(Thistopicwasnotincludedinthecoursein2013,buttheformu-
lasarenonethelessincludedhereforlogicalcompleteness.They
willnotberelevanttoQuiz3.TheyarerelevanttoProblem13
intheseReviewProblems,whichisalsonotrelevanttoQuiz3.
Pleaseenjoylookingattheseitems,orenjoyignoringthem!)
IdealGasofClassicalNonrelativisticParticles:
ni=gi (2�mi kT)3=2
(2��h)3
e(�i�
mi c2)=kT
:
whereni=numberdensityofparticle
gi=numberofspinstatesofparticle
mi=massofparticle
�i=chemicalpotential
Foranyreaction,thesumofthe�iontheleft-handsideofthe
reactionequationmustequalthesumofthe�iontheright-hand
side.Formulaassumesgasisnonrelativistic(kT�mi c2)and
dilute(ni �(2�mi kT)3=2=(2��h)3).
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.9
PROBLEM
LIST
1.DidYouDotheReading(2007)?
............
10(Sol:24)
*2.DidYouDotheReading(2009)?
............
11(Sol:26)
3.DidYouDotheReading(2013)?
............
13(Sol:28)
*4.NumberDensitiesintheCosmicBackgroundRadiation...
14(Sol:30)
*5.PropertiesofBlack-BodyRadiation
...........
15(Sol:31)
6.ANewSpeciesofLepton
................
15(Sol:33)
*7.ANewTheoryoftheWeakInteractions
.........
16(Sol:36)
*8.DoublingofElectrons
.................
17(Sol:42)
9.TimeScalesinCosmology
...............
18(Sol:44)
*10.EvolutionofFlatness..................
18(Sol:44)
*11.TheSloanDigitalSkySurveyz=5:82Quasar.......
19(Sol:45)
12.SecondHubbleCrossing
................
20(Sol:51)
13.NeutrinoNumberandtheNeutron/ProtonEquilibrium
...
21(Sol:53)
*14.TheEventHorizonforOurUniverse...........
23(Sol:56)
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.10
PROBLEM
1:DID
YOU
DO
THEREADING?(25points)
Thefollowingproblem
wasProblem
1,Quiz3,in2007.Eachpartwasworth5
points.
(a)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBisnot
correct:
(i)AfterthedipoledistortionoftheCMBissubtractedaway,themeantem-
peratureaveragingovertheskyishTi=2:725K.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,therootmean
squaretemperature uctuationis D�ÆTT �2 E1=2
=1:1�10�
3.
(iii)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionof
theobserverrelativetoaframeofreferenceinwhichtheCMBisisotropic.
(iv)Intheirgroundbreakingpaper,WilsonandPenziasreportedthemeasure-
mentofanexcesstemperatureofabout3.5Kthatwasisotropic,unpolar-
ized,andfreefromseasonalvariations.Inacompanionpaperwrittenby
Dicke,Peebles,RollandWilkinson,theauthorsinterpretedtheradiation
tobearelicofanearly,hot,dense,andopaquestateoftheuniverse.
(b)(CMBexperiments)ThecurrentmeanenergyperCMBphoton,about6�
10�
4
eV,iscomparabletotheenergyofvibrationorrotationforasmall
moleculesuchasH2 O.
Thusmicrowaveswithwavelengthsshorterthan
��3cm
arestronglyabsorbedbywatermoleculesintheatmosphere.To
measuretheCMBat�<3cm,whichoneofthefollowingmethodsisnota
feasiblesolutiontothisproblem?
(i)MeasureCMBfromhigh-altitudeballoons,e.g.MAXIMA.
(ii)MeasureCMBfromtheSouthPole,e.g.DASI.
(iii)MeasureCMBfromtheNorthPole,e.g.BOOMERANG.
(iv)MeasureCMBfrom
asatelliteabovetheatmosphereoftheEarth,e.g.
COBE,WMAPandPLANCK.
(c)(Temperature uctuations)Thecreationoftemperature uctuationsinCMB
byvariationsinthegravitationalpotentialisknownastheSachs-Wolfee�ect.
Whichoneofthefollowingstatementsisnotcorrectconcerningthise�ect?
(i)ACMBphotonisredshiftedwhenclimbingoutofagravitationalpotential
well,andisblueshiftedwhenfallingdownapotentialhill.
(ii)Atthetimeoflastscattering,thenonbaryonicdarkmatterdominatedthe
energydensity,andhencethegravitationalpotential,oftheuniverse.
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.11
(iii)Thelarge-scale uctuationsinCMBtemperaturesarisefrom
thegrav-
itationale�ectofprimordialdensity uctuationsinthedistributionof
nonbaryonicdarkmatter.
(iv)Thepeaksintheplotoftemperature uctuation�T
vs.multipolelare
duetovariationsinthedensityofnonbaryonicdarkmatter,whilethe
contributionsfrombaryonsalonewouldnotshowsuchpeaks.
(d)(Darkmattercandidates)Whichoneofthefollowingisnotacandidateof
nonbaryonicdarkmatter?
(i)massiveneutrinos
(ii)axions
(iii)mattermadeoftopquarks(atypeofquarkswithheavymassofabout
171GeV).
(iv)WIMPs(WeaklyInteractingMassiveParticles)
(v)primordialblackholes
(e)(Signaturesofdarkmatter)Bywhatmethodscansignaturesofdarkmatter
bedetected?Listtwomethods.(Grading:3pointsforonecorrectanswer,
5pointsfortwocorrectanswers.Ifyougivemorethantwoanswers,your
scorewillbebasedonthenumberofrightanswersminusthenumberofwrong
answers,withalowerboundofzero.)
�
PROBLEM
2:DID
YOU
DO
THEREADING?(25points)
ThisproblemwasProblem1,Quiz3,2009.
(a)(10points)Thisquestionconcernssomenumbersrelatedtothecosmicmi-
crowavebackground(CMB)thatoneshouldneverforget.Statethevaluesof
thesenumbers,towithinanorderofmagnitudeunlessotherwisestated.Inall
casesthequestionreferstothepresentvalueofthesequantities.
(i)TheaveragetemperatureToftheCMB(towithin10%).
(ii)ThespeedoftheLocalGroupwithrespecttotheCMB,expressedasa
fractionv=cofthespeedoflight.(ThespeedoftheLocalGroupisfound
bymeasuringthedipolepatternoftheCMBtemperaturetodetermine
thevelocityofthespacecraftwithrespecttotheCMB,andthenremoving
spacecraftmotion,theorbitalmotionoftheEarthabouttheSun,theSun
aboutthegalaxy,andthegalaxyrelativetothecenterofmassoftheLocal
Group.)
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.12
(iii)Theintrinsicrelativetemperature uctuations�T=T,afterremovingthe
dipoleanisotropycorrespondingtothemotionoftheobserverrelativeto
theCMB.
(iv)Theratioofbaryonnumberdensitytophotonnumberdensity,�=
nbary =n .
(v)Theangularsize�H,indegrees,correspondingtowhatwastheHubble
distancec=Hatthesurfaceoflastscattering.Thisanswermustbewithin
afactorof3tobecorrect.
(b)(3points)Becausephotonsoutnumberbaryonsbysomuch,theexponential
tailofthephotonblackbodydistributionisimportantinionizinghydrogen
wellafterkT fallsbelowQH
=13:6eV.WhatistheratiokT =QH
whenthe
ionizationfractionoftheuniverseis1=2?
(i)1=5
(ii)1=50
(iii)10�
3
(iv)10�
4
(v)10�
5
(c)(2points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearredderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearbluerbecauseoftheDopplere�ect.
(iii)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
redderbecausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
bluerbecausetheymustclimboutofthegravitationalpotentialwell.
(v)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
redderbecauseofabsorptionintheintergalacticmedium.
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
bluerbecauseofabsorptionintheintergalacticmedium.
(d)(10points)Foreachofthefollowingstatements,saywhetheritistrueorfalse:
(i)Darkmatterinteractsthroughthegravitational,weak,andelectromag-
neticforces.
T
orF?
(ii)Thevirialtheoremcanbeappliedtoaclusterofgalaxiesto�nditstotal
mass,mostofwhichisdarkmatter.
T
orF?
(iii)Neutrinosarethoughttocompriseasigni�cantfractionoftheenergyden-
sityofdarkmatter.
T
orF?
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.13
(iv)Magneticmonopolesarethoughttocompriseasigni�cantfractionofthe
energydensityofdarkmatter.
T
orF?
(v)LensingobservationshaveshownthatMACHOscannotaccountforthe
darkmatteringalactichalos,butthatasmuchas20%ofthehalomass
couldbeintheformofMACHOs.
T
orF?
PROBLEM
3:DID
YOU
DO
THEREADING?(35points)
ThiswasProblem1ofQuiz3,2013.
(a)(5points)RydensummarizestheresultsoftheCOBEsatelliteexperimentfor
themeasurementsofthecosmicmicrowavebackground(CMB)intheformof
threeimportantresults.The�rstwasthat,inanyparticulardirectionofthe
sky,thespectrumoftheCMBisveryclosetothatofanidealblackbody.The
FIRASinstrumentontheCOBEsatellitecouldhavedetecteddeviationsfrom
theblackbodyspectrumassmallas��=��10�
n,wherenisaninteger.To
within�1,whatisn?
(b)(5points)Thesecondresultwasthemeasurementofadipoledistortionof
theCMBspectrum;thatis,theradiationisslightlyblueshiftedtohighertem-
peraturesinonedirection,andslightlyredshiftedtolowertemperaturesin
theoppositedirection.Towhatphysicale�ectwasthisdipoledistortionat-
tributed?
(c)(5points)Thethirdresultconcernedthemeasurementoftemperature uctu-
ationsafterthedipolefeaturementionedabovewassubtractedout.De�ning
ÆTT(�;�)�T(�;�)�hTi
hTi
;
wherehTi=2:725K,theaveragevalueofT,theyfoundarootmeansquare
uctuation,
*�ÆTT �2 +1=2
;
equaltosomenumber.Towithinanorderofmagnitude,whatwasthatnum-
ber?
(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearredderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearbluerbecauseoftheDopplere�ect.
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.14
(iii)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
redderbecausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
bluerbecausetheymustclimboutofthegravitationalpotentialwell.
(v)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
redderbecauseofabsorptionintheintergalacticmedium.
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
bluerbecauseofabsorptionintheintergalacticmedium.
(e)(5points)The atnessproblemreferstotheextreme�ne-tuningthatisneeded
inatearlytimes,inorderforittobeascloseto1todayasweobserve.
Startingwiththeassumptionthattodayisequalto1withinabout1%,one
concludesthatatonesecondafterthebigbang,
j�1jt=1sec<10�
m
;
wheremisaninteger.Towithin�3,whatism?
(f)(5points)Thetotalenergydensityofthepresentuniverseconsistsmainlyof
baryonicmatter,darkmatter,anddarkenergy.Givethepercentagesofeach,
accordingtothebest�tobtainedfromthePlanck2013data.Youwillgetfull
creditifthe�rst(baryonicmatter)isaccurateto�2%,andtheothertwoare
accuratetowithin�5%.
(g)(5points)Withintheconventionalhotbigbangcosmology(withoutin ation),
itisdiÆculttounderstandhowthetemperatureoftheCMBcanbecorrelated
atangularseparationsthataresolargethatthepointsonthesurfaceoflast
scatteringwasseparatedfromeachotherbymorethanahorizondistance.Ap-
proximatelywhatangle,indegrees,correspondstoaseparationonthesurface
lastscatteringofonehorizonlength?Youwillgetfullcreditifyouransweris
righttowithinafactorof2.
�
PROBLEM
4:
NUMBER
DENSITIESIN
THE
COSMIC
BACK-
GROUND
RADIATION
Todaythetemperatureofthecosmicmicrowavebackgroundradiationis2:7ÆK.
Calculatethenumberdensityofphotonsinthisradiation.Whatisthenumber
densityofthermalneutrinosleftoverfromthebigbang?
8.286QUIZ3REVIEW
PROBLEMS,FALL2016
p.15
�
PROBLEM
5:PROPERTIESOF
BLACK-BODY
RADIATION
(25
points)
ThefollowingproblemwasProblem4,Quiz3,1998.
Inansweringthefollowingquestions,rememberthatyoucanrefertothefor-
mulasatthefrontoftheexam.Sinceyouwerenotaskedtobringcalculators,you
mayleaveyouranswersintheformofalgebraicexpressions,suchas�32= p5�(3).
(a)(5points)Fortheblack-bodyradiation(alsocalledthermalradiation)ofpho-
tonsattemperatureT,whatistheaverageenergyperphoton?
(b)(5points)Forthesameradiation,whatistheaverageentropyperphoton?
(c)(5points)Nowconsidertheblack-bodyradiationofamasslessbosonwhichhas
spinzero,sothereisonlyonespinstate.Wouldtheaverageenergyperparticle
andentropyperparticlebedi�erentfromtheanswersyougaveinparts(a)
and(b)?Ifso,howwouldtheychange?
(d)(5points)Nowconsidertheblack-bodyradiationofelectronneutrinos.These
particlesarefermionswithspin1/2,andwewillassumethattheyaremassless
andhaveonlyonepossiblespinstate.Whatistheaverageenergyperparticle
forthiscase?
(e)(5points)Whatistheaverageentropyperparticlefortheblack-bodyradiation
ofneutrinos,asdescribedinpart(d)?
PROBLEM
6:A
NEW
SPECIESOFLEPTON
ThefollowingproblemwasProblem2,Quiz3,1992,worth25points.
Supposethecalculationsdescribingtheearlyuniverseweremodi�edbyinclud-
inganadditional,hypotheticallepton,calledan8.286ion.The8.286ionhasroughly
thesamepropertiesasanelectron,exceptthatitsmassisgivenbymc2=0:750
MeV.P
arts(a)-(c)ofthisquestionrequirenumericalanswers,butsinceyouwere
nottoldtobringcalculators,youneednotcarryoutthearithmetic.Youranswer
shouldbeexpressed,however,in\calculator-ready"form|
thatis,itshouldbean
expressioninvolvingpurenumbersonly(nounits),withanynecessaryconversion
factorsincluded.(Forexample,ifyouwereaskedhowmanymetersalightpulsein
vacuumtravelsin5minutes,youcouldexpresstheansweras2:998�108�5�60.)
a)(5points)Whatwouldbethenumberdensityof8.286ions,inparticlesper
cubicmeter,whenthetemperatureTwasgivenbykT=3MeV?
b)(5points)Assuming(asinthestandardpicture)thattheearlyuniverseis
accuratelydescribedbya at,radiation-dominatedmodel,whatwouldbethe
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valueofthemassdensityatt=:01sec?Youmayassumethat0:75MeV�
kT�100MeV,sotheparticlescontributingsigni�cantlytotheblack-body
radiationincludethephotons,neutrinos,e+-e�
pairs,and8.286ion-anti8286ion
pairs.Expressyouranswerintheunitsofg/cm3.
c)(5points)Underthesameassumptionsasin(b),whatwouldbethevalueof
kT,inMeV,att=:01sec?
d)(5points)Whennucleosynthesiscalculationsaremodi�edtoincludethee�ect
ofthe8.286ion,istheproductionofheliumincreasedordecreased?Explain
youranswerinafewsentences.
e)(5points)SupposetheneutrinosdecouplewhilekT
�0:75MeV.Ifthe
8.286ionsareincluded,whatdoesonepredictforthevalueofT�=T
today?
(HereT�
denotesthetemperatureoftheneutrinos,andT
denotesthetem-
peratureofthecosmicbackgroundradiationphotons.)
�
PROBLEM
7:ANEW
THEORYOFTHEWEAK
INTERACTIONS
(40points)
ThisproblemwasProblem3,Quiz3,2009.
SupposeaNewTheoryoftheWeakInteractions(NTWI)wasproposed,which
di�ersfromthestandardtheoryintwoways.First,theNTWIpredictsthatthe
weakinteractionsaresomewhatweakerthaninthestandardmodel.Inaddition,
thetheoryimpliestheexistenceofnewspin-12
particles(fermions)calledtheR+
andR�
,witharestenergyof50MeV(where1MeV=106eV).Thisproblemwill
dealwiththecosmologicalconsequencesofsuchatheory.
TheNTWIwillpredictthattheneutrinosintheearlyuniversewilldecouple
atahighertemperaturethaninthestandardmodel.Supposethatthisdecoupling
takesplaceatkT�200MeV.Thismeansthatwhentheneutrinosceasetobe
thermallycoupledtotherestofmatter,thehotsoupofparticleswouldcontain
notonlyphotons,neutrinos,ande+-e�
pairs,butalso�+,��
,�+,��
,and�0
particles,alongwiththeR+-R�
pairs.(Themuonisaparticlewhichbehaves
almostidenticallytoanelectron,exceptthatitsrestenergyis106MeV.Thepions
arethelightestofthemesons,withzeroangularmomentumandrestenergiesof
135MeVand140MeVfortheneutralandchargedpions,respectively.The�+
and
��
areantiparticlesofeachother,andthe�0isitsownantiparticle.Zeroangular
momentumimpliesasinglespinstate.)Youmayassumethattheuniverseis at.
(a)(10points)Accordingtothestandardparticlephysicsmodel,whatisthemass
density�oftheuniversewhenkT�200MeV?Whatisthevalueof�at
thistemperature,accordingtoNTWI?Useeitherg/cm3orkg/m3.(Ifyou
wish,youcansavetimebynotcarryingoutthearithmetic.Ifyoudothis,
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however,youshouldgivetheanswerin\calculator-ready"form,bywhichI
meananexpressioninvolvingpurenumbers(nounits),withanynecessary
conversionfactorsincluded,andwiththeunitsoftheanswerspeci�edatthe
end.Forexample,ifaskedhowfarlighttravelsin5minutes,youcouldanswer
2:998�108�5�60m.)
(b)(10points)Accordingtothestandardmodel,thetemperaturetodayofthe
thermalneutrinobackgroundshouldbe(4=11)1=3T
,whereT isthetempera-
tureofthethermalphotonbackground.WhatdoestheNTWIpredictforthe
temperatureofthethermalneutrinobackground?
(c)(10points)Accordingtothestandardmodel,whatistheratiotodayofthe
numberdensityofthermalneutrinostothenumberdensityofthermalphotons?
WhatisthisratioaccordingtoNTWI?
(d)(10points)Sincethereactionswhichinterchangeprotonsandneutronsinvolve
neutrinos,thesereactions\freezeout"atroughlythesametimeastheneutrinos
decouple.Atlatertimestheonlyreactionwhiche�ectivelyconvertsneutrons
toprotonsisthefreedecayoftheneutron.Despitethefactthatneutrondecay
isaweakinteraction,wewillassumethatitoccurswiththeusual15minute
meanlifetime.WouldtheheliumabundancepredictedbytheNTWIbehigher
orlowerthanthepredictionofthestandardmodel?Towithin5or10%,what
wouldtheNTWIpredictforthepercentabundance(byweight)ofheliumin
theuniverse?(Asinpart(a),youcaneithercarryoutthearithmetic,orleave
theanswerincalculator-readyform.)
Usefulinformation:Theprotonandneutronrestenergiesaregivenbympc2=
938:27MeVandmnc2=939:57MeV,with(mn �mp )c2=1.29MeV.The
meanlifetimefortheneutrondecay,n!p+e�
+��e,isgivenby�=886s.
�
PROBLEM
8:DOUBLING
OFELECTRONS(10points)
ThefollowingwasonQuiz3,2011(Problem4):
Supposethatinsteadofonespeciesofelectronsandtheirantiparticles,suppose
therewasalsoanotherspeciesofelectron-likeandpositron-likeparticles.Suppose
thatthenewspecieshasthesamemassandotherpropertiesastheelectronsand
positrons.Ifthiswerethecase,whatwouldbetheratioT�=T ofthetemperature
todayoftheneutrinostothetemperatureoftheCMBphotons.
8.286QUIZ3REVIEW
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PROBLEM
9:TIMESCALESIN
COSMOLOGY
Inthisproblemyouareaskedtogivetheapproximatetimesatwhichvarious
importanteventsinthehistoryoftheuniversearebelievedtohavetakenplace.
Thetimesaremeasuredfromtheinstantofthebigbang.Toavoidambiguities,
youareaskedtochoosethebestanswerfromthefollowinglist:
10�
43sec.
10�
37sec.
10�
12sec.
10�
5sec.
1sec.
4mins.
10,000{1,000,000years.
2billionyears.
5billionyears.
10billionyears.
13billionyears.
20billionyears.
Forthisproblem
itwillbesuÆcienttostateananswerfrom
memory,without
explanation.Theeventswhichmustbeplacedarethefollowing:
(a)thebeginningoftheprocessesinvolvedinbigbangnucleosynthesis;
(b)theendoftheprocessesinvolvedinbigbangnucleosynthesis;
(c)thetimeofthephasetransitionpredictedbygranduni�edtheories,which
takesplacewhenkT�1016GeV;
(d)\recombination",thetimeatwhichthematterintheuniverseconverted
fromaplasmatoagasofneutralatoms;
(e)thephasetransitionatwhichthequarksbecamecon�ned,believedto
occurwhenkT�300MeV.
Sincecosmologyisfraughtwithuncertainty,insomecasesmorethanonean-
swerwillbeacceptable.Youareasked,however,togiveONLY
ONEofthe
acceptableanswers.
�
PROBLEM
10:EVOLUTION
OFFLATNESS(15points)
ThefollowingproblemwasProblem3,Quiz3,2004.
The\ atnessproblem"isrelatedtothefactthatduringtheevolutionofthe
standardcosmologicalmodel,isalwaysdrivenawayfrom1.
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(a)(9points)Duringaperiodinwhichtheuniverseismatter-dominated(meaning
thattheonlyrelevantcomponentisnonrelativisticmatter),thequantity
�1
growsasapoweroft.Showthatthisistrue,andderivethepower.(Stating
therightpowerwithoutaderivationwillbeworth3points.)
(b)(6points)Duringaperiodinwhichtheuniverseisradiation-dominated,the
samequantitywillgrowlikeadi�erentpoweroft.Showthatthisistrue,and
derivethepower.(Statingtherightpowerwithoutaderivationwillagainbe
worth3points.)
Ineachpart,youmayassumethattheuniversewasalwaysdominatedbythe
speci�edformofmatter.
�
PROBLEM
11:THE
SLOAN
DIGITAL
SKY
SURVEY
z
=
5:82
QUASAR
(40points)
ThefollowingproblemwasProblem4,Quiz3,2004.
OnApril13,2000,theSloanDigitalSkySurveyannouncedthediscoveryof
whatwasthenthemostdistantobjectknownintheuniverse:aquasaratz=5:82.
Toexplaintothepublichowthisobject�tsintotheuniverse,theSDSSpostedon
theirwebsiteanarticlebyMichaelTurnerandCraigWiegerttitled\HowCanAn
ObjectWeSeeTodaybe27BillionLightYearsAwayIftheUniverseisonly14
BillionYearsOld?"UsingamodelwithH0=65km-s�
1-Mpc�
1,m
=0:35,and
�
=0:65,theyclaimed
(a)thattheageoftheuniverseis13.9billionyears.
(b)thatthelightthatwenowseewasemittedwhentheuniversewas0.95billion
yearsold.
(c)thatthedistancetothequasar,asitwouldbemeasuredbyarulertoday,is
27billionlight-years.
(d)thatthedistancetothequasar,atthetimethelightwasemitted,was4.0
billionlight-years.
(e)thatthepresentspeedofthequasar,de�nedastherateatwhichthedistance
betweenusandthequasarisincreasing,is1.8timesthevelocityoflight.
Thegoalofthisproblem
istocheckalloftheseconclusions,althoughyouare
ofcoursenotexpectedtoactuallyworkoutthenumbers.Youranswerscanbe
expressedintermsofH0 ,m,�,andz.De�niteintegralsneednotbeevaluated.
8.286QUIZ3REVIEW
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Notethatm
representsthepresentdensityofnonrelativisticmatter,expressed
asafractionofthecriticaldensity;and�
representsthepresentdensityofvacuum
energy,expressedasafractionofthecriticaldensity.Inansweringeachofthe
followingquestions,youmayconsidertheanswertoanypreviouspart|
whether
youanswereditornot|
asagivenpieceofinformation,whichcanbeusedinyour
answer.
(a)(15points)Writeanexpressionfortheaget0ofthismodeluniverse?
(b)(5points)Writeanexpressionforthetimeteatwhichthelightwhichwenow
receivefromthedistantquasarwasemitted.
(c)(10points)Writeanexpressionforthepresentphysicaldistance`phys;0tothe
quasar.
(d)(5points)Writeanexpressionforthephysicaldistance`phys;ebetweenusand
thequasaratthetimethatthelightwasemitted.
(e)(5points)Writeanexpressionforthepresentspeedofthequasar,de�nedas
therateatwhichthedistancebetweenusandthequasarisincreasing.
PROBLEM
12:SECOND
HUBBLECROSSING
(40points)
ThisproblemwasProblem3,Quiz3,2007.In2016wehavenotyettalkedabout
Hubblecrossingsandtheevolutionofdensityperturbations,sothisproblemwould
notbefairasworded.Actually,however,youhavelearnedhowtodothesecalcu-
lations,sotheproblemwouldbefairifitdescribedinmoredetailwhatneedstobe
calculated.
InProblem
Set9(2007)wecalculatedthetimetH1 (�)ofthe�rstHubble
crossingforamodespeci�edbyits(physical)wavelength�atthepresenttime.
InthisproblemwewillcalculatethetimetH2 (�)ofthesecondHubblecrossing,
thetimeatwhichthegrowingHubblelengthcH�
1(t)catchesuptothephysical
wavelength,whichisalsogrowing.AtthetimeofthesecondHubblecrossingforthe
wavelengthsofinterest,theuniversecanbedescribedverysimply:itisaradiation-
dominated atuniverse.However,since�isde�nedasthepresentvalueofthe
wavelength,theevolutionoftheuniversebetweentH2 (�)andthepresentwillalso
berelevanttotheproblem.Wewillneedtousemethods,therefore,thatallowfor
boththematter-dominatederaandtheonsetofthedark-energy-dominatedera.As
inProblemSet9(2007),themodeluniversethatweconsiderwillbedescribedby
theWMAP3-yearbest�tparameters:
Hubbleexpansionrate
H0
=
73:5km�s�
1�Mpc�
1
Nonrelativisticmassdensity
m
=
0.237
Vacuummassdensity
vac
=
0.763
CMBtemperature
T ;0
=
2.725K
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Themassdensitiesarede�nedascontributionsto,andhencedescribethemass
densityofeachconstituentrelativetothecriticaldensity.Notethatthemodel
isexactly at,soyouneednotworryaboutspatialcurvature.Hereyouarenot
expectedtogiveanumericalanswer,sotheabovelistwillserveonlytode�nethe
symbolsthatcanappearinyouranswers,alongwith�andthephysicalconstants
G,�h,c,andk.
(a)(5points)Foraradiation-dominated atuniverse,whatistheHubblelength
`H(t)�cH�
1(t)asafunctionoftimet?
(b)(10points)ThesecondHubblecrossingwilloccurduringtheinterval
30sec�t�50;000years,
whenthemassdensityoftheuniverseisdominatedbyphotonsandneutrinos.
Duringthiseratheneutrinosarealittlecolderthanthephotons,withT�=
(4=11)1=3T
.Thetotalenergydensityofthephotonsandneutrinostogether
canbewrittenas
utot=g1�2
30(kT )4
(�hc)3
:
Whatisthevalueofg1 ?(Forthefollowingpartsyoucantreatg1asagiven
variablethatcanbeleftinyouranswers,whetherornotyoufoundit.)
(c)(10points)Fortimesintherangedescribedinpart(b),whatisthephoton
temperatureT (t)asafunctionoft?
(d)(15points)Finally,wearereadyto�ndthetimetH2 (�)ofthesecondHubble
crossing,foragivenvalueofthephysicalwavelength�today.Makinguseof
thepreviousresults,youshouldbeabletodeterminetH2 (�).Ifyouwerenot
abletoanswersomeofthepreviousparts,youmayleavethesymbols`H(t),
g1 ,and/orT (t)inyouranswer.
PROBLEM
13:NEUTRINONUMBERANDTHENEUTRON/PRO-
TON
EQUILIBRIUM
(35points)
Thefollowingproblem
was1998Quiz4,Problem
4.ThiswouldNOTbeafair
problem
for2016,asthisyearwehavenotdiscussedbigbangnucleosynthesisat
thislevelofdetail.ButIamincludingtheproblemanyway,asyoumight�ndit
interesting.
Inthestandardtreatmentofbigbangnucleosynthesisitisassumedthatat
earlytimestheratioofneutronstoprotonsisgivenbytheBoltzmannformula,
nn
np
=e�
�E=kT
;
(1)
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wherekisBoltzmann'sconstant,Tisthetemperature,and�E=1:29MeVis
theproton-neutronmass-energydi�erence.Thisformulaisbelievedtobevery
accurate,butitassumesthatthechemicalpotentialforneutrons�nisthesameas
thechemicalpotentialforprotons�p .
(a)(10points)GivethecorrectversionofEq.(1),allowingforthepossibilitythat
�n 6=�p .
Theequilibrium
betweenprotonsandneutronsintheearlyuniverseissustained
mainlybythefollowingreactions:
e+
+n !p+��e
�e+n !p+e�
:
Let�eand��denotethechemicalpotentialsfortheelectrons(e�
)andtheelectron
neutrinos(�e )respectively.Thechemicalpotentialsforthepositrons(e+)andthe
anti-electronneutrinos(��e )arethen{�e
and{�� ,respectively,sincethechemi-
calpotentialofaparticleisalwaysthenegativeofthechemicalpotentialforthe
antiparticle.*
(b)(10points)Expresstheneutron/protonchemicalpotentialdi�erence�n ��p
intermsof�eand��.
Theblack-bodyradiationformulasatthebeginningofthequizdidnotallowforthe
possibilityofachemicalpotential,buttheycaneasilybegeneralized.Forexample,
theformulaforthenumberdensityni(ofparticlesoftypei)becomes
ni=g�i�(3)
�2
(kT)3
(�hc)3e�i =kT
:
(c)(10points)Supposethatthedensityofanti-electronneutrinos�n�intheearly
universewashigherthanthedensityofelectronneutrinosn� .Expressthe
thermalequilibriumvalueoftherationn=npintermsof�E,T,andeitherthe
ratio�n�=n�ortheantineutrinoexcess�n=�n� �n�.(Youranswermayalso
containfundamentalconstants,suchask,�h,andc.)
(d)(5points)Wouldanexcessofanti-electronneutrinos,asconsideredinpart(c),
increaseordecreasetheamountofheliumthatwouldbeproducedintheearly
universe?Explainyouranswer.
*Thisfactisaconsequenceoftheprinciplethatthechemicalpotentialofa
particleisthesumofthechemicalpotentialsassociatedwithitsconservedquanti-
ties,whileparticleandantiparticlealwayshavetheoppositevaluesofallconserved
quantities.
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PROBLEM
14:THE
EVENT
HORIZON
FOR
OUR
UNIVERSE
(25
points)
ThefollowingproblemwasProblem3fromQuiz3,2013.
Wehavelearnedthattheexpansionhistoryofouruniversecanbedescribed
intermsofasmallsetofnumbers:m;0 ,thepresentcontributiontofrom
nonrelativisticmatter;rad;0 ,thepresentcontributiontofromradiation;vac ,
thepresentcontributiontofromvacuumenergy;andH0 ,thepresentvalueofthe
Hubbleexpansionrate.Thebestestimatesofthesenumbersareconsistentwitha
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,andthat
atthepresenttimet0weemitasphericalpulseoflight.Itturnsoutthatthere
isamaximumcoordinateradiusr=rmax
thatthispulsewilleverreach,no
matterhowlongwewait.(Thepulsewillneveractuallyreachrmax ,butwill
reachallrsuchthat0<r<rmax .)rmax
isthecoordinateofwhatiscalled
theeventhorizon:eventsthathappennowatr�rmax
willneverbevisible
tous,assumingthatweremainattheorigin.Assumingforthispartthatthe
functiona(t)isaknownfunction,writeanexpressionforrmax .Youranswer
shouldbeexpressedasanintegral,whichcaninvolvea(t),t0 ,andanyofthe
parametersde�nedinthepreamble.[Advice:Ifyoucannotanswerthis,you
shouldstilltrypart(c).]
(b)(10points)Sincea(t)isnotknownexplicitly,theanswertothepreviouspart
isdiÆculttouse.Show,however,thatbychangingthevariableofintegration,
youcanrewritetheexpressionforrmaxasade�niteintegralinvolvingonlythe
parametersspeci�edinthepreamble,withoutanyreferencetothefunctiona(t),
exceptperhapstoitspresentvaluea(t0 ).Youarenotexpectedtoevaluatethis
integral.[Hint:Onemethodistouse
x=
a(t)
a(t0 )
asthevariableofintegration,justaswedidwhenwederivedthe�rstofthe
expressionsfort0shownintheformulasheets.]
(c)(10points)Astronomersoftendescribedistancesintermsofredshifts,soit
isusefulto�ndtheredshiftoftheeventhorizon.Thatis,ifalightraythat
originatedatr=rmaxarrivedatEarthtoday,whatwouldbeitsredshiftzeh
(eh=eventhorizon)?
Youarenotaskedto�ndanexplicitexpressionfor
zeh ,butinsteadanequationthatcouldbesolvednumericallytodetermine
zeh .Forthispartyoucantreatrmax
asgiven,soitdoesnotmatterifyou
havedoneparts(a)and(b).Youwillgethalfcreditforacorrectanswerthat
involvesthefunctiona(t),andfullcreditforacorrectanswerthatinvolvesonly
explicitintegralsdependingonlyontheparametersspeci�edinthepreamble,
andpossiblya(t0 ).
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SOLUTIONS
PROBLEM
1:DID
YOU
DO
THEREADING?(25points)
Thefollowingpartsareeachworth5points.
(a)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBisnot
correct:
(i)AfterthedipoledistortionoftheCMBissubtractedaway,themeantem-
peratureaveragingovertheskyishTi=2:725K.
(ii)AfterthedipoledistortionoftheCMBissubtractedaway,therootmean
squaretemperature uctuationis D�ÆTT �2 E1=2
=1:1�10�
3.
(iii)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionof
theobserverrelativetoaframeofreferenceinwhichtheCMBisisotropic.
(iv)Intheirgroundbreakingpaper,WilsonandPenziasreportedthemeasure-
mentofanexcesstemperatureofabout3.5Kthatwasisotropic,unpolar-
ized,andfreefromseasonalvariations.Inacompanionpaperwrittenby
Dicke,Peebles,RollandWilkinson,theauthorsinterpretedtheradiation
tobearelicofanearly,hot,dense,andopaquestateoftheuniverse.
Explanation:Aftersubtractingthedipolecontribution,thetemperature
uctuationisabout1:1�10�
5.
(b)(CMBexperiments)ThecurrentmeanenergyperCMBphoton,about6�
10�
4
eV,iscomparabletotheenergyofvibrationorrotationforasmall
moleculesuchasH2 O.
Thusmicrowaveswithwavelengthsshorterthan
��3cm
arestronglyabsorbedbywatermoleculesintheatmosphere.To
measuretheCMBat�<3cm,whichoneofthefollowingmethodsisnota
feasiblesolutiontothisproblem?
(i)MeasureCMBfromhigh-altitudeballoons,e.g.MAXIMA.
(ii)MeasureCMBfromtheSouthPole,e.g.DASI.
(iii)MeasureCMBfromtheNorthPole,e.g.BOOMERANG.
(iv)MeasureCMBfrom
asatelliteabovetheatmosphereoftheEarth,e.g.
COBE,WMAPandPLANCK.
Explanation:TheNorthPoleisatsealevel.Incontrast,theSouthPole
isnearly3kilometersabovesealevel.BOOMERANGisaballoon-borne
experimentlaunchedfromAntarctica.
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(c)(Temperature uctuations)Thecreationoftemperature uctuationsinCMB
byvariationsinthegravitationalpotentialisknownastheSachs-Wolfee�ect.
Whichoneofthefollowingstatementsisnotcorrectconcerningthise�ect?
(i)ACMBphotonisredshiftedwhenclimbingoutofagravitationalpotential
well,andisblueshiftedwhenfallingdownapotentialhill.
(ii)Atthetimeoflastscattering,thenonbaryonicdarkmatterdominatedthe
energydensity,andhencethegravitationalpotential,oftheuniverse.
(iii)Thelarge-scale uctuationsinCMBtemperaturesarisefrom
thegrav-
itationale�ectofprimordialdensity uctuationsinthedistributionof
nonbaryonicdarkmatter.
(iv)Thepeaksintheplotoftemperature uctuation�T
vs.multipolelare
duetovariationsinthedensityofnonbaryonicdarkmatter,whilethe
contributionsfrombaryonsalonewouldnotshowsuchpeaks.
Explanation:Thesepeaksareduetotheacousticoscillationsinthephoton-
baryon uid.
(d)(Darkmattercandidates)Whichoneofthefollowingisnotacandidateof
nonbaryonicdarkmatter?
(i)massiveneutrinos
(ii)axions
(iii)mattermadeoftopquarks(atypeofquarkswithheavymassofabout
171GeV).
(iv)WIMPs(WeaklyInteractingMassiveParticles)
(v)primordialblackholes
Explanation:Mattermadeoftopquarksissounstablethatitisseenonly
eetinglyasaproductinhighenergyparticlecollisions.
(e)(Signaturesofdarkmatter)Bywhatmethodscansignaturesofdarkmatter
bedetected?Listtwomethods.(Grading:3pointsforonecorrectanswer,
5pointsfortwocorrectanswers.Ifyougivemorethantwoanswers,your
scorewillbebasedonthenumberofrightanswersminusthenumberofwrong
answers,withalowerboundofzero.)
Answers:
(i)Galaxyrotationcurves.(I.e.,measurementsoftheorbitalspeedofstars
inspiralgalaxiesasafunctionofradiusRshowthatthesecurvesremain
atatradiifarbeyondthevisiblestellardisk.Ifmostofthematterwere
containedinthedisk,thenthesevelocitiesshouldfallo�as1= pR.)
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(ii)Usethevirialtheorem
toestimatethemassofagalaxycluster.(For
example,thevirialanalysisshowsthatonly2%ofthemassoftheComa
clusterconsistsofstars,andonly10%consistsofhotintraclustergas.
(iii)Gravitationallensing.(Forexample,themassofaclustercanbeestimated
fromthedistortionoftheshapesofthegalaxiesbehindthecluster.)
(iv)CMBtemperature uctuations.(I.e.,theanalysisoftheintensityofthe
uctuationsasafunctionofmultipolenumbershowsthattot �1,and
thatdarkenergycontributes�
�0:7,baryonicmattercontributesbary �
0:04,anddarkmattercontributesdarkmatter �0:26.)
Thereareotherpossibleanswersaswell,butthesearetheonesdiscussedby
RydeninChapters8and9.
PROBLEM
2:DID
YOU
DO
THEREADING?(25points)
(a)(10points)Thisquestionconcernssomenumbersrelatedtothecosmicmi-
crowavebackground(CMB)thatoneshouldneverforget.Statethevaluesof
thesenumbers,towithinanorderofmagnitudeunlessotherwisestated.Inall
casesthequestionreferstothepresentvalueofthesequantities.
(i)TheaveragetemperatureToftheCMB(towithin10%).2:725K
(ii)ThespeedoftheLocalGroupwithrespecttotheCMB,expressedasa
fractionv=cofthespeedoflight.(ThespeedoftheLocalGroupisfound
bymeasuringthedipolepatternoftheCMBtemperaturetodetermine
thevelocityofthespacecraftwithrespecttotheCMB,andthenremoving
spacecraftmotion,theorbitalmotionoftheEarthabouttheSun,theSun
aboutthegalaxy,andthegalaxyrelativetothecenterofmassoftheLocal
Group.)
Thedipoleanisotropycorrespondstoa\peculiarvelocity"(thatis,velocity
whichisnotduetotheexpansionoftheuniverse)of630�20kms�
1,or
intermsofthespeedoflight,v=c�2�10�
3.
(iii)Theintrinsicrelativetemperature uctuations�T=T,afterremovingthe
dipoleanisotropycorrespondingtothemotionoftheobserverrelativeto
theCMB.1:1�10�
5
(iv)Theratioofbaryonnumberdensitytophotonnumberdensity,�=
nbary =n .
TheWMAP5-yearvaluefor�=
nb =n
=
(6:225�0:170)�10�
10,
whichtoclosestorderofmagnitudeis10�
9.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.27
(v)Theangularsize�H,indegrees,correspondingtowhatwastheHubble
distancec=Hatthesurfaceoflastscattering.Thisanswermustbewithin
afactorof3tobecorrect.�1Æ
(b)(3points)Becausephotonsoutnumberbaryonsbysomuch,theexponential
tailofthephotonblackbodydistributionisimportantinionizinghydrogen
wellafterkT fallsbelowQH
=13:6eV.WhatistheratiokT =QH
whenthe
ionizationfractionoftheuniverseis1=2?
(i)1=5
(ii)1=50
(iii)10�
3
(iv)10�
4
(v)10�
5
Thisisnotanumberonehastocommittomemoryifonecanremember
thetemperatureof(re)combinationineV,orifonlyinKalongwiththe
conversionfactor(k�10�
4eVK�
1).Onecanthencalculatethatnear
recombination,kT =QH
�(10�
4eVK�
1)(3000K)=(13:6eV)�1=45.
(c)(2points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearredderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearbluerbecauseoftheDopplere�ect.
(iii)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
redderbecausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
bluerbecausetheymustclimboutofthegravitationalpotentialwell.
(v)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
redderbecauseofabsorptionintheintergalacticmedium.
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
bluerbecauseofabsorptionintheintergalacticmedium.
Explanation:Denserregionshaveadeeper(morenegative)gravitational
potential.Photonswhichtravelthroughaspatiallyvaryingpotentialac-
quirearedshiftorblueshiftdependingonwhethertheyaregoingupordown
thepotential,respectively.Photonsoriginatinginthedenserregionsstart
atalowerpotentialandmustclimbout,sotheyendupbeingredshifted
relativetotheiroriginalenergies.
(d)(10points)Foreachofthefollowingstatements,saywhetheritistrueorfalse:
(i)Darkmatterinteractsthroughthegravitational,weak,andelectromag-
neticforces.
T
orF?
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.28
(ii)Thevirialtheoremcanbeappliedtoaclusterofgalaxiesto�nditstotal
mass,mostofwhichisdarkmatter.
T
orF?
(iii)Neutrinosarethoughttocompriseasigni�cantfractionoftheenergyden-
sityofdarkmatter.
T
orF?
(iv)Magneticmonopolesarethoughttocompriseasigni�cantfractionofthe
energydensityofdarkmatter.
T
orF?
(v)LensingobservationshaveshownthatMACHOscannotaccountforthe
darkmatteringalactichalos,butthatasmuchas20%ofthehalomass
couldbeintheformofMACHOs.
T
orF?
PROBLEM
3:DID
YOU
DO
THEREADING?(35points)
(a)(5points)RydensummarizestheresultsoftheCOBEsatelliteexperimentfor
themeasurementsofthecosmicmicrowavebackground(CMB)intheformof
threeimportantresults.The�rstwasthat,inanyparticulardirectionofthe
sky,thespectrumoftheCMBisveryclosetothatofanidealblackbody.The
FIRASinstrumentontheCOBEsatellitecouldhavedetecteddeviationsfrom
theblackbodyspectrumassmallas��=��10�
n,wherenisaninteger.To
within�1,whatisn?
Answer:n=4
(b)(5points)Thesecondresultwasthemeasurementofadipoledistortionof
theCMBspectrum;thatis,theradiationisslightlyblueshiftedtohighertem-
peraturesinonedirection,andslightlyredshiftedtolowertemperaturesin
theoppositedirection.Towhatphysicale�ectwasthisdipoledistortionat-
tributed?
Answer:ThelargedipoleintheCMBisattributedtothemotionofthesatellite
relativetotheframeinwhichtheCMBisverynearlyisotropic.(Theentire
LocalGroupismovingrelativetothisframeataspeedofabout0.002c.)
(c)(5points)Thethirdresultconcernedthemeasurementoftemperature uctu-
ationsafterthedipolefeaturementionedabovewassubtractedout.De�ning
ÆTT(�;�)�T(�;�)�hTi
hTi
;
wherehTi=2:725K,theaveragevalueofT,theyfoundarootmeansquare
uctuation,
*�ÆTT �2 +1=2
;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.29
equaltosomenumber.Towithinanorderofmagnitude,whatwasthatnum-
ber?
Answer:
*�ÆTT �2 +1=2
=1:1�10�
5:
(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?
(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearredderbecauseoftheDopplere�ect.
(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsight
appearbluerbecauseoftheDopplere�ect.
(iii)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
redderbecausetheymustclimboutofthegravitationalpotentialwell.
(iv)Photonsfrom
overdenseregionsatthesurfaceoflastscatteringappear
bluerbecausetheymustclimboutofthegravitationalpotentialwell.
(v)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
redderbecauseofabsorptionintheintergalacticmedium.
(vi)Photonstravelingtowardusfrom
thesurfaceoflastscatteringappear
bluerbecauseofabsorptionintheintergalacticmedium.
(e)(5points)The atnessproblemreferstotheextreme�ne-tuningthatisneeded
inatearlytimes,inorderforittobeascloseto1todayasweobserve.
Startingwiththeassumptionthattodayisequalto1withinabout1%,one
concludesthatatonesecondafterthebigbang,
j�1jt=1sec<10�
m
;
wheremisaninteger.Towithin�3,whatism?
Answer:m=18.(SeethederivationinLectureNotes8.)
(f)(5points)Thetotalenergydensityofthepresentuniverseconsistsmainlyof
baryonicmatter,darkmatter,anddarkenergy.Givethepercentagesofeach,
accordingtothebest�tobtainedfromthePlanck2013data.Youwillgetfull
creditifthe�rst(baryonicmatter)isaccurateto�2%,andtheothertwoare
accuratetowithin�5%.
Answer:Baryonicmatter:5%.Darkmatter:26.5%.Darkenergy:68.5%.
ThePlanck2013numbersweregiveninLectureNotes7.Totherequested
accuracy,however,numberssuchasRyden'sBenchmarkModelwouldalsobe
satisfactory.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.30
(g)(5points)Withintheconventionalhotbigbangcosmology(withoutin ation),
itisdiÆculttounderstandhowthetemperatureoftheCMBcanbecorrelated
atangularseparationsthataresolargethatthepointsonthesurfaceoflast
scatteringwasseparatedfromeachotherbymorethanahorizondistance.Ap-
proximatelywhatangle,indegrees,correspondstoaseparationonthesurface
lastscatteringofonehorizonlength?Youwillgetfullcreditifyouransweris
righttowithinafactorof2.
Answer:Rydengives1Æ
astheanglesubtendedbytheHubblelengthonthe
surfaceoflastscattering.Foramatter-dominateduniverse,whichwouldbe
agoodmodelforouruniverse,thehorizonlengthistwicetheHubblelength.
Anynumberfrom1Æ
to5Æ
wasconsideredacceptable.
PROBLEM
4:
NUMBER
DENSITIES
IN
THE
COSMIC
BACK-
GROUND
RADIATION
Ingeneral,thenumberdensityofaparticleintheblack-bodyradiationisgiven
by
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�nds
n =399=cm3:
Fortheneutrinos,
g��=2�34
=32
perspecies.
Thefactorof2istoaccountfor�and��,andthefactorof3/4arisesfromthe
Pauliexclusionprinciple.Soforthreespeciesofneutrinosonehas
g��=92
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.31
Usingtheresult
T3�
=
411T3
fromProblem8ofProblemSet3(2000),one�nds
n�= �g��
g� ��T�
T �
3n
= �94 ��411 �
399cm�
3
=)
n�=326=cm3(forallthreespeciescombined).
PROBLEM
5:PROPERTIESOFBLACK-BODY
RADIATION
(a)Theaverageenergyperphotonisfoundbydividingtheenergydensitybythe
numberdensity.Thephotonisabosonwithtwospinstates,sog=g�
=2.
Usingtheformulasonthefrontoftheexam,
E=
g�2
30(kT)4
(�hc)3
g�
�(3)
�2
(kT)3
(�hc)3
=
�4
30�(3)kT:
Youwerenotexpectedtoevaluatethisnumerically,butitisinterestingto
knowthat
E=2:701kT:
Notethattheaverageenergyperphotonissigni�cantlymorethankT,which
isoftenusedasaroughestimate.
(b)Themethodisthesameasabove,exceptthistimeweusetheformulaforthe
entropydensity:
S=
g2�2
45
k4T3
(�hc)3
g�
�(3)
�2
(kT)3
(�hc)3
=
2�4
45�(3)k:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.32
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,theaverageenergy
perparticleisuna�ected|
theenergydensityandthetotalnumberdensity
arebothdoubled,buttheirratioisunchanged.
Notethattheenergyperparticleishigherforfermionsthanitisforbosons.
Thisresultcanbeunderstoodasanaturalconsequenceofthefactthatfermions
mustobeytheexclusionprinciple,whilebosonsdonot.Largenumbersof
bosonscanthereforecollectinthelowestenergylevels.Infermionsystems,
ontheotherhand,thelow-lyinglevelscanaccommodateatmostoneparticle,
andthenadditionalparticlesareforcedtohigherenergylevels.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.33
(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
6:A
NEW
SPECIESOFLEPTON
a)Thenumberdensityisgivenbytheformulaatthestartoftheexam,
n=g�
�(3)
�2
(kT)3
(�hc)3
:
Sincethe8.286ionisliketheelectron,ithasg�
=3;thereare2spinstates
fortheparticlesand2fortheantiparticles,giving4,andthenafactorof3/4
becausetheparticlesarefermions.So
Then
Answer=3�(3)
�2
� �
3�106�102
6:582�10�
16�2:998�1010 �
3
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.34
Youwerenotaskedtoevaluatethisexpression,buttheansweris1:29�1039.
b)Fora atcosmology�=0andoneoftheEinsteinequationsbecomes
�_aa �2
=8�3
G�:
Duringtheradiation-dominatederaa(t)/t1=2,asclaimedonthefrontcover
oftheexam.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,
doingsogives�=4:47�109g=cm3.
c)Themassdensity�=u=c2,whereuistheenergydensity.Theenergydensity
forblack-bodyradiationisgivenintheexam,
u=�c2=g�2
30(kT)4
(�hc)3
:
WecanusethisinformationtosolveforkTintermsof�(t)whichwefound
aboveinpart(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,FALL2016
p.35
gtot=1414
:
SolvingforkTintermsof�gives
kT= �30
�2
1gtot �h
3c5� �1=4
:
Usingtheresultfor�frompart(b)aswellasthelistoffundamentalconstants
fromthecoversheetoftheexamgives
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.Puttinginthenumbersyields
kT=8:02MeV.
d)Theproductionofhelium
isincreased.Atanygiventemperature,theaddi-
tionalparticleincreasestheenergydensity.SinceH
/�1=2,theincreased
energydensityspeedstheexpansionoftheuniverse|
theHubbleconstantat
anygiventemperatureishigheriftheadditionalparticleexists,andthetem-
peraturefallsfaster.Theweakinteractionsthatinterconvertprotonsandneu-
trons\freezeout"whentheycannolongerkeepupwiththerateofevolution
oftheuniverse.Thereactionratesatagiventemperaturewillbeuna�ected
bytheadditionalparticle,butthehighervalueofH
willmeanthatthetem-
peratureatwhichtheseratescannolongerkeeppacewiththeuniversewill
occursooner.Thefreeze-outwillthereforeoccuratahighertemperature.The
equilibriumvalueoftheratioofneutrontoprotondensitiesislargerathigher
temperatures:nn=np
/exp(��mc2=kT),wherenn
andnp
arethenumber
densitiesofneutronsandprotons,and�mistheneutron-protonmassdi�er-
ence.Consequently,therearemoreneutronspresenttocombinewithprotons
tobuildheliumnuclei.Inaddition,thefasterevolutionrateimpliesthatthe
temperatureatwhichthedeuteriumbottleneckbreaksisreachedsooner.This
impliesthatfewerneutronswillhaveachancetodecay,furtherincreasingthe
heliumproduction.
e)Aftertheneutrinosdecouple,theentropyintheneutrinobathisconserved
separatelyfromtheentropyintherestoftheradiationbath.Justafterneu-
trinodecoupling,alloftheparticlesinequilibriumaredescribedbythesame
temperaturewhichcoolsasT/1=a.Theentropyinthebathofparticlesstill
inequilibriumjustaftertheneutrinosdecoupleis
S/grest T3(t)a3(t)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.36
wheregrest=gtot �g�
=9.Bytoday,thee+
�e�
pairsandthe8.286ion-
anti8.286ionpairshaveannihilated,thustransferringtheirentropytothepho-
tonbath.Asaresultthetemperatureofthephotonbathisincreasedrelative
tothatoftheneutrinobath.Fromconservationofentropywehavethatthe
entropyafterannihilationsisequaltotheentropybeforeannihilations
g T3 a3(t)=grest T3(t)a3(t):
So,
T
T(t)= �grest
g �
1=3
:
Sincetheneutrinotemperaturewasequaltothetemperaturebeforeannihila-
tions,wehavethat
T�
T
= �29 �1=3
:
PROBLEM
7:A
NEW
THEORY
OFTHEWEAK
INTERACTIONS
(40points)
(a)Inthestandardmodel,theblack-bodyradiationatkT�200MeVcontains
thefollowingcontributions:
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,FALL2016
p.37
Checkingtheunits,
[�]=
J4
J3-s3-m5-s�
5
=J-s2
m5
= �kg-m2-s�
2 �s2
m5
=kg/m3:
So,the�nalanswerwouldbe
�= �1714 ��2
30 �2
00�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,FALL2016
p.38
whichagreeswiththeanswerabove.
Note:
A
common
mistake
was
to
leave
out
the
conversion
factor
1:602�10�
19
J/eV
(or1:602�10�
12
erg/eV),and
instead
to
use
�h=6:582�10�
16eV-s.Butifoneworksouttheunitsofthisanswer,they
turnouttobeeV-sec2/m5(oreV-sec2/cm5),whichisamostpeculiarsetof
unitstomeasureamassdensity.
IntheNTWI,wehaveinadditionthecontributiontothemassdensityfrom
R+-R�
pairs,whichwouldactjustlikee+-e�
pairsor�+-��
pairs,withg=
312 .ThusgTOT
=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 �2
00�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.The
entropyinagivenvolumeofthecomovingcoordinatesystemis
a3(t)sVcoord
;
wheresistheentropydensityanda3V
coordisthephysicalvolume.So
a3(t)s
isconserved.Aftertheneutrinosdecouple,
a3s
�
and
a3s
other
areseparatelyconserved,wheresotheristheentropyofeverythingexceptneu-
trinos.
Notethatscanbewrittenas
s=gAT3
;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.39
whereAisaconstant.Beforethedisappearanceofthee;�,R,and�particles
fromthethermalequilibriumradiation,
s�= �514 �
AT3
sother= �1512 �
AT3
:
So
s�
sother=
514
1512
:
Ifa3s
�anda3s
otherareconserved,thensoiss�=sother .Bytoday,theentropy
previouslysharedamongthevariousparticlesstillinequilibriumafterneutrino
decouplinghasbeentransferedtothephotonssothat
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.Inthestandard
model,onehastoday
n�
n
=g�� T3�
g� T3
= �412 �2
411=
911:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.40
IntheNTWI,
n�
n
= �412 �2
431=
931:
(d)AtkT=200MeV,thethermalequilibrium
ratioofneutronstoprotonsis
givenby
nn
np
=e�
1:29MeV=200MeV
�1:
Inthestandardtheorythisratiowoulddecreaserapidlyastheuniversecooled
andkTfellbelowthep-nmassdi�erenceof1.29MeV,butintheNTWIthe
ratiofreezesoutatthehightemperaturecorrespondingtokT=200MeV,
whentheratioisabout1.WhenkTfallsbelow200MeVintheNTWI,the
neutrinointeractions
n+�e $p+e�
and
n+e+
$p+��e
thatmaintainthethermalequilibriumbalancebetweenprotonsandneutrons
nolongeroccuratasigni�cantrate,sotheration= npisnolongercontrolledby
thermalequilibrium.AfterkTfallsbelow200MeV,theonlyprocessthatcan
convertneutronstoprotonsistheratherslowprocessoffreeneutrondecay,
withadecaytime�d
ofabout890s.Thus,whenthedeuteriumbottleneck
breaksatabout200s,thenumberdensityofneutronswillbeconsiderably
higherthaninthestandardmodel.Sinceessentiallyalloftheseneutronswill
becomeboundintoHenuclei,thehigherneutronabundanceoftheNTWI
impliesa
higherpredictedHeabundance:
ToestimatetheHeabundance,notethatifwetemporarilyignorefreeneutron
decay,thentheneutron-protonratiowouldbefrozenatabout1andwould
remain1untilthetimeofnucleosynthesis.Atthetimeofnucleosynthesis
essentiallyalloftheseneutronswouldbeboundintoHenuclei(eachwith2
protonsand2neutrons).Foraninitial1:1ratioofneutronstoprotons,all
theneutronsandprotonscanbeboundintoHenuclei,withnoprotonsleft
overintheformofhydrogen,soYwouldequal1.However,thefreeneutron
decayprocesswillcausetherationn=np
tofallbelow1beforethestartof
nucleosynthesis,sothepredictedvalueofYwouldbelessthan1.
Tocalculatehowmuchless,notethatRydenestimatesthestartofnucleosyn-
thesisatthetimewhenthetemperaturereachesTnuc ,whichisthetemperature
forwhichathermalequilibriumcalculationgivesnD=nn=1.Thiscorresponds
8.286QUIZ3REVIEW
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SOLUTIONS,FALL2016
p.41
towhatWeinbergreferstoasthebreakingofthedeuteriumbottleneck.The
temperatureTnuciscalculatedintermsof�=nB=n andphysicalconstants,
soitwouldnotbechangedbytheNTWI.Thetimewhenthistemperatureis
reached,however,wouldbechangedslightlybythechangeintheratioT�=T .
Sincethise�ectisrathersubtle,nopointswillbetakeno�ifyouomittedit.
However,tobeasaccurateaspossible,oneshouldrecognizethatnucleosynthe-
sisoccursduringtheradiation-dominatedera,butlongafterthee+-e�
pairs
havedisappeared,sotheblack-bodyradiationconsistsofphotonsattempera-
tureT andneutrinosatalowertemperatureT� .Theenergydensityisgiven
by
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 ��431 �4=3
:
Therelationbetweentimeandtemperatureina atradiation-dominateduni-
verseisgivenintheformulasheetsas
kT= �45�h3c5
16�3gG �
1=4
1pt:
Thus,
t/
1g1=2
e�
T2
:
InthestandardmodelRydenestimatesthetimeofnucleosynthesisastsmnu
c �
200s,sointheNTWIitwouldbelongerbythefactor
tNTWI
nuc
= sgsm
e�
gNTWI
e�
tsmnu
c:
Whileofcoureyouwerenotexpectedtoworkoutthenumerics,thisgives
tNTWI
nuc
=1:20tsmnu
c:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.42
NotethatRydengivestnuc �200s,whileWeinbergplacesitat334
minutes
�225s,whichiscloseenough.
Tofollowthee�ectofthisfreedecay,itiseasiesttodoitbyconsideringthe
rationeutronstobaryonnumber,nn=nB,sincenB
doesnotchangeduringthis
period.Atfreeze-out,whenkT�200MeV,
nn
nB
�12
:
Justbeforenucleosynthesis,attimetnuc ,theratiowillbe
nn
nB
�12
e�
tnuc=�d
:
Iffreedecayisignored,wefoundY=1.Sinceallthesurvivingneutronsare
boundintoHe,thecorrectedvalueofYissimplydeceasedbymultiplyingby
thefractionofneutronsthatdonotundergodecay.Thus,thepredictionof
NTWIis
Y=e�
tnuc =�d
=exp 8<:� q
gsm
eff
gNTW
I
eff
200
890
9=;;
wheregsm
e�
andgNTWI
e�
aregivenabove.Whenevaluatednumerically,thiswould
give
Y=PredictedHeabundancebyweight�0:76:
PROBLEM
8:DOUBLING
OFELECTRONS(10points)
Theentropydensityofblack-bodyradiationisgivenby
s=g �2�2
45
k4
(�hc)3 �T3
=gCT3;
whereCisaconstant.Atthetimewhentheelectron-positronpairsdisappear,
theneutrinosaredecoupled,sotheirentropyisconserved.Alloftheentropy
fromelectron-positronpairsisgiventothephotons,andnonetotheneutrinos.
Thesamewillbetruehere,forbothspeciesofelectron-positronpairs.
TheconservedneutrinoentropycanbedescribedbyS�
�a3s
�,whichindi-
catestheentropypercubicnotch,i.e.,entropyperunitcomovingvolume.We
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.43
introducethenotationn�
andn+
forthenewelectron-likeandpositron-like
particles,andalsotheconventionthat
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,theirentropyisaddedtothepho-
tons:
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�
annihilationtheneutrinoswereinthermalequilibrium
withthe
photons,soT =T� .Byconsideringthetwoboxedequationsabove,onehas
T0�= �29 �1=3
T0 :
Thisratiowouldremainunchangeduntilthepresentday.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.44
PROBLEM
9:TIMESCALESIN
COSMOLOGY
(a)1sec.[Thisisthetimeatwhichtheweakinteractionsbeginto\freezeout",
sothatfreeneutrondecaybecomestheonlymechanismthatcaninterchange
protonsandneutrons.Fromthistimeonward,therelativenumberofprotons
andneutronsisnolongercontrolledbythermalequilibriumconsiderations.]
(b)4mins.[Bythistimetheuniversehasbecomesocoolthatnuclearreactions
arenolongerinitiated.]
(c)10�
37sec.[WelearnedinLectureNotes7thatkTwasabout1MeVatt=1
sec.Since1GeV=1000MeV,thevalueofkTthatwewantis1019
times
higher.Intheradiation-dominatederaT/a�
1/t�
1=2,soweget10�
38sec.]
(d)10,000{1,000,000years.[ThisnumberwasestimatedinLectureNotes7as
200,000years.]
(e)10�
5sec.[Asin(c),wecanuset/T�
2,withkT�1MeVatt=1sec.]
PROBLEM
10:EVOLUTION
OFFLATNESS(15points)
(a)WestartwiththeFriedmannequationfromtheformulasheetonthequiz:
H2= �_aa �2
=8�3
G��kc2
a2
:
Thecriticaldensityisthevalueof�correspondingtok=0,so
H2=8�3
G�c:
UsingthisexpressiontoreplaceH2
ontheleft-handsideoftheFriedmann
equation,andthendividingby8�G=3,one�nds
�c=��3kc2
8�Ga2
:
Rearranging,
���c
�
=
3kc2
8�Ga2�:
Ontheleft-handsidewecandividethenumeratoranddenominatorby�c ,and
thenusethede�nition��=�ctoobtain
�1
=
3kc2
8�Ga2�:
(1)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.45
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.
Foraradiation-dominateduniverseweknowthat�/1=a4(t),so
�1
/a2(t):
Iftheuniverseisnearly atthena(t)/t1=2,so
�1
/t:
PROBLEM
11:
THE
SLOAN
DIGITAL
SKY
SURVEY
z
=
5:82
QUASAR
(40points)
(a)Sincem
+�
=0:35+0:65=1,theuniverseis at.Itthereforeobeysa
simpleformoftheFriedmannequation,
H2= �_aa �2
=8�3
G(�m
+��);
wheretheoverdotindicatesaderivativewithrespecttot,andthetermpro-
portionaltokhasbeendropped.Usingthefactthat�m
/1=a3(t)and��
=
const,theenergydensitiesontheright-handsidecanbeexpressedintermsof
theirpresentvalues�m;0and��
���;0 .De�ning
x(t)�a(t)
a(t0 );
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.46
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;
whichinturnimpliesthatd
t=
1H0
dx
x qm
;0
x3
+�;0
:
Usingthefactthatxchangesfrom0to1overthelifeoftheuniverse,this
relationcanbeintegratedtogive
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,FALL2016
p.47
whereinthelastanswerIchangedthevariableofintegrationusing
x=
11+z;
dx=�dz
(1+z)2
:
Notethattheminussignintheexpressionfordxiscanceledbytheinterchange
ofthelimitsofintegration:x=0correspondstoz=1,andx=1corresponds
toz=0.
Youranswershouldlooklikeoneoftheaboveboxedanswers.Youwerenot
expectedtocompletethenumericalcalculation,butforpedagogicalpurposes
Iwillcontinue.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.Thedynamicsoftheuniverseis
ofcoursethesame,andthequestionisonlyslightlydi�erent.Inpart(a)we
foundtheamountoftimethatittookforxtochangefrom0to1.Thelight
fromthequasarthatwenowreceivewasemittedwhen
x=
11+z;
sincethecosmologicalredshiftisgivenby
1+z=a(tobserved )
a(temitted ):
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.48
Usingtheexpressionfordtfrompart(a),theamountoftimethatittookthe
universetoexpandfromx=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.Con-
tinuing,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�gureouthowfarlight
cantravelfromz=5:82tothepresent.Sincewewantthepresentdistance,
wemultiplythecoordinatedistancebya(t0 ).Forthe atmetric
ds2=�c2d�2=�c2dt2+a2(t) �dr2+r2(d�2+sin2�d�2) ;
thecoordinatevelocityoflight(intheradialdirection)isfoundbysetting
ds2=0,giving
dr
dt=
ca(t):
Sothetotalcoordinatedistancethatlightcantravelfromtetot0is
`c= Z
t0
te
ca(t)dt:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.49
Thisisnotthe�nalanswer,however,becausewedon'texplicitlyknowa(t).
Wecan,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,thistimetheintegralhasnoanalytic
form,sofarasIknow.Integratingnumerically,
Z5:82
0
dz0
p0:35(1+z0)3+0:65=1:8099;
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.50
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,FALL2016
p.51
PROBLEM
12: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,
soweneedtoaddtogetherthesetwocontributionstotheenergydensity.For
photons,theformulasheetremindsusthatg =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,FALL2016
p.52
(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.Thus
anyHubblecrossingduringtheradiation-dominatederamustbethesecond
Hubblecrossing.
If�isthepresentphysicalwavelengthofthedensityperturbationsunderdis-
cussion,thewavelengthattimetisscaledbythescalefactora(t):
�(t)=
a(t)
a(t0 )�:
BetweenthesecondHubblecrossingandnow,therehavebeennofreeze-outs
ofparticlespecies.Todaytheentropyoftheuniverseisstilldominatedby
photonsandneutrinos,sotheconservationofentropyimpliesthataT
has
remainedessentiallyconstantbetweenthenandnow.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,FALL2016
p.53
Solvingfort,thetimeofsecondHubblecrossingisfoundtobe
tH2 (�)=(kT ;0 �)2 ��3g
1 G
45�h3c9 �
1=2
:
Extension:Youwerenotaskedtoinsertnumbers,butitisofcourseinteresting
toknowwheretheaboveformulaleads.Ifwetake�=106lt-yr,itgives
tH2 (106lt-yr)=1:04�107s=0:330year:
For�=1Mpc,
tH2 (1Mpc)=1:11�108s=3:51year:
Taking�=
2:5�106
lt-yr,thedistancetoAndromeda,thenearestspiral
galaxy,
tH2 (2:5�106lt-yr)=6:50�107sec=2:06year:
PROBLEM
13:NEUTRINO
NUMBER
AND
THENEUTRON/PRO-
TON
EQUILIBRIUM
(a)Fromthechemicalequilibriumequationonthefrontoftheexam,thenumber
densitiesofneutronsandprotonscanbewrittenas
nn=gn(2�mnkT)3=2
(2��h)3
e(�n�
mnc2)=kT
np=gp(2�mp kT)3=2
(2��h)3
e(�p�
mpc2)=kT
;
wheregn=gp=2.Dividing,
nn
np
= �mn
mp �
3=2
e�
(�E+�p�
�n)=kT
;
where�E=(mn �mp )c2istheproton-neutronmass-energydi�erence.Ap-
proximatingmn=mp �1,onehas
nn
np
=e�
(�E+�p�
�n)=kT
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.54
Theapproximationmn=mp
�1isveryaccurate(0.14%),butisclearlynot
necessary.Fullcreditwasgivenwhetherornotthisapproximationwasused.
(b)Foranyallowedchemicalreaction,thesumofthechemicalpotentialsonthe
twosidesmustbeequal.So,from
e+
+n !p+��e;
wecaninferthat
��e+�n=�p ���;
whichimpliesthat
�n ��p=�e ���:
(c)Applyingtheformulagivenintheproblemtothenumberdensitiesofelectron
neutrinosandthecorrespondingantineutrinos,
n�=g���(3)
�2
(kT)3
(�hc)3e��=kT
�n�=g���(3)
�2
(kT)3
(�hc)3e�
��=kT
;
sincethechemicalpotentialfortheantineutrinos(��)isthenegativeofthe
chemicalpotentialforneutrinos.Aneutrinohasonlyonespinstate,sog�=
3=4,wherethefactorof3/4arisesbecauseneutrinosarefermions.Setting
x�e�
��=kT
and
A�34�(3)
�2
(kT)3
(�hc)3
;
thenumberdensityequationscanbewrittencompactlyas
n�=Ax
;
�n�=xA:
Toexpressxintermsoftheratio�n� =n� ,dividethesecondequationbythe
�rsttoobtain
�n�
n�
=x2
=)
x= r�n�
n�
:
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.55
Alternatively,xcanbeexpressedintermsofthedi�erenceinnumberdensities
�n� �n�bystartingwith
�n=�n� �n�=xA�Ax
:
Rewritingtheaboveformulaasanexplicitquadratic,
Ax2��nx�A=0;
one�nds
x=�n�p
�n2+4A2
2A
:
Sincethede�nitionofximpliesx>0,onlythepositiverootisrelevant.Since
thenumberofelectronsisstillassumedtobeequaltothenumberofpositrons,
�e=0,sotheanswerto(b)reducesto�n ��p=��� .From(a),
nn
np
=e�
(�E+�p�
�n)=kT
=e�
(�E+��)=kT
=xe�
�E=kT
=
r�n�
n�e�
�E=kT
:
Alternatively,onecanwritetheansweras
nn
np
=p
�n2+4A2+�n
2A
e�
�E=kT
;
where
A�34�(3)
�2
(kT)3
(�hc)3
:
(d)For�n>0,theanswerto(c)impliesthattherationn=npwouldbelarger
thanintheusualcase(�n=0).Thisisconsistentwiththeexpectationthat
anexcessofantineutrinoswilltendtocausep'stoturninton'saccordingto
thereaction
p+��e �!e+
+n:
Sincetheamountofhelium
producedisproportionaltothenumberofneu-
tronsthatsurviveuntilthebreakingofthedeuterium
bottleneck,starting
withahigherequilibriumabundanceofneutronswillincreasetheproduction
ofhelium.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.56
PROBLEM
14:THE
EVENT
HORIZON
FOR
OUR
UNIVERSE
(25
points)
(a)Inasphericalpulseeachlightrayismovingradiallyoutward,sod�=d�=0.
Alightraytravelsalonganulltrajectory,meaningthatds2=0,sowehave
ds2=�c2dt2+a2(t)dr2=0:
(3.1)
fromwhichitfollowsthat
dr
dt=�c
a(t):
(3.2)
Weareinterestedinaradialpulsethatstartsatr=0attimet=t0 ,sothe
limitingvalueofrisgivenbyr
max= Z
1t0
ca(t)dt:
(3.3)
(b)Changingvariablesofintegrationto
x=
a(t)
a(t0 );
(3.4)
theintegralbecomes
rmax= Z
11
ca(t)
dt
dxdx=
ca(t0 ) Z
11
1xdt
dxdx;
(3.5)
whereweusedthefactthatt=t0
correspondstox=a(t0 )=a(t0 )=1.As
giventousontheformulasheet,the�rst-orderFriedmannequationcanbe
writtenas
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.
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.57
(c)To�ndthevalueoftheredshiftforthelightthatwearepresentlyreceivingfrom
coordinatedistancermax ,wecanbeginbynoticingthatthetimeofemissionte
canbedeterminedbytheequationwhichimpliesthatthecoordinatedistance
traveledbyalightpulsebetweentimeste
andt0
mustequalrmax .Using
Eq.(3.2)forthecoordinatevelocityoflight,thisequationreads
Zt0
te
ca(t)dt=rmax:
(3.8)
The\half-credit"answertothequizproblemwouldincludetheaboveequation,
followedbythestatementthattheredshiftzehcanbedeterminedfrom
z=a(t0 )
a(te ) �1:
(3.9)
The\full-credit"answerisobtainedbychangingthevariableofintegrationas
inpart(b),soEq.(3.8)becomes
rmax= Z
1xe
ca(t)
dt
dxdx
=
ca(t0 ) Z
1xe
1xdt
dxdx;
(3.10)
wherexeisthevalueofxcorrespondingtot=te .ThenusingEq.(3.6)with
k;0=0,we�nd
rmax=
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,butonecould
gofurtherandreplacermaxusingEq.(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)
8.286QUIZ3REVIEW
PROBLEM
SOLUTIONS,FALL2016
p.58
InthisformtheanswerdependsonlyonthevaluesofX;0 .
Youwereofcoursenotaskedtoevaluatethisformulanumerically,butyou
mightbeinterestedinknowingthatthePlanck2013valuesm;0
=
0:315,
vac;0=0:685,andrad;0=9:2�10�
5
leadtozeh
=1:87.Thus,noevent
thatishappeningnow(i.e.,atthesamevalueofthecosmictime)inagalaxy
atredshiftlargerthan1.87willeverbevisibletousorourdescendants,even
inprinciple.