1192020 Chris Pearson Fundamental Cosmology 6 Cosmological world

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 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 1

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 1: Cosmological World Models • What describes a universe ? • COSMOLOGICAL WORLD MODELS We want to classify the various cosmological models R W=0 W<1 W=0 Open universe expands forever W<1 Open universe expands forever W>1 Closed universe collapses W>1 W=1 Closed universe limiting case t from Friedmann eqn. Defined the density parameter Matter 1 Cosmological Constant 2 Curvature 3 W 2

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 1: Cosmological World Models L=0 World Models Lets think about life without Lambda 3

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 2: Curvature Dominated World Models • The Milne Universe Friedmann Equation • 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 R Age: to=Ho-1 Useful model for • Universes with Wm<<1 • open Universes at late times t 4

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 3: Flat World Models • General Flat Models Friedmann Equation • Flat, k=0 universe • Assume only single dominant component • r= ro(Ro/R)3(1 -w) • W=1 Solution Age of Universe 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 5

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 3: Flat World Models • The Einstein De Sitter Universe Friedmann Equation • Flat, k=0 universe • Matter dominated r= ro(Ro/R)3 • No Cosmological Constant : L=0, WL=0 • W=1 Universe expands uniformly and monotomically but at an ever decreasing rate: R Age: W=0 W=1 to=(2/3 Ho ) • Radiation dominated r= ro(Ro/R)4 • Radiation dominated to=(1/2 Ho ) t=0 Until relatively recently, the Ede. S Universe was the “most favoured model” Ho-1 to t 6

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 4: Matter Curvature World Models Matter + Curvature Friedmann Equation 1 Matter Dominated 2 2 3 4 5 COSMOLOGICAL WORLD MODELS 1 Density Parameter 3 Curvature Density Parameter 4 Curvature & Matter 5 6 6 Equation for the evolution of the scale factor independent of the explicit curvature 7

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 4: Matter Curvature World Models The Einstein Lemaitre Closed Model Friedmann Equation • 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: R(q), t(q) are characteristic of a cycloid parameterization cos(q), sin(q) are Circular Parametric Functions Cycloid Parametization: • q=0 t=0 • q=p d. R/dt=0 Rmax • Universe will contract when q=2 p t For the case of W=2, the Universe will be at half lifetime at maximum expansion 8

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 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 W=0 R • High W Age universe decreases (start point gets closer to Origin) W=4 W=2 • Universe evolves faster for higher values of W W=10 Ho-1 to t 9

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 4: Matter Curvature World Models The Einstein Lemaitre open Model Friedmann Equation • 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: R(f), t(f) are characteristic of a hyperbola parameterization cosh(f), sinh(f) are Hyperbolic Parametric Functions Hyperbolic Parametization: • f=0 t=0 • f R • Universe will become similar and similar to the Milne Model ( W=0) as t 10

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 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: W=0. 2 R • Models normalized at tangent to Milne Universe at present time W=0. 4 • Low W Age universe increases (start point gets farther from origin) Oldest universe is Milne Universe • Universe evolves faster for lower values of W Ho-1 to t 11

11/9/2020 COSMOLOGICAL WORLD MODELS 6. 4: Matter Curvature World Models Summary 1. 0 Einstein De Sitter 0. 5 Open Cosmologies Age (Ho-1) • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 Closed Cosmologies 0 1 2 4 W 0 6 8 12

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 4: Matter Curvature World Models Summary Universe Type Parameters Fate k W r q L Milne Special Relativistic Open -1 0 0 Expand forever Friedmann Lemaitre open Hyperbolic Open -1 <1 < rc < 0. 5 0 Expand Forever Einstein De Sitter Flat Closed 0 =1 = rc = 0. 5 0 t R Friedmann Lemaitre closed Spherical Closed 1 >1 > rc > 0. 5 0 Topology R/t Re-contract Big Crunch 13

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: • L COSMOLOGICAL WORLD MODELS World Models L 0 World Models Lets think about life with Lambda 14

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: • L COSMOLOGICAL WORLD MODELS World Models Living with Lambda 1 The Friedmann Equations including Cosmological Constant L 2 • Modifies gravity at large distances • Repulsive Force (L>0) • Repulsion proportional to distance (from acceleration eqn. ) For a static universe There exists a critical size RC (=RE) where Friedmann eqns =0 1 2 Consider the following scenarios • The Einstein Static Universe • L < 0 universes • L > 0 universes • k < 0, k=0 • L > LC • L ~ LC • L < LC 15

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: • L COSMOLOGICAL WORLD MODELS World Models The Einstein Static Universe • (k=+1, r>0, L>0) For a static universe The is possibility of a Static Universe with R=Rc=RE, L=Lc for all t R Einstein Static Model • original assumed solution to field equations • Problem: • no big bang • no redshift RE 0 t 16

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: • • L COSMOLOGICAL WORLD MODELS World Models Oscillating Models (L < 0) To ensure a real When R=RC universe contracts R RC • Universe is Oscillatory • Oscillatory independent of L<0 k 0 t 17

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: • L COSMOLOGICAL WORLD MODELS World Models The De Sitter Universe • (k=0, r=0, L>0) For k 0 Monotomically expanding Universe, at large R Special Case k = 0, r=0 De Sitter Model R De Sitter L > 0 RC • Does have a Big Bang • But is infinitely old L<0 0 t 18

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: L COSMOLOGICAL WORLD MODELS 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 19

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: L COSMOLOGICAL WORLD MODELS World Models k=+1 , L>0 Eddington Lemaitre Models • (k=+1, L ~ LC= LC+e) 3 Models • Special Case Einstein Static Model R 1 Eddington Lemaitre 1 (EL 1) : • Big Bang • R Einstein Static Universe as t RC L = LC (EL 2) L = LC (EL 1) • k=+1, R= finite, can see around the universe !! • Ghost Milky Way 0 • (normally light doesn’t have time to make this journey inside the horizon) 2 t Eddington Lemaitre 2 (EL 2) : • Expands gradually from Einstein Static universe from t =- • Becomes exponential • No Big Bang (infinitely old) : But there exists a maximum redshift ~Ro/Rc 20

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: L COSMOLOGICAL WORLD MODELS World Models k=+1 , L>0 Lemaitre Models • (k=+1, L ~ LC= LC+e) 3 Models • 3 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 RC 0 L = LC + e t Long Coast period Concentration of objects at a particular redshift (1+z=Ro/Rcoast) (c. f. QSO at z=2) 21

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: L COSMOLOGICAL WORLD MODELS 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 R 1, R 2 (R 1<R 2) for which no real solutions exist No solution R 1<R<R 2 because (d. R/dt)2<0 R 1 R>R 2 : Bounce Solution R<R 1 : Oscillatory Solution • Universe expands to a maximum size • Contracts to Big Crunch R 2 R 1 R<R 1 : Oscillatory Solution 2 R>R 2 : Bounce Solution 0 • Initial Contraction from finite R (universe is infinitely old) • Bounce under Cosmic repulsion there exists a maximum redshift ~Ro/Rmin • Expands monotomically t 22

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: L COSMOLOGICAL WORLD MODELS World Models • Summary of L models Name Dynamics Evolution L k <0 >0 0 De Sitter monotomically expanding LC 1 Einstein Static t at R=RE with L= Lc >LC 1 LC+e 1 Eddington Lemaitre (EL 1) Big Bang Einstein Static universe LC+e 1 Eddington Lemaitre (EL 2) expand from Einstein Static LC+e 1 Lemaitre Long coasting period at R=RE 0<L < LC 1 Oscillatory (1 st kind) contract back to R=0 (oscillatory) 0<L < LC 1 Oscillatory (2 nd kind) Universe bounces at RB k Oscillatory (1 st kind) contract back to R=0 (oscillatory) monotomically expanding 23

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: L • Summary of L models COSMOLOGICAL WORLD MODELS World Models 3 BOUNCE MODEL 2 ST A CO WL, 0 L E OD M COLD DEATH 1 k= 0 0 k=+1 k=-1 BIG CRUNCH -1 0 1 Wm, 0 2 24

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 6. 5: Summary of L models L COSMOLOGICAL WORLD MODELS World Models L >LC De Sitter Einstein Static Eddington Lemaitre 1 (EL 1) Eddington Lemaitre 2 (EL 2) Lemaitre Oscillatory (1 st kind) Oscillatory (2 nd kind) - Bounce L = LC L <LC 0 R 1 R 2 R • L <0 models all have a “big crunch” • L >0 models depenent on k • Expansion to if k 0 : L becomes dominant • k>0 and L > 0 multiple solutions. • Our Universe……. ? 25

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 6: Alternative Cosmologies 変な宇宙論 There a lot of strange theories out there ! 26

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 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 Observers At All Times Density of Matter = constant continuous creation of matter at steady rate / volume 27

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 6: Alternative Cosmologies Steady State Cosmology Metric for De Sitter Model The metric given by : Curvature : 3 D Gaussian (k/R(t)2) dependent on t if k 0 Additional term in General Relativity field Equations Creation of matter!! ~ 10 x mass found in galaxies Intergalactic Hydrogen at creation rate ~10 -44 kg/m 3/s Problems: 8 Magnitude-Redshift Relation qo=-1 De Sitter Model • qo=-1 not consistent with observation. 8 Galaxy Source Counts 8 2. 7 K Cosmic Microwave Background Moreover no evolution is permitted Corresponding N(S) slope flatter < -1. 5 Inconsistent with observation ~~~ No explanation 28

11/9/2020 • Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 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 • Stars L G 7 G as as t continents drift apart. t stars brighter in the past. • Earth is moving away from the Sun if • G G as t T t 9 n/4 inconsistent with Earth history G(t) Perturbations in moon & planet orbits (constraints (d. G/dt)/G<3 x 10 -11 yr-1 ) • Light Elements Abundance (d. G/dt)/G<3 x 10 -12 yr-1 29

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 6: Alternative Cosmologies • Changing Gravitational Constant • Variation on the variation of Brans & Dicke Cosmology G Theory • As well as the Gravitational Tensor field there is an additional Scalar field • L=0, Mach Principle G-1~Sm/rc 2 G(t) w= coupling constant between scalar field and the geometry Such that Grt 2 = constant Diracs original 1937 theory w=-2/3 nucleosynthesis w>100 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/G 0 are shown. Copi et al. Astroph/0311334 Analysis of lunar data for Nordtvedt effect w>29 d. G/dt)/G<10 -12 yr-1 30

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 6: Alternative Cosmologies • Other Cosmological Theories • 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 -23 s) creation of particle pairs at expense of gravitational energy But: initial fluctuations HAVE been observed and explainations are available ! 31

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • What Kind of Universe do we live in then ? Lets think about Our Universe 32

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 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. accelerating empty critical 33

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 1: Supernova Cosmology Project 34

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 6: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 2: Hubble Key Project Mould et al. 2000; Freedman et al. 2000 H 0 = 71 6 km s-1 Mpc-1 t 0 = 1. 3 1010 yr 35

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 6: Our Universe - The Concordance Model • What Universe do we live in ? • Evidence 2: Hubble Key Project H 0 = 71 6 km s-1 Mpc-1 t 0 = 1. 3 1010 yr 36

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • • What Universe do we live in ? Evidence 3: WMAP Wilkinson Microwave Anisotropy Probe (2001 at L 2) Detailed full-sky map of the oldest light in Universe. It is a "baby picture" of the 380, 000 yr old Universe Red - warm Blue - cool fundemental 1 st harmonic • 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 37

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 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 ~ 1 sq. 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) 38

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS Scale Factor (Size) • • • What Universe do we live in ? Evidence 3: WMAP maps and geometry Wm WL 0. 3 0. 7 0. 3 0 1 0 2 0 http: //map. gsfc. nasa. gov/ 6. 7: Our Universe - The Concordance Model W=1 W<1 W=1 W>1 t 0 time 39

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • • What Universe do we live in ? Evidence 4: WMAP +SDSS Tegmark et al. 2003 40

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • What Universe do we live in ? • Approximately Flat (k=0) q CMB measurements • WL=0. 6 -0. 7 q Type Ia supernovae • There is also evidence that Wm~0. 3 q Structure formation, clusters • H 0=72 km s-1 Mpc-1 q Cepheid distances HST key program • Currently matter dominated Concordance Model Wtot = 1. 0 WL= 0. 7 Wm=0. 3 Wb=0. 02 H 0=72 km s-1 Mpc-1 k=0, L>0 41

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • The Evolution of the Concordance Model - The Evolution of Our Universe L>0, k = 0 Monotonic expansion t Universe De Sitter Universe • Early times Universe is decelerating • Later times L dominates Universe accelerates 42

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • The Evolution of the Concordance Model - The Evolution of Our Universe e Th re he d an lg(R) w no Matter Dominated Dark Energy Dominated Why do we live at a special epoch ? ? tr=m lg(t) tm=L t 0 43

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 7: Our Universe - The Concordance Model • The Evolution of the Concordance Model - The Evolution of Our Universe http: //map. gsfc. nasa. gov/ 44

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 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 L 0 Models L = 0 Models W=0 W<1 W=1 lg(R) R Concordance Model W>1 lg(t) t Parameters of Concordance Model Wtot = 1. 0 WL= 0. 7 Wm=0. 3 Wb=0. 02 H 0=72 km s-1 Mpc-1 k=0, L>0 45

11/9/2020 Chris Pearson : Fundamental Cosmology 6: Cosmological world Models ISAS -2003 COSMOLOGICAL WORLD MODELS 6. 8: SUMMARY • Summary Fundamental Cosmology 6. Cosmological World Models 終 Fundamental Cosmology 7. Big Bang Cosmology 次： 46