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Cosmology & Culture Astronomy 233 Winter 2009 Physical Cosmology Week 4 Distances and BBN Start Joel Primack University of California, Santa Cruz Picturing the History of the Universe: The Backward Lightcone Big Bang Cosmic Horizon: tangent to backward lightcone at Big Bang From E. Harrison, Cosmology (Cambridge UP, 2000). Picturing the History of the Universe: The Backward Lightcone tangent to backward lightcone at big bang From E. Harrison, Cosmology (Cambridge UP, 2000). Distances in an Expanding Universe Proper distance = physical distance = dp time (our time (galaxy worldline) worldline) t1 dp(t0) = (physical distance at t0) = a(t0) re = re t0 χ(te) = (comoving distance of galaxy dp(t0) = re emitting at time te) re t0 1 2 χ(te) = ∫ dr = re = c∫ dt/a = c∫ da/(a H) χ 0 t a e e t because e dt = (dt/da) da = (a dt/da) da/a = da/(aH) dp(te) dp(te) = (physical distance at te) = a(te) re = ae re -1 Cosmic Horizon: The Hubble radius dH = c H0 = tangent to backward -1 -1 = 4.29 h70 Gpc = 13.97h70 Glyr lightcone at Big Bang -3/2 For E-dS, where H = H0 a , 1/2 From E. Harrison, Cosmology (te) = re = dp(t0) = 2dH (1-ae ) Big Bang χ (Cambridge UP, 2000). 1/2 dp(te) = 2dH ae (1-ae ) Distances in reception distance an Expanding in units of dH Universe E-dS -1 Hubble radius dH = c H0 = dp(t0) -1 = 4.29 h70 Gpc -1 = 13.97h70 Glyr emission distance dp(te) in units of dH -3/2 For E-dS, where H = H0 a , 1/2 E-dS χ(te) = re = dp(t0) = 2dH (1-ae ) 1/2 dp(te) = 2dH ae (1-ae ) Barbara Ryden, Introduction to Cosmology (Addison-Wesley, 2003) Our Particle Horizon 2 2 2 2 2 2 2 2 2 2 FRW: ds = -c dt + a(t) [dr + r dθ + r sin θ dφ ] for curvature K=0 so √grr =a(t) Particle Horizon dp(horizon) = (physical distance at time t0) = a(t0) rp = rp rhorizon t0 1 2 dp(horizon) = ∫ dr = rhorizon = c∫ dt/a = c∫ da/(a H) 0 0 0 -3/2 For E-dS, where H = H0 a , 1/2 rhorizon = lim 2dH (1-ae ) = 2dH = ae➝0 tangent to backward -1 -1 lightcone at big bang = 8.58 h70 Gpc = 27.94 h70 Glyr For the Benchmark Model with h=0.70, rhorizon = 13.9 Gpc = 45.2 Glyr. For the WMAP5 parameters h = 0.70, M = 0.28, K = 0, rhorizon = 14.3 Gpc = 46.5 Glyr. Velocities in an Expanding Universe The velocity away from us now of a galaxy whose light we receive with redshift ze, corresponding to scale factor ae = 1/(1 + ze), is v(te) = v(t0) = H0 dp(t0) The velocity away from us that this galaxy had when it emitted the light v(t0) = we receive now is v(te) = He dp(te) The graph at right shows v(t0) and v(te) for the E-dS cosmology. E-dS cosmology -3/2 For E-dS, where H = H0 a , 1/2 v(t0) = H0 dp(t0) = 2c (1-a ) v(te) = He dp(te) -3/2 1/2 -1/2 = H0 ae ae 2c (1-a )/H0 = 2c (a - 1) Velocities in an Expanding Universe From E. Harrison, Cosmology (Cambridge UP, 2000). ● ● Y Distances in an Expanding Universe Angular Diameter Distance D From the FRW metric, the distance D across a source at comoving distance r = re dθ which subtends an angle dθ = θ1 - θ2 is D = a(t) r dθ, or dθ = D/[ a(t) r] . The angular diameter distance dA is defined by dA = D/dθ, so dA = a(te) re = re/(1+ze) = dp(te) . This has a maximum, and dθ a minimum. Barbara Ryden, Introduction to Cosmology (Addison-Wesley, 2003) For the Benchmark Model redshift z D↔1 arcsec max at z=1.6 0.1 1.8 kpc 0.2 3.3 0.5 6.1 1 8.0 2 8.4 max at z=1.25 3 7.7 4 7.0 6 5.7 Distances in an Expanding Universe In Euclidean space, the luminosity L of a source at distance d is related to the apparent luminosity l by l = Power / Area = L / 4πd2 The luminosity distance dL is defined by 1/2 dL = (L / 4πl) . Weinberg, Cosmology, pp. 31-32, shows that in FRW 2 l = Power/Area = L / 4πdL fraction of photons reaching unit area at t0 2 2 2 2 2 2 = L [a(t1)/a(t0)] / [4πdp(t0) ] = L a(t1) / 4πr1 = L / 4πr1 (1+z1) Thus (redshift of each photon)(delay in arrival) 2 dL = r1/a(t1) = r1 (1+z1) = dp(t0) (1+z1) = dA (1+z1) E-dS Barbara Ryden, Introduction to Cosmology (Addison-Wesley, 2003) redshift z → Summary: Distances in an Expanding Universe 2 2 2 2 2 2 2 2 2 2 FRW: ds = -c dt + a(t) [dr + r dθ + r sin θ dφ ] for curvature K=0 so √grr =a(t) r1 t0 adding distances at time t1 χ(t1) = (comoving distance at time t1) = ∫ dr = r1 = ∫ dt/a 0 t1 d(t1) = (physical distance at t1) = a(t1)χ(t1) = a1 r1 χ(t1) = (comoving distance at time t0) = r1 r1 t0 dp = (physical distance at time t0) = a(t0) rp = rp since a(t0) =1 From the FRW metric above, the distance D across a source t1 at comoving distance r1 which subtends an angle dθ is D=a(t1) r1 dθ. The angular diameter distance dA is a1 r1 defined by dA = D/dθ, so dA = a(t1) r1 = r1/(1+z1) In Euclidean space, the luminosity L of a source at distance d is related to the apparent luminosity l by χ(t ) = (comoving distance at time t ) = r l = Power/Area = L/4πd2 1 0 p 1/2 so the luminosity distance dL is defined by dL = (L/4πl) . Weinberg, Cosmology, pp. 31-32, shows that in FRW 2 2 2 -1 2 l = Power/Area = L [a(t1)/a(t0)] [4πa(t0) r1 ] = L/4πdL Thus fraction of photons reaching unit area at t0 dL = r1/a(t1) = r1 (1+z1) (redshift of each photon)(delay in arrival) Distances in an Expanding Universe FRW: ds2 = -c2 dt2 + a(t)2 [dr2 + r2 dθ2 + r2 sin2θ dφ2] for curvature k=0 =a(t1) for k=0 r1 adding distances at time t1 χ(t1) = (comoving distance at time t1) = ∫ dr √grr = a(t1) r1 0 = (comoving distance at time t ) = r [since a(t )=1] t0 χ 0 1 0 χ From the FRW metric above, the distance D across a t1 source at distance r1 which subtends an angle dθ is D=a(t1) r1 dθ. The angular diameter distance dA is χ(t1) defined by dA = D/dθ, so dA = a(t1) r1 = r1/(1+z1) In Euclidean space, the luminosity L of a source at distance d is related to the apparent luminosity l by l = Power/Area = L/4πd2 1/2 so the luminosity distance dL is defined by dL = (L/4πl) . Weinberg, Gravitation and Cosmology, pp. 419-421, shows that in FRW = Power/Area = L [a(t )/a(t )]2 [4πa(t )2 r 2]-1 = L/4πd 2 l 1 0 fraction0 1 of photons reachingL unit area at t Thus 0 (redshift of each photon)(delay in arrival) dL = r1/a(t1) = r1 (1+z1) Distances in a Flat (k=0) Expanding Universe upper curves: Benchmark Model lower curves: Einstein - de Sitter Scott Dodelson, Modern Cosmology (Academic Press, 2003) Distances in the Expanding Universe Dnow = proper distance, DL = luminosity distance, DA = angular diameter distance, Dltt = c(t0 – tz) http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH Distances in the Expanding Universe: Ned Wright’s Javascript Calculator H0DL(z=0.83) =17.123/13.97 =1.23 http://www.astro.ucla.edu/~wright/CosmoCalc.html See also David W. Hogg, “Distance Measures in Cosmology” http://arxiv.org/abs/astro-ph/9905116 Neutrino Decoupling and Big Bang Nucleosynthesis, Photon Decoupling, and WIMP Annihilation These will be the next topics covered, although we will probably only be able to start them Wednesday. See Ch. 3 of Dodelson and/or Ch. 3 of Weinberg. Neutrino Decoupling and Big Bang Nucleosynthesis, Photon Decoupling, and WIMP Annihilation Börner, Early Universe 4th Ed, p. 152 Big Bang Nucleosynthesis BBN was conceived by Gamow in 1946 as an explanation for the formation of all the elements, but the absence of any stable nuclei with A=5,8 makes it impossible for BBN to proceed past Li. The formation of carbon and heavier elements occurs instead through the triple-α process in the centers of 2 -29 2 3 -3 red giants (Burbidge , Fowler, & Hoyle 57). At the BBN baryon density of 2×10 Ωb h (T/T0) g cm ≈ 2 ×10-5 g cm-3, the probability of the triple-α process is negligible even though T ≈ 109K. time time Kolb & Turner Thermal equilibrium between n and p is maintained by weak interactions, which keeps n/p = exp(-Q/T) (where Q = mn–mp = 1.293 MeV) until about t ≈ 1 s. But because the neutrino mean free time -1 2 3 -5 -5 -2 tν ≈ σν ne± ≈ (GFT) (T ) is increasing as tν ∝T (here the Fermi constant GF ≈10 GeV ), while the -½ -2 horizon size is increasing only as tH ≈ (Gρ) ≈ MPl T , these interactions freeze out when T drops below about 0.8 MeV. This leaves n/(p+n) ≈ 0.14.
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