1 / 46

Exploring Cosmological World Models

Delve into diverse cosmological models from Friedmann equations discussing density parameters, universe classifications, and universe evolution rates.

marilyns
Download Presentation

Exploring Cosmological World Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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：

More Related