Astronomy, Astrophysics, And Cosmology

Astronomy, Astrophysics, and CosmologyLuis A. AnchordoquiDepartment of Physics and AstronomyLehman College, City University of New YorkLesson VIIMarch 29, 2016arXiv:0706.1988L. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-20161 / 22

Table of Contents1Expansion of the UniverseAge and size of the UniverseAngular diameter and luminosity distances2The force awakensSupernova CosmologyL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-20162 / 22

Expansion of the UniverseAge and size of the UniverseRate of change for proper distance between us and distant galaxyȧd p ȧr dpa(1)@ present time (t t0 ) there is linear relationbetween proper distance to galaxy and its recession speedv(t0 ) H0 dp (t0 )(2)v(t0 ) d p (t0 )(3) ȧH0 a t t0(4)whereandin agreement with Hubble’s LawL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-20163 / 22

Expansion of the UniverseAge and size of the UniverseIn expanding universe wavelength of radiation is proportional to aλ0 /λem a0 /a(tem )(5)Redshift of a galaxyz λ0 λema0 1λema(tem )(6)expresses how much scale factor changed since light was emittedLight detected today was emitted at some time temAccording to (6) 1-to-1 correspondence between z and temz can be used instead of t to parametrize history of universeA given z corresponds to timewhen our universe was 1 z times smaller than nowL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-20164 / 22

Expansion of the UniverseAge and size of the UniverseExpressions for a(t) are rather complicatedOne cannot directly invert (6) to express t tem in terms of zIt is useful to derive general integral expression for t(z)Differentiating (6)a0dz 2 ȧ(t)dt (1 z) H (t)dta (t)(7)from which follows thatt Z zdzH (z)(1 z)(8)Integration constant has been chosenso that z corresponds to initial moment of t 0Since photons travel on null geodesics of zero proper timewe see directly from FRW metricr L. A. Anchordoqui (CUNY)Zcdt a(t)Zdtc (1 z ) cdzAstronomy, Astrophysics, and CosmologyZdzH (z)(9)3-29-20165 / 22

Expansion of the UniverseAge and size of the UniverseFor the moment set c 1 and take ρ ρm c2To obtain expression for H (z, H0 , Ωm,0 ) rewrite Friedmannequationρm (z)kH 2 (z) 2 (1 z)2 Ωm,0 H02ρm,0a0 R0(10)At z 0 this reduces tok (Ωm,0 1) H02a0 R0(11)allowing to express current value a0 R0in a spatially curved universe (k 6 0)in terms of H0 and Ωm,0Taking this into account H (z) H0L. A. Anchordoqui (CUNY)ρm (z)(1 Ωm,0 )(1 z) Ωm,0ρm,02Astronomy, Astrophysics, and Cosmology 1/2(12)3-29-20166 / 22

Expansion of the UniverseAge and size of the UniverseHubble’s Law approximation for small redshiftTaylor expansion1a(t) a(t0 ) (t t0 ) ȧ(t0 ) (t t0 )2 ä(t0 ) · · ·2 122 a(t0 ) 1 (t t0 ) H0 (t t0 ) q0 H0 · · · (13)2q0 ä(t0 ) a(t0 )/ ȧ2 (t0 ) deceleration parameterIf expansion is slowing down ä 0 and q0 0For not too large time-differenceswe can use Taylor expansion of a(t) and write1 z 1a(t) 1 (t t0 ) H01 za ( t0 )(14)Hubble’s law z (t0 t) H0 d/cH0is valid as long as z H0 (t0 t) 1Deviations from its linear form arises for z & 1and can be used to determine q0L. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-20167 / 22

Expansion of the UniverseAngular diameter and luminosity distances( 0 ,(t0 , %0 0) θobserverχ 0t t0ϕ0 const0 ) θ0 const l ϕ0θ0 θ( 0 ,0)(t1tχem,em%1 )Fig. 2.11.Light source of size at 1 and t t1subtending θ at( angular0, t t0 )tem propagate along radialgeodesics andanglearrive todaywithoriginan apparentseparation!θ. The propersize of theobject,l, is ofequalto the intervalbetweentothe θ byProperdistancebetweentwoendsobjectis relatedemission events at the endpoints: !2 a(tem ) #(χem ) !θ,l !s θa ( t1 ) 1(2.68)as obtained from metric (2.2). The angle subtended by the object is thenAngulardiameter distance lld A ,(2.69)!θ a(tem ) #(χem )a(η θ0 χem ) #(χem ) 1 physicala(t1 ) timed A theso thatwhere we have used the fact that1 tem corresponds to the conformal1 is, χzem η0 , thentime ηem η0 χem . If the object is close to us, thatL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-2016(15)(16)(17)8 / 22

Expansion of the UniverseAngular diameter and luminosity distancesAngular diamter distance as function of redshiftRecall light travel on null geodesics following similar derivation pRzc f k H0 Ωk,0 0 dz/H (z)p(18)d A (z) H0 Ωk,0 (1 z)with sin xxf k (x) sinh xfor k 1for k 0for k 1(19)For flat universe filled with dust angular diameter is θ (z) L. A. Anchordoqui (CUNY) H0 (1 z)3/22c (1 z)1/2 1Astronomy, Astrophysics, and Cosmology(20)3-29-20169 / 22

Expansion of the UniverseAngular diameter and luminosity distancesAt low redshifts (z 1) θ decreases in inverse proportion to z2.5 Kinematic testsreaches a minimum at z 5/4 and then scales as z for z 1 θz 5/4L. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmologyz3-29-201610 / 22

Expansion of the UniverseAngular diameter and luminosity distancesRelation between monochromatic flux density and luminosityAssume isotropic emission photons emitted by sourcepass with uniform flux density any sphere surrounding sourceShift origin and consider FRW metric as being centred on sourceBecause of homogeneitysame comoving distance 1 between source and observerPhotons from source pass through sphere of proper surface area4πa20 21Redshift still affects flux density in four further waysPhoton energies are redshiftedreducing flux density by factor 1 zPhoton arrival rates are time dilatedreducing flux density by factor 1 zBandwidth dν is reduced by a factor 1 zincreasing energy flux per unit bandwidth by one power of 1 zObserved photons at frequency ν0were emitted at frequency (1 z)ν0L. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201611 / 22

Expansion of the UniverseAngular diameter and luminosity distancesOverall flux density is luminosity at frequency (1 z)ν0divided by total area and divided by (1 z)Fν (ν0 ) Lν (ν0 )Lν ([1 z]ν0 ) 2224πa0 1 (r )(1 z)4πa0 21 (1 z)1 α(21)(second expression assumes power-law spectrum L ν α )Integrate over ν0 to obtain bolometric formulaeF L(22)4πa20 21 (1 z)2Luminosity distance d L is defined to satisfy relationF L4πd2L(23)If we normalize scale factor today a0 1d L (1 z ) 1 (1 z )2 d AL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology(24)3-29-201612 / 22

The force awakensL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201613 / 22

The force awakensIndependent cosmological observations have unmaskedpresence of some unknown form of energy densityrelated to otherwise empty space which appears to dominaterecent gravitational dynamics of universeand yields a stage of cosmic accelerationWe still have no solid clues as to nature of such dark energy(or perhaps more accurately dark pressure)Cosmological constant is simplest possible form of dark energybecause it is constant in both space and timeand provides good fit to experimental data as of todayL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201614 / 22

The force awakensSupernova CosmologyExpansion history determined using as standard candleany class of objects of known intrinsic brightnessthat can be identified over a wide distance rangeAs light from such beacons travels to Earthexpansion stretches not only distances between galaxy clustersbut also wavelengths of photons en routeRecorded redshift and brightness of each these candlesprovide a measurement of total integrated exansionsince time light was emittedCollection of measurements over sufficient range of distanceswould yield an entire historical record of universe’s expansionL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201615 / 22

The force awakensSupernova CosmologySNe Ia are best cosmological yard sticks in marketThey are precise distance indicatorsbecause they have a uniform intrinsic brightnessdue to similarity of triggering white dwarf mass MCh Mand consequently amount of nuclear fuel available to burnThis makes SNe Ia best (or at least most practical)example of standardizable candles in distant universeL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201616 / 22

The force awakensSupernova CosmologyApparent magnitude m of celestial objectmeasure of its apparent brightness as seen by observer on EarthThe smaller the magnitude the brighter a star appearsMagnitude scale originates in Hellenistic practiceof dividing stars visible to the naked eye into six magnitudesBrightest stars in night sky were said to be m 1faintest were m 6 (limit of human visual perception)Pogson formalized system apparent magnitude in band xm x m x,0 2.5 log10 (F x /F x,0 )(25)Difference in magnitudes m m1 m2can be converted to relative brightnessI2 2.5 mI1L. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201617 / 22

The force awakensSupernova CosmologySupernova CosmologyProjectHigh-Z SupernovaSearch240.001220.010.11Hamuy et al.200.2180.4 120vaumEmMass density pty260.0001OBSERVED MAGNITUDEed magnitudet is plotted fordistant12,13 andby7 type Ia surity, measureme redshift aredshifts beyonds greater thanears), the cosdictions (indirves) begin toing on the asic densities ofm energy. Theresent modelsm energy andnging from thedown to zeros). The best fitsumes a massut rc /3 plus adensity twiceying an accelmic expansion.RELATIVE BRIGHTNESSObserved magnitude (and relative brightness) versus redshift0.20.4REDSHIFT z0.61.00.80.70.60.5LINEAR SCALE OF THE UNIVERSE RELATIVE TO TODAYfor well-measured distant and (in the inset) nearby SNe Iapproach also made it possible to use theBy the end of the year, the error bars began to tighten,escope forlight-curveas bothAstrophysics,groups now andsubmittedpapers with a few moresuL. A.follow-upAnchordoqui(CUNY) observaAstronomy,Cosmology3-29-201618 / 22

The force awakensSupernova CosmologyFaintness (or distance) of high-redshift supernovaecomes as a dramatic surpriseIn (simplest) standard cosmological modelsexpansion history determined entirely by its mass densityThe greater density the more expansion is slowed by gravityIn past high-mass-density universewould have been expanding much faster than it does todayWe shouldn’t have to look far back in time to distant (faint) SNe Iato find given integrated expansion (redshift)Conversely in low-mass-density universewe would have to look farther backBut there is a limit to how low mean mass density could beAfter all we are here and stars and galaxies are hereAll that mass surely puts a lower limit on how far-that isto what level of faintness we must look to find a given redshiftHowever high-redshift SNe Iaare fainter than would be expected even for empty cosmosL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201619 / 22

The force awakensSupernova CosmologyIf these data are correctobvious implication is that three simplest models of cosmologymust be too simpleNext-2-simplest model include expansionary term in eq. of motiondriven by the cosmological constant Λwhich competes against gravitational collapseBest fit to 1998 supernova data impliesin present epoch vacuum energy density ρΛis larger than energy density attributable to mass ρmCosmic expansion is now acceleratingL. A. Anchordoqui (CUNY)Astronomy, Astrophysics, and Cosmology3-29-201620 / 22

The force awakensSupernova Cosmology1.5}Eternal expansion}0.010.110.00010.001RELATIVE BRIGHTNESS OF SUPERNOVAE1.00–20L. A. Anchordoqui (CUNY)REDSHIFT zsdecelerat esteraalwaystfirsionsnpaeraceledlecceaen, thtes0.511.523. . . orEx0.50.0Eventual collapseLINEAR SCALE OF UNIVERSE RELATIVE TO TODAYHistory of cosmic expansion as measured by high-redshift SNe Ia–100Astronomy,Astrophysics,and FROMCosmologyBILLIONSOF YEARSTODAY 103-29-201621 / 22Fehdguecrwvctsrdssedsrfpv