Clusters are Very Large Magnets U NM December 1, 2009
Carilli & Taylor 2002 (ARA&A) B ~ 10 µg
Hydra A Faraday Rota<on Measures B ~ 30 µg
Coma X-ray + radio 6
Results B 0 = 4.6 +/- 0.7 µg scale lengths between 2 and 30 kpc average B ~ 2 µg Bonafede et al. 2009 7
Astronomy 422 Lecture 19: Newtonian Cosmology
Key concepts: Scale factor - expansion of the universe Age of the universe Lookback time
Cosmology The origin and evolution of the Universe. Recall two concepts: Olbers's paradox and the Cosmological Principle. Olbers's paradox: Why is the sky dark at night? If the Universe is spatially infinite infinitely old and not evolving then the night sky should be bright.
Solution: The Universe has finite age, so the light from most stars have not had time to reach us yet.
The Cosmological Principle Modern cosmology based on the assumption that the Universe is Homogeneous Isotropic These assumptions are the cosmological principle. These tenets seem to hold on large scales (>100 Mpc), but definitely not on small scales
Hubble's law a natural outcome of this? Definition of redshift, given from spectra. The Hubble constant Luminosity distance In observational astronomy, the term recessional velocity, V is used. At lowz:
NB: To determine the distance d to a quasar with z>1, it is NOT correct to use the Doppler formula and then Hubble's Law relating the recessional velocity V to d. V is not really the radial velocity of the quasar, it is the expansion velocity of space time. A cosmological model is required to relate z to d.
The Doppler shift is a linear effect due to the relative motion of two objects The cosmological redshift is due to stretching of the photon wavelength due to the expansion of the Universe. At small redshifts, these agree, but not at high redshifts (not Euclidean any longer!) Recessional velocity depends on the scale factor (more later). So at z = 3, R(t 0 )/R(t em ) = 4, universe was ¼ of the size it is today
Naïve interpretation: age of the Universe The most naïve interpretation is that there is an explosion at a single point with us at the center, and no acceleration of deceleration. It t is the time passed between moment of Big Bang and now, then r=vt is the distance of any object traveling at speed v. Hubble's Law if t=1/h Thus, this simple interpretation gives current age of the Universe as t=1/h 0 = 14Gyrs.
Problems with this interpretation Needs us to be at the center of explosion What happens at large r, when Hubble's Law predict v>c? Deceleration produced by gravitation of all the matter contained in galaxies? Adequate resolution of these difficulties requires GR combined with the cosmological principle. Cosmological principle illustrated as a raisin cake. On average the universe is isotropic and homogeneous when viewed at any instant by any observer in a typical galaxy.
Newtonian Cosmology Birkhoff's rule: For a homogeneous density distribution ρ(t) the effect of gravitation on a galaxy m at a small distance r from an observer in any other galaxy can be computed by Newtonian physics considering only the mass M r within radius dr. r
Anticipating GR, we will write k is a constant with units length -2, related to curvature of space in GR. = constant, describing the shell radius at current time t 0, = r(t 0 ) k will determine the fate of the Universe, basically three cases.
Expansion will stop, the Universe is bound. Expansion continues forever, the Universe is unbound. Expansion stops at t=.
Now let us introduce the scale factor, R(t). Recall r(t) is the coordinate distance. Same for all shells = constant as shell expands, called the comoving coordinate. Now, R(t 0 )=1. During the expansion, M r = constant: = present density
Relating R(t) to the Hubble Law:
Now use the expressions for r(t) and v(t) into the expression for the total energy: Expansion Law of the Universe
Critical Density Dividing line between expansion and eventual contraction. : E>0 Critical: E=0 Bound: E<0
Set k=0 in the equation for the total energy:
Comments on actual density and critical density The critical density is ~29 orders of magnitude smaller than the density of water - but 20-25 times larger than the density of luminous matter if this matter was spread out uniformly throughout space. For the Universe to be critical, there has to be a lot of unseen matter or energy (dark matter/dark energy?) Empirical evidence for dark matter concentrated in halos of galaxies appears in rotation curves of spiral galaxies. a 1oo I A=1 I Rich clusters (med) a Morgan groups (med) LI Hickson groups (med) Q CFA groups (med) A x-ruy group, jf The Local Group *M101, M31, MilkyWay * Spirals (med) 0 Ellipticals (med) 0 Cor Bor supercluster I Shapley supercluster X Cosmic virial theorem I Least action method f Virgo inlall (range) : Bulk flows (range) 0.01 0.1 1 10 Characteristic size (Mpc)
Density parameter At present: Using the definitions we have for the Expansion Law gives: Useful later in GR: A flat universe (no acceleration case).
Flatness problem: Behavior of Ω with redshift: Ω = 1 + (Ω 0-1)/(1+ Ω 0 z) This presents us with the flatness problem. Ω has to be tuned very close to 1 in the early universe so that we can observe it close to 1 now.
Age for a critical universe R vs. t, for Use Expansion law: Integrate to get R: Use and to get: age of universe
k > 0: R oscillates between 0 and R max k < 0: R increases forever C&O give the expressions for R. Verify that they are solutions to expansion law!
Redshift and scale factor As the universe expands, the photon wavelength grows: True for any
Age of universe observed at redshift z For k=0 - see C&O for k>0 and k<0. for example: z=1: t=0.24t h = 3 Gyr z=6: t=0.035t h = 500 Myr z= 50: t=0.002t h = 26 Myr Note also Ned Wright s cosmology calculator at: http://www.astro.ucla.edu/~wright/cosmocalc.html
Lookback time How far back in time are we looking? For k=0, plugging in for t 0, t(z): C&O show, for any k, that at large z: expansion law same, regardless of k.
Difficulties with Newtonian Cosmology Arbitrary nature of Birkhoff's rule (in context of Newtonian theory) to account for 'local' effects of gravitation. Globally we cannot avoid the issue of galaxies separating from each other at very high speeds. E=0, the critical case, is preferred. Otherwise, we would need to explain from where the energy of the expansion came? Then, the age of the Universe is 9Gyr, shorter than the age of oldest stars in the Milky Way. We will need relativistic cosmology (next time).
Next time: Chapter 29.2-4