Cosmology ASTR 2120 Sarazin. Hubble Ultra-Deep Field
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1 Cosmology ASTR 2120 Sarazin Hubble Ultra-Deep Field
2 Cosmology - Da Facts! 1) Big Universe of Galaxies 2) Sky is Dark at Night 3) Isotropy of Universe Cosmological Principle = Universe Homogeneous 4) Hubble Expansion of Universe
3 Cosmology - Da Facts! (Cont.) 4) Hubble Expansion of Universe Hubble Law is only homogeneous expanion law Universe is expanding everywhere, not away from a single point
4 Cosmology - Da Facts! (Cont.) 5) Cosmic Microwave Background (CMB) (Penzias & Wilson 1965)
5 Cosmology - Da Facts! (Cont.) 5) Cosmic Microwave Background (CMB) (Penzias & Wilson 1965) Predicted by Gamow in 1940 s Hot Big Bang explained fusion of H & He in early Universe Heat of Big Bang explosion T = ± K (from COBE satellite) MEMORIZE Redshifted today very hot in past
6 Cosmology - Da Facts! (Cont.) 5) Cosmic Microwave Background (CMB) Cold, but everywhere a) Most of photons in Universe b) Most of known heat and free energy in Universe (not including rest mass energy and Dark Energy) c) Most of entropy in Universe Spectrum and intensity = blackbody to 10-5 Agrees with prediction
7 CMB Spectrum COBE Satellite CMB Spectrum
8 CMB Distribution on Sky 1) Extremely Uniform (part in 10 3 ) 2) Dipole due to Doppler shift from Earth s motion Around Sun, Sun around Milky Way, Milky Way and Local Group (630 km/s) 3) Fluctuations at 10-5 level
9 Cosmology - Da Facts! (Cont.) 6) Composition of Universe a) Ordinary, baryonic matter p, n, e, most of observed mass i. Matter, not antimatter ii. Why? Aren t laws of physics symmetric? Mainly H (75% of mass, 90% of atoms) & He (25% of mass, 10% of atoms) b) Photons (CMB, most of free energy?) c) Dark Matter (most of mass, what is it?) d) Dark Energy (come back to later, most of energy?)
10 Composition of Universe
11 Cosmology - Da Facts! 1) Big Universe of Galaxies 2) Sky is Dark at Night 3) Isotropy of Universe Cosmological Principle = Universe Homogeneous 4) Hubble Expansion of Universe 5) Cosmic Microwave Background (CMB) 6) Composition of Universe
12 Cosmology - Standard Models ASTR 2120 Sarazin Positive and Negative Curvature
13 No Dark Energy Work started with Einstein, after development of General Relativity (~1920)
14 No Dark Energy Work started with Einstein, after development of General Relativity (~1920) Contributions from desitter, Lemaitre, Friedmann, others
15 No Dark Energy Gravity decelerates expansion v = H (t) r dh dt 0
16 No Dark Energy Standard Models assume: 1) Cosmological Principle Averaged over a large volume <density>, etc. the same everywhere at the same time Co-moving frame: Measure properties, time in frame moving at <v> for that location (= rest frame of CMB) Cosmic time t: measure in co-moving frame start t = 0 at Big Bang
17 No Dark Energy Standard Models assume: 1) Cosmological Principle History is universal <ρ> = ρ(t) is the same everywhere
18 No Dark Energy Standard Models assume: 1) Cosmological Principle 2) Gravity = General Relativity Gravity due to curvature of spacetime Note: in HW, you show that there is NO solution in Newtonian gravity
19 No Dark Energy Standard Models assume: 1) Cosmological Principle 2) Gravity = General Relativity 3) Only force is gravity Only long range force known
20 No Dark Energy Standard Models assume: 1) Cosmological Principle 2) Gravity = General Relativity 3) Only force is gravity 4) Matter dominates over energy (Mass) c 2 >> Energy Universe is a cold gas of galaxies (now, Dark Matter)
21 No Dark Energy Standard Models assume: 1) Cosmological Principle 2) Gravity = General Relativity 3) Only force is gravity 4) Matter dominates over energy All consistent with all known physics, but... Observed expansion implies Dark Energy (accelerating expansion) - at least one assumption is wrong
22 General Relativity tensor calculus, differential geometry tough mathematics Then, a miracle occurs... Oh no, my head is going to explode!!
23 1) Cosmological Principle A Wonderful Trick a) Every small piece of Universe reflects history and dynamics of whole Universe b) The Universe is spherically symmetric about any point 2) Birkoff s Theorem (General Relativity), & Newton s Theorem Spherical symmetry solution only depends on interior mass Consider a small sphere Throw away the rest of Universe!!
24 A Wonderful Trick (Cont.) Dynamics of entire Universe is contained in the expansion of a small, isolated, uniform density sphere!! 3) Correspondence Principle: General Relativity Newtonian physics if v << c PE << rest mass energy 4) Small sphere: r << d H c / H 0 (~ size of observable Universe) v = H r H 0 r << c GM/r << c 2 General Relativity Newtonian physics M r
25 A Wonderful Trick (Cont.) Dynamics of entire Universe is contained in the Newtonian expansion of a small, isolated, uniform density sphere!! Consider a small test mass m on the surface of the sphere All mass interior, acts like at center, fixed potential Symmetry: motion is r purely radial M m r
26 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. ma = m dv dt = GMm r 2 Conservation of energy : 1 2 mv2 GMm = constant r v 2 2GM = 2 constant r m = constan t $ = Kr 2 o c 2 Let "o" denote present - time values. v 2 = 2GM r Kr o 2 c 2 Great - mass drops out M m r
27 No Dark Energy v 2 = 2GM r The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Kr o 2 c 2 M = 4π 3 r 3 ρ = 4π 3 r o3 ρ o = constant M r v = H(t)r, H o 2 r o 2 = 2G r o H o = v o = H o r o 4π 3 r 3 o ρ o Kr 2 o c 2 8πG 3 ρ o Kc2 Great - r o drops out gets Hubble law correct!
28 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. H o = 8πG 3 ρ o Kc 2 Constant K = 0 if ρ o = ρ crit, where 8πG 3 ρ 2 crit = H o ρ crit 3H 2 o 8πG ' H = 0.95 o * 10 29), ( 71 km/s/mpc+ Interesting - not too far from actual density! 2 gm/cm 3
29 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Define Ω M ρ o ρ crit K = H 2 o c 2 ( Ω M 1) = curvature of space K has units 1/cm 2, K ~ 1 d H 2 (from General Relativity)
30 Curvature - Review K K > K < Plane K = 0 Sphere K > 0 Saddle-shape K < 0 (hyperbolic)
31 No Dark Energy Going back : The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Going back : Kr 2 o c 2 v 2 = 2GM r v = dr dt % ' & dr dt ( * ) 2 M = 4π 3 r 3 o ρ o = 8πGr 3 o ρ o 3r Divide by r o 2 +, - d(r /r o ) dt. / 0 2 = 8πGρ o 3 Kr o 2 c 2 r o r Kc2 8πG 3 ρ = H 2 crit o 8πG 2 3 = H o Ω M ρ o ρ crit ρ o = Ω M ρ crit K = H 2 o c 2 ( ) * d(r /r o ) dt ( Ω M 1) +, - 2 ρ crit 2 r = Ω M H o o r + H 2 o ( 1 Ω M )
32 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. " # $ d(r /r o ) dt % & ' Take square root d(r /r o ) dt 2 2 r = Ω M H o o r + H 2 o ( 1 Ω M ) = H o Ω M (r /r o ) +1 Ω M Equation of Cosmic Dynamics Friedmann Equation Depends only on (r / r o ) small sphere drops out, (r / r o ) = any size or distance in Universe relative to present H o t = t / t H, time scale is Hubble time
33 Metric - Review Distance: τ proper time time measured by observer following path in spacetime s c τ proper distance metric connects changes in s or τ to small changes in coordinates ct r x
34 Robertson-Walker Metric Assuming only Cosmological Principle (spherical coordinates) ( ds) 2 = ( cdt) 2 # % $ dr 1 K r 2 & ( ' 2 ( r dθ) 2 ( r sinθ dφ) 2 K = K(t) curvature of space, function of cosmic time
35 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Returning to the Standard Models and the Equation of Cosmic Dynamics: d(r /r o ) dt = H o Ω M (r /r o ) +1 Ω M Ω M ρ o ρ crit K = H 2 o c 2 ( Ω M 1) Three cases : Ω M =1, Ω M <1, Ω M >1
36 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Ω M =1, K = 0, ρ o = ρ crit d(r /r o ) dt Ω = H M o (r /r o ) +1 Ω = H (r /r M o o ) 1/2 (r /r o ) 1/2 d(r /r o ) = H o dt 2 3(r /r o ) 3/2 = H o t + constant = H o t Cosmic time r = 0 at t = 0 & r ) ( + = 3H t 2/3 & ) o ( + ' r o * ' 2 * Today r = r o t o = ( 2 3)H 1 0 = ( 2 3)t H 1 1
37 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Ω M <1, K < 0, ρ o < ρ crit $ & % r r o ' ) = Ω $ M 1 sinh 2 & ( 1 Ω M % 2 θ ' ) ( H o t = Ω M sinh( θ) θ ( ) 3/2 2 1 Ω M [ ] parametric solution Today ( 2 3)t H < t o t H
38 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Ω M >1, K > 0, ρ o > ρ crit $ & % r r o ' ) = Ω $ M 1 ( Ω M 1 sin2 & % 2 θ ' ) ( H o t = Ω M 2 Ω M 1 ( ) Today t o < ( 2 3)t H θ sin θ 3/2 [ ( )] parametric solution Note: Ω M <1 Ω M >1 with θ iθ
39 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.
40 Expansion The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Ω M < 1 Ω M = 1 Ω M > 1
41 Expansion The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Analogy to ballistic motion on Earth (Mass = constant, sphere and Earth) v < v esc Ω M > 1 v = v esc Ω M = 1 v > v esc Ω M < 1
42 Deceleration Parameter Another notation q = deceleration parameter The image cannot be displayed. Your computer may have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. $ r(t) d 2 r(t) ' & dt 2 ) % ( q(t) 2 $ d r(t) ' % & dt ( ) q o present - time value Standard models w/o Dark Energy q o = Ω M 2 Ω M <1 q o <1 2 Ω M =1 q o =1 2 Ω M >1 q o >1 2
43 Current Age The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Given current expansion: higher mass more deceleration faster expansion in past smaller age
44 Current Age The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Age/Hubble time 2/3
45 Curvature and Shape The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Ω M > 1, K > 0 Uniform positive curvature space Sphere Finite, closed Universe Ω M < 1, K < 0 Uniform negative curvature space Hyperbolic Infinite, open Universe Ω M = 1, K = 0 Flat space, no curvature Infinite, open Universe
46 No Dark Energy The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Which model (if any) is correct?
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