Class11:Cosmology

InthisclasswewillexplorehowGeneralRelativitycanbeappliedtotheexpansionoftheUniverse,linkingitsdevelopmentfromtheBig

Bangtothematter-energyitcontains

Class11:Cosmology

Attheendofthissessionyoushouldbeableto…

• … understandhowthecosmologicalprincipleleadstothespace-timemetricofahomogeneousandisotropicUniverse

• … usegeodesicsoflightraystoshowthatlightisredshiftedasitpropagatesinanexpandingUniverse

• … describedifferentdefinitionsofcosmologicaldistance

• … linktheexpansionoftheUniversetoitsmatter-energycontent andcurvature usingEinstein’sequation

• … calculatethelook-backtimeanddistance toagalaxy

Thecosmologicalprinciple

• ThereisanothersituationinwhichtheequationsofGRcanbesolved:cosmologicalobservationsoftheUniverse

https://en.wikipedia.org/wiki/Chronology_of_the_universe

TheUniverseisexpandingfromaHotBigBang!

Thecosmologicalprinciple

• AlthoughtheUniverseisirregularindetail,wecan“smoothout”itscontentsintoasetofspace-fillingparticles whichareuniformlyexpandinginalldirections

• Thesearetheassumptionsofhomogeneityandisotropy,togetherknownasthecosmologicalprinciple

http://wise2.ipac.caltech.edu/staff/jarrett/ngss/wise_LSS.html http://background.uchicago.edu/~whu/intermediate/map2.html

Thecosmologicalprinciple

• HomogeneitymeansthatthepropertiesoftheUniversearethesameineverylocation

• IsotropymeansthatthepropertiesoftheUniversearethesameineverydirection

Homogeneousbutnotisotropic Isotropicbutnothomogeneoushttp://www.astro.ucla.edu/~wright/cosmo_01.htm

Thecosmologicalprinciple

• Whyisthissuchapowerfulassumption?

• Supposeeachofthesespace-fillingparticlescarriesa“fundamentalobserver”– byhomogeneity,theexperienceofeachoftheseobserversisidentical

• AnypartoftheUniverseisrepresentativeofthewhole–homogeneousUniversescanbestudiedlocally

• Thereisanabsolutecosmictime– thepropertimeforeachfundamentalobserver– forwhichtheUniverseitselfactsasthesynchronizationagent

ThemetricoftheUniverse

• Whatdothesesymmetriesimplyaboutthespace-timemetricoftheexpandingUniverse?Itmusttaketheform…

𝑑𝑠# = −𝑐#𝑑𝑡# + 𝑎(𝑡)#𝑑𝑙#

Thispiecebecausecosmictime𝑡 isjustpropertime𝜏forfundamentalobservers(whichhave𝑑𝑙 = 0)

ThispieceensuresthattheUniverseremainshomogeneousandisotropicasitexpands,withalldistancessimplyscalingas𝑎(𝑡),thecosmicscalefactor

ThemetricoftheUniverse

• Nowlet’sthinkabouttheformoftheproperseparation𝑑𝑙

• Homogeneityandisotropyimplythatthecurvatureofspacemusteverywherebeequalandindependentoforientation,socanbewrittenasasinglenumber𝐾

• Thecurvaturecanbeflat(𝐾 = 0),positiveornegative

http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html

ThemetricoftheUniverse

• Wecanderivetheformof𝑑𝑙 byanalogywiththe2Dsurfaceofasphereembeddedina3DEuclideanspace,whichsatisfiestheequation𝑥# + 𝑦# + 𝑧# = 1/𝐾

• Foraconstantcurvature3Dsurfaceembeddedina4DEuclideanspace:𝑥# + 𝑦# + 𝑧# + 𝑤# = 1/𝐾

• Inthe4DEuclideanspace:𝑑𝑙# = 𝑑𝑥# + 𝑑𝑦# + 𝑑𝑧# + 𝑑𝑤#

• Transform(𝑥, 𝑦, 𝑧) tosphericalpolarco-ordinates(𝑟, 𝜃, 𝜙)

• Fromabove:𝑤# = 1/𝐾 − 𝑟#,hence𝑑𝑤# = :;=:;

>/?@:;

• Puttingitalltogether:𝑑𝑙# = =:;

>@?:;+ 𝑟# 𝑑𝜃# + sin 𝜃 𝑑𝜙 #

ThemetricoftheUniverse

• Theassumptionsofhomogeneityandisotropy,allowingforconstantcurvature𝐾,producetheRobertson-Walkermetric

• Theco-ordinates(𝑟, 𝜃, 𝜙) arefixedfor“fundamentalobservers”whomeasurepropertime𝑡

• Thesearealsoknownasco-movingobservers,sincetheyexpandwiththeUniverse

• Thenon-zerometricelementsarehence𝑔EE = −1,𝑔:: =F(E);

>@?:;,𝑔GG = 𝑎(𝑡)#𝑟# and𝑔HH = 𝑎(𝑡)#𝑟# sin 𝜃 #

𝑑𝑠# = −𝑐#𝑑𝑡# + 𝑎(𝑡)#𝑑𝑟#

1 − 𝐾𝑟# + 𝑟# 𝑑𝜃# + sin 𝜃 #𝑑𝜙#

Hubble’sLaw

• Theproperdistance𝐿 betweenany2fundamentalobservers,separatedbyco-ordinatedistance𝑙,increasesas𝐿 𝑡 = 𝑎 𝑡 𝑙

• Therateofincreaseofthisdistanceis=J=E= =F

=E𝑙 = =F/=E

F(E)𝐿

• Hencethespeedofrecessionisproportionaltodistance

http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/hubble.html

ThisrelationisknownasHubble’sLaw– thecoefficientofproportionalityisHubble’sconstant,

𝐻 = ��/𝑎

Hubble’sLaw

• Hubble’sLawhasbeenbeautifullyconfirmedbymeasuringdistancesandrecessionvelocitiesofnearbygalaxies:

• ThevelocitiesarededucedfromDopplershiftinspectrallines

https://ned.ipac.caltech.edu/level5/Tyson/Tyson2.html

Redshifting

• Thefrequency𝜔 ofalightray,travellinginexpandingspacewithscalefactor𝑎(𝑡),changessuchthat𝜔 ∝ 1/𝑎

• Thisbehaviouriscalledredshifting oflight,whereredshift𝑧isdefinedastheratioofemission/observation frequencies

1 + 𝑧 =𝜔OPQEEO=𝜔RSTO:UO=

=𝑎RSTO:UO=𝑎OPQEEO=

Redshiftoflightfromdistantgalaxiescanbemeasuredfromthefrequenciesof

spectrallineshttp://www.space-exploratorium.com/doppler-shift.htm

Redshifting

• Inacrudeanalogy,thewavelengthoflightisstretchingwiththeexpansionoftheUniverse

• Moreaccurately,wecanthinkofthelightundergoingmanysmallDopplershiftsbetweenpairsoffundamentalobservers,ortravellingalongageodesicinexpandingspace-time

http://scienceblogs.com/startswithabang/2014/11/01/ask-ethan-60-why-is-the-universes-energy-disappearing-synopsis/

Distancesinexpandingspace

• Nowlet’sconsiderhowtomeasuredistancesintheexpandingUniverse,from𝑟 = 0 toagalaxyatco-ordinate𝑟

https://www.nasa.gov/mission_pages/spitzer/multimedia/pia15818.html

Distancesinexpandingspace

• Nowlet’sconsiderhowtomeasuredistancesintheexpandingUniverse,from𝑟 = 0 toagalaxyatco-ordinate𝑟

• Theproperdistance𝐿 isthedistancemeasuredif,atthesamecosmictime𝑡,achainoffundamentalobserverstothegalaxyaddeduptheirinfinitesimalproperseparations

• 𝑑𝐿 = 𝑔::� 𝑑𝑟 = F E =:>@?:;� (since𝑑𝜃 = 𝑑𝜙 = 0)

• Integrating: 𝐿 = 𝑎 𝑡 sin@> 𝑟 𝐾� / 𝐾�

𝐿 = 𝑎 𝑡 𝑟

𝐿 = 𝑎 𝑡 sinh@> 𝑟 −𝐾� / −𝐾�

𝐾 > 0

𝐾 < 0

𝐾 = 0

Distancesinexpandingspace

• Theproperdistanceisnotveryusefulinobservationalcosmology,sincewecan’tmeasureit!Moreusefulmeasuresaretheluminositydistanceandangulardiameterdistance

https://www.nasa.gov/mission_pages/galex/pia14095.html

Luminositydistance

Angulardiameterdistance

Distancesinexpandingspace

𝑑Z

𝐷∆𝜃

• Angulardiameterdistance:alightsourceatco-ordinate𝑟hasproperdiameter𝐷 andapparentangularsize∆𝜃

• Theangulardiameterdistanceisdefinedas:𝑑Z = 𝐷/∆𝜃

• Fromthemetric:𝐷 = ∆𝐿 = 𝑎 𝑡 𝑟∆𝜃

• Hence𝑑Z = 𝑎 𝑡OP 𝑟,where𝑡OP isthelightemissiontime

Distancesinexpandingspace

• Luminositydistance:considerphotonsemittedbyadistantgalaxy,travellingtoourtelescopes.Whatistheequivalentofthe“inversesquarelaw”?

• Sincefrequency𝜔 ∝ 1/𝑎,photonsloseenergyastheytravel

• Theco-ordinatetimeintervalchangesbetweenthephotons

• Why? Since𝑑𝑠 = 0,weknowthat∫ =EF(E)

E^_`Eab

= >c; ∫

=:>@?:;�

d:ab

Distancesinexpandingspace

• So,∫ =EF(E)

E^_`Eab

= constant= ∫ =EF(E)

E^_`efE^_`EabefEab

forthe2nd photon

• Thisimpliesthat fE^_`F E^_`

= fEabF Eab

• Eachphotondecreasesinenergyby𝒂(𝒕𝒆𝒎)/𝒂(𝒕𝒐𝒃𝒔),andthetimebetweenthemincreasesbythesamefactor

• Thefluxofenergyreceivedis𝑓 = JF(Eab);

op:;

• Theluminositydistanceisdefinedby𝑑J =Jopq

� = :F(Eab)

Matter-energyintheUniverse

• WenowrelatetheexpansionoftheUniversetoitsmatter-energycontent

https://www.spacetelescope.org/images/opo9919k/

Matter-energyintheUniverse

• ThisisdoneviaEinstein’sequationfromthepreviousclass:

• Themetric𝑔rs isgivenby

• Wewon’tgothroughthealgebraofcomputingtheRiccitensor𝑅rs from𝑔rs,butthenon-zerocomponentsare:

𝑅rs −12𝑅𝑔rs =

8𝜋𝐺𝑐o 𝑇rs

𝑑𝑠# = −𝑐#𝑑𝑡# + 𝑎(𝑡)#𝑑𝑟#

1 − 𝐾𝑟# + 𝑟# 𝑑𝜃# + sin 𝜃 #𝑑𝜙#

𝑅EE = −3𝑐#��𝑎

𝑅QQ =��𝑎 + 2

��𝑎

#+2𝐾𝑐#

𝑎#𝑔QQ𝑐#

Matter-energyintheUniverse

• Whatistheenergy-momentumtensorofthehomogeneousandisotropicUniverse?Wewillconsider2components…

• First,thesmoothlydistributedparticlesfillingtheUniversecontainanenergydensity,butnopressure,suchthat

• Second,itturnsoutthatemptyspacecontainsanenergydensity (the“cosmologicalconstant”)thatisrequiredtodescribeourobservations

𝑇dd = 𝜌 𝑡 𝑐# 𝑇RE}O:T = 0

𝑇rs = Λ𝑔rs

TheFriedmann equation

• WecanuseEinstein’sEquationtoderivetheFriedmannEquation,whichdescribestheexpansionoftheUniverseintermsofitsmatter-energycontent:

��𝑎

#=8𝜋𝐺𝜌(𝑡)

3 −𝐾𝑐#

𝑎# +Λ𝑐#

3

Effectofmatter

Effectofcurvature

Effectofcosmologicalconstant

Expansionrate(where��/𝑎 istheHubbleparameter)

Densityparameters

• Wenormalizethescalefactorsuchthattoday,𝒂 = 𝟏,withtoday’sdensityas𝜌d andHubbleparameteras𝐻d

• TheFriedmann equationtoday:𝐻d# =�p����

− 𝐾𝑐# + �c;

�

• It’sconvenienttodefinethecriticaldensity𝜌c:QE =���;

�p�and

thedimensionlessdensityparameters

• ThesearehencerelatedconvenientlybyΩP + Ω? + Ω� = 1

ΩP =𝜌

𝜌c:QE=8𝜋𝐺𝜌3𝐻d#

Ω? = −𝐾𝑐#

𝐻d#Ω� =

Λ𝑐#

3𝐻d#

Densityparameters

• WealsonotethatastheUniverseexpands,thematterdensitydilutes:𝜌 𝑡 = 𝜌d/𝑎� – theΛ densitydoesn’t!

• TheFriedmann equationisthen:>��;

FF

#= �b

F�+ ��

F;+ Ω�

ΩP Ω�

TheexpansionhistoryoftheUniversedependsonthevaluesofΩP andΩ�(withΩ? = 1 − ΩP − Ω�)

http://www.preposterousuniverse.com/blog/2006/01/26/the-future-of-the-universe/