stuff.mit.edu€¦ · MASSA CHUSETTS INSTITUTE OF TECHNOLOGY Ph ysics Departmen t Ph ysics 8.286:...

MASSACHUSETTSINSTITUTEOFTECHNOLOGY

PhysicsDepartment

Physics8.286:TheEarlyUniverse

November30,2018

Prof.AlanGuth

REVIEW

PROBLEMSFOR

QUIZ3

QUIZDATE:Wedneday,December5,2018,duringthenormalclasstime.

COVERAGE:LectureNotes6(pp.12{end),LectureNotes7and8.ProblemSets7

and8;StevenWeinberg,TheFirstThreeMinutes,Chapter8andtheAfterword;

BarbaraRyden,IntroductiontoCosmology,Chapters9(TheCosmicMicrowave

Background)and11(In ationandtheVeryEarlyUniverse);AlanGuth,In ation

andtheNewEraofHigh-PrecisionCosmology,

http://web.mit.edu/physics/news/physicsatmit/physicsatmit_

02_

cosmology.pdf.

Oneoftheproblemsonthequizwillbetakenverbatim

(oratleastalm

ost

verbatim)from

eitherthehomeworkassignments,orfrom

thestarred

problemsfrom

thissetofReview

Problems.Thestarredproblemsarethe

onesthatIrecommendthatyoureviewmostcarefully:Problems2,6,8,11,12,15,

17,and18.

PURPOSE:Thesereviewproblemsarenottobehandedin,butarebeingmadeavail-

abletohelpyoustudy.Theycomemainlyfromquizzesinpreviousyears.Insome

casesthenumberofpointsassignedtotheproblemonthequizislisted|

inall

suchcasesitisbasedon100pointsforthefullquiz.

Inadditiontothissetofproblems,youwill�ndonthecoursewebpagetheactual

quizzesthatweregivenin1994,1996,1998,2000,2002,2004,2007,2009,2011,2013,

and2016.Therelevantproblemsfromthosequizzeshavemostlybeenincorporated

intothesereviewproblems,butyoustillmaybeinterestedinlookingatthequizzes,

justtoseehowmuchmaterialhasbeenincludedineachquiz.Thecoverageofthe

upcomingquizwillnotnecessarilymatchthecoverageofanyofthequizzesfrom

previousyears.Thecoverageforeachquizinrecentyearsisusuallydescribedatthe

startofthereviewproblems,asIdidhere.

REVIEW

SESSION

AND

OFFICE

HOURS:Tohelpyoustudyforthequiz,

HonggeunKim

willholdareviewsessiononSunday,December2,at8:00pm,in

Room4-153.Inaddition,bothHonggeunKimandIwillbemovingouroÆcehours

forthetwolastweeksoftheterm.I(Alan)willholdmyoÆcehoursonMondays

(December3and10)at7:30pminmyoÆce,6-322,andHonggeunwillholdhis

oÆcehoursonTuesdays(December4and11)inRoom8-308.

8.286QUIZ3REVIEW

PROBLEMS,FALL2018

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

:

KINEMATICSOFAHOMOGENEOUSLY

EXPANDING

UNI-

VERSE:

Hubble'sLaw:v=Hr,

wherev=

recessionvelocityofadistantobject,H

=

Hubble

expansionrate,andr=distancetothedistantobject.

8.286QUIZ3REVIEW

PROBLEMS,FALL2018

p.3

PresentValueofHubbleExpansionRate(Planck2018):

H0=67:66�0:42km-s�

1-Mpc�

1

ScaleFactor:`p (t)=a(t)`c;

where`p (t)isthephysicaldistancebetweenanytwoobjects,a(t)

isthescalefactor,and`cisthecoordinatedistancebetweenthe

objects,alsocalledthecomovingdistance.

HubbleExpansionRate:H(t)=

1a(t)

da(t)

dt

.

LightRaysinComovingCoordinates:Lightraystravelinstraight

lineswithphysicalspeedcrelativetoanyobserver.InCartesian

coordinates,coordinatespeeddxd

t=

ca(t).Ingeneral,ds2

=

g�� dx�dx�=0:

HorizonDistance:

`p;horizon (t)=a(t) Z

t0

ca(t0)dt0

= �3ct

( at,matter-dominated),

2ct

( at,radiation-dominated).

COSMOLOGICALEVOLUTION:

H2= �_aa �2

=8�3

G��kc2

a2

;

�a=�4�3

G ��+3pc

2 �a;

�m(t)=a3(t

i )

a3(t)�m(ti )(matter);

�r (t)=a4(t

i )

a4(t)�r (ti )(radiation):

_�=�3_aa �

�+

pc2 �;��=�c;where�c=3H2

8�G

:

8.286QUIZ3REVIEW

PROBLEMS,FALL2018

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:

MINKOWSKIMETRIC(SpecialRelativity):

ds2��c2d�2=�c2dt2+dx2+dy2+dz2:

ROBERTSON-WALKER

METRIC:

ds2��c2d�2=�c2dt2+a2(t) �dr2

1�kr2+r2 �d�2+sin2�d�2 � �;

wheretheuniverseisclosed/open/ atifk>0/k<0/k=0.kcanbe

takenas+1(�1)iftheuniverseisclosed(open).

Alternatively,fork>0,wecande�ner=sin p

k,andthen

ds2��c2d�2��c2dt2+~a2(t) �d 2+sin2 �d�2+sin2�d�2 �;

where~a(t)=a(t)= pk.Fork<0wecande�ner=sinh

p�k,andthen

ds2��c2d�2=�c2dt2+~a2(t) �d 2+sinh2 �d�2+sin2�d�2 �;

where~a(t)=a(t)= p�k.Notethat~acanbecalledaifthereisnoneed

torelateittothea(t)thatappearsinthe�rstequationabove.

8.286QUIZ3REVIEW

PROBLEMS,FALL2018

p.5

SCHWARZSCHILD

METRIC:

ds2��c2d�2=� �1�2GM

rc2 �c2dt2+ �1�2GM

rc2 �

�

1dr2

+r2d�2+r2sin2�d�2;

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)

whereg� (

1perspinstateforbosons(integerspin)

7/8perspinstateforfermions(half-integerspin)

g�� (1perspinstateforbosons

3/4perspinstateforfermions,

8.286QUIZ3REVIEW

PROBLEMS,FALL2018

p.6

and

�(3)=

113+

123+

133+��1:202:

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

UNI-

VERSE:

�=

3

32�Gt2

kT= �45�h3c5

16�3gG �

1=4

1pt

Form�=106MeV�kT�me=0:511MeV,g=10:75andthen

kT=

0:860MeV

pt(insec) �

10:75

g �1=4

Afterthefreeze-outofelectron-positronpairs,

T�

T

= �411 �1=3

:

COSMOLOGICALCONSTANT:

uvac=�vac c2=

�c4

8�G

;

8.286QUIZ3REVIEW

PROBLEMS,FALL2018

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,FALL2018

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,thePlanckmass

mP,andthePlanckenergyEparegivenby

`P

= rG�h

c3

=1:616�10�

35m;

=1:616�10�

33cm;

tP

= r�hGc

5

=5:391�10�

44s;

mP

= r�hcG

=2:177�10�

8kg;

=2:177�10�

5g;

EP

= r�hc5

G

=1:221�1019GeV:

CHEMICALEQUILIBRIUM:

(Thistopicwasnotincludedinthecoursein2018,buttheformulas

arenonethelessincludedhereforlogicalcompleteness.Theywillnot

berelevanttoQuiz3.TheyarerelevanttoProblem14intheseReview

Problems,whichisalsonotrelevanttoQuiz3.Pleaseenjoylooking

attheseitems,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,thesum

ofthe�i

ontheleft-handsideofthe

reactionequationmustequalthesum

ofthe�iontheright-hand

side.Formulaassumesgasisnonrelativistic(kT�mi c2)anddilute

(ni �(2�mi kT)3=2=(2��h)3).

8.286QUIZ3REVIEW

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p.9

PROBLEM

LIST

1.DidYouDotheReading(2007)?

...............10(Sol:28)

*2.DidYouDotheReading(2009)?

...............11(Sol:30)


...............13(Sol:32)


...............14(Sol:34)

5.NumberDensitiesintheCosmicBackgroundRadiation......16(Sol:36)

*6.PropertiesofBlack-BodyRadiation

..............16(Sol:38)

7.ANewSpeciesofLepton

...................17(Sol:40)

*8.ANewTheoryoftheWeakInteractions

............18(Sol:43)

9.DoublingofElectrons

....................19(Sol:49)

10.TimeScalesinCosmology

..................19(Sol:51)

*11.EvolutionofFlatness.....................20(Sol:51)

*12.TheSloanDigitalSkySurveyz=5:82Quasar..........20(Sol:52)

13.SecondHubbleCrossing

...................21(Sol:57)

14.NeutrinoNumberandtheNeutron/ProtonEquilibrium

......23(Sol:60)

*15.TheEventHorizonforOurUniverse..............24(Sol:63)

16.TheE�ectofPressureonCosmologicalEvolution

........25(Sol:65)

*17.TheFreeze-outofaFictitiousParticleX

............26(Sol:67)

*18.TheTimeofDecoupling

...................26(Sol:70)

8.286QUIZ3REVIEW

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p.10

PROBLEM

1:DID

YOU

DO

THEREADING?(25points)

ThefollowingproblemwasProblem1,Quiz3,in2007.Eachpartwasworth5points.

(a)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBisnotcorrect:

(i)AfterthedipoledistortionoftheCMBissubtractedaway,themeantemperature

averagingovertheskyishTi=2:725K.

(ii)AfterthedipoledistortionoftheCMBissubtractedaway,therootmeansquare

temperature uctuationis D�ÆTT �2 E1=2

=1:1�10�

3.

(iii)ThedipoledistortionisasimpleDopplershift,causedbythenetmotionofthe

observerrelativetoaframeofreferenceinwhichtheCMBisisotropic.

(iv)Intheirgroundbreakingpaper,WilsonandPenziasreportedthemeasurement

ofanexcesstemperatureofabout3.5Kthatwasisotropic,unpolarized,and

freefromseasonalvariations.InacompanionpaperwrittenbyDicke,Peebles,

RollandWilkinson,theauthorsinterpretedtheradiationtobearelicofan

early,hot,dense,andopaquestateoftheuniverse.

(b)(CMBexperiments)ThecurrentmeanenergyperCMBphoton,about6�10�

4eV,

iscomparabletotheenergyofvibrationorrotationforasmallmoleculesuchasH2 O.

Thusmicrowaveswithwavelengthsshorterthan��3cmarestronglyabsorbedby

watermoleculesintheatmosphere.TomeasuretheCMBat�<3cm,whichone

ofthefollowingmethodsisnotafeasiblesolutiontothisproblem?

(i)MeasureCMBfromhigh-altitudeballoons,e.g.MAXIMA.

(ii)MeasureCMBfromtheSouthPole,e.g.DASI.

(iii)MeasureCMBfromtheNorthPole,e.g.BOOMERANG.

(iv)MeasureCMBfromasatelliteabovetheatmosphereoftheEarth,e.g.COBE,

WMAPandPLANCK.

(c)(Temperature uctuations)Thecreationoftemperature uctuationsinCMBby

variationsinthegravitationalpotentialisknownastheSachs-Wolfee�ect.Which

oneofthefollowingstatementsisnotcorrectconcerningthise�ect?

(i)ACMBphotonisredshiftedwhenclimbingoutofagravitationalpotentialwell,

andisblueshiftedwhenfallingdownapotentialhill.

(ii)Atthetimeoflastscattering,thenonbaryonicdarkmatterdominatedtheen-

ergydensity,andhencethegravitationalpotential,oftheuniverse.

(iii)Thelarge-scale uctuationsinCMBtemperaturesarisefromthegravitational

e�ectofprimordialdensity uctuationsinthedistributionofnonbaryonicdark

matter.

8.286QUIZ3REVIEW

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(iv)Thepeaksintheplotoftemperature uctuation�T

vs.multipolelaredueto

variationsinthedensityofnonbaryonicdarkmatter,whilethecontributions

frombaryonsalonewouldnotshowsuchpeaks.

(d)(Darkmattercandidates)Whichoneofthefollowingisnotacandidateofnonbary-

onicdarkmatter?

(i)massiveneutrinos

(ii)axions

(iii)mattermadeoftopquarks(atypeofquarkswithheavymassofabout171

GeV).

(iv)WIMPs(WeaklyInteractingMassiveParticles)

(v)primordialblackholes

(e)(Signaturesofdarkmatter)Bywhatmethodscansignaturesofdarkmatterbe

detected?Listtwomethods.(Grading:3pointsforonecorrectanswer,5pointsfor

twocorrectanswers.Ifyougivemorethantwoanswers,yourscorewillbebased

onthenumberofrightanswersminusthenumberofwronganswers,withalower

boundofzero.)

�

PROBLEM

2:DID

YOU

DO


ThisproblemwasProblem1,Quiz3,2009.

(a)(10points)Thisquestionconcernssomenumbersrelatedtothecosmicmicrowave

background(CMB)thatoneshouldneverforget.Statethevaluesofthesenumbers,

towithinanorderofmagnitudeunlessotherwisestated.Inallcasesthequestion

referstothepresentvalueofthesequantities.

(i)TheaveragetemperatureToftheCMB(towithin10%).

(ii)ThespeedoftheLocalGroupwithrespecttotheCMB,expressedasafraction

v=cofthespeedoflight.(ThespeedoftheLocalGroupisfoundbymeasuring

thedipolepatternoftheCMBtemperaturetodeterminethevelocityofthe

spacecraftwithrespecttotheCMB,andthenremovingspacecraftmotion,the

orbitalmotionoftheEarthabouttheSun,theSunaboutthegalaxy,andthe

galaxyrelativetothecenterofmassoftheLocalGroup.)

(iii)Theintrinsicrelativetemperature uctuations�T=T,afterremovingthedipole

anisotropycorrespondingtothemotionoftheobserverrelativetotheCMB.

(iv)Theratioofbaryonnumberdensitytophotonnumberdensity,�=nbary =n .

8.286QUIZ3REVIEW

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p.12

(v)Theangularsize�H,indegrees,correspondingtowhatwastheHubbledistance

c=H

atthesurfaceoflastscattering.Thisanswermustbewithinafactorof3

tobecorrect.

(b)(3points)Becausephotonsoutnumberbaryonsbysomuch,theexponentialtailof

thephotonblackbodydistributionisimportantinionizinghydrogenwellafterkT

fallsbelowQH

=13:6eV.WhatistheratiokT =QH

whentheionizationfraction

oftheuniverseis1=2?

(i)1=5

(ii)1=50

(iii)10�

3

(iv)10�

4

(v)10�

5

(c)(2points)WhichofthefollowingdescribestheSachs-Wolfee�ect?

(i)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear

redderbecauseoftheDopplere�ect.

(ii)Photonsfrom uidwhichhadavelocitytowardusalongthelineofsightappear

bluerbecauseoftheDopplere�ect.

(iii)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearredder

becausetheymustclimboutofthegravitationalpotentialwell.

(iv)Photonsfromoverdenseregionsatthesurfaceoflastscatteringappearbluer


(v)Photonstravelingtowardusfromthesurfaceoflastscatteringappearredder

becauseofabsorptionintheintergalacticmedium.

(vi)Photonstravelingtowardusfrom

thesurfaceoflastscatteringappearbluer


(d)(10points)Foreachofthefollowingstatements,saywhetheritistrueorfalse:

(i)Darkmatterinteractsthroughthegravitational,weak,andelectromagnetic

forces.

T

orF?

(ii)Thevirialtheoremcanbeappliedtoaclusterofgalaxiesto�nditstotalmass,

mostofwhichisdarkmatter.

T

orF?

(iii)Neutrinosarethoughttocompriseasigni�cantfractionoftheenergydensityof

darkmatter.

T

orF?

(iv)Magneticmonopolesarethoughttocompriseasigni�cantfractionoftheenergy

densityofdarkmatter.

T

orF?

(v)LensingobservationshaveshownthatMACHOscannotaccountforthedark

matteringalactichalos,butthatasmuchas20%ofthehalomasscouldbein

theformofMACHOs.

T

orF?

8.286QUIZ3REVIEW

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p.13

PROBLEM

3:DID

YOU

DO


ThiswasProblem1ofQuiz3,2013.

(a)(5points)RydensummarizestheresultsoftheCOBEsatelliteexperimentforthe

measurementsofthecosmicmicrowavebackground(CMB)intheform

ofthree

importantresults.The�rstwasthat,inanyparticulardirectionofthesky,the

spectrum

oftheCMBisveryclosetothatofanidealblackbody.TheFIRAS

instrumentontheCOBEsatellitecouldhavedetecteddeviationsfromtheblackbody

spectrumassmallas��=��10�

n,wherenisaninteger.Towithin�1,whatisn?

(b)(5points)ThesecondresultwasthemeasurementofadipoledistortionoftheCMB

spectrum;thatis,theradiationisslightlyblueshiftedtohighertemperaturesinone

direction,andslightlyredshiftedtolowertemperaturesintheoppositedirection.To

whatphysicale�ectwasthisdipoledistortionattributed?

(c)(5points)Thethirdresultconcernedthemeasurementoftemperature uctuations

afterthedipolefeaturementionedabovewassubtractedout.De�ning

ÆTT(�;�)�T(�;�)�hTi

hTi

;

wherehTi=2:725K,theaveragevalueofT,theyfoundarootmeansquare uctu-

ation,

*�ÆTT �2 +1=2

;

equaltosomenumber.Towithinanorderofmagnitude,whatwasthatnumber?

(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?














8.286QUIZ3REVIEW

PROBLEMS,FALL2018

p.14

(e)(5points)The atnessproblemreferstotheextreme�ne-tuningthatisneededin

atearlytimes,inorderforittobeascloseto1todayasweobserve.Startingwith

theassumptionthattodayisequalto1withinabout1%,oneconcludesthatat

onesecondafterthebigbang,j

�1jt=1sec<10�

m

;

wheremisaninteger.Towithin�3,whatism?

(f)(5points)Thetotalenergydensityofthepresentuniverseconsistsmainlyofbaryonic

matter,darkmatter,anddarkenergy.Givethepercentagesofeach,accordingto

thebest�tobtainedfromthePlanck2013data.Youwillgetfullcreditifthe�rst

(baryonicmatter)isaccurateto�2%,andtheothertwoareaccuratetowithin�5%.

(g)(5points)Withintheconventionalhotbigbangcosmology(withoutin ation),it

isdiÆculttounderstandhowthetemperatureoftheCMBcanbecorrelatedat

angularseparationsthataresolargethatthepointsonthesurfaceoflastscattering

wasseparatedfrom

eachotherbymorethanahorizondistance.Approximately

whatangle,indegrees,correspondstoaseparationonthesurfacelastscatteringof

onehorizonlength?Youwillgetfullcreditifyouranswerisrighttowithinafactor

of2.

PROBLEM

4:DID

YOU

DO

THEREADING?(2016)(25points)

Exceptforpart(d),youshouldanswerthesequestionsbycirclingtheonestatementthat

iscorrect.

(a)(5points)IntheEpilogueofTheFirstThreeMinutes,SteveWeinbergwrote:\The

moretheuniverseseemscomprehensible,themoreitalsoseemspointless."

The

sentencewasquali�ed,however,byaclosingparagraphthatpointsoutthat

(i)thequestofthehumanracetocreateabetterlifeforallcanstillgivemeaning

toourlives.

(ii)iftheuniversecannotgivemeaningtoourlives,thenperhapsthereisanafterlife

thatwill.

(iii)thecomplexityandbeautyofthelawsofphysicsstronglysuggestthatthe

universemusthaveapurpose,evenifwearenotawareofwhatitis.

(iv)thee�orttounderstandtheuniversegiveshumanlifesomeofthegraceof

tragedy.

(b)(5points)IntheAfterwordofTheFirstThreeMinutes,Weinbergdiscussesthe

baryonnumberoftheuniverse.(Thebaryonnumberofanysystem

isthetotal

numberofprotonsandneutrons(andcertainrelatedparticlesknownashyperons)

8.286QUIZ3REVIEW

PROBLEMS,FALL2018

p.15

minusthenumberoftheirantiparticles(antiprotons,antineutrons,antihyperons)

thatarecontainedinthesystem.)Weinbergconcludedthat

(i)baryonnumberisexactlyconserved,sothetotalbaryonnumberoftheuniverse

mustbezero.Whilenucleiinourpartoftheuniversearecomposedofprotons

andneutrons,theuniversemustalsocontainantimatterregionsinwhichnuclei

arecomposedofantiprotonsandantineutrons.

(ii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe

partoftheuniversewecanobserve,andhenceapositivedensityofbaryon

number.Sincebaryonnumberisconserved,thiscanonlybeexplainedby

assumingthattheexcessbaryonswereputinatthebeginning.

(iii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe

partoftheuniversewecanobserve,andhenceapositivedensityofbaryonnum-

ber.Thiscanbetakenasapositivehintthatbaryonnumberisnotconserved,

whichcanhappenifthereexistasyetundetectedheavy\exotic"particles.

(iv)itispossiblethatbaryonnumberisnotexactlyconserved,butevenifthatisthe

case,itisnotpossiblethattheobservedexcessofmatteroverantimattercanbe

explainedbytheveryrareprocessesthatviolatebaryonnumberconservation.

(c)(5points)IndiscussingtheCOBEmeasurementsofthecosmicmicrowaveback-

ground,Rydendescribesadipolecomponentofthetemperaturepattern,forwhich

thetemperatureoftheradiationfromonedirectionisfoundtobehotterthanthe

temperatureoftheradiationdetectedfromtheoppositedirection.

(i)Thisdiscoveryisimportant,becauseitallowsustopinpointthedirectionofthe

pointinspacewherethebigbangoccurred.

(ii)ThisisthelargestcomponentoftheCMBanisotropies,amountingtoa10%

variationinthetemperatureoftheradiation.

(iii)Inadditiontothedipolecomponent,theanisotropiesalsoincludescontributions

fromaquadrupole,octupole,etc.,allofwhicharecomparableinmagnitude.

(iv)ThispatternisinterpretedasasimpleDopplershift,causedbythenetmotion

oftheCOBEsatelliterelativetoaframeofreferenceinwhichtheCMBis

almostisotropic.

(d)(5points)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBis

notcorrect:





=1:1�10�

3.

8.286QUIZ3REVIEW

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(e)(5points)In ationisdrivenbya�eldthatisbyde�nitioncalledthein aton�eld.

Instandardin ationarymodels,the�eldhasthefollowingproperties:

(i)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe

universeisdominatedbyitspotentialenergy.

(ii)Thein atonisavector�eld,andduringin ationtheenergydensityofthe


(iii)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe

universeisdominatedbyitskineticenergy.

(iv)Thein atonisavector�eld,andduringin ationtheenergydensityofthe


(v)Thein atonisatensor�eld,whichisresponsibleforonlyasmallfractionof

theenergydensityoftheuniverseduringin ation.

PROBLEM

5:NUMBER

DENSITIESIN

THECOSMIC

BACKGROUND

RADIATION

Todaythetemperatureofthecosmicmicrowavebackgroundradiationis2:7ÆK.

Calculatethenumberdensityofphotonsinthisradiation.Whatisthenumberdensity

ofthermalneutrinosleftoverfromthebigbang?

�

PROBLEM

6:PROPERTIESOFBLACK-BODYRADIATION(25points)

ThefollowingproblemwasProblem4,Quiz3,1998.

Inansweringthefollowingquestions,rememberthatyoucanrefertotheformulas

atthefrontoftheexam.Sinceyouwerenotaskedtobringcalculators,youmayleave

youranswersintheformofalgebraicexpressions,suchas�32= p5�(3).

(a)(5points)Fortheblack-bodyradiation(alsocalledthermalradiation)ofphotonsat

temperatureT,whatistheaverageenergyperphoton?

(b)(5points)Forthesameradiation,whatistheaverageentropyperphoton?

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(c)(5points)Nowconsidertheblack-bodyradiationofamasslessbosonwhichhasspin

zero,sothereisonlyonespinstate.Wouldtheaverageenergyperparticleand

entropyperparticlebedi�erentfromtheanswersyougaveinparts(a)and(b)?If

so,howwouldtheychange?

(d)(5points)Nowconsidertheblack-bodyradiationofelectronneutrinos.Theseparti-

clesarefermionswithspin1/2,andwewillassumethattheyaremasslessandhave

onlyonepossiblespinstate.Whatistheaverageenergyperparticleforthiscase?

(e)(5points)Whatistheaverageentropyperparticlefortheblack-bodyradiationof

neutrinos,asdescribedinpart(d)?

PROBLEM

7:A

NEW

SPECIESOFLEPTON

ThefollowingproblemwasProblem2,Quiz3,1992,worth25points.

Supposethecalculationsdescribingtheearlyuniverseweremodi�edbyincludingan

additional,hypotheticallepton,calledan8.286ion.The8.286ionhasroughlythesame

propertiesasanelectron,exceptthatitsmassisgivenbymc2=0:750MeV.

Parts(a)-(c)ofthisquestionrequirenumericalanswers,butsinceyouwerenot

toldtobringcalculators,youneednotcarryoutthearithmetic.Youranswershould

beexpressed,however,in\calculator-ready"form|

thatis,itshouldbeanexpression

involvingpurenumbersonly(nounits),withanynecessaryconversionfactorsincluded.

(Forexample,ifyouwereaskedhowmanymetersalightpulseinvacuumtravelsin5

minutes,youcouldexpresstheansweras2:998�108�5�60.)

a)(5points)Whatwouldbethenumberdensityof8.286ions,inparticlespercubic

meter,whenthetemperatureTwasgivenbykT=3MeV?

b)(5points)Assuming(asinthestandardpicture)thattheearlyuniverseisaccurately

describedbya at,radiation-dominatedmodel,whatwouldbethevalueofthemass

densityatt=:01sec?Youmayassumethat0:75MeV�kT�100MeV,sothe

particlescontributingsigni�cantlytotheblack-bodyradiationincludethephotons,

neutrinos,e+-e�

pairs,and8.286ion-anti8286ionpairs.Expressyouranswerinthe

unitsofg/cm3.

c)(5points)Underthesameassumptionsasin(b),whatwouldbethevalueofkT,in

MeV,att=:01sec?

d)(5points)Whennucleosynthesiscalculationsaremodi�edtoincludethee�ectofthe

8.286ion,istheproductionofheliumincreasedordecreased?Explainyouranswer

inafewsentences.

e)(5points)SupposetheneutrinosdecouplewhilekT�0:75MeV.Ifthe8.286ions

areincluded,whatdoesonepredictforthevalueofT�=T today?(HereT�denotes

thetemperatureoftheneutrinos,andT

denotesthetemperatureofthecosmic

backgroundradiationphotons.)

8.286QUIZ3REVIEW

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�

PROBLEM

8:A

NEW

THEORY

OFTHEWEAK

INTERACTIONS(40

points)

ThisproblemwasProblem3,Quiz3,2009.

SupposeaNewTheoryoftheWeakInteractions(NTWI)wasproposed,whichdif-

fersfrom

thestandardtheoryintwoways.First,theNTWIpredictsthattheweak

interactionsaresomewhatweakerthaninthestandardmodel.Inaddition,thetheory

impliestheexistenceofnewspin-12particles(fermions)calledtheR+

andR�

,witharest

energyof50MeV(where1MeV=106eV).Thisproblemwilldealwiththecosmological

consequencesofsuchatheory.

TheNTWIwillpredictthattheneutrinosintheearlyuniversewilldecoupleat

ahighertemperaturethaninthestandardmodel.Supposethatthisdecouplingtakes

placeatkT�200MeV.Thismeansthatwhentheneutrinosceasetobethermally

coupledtotherestofmatter,thehotsoupofparticleswouldcontainnotonlyphotons,

neutrinos,ande+-e�

pairs,butalso�+,��

,�+,��

,and�0particles,alongwiththe

R+-R�

pairs.(Themuonisaparticlewhichbehavesalmostidenticallytoanelectron,

exceptthatitsrestenergyis106MeV.Thepionsarethelightestofthemesons,with

zeroangularmomentumandrestenergiesof135MeVand140MeVfortheneutraland

chargedpions,respectively.The�+

and��

areantiparticlesofeachother,andthe�0

isitsownantiparticle.Zeroangularmomentumimpliesasinglespinstate.)Youmay

assumethattheuniverseis at.

(a)(10points)Accordingtothestandardparticlephysicsmodel,whatisthemass

density�oftheuniversewhenkT�200MeV?Whatisthevalueof�atthis

temperature,accordingtoNTWI?Useeitherg/cm3orkg/m3.(Ifyouwish,youcan

savetimebynotcarryingoutthearithmetic.Ifyoudothis,however,youshould

givetheanswerin\calculator-ready"form,bywhichImeananexpressioninvolving

purenumbers(nounits),withanynecessaryconversionfactorsincluded,andwith

theunitsoftheanswerspeci�edattheend.Forexample,ifaskedhowfarlight

travelsin5minutes,youcouldanswer2:998�108�5�60m.)

(b)(10points)Accordingtothestandardmodel,thetemperaturetodayofthethermal

neutrinobackgroundshouldbe(4=11)1=3T

,whereT

isthetemperatureofthe

thermalphotonbackground.WhatdoestheNTWIpredictforthetemperatureof

thethermalneutrinobackground?

(c)(10points)Accordingtothestandardmodel,whatistheratiotodayofthenumber

densityofthermalneutrinostothenumberdensityofthermalphotons?Whatis

thisratioaccordingtoNTWI?

(d)(10points)Sincethereactionswhichinterchangeprotonsandneutronsinvolveneu-

trinos,thesereactions\freezeout"atroughlythesametimeastheneutrinosdecou-

ple.Atlatertimestheonlyreactionwhiche�ectivelyconvertsneutronstoprotons

isthefreedecayoftheneutron.Despitethefactthatneutrondecayisaweakinter-

action,wewillassumethatitoccurswiththeusual15minutemeanlifetime.Would

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theheliumabundancepredictedbytheNTWIbehigherorlowerthantheprediction

ofthestandardmodel?Towithin5or10%,whatwouldtheNTWIpredictforthe

percentabundance(byweight)ofheliumintheuniverse?(Asinpart(a),youcan

eithercarryoutthearithmetic,orleavetheanswerincalculator-readyform.)

Usefulinformation:Theprotonandneutronrestenergiesaregivenbympc2

=

938:27MeVandmnc2=939:57MeV,with(mn �mp )c2=1.29MeV.Themean

lifetimefortheneutrondecay,n!p+e�

+��e,isgivenby�=886s.

PROBLEM

9:DOUBLING

OFELECTRONS(10points)

ThefollowingwasonQuiz3,2011(Problem4):

Supposethatinsteadofonespeciesofelectronsandtheirantiparticles,supposethere

wasalsoanotherspeciesofelectron-likeandpositron-likeparticles.Supposethatthenew

specieshasthesamemassandotherpropertiesastheelectronsandpositrons.Ifthis

werethecase,whatwouldbetheratioT� =T ofthetemperaturetodayoftheneutrinos

tothetemperatureoftheCMBphotons.

PROBLEM

10:TIMESCALESIN

COSMOLOGY

Inthisproblemyouareaskedtogivetheapproximatetimesatwhichvariousim-

portanteventsinthehistoryoftheuniversearebelievedtohavetakenplace.Thetimes

aremeasuredfromtheinstantofthebigbang.Toavoidambiguities,youareaskedto

choosethebestanswerfromthefollowinglist:

10�

43sec.

10�

37sec.

10�

12sec.

10�

5sec.

1sec.

4mins.

10,000{1,000,000years.

2billionyears.

5billionyears.

10billionyears.

13billionyears.

20billionyears.

ForthisproblemitwillbesuÆcienttostateananswerfrommemory,withoutexplanation.

Theeventswhichmustbeplacedarethefollowing:

(a)thebeginningoftheprocessesinvolvedinbigbangnucleosynthesis;

(b)theendoftheprocessesinvolvedinbigbangnucleosynthesis;

8.286QUIZ3REVIEW

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(c)thetimeofthephasetransitionpredictedbygranduni�edtheories,whichtakes

placewhenkT�1016GeV;

(d)\recombination",thetimeatwhichthematterintheuniverseconvertedfrom

aplasmatoagasofneutralatoms;

(e)thephasetransitionatwhichthequarksbecamecon�ned,believedtooccur

whenkT�300MeV.

Sincecosmologyisfraughtwithuncertainty,insomecasesmorethanoneanswerwill

beacceptable.Youareasked,however,togiveONLY

ONEoftheacceptableanswers.

�

PROBLEM

11:EVOLUTION

OFFLATNESS(15points)


The\ atnessproblem"isrelatedtothefactthatduringtheevolutionofthestandard

cosmologicalmodel,isalwaysdrivenawayfrom1.

(a)(9points)Duringaperiodinwhichtheuniverseismatter-dominated(meaningthat

theonlyrelevantcomponentisnonrelativisticmatter),thequantity

�1

growsasapoweroft.Showthatthisistrue,andderivethepower.(Statingthe

rightpowerwithoutaderivationwillbeworth3points.)

(b)(6points)Duringaperiodinwhichtheuniverseisradiation-dominated,thesame

quantitywillgrowlikeadi�erentpoweroft.Showthatthisistrue,andderivethe

power.(Statingtherightpowerwithoutaderivationwillagainbeworth3points.)

Ineachpart,youmayassumethattheuniversewasalwaysdominatedbythespeci�ed

formofmatter.

�

PROBLEM

12:THESLOANDIGITALSKYSURVEYz

=5:82QUASAR

(40points)


OnApril13,2000,theSloanDigitalSkySurveyannouncedthediscoveryofwhat

wasthenthemostdistantobjectknownintheuniverse:aquasaratz=5:82.Toexplain

tothepublichowthisobject�tsintotheuniverse,theSDSSpostedontheirwebsitean

articlebyMichaelTurnerandCraigWiegerttitled\HowCanAnObjectWeSeeToday

be27BillionLightYearsAwayIftheUniverseisonly14BillionYearsOld?"Usinga

modelwithH0=65km-s�

1-Mpc�

1,m

=0:35,and�

=0:65,theyclaimed

(a)thattheageoftheuniverseis13.9billionyears.

8.286QUIZ3REVIEW

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(b)thatthelightthatwenowseewasemittedwhentheuniversewas0.95billionyears

old.

(c)thatthedistancetothequasar,asitwouldbemeasuredbyarulertoday,is27

billionlight-years.

(d)thatthedistancetothequasar,atthetimethelightwasemitted,was4.0billion

light-years.

(e)thatthepresentspeedofthequasar,de�nedastherateatwhichthedistance

betweenusandthequasarisincreasing,is1.8timesthevelocityoflight.

Thegoalofthisproblemistocheckalloftheseconclusions,althoughyouareofcourse

notexpectedtoactuallyworkoutthenumbers.Youranswerscanbeexpressedinterms

ofH0 ,m,�,andz.De�niteintegralsneednotbeevaluated.

Notethatm

representsthepresentdensityofnonrelativisticmatter,expressedas

afractionofthecriticaldensity;and�

representsthepresentdensityofvacuumenergy,

expressedasafractionofthecriticaldensity.Inansweringeachofthefollowingquestions,

youmayconsidertheanswertoanypreviouspart|

whetheryouanswereditornot|

asagivenpieceofinformation,whichcanbeusedinyouranswer.

(a)(15points)Writeanexpressionfortheaget0ofthismodeluniverse?

(b)(5points)Writeanexpressionforthetimeteatwhichthelightwhichwenowreceive

fromthedistantquasarwasemitted.

(c)(10points)Writeanexpressionforthepresentphysicaldistance`phys;0tothequasar.

(d)(5points)Writeanexpressionforthephysicaldistance`phys;ebetweenusandthe

quasaratthetimethatthelightwasemitted.

(e)(5points)Writeanexpressionforthepresentspeedofthequasar,de�nedasthe

rateatwhichthedistancebetweenusandthequasarisincreasing.

PROBLEM

13:SECOND

HUBBLECROSSING

(40points)

ThisproblemwasProblem3,Quiz3,2007.In2018wehavenotyettalkedaboutHubble

crossingsandtheevolutionofdensityperturbations,sothisproblemwouldnotbefairas

worded.Actually,however,youhavelearnedhowtodothesecalculations,sotheproblem

wouldbefairifitdescribedinmoredetailwhatneedstobecalculated.

InProblemSet9(2007)wecalculatedthetimetH1 (�)ofthe�rstHubblecrossing

foramodespeci�edbyits(physical)wavelength�atthepresenttime.Inthisproblem

wewillcalculatethetimetH2 (�)ofthesecondHubblecrossing,thetimeatwhichthe

growingHubblelengthcH�

1(t)catchesuptothephysicalwavelength,whichisalso

growing.AtthetimeofthesecondHubblecrossingforthewavelengthsofinterest,the

universecanbedescribedverysimply:itisaradiation-dominated atuniverse.However,

since�isde�nedasthepresentvalueofthewavelength,theevolutionoftheuniverse

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betweentH2 (�)andthepresentwillalsoberelevanttotheproblem.Wewillneedto

usemethods,therefore,thatallowforboththematter-dominatederaandtheonsetof

thedark-energy-dominatedera.AsinProblemSet9(2007),themodeluniversethatwe

considerwillbedescribedbytheWMAP3-yearbest�tparameters:

Hubbleexpansionrate

H0

=

73:5km�s�

1�Mpc�

1

Nonrelativisticmassdensity

m

=

0.237

Vacuummassdensity

vac

=

0.763

CMBtemperature

T ;0

=

2.725K

Themassdensitiesarede�nedascontributionsto,andhencedescribethemassdensity

ofeachconstituentrelativetothecriticaldensity.Notethatthemodelisexactly at,

soyouneednotworryaboutspatialcurvature.Hereyouarenotexpectedtogivea

numericalanswer,sotheabovelistwillserveonlytode�nethesymbolsthatcanappear

inyouranswers,alongwith�andthephysicalconstantsG,�h,c,andk.

(a)(5points)Foraradiation-dominated atuniverse,whatistheHubblelength`H(t)�

cH�

1(t)asafunctionoftimet?

(b)(10points)ThesecondHubblecrossingwilloccurduringtheinterval

30sec�t�50;000years,

whenthemassdensityoftheuniverseisdominatedbyphotonsandneutrinos.During

thiseratheneutrinosarealittlecolderthanthephotons,withT�=(4=11)1=3T

.

Thetotalenergydensityofthephotonsandneutrinostogethercanbewrittenas

utot=g1�2

30

(kT )4

(�hc)3

:

Whatisthevalueofg1 ?(Forthefollowingpartsyoucantreatg1asagivenvariable

thatcanbeleftinyouranswers,whetherornotyoufoundit.)

(c)(10points)Fortimesintherangedescribedinpart(b),whatisthephotontemper-

atureT (t)asafunctionoft?

(d)(15points)Finally,wearereadyto�ndthetimetH2 (�)ofthesecondHubble

crossing,foragivenvalueofthephysicalwavelength�today.Makinguseofthe

previousresults,youshouldbeabletodeterminetH2 (�).Ifyouwerenotableto

answersomeofthepreviousparts,youmayleavethesymbols`H(t),g1 ,and/or

T (t)inyouranswer.

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PROBLEM

14:NEUTRINO

NUMBER

AND

THENEUTRON/PROTON

EQUILIBRIUM

(35points)

Thefollowingproblemwas1998Quiz4,Problem4.ThiswouldNOTbeafairproblem

for2018,asthisyearwehavenotdiscussedbigbangnucleosynthesisatthislevelofdetail.

ButIamincludingtheproblemanyway,asyoumight�nditinteresting.

Inthestandardtreatmentofbigbangnucleosynthesisitisassumedthatatearly

timestheratioofneutronstoprotonsisgivenbytheBoltzmannformula,

nn

np

=e�

�E=kT

;

(1)

wherekisBoltzmann'sconstant,Tisthetemperature,and�E

=1:29MeVisthe

proton-neutronmass-energydi�erence.Thisformulaisbelievedtobeveryaccurate,

butitassumesthatthechemicalpotentialforneutrons�n

isthesameasthechemical

potentialforprotons�p .

(a)(10points)GivethecorrectversionofEq.(1),allowingforthepossibilitythat

�n 6=�p .

Theequilibriumbetweenprotonsandneutronsintheearlyuniverseissustainedmainly

bythefollowingreactions:

e+

+n !p+��e

�e+n !p+e�

:

Let�eand��

denotethechemicalpotentialsfortheelectrons(e�

)andtheelectron

neutrinos(�e )respectively.Thechemicalpotentialsforthepositrons(e+)andtheanti-

electronneutrinos(��e )arethen{�eand{�� ,respectively,sincethechemicalpotential

ofaparticleisalwaysthenegativeofthechemicalpotentialfortheantiparticle.*

(b)(10points)Expresstheneutron/protonchemicalpotentialdi�erence�n ��pinterms

of�eand��.

Theblack-bodyradiationformulasatthebeginningofthequizdidnotallowforthe

possibilityofachemicalpotential,buttheycaneasilybegeneralized.Forexample,the

formulaforthenumberdensityni(ofparticlesoftypei)becomes

ni=g�i�(3)

�2

(kT)3

(�hc)3e�i =kT

:

(c)(10points)Supposethatthedensityofanti-electronneutrinos�n�intheearlyuni-

versewashigherthanthedensityofelectronneutrinosn� .Expressthethermal

*Thisfactisaconsequenceoftheprinciplethatthechemicalpotentialofaparticleis

thesumofthechemicalpotentialsassociatedwithitsconservedquantities,whileparticle

andantiparticlealwayshavetheoppositevaluesofallconservedquantities.

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equilibriumvalueoftherationn=npintermsof�E,T,andeithertheratio�n� =n�

ortheantineutrinoexcess�n=�n� �n� .(Youranswermayalsocontainfundamental

constants,suchask,�h,andc.)

(d)(5points)Wouldanexcessofanti-electronneutrinos,asconsideredinpart(c),

increaseordecreasetheamountofhelium

thatwouldbeproducedintheearly

universe?Explainyouranswer.

PROBLEM

15:THEEVENTHORIZON

FOROURUNIVERSE(25points)

ThefollowingproblemwasProblem3fromQuiz3,2013.

Wehavelearnedthattheexpansionhistoryofouruniversecanbedescribedinterms

ofasmallsetofnumbers:m;0 ,thepresentcontributiontofromnonrelativisticmatter;

rad;0 ,thepresentcontributiontofromradiation;vac ,thepresentcontributionto

fromvacuumenergy;andH0 ,thepresentvalueoftheHubbleexpansionrate.The

bestestimatesofthesenumbersareconsistentwitha atuniverse,sowecantakek=0,

m;0+rad;0+vac=1,andwecanusethe atRobertson-Walkermetric,

ds2=�c2dt2+a2(t) �dr2+r2 �d�2+sin2�d�2 ��:

(a)(5points)Supposethatweareattheoriginofthecoordinatesystem,andthatat

thepresenttimet0weemitasphericalpulseoflight.Itturnsoutthatthereisa

maximumcoordinateradiusr=rmaxthatthispulsewilleverreach,nomatterhow

longwewait.(Thepulsewillneveractuallyreachrmax ,butwillreachallrsuchthat

0<r<rmax .)rmax

isthecoordinateofwhatiscalledtheeventhorizon:events

thathappennowatr�rmaxwillneverbevisibletous,assumingthatweremainat

theorigin.Assumingforthispartthatthefunctiona(t)isaknownfunction,write

anexpressionforrmax .Youranswershouldbeexpressedasanintegral,whichcan

involvea(t),t0 ,andanyoftheparametersde�nedinthepreamble.[Advice:Ifyou

cannotanswerthis,youshouldstilltrypart(c).]

(b)(10points)Sincea(t)isnotknownexplicitly,theanswertothepreviouspartis

diÆculttouse.Show,however,thatbychangingthevariableofintegration,youcan

rewritetheexpressionforrmaxasade�niteintegralinvolvingonlytheparameters

speci�edinthepreamble,withoutanyreferencetothefunctiona(t),exceptperhaps

toitspresentvaluea(t0 ).Youarenotexpectedtoevaluatethisintegral.[Hint:One

methodistouse

x=

a(t)

a(t0 )

asthevariableofintegration,justaswedidwhenwederivedthe�rstoftheexpres-

sionsfort0shownintheformulasheets.]

(c)(10points)Astronomersoftendescribedistancesintermsofredshifts,soitisuseful

to�ndtheredshiftoftheeventhorizon.Thatis,ifalightraythatoriginatedat

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r=rmaxarrivedatEarthtoday,whatwouldbeitsredshiftzeh(eh=eventhorizon)?

Youarenotaskedto�ndanexplicitexpressionforzeh,butinsteadanequationthat

couldbesolvednumericallytodeterminezeh .Forthispartyoucantreatrmax

asgiven,soitdoesnotmatterifyouhavedoneparts(a)and(b).Youwillget

halfcreditforacorrectanswerthatinvolvesthefunctiona(t),andfullcreditfora

correctanswerthatinvolvesonlyexplicitintegralsdependingonlyontheparameters

speci�edinthepreamble,andpossiblya(t0 ).

PROBLEM

16:THEEFFECTOFPRESSUREONCOSMOLOGICALEVO-

LUTION

(25points)

ThefollowingproblemwasProblem2ofQuiz3,2016.ItwasalsoProblem2ofProblem

Set7(2016),exceptthatsomenumericalconstantshavebeenchanged,sotheanswers

willnotbeidentical.

Aradiation-dominateduniversebehavesdi�erentlyfromamatter-dominateduni-

versebecausethepressureoftheradiationissigni�cant.Inthisproblemweexplorethe

roleofpressureforseveral�ctitiousformsofmatter.

(a)(8points)Forthe�rst�ctitiousformofmatter,themassdensity�decreasesasthe

scalefactora(t)grows,withtherelation

�(t)/1

a8(t):

Whatisthepressureofthisformofmatter?[Hint:theanswerisproportionalto

themassdensity.]

(b)(9points)Findthebehaviorofthescalefactora(t)fora atuniversedominated

bytheformofmatterdescribedinpart(a).Youshouldbeabletodeterminethe

functiona(t)uptoaconstantfactor.

(c)(8points)Nowconsiderauniversedominatedbyadi�erentformof�ctitiousmatter,

withapressuregivenby

p=23

�c2:

Astheuniverseexpands,themassdensityofthisformofmatterbehavesas

�(t)/1

an(t):

Findthepowern.

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�

PROBLEM

17:THE

FREEZE-OUT

OFA

FICTITIOUSPARTICLE

X

(25points)

ThefollowingproblemwasProblem3ofQuiz3,2016.

Supposethat,inadditiontotheparticlesthatareknowntoexist,therealsoexisteda

familyofthreespin-1particles,X+,X�

,andX0,allwithmasses0.511MeV/c2,exactly

thesameastheelectron.TheX�

istheantiparticleoftheX+,andtheX0isitsown

antiparticle.SincetheX'sarespin-1particleswithnonzeromass,eachparticlehasthree

spinstates.

TheX'sdonotinteractwithneutrinosanymorestronglythantheelectronsand

positronsdo,sowhentheX'sfreezeout,alloftheirenergyandentropyaregiventothe

photons,justliketheelectron-positronpairs.

(a)(5points)InthermalequilibriumwhenkT�0:511MeV/c2,whatisthetotalenergy

densityoftheX+,X�

,andX0particles?

(b)(5points)Inthermalequilibrium

whenkT

�0:511MeV/c2,whatisthetotal

numberdensityoftheX+,X�

,andX0particles?

(c)(10points)TheXparticlesandtheelectron-positronpairsfreezeoutofthethermal

equilibriumradiationatthesametime,askTdecreasesfromvalueslargecompared

to0.511MeV/c2tovaluesthataresmallcomparedtoit.IftheX's,electron-positron

pairs,photons,andneutrinoswereallinthermalequilibriumbeforethisfreeze-out,

whatwillbetheratioT�=T ,theratiooftheneutrinotemperaturetothephoton

temperature,afterthefreeze-out?

(d)(5points)IfthemassoftheX'swas,forexample,0.100MeV/c2,sothattheelectron-

positronpairsfrozeout�rst,andthentheX'sfrozeout,wouldthe�nalratioT� =T

behigher,lower,orthesameastheanswertopart(c)?Explainyouranswerina

sentenceortwo.

�

PROBLEM

18:THETIMEtd

OFDECOUPLING

(25points)

ThefollowingproblemwasProblem4ofQuiz3,2016.

Theprocessbywhichthephotonsofthecosmicmicrowavebackgroundstopscatter-

ingandbegintotravelonstraightlinesiscalleddecoupling,andithappensataphoton

temperatureofaboutTd

�3;000K.InLectureNotes6weestimatedthetimetd

of

decoupling,workingintheapproximationthattheuniversehasbeenmatter-dominated

fromthattimetothepresent.Wefoundavalueof370,000years.Inthisproblemwe

willremovethisapproximation,althoughwewillnotcarryoutthenumericalevaluation

neededtocomparewiththepreviousanswer.

8.286QUIZ3REVIEW

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(a)(5points)Letusde�ne

x(t)�a(t)

a(t0 );

asontheformulasheets,wheret0isthepresenttime.Whatisthevalueofxd �

x(td )?Assumethattheentropyofphotonsisconservedfromtimetdtothepresent,

andletT0denotethepresentphotontemperature.

(b)(5points)Assumethattheuniverseis at,andthatm;0 ,rad;0 ,andvac;0denote

thepresentcontributionstofromnonrelativisticmatter,radiation,andvacuum

energy,respectively.LetH0denotethepresentvalueoftheHubbleexpansionrate.

Writeanexpressionintermsofthesequantitiesfordx=dt,thederivativeofxwith

respecttot.Hint:youmayuseformulasfromtheformulasheetwithoutderivation,

sothisproblemshouldrequireessentiallynowork.Toreceivefullcredit,youranswer

shouldincludeonlytermsthatmakeanonzerocontributiontotheanswer.

(c)(5points)Writeanexpressionfortd .Ifyouranswerinvolvesanintegral,youneed

nottrytoevaluateit,butyoushouldbesurethatthelimitsofintegrationareclearly

shown.

(d)(10points)Nowsupposethatinadditiontotheconstituentsdescribedinpart(b),

theuniversealsocontainssomeofthe�ctitiousmaterialfrompart(a)ofProblem

2,with

�(t)/1

a8(t):

Denotethepresentcontributiontofrom

this�ctitiousmaterialasf;0 .The

universeisstillassumedtobe at,sothenumericalvaluesofm;0 ,rad;0 ,and

vac;0

mustsum

toasmallervaluethaninparts(b)and(c).Withthisextra

contributiontothemassdensityoftheuniverse,whatisthenewexpressionfortd ?

8.286QUIZ3REVIEW

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p.28

SOLUTIONS

PROBLEM

1:DID

YOU

DO


Thefollowingpartsareeachworth5points.

(a)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBisnotcorrect:





=1:1�10�

3.








Explanation:Aftersubtractingthedipolecontribution,thetemperature uctua-

tionisabout1:1�10�

5.

(b)(CMBexperiments)ThecurrentmeanenergyperCMBphoton,about6�10�

4eV,

iscomparabletotheenergyofvibrationorrotationforasmallmoleculesuchasH2 O.

Thusmicrowaveswithwavelengthsshorterthan��3cmarestronglyabsorbedby

watermoleculesintheatmosphere.TomeasuretheCMBat�<3cm,whichone

ofthefollowingmethodsisnotafeasiblesolutiontothisproblem?

(i)MeasureCMBfromhigh-altitudeballoons,e.g.MAXIMA.

(ii)MeasureCMBfromtheSouthPole,e.g.DASI.

(iii)MeasureCMBfromtheNorthPole,e.g.BOOMERANG.

(iv)MeasureCMBfromasatelliteabovetheatmosphereoftheEarth,e.g.COBE,

WMAPandPLANCK.

Explanation:TheNorthPoleisatsealevel.Incontrast,theSouthPoleis

nearly3kilometersabovesealevel.BOOMERANGisaballoon-borneexperi-

mentlaunchedfromAntarctica.

(c)(Temperature uctuations)Thecreationoftemperature uctuationsinCMBby

variationsinthegravitationalpotentialisknownastheSachs-Wolfee�ect.Which

oneofthefollowingstatementsisnotcorrectconcerningthise�ect?

8.286QUIZ3REVIEW

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(i)ACMBphotonisredshiftedwhenclimbingoutofagravitationalpotentialwell,

andisblueshiftedwhenfallingdownapotentialhill.

(ii)Atthetimeoflastscattering,thenonbaryonicdarkmatterdominatedtheen-

ergydensity,andhencethegravitationalpotential,oftheuniverse.

(iii)Thelarge-scale uctuationsinCMBtemperaturesarisefromthegravitational

e�ectofprimordialdensity uctuationsinthedistributionofnonbaryonicdark

matter.

(iv)Thepeaksintheplotoftemperature uctuation�T

vs.multipolelaredueto

variationsinthedensityofnonbaryonicdarkmatter,whilethecontributions

frombaryonsalonewouldnotshowsuchpeaks.

Explanation:Thesepeaksareduetotheacousticoscillationsinthephoton-

baryon uid.

(d)(Darkmattercandidates)Whichoneofthefollowingisnotacandidateofnonbary-

onicdarkmatter?

(i)massiveneutrinos

(ii)axions

(iii)mattermadeoftopquarks(atypeofquarkswithheavymassofabout171

GeV).

(iv)WIMPs(WeaklyInteractingMassiveParticles)

(v)primordialblackholes

Explanation:Mattermadeoftopquarksissounstablethatitisseenonly eet-

inglyasaproductinhighenergyparticlecollisions.

(e)(Signaturesofdarkmatter)Bywhatmethodscansignaturesofdarkmatterbe

detected?Listtwomethods.(Grading:3pointsforonecorrectanswer,5pointsfor

twocorrectanswers.Ifyougivemorethantwoanswers,yourscorewillbebased

onthenumberofrightanswersminusthenumberofwronganswers,withalower

boundofzero.)

Answers:

(i)Galaxyrotationcurves.(I.e.,measurementsoftheorbitalspeedofstarsinspiral

galaxiesasafunctionofradiusRshowthatthesecurvesremain atatradii

farbeyondthevisiblestellardisk.Ifmostofthematterwerecontainedinthe

disk,thenthesevelocitiesshouldfallo�as1= pR.)

(ii)Usethevirialtheoremtoestimatethemassofagalaxycluster.(Forexample,

thevirialanalysisshowsthatonly2%ofthemassoftheComaclusterconsists

ofstars,andonly10%consistsofhotintraclustergas.

8.286QUIZ3REVIEW

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(iii)Gravitationallensing.(Forexample,themassofaclustercanbeestimatedfrom

thedistortionoftheshapesofthegalaxiesbehindthecluster.)

(iv)CMBtemperature uctuations.(I.e.,theanalysisoftheintensityofthe uc-

tuationsasafunctionofmultipolenumbershowsthattot �1,andthatdark

energycontributes�

�0:7,baryonicmattercontributesbary

�0:04,and

darkmattercontributesdarkmatter �0:26.)

Thereareotherpossibleanswersaswell,butthesearetheonesdiscussedbyRyden

inChapters8and9.

PROBLEM

2:DID

YOU

DO


(a)(10points)Thisquestionconcernssomenumbersrelatedtothecosmicmicrowave

background(CMB)thatoneshouldneverforget.Statethevaluesofthesenumbers,

towithinanorderofmagnitudeunlessotherwisestated.Inallcasesthequestion

referstothepresentvalueofthesequantities.

(i)TheaveragetemperatureToftheCMB(towithin10%).2:725K

(ii)ThespeedoftheLocalGroupwithrespecttotheCMB,expressedasafraction

v=cofthespeedoflight.(ThespeedoftheLocalGroupisfoundbymeasuring

thedipolepatternoftheCMBtemperaturetodeterminethevelocityofthe

spacecraftwithrespecttotheCMB,andthenremovingspacecraftmotion,the

orbitalmotionoftheEarthabouttheSun,theSunaboutthegalaxy,andthe

galaxyrelativetothecenterofmassoftheLocalGroup.)

Thedipoleanisotropycorrespondstoa\peculiarvelocity"(thatis,velocitywhich

isnotduetotheexpansionoftheuniverse)of630�20kms�

1,orintermsof

thespeedoflight,v=c�2�10�

3.

(iii)Theintrinsicrelativetemperature uctuations�T=T,afterremovingthedipole

anisotropycorrespondingtothemotionoftheobserverrelativetotheCMB.

1:1�10�

5

(iv)Theratioofbaryonnumberdensitytophotonnumberdensity,�=nbary =n .

TheWMAP5-yearvaluefor�=nb =n =

(6:225�0:170)�10�

10,whichto

closestorderofmagnitudeis10�

9.

(v)Theangularsize�H,indegrees,correspondingtowhatwastheHubbledistance

c=H

atthesurfaceoflastscattering.Thisanswermustbewithinafactorof3

tobecorrect.�1Æ

(b)(3points)Becausephotonsoutnumberbaryonsbysomuch,theexponentialtailof

thephotonblackbodydistributionisimportantinionizinghydrogenwellafterkT

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.31

fallsbelowQH

=13:6eV.WhatistheratiokT =QH

whentheionizationfraction

oftheuniverseis1=2?

(i)1=5

(ii)1=50

(iii)10�

3

(iv)10�

4

(v)10�

5

Thisisnotanumberonehastocommittomemoryifonecanrememberthe

temperatureof(re)combinationineV,orifonlyinKalongwiththeconversion

factor(k�10�

4eVK�

1).Onecanthencalculatethatnearrecombination,

kT =QH

�(10�

4eVK�

1)(3000K)=(13:6eV)�1=45.

(c)(2points)WhichofthefollowingdescribestheSachs-Wolfee�ect?














Explanation:Denserregionshaveadeeper(morenegative)gravitationalpoten-

tial.Photonswhichtravelthroughaspatiallyvaryingpotentialacquirearedshift

orblueshiftdependingonwhethertheyaregoingupordownthepotential,re-

spectively.Photonsoriginatinginthedenserregionsstartatalowerpotential

andmustclimbout,sotheyendupbeingredshiftedrelativetotheiroriginal

energies.

(d)(10points)Foreachofthefollowingstatements,saywhetheritistrueorfalse:

(i)Darkmatterinteractsthroughthegravitational,weak,andelectromagnetic

forces.

T

orF?

(ii)Thevirialtheoremcanbeappliedtoaclusterofgalaxiesto�nditstotalmass,

mostofwhichisdarkmatter.

T

orF?

(iii)Neutrinosarethoughttocompriseasigni�cantfractionoftheenergydensityof

darkmatter.

T

orF?

(iv)Magneticmonopolesarethoughttocompriseasigni�cantfractionoftheenergy

densityofdarkmatter.

T

orF?

8.286QUIZ3REVIEW

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(v)LensingobservationshaveshownthatMACHOscannotaccountforthedark

matteringalactichalos,butthatasmuchas20%ofthehalomasscouldbein

theformofMACHOs.

T

orF?

PROBLEM

3:DID

YOU

DO


(a)(5points)RydensummarizestheresultsoftheCOBEsatelliteexperimentforthe

measurementsofthecosmicmicrowavebackground(CMB)intheform

ofthree

importantresults.The�rstwasthat,inanyparticulardirectionofthesky,the

spectrum

oftheCMBisveryclosetothatofanidealblackbody.TheFIRAS

instrumentontheCOBEsatellitecouldhavedetecteddeviationsfromtheblackbody

spectrumassmallas��=��10�

n,wherenisaninteger.Towithin�1,whatisn?

Answer:n=4

(b)(5points)ThesecondresultwasthemeasurementofadipoledistortionoftheCMB

spectrum;thatis,theradiationisslightlyblueshiftedtohighertemperaturesinone

direction,andslightlyredshiftedtolowertemperaturesintheoppositedirection.To

whatphysicale�ectwasthisdipoledistortionattributed?

Answer:ThelargedipoleintheCMBisattributedtothemotionofthesatellite

relativetotheframeinwhichtheCMBisverynearlyisotropic.(TheentireLocal

Groupismovingrelativetothisframeataspeedofabout0.002c.)

(c)(5points)Thethirdresultconcernedthemeasurementoftemperature uctuations

afterthedipolefeaturementionedabovewassubtractedout.De�ning

ÆTT(�;�)�T(�;�)�hTi

hTi

;

wherehTi=2:725K,theaveragevalueofT,theyfoundarootmeansquare uctu-

ation,

*�ÆTT �2 +1=2

;

equaltosomenumber.Towithinanorderofmagnitude,whatwasthatnumber?

Answer:

*�ÆTT �2 +1=2

=1:1�10�

5:

(d)(5points)WhichofthefollowingdescribestheSachs-Wolfee�ect?



8.286QUIZ3REVIEW

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p.33












(e)(5points)The atnessproblemreferstotheextreme�ne-tuningthatisneededin

atearlytimes,inorderforittobeascloseto1todayasweobserve.Startingwith

theassumptionthattodayisequalto1withinabout1%,oneconcludesthatat

onesecondafterthebigbang,j

�1jt=1sec<10�

m

;

wheremisaninteger.Towithin�3,whatism?

Answer:m=18.(SeethederivationinLectureNotes8.)

(f)(5points)Thetotalenergydensityofthepresentuniverseconsistsmainlyofbaryonic

matter,darkmatter,anddarkenergy.Givethepercentagesofeach,accordingto

thebest�tobtainedfromthePlanck2013data.Youwillgetfullcreditifthe�rst

(baryonicmatter)isaccurateto�2%,andtheothertwoareaccuratetowithin�5%.

Answer:Baryonicmatter:5%.Darkmatter:26.5%.Darkenergy:68.5%.The

Planck2013numbersweregiveninLectureNotes7.Totherequestedaccuracy,

however,numberssuchasRyden'sBenchmarkModelwouldalsobesatisfactory.

(g)(5points)Withintheconventionalhotbigbangcosmology(withoutin ation),it

isdiÆculttounderstandhowthetemperatureoftheCMBcanbecorrelatedat

angularseparationsthataresolargethatthepointsonthesurfaceoflastscattering

wasseparatedfrom

eachotherbymorethanahorizondistance.Approximately

whatangle,indegrees,correspondstoaseparationonthesurfacelastscatteringof

onehorizonlength?Youwillgetfullcreditifyouranswerisrighttowithinafactor

of2.

Answer:Rydengives1Æ

astheanglesubtendedbytheHubblelengthonthesurface

oflastscattering.Foramatter-dominateduniverse,whichwouldbeagoodmodel

forouruniverse,thehorizonlengthistwicetheHubblelength.Anynumberfrom

1Æ

to5Æ

wasconsideredacceptable.

8.286QUIZ3REVIEW

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p.34

PROBLEM

4:DID

YOU

DO


Exceptforpart(d),youshouldanswerthesequestionsbycirclingtheonestatementthat

iscorrect.

(a)(5points)IntheEpilogueofTheFirstThreeMinutes,SteveWeinbergwrote:\The

moretheuniverseseemscomprehensible,themoreitalsoseemspointless."

The

sentencewasquali�ed,however,byaclosingparagraphthatpointsoutthat

(i)thequestofthehumanracetocreateabetterlifeforallcanstillgivemeaning

toourlives.

(ii)iftheuniversecannotgivemeaningtoourlives,thenperhapsthereisanafterlife

thatwill.

(iii)thecomplexityandbeautyofthelawsofphysicsstronglysuggestthatthe

universemusthaveapurpose,evenifwearenotawareofwhatitis.

(iv)thee�orttounderstandtheuniversegiveshumanlifesomeofthegraceof

tragedy.

(b)(5points)IntheAfterwordofTheFirstThreeMinutes,Weinbergdiscussesthe

baryonnumberoftheuniverse.(Thebaryonnumberofanysystem

isthetotal

numberofprotonsandneutrons(andcertainrelatedparticlesknownashyperons)

minusthenumberoftheirantiparticles(antiprotons,antineutrons,antihyperons)

thatarecontainedinthesystem.)Weinbergconcludedthat

(i)baryonnumberisexactlyconserved,sothetotalbaryonnumberoftheuniverse

mustbezero.Whilenucleiinourpartoftheuniversearecomposedofprotons

andneutrons,theuniversemustalsocontainantimatterregionsinwhichnuclei

arecomposedofantiprotonsandantineutrons.

(ii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe

partoftheuniversewecanobserve,andhenceapositivedensityofbaryon

number.Sincebaryonnumberisconserved,thiscanonlybeexplainedby

assumingthattheexcessbaryonswereputinatthebeginning.

(iii)thereappearstobeacosmicexcessofmatteroverantimatterthroughoutthe

partoftheuniversewecanobserve,andhenceapositivedensityofbaryonnum-

ber.Thiscanbetakenasapositivehintthatbaryonnumberisnotconserved,

whichcanhappenifthereexistasyetundetectedheavy\exotic"particles.

(iv)itispossiblethatbaryonnumberisnotexactlyconserved,butevenifthatisthe

case,itisnotpossiblethattheobservedexcessofmatteroverantimattercanbe

explainedbytheveryrareprocessesthatviolatebaryonnumberconservation.

Explanation:Allstudentsweregivencreditforthispart,whethertheyansweredit

correctlyornot.IwasinSanFranciscowhenImadeupthisquiz,anddue

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.35

topoorplanningIdidnothavemycopyofTheFirstThreeMinutes.SoI

foundaversiononline,butIcouldonly�ndtheBritishversion,publishedby

Flamingo/FontanaPaperbacks,ratherthantheUSversionpublishedbyBasic

Books.Iassumedthatthe\Afterword"inthetwoversionswouldbethesame,

butIwaswrong!Sothisquestionwasbasedonadi�erent\Afterword"than

theonethatyouread.55%ofyoustillgotitright,butobviouslythequestion

wasnotfair.Apologies.

(c)(5points)IndiscussingtheCOBEmeasurementsofthecosmicmicrowaveback-

ground,Rydendescribesadipolecomponentofthetemperaturepattern,forwhich

thetemperatureoftheradiationfromonedirectionisfoundtobehotterthanthe

temperatureoftheradiationdetectedfromtheoppositedirection.

(i)Thisdiscoveryisimportant,becauseitallowsustopinpointthedirectionofthe

pointinspacewherethebigbangoccurred.

(ii)ThisisthelargestcomponentoftheCMBanisotropies,amountingtoa10%

variationinthetemperatureoftheradiation.

(iii)Inadditiontothedipolecomponent,theanisotropiesalsoincludecontributions

fromaquadrupole,octupole,etc.,allofwhicharecomparableinmagnitude.

(iv)ThispatternisinterpretedasasimpleDopplershift,causedbythenetmotion

oftheCOBEsatelliterelativetoaframeofreferenceinwhichtheCMBis

almostisotropic.

Explanation:(i)isnonsense,sincetheconventionalbigbangtheorydescibesacom-

pletelyhomogeneousuniverse,whichhasnosinglepointatwhichthebigbang

occurred.(ii)iswrong,becausethevariationsinthetemperatureoftheCMB

aremuchsmallerthan10%.Thedipoletermhasamagnitudeofabout1/1000

ofthemeantemperature.(iii)iswrongbecausethedipoleisnotcomparableto

theotherterms,becausetheyhavemagnitudesofonlyabout1/100,000ofthe

mean.

(d)(5points)(CMBbasicfacts)WhichoneofthefollowingstatementsaboutCMBis

notcorrect:





=1:1�10�

3.



8.286QUIZ3REVIEW

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p.36






Explanation:Therightvalueis*�

ÆTT �2 +1=2

=1:1�10�

5:

(e)(5points)In ationisdrivenbya�eldthatisbyde�nitioncalledthein aton�eld.

Instandardin ationarymodels,the�eldhasthefollowingproperties:

(i)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe


(ii)Thein atonisavector�eld,andduringin ationtheenergydensityofthe


(iii)Thein atonisascalar�eld,andduringin ationtheenergydensityofthe


(iv)Thein atonisavector�eld,andduringin ationtheenergydensityofthe


(v)Thein atonisatensor�eld,whichisresponsibleforonlyasmallfractionof

theenergydensityoftheuniverseduringin ation.

Explanation:ThesefactswerementionedinbothSection11.5(ThePhysicsofIn-

ation)ofRyden'sbook,andalsointhearticlethatyouwereaskedtoread

calledIn ationandtheNewEraofHigh-PrecisionCosmology,writtenbyme

forthePhysicsDepartment2002newsletter.

PROBLEM

5:NUMBER

DENSITIESIN

THECOSMIC

BACKGROUND

RADIATION

Ingeneral,thenumberdensityofaparticleintheblack-bodyradiationisgivenby

n=g�

�(3)�2 �kT�h

c �3

8.286QUIZ3REVIEW

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Forphotons,onehasg�

=2.Then

k=1:381�10�

16erg=ÆK

T=2:7ÆK

�h=1:055�10�

27erg-sec

c=2:998�1010cm/sec 9>>>>>=>>>>>;

=)

�kT�h

c �3

=1:638�103cm�

3:

Thenusing�(3)'1:202,one�ndsn

=399=cm3:

Fortheneutrinos,

g��=2�34

=32

perspecies.

Thefactorof2istoaccountfor�and��,andthefactorof3/4arisesfromthePauli

exclusionprinciple.Soforthreespeciesofneutrinosonehas

g��=92

:

Usingtheresult

T3�

=

411T3

fromProblem8ofProblemSet3(2000),one�nds

n�= �g��

g� ��T�

T �

3n

= �94 ��411 �

399cm�

3

=)

n�=326=cm3(forallthreespeciescombined).

8.286QUIZ3REVIEW

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PROBLEM

6:PROPERTIESOFBLACK-BODY

RADIATION

(a)Theaverageenergyperphotonisfoundbydividingtheenergydensitybythenumber

density.Thephotonisabosonwithtwospinstates,sog=g�

=2.Usingthe

formulasonthefrontoftheexam,

E=

g�2

30(kT)4

(�hc)3

g�

�(3)

�2

(kT)3

(�hc)3

=

�4

30�(3)kT:

Youwerenotexpectedtoevaluatethisnumerically,butitisinterestingtoknowthat

E=2:701kT:

Notethattheaverageenergyperphotonissigni�cantlymorethankT,whichisoften

usedasaroughestimate.

(b)Themethodisthesameasabove,exceptthistimeweusetheformulafortheentropy

density:

S=

g2�2

45

k4T3

(�hc)3

g�

�(3)

�2

(kT)3

(�hc)3

=

2�4

45�(3)k:

Numerically,thisgives3:602k,wherekistheBoltzmannconstant.

(c)Inthiscasewewouldhaveg=

g�

=

1.Theaverageenergyperparticleand

theaverageentropyparticledependsonlyontheratiog=g�,sotherewouldbe

nodi�erencefromtheanswersgiveninparts(a)and(b).

(d)Forafermion,gis7/8timesthenumberofspinstates,andg�

is3/4timesthe

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.39

numberofspinstates.Sotheaverageenergyperparticleis

E=

g�2

30

(kT)4

(�hc)3

g�

�(3)

�2

(kT)3

(�hc)3

=

78�2

30

(kT)4

(�hc)3

34�(3)

�2

(kT)3

(�hc)3

=

7�4

180�(3)kT:

Numerically,E=3:1514kT.

Warning:theMathematicianGeneralhasdetermined

thatthememorizationofthisnumbermayadversely

a�ectyourabilitytorememberthevalueof�.

Ifonetakesintoaccountbothneutrinosandantineutrinos,theaverageenergyper

particleisuna�ected|

theenergydensityandthetotalnumberdensityareboth

doubled,buttheirratioisunchanged.

Notethattheenergyperparticleishigherforfermionsthanitisforbosons.This

resultcanbeunderstoodasanaturalconsequenceofthefactthatfermionsmust

obeytheexclusionprinciple,whilebosonsdonot.Largenumbersofbosonscan

thereforecollectinthelowestenergylevels.Infermionsystems,ontheotherhand,

thelow-lyinglevelscanaccommodateatmostoneparticle,andthenadditional

particlesareforcedtohigherenergylevels.

8.286QUIZ3REVIEW

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(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

7:A

NEW

SPECIESOFLEPTON

a)Thenumberdensityisgivenbytheformulaatthestartoftheexam,

n=g�

�(3)

�2

(kT)3

(�hc)3

:

Sincethe8.286ionisliketheelectron,ithasg�

=3;thereare2spinstatesforthe

particlesand2fortheantiparticles,giving4,andthenafactorof3/4becausethe

particlesarefermions.So

Then

Answer=3�(3)

�2

� �

3�106�102

6:582�10�

16�2:998�1010 �

3

:

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.41

Youwerenotaskedtoevaluatethisexpression,buttheansweris1:29�1039.

b)Fora atcosmology�=0andoneoftheEinsteinequationsbecomes

�_aa �2

=8�3

G�:

Duringtheradiation-dominatederaa(t)/t1=2,asclaimedonthefrontcoverofthe

exam.So,

_aa=

12t:

Usingthisintheaboveequationgives

14t2

=8�3

G�:

Solvethisfor�,

�=

3

32�Gt2

:

Thequestionasksthevalueof�att=

0:01sec.

WithG

=

6:6732�

10�

8cm3sec�

2g�

1,then�

=

3

32��6:6732�10�

8�(0:01)2

inunitsofg=cm3.Youweren'taskedtoputthenumbersin,but,forreference,doing

sogives�=4:47�109g=cm3.

c)Themassdensity�=u=c2,whereuistheenergydensity.Theenergydensityfor

black-bodyradiationisgivenintheexam,

u=�c2=g�2

30

(kT)4

(�hc)3

:

WecanusethisinformationtosolveforkTintermsof�(t)whichwefoundabove

inpart(b).Atatimeof0.01sec,ghasthefollowingcontributions:

Photons:

g=2

e+e�

:

g=4�78=312

�e ;��;��:

g=6�78=514

8:286ion�anti8:286ion

g=4�78=312

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.42

gtot=1414

:

SolvingforkTintermsof�gives

kT= �30

�2

1gtot �h

3c5� �1=4

:

Usingtheresultfor�frompart(b)aswellasthelistoffundamentalconstantsfrom

thecoversheetoftheexamgives

kT= �90�(1:055�10�

27)3�(2:998�1010)5

14:24�32�3�6:6732�10�

8�(0:01)2 �

1=4

�

1

1:602�10�

6

wheretheanswerisgiveninunitsofMeV.PuttinginthenumbersyieldskT=8:02

MeV.

d)Theproductionofhelium

isincreased.Atanygiventemperature,theadditional

particleincreasestheenergydensity.SinceH/�1=2,theincreasedenergydensity

speedstheexpansionoftheuniverse|theHubbleconstantatanygiventemperature

ishigheriftheadditionalparticleexists,andthetemperaturefallsfaster.The

weakinteractionsthatinterconvertprotonsandneutrons\freezeout"whenthey

cannolongerkeepupwiththerateofevolutionoftheuniverse.Thereaction

ratesatagiventemperaturewillbeuna�ectedbytheadditionalparticle,butthe

highervalueofHwillmeanthatthetemperatureatwhichtheseratescannolonger

keeppacewiththeuniversewilloccursooner.Thefreeze-outwillthereforeoccur

atahighertemperature.Theequilibrium

valueoftheratioofneutrontoproton

densitiesislargerathighertemperatures:nn=np /exp(��mc2=kT),wherennand

nparethenumberdensitiesofneutronsandprotons,and�mistheneutron-proton

massdi�erence.Consequently,therearemoreneutronspresenttocombinewith

protonstobuildhelium

nuclei.Inaddition,thefasterevolutionrateimpliesthat

thetemperatureatwhichthedeuteriumbottleneckbreaksisreachedsooner.This

impliesthatfewerneutronswillhaveachancetodecay,furtherincreasingthehelium

production.

e)Aftertheneutrinosdecouple,theentropyintheneutrinobathisconservedseparately

fromtheentropyintherestoftheradiationbath.Justafterneutrinodecoupling,

alloftheparticlesinequilibriumaredescribedbythesametemperaturewhichcools

asT/1=a.Theentropyinthebathofparticlesstillinequilibriumjustafterthe

neutrinosdecoupleis

S/grest T3(t)a3(t)

wheregrest=gtot �g�=9.Bytoday,thee+�e�

pairsandthe8.286ion-anti8.286ion

pairshaveannihilated,thustransferringtheirentropytothephotonbath.Asaresult

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.43

thetemperatureofthephotonbathisincreasedrelativetothatoftheneutrinobath.

Fromconservationofentropywehavethattheentropyafterannihilationsisequal

totheentropybeforeannihilations

g T3 a3(t)=grest T3(t)a3(t):

So,

T

T(t)= �grest

g �

1=3

:

Sincetheneutrinotemperaturewasequaltothetemperaturebeforeannihilations,

wehavethat

T�

T

= �29 �1=3

:

PROBLEM

8:A

NEW

THEORY

OF

THE

WEAK

INTERACTIONS(40

points)

(a)Inthestandardmodel,theblack-bodyradiationatkT�200MeVcontainsthe

followingcontributions:

Photons:

g=2

e+e�

:

g=4�78=312

�e ;��;�� :

g=6�78=514

�+��

:

g=4�78=312

�+��

�0

g=3

9>>>>>>>=>>>>>>>;gTOT

=1714

Themassdensityisthengivenby

�=

uc2=gTOT�2

30(kT)4

�h3c5

:

Inkg/m3,onecanevaluatethisexpressionby

�= �1714 ��2

30 �200�106eV�1:602�10�

19J

eV

�4

(1:055�10�

34J-s)3(2:998�108m/s)5

:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.44

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,FALL2018

p.45

whichagreeswiththeanswerabove.

Note:Acommonmistakewastoleaveouttheconversionfactor1:602�10�

19J/eV

(or1:602�10�

12erg/eV),andinsteadtouse�h=6:582�10�

16eV-s.Butifone

worksouttheunitsofthisanswer,theyturnouttobeeV-sec2/m5(oreV-sec2/cm5),

whichisamostpeculiarsetofunitstomeasureamassdensity.

IntheNTWI,wehaveinadditionthecontributiontothemassdensityfromR+-R�

pairs,whichwouldactjustlikee+-e�

pairsor�+-��

pairs,withg=312.Thus

gTOT

=2034 ,so

�= �2034 ��2

30 �200�106�1:602�10�

19 �4

(1:055�10�

34)3(2:998�108)5kg

m3

or

�= �2034 ��2

30 �200�106�1:602�10�

12 �4

(1:055�10�

27)3(2:998�1010)5

gcm3

:

Numerically,theanswerinthiscasewouldbe

�NTWI=2:53�1018kg/m3=2:53�1015g/cm3:

(b)Aslongastheuniverseisinthermalequilibrium,entropyisconserved.Theentropy

inagivenvolumeofthecomovingcoordinatesystemis

a3(t)sVcoord

;

wheresistheentropydensityanda3V

coordisthephysicalvolume.So

a3(t)s

isconserved.Aftertheneutrinosdecouple,

a3s

�

and

a3s

other

areseparatelyconserved,wheresotheristheentropyofeverythingexceptneutrinos.

Notethatscanbewrittenas

s=gAT3

;

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.46

whereAisaconstant.Beforethedisappearanceofthee;�,R,and�particlesfrom

thethermalequilibriumradiation,s

�= �514 �

AT3

sother= �1512 �

AT3

:

So

s�

sother=

514

1512

:

Ifa3s

�anda3s

otherareconserved,thensoiss�=sother .Bytoday,theentropyprevi-

ouslysharedamongthevariousparticlesstillinequilibriumafterneutrinodecoupling

hasbeentransferedtothephotonssothat

sother=sphotons=2AT3

:

Theentropyinneutrinosisstill

s�= �514 �

AT3�

:

Sinces�=sotherisconstantweknowthat

�514 �T3�

2T3

=

s�

sother=

514

1512

=)

T�= �43

1 �1=3

T

:

(c)Onecanwrite

n=g�BT3

;

whereBisaconstant.Hereg� =2,andg��=6�34=412 .Inthestandardmodel,

onehastoday

n�

n

=g�� T3�

g� T3

= �412 �2

411=

911:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.47

IntheNTWI,

n�

n

= �412 �2

431=

931:

(d)AtkT=200MeV,thethermalequilibriumratioofneutronstoprotonsisgivenby

nn

np

=e�

1:29MeV=200MeV

�1:

Inthestandardtheorythisratiowoulddecreaserapidlyastheuniversecooledand

kTfellbelowthep-nmassdi�erenceof1.29MeV,butintheNTWItheratiofreezes

outatthehightemperaturecorrespondingtokT=200MeV,whentheratiois

about1.WhenkTfallsbelow200MeVintheNTWI,theneutrinointeractions

n+�e $p+e�

and

n+e+

$p+��e

thatmaintainthethermalequilibrium

balancebetweenprotonsandneutronsno

longeroccuratasigni�cantrate,sotheration= npisnolongercontrolledbyther-

malequilibrium.AfterkTfallsbelow200MeV,theonlyprocessthatcanconvert

neutronstoprotonsistheratherslowprocessoffreeneutrondecay,withadecay

time�dofabout890s.Thus,whenthedeuteriumbottleneckbreaksatabout200

s,thenumberdensityofneutronswillbeconsiderablyhigherthaninthestandard

model.SinceessentiallyalloftheseneutronswillbecomeboundintoHenuclei,the

higherneutronabundanceoftheNTWIimpliesa

higherpredictedHeabundance:

ToestimatetheHeabundance,notethatifwetemporarilyignorefreeneutrondecay,

thentheneutron-protonratiowouldbefrozenatabout1andwouldremain1until

thetimeofnucleosynthesis.Atthetimeofnucleosynthesisessentiallyallofthese

neutronswouldbeboundintoHenuclei(eachwith2protonsand2neutrons).For

aninitial1:1ratioofneutronstoprotons,alltheneutronsandprotonscanbebound

intoHenuclei,withnoprotonsleftoverintheformofhydrogen,soYwouldequal

1.However,thefreeneutrondecayprocesswillcausetherationn=nptofallbelow

1beforethestartofnucleosynthesis,sothepredictedvalueofYwouldbelessthan

1.

Tocalculatehowmuchless,notethatRydenestimatesthestartofnucleosynthe-

sisatthetimewhenthetemperaturereachesTnuc ,whichisthetemperaturefor

whichathermalequilibriumcalculationgivesnD=nn=1.Thiscorrespondstowhat

Weinbergreferstoasthebreakingofthedeuteriumbottleneck.Thetemperature

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.48

Tnuciscalculatedintermsof�=nB=n andphysicalconstants,soitwouldnotbe

changedbytheNTWI.Thetimewhenthistemperatureisreached,however,would

bechangedslightlybythechangeintheratioT�=T .Sincethise�ectisrather

subtle,nopointswillbetakeno�ifyouomittedit.However,tobeasaccurate

aspossible,oneshouldrecognizethatnucleosynthesisoccursduringtheradiation-

dominatedera,butlongafterthee+-e�

pairshavedisappeared,sotheblack-body

radiationconsistsofphotonsattemperatureT andneutrinosatalowertemperature

T� .Theenergydensityisgivenby

u=�2

30(kT )4

(�hc)3 "2+ �214 ��T�

T �

4 #�ge��2

30

(kT )4

(�hc)3

;

where

ge�

=2+ �214 ��T�

T �

4

:

Forthestandardmodel

gsm

e�

=2+ �214 ��41

1 �4=3

;

andfortheNTWI

gNTWI

e�

=2+ �214 ��43

1 �4=3

:

Therelationbetweentimeandtemperatureina atradiation-dominateduniverse

isgivenintheformulasheetsas

kT= �45�h3c5

16�3gG �

1=4

1pt:

Thus,

t/

1g1=2

e�

T2

:

InthestandardmodelRydenestimatesthetimeofnucleosynthesisastsmnu

c �200s,

sointheNTWIitwouldbelongerbythefactor

tNTWI

nuc

= sgsm

e�

gNTWI

e�

tsmnu

c:

Whileofcoureyouwerenotexpectedtoworkoutthenumerics,thisgives

tNTWI

nuc

=1:20tsmnu

c:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.49

NotethatRydengivestnuc �200s,whileWeinbergplacesitat334minutes�225s,

whichiscloseenough.

Tofollowthee�ectofthisfreedecay,itiseasiesttodoitbyconsideringtheratio

neutronstobaryonnumber,nn=nB,sincenB

doesnotchangeduringthisperiod.

Atfreeze-out,whenkT�200MeV,

nn

nB

�12

:

Justbeforenucleosynthesis,attimetnuc ,theratiowillbe

nn

nB

�12

e�

tnuc=�d

:

Iffreedecayisignored,wefoundY=1.Sinceallthesurvivingneutronsarebound

intoHe,thecorrectedvalueofYissimplydeceasedbymultiplyingbythefraction

ofneutronsthatdonotundergodecay.Thus,thepredictionofNTWIis

Y=e�

tnuc =�d

=exp 8<:� q

gsm

eff

gNTW

I

eff

200

890

9=;;

wheregsm

e�

andgNTWI

e�

aregivenabove.Whenevaluatednumerically,thiswouldgive

Y=PredictedHeabundancebyweight�0:76:

PROBLEM

9:DOUBLING

OFELECTRONS(10points)

Theentropydensityofblack-bodyradiationisgivenby

s=g �2�2

45

k4

(�hc)3 �T3

=gCT3;

whereC

isaconstant.Atthetimewhentheelectron-positronpairsdisappear,

theneutrinosaredecoupled,sotheirentropyisconserved.Alloftheentropyfrom

electron-positronpairsisgiventothephotons,andnonetotheneutrinos.Thesame

willbetruehere,forbothspeciesofelectron-positronpairs.

TheconservedneutrinoentropycanbedescribedbyS� �a3s

� ,whichindicatesthe

entropypercubicnotch,i.e.,entropyperunitcomovingvolume.Weintroducethe

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.50

notationn�

andn+

forthenewelectron-likeandpositron-likeparticles,andalso

theconventionthat

Primedquantities:

valuesaftere+e�

n+n�

annihilation

Unprimedquantities:

valuesbeforee+e�

n+n�

annihilation.

Fortheneutrinos,

S0�=S�

=)

g� C(a0T0� )3=g�C(aT�)3

=)

a0T0�=aT�:

Forthephotons,beforee+e�

n+n�

annihilationwehave

T =Te+

e�

n+n�

=T�;

g =2;ge+

e�

=gn+n�

=7=2:

Whenthee+e�

andn+n�

pairsannihilate,theirentropyisaddedtothephotons:

S0 =Se+

e�

+Sn+

n�

+S

=)

2C �a0T0 �3

= �2+2�72 �

C(aT )3

=)

a0T0 = �92 �1=3

aT ;

soaT increasesbyafactorof(9=2)1=3.

Beforee+e�

annihilationtheneutrinoswereinthermalequilibriumwiththephotons,

soT =T� .Byconsideringthetwoboxedequationsabove,onehas

T0�= �29 �1=3

T0 :

Thisratiowouldremainunchangeduntilthepresentday.

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.51

PROBLEM

10:TIMESCALESIN

COSMOLOGY

(a)1sec.[Thisisthetimeatwhichtheweakinteractionsbeginto\freezeout",sothat

freeneutrondecaybecomestheonlymechanismthatcaninterchangeprotonsand

neutrons.Fromthistimeonward,therelativenumberofprotonsandneutronsisno

longercontrolledbythermalequilibriumconsiderations.]

(b)4mins.[Bythistimetheuniversehasbecomesocoolthatnuclearreactionsareno

longerinitiated.]

(c)10�

37sec.[WelearnedinLectureNotes7thatkTwasabout1MeVatt=1sec.

Since1GeV=1000MeV,thevalueofkTthatwewantis1019timeshigher.Inthe

radiation-dominatederaT/a�

1/t�

1=2,soweget10�

38sec.]

(d)10,000{1,000,000years.[ThisnumberwasestimatedinLectureNotes7as200,000

years.]

(e)10�

5sec.[Asin(c),wecanuset/T�

2,withkT�1MeVatt=1sec.]

PROBLEM

11:EVOLUTION

OFFLATNESS(15points)

(a)WestartwiththeFriedmannequationfromtheformulasheetonthequiz:

H2= �_aa �2

=8�3

G��kc2

a2

:

Thecriticaldensityisthevalueof�correspondingtok=0,so

H2=8�3

G�c:

UsingthisexpressiontoreplaceH2ontheleft-handsideoftheFriedmannequation,

andthendividingby8�G=3,one�nds

�c=��3kc2

8�Ga2

:

Rearranging,

��c

�

=

3kc2

8�Ga2�:

Ontheleft-handsidewecandividethenumeratoranddenominatorby�c ,andthen

usethede�nition��=�ctoobtain

�1

=

3kc2

8�Ga2�:

(1)

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.52

Foramatter-dominateduniverseweknowthat�/1=a3(t),andso

�1

/a(t):

Iftheuniverseisnearly atweknowthata(t)/t2=3,so

�1

/t2=3:

(b)Eq.(1)aboveisstilltrue,soouronlytaskistore-evaluatetheright-handside.For

aradiation-dominateduniverseweknowthat�/1=a4(t),so

�1

/a2(t):

Iftheuniverseisnearly atthena(t)/t1=2,so

�1

/t:

PROBLEM

12:THESLOAN

DIGITALSKY

SURVEY

z

=

5:82QUASAR

(40points)

(a)Sincem

+�

=0:35+0:65=1,theuniverseis at.Itthereforeobeysasimple

formoftheFriedmannequation,

H2= �_aa �2

=8�3

G(�m

+��);

wheretheoverdotindicatesaderivativewithrespecttot,andthetermproportional

tokhasbeendropped.Usingthefactthat�m

/1=a3(t)and��

=const,theenergy

densitiesontheright-handsidecanbeexpressedintermsoftheirpresentvalues

�m;0and��

��;0 .De�ning

x(t)�a(t)

a(t0 );

onehas

�_xx �2

=8�3

G ��m;0

x3

+��

=8�3

G�c;0 �m;0

x3

+�;0 �

=H20 �m;0

x3

+�;0 �:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.53

Hereweusedthefactsthat

m;0 ��m;0

�c;0;

�;0 ��

�c;0

;

and

H20=8�3

G�c;0:

Theequationabovefor(_x=x)2impliesthat

_x=H0x rm;0

x3

+�;0;

whichinturnimpliesthat

dt=

1H0

dx

x qm

;0

x3

+�;0

:

Usingthefactthatxchangesfrom0to1overthelifeoftheuniverse,thisrelation

canbeintegratedtogive

t0= Z

t0

0

dt=

1H0 Z

10

dx

x qm

;0

x3

+�;0

:

Theanswercanalsobewrittenas

t0=

1H0 Z

10

xdx

pm;0 x+�;0 x4

or

t0=

1H0 Z

10

dz

(1+z) pm;0 (1+z)3+�;0

;

whereinthelastanswerIchangedthevariableofintegrationusing

x=

11+z;

dx=�dz

(1+z)2

:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.54

Notethattheminussignintheexpressionfordxiscanceledbytheinterchangeof

thelimitsofintegration:x=0correspondstoz=1,andx=1correspondsto

z=0.

Youranswershouldlooklikeoneoftheaboveboxedanswers.Youwerenotexpected

tocompletethenumericalcalculation,butforpedagogicalpurposesIwillcontinue.

Theintegralcanactuallybecarriedoutanalytically,giving

Z1

0

xdx

pm;0 x+�;0 x4=

2

3 p�;0ln pm

+�;0+ p�;0

pm

!:

Using

1H0

=9:778�109

h0

yr;

whereH0=100h0km-sec�

1-Mpc�

1,one�ndsforh0=0:65that

1H0=15:043�109yr:

Thenusingm

=0:35and�;0=0:65,one�nds

t0=13:88�109yr:

SotheSDSSpeoplewererightontarget.

(b)Havingdonepart(a),thispartisveryeasy.Thedynamicsoftheuniverseisof

coursethesame,andthequestionisonlyslightlydi�erent.Inpart(a)wefoundthe

amountoftimethatittookforxtochangefrom0to1.Thelightfromthequasar

thatwenowreceivewasemittedwhen

x=

11+z;

sincethecosmologicalredshiftisgivenby

1+z=a(tobserved )

a(temitted ):

Usingtheexpressionfordtfrompart(a),theamountoftimethatittooktheuniverse

toexpandfromx=0tox=1=(1+z)isgivenby

te= Z

te

0

dt=

1H0 Z

1=(1+z)

0

dx

x qm

;0

x3

+�;0

:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.55

Againonecouldwritetheanswerotherways,including

t0=

1H0 Z

1z

dz0

(1+z0) pm;0 (1+z0)3+�;0

:

Againyouwereexpectedtostopwithanexpressionliketheoneabove.Continuing,

however,theintegralcanagainbedoneanalytically:

Zxmax

0

dx

x qm

;0

x3

+�;0

=

2

3 p�;0ln pm

+�;0 x3m

ax+ p�;0x3=2

max

pm

!:

Usingxmax=1=(1+5:82)=:1466andtheothervaluesasbefore,one�nds

te=0:06321

H0

=0:9509�109yr:

SoagaintheSDSSpeoplewereright.

(c)To�ndthephysicaldistancetothequasar,weneedto�gureouthowfarlightcan

travelfromz=5:82tothepresent.Sincewewantthepresentdistance,wemultiply

thecoordinatedistancebya(t0 ).Forthe atmetric

ds2=�c2d�2=�c2dt2+a2(t) �dr2+r2(d�2+sin2�d�2) ;

thecoordinatevelocityoflight(intheradialdirection)isfoundbysettingds2=0,

giving

dr

dt=

ca(t):

Sothetotalcoordinatedistancethatlightcantravelfromtetot0is

`c= Z

t0

te

ca(t)dt:

Thisisnotthe�nalanswer,however,becausewedon'texplicitlyknowa(t).We

can,however,changevariablesofintegrationfromttox,using

dt=

dt

dxdx=dx_x

:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.56

So

`c=

ca(t0 ) Z

1xe

dx

x_x;

wherexeisthevalueofxatthetimeofemission,soxe=1=(1+z).Usingthe

equationfor_xfrompart(a),thisintegralcanberewrittenas

`c=

c

H0 a(t0 ) Z

11=(1+z)

dx

x2 qm

;0

x3

+�;0

:

Finally,then

`phys;0=a(t0 )`c=

cH0 Z

11=(1+z)

dx

x2 qm

;0

x3

+�;0

:

Alternatively,thisresultcanbewrittenas

`phys;0=

cH0 Z

11=(1+z)

dx

pm;0x+�;0x4

;

orbychangingvariablesofintegrationtoobtain

`phys;0=

cH0 Z

z0

dz0

pm;0(1+z0)3+�;0

:

Continuingforpedagogicalpurposes,thistimetheintegralhasnoanalyticform,so

farasIknow.Integratingnumerically,

Z5:82

0

dz0

p0:35(1+z0)3+0:65=1:8099;

andthenusingthevalueof1=H0frompart(a),

`phys;0=27:23light-yr:

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.57

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.

PROBLEM

13:SECOND

HUBBLECROSSING

(40points)

(a)Fromtheformulasheets,weknowthatfora atradiation-dominateduniverse,

a(t)/t1=2:

Since

H=

_aa;

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.58

(whichisalsoontheformulasheets),H

=

12t:

Then

`H(t)�cH�

1(t)=

2ct:

(b)Wearetoldthattheenergydensityisdominatedbyphotonsandneutrinos,sowe

needtoaddtogetherthesetwocontributionstotheenergydensity.Forphotons,the

formulasheetremindsusthatg =2,so

u =2�2

30(kT )4

(�hc)3

:

Forneutrinostheformulasheetremindsusthat

g�=

78|{z}

Fermion

factor

�

3|{z}

3species

�e;��;��

2|{z}

Particle=

antiparticle �

1|{z}

Spinstates

=

214;

so

u�=214�2

30

(kT� )4

(�hc)3

:

CombiningthesetwoexpressionsandusingT�=(4=11)1=3T ,onehas

u=u +u�= "2+214 �411 �4=3 #�2

30(kT )4

(�hc)3

;

so�nally

g1=2+214 �41

1 �4=3

:

(c)TheFriedmannequationtellsusthat,fora atuniverse,

H2=8�3

G�;

whereinthiscaseH=1=(2t)and

�=

uc2=g1�2

30

(kT )4

�h3c5

:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.59

Thus

�12

t �2

=8�G3

g1�2

30(kT )4

�h3c5

:

SolvingforT ,

T =1k �45�h3c5

16�3g1 G �

1=4

1pt:

(d)TheconditionforHubblecrossingis

�(t)=cH�

1(t);

andthe�rstHubblecrossingalwaysoccursduringthein ationaryera.Thusany

Hubblecrossingduringtheradiation-dominatederamustbethesecondHubble

crossing.

If�isthepresentphysicalwavelengthofthedensityperturbationsunderdiscussion,

thewavelengthattimetisscaledbythescalefactora(t):

�(t)=

a(t)

a(t0 )�:

BetweenthesecondHubblecrossingandnow,therehavebeennofreeze-outsof

particlespecies.Todaytheentropyoftheuniverseisstilldominatedbyphotonsand

neutrinos,sotheconservationofentropyimpliesthataT

hasremainedessentially

constantbetweenthenandnow.Thus,

�(t)=

T ;0

T (t)�:

UsingthepreviousresultsforcH�

1(t)andforT (t),thecondition�(t)=cH�

1(t)

canberewrittenas

kT ;0 �16�3g

1 G

45�h3c5 �

1=4p

t�=2ct:

Solvingfort,thetimeofsecondHubblecrossingisfoundtobe

tH2 (�)=(kT ;0 �)2 ��3g

1 G

45�h3c9 �

1=2

:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.60

Extension:Youwerenotaskedtoinsertnumbers,butitisofcourseinterestingto

knowwheretheaboveformulaleads.Ifwetake�=106lt-yr,itgives

tH2 (106lt-yr)=1:04�107s=0:330year:

For�=1Mpc,

tH2 (1Mpc)=1:11�108s=3:51year:

Taking�=2:5�106lt-yr,thedistancetoAndromeda,thenearestspiralgalaxy,

tH2 (2:5�106lt-yr)=6:50�107sec=2:06year:

PROBLEM

14:NEUTRINO

NUMBER

AND

THENEUTRON/PROTON

EQUILIBRIUM

(a)Fromthechemicalequilibriumequationonthefrontoftheexam,thenumberden-

sitiesofneutronsandprotonscanbewrittenas

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.Approxi-

matingmn=mp �1,onehas

nn

np

=e�

(�E+�p�

�n)=kT

:

Theapproximationmn=mp �1isveryaccurate(0.14%),butisclearlynotnecessary.

Fullcreditwasgivenwhetherornotthisapproximationwasused.

(b)Foranyallowedchemicalreaction,thesumofthechemicalpotentialsonthetwo

sidesmustbeequal.So,from

e+

+n !p+��e;

8.286QUIZ3REVIEW

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p.61

wecaninferthat

��e+�n=�p ��;

whichimpliesthat

�n ��p=�e ��:

(c)Applyingtheformulagivenintheproblemtothenumberdensitiesofelectronneu-

trinosandthecorrespondingantineutrinos,

n�=g��(3)

�2

(kT)3

(�hc)3e��=kT

�n�=g��(3)

�2

(kT)3

(�hc)3e�

��=kT

;

sincethechemicalpotentialfortheantineutrinos(��)isthenegativeofthechemical

potentialforneutrinos.Aneutrinohasonlyonespinstate,sog�=3=4,wherethe

factorof3/4arisesbecauseneutrinosarefermions.Setting

x�e�

��=kT

and

A�34�(3)

�2

(kT)3

(�hc)3

;

thenumberdensityequationscanbewrittencompactlyas

n�=Ax

;

�n�=xA:

Toexpressxintermsoftheratio�n� =n�,dividethesecondequationbythe�rstto

obtain

�n�

n�

=x2

=)

x= r�n�

n�

:

Alternatively,xcanbeexpressedintermsofthedi�erenceinnumberdensities

�n� �n�bystartingwith

�n=�n� �n�=xA�Ax

:

Rewritingtheaboveformulaasanexplicitquadratic,

Ax2��nx�A=0;

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.62

one�nds

x=�n�p

�n2+4A2

2A

:

Sincethede�nitionofximpliesx>0,onlythepositiverootisrelevant.Sincethe

numberofelectronsisstillassumedtobeequaltothenumberofpositrons,�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=npwouldbelargerthanin

theusualcase(�n=0).Thisisconsistentwiththeexpectationthatanexcessof

antineutrinoswilltendtocausep'stoturninton'saccordingtothereaction

p+��e �!e+

+n:

Sincetheamountofhelium

producedisproportionaltothenumberofneutrons

thatsurviveuntilthebreakingofthedeuteriumbottleneck,startingwithahigher

equilibriumabundanceofneutronswillincreasetheproductionofhelium.

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.63

PROBLEM

15:THEEVENTHORIZON

FOROURUNIVERSE(25points)

(a)Inasphericalpulseeachlightrayismovingradiallyoutward,sod�=d�=0.A

lightraytravelsalonganulltrajectory,meaningthatds2=0,sowehave

ds2=�c2dt2+a2(t)dr2=0:

(3.1)

fromwhichitfollowsthat

dr

dt=�c

a(t):

(3.2)

Weareinterestedinaradialpulsethatstartsatr=0attimet=t0 ,sothelimiting

valueofrisgivenby

rmax= Z

1t0

ca(t)dt:

(3.3)

(b)Changingvariablesofintegrationto

x=

a(t)

a(t0 );

(3.4)

theintegralbecomes

rmax= Z

11

ca(t)

dt

dxdx=

ca(t0 ) Z

11

1xdt

dxdx;

(3.5)

whereweusedthefactthatt=t0correspondstox=a(t0 )=a(t0 )=1.Asgivento

usontheformulasheet,the�rst-orderFriedmannequationcanbewrittenas

xdxd

t=H0 qm;0 x+rad;0+vac;0 x4+k;0 x2:

(3.6)

Usingthissubstitution,

rmax=

c

a(t0 )H0 Z

11

dx

pm;0 x+rad;0+vac;0 x4

;

(3.7)

wherewehaveusedk;0=0,sincetheuniverseistakentobe at.

(c)To�ndthevalueoftheredshiftforthelightthatwearepresentlyreceivingfrom

coordinatedistancermax ,wecanbeginbynoticingthatthetimeofemissiontecan

bedeterminedbytheequationwhichimpliesthatthecoordinatedistancetraveled

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.64

byalightpulsebetweentimesteandt0mustequalrmax .UsingEq.(3.2)forthe

coordinatevelocityoflight,thisequationreads

Zt0

te

ca(t)dt=rmax:

(3.8)

The\half-credit"answertothequizproblem

wouldincludetheaboveequation,

followedbythestatementthattheredshiftzehcanbedeterminedfrom

z=a(t0 )

a(te ) �1:

(3.9)

The\full-credit"answerisobtainedbychangingthevariableofintegrationasin

part(b),soEq.(3.8)becomesr

max= Z

1xe

ca(t)

dt

dxdx

=

ca(t0 ) Z

1xe

1xdt

dxdx;

(3.10)

wherexe

isthevalueofxcorrespondingtot=

te .ThenusingEq.(3.6)with

k;0=0,we�ndr

max=

c

a(t0 )H0 Z

1xe

dx


:

(3.11)

Tocompletetheanswerinthislanguage,weuse

z=

1xe �1:

(3.12)

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

furtherandreplacermaxusingEq.(3.7),�nding

Z1

1

dx


= Z1

xe

dx


:

(3.13)

InthisformtheanswerdependsonlyonthevaluesofX;0 .

Youwereofcoursenotaskedtoevaluatethisformulanumerically,butyoumight

beinterestedinknowingthatthePlanck2013valuesm;0=0:315,vac;0=0:685,

andrad;0=9:2�10�

5leadtozeh=1:87.Thus,noeventthatishappeningnow

(i.e.,atthesamevalueofthecosmictime)inagalaxyatredshiftlargerthan1.87

willeverbevisibletousorourdescendants,eveninprinciple.

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.65

PROBLEM

16:THEEFFECTOFPRESSUREONCOSMOLOGICALEVO-

LUTION

(25points)

(a)(8points)Thisproblemisansweredmosteasilybystartingfromthecosmological

formulaforenergyconservation,whichIremembermosteasilyintheformmotivated

bydU=�pdV.Usingthefactthattheenergydensityuisequalto�c2,theenergy

conservationrelationcanbewritten

dUd

t=�pdVd

t

=)

ddt ��c2a3 �=�pdd

t �a3 �:

Setting

�=

�a8

forsomeconstant�,theconservationofenergyformulabecomes

ddt �

�c2

a5 �=�pdd

t �a3 �;

whichimplies

�5�c2

a6

dad

t=�3pa2dad

t:

Thus

p=53�c2

a8

=

53�c2:

Alternatively,onemaystartfromtheequationforthetimederivativeof�,

_�=�3_aa �

�+

pc2 �:

Since�=

�a8 ,wetakethetimederivativeto�nd_�=�8(_a=a)�,andtherefore

�8_aa

�=�3_aa �

�+

pc2 �;

andtherefore

p=53

�c2:

(b)(9points)Fora atuniverse,theFriedmannequationreducesto

�_aa �2

=8�3

G�:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.66

Using�/1=a8,thisimpliesthat

_a=

�a3

;

forsomeconstant�.Rewritingthisas

a3da=�dt;

wecanintegratetheequationtogive

14a4=�t+const;

wheretheconstantofintegrationhasnoe�ectotherthantoshifttheoriginofthe

timevariablet.Usingthestandardbigbangconventionthata=0whent=0,the

constantofintegrationvanishes.Thus,

a/t1=4:

Thearbitraryconstantofproportionalityinthisanswerisconsistentwiththe

wordingoftheproblem,whichstatesthat\Youshouldbeabletodeterminethe

functiona(t)uptoaconstantfactor."

Notethatwecouldhaveexpressedthe

constantofproportionalityintermsoftheconstant�thatweusedinpart(a),

buttherewouldnotreallybeanypointindoingthat.Theconstant�wasnota

givenvariable.Ifthecomovingcoordinatesaremeasuredin\notches,"thenais

measuredinmeterspernotch,andtheconstantofproportionalityinouranswercan

bechangedbychangingthearbitraryde�nitionofthenotch.

(c)(8points)Westartfromtheconservationofenergyequationintheform

_�=�3_aa �

�+

pc2 �:

Substituting_�=�n(_a=a)�andp=(2=3)�c2,wehave

�nH�=�3H �53

� �

andtherefore

n=5:

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.67

PROBLEM

17:THEFREEZE-0UTOFA

FICTITIOUSPARTICLEX

(25

points)

(a)(5points)Theformulasheettellsusthattheenergydensityofblack-bodyradiation

is

u=g�2

30

(kT)4

(�hc)3

;

where

g� �1perspinstateforbosons(integerspin)

7/8perspinstateforfermions(half-integerspin).

SincetheX

isspin-1,and1isaninteger,theX

particlesarebosonsandg=1per

spinstate.Thereare3species,X+,X�

,andX0,andeachspecieswearetoldhas

threespinstates,sothereareatotalof9spinstates,sog=9.Thus,

u=9�2

30

(kT)4

(�hc)3

:

Alternatively,onecouldcounttheX+

andX�

asonespecieswithadistinct

particleandantiparticle,sogX+X�

isgivenby

gX+X�

=

1|{z}

Fermion

factor

�1

|{z}Species �

2|{z}

Particle=

antiparticle �

3|{z}

Spinstates

=

6:

TheX0isitsownantiparticle,whichmeansthattheparticle/antiparticlefactoris

one,so

gX0

=

1|{z}

Fermion

factor

�1

|{z}Species �

1|{z}

Particle=

antiparticle �

3|{z}

Spinstates

=

3;

sothetotalgforX+,X�

,andX0isagainequalto9.

(b)(5points)Theformulasheettellsusthatthenumberdensityofparticlesinblack-

bodyradiationis

n=g�

�(3)

�2

(kT)3

(�hc)3

;

where

g�� 1perspinstateforbosons

3/4perspinstateforfermions.

8.286QUIZ3REVIEW

PROBLEM

SOLUTIONS,FALL2018

p.68

Forbosonsg�

=g,sog�

fortheXparticlesis9.Then

nX

=9�(3)

�2

(kT)3

(�hc)3

:

(c)(10points)Wearetoldthat,whentheXparticlesfreezeout,alloftheirenergyand

entropyisgiventothephotons.Weuseentropyratherthanenergytodetermine

the�naltemperatureofthephotons,becausetheentropyinacomovingvolumeis

simplyconserved,whiletheenergydensityvariesas

_�=�3_aa �

�+

pc2 �:

Thus,totracktheenergy,weneedtoknowexactlyhowpbehaves,andthebehavior

ofpduringfreeze-outiscomplicated,andwehavenotcalculateditinthiscourse.

Theformulasheettellsusthattheentropydensityofaconstituentofblack-body

radiationisgivenby

s=g2�2

45

k4T3

(�hc)3

:

Ifweconsidersome�xedcoordinatevolumeVcoord ,thecorrespondingphysicalvol-

umeisVphys=

Vcoorda3(t),wherea(t)isthescalefactor.Thetotalentropyof

neutrinosinVcoordisthenS

�=g�2�2

45

k4T3� (t)

(�hc)3

Vcoorda3(t):

ThequantitiesT� (t)anda(t)dependontime,buttheexpressionontheright-hand-

sidedoesnot,sinceentropyisconserved.ForbrevityIwillwrite

S�=g� A(t)T3�(t);

(1)

where

A(t)�2�2

45

k4

(�hc)3Vcoorda3(t):

Thee+e�

pairsandtheX'scontributetotheblack-bodyradiationonlybefore

thefreeze-out,whenkT�0:511MeV/c2.Lettbdenoteanytimebeforethefreeze-

out.Beforethefreeze-out,thetotalentropyofphotons,e+e�

pairs,andXparticles

isgivenby

Sbefore; eX

=(g +ge+

e�

+gX)A(tb )T3 (tb ):

(2)

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.69

IcancallthetemperatureT ,becausethee+e�

pairsandtheX's(aswellasthe

neutrinos)areallinthermalequilibrium

atthispoint,sotheyallhavethesame

temperature.

Usingta

todenoteanarbitrarytimeafterthefreeze-out,theentropyofthe

photonsduringthistimeperiodcanbewritten

Safter; =g A(ta )T3 (ta ):

(3)

Butsincethee+e�

pairsandX

particlesgivealltheirentropytothephotons,we

have

Safter; =Sbefore; eX

:

(4)

ThenusingEqs.(2)and(3)we�nd

g A(ta)T3 (ta )=(g +ge+e�

+gX)A(tb )T3 (tb ):

(5)

WecanrewritethelastfactorinEq.(5)byrememberingthatEq.(1)holdsatall

times,andthatT�(tb )=T (tb ).So,

A(tb )T3 (tb )=A(tb )T3�(tb )=S�

g�

=A(ta)T3�(ta ):

(6)

SubstitutingEq.(6)intoEq.(5),wehave

g A(ta )T3 (ta )=(g +ge+

e�

+gX)A(ta )T3�(ta );

fromwhichweseethat

T3 (ta)=g +ge+e�

+gX

g

T3�(ta );

andtherefore

T� (ta )

T (ta )= �g

g +ge+

e�

+gX �

1=3

= �2

2+72+9 �

1=3

=

�429 �1=3

:

(d)(5points)Theanswerwouldbethesame,sinceitwascompletelydeterminedbythe

conservationequation,Eq.(4)intheaboveanswer.Regardlessoftheorderinwhich

thefreeze-outsoccurred,thetotalentropyfromthee+e�

pairsandtheX'swould

ultimatelybegiventothephotons,sotheamountofheatingofthephotonswould

bethesame.

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.70

PROBLEM

18:THETIMEtd

OFDECOUPLING

(25points)

(a)(5points)Iftheentropyofphotonsisconserved,thentheentropydensityfallsas

s/1

a3(t):

Sinces/T3,itfollowsthat

T/1

a(t):

Thus,theratioofthescalefactorsisequaltotheinverseoftheratiotemperatures:

xd=T0

Td

:

(b)(5points)Theformulasheetremindsusthat

xdxd

t=H0 qm;0 x+rad;0+vac;0 x4+k;0 x2;

where

k;0 ��

kc2

a2(t0 )H20

=1�m;0 �rad;0 �vac;0:

Sofora atuniversek;0=0,andwehave

dxd

t=H0

x qm;0 x+rad;0+vac;0 x4:

(c)(5points)Theanswertopart(b)canberewrittenas

dt=

xdx

H0 pm;0 x+rad;0+vac;0 x4

:

tdisthetimethatelapsesfromwhentheuniversehasx=0towhenithasx=xd ,

so

td=

1H0 Z

xd

0

xdx


:

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.71

Youwereofcoursenotaskedtoevaluatethisintegralnumerically,butwewill

dothatnow.WetakeT0

=2:7255KfromFixsenetal.(citedinLectureNotes

6)andthePlanck2015best�tvaluesofH0=67:7km-s�

1-Mpc�

1,m;0=0:309,

vac;0=0:691.Theenergydensityofradiation(photonsplusneutrinos)canthen

becalculatedtogiverad;0

=9:2�10�

5

(seeEq.(6.23)ofLectureNotes6and

thetextofthe2ndparagraphofp.12ofLectureNotes7).Tokeepourmodel

universeexactly at,Iammodifyingvac;0tosetitequalto0:691�rad;0 ,whichis

wellwithintheuncertainties.Numericalintegrationthengives366,000years,very

closetoouroriginalestimate.Ofcoursethisnumberisstillapproximate,sincewe

startedwithTd �3000K.Inanycase,thedecouplingofthephotonsintheCMBis

actuallyagradualprocess.In2003Imodi�edastandardprogramcalledCMBFast

tocalculatetheprobabilitydistributionofthetimeoflastscattering(publishedin

https://arxiv.org/abs/astro-ph/0306275),withthefollowingresults:

Theparametersusedwerevac;0=0:70,m;0=0:30,H0

=68km-s�

1-Mpc�

1.The

peakofthecurveisat367,000years,andthemedianisat388,000years.

(d)(10points)Thederivationstartswiththe�rst-orderFriedmannequation.Sincewe

aredescribinga atuniverse,wecanstartwiththeFriedmannequationfora at

universe,

H2=8�3

G�:

Nowweusethefactsthat�m

/1=a3,�rad /1=a4,�vac /1,and�f /1=a8towrite

H2=8�3

G h�m;0

x3

+�rad;0

x4

+�vac;0+�f;0

x8 i:

Thenweuse

�m;0=�c m;0=3H20

8�Gm;0 ;

8.286QUIZ3REVIEW

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SOLUTIONS,FALL2018

p.72

withsimilarrelationsfortheothercomponentsofthemassdensity,torewritethe

Friedmannequationas

H2=H20 �m;0

x3

+rad;0

x4

+vac;0+f;0

x8 �:

NextwerewriteH2as

H2= �_aa �2

= �_xx �2

;

so

�_xx �2

=H20 �m;0

x3

+rad;0

x4

+vac;0+f;0

x8 �;

whichcanberewrittenas

xdxd

t=H0 rm;0 x+rad;0+vac;0 x4+f;0

x4

:

Fromherethederivationisidenticaltothatinpart(c),leadingto

td=

1H0 Z

xd

0

xdx

qm;0 x+rad;0+vac;0 x4+f;0

x4

;

whichcanalsobewrittenmoreneatlyas

td=

1H0 Z

xd

0

x3dx

pm;0 x5+rad;0 x4+vac;0 x8+f;0

:

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