1 2 1–2 1–3 C high...still: a “must 0 0 H H very Λ age is commended. he level of this lecture. and no lecture Ω (English edition also available) , Cambridge: Cambridge Univ. C , Berlin: Springer, 53.45 11.12. 05 13.11. Distances, 07 27.11. Nucleosynthesis 08 04.12. Inflation 10 08.01. Dark11 Matter 15.01. Large Scale12 Structures 22.01. Structure Formation13 29.01. Structure Formation 01 16.10. Introduction/History 02 23.10. Basic Facts 03 30.10. World Models 04 06.11. Distances, 06 20.11. Hot Big Bang Model 14 05.02. Wrap Up 09 18.12. Schedule Literature Galaxy Formation Cosmological Physics Einführung in die Extragalaktische Astronomie und C , Heidelberg: Springer, 59.95 , P., 2005, Introduction World Models Classical Cosmology The Early Universe Large Scale Structures Summary , J.A., 1999, , M.S., 1998,

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Well written introduction to cosmology, approximatelyRecommended. at t Very exhaustive, but difficult to read since the entropy per p Clear and pedagogical treatment of structure formation, re Kosmologie buy”. Press, 49.50 F F

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A A C C A A S P L Introduction 1. Cosmology Textbooks Introduction

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D A C A Friedrich−Alexander−Universität Erlangen−Nürnberg Jörn Wilms Introduction Tel.: (0951) 95222-13 Wintersemester 2007/2008 Observational Cosmology  Büro: Dr. Karl Remeis-Sternwarte, Bamberg Email:

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D D A A C C A A 5 1–6 2–1 C , San Gravitation , Cambridge: .Uses a weird notation. The very short, though. s. Nice section on classical , New York: Wiley, 129 , J.A., 1973, , Chicago: Univ. Chicago Press (only HEELER History Literature C C , K.S. & W Gravitation and Cosmology A First Course in General Relativity General Relativity HORNE $40) ∼ , S., 1972, , B.F., 1985, , C.W., T , R.M., 1984,

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R R Cambridge Univ. Press, 45.90 Classical textbook on GR, still onecosmology. of the best introduction Nice and modern introduction to GR. The cosmology section is Commonly called “MTW”, this book iscosmology as section heavy is as outdated. the subject.. Modern introduction to GR for the mathematically inclined. Francisco: Freeman, 104.90 antiquarian, F F EINBERG

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A A C C A A W S M W 2. Textbooks on General Relativity Introduction 3 4 , 1–4 1–5 , , Cambridge: , Cambridge: , Princeton: Princeton , Reading: ivity , in some parts quite readable, t to read for beginners. t. tion of structure ...Less gh energy astrophysics, focusing ogy, recently revised. nflation in the early universe is solutions) ranging from radiation Cosmology and Particle Astrophysics The Early Universe Literature Literature C C , A., 1999, Cosmology and Astrophysics Through Problems Structure Formation in the Universe C C Principles of Physical Cosmology , M.S., 1990, OOBAR An Introduction to Mathematical Cosmology URNER , T., 1996, , T., 1993, ,L.&G , P.J.E., 1993, , J.N., 2002, , E.W. & T ALEX ALEX O AN O AN IC D IC D R R R R

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New York: Wiley, 47.90 on concepts. Less detailed than Peacock, but easier to diges processes and hydrodynamics to cosmology and general relat astrophysical than the book by Longair. especially recommended. however, many forward references make the book very difficul Nice description of the physics relevant to cosmology and hi Large collection of standard astrophysical problems (with Mathematical treatment of cosmology, focusing on the forma Useful summary of the facts of classical theoretical cosmol Graduate-level text, the section on phase transitions and i 700p introduction to modern cosmology by one of its founders Cambridge Univ. Press, 46.50 Cambridge: Cambridge Univ. Press, $36.95 Cambridge Univ. Press, 42.50 Univ. Press (antiquarian only, do not pay more than $30!) Addison-Wesley, 49.90 F F

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A A C C A A B P P I K P Introduction Introduction 3 4 2–4 2–5 Hesiod First real model ). Constellations. en hidden by the Sun. Egyptian coffin lid showing two assistant astronomers, 2000...1500BC; hieroglyphs list (“decans”) whose rise defines the start of each hourthe of night. (Aveni, 1993, p. 42) ! : folk tale ( Works and Days (500–428 BC): is flat, floats in (408–355 BC): Geocentric, planets affixed Egypt (624–547 BC): Earth is flat, surrounded by 30d plus 5d extra), fixed to Nile flood (heliacal water. to concentric crystalline spheres. (730?–? BC), for planetary motions nothingness, stars are far away, fixed onrotating sphere around us. Lunar eclipses:shadow, due Sun to is Earth’s hot iron sphere Greek/Roman, I × Early Greek astronomy Thales Anaxagoras Eudoxus first appearance of in eastern sky at dawn, after it has be

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D D A A C C A A rising of ), star clocks. heliacal rising: ∼ History Atlas Farnese, 2c A.D., Museo Archeologico Nazionale, Napoli History 1 2 2–2 2–3 no written [360:60:60], 24h 360d 4cm); Back side shows . : Earliest astronomy 1100BC): Universe is 1 ∼ × ∼ 8cm . 30d year,... myth ( × through the babylonian calendar. Observations of the sky must have kr. Ulm; 3 Mul.Apin cuneiform tablet (British Museum, day, 12 sexagesimal system Pre-Babylonian astronomy: been important! records known But: 690 BC. ⇒ Babylonian astronomy Enuma Elish constellations place of battle between Earth andfrom Sky, world born parents. Note similar myth in the Genesis... Image: BM 86378, 8cm high), describes rising and setting of Summarizes astronomical knowledge as of∼ before = with influence on us: Babylon Prehistory

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D D A A C C A A History History marks which have been interpreted as a lunar calendar. “Adorant” from the Geißenklösterle cave near Blaubeuren (L 7 8 de . 2–8 2–9 ). predictions (384–322 BC, ): Refinement of Universe filled with 300d vs. / ake ⇒ 1 nature abhors caelo Eudoxus model: add spheres to ensure smooth motion = crystalline spheres ( vacuum Ether in celestial spheres, not on Earth (everything falls, except for planets and stars); Stars are very distant since they do not show . − Aristotle 25 . pical year [365 ) Hipparcus 1450AD! Greek/Roman, IV ∼ trigonometry . 127BC): Refinement of geocentric Aristotelian model into ∼ precession of 850 stars predictions 400d], through comparison with babylonian measurements (?? – / 1 + 25 . ALEX ALEX O AN O AN C D C D I R I R R Central philosophy until R

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Table of “chords” (=early Discovery of conversion of of Aristotele into a tool to m Catalogue magnitudes lunar Difference between the durations of the siderial and the tro different duration of seasons 365 F F

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D D A A C C A A History Hipparchus History tool to make = 5 6 2–6 2–7 : von en the we do not know ⇒ =

Cyrene (276–196 BC): Eratosthenes Measurement of the radius of the Earth Distance between Cyrene (Assuan) and Alexandria, diameter of Earth is 250000 stadia The length of a stadiumunknown is how precise he was. Sun : Cyrene farther away than the Moon Greek/Roman, II Greek/Roman, III × scale of the universe Alexandria (310–230 BC): Determination of the relative distance betwe × 87deg

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D D A A C C A A reality: 400 Moon and the Sun: Sun is 20 First attempts to measure Aristarch History History 11 12 2–12 2–12 (1473–1543): (1473–1543): “In no other way do we : perceive the clear harmonious linkage between the motions of the planets and the sizes of their orbs.” Nicolaus Copernicus Earth centred Ptolemaic system is too complicated, a Sun-centred system is more elegant. Nicolaus Copernicus Earth centred Ptolemaic system is too complicated, a Sun-centred system is more elegant: De revolutionibus orbium coelestium Renaissance Renaissance

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D D A A C C A A History (Gingerich, 1993, p. 165) History 9 2–10 (aka . Syntaxis 140AD): ∼ ): Refinement of Aristotelian ( Ptolemaic System ⇒ After Hipparcus and Ptolemy: end of the golden age of early astronomy. Greek works are continued by arabs and further refined. Aristotele’s philosophy remains foundation of science of medieval ages and is not questioned (in Europe). Ptolemy Almagest Foundation of astronomy until Copernicus = theory into model useable for computations Ptolemy, I

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D A C A (Aveni, 1993, p. 58) History h” (Gingerich, 2005) (Gingerich, 2005) Deleted: “Indeed, large is the work of ...God” “On the hypothesis of the triple motion of the Earth” ⇒ = The “censored” copy of Galileo’s “de revolutionibus” Changed: “On the explanation of the triple motion of the Eart Distribution of the censored copies of “De revolutionibus” 13 2–12 (1473–1543): . Florenz) “In no other way do we : Copernican principle: The Earth is not at the center ofuniverse. the perceive the clear harmonious linkage between the motions of the planets and the sizes of their orbs.” Nicolaus Copernicus Earth centred Ptolemaic system is too complicated, a Sun-centred system is more elegant: De revolutionibus orbium coelestium Renaissance The “censored” copy owned by Galileo (Gingerich, 2005, Bibl

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Tycho Brahe (1546–1601): Visual planetary positions of highest precision reveal flaws in Ptolemaic positions.

(Gingerich, 1993) The error in the Copernican position of ...

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Johannes Kepler (1571–1630): • 27.12.1571, Weil der Stadt • Studies in Tübingen with Maestlin • 1594–1600: Graz • 1596: Mysterium Cosmographicum • 1600–1612: Prag, with Brahe, ...is not smaller than the error in the ptolemaic Alfonsinian Tables court astrologer, theory of planets, discovery of the supernova of 1604,... • 1609: Astronomia Nova

History 20 23 2–22 , 1627 !) ′ 5 ∼ extreme improvement! ⇒ Comparison of positions, Kepler vs. copernican theory = (Gingerich, 1993) Best planetary positions (error only (Gingerich, 2005) Tabulae Rudolphinae Renaissance

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D A C A History 21 2–20 (Prag, ! Astronomia nova inelegant ⇒ Kepler’s theory of planetary motion: 1609) Critique of epicycles: “panis quadragesimalis” (Osterbrezel) = Astronomia Nova, chapter 1: Motion of Mars in the theory of epicycles Kepler’s laboratory book Drawing of Mars in opposition highlighted: one of the few positionsMars of done by Brahe which Keplerallowed was to use (Gingerich, 1993) Renaissance

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D A C A History 3 4 2–26 2–27 on!). (Il Saggiatore, 1623) , III Galileo Galilei, IV phases of Venus Discovery of the

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D D A A C C A A Moon has surface features, shadows, and “wiggles” (librati Galilei Galilei 1 2 2–24 2–25 (1564–1642): Telescope Siderius Nuncius (1610) Observations! ⇒ ⇒ = Galileo Galilei = Galileo Galilei, I Galileo Galilei, II similar to the heliocentric model!)... ⇒ = ( The moons of Jupiter move around Jupiter

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D D A A C C A A Galilei Galilei 7 8 2–30 2–31 see (1738–1822): Milky Way is a (1784–1846): Distance to 61 Cyg (1724–1804): “Nebulae are (1792–1871): General Catalogue (1852–1926): NGC+IC (1879–1955): Theory of ). : Stars are distant suns : Milky Way consists of stars. flattened disk of stars, Sun is at center ( of Galaxies (1864, 5079 Objects) (1838), positions of 50000 stars galaxies” (disputed until the 1910s). (15000 Objects) figure Galileo Newton William Herschel Immanuel Kant Friedrich Bessel John Dreyer gravitation, applicability of theory to evolutionthe of universe as a whole. Albert Einstein Modern Cosmology Modern Cosmology

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D D A A C C A A Galilei Galilei 5 6 2–28 2–29 Principia be explained by the . (1642–1727): Newton’s modern physics based , 1687). cannot Begin of astronomy ⇒ De Philosophiae Naturalis Isaac Newton laws, physical cause for shape ofis orbits gravitation ( Mathematica = Newton Galileo Galilei, V

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D D A A C C A A geocentric theory, only by a heliocentric theory. The observed sequence of the phases of Venus Galilei (Newton, 1730) Galilei 1 3–1 3–2 . erse as a whole. How came the universe into being? cosmological principle . Basic Facts attempt to answer is: Basic Facts , , The universe is habitable to humans. homogeneous does not • . and expanding isotropic How did the universe evolve into what it is now? need to be taken into account: • • • Realm of theology! ⇒ = four major facts anthropic principle

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The universe is:

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D D A A C C A A Cosmology deals with answering the questionsThe about main the question univ is: For this, The isotropy and homogeneity of the universe is called the i.e., the Basic Facts Perhaps (for us) the most important fact is: The one question cosmology 1 2–32 universe is expanding galaxies as being (1889–1953): outside of the Milky Way Realization of Discovery that Founder of modern extragalactic astronomy • • Edwin Hubble Edwin Hubble oks) , Kepler, (New York: American Institute of Physics) : Smithsonian Books) ), reprint: Dover Publications, 1952

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D A C A Gingerich, O., 1993, The Eye of Heaven – Ptolemy,Gingerich, Copernicus O., 2005, The book nobody read, (London: arrow bo Christianson, 1995, p. 165 Edwin Hubble 2–32 Aveni, A. F., 1993, Ancient Astronomers, (Washington, D.C. Newton, I., 1730, Opticks, Vol. 4th, (London: William Innys

Wavelength Redshift of Source of Redshift 2 ) 3–3 δ , (3.1) ◦ λ/λ δ 48 cos v is = α sin r b = ∆ , Z ◦ z v cos X + 264 := v δ = + l different parts of the v/c cos r z 0 t α H sin towards 1 ) = Y − r v ( v + for galaxy at distance Hubble (1929): Velocity (defined as 350kms ∼ component of velocity due to Expansion, I const. (cf. Eq. (4.37)). intrinsic 6 = courtesy 2dF QSO Redshift survey 0 H spectrum of a distant source are visible. of the universe. velocity due to motion of ( ) Z : “Hubble parameter”; ,v 0 ALEX O AN IC D Y R R E I

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A C A v ( Bennet et al., 1996) (Hubble, 1929, Fig. 1) Old usage: “Hubble constant”, but Basic Facts As a consequence of the cosmological redshift, for differen 8 9 o r 3–9 0 3–10 of H )= ′ : r unchanged − is r : Observed ( ′ 0 r v , observed velocity 0 ′ H r homogeneity H = translation − ′ = r r 0 v . Because of the Hubble law, = and H ′ o v . − r and velocity at the center of the − Observations from place with ′ r 0 r v Trivial. H not = rotation = o ! o v v is position distance is This isomorphism is a direct consequence of the Proof: Rotation: Translation: Expansion law under isomorphism the universe. Expansion, VII Homogeneity and Isotropy, I Copernicus principle still holds. ⇒ = homogeneity does not imply isotropy

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Despite everything receding from us, weuniverse are

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D D A A C C A A Basic Facts after Silk (1997, p. 8). Note that Basic Facts 6 7 75 . h 3–7 3–8 (3.3) (3.4) (3.5) 1 · h (3.2) − · 1 2 1 − / . 2 1 / − 1 1  Mpc −  1 . Mpc L πf − Mpc 1 L πf 4 s ) 4 1 Mpc −  1 − 1  0 M − − H = − ; L ∝ m d ( z b is known, the Hubble law 10km observed flux , i.e., objects where the 100kms 75kms + L 75kms ∼ are defined via ⇒ a = = ⇒ = ⇐⇒ = ∼ is 0 0 : = luminosity distance f 0 0 z H H 2 L The systematic uncertainty of H Parameterize uncertainty in formulae by defining Currently accepted value: H Hubble-Time log cz log L is the πd : 5 . = 4 L 2 d L /h magnitudes = d ⇒ 0 standard candles f ∝− H dV 1980 For Using the Hubble law eq. (3.1) absolute luminosity can be written using observed quantitiesEuclidean only: space where Since m ) = 78 Gyr f . ST 9 log dV 1970 = − dV 1 Expansion, V L Expansion, VI A − 0 H S 1960 McV (log : . 2 1 HMS T B Year B + 0 time 1950 H M log ∝ 1940 H z H distance modulus log H H H : has units of 1930 L M 1 75, the Hubble-Time is 13 Gyr. . − 0 − 0 0 H

600 500 400 300 200 100

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D D A A C C A A for where (after Trimble, 1997) Basic Facts Basic Facts Note: gue (larger structures, not [yet] . superclusters structure 220000galaxies total ∼ (Jarrett, 2004, Fig. 1) gravitationally bound). Below that: (gravitationally bound) and 2dF Survey, 100Mpc the universe looks indeed the same. ≫ galaxy clusters On scales Distribution of Galaxy redshifts in the 2MASS galaxy catalo Structures seen are 10 11 3–11 3–12 ! . imply homogeneity The universe looks the same everywhere in Homogeneity, I ⇐⇒ Homogeneity and Isotropy, II spatial distribution of galaxies around one point 220000galaxies total assumptions need to be tested. ∼

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D D A A C C A A Neither does isotropy Basic Facts 2dF Survey, The universe is homogeneous Testable by observing Basic Facts = space 3–16 4–1 World Models erican Library 53, (New York: W. H. Freeman) Princeton: Princeton Univ. Press)

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D A C A Bennet, C. L., et al., 1996, ApJ, 464, L1 Hubble, E. P., 1929, Proc. Natl. Acad. Sci. USA, 15,Jarrett, T., 168 2004, Proc. Astron. Soc. Aust., 21, 396 Peebles, P. J. E., 1993, Principles of Physical Cosmology, ( Trimble, V., 1997, Space Sci. Rev., 79, 793 Silk, J., 1997, A Short History of the Universe, Scientific Am 4 − 14 15 10 3–15 3–16 . 6 cm from 1) T/T = ∆ are mainly & λ z . ◦ due to motion of Sun The universe looks the Clear isotropy. Sample large space volume ( to 180 for isotropy: Intensity of ′′ ⇒ ⇒ quasars = same in all directions Radio galaxies Peebles (1993): Distribution of 31000 objects at = the Greenbank Catalogue. Anisotropy in the image: galactic plane, exclusion region around Cyg A, Cas A, and the north celestial pole. The universe is isotropic ⇐⇒ : structure in CMB due to structure of 5 − dipole anisotropy Best evidence 3 K Cosmic Microwave Background (CMB) radiation. First: on scales from 10 surface of last scattering of thestructure CMB at photons, the i.e., time when Hydrogen recombined. At level of 10 (see slide 3–3), after subtraction: Isotropy Isotropy

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D D A A C C A A Basic Facts Basic Facts 2 3 4–4 4–5 (4.3) (4.1) (4.2) 2D spaces ν at x d µ two-dimensional spaces: x d µν g 1 0 2 2 : =: 2 x = = d ν Ê 22 12 x + g g d 2 1 ) µ ◦ x curved x curved d for these spaces looks like. d 180 0 1 = µν < 2 = = g 2D Metrics 2D Metrics ) s d , or 21 11 metric ν g g = X , positively zero curvature negatively > ) isotropic and homogeneous µ 2 X of , in Euclidean space, = H s , is defined through 2 µν s ) g d ) 2 , 2 Ê , S 2 The metric describes the local geometry of a space. Ê angles in triangle P summation convention three classes ≈ -plane ( y - ALEX ALEX O AN O AN C D C D x hyperbolic plane ( I R 2-sphere ( I R R R metric tensor

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D D A A C C A A We will now calculate what the Before describing the 4D geometry of the universe: first look (easier to visualize). After Silk (1997, p. 107) There are (curvature FRW Metric Differential distance, d Thus, for the FRW Metric The (Einstein’s 1 1 4–2 4–3 and physics. : iple. ertial systems. homogeneous separately coordinate-system . terature (Weinberg or MTW). (i.e., locally, SRT holds). Brief introduction to assumptions of holds: The universe is (Einstein field equation). of universe using Einstein field equations. Structure equation of state : There is no experiment by which one can GRT vs. Newton must be formulated in a , might be curved locally Minkowski . modifies space obeying cosmological principle. . way. physical laws cosmological principle equation for evolution metric General Relativity Thermodynamics • • • (=Energy) . Solve equations ALEX ALEX O AN O AN IC D IC D R combination of R R Complicated! R I I

Need theoretical framework obeying the cosmological princ

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Covariance: At each point, space is distinguish between free falling coordinate systems and in independent Space is 4-dimensional Matter Strong equivalence principle F F

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D D ⇒ A A C C A A Assumptions of GRT: = Before we can start to thinkgeneral about relativity universe: = isotropic = For 99% of the work, the above points can be dealt with Introduction FRW Metric Observations: = Use 6 7 4–8 4–9 (4.4) (4.9) (4.10) (4.12) (4.11) (4.13) (4.14) θ d edious algebra θ cos ′ r 2 + ′ 2 ) r S 2 d 2 2

x 2 x θ d on φ θ 2 − d x 2 1 2 sin φ θ x ′ φ = sin + ′ r ) 2 1 2 r 2 d − x x θ 2 sin cos 2 d ′ d =: θ R r is θ θ 1 2 2 2 2 : r x + sin − ( R 2 2 S + θx cos sin sin and d + θ 2 R 2 2 d R R R r 2 cos -plane x θ r + ′  3 d − d d r = = = 2 2 x 2D Metrics 2D Metrics θ 1 θ + 3 2 1 1) results in d R in =: 2 1 ( x x x 2 1 x θ 2 ′ sin ≤ ′ = r d , x ′ R spherical coordinates r r r 2 = = s = − ≤ 2 2 ′ d s s r d d d θ ). ] (i.e., 0 π 2 , /R = cos ′ 0 r 1 [ x = d ∈ polar coordinates r φ , ] are only unique in upper or lower half-sphere) θ , π , ′ 0 ALEXA r ALEXA

O N O N [ IC D IC D R R R R

E I E I

N D A N D A I I

E E

R R

F F

S S ∈

I I

G G E E

I I A A

I I L L

L L

M M

V V

E E

M M

D D

A A C C A A θ In cartesian coordinates, the length element on Going through the same steps as before, we obtain after some t (note: inserting eq. (4.9) gives after some algebra ( The differentials are given by Introduce again Alternatively, we can work in FRW Metric FRW Metric finally, defining 4 5 4–6 4–7 (4.7) (4.8) (4.5) (4.4) (4.6) 2 2 2 x x d − 2 3 2 2 2 2 1 x x 2 2 x d } x θ 2 + θ R θ d − + − θ 1 2 2 1 2 2 2 = 2 x d ′ x x x d 2 3 sin cos , then gives d r R 2 d : ′ ′ x 1 2 3 3 − r r r q x + + Ê : 2 + Rr 3 ! ∂x ∂x 2 1 + − 2 2 2 R ′ Ê in x =: =: x 2 = r + = d q r R 2 1 ′ d 1 + d r = x x = x 2 1 { d = 2 3 x s x 1 3 R 2 d s ∂x ∂x = d 2 = s 3 d x d Performing a change of scale by substituting Eq. (4.7) gives it is easy to see that such that Length element of . 2 x 1 Two-sphere with radius x , defined by 2D Metrics 2D Metrics θ , ′ r ds space is curved R dr´ θ r´ φ r´ r´d 3 x θ θ polar coordinates θ surface of three-dimensional sphere (a two-sphere). θ d 1 x

ALEX ALEX O AN O AN

IC D Other coordinate-systems are also possible in the plane IC D R R R R

E I E I

N D A N D A I I

E E

R R

F F

S S

I I

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L

L 2

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D D A A C C A A x But: Changing to A more complicated case occurs if Easiest case: FRW Metric After Kolb & Turner (1990, Fig. 2.1) FRW Metric 10 4–12 (4.20) (4.19) are given by se coordinates give

2 φ d θ φ φ 2 cos sin θ θ θ . + sinh 2 sinh sinh cosh θ d R R R  = = = 2D Metrics 2 1 2 3 R x x x = infinite volume ). ] 2 s π d 2 , 0 [ ∈ φ and has an , ] ∞ + , unbound −∞ ALEXA [ O N is IC D R R

E I D A N I

E 2

R

F

S ∈

I

G E

I A

I L

L

M

V

E

M D

A C A θ The analogy to spherical coordinates on the hyperbolic plane Remark: H FRW Metric 4–12 Transcript of Maple session to obtain Eq. (4.20): ( A session with Maple (see handout) will convince you that the 8 9 , R 4–10 4–11 (4.18) (4.15) (4.16) (4.17) , they R finite volume ⇒ = different on the value of R 2 very ) . 2 2 2 2 x x d S look  2 + 2 x θ 2 1 , but has still a 2 d independent variation of x + , 2 R 1 θ + r − x 2 d + = R 2 3 1 2 and scale factor 2 3 x r x 2 r ( d x r + 1) to obtain same form as for sphere d comoving coordinates − − − 1 − that’s why physics should be covariant, i.e., 2 2 2 2 2 2 no boundaries  √ x x x ⇒ 2 d d 2D Metrics 2D Metrics = = + R + + 2 1 i , has 2 1 2 1 x = is called the x x are called 2 d d R s , is defined by θ space 2 d = = of volumes and distances on Eq. (4.10), (4.12), and (4.14) space, where (where 2 H s and no edges , d iR are defined, e.g., by r same 2 scale → metrics S R . Minkowski 2 on πR 4 describe the

ALEX = ALEX O AN O AN C D C D I R I R R R hyperbolic plane I I

substitute

E E (eq. 4.11)!

N D A N D A I I

E E

R R

V still independent of the coordinate system! determines the Positions F F

S S

I I

G G E E

I I A A

I I L L

L

L ⇒

M M

V V

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D D A A C C A A 2. Expansion or contraction of sphere caused by 3. 4. Although the 1. The 2-sphere has (Important) remarks: The FRW Metric FRW Metric If we work in then = Therefore, 13 14 (4.25) (4.26) . 4–15 4–16 (4.27) . metrics (RW) : i 2 ψ  d 2 comoving coordinates ) ψ cosmical coordinate system r d ( 2 2 ) g φ r d ( + tropic, clocks can be synchronized, e.g., by 2 k θ 2 r 2 S d ) + r Robertson-Walker 2 ( + sin 2 r spherically symmetric 2 f d θ h  freely expanding d ) ) t on sky, as seen from the arbitrary center of ( t := 2 ( 2 2 ⇒∃ R can be interpreted as a radial coordinate. RW Metric RW Metric = measure distances using ψ R r − d 2 ⇒ − t d spatial part is 2 )= t 2 t c d directions ( ⇒ = 2 R = expansion c , 2 s + = d 2 s d describe are arbitrary. fundamental observers ) φ r =: scale factor ( g ∃ cosmic time and =: and θ ) comoving coordinates r ( Metric has temporal and spatial part. : scale factor, containing the physics Observers Time ALEXA f ALEXA

O N O N was defined in Eq. (4.24). IC D IC D φ R R R R : I I

E ⇒ E

– –

N D N D

A A ) metric looks like )

I I ,

E E

R R the coordinate system (=us), adjusting time to the local density of the universe. = This is the coordinate system in which the 3K radiation is iso This also follows directly from the equivalence principle. Expansion: Homogeneity and isotropy Cosmological principle F F

r t

S S

θ

I I

cosmic time G G

E E

(

I I ( A A

I I L

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M M

V

V k E E

M

M ,

⇒

D D A A C C A A : S where where FRW Metric Metrics of the form of eq.(introduced (4.26) in are 1935). called Previously studied by Friedmann and Lemaître... One common choice is R t r Remark: FRW Metric = 11 12 (4.6) (4.6) (4.24) (4.22) (4.21) (4.20) (4.23) (4.12) (4.18) (4.14) 4–13 4–14 1 0 1 − = =+ = k k k for for θ θ cosh 1 for cos          )= ) o θ o ) 2 ) 2 ( o 2 2 2 θ 2 2 k φ θ φ θ d d φ d d d o 2 d o kS 2 θ 2 ) 2 r 2 θ θ 2 r θ r φ − d ( 2 + 2 d es have been renamed. This is confusing, but k + + 2 1 2 2 S r 2 θ 2 r q r r 2 2 2 + + k + sin + + sinh r r r 2 2 − + 2 2 2 d d d − θ r )= θ θ θ 1 hyperbolic 1 spherical 0 planar 1 1 d d θ d d d 1 ( + − n ( n ( n n n k ( 2 2 2 2          2 2 2 2 C 2D Metrics 2D Metrics R R R R R R R R ======k = 2 2 2 2 2 2 2 2 and s s s s s s s s d d 1 1 0 − , will be needed later k = = =+ C k k k , k S :: d d : d ::: d d d for for for θ θ Plane Plane Sphere Sphere : sinh θ sin          defines the geometry: )=

ALE k ALE

X X θ O AN O AN C D C D I R I R R R ( I I

E Hyperbolic Plane E

Hyperbolic Plane N D A N D A All three metrics can be written as All three metrics can be written as

I I k

E E

R R

F F

S S

summarize

S

I I

G G E E

I I A A

I I L L

L L

M M

V V

E E

M

M ⇒ ⇒

D D A A C C A A Note that, compared to the earlier formulae, some coordinat legal. . . The cos-like analogue of = To For “spherical coordinates” we found: where FRW Metric where FRW Metric = ), 17 18 0 4–19 4–20 (4.31) (4.33) (4.32) c/H = non-trivial isotropy! r where we ⇒ ) t = ( a ,  ) 2 t (

ψ 2 d  2 ψ 2 t/R d r ψ d 2 d r Cosmologies with a + Equivalence principle! 2 = 0, i.e., a flat space, the RW 2 r + ) η ⇒ is also possible: = 2 isotropic form r ⇒ = − r 0 r ,d k d R ) = 2 (  kr 2 η r · ( 2 r d −→ 2 r − d k ) ) R 1 4 r ) − ( t × ( k − k/ 1 S 2 R RW Metric RW Metric Used for observations!  η +( ) d t 1 ⇒ conformal time (  2 = ) a − η is just the line element of a 3d-sphere 2 ( , by − t 2 t } , d 2 R 2 t c d ... = { 2 are Mpc = homogeneity and isotropy 2 c Be careful! the substitution r s 2 0 s d = ⇒ d R 2 = homogeneity and isotropy (i.e., within a Hubble radius, and s global d ) Minkowski line element t ( = a are possible (e.g., also with more dimensions. . . ). , etc. ) conformal metric t There are as many notations as authors, e.g., some use ( ALEX ALEX O AN O AN IC D IC D R R R R ⇒

E I E I R

N D A N D A I I

E E

R R

The units of Here, the term in Theoretical importance of this metric: For metric = F F

S S

I I

G G E E

I I A A

I I L L

L L

M M

V V

E E

M M

D D A A C C A A 3. Using 4. Replace cosmic time, 5. Finally, the metric can also be written in the use Note: do not imply FRW Metric Note 2: Local topology FRW Metric 15 16 4–17 4–18 (4.30) (4.28) (4.29) . d R(t2) pansion of . B(x2,y2)   2 2 ψ ψ d d A(x1,y1) ) ) 2 t 0 r ( r 0 2 0 R , R R + R ( 2 k 2 = , S ). evolution of universe 2 ) kr ) r 0 r t + t ( d ( − 2 R r R 1 d  .  ) ) t ) := RW Metric RW Metric t ( describes t ( 2 . 1), gives proper distance ( ) 2 are measured in Mpc) d R(t1) a t R a ) ( t ( ) = − R gives − R 0 2 · t 2 r t ( t d d d a 2 2 and c c −→ d B(x2,y2) ) is called the = = r dimensionless scale factor ) ( 2 2 t comoving distance k s ( s A(x1,y1) Scale factor d S d R · =today, i.e., d 0 t is unitless, i.e., R ) := t ALEXA ALEXA O N ( O N IC D is called the IC D R R R R

E I E I

N D A N D A I I

E E

R R

D d (i.e., other definition of comoving radius (where F F

S S

I I

G G E E

I I A A

I I L L

• •

L L

M M

V V

E E

M M

D D A A C C A A 1. Substitution 2. A metric with a The RW metric defines an universalspace: coordinate system tied to ex Other forms of the RW metric are also used: FRW Metric FRW Metric (note that