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Early Models and Expanding Universe: Exploring Cosmology and Subsequent Developments

Delve into the historical evolution of cosmic models from Friedmann to Lemaître, understanding cosmological parameters, consequences of universe geometry, and the dynamics of an expanding universe. Explore key concepts such as dark energy, curvature of space, and the implications of Einstein's theories on the expanding universe, all discussed in this informative astronomy seminar.

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Early Models and Expanding Universe: Exploring Cosmology and Subsequent Developments

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  1. Early models of an expanding Universe Paramita Barai Astr 8900 : Astronomy Seminar 5th Nov, 2003

  2. Contents • Introduction • Discuss papers : • 1922 : Friedmann • 1927 : Lemaître • 1932 : Einstein & De Sitter • Present cosmological picture • Some results • SN project, WMAP, SDSS

  3. Cosmological foundations • Cosmological principle • Universe is Homogeneous & Isotropic on large scales (> 100Mpc) • Universe (space itself) expanding, dD/dt ~ D (Hubble Law) • Universe expanded from a very dense, hot initial state (Big Bang) • Expansion of universe – mass & energy content – explained by laws of GTR Dynamics of universe • Structure formation in small scales (<10-100 Mpc) by gravitational self organization • WHAT IS THE GEOMETRY OF OUR UNIVERSE, & IT’S CONSEQUENCES ??

  4. Cosmological parameters • R – Scale factor of Universe • Critical density , C – density to make universe flat (it just stops expanding) • Density parameter,  =  / C • H = Hubble constant = v / r •  = Cosmological Constant (still speculative!!) • Dark Energy • Repulsive force, opposing gravity

  5. Curvature of space • Positive curvature – Closed – contract in future •  > C •  > 1 • Zero curvature – Flat – stop expansion in future & stationary •  = C •  = 1 • Negative curvature – Open – expand forever •  < C •  < 1

  6. Timeline • 1905 – Einstein’s STR, 1915 – GTR • 1917 – Einstein & De Sitter static cosmological models with  • 1922 – Friedmann • First non-static model • Universe contracts / expands (with ) • 1927 – Lemaître – expanding universe • 1930 – Hubble: expanding universe, Einstein drops  (“biggest blunder”) • 1932 – Einstein & de Sitter • Expanding universe of zero curvature

  7. Timeline – cont’d… • 1948 – Particle theory (QED) predicts non zero vacuum energy , but QED = 10120other • 1965 – CMBR • Early 1980’s: LUM << C Open universe • 1980’s – • Inflation theory  Flat universe (TOT = 1) • Dark matter • 1990’s - LUM~ 0.02-0.04, DARK ~ 0.2-0.4, REST= ? • 1998 – Accelerating universe • Present model – universe very near to flat (with matter and vacuum energy)

  8. Matter density = 0 Advantage : Explains naturally observed radial receding velocities of extra galactic objects From consequence of gravitational field Without assuming we are at special position Parameters c = velocity of light  = Cosmological constant  = Density of universe Two first models of universe:De Sitter

  9. Non zero matter density Relation between density & radius of universe  masses much greater than known in universe at that time Can’t explain receding motion of galaxies Advantage Explains existence of matter Parameters  = Einstein constant = 1.8710-27 (cgs) Einstein universe

  10. Curvature of space Aleksandr Friedman Zeitschrift fur Physik 10, 377-386, 1922

  11. Summary • First non static model of universe • Work immediately not noticed, but found important later … • R independent of t : • Stationary worlds of Einstein & de Sitter • R depends on time only : • Monotonically expanding world • Periodically oscillating world • depending on  chosen

  12. Goal of the paper • Derive the worlds of Einstein & de Sitter from more general considerations

  13. Assumptions of 1st class • Same as Einstein & de Sitter • Gravitational potentials obey Einstein field equations with cosmological term • Matter is at relative rest

  14. Assumptions of 2nd class • Space curvature is constant wrt 3 space coordinates; but depends on time • Metric coefficients: g14, g24, g34 = 0, suitable choice of time coordinate

  15. R(x4) = 0 M = M0 = constant Cylindrical world Einstein’s results M = (A0x4+B0) cos x1 Transform x4  De Sitter spherical world (M=cos x1) Solutions: Einstein & de Sitter worlds as special cases Stationary world

  16. R(x4)  0 M = M(x4) But – suitable x4 – M = 1 Non stationary world

  17. R ( > 0 ) Increases with t Initial value, R = R0 (>0) at t = t0 R = 0, at t = t t = Time since creation of world Monotonic world of first kind  > 4c2/9A2

  18. Time since creation of world, t R increases with t Initial R = x0 x0 & x0 are roots of equation: A-x+(x3/3c2) = 0 Monotonic world of second kind 0 <  < 4c2/9A2

  19. R – periodic function of t World Period = t Periodic World t  if   Small , approximate  - <  < 0

  20. Possible universes of Friedmann • Monotonic worlds •  > 4c2/9A2 • First kind • 0 <  < 4c2/9A2 • Second kind • Periodic universe • - <  < 0

  21. Conclusions • Insufficient data to conclude which world our universe is … • Cosmological constant,  is undetermined … • If  = 0, M = 5  1021 M • Then, world period = 10 billion yrs • But this only illustrates calculation

  22. A Homogeneous universe of Constant Mass & Increasing Radius accounting for the Radial Velocity of Extra – Galactic NebulaeAbbe Georges Lemaître • Annales de la Société scientifique de Bruxelles, A47, 49, 1927 • English translation in MNRAS, 91, 483-490, 1931

  23. Summary • Dilemma between de Sitter & Einstein world models • Intermediate solution – advantages of both • R = R(t) • R(t)  as t  • Similar differential equation of R(t) as Friedmann

  24. Summary cont’d.… • Accounted the following: • Conservation of energy • Matter density • Radiation pressure • Role in early stages of expansion of universe • First idea: • Recession velocities of galaxies are results of expansion of universe • Universe expanding from initial singularity, the ‘primeval atom’

  25. Intermediate model • Solution intermediate to Einstein & De Sitter worlds • Both material content & explaining recession of galaxies • Look for Einstein universe • Radius varying with time arbitrarily

  26. Universe ~ Sparsely dense gas Molecules ~ galaxies Uniformly distributed Density – uniform in space, time variable Ignore local condensation Internal stresses ~ Pressure p = (2/3) K.E. Negligible w.r.t energy of matter Radiation pressure of E.M. wave Weak Evenly distributed Keep p in general eqn For astronomical applications, p = 0 Assumptions of model

  27. Field equations : conservation of energy • Einstein field equations •  = Cosmological Constant (unknown) •  = Einstein Constant • Total energy change + Work done by radiation pressure in the expanding universe = 0

  28.  = Total density  = Matter density  =  - 3p Mass, M = V = constant  = constant  = integration constant Equations: Universe of constant mass

  29. De Sitter world  = 0  = 0 Einstein world  = 0 R = constant Existing solutions

  30. R0 = Initial radius of universe (from which expanding) R = Lemaître distance scale at time t RE = Einstein distance scale at t For  = 0 &  = 2R0 Lemaître solution

  31. Solution

  32. Cosmological Redshift • R1, R2 = Radius of Universe at times of emission & observation of light • Apparent Doppler effect • If nearby source, r = distance of source

  33. Einstein radius of universe: by Hubble from mean density RE = 2.7  1010 pc If R0 from radial velocities of galaxies R from R3 = RE2 R0 From data R/R = 0.6810-27 cm-1 R0/R = 0.0465 R = 0.215RE = 6  109 pc R0 = 2.7  108 pc = 9  108 LY Values Calculated

  34. Mass of universe – constant Radius of universe – increases from R0 (t = -) Galaxies recede as effect of expansion of universe Advantage of both Einstein & de Sitter solutions Conclusions

  35. Expanding space Possible universe of Lemaître

  36. 100 Mt. Wilson telescope range: 5  107 pc = R / 200 Doppler effect – 3000 km/s Visible spectrum displaced to IR Why universe expands? Radiation pressure does work during expansion  expansion set up by radiation itself Limitations & Further scopes

  37. On the relation between Expansion & mean density of universeAlbert Einstein& Wilhelm de sitter(Proceedings of the National Academy of Sciences 18, 213 – 214, 1932)

  38. Summary • After Hubble discovered expansion of universe: Einstein & de Sitter withdrew  • Expanding universe – without space curvature • If matter = C= 3H2/(8G) • Euclidean geometry • Flat, infinite universe • Using H0 ~ 10  H0 today • G(optically visible galaxies) ~ C Flat space

  39. Motivation • Observational data for curvature • Mean density • Expansion  Universe – non static • Can’t find curvature sign or value • If can explain observation without curvature ??

  40.   to explain finite mean density in static universe Dynamic universe – without   = 0 Line element: R = R(t) Neglect pressure (p) Field equation => 2 differential eqns Zero curvature

  41. From observation H - coefficient of expansion  - mean density From H = 500 km sec-1 Mpc-1 or, RB = 2  1027 cm Get RA = 1.63  1027 cm  = 4  10-28 g cm-3 Coincide exactly with theoretical upper limit of density for Flat space Solutions

  42. H – depends on measured redshifts Density – depends on assumed masses of galaxies & distance scale Extragalactic distances Uncertain H2 /  or RA2/RB2 ~ /M  = Side of a cube containing 1 galaxy = 106 LY M = average galaxy mass = 2  1011 M ~ close to Dr. Oort’s estimate of milky way mass Confidence limit of solution

  43.  - higher limit Correct magnitude order Possible to describe universe without curvature of 3-D space However, curvature is determinable More precise data Fix curvature sign Get curvature value Conclusions

  44. Present status of cosmological model • Search for cosmological parameters determining dynamics of universe: • Hubble constant, H0 • TOT = M +  + K • M = M/C • Matter (visible+dark) •  =  / 3H02 • Vacuum energy • K = -k / R02H02 • Curvature term • If flat k = 0

  45. H0 Hubble key project WMAP H0 = (71  3) km/s/Mpc M Cluster velocity dispersion Weak gravitational lens effect visible ~ 0.02 – 0.04 dark ~ 0.25 M ~ 0.3 Current values

  46.  • Energy density of vacuum • Discrepancy of > 120 orders of magnitude with theory •  ~ 0.7 • SN Type Ia • WMAP • Age of universe: • t0 = 13.7 G yr

  47. SN Type Ia • Giant star accreting onto white dwarf • Standard candle • Compare observed luminosity with predicted • Far off SN fainter than expected •  Expansion of Universe is accelerating

  48. Hubble diagram for SN type Ia

  49. Microwave background fluctuations • Brightest microwave background fluctuations (spots): 1 deg across • Ground & balloon based experiments • Flat – 15 % accuracy • WMAP • Measures basic parameters of Big Bang theory & geometry of universe • Flat – 2 % accuracy

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