Cosmology & the Big Bang

1. Cosmology & the Big Bang AY16 Lecture 20, April 15, 2008 Mathematical Cosmology, con’t Determination of Cosmological Parameters Inflation & the Big Bang

2. Einstein’s Equations: (dR/dt)2/R2 + kc2/R2 = 8pGe/3c2+Lc2/3 energy density CC 2(d2R/dt2)/R + (dR/dt)2/R + kc2/R2 = -8pGP/c3+Lc2 pressure term

3. And Friedmann’s Equations: (dR/dt)2 = 2GM/R + Lc2R2/3 – kc2  kc2 = Ro2[(8pG/3)ro – Ho]2 if L = 0 (no Cosmological Constant) or (dR/dt)2/R2- 8pGro /3 =Lc2/3 – kc2/R2 which is known as Friedmann’s Equation

4. 4πG Note that if we assume Λ = 0, we have (d2R/dt2)/R = - (ρ + 3P) and in a matter dominated Universe, ρ >> P So we can define a critical density by combining the cosmological equations: ρC = = 3 . 3H02 3 R2 8πG R2 8πG

5. And we define the ratio of the density to the critical density as the parameter Ω ≡ ρ/ρC Fora matter dominated, Λ=0 cosmology, Ω > 1 = closed Ω = 1 = flat, just bound Ω < 1 = open There are many possible forms of R(t), especially when Λ and P are reintroduced. Its our job to find the right one!

6. Λ = 0

9. Some of possible forms are: Big Bang Models: Einstein-deSitter k=0 flat, open & infinite expands Friedmann-Lemaitre k=-1 hyperbolic “ “ “ k=+1 spherical, closed finite, collapses Leimaitre Λ ≠0 k=+1 spherical, closed finite, expands

10. Non-Big Bang Models Eddington-Lemaitre Λ≠0 k=+1 spherical, closed, finite, static then expands Steady State k=0 flat, open, infinite, stationary deSitter k=0 empty, no singularity, open, infinite k = ≡ Radius of Curvature of the Universe H02 (Ω0 – 1) + 1/3 Λ0 c2

11. R(t) E-L A Child’s Garden F-L,0 EdS of Cosmological Models SS,dS L F-L,C t

14. Cosmology is now the search for three numbers: The Expansion Rate = Hubble’s Constant = H0 The Mean Matter Density = Ωmatter The Cosmological Constant = ΩΛ Taken together, these three numbers describe the geometry of space-time and its evolution. They also give you the Age of the Universe.

17. Lookback Time For a Friedmann-Lemaitre Big-Bang Model, the lookback time as a function of redshift is τL = H0-1()for q0=0; Λ=0 = 2/3 H0-1 [1 – (1 + z)-3/2] for q0=1/2, Λ=0 z 1+z

18. The Hubble Constant: H0 = *current* expansion rate = (velocity) / (distance) = (km/s) / (Megaparsecs) named after Edwin Hubble who discovered the relation in 1929.

19. The story of the Hubble Constant (never called that by Hubble!) is the “Cosmological Distance Ladder” or the “Extragalactic Distance Scale” Basically, we need distances & velocities to galaxies and other things. Velocities are easy --- pick a galaxy, any galaxy, get spectrum with moderate resolution, R ~ 1000 (i.e λ/R ~ 5Å) N.B. R = Linear Reciprocal Dispersion, get line centroids to ~ 1/10 R ~ 0.5Å/5000Å ~ 1 part in 104 ~ 30 km/s

20. Spectral features in galaxies

21. Velocity Measurement Radial Velocities (stars, galaxies) now usually measured by cross-correlation techniques pioneered by Simkin (1973), Schechter (1976) & Tonry & Davis (1979). Accuracy depends on Signal-to-Noise and resolution. Typically, for S/N > ~ 20, errors are ~ 10% of Δλ, where (remember) R = λ/Δλ

22. Distances are Hard! Hubble’s original estimates of galaxy distances were based on brightest stars which were based on Cepheid Variables • Distances to the LMC, SMC, NGC6822 & eventually M31 from Cepheids. • Find the brightest stars and assume they’re the same (independent of galaxy type, etc.)

23. Cepheids Pretty Good Distance Indicators --- Standard Candles from the Period-Luminosity (PL) relation: L ≈ P3/2 PLC relation • MV = -2.61 - 3.76 log P +2.60 (B-V) • but ya gotta find them! H0 circa 1929 ~ 600 km/s/Mpc Wrong! 1. Hubble’s galactic calibrators not classical Cepheids. 2. At large distances, brightest stars confused with star clusters. 3. Hubble’s magnitude scale was off.

24. P-L Relation, LMC

25. Cosmological Distance Ladder deVaucouleurs ‘76

26. Cosmological Distance Ladder Find things that work as distance indicators (standard candles, standard yardsticks) to greater and greater distances. Locally: Primary Indicators Cepheids MB ~ -2 to -6 RR Lyrae Stars MB ~ 0 Novae MB ~ -6 to -9

27. Calibrate Cepheids via parallax, moving cluster = convergent point method, expansion parallax Baade-Wesselink, main sequence (HR diagram) fitting. Secondary Distance Indicators Brightest Stars (XX??) Tully-Fisher (+ IRTF) Planetary Nebulae LF Globular Cluster LF

32. Supernovae of type Ia Supernovae of type II (EPM) Fundamental Plane (Dn-σ) Faber-Jackson Surface Brightness Fluctuations Red Giant Branch Tip Luminosity Classes (XXX) HII Region Diameters (XXX) HII Region Luminosities (???)

33. Lemaitre 1927 Hubble 1929 Oort 1932 Baade 1952

34. Tully-Fisher

37. Surface Brightness Fluctuations Tonry & Schneider

39. Baade-Wesselink --- EPM EPM = Expanding Photospheres Method Basically observe and expanding/contracting object at two (multiple) times. Get redshift and get SED. Then L1 = 4πR12σT14 &L2 = 4πR22σT24 and R2 = R1 + v δt (or better ∫ vdt)

41. Fukugita, Hogan & Peebles 1993

46. HST H0 Key Project Team

47. WFPC2 footprint

49. Cepheid Light Curves N1326a

50. Matching P-L RelationsIC4182 (HST) MW (Ground)