1	ASTR 5460, Mon. Nov. 15, 2004 “Cosmology” Combes et al. chapter 13, also Longair (chapters 5-9), plus supplemental sources (Hogg papers, Ned Wright and Wayne Hu webpages, WMAP results) Other: Proposal Project prelims Mini-TAC exercise Tuesday 5pm at Library
2	Supplemental Web Sources
3	Basic Cosmology Assumptions Homogeneity – matter is uniformly spread across the universe on large scales Isotropy – the universe looks the same in all directions, again strictly true on large scales Universality – laws of physics apply everywhere in the universe (being challenged!) These lead to the “cosmological principle” which says that any observer in any galaxy in the universe should see essentially the same features of the universe.
4	Homogeneity and Isotropy Formalism is built on notion that the universe is homogeneous and isotropic, at least on the largest scales. Is it??? Locally the universe is not homogeneous. Is this an issue? Keep in mind. Characteristic gravitational timescale: t_c= 1/(Gρ)^1/2~ 10¹⁰years Also light speed gives us a natural distance scale: L ~ c/(Gρ)^1/2~ 3000 Mpc
5	Homogeneity and Isotropy Can measure isotropy and homogeneity on different scales. ‘Nearby’ using galaxies, at early epochs using the cosmic microwave background radiation (CMBR). Some more on this later, but note that temperature inhomogeneity corresponds to a density inhomogeniety via Einstein effect (through gravitational frequency shift, verified by Pounds and REBKA):
6	Homogeneity and Isotropy Also note that there may be apparent inhomogeneities from the radiation passing through an inhomogeneous, expanding medium, so there is a differential effect.
7	Robertson-Walker Metric Homogeneous and Isotropic universe is described geometrically by a metric, or line element, in GR, for “pressureless dust”. A nice webpage about the Friedman equation and the Robertson-Walker Metric: http://www.jb.man.ac.uk/~jpl/cosmo/friedman.html Textbook (Combes) is alternatively light and heavy on these topics. Anyway, R-W metric:
8	Geometry of the universe R-W metric in spherical coordinates: The curvature k=+1 for spherical, 0 for flat, and -1 for hyperbolic spacetime. Spherical spaces are finite volumes, other geometries may or may not be finite. Note, photon trajectories obey ds²=0. GR has a tensor formalism we won’t get into.
9	Geometry of the universe and redshift z ds²=0 is what is called a “null geodesic” and here we consider the radial case of an observer at r=0: Coordinate of a light source, r1, is then
10	Geometry of the universe and redshift z Now for two events separated by dt1: And from this we get So if a source emits at a frequency ν1=1/dt1
11	Distances (see also Hogg) Many different ways of defining distance Rulers, parallax, angular diameter of object of known size, measurement of apparent brightness of a standard source, radar echo, etc. All do not give the same result in cosmological theory. Can write all the distances as d = z/H₀+O(z²) I suspect the Hogg et al. article will be clearer than the text…or me.
12	Distances in Cosmology Left, illustrating “angular diameter distance” and right, illustrating “luminosity distance.”
13	Angular Distance D_θ = D/θ where D is the real size of the object (e.g., standard ruler) and theta is the apparent angle on the sky (assuming no projection effects). Distance between ends of the ruler is given by D² = -ds² = +R²(t_s)r₁² θ² (see figure) so D = R(t_s)r₁θ, so D_θ = D/θ = R(t_s)r₁where r1 depends on the cosmology as before and is modified for spherical or hyperbolic cosmologies from the flat case (and keep in mind R(t) varies)
14	Luminosity Distance Luminosity distance is the distance that makes the standard flux-luminosity-distance equation work (if you know what the luminosity is, and measure a flux, what is the luminosity distance? Just the flux time 4πD². Now…in terms of the scale factor… Keep in mind cosmological time dilation and spatial expansion, which will give factors of 1+z each.
15	Luminosity Distance Energy received at telescope: Talking about the differential time. Keep in mind cosmological time dilation and spatial expansion, which will give factors of 1+z each.
16	Luminosity Distance Apparent luminosity is:
17	Distances in Cosmology Left, illustrating “angular diameter distance” and right, illustrating “luminosity distance.”
18	Luminosity Distance Be aware we’re talking about bolometric luminosity. Things get more complicated when you keep in mind that frequency changes with redshift. Monochromatic luminosities/fluxes will need to take this into consideration. This effect is related to k-corrections. The other Hogg paper may be useful.
19	Proper Distance This is a more theoretical construct, less subject to observation. It is the “same-time” distance. Point here really is that there are several distances in question, and that distance is not absolute.
20	Microwave Background Temperature Spectrum Best Blackbody in nature!!!
21	Microwave Background Temperature Spectrum Textbook goes through the argument about how conservation of photons per volume element leads to the preservation of a blackbody, albeit at lower temperature.
22	Friedman Equation Homogeneous and Isotropic universe is described geometrically by a metric, or line element, in GR, for “pressureless dust”. Einstein field equations yield the Friedman Equation (no cosmological constant): Term on left is Hubble parameter: or H = 1/R(dR/dt), and k is the curvature term from the R-W metric (-1,0, or +1)
23	Friedman Equation Einstein field equations yield the Friedman Equation (with cosmological constant):
24	Summary of key parameters You should know these. All functions of time, too.
25	A Bunch of Stuff, Equations This will get filled in by Wednesday’s lecture. If desperate, read the text or go to the Longair lecture slides.
26	Scale Factor and Curvature In zero-Λ cosmologies, you get this well known behavior.
27	Scale Factor in non-zero Λ Cosmology, function of k, Ω_o Behavior can be rather complicated, and…
28	Second prediction from Big Bang Model: Abundance of the light elements Big Bang Nucleosynthesis T, r both high enough at start to fuse protons into heavier elements T, r both dropping quickly so only have time enough to fuse a certain amount. Simple models of expansion predict 25% abundance He 25% is the amount of He observed Abundance of ²H, ³He, ⁷Li depends on r_{normal matter} Suggests r_{normal matter} is only 5% of r_critical But we need to also consider “dark matter” and its gravity
29	Primordial Light Element Abundances
30	Neutrino Species
31	Wilkinson Microwave Anisotropy Probe
32	Wilkinson Microwave Anisotropy Probe
33	CMB Power Spectrum
34	Main Tests of the Big Bang Hubble Expansion (not a test really, inspiration) Cosmic Microwave Background Abundance of light elements Refinements of Big Bang Still Being Tested Possible “cosmological constant” Very early history: particle/antiparticle asymmetry “inflation” -- Details of very early very rapid expansion small r, T fluctuations which lead to galaxies
35	Critical points with time running forward 10^-45 sec Quantum gravity? Physics not understood 10^-34 sec 10²⁶ K Nuclear strong force/electro weak force separate (inflation, matter/antimatter asymmetry) 10^-7 sec 10¹⁴K Protons, AntiprotonsÛphotons 10^-4 sec 10¹² K Number of protons frozen 4 sec 10¹⁰ K Number of electrons frozen 2 min Deuterium nuclei begins to survive 3 min 10⁹ K Helium nuclei begin to survive 30 min 10⁸ K T, r too low for more nuclear reactions (frozen number of D, He – critical prediction) 300,000 yr 10⁴ K Neutral H atoms begin to survive (frozen number of photons – critical prediction) ~1 billion yr Galaxies begin to form 13.7 billion yr Present time