Big Bang Cosmology Big Bang vs. Steady State Perfect cosmological principle: universe is unchanging in space and time => Steady-State universe - Bondi, Hoyle, Gold. True? No! • Hubble’s Law => expansion means no steady state, unless matter continually created to preserve density • preference of AGN/quasars for large distances (early times) • cosmic microwave background - consistent with Big Bang (BB) • predominance of light elements (e.g., H, He) consistent with early hot universe • Olbers’ paradox (why is night sky dark?) resolved with BB model Cosmology
Cosmological Principle: At any instant in time, universe is homogeneous (same at all locations) and isotropic (same in all directions), i.e., the universe looks the same to all observers. This despite superclustering on scales up to ~ 100 Mpc; distribution apparently smoother on larger scales.
It turns out that the cosmological principle is completely consistent with the Hubble expansion. Cosmology
If vBA=H0rBA and vCA=H0rCA ,
then vBC= vBA- vCA = H0(rBA- rCA)
\ Hubble’s Law applies to every galaxy if it applies to just one. Cosmological principle <=> Hubble’s Law
where H0 can be positive, negative, or zero. Expansion of the Universe Cosmological principle => no boundary. How to understand this? Answer: General relativity introduces the idea of space-time curvature. Curvature allows us to envision a boundary-free universe that is not infinite. A 3-D universe curved into a 4th dimension. Make analogy to a 2-D universe on surface of a 3-D sphere. If the sphere expands in 3-D, the 2-D surface area expands. A 2-D observer on the surface infers Hubble’s Law.
Expansion and subsequent contraction of a hypothetical 2-D universe on the surface of a sphere. Expansion of the Universe In GR cosmology, 3-D space itself expands. All lengths, e.g., distance between galaxies, wavelength of light, etc. expand with the universe. However, this expansion can be opposed locally by various forces.
Therefore, expansion of space itself is the real explanation for cosmological redshifts z, not the Doppler effect. General Relativity Gravity understood in a new light. Compare with old view. Newton: Gravitational force causes matter to accelerate. Matter exerts gravitational force. Einstein: Gravitational acceleration is due to curved space-time. Curvature of space-time due to mass-energy (recall E = mc2). Geometry of Space and Time Newton: Euclidean geometry where ds2 = dx2 + dy2 + dz 2 is invariant, i.e., absolute space (and time).
Einstein (SR): No absolute space or time, but
ds2 = c2dt2 - (dx2 + dy2 + dz2 ) is invariant. Einstein (GR): Curvature of space time due to mass-energy yields
4 4 2 m n dx1,2,34==dx,,dydz, and dxcdt. ds = å å gmn dx dx where m =1 n =1
gmn is a tensor containing information about curvature. Specifically, where no curvature, get back to SR limit
gmn = 0 for m ¹ n and g11 = g22 = g33 = -1, g44 = 1. Some Effects of Space-Time Curvature
• deflection of light around massive object, e.g., “gravitational lensing” • Euclidean geometry not valid on large scale • Large-scale structure and evolution of universe affected by curvature Expansion of Curved Universe Theory of General Relativity yields an equation for radius of curvature R, 1 2GM 1 R& 2 - = - c2 2 3p R 2
Solutions of this equation, R(t), yield evolution of the universe.
Our measurements of H0 yield current value of R& / R. Expansion of the Universe Follow a simpler Newtonian model. Imagine expansion of a spherical region of radius R(t). GM (R) F = ma Þ R&& = - . R2 Multiply by R&. GM (R) d æ 1 ö d æ GM (R) ö R&R&& = - R& Þ ç R& 2 ÷ + ç- ÷ = 0 R2 dt è 2 ø dt è R ø 1 GM (R) Þ R& 2 - = E = constant, 2 R
i.e., conservation of energy. Expansion of the Universe Does the universe expand forever? Analogy to earlier escape velocity calculation. Universe unbound (open) if KE > |PE|, i.e., E > 0. marginal (flat) if KE = |PE|, i.e., E = 0. bound (closed) if KE < |PE|, i.e., E < 0. Rewrite in terms of Hubble constant and density: v = R& = HR, M (R) = 4 3p R3r, and note that r = r(t), H = H(t). 2 Open universe => r < r crit º 3H 8p G.
2 Flat universe => r = rcrit º 3H 8p G. 2 Closed universe => r > rcrit º 3H 8p G.
At current epoch (t = t0), we measure r0 , H0. Expansion of the Universe Key question in cosmology: 8p G 8p G What is the value of W = r rcrit = r 2 = r0 2 ? 3H 3H0 Important parameters r (1) W = r crit R& (2)expansion parameter H(t) = , currently H R 0 R&&R 1 r (3)deceleration parameter q(t) = - 2 = , currently q0 R& 2 rcrit Current status of W: Observed luminous matter W << 1. Observed matter and inferred dark matter W £ 0.2. Theory of early universe W = 1. Age of the Universe
-1 Earlier, we argued t0 < H0 if universe decelerating.
-1 Solve expansion equations for W = 1 => find t0 = 2/3 H0 .
-1 -1 \ open universe W <1 Þ 2 3 H0 < t0 < H 0 -1 flat universe W =1 Þ t0 = 2 3H0 closed universe -1 W > 1 Þ 0 < t0 < 2 3H 0 Ultimate fate? If W £1, expansion continues; all stars eventually die, > 1012 yr; universe becomes dark.
If W > 1, recontraction of universe; followed by rebound? Olbers’ Paradox In an infinite static universe, every line of sight eventually intercepts a star => night sky is everywhere bright!
Resolution in Big Bang model:
Finite age => can’t see beyond a distance r = cDt. Also, light from within this distance is increasingly redshifted as we approach the edge, the “cosmic event horizon”. Light Elements Can trace expansion back to an early hot dense state. At high energies, particles exist in an unbound state. As universe expands and cools, synthesis of elements, then atoms. Given presence of protons (1H) and neutrons, light elements 2H, 3He, 4He, 6Li, 7Li produced in early universe - these elements are also not produced efficiently in stars.
Cosmic abundances: 75% H, 25% He, trace Li, Be are all explained by Big Bang model. High H, He content implies a high temperature past, since such matter prefers less binding energy, more light elements. Cosmic Microwave Background Back in time, at z ~ 103 (when T ~ 3000 K), electrons and protons combine to form H atoms => matter is no longer opaque to radiation, since free electrons were good at absorbing photons.
Blackbody radiation from this epoch flies out unhindered by matter. Should see this relic radiation, but redshifted so that l T max,0 = =1+ z »103 First observed by Penzias & Wilson lmax T0 (1965). Newer data from COBE T satellite (1992). Þ T = » 3 K. 0 1+ z Note: we can see galaxies/quasars back to z < 5, but CMB comes from z ~ 103! Cannot “see” any further back. Cosmic Microwave Background COBE’s measurement of the CMB spectrum. T = 2.726± 0.005 K.
COBE all-sky map of CMB. See fluctuations DT ~10-5 T which could have lead to supercluster structure. Seeing Through the Distance Extrapolating to Earliest Phases Before z ~ 103, guided only by theory. However, cannot go back arbitrarily far. Limit of current knowledge: Gm Gravity => can’t detect events within DL ~ . c2 h h Quantum Mechanics => can’t observe within DL < = . Dp mc Equate two DL’s. 1/2 photons GM h æ hc ö Planck mass, a combination of 3 ~ Þ m = mp = ç ÷ , c 2 mc è G ø fundamental constants. Gm 1/2 Also, h p æ Gh ö Lp = = 2 = ç 3 ÷ , Planck length. mpc c è c ø 1/ 2 Lp æ Gh ö Planck time. Plug in #’s => t = = ç ÷ , p c c5 -43 è ø t p =1.35´10 s. Can’t describe t £ t p . Big Bang Model Expansion from highly condensed initial state. Theory combines general relativity and particle physics.
Four fundamental forces: strong nuclear - weak nuclear - electromagnetic - gravity
electroweak at high energies
combines at even higher energies
? Nucleons composed of quarks. Quarks composed of …? All particles have corresponding antiparticles. Big Bang Model Brief history: time
-43 10 s Planck time tp. Don’t know what precedes this. Need a quantum theory of gravity. 10-35 s Strong nuclear force decouples from electroweak. Inflation begins - rapid exponential growth. Most quarks and antiquarks annihilate. Small asymmetry => some quarks remain. Baryon (made of quarks) to photon ratio 10-9-10-10. 10-32 s Inflation ends. Observable universe went from 10-23 cm to 10 cm. 10-12 s Weak and electromagnetic force separate.
10-6 s Nucleons form. Big Bang Model Brief history: time 102 -103 s Cosmic nucleosynthesis - light nuclei form, e.g., He, Li.
1013 s (z ~ 103) electrons and protons combine => atoms form. Photons now able to stream freely. 1016 s Galaxies, stars, planets begin to form. 1016 s The present.
1040 s Protons decay (perhaps). Atomic matter ceases to exist. Universe heads toward darkness/heat death. Last Word Observed luminous matter W < 0.01. Dark matter W » 0.2. Inflation theory W = 1. Where is the rest of the mass-energy? (1) In matter? Nucleons (e.g., brown dwarfs, white dwarfs, planets, rocks) can only account for up to W = 0.2, according to cosmic nucleosynthesis arguments. So look for exotic particles: massive n’s, axions, supersymmetric particles. (2) In energy? Dark energy may make up the required deficit. An
unknown energy so that Weff= W + L = 1, where L is the cosmological constant (dark energy term) that makes the Hubble expansion accelerate at the current epoch! Recent evidence supports this hypothesis.