the expanding universe - research group of ignacio taboada · phys 8803 – special topics on...

Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada

TheExpandingUniverse

DistanceLadder&Hubble’sLawRobertson-WalkermetricFriedmanequa7onsEinstein–DeSiKersolu7onsCosmologicaldistanceObservedproper7esoftheUniverse

1


Thedistanceladder

Measuringdistancesinastronomyishard.Themostdirectwayofmeasuringdistanceisparallax.Theparsecisthedistanceatwhichaparallaxof1”isobserved.1pc=2.06x105AU=3.26light-year=3.09x1018cmStateoftheartinmeasuringparallaxesisspacebased(Gaiasatellite–successortoHipparchus).Upto10kpcdistancescanbemeasuredwith10%precision

d =1 AU

tan p⇡ 1

pAU

2


Cepheidvariablestars

Cepheidstarsarevariablestars.Theyarenotmainsequencestars.Insteadtheyareredgiantsthatundergooscilla7onsinsize,andhenceinluminosity.Theperiodoftheoscilla7onsvariesfromafewdaystoafewmonths.HenrieKaLeaviKno7cedarela7onshipbetweenaverageluminosityandperiodthiswasdonewithCepheidstarsintheSMCandLMC.CepheidswereusedbyEdwinHubbletomeasurethedistancetonearbygalaxies.

L = �AT 4Stefan-Boltzman

3


Hubble’slawandredshi:

Hubbleobservedthattheredshib(recedingspeed)ofgalaxieswaslinearlycorrelatedwithdistance.Thisrela7onshipisHubble’slaw:.WithCepheids,andusingtheHubblespacetelescope:H0,Hubble’sconstant,is73.00±1.75km.s-1.Mpc.(arXiv:1604.01424)BecausetheHubbleconstanthas(tradi7onally)beenuncertain,itiscommontoseethedefini7on,withH=100km.s-1.Mpc.

RedshibisnowadistanceindicatorTheUniverseisexpanding.

v = H0D

H0 = hH4

�0 = �(1 + z)

�0 = �

s1 + �

1� �

�0 ! �(1 + �) (nearby)


Moreonthedistanceladder

Therearemanymorestepsinthedistanceladder…

1 10 102 103 104 105 106 107 108 109 1010 d(pc)

Redshib

TullyFisher

ParallaxProperMo7on

MainSequenceFijngCepheids&RRLyrae

SNeIa

GravLensQuasarsProxim

aCe

ntauri LMC

M31

VirgoCluster

Galac7cCe

nter

z=1

(insomedefini7onofdistance)

5


Distances

6


FromSpecialtoGeneralrelaBvitySpecialrela7vity–Iner7alframes:Takeaniner7alframeSandanacceleratedframeS’.S’hassmallaccelera7onwithrespecttoS.ThenNowSoForaclockinS’:NowallthewayGeneralrela7vity:

aı

x

0 = x� 1/2at2

ds

2 = (c2 � a

2t

2)dt2 � 2atdx0dt� dx

02 � dy

02 � dz

02

dx =@x

@x

0 dx0 +

@x

@t

dt = dx

0 + at dt

S : x, y, z, t S

0 : x

0, y

0, z

0, t

0

ds

2 = gµ⌫dxµdx

⌫

7

ds2 = c2d⌧2 = c2dt02 = (c2 � 2ad)dt2

ds

2 = c

2d⌧

2 = c

2dt

2 � dx

2 � dy

2 � dz

2 = c

2dt

2 � dr

2 � r

2(d✓ + sin2 ✓d�2)


TheRobertson-Walkermetric

GeneralRela7vityprovidesaframeworktodescribetheUniverseexpansion.TheRobertson-WalkermetricisHerea(t)isdimensionlessandknownastheexpansionparameter.TheconstantKistheGaussiancurvatureandhasunits1/length2.K=0isforanEuclideanuniverse.K>0posi7vecurvature,K<0nega7vecurvature.Theabovemetricusescomovingcoordinates,whicharethecoordinatesonwhichtheCosmologicalPrincipleapplies.CosmologicalPrinciple:theUniverseishomogeneousandisotropic,atleasttofirstapproxima7on.Comoving7meisthe7mesincetheBigBangasmeasuredinaclockthatfollowstheHubbleflow.

ds2 = (cdt)2 � a(t)2

dr2

1�Kr2+ r2(d✓2 + sin2 ✓d�2)

�

8


FriedmanCosmologicalEquaBons

UsingEinstein’sequa7onandassumingtheUniverseisfilledwithahomogeneousandisotropicfluidofdensityρc2andpressurep,resultsin:Intheequa7onsabove,bothpandρarefunc7onsofthecomoving7metoo.Ingeneral,anequa7onofstaterela7ngpressureanddensityisalsoneeded.Adiaba7cexpansionoftheUniverseisassumedsoNotethattheexpansionoftheUniverseismeasuredbytheHubbleparameter(notaconstantanymore)

H(t) =a(t)

a(t)

✓a

a

◆2

=8

3⇡G⇢� K

a2+

⇤

3

a

a=

⇤

3� 4⇡G

3(⇢+ 3

p

c)

d(⇢c2a3)/da3 = �p

9


CriBcalDensity

InthecaseofaflatUniverse(K=0)andnocosmologicalconstantThedensitytakesthevalue:Thisisknownasthecri7caldensity.It’svalueis:Thepresentes7mateofthebaryonicdensityintheUniverseis:

⇢c(t) =3H2(t)

8⇡G

⇢c,0 =3H2

0

8⇡G= 1.88⇥ 10�29h2 g · cm�3

⇢B,0 = 3⇥ 10�31 g · cm�3

✓a

a

◆2

� 8⇡G⇢

3= 0

10


Densityparameter–ΛCDMcosmology

Thera7oofthedensitytothecri7caldensityisthedensityparameterThecurrentbaryonicmaKerdensityparameterisThereisa(present)contribu7onfromdarkmaKer:Anda(present)contribu7onfromdarkenergy:Withh=0.673–theUniverseisflat.(Notethathhadadifferentvalueinapreviousslide)

⌦(t) =⇢(t)

⇢c(t)

11

PlanckCosmologicalparameters:arXiv1502.01589

⌦Bh2 = 0.02205

⌦cdmh2 = 0.1199

⌦⇤ = 0.6866


Einstein–DeSiJerModel

Equa7on(nocosmologicalconstant)canbere-wriKenthus:Agenericequa7onofstatetakestheformω=0representspressure-lessmaterial(dust)thisisalsoagoodrepresenta7onofanon-rela7vis7cidealgas.ω=1/3representsaradia7vefluid(e.g.photons)ω=-1representsdarkenergy–thecosmologicalconstant

a2 +Kc2 =8⇡G

3⇢a2

p = !⇢c2

12

✓a

a

◆2

� 8⇡G

3⇢

✓a

a0

◆2

= �Kc2

a20


Einstein–DeSiJerModel

FromandwegetWecanusethisinequa7on(nocosmologicalconstant)SothatInthecaseofK=0,whichimplies

d(⇢c2a3) = �pda3 p = !⇢c2

⇢a3(1+!) = ⇢0a3(1+!)0

✓a

a

◆2

� 8⇡G

3⇢

✓a

a0

◆2

= �Kc2

a20

✓a

a

◆2

� ⇢

⇢cH2

0 =

✓a

a

◆2

� ⇢0⇢c

H20

⇣a0a

⌘3(1+!)= �Kc2

a20

⌦0 = ⇢0/⇢c = 1✓a

a

◆2

�H20

⇣a0a

⌘3(1+!)= 0

13


EinsteinDeSiJerModel

TheK=0solu7onis:ForΩ<1(openUniverse,K=-1)andω=0theparametricsolu7onis:AndforΩ>1(closedUniverse,K=1)andω=0theparametricsolu7onis:

a(x) = a0⌦

2(1� ⌦)

(coshx� 1)

t(x) =1

2H0

⌦

(1� ⌦)

3/2(sinhx� x)

a(x) = a0⌦

2(1� ⌦)

(1� cosx)

t(x) =1

2H0

⌦

(1� ⌦)

3/2(x� sinx)

a(t) = a0

✓t

t0

◆2/3(1+!)

14


DarkMaJerdominatedUniverse

ForK=0andω=-1ω=1isnottheonlypossibilityanditdoesn’tevenhavetobeconstantwithproper7met.Thebestmeasurementofw,assumingthatitisconstantisω=1.00±0.06.

a(t) / ep

⇤/3t

15


Einstein–DeSiJerUniverse

Theω=1/3generalsolu7onis:

a(t) = a0(2H0⌦1/20 t)1/2

1 +

1� ⌦0

2⌦1/20 H0t

!1/2

t

a(t)

16


Luminosityandproperdistance

Let’sreturntotheRobertson-WalkermetricTheproperdistanceassumethatthetwoendsaremeasuredsimultaneouslyinproper7me.Ifatt0twoeventshaveaproperdistancer0,thenat7mettheyhaveaproperdistancea(t)/a(t0)r0.Butproperdistanceisnotdirectlymeasurableinprac7ce.Let’slookatadifferentdefini7onofdistance.Theredshibofaphotonisdefinedas:WhereλeistheemiKedwavelengthandλ0istheobservedwavelength.

ds2 = (cdt)2 � a(t)2

dr2

1�Kr2+ r2(d✓2 + sin2 ✓d�2)

�

17

z =�o

� �e

�e


Luminositydistance

Photonstravelalongnullgeodesics,ds2=0.Thisimplies:LightemiKedthesourceatisobservedat.AssumingsmallΔteandΔt0,f(r)doesnotchange.HereΔtecanrepresent,e.g.thefrequencyofemiKedlight.Then,ForsmallΔteandΔt0,youfindAphotonfrequencywillthusbe

t0e = te +�te

t00 = t0 +�t0

�t0a(t0)

=�tea(te)

⌫ea(te) = ⌫0a(t0)

Z t0

te

cdt

a=

Z t00

t0e

cdt

a

18

Z t0

te

cdt

a(t)=

Z 0

R

drp1�Kr2

= f(r)


Luminositydistance

Fromwhichitfollows, .Andusingthedefini7onofredshib:LetPbethepoweremiKedbyasourceat7metandco-movingdistancer.Letp0bethepowerperunitarea(flux)observedat7met0.WewantLuminositydistancetobesuchthat:Theareaofaspherecenteredatthesource,at7met0is.

a(te)

�e=

a(t0)

�0

1 + z = a(t0)/a(te)

4⇡a20r20

19

p0 =P

4⇡d2L


Luminositydistance

PhotonsemiKedbythesourceareobservedwitharedshiba/a0AndphotonsemiKedwithinasmall7meΔteareobservedwithina7meΔt0=a(t)/a0Δte.Thus:Thereareotherdefini7onsofdistanceincosmology–Iskipthat.

20

p0 =P

4⇡a20r2

✓a

a0

◆2

=P

4⇡d2L

Twofactors:oneforenergy,anotherfor7me

dL = a20r

a


TypeIaSupernovae:Luminositydistance

TopplotshowstheHubblediagramobtainedwithSNtypeIa–ThisconfirmsthatHubble’slawisgoodandthatSNIaaregoodstandardcandlesThisisthesameplot,butshows.ThedataisbestdescribedforK=0andΩm=0.3.BecauseK=0canonlybeachievedforΩtotal=1,thisimpliesthatthereisacosmologicalconstant.

21


CosmicMicrowavebackground.

It’seasytoseethattheearlyUniverse,wasdenserandhoKerthatitistoday.ThehighdensitycoupledphotonstomaKer.Atsomepastredshib(z~1100)theUniversebecametransparenttophotons.Thisiseraof“lastscaKering”.Thisbackgroundofphotonscon7nuestofilltheUniverseandithasredshibed,duetoexpansion.It’sextremelywelldescribedbyaPlanckfunc7onwithT=2.7255±0.0006K

22

Planck(satellite)


DensityfluctuaBons

Theuniverseisveryhomogeneous.Butinhomogeneityisobservablefromindividualgalaxiestothe100Mpcscale.Theinhomogenei7esseenintheCMBgrowintotoday’sdensityinhomogenei7es.Thedensityfluctua7onsaredescribedas:Itisconven7onaltodescribethedensityfluctua7onsasaFourierexpansion(periodicboundarycondi7onsareassumedoveraverylargevolumeV).Thevarianceoffluctua7onsis:(Therenopreferreddirec7onforinhomogenei7es)

�(~x) =X

�

~

k

e

�i

~

k.~x

< �(~x)2 >=X

|�~k|2 =

XP (k)

�(~x) =⇢(~x)� < ⇢ >

< ⇢ >

23


DensityfluctuaBons

GalaxypowerspectrummeasuredbySDSS.ThebestfitofΛCMDisshown.BaryonAcous7cOscilla7onsareinset.

24


CMBpowerspectrum

PowerspectrumoftheCMBasmeasuredbyPlanck.

25


CMB/SneTypeIa/acousBcbaryonoscillaBons

BestfitregionforPlanck,SNtypeIadataandacous7cbaryonoscilla7ons.Contourlinesarefor68.3%and95.4%confidencelevels(1,2sigma)

26

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