1Angular DiametersGalaxy’s diameter is proper distance linear diameter: but must use R(te)Using ϖcoordinate Î Looks like Euclidean result, regardless of curvature of space.θR(te)ϖRW metric:What is angular size of galaxy at co-moving distance ϖ?dt = dϖ= dφ= 0DMore angular diameterIn practice(because of that @#$% cosmological constant)[CO Fig. 29.30]29.193)121)(1()1(0002200−+−−+=zqqzqzqcDHθFor Λ= 0:LdzD2)1( +=θSurprise!Even for flat, Λ= 0 universe, θfirst decreases but then increases with increasing z.Gal.1Gal.2Gal.3Gal.3 at large lookback timeCompetingEffects:• Distance• Expansionθc/H0Dθc/H0D2Kellerman (1993) Gurvits (1994) VLBI measurements of compact radio sources:Gurvits, Kellerman & Frey (1999)Authors say “consistent with”qo= 0.5, no evolution.qo0.50.Just sources with flat radio spectrum + high radio LRedshift ÆAng. Size ÆAng. Size ÆAng. Size ÆRedshift ÆRedshift Æθc/H0Dqo1.00.50.20.1SSqo2.51.00.50.1SS1/zThe Concordance Cosmology= ΛCDM = LCDMDark EnergyColdDark MatterConcordance between:• CMB fluctuations.• Supernovae.• Galaxy cluster growth rate• Globular cluster ages• Power spectrum of large-scale structure.•H0: HST key project vs. WMAP.• Baryon density: primordial nucleosynthesisvs. WMAP.• Ωmfrom Ωbaryon× (ρdark matter /ρbaryon)3ΩBaryons•d, 7Li, 3He ÎΩB= = 0.02 – 0.05 • But better determination now from CMB fluctuations (WMAP)ΩB= 0.044ρB,oρc,o[CO Fig 29.14]Consistentwith obs.Big Bang Nucleosynthesis (Oct. 14 lecture)The Concordance Cosmology= ΛCDM = LCDMDark EnergyColdDark MatterclosedflatopenoodtdRHdtdRRH==1HConcordance= ΛCDM4Definitions, results, etc.PhysicsGtHtπ8)(3)(ρ2c=)(ρ)(ρ)(tttc=ΩP = wu = wρc2dU = -PdV2cu=ρ0,4,3,RRRorormomΛΑ−−===ρρρρρρMatter:Radiation:Dark energy:r = R(t) ϖdtdRRH1=222ρπ381kcRGdtdRR−=−Per unit mass:K.E. + potential E. = Total EnergyDensities:Temp. of radiation field:Curvature 21ℜ=kCosmological Constant(a.k.a. Dark Energy)+1x 0-1[][]222/)(/)()()(dttdRdttRdtRtq −=**= you should be able to write these down from
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