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Basic cosmology - Haus der Astronomie
Basic cosmology Heraeus Summer School 2013 Markus Pössel Haus der Astronomie August 19, 2013 Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Contents 1 Introduction 2 Expansion with a scale factor 3 The Friedmann-Lemaı̂tre-Robertson-Walker (FLRW) metric 4 Dynamics of FLRW universes 5 Observing the expanding universe Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The framework behind cosmology Modern framework for cosmology: homogeneous, isotropic models based on general relativity • A surprising amount can be derived in a Newtonian way! (cf. tutorial) • Inaccessible to Newton: • consistent theory of the behavior of light • non-flat spaces • some sources of gravity ⇒ general relativity needed! Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Goals of this lecture • Introduce Friedmann-Lemaı̂tre-Robertson-Walker models • Distinguish between kinematic (scale factor, metric) and dynamic (Friedmann equations) effects • Introduce concepts, motivation and nomenclature needed in later lectures Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations General relativity • Einstein’s theory (1915) linking space-time-geometry and gravity • Central model-building element: space-time metric gµν (or line element ds 2 ) • Light and freely falling particles move along the straightest-possible space-time paths (geodesics) • Einstein’s equations link space-time geometry (“Einstein tensor Gµν ”) and sources of gravity (“Stress-energy tensor Tµν ) • Physical models: solutions of Einstein’s equations Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations ...for our purposes: • What does it mean to have a universal scale factor? • Heuristic introduction of the metric • (Newtonian) derivation of dynamic equations (⇒ tutorial) • Basic parameters • Observing the expanding universe Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Expansion with a scale factor Deduced from Hubble’s (and later) results: scale factor expansion Idea: cosmic substratum (i.e. “galaxy dust”) floating in space. All distances between substratum particles change proportional to a (t ), e.g. distance d12 between galaxy 1 and galaxy 2: d12 (t1 ) = Markus Pössel a (t1 ) · d12 (t0 ). a (t0 ) Basic cosmology ph en we g Scale-factor expansion 16a 6 Heidelberg Suites FLRW spacetimes 18b 20 26 8 Heidelberg College 16 18 Introduction 19 hilosoph enweg 18a FLRW dynamics Observations L 534 22 Phil oso 24 28 Expansion with a scale factor 32 36 38 30 fad Leinp traße nds er La nheim Neue 34 eberle-Straße 38/2 Alte Brücke Alb ert -U Neckar Neckar r Necka 40 Unitas Haus 40/1 44 M önc hg as se L 534 eufel Am Hackt Neckarstaden 2 e 12 14 Kettengasse Ke 7 3 Dokumentationsund Kulturzentrum Deutscher Sinti und Roma 2 7a 7b ue Ne Oberer Fauler Pelz h Sc e raß ssst hlo e Sc Neu 41 18 39 e straß loss Sch ue 22 Ne 17 9 27 21 11 13 15 berg Schloss 3a 23 29 31 33 35 g ossber Schl ue Ne 6 erg ossb Schl el nn tu rg be ss hlo Sc 10 lz Pe ler au rF ere Ob 37 e raß sst los Sch aße lossstr e Sch Neu 10 42 16 4 16a Klingentor 6c eg Rudolf Steiner Haus gw 6a 10 Klinge nteic hst raß e 6b 34 36 13 lz r Pe Faule rer Unte tte ngas se 2 Arthotel Heidelberg 8 32 e gass rbad Obe 4 10 5 Hotel Anlage 1 Sporthalle Klingenteich r be aim Gr 3 6 t ark rnm Ko Piccolo Mondo e hgass Mönc 4 28 1 traße 3 ims Ingr 15 30 sse adga Mittelb 2 e ass badg Ober 66 platz Markt 6 1 64 13 11 22 24 26 5 8 7 10 12 14 9 16 ergasse Kräm Hotel Acor rt-Anlage h-Ebe Friedric sse erga Apothek e ert-Anlag Friedrich-Eb 53 53a 53b rgasse Fische rgasse Kräme traße inars Sem 3 3 rsität Unive Neue -Anlage h-Ebert Friedric t-Anlage Friedrich-Eber tplatz Mark Fischmarkt Parkhaus am Theater 62 lage rt-An -Ebe drich Frie Seminar for Roman Languages Peterskirche 56a 56a 6 9 3 Tangente Zuckerladen Marakesch Friedrich-Ebert-Anlage Steingasse Markus Pössel Riegler Europahaus III Chili's Raja Rani Plöck 4 3 engasse Kett age t-Anl Eber richFried Plöck e gass Sand asse Märzg Platz h-EbertFriedric Sporthalle Hölderlin-Gymnasium Schießtorstraße Plöck Hölderlin-Gymnasium Göbes t rsitä nive ue U Ne rsität Unive Neue engasse Grab rstraße Theate e hstraß Friedric asse Märzg Plöck Coffee Inn Schulgasse sse Märzga 71 61 57 Plöck Plöck 4 Zwinger1 und Zwinger3 Figurenhof Zentrum für Lehrerbildung 1 1 2 1 12 ße tra ers ing Zw 12 Philosophisches Seminar, Slavisches Institut Universitätsbibliothek Plöck 1 rsität Unive Neue Friedrich-Ebert-Grundschule Sporthalle Theodor-Heuss-Realschule 6 Neue Universität 14 9 57a Sprachlabor Yufka's Kebap & Pizza aße Ingrimstr 6 Universitätsbibliothek Hochschule für Jüdische Studien 5 5 Prinz Friedrich 4 9 ät ersit Univ Neue Triplex-Mensa traße frieds Land e straß fried Land Falafel 7 191-201 traße Haupts 200202 190 Stefansbäck Goldene Sonne 6-8 Rathaus 10 Marktplatz e Hauptstraß 2 sse enga Grab 3 Regie 16 Café Moro Coffee & Kiss ra ß Merianst Der Kleine Gundel Städtische Bühne Bäckerei Grimm 22 2 Zimmertheater Heiliggeistkirche Schmidts 168 Gloria & Gloriette Altstadt Hallhuber Hauptstraße MoschMosch Alte Universität 4 6 Kaiser Avatar Foods Cafe Romantic 120 e gass Sand 74 La Fée Bar Café Schiller Korea 8 Mel's Cenmoro tz Marktpla Universitäts-Apotheke rstraße Theate Schmelzpunkt traße dwig-S Karl-Lu 72 126 Marktstube Floringasse Sayuki aße ichstr Friedr 86 82 70 La Flamm-Pâtisserie Weisser Schwan 9 6 3 Fischm arkt aße Untere Str Alfredo OSTERIA Heugasse straße Bienen e ass elg 97 78 Unisex 57 e tstraß 64 Haup 58 Hörnchen 124 traße Haupts 114 Hotel The Dubliner Big Pommes Perkeo 71 aße Untere Str Heumarkt Kupferkanne Institutsgebäude Kamps Schnitzelbank Chocolaterie YilliY Weinloch 3 Hotel Weisser Bock 135Starbucks 12 C&A 6 se Haspelgas Bauamtsgasse se Schiffgas Konomi e fengass Karp Zie g 56 sse Pfaffenga 54 Vinothek Goldener Anker Thai Restaurant Phuket 14 13 25 21 17 ßchen Küchengä ße rstra cka e Ne Unter Jubiläumsplatz Zum Binsebub Europahaus IV straße Marstall B 37 Kollegiengebäude asse Kraneng Kongresshaus Stadthalle Heidelberg raße rst ka e Nec Unter Wirtshaus Spreisel ße nigstra Dreikö taden Neckars 13 Kilimanjaro Bussemgasse traße ckars e Ne Unter 7-11 17 Montpellierplatz Kleine Mantelgasse Wohnheim Wartburg lgasse Große Mante Marstallcafé 32 24 ße Marstallstra Zeughaus-Mensa n stade Neckar Semann's lsgasse Semme Goldener LENOX Stern Hörsaalgebäude r Necka Brem enec kgas se 100 m re Neckarfäh Nectar Heidelberger Brückenaffe Am Brückentor Karl Lauerstraße kteufel Neckarstaden Karl Theodor Am Hac Neckarstaden Neckarstaden rstaden Necka Hotel Villa Marstall lo ss st ra ße ar Neck 8 Basic cosmology Moore Haus 43 ph en we g Scale-factor expansion 16a 6 Heidelberg Suites FLRW spacetimes 18b 20 26 8 Heidelberg College 16 18 Introduction 19 hilosoph enweg 18a FLRW dynamics Observations L 534 22 Phil oso 24 28 Expansion with a scale factor 32 36 38 30 fad Leinp traße nds er La nheim Neue 34 eberle-Straße 38/2 Alte Brücke Alb ert -U Neckar Neckar r Necka 40 Unitas Haus 40/1 44 M önc hg as se L 534 eufel Am Hackt Neckarstaden 2 e 12 14 Kettengasse Ke 7 3 Dokumentationsund Kulturzentrum Deutscher Sinti und Roma 2 7a 7b ue Ne Oberer Fauler Pelz h Sc e raß ssst hlo e Sc Neu 41 18 39 e straß loss Sch ue 22 Ne 17 9 27 21 11 13 15 berg Schloss 3a 23 29 31 33 35 g ossber Schl ue Ne 6 erg ossb Schl el nn tu rg be ss hlo Sc 10 lz Pe ler au rF ere Ob 37 e raß sst los Sch aße lossstr e Sch Neu 10 42 16 4 16a Klingentor 6c eg Rudolf Steiner Haus gw 6a 10 Klinge nteic hst raß e 6b 34 36 13 lz r Pe Faule rer Unte tte ngas se 2 Arthotel Heidelberg 8 32 e gass rbad Obe 4 10 5 Hotel Anlage 1 Sporthalle Klingenteich r be aim Gr 3 6 t ark rnm Ko Piccolo Mondo e hgass Mönc 4 28 1 traße 3 ims Ingr 15 30 sse adga Mittelb 2 e ass badg Ober 66 platz Markt 6 1 64 13 11 22 24 26 5 8 7 10 12 14 9 16 ergasse Kräm Hotel Acor rt-Anlage h-Ebe Friedric sse erga Apothek e ert-Anlag Friedrich-Eb 53 53a 53b rgasse Fische rgasse Kräme traße inars Sem 3 3 rsität Unive Neue -Anlage h-Ebert Friedric t-Anlage Friedrich-Eber tplatz Mark Fischmarkt Parkhaus am Theater 62 lage rt-An -Ebe drich Frie Seminar for Roman Languages Peterskirche 56a 56a 6 9 3 Tangente Zuckerladen Marakesch Friedrich-Ebert-Anlage Steingasse Markus Pössel Riegler Europahaus III Chili's Raja Rani Plöck 4 3 engasse Kett age t-Anl Eber richFried Plöck e gass Sand asse Märzg Platz h-EbertFriedric Sporthalle Hölderlin-Gymnasium Schießtorstraße Plöck Hölderlin-Gymnasium Göbes t rsitä nive ue U Ne rsität Unive Neue engasse Grab rstraße Theate e hstraß Friedric asse Märzg Plöck Coffee Inn Schulgasse sse Märzga 71 61 57 Plöck Plöck 4 Zwinger1 und Zwinger3 Figurenhof Zentrum für Lehrerbildung 1 1 2 1 12 ße tra ers ing Zw 12 Philosophisches Seminar, Slavisches Institut Universitätsbibliothek Plöck 1 rsität Unive Neue Friedrich-Ebert-Grundschule Sporthalle Theodor-Heuss-Realschule 6 Neue Universität 14 9 57a Sprachlabor Yufka's Kebap & Pizza aße Ingrimstr 6 Universitätsbibliothek Hochschule für Jüdische Studien 5 5 Prinz Friedrich 4 9 ät ersit Univ Neue Triplex-Mensa traße frieds Land e straß fried Land Falafel 7 191-201 traße Haupts 200202 190 Stefansbäck Goldene Sonne 6-8 Rathaus 10 Marktplatz e Hauptstraß 2 sse enga Grab 3 Regie 16 Café Moro Coffee & Kiss ra ß Merianst Der Kleine Gundel Städtische Bühne Bäckerei Grimm 22 2 Zimmertheater Heiliggeistkirche Schmidts 168 Gloria & Gloriette Altstadt Hallhuber Hauptstraße MoschMosch Alte Universität 4 6 Kaiser Avatar Foods Cafe Romantic 120 e gass Sand 74 La Fée Bar Café Schiller Korea 8 Mel's Cenmoro tz Marktpla Universitäts-Apotheke rstraße Theate Schmelzpunkt traße dwig-S Karl-Lu 72 126 Marktstube Floringasse Sayuki aße ichstr Friedr 86 82 70 La Flamm-Pâtisserie Weisser Schwan 9 6 3 Fischm arkt aße Untere Str Alfredo OSTERIA Heugasse straße Bienen e ass elg 97 78 Unisex 57 e tstraß 64 Haup 58 Hörnchen 124 traße Haupts 114 Hotel The Dubliner Big Pommes Perkeo 71 aße Untere Str Heumarkt Kupferkanne Institutsgebäude Kamps Schnitzelbank Chocolaterie YilliY Weinloch 3 Hotel Weisser Bock 135Starbucks 12 C&A 6 se Haspelgas Bauamtsgasse se Schiffgas Konomi e fengass Karp Zie g 56 sse Pfaffenga 54 Vinothek Goldener Anker Thai Restaurant Phuket 14 13 25 21 17 ßchen Küchengä ße rstra cka e Ne Unter Jubiläumsplatz Zum Binsebub Europahaus IV straße Marstall B 37 Kollegiengebäude asse Kraneng Kongresshaus Stadthalle Heidelberg raße rst ka e Nec Unter Wirtshaus Spreisel ße nigstra Dreikö taden Neckars 13 Kilimanjaro Bussemgasse traße ckars e Ne Unter 7-11 17 Montpellierplatz Kleine Mantelgasse Wohnheim Wartburg lgasse Große Mante Marstallcafé 32 24 ße Marstallstra Zeughaus-Mensa n stade Neckar Semann's lsgasse Semme Goldener LENOX Stern Hörsaalgebäude r Necka Brem enec kgas se 200 m re Neckarfäh Nectar Heidelberger Brückenaffe Am Brückentor Karl Lauerstraße kteufel Neckarstaden Karl Theodor Am Hac Neckarstaden Neckarstaden rstaden Necka Hotel Villa Marstall lo ss st ra ße ar Neck 8 Basic cosmology Moore Haus 43 Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Distances between galaxies Consider galaxies in the Hubble flow: d (t0 ) t = t0 All distances change as d (t ) = Markus Pössel a (t ) · d (t0 ). a (t0 ) Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Distances between galaxies Consider galaxies in the Hubble flow: d (t1 ) t = t1 All distances change as d (t ) = Markus Pössel a (t ) · d (t0 ). a (t0 ) Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Expansion with a scale factor Signals emitted at time te with time difference ∆te , initial spatial separation v · ∆t 1 2 v ∆t ⇒ if separation is stretched with the expansion factor and signals arrive at time t1 , then ∆t0 = Markus Pössel a (t0 ) · ∆te . a (te ) Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Cosmological redshift Wavelength scaling with scale factor: wavelength Redshift defined as z= a (t0 ) λ0 − λ1 −1 = λ1 a (te ) or 1+z = a (t0 ) a (te ) For co-moving galaxies: z is directly related to re . For monotonous a (t ): distance measure. Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Cosmological redshift Redshift for a (t0 ) > a (te ); blueshift for a (t0 ) < a (te ) Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Taylor expansion of the scale factor Generic Taylor expansion: 1 ä · (t − t0 )2 + . . . 2 Re-define the Taylor parameters by introducing two functions a (t ) = a (t0 ) + ȧ (t0 )(t − t0 ) + H (t ) ≡ ȧ (t ) ä (t )a (t ) and q(t ) ≡ − a (t ) ȧ (t )2 and corresponding constants H0 ≡ H (t0 ) and q0 ≡ q(t0 ) " a (t ) = a0 1 + (t − t0 )H0 − Markus Pössel 1 q0 H02 (t − t0 )2 + . . . 2 # Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Some nomenclature and values 1/2 t0 is the standard symbol for the present time. If coordinates are chosen so cosmic time t = 0 denotes the time of the big bang (phase), then t0 is the age of the universe. Sometimes, the age of the universe is denoted by τ. H (t ) is the Hubble parameter (sometimes misleadingly Hubble constant) H0 ≡ H (t0 ) is the Hubble constant. Current values around H0 = 70 Markus Pössel km/s . Mpc Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Some nomenclature and values 2/2 Sometimes, the Hubble constant is written as H0 = h · 100 km/s Mpc (keep your options open!) with h the dimensionless Hubble constant. The inverse of the Hubble constant is the Hubble time (cf. the linear case and the models later on). 1 /s h · 100 km Mpc Markus Pössel ≈ h −1 · 1010 a. Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations For “nearby” galaxies. . . . . . use the Taylor expansion a (t ) = a (t0 )[1 + H0 (t − t0 ) + O ((t − t0 )2 )]: 1−z ≈ 1 1+z = a (te ) ≈ 1 + H0 (te − t0 ) a (t0 ) or, using the classical Doppler formula v = cz and defining distance d = c ∆t for light-travel time ∆t, v = cz ≈ H0 c (t0 − te ) ≈ H0 d for small z, small t0 − te . This is Hubble’s law. Originally discovered by Alexander Friedmann (cf. Stigler’s law). Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The Friedmann-Lemaı̂tre-Robertson-Walker metric Next step: Re-visit scale factor expansion within the framework of general relativity. Space-time described by a metric . . . say what? Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The meaning of the metric: the flat/Cartesian case Ordinary Cartesian coordinates: Pythagoras! ds 2 = dx 2 + dy 2 Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The meaning of the metric: spherical coordinates Introduce spherical coordinates r , θ, φ: x = r · sin θ · cos φ y = r · sin θ · sin φ z = r · cos θ The line element is given by ds 2 = dr 2 + r 2 (dθ2 + sin2 θ dφ2 ) ≡ dr 2 + r 2 dΩ. Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations A different kind of metric: Special relativity t [s] 1 0,97 s 0,88 s 0,71 s 0,42 s 0,42 Ls = 125000 km 0,71 Ls = 212000 km 0,88 Ls = 262000 km 0,97 Ls = 291000 km x [Ls = 300000 km] −1 0 1 ds 2 = −c 2 dτ2 = d~x 2 − c 2 dt 2 . Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Cosmological model-building: strategy Two-step model-building strategy guided by the cosmological principle: 1 Build idealized exactly homogeneous and isotropic models: Friedmann-Lemaı̂tre-Robertson-Walker, FLRW (exact family of solutions; this lecture) 2 Add inhomogeneities on smaller scales as perturbations (Björn Malte Schäfer on Saturday) Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The Friedmann-Robertson-Walker Metric Only possibility of describing an isotropic and homogeneous universe (up to freedom of choosing different coordinates): " 2 2 2 ds = −c dt + a (t ) 2 dr 2 + r 2 dΩ 1 − Kr 2 # with a (t ) the cosmic scale factor. From now on: convention c ≡ 1. K a parameter that describes spatial geometry: +1 spherical space 0 Euclidean space K = −1 hyperbolical space Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations A caveat: global vs. local Globally, there is topology to consider — e.g. a flat metric can belong to infinite Euclidean space, but also, say, to a torus (a patch of Euclidean space with certain identifications). • K = 0: 18 topologically different forms of space. Some inifinite, some finite. • K = +1: inifinitely many topologically different forms. All are finite. • K = −1: infinitely many topologically different forms of space. Some inifinite, some finite. Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Light in an FRW universe Great advantage of GR: Light propagation along geodesic (straightest-possible) lines! As in special relativity: for light, ds 2 = 0 (alternative: geodetic equation). Also: use symmetries! Move origin of your coordinate system wherever convenient. Look only at radial movement. " ds 2 = −dt 2 + a (t )2 dr 2 + r 2 dΩ 1 − Kr 2 # becomes a (t ) dr dt = ± √ . 1 − Kr 2 ⇒ this will become important in Andreas Just’s lecture Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The evolution of the scale factor Up to now, all our conclusions drawn from metric — derived by symmetry. To find the explicit form of a (t ) in a rigorous way, we need Einstein’s equations, Gµν = 8πG Tµν with a perfect-fluid stress-energy tensor T µν = (ρ + p ) uµ uν + pg µν — both are beyond the scope of this lecture. Here, we will directly skip to the solution — in the tutorial, we will find sort-of-Newtonian answers Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Solving Einstein’s equations for FRW 00 component of Einstein’s eq.: 3 ȧ 2 + K = 8πG ρ a2 i0 components vanish. ij components give 2 ä ȧ 2 + K = −8πGp + a a2 These are the Friedmann equations. Their solutions are the Friedmann-Lemaı̂tre-Robertson-Walker (FLRW) universes. Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Re-casting the Friedmann equations 8πG ρ ȧ 2 + K = 2 3 a and for ȧ , 0 (by differentiating the above and inserting the ij-equation) ȧ a ρ̇ = −3 (ρ + p ) = −3H (t )(ρ + p ). (in tutorial, this will be linked to energy conservation). Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations How does density change with the scale factor? ȧ a links density changes to expansion. Details depend on equations of state. In cosmology, most important examples satisfy p = w ρ: ρ̇ = −3 (ρ + p ) = −3H (t )(ρ + p ). 1 Dust: w = 0 ⇒ ρ ∼ 1/a 3 2 Radiation: w = 1/3 ⇒ ρ ∼ 1/a 4 3 Dark energy/cosmological constant: w = −1 ρ =const. Whenever these are the only important components, a universe can have different phases — depending on size, different components will dominate. Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Different eras depending on the scale factor 1021 Density ρ 1014 107 Dust 100 Dark energy 10−7 10−14 −6 10 Markus Pössel Radiation 10−5 10−4 10−3 10−2 10−1 Scale factor a 100 101 102 Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Deceleration from the Friedmann equations Recombine Friedmann equations to give equation for ä: ä 4πG =− (ρ + 3p ) a 3 ⇒ q0 = 4πG 3H02 (ρ0 + 3p0 ) (with ρ0 and p0 the present density/pressure). Recall equation of state p = w ρ: Ordinary matter decelerates, but w = −1 accelerates via its pressure! Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The initial singularity Shows that, for universes where dark energy (w = −1) does not dominate, ä /a ≤ 0. But by our earlier considerations (eras), this is true for small a, that is, in the early universe! a t0 t Initial singularity — special case of Hawking-Penrose theorems. Convenient: Choose as zero point for cosmic time. Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The critical density and the Ω parameters Rescale all present densities (M Matter, R Radiation, Λ dark energy) in terms of the present critical density, ρc0 ≡ 3H02 8πG , and re-scale K accordingly: ΩΛ = ρλ (t0 )/ρc0 , ΩM = ρM (t0 )/ρc0 , ΩR = ρR (t0 )/ρc0 , ΩK = −K /(a0 H0 )2 . Re-write the present-day Friedmann equation as ΩΛ + ΩM + ΩR + ΩK = 1. ⇒ link densities and spatial geometry (important later on for CMB) Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations General considerations for FLRW models Scaling behaviour of the different densities means that ρ(t ) = 3H02 8πG ΩM a0 a (t ) !3 + ΩR a0 a (t ) !4 + ΩΛ . Substitute back into the Friedmann equation ȧ 2 + K 8πG ρ = 2 3 a and substitute x (t ) ≡ a (t )/a0 = 1/(1 + z ) to obtain dt = Markus Pössel dx . p H 0 x Ω Λ + Ω K x −2 + Ω M x −3 + Ω R x −4 Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The age of the universe in FLRW models Defining t = 0 by a (0) = 0 [initial singularity] corresponds to z → ∞ or x = 0. With this zero point, emission time tE (z ) of light reaching us with redshift z: 1 tE (z ) = H0 1/(1+z ) Z 0 dx . p x Ω Λ + Ω K x −2 + Ω M x −3 + Ω R x −4 Special case z = 0 corresponds to the present time — gives the age of the universe τ, 1 τ= H0 Markus Pössel 1 Z 0 dx . p x Ω Λ + Ω K x −2 + Ω M x −3 + Ω R x −4 Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The acceleration parameter q0 Present pressure: p0 = 3H02 8πG inserting in q0 = we find that q0 = Markus Pössel 1 3 (−ΩΛ + ΩR ). 4πG 3H02 (ρ0 + 3p0 ), 1 (ΩM − 2ΩΛ + 2ΩR ). 2 Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The fate of the universe Rewrite ȧ 2 + K 8πG ρ = 2 3 a as h ȧ 2 = (H0 a0 )2 ΩΛ x 2 + ΩM x −1 + ΩR x −2 + ΩK i where x = a /a0 . Forget about ΩR . If we want a re-collapse, we must have ȧ = 0 at some time, in other words: ΩΛ x 3 + Ω K x + Ω M = 0 . Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The fate of the universe Discussion of ΩΛ x 3 + Ω K x + Ω M = 0 • We know that, for x = 1, this expression is +1 • For ΩΛ < 0, for sufficiently large x, the expression will become negative ⇒ must have a zero • For ΩΛ = 0, we know recollapse for ΩM > 1, requiring K = +1 • For ΩΛ > 0, recollapse if ΩK sufficiently negative (again, K = +1). Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Overview of FLRW solutions Image from: d’Inverno, Introducing Einstein’s Relativity, ch. 22.3 Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations FLRW model with K = 0, Λ > 0 scale factor a This is the special case that is probably our own universe: cosmic time t Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Observing the expanding universe • Age of the universe (current best: 13.8 billion a) – upper limit! • Tracing a (t ) gives H0 and constraints on Ω parameters • Early universe (nucleosynthesis, cosmic background radiation) • Geometry from CMB and large-scale structure • Consistency checks: SN time dilation, surface brightness, energy inventory Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Reconstructing cosmic history 1.5 } Eternal expansion } 0.01 0.1 1 0.0001 0.001 RELATIVE BRIGHTNESS OF SUPERNOVAE 1.0 0 0.0 –20 Markus Pössel de cel e ra t es REDSHIFT z s alw ays t firs ion s n pa era cel de te ra –10 0 BILLIONS OF YEARS FROM TODAY 0.5 1 1.5 2 3 . . . or Ex 0.5 en , th tes le ce ac Eventual collapse LINEAR SCALE OF UNIVERSE RELATIVE TO TODAY Perlmutter, Physics Today 2003 +10 Figure 4. expansion high-reds data poin geometry universe ent, so it curves in represent which the vacuum e comes th the mass sume vac ranging fr down to 0 shaded re sent mod expansion due to hig sume ma right) from fact, for th pansion e verses int Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Supernova Cosmology Project Plot Suzuki et al. 2011 Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations ...directly related to our Ω formulae: Sensitivity of SNe and CMB observations directly linked to q0 = 1 (ΩM − 2ΩΛ + 2ΩR ). 2 and ΩΛ + ΩM + ΩR + ΩK = 1. Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations The matter content of our universe ( ΩM = Ωr Ωb Ωd = 4.9% = 26.8% ) = 31.7% = 0.005% ΩΛ = 68.3% Image: ESA/Planck Collaboration Where Ωb = ordinary matter (protons, neutrons, . . . ), Ωd = dark matter (no interaction with light) ΩΛ = (accelerating) dark energy Markus Pössel Basic cosmology Introduction Scale-factor expansion FLRW spacetimes FLRW dynamics Observations Literature d’Inverno, Ray: Introducing Einstein’s Relativity. Oxford University Press 1992. Weinberg, Steven: Cosmology. Oxford University Press 2008 Wright, Ned: Cosmology tutorial at http://www.astro.ucla.edu/ wright/cosmolog.htm Markus Pössel Basic cosmology
