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Fundamental Cosmology: 6. Cosmological World Models

Fundamental Cosmology: 6. Cosmological World Models. “ This is the way the World ends, Not with a Bang , But a whimper ” T.S. Elliot. R. W =0. W <1. W =0 Open universe expands forever. W =1. W <1 Open universe expands forever.

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Fundamental Cosmology: 6. Cosmological World Models

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  1. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Fundamental Cosmology: 6.Cosmological World Models “ This is the way the World ends, Not with a Bang, But a whimper ” T.S. Elliot

  2. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 R W=0 W<1 W=0Open universe expands forever W=1 W<1 Open universe expands forever W>1Closed universe collapses W>1 W=1Closed universe limiting case 3 2 1 t Matter Cosmological Constant Curvature 6.1: Cosmological World Models • What describes a universe ? • We want to classify the various cosmological models from Friedmann eqn. Defined the density parameter W

  3. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.1: Cosmological World Models • L=0 World Models Lets think about life without Lambda

  4. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 R t 6.2: Curvature Dominated World Models Friedmann Equation • The Milne Universe • Special relativistic Universe • negliable matter / radiation : r~0, Wm<<1 • No Cosmological Constant : L=0, WL=0 • Curvature, k=-1 Universe expands uniformly and monotomically : Rt Age: to=Ho-1 • Useful model for • Universes with Wm<<1 • open Universes at late times

  5. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Solution Age of Universe 6.3: Flat World Models Friedmann Equation • General Flat Models • Flat, k=0 universe • Assume only single dominant component • r= ro(Ro/R)3(1-w) • W=1 • For spatially flat universe : • Universes with w>-1/3 - Universe is younger than the Hubble Time • Universes with w<-1/3 - Universe is older than the Hubble Time

  6. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Universe expands uniformly and monotomically but at an ever decreasing rate: W=0 R W=1 • Radiation dominated r= ro(Ro/R)4 • Radiation dominated to=(1/2Ho ) t=0 t Ho-1 to 6.3: Flat World Models Friedmann Equation • The Einstein De Sitter Universe • Flat, k=0 universe • Matter dominated r= ro(Ro/R)3 • No Cosmological Constant : L=0, WL=0 • W=1 Age: to=(2/3Ho ) Until relatively recently, the EdeS Universe was the “most favoured model”

  7. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Density Parameter 3 Friedmann Equation Curvature Density Parameter 4 Matter Dominated Curvature & Matter 5 5 2 2 6 6 1 1 3 4 Equation for the evolution of the scale factor independent of the explicit curvature 6.4: Matter Curvature World Models • Matter + Curvature

  8. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 R(q), t(q) are characteristic of a cycloid parameterization cos(q), sin(q) are Circular Parametric Functions Cycloid Parametization: • q=p  dR/dt=0  Rmax • Universe will contract when q=2p  t 6.4: Matter Curvature World Models Friedmann Equation • The Einstein Lemaitre Closed Model • Closed, k=+1 universe • Matter dominated r= ro(Ro/R)3 • No Cosmological Constant : L=0, WL=0 • W>1 The Scale Factor has parametric Solutions: • q=0  t=0 For the case of W=2, the Universe will be at half lifetime at maximum expansion

  9. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 W=0 R W=2 W=4 W=10 t Ho-1 to 6.4: Matter Curvature World Models • The Einstein Lemaitre Closed Model • Closed, k=+1 universe • No Cosmological Constant : L=0, WL=0 • W>1 Age: • Models normalized at tangent to Milne Universe at present time • High W Age universe decreases (start point gets closer to Origin) • Universe evolves faster for higher values of W

  10. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 R(f), t(f) are characteristic of a hyperbola parameterization cosh(f), sinh(f) are Hyperbolic Parametric Functions Hyperbolic Parametization: 6.4: Matter Curvature World Models Friedmann Equation • The Einstein Lemaitre open Model • Open, k=-1 universe • Matter dominated r= ro(Ro/R)3 • No Cosmological Constant : L=0, WL=0 • W<1 The Scale Factor has parametric Solutions: • f=0  t=0 • f R  • Universe will become similar and similar to the Milne Model (W=0) as t

  11. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 W=0 W=0.2 R W=0.4 t to Ho-1 6.4: Matter Curvature World Models • The Einstein Lemaitre open Model • Open, k=-1 universe • No Cosmological Constant : L=0, WL=0 • W<1 Age: • Models normalized at tangent to Milne Universe at present time • Low W  Age universe increases (start point gets farther from origin)Oldest universe is Milne Universe • Universe evolves faster for lower values of W

  12. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 1.0 Einstein De Sitter Age (Ho-1) 0.5 0 8 6 1 2 4 W0 6.4: Matter Curvature World Models • Summary Open Cosmologies Closed Cosmologies

  13. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.4: Matter Curvature World Models • Summary

  14. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.5: L World Models • L  0 World Models Lets think about life with Lambda

  15. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 For a static universe 2 2 1 1 There exists a critical size RC (=RE) where Friedmann eqns =0 6.5: L World Models • Living with Lambda The Friedmann Equations including Cosmological Constant L • Modifies gravity at large distances • Repulsive Force (L>0) • Repulsion proportional to distance (from acceleration eqn.) Consider the following scenarios • The Einstein Static Universe • L < 0 universes • L > 0 universes • k < 0, k=0 • L > LC • L ~ LC • L < LC

  16. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 The is possibility of a Static Universe with R=Rc=RE,L=Lc for all t R RE Einstein Static Model 0 t 6.5: L World Models • The Einstein Static Universe • (k=+1, r>0, L>0) For a static universe • original assumed solution to field equations • Problem: • no big bang • no redshift

  17. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 To ensure a real R RC L < 0 0 t 6.5: L World Models • Oscillating Models • (L < 0) When R=RC universe contracts • Universe is Oscillatory • Oscillatory independent of k

  18. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 R De Sitter L > 0 RC L < 0 0 t 6.5: L World Models • The De Sitter Universe • (k=0, r=0, L>0) For k  0 Monotomically expanding Universe, at large R De Sitter Model Special Case k = 0, r=0 • Does have a Big Bang • But is infinitely old

  19. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.5: L World Models • k=+1 , L>0 World Models • (k=+1, L> LC) For k=+1, L> LC Monotomically expanding Universe, at large R  De Sitter Universe

  20. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Special Case R L = LC (EL2) Eddington Lemaitre 1 (EL1) : • Big Bang • R Einstein Static Universe ast  • k=+1, R= finite, can see around the universe !! • Ghost Milky Way • (normally light doesn’t have time to make this journey inside the horizon) 2 1 L = LC (EL1)  RC Einstein Static Model 0 t Eddington Lemaitre 2 (EL2) : • Expands gradually from Einstein Static universe from t =- • Becomes exponential • No Big Bang (infinitely old) : But there exists a maximum redshift ~Ro/Rc 6.5: L World Models • k=+1 , L>0 Eddington Lemaitre Models • (k=+1, L ~ LC= LC+e) 3 Models

  21. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Lemaitre Models : • Long Period of Coasting at R~Rc • Repulsion & Attraction in balance • Finally repulsion wins and universe expands • Lemaitre Models permit ages longer than the Hubble Time (Ho-1) R 3 L = LC +e RC 0 t 6.5: L World Models • k=+1 , L>0 Lemaitre Models • (k=+1, L ~ LC= LC+e) 3 Models Long Coast period  Concentration of objects at a particular redshift (1+z=Ro/Rcoast) (c.f. QSO at z=2)

  22. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 R R>R2 : BounceSolution R<R1: OscillatorySolution • Universe expands to a maximum size • Contracts to Big Crunch R2 2 1 R1 R<R1 : OscillatorySolution R>R2: BounceSolution • Initial Contraction from finite R (universe is infinitely old) • Bounce under Cosmic repulsion  there exists a maximum redshift ~Ro/Rmin • Expands monotomically 0 t 6.5: L World Models • k=+1 , L>0 Oscillatory and Bounce Models • (k=+1, 0<L < LC) 2 sets of solutions for 0<L < LC separated by R1, R2 (R1<R2) for which no real solutions exist No solution R1<R<R2because (dR/dt)2<0

  23. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.5: L World Models • Summary of L models

  24. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 3 BOUNCE MODEL COAST MODEL 2 WL,0 1 k=0 0 BIG CRUNCH -1 Wm,0 0 1 2 6.5: L World Models • Summary of L models COLD DEATH k=+1 k=-1

  25. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 De Sitter Einstein Static Eddington Lemaitre 1 (EL1) Eddington Lemaitre 2 (EL2) Lemaitre Oscillatory (1st kind) Oscillatory (2nd kind) - Bounce L >LC L = LC L <LC 0 R R2 R1 6.5: L World Models • Summary of L models • L<0models all have a “big crunch” • L>0 models depenent on k • Expansion toif k 0 : Lbecomes dominant • k>0 andL > 0 multiple solutions. • Our Universe…….?

  26. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.6: Alternative Cosmologies • 変な宇宙論 There are a lot of strange theories out there !

  27. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.6: Alternative Cosmologies • Steady State Cosmology • Bondi & Gold 1948 (Narliker, Hoyle) • 1948 Ho-1 = to< age of Galaxies • 注意 Steady State  Static Recall: PERFECT COSMOLOGICAL PRINCIPAL The Universe appears Homogeneous & Isotropic to all Fundamental ObserversAt All Times Density of Matter = constant  continuous creation of matter at steady rate / volume

  28. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003  Metric for De Sitter Model The metric given by : Additional term in General Relativity field Equations  Creation of matter!!  qo=-1  De Sitter Model • Magnitude-Redshift Relation • qo=-1 not consistent with observation. Moreover no evolution is permitted Corresponding N(S) slope flatter < -1.5 Inconsistent with observation • Galaxy Source Counts ~~~ No explanation • 2.7K Cosmic Microwave Background 6.6: Alternative Cosmologies • Steady State Cosmology Curvature : 3D Gaussian (k/R(t)2)  dependent on t if k0 ~ 10x mass found in galaxies  Intergalactic Hydrogen at creation rate ~10-44 kg/m3/s Problems:

  29. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.6: Alternative Cosmologies • Changing Gravitational Constant • Milne, Dirac, Jordan (Brans & Dicke, Hoyle & Narliker) • G decreases with time • e.g. Earth’s Continents fitted together as Pangea Gas t continents drift apart. • Stars LG7Gas t  stars brighter in the past. • Earth is moving away from the Sun if Gas t  Tt9n/4inconsistent with Earth history • G(t) Perturbations in moon & planet orbits (constraints (dG/dt)/G<3x10-11 yr-1 ) • Light Elements Abundance (dG/dt)/G<3x10-12 yr-1

  30. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Observational limits and theoretical expections for D/H versus . The one (light shading) and 2 (dark shading) sigma observational uncertainties for D/H and are shown. They do not appear as ellipses due to the linear scale in D/H but logarithmic uncertainties from the observations. The BBN predictions are shown as the solid curves where the width is the 3% theoretical uncertainties. Three different values of GBBN/G0 are shown. Copi et al. Astroph/0311334 6.6: Alternative Cosmologies • Changing Gravitational Constant Brans & Dicke Cosmology • Variation on the variation of G Theory • As well as the Gravitational Tensor field there is an additional Scalar field G(t) • L=0, Mach Principle G-1~Sm/rc2 • = coupling constant between scalar field and the geometry Such thatGrt2 = constant Diracs original 1937 theory w=-2/3 • nucleosynthesis  w>100 •  Analysis of lunar data for Nordtvedt effect  w>29 dG/dt)/G<10-12 yr-1

  31. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.6: Alternative Cosmologies • Other Cosmological Theories • Anisotropic Cosmologies Anisotropic Cosmologies : • Universe is homogeneous and isotropic on the largest scales (CMB) • Obviously anisotropic on smaller scales  Clusters • Quiescent Cosmology • Universe is smooth except for inevitable statistical fluctuations that grow • Chaotic Cosmology (Misner) • Whatever the initial conditions, the Universe would evolve to what we observe today • Misner - neutrinos damp out initial anisotropies • Zeldovich - rapidly changing gravitational fields after Planck time (10-43-10-23s)  creation of particle pairs at expense of gravitational energy But: initial fluctuations HAVE been observed and explainations are available !

  32. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.7: Our Universe - The Concordance Model • What Kind of Universe do we live in then ? Lets think about Our Universe

  33. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 accelerating empty critical 6.7: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 1: Supernova Cosmology Project • Type Ia supernovae : Absolute luminosity depends on decay time  "standard candles” • Apparent magnitude (a measure of distance) • Redshifts (recession velocity). • Different cosmologies - different curves.

  34. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.7: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 1: Supernova Cosmology Project

  35. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Mould et al. 2000; Freedman et al. 2000 H0 = 716 km s-1 Mpc-1 t0 = 1.3  1010 yr 6.6: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 2: Hubble Key Project

  36. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 H0 = 716 km s-1 Mpc-1 t0 = 1.3  1010 yr 6.6: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 2: Hubble Key Project

  37. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Red - warm Blue - cool fundemental 1st harmonic 6.7: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 3: WMAP Wilkinson Microwave Anisotropy Probe (2001 at L2) Detailed full-sky map of the oldest light in Universe. It is a "baby picture" of the 380,000yr old Universe • Temperature fluctuations over angular scales in CMB correspond to variations in matter/radiation density • Temperature fluctuations imprinted on CMB at surface of last scattering • Largest scales ~ sonic horizon at surface of last scattering • Flat universe this scale is roughly 1 degree (l=180) • Relative heights and locations of these peaks  signatures of properties of the gas at this time

  38. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.7: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 3: WMAP http://map.gsfc.nasa.gov/ • WMAP - fingerprint of our Universe • Flat Universe - sonic horizon ~ 1sq. Deg. (l=180) • Open Universe - photons move on faster diverging pathes => angular scale is smaller for a given size • Peak moves to smaller angular scales (larger values of l)

  39. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 WL Wm W=1 0.3 0.7 W<1 0.3 0 W=1 1 0 Scale Factor (Size) W>1 2 0 t0 time 6.7: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 3: WMAP • WMAP maps and geometry http://map.gsfc.nasa.gov/

  40. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.7: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 4: WMAP +SDSS Tegmark et al. 2003

  41. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Concordance Model Wtot= 1.0 WL= 0.7 Wm=0.3 Wb=0.02 H0=72 km s-1 Mpc-1 k=0,L>0 6.7: Our Universe - The Concordance Model • What Universe do we live in ? • Approximately Flat (k=0) • CMB measurements • WL=0.6-0.7 • Type Ia supernovae • There is also evidence that Wm~0.3 • Structure formation, clusters • H0=72 km s-1 Mpc-1 • Cepheid distances HST key program • Currently matter dominated

  42. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003  Monotonic expansion tUniverse  De Sitter Universe 6.7: Our Universe - The Concordance Model • The Evolution of the Concordance Model - The Evolution of Our Universe L>0, k = 0 • Early times Universe is decelerating • Later times L dominates Universe accelerates

  43. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Dark Energy Dominated The here and now Matter Dominated lg(R) tm=L tr=m t0 lg(t) 6.7: Our Universe - The Concordance Model • The Evolution of the Concordance Model - The Evolution of Our Universe Why do we live at a special epoch ??

  44. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.7: Our Universe - The Concordance Model • The Evolution of the Concordance Model - The Evolution of Our Universe http://map.gsfc.nasa.gov/

  45. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 R W=0 W<1 W=1 L = 0 Models L  0 Models Concordance Model W>1 lg(R) t lg(t) Wtot= 1.0 WL= 0.7 Wm=0.3 Wb=0.02 H0=72 km s-1 Mpc-1 k=0,L>0 Parameters of Concordance Model 6.8: SUMMARY • Summary • Used the Friedmann Equations to derive Cosmological Models depending on the density W • Have discovered a large family of cosmological World Models

  46. Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6.8: SUMMARY • Summary 終 Fundamental Cosmology 6. Cosmological World Models Fundamental Cosmology 7. Big Bang Cosmology 次：

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