The Expanding Universe - Research Group of Ignacio Taboada · Phys 8803 – Special Topics on...
Transcript of The Expanding Universe - Research Group of Ignacio Taboada · Phys 8803 – Special Topics on...
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
TheExpandingUniverse
DistanceLadder&Hubble’sLawRobertson-WalkermetricFriedmanequa7onsEinstein–DeSiKersolu7onsCosmologicaldistanceObservedproper7esoftheUniverse
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
Thedistanceladder
Measuringdistancesinastronomyishard.Themostdirectwayofmeasuringdistanceisparallax.Theparsecisthedistanceatwhichaparallaxof1”isobserved.1pc=2.06x105AU=3.26light-year=3.09x1018cmStateoftheartinmeasuringparallaxesisspacebased(Gaiasatellite–successortoHipparchus).Upto10kpcdistancescanbemeasuredwith10%precision
d =1 AU
tan p⇡ 1
pAU
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
Cepheidvariablestars
Cepheidstarsarevariablestars.Theyarenotmainsequencestars.Insteadtheyareredgiantsthatundergooscilla7onsinsize,andhenceinluminosity.Theperiodoftheoscilla7onsvariesfromafewdaystoafewmonths.HenrieKaLeaviKno7cedarela7onshipbetweenaverageluminosityandperiodthiswasdonewithCepheidstarsintheSMCandLMC.CepheidswereusedbyEdwinHubbletomeasurethedistancetonearbygalaxies.
L = �AT 4Stefan-Boltzman
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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)
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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)
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Distances
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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
⌫
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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)
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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)
�
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
Densityparameter–ΛCDMcosmology
Thera7oofthedensitytothecri7caldensityisthedensityparameterThecurrentbaryonicmaKerdensityparameterisThereisa(present)contribu7onfromdarkmaKer:Anda(present)contribu7onfromdarkenergy:Withh=0.673–theUniverseisflat.(Notethathhadadifferentvalueinapreviousslide)
⌦(t) =⇢(t)
⇢c(t)
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PlanckCosmologicalparameters:arXiv1502.01589
⌦Bh2 = 0.02205
⌦cdmh2 = 0.1199
⌦⇤ = 0.6866
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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
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✓a
a
◆2
� 8⇡G
3⇢
✓a
a0
◆2
= �Kc2
a20
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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+!)
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
DarkMaJerdominatedUniverse
ForK=0andω=-1ω=1isnottheonlypossibilityanditdoesn’tevenhavetobeconstantwithproper7met.Thebestmeasurementofw,assumingthatitisconstantisω=1.00±0.06.
a(t) / ep
⇤/3t
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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)
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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)
�
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z =�o
� �e
�e
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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
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Z t0
te
cdt
a(t)=
Z 0
R
drp1�Kr2
= f(r)
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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
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p0 =P
4⇡d2L
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
Luminositydistance
PhotonsemiKedbythesourceareobservedwitharedshiba/a0AndphotonsemiKedwithinasmall7meΔteareobservedwithina7meΔt0=a(t)/a0Δte.Thus:Thereareotherdefini7onsofdistanceincosmology–Iskipthat.
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p0 =P
4⇡a20r2
✓a
a0
◆2
=P
4⇡d2L
Twofactors:oneforenergy,anotherfor7me
dL = a20r
a
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
TypeIaSupernovae:Luminositydistance
TopplotshowstheHubblediagramobtainedwithSNtypeIa–ThisconfirmsthatHubble’slawisgoodandthatSNIaaregoodstandardcandlesThisisthesameplot,butshows.ThedataisbestdescribedforK=0andΩm=0.3.BecauseK=0canonlybeachievedforΩtotal=1,thisimpliesthatthereisacosmologicalconstant.
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
CosmicMicrowavebackground.
It’seasytoseethattheearlyUniverse,wasdenserandhoKerthatitistoday.ThehighdensitycoupledphotonstomaKer.Atsomepastredshib(z~1100)theUniversebecametransparenttophotons.Thisiseraof“lastscaKering”.Thisbackgroundofphotonscon7nuestofilltheUniverseandithasredshibed,duetoexpansion.It’sextremelywelldescribedbyaPlanckfunc7onwithT=2.7255±0.0006K
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Planck(satellite)
Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
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)� < ⇢ >
< ⇢ >
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
DensityfluctuaBons
GalaxypowerspectrummeasuredbySDSS.ThebestfitofΛCMDisshown.BaryonAcous7cOscilla7onsareinset.
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
CMBpowerspectrum
PowerspectrumoftheCMBasmeasuredbyPlanck.
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Phys8803–SpecialTopicsonAstropar7clePhysics–IgnacioTaboada
CMB/SneTypeIa/acousBcbaryonoscillaBons
BestfitregionforPlanck,SNtypeIadataandacous7cbaryonoscilla7ons.Contourlinesarefor68.3%and95.4%confidencelevels(1,2sigma)
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