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An overview of Cosmology

An overview of Cosmology. CERN Student Summer School 10 - 12 August 200 5 Julien Lesgourgues (LAPTH, Annecy). What is Cosmology?. Astrophysics  detailed description of « small » structures Cosmology  Universe as a whole  Is it static? Expanding ? Is it flat, open or closed ?

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An overview of Cosmology

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  1. An overview of Cosmology CERN Student Summer School 10-12 August 2005 Julien Lesgourgues (LAPTH, Annecy) An overview of cosmology

  2. What is Cosmology? • Astrophysics detailed description of « small » structures • Cosmology Universe as a whole  • Is it static? Expanding ? • Is it flat, open or closed ? • What is it composed of ? • What about its past and future ? An overview of cosmology

  3. Geometry and abstraction … • Part I :the expanding Universe • Hubble law • Newtonian gravity • General relativity • Friedmann-Lemaître model • Part II :the standard cosmological model • Hot Big bang scenario • Cosmological perturbations • Cosmological parameters • Inflation & Quintessence Concrete predictions, results, observations !! An overview of cosmology

  4. Part I :The Expanding Universe An overview of cosmology

  5. Part I : (1) - The Hubble Law • First step in understanding the Universe… • first telescopes: observation of nebulae • 1750 : T. Wright : Milky way = thin plate of stars? • 1752 : E. Kant : nebulae = other galaxies? • Galactic structure not tested before… 1923! An overview of cosmology

  6. redshift : z = D l / l = v . n / c • 1842 : Doppler effect for sound and light • 1868 : Huggins finds redshift of star spectral lines • 1868 – 1920 : observation of many redshift of stars and nebulae • random distribution • late observations : excess of z>0 for nebulae An overview of cosmology

  7. 1920’s : Leavitt & Shapley : • cepheids  period / absolute luminosity relation • measurement of distances of stars inside the Milky Way ( ~ 80.000 lightyear) absolute luminosity L apparent luminosity l = dL/ds = L/(4pr2) An overview of cosmology

  8. 1923 : Edwin Hubble : • 2,50 m telescope at Mount Wilson (CA) • cepheids in Andromeda • distance of nearest galaxy = 900.000 lyr (in fact 2 Mlyr) first probe of galactic structure !!! • so : excess of redshifted galaxies  Universe expansion ??? An overview of cosmology

  9. IN GENERAL : expansion  center Against « cosmological principle »  (Milne): Universe homogeneous … no privileged point! QUESTION : is any expansion a proof against homogeneity?  An overview of cosmology

  10. … like infinite rubber grid stretched in all directions … Proof that linear expansion is the only possible homogeneous expansion : vB/A = vC/B homogeneity vC/A = vC/B + vB/A = 2 vB/Alinearity ANSWER : not if v = H r  linear expansion An overview of cosmology

  11. 1000 km.s-1 • 1929 : Hubble gives the first velocity / distance diagram : H = v / r = 500 km.s-1.Mpc-1 for Hubble (  70km.s-1.Mpc-1 for us ) 1 Mpc = 3.106 lyr = 3.1022 m 0 km.s-1 0 Mpc 2 Mpc An overview of cosmology

  12. 1929 : starting of modern cosmology … Remark : what do we mean by « the Universe is homogeneous » (cosmological principle) ?  THE UNIVERSE IS IN HOMOGENEOUS EXPANSION An overview of cosmology

  13. v r • example of structure homogeneous after smoothing: • today : data on very large scales  confirmation of homogeneity beyond ~ 30 – 40 Mpc • local inhomogeneities  scattering An overview of cosmology

  14. Part I : (2) – Universe expansion from Newtonian gravity • on cosmic scales, only gravitation • Newton’s law = limit of General Relativity(GR) • Newton’s law should describe expansion at small distances with v = H r << c … • but historically, GR proposed the first predictions / explanations !!! when v << c F = G m1 m2 / r2 speed of object OR speed of liberation An overview of cosmology

  15. Newton: finite Universe infinite Universe but how to deal with infinity? Gauss theorem : r = - G Mr / r2 Mr = constant = (4/3) p r3 rmass  r2 = 2 G Mr / r - k = (8/3) pGrmass r2 - k .. . r(t) An overview of cosmology

  16. rmass(t)  r(t)-3 same motion as a two-body problem : rmass< rmass= rmass> k  0  non-homogeneous expansion ??? v = H r and v << c  r < RH c / H Newtonian expansion law : ( r / r )2 = (8pG/3) rmass - k/r2 . r(t) v . GRAVITY/ INERTIA 3 ( r / r )2 8 pG An overview of cosmology

  17. Part I : (3) – General Relativity and the Friedmann-Lemaître Universe Newtonian gravity  General Relativity (Einstein 1916) invariant speed of light no more Fgrav(matter distribution Fgrav  E = Fgrav) three basic principles : • space-time (t,x,y,z) is curved • curvature  matter • free-falling bodies follow geodesics An overview of cosmology

  18. how can we define the curvature : • of a 2-D surface ? • embedded in 3D • stay in 2D, and use angles : • stay in 2D, and use a scaling law : dl(x1,x2) ex: sphere projected on ellipse, dl(q) hyperboloid plane sphere An overview of cosmology

  19. OR + scale(t,x,y,z) y t x t x • of 3-D space ? • embedded in 4D : x12 + x22 + x32 + x42 = R2 • stay in 3D, but provide a scaling law, like on a planisphere :dl(x1,x2,x3) • of 4-D space-time ? • one more dimension • time different from space (special relativity : - + + + ) intuitive representations: cuts (t,x), (x,y), etc., Euclidian coordinates embedded in 3D + scaling law An overview of cosmology

  20. curvature  matter: mathematical formulation = Einstein equation An overview of cosmology

  21. free-falling bodies follow geodesics feel only gravity given one point and one direction (not E.M., etc.)  one single line such that : e.g. galaxies, light…  A, B, [AB] = shortest trajectory • example 1 : geodesics a sphere : parallel great circle NO YES An overview of cosmology

  22. example 2 : gravitational lensing : A sees an image of C lensed by B : An overview of cosmology

  23. Newtonian gravity versus G.R. : two different theories of gravity, i.e. two ways of describing how the presence of matter affects the trajectories of surrounding bodies… Newton Einstein matter gravitational potential curvature tensor trajectories An overview of cosmology

  24. applying G.R. to the Universe: some history • 1916 : Einstein has formulated G.R. • 1917 : Einstein, De Sitter try to build the first cosmological models (PREJUDICE : STATIC / STATIONNARY UNIVERSE) • 1922 : A. Friedmann (Ru) investigate most general • 1927 : G. Lemaître (B) HOMOGENEOUS, ISOTROPIC, • 1933 : Robertson, NON-STATIONNARY Walker (USA) solutions of G.R. equations • 1929 : Hubble’s law (first confirmation) • 1930-65 : accumulation of proofs in favour of FLRW • 1965 : CMB discovery : full confirmation An overview of cosmology

  25. An overview of cosmology

  26. summary of the situation : NEWTON  matter distribution  Fgrav  forces changing the trajectories General Relativity (GR) three basic principles : • space-time (t,x,y,z) is curved • free-falling bodies follow geodesics • curvature  matter Friedmann-Lemaitre model = application of GR to homogeneous Universe An overview of cosmology

  27. the curvature of the FLRW Universe : • Universe space-time (t,x,y,z) curved by its own homogeneous matter density r(t) • HOMOGENEITY  decomposition of curvature in : • spatial curvature of (x,y,z) at fixed t 3-D space is maximally symmetric : • 2-D space-time curvatureof (t,x)  (t,y)  (t,z) accounts for the expansion 3-plane 3-sphere 3-hyperboloid FLAT CLOSED  RC(t)  OPEN An overview of cosmology

  28. (r+dr, q+dq, f+df) dl (r, q, f) O • scale as a function of coordinates ? • COMOVING COORDINATES (t, r, q, f) • for Euclidian space : dl2 = dr2 + r2 (dq2 + sin2q df2 ) for FLRW : dl2 = a2(t) + r2 (dq2 + sin2q df2 ) • a(t)  scale factor  2-D space-time curvature • k  spatial curvature • k = 0 : FLAT • k > 0 : CLOSED, RC(t) = a(t) / k1/2, 0  r < 1/ k1/2 • k < 0 : OPEN , RC(t) = a(t) / (-k)1/2 dr2 (1-k r2) An overview of cosmology

  29. the geodesics in the FLRW Universe photons : v = c ULTRA-RELATIVISTIC •  for ordinary matter : v << c NON-RELATIVISTIC (e.g. galaxies) • non-relativistic matter: dl = 0  (r, q, f) = constant • galaxies are still in coordinate space … • … but all distances are proportional to a(t) a(t) gives the expansion between galaxies (although they are still !!!) • like an inflated rubber balloon with points drawn on its surface … An overview of cosmology

  30. dr2 (1-k r2)  • Relativistic matter : straight line in 3-D space, dl = c dt c2dt2 = a2(t) + r2 (dq2 + sin2q df2 ) EQUATION OF PROPAGATION OF LIGHT • So Dl = c Dt is WRONG:  « bending of light in the Universe » (one of the two most fundamental equations in cosmology)  various important consequences … (r2 , t2) y (r1 , t1) O x An overview of cosmology

  31. t0 te re qe we are here • definition of the past light-cone : • in Euclidian space : • q = constant • re = c (t0 – te) • in Friedmann universe : • q = constant • dr c (1-kr)1/2 dt a(t) approximately linear region = An overview of cosmology

  32. observable consequences of propagation of light equation: • the redshift : z = Dl / l = l0 / le – 1 z = a(t0) / a(te) – 1 • remark 1 : • Newtonian : z = v / c  1 • G.R. : no limit, as observed … An overview of cosmology

  33. t0 te re qe ZOOM • remark 2 : at short distance, we can recover the Hubble law ( z = v / c = H r / c ) • then : • so : Hubble parameter = ( H0 = 70 km.s-1.Mpc –1) nearby galaxy t0 – te = dt = dl / c An overview of cosmology

  34. dl dq r • the angular diameter-redshift relation • Euclidian space : dl = r dqwith r = v / H = z c / H • G.R. : dl = a(te) re dqwith refrom if dl is known, measurement of (dq, z)  k, a(t) An overview of cosmology

  35. CLOSED UNIVERSE :objects seen under larger angle • OPEN UNIVERSE :objects seen under smaller angle An overview of cosmology

  36. L absolute luminosity r l apparent luminosity • the luminosity distance-redshift relation • Euclidian space :with r = z c / H • G.R. :with refrom prop. of light if L is known, measurement of (l, z)  k, a(t) An overview of cosmology

  37. relation between matter and curvature : FRIEDMANN LAW Remark : for non-relativistic matter, E = m c2 r / c2 = rmass  then Friedmann law looks similar to Newtonian expansion law, but CRUCIAL DIFFERENCES : • a(t)  r(t) : very different interpretation • k  0 not in contradiction with homogeneity • accounts for non-relativistic and relativistic matter An overview of cosmology

  38. NON-RELATIVISTIC v << c E = m c2 sphere with fixed comobile r fixed particle number EV = constant r = EV / V  a(t)-3 ULTRA-RELATIVISTIC v = c E = ħ n = ħ c / l V = 4/3 p r3 a(t)3 EV  1/ l 1 / a(t) r = EV / V  a(t)-4 r FRIEDMANN LAW is the same but DILUTION RATE is different An overview of cosmology

  39. in fact, in G.R., curvature  matter relation given by EINSTEIN EQUATION Gmn = 8pG Tmn • in the FLRW solution : Friedmann law EINSTEIN EQUATION conservation equation : r = - 3 (a / a) ( r + p ) • non-relativistic : v << c  p  0  r / r = - 3 a / a r a(t)-3 • ultra-relativistic : v = c  p = r / 3  r / r = - 4 a / a r a(t)-4 • in QFT : vacuum with p = - r  r = 0 r= constant  « cosmological constant » . . . . . . . An overview of cosmology

  40. summary of the situation : An overview of cosmology

  41. Part II :The Standard Cosmological Model An overview of cosmology

  42. _ _ • decomposition of quantities: • r(t,r) = r(t) + dr(t,r) • p(t,r) = p(t) + dp(t,r) • curvature = {a(t), k} + {F(t,r), etc.} HOMOGENEOUS PERTURBATIONS BACKGROUND THEORY OF LINEAR PERTURBATIONS Initial conditions INFLATION part II - 4 HOMOGENEOUS COSMOLOGY part II - 1 LINEAR PERT. part II - 2 NON-LINEAR PERT. An overview of cosmology

  43. Part II : (1) – Homogeneous cosmology • the evolution of the Universe depends : • onSPATIAL CURVATURE • on the density of : • RADIATION : ultra-relativistic particles p = r / 3 r a-4 ( photons, massless n’s, … ) • MATTER : non-relativistic bodies p = 0 r a-3 ( galaxies, gas clouds, … ) • COSMOLOGICAL CONSTANTL[t-2] p = - rr = constant = L c2 / (8pG) ( vacuum ? … ? ) • … An overview of cosmology

  44. BIG BANG Friedmann law: • most « complete » scenario : • phases can be skipped, but order cannot change • RADIATION DOMINATION : a  t1/2H = 1 / 2 t • MATTER DOMINATION : a  t2/3H = 2 / 3 t • CURVATURE DOMINATION : • k < 0 (open) : a  t H = 1 / t • k > 0 (closed) : a  0, then a < 0 or  • VACUUM DOMINATION : a  exp(L/3t)1/2 H  constant . . An overview of cosmology

  45. Future of the Universe: • if L = 0 : • if k < 0 or k = 0 indefinite decelerated expansion • if k > 0  recollapse (BIG CRUNCH) • if L 0 : •  k  indefinite accelerated expansion BIG BANG Friedmann law: • most « complete » scenario : • phases can be skipped, but order cannot change • RADIATION DOMINATION : a  t1/2H = 1 / 2 t • MATTER DOMINATION : a  t2/3H = 2 / 3 t • CURVATURE DOMINATION : • k < 0 (open) : a  t H = 1 / t • k > 0 (closed) : a  0, then a < 0 or  • VACUUM DOMINATION : a  exp(L/3t)1/2 H  constant . . An overview of cosmology

  46. MATTER BUDGET EQUATION • the matter budget : • if we can measure {rR, rM, k, L} today, we can extrapolate back … • today : • flatness condition : W0WR + WM + WL = 1 • then : rR0 + rM0 + rL0 rc0  WX = rX / rc0 • so far : COSMOLOGICAL4 independent parameters SCENARIOS{WR , WM , WL, H0} 3 H02 c2 8pG An overview of cosmology

  47. COLD or HOT BIG BANG ??? • 1929–65 :no decisive observation in favour of Friedmann model (apart from accumulation of redshifts)  works in cosmology remain marginal • but spectacular progress in particle physics… • studies based on the most simple possible scenario: • Universe contains only non-relativistic matter • evolution under the laws of nuclear physics between Big Bang and today  COLD BIG BANG SCENARIO An overview of cosmology

  48. e- p n • COLDBIG BANG : • H2 = (8pG/3c2) rM r a-3  t-2 • NUCLEOSYNTHESIS : • ensemble of nuclear reactions p H n D e- 3He , 4He n Li , etc. • freeze-out due to expansion NUCLEOSYNTHESIS RECOMBINATION time An overview of cosmology

  49. pioneering works on nucleosynthesis : • 1940 : Gamow et al. (USSR  USA) 1964 : Zel’dovitch et al. (USSR) 1965 : Hoyle & Taylor (UK) 1965 : Peebles et al. (USA) • COLD BIG BANG  no hydrogen  need to change H(tnucleo) add relativistic matter (photons) with rR >> rM HOT BIG BANG!!! An overview of cosmology

  50. H2 = (8pG/3c2) (rR + rM) p n & recombination • HOT BIG BANG : An overview of cosmology

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