Massachusetts Institute of Technologyweb.mit.edu/course/8/8.286/www/quiz16/q3rp-euf16-2up.pdf ·...

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

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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

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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


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

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(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


bluerbecausetheymustclimboutofthegravitationalpotentialwell.

(v)Photonstravelingtowardusfrom

thesurfaceoflastscatteringappear

redderbecauseofabsorptionintheintergalacticmedium.

(vi)Photonstravelingtowardusfrom


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

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(iv)Magneticmonopolesarethoughttocompriseasigni�cantfractionofthe

energydensityofdarkmatter.

T

orF?

(v)LensingobservationshaveshownthatMACHOscannotaccountforthe

darkmatteringalactichalos,butthatasmuchas20%ofthehalomass

couldbeintheformofMACHOs.

T

orF?

PROBLEM

3:DID

YOU

DO


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?





8.286QUIZ3REVIEW

PROBLEMS,FALL2016

p.14

(iii)Photonsfrom



(iv)Photonsfrom









(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

8.286QUIZ3REVIEW

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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,

8.286QUIZ3REVIEW

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p.17

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)


The\ atnessproblem"isrelatedtothefactthatduringtheevolutionofthe

standardcosmologicalmodel,isalwaysdrivenawayfrom1.

8.286QUIZ3REVIEW

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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)


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

8.286QUIZ3REVIEW

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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)

8.286QUIZ3REVIEW

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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.

8.286QUIZ3REVIEW

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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 ).

8.286QUIZ3REVIEW

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SOLUTIONS

PROBLEM

1:DID

YOU

DO


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.

8.286QUIZ3REVIEW

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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.)

8.286QUIZ3REVIEW

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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


(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?





(iii)Photonsfrom



(iv)Photonsfrom









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


(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?





(iii)Photonsfrom



(iv)Photonsfrom









(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

PROBLEM

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


:

(3.11)

Tocompletetheanswerinthislanguage,weuse

z=

1xe �1:

(3.12)

Eqs.(3.11)and(3.12)constituteafullanswertothequestion,butonecould

gofurtherandreplacermaxusingEq.(3.7),�nding

Z1

1

dx


= Z1

xe

dx


:

(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.

Massachusetts Institute of Technologyweb.mit.edu/course/8/8.286/www/quiz16/q3rp-euf16-2up.pdf ·...

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