Cosmology primer The Cosmological principle The standard model: (The Robertson-Walker metric & General Relativity) Newtonian derivation of Friedmann eqns Proper, angular, and luminosity distances Derivation of the proper distance The Cosmological principle • Axiom: “The Universe is both homogeneous and isotropic” Same in all locations Same in all directions Laws of Physics Universal • Supporting evidence: Cosmic Microwave Background Deep galaxy counts Large scale galaxy surveys Page 1 The Cosmic Microwave Background from WMAP year 5 data. 2dFGRS large scale galaxy structure north and south Page 2 Homogeneity and Isotropy homogeneous isotropic not isotropic not homogeneous Universal expansion is the only motion allowed by Cosmological Principle Standard Cosmological Model A metric system for defining the geometry = spacetime-metric A census of the key contents= vacuum energy, matter, radiation A model for how the contents modify the geometry = gen. rel. Observational Cosmology: Can measure geometry and use model to constrain contents: - Standard candles - Baryon Acoustic Oscillations - CMB Anisotropy Peak Can obtain direct measurements of contents (matter, radiation): - Mass density via nearby studies - Radiation density via CMB Can use orthogonal evidence: - Age of galaxy, stars, Earth < Age of Universe - Abundances via Big Bang Nucleosynthesis Page 3 Density - Evolution - Geometry ρ < ρc Open ρ =ρ € c Flat Closed ρ >ρ c Page 4 Big Bang Nucleosynthesis Results: Page 5 The Robertson-Walker metric Space-time interval/metric Normal spatial polar coords 2 2 2 2 2 dr 2 2 2 2 ds = c dt − R (t) 2 + r (dθ + sin θdφ ) 1− k r Light travel distance Curvature Term (open, closed, flat) Expansion factor € This metric is forced by the Cosmological Principle (universal expansion) Spacetime interval for a photon = 0 (from special relativity) If we consider photons only on radial paths in a flat Universe (k=0): c 2dt 2 = R2(t)dr2 € Einstein’s General Relativity • Spacetime geometry tells matter how to move – gravity = effect of curved spacetime – free particles follow geodessic trajectories – ds2 > 0 v < c time-like massive particles – ds2 = 0 v = c null massless particles (photons) – ds2 < 0 v > c space-like tachyons (not observed) • Matter (+energy) tells spacetime how to curve – Einstein field equations – nonlinear – second-order derivatives of metric with respect to space/time coordinates Page 6 Einstein Field Equations Gµν = 8π G Tµν + Λ gµν 2 µ ν gµν = spacetime metric ( ds = gµν dx dx ) Gµν = Einstein tensor (spacetime curvature) G = Netwon's gravitational constant Tµν = energy - momentum tensor Λ = cosmological constant € The Friedmann eqns: • Adopt standard GR G = 8πGT + Λg • Assume µν µν µν – Homogeneity and Isotropy, i.e., uniform fluid – Adiabatic expansion (lossless!) δE = −pδV € Hubble constant or H(t) > Friedmann eqns: • R 8πGρ kc 2 Λ = Cosmological constant Λ = − 2 + = vacuum energy € R 3 R 3 = dark energy = “Greatest blunder” •• R 4πG 3p Λc 2 = Inflation = − ρ + + = Useful Fudge factor? R 3 c 2 3 € Page 7 Newtonian Analogy m • 2 G M m 4π E = R − , M = R3ρ 2 R 3 m • 2 8π G 2E R R = ρ R2 + 3 m ρ Friedmann equation: 2 € € • 8π G ρ + Λ R = R2 − k c 2 3 Λ 2E same equation if ρ → ρ + , → −k c2 8π G m € € Newtonian Analogy m G M m E = R˙ 2 − 2 R 2 G M V = esc R E > 0 V > Vesc R → ∞ V∞ > 0 € E = 0 V = Vesc R → ∞ V∞ = 0 E < 0 V < Vesc R → 0 € Page 8 Critical Density • Derive using Newtonian analogy: escape velocity : 3 2 2 2 G M 2 G 4π R ρ 8π G R ρ Vesc = = = R R 3 3 Hubble expansion : V = R˙ = H R R 0 ρ critical density : 2 Vesc 8π G ρ ρ = 2 ≡ V 3 H ρc 0 € 2 8πGρ 3 H0 ρ Subscript, o ρc = , ΩM = = 2 = today’s values 8π G ρc 3H € Cosmological Distances • Proper distance (dp) – Actual distance to object today (if one could stop time and measure it with a ruler). • Angular diameter distance (da) – Proper distance at time of photon emission. • Luminosity distance (dl) – Distance the photon feels it has travelled. d d = (1+ z)d = l p a (1+ z) Why? € Page 9 Proper distance • co-moving radial distance 0 ro – now ( when photon received ): dz= 0 = R(t0 ) ro = R0 ro – when photon emitted: dz= 0 € to R(t ) R r d d = R(t ) r = e 0 o = z= 0 z e o R 1 z € 0 + te € dz λ R(t ) ( 1+ z) = observed = o λlaboratory R(te ) • Proper distance (for flat universe):€ dp = dz= 0 = Roro • Angular diameter distance: € d d = R r = p a e o (1+ z) € € Angular diameter distance Angles frozen at emission. If structure bound need to Use distance at emission to convert angular sizes to r physical sizes θ r = da tan(θ) Proper distance Angular diameter distance € For non bound objects use proper distances to work out tangential sizes Page 10 Luminosity Distance N h νe – Luminosity ( erg s-1 ) L = 0 ro Δte – area of photon sphere 2 A = 4π dP t0 – redshift: Δto λo = λe (1+ z) € € νo = νe (1+ z) te Δte – time dilation:€ lower photon arrival rate € € Δto = Δte (1+ z) – observed€ flux ( erg cm-2 s-1 ) € N h ν L € L F = o = = A Δt 4 d 2 1 z 2 4π d 2 € o π P ( + ) L • Luminosity distance = (1+ z) Sources look d L d P fainter. € € Surface Brightness 2 • Solid angle Ω = A/ D A • Surface brightness D A A – Flux per solid angle (erg s-1 cm-2 arcsec-2) 2 F L DA L Σ ≡ = 2 = 4 Ω 4π DL A 4π A (1+ z) – decreases very rapidly with z because: – expansion spreads out the photons – decreases their energy € – decreases their arrival rate Page 11 Angular Diameter Distance ΩΛ ΩM € ΩM +ΩΛ = 1 € ΩΛ = 0 € € Evaluating the Proper Distance ro tr cdt d = R r = R dr = R (From metric : cdt = Rdr) p o o o ∫ o ∫ R o te Want dp in terms of z and H(1+z) (z dependent Hubble constant) : R d R d o R dR x =1+ z = 0 = R = − o = −(1+ z)H dt R dt R2 dt −dx i.e., dt = x H(x) 1+z cdx dp = Ro ∫ 1 xH(x) Now need expression for H in terms of (1+z) € Page 12 Evaluating the Proper Distance From Friedmann eqn : 8πGρ 8πGρ kc 2 Λ H 2 = M + R − + 3 3 R2 3 2 Set to todays values and divide by Ho : 2 8πGρM ,o 8πGρR,o kc Λ 1= 2 + 2 − 2 2 + 2 3Ho 3Ho Ro Ho 3Ho Simplify by defining : ΩM ,o,ΩR,o,ΩΛ,o,Ωo 2 2 2 2 2 kc = Ro Ho (ΩM ,0 + ΩR,0 + ΩΛ,0 −1) = Ro Ho (Ω0 −1) 2 2 Sub for kc term in Fr. eqn. and divide by Ho : 2 2 H 8πGρM 8πGρR Ro Λ 2 = 2 + 2 − 2 (Ω −1) + 2 Ho 3Ho 3Ho R 3Ho € Evaluating the Proper Distance 3 Matter scales with z as: ρz = ρo (1+ z) (think of a box) 4 Radiation scales withz as: ρz = ρo(1+ z) (think of a box +loss of energy) Cosmological constant does not scale with z Ro Sub for densities and use Ω notation and recall (1+ z) = R : 2 2 3 4 2 ⇒ H = Ho (ΩM ,0 (1+ z) + ΩR,0 (1+ z) − (Ω0 −1)(1+ z) + ΩΛ,0) Proper distance : t0 1+z 1+z R0 c dx c dx € dp = R0 ro = c dt = = ∫ R(t) ∫ x H(x) H ∫ 4 3 2 te 1 0 1 ΩR,0 x + ΩM ,0 x + ΩΛ,0 + (1− Ω0 ) x Depends only on redshift plus Ho plus contents (Ω’s) € Page 13 Page 14 .