Observing the Big Bang

Matthias Bartelmann Institut fur¨ Theoretische Astrophysik, Universitat¨ Heidelberg and Eric Bell Max-Planck-Institut fur¨ Astronomie, Heidelberg

January 15, 2009 Contents

1 Preamble 1 1.1 Purpose ...... 1 1.2 Schedule ...... 1 1.3 Assigned reading ...... 3 1.4 Contact Details ...... 4

2 The cosmological standard model 5 2.1 Introduction ...... 5 2.2 Observational overview; the basics ...... 5 2.3 A brief history of time ...... 6 2.4 Friedmann models ...... 7 2.4.1 The metric ...... 7 2.4.2 Redshift and expansion ...... 9 2.4.3 Age and distances ...... 10 2.4.4 The radiation-dominated phase ...... 11 2.5 Structures ...... 12 2.5.1 Structure growth ...... 12 2.5.2 The power spectrum ...... 13 2.5.3 Non-linear evolution ...... 14

3 The age of the Universe 16 3.1 Nuclear cosmo-chronology ...... 16 3.1.1 The age of the Earth ...... 17 3.1.2 The age of the Galaxy ...... 18

i CONTENTS ii

3.2 Stellar ages ...... 20 3.3 Cooling of white dwarfs ...... 23 3.4 Summary ...... 23

4 The Hubble Constant 25 4.1 Hubble constant from Hubble’s law ...... 25 4.1.1 Hubble’s law: history ...... 25 4.1.2 Hubble’s law: the challenge ...... 26 4.1.3 The distance ladder: the ﬁrst 20Mpc ...... 27 4.1.4 Distance Ladder: extending beyond 20Mpc . . 30 4.1.5 The HST Key Project ...... 32 4.2 Gravitational Lensing ...... 33 4.3 The Sunyaev-Zel’dovich eﬀect ...... 34 4.4 Summary ...... 36

5 Big-Bang Nucleosynthesis 38 5.1 Key concepts ...... 38 5.2 The origin of Helium-4 and the other light elements . . 39 5.2.1 The riddle of Helium ...... 39 5.2.2 Elementary considerations ...... 40 5.2.3 The Gamow criterion ...... 41 5.2.4 Elements produced ...... 42 5.2.5 Predicted approximate Helium abundance . . . 43 5.2.6 Expected abundances and abundance trends . . 44 5.3 Observed element abundances ...... 45 5.3.1 Principles ...... 45 5.3.2 Evolutionary corrections ...... 46 5.3.3 Speciﬁc results ...... 46 5.3.4 Summary of results ...... 48

6 The Matter Density in the Universe 50 CONTENTS iii

6.1 Key concepts ...... 50 6.2 Baryonic mass in galaxies ...... 51 6.2.1 Stars ...... 51 6.2.2 Cold gas ...... 52 6.2.3 Warm/hot gas in between galaxies ...... 52 6.3 Total mass in galaxies ...... 53 6.3.1 Galaxies ...... 53 6.3.2 Mass in galaxy clusters: kinematic masses . . . 55 6.3.3 Mass in galaxy clusters: the hot intracluster gas 55 6.3.4 Alternative cluster mass estimates ...... 57 6.4 Mass density from cluster evolution ...... 58 6.5 Musings on the nature of the dark matter ...... 60

7 The Cosmic Microwave Background 62 7.1 The isotropic CMB ...... 62 7.1.1 Thermal history of the Universe ...... 62 7.1.2 Mean properties of the CMB ...... 64 7.1.3 Decoupling of the CMB ...... 65 7.2 Structures in the CMB ...... 67 7.2.1 The dipole ...... 67 7.2.2 Expected amplitude of CMB ﬂuctuations . . . 67 7.2.3 Expected CMB ﬂuctuations ...... 68 7.2.4 CMB polarisation ...... 71 7.2.5 The CMB power spectrum ...... 72 7.2.6 Microwave foregrounds ...... 72 7.2.7 Measurements of the CMB ...... 74

8 Inflation and Dark Energy 80 8.1 Cosmological inflation ...... 80 8.1.1 Motivation ...... 80 8.1.2 The idea of inflation ...... 82 CONTENTS iv

8.1.3 Slow roll, structure formation, and observational constraints ...... 84 8.2 Dark energy ...... 86 8.2.1 Motivation ...... 86 8.2.2 Observational constraints? ...... 87

9 Cosmic Structures 90 9.1 Quantifying structures ...... 90 9.1.1 Introduction ...... 90 9.1.2 Power spectra and correlation functions . . . . 91 9.1.3 Measuring the correlation function ...... 92 9.1.4 Measuring the power spectrum ...... 93 9.1.5 Biasing ...... 96 9.1.6 Redshift-space distortions ...... 97 9.1.7 Baryonic acoustic oscillations ...... 98 9.2 Measurements and results ...... 98 9.2.1 The power spectrum ...... 98

10 Cosmological Weak Lensing 102 10.1 Cosmological light deﬂection ...... 102 10.1.1 Deﬂection angle, convergence and shear . . . . 102 10.1.2 Power spectra ...... 105 10.1.3 Correlation functions ...... 106 10.2 Cosmic-shear measurements ...... 108 10.2.1 Typical scales and requirements ...... 108 10.2.2 Ellipticity measurements ...... 110 10.2.3 Results ...... 112

11 The Normalisation of the Power Spectrum 114 11.1 Introduction ...... 114 11.2 Fluctuations in the CMB ...... 115 11.2.1 The large-scale ﬂuctuation amplitude . . . . . 115 CONTENTS v

11.2.2 Translation to σ8 ...... 117 11.3 Cosmological weak lensing ...... 119 11.4 Galaxy clusters ...... 120 11.4.1 The mass function ...... 120 11.4.2 What is a cluster’s mass? ...... 122

12 Supernovae of Type Ia 125 12.1 Standard candles and distances ...... 125 12.1.1 The principle ...... 125 12.1.2 Requirements and degeneracies ...... 126 12.2 Supernovae ...... 129 12.2.1 Types and classiﬁcation ...... 129 12.2.2 Observations ...... 131 12.2.3 Potential problems ...... 134

13 Galaxies in a cosmological context 137 13.1 Predictions for galaxy formation in a cosmological context138 13.1.1 Evolution of the dark matter framework . . . . 138 13.1.2 Evolution of the baryonic content of dark matter halos ...... 138 13.2 Major galaxy merging ...... 139 13.3 The accretion of small galaxies through minor mergers 141 13.4 Challenges to ΛCDM on small scales ...... 142 13.4.1 Rotation curve problem ...... 142 13.4.2 The substructure problem ...... 143 13.4.3 The bulgeless galaxy problem ...... 144

14 Appendix 145 14.1 Cosmological parameters ...... 145 14.2 Cosmic time, lookback time and redshift ...... 146 14.3 Linear growth factor ...... 147 14.4 Distances ...... 148 CONTENTS vi

14.5 Density and Hubble parameters ...... 149 14.6 The CDM power spectrum ...... 150 Chapter 1

Preamble

1.1 Purpose

The intention of this course is twofold. The ﬁrst goal is to develop a deeper understanding of the Universe, and in particular those aspects of our Universe which contain large amounts of information about the evolution of the cosmos. The second goal is to develop a sense of as- tonishment that one can construct a theoretical framework in which to interpret the vast and diverse observational cosmological dataset, and that this theoretical framework appears to be remarkably consistent with the data. We will discuss the successes of this framework, and we will discuss also challenges that lay ahead.

1.2 Schedule

There are no examinations. There are two workshops and one home- work:

• We will have a distance ladder workshop, where each student chooses some aspect of distance ladder to study and discuss in a little more detail in class [∼10 minutes per student].

• We will have a dark energy workshop, where each student chooses a future initiative to explore cosmological parameters [∼10 minutes per student]

• An assignment about the cosmic microwave background (CMB). The idea is to collect ‘10 fascinating pieces of information about the CMB’ in note form, http://background.uchicago.edu/∼whu/ is a particularly useful reference.

1 CHAPTER 1. PREAMBLE 2

Furthermore, the last few lectures will take the form of a discussion, where we have an interactive discussion about some key questions in using cosmic structures, weak lensing, and supernovae to understand cosmology. This part draws on the course notes and on the Fundamen- tal Cosmology joint ESA-ESO report (Peacock & Schneider) in equal measure. http://www.stecf.org/coordination/esa eso/cosmology/report cover.pdf You may also find the following resources useful: these are discussions of individual mission concepts to study cosmology (and dark energy in particular). These give a flavor of the future possible trajectory of this field. http://de.arxiv.org/pdf/astro-ph/0507460 http://de.arxiv.org/pdf/astro-ph/0507459 http://de.arxiv.org/pdf/astro-ph/0507458 http://de.arxiv.org/ftp/astro-ph/papers/0507/0507457.pdf http://de.arxiv.org/pdf/astro-ph/0507043 http://de.arxiv.org/pdf/astro-ph/0507013

The class takes place in the Kleiner Horsaal¨ on Philisophenweg, unless stated otherwise.

• 6 October 2008 - 15:15-17:00 Preamble and Introduction • 13 October 2008 - 14:15-16:00 Cosmic Ages • 20 October 2008 - 14:15-16:00 The Hubble Constant • 27 October 2008 - 14:15-16:00 Distance Ladder Workshop and revision of themes so far • 3 November 2008 - 14:15-16:00 Big Bang Nucleosynthesis • 10 November 2008 - 14:15-16:00 Cosmic Matter • 17 November 2008 - 14:15-16:00 CMB • 24 November 2008 - 14:15-16:00 Inﬂation and Dark Energy • 1 December 2008 - meet at 12:50 at Bismarkplatz, at bus stop for the 39 bus - Visit to MPIA • 8 December 2008 - 14:15-16:00 Cosmic Structures • 15 December 2008 - 14:15-16:00 Weak Lensing • 12 January 2009 - 14:15-16:00 Supernovae • 19 January 2009 - 14:15-16:00 Challenges for LCDM on Galaxy Scales • 26 January 2009 - 14:15-16:00 Dark Energy Workshop CHAPTER 1. PREAMBLE 3

1.3 Assigned reading

The Fundamental Cosmology joint ESO/ESA report is an excellent re- source for understanding the state of the art in the community’s understanding of the basic issues needing to be addressed in order to make progress in understanding cosmic acceleration. http://www.stecf.org/coordination/esa eso/cosmology/report cover.pdf It is worth reading the whole report through, as there are issues not touched on in this lecture series: e.g., constraints from varying fundamental constants, and gravity wave cosmology.

• For chapter 9, look at Chapters 5 and 9 of the Fundamental Cos- mology report. 1. Why does the general break scale of the power spectrum give the scale of the horizon at matter-radiation equality?

2. How can σ8 be measured from galaxy clustering (according to the Fundamental cosmology report?) 3. For Baryon Acoustic Oscillation surveys, what is the trade- oﬀ between spectroscopic redshift and photometric redshift surveys? 4. How are Lyman α ﬂux distributions observed? 5. What is/are the main use/uses of Lyman α power spectra for cosmology? How do observations relate to these astrophysical constraints? • For chapter 10, look at Chapter 7 of the Fundamental Cosmology report. 1. How does weak lensing give access to cosmological parameters? 2. How mature is weak lensing, as a method? 3. Why is there so much interest in weak lensing? 4. What are some of the critical challenges observationally, and theoretically? • For chapter 11, review chapters 5 and 7, and read through chapter 6 of the Fundamental Cosmology report. 1. What are the main cosmological parameters which are ac- cessible via galaxy clusters? 2. What are some of the systematic uncertainties? 3. Do you agree with the message of the Fundamental Cosmol- ogy report that the uncertainties will be able to be calibrated out well? CHAPTER 1. PREAMBLE 4

• For chapter 12, look through Chapter 8 of the Fundamental Cos- mology report.

1. What are the main strengths of supernovae (what are the main cosmological paramters which can be accessed using this method)? 2. How are SNe transformed into standard candles? 3. Be ready to discuss the main sources of systematic error, and ideas/proposals on how to reduce them. 4. What are the main observational requirements for next generation surveys?

1.4 Contact Details

Eric Bell Max-Planck-Institut fur¨ Astronomie Konigstuhl¨ 17 69117 Heidelberg

Tel: 06221 528 263 Fax: 06221 528 369 [email protected] http://www.mpia-hd.mpg.de/homes/bell/ Course website, with links to all materials — http://www.mpia- hd.mpg.de/homes/bell/obsbb.html Chapter 2

The cosmological standard model

2.1 Introduction

• One of the landmark achievements of the last decade of astronomical study is the establishment of a ‘cosmological standard model’. By this term, we mean a consistent theoretical background which is at the same time simple and broad enough to oﬀer coherent explanations for the vast majority of cosmoogical phenomena.

• This lecture will explain and discuss the empirical evidence to which this cosmological standard model owes its convincing power. The construction of homogeneous and isotropic cosmolo- gies from general relativity, and the study of their physical properties and evolution, is treated elsewhere (see, e.g. the separate lecture scripts on general relativity and on cosmology).

• We will start with a brief overview of a few relevant observational facts about the Universe which played a critical role in motivating our current cosmological picture, continue with a short timeline of how it is currently imagined that the Universe evolved, and we will review a few key aspects of the cosmological model. The bulk of the course will discuss in depth a number of observations/methods, and how they ﬁt in to our current cosmological picture.

2.2 Observational overview; the basics

• Galaxies exist - the Universe is ﬁlled with hundreds of billions of galaxies, with a wide range in properties. In their inner parts,

5 CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 6

their mass is dominated by cold gas (mostly neutral or molecular hydrogen) and stars (really dense hydrogen!).

• Exapanding Universe - Slipher and Hubble demonstrated con- vincingly in the 20s and 30s that the Universe is expanding, and that the redshift is proportional to distance (at least locally). The truly astounding result, initially determined in the 1990s using Supernova Ia, is that the expansion of the Universe is accelerating at the present time. Na¨ıvely, this is counterintuitive, inasmuch as the matter content of the Universe should be decelerating its own expansion.

• Dark Matter - both inside galaxies and in galaxy clusters, motions of gas and stars imply gravitational masses dramatically in excess√ of any mass plausibly associated with stars and gas, i.e., v GMvisible/R. This discrepancy was ﬁrst noted in the 1940s by Fritz Zwicky, and became really obvious in the 1970s in the study of spiral galaxy rotation curves.

• Cosmic Microwave Background - There is a cosmic microwave background with T ∼ 2.73K. Its main deﬁning feature is its astonishing level of homogeneity – it is ﬂat to 1 part in 105.

• Helium - There is way too much Helium in the Universe to have been made in just stars (or, put diﬀerently, the ratio between He- lium and the other, heavier elements is just too high to be ex- plained by being made in stars alone).

There are a large number of other important observations, but these are the key aspects which are useful to bear in mind when trying to parse the elements of our current cosmological picture.

2.3 A brief history of time

Our cosmological picture uniﬁes ‘known physics’ with a few novel ingredients motivated entirely by astronomical observation.

• Inﬂation - it is currently postulated that the Universe had very early in its evolution a phase of exponential expansion (roughly 60 e-foldings) which took microscopically small parts of the Uni- verse and boosted them in scale to giant, macroscopic scales (∼ galaxy scales and larger). This has two main advantages. It solves the ﬂatness problem (the Universe would recollapse or expand very rapidly in the case of Ωinitial > 1 or Ωinitial < 1. Ωinitial needs to be within 1 part in 1060 of unity to ensure that Ω ∼ 1 today (as CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 7

17 |Ωtotal −1| ∝ t, and given that the age of the Universe is ∼ 10 s to- −43 day and evaluating the Ωinitial at the Planck time tPlanck ∼ 10 s). There is the advantage also that it solves the horizon problem, and blows up tiny quantum fluctuations in the initial density field into density perturbations that later seed structure formation. Inflation is completely motivated by cosmology, although physics theorists are happy about it.

• Radiation Dominated Era - At this time T ∝ 1/a; i.e., as the Universe expands it cools also. Protons and neutrons freeze out of the mix as T 1 GeV. The ratio of p to n is determined by their mass diﬀerence coupled with the decay half life of neutrons, ending up in a n/p number ratio of 1/7. When temperatures are low enough T ∼ 1 MeV nucleosynthesis can proceed, forming 2H, 3H, 4He, 7Li, etc.

• Recombination - At T ∼ 3000K, the hydrogen can no longer stay ionised, so it recombines. There are no more electrons to Thomp- son scatter oﬀ of, so the Universe suddenly becomes transparent and photons can freely propagate. The CMB is a relic of this transition, and oﬀers a direct view of the structure of the Universe at this time.

• Matter Dominated Era - structure formation, galaxy formation, expansion of the Universe. A ‘boring’ time for cosmologists (who want to measure the formation of structure and the expansion history to measure cosmological parameters), and the most exciting time for astrophysicists (galaxy formation, star formation, planet formation, life, etc - minor details!) The main astrophysically- motivated ingredients here are dark matter and dark energy / cosmological constant.

2.4 Friedmann models

2.4.1 The metric

• Cosmology deals with the physical properties of the Universe as a whole. The only of the four known interactions which can play a role on cosmic length scales is gravity. Electromagnetism, the only other interaction with inﬁnite range, has sources of opposite charge which tend to shield each other on comparatively very small scales. Cosmic magnetic ﬁelds can perhaps reach coher- ence lengths on the order of & 10 Mpc, but their strengths are far too low for them to be important for the cosmic evolution. The weak and the strong interaction, of course, have microscopic range and must thus be unimportant for cosmology as a whole. CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 8

• The best current theory of gravity is Einstein’s theory of general relativity, which relates the geometry of a four-dimensional space-time manifold to its material and energy content. Cosmo- logical models must thus be constructed as solutions of Einstein’s field equations. • Symmetry assumptions greatly simplify this process. Guided by observations to be specified later, we assume that the Universe appears approximately identically in all directions of observation, in other words, it is assumed to be isotropic on average. While this assumption is obviously incorrect in our cosmological neighbourhood, it holds with increasing precision if observations are averaged on increasingly large scales. • Strictly speaking, the assumption of isotropy can only be valid in a prefered reference frame which is at rest with respect to the mean cosmic motion. The motion of the Earth within this rest frame must be subtracted before any observation can be expected to appear isotropic. • The second assumption holds that the Universe should appear equally isotropic about any of its points. Then, it is homogeneous. Searching for isotropic and homogeneous solutions for Einstein’s field equations leads uniquely to line element of the Robertson-Walker metric, " # dr2 ds2 = −c2dt2 + a2(t) + r2 dθ2 + sin2 θdφ2 , (2.1) 1 − kr2 in which r is a radial coordinate, k is a parameter quantifying the curvature, and the scale factor a(t) isotropically stretches or shrinks the three-dimensional spatial sections of the four- dimensional space-time; the scale factor is commonly normalised such that a0 = 1 at the present time; • as usual, the line element ds gives the proper time measured by an observer who moves by (dr, rdθ, r sin θdφ) within the coordinate time interval dt; for light, in particular, ds = 0; • coordinates can always be scaled such that the curvature parameter k is either zero or ±1; • by a suitable transformation of the radial coordinate r, we can rewrite the metric in the form 2 2 2 2 h 2 2 2 2 2i ds = −c dt + a (t) dw + fk (w) dθ + sin θdφ , (2.2)

where the radial function fk(w) is given by  sin(w)(k = 1)  fk(w) = w (k = 0) ; (2.3)  sinh(w)(k = −1) CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 9

sometimes one or the other form of the metric is more convenient;

2.4.2 Redshift and expansion

• the changing scale of the Universe gives rise to the cosmological redshift z; the wavelength of light from a distant source seen by an observer changes by the same amount as the Universe changes its scale while the light is travelling; thus, if λe and λo are the emitted and observed wavelengths, respectively, they are given by λ a 1 o = 0 = , (2.4) λe a a

where a is the scale factor at the time of emission and a0 is normalised to unity; the relative wavelength change is the redshift, λ − λ 1 z ≡ o e = − 1 , (2.5) λe a and thus 1 1 1 + z = , a = ; (2.6) a 1 + z • when inserted into Einstein’s ﬁeld equations, two ordinary diﬀer- ential equations for the scale factor a(t) result; when combined, they can be brought into the form "Ω Ω 1 − Ω − Ω − Ω # H(a)2 = H2 m,0 + r,0 + Ω + m,0 r,0 Λ 0 a3 a4 Λ a2 2 2 ≡ H0 E (a); (2.7) this is Friedmann’s equation, in which the relative expansion rate a˙/a ≡ H(a) is replaced by the Hubble function whose present value is the Hubble constant, and the matter-energy content is described by the three density parameters Ωr,0, Ωm,0 and ΩΛ,0;

• the dimension-less parameters Ωm,0 and Ωr,0 describe the densities of matter and radiation in units of the critical density 3H2 ρ ≡ 0 ; (2.8) cr,0 8πG matter and radiation are distinguished by their pressure; for matter, the pressure p is neglected because it is very small compared to the energy density ρc2, while radiation is characterised by p = ρc2/3; • a Robertson-Walker metric whose scale factor satisﬁes Fried- mann’s equation is called a Friedmann-Lemaˆıtre-Robertson- Walker metric; the cosmological standard model asserts that the Universe at large is described by such a metric, and is thus characterised by the four parameters Ωm,0, Ωr,0, ΩΛ and H0; CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 10

• since the critical density evolves in time, so do the density parameters; their evolution is given by Ω a m,0 Ωm( ) = 3 (2.9) a + Ωm,0(1 − a) + ΩΛ,0(a − a) for the matter-density parameter and

Ω a3 a Λ,0 ΩΛ( ) = 3 (2.10) a + Ωm,0(1 − a) + ΩΛ,0(a − a) for the cosmological constant; in particular, these two equations show that Ωm(a) → 1 and ΩΛ(a) → 0 for a → 0, independent of their present values, and that Ωm(a)+ΩΛ(a) = 1 if Ωm,0 +ΩΛ,0 = 1 today;

• this lecture is devoted to answering two essential questions: (1) What are the values of the parameters deﬁning characterising Friedmann’s equation? (2) How can we understand the deviations of the real universe from a purely homogeneous and isotropic space-time?

2.4.3 Age and distances

• since Friedmann’s equation gives the relative expansion ratea ˙/a, we can use it to infer the age of the Universe,

Z t Z 1 da Z 1 da 1 Z 1 da t = dt0 = = = , (2.11) 0 0 a˙ 0 aH(a) H0 0 aE(a) which illustrates that the age scale is the inverse Hubble constant −1 H0 ; a simple example is given by the Einstein-de Sitter model, which (unrealistically, as we shall see later) assumes Ωm,0 = 1, −3/2 Ωr,0 = 0 and ΩΛ = 0; then, E(a) = a and

1 Z 1 √ 2 t = ada = ; (2.12) H0 0 3H0

• distances can be deﬁned in many ways which typically lead to different expressions; we summarise the most common deﬁnitions here; the proper distance Dprop is the distance measured by the light-travel time, thus c Z da dDprop = cdt ⇒ Dprop = , (2.13) H0 aE(a) where the integral has to be evaluated between the scale factors of emission and observation of the light signal; CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 11

• the comoving distance Dcom is simply deﬁned as the distance measured along a radial light ray ignoring changes in the scale factor, thus dDcom = dw; since light rays propagate with zero proper time, ds = 0, which gives cdt c Z da c Z da D w d com = d = = = 2 ; (2.14) a H0 aa˙ H0 a E(a)

• the angular-diameter distance Dang is deﬁned such that the same relation as in Euclidean space holds between the physical size of an object and its angular size; it turns out to be

Dang(a) = a fK[w(a)] = a fK[Dcom(a)] , (2.15)

where fK(w) is given by (2.3);

• the luminosity distance Dlum is analogously deﬁned to reproduce the Euclidean relation between the luminosity of an object and its observed ﬂux; this gives

Dang(a) f [w(a)] f [D (a)] D (a) = = K = K com , (2.16) lum a2 a a

• these distance measures can vastly differ at scale factors a 1; for small distances, i.e. for a ∼ 1, they all reproduce the linear relation cz D(z) = . (2.17) H0 • since time is finite in a universe with Big Bang, any particle can only be influenced by, and can only influence, events within a finite region; such regions are called horizons; several different definitions of horizons exist; they are typically characterised by some speed, e.g. the light speed, times the inverse Hubble function which sets the time scale;

2.4.4 The radiation-dominated phase

• it is an empirical fact that the Universe is expanding; earlier in time, therefore, the scale factor must have been smaller than today, a < 1; in principle, it is possible for Friedmann models that they had a finite minimum size at a finite time in the past and thus never reached a vanishing radius, a = 0; however, it turns out that a few crucial observational results rule out such “bouncing” models; this implies that a Unniverse like ours which is expanding today must have started from a = 0 a finite time ago, in other words, there must have been a Big Bang; CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 12

• equation (2.7) shows that the radiation density increases like a−4 as the scale factor decreases, while the matter density increases with one power of a less; even though the radiation density is very much smaller today than the matter density, this means that there has been a period in the early evolution of the Universe in which radiation dominated the energy density; this radiation-dominated era is very important for several observational aspects of the cosmological standard model;

• since the radiation retains the Planckian spectrum which it ac- quired in the very early Universe in the intense interactions with charged particles, its energy density is fully characterised by its temperature T; since the energy density is both proportional to T 4 and a−4, its temperature falls like T ∝ a−1;

2.5 Structures

2.5.1 Structure growth

• the hierarchy of cosmic structures is assumed to have grown from primordial seed ﬂuctuations in the process of gravitational collapse: overdense regions attract material and grow; they are described by the density contrast δ, which is the density ﬂuctuation relative to the mean densityρ ¯, ρ − ρ¯ δ ≡ ; (2.18) ρ¯

• linear perturbation theory shows that the density contrast δ is described by the second-order diﬀerential equation

δ¨ + 2Hδ˙ − 4πGρδ¯ = 0 (2.19)

if the dark matter is cold, i.e. if its constituens move with negligible velocities; notice that this is an oscillator equation with an imaginary frequency and a characteristic time scale (4πGρ¯)−1/2, and a damping term 2Hδ˙ which shows that the cosmic expansion slows down the gravitational instability;

• equation (2.19) has two solutions, a growing and a decaying mode; while the latter is irrelevant for structure growth, the growing mode is described by the growth factor D+(a), deﬁned such that the density contrast at the scale factor a is related to an initial density contrast δi by δ(a) = D+(a)δi; in most cases of practical relevance, the growth factor is accurately described by the ﬁtting CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 13

formula " ! !#−1 5a 1 1 D (a) = Ω Ω4/7 − Ω + 1 + Ω 1 + Ω , + 2 m m Λ 2 m 70 Λ (2.20) where the density parameters have to be evaluated at the scale factor a; • a very important length scale for cosmic structure growth is set by the horizon size at the end of the radiation-dominated phase; structures smaller than that became causally connected while radiation was still dominating; the fast expansion due to the radiation density inhibited further growth of such structures until the matter density became dominant; small structures are therefore suppressed compared to large structures which became causally connected only after radiation domination; the horizon size at the end of the radiation-dominated era thus divides between larger structures which could grow without inhibition, and smaller structures which were suppressed during radiation domination; it turns out to be 3/2 c aeq req = p ; (2.21) H0 2Ωm,0

2.5.2 The power spectrum

• it is physically plausible that the density contrast in the Universe is a Gaussian random field, i.e. that the probability for finding a value between δ and δ + dδ is given by a Gaussian distribution; the principal reason for this is the central limit theorem, which holds that the distribution of a quantity which is obtained by superposition of random contributions which are all drawn from the same probability distribution (with finite variance) turns into a Gaussian in the limit of infinitely many contribtions; • a Gaussian random process is characterised by two numbers, the mean and the variance; by construction, the mean of the density contrast vanishes, such that the variance defines it completely; • in linear approximation, density perturbations grow in place, as eq. (2.19) shows because the density contrast at one position ~x does not depend on the density contrast at another; as long as structures evolve linearly, their scale will be preserved, which implies that it is advantageous to study structure growth in Fourier rather than in configuration space; • the variance of the density contrast δˆ(~k) in Fourier space is called the power spectrum D ∗ 0 E 3 0 δˆ(~k)δˆ (~k ) ≡ (2π) Pδ(k) δD(~k − ~k ) , (2.22) CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 14

where the Dirac δ function ensures that modes with diﬀerent wave vectors are independent;

• once the power spectrum is known, the statistical properties of the linear density contrast are completely speciﬁed; it is a remarkable fact that two simple assumptions about the nature of the cosmic structures and the dark matter constrain the shape of Pδ(k) completely; if the mass contained in ﬂuctuations of horizon size is independent of time, and if the dark matter is cold, the power spectrum will behave as  k (k k0) P k ∝  , δ( )  −3 (2.23) k (k k0)

where k0 = 2πreq is the wave number of the horizon size at the end of radiation domination. The rise of the amount of structure towards larger k (for small k) comes from the fact that the horizon size is very small at early times and becomes larger with time, allowing a head-start for growth of smaller scale modes; the steep decline for structures smaller than req reﬂects the suppression of structure growth during radiation domination;

The linear CDM power spectrum with its characteristic shape (red), 2.5.3 Non-linear evolution and the deformation by non-linear evolution at the small-scale end • as the density contrast approaches unity, its evolution becomes (green). non-linear; the onset of non-linear evolution can be described by the so-called Zel’dovich approximation, which gives an approximate description of particle trajectories;

• although the Zel’dovich approximation breaks down as the non- linear evolution proceeds, it is remarkable for two applications; first, it allows a computation of the shapes of collapsing dark- matter structures and arrives at the conclusion that the collapse must be anisotropic, leading to the formation of sheets and filaments; second, it provides an explanation for the origin of the angular momentum of cosmic structures; filamentary structures thus appear as a natural consequence of gravitational collapse in a Gaussian random field;

• in the course of non-linear evolution, overdensities contract, which implies that matter is transported from larger to smaller scales; the linear result that density fluctuations grow in place therefore becomes invalid, and power in the density fluctuation field is transported towards smaller modes, or towards larger wave number k; this mode coupling process deforms the power spectrum on small scales, i.e. for large k; CHAPTER 2. THE COSMOLOGICAL STANDARD MODEL 15

• detailed studies of the non-linear evolution of cosmic structures require numerical simulations, which need to cover large scales and to resolve small scales well at the same time; much progress has been achieved in this ﬁeld within the last two decades due to the fortunate combination of increasing computer power with highly sophisticated numerical algorithms, such as particle-mesh and tree codes, and adaptive mesh reﬁnement techniques; Chapter 3

The age of the Universe

A key constraint on cosmology is the ‘observed’ age of the Universe. We have no direct way to measure how long ago the Big Bang happened, but there are various ways to set lower limits to the age of the Universe. They are all based on the same principle: since the Universe cannot be younger than any of its parts, it must be older than the oldest objects it contains. Three methods for age determination have been developed. One is based on the radioactive decay of long-lived isotopes, another constrains the age of globular clusters, and the third is based on the age of white dwarfs. We shall discuss them in turn to ﬁnd out how old the Universe should be at least. It turns out that these arguments, because of astrophysical complications, no longer provide the strongest constraints on cosmological parameters. They are included here for three reasons: (1) historical importance - age of the Universe constraints played an important role historically, (2) it is an important consistency check, and (3) the fact that the age of the oldest constituents of the Universe that we can observe is ﬁnite (and not that old) is a fundamental argument for the Big Bang paradigm.

3.1 Nuclear cosmo-chronology

• nuclear cosmo-chronology compares the measured abundance of certain radioactive isotopes with their initial abundance.

• This method can in principle be a very powerful one. Since radioactive decay is described N˙ = −λN, where N is the number of decaying nuclei in a closed sample and λ is the decay rate, integration gives −λt N(t) = N0e (3.1)

• Since N and λ are measurable, the only barrier to measuring t is

16 CHAPTER 3. THE AGE OF THE UNIVERSE 17

knowing the initial abundance of nuclei N0. • In some well-posed cases (e.g., measuring the age of the Earth) the initial abundance can be estimated with some accuracy through the comparison of the abundances of diﬀerent isotopes of both the decaying nuclei and the decay products. • In other cases, the initial abundance can only be crudely estimated, and the constraints are considerably less robust (e.g., estimating the age of the Galactic disk).

3.1.1 The age of the Earth

• to give a speciﬁc example, consider the two uranium isotopes 235U and 238U; they both decay into stable lead isotopes, 235U → 207Pb through the actinium series and 238U → 206Pb through the radium series; the abundance of any of these two lead isotopes is the sum of the initial abundance, plus the amount produced by the uranium decay; • since the radioactive decay is described N˙ = −λN, where N is the number of decaying nuclei in a closed sample and λ is the decay rate, integration gives −λt N(t) = N0e (3.2)

for the remaining number of initially N0 radioactive nuclei, or −λt λt N¯ = N0 1 − e = N(t) e − 1 (3.3) for the number of nuclei of the stable decay product; • thus, the present abundance of 207Pb nuclei is its primordial abundance N207,0 plus the amount produced, λ235t N207 = N207,0 + N235 e − 1 , (3.4)

235 where N235 is the abundance of U nuclei today; a similar equation with 235 replaced by 238 and 207 replaced by 206 holds for the decay of 238U to 206Pb; the decay constants for the two uranium isotopes are measured as −1 −1 λ235 = (1.015 Gyr) , λ238 = (6.45 Gyr) ; (3.5)

• the idea is now that ores on Earth or meteorites formed during a period which was very short compared to the age of the Earth te, so that their abundances can be assumed to have been locked up instantaneously and simultaneously a time te ago; chemical frac- tionation has given different abundances to different samples, but could not distinguish between different isotopes of the same element; thus, we expect different samples to show different isotope abundances, but identical abundance ratios of different isotopes; CHAPTER 3. THE AGE OF THE UNIVERSE 18

• the stable lead isotope 204Pb has no long-lived parents and is therefore a measure for the primordial lead abundance; thus, the abundance ratios between 207Pb and 208Pb to 204Pb calibrate the abundances in diﬀerent samples;

• suppose we have two independent samples a and b. The key thing here is that we assume i) that the isotopic ratios in the initial samples were the same (i.e., the initial 235U to 238U ratios, and the initial ratios of 204Pb to 206Pb to 207Pb were independent of sample), and ii) that the two samples had diﬀerent initial U to Pb ratios. Then, the abundance ratios

N206 N207 R206 ≡ and R207 ≡ (3.6) N204 N204 are measured; according to (3.3), they are N 238 λ238te R206 = R206,0 + e − 1 , N204 N 235 λ235te R207 = R207,0 + e − 1 ; (3.7) N204

the lead abundance ratios R206,0 and R207,0 should be the same in the two samples and cancel when the diﬀerence between the ratios in the two samples is taken; then, the ratio of diﬀerences can be written as a b R − R N eλ235te − 1 207 207 = 235 ; (3.8) a b λ238te − R206 − R206 N238 e 1 once the lead abundance ratios have been measured in the two samples, and the present uranium isotope ratio N 235 = 0.00725 (3.9) N238

is known, the age of the Earth te is the only unknown in (3.8); this method yields te = 4.6 ± 0.1 Gyr ; (3.10)

3.1.2 The age of the Galaxy

• a variant of this method can be used to estimate the age of the Galaxy, but this requires a model for how the radioactive elements were formed during the lifetime of the galaxy until they were locked up in samples where we can measure their abundances today;

• suppose there was an instantaneous burst of star formation and subsequent supernova explosions a time tg ago and no further production thereafter; then, the radioactive elements found on Earth CHAPTER 3. THE AGE OF THE UNIVERSE 19

today decayed for the time tg − te until they were locked up when the Solar System formed; if we can infer from supernova theory what the primordial abundance ratio 235U/238U is, we can conclude from its present value (3.9) and the age of the Earth what the age of the Galaxy must be; • the situation is slightly more complicated because element production did not stop after the initial burst; suppose that a fraction f of the heavy elements locked up in the Solar System was produced in a burst at t = 0, and the remaining fraction 1 − f was added at a steady rate until t = tg − te when the Earth was formed (i.e., we ignore all elements produced after the formation of the Earth); • the diﬀerential equation we have to solve now is N˙ = −λN + p , (3.11) where p is the constant production rate; we solve it by variation of constants, starting from the ansatz N = C(t)e−λt (3.12) which solves (3.11) if p C = eλt + D (3.13) λ with a constant D; thus, the abundance of a radioactive element with decay constant λ is p N = De−λt + (3.14) λ

before tg − te, and −λ[t−(tg−te)] N = N0e (3.15)

thereafter, where N0 is the abundance of elements locked up in the Solar System, as before;

• now, let Np be the total amount produced, then the initial conditions require that p N(0) = D + = f N , (3.16) λ p and thus p N(t − t ) = e−λ(tg−te) f N + eλ(tg−te) − 1 (3.17) g e p λ when the Earth formed, and p N(t ) = e−λtg f N + eλ(tg−te) − 1 (3.18) g p λ now on the Earth. CHAPTER 3. THE AGE OF THE UNIVERSE 20

• the production rate must be

(1 − f )Np p = , (3.19) tg − te which gives the present abundance " # (1 − f ) −λtg λ(tg−te) N = Npe f + e − 1 (3.20) λ(tg − te)

in terms of the produced abundance Np; • supernova theory says that the produced abundance ratio of the isotopes 235U and 238U is

N235,p = 1.4 ± 0.2 ; (3.21) N238,p taking the ratio of (3.20) for the present abundances of 235U and 238U, inserting the decay constants from (3.5), the abundance ratios from (3.9) and (3.21), and the age of the Earth te from (3.10) yields an equation which contains only the age of the galaxy tg in terms of the assumed fraction f ; this gives  6.3 ± 0.2 Gyr f = 1(all in burst)  tg = 8.0 ± 0.6 Gyr f = 0.5 (3.22)  12 ± 2 Gyr f = 0(constant)

• of course, the Universe must be older than the Galaxy; common assumptions and results from galaxy-formation theory assert that there at least 1 Gyr is necessary before galactic disks could have been assembled; therefore, nuclear cosmochronology constrains the age of the Universe to fall within

7 Gyr . t0 . 13 Gyr ; (3.23)

3.2 Stellar ages

• another method for measuring the age of the Universe caused much trouble for cosmologists for a long time; it is based on stellar evolution and exploits the fact that the time spent by stars on the main sequence of the Hertzsprung-Russell diagram depends sensitively on their mass and thus on their color; • Using this, if one can find collections of stars which are relatively ancient, it offers a chance to put a stringent lower limit on the age of the Universe. Globular clusters offer access to such populations (ancient and reasonably metal poor). In what follows, we will explore age-dating globular cluster stellar populations. CHAPTER 3. THE AGE OF THE UNIVERSE 21

• stars are described by the stellar-structure equations, which relate the mass M, the density ρ and the pressure P to the radius r and specify the temperature T and the luminosity L; they read dP GMρ dM = − , = 4πr2ρ , (3.24) dr r2 dr which simply state hydrostatic equilibrium and mass conserva- tion, and dT 3Lκρ dL = , = 4πr2ρ , (3.25) dr 4πr2acT 3 dr which describe energy transport and production; κ is the opacity of the stellar material, is the energy production rate per mass, and a is the Stefan-Boltzmann constant;

• The goal is to understand how luminosity and temperature depend on lifetime and opacity (i.e., age and metallicity).

• What we’ll do to get a ﬂavor for the problem is to do a dimensional analysis (i.e., drop physical constants and equate dP/dr ∼ P/R, etc.) We will explicitly keep track of the opacity κ in what follows. Using the equation for hydrostatic equilibrium and the ideal gas law P = ρkT one ﬁnds M ∝ TR/µ, where µ is the mean µmH molecular mass in units of the hydrogen atom mass.

• Using then the equation for temperature change as a function of luminosity, density, radius, temperature and opacity, one derives L ∝ µ4 M3κ−1. Thus, given that the lifetime τ ∼ M/L, τ ∝ M−2, τ ∝ T −4, and L ∝ κ1/2τ−3/2. (Please work through this derivation; it is informative).

• Using then the Stefan-Boltzmann equation L ∝ R2T 4, one can use the above to determine T ∝ (τκ)−1/4.

• There are a few key results here. L ∝ M3 and L ∝ T 4 describe (in a very approximate fashion) the stellar main sequence. Fur- thermore, lifetime is a very strong function of L and T; hot, luminous high-mass stars have short lifetimes whereas lower mass stars have very long lifetimes (in the case of stars considerably less massive than the Sun, the lifetime exceeds the Hubble time)1.

• The key point for the purpose of age-dating the globular cluster population is that as a coeval stellar population ages, the point in its Hertzsprung-Russell diagram up to which the main sequence remains populated moves towards lower luminosities and temperatures as (L, T) ' (κ1/2τ−3/2, [κτ]−1/4);

1In a crude sense, this is used by astronomers to age-date light from integrated stellar populations: blue light is from young populations whereas red light is from older populations. CHAPTER 3. THE AGE OF THE UNIVERSE 22

• Thus, the main sequence turn-off points of the populations in globular clusters can be used to derive lower limits to the age of the Galaxy and the Universe; • in practice, such age determinations proceed by adapting simulated stellar-evolution tracks to the Hertzsprung-Russell dia- grams of globular clusters and assigning the age of the best-fitting stellar-evolution model to the cluster; • The key uncertainty in this game is distance. Since observations cannot tell the luminosity of the turn-off point on the main sequence, but only its apparent brightness, age determinations from Colour-magnitude diagram of a globular clusters require that the cluster distances be known; there globular cluster. The turn-off point are several ways for estimating cluster distances; one uses the in the main sequence is clearly visi- period-luminosity relation of certain classes of variable stars; an- ble, but not very well defined. other method uses that the horizontal branch has a typical luminosity and can thus be used to calibrate the cluster distance; • therefore, uncertainties in the distance determinations directly translate to uncertainties in age the determinations; if the distance is overestimated, so is the luminosity, which implies that the age is underestimated, and vice versa; • Another major uncertainty is reddening: this causes the observed Hertzsprung-Russell diagram to shift along a well-known vector towards lower luminosities and lower temperatures (“redder” colours); it can be corrected for to a certain extent using other information (typically foreground thermal IR emission or H column density, assuming a dust-to-IR or dust-to-gas ratio), and using other well-defined features of the diagram like the red giant or horizontal branches as a consistency check; • several other difficulties are typically met: the simulated stellar- evolution tracks depend on the assumed metallicity of the stellar material, which changes the opacity and thus the energy transport through the stars (the dependence on opacity can be seen above); the light from the clusters is reddened and attenuated by interstellar absorption; the stellar population tracks are uncertain (because of opacity uncertainties, convection uncertainties, etc); and other difficulties... • globular clusters typically gave age determinations which were well above estimates based on the cosmological parameters assumed; in the past decade or so, this has changed because im- provements in stellar-evolution theory have lowered the globular- cluster ages, while recently determined cosmological parameters now yield a higher age for the Universe as assumed before; now, globular-cluster ages imply t & 12 Gyr (3.26) CHAPTER 3. THE AGE OF THE UNIVERSE 23

for the age of the Universe;

3.3 Cooling of white dwarfs

• a key method for cosmic age determinations is based on the cooling of white dwarfs. White dwarfs oﬀer a number of advantages: no energy generation (except for some latent heat lost as the white dwarf rearranges its internal structure during some phases of cooling); the gas is degenerate making the interior approximately isothermal (because electrons have essentially inﬁnite mean free path) and with a simple equation of state.

• White dwarfs have L ∝ 1/t approximately, with some mass de- From Hansen et al. 2004 pendence.

• What are white dwarf masses? – if there is mass dependence, we need to know the masses! Hydrostatic equilibrium gives P ∝ M2/r4, and the equation of state of a non-relativitsic degenerate gas is P ∝ ρ5/3 ∝ M5/3/r5, giving then r ∝ M−1/3 (i.e., as mass increases, radius decreases).

• The surface gravity g ∝ M/r2 ∝ M5/3, i.e., surface gravity measurements (from the gravitational redshift of spectral lines) gives then mass estimates. Result: remarkably, white dwarfs almost all have masses between 0.55 M and 0.6 M (that’s just the way mass loss during stellar evolution works).

• Thus, given M, we can use the T and L of white dwarf cooling sequences to get an age. The key uncertainties are distance and reddening (for the same reasons given above for main sequence- derived ages), along with uncertainties in white dwarf mass and composition (through heat capacities).

• Results so far tglobularclusters ∼ 12 Gyr (3.27) for the age of Galactic globular clusters (with a limit of 10Gyr), and ages of > 7 Gyr for the galactic disk. See Hansen et al. 2004, ApJS, 155, 551 for a beautiful discussion of the method and sources of uncertainty.

3.4 Summary

• combining results, we see that the age of the Universe, as measured by its supposedly oldest parts, is at least & 11 Gyr, and this places serious cosmological constraints; in the framework of the CHAPTER 3. THE AGE OF THE UNIVERSE 24

Friedmann-Lemaˆıtre models, this can be interpreted as limits on the cosmological parameters;

• suppose we live in an Einstein-de Sitter universe with Ωm,0 = 1 and ΩΛ = 0; then, we know from (2.12) that

2 −18 t0 = & 11 Gyr ⇒ H0 . 2 × 10 s , (3.28) 3H0 which reads −1 −1 H0 . 61 km s Mpc (3.29) in conventional units; Constraints on the cosmic age • as we shall see in the next chapter, the Hubble constant is mea- have meaningful implications sured to be larger than this, which can immediately be interpreted on the cosmological parame- as an indication that we are not living in an Einstein-de Sitter uni- ters, in particular on the cosmic verse; density parameter. The three curves for each cosmological model are obtained assuming −1 −1 H0 = (64, 72, 80) km s Mpc . Chapter 4

The Hubble Constant

The Hubble constant,a ˙/a ≡ H(a)1, is one of the key parameters de- scribing the Universe. It is largely a scaling parameter, setting (along with other parameters to a lesser extent) the age of the Universe, and the absolute values of luminosities and sizes. There are a number of ways of measuring Hubble’s constant, which we will skim in this lecture. The ﬁrst, and so far most important method, uses the distance–recession velocity relation to estimate Hubble’s constant. Other methods are then discussed, including gravitational lensing and distances from cluster Sunayev-Zel’dovich eﬀect measurements. Then the results are summarised.

4.1 Hubble constant from Hubble’s law

4.1.1 Hubble’s law: history

• Vesto Slipher discovered in the 1920s that distant galaxies typically move away from us. Edwin Hubble found that their recession velocity grows with distance,

v = H0 D , (4.1)

and determined the constant of proportionality as H0 ≈ 570 km s−1 Mpc−1. This value of the Hubble constant is very high, primarily because the absolute magnitude of the Cepheid variable stars had been dramatically overestimated (i.e., the luminosity was underestimated; they were mixed up with W Virginis stars, which show similar variability but dramatically fainter luminosities). A luminosity–eﬀective temperature 1Only astronomers would call something that is manifestly not constant a constant. relation, with the shaded areas de- noting the instability strip.

25 CHAPTER 4. THE HUBBLE CONSTANT 26

• we had seen in (2.17) before that all distance measures in a Friedmann-Lemaˆıtre universe follow the linear relation cz D = (4.2) H0 to ﬁrst order in z 1; since cz = v is the velocity according to the linearised relation for the Doppler shift, c + v v 1 + z = ≈ 1 + , (4.3) c − v c (4.2) is exactly the relation that Hubble found; The relation between recession ve- • there is little doubt that (4.1) is the result that Hubble wanted locity and distance originally pub- to ﬁnd because he wanted his measurements to support the lished by Hubble and Humason in Friedmann-Lemaˆıtre cosmology; he even left out data points from 1931. Note the value of the Hubble the analysis that did not support his conclusion; constant!

4.1.2 Hubble’s law: the challenge

• There are multiple problems with measuring the Hubble constant. Firstly, we have no way (yet) of measuring a direct (trigonometric parallax) measurement to (any) galaxy, making a trigonometric distance impossible to measure (except for masers; see later). • Thus, a sequence of distance indicators must be used to estimate the distances to (typically) star clusters; these star clusters are used to calibrate other distance indicators which in turn are used to calibrate other distance indicators which give a value for the Hubble constant. • A second problem is that while (4.2) holds for an idealised, homogeneous and isotropic universe, real galaxies have peculiar motions on top of their Hubble velocity which are caused by the at- traction from local density inhomogeneities; for instance, galaxies in our neighbourhood feel the gravitational pull of a cosmologically nearby supercluster called the Great Attractor and accelerate towards it; the galaxy M 31 in Andromeda and the Milky Way approach each other at ∼ 100 km s−1; • thus, the peculiar velocities of the galaxies must either be known well enough, for which a model for the velocity ﬁeld is necessary, or they must be observed at so large distances that any peculiar motion is unimportant compared to their Hubble velocity; requir- ing that peculiar velocities of order 300 ... 600 km s−1 be less than 10% of the Hubble velocity, galaxies with redshifts 300 ... 600 km s−1 z & 10 × ≈ 0.01 ... 0.02 (4.4) c must be observed; this is already so distant that it is hard or impossible to apply direct distant estimators; CHAPTER 4. THE HUBBLE CONSTANT 27

• this illustrates why accurate measurements of the Hubble constant are so diﬃcult: nearby galaxies, whose distances are more accurately measurable, do not follow the Hubble expansion well, but the distances to galaxies far enough to follow the Hubble law are very hard to measure;

4.1.3 The distance ladder: the ﬁrst 20Mpc

• measurements of the Hubble constant from Hubble’s law thus require accurate distance measurements out to cosmologically relevant distance scales; since this is impossible in one step, the so- called distance ladder must be applied, in which each step in the ladder calibrates the next;

• There are a number of key steps in the distance ladder.

• Trigonometric parallaxes: a key direct distance measurement is the trigonometric parallax caused by the annual motion of the Earth around the Sun; by deﬁnition, a star at a distance of a parsec perpendicular to the Earth’s orbital plane has a parallax of an arc second; astrometric measurement accuracies of order 10−3 00 are thus necessary to measure distances of order 100 pc at 10σ;

• in such a way, the distances to local stars and star clusters (notably the Hyades) have been measured (most accurately by the European satellite Hipparcos; see http://www.rssd.esa.int/index.php?project=HIPPARCOS&page=Hyades for some information on the Hyades from Hipparcos, and http://www.rssd.esa.int/index.php?project=HIPPARCOS&page=index for more general information on Hipparcos), allowing measurement of the absolute magnitude of main sequence stars: this in turn allows the distances to much more distant clusters of stars to be measured.

• Cluster distances from convergent points: The distance to nearby clusters can be worked out from their proper motions on the sky and their radial velocities. Just as parallel train tracks (or the edges of roads, or meteors) appear to converge towards a point, so do the paths of stars in a star cluster. The distance to a cluster can be worked out by using the angle to the convergent point θ and the radial velocity vr and the proper motion µ: v (km/s) tan θ D(pc) = r . (4.5) 4.74µ(”/yr) Please see http://www.astro.washington.edu/labs/clearinghouse/labs/Hyades/disthyad.html for a great discussion of this point at length and an example that you can worth through. CHAPTER 4. THE HUBBLE CONSTANT 28

• Distances from variable stars: Star clusters allow access to one of the key distance indicators: Cepheid variable stars. Cepheids are high-mass stars in late evolutionary stages which undergo periodic variability (RR Lyraes are also critical distance indicators for old populations); the underlying instability is driven by the temperature dependence of the atmospheric opacity in these stars, which is caused by the transition between singly and doubly ionised Helium;

• the cosmologically important aspect of the Cepheids is that their variability period τ and their luminosity L are related,

L ∝ τ1.3 , (4.6) Some examples for Cepheid lightcurves hence their luminosity can be inferred from their period, and their distance from the ratio of their luminosity to the flux S observed from them, r L D = ; (4.7) S at the relevant distances, any distinction between differently defined distance measures is irrelevant; In some Cepheids, overtones of the • it is of crucial importance here that the calibration of the period- pulsation are excited rather than the luminosity relation depends on the metallicity of the Cepheids, fundamental mode. and thus on the stellar population they belong to; Hubble’s originally much too high result for H0 was corrected when Baade realised that stars in the Galactic disk belong to another stellar population than in the halo;

• by measuring the periods of Cepheids and comparing their apparent brightnesses in star clusters of known distance (from typically main sequence ﬁtting distances, which in turn were calibrated using very nearby clusters with trigonometric distances) and Cepheids in the LMC it was possible to determine the distance to the Large Magellanic Cloud as DLMC = (50.1 ± 3) kpc; • measuring the periods of Cepheids in the LMC and comparing their apparent brightnesses in diﬀerent galaxies, it is thus possible to determine the relative distances to the galaxies; for example, comparisons between Cepheids in the LMC and the Andromeda galaxy M 31 show D M 31 = 15.28 ± 0.75 , (4.8) DLMC while Cepheids in the member galaxies of the Virgo cluster yield

DVirgo = 316 ± 25 ; (4.9) DLMC CHAPTER 4. THE HUBBLE CONSTANT 29

• of course, for the Cepheid method to be applicable, it must be possible to resolve at least the outer parts of distant galaxies into individual stars and to reliably identify Cepheids among them; this was one reason why the Hubble Space Telescope was proposed, to apply the superb resolution of an orbiting telescope to the measurement of H0; Cepheid distance measurements are possible to distances . 20 Mpc; • Masers: There is a critical consistency check with the trigonet- ric parallax/main sequence ﬁtting/Cepheid distance scale: water masers orbiting the central regions of distant galaxies. The principle is simple: if there is perfect circular motion in the maser ring and the inclination is known and not face on, then the proper motion of the maser clouds can be combined with the radial velocity of the clouds at the orbit tangent points to provide a distance: v (km/s) D(Mpc) = r . (4.10) 4.74 cos(90 − i)µ(µas/yr) In practice, the acceleration of the masers can also be measured using radio interferometric observations with VLBI. The disk rotation of ∼ 1000 km/s coupled with 30µas/yr gives a distance of 7.3±0.3Mpc; a direct geometric distance to a galaxy with Cepheids.

• Eclipsing binary stars: Eclipsing binaries provide an accurate method of measuring distances to nearby galaxies with an accuracy of 5%. A review of the method can be found from Paczynski (1997). The method requires both photometry and spectroscopy of an eclipsing binary. From the light and radial velocity curve the fundamental parameters of the stars can be determined accurately. The light curve provides the fractional radii of the stars, which are then combined with the spectroscopy to yield the physical radii and effective temperatures. The velocity semi-amplitudes determine both the mass ratio and the sum of the masses, thus the individual masses can be solved for. Furthermore, by fitting synthetic spectra to the observed ones, one can infer the effective temperature, surface gravity and luminosity. Comparison of the luminosity of the stars and their observed brightness yields the reddening of the system and distance. Measuring distances with eclipsing binaries is an essentially geometric method and thus accurate and independent of any intermediate calibration steps. With the ad- vent of 8 m class telescopes, eclipsing binaries have been used to obtain accurate distance estimates to the LMC, SMC, M31 and M33.

• Light echos: With SN1987A, it was possible to measure the distance to the LMC using a light echo. The idea is that if one has a spherical shell or ring which existed before the supernova (ring CHAPTER 4. THE HUBBLE CONSTANT 30

in this case), then it lights up after a certain time when hit by the light from SN1987A (the front side), and will remain illuminated until the echo has faded from the back side. Given an angular diameter at that time from direct measurements, the distance to the LMC can be measured.

4.1.4 Distance Ladder: extending beyond 20Mpc

• Fundamental plane: scaling relations within classes of galaxies provide additional distance indicators; in the three-dimensional parameter space spanned by the velocity dispersion σv, the effective radius Re and the surface brightness Ie at the effective radius, elliptical galaxies populate the tight fundamental plane defined by 1.4 −0.85 Re ∝ σv Ie ; (4.11) since the luminosity is evidently

2 L ∝ IeRe , (4.12)

the fundamental-plane relation implies

2.8 −0.7 L ∝ σv Ie ; (4.13)

such a relation follows directly from the virial theorem if the mass-to-light ratio in elliptical galaxies increases gently with mass, M ∝ M0.2 ; (4.14) L

• thus, if it is possible to measure the effective surface brightness Ie (which does not depend on distance, if one neglects cosmological surface brightness dimming and k-corrections) and the velocity dispersion σv of an elliptical galaxy, the fundamental plane gives the luminosity, which can be compared to the flux to find the distance; such distances are accurate to 11% in the best cases (i.e., 22% intrinsic scatter in luminosity).

• Tully-Fisher relation: a relation similar to (4.13), the Tully- Fisher relation, holds for spiral galaxies if the velocity dispersion σv is replaced by the rotational velocity vrot and if surface brightness is neglected; however, spiral galaxies avoid galaxy clusters, and it is therefore more diﬃcult to decide whether they belong to a galaxy cluster such as Virgo or Coma;

α • The form of the relation is L ∝ vrot where α is between 2.5 (in blue bands) and 4 (in the near-infrared). The scatter in the Tully-Fisher relation can be as little as 0.2 mags or less in carefully-selected samples in the far red and near-infrared. CHAPTER 4. THE HUBBLE CONSTANT 31

• Surface Brightness Fluctuations: an interesting way for determining distances to galaxies uses the fluctuations in their surface brightness; the idea behind this method is that the fluctuations in the surface brightness will be dominated by the rare brightest stars, and that the optical luminosity of the entire galaxy will be proportional to the number N of such stars; assuming Pois-√ son statistics, the fluctuation level will be proportional to N, from which N and L ∝ N can be determined once the method has been calibrated with galaxies whose distance is known otherwise; again, the distance is then found by comparing the flux to the luminosity;

• Planetary Nebulae: Planetary nebulae, which are late stages in the evolution of stars, have a luminosity function with a steep upper cut-off; moreover, their spectra are dominated by sharp neb- ular emission lines which facilitate their detection even at large distances because they appear as bright objects in narrow-band filters tuned to the emission lines; since the cut-off luminosity is known, it can be converted to a distance as usual;

• Supernova Type Ia: One of the most important classes of distance indicators are supernovae of type Ia; they occur in binary systems in which one of the components is a white dwarf accret- ing mass from an overflowing companion; since the electron degeneracy pressure in the cores of white dwarfs can stabilise them only up to the Chandrasekhar mass of ≈ 1.4 M , the white dwarf suddenly collapses once mass accretion drives it over this limit; in the ensuing supernova explosion, part of the white dwarf’s material is converted to elements of the iron group; since the amount of nuclear fuel is fixed by the Chandrasekhar mass, the explosion energy is also fixed, and thus so is the luminosity;

• this idealised picture needs to be modiﬁed because the amount of energy released depends on the opacity of the material surrounding the supernovae explosion; this leads to a scatter in the peak luminosities, but this scatter can be corrected applying the empirical Philipps relation, which relates the peak luminosity L to the time scale τ of the light-curve decay,

L ∝ τ1.7 ; (4.15)

when this correction is applied, type-Ia supernova are turned into precise standard candles with a dispersion of only 6%;

• Because this is one of the brightest standard candles, it has been applied out to redshifts of 1.5 (with diﬃculty); it was the ﬁrst convincing and is still one of the most important observations indicating the accelerating expansion of the Universe (see later for a deeper discussion). CHAPTER 4. THE HUBBLE CONSTANT 32

• Type II supernovae: although they are not standard (or standardisable) candles, core-collapse supernovae of type II can also be used as distance indicators through the Baade-Wesselink method; suppose the spectrum of the supernova photosphere can be approximated by a Planck curve whose temperature can be determined from the spectral lines; then, the Stefan-Boltzmann law says that the total luminosity is

L = aR2T 4 , (4.16)

where a is again the Stefan-Boltzmann constant from (??); the photospheric radius, however, can be inferred from the expansion velocity of the photosphere, which is measurable by the Doppler shift in the emission lines, times the time after the explosion; when applied to the supernova SN 1987A in the Large Magel- lanic Cloud, the Baade-Wesselink method yields a distance of

DLMC = (44 ... 50) kpc , (4.17)

which agrees with other distance measurements (Cepheids, eclipsing binaries, etc).

4.1.5 The HST Key Project

• all these distance indicator were used by the HST Key Project to determine accurate distances to 26 galaxies between 3 ... 25 Mpc, and five very nearby galaxies2 for testing and calibration; Hubble laws as measured by the • double-blind photometry was applied to the identified distance in- Hubble Key Project in different dicators; since Cepheids tend to lie in star-forming regions and are wave bands (top to bottom) and in thus attenuated by dust, and since their period-luminosity relation different stages of correction (left to depends on metallicity, respective corrections had to be carefully right). applied;

• then, the measured velocities had to be corrected by the peculiar velocities, which were estimated by a model for the ﬂow ﬁeld;

• the estimated luminosities of the distance indicators could then be compared with the extinction-corrected fluxes to determine distances, whose proportionality with the velocities corrected by the peculiar motions finally gave the Hubble constant; a weighted average over all distance indicators is Probability distributions for H0 ob- −1 −1 H0 = (72 ± 8) km s Mpc , (4.18) tained with different measurement techniques applied in the Hubble where the error is the square root of the systematic and statistical Key Project, and the combined dis- errors summed in quadrature; tribution. 2see http://www.ipac.caltech.edu/H0kp/H0KeyProj.html CHAPTER 4. THE HUBBLE CONSTANT 33

4.2 Gravitational Lensing

• a totally different method for determining the Hubble constant uses gravitational lensing; masses bend passing light paths towards themselves and therefore act in a similar way as convex glass lenses; as in ordinary geometrical optics, this effect can be described applying Fermat’s principle to a medium with an index of refraction 2Φ n = 1 − , (4.19) c2 where Φ is the Newtonian gravitational potential; • if it is strong enough, the curvature of the light paths causes multiple images to appear from single sources; compared to the straight light paths in absence of the deflecting mass distribution, the curved paths are geometrically longer, and they have to addi- tionally propagate through a medium whose index of refraction is n > 1; this gives rise to a time delay which has a geometrical and a gravitational component,

1 2 τ = ~θ − ~β − ψ(~θ) , (4.20) 2 where ~θ are angular coordinates on the sky and ~β is the angular position of the source; ψ is the appropriately scaled Newtonian potential of the deﬂector, projected along the line-of-sight; according to Fermat’s principle, images occur where τ is extremal,

i.e. ∇~ θτ = 0; • the projected lensing potential ψ is related to the surface-mass density Σ of the deﬂector by Σ ∇~ 2ψ = 2 ≡ 2κ , (4.21) Σcr where the critical surface-mass density

2 c Ds Σcr ≡ (4.22) 4πG DdDds

contains the distances Dd,s,ds from the observer to the deflector, the source, and from the deflector to the source, respectively; • the dimension-less time delay τ from (4.20) is related to the true physical time delay t by τ t ∝ , (4.23) H0 where the proportionality constant is a dimension-less combina- −1 tion of the distances Dd,s,ds with the Hubble radius cH0 and the deflector redshift 1 + zd; (4.23) shows that the true time delay is proportional to the Hubble time, as it can intuitively be expected;

The quasars MG 0414 (top) and PG 1115 are quadruply gravitationally lensed by galaxies along the line-of-sight. Time delays between diﬀerent images allow measurements of the Hubble constant if a plausible mass model for the lensing galaxy exists. CHAPTER 4. THE HUBBLE CONSTANT 34

• time delays are measurable in multiple images of a variable source; the variable signal arrives after different times in the images seen by the observer, and if the deflector is a galaxy, time delays are typically of order days to months and therefore observable with a reasonable monitoring effort;

• interestingly, it can be shown in an elegant, but lengthy calculation that measured time delays can be inverted to ﬁnd the Hubble constant from the approximate equation

H0 ≈ A(1 − hκi) + Bhκi(η − 1) , (4.24)

where A and B are constants depending on the measured image positions and time delays, hκi is the mean scaled surface-mass density of the deflector averaged within an annulus bounded by the image positions, and η ≈ 2 is the logarithmic slope of the deflector’s density profile;

• therefore, if a model exists for the gravitationally-lensing galaxy, the Hubble constant can be found from the positions and time delays of the images; applying this technique to ﬁve diﬀerent lens systems3, Kochanek (2002) found

−1 H0 = (73 ± 8) km s (4.25)

assuming that the lensing galaxies have radially constant mass- to-light ratios;

• this result is highly remarkable because it was obtained in one step without any reference to the extragalactic distance ladder; although there is the remaining ambiguity from the mass model for the lensing galaxies, the perfect agreement between the results from lensing time delays and the HST Key Project is a very reassuring confirmation of the cosmological standard model; Values for the Hubble constant obtained with alternative methods 4.3 The Sunyaev-Zel’dovich effect (gravitational lensing, GL, and the thermal Sunyaev-Zel’dovich effect, SZ) not depending on the distance • another method should finally be mentioned because it is physi- ladder. cally interesting and conceptually elegant, although it will probably never become competitive; it is based on two different types of observations of the hot gas in massive galaxy clusters;

• galaxy clusters contain diﬀuse, fully ionised plasma with temperatures of order (1 ... 10) keV which emits X-rays by the thermal bremsstrahlung (free-free emission) of the electrons scattering oﬀ

3these are: PG 1115 + 80, SBS 1520 + 530, B 1600 + 434, PKS 1830 − 211 and HE 2149 − 2745 CHAPTER 4. THE HUBBLE CONSTANT 35

the ions; as a two-body process, the bremsstrahlung emissivity jX is proportional to the product of the electron and ion densities ne and ni, times the square root of the temperature T, √ √ 2 jX ∝ neni T = CXne T , (4.26)

where CX is a constant whose value is irrelevant for our current purposes; moreover, we have used that the ion density will be proportional to the electron density ne; • since the emissivity is the energy released per volume per time, the energy emitted by the galaxy cluster per surface-area element dA is Z dE = dA dl jX , (4.27) where the integral extends along the line-of-sight; the energy ﬂux seen by the observer from this surface-area element is R dE dA dl jX dS = 2 = 2 , (4.28) 4πDlum 4πDlum

• by deﬁnition of the angular-diameter distance, the surface-area 2 element dA spans the solid angle element dΩ = dA/Dang, so the X-ray ﬂux per unit solid angle, or the X-ray surface brightness, is

dS D2 Z 1 Z I ang l j l j , = = 2 d X = 4 d X (4.29) dΩ 4πDlum 4π(1 + z) where we have used the remarkable Etherington relation between the angular-diameter and luminosity distances,

2 Dlum = (1 + z) Dang , (4.30) which holds in any space-time; • the hot electrons in the galaxy clusters scatter microwave background photons passing by to much higher energies by inverse Compton scattering; this process will neither create nor destroy photons, but transport the photons to higher energy; thus, if the CMB is observed towards a galaxy cluster, its intensity at low photon energies will appear reduced, and increased at high energies; this is the so-called thermal Sunyaev-Zel’dovich effect: clusters cast shadows on the CMB at low frequencies, and appear as sources at high frequencies, where the division line lies at 217 GHz; • the amplitude of the thermal Sunyaev-Zel’dovich effect is quantified by the Compton-y parameter, Z kT y l σ n , = d 2 T e (4.31) mec CHAPTER 4. THE HUBBLE CONSTANT 36

where me is the electron rest-mass and σT is the Thomson scattering cross section; the total Compton-y parameter of a galaxy cluster, integrated over the entire solid angle of the cluster, is thus Z 1 Z 1 Z kT Y y A l y V σ n , = dΩ = 2 d d = 2 d 2 T e (4.32) Dang Dang mec i.e. it is determined by a volume integral over the cluster divided by the squared angular-diameter distance;

• the comparison between the two observables discussed here, the X-ray surface brightness (4.29) and the integrated Compton-y parameter (4.32), shows that they both depend on the distribution of temperature and electron density within the cluster, and on the squared angular-diameter distance to the cluster; assuming a model for radial T and ne proﬁles then allows combining the two types of measurement to ﬁnd the cluster’s angular-diameter dis- −1 tance, which is proportional to the Hubble length cH0 and thus to the inverse Hubble constant;

• in this way, it is possible to estimate the Hubble constant by combining X-ray and thermal Sunyaev-Zel’dovich measurements on galaxy clusters; typical values for H0 derived in this way are substantially lower than the values discussed above, which is probably due to overly simpliﬁed assumptions about the temperature and electron-density distributions in the clusters;

4.4 Summary

• if we accept the result of the Hubble Key Project for now,

−1 −1 H0 = (72 ± 8) km s Mpc , (4.33)

we can calibrate several important numbers that scale with some power of the Hubble constant;

• ﬁrst, in cgs units, the Hubble constant can be written

−18 H0 = (2.3 ± 0.3) × 10 s , (4.34)

which implies the Hubble time, i.e. the inverse of the Hubble constant 1 = (13.6 ± 1.5) Gyr (4.35) H0 and the Hubble radius c = (1.3 ± 0.1) × 1028 cm = (4.1 ± 0.5) Gpc ; (4.36) H0 CHAPTER 4. THE HUBBLE CONSTANT 37

the critical density of the Universe turns out to be

3H2 ρ = 0 = (9.65 ± 2.1) × 1030 g cm−3 ; (4.37) cr 8πG

• the uncertainty in H0 is conventionally expressed in terms of −1 −1 the dimension-less parameter h ≡ H0/100 km s Mpc ; since lengths in the Universe are typically measured with respect to the Hubble length, they are often given in units of h−1Mpc; similarly, luminosities are typically obtained by multiplying ﬂuxes with squared luminosity distances and are thus often given in −2 units of h L ; we avoid this notation in the following and insert h = 0.72 where needed; Chapter 5

Big-Bang Nucleosynthesis

5.1 Key concepts

• Impossible to produce observed Helium through stellar nucleosynthesis: need primordial generation

• Critical step in nucleosynthesis chain is p+n → d +γ; other reactions in the chain are fast, converting almost all d into Helium-4.

• The Gamow criterion quantiﬁes the ’sweet spot’ that one needs to hit to generate Helium: not too little (because no d to generate Helium). The criterion goes nBhσvit ∼ 1; given nB estimates and known velocity-averaged cross section one can estimate t and therefore T, and predict the temperature of the CMB (the answer comes out at ∼5K).

• The abundances of d, 3He, 4He and 7Li can be estimated (from very precise spectral measurements of stars and high-redshift absorption-line systems) and yield important constraints on the baryon to photon ratio; because the photon density of the CMB is known to exquisite accuracy such an exercise yields an excellent estimate of baryon density (where most of the power comes from d).

• useful website for more is : http://www.astro.ucla.edu/∼wright/BBNS.html

38 CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 39

5.2 The origin of Helium-4 and the other light elements

5.2.1 The riddle of Helium

• since conversions between temperatures and energies will occur frequently in this chapter, recall that a thermal energy of 1 eV corresponds to a temperature of 1.16 × 104 K;

• stellar spectra show that the abundance of Helium-4 in stellar atmospheres is of order Y = 0.25 by mass, i.e. about a quarter of the baryonic mass in the Universe is composed of Helium-4;

• Helium-4 is produced in stars in the course of hydrogen burning; per 4He nucleus, the amount of energy released corresponds to 0.7% of the masses involved, or

2 2 2 ∆E = ∆mc = 0.007 (2mp + 2mn)c ≈ 0.028 mpc ≈ 26 MeV ≈ 4.2 × 10−5erg ; (5.1)

• suppose a galaxy such as ours, the Milky Way, shines with a lu- 10 43 −1 minosity of L ≈ 10 L ≈ 3.8 × 10 erg s for a good fraction of the age of the Universe, say for τ = 1010 yr ≈ 3 × 1017 s; then, it releases a total energy of

61 Etot ≈ Lτ ≈ 1.1 × 10 erg ; (5.2)

• the number of 4He nuclei required to produce this energy is

E 1.1 × 1061 ∆N = ≈ ≈ 2.8 × 1065 , (5.3) ∆E 4.2 × 10−5 which amounts to a Helium-4 mass of

42 MHe ≈ 4mp∆N ≈ 1.9 × 10 g ; (5.4)

• assume further that the galaxy’s stars were all composed of pure hydrogen initially, and that they are all more or less similar to the 10 43 Sun; then, the mass in hydrogen was MH ≈ 10 M ≈ 2 × 10 g initially, and the ﬁnal Helium-4 abundance by mass expected from the energy production amounts to

1.9 × 1042 Y ≈ ≈ 10% , (5.5) ∗ 2 × 1043 which is much less than the Helium-4 abundance actually observed; CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 40

• this discrepancy is exacerbated by the fact that 4He is destroyed in later stages of the evolution of massive stars, and that most of this Helium should be locked up in the centres of stellar remnants (mostly white dwarfs). From Izotov et al. 1997; the • Another key argument against stellar nucleosynthesis as the main key point is that Helium is a very production route of Helium is that it is observed that Helium-4 weak function of metallicity, argu- abundance is a very weak function of metallicity (i.e., Helium-4 ing powerfully against stellar nucle- does not increase in lockstep with the other elements produced by osynthesis as its main production stellar nucleosynthesis). route. • we thus see that the amount of 4He observed in stars can by no means have been produced by these stars themselves under reasonable assumptions during the lifetime of the galaxies; we must therefore consider that most of the 4He which is now observed must have existed already before the galaxies formed;

5.2.2 Elementary considerations

• nuclear fusion of 4He and similar light nuclei in the early Uni- verse is possible only if the Universe was hot enough for a suf- ﬁciently long period during its early evolution; the nuclear binding energies of order ∼ MeV imply that at least temperatures of T ∼ 106 × 1.16 × 104 K ≈ 1.2 × 1010 K must have been reached; since the temperature of the (photon background in the) Universe is now T0 ∼ 3 K as we shall see later, this corresponds to times when the scale factor of the Universe was 3 a ∼ ≈ 2.5 × 10−10 ; (5.6) nuc 1.2 × 1010

• at times so early, the actual mass density and a possible cosmological constant are entirely irrelevant for the expansion of the Universe, which is only driven by the radiation density; thus, the 1/2 −2 expansion function can be simpliﬁed to read E(a) = Ωr,0 a , and we ﬁnd for the cosmic time according to (2.11)

1 Z a a2 0 0 ≈ × 19 2 t(a) = 1/2 a da = 1/2 4.3 10 a s , (5.7) Ωr,0 H0 0 2Ωr,0 H0 where we have inserted the Hubble constant from (4.18) and the −5 radiation-density parameter today Ωr,0 ≈ 2.5 × 10 , which will be justiﬁed later;

• inserting anuc from (5.6) into (5.7) yields a time scale for nucleosynthesis of order a few seconds; we shall argue later that it is in fact delayed until a few minutes after the Big Bang; CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 41

• it is instructive for later purposes to establish a relation between time and temperature based on (5.7); using T = T0/a, we substi- tute a = T0/T to obtain T 2 T −2 t = 4.3 × 1019 0 s ≈ 1.6 s ; (5.8) T MeV

5.2.3 The Gamow criterion

• a crucially important step in the fusion of 4He is the fusion of deuterium 2H or d, p + n → d + γ (5.9) because the direct fusion of 4He from two neutrons and two protons is extremely unlikely; • If too little deuterium is produced, no 4He is produced because deuterium forms a necessary intermediate step; realising this, Gamow suggested that the amount of deuterium produced has to be “just right”, which he translated into the intuitive criterion

nBhσvit ≈ 1 , (5.10)

where np is the baryon number density, hσvi is the velocity- averaged cross section for the reaction (5.9), and t is the available time for the fusion, which we have seen in (5.8) to be set by the present temperature of the cosmic radiation background, T0, and the temperature T required for deuterium fusion;

• thus, from an estimate of the baryon density nB in the Universe, from the known velocity-averaged cross section hσvi, and from the known temperature required for deuterium fusion, Gamow’s criterion allows us to estimate the present temperature T0 of the cosmic radiation background; already in the 1940’s, Gamow was able to predict T0 ≈ 5 K! • summarising, we have arrived at two remarkable arguments so far; first, the observation that the 4He abundance is Y ≈ 25% by mass shows that stars alone are insufficient for the production of light nuclei in the Universe, so we are guided to suggest that the early Universe must have been hot enough for nuclear fusion processes to be efficient; in other words, the observed abundance of 4He indicates that there should have been a hot Big Bang; second, the crucially important intermediate step of deuterium fusion allows an estimate of the present temperature of the cosmic radiation background which lead Gamow already in 1942 to predict that it should be of order a few Kelvin; • after these remarkably simple and far-reaching conclusions, we shall now study primordial nucleosynthesis and consequences thereof in more detail; CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 42

5.2.4 Elements produced

• the fusion of deuterium (5.9) is the crucial ﬁrst step; since the pho- todissociation cross section of d is large, destruction of d is very likely because of the intense photon background until the temperature has dropped way below the binding energy of d, which is only 2.2 MeV, corresponding to 2.6 × 1010 K; in fact, substantial d fusion is delayed until the temperature falls to T = 9 × 108 K or kT ≈ 78 keV! as (5.8) shows, this happens t ≈ 270 s after the Big Bang; • from there, Helium-3 and tritium (3H or t) can be built, which can both be converted to 4He; these reactions are now fast, immediately converting the newly formed d; in detail, these reactions are

d + p → 3He + γ , d + d → 3He + n , d + d → t + p , and 3He + n → t + p , (5.11)

followed by

3He + d → 4He + p and t + d → 4He + n ; (5.12)

Nuclear fusion reactions responsi- • fusion reactions with neutrons are irrelevant because free neu- ble for primordial nucleosynthesis trons are immediately locked up in deuterons once deuterium fusion begins, and passed on to t, 3He and 4He in the further fusion steps; • since there are no stable elements with atomic weight A = 5, addition of protons to 4He is unimportant; fusion with d is unimportant because its abundance is very low due to the eﬃcient follow- up reactions; we can therefore proceed only by fusing 4He with t and 3He to build up elements with A = 7,

t + 4He → 7Li + γ , 3He + 4He → 7Be + γ , followed by 7 − 7 Be + e → Li + νe ; (5.13) some 7Li is destroyed by

7Li + p → 2 4He ; (5.14)

the fusion of two 4He nuclei leads to 8Be, which is unstable; further fusion of 8Be in the reaction

8Be + 4He → 12C + γ (5.15) CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 43

is virtually impossible because the low density of the reaction partners essentially excludes that a 8Be nucleus meets a 4He nucleus during its lifetime; • thus, while the reaction (5.15) is possible and extremely important in stars, it is suppressed below any importance in the early Uni- verse; this shows that the absence of stable elements with A = 8 prohibits any primordial element fusion beyond 7Li;

5.2.5 Predicted approximate Helium abundance

• once stable hadrons can form from the quark-gluon plasma in the very early universe, neutrons and protons are kept in thermal equilibrium by the weak interactions

− p + e ↔ n + νe , + n + e ↔ p + ν¯e (5.16) until the interaction rate falls below the expansion rate of the Uni- verse;

• while equilibrium is maintained, the abundances nn and np are controlled by the Boltzmann factor

!3/2 n m Q Q n = n exp − ≈ exp − , (5.17) np mp kT kT where Q = 1.3 MeV is the energy equivalent of the mass diﬀer- ence between the neutron and the proton; • the weak interaction freezes out when T ≈ 1010 K or kT ≈ 0.87 MeV, which is reached t ≈ 2 s after the Big Bang; at this time, the n abundance by mass is

" !#−1 nnmn nn Q Xn(0) ≡ ≈ = 1 + exp ≈ 0.17 ; nnmn + npmp nn + np kTn (5.18) detailed calculations show that this value is kept until tn ≈ 20 s 9 after the Big Bang, when Tn ≈ 3.3 × 10 K; Light-element abundances as a • afterwards, the free neutrons undergo β decay with a half life of function of cosmic time during pri- ± τn = 886.7 1.9 s, thus mordial nucleosynthesis ! t − t n −t/τn Xn = Xn(0) exp − ≈ Xn(0)e ; (5.19) τn

when d fusion ﬁnally sets in at td ≈ 270 s after the Big Bang, the neutron abundance has dropped to

−td/τn Xn(td) ≈ Xn(0)e ≈ 0.125 ; (5.20) CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 44

now, essentially all these neutrons are collected into 4He because the abundances of the other elements can be neglected to ﬁrst order; this yields a 4He abundance by mass of

Y ≈ 2Xn(td) = 0.25 (5.21)

because the neutrons are locked up in pairs to form 4He nuclei;

• the Big-Bang model thus allows the prediction that 4He must have been produced such that its abundance is approximately 25% by mass, which is in remarkable agreement with the observed abundance and thus a strong conﬁrmation of the Big-Bang model;

5.2.6 Expected abundances and abundance trends

• the detailed abundances of the light elements as produced by the primordial fusion must be calculated solving rate equations based on the respective fusion cross sections; uncertainties involved concern the exact values of the cross sections and their energy dependence, and the precise life time of the free neutrons;

• since primordial nucleosynthesis happens during the radiation era (which we shall conﬁrm later on), the expansion rate is exclusively set by the radiation density; then, the only other parameter controlling the primordial fusion processes is the baryon density;

• in fact, the only relevant parameter deﬁning the primordial abundances is the ratio between the number densities of baryons and photons; since both densities scale like a−3 or, equivalently, like T 3, their ratio η is constant; anticipating the photon number density to be determined from the temperature of the CMB,

nB 10 2 η = = 10 η10 , η10 ≡ 273ΩBh ; (5.22) nγ thus, once we know the photon number density, and once we can determine the parameter η from the primodial element abundances, we can infer the baryon number density;

• typical 2-σ uncertainties from cross sections and neutron half-life 4 are, at a ﬁducial η parameter of η10 = 5, 0.4% for He, 15% for d Dependence of the 4He, d, and 7Li 3 7 and He, and 42% for Li; abundances on the parameter η • the 4He abundance depends only very weakly on the η because the largest fraction of free neutrons is swept up into 4He without strong sensitivity to the detailed conditions;

• the principal eﬀects determining the abundances of d, 3He and 7Li are the following: with increasing η, they can more easily be CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 45

burned to 4He, and so their abundances drop as η increases; at low η, an increase in the proton density causes 7Li to be destroyed by the reaction (5.14), while the precursor nucleus 7Be is more easily produced if the baryon density increases further; this creates a characteristic “valley” of the predicted 7Li abundance near η ≈ (2 ... 3) × 10−10;

5.3 Observed element abundances

5.3.1 Principles

• of course, the main problem with any comparison between light- element abundances predicted by primordial nucleosynthesis and their observed values is that much time has passed since the primordial fusion ceased, and further fusion processes have happened since;

• seeking to determine the primordial abundances, observers must therefore either select objects in which little or no contamination by later nucleosynthesis can reasonably be expected, in which the primordial element abundance may have been locked up and separated from the surroundings, or whose observed element abundances can be corrected for their enrichment during cosmic history in some way; Deuterium signature in the wing of • deuterium can be observed in cool, neutral hydrogen gas (HI re- a damped (saturated) hydrogen ab- gions) via resonant UV absorption from the ground state, or in sorption line in a QSO spectrum radio wavebands via the hyperﬁne spin-ﬂip transition, or in the sub-millimetre regime via DH molecule lines; these methods all employ the fact that the heavier d nucleus causes small changes in the energy levels of electrons bound to it;

• Helium-3 is observed through the hyperﬁne transition in its ion 3He+ in radio wavebands, or through its emission and absorption lines in HII regions;

• Helium-4 is of course most abundant in stars, but the fusion of 4He in stars is virtually impossible to correct precisely; rather, 4He is probed via the emission from optical recombination lines in HII regions;

• measurements of Lithium-7 must be performed in old, local stellar populations; this restricts observations to cool, low-mass stars because of their long lifetime, and to stars in the Galactic halo to allow precise spectra to be taken despite the low 7Li abundance; CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 46

5.3.2 Evolutionary corrections

• stars brooded heavy elements as early as z ∼ 6 or even higher; any attempts at measuring primordial element abundances must therefore concentrate on gas with as low a metal abundance as possible; the dependence of the element abundances on metallicity allows extrapolations to zero enrichment; • such evolutionary corrections are low for deuterium because it is observed in the Lyman-α forest lines, which arise from absorption in low-density, cool gas clouds at high redshift; likewise, they are low for the measurements of Helium-4 because it is observed in low-metallicity, extragalactic HII regions; • probably, little or no correction is required for the Lithium-7 abundances determined from the spectra of very metal-poor halo stars (there is quite a debate about this point); • inferences from Helium-3 are diﬀerent because 3He is produced from deuterium in stars during the pre-main sequence evolution; it is burnt to 4He during the later phases of stellar evolution in stellar cores, but conserved in stellar exteriours; observations indicate that a net destruction of 3He must happen, possibly due to extra mixing in stellar interiours; for these uncertainties, 3He commonly excluded from primordial abundance measurements;

5.3.3 Speciﬁc results

• due to the absence of strong evolutionary eﬀects and its steep monotonic abundance decrease with increasing η, deuterium is the ideal baryometer; since it is produced in the early Universe and destroyed by later fusion in stars, all d abundance determinations are lower bounds to its primordial abundance; • measurements of the deuterium abundance at high redshift are possible through absorption lines in QSO spectra, which are likely to probe gas with primordial element composition or close to it; • such measurements are challenging in detail because the tiny isotope shift in the d lines needs to be distinguished from velocity- shifted hydrogen lines, H abundances from saturated H lines need to be corrected by comparison with higher-order lines, and high- resolution spectroscopy is required for accurate continuum subtraction; • at high redshift, a deuterium abundance of n D = 3.4 × 10−5 (5.23) nH CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 47

relative to hydrogen is consistent with all relevant QSO spectra at 95% conﬁdence level; a substantial depletion from the primordial value is unlikely because any depletion should be caused by d fusion and thus be accompanied by an increase in metal abundances, which should be measurable;

• some spectra which were interpreted as having . 10 times the d abundance from (5.24) may be due to lack of spectral resolution; the d abundance in the local interstellar medium is typically lower n d ∼ (1 ... 1.5) × 10−5 (5.24) nH which is consistent with d consumption due to fusion processes; conversely, the d abundance in the Solar System, is higher because d is locked up in the ice on the giant planets;

• in low-metallicity systems, 4He should be near its primordial abundance, and a metallicity correction can be applied; possible systematic uncertainties are due to modiﬁcations by underlying stellar absorption, collisional excitation of observed recombination lines, and the exact regression towards zero metallicity;

• a conservative range is 0.228 ≤ Yp ≤ 0.248, and a high value is likely, Yp = 0.2452 ± 0.0015; • observations of the Lithium-7 abundance aim at stars in the stellar halo with very low metallicity; they should have locked up very nearly primordial gas, but may have processed it;

• cool stellar atmospheres are diﬃcult to model, and 7Li may have been produced by cosmic-ray spallation on the interstellar medium; The Spite plateau in the 7Li abun- 7 • in the limit of low stellar metalicity, the observed Li abundance dance as a function of the metalicity turns towards the Spite plateau, which is asymptotically independent of metalicity,

7 A( Li) = 12 + log(nLi/nH) = 2.2 ± 0.1 , (5.25)

and shows very little dispersion; stellar rotation is important because it increases mixing in stellar interiors;

• the Spite plateau is unlikely to reﬂect the primordial 7Li abundance, but a corrections are probably moderate; a possible increase of 7Li with the iron abundance indicates low production of 7Li, but the probable net eﬀect is a depletion with respect to the primordial abundance by no more than ∼ 0.2 dex; a conservative estimate yields 2.1 ≤ A(7Li) ≤ 2.3 ; (5.26) CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 48

• in absence of depletion, this value falls into the valley expected in the primordial 7Li at the boundary between destruction by protons and production from 8Be; however, if 7Li was in fact depleted, its primordial abundance was higher than the value (5.26), and then two values for η10 are possible;

5.3.4 Summary of results

2 • through the relation η10 = 273 ΩBh , the density of visible baryons alone implies η10 ≥ 1.5; • the deuterium abundance derived from absorption systems in the spectra of high-redshift QSOs indicates η10 = 4.2 ... 6.3;

7 7 • the Li abundance predicted from this value of η is then A( Li)p = 2.1 ... 2.8 which is fully consistent with the observed value A(7Li) = 2.1 − 2.3, even if a depletion by 0.2 dex due to stellar destruction is allowed;

• the predicted primordial abundance of helium-4 is then Yp = 0.244 ... 0.250, which overlaps with the measured value YP = 0.228 ... 0.248; thus, the light-element abundances draw a consistent picture for low deuterium abundance; however, this is also true for high deuterium abundance: if η10 = 1.2 ... 2.8, the lithium-7 and helium-4 abundances are A(7Li) = 1.9 ... 2.7 and YP = 0.225 ... 0.241, which are also compatible with the observations; Predicted primordial element abun- • we thus ﬁnd that Big-Bang nucleosynthesis alone implies dances as a function of η, over- laid with the measurements (boxes). Ω h2 = 0.019 ± 0.0024 or Ω = 0.037 ± 0.009 (5.27) B B The η parameter compatible with all at 95% conﬁdence level if conclusions are predominantly based measurements is marked by the ver- on the deuterium abundance in high-redshift absorption systems; tical bar. we shall later see that this result is in fantastic agreement with independent estimates of the baryon density obtained from the analysis of structures in the CMB;

• a historically very important application of Big-Bang nucleosynthesis begins with the realisation that, at ﬁxed baryon density, the light-element abundances are set by the cosmic expansion rate while the Universe was hot enough to allow nuclear fusion, and that the expansion rate in turn depends on the density of relativistic particle species; a larger number of relativistic species, as could be provided by a number of lepton ﬂavours larger than three, gave rise to a faster expansion, which allowed fewer neutrons to decay until the Universe became too cool for fusion, and thus implied a higher number of neutrons per proton, leading to CHAPTER 5. BIG-BANG NUCLEOSYNTHESIS 49

a higher abundance of 4He; in this way, the 4He abundance was found to limit the number of lepton families to three; Chapter 6

The Matter Density in the Universe

We saw in the last section that Ωb ∼ 0.04, i.e., 4% of the closure density is in baryons; we will see later that Ωm ∼ 0.25 − 0.3. In this section, we will try to understand where these baryons are at the present day and try to estimate the present-day dark matter density.

6.1 Key concepts

• Constructing a satisfying census of cosmic mass is an incredi- bly diﬃcult task, and at the end of the lecture no-one will be happy; a key reference for anyone who is interested is Fukugita, Hogan and Peebles 1998, ApJ, 503, 518, and can be downloaded at http://www.mpia-hd.mpg.de/homes/bell/teaching/fhp.pdf

• Almost all baryonic mass is in plasma at the present day.

• Only ∼ 10% of baryons are in stars or cold (H or H2) gas. • The other 90% is in warm/hot plasma, either in hot galactic halos, intergalactic space, or in galaxy clusters. While in this state, it is actually pretty hard to observe, so we have only a very rough census of these baryons.

• We will see later that there are a variety of ways to estimate the present-day dark matter density; local galaxy motions provide a rough method for their measurement, and galaxy clusters allow two quasi-independent estimates to be made, both suggesting that Ωm ∼ 0.3 (dark and baryonic mass density).

50 CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 51

6.2 Baryonic mass in galaxies

6.2.1 Stars

• Given the luminosity of a stellar population, what is its mass? The astronomical community has more-or-less settled on an approach to attack this problem, but there are a number of subtleties/debates in the literature about this issue.

• A key concept is that of the stellar “initial mass function”, which describes the distribution of stellar masses of a newly-formed stellar population. A convenient form is the Kroupa (2001) IMF: dN ∝ M−0.3 0.1 < M/M < 0.5 (6.1) d ln M dN ∝ M−1.3 0.5 < M/M < 120. (6.2) d ln M The limits are somewhat arbitrary: the lower mass limit for a star is 0.08M because nuclear hydrogen burning cannot take place below that mass. The upper mass limit of 120M is a number that is widely debated, but for our purposes is not important so long as the upper mass limit is 10M .

• Recall from earlier that L ∝ M3 on the main sequence and L ∝ T 6; thus one can see that M/L is a strong function of temperature (i.e., color);

• Thus, if one assumes a universally-applicable form of the stellar IMF (this is a strong assumption), one can use the colors or spectral line indices (which are sensitive to temperature) to estimate stellar M/L The relationship between g − r • For example, Bell & de Jong (2001); Bell et al. (2003) find: color and stellar mass, assuming a universally-applicable stellar IMF. log M/L ∼ −0.4 + 1.1(g − r) (6.3) 10 The arrow shows the effect of a where the M/Ls are in solar units and g and r are magnitudes from large amount of obscuring dust; the the SDSS k-corrected to z = 0, and the g and r absolute magnitude solid line shows schematically the of the Sun is 5.15 and 4.67 respectively. result of a different method using spectral line indices. • galaxy luminosities and stellar masses are observed to be distributed approximately according to the Schechter function,

!−α ! dN Φ L L = ∗ exp − , (6.4) dL L∗ L∗ L∗

−3 −3 where the normalising factor is Φ∗ ≈ 3 × 10 Mpc , the scale 11 mass is M∗ ≈ 10 M and the power-law exponent is α ≈ 1.1; CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 52

• irrespective of what physical processes this distribution originates from, it turns out to characterise mixed galaxy populations very well, even in galaxy clusters; • the stellar mass density in galaxies is easily found to be Z ∞ Z ∞ dN 1−α −m Mg = M dM = Φ∗ M∗ m e dm 0 dM 0 8 M = Γ(2 − α)Φ∗ M∗ ≈ Φ∗ M∗ ≈ 3 × 10 ; (6.5) Mpc3

30 −3 or in terms of Ω = ρ∗/ρcr with ρcr = 9.65 × 10 g cm , one obtains Ω∗ ∼ 0.002.

6.2.2 Cold gas

• Galaxies also contain cold gas (especially spiral galaxies), primarily in the form of H and H2 (of course with Helium and met- als). • Direct blind H surveys can yield an estimate of H mass density (with e.g., Parkes or Aricebo). • Molecular hydrogen is a much stickier problem — there are no observable transitions of the cool H2 gas that dominates the gas mass (there are some IR transitions of warm H2 gas with T > 100K, but these are pretty useless as such a small fraction of H2 is at such warm temperatures). So instead, historically, one has used CO masses (which one can measure) plus a CO-to-H2 ratio which has been calibrated in the Milky Way (in molecular clouds by comparing the CO mass and a virial mass M ∝ σ2r). One then measures the CO ﬂuxes from galaxies (this is hard, the CO line is faint and telescopes that work at this wavelength are not that big, yet) and one uses this (hopefully) representative sample of galaxies to estimate the cosmic H2 mass. Obviously, this is not a particularly well-posed problem. • Using up-to-date surveys, incorporated with statistical methods, I estimated (in 2003) a ’cold gas in galaxies’ density of Ωcoldgas ∼ 0.0004.

• Thus, cold gas and stars have Ω∗,gas ∼ 0.0024, or < 10% of the available Ωb ∼ 0.04.

6.2.3 Warm/hot gas in between galaxies

• We have accounted for < 10% of the baryons expected at z = 0 (and even this involved using some dodgy assumptions). CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 53

• Where is the rest? Our current idea is that it is in warm/hot intergalactic medium. This is diffuse, ionised filamentary gas that fills out the spaces between galaxies. In clusters of galaxies, the temperatures and densities are hot enough that it is possible to detect via its X-ray emission (see next section). In filaments, the gas is neither hot nor dense enough to emit much in X-rays, and instead must be constrained by detection of absorption line systems in the far-UV or X-ray (very highly ionised oxygen or nitrogen).

• A huge breakthrough in recent times has been the detection of 6-times-ionised oxygen and nitrogen from filaments of the IGM (see attached article by Nicastro 2004) from which > 1/2 of the A schematic diagram of the warm- baryonic density of the Universe has been inferred. These lines hot intergalactic medium; the bulk are *so* faint that one has to wait until a bright flare from a blazar of the gas is in filaments which con- happens to take the spectra (otherwise one needs to integrate on a nect galaxies ‘normal’ bright X-ray source for months).

• I want to give an idea of how extreme this extrapolation is. At typical column densities for detection of ∼ 1015cm−2, and for an ∼AU/pc-sized source, one estimates around 1042 or 1052 ions were along the line of sight that were detected, corresponding to ∼ the mass of an asteroid / the mass of Jupiter. From this small amount of (more-or-less) detected matter, one has extrapolated more than 1/2 of the baryonic density of the Universe!

6.3 Total mass in galaxies

6.3.1 Galaxies

• the rotation velocities of stars orbiting in spiral galaxies are observed to rise quickly with radius and then to remain roughly constant; if measurements are continued with neutral hydrogen beyond the radii out to which stars can be seen, these rotation curves are observed to continue at an approximately constant level; After a quick rise, stellar velocities • in a spherically-symmetric mass distribution, test particles on cir- in spiral galaxies remain approxi- cular orbits have orbital velocities of mately constant with radius. (The GM(r) galaxy shown is NGC 3198.) v2 (r) = ; (6.6) rot r flat rotation curves thus imply that M(r) ∝ r; based on the conti- nuity equation dM = 4πr2ρdr, this requires that the density falls off as ρ(r) ∝ r−2 (theory predicts a r−3 fall-off at large radii); this is much flatter than the light distribution, which shows that spiral galaxies are characterised by an increasing amount of dark matter as the radius increases; CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 54

• a mass distribution with ρ ∝ r−2 has formally infinite mass, which is physically implausible; however, at finite radius, the density of the galaxy falls below the mean density of the surrounding universe; the spherical collapse model often invoked in cosmology shows that a spherical mass distribution can be considered in dynamical equilibrium if its mean overdensity is approximately 200 times higher than the mean densityρ ¯; • let R be the radius enclosing this overdensity, and M the mass enclosed, then M 3M M 800πρ¯R2 = = 200¯ρ ⇒ = ; (6.7) V 4πR3 R 3 at the same time, (6.6) needs to be satisfied, hence !1/2 800πρ¯R2 v2 3v2 = rot ⇒ R = rot ; (6.8) 3 G 800πGρ¯ inserting a typical numbers yields v R = 290 kpc rot ; (6.9) 200 km s−1 with (6.6), this implies Rv2 v 3 M = rot = 2.7 × 1012 M rot ; (6.10) G 200 km s−1 The actual normalisation constant is somewhat lower because of the r−3 fall-off (roughly 1012) but this gives the flavour of the line of argument. Far beyond the stars, flat rotation • typical luminosities of spiral galaxies are given by the Tully- curves are inferred from the motion Fisher relation, of neutral-hydrogen clouds (blue; v 3...4 L = L rot , (6.11) the galaxy shown is NGC 2915). ∗ 220 km s−1 10 with L∗ ≈ 2.4 × 10 L , or the normalising mass is roughly M∗ ≈ 10 6×10 M ; thus, the mass-to-light ratio of a massive spiral galaxy is found to be m ≈ 60 (6.12) l in solar units, or the mass-to-stellar mass ratio is m ≈ 25; (6.13) m∗ evidently, this exceeds the stellar mass-to-light ratio by far, but the details of the measurement depend on the maximum radius assumed... • The same analysis can be run with elliptical galaxies (using other methods to estimate dynamical masses, using either velocity dis- persions or weak lensing masses), typical values of m ∼ 45 are m∗ typically found. CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 55

6.3.2 Mass in galaxy clusters: kinematic masses

• the next step upward in the cosmic hierarchy are galaxy clusters, which were first identified as significant galaxy overdensities in relatively small areas of the sky;

• rich galaxy clusters contain several hundred galaxies, which by 13 themselves contain a total amount of stellar mass sim10 M ;

• Yet, the galaxies in rich galaxy clusters move with typical velocities of order . 103 km s−1 which are measured based on redshifts in galaxy spectra; therefore, only one component of the galaxy velocity is observed; its distribution is characterised by the velocity dispersion σv; Galaxies move so fast in galaxy • if these galaxies were not gravitationally bound to the clusters, clusters (here the Coma cluster) that the clusters would disperse within . 1 Gyr; since they exist over much more than the visible mass is cosmological time scales, clusters need to be (at least in some needed to keep them gravitationally sense) gravitationally stable; bound; this was the ﬁrst argument • assuming virial equilibrium, the potential energy of the cluster for dark matter, as put forward by galaxies should be minus two times the kinetic energy, or Zwicky in the 1930s. GM ≈ 3σ2 , (6.14) R v where M and R are the total mass and the virial radius of the cluster, and the factor three arises because the velocity dispersion represents only one of three velocity components;

• we combine (6.14) with (6.7) to ﬁnd

9σ2 !1/2 R = v ≈ 2.5 Mpc , (6.15) 800πGρ¯ Galaxy clusters are the most lumi- and, with (6.14), nous emitters of diﬀuse X-ray radi- M ≈ 2 × 1015 M ; (6.16) ation. The ﬁgure shows the X-ray hence, the mass required to keep cluster galaxies bound despite emission of the Coma cluster ob- their high velocities exceeds the mass in galaxies by 1-2 orders of served with the Rosat satellite. magnitude;

6.3.3 Mass in galaxy clusters: the hot intracluster gas

• galaxy clusters are diﬀuse sources of thermal X-ray emission; their X-ray continuum is caused by thermal bremsstrahlung, whose bolometric volume emissivity is √ 2 2 jX = Z gﬀCX n T (6.17) CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 56

in cgs units, where Z is the ion charge, gﬀ is the Gaunt factor, n is the ion number density, T is the gas temperature, and

−24 CX = 2.68 × 10 (6.18) in cgs units, if T is measured in keV; • a common simple, axisymmetric model for the gas-density distribution in clusters is n r n x 0 , x ≡ , ( ) = 2 3β/2 (6.19) (1 + x ) rc

where rc is the core radius; • the line-of-sight projection of the X-ray emissivity yields the X- ray surface brightness as a function of the projected radius ρ, √ Z ∞ √ 2 2 πΓ(3β − 1/2) Z gffCX Tn S ρ j z 0 , X( ) = Xd = 2 3β−1/2 (6.20) −∞ Γ(3β) (1 + ρ ) where we have assumed for simplicity that the cluster is isothermal, so T does not change with radius; • the latter equation shows that two parameters of the density profile (6.19), namely the slope β and the core radius rc, can be read off the observable surface-brightness profile; • the missing normalisation constant can then be obtained from the X-ray luminosity, √ Z ∞ √ 3 2 3 2 2 πΓ(3β − 3/2) LX = 4πrc jX x dx = 4πrc Z gffCX Tn0 , 0 4Γ(3β) (6.21) and a spectral determination of the temperature T; • finally, the total mass of the X-ray gas enclosed in spheres of radius R is Z R/rc 3 2 MX(R) = 4πrc n(x)x dx , (6.22) 0 which is a complicated function for general β; for β = 2/3, which is a commonly measured value, ! 3 R R MX(R) = 4πrc n0 − arctan , (6.23) rc rc which is of course formally divergent for R → ∞ because the density falls off ∝ r−2 for β = 2/3 and r → ∞;

• inserting typical numbers, we√ ﬁrst set Z = 1 = gﬀ and β = 2/3 as above, then use Γ(1/2) = π, Γ(1) = 1 = Γ(2), and assume a 45 −1 hypothetic cluster with LX = 10 erg s , a temperature of kT = 23 10 keV and a core radius of rc = 250 kpc = 7.75 × 10 cm; CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 57

• then, (6.21) yields the central ion density

−3 −3 n0 = 5 × 10 cm (6.24) and thus the central gas mass density

−27 −3 ρ0 = mpn0 = 8.5 × 10 g cm ; (6.25)

• based on the virial radius (6.15) and on the mass (6.23), we ﬁnd the total gas mass

14 MX = 1.0 × 10 M ; (6.26) this is of the same order as the cluster mass in galaxies, and approximately one order of magnitude less than the total cluster mass; • it is reasonable to believe that clusters are closed systems in the sense that there cannot have been much material exchange between their interior and their surroundings; if this is indeed the case, and the mixture between dark matter and baryons in clusters is typical for the entire universe, the density parameter in dark matter should be M Ωdm,0 ≈ Ωb,0 ≈ 10Ωb,0 ≈ 0.4 ; (6.27) M∗ + MX more precise estimates based on detailed investigations of individual clusters yield Ωdm,0 ≈ 0.3 ; (6.28)

6.3.4 Alternative cluster mass estimates

• cluster masses can be estimated in several other ways; one of them is directly related to the X-ray emission discussed above; the hydrostatic Euler equation for an isothermal gas sphere in equilibrium with the spherically-symmetric gravitational potential of a mass M(r) requires 1 dp GM(r) = − , (6.29) ρ dr r2 where ρ and p are the gas density and pressure, respectively; assuming an ideal gas, the equation of state is p = nkT, where n = ρ/mp is the particle density of the gas and T is its temperature; if we further simplify the problem assuming an isothermal gas distribution, we can write kT dρ GM(r) − = 2 (6.30) mpρ dr r CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 58

or, solving for the mass rkT d ln ρ M(r) = − ; (6.31) Gmp d ln r

• for the β model introduced in (6.19), the logarithmic density slope is d ln ρ d ln n r2 = = −3β , (6.32) d ln r d ln r 1 + r2 thus the cluster mass is determined from the slope of the X-ray surface brightness and the cluster temperature, 3βrkT r2 M r ( ) = 2 ; (6.33) Gmp 1 + r • with the typical numbers used before, i.e. R ≈ 2.5 Mpc, β ≈ 2/3 and kT = 10 keV, the X-ray mass estimate gives 15 M(R) ≈ 1.1 × 10 M , (6.34) in reassuring agreement with the mass estimate (6.16) from galaxy kinematics; Strong gravitational lensing in • a third, completely independent way of measuring cluster masses galaxy clusters can cause strong is provided by gravitational lensing; without going into any de- distortions of background galaxies tail on the theory of light deﬂection, we mention here that it can into arcs (shown is the large arc in generate image distortions from which the projected lensing mass the cluster Abell 370). They allow distribution can be reconstructed; mass estimates obtained in this independent cluster-mass estimates. way by and large conﬁrm those from X-ray emission and galaxy kinematics, although interesting discrepancies exist in detail; • none of the cluster mass estimates is particularly reliable because they are all to some degree based on stability and symmetry assumptions; for mass estimates based on galaxy kinematics, for instance, assumptions have to be made on the shape of the galaxy orbits, the symmetry of the gravitational potential and the me- chanical equilibrium between orbiting galaxies and the body of the cluster; numerous assumptions also enter X-ray based mass determinations, such as hydrostatic equilibrium, spherical symmetry and, in some cases, isothermality of the intracluster gas; gravitational lensing does not need any equilibrium assumption, but inferences from strongly distorted images depend very sensitively on the assumed symmetry of the mass distribution;

6.4 Mass density from cluster evolution

• a very interesting constraint on the cosmic mass density is based on the evolution of cosmic structures; Abell’s cluster catalog covers the redshift range 0.02 . z . 0.2, which encloses a volume of CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 59

≈ 9 × 108 Mpc3; of the 2712 clusters in the catalog, 818 fall into (the poorest) richness class 0; excluding those, there are 1894 clusters with richness class ≥ 1 in that volume, which yields an estimate for the spatial cluster density of

−6 −3 nc ≈ 2 × 10 Mpc ; (6.35)

• it is a central assumption in cosmology that structures formed from Gaussian random density fluctuations; the spherical collapse model then says that gravitationally bound objects form where the linear density contrast exceeds a critical threshold of δc ≈ 1.686, quite independent of cosmology; the probability for this to happen in a Gaussian random field with a (suitably chosen) standard deviation σ(z) is ! 1 δc pc(z) = erfc √ , (6.36) 2 2σ(z) where σ(z) = σ0D+(z) (6.37) because the linear growth of the density contrast is determined by the growth factor, a fitting formula for which was given in (2.20); Cluster probability as a function of • now, the present-day standard deviation σ0 must be chosen such σ for two different values of Ωm0. as to reproduce the observed number density of clusters given in (6.35); the measured probability for finding a cluster is approximated by 0 Mnc −3 −1 pc = ≈ 3 × 10 Ωm0 ; (6.38) ρcΩm the standard deviation σ in (6.36) must now be chosen such that this number is reproduced, which yields   0.61 Ωm0 = 1.0 σ0 ≈  ; (6.39) 0.72 Ωm0 = 0.3

• equations (6.36) and (6.37) can now be used to estimate how the Evolution of the cluster abundance, cluster abundance should change with redshift; simple evaluation depending on the density parameter reveals that the cluster abundance is expected to drop very rapidly Ωm0. with increasing redshift if Ωm0 is high, and much more slowly if Ωm0 is low; • qualitatively, this behaviour is easily understood; if, in a low- density universe, cluster do not form early, they cannot form at all because the rapid expansion due to the low matter density pre- vents them from growing late in the cosmic evolution; CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 60

• from the observed slow evolution of the cluster population as a whole, it can be concluded that the matter density must be low; estimates arrive at Ωm0 ≈ 0.3 , (6.40) in good agreement with the preceding determinations;

6.5 Musings on the nature of the dark matter

• The preceding discussion should have demonstrated that the matter density in the Universe is I) considerably less than its critical value, approximately one third of it. However, II) only a small fraction of this matter is visible; thus we call the remaining invis- ible majority dark matter.

• What is this dark matter composed of? Can it be baryons? Tight limits are set by primordial nucleosynthesis, which predicts that the matter density in baryonic matter should be ΩB ≈ 0.04, cf. (5.27). In the framework of the Friedmann-Lemaˆıtre models, the baryon density in the Universe can be higher than this only if baryons are locked up in some way before nucleosynthesis commences. They could form black holes before, but their mass is limited by the mass enclosed within the horizon at, say, up to one minute after the Big Bang. According to (2.6), the scale factor at this time was a ≈ 10−10, and thus the matter den- 30 −3 sity was of order ρm ≈ 10 ρcr ≈ 10 g cm . The horizon size is 12 rH ≈ ct ≈ 1.8 × 10 cm, thus the mass enclosed by the horizon 4 is ≈ 3 × 10 M , which limits possible black-hole masses from above.

• It is expected that quantum eﬀects cause black holes to radiate, thus to convert their mass to radiation energy and to “evaporate”. The estimated time scale for complete evaporation is

!3 70 M τbh ≈ 4 × 10 s , (6.41) M which is shorter than the Hubble time (4.35) if

M . 4 × 1015 g . (6.42)

Black holes formed very early in the Universe should thus have disappeared by now.

• Gravitational microlensing was used to constrain the amount of dark, compact objects in the halo of the Milky Way. Although they were found to contribute part of the mass, they cannot account for all of it. In particular, black holes with masses of the CHAPTER 6. THE MATTER DENSITY IN THE UNIVERSE 61

3...4 order 10 M should have been found through their microlensing eﬀect.

• We are thus guided to the conclusion that the dark matter is most probably not baryonic and not composed of compact dark objects. We shall see later that and why the most favoured hypothesis now holds that it is composed of weakly interacting massive particles. Neutrinos, however, are ruled out because their total mass has been measured to be way too low. Chapter 7

The Cosmic Microwave Background

7.1 The isotropic CMB

7.1.1 Thermal history of the Universe

• How does the Universe evolve thermally? We have seen earlier that the abundance of 4He shows that the Universe must have gone through an early phase which was hot enough for the nuclear fusion of light elements. But was there thermal equilibrium? Thus, can we speak of the “temperature of the Universe”?

• from isotropy, we must conclude that the Universe expanded adiathermally: no heat can have flowed between any two volume elements in the Universe because any flow would have defined a preferred direction, which is forbidden by isotropy;

• an adiathermal process is adiabatic if it proceeds slow enough for equilibrium to be maintained; then, it is also reversible and isentropic;

• of course, irreversible processes such as particle annihilations must have occurred during the evolution of the Universe; however, as we shall see later, the entropy in the Universe is so ab- solutely dominated by the photons of the microwave background radiation that no entropy production by irreversible processes can have added a signiﬁcant amount of entropy; thus, we assume that the Universe has in fact expanded adiabatically;

• the next question concerns thermal equilibrium; of course, as the Universe expands and cools, particles are diluted and interaction rates drop, so thermal equilibrium must break down at some point for any particle species because collisions become to rare; very

62 CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 63

early in the Universe, however, the expansion rate was very high, and it is important to check whether thermal equilibrium can have been maintained despite the rapid expansion;

• the collision probability between any two particle species will be proportional to their number densities squared, ∝ n2, because collisions are dominated by two-body encounters; the collision rate, i.e. the number of collisions experienced by an individual particle with others will be ∝ n, which is ∝ a−3 for non-relativistic 3 particles; thus, the collision time scale was τcoll ∝ a ; • according to Friedmann’s equation, the expansion rate in the very early Universe was determined by the radiation density, and thus proportional to ∝ a˙/a ∝ a−2, and the expansion time scale was 2 τexp ∝ a ; • equilibrium could be maintained as long as the collision time scale was suﬃciently shorter than the expansion time scale,

τcoll τexp , (7.1)

which is easily achieved in the early Universe when a 1; thus, even though the expansion rate was very high in the early Uni- verse, the collision rates were even higher, and thermal equilibrium can have been maintained;

• the final assumption is that the components of the cosmic fluid behave as ideal gases; by definition, this requires that their particles interact with a very short-ranged force, which implies that partition sums can be written as powers of one-particle partition sums and that the internal energy of the fluids does not depend on the volume occupied; this is a natural assumption which holds even for charged particles because they shield opposite charges on length scales small compared to the size of the observable universe;

• it is thus well justiﬁed to assume that there was thermal equilibrium between all particle species in the early universe, that the constituents of the cosmic “ﬂuid” can be described as ideal gases, and that the expansion of the universe can be seen as an adiabatic process; in later stages of the cosmic evolution, particle species will drop out of equilibrium when their interaction rates fall below the expansion rate of the Universe;

• as long as all species in the Universe are kept in thermodynamic equilibrium, there is a single temperature characterising the cosmic ﬂuid; once particle species freeze out, their temperatures will begin deviating; even then, we characterise the thermal evolution of the Universe by the temperature of the photon background; CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 64

7.1.2 Mean properties of the CMB

• as discussed before, the CMB had been predicted in order to explain the abundance of the light elements, in particular of 4He; it was serendipitously discovered by Penzias and Wilson in 1965;

• measurements of the energy density in this radiation were mostly undertaken at long (radio) wavelengths, i.e. in the Rayleigh-Jeans part of the CMB spectrum; to ﬁrmly establish that the spectrum is indeed the Planck spectrum expected for thermal black-body radiation, the FIRAS experiment was placed on-board the COBE satellite, where it measured the best realisation of a Planck spectrum ever observed;

• we shall see shortly that the mere fact that the CMB does indeed have a Planck spectrum lends strong support to the cosmological standard model; the temperature of the Planck curve best ﬁt- ting the latest measurement of the CMB spectrum by the WMAP satellite is T0 = 2.726 K , (7.2) which implies a mean number density of CMB photons of

−3 nCMB = 405 cm (7.3)

and an energy density in the CMB of

−13 −3 uCMB = 4.2 × 10 erg cm , (7.4)

which corresponds to a mass density of

−34 −3 ρCMB = 4.7 × 10 g cm ; (7.5)

• the density parameter of the CMB radiation is thus

−5 Ωr0 = 4.8 × 10 , (7.6)

which shows that the scale factor at matter-radiation equality was

Ωm0 1/aeq = ≈ 6200 ; (7.7) Ωr0

• the number density of baryons in the Universe is approximately

ΩB0ρcr −7 −3 nB ≈ ≈ 2.3 × 10 cm , (7.8) mp conﬁrming that the photon-to-baryon ratio is extremely high, 1 405 ≈ ≈ 1.8 × 109 ; (7.9) η 2.3 × 10−7 CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 65

7.1.3 Decoupling of the CMB

• When and how was the CMB set free? While the Universe was sufficiently hot to keep electrons and protons separated (we ne- glect heavier elements here for simplicity), the photons scattered off the charged particles, their mean free path was short, and the photons could not propagate. When the Universe cooled below a certain temperature, electrons and protons combined to form hydrogen, free charges disappeared, the mean free path became virtually infinite and photons could freely propagate.

• the recombination reaction

p + e− → H + γ (7.10)

can thermodynamically be described by minimising the free energy N  p Ne NH  Zp Ze ZH  F = −kT ln Z = −kT ln   , (7.11) Np! Ne! NH! where Z is the canonical partition sum of the mixture of protons, electrons and hydrogen atoms, while Zp,e,H are the grand- canonical one-particle partition sums of the protons, the electrons and the hydrogen atoms, respectively, and Np,e,H are their numbers in a closed subvolume;

• the constant number of baryons is NB = Np + NH and the number of electrons is Ne = Np, thus NH = NB − Ne, and we can express the numbers of all particle species by the number of electrons Ne; ﬁnally, the chemical potentials must sum to zero in equilibrium, µp + µe − µH = 0; • then, the equilibrium state is found by extremising the free energy, ∂F = 0 , (7.12) ∂Ne

and solving for the electron number Ne or, equivalently, for the ionisation fraction x = Ne/NB; the result is Saha’s equation

√ !3/2 !3/2 x2 π m c2 0.26 m c2 = √ e e−χ/kT ≈ e e−χ/kT , 1 − x 4 2ζ(3)η kT η kT (7.13) where χ is the ionisation energy of hydrogen, χ = 13.6 eV, and ζ is the Riemann Zeta function; Once recombination sets in, the • notice that Saha’s equation contains the inverse of the η param- ionisation fraction x drops very eter (7.9), which is a huge number due to the high photon-to- quickly. baryon ratio in the Universe; this counteracts the exponential which would otherwise guarantee that recombination happens CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 66

when kT ≈ χ, i.e. at T ≈ 1.6 × 105 K; recombination is thus delayed by the high photon number, which illustrates that newly formed hydrogen atoms are eﬀectively reionised by suﬃciently energetic photons until the temperature has dropped well below the ionsation energy (we saw this also for big bang nucleosynthesis; the epoch of actual nucleosynthesis was delayed until a very long time after the mean temperature of the Universe was ∼1 MeV). Putting x ≈ 0.5 in (7.13) yields a recombination temperature of kTrec ≈ 0.3 eV , Trec ≈ 3500 K (7.14)

and thus a recombination redshift of zrec ≈ 1280; • this is well in the matter-dominated phase, and therefore we can estimate the age of the Universe using

Z a da0 1 Z a √ 2a3/2 ≈ 0 0 t = 0 0 √ a da = √ 0 a H(a ) H0 Ωm0 0 3H0 Ωm0 ≈ 360, 000 yr ; (7.15)

• recombination does not proceed instantaneously; the ionisation fraction x drops from 0.9 to 0.1 within a temperature range of approximately 200 K, corresponding to a redshift range of ! dz d T ∆T ∆z ≈ ∆T ≈ − 1 ∆T ≈ ≈ 75 ; (7.16) dT z dT T0 T0 rec zrec or a time interval of ∆a ∆z ∆t ≈ ≈ √ ≈ 35, 000 yr ; (7.17) 5/2 aH H0 Ωm0(1 + z)

• we are thus led to conclude that the CMB was released when the Universe was approximately 360,000 years old, during a phase that lasted approximately 35,000 years; we have derived this result merely using the present temperature of the CMB, the photon-to-baryon ratio, the Hubble constant and the matter density parameter Ωm0; the cosmological constant or a possible curvature of the Universe do not matter here; The FIRAS instrument on-board the • the fact that the temperature of the Universe dropped by ≈ 200 K COBE satellite conﬁrmed that the while the CMB was released leads to another remarkable reali- CMB has the most perfect Planck sation: How can the CMB have a Planck spectrum with a single spectrum ever measured. temperature if it was released from a plasma with a fairly broad range of temperatures? In a Friedmann-Lemaˆıtre model universe, this is easy to understand: Photons released from higher- temperature plasma were released somewhat earlier and were subsequently redshifted by a somewhat larger amount. The range CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 67

of temperatures is thus precisely compensated by the redshift, which conﬁrms the expectation that T ∝ a−1 in Friedmann- Lemaˆıtre models. Thus, the fact that the CMB has a Planck spectrum with a single temperature indirectly conﬁrms that we are living in a Friedmann-Lemaˆıtre universe.

7.2 Structures in the CMB

7.2.1 The dipole

• the Earth is moving around the Sun, the Sun is orbiting around the Galactic centre, the Galaxy is moving within the Local Group, which is falling towards the Virgo cluster of galaxies; we can thus not expect that the Earth is at rest with respect to the CMB; we denote the net velocity of the Earth with respect to the CMB rest frame by v⊕; The Earth’s motion with respect to • Lorentz transformation shows that, to lowest order in v⊕/c, the the CMB rest frame imprints a dipo- Earth’s motion imprints a dipolar intensity pattern on the CMB lar temperature pattern on the CMB with an amplitude of with milli-Kelvin ampitude. ∆T v = ⊕ ; (7.18) T0 c the dipole’s amplitude has been measured to be ≈ 1.24 mK, from which the Earth’s velocity is inferred to be

−1 v⊕ ≈ 371 km s ; (7.19)

• this is the highest-order deviation from isotropy in the CMB, but it is irrelevant for our purposes since it does not allow any conclusions on the Universe at large;

7.2.2 Expected amplitude of CMB ﬂuctuations

• it is reasonable to expect that density fluctuations in the CMB should match density fluctuations in the matter because photons were tightly coupled to baryons by Compton scattering before recombination; since the radiation density is ∝ T 4, a density contrast δ is expected to produce relative temperature fluctuations of order δρ 4T 3δT δT 1 + δ δ = − 1 ≈ − 1 ⇒ ≈ ; (7.20) ρ T 4 T 4

• obviously, there are large-scale structures in the Universe today whose density contrast reaches or even substantially exceeds unity; assuming linear structure growth on large scales, and CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 68

knowing the scale factor of recombination, we can thus infer that relative temperature fluctuations of order ! δT 1 1 1 ≈ 1 + ≈ ≈ 10−3 (7.21) T 4 D+(arec) 4arec should be seen in the CMB, i.e. fluctuations of order mK, similar to the dipole; such fluctuations, however, were not observed, although cosmologists kept searching increasingly desperately for decades after 1965;

• Why do they not exist? The estimate above is valid only under the assumption that matter and radiation were tightly coupled. Should this not have been the case, density ﬂuctuations did not need to leave such a pronounced imprint on the CMB. In order to avoid the tight coupling, the majority of matter must not interact electromagnetically. Thus, we conclude from the absence of mK ﬂuctuations in the CMB that matter in the Universe must be dominated by something that does not interact with light. This is perhaps the strongest argument in favour of dark matter.

7.2.3 Expected CMB ﬂuctuations

• before we come to the results of CMB observations and their sig- niﬁcance for cosmology, let us summarise which physical eﬀects we expect to imprint structures on the CMB;

• the basic hypothesis is that the cosmic structures that we see today formed via gravitational instability from seed fluctuations in the early Universe, whose origin is yet unclear; this implies that there had to be density fluctuations at the time when the CMB was released; via Poisson’s equation, these density fluctuations were related to fluctuations in the Newtonian potential;

Sachs-Wolfe Eﬀect

• photons released in a potential fluctuation δΦ lost energy if the fluctuation was negative, and gained energy when the fluctuation was positive; this energy change can be translated to the temperature change δT 1 δΦ = , (7.22) T 3 c2 which is called the Sachs-Wolfe effect after the people who first described it; CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 69

• let us briefly look into the expected statistics of the Sachs-Wolfe effect; we introduced the power spectrum of the density fluctuations in (2.22) as the variance of the density contrast in Fourier space; Fourier-transforming Poisson’s equation, we see that

δˆ δΦˆ ∝ − , (7.23) k2 and thus the power spectrum of the temperature ﬂuctuations due to the Sachs-Wolfe eﬀect is determined by

∗  −3 hδˆδˆ i Pδ k k k0 P ∝ P ∝ ∝ ∝  T Φ 4 4  −7 (7.24) k k k k k0

according to (2.23); this shows that the Sachs-Wolfe eﬀect can only be important at small k, i.e. on large scales, and dies oﬀ quickly towards smaller scales;

Acoustic Oscillations

• the cosmic fluid consisted of dark matter, baryons and photons; overdensities in the dark matter compressed the fluid due to their gravity until the rising pressure in the tightly coupled baryon- photon fluid was able to counteract gravity and drove the fluctuations apart; in due course, the pressure sank and gravity won again, and so forth: the cosmic fluid thus underwent acoustic oscillations;

• since the pressure was dominated by the photons, whose pressure is a third of their energy density, the sound speed was r p c cs ≈ = √ ≈ 0.58 c ; (7.25) ρ 3

• only such density ﬂuctuations could undergo acoustic oscillations which were small enough to be crossed by sound waves in the available time; we saw before that recombination happened when the Universe was ≈ 360, 000 yr old, so the largest length that could be traveled by sound wave was the sound horizon c rs ≈ 360, 000 yr × √ ≈ 63 kpc ; (7.26) 3 larger-scale density ﬂuctuations could not oscillate;

• we saw in (2.15) that the angular-diameter distance from today to scale factor a 1 is ca Z 1 dx D a ang( ) = 2 (7.27) H0 a x E(x) CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 70

if we assume for simplicity that the universe is spatially ﬂat; then, the denominator in the integrand is s p − 2 4 p 1 Ωm0 3 x E(x) = Ωm0 x + (1 − Ωm0)x = Ωm0 x 1 + x Ωm0 (7.28) • inserting, we ﬁnd that we can approximate the angular-diameter distance for a 1 by ! 2carec 1 − Ωm0 Dang(arec) ≈ √ 1 − ≈ 7.3 Mpc , (7.29) H0 Ωm0 6Ωm0 and the sound horizon sets an angular scale of

2rs ◦ θs = ≈ 1 , (7.30) Dang(arec) to which we shall shortly return; • inserting the time directly from (7.15), the sound speed from (7.25) and the angular-diameter distance from (7.29) reveals a weak dependence of θs on Ωm0 even for a ﬂat Universe,

1/2 ! 2arec 1 − Ωm0 θs ≈ √ 1 + ; (7.31) 3 3 6Ωm0

Silk Damping

• a third effect influencing structures in the CMB is caused by the fact that, as recombination proceeds, the mean-free path of the photons increases; if ne = xnB is the electron number density and σT is the Thomson cross section, the mean-free path is 1 λ ≈ ; (7.32) xnBσT as the ionisation fraction x drops towards zero, the mean-free path aproaches infinity; • after N scatterings, the photons will have diffused by √ λD ≈ Nλ ; (7.33) the number of scatterings per unit time is

dN ≈ xnBσTcdt , (7.34) and thus the diﬀusion scale is given by Z Z 2 2 cdt λD ≈ λ dN ≈ ; (7.35) xnBσT CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 71

• the latter integral is dominated by the short recombination phase during which x drops to zero; inserting x ≈ 1/2 as a typical value, we can thus approximate

2 2c∆t λD ≈ ; (7.36) nBσT

• around recombination, the baryon number density is Ω ρ ≈ B0 cr ≈ −3 nB −3 500 cm , (7.37) mparec and we ﬁnd λ ≈ 2 kpc and λD ≈ 4.5 kpc (7.38)

from Eqs. (7.32) and (7.36), respectively; λD thus corresponds to 0 an angular scale of θD ≈ 5 on the sky; this damping mechanism is called Silk damping after its discoverer;

• we thus expect three mechanisms to shape the appearance of the microwave sky: the Sachs-Wolfe eﬀect on the largest scales, the acoustic oscillations on scales smaller than the sound horizon, and Silk damping on scales smaller than a few arc minutes;

7.2.4 CMB polarisation

• if the CMB does indeed arise from Thomson scattering, interesting eﬀects must arise from the fact that the Thomson scattering cross section is polarisation sensitive and can thus produced linearly polarised from unpolarised radiation;

• suppose an electron is illuminated by unpolarised radiation from the left, then the radiation scattered towards the observer will be linearly polarised in the perpendicular direction; likewise, unpolarised radiation incoming from the top will be linearly polarised horizontally after being scattered towards the observer;

• thus, if the electron is irradiated by a quadrupolar intensity pattern, the scattered radiation will be partially linearly polarised; the polarised intensity is expected to be of order 10% of the total intensity; The anisotropy of Thomson scatter- • the polarised radiation must reflect the same physical effects as ing causes the CMB to be partially the unpolarised radiation, and the two must be cross-correlated; linearly polarised. much additional information on the physical state of the early Universe should thus be contained in the polarised component of the CMB, besides that a detection of the polarisation would add confirmation to the physical picture of the CMB’s origin; CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 72

7.2.5 The CMB power spectrum

• Fourier transformation is not possible on the sphere, but the analysis of the CMB proceeds in a completely analogous way by decomposing the relative temperature fluctuations into spherical harmonics, finding the spherical-harmonic coefficients Z The Earth’s surface, and its 2 δT lowest-order multipoles: dipole, alm = d θ Ylm(~θ) , (7.39) T quadrupole and octupole (left and from them the power spectrum column below the map), and the multipoles with l = 4 ... 7 (right l 1 X column). C ≡ |a |2 , (7.40) l 2l + 1 lm m=−l which is equivalent to the matter power spectrum (2.22) on the sphere; the average over m expresses the expectation of statistical isotropy; • the shape of the CMB power spectrum reflects the three physical mechanisms identified above: at small l (on large scales), the Sachs-Wolfe effect causes a feature-less plateau, followed by pronounced maxima and minima due to the acoustic oscillations, damped on the smallest scales (largest l) by Silk damping; • the detailed shape of the CMB power spectrum depends sensitively on the cosmological parameters, which can in turn be determined by fitting the theoretically expected to the measured Cl; this is the main reason for the detailed and sensitive CMB measurements pioneered by COBE, continued by ground-based and balloon experiments, and culminating recently in the spectacular results obtained by the WMAP satellite; The CMB power spectrum is characterised by three physical effects: 7.2.6 Microwave foregrounds the Sachs-Wolfe effect, acoustic oscillations, and Silk damping. • by definition, the CMB is the oldest visible source of photons because all possible earlier sources could not shine through the hot cosmic plasma; therefore, every source that produced microwave photons since, or that produced photons which became redshifted into the microwave regime by now, must appear superposed on the CMB; the CMB is thus hidden behind curtains of foreground emission that have to be opened before the CMB can be observed; • broadly, the CMB foregrounds can be grouped into point sources and diffuse sources; the most important among the point sources are infrared galaxies at high redshift, galaxy clusters affecting the CMB through the Sunyaev-Zel’dovich effect, and bodies in the Solar System such as the major planets, but even some of the asteroids; CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 73

• the population of infrared sources at high redshift is poorly known, but the angular resolution of CMB measurements has so far been too low to be signiﬁcantly contaminated by them; future CMB observations will have to remove them carefully;

• the Sunyaev-Zel’dovich effect was introduced under 3.3 before; once the angular resolution of CMB detectors will drop towards a few arc minutes, a large number of galaxy clusters are expected to be discovered by their peculiar spectral signature, casting a shadow below, emitting above, and vanishing at 217 GHz; the Sunyaev-Zel’dovich effect comes in two variants; one is the thermal effect discussed above, the other is the kinetic effect caused Galaxy clusters appear as character- by the bulk motion of the cluster as a whole, which causes CMB istic point-like sources on the CMB. radiation to be scattered by the electrons moving with the cluster; very few clusters have so far been detected in CMB data, but thousands are expected to be found in future missions;

• microwave radiation from bodies in the Solar System has so far been used to calibrate microwave detectors; CMB observations at an angular resolution below ∼ 100 are expected to detect hundreds of minor planets;

• diffuse CMB foregrounds are mainly caused by our Galaxy itself; there are three main components: synchrotron emission, emission from warm dust, and bremsstrahlung; Relativistic electrons gyrating in the • synchrotron radiation is emitted by relativistic electrons in the Galaxy’s magnetic field emit syn- Galaxy’s magnetic field; it is highly linearly polarised and has chrotron radiation. a power-law spectrum falling steeply from radio towards microwave frequencies; it is centred on the Galactic plane, but shows filamentary extensions from the Galactic centre towards the Galactic poles;

• the dust in the Milky Way is also concentrated in the Galactic plane; it is between 10 ... 20 K warm and therefore substantially Warm dust in the galaxy also adds warmer than the CMB itself; it has a Planck spectrum which is to the microwave foregrounds. self-absorbed due to the high optical depth of the dust; due to its higher temperature, the dust has a spectrum rising with increasing frequency in the frequency window in which the CMB is usually observed;

• bremsstrahlung radiation is emitted by ionised hydrogen clouds (HII regions) in the Galactic plane; it has the typical, exponentially-falling spectrum of thermal free-free radiation; further sources of microwave radiation in the Galaxy are less promi- nent; among them are line emission from CO molecules embed- ded in cool gas clouds; CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 74

• the falling spectra of the synchrotron and free-free radiation, and the rising spectrum of the dust create a window for CMB observations between ∼ 100 ... 200 GHz; the diﬀerent spectra of the foregrounds, and their non-Planckian character, are crucial for their proper removal from the CMB data; therefore, CMB measurements have to be carried out in multiple frequency bands;

7.2.7 Measurements of the CMB

• Wien’s law shows that the CMB spectrum peaks at λmax ≈ 0.11 cm, or at a frequency of νmax ≈ 282 GHz; • as we saw, Silk damping sets in below a few arc minutes, thus most of the structures in the CMB are resolvable for rather small telescopes; according to the formula λ ∆θ ≈ 1.44 (7.41) D relating the angular resolution ∆θ to the ratio between wavelength and telescope diameter D, we ﬁnd that mirrors with λ D . 1.44 max ≈ 75 cm (7.42) θD

are suﬃcient (recall that θD needs to be inserted in radians here); • thus, detectors are needed which are sensitive to millimetre and sub-mm radiation and reach µK sensitivity, while the telescope optics can be kept rather small and simple;

• two types of detector are commonly used; the ﬁrst are bolometers, which measure the energy of the absorbed radiation by the temperature increase it causes; therefore, they have to be cooled to very low temperatures typically in the mK regime; the second are so-called high electron mobility detectors (HEMTs), in which the currents caused by the incoming electromagnetic ﬁeld are measured directly; the latter detectors measure amplitude and phase of the waves and are thus polarisation-sensitive by construction, which bolometers are not; polarisation measurements with bolometers is possible with suitably shaped so-called feed horns guiding the radiation into the detectors;

• since water vapor in the atmosphere both absorbs and emits microwave radiation through molecular lines, CMB observations need to be carried out either at high, dry and cold sites on the ground (e.g. in the Chilean Andes or at the South Pole), or from balloons rising above the troposphere, or from satellites in space;

Microwave observations from the ground require cold and dry sites. The Boomerang experiment was carried around the South Pole by a balloon (the top ﬁgure shows the balloon before launch, with Mt. Erebus in the background). The CMB polarisation was ﬁrst measured with the DASI interferometer (middle, bottom), also at the South Pole. CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 75

• after the breakthrough achieved with COBE, progress was made with balloon experiments such as Boomerang and Maxima, and with ground-based interferometers such as Dasi (Degree Angular Scale Interferometer), VSA (Very Small Array) and CBI (Cosmic Background Imager); the balloons covered a small fraction of the sky (typically ∼ 1%) at frequencies between 90 and 400 GHz, while the interferometers observe even smaller ﬁelds at somewhat lower frequencies (typically around 30 GHz);

• the ﬁrst discovery of the CMB polarisation and its cross- correlation with the CMB temperature was achieved in 2003 with the Dasi interferometer;

• the existence, location and height of the ﬁrst acoustic peak had been ﬁrmly established before the NASA satellite Wilkinson Mi- crowave Anisotropy Probe (WMAP for short) was launched, but the increased sensitivity and the full-sky coverage of WMAP produced breath-taking results; WMAP is still operating, measuring the CMB temperature at frequencies between 23 and 94 GHz with an angular resolution of & 150; the sensitivity of WMAP is barely high enough for polarisation measurements;

• by now, data from the first five years of operation have been published, and cosmological parameters have been obtained fitting theoretically expected to the measured temperature-fluctuation power spectrum and the temperature-polarisation power spectrum; results are given in the following table:

CMB temperature TCMB 2.728 ± 0.004 K +0.01 total density Ωtot 0.99−0.01 +0.01 matter density Ωm0 0.25−0.01 +0.001 baryon density Ωb0 0.043−0.001 cosmological constant ΩΛ0 0.74 ± 0.03 decoupling redshift zdec 1089 ± 1 age of the Universe t0 13.7 ± 0.2 Gyr +8 age at decoupling tdec 379−7 kyr +0.04 power-spectrum normalisation σ8 0.80−0.04

• the Hubble constant is not an independent measurement from the CMB alone; only by assuming a ﬂat universe, it can be inferred from the location of the ﬁrst acoustic peak in the CMB power −1 −1 spectrum to be H0 = 73 ± 3 km s Mpc , which agrees perfectly with the results of the Hubble Key Project and gravitational-lens time delays;

• a European CMB satellite mission is under way: ESA’s Planck satellite is expected to be launched in 2009; it will observe the microwave sky in ten frequency bands between 30 and 857 GHz

The COBE (top) and WMAP satellites. CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 76

with about ten times higher sensitivity than WMAP, and an angular resolution of & 50; its wide frequency coverage will be very important for substantially improved foreground subtraction; also, it will have suﬃcient sensitivity to precisely measure the CMB polarisation in some of its frequency bands; moreover, it is expected that Planck will detect of order 10,000 galaxy clusters through their thermal Sunyaev-Zel’dovich eﬀect;

Figure 7.1: The European Planck satellite, to be launched in 2008 (left), and the expected error bars on the temperature and polarisation power spectra (right). CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 77

Figure 7.2: Left: Comparison between the WMAP temperature maps obtained after one (top) and three years of measurement. Right: De- composition of the WMAP 3-year temperature map into low-order multipoles.

Figure 7.3: Power spectrum of CMB temperature ﬂuctuations as measured from the 3-year data of WMAP and several additional ground- based experiments. CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 78

Figure 7.4: Top: Polarisation map obtained by WMAP. Bottom: The temperature-ﬂuctuation power spectrum (top) and the temperature- polarisation cross-power spectrum determined from the WMAP 3-year data. CHAPTER 7. THE COSMIC MICROWAVE BACKGROUND 79

Figure 7.5: Left: Constraints on cosmological parameters derived from WMAP 3-year data alone (black contours), and combined with other cosmological data sets (red islands). Right: Constraints on the baryon density from primordial nucleosynthesis (vertical grey bar) and from the CMB. The agreement is extraordinary. Chapter 8

Inﬂation and Dark Energy

8.1 Cosmological inﬂation

8.1.1 Motivation

• In the preceding chapters, we have seen the remarkable success of the cosmological standard model, which is built upon the two symmetry assumptions underlying the class of Friedmann- Lemaˆıtre-Robertson-Walker models which experienced a Big Bang a ﬁnite time ago. We shall now discuss a fundamental problem of these models, and a possible way out.

• Historically, the problem was raised in a diﬀerent way, but it becomes obvious with the very straightforward realisation that it is by no means obvious why the CMB should appear as isotropic as it is, and why there should be large coherent structures in it.

• Let us begin with the so-called comoving particle horizon, which is the distance that light can travel between the Big Bang and time t. Since light travels on null geodesics, ds = 0, a radial light ray propagates according to cdt = adw [cf. (2.2)]. Therefore, Z t Z t dt0 Z t da w(t) = dw = c = c . (8.1) 0 0 a 0 aa˙

• Between the Big Bang and the recombination time trec, the integrand in (8.1) can be approximated by the expansion rate for a matter-dominated universe, or a˙2 = H2Ω a−3 (8.2) a2 0 m0 according to (2.7). Thus, p aa˙ = H0 Ωm0a , (8.3)

80 CHAPTER 8. INFLATION AND DARK ENERGY 81

the comoving particle horizon becomes √ Z arec c da 2c arec w(trec) = √ √ = √ , (8.4) H0 Ωm0 0 a H0 Ωm0 and the physical particle horizon at the time of recombination is rrec = arecwrec. • On the other hand, we have seen in (7.29) that the angular- diameter distance to the CMB is

2carec Dang(arec) ≈ √ , (8.5) H0 Ωm0 which implies that the angular size of the particle horizon is

rrec √ ◦ θrec = ≈ arec ≈ 2 . (8.6) Dang(arec)

• The physical meaning of the particle horizon is that no event be- Illustration of the causality prob- tween the Big Bang and recombination can exert any influence lem: The particle horizon at CMB on a given particle if it is more than the horizon length away. decoupling corresponds to a circle ◦ Our simple calculation thus shows that we cannot understand how of ∼ 2 radius. causal processes could establish identical physical conditions in patches of the sky with radius a few degrees. Points on the CMB separated by larger angles were never causally connected before the CMB was released. It is therefore not at trivial that the CMB could have attained almost the same temperature across the entire sky! The simple fact that the CMB is almost entirely isotropic across the sky thus poses a problem which the standard cosmological model is apparently unable to solve. Moreover, the formation of coherent structures larger than the particle horizon remains mysterious. This is one way to state the horizon problem. • It is sometimes called the causality problem: How can coherent structures in the CMB be larger than the particle horizon was at recombination? • Another uncomfortable problem of the standard cosmological model is the flatness, or at least the near-flatness, of spatial hypersurfaces of our Universe. To see this, we write Friedmann’s equation in the form 8πG Λ Kc2 H2(a) = ρ + − 3 3 a2 " # Kc2 = H2(a) Ω (a) − , (8.7) total a2H2 from which we conclude Kc2 |Ω (a) − 1| = . (8.8) total a2H2 CHAPTER 8. INFLATION AND DARK ENERGY 82

• According to (8.3), we have, in the matter-dominated era, √ ada dt = √ ⇒ a ∝ t2/3 , (8.9) H0 Ωm0 hence a2H2 = a˙2 ∝ t−2/3 (8.10) or, from (8.8), 2/3 |Ωtotal(a) − 1| ∝ t . (8.11)

• This shows that any deviation of the total density parameter Ωtotal from unity tends to grow with time. Thus, (spatial) flatness is an unstable property. If it is not very precisely flat in the beginning, the Universe will develop away from flatness. Since we know that spatial hypersurfaces are now almost flat, |Ωtotal(a)−1| . 1%, say, the deviation from flatness must have been at most !2/3 4 × 105 |Ω (a ) − 1| . 1% ≈ 10−5 , (8.12) total rec 1.4 × 1010 or ten parts per million at the time of recombination. Clearly, this requires enormous fine-tuning. This is called the flatness problem: How can we understand flatness in the late universe without assuming an extreme degree of fine-tuning at early times?

8.1.2 The idea of inﬂation

• Since the c/H is the Hubble radius, the quantity rH ≡ c/(aH) is the comoving Hubble radius.√ During the matter-dominated era, −3/2 −2 H ∝ a and thus rH ∝ a, while H ∝ a and rH ∝ a during the radiation-dominated era. Therefore, the comoving Hubble radius typically grows with time. Since we can write (8.8) as

2 |Ωtotal(a) − 1| = KrH , (8.13) this is equivalent to the flatness problem. The universe can be driven into flat- • This motivates the idea that at least the flatness problem would ness (top) if the comoving Hub- be solved if the comoving Hubble radius could, at least for some ble radius can shrink for sufficiently sufficiently long period, shrink with time. If that could be ar- long time (middle). This can also ranged, any deviation of Ωtotal(a) from unity would be driven to- solve the causality problem (bot- wards zero. tom). • Conveniently, such an arrangement would also remove or at least alleviate the causality problem. Since the Hubble length charac- terises the radius of the observable universe, it could be driven inside the horizon and thus move the entire observable universe into a causally-connected region. When the hypothesised epoch CHAPTER 8. INFLATION AND DARK ENERGY 83

of a shrinking comoving Hubble radius is over, it starts expanding again, but if the reduction was sufficiently large, it could remain within the causally-connected region at least until the present. • How could such a shrinking comoving Hubble radius be ar- ranged? Obviously, we require ! d c c c a˙2 a˙2 = − (˙aH + aH˙ ) = − + a¨ − < 0 , dt aH (aH)2 (aH)2 a a (8.14) which is possible if and only ifa ¨ > 0, in other words, if the expansion of the Universe accelerates. • This appears counter-intuitive because the cosmic expansion is dominated by gravity, which should be attractive and thus necessarily decelerate the expansion. The first law of thermodynamics implies the matter condition ρc2 ρc2 + 3p < 0 ⇒ p < − . (8.15) 3 In other words, cosmic acceleration is possible if the dominant ingredient of the cosmic fluid has sufficiently negative pressure. • When applied to a cosmic sub-volume V = a3, the first law of thermodynamics dE + pdV = 0 ⇒ d(ρc2a3) + pda3 = 0 (8.16) because any heat current would violate isotropy. We thus obtain the equation (˙ρa3 + 3ρa2a˙)c2 + 3pa2a˙ = 0 , (8.17) which implies the density evolution a˙ p ρ˙ = −3 ρ + . (8.18) a c2 • The cosmological constant must haveρ ˙ = 0 and therefore p = −ρ/c2. It has a suitable equation-of-state for cosmic acceleration. (We will see later that Type Ia supernovae, and other observations, motivate a cosmological constant and thus cosmic acceleration). • If we bring Friedmann’s equation (2.7) into the form

2 2 2 h −2 −1 2i a H = H0 Ωm0a + Ωm0a − ΩK + ΩΛ0a , (8.19) it is obvious that a cosmological constant dominates quickly once it becomes comparable to the other density components, because it has the highest power of the scale factor a attached. Once it dominates, (8.19) becomes p p a˙ = H0 ΩΛ0a ⇒ a ∝ exp H0 ΩΛ0t , (8.20) and the universe enters into exponential expansion. CHAPTER 8. INFLATION AND DARK ENERGY 84

8.1.3 Slow roll, structure formation, and observational constraints

• We have seen that we need inflation to solve the flatness and causality problems, and inflation needs a form of matter with negative pressure. What could that be? Fortunately, conditions like that are not hard to arrange for particle physics.

• Consider a scalar field φ with a self-interaction potential V(φ). Then, field theory shows that pressure and density of the scalar field are related by the equation of state

1 ˙2 − 2 2 φ V pφ = wρφc with w ≡ . (8.21) 1 ˙2 2 φ + V Evidently, negative pressure is possible if the kinetic energy of the scalar ﬁeld is suﬃciently smaller than its potential energy. For the cosmological-constant case, φ˙ = 0, we have w = −1 or p = −ρc2, in agreement with the conclusion from (8.18).

• In other words, a suitably strongly self-interacting scalar field has exactly the properties we need. Inflation, i.e. accelerated expansion, broadly requires φ˙2 to be sufficiently smaller than V.

• Moreover, we need inﬂation to operate long enough to drive the total matter density parameter suﬃciently close to unity for it to remain there to the present day. These two conditions are conventionally cast into the form ! 1 V0 1 V00 ≡ 1 and η ≡ 1 . (8.22) 24πG V 8πG V

They are called the slow-roll conditions. The first assures that inflation can set in, because if it is satisfied, the potential has a small gradient and cannot drive rapid rapid changes in the scalar field. The second restricts the curvature of the potential and thus assures that the inflationary condition is satisfied long enough. The slow-roll conditions mean that • Estimates show that inflation needs to expand the Universe by the potential must be sufficiently flat 50...60 ∼ 50 ... 60 e-foldings (i.e. by a factor of e ) for solving the for inflation to set in, and gently causality and flatness problems. curved for it to last long enough. • Inflation ends once the slow-roll conditions are violated. By then, the Universe will have become extremely cold. While the density of the inflaton field will be approximately the same as at the onset of inflation (as for the cosmological constant, this is a consequence of the negative pressure), all other matter and radiation fields will have their energy densities lowered by factors of a−3...4. CHAPTER 8. INFLATION AND DARK ENERGY 85

• Once approaches unity, the kinetic term φ˙2 will dominate the potential, and the scalar field will start oscillating rapidly. It is assumed that the scalar field then decays into ordinary matter which fills or reheats the Universe after inflation is over.

• It is an extremely interesting aspect of inflation that it also provides a mechanism for seeding structure formation. As any other quantum field, the inflaton field φ must have undergone vacumm oscillations because the zero-point energy of a quantum harmonic oscillator cannot vanish due to Heisenberg’s uncertainty principle. Inflation may expand quantum fluc- • These vacuum oscillations cause the spontaneous creation and tuations to cosmological scales. It annihilation of particle-antiparticle pairs. Once inflation sets in, is possible that the large-scale struc- vacuum fluctuation modes are quickly driven out of the horizon ture in the universe originated from and loose causal connection. Then, they cannot decay any more inevitable quantum fluctuations in and “freeze in”. Thus, inflation introduces the breath-taking no- the very early universe. tion that density fluctuations in our Universe today may have been seeded by vacuum fluctuations of the inflaton field before inflation set in and enlarged them to cosmological scales.

• This idea has precisely quantifiable consequences. First, by the central limit theorem, it demands that linear density fluctuations in the Universe should be a Gaussian random field. This is because they arise from incoherent superposition of extremely many independent fluctuation modes whose amplitude and wave number are all drawn from the same probability distribution. Under these circumstances, the central limit theorem shows that the result, i.e. the superposition of all these modes, must be a Gaussian random field.

• Second, it implies that the statistics of density fluctuations in the Universe today must be explicable by the statistics of vacuum fluctuations in a scalar quantum field. This is indeed the case. The power spectrum resulting from this consideration is very close to the scale-free Harrison-Zel’dovich-Peebles shape introduced in Sect. 1.2.2, n Pδ(k) ∝ k , (8.23) where n ≈ 1.

• The spectral index n would be precisely unity if inﬂation lasted forever. Since this was obviously not so, n must deviate slightly from unity, and detailed calculations show that it must be slightly smaller, n = 1 + 2η − 6 . (8.24)

The latest WMAP measurements do in fact show that

+0.014 n = 0.963−0.015 . (8.25) CHAPTER 8. INFLATION AND DARK ENERGY 86

The completely scale-invariant spectrum, n = 1, is excluded at 2.5σ.

• The measured deviation of n from unity also restricts the number N of e-foldings completed by inﬂation; a value of n = 0.963 is consistent with 60 e-foldings.

• Another prediction of inflation is that it may excite not only scalar, but also tensor perturbations. Scalar perturbations lead to the density fluctuations, tensor perturbations correspond to gravitational waves. Vector perturbations do not play any role because they decay quickly as the universe expands. Inflation predicts that the ratio r between the amplitudes of tensor and scalar perturbations, taken in the limit of small wave numbers, is

r = 16 . (8.26)

• An inflationary background of gravitational waves is in principle detectable through the polarisation of the CMB. Limits of order r . 0.05 are expected from the upcoming Planck satellite. To- gether with the result n , 1 from WMAP, it will then be possible to constrain viable inflation models, i.e. to constrain the shape of the inflaton potential.

8.2 Dark energy

8.2.1 Motivation

• The CMB shows us that the Universe is at least nearly spatially flat. Constraints from the CMB, and from kinematics and cluster evolution (we will discuss this later) show that the matter density alone cannot be responsible for flattening space, and primordial nucleosynthesis and the CMB show that baryons contribute at a very low level only. Something is missing, and it even dominates today’s cosmic fluid.

• From structure formation, we know that this remaining con- stituent cannot clump on the scales covered by the galaxy surveys and below. It is thus diﬀerent from dark matter. We call it dark energy. The type-Ia supernovae (later) tell us that it behaves at least very similar to a cosmological constant.

• Maybe the dark energy is a cosmological constant? Nothing currently indicates any deviation from this “simplest” assumption. So far, the cosmological constant is a perfectly viable description for all observational evidence we have. CHAPTER 8. INFLATION AND DARK ENERGY 87

• However, this is deeply unsatisfactory from the point of view of theoretical physics. The problem is the value of ΩΛ0. As we have seen above, a self-interacting scalar ﬁeld with negligible kinetic energy behaves like a cosmological constant. Then, its density should simply be given by its potential V. Simple arguments suggest that V should be the fourth power of the Planck mass, which turns out to be 120 orders of magnitude larger than the cosmological constant derived from observations. Since this fails, it seems natural to expect that the cosmological constant should vanish, but it does not. The main problem with the cosmological constant is therefore, why is it not zero if it is so small?

• The explanation of inflation by means of an inflaton field suggests one way out. As we have seen there, accelerated expansion can be driven by a self-interacting scalar field while its potential energy dominates. Moreover, it can be shown that if the potential V has an appropriate shape, the dark energy has attractor properties in the sense that a vast range of initial density values can evolve towards the same value today. Such models for a dynamical dark energy are theoretically very attractive.

8.2.2 Observational constraints?

• If the dark energy is indeed dynamical and provided by a self- interacting scalar ﬁeld, how can we ﬁnd out more about it? Re- viewing the cosmological measurements we have discussed so far, it becomes evident that they are all derived from constraints on

– cosmic time, as in the age of the Galaxy or of globular clusters, or in primordial nucleosynthesis; – distances, as in the spatial ﬂatness derived from the CMB, the type-Ia supernovae or the geometry of cosmological weak lensing; or – the growth of cosmic structures, as in the acoustic oscillations in the CMB, the evolution of the cluster population, the structures in the galaxy distribution or the source of cosmological weak-lensing eﬀects.

• We must therefore seek to constrain the dark energy by measurements of distances, times, and structure growth. Since they can all be traced back to the expansion behaviour of the universe as described by the Friedmann equation, we must see how the dark energy enters there, and what eﬀects it can seed through it. CHAPTER 8. INFLATION AND DARK ENERGY 88

• Let us therefore assume that the dark energy is a suitably self- interacting, homogeneous scalar ﬁeld. Then, its pressure can be described by p = w(a)ρc2 , (8.27) where the equation-of-state parameter w is some function of a. According to (8.15), accelerated expansion needs w < −1/3, and the cosmological constant corresponds to w = −1. Since all cosmological measurements to date are in agreement with the assumption of a cosmological constant, we need to arrange things such that w → −1 today. • Suppose we have some function w(a), which could either be obtained from a phenomenological choice, a model for the self- interaction potential V(φ) through (8.21) or from a simple ad-hoc parameterisation. Then, (8.18) implies ρ˙ a˙ = −3(1 + w) , (8.28) ρ a or ( ) Z a da0 − 0 ≡ ρ(a) = ρ0 exp 3 [1 + w(a )] 0 ρ0 f (a) . (8.29) 1 a

• If w = const., this simpliﬁes to

−3(1+w) ρ(a) = ρ0 exp [−3(1 + w) ln a] = ρ0a . (8.30)

If w = −1, we recover the cosmological-constant case ρ = ρ0 = const., for pressure-less material, w = 0 and ρ ∝ a−3, and for radiation, w = 1/3 and ρ ∝ a−4. • Therefore, we can take account of the dynamical dark energy by replacing the term ΩΛ0 in the Friedmann equation (2.7) by ΩDE0 f (a), and the expansion function E(a) turns into

h −4 −3 −2i1/2 E(a) = Ωr0a + Ωm0a + ΩDE0 f (a) + ΩK0a , (8.31)

where ΩK0 = 1 − Ωr0 − Ωm0 − ΩDE0 is the curvature density parameter. • We thus see that the equation-of-state parameter enters the expansion function in integrated form. Since all cosmological observables are integrals over the expansion function, including the growth factor D+(a) satisfying (2.19), this implies that cosmological observables measure integrals over the integrated equation- of-state function w(a). Needless to say, the dependence of cosmological measurements on the exact form of w(a) will be extremely weak, which in turn implies that extremely accurate measurements will be necessary for constraining the nature of the dark energy. Logarithmic derivatives of the angular-diameter distance and the growth factor with respect to the equation-of-state parameter. CHAPTER 8. INFLATION AND DARK ENERGY 89

• In order to illustrate the required accuracies, let us consider by how much the angular-diameter distance and the growth factor change compared to ΛCDM upon changes in w away from −1,

d ln Dang(z) d ln D (z) , + , (8.32) dw dw

as a function of redshift z. Assuming Ωm0 = 0.3 and ΩΛ0 = 0.7, we ﬁnd typical values between −0.1 and −0.2 at most. Since we currently expect deviations of w from −1 at most at the ∼ 10% level, accurate constraints on the dark energy require relative accuracies of distances and the growth factor at the per-cent level.

• It seems clear that all suitable cosmological information will need to be combined in order to make any progress. In the next chapters, we will study weak lensing, supernovae, and the growth of cosmic structures as probes of dark energy. All of these are com- plementary and powerful ways to measure and exploit the expansion history and structure formation history. Chapter 9

Cosmic Structures

9.1 Quantifying structures

9.1.1 Introduction

• We have seen before that there is a very specific prediction for the power spectrum of density fluctuations in the Universe, characterised by (2.23). Recall that its shape was inferred from the simple assumption that the mass of density fluctuations entering the horizon should be independent of the time when they enter the horizon, and from the fact that perturbation modes entering during the radiation era are suppressed until matter begins dominating. • Given the simplicity of the argument, and the corresponding strength of the prediction, it is very important for cosmology to find out whether the actual power spectrum of matter density fluctuations does in fact have the expected shape, and furthermore to determine the only remaining parameter, namely the normalisation of the power spectrum (this is discussed later in chapter 11).

• Since the location k0 of the maximum in the power spectrum is determined by the horizon radius at matter-radiation equality (2.21), √ √ 2π 2 2πH0 Ωm0 k0 = = 3/2 , (9.1) req c aeq and the scale factor at equality is

Ωr0 aeq = , (9.2) Ωm0 the peak scale provides a measure of the matter-density parameter, k0 ∝ Ωm0. A measurement of k0 would thus provide an independent and very elegant determination of Ωm0.

90 CHAPTER 9. COSMIC STRUCTURES 91

• Since the power spectrum is defined in Fourier space, it is not obvious how it can be measured. In a brief digression, we shall first summarise the relation between the power spectrum and the correlation function in configuration space, and clarify the meaning of the correlation function.

9.1.2 Power spectra and correlation functions

• The definition (2.22) shows that the power spectrum is given by an average over the Fourier modes of the density contrast. This average extends over all Fourier modes with a wave number k, i.e. it is an average over all directions in Fourier space keeping k constant. In other words, Fourier modes are averaged within spherical shells of radius k. • In configuration space, structures can be quantified by the (two- point) correlation function

ξ(x) ≡ δ(~y)δ(~x + ~y) , (9.3)

where the average is now taken over all positions ~y and all orientations of the separation vector ~x, assuming homogeneity and isotropy. • Inserting the Fourier expansion Z d3k δ(~x) = δˆ(~k)e−i~k~x (9.4) (2π)3 of the density contrast into (9.3), using the deﬁnition (2.22) of the power spectrum and taking into account that the Fourier transform δˆ must obey δˆ(−~k) = δˆ∗(~k) because δ is real, it is straightforward to show that the correlation function ξ is the Fourier transform of the power spectrum, Z d3k ξ(x) = P(k)e−i~k~x . (9.5) (2π)3

• Assuming isotropy, the integral over all relative orientations between ~x and ~k can be carried out, yielding 1 Z ∞ sin kx ξ x P k k2 k , ( ) = 2 ( ) d (9.6) 2π 0 kx whose inverse transform is Z ∞ sin kx P(k) = 4π ξ(x) x2dx . (9.7) 0 kx This indicates one way to determine the power spectrum via measuring the correlation function ξ(x). CHAPTER 9. COSMIC STRUCTURES 92

9.1.3 Measuring the correlation function

• How can the correlation function be measured? Obviously, we cannot measure the correlation function of the density field directly. All we can do is using galaxies as tracers of the underlying density field and use their correlation function as an estimate for that of the matter. • Suppose we divide space into cells of volume dV small enough to contain at most a single galaxy. Then, the probability of finding one galaxies in dV1 and another galaxy in dV2 is The correlation function quantifies dP = hn(~x )n(~x )idV dV , (9.8) 1 2 1 2 the probability to find a galaxy in where n is the number density of the galaxies as a function of the small volume dV2 if there is a position. galaxy in the small volume dV1, a distance r = |~r2 − ~r1| away. • If we introduce a density contrast for the galaxies in analogy to the density contrast for the matter, n δn ≡ − 1 , (9.9) n¯ and assume for now that δn = δ, we find from (9.8) with n = n¯(1 + δ)

2 2 dP = n¯ h(1 + δ1)(1 + δ2)idV1dV2 = n¯ [1 + ξ(x)]dV1dV2 , (9.10) where x is the distance between the two volume elements. This shows that the correlation function quantifies the excess probability above random for finding galaxy pairs at a given distance. Correlations between points can be • Thus, the correlation function can be measured by counting determined by counting pairs. galaxy pairs and comparing the result to the Poisson expectation, i.e. to the pair counts expected in a random point distribution. Symbolically, hDDi 1 + ξ = , (9.11) 1 hRRi where D and R represent the data and the random point set, respectively. • Several other ways of measuring ξ have been proposed, such as hDDi 1 + ξ = , 2 hDRi hDDihRRi 1 + ξ = , 3 hDRi2 h(D − R)2i 1 + ξ = 1 + , (9.12) 4 hRRi2 which are all equivalent in the ideal situation of an infinitely extended point distribution. For finite point sets, ξ3 and ξ4 are supe- rior to ξ1 and ξ2 due to their better noise properties. CHAPTER 9. COSMIC STRUCTURES 93

• The recipe for measuring ξ(x) is thus to count pairs separated by x in the data D and in the random point set R, or between the data and the random point set, and to use one of the estimators given above.

• The obvious question is then how accurately ξ can be determined. The simple expectation in the absence of clustering is 1 hξi = 0 , hξ2i = , (9.13) Np

where Np is the number of pairs found. Thus, the Poisson error on the correlation function is ∆ξ 1 = p . (9.14) 1 + ξ Np

• This is a lower limit to the actual error, however, because the galaxies are in fact correlated. It turns out that the result (9.14) should be multiplied with 1 + 4πnJ¯ 3, where J3 is the volume integral over ξ within the galaxy-survey volume. The true error bars on ξ are therefore hard to estimate.

• Having measured the correlation function, it would in principle suffice to carry out the Fourier transform (9.7) to find P(k), but this is difficult in reality because of the inevitable sample limita- tions. Consider (9.6) and an underlying power spectrum of CDM shape, falling off ∝ k−3 for large k, i.e. on small scales. For fixed x, the integrand in (9.6) falls off very slowly, which means that a considerable amount of small-scale power is mixed into the correlation function. Since ξ at large x is small and most affected by measurement errors, this shows that any uncertainty in the large- scale correlation function is propagated to the power spectrum even on small scales.

• A further problem is the uncertainty in the mean galaxy number densityn ¯. Since 1 + ξ ∝ n¯−1 according to (9.10), the uncertainty in ξ due to an uncertainty inn ¯ is ∆ξ ∆n¯ ≈ ∆ξ = , (9.15) 1 + ξ n¯ showing that ξ cannot be measured with an accuracy better than the relative accuracy of the mean galaxy density.

9.1.4 Measuring the power spectrum

• Given these problems with real data, it seems appropriate to estimate the power spectrum directly. The function to be transformed CHAPTER 9. COSMIC STRUCTURES 94

is the density ﬁeld sampled by the galaxies, which can be represented by a sum of Dirac delta functions centred on the locations of the N galaxies,

XN n(~x) = δD(~x − ~xi) . (9.16) i=1

• The Fourier transform of the density contrast is then

N 1 X ~ δˆ(~k) = eik~xi . (9.17) N i=1 In the absence of correlations, the Fourier phases of the individual terms are independent, and the variance of the Fourier amplitude for a single mode becomes

N 1 X ~ ~ 1 hδˆ(~k)δˆ∗(~k)i = eik~xi e−ik~xi = . (9.18) N2 N i=1 This is the so-called shot noise present in the power spectrum due to the discrete sampling of the density ﬁeld.

• The shot-noise contribution needs to be subtracted from the power spectrum of the real, correlated galaxy distribution, 1 X 1 P(k) = |δˆ(~k)|2 − , (9.19) m N where the sum extends over all m modes contained in the survey with wave number k.

• This is not the ﬁnal result yet, because any real survey typically covers an irregularly shaped volume from which parts need to be excised because they are overshone by stars or unusable for any other reasons. The combined eﬀect of mask and irregular survey volume is described by a window function f (~x) which multiplies the galaxy density,

n(~x) → f (~x)n(~x) , (1 + δ) → f (~x)(1 + δ) , (9.20)

implying that the Fourier transform of the mask needs to be subtracted.

• Moreover, the Fourier convolution theorem says that the Fourier transform of the product f (~x)δ(~x) is the convolution of the Fourier transforms fˆ(~k) and δˆ(~k), Z fbδ = fˆ ∗ δˆ ≡ fˆ(~k0)δˆ(~k0 − ~k)d3k0 . (9.21) CHAPTER 9. COSMIC STRUCTURES 95

If the Fourier phases of fˆ and δˆ are uncorrelated, which is the case if the survey volume is large enough compared to the size 2π/k of the density mode, this translates to a convolution of the power spectrum, 2 Pobs = Ptrue ∗ | fˆ(~k)| . (9.22)

• This convolution typically has two eﬀects; ﬁrst, it smooths the observed compared to the true power spectrum, and second, it changes its amplitude. The corresponding correction is given by R ( f d3 x)2 P(k) → P(k) R R . (9.23) f 2d3 x d3 x

• If the Poisson error dominates in the survey, the diﬀerent modes ˆ ~ δ(k) can be shown to be uncorrelated, and the standard deviation√ after summing over the m modes with wave number k is 2m/N, which yields the minimal error bar to be attached to the power spectrum.

• Thus, the shot noise contribution and the Fourier transform of the window function need to be subtracted, the window function needs to be deconvolved, and the amplitude needs to be corrected for the eﬀective volume covered by the window function before the measured power spectrum can be compared to the theoretical expectation.

• Finally, it is usually appropriate to assign weights 0 ≤ wi ≤ 1 to the individual galaxies to account for their varying density. The optimal weight for the ith galaxy sampling a Fourier mode with wave number k has been determined to be 1 wi(k) = , (9.24) 1 + n¯iP(k)

wheren ¯i is the local mean density around the ith galaxy, and P(k) is the power spectrum. If the density is low, the galaxies are weighted equally, and less if the local density is very high, because the many galaxies from a dense environment might otherwise suppress information from galaxies in less dense regions.

• Including weights, eqs. (9.17) and (9.18) become

P ~ P 2 w eik~xi w δˆ ~k i , h|δˆ ~k |2i i . ( ) = P ( ) = P 2 (9.25) wi ( wi)

• A ﬁnal problem due to the ﬁnite size of the survey regards the normalisation of the power spectrum. The mean density estimate within the survey volume does not necessarily equal the true mean CHAPTER 9. COSMIC STRUCTURES 96

density. Since, by deﬁnition, the mean of the density contrast δ0 within the survey vanishes, we must have Z δ0 = δ − f (~x)δ(~x)d3 x , (9.26)

where δ is the true density contrast. Thus, the constant mean value of δ within the (masked) survey volume is subtracted.

• Subtracting a constant gives rise to a delta-function peak at k = 0 in the Fourier-transformed density contrast, and thus also in the power spectrum P0 estimated from the survey.

• The observed power spectrum, however, is a convolution of the true power spectrum, as shown in (9.22). Thus, the delta-function peak caused by the misestimate of the mean density also needs to be convolved, giving rise to a contribution P(0) ∗ | fˆ(~k)|2 in the observed power spectrum.

• Since the mean density contrast δ0 within the survey is zero, the observed power spectrum at k = 0 must vanish, thus

0 ˆ ~ 2 Pobs(k) = Pobs(k) − Pobs(0) ∗ | f (k)| . (9.27)

9.1.5 Biasing

• What we have determined so far is the power spectrum of the galaxy number-density contrast δn rather than that of the matter density contrast δ. Simple models for the relation between both assume that there is a so-called bias factor b(k) between them, such that δbn(~k) = b(k)δˆ(~k) , (9.28) where b(k) may or may not be more or less constant as a function of scale.

• Clearly, different types of objects sample the underlying matter density field in different ways. Galaxy clusters, for instance, are much more rare than galaxies and are thus expected to have a substantially higher bias factor than galaxies.

• Obviously, the bias factor enters squared into the power spectrum, e.g. 2 Pgal = bgal(k) P(k) . (9.29) It constitutes a major uncertainty in the determination of the matter power spectrum from the galaxy power spectrum. CHAPTER 9. COSMIC STRUCTURES 97

9.1.6 Redshift-space distortions

• Of course, for the estimate (9.17) of the Fourier-transformed (galaxy) density contrast, the three-dimensional positions ~xi of the galaxies in the survey need to be known. Distances can be inferred only from the galaxy redshifts and thus from galaxy velocities. • These, however, are composed of the Hubble velocities, from which the distances can be determined, and the peculiar velocities, v = vHubble + vpec , (9.30) which are caused by local density perturbations and are unrelated to the galaxy densities. • Since observations of individual galaxies do not allow any separation between the two velocity components, distances are inferred from the total velocity v rather than the Hubble velocity as it should be,

v vHubble + vpec D = = = Dtrue + ∆D , (9.31) H0 H0

giving rise to a distance error δD = vpec/H0, the so-called redshift- space distortion. Peculiar velocities give rise to • Fortunately, the redshift-space distortions have a peculiar pat- redshift-space distortions, whose tern through which they can be corrected. Consider a matter characteristic shape constrains the overdensity such as a galaxy cluster, containing galaxies moving bias. with random virial velocities in it. The virial velocities of order 1000 km s−1 scatter around the systemic cluster velocity and thus widen the redshift distribution of the cluster galaxies. In redshift space, therefore, the cluster appears stretched along the line-of- sight, which is called the finger-of-god effect. • In addition, the cluster is surrounded by an infall region, in which the galaxies are not virialised yet, but move in an ordered, radial pattern towards the cluster. Galaxies in front of the cluster thus have higher, and galaxies behind the cluster have lower recession velocities compared to the Hubble velocity, leading to a flattening of the infall region in redshift space. • A detailed analysis shows that the redshift-space power spectrum Pz is related to the real-space power spectrum P by

22 Pz(k) = P(k) 1 + βµ , (9.32) where µ is the direction cosine between the line-of-sight and the wave vector ~k, and β is related to the bias parameter b through f (Ω ) β ≡ m , (9.33) b CHAPTER 9. COSMIC STRUCTURES 98

and f (Ωm) is the logarithmic derivative of the growth factor D+(a), d ln D (a) f (Ω ) ≡ + ≈ Ω0.6 . (9.34) m d ln a m • Thus, the characteristic pattern of the redshift-space distortions around overdensities allows a measurement of the bias factor. Another way of measuring b is based upon gravitational lensing. Corresponding measurements of b show that it is in fact almost constant or only weakly scale-dependent, and that it is very close to unity for “ordinary” galaxies.

9.1.7 Baryonic acoustic oscillations

• As we have seen in the discussion of the CMB, acoustic oscillations in the cosmic ﬂuid have left density waves in the cosmic baryon distribution. Their characteristic wave length is set by the sound horizon at decoupling (7.26), rs ≈ 63 kpc. By now, this was increased by the cosmic expansion to 1280 × 63 kpc ≈ 80.6 Mpc, −1 or k0 ≈ 0.078 Mpc . • This must be compared to the horizon size at matter-radiation equality (2.21). With aeq ≈ 6200 from (7.7), we ﬁnd req ≈ 11.0 kpc, which was stretched by now to 6200 × 11.0 kpc ≈ −1 68.3 Mpc, or ks ≈ 0.092 Mpc . • Thus, the peak scale of the power spectrum and the wavelength of the fundamental mode of the baryonic acoustic oscillations are of comparable size. Near the peak of the power spectrum, we thus expect a weak wave-like imprint on top of the otherwise smooth dark-matter power spectrum.

9.2 Measurements and results

9.2.1 The power spectrum Top: The telescope dedicated to the Sloan Digital Sky Survey. Bot- tom: The two-degree ﬁeld camera • Spectacularly successful measurements of the power spectrum in the prime focus of the Anglo- became recently possible with the two largest galaxy surveys to Australian Telescope. date, the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS).

• As expected from the preceding discussion, an enormous eﬀort has to be made to identify galaxies, measure their redshifts, selecting homogeneous galaxy subsamples as a function of redshift by luminosity and colour so as not to compare and correlate ap- ples with oranges, estimating the window function of the survey, CHAPTER 9. COSMIC STRUCTURES 99

determining the average galaxy number density, correcting for the convolution with the window function and for the bias, and so forth.

• Moreover, calibration experiments have to be carried out in which all measurement and correction techniques are applied to simulated data in the same way as to the real data to determine reliable error estimates and to test whether the full sequence of analysis steps ultimately yields an unbiased result.

• Based on 221, 414 galaxies, the 2dFGRS consortium derived a power spectrum of superb quality. First and foremost, it conﬁrms the power-spectrum shape expected for cold dark matter on the small-scale side of the peak. On its own, this is a highly remarkable result.

• Next, the 2dFGRS power spectrum clearly shows a turn-over towards larger scales, signalling the peak. The survey is still not quite large enough to show the peak, but the peak location can be estimated from the ﬂattening of the power spectrum. Its proportionality to Ωm0 allows an independent determination of the matter density parameter.

• Finally, and most spectacularly, the power spectrum shows the baryonic acoustic oscillations, whose amplitude allows an independent determination of the ratio between the density parameters of baryons and dark matter.

• Apart from the fact that the CDM shape of the power spectrum is conﬁrmed on small scales, the results obtained from the 2dFGRS can be summarised as follows: Top: Geometry of the 2dFGRS survey volume. Middle: Galaxy dis- Ωm0 0.233 ± 0.022 tribution therein. Bottom: The area Ωb0/Ωm0 0.185 ± 0.046 covered by the 2dFGRS on the sky.

The Hubble constant of h = 0.72 is assumed here. Indirectly, the baryon density is constrained to be Ωb0 ≈ 0.04, which is in perfect agreement with the value derived from primordial nucleosynthesis and the measured abundances of the light elements; the SDSS gives a value of Ωm0 = 0.24 ± 0.02, in agreement with 2dF and WMAP5. CHAPTER 9. COSMIC STRUCTURES 100

Figure 9.1: Left: Power spectrum of the 2dFGRS galaxy distribution (top) and after division by the smooth ΛCDM expectation (bottom). Right: Separate power spectra of red and blue galaxies (top) and their ratio (bottom). CHAPTER 9. COSMIC STRUCTURES 101

Figure 9.2: The galaxy power spectrum obtained from the SDSS (bottom), and the ratio between the power spectra of red and blue galaxies (top). Chapter 10

Cosmological Weak Lensing

10.1 Cosmological light deﬂection

10.1.1 Deﬂection angle, convergence and shear

• Gravitational lensing was mentioned two times before: first in Sect. 3.2 as a means for measuring the Hubble constant through the time delay caused by gravitational light deflection, and second as a means for measuring cluster masses in Sect. 5.2.3. For cosmology as a whole, gravitational lensing has also developed into an increasingly important tool. Density inhomogeneities along the • Matter inhomogeneities deflect light. Working out this effect in way deflect light rays. the limit that the Newtonian gravitational potential is small, Φ c2 leads to the deflection angle 2 Z w f (w − w0) α~ ~θ w0 k ∇~ f w0 ~θ . ( ) = 2 d ⊥Φ[ k( ) ] (10.1) c 0 fk(w) It is determined by the weighted integral over the gradient of the Newtonian gravitational potential Φ perpendicular to the line-of- sight into direction θ on the observer’s sky, and the weight is given by the comoving angular-diameter distance fk(w) defined in (2.3). The integral extends along the comoving radial distance w0 along the line-of-sight to the distance w of the source.

• Equation (10.1) can be intuitively understood. Light is deflected due to the pull of the dimension-less Newtonian gravitational 2 field ∇~ ⊥Φ/c perpendicular to the otherwise unperturbed line-of- sight, and the effect is weighted by the ratio between the angular- diameter distances from the deflecting potential to the source, 0 fk(w − w ), and from the observer to the source, fk(w). Thus, a lensing mass distribution very close to the observer gives rise to a large deflection, while a lens near the source, w0 ≈ w, has very

102 CHAPTER 10. COSMOLOGICAL WEAK LENSING 103

little eﬀect. The factor of two is a relic from general relativity and is due to space-time curvature, which is absent from Newtonian gravity.

• It is important to realise that the deflection itself is not observable. If all light rays emerging from a source would be deflected by the same angle on their way to the observer, no noticeable effect would remain. What is important, therefore, is not the deflection angle itself, but its change from one light ray to the next. This is quantified by the derivative of the deflection angle with respect to the direction ~θ,

∂α 2 Z w f (w − w0) f (w0) ∂2Φ i w0 k k f w0 ~θ . = 2 d [ k( ) ] (10.2) ∂θ j c 0 fk(w) ∂xi∂x j

0 The additional factor fk(w ) in the weight function arises because the derivative of the potential is taken with respect to comoving coordinates xi rather than the angular components θi. • Obviously, the complete weight function f (w − w0) f (w0) W(w0, w) ≡ k k (10.3) fk(w) vanishes at the observer, w0 = 0, and at the source, w0 = w, and peaks approximately half-way in between.

• For applications of gravitational lensing, it is important to distinguish between the trace-free part of the matrix (10.2) and its trace,

∂α 2 Z w ∂2Φ i w0 W w0, w f w0 ~θ , tr = 2 d ( ) 2 [ k( ) ] (10.4) ∂θ j c 0 ∂xi where the sum over i is implied. Therefore, the derivatives of Φ can be combined to the two-dimensional Laplacian, which can then be replaced by the three-dimensional Laplacian because the derivatives along the line-of-sight do not contribute to the integral (10.4). Thus, we ﬁnd ∂α 2 Z w i w0 W w0, w . tr = 2 d ( ) ∆Φ (10.5) ∂θ j c 0

• Next, we can use Poisson’s equation to replace the Laplacian of Φ by the density. In fact, we have to take into account that light de- ﬂection is caused by density perturbations, and that we need the Laplacian in terms of comoving rather than physical coordinates. Thus, 1 ∆Φ = 4πGρδ¯ , (10.6) a2 CHAPTER 10. COSMOLOGICAL WEAK LENSING 104

where δ is the density contrast and 3H2 ρ¯ = ρ¯ a−3 = ρ Ω = 0 Ω a−3 (10.7) 0 cr m0 8πG m0 is the mean matter density. • Thus, Poisson’s equation reads 3 δ ∆Φ = H2Ω , (10.8) 2 0 m0 a and (10.5) becomes 2 Z w ∂α 3H Ωm0 δ i 0 w0 W w0, w ≡ κ , tr = 2 d ( ) 2 (10.9) ∂θ j c 0 a where we have introduced the (eﬀective) convergence κ. The gravitational tidal ﬁeld (shear) • The trace-free part of the matrix (10.2) is of large-scale structures distorts the ! ∂αi 1 ∂αi ∂αi γ1 γ2 images of background galaxies (ex- − δi jtr = − δi jκ ≡ , (10.10) aggerated). ∂θ j 2 ∂θ j ∂θ j γ2 −γ1

which defines the so-called shear components γi. Specifically, ! 1 Z w ∂2Φ ∂2Φ γ w0 W w0, w − , 1 = 2 d ( ) 2 2 c 0 ∂x1 ∂x2 ! 2 Z w ∂2Φ γ w0 W w0, w . 2 = 2 d ( ) (10.11) c 0 ∂x1∂x2 • Combining the results, we can write the matrix of deflection- angle derivatives as ! ∂α κ + γ γ i = 1 2 . (10.12) ∂θ j γ2 κ − γ1 This matrix contains the important information on how an image is magnified and distorted. In the limit of weak gravitational lensing, the size of a lensed image is changed by the relative magnification δµ = 2κ , (10.13) while the image distortion is given by the shear components. • In fact, an originally circular source with radius r will appear as an ellipse with major and minor axes r r a = , b = , (10.14) 1 − κ − γ 1 − κ + γ

2 2 1/2 where γ ≡ (γ1 + γ2) . The ellipticity of the observed image of a circular source thus provides an estimate for the shear, a − b γ ≡ = ≈ γ . (10.15) a + b 1 − κ CHAPTER 10. COSMOLOGICAL WEAK LENSING 105

10.1.2 Power spectra

• Of course, the exact light deflection expected along a particular line-of-sight cannot be predicted because the mass distribution along that light path is unknown. However, we can predict the statistical properties of weak lensing from those of the density- perturbation field. • We are thus led to the following problem: Suppose the power spectrum P(k) of a Gaussian random density-perturbation field δ is known, what is the power spectrum of any weighted projection of δ along the line-of-sight? • The answer is given by Limber’s equation. Suppose the weight function is q(w) and the projection is Z w 0 0 0 g(~θ) = dw q(w )δ[ fk(w )~θ] . (10.16) 0 If q(w) is smooth compared to δ, i.e. if the weight function changes on scales much larger than typical scales in the density contrast, then the power spectrum of g is ! Z w q2(w0) l P l w0 P , g( ) = d 2 0 0 (10.17) 0 fk (w ) fk(w ) where ~l is a two-dimensional wave vector which is the Fourier conjugate variable to the two-dimensional position ~θ on the sky. • Strictly speaking, Fourier transforms are inappropriate because the sky is not an infinite, two-dimensional plane. The appropriate set of orthonormal base functions are the spherical harmonics instead. However, lensing effects are usually observed in areas whose solid angle is very small compared to the full sky. If this is so, the survey area can be approximated by a section of the local tangential plane to the sky, and then Fourier transforms can be used. This is the so-called flat-sky approximation. • Equation (10.9) is clearly of the form (10.16) with the weight function 3 H2 W(w0, w) q(w0) = Ω 0 , (10.18) 2 m0 c2 a thus the power spectrum of the convergence is, according to Lim- ber’s equation,

2 4 ! 9Ω H Z w l P l m0 0 w0 W¯ 2 w0, w P , κ( ) = 4 d ( ) 0 (10.19) 4 c 0 fk(w ) with a new weight function W(w0, w) ¯ 0 ≡ W(w , w) 0 . (10.20) a fk(w ) CHAPTER 10. COSMOLOGICAL WEAK LENSING 106

• While it is generally difficult or impossible to observe the differential magnification δµ or the convergence κ, image distortions can in principle be measured. With a brief excursion through Fourier space, it can easily be shown that the power spectrum of the shear is exactly identical to that of the convergence,

Pγ(l) = Pκ(l) . (10.21)

Thus, the statistics of the image distortions caused by cosmological weak lensing contains integral information on the power spectrum of the matter ﬂuctuations. The power spectrum of the weak- • Since the shear is deﬁned on the two-dimensional sphere (the ob- lensing convergence κ for three dif- server’s sky), its power spectrum is related to its correlation func- ferent source redshifts. tion ξγ through the two-dimensional Fourier transform

Z d2l Z ∞ ldl ξ φ P l iφ~~l P l lφ , γ( ) = 2 γ( )e = γ( )J0( ) (10.22) (2π) 0 2π

where Jν is the ordinary Bessel function of order ν.

10.1.3 Correlation functions

• In principle, shear correlation functions are measured by comparing the ellipticity of one galaxy with the ellipticity of other galaxies at an angular distance φ from the ﬁrst.

• Ellipticities are oriented, of course, and one has to specify against what other direction the direction of, say, the major axis of a given ellipse is to be compared to. Since correlation functions are measured by counting pairs, a preferred direction is deﬁned by the line connecting the two galaxies of the pair under consideration.

• Let α be the angle between this direction and the major axis of the ellipse, then the tangential and cross components of the shear are deﬁned by

γ+ ≡ γ cos 2α , γ× ≡ γ sin 2α . (10.23)

The factor two is important because it accounts for the fact that an ellipse is mapped onto itself when rotated by an angle π. This illustrates that the shear is a spin-2 ﬁeld: It returns into its original orientation when rotated by π rather than 2π.

• The correlation functions of the tangential and cross components of the shear are Z ∞ 1 ldl ξ++(φ) = hγ+(θ)γ+(θ + φ)i = Pκ(l) J0(lφ) + J4(lφ) 2 0 2π (10.24) CHAPTER 10. COSMOLOGICAL WEAK LENSING 107

and Z ∞ 1 ldl ξ××(φ) = hγ×(θ)γ×(θ + φ)i = Pκ(l) J0(lφ) − J4(lφ) , 2 0 2π (10.25) while the cross-correlation between the tangential and cross components must vanish, ξ+×(φ) = 0 . (10.26) The convergence (or shear) corre- • This suggests to deﬁne the correlation functions ξ± = ξ++ ± ξ××, which are related to the power spectrum through lation function for three diﬀerent source redshifts. Z ∞ ldl ξ+ = Pκ(l)J0(lφ) , 0 2π Z ∞ ldl ξ− = Pκ(l)J4(lφ) . (10.27) 0 2π • Yet another measure for cosmological weak lensing is given by the absolute value of the shear averaged within a circular mask (or aperture) of radius θ, Z θ d2ϑ γ θ ≡ γ ϑ~ , ¯( ) 2 ( ) (10.28) 0 πθ which is related to the power spectrum by Z ∞ " #2 2 ldl 2J1(lθ) h|γ¯(θ)| i = Pκ(l) . (10.29) 0 2π lθ

• The principle of all these measures for cosmic shear is the same: The power of cosmological weak They are integrals of the weak-lensing power spectrum times so- lensing as a function of angular called filter functions which describe the detailed response of the scale. measurement to the underlying power spectrum of density fluctuations. The width of the filter functions controls the range of density-perturbation modes ~k that contribute to one specific mode ~l of weak-lensing on the sky. • We can now estimate typical numbers for the cosmological weak- lensing effect. The power ∆κ in the weak-lensing quantities such as the cosmic shear is given by the power spectrum Pκ(l) found in (10.19), times the volume in l-space, 2 ∆κ(l) = l Pκ(l) . (10.30)

• Assuming a cosmological model with Ωm0 = 0.3 and ΩΛ0 = 0.7, the CDM power spectrum and a reasonable source redshift dis- 1/2 tribution, ∆κ(l) is found to peak on scales l corresponding to angular scales 2π/l of 20 ... 30, and the peak reaches values of 0.04 ... 0.05. This shows that cosmological weak lensing will typically cause source ellipticities of a few per cent, and they have a typical angular scale of a few arc minutes. Details depend on the measure chosen through the ﬁlter function. CHAPTER 10. COSMOLOGICAL WEAK LENSING 108

10.2 Cosmic-shear measurements

10.2.1 Typical scales and requirements

• How can cosmic gravitational lensing effects be measured? As shown in (10.15), the ellipticity of a hypothetic circular source is a direct measure, a so-called unbiased estimator for the shear. But typical sources are not circular, but to first approximation elliptical themselves. Thus, measuring their ellipticities yields their intrinsic ellipticities in the first place. • Let (s) be the intrinsic source ellipticity. It is a two-component quantity because an ellipse needs two parameters to be described (e.g. an axis ratio and an orientation), and it is a spin-2 quantity because it is mapped onto itself upon a rotation by 2π/2 = π. The cosmic shear adds to that ellipticity, such that the observed ellipticity is ≈ (s) + γ (10.31) in the weak-lensing approximation. What is observed is therefore the sum of the signal, γ, and the intrinsic noise component (s). • On sufficiently deep observations, some 30 galaxies per square arc minute are detected. Since the full moon has half a degree diameter, it covers a solid angle of 152π = 700 square arc minutes, or 21, 000 of such distant, faint galaxies! From this point of view, the sky is covered by densely patterned “wall paper” of distant galaxies. • Thus, it is possible to average observed galaxy ellipticities. As- suming their shapes are intrinsically independent, the intrinsic ellipticities will average out, and the shear will remain, hi ≈ h(s)i + hγi ≈ hγi . (10.32)

• It is a fortunate coincidence that the typical angular scale of cosmic lensing, which we found to be of order a few arc minutes, is large compared to the mean√ distance between background galaxies, which is of order 1/30 ≈ 0.20. This allows averaging over background galaxies without cancelling the cosmic shear signal. If γ varied on scales comparable to or smaller than the mean galaxy separation, any average over galaxies would remove the lensing signal. • The intrinsic ellipticities of the faint background galaxies have a distribution with a standard deviation of σ ≈ 0.3. Averaging over N of them, and assuming Poisson statistics, gives expectation values of σ h(s)i = 0 , δ = h((s))2i1/2 = √ (10.33) N CHAPTER 10. COSMOLOGICAL WEAK LENSING 109

for the mean and its intrinsic ﬂuctuation.

• A rough estimate for the signal-to-noise ratio of a cosmic shear measurement can proceed as follows. Suppose the correlation function ξ is measured by counting pairs of galaxies with a separation within δθ of θ. As long as θ is small compared to the side length of the survey area A, the number of pairs will be 1 N = 2πn2Aθδθ , (10.34) p 2 and thus the Poisson noise due to the intrinsic ellipticities will be 2σ noise ≈ √ , (10.35) n πAθδθ where the factor of two arises because of the two galaxies involved in each pair.

• The signal is the square root of the correlation function ξ, which we can approximate as δl δθ ξ ≈ l2P (l)δ ln l ≈ l2P (l) ≈ l2P (l) , (10.36) κ κ l κ θ where we have used in the last step that θ = 2π/l. The estimated signal-to-noise ratio • Thus, the signal-to-noise ratio turns out to be of weak-lensing measurements for √ √ p a hypothetical survey on an area of S ξ lnδθ πAP n π3AP δθ ≈ ≈ κ = κ . (10.37) one square degree. N noise 2σ σ θ Evidently, the signal-to-noise ratio, and thus the signiﬁcance of any cosmic-lensing detection, grows with the survey area and decreases with the intrinsic ellipticity of the source galaxies.

• In evaluating (10.37) numerically, we have to take into account 2 that l Pκ(l) must be a dimension-less number, which implies that the power spectrum Pκ must have the dimension steradian. There- fore, either the survey area A and the number density n in (10.37) must be converted to steradians, or Pκ must be converted to square arc minutes ﬁrst.

• The signal-to-noise ratio increases approximately linearly with scale. Assuming δθ/θ = 0.1, it is S/N ≈ 1.5 on a scale of 0.10 for a survey of one square degree area. This shows that, if the cosmic shear should be measured on such small scales with an accuracy of, say, ﬁve per cent, a survey area of A ≈ (20/1.5)2 ≈ 180 square degrees is needed since the signal-to-noise ratio scales like the square root of the survey area. On such an area, the ellipticities of 180 × 3600 × 30 ≈ 2 × 107 background galaxies have to be accurately measured. CHAPTER 10. COSMOLOGICAL WEAK LENSING 110

• Matters are more complicated in reality, but the orders-of- magnitude are well represented by this rough estimate. Bear- ing in mind that typical ﬁelds-of-view of telescopes which are large enough to detect suﬃciently many faint background galaxies reach one to ten per cent of a square degree, and that typical exposure times are of order half an hour for that purpose, the total amount of telescope time for a weak-lensing survey like that is estimated to be several thousand telescope hours. With perhaps eight hours of telescope time per night, and perhaps half of the nights per year usable, it is easy to see that the time needed for such surveys is measured in years.

• Since the faint background galaxies have typical sizes of arc seconds, shape measurements require a pixel resolution of, say, 0.100. The total survey area of 180 square degrees must therefore be resolved into 180 × 3600 × 3600/0.12 ≈ 2.3 × 1011 pixels. Stor- ing only one 4-byte number per pixel (i.e. the photon count), this amounts to 4.6 × 1011/240 = 0.8 TBytes.

10.2.2 Ellipticity measurements

• The determination of image ellipticities is straightforward in prin- The sobering appearance of real ciple, but diﬃcult in practice. Usually, the surface-brightness data. quadrupole R 2 I(~x)xi x jd x Qi j = R (10.38) I(~x)d2 x is measured, from whose principal axes the ellipticity can be read oﬀ.

• Real galaxy images, however, are typically far from ideally elliptical. They are structured or otherwise irregular. In addition, if they are small, they are coarsely resolved into just a few pixels, so that only a crude approximation to the integral in (10.38) can be found. The compatibility of the lower data • Even if the surface-brightness quadrupole of the image on the points signals the almost complete detector can be accurately determined, the image appears aﬀected absence of systematic eﬀects in the by imperfections of the telescope optics and by the turbulence in data show above. the atmosphere, the so-called seeing.

• Due to the wave nature of light and the ﬁnite size of the telescope mirror, the telescope will have ﬁnite resolution. The angular resolution limit is given by λ ∆θ ≈ 1.44 (10.39) D CHAPTER 10. COSMOLOGICAL WEAK LENSING 111

as mentioned in (7.41) before. With λ ≈ 6 × 10−5 cm and D = 400 cm, the angular resolution is ∆θ ≈ 0.0400, much smaller than needed for our purposes.

• The turbulence of the Earth’s atmosphere eﬀectively convolves images with a Gaussian whose width depends on the site, the weather and other conditions. Typical seeing ranges around 100. Under very good conditions, it can shrink to ∼ 0.500 or less. Clearly, if an image of approximately one arc second size is convolved with a Gaussian of similar width, any ellipticity is substantially reduced.

• How the image of a point-like source, such as a star, appears on the detector is described by the so-called point-spread function (PSF). The PSF may be anisotropic if the telescope optics The point-spread function of the is slightly astigmatic, and this anisotropy may, and will in gen- Canada-France-Hawaii telescope. eral, depend on the location on the focal plane. The image is a convolution of the ideal image shape before any distortion by the atmosphere and the telescope optics and the PSF. Any accurate measurement of image ellipticities requires a PSF deconvolution, for which the PSF must of course be known. It is measured by ﬁtting elliptical Gaussians to stellar images on the exposure.

• Many other eﬀects may distort images in systematic ways. For instance, if the CCD chips are not exactly perpendicular to the optical axis of the telescope, or if the individual chips of a CCD Illustration of systematic image dis- mosaic are not exactly in the same plane, or if the telescope is tortions in the CFHTLS and their slightly out of focus, systematic image deformations may result correction. which typically vary across the focal plane. They have to be measured and corrected. This is commonly achieved by ﬁtting the parameters of a model PSF to a low-order, two-dimensional poly- nomial on the focal plane. Since part of the image distortions may depend on time due to thermal deformation, changing atmospheric conditions and such, PSF corrections will also typically depend on time and have to be determined and applied with much care.

• Systematic effects may remain which need to be detected and quantified. Any coherent image distortions caused by gravita- E- and B-mode distortion patterns. tional lensing must be describable by the tidal gravitational field, i.e. by second-order derivatives of a scalar potential. In analogy to the E~-field in electromagnetism, such distortion patterns are called E-modes. Similarly, distortion patterns which are the curl of a vector field are called B-modes. They cannot be due to gravitational lensing and thus signal systematic effects remaining in the data. Such B-mode contaminations could recently be strongly reduced or suppressed by improved algorithms for PSF correction. CHAPTER 10. COSMOLOGICAL WEAK LENSING 112

10.2.3 Results

• Despite the smallness of the effect and the many difficulties in measuring it, much progress in cosmic-shear observations has been achieved in the past few years. Current and ongoing surveys, in particular the Canada-France-Hawaii Legacy Survey, combined with well-developed, largely automatic data-analysis pipelines, have managed to produce cosmic-shear correlation functions with very small error bars covering angular scales from The first published measurements below an arc minute to several degrees. The best correlation func- of the cosmic-shear correlation tions could be shown to be at most negligibly contaminated by function. B-modes.

• The power spectrum Pκ(l) depends crucially on the non-linear evolution of the dark-matter power spectrum. This, and the exact redshift distribution of the background galaxies, are the major uncertainties now remaining in the interpretation of cosmic-shear surveys. Apart from that, the measured cosmic-shear correlation functions agree very well with the theoretical expectation from CDM density ﬂuctuations in a spatially-ﬂat, low-density universe.

• As (10.19) shows, the weak-lensing power spectrum Pκ(l) de- 2 pends on the product of a factor Ωm0 due to the Poisson equation, times the amplitude A of the matter power spectrum. An additional weak dependence on cosmological parameters is caused by the geometric weight function W¯ (w0, w), but this is not very important. By and large, therefore, the cosmic-shear correlation 2 function measures the product AΩm0, which means that the amplitude of the power spectrum is (almost) precisely degenerate with the matter density parameter. Only if it is possible to constrain Ωm0 or A in any other way can the degeneracy be broken. • We shall see later how this may work. The amplitude of the power 2 spectrum A is conventionally described by a parameter σ8 which will be defined and described in more detail later. Weak lensing thus measures the product σ8Ωm0, and current measurements find σ8Ωm0 ≈ 0.2. • Weak gravitational lensing is a fairly new field of cosmological research. Within a few years, it has considerably matured and re- The CFHT dome (top) and the turned cosmologically interesting constraints. Considerable po- Mega-Prime Camera in its prime fo- tential is expected from weak lensing in wide-area surveys in par- cus (bottom). ticular when combined with photometric redshift information. CHAPTER 10. COSMOLOGICAL WEAK LENSING 113

Figure 10.1: Recent constraints in the Ωm0 − σ8 plane obtained from weak-lensing measurements. The Universe is assumed spatially ﬂat here. Chapter 11

The Normalisation of the Power Spectrum

11.1 Introduction

• We saw in Chapter 9 that the measured power spectrum of the galaxy distribution follows the CDM expectation in the range of wave numbers where current large surveys allow it to be determined. This range can be extended to some degree towards smaller scales by measuring the autocorrelation of hydrogen absorption lines in the spectra of distant quasars. Such observations of the power spectrum of the so-called Lyman-α forest lines show that the power spectrum does indeed turn towards the asymptotic behaviour ∝ k−4. In addition, we have seen that the peak location agrees with the expectation for universe with Ωm0 ≈ 0.3 and h ≈ 0.72. This indicates that the CDM expectation for the dark- matter power spectrum is indeed at least very close to its real shape, which is a remarkable success.

• Although the shape of the power spectrum could thus be quite well established, its amplitude still poses a surprisingly obstinate problem. We shall see in this section why it is so diﬃcult to measure. For this purpose, we shall discuss three ways of measuring σ8; the amplitude of large-scale temperature ﬂuctuations in the CMB, the cosmic-shear autocorrelation function, and the abundance and evolution of the galaxy-cluster population.

• For historical reasons, the amplitude of the dark-matter power spectrum is characterised by the density-fluctuation variance within spheres of 8 h−1Mpc radius. This is because in the first measurement of the fluctuation amplitude in the galaxy distribution, Davis & Peebles found that it reached unity in such spheres.

• More generally, one imagines randomly placing spheres of radius

114 CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM115

R and measuring the density-contrast variance within them. Since the variance in Fourier space is characterised by the power spectrum, it can be written as

Z ∞ d3k σ2 P k W2 k , R = 3 δ( ) R( ) (11.1) 0 (2π)

where WR(k) is a window function selecting the k modes con- tributing to the variance.

• Imagining spheres of radius R in real space, the window function should be the Fourier transform of a step function, which is, however, inconvenient because it extends to inﬁnite wave numbers. It is thus more common to use either Gaussians, since they Fourier transform into Gaussians, or step functions in Fourier space. For simplicity of the illustrative calculations that will follow, we use the latter choice, thus ! 2π W (k) = Θ(k − k) = Θ − k . (11.2) R R R

This is a step function dropping to zero for k > 2π/R.

• Inserting this into (11.1), we ﬁnd

Z 2π/R k2dk σ2 P k . R = 2 δ( ) (11.3) 0 2π In other words, all modes larger than R contribute to the density ﬂuctuations in spheres of radius R because all smaller modes average to zero.

• The normalisation of the power spectrum is usually expressed in terms of σ8.

11.2 Fluctuations in the CMB

11.2.1 The large-scale ﬂuctuation amplitude

• We saw in Chapter 6 that the long-wavelength (low-k) tail of the CMB power spectrum is caused by the Sachs-Wolfe effect, giving rise to relative temperature fluctuations of δT Φ ≡ τ = (11.4) T 3c2 in terms of the Newtonian potential fluctuations Φ; see also Eq. (7.22). CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM116

• The three-dimensional temperature-ﬂuctuation power spectrum is then 1 P (k) = P (k) . (11.5) τ 9c4 Φ The Poisson equation in its form (10.8) implies that the power spectra of potential- and density ﬂuctuations are related through

4 !2 9H D (a) P (k) P (k) = 0 Ω2 + δ , (11.6) Φ 3 m0 a k4

where the linear growth factor D+(a) was introduced to relate the potential-fluctuation power spectrum at the time of decoupling to the present density-fluctuation power spectrum Pδ(k). • Now, we need to account for projection effects. A three- dimensional mode with wave number k and wavelength λ = 2π/k appears under an angle θ = λ/D, where D is the angular-diameter distance to the CMB. We saw in (7.29) that 2ca 1 D ≈ √ ∝ √ (11.7) H0 Ωm0 H0 Ωm0

to ﬁrst order in Ωm0. Thus, the angular wave number under which the mode appears is 2π l ≈ ≈ Dk . (11.8) θ • Expressing now the power spectrum (11.5) in terms of the angular wave number l yields

!2 ! H 4 D (a) 1 D4 l P (l) ∝ 0 Ω2 + P , (11.9) τ c m0 a D2 l4 δ D where the factor D−2 arises because of the transformation from spatial to angular wave numbers l, and the factor D4/l4 expresses the factor k−4 from the squared Laplacian. • Let us now insert a highly simpliﬁed model for the power spectrum,   n k (k < k0) Pδ(k) = A . (11.10) kn−4 else n Inserting its long-wave limit, Pδ(k) = Ak , into (11.9) yields

!2 − H 4 D (a) 1 D4 n P (l) ∝ A 0 Ω2 + . (11.11) τ c m0 a D2 l

• This shows that the temperature-ﬂuctuation power spectrum depends on the cosmological parameters in various subtle ways; through the Poisson equation, the projection, the angular- diameter distance, the growth factor and the power-spectrum exponent n. CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM117

• Taking all dependences on H0 and Ωm0 into account shows that the amplitude A of the dark-matter power spectrum depends on the cosmological parameters through !−2 D (a) A ∝ Ω−1−n/2h−2−n + P (l) . (11.12) m0 a τ

In other words, the measured power Pτ(l) in the CMB temperature ﬂuctuations can only be translated into the amplitude of the dark-matter power spectrum A if the cosmological parameters are known well enough.

11.2.2 Translation to σ8

• Regarding σ8, we are not done yet. Inserting the model power spectrum (11.10) into the deﬁnition (11.3) gives

 n  kn−1 kn−1  A  k +3  8 − 0 (n 1) σ2  0  n−1 n−1 ,  , 8 = 2  +  k  (11.13) 2π n + 3 ln 8 (n = 1) k0 −1 where k8 = 2π/(8 h Mpc). The translation of the CMB temper- • n ≈ Since 1, the second term is close to logarithmic and thus ature fluctuations depends on cos- weakly dependent on the cosmological parameters in k0. Then, mological parameters, e.g. on Ωm0. we see by combining (11.13) with (11.12) that D (a) σ ∝ Ω1+n/4h2+n/2 + . (11.14) 8 m0 a Note that this is an approximate result which is meant to illustrate the principle. It shows that a measurement of the temperature fluctuations in the CMB can only be translated into σ8 if the matter-density parameter, the Hubble constant, the growth factor and the shape of the power spectrum are accurately known. • Of course, one could also use the small-scale part of the CMB power spectrum for normalising the dark-matter power spectrum. Due to the acoustic oscillations, however, this part depends in a much more complicated way on additional cosmological parameters, such as the baryon density. Reading σ8 off the low-order multipoles is thus a safer procedure. • Even if the cosmological parameters are now known well enough to translate the low-order CMB multipoles to σ8, an additional uncertainty remains. We know that, although the Universe became neutral ∼ 400, 000 years after the Big Bang, it must have been reionised after the first stars and other sources of UV radiation formed. Since then, CMB photons are travelling through ionised material again and experience Thomson (or Compton) scattering. CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM118

• The optical depth for Thomson scattering is Z τ = dx neσT , (11.15)

where ne is the number density of free electrons and σT is the Thomson scattering cross section. After propagating through the optical depth τ, the CMB ﬂuctuation amplitude is reduced by exp(−τ).

• Of course, the CMB photons cannot disappear through Thom- son scattering, thus its overall intensity cannot be changed in this way, but the ﬂuctuation amplitudes are lowered in this diﬀusion process.

• The optical depth τ depends on the path length through ionised material. In view of the CMB, this means that the degree of ﬂuc- tuation damping depends on the reionisation redshift, i.e. the redshift after which the cosmic baryons were transformed back into a plasma. Unless the reionisation redshift is known, we cannot know by how much the CMB ﬂuctuations were suppressed.

• So far, the reionisation redshift can be estimated in two ways. First, as discussed in Sect. 6.2.4, Thomson scattering creates linear polarisation. Of course, the polarisation due to reionised material appears superposed on the primordial polarisation, but on diﬀerent angular scales. The characteristic scale for secondary polarisation is the horizon size at the reionisation redshift, which is much larger than the typical scales of the primordial polarisation. Thus, the reionisation redshift can be inferred from large- scale features in the CMB polarisation, provided the cosmological parameters are known well enough to translate angular scales into physical scales.

• Unfortunately, this is aggravated by the polarised microwave radiation from the Milky Way. Synchrotron and dust emission can be substantially polarised and mask the CMB polarisation, which can only be measured reliably if the foregrounds of Galactic origin can be accurately subtracted. Thus, the degree to which the foreground polarisation is known directly determines the accuracy of the σ8 parameter derived from the CMB ﬂuctuations. This is the main reason for a considerable remaining uncertainty in the σ8 derived from the 3-year WMAP data given in the table in Sect. 6.2.7.

• The other way to constrain the reionisation redshift uses the spectra of distant quasars. Light with wavelengths shorter than the Lyman-α wavelength cannot propagate through neutral hydrogen because it is immediately absorbed. Therefore, quasar spectra CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM119

released before the reionisation redshift must be completely absorbed blueward of the Lyman-α emission line. The appearance of this so-called Gunn-Peterson eﬀect at high redshift thus signals the transition from ionised into neutral material. Using this technique, the reionisation redshift was found to be ∼ 6.5 ... 7, which now agrees well with the estimates from the secondary polarisation of the CMB.

11.3 Cosmological weak lensing

• Compared to the outlined procedure to obtain σ8 from the CMB, it appears completely straightforward to derive it from the cosmic-shear measurements. As we have seen in (10.19), the 2 cosmic-shear power spectrum is proportional to Ωm0 times the amplitude A of the dark-matter power spectrum, which leads to the approximate degeneracy Ωm0σ8 ≈ const. between σ8 and the matter-density parameter Ωm0.

• A more subtle dependence on Ωm0 and to some degree also on other cosmological parameters is introduced by the geometrical weight function W¯ (w0, w) shown in (10.20), and by the growth of the power spectrum along the line-of-sight. This slightly modiﬁes the form of the σ8-Ωm0 degeneracy, but does not lift it.

• However, knowing Ωm0 well enough, we should be able to read σ8 oﬀ the cosmic-shear correlation function. However, there are three problems associated with that.

• First, the cosmic shear measured on angular scales below ∼ 100 is heavily influenced by the onset of non-linear structure growth and the effect this has on the dark-matter power spectrum. While the linear growth factor can be straightforwardly calculated analyti- cally, non-linear growth can only be quantified by means of large numerical simulations and recipes derived from them. Insuffi- cient knowledge of the non-linear dark-matter power spectrum is a major uncertainty in the cosmological interpretation of cosmic shear.

• Second, the amplitude of cosmological weak-lensing eﬀects depends on the redshift distribution of the sources used for measuring ellipticities. Since these background galaxies are typically very faint, it is demanding to measure their redshifts. Two methods have typically been used. One adapts the known redshift distribution of sources in small, very deep observations such as the Hubble Deep Field to the characteristics of the observation to be analysed. The other relies on photometric redshifts, i.e. redshift CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM120

estimates based on multi-band photometry. Yet, the precise redshift distribution of the background sources adds additional uncertainty to estimates of σ8. • Third, it is possible that systematic effects remain in weak-lensing measurements because the effect is so small, and many corrections have to be applied to measured ellipticities before the cosmic shear can be extracted. Advanced correction methods have been developed which made the B-mode contamination almost or completely disappear. This is good news, but it does not yet guarantee the absence of other systematic effects in the data.

• Still, cosmic lensing, combined with estimates of the matter- density parameter, is perhaps the most promising method for precisely determining σ8. Table 11.1 lists values of σ8 derived from some cosmic-shear measurements under the assumption of Ωm0 = 0.3 in a spatially-ﬂat universe.

σ8 data reference +0.09 0.86−0.13 RCS Hoekstra et al. 2002 +0.12 0.71−0.16 CTIO Jarvis et al. 2003 0.72 ± 0.09 Combo-17 Brown et al. 2003 0.97 ± 0.13 Keck-II Bacon et al. 2003 1.02 ± 0.16 HST/STIS Rhodes et al. 2004 0.83 ± 0.07 Virmos-Descart van Waerbeke et al. 2005 0.68 ± 0.13 GEMS Heymans et al. 2005 0.85 ± 0.06 CFHTLS Hoekstra et al. 2006

Table 11.1: Values for σ8 derived from cosmic-shear measurements under the assumption of a spatially-ﬂat universe with Ωm0 = 0.3.

11.4 Galaxy clusters

11.4.1 The mass function

• Based on the assumption that the density contrast is a Gaussian random field and the spherical-collapse model, Press & Schechter in 1974 derived a mass function for dark-matter halos. It compares the standard deviation σR of the density-fluctuation field to the linear density-contrast threshold δc ≈ 1.686 for collapse in the spherical-collapse model. The mean mass contained in spheres of radius R sets the halo mass, which brings the mean (dark-) matter densityρ ¯ into the game. CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM121

• The standard deviation σR is related to the power spectrum. For convenience, we introduce an eﬀective slope d ln P(k) n = (11.16) d ln k for the power spectrum, which will of course be scale-dependent. On large scales, n ≈ 1, while n → −3 on small scales, i.e. for small halo masses. For galaxy clusters, n ≈ −1.

• We introduce the non-linear mass scale M∗ as the mass contained in spheres of radius R such that σR = 1. Since σR grows with the linear growth factor D+(a), the non-linear mass grows with time. It is convenient here to express the amplitude of the power spectrum, and thus σ8, in terms of M∗. It is straightforward to show that M α σ = ∗ , (11.17) R M with 1 n α ≡ 1 + . (11.18) 2 3

• In terms of the dimensionless mass m ≡ M/M∗, the Press- Schechter mass function can then be written in the form r ! 2 ρδ¯ δ2 N m, a m c αmα−2 − c m2α m . ( )d = 2 exp 2 d π M∗ D+(a) 2D+(a) (11.19)

• The Press-Schechter mass function, and some improved variants of it, have been spectacularly conﬁrmed by numerical simulations. It shows that the mass function is a power law with an exponential cut-oﬀ near the non-linear mass scale M∗. For galaxy clusters, n ≈ −1, thus α ≈ 1/3, and δ2 ! ∝ −5/3 − c 2/3 N(m, a)dm m exp 2 m dm , (11.20) 2D+(a)

with an amplitude characterised by M∗, the mean dark-matter densityρ ¯, and the growth factor D+(a). The X-ray flux (top) or luminosity • This opens a way to constrain cosmological parameters as well functions of galaxy clusters can be as σ8 with galaxy clusters: if the abundance and evolution of the converted to a mass function if it is cluster mass function can be measured, they can be determined possible to measure cluster masses from the mass scale of the exponential cut-off and the amplitude sufficiently accurately. of the power-law end. Today, the non-linear mass scale is a few 13 times 10 M . Therefore, the exponential cut-off in the halo mass will not be seen in the galaxy mass function. Clusters, however, show the exponential cut-off very well, and thus their population is very sensitive to changes in σ8. In principle, therefore, σ8 should be very well constrained by the cluster population. CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM122

11.4.2 What is a cluster’s mass?

• The main problem here is how observable cluster properties should be related to quantities used in theory. Strictly speaking, the cluster mass, as used in the theoretical mass function (11.20), is not an observable. Global cluster observables are the X-ray temperature and flux, the optical luminosity and the velocity distribution of their galaxies, and their gravitational-lensing effects. Before we discuss their relation to mass, let us first see what the “mass of a galaxy cluster” could be.

• It is easy to define masses of gravitationally bound, well localised objects, such as planets or stars. They have a well-defined boundary, e.g. the planetary surfaces or the stellar photospheres. This is markedly different for objects like galaxies and galaxy clusters. As far as we know, their densities drop smoothly towards zero like power laws, ∝ r−(2...3). Thus, although they are gravitationally bound, it is less obvious what should be seen as their outer boundary. Strictly speaking, there is none.

• The only way out is then to define an outer boundary in such a way that it is well-defined in theory and identifiable in observational data. A common choice was introduced in Sect. 5.1.2: it defines the boundary by the mean overdensity it encloses. Although this is problematic as well, it may be as good as it gets. Three im- mdiately obvious problems created by this definition are that objects like galaxy clusters are often irregularly shaped rather than spherical, that the overdensity of 200 is quite arbitrary, even if it is inspired by virial equilibrium in the spherical-collapse model, and that its measurement requires a sufficiently accurate density profile to be known or assumed.

• How could standardised radii such as R200 be measured? This could for instance be achieved applying equations such as (6.33) after measuring the slope β and the core radius of the X-ray surface brightness profile together with the X-ray temperature, by calibrating an assumed density profile with galaxy kinematics based on the virial theorem, or by constraining the cluster mass profile with gravitational lensing.

• Obviously, all these measurements have their own problems. Be- ing sensitive to all mass along the line-of-sight, gravitational lensing cannot distinguish between mass bound to a cluster or just projected onto it. Any measurement based on the virial theorem must of course rely on virial equilibrium, which takes time to be established and is often perturbed in real clusters because of merging and accretion. The common interpretation of X-ray measurements requires the assumption that the X-ray gas be in CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM123

hydrostatic equilibrium with the host cluster’s gravitational potential.

• This illustrates that it may be fair to say that there is no such thing as the mass of a galaxy cluster. Even if measurements of cluster “radii” were less dubious, it remained unclear whether they mean the same as those assumed in theory, which are related to the spherical-collapse model. Interestingly, but not surprisingly, cluster masses obtained from numerical simulations suﬀer from the same poor deﬁnition of the concept of a “cluster radius”.

• How can we make progress then? Observables such as the cluster temperature TX or its X-ray luminosity LX should be related to the depth of the gravitational-potential well they are embed- ded in, which should in turn be related to some measure of the total mass. If we can calibrate such expected temperature-mass or luminosity-mass relations, e.g. using numerical simulations of galaxy clusters, a direct comparison between theory and observations seems possible. This is sometimes called an external calibration of the required relations.

• Internal calibrations, i.e. calibrations based on cluster data alone, have become increasingly fashionable over the past years. Here, empirical temperature-mass and luminosity-mass relations are obtained based on one or more estimates of the mass estimates sketched above.

σ8 data reference 1.02 ± 0.07 M-T relation Pierpaoli et al. 2001 0.77 ± 0.07 M-T relation Seljak 2002 0.75 ± 0.16 lensing masses Smith et al. 2003 +0.06 0.79−0.07 luminosity function Pierpaoli et al. 2003 +0.05 0.77−0.04 temperature function Pierpaoli et al. 2003 0.69 ± 0.03 lensing masses Allen et al. 2003 0.78 ± 0.17 optical richness Eke et al. 2006 +0.04 0.67−0.05 lensing masses Dahle 2006

Table 11.2: Values of σ8 derived from the galaxy-cluster population based on diﬀerent observational data.

Several recent determinations of σ8. • The result of both procedures is qualitatively the same. It allows the conversion of observables to mass, and thus of the observed cluster temperature or luminosity functions to mass functions, which can then compared to theory. The shape and amplitude of the power spectrum and the growth factor can then be adapted to optimise the agreement between observed and expected mass CHAPTER 11. THE NORMALISATION OF THE POWER SPECTRUM124

functions. Clusters at moderate or high redshift constrain the evolution of the mass function and allow an independent estimate of the matter-density parameter Ωm0, as sketched in Sect. 5.3 before. • In view of the many diﬃculties listed, it is an astonishing fact that, when applied not to cluster samples rather than individual clusters, the determination of the cluster mass function and its evolution seems to work very well. Values for σ8 derived there- from are given in Tab. 11.2. Chapter 12

Supernovae of Type Ia

12.1 Standard candles and distances

12.1.1 The principle

• Before starting with the details of supernovae, their type Ia and their cosmological relevance, let us set the stage with a few illustrative considerations.

• Suppose we had a standard candle whose luminosity, L, we know precisely. Then, according to the deﬁnition of the luminosity distance in (2.16), the distance can be inferred from the measured ﬂux, f , through s L D = . (12.1) lum 4π f

• Besides the redshift z, the luminosity distance will depend on the cosmological parameters,

Dlum = Dlum(z; Ωm0, ΩΛ0, H0,...) , (12.2)

which can be used in principle to determine cosmological parameters from a set of distance measurements from a class of standard candles.

• For this to work, the standard candles must be at a suitably high redshift for the luminosity distance to depend on the cosmological model. As we have seen in (2.17), all distance measures tend to cz D ≈ (12.3) H0 at low redshift and lose their sensitivity to all cosmological parameters except H0.

125 CHAPTER 12. SUPERNOVAE OF TYPE IA 126

• In reality, we rarely know the absolute luminosity L even of cosmological standard candles. The problem is that they need to be calibrated ﬁrst, which is only possible from a ﬂux measurement once the distance is known by other means, such as from parallaxes in case of the Cepheids.

• Supernovae, however, which are the subject of this chapter, are typically found at distances which are way too large to allow direct distance measurements. Therefore, the only way out is to combine distant supernovae with local ones, for which the approximate distance relation (12.3) holds.

• Any measurement of ﬂux fi and redshift zi of the i-th standard candle in a sample then yields an estimate for the luminosity L in terms of the squared inverse Hubble constant,

!2 czi L = 4π fi . (12.4) H0 Since all cosmological distance measures are proportional to the Hubble length c/H0, the dependences on H0 on both sides of (12.1) cancels, and the determination of cosmological parameters other than the Hubble constant becomes possible. Thus, the ﬁrst lesson to learn is that cosmology from distant supernovae requires a sample of nearby supernovae for calibration.

• Of course, this nearby sample must satisfy the same criterion as the distance indicators used for the determination of the Hubble constant: their redshifts must be high enough for the peculiar velocities to be negligible, thus z & 0.02. On the other hand, the redshifts must be low enough for the linear appoximation (12.3) to hold.

• It is important to note that it is not necessary to know the absolute luminosity L even up to the uncertainty in H0. If L is truly independent of redshift, cosmological parameters could still be determined through (12.1) from the shape of the measured relation between ﬂux and redshift even though its precise amplitude may be unknown. It is only important that the objects used are standard candles, but not how bright they are.

12.1.2 Requirements and degeneracies

• Let us now collect several facts about cosmological inference from standard candles. Since we aim at the determination of cosmological parameters, say Ωm0, it is important to estimate the accuracy that we can achieve from measurements of the luminosity distance. CHAPTER 12. SUPERNOVAE OF TYPE IA 127

• Suppose we restrict the attention to spatially ﬂat cosmological models, for which ΩΛ0 = 1 − Ωm0. Then, because the dependence on the Hubble constant was canceled, Ωm0 is the only remaining relevant parameter. We estimate the accuracy through ﬁrst-order Taylor expansion,

dDlum ∆Dlum ≈ ∆Ωm0 , (12.5) dΩm0

about a fiducial model, such as a ΛCDM model with Ωm0 = 0.3. • At a fiducial redshift of z ≈ 0.8, we find numerically Logarithmic derivative of the lumi- d ln Dlum nosity distance with respect to Ω . ≈ −0.5 , (12.6) m0 dΩm0 which shows that a relative distance accuracy of

∆Dlum ≈ −0.5∆Ωm0 (12.7) Dlum

is required to achieve an absolute accuracy of ∆Ωm0. For ∆Ωm0 ≈ 0.02, say, distances thus need to be known to ≈ 1%. • This accuracy requires suﬃciently large supernova samples. As- suming Poisson statistics for simplicity and distance measurements to N supernovae, the combined accuracy is

2 ∆Dlum |∆Ωm0| ≈ √ . (12.8) N Dlum

That is, an accuracy of ∆Ωm0 ≈ 0.02 can be achieved from ≈ 100 supernovae whose individual distances are known to ≈ 10%. • Anticipating physical properties of type-Ia supernovae, their intrinsic peak luminosities in blue light are L ≈ 3.3 × 1043 erg s−1, with a relative scatter of order 10%. (As we shall see later, type-Ia supernovae are standardisable rather than standard candles, and the standardising procedure is currently not able to reduce the scatter further.) • Given uncertainties in the luminosity L and in the ﬂux measurement, error propagation on (12.1) yields the distance uncertainty

 !2 !2 1/2  dDlum 2 dDlum 2 Dlum =  ∆L + ∆ f  , (12.9)  dL d f  or the relative uncertainty

 !2 !21/2 ∆Dlum 1  ∆L ∆ f  =  +  . (12.10) Dlum 2 L f Even if the ﬂux could be measured precisely, the intrinsic luminosity scatter currently forbids distance determinations to better than 10%. CHAPTER 12. SUPERNOVAE OF TYPE IA 128

• Fluxes have to be inferred from photon counts. For various reasons to be clariﬁed later, supernova light curves should be determined until ∼ 35 days after the peak, when the luminosity has typically dropped to ≈ 2.5 × 1042 erg s−1. The luminosity distance to z ≈ 0.8 is ≈ 5 Gpc, which implies ﬂuxes f ≈ 1.1 × 10−14 erg s−1 cm−2 at peak and f ≈ 8.7 × 10−16 erg s−1 cm−2 35 days later.

• Dividing by an average photon energy of 5 × 10−12 erg, multiplying with the area of a typical telescope mirror with 4 m diameter, and assuming a total quantum efficiency of 30%, we find detected −1 −1 photon fluxes of fγ ≈ 85 s at peak and fγ ≈ 7 s 35 days af- The luminosity distance in a uni- terwards. These fluxes are typically distributed over a few CCD verse with Ωm0 = 0.3 and ΩΛ0 = pixels. 0.7 with Hubble constant h = 0.72.

• Supernovae occur in galaxies, which means that their fluxes need to be measured on the background of the galactic light. On the area of a distant supernova image, the photon flux from the host galaxy is comparable to the flux from the supernova. Therefore, an estimate for the signal-to-noise ratio for the detection is √ S N N ≈ √ = , (12.11) N 2 N 2 where N is the number of photons per pixel detected from supernova and host galaxy during the exposure time. Signal-to-noise ratios of & 10 up to 35 days after the maximum thus require N ≈ 400 photons per pixel. Assuming that the supernova appears on typically ∼ 4 pixels, this implies exposure times of order 4 × 400/7 ≈ 230 s, or a few minutes. Typical exposure times are of order 15 ... 30 minutes to capture supernovae out to redshifts z ∼ 1. Then, the photometric error around peak luminosity is cer- tainly less than the remaining scatter in the intrinsic luminosity, and relative distance accuracies of order 10% are within reach.

• However, a major difficulty is the fact that the identification of type-Ia supernovae requires spectroscopy. Sufficiently accurate spectra typically require long exposures on the world’s largest telescopes, such as ESO’s Very Large Telescope which consists of four individual mirrors with 8 m diameter each.

• In order to see what we can hope to constrain by measuring angular-diameter distances, we form the gradient of Dlum in the Ωm0-ΩΛ0 plane, !t ∂D ∂D ~g ≡ lum , lum , (12.12) ∂Ωm0 ∂ΩΛ0

at a ﬁducial ΛCDM model with Ωm0 = 0.3. When normalised to CHAPTER 12. SUPERNOVAE OF TYPE IA 129

unit length, it turns out to point into the direction ! −0.76 ~g = . (12.13) 0.65

• This vector rotated by 90◦ then points into the direction in the Ωm0-ΩΛ0 plane along which the luminosity distance does not change. Thus, near the fiducial ΛCDM model, the parameter combination ! Ωm0 P ≡ ~g · = −0.76 Ωm0 + 0.65 ΩΛ0 (12.14) ΩΛ0 is degenerate. The degeneracy direction, characterised by the vector R(π/2)~g = (0.65, 0.76)t, points under an angle of ◦ arctan(0.76/0.65) = 49.5 with the Ωm0 axis, almost along the diagonal from the lower left to the upper right corner of the parameter plane. Thus, it is almost perpendicular to the degeneracy direction obtained from the curvature constraint due to the CMB. This illustrates how parameter degeneracies can very efficiently be broken by combining suitably different types of measurement.

12.2 Supernovae

12.2.1 Types and classiﬁcation

• Supernovae are “eruptively variable” stars. A sudden rise in Supernova 1994d in its host galaxy. brightness is followed by a gentle decline. They are unique events which at peak brightness reach luminosities comparable to those 10 11 of an entire galaxy, or 10 ... 10 L . They reach their maxima within days and fade within several months.

• Supernovae are traditionally characterised according to their early spectra. If hydrogen lines are missing, they are of type I, otherwise of type II. Type-Ia supernovae show silicon lines, un- like type-Ib/c supernovae, which are distinguished by the promi- nence of helium lines. Normal type-II supernovae have spectra dominated by hydrogen. They are subdivided according to their lightcurve shape into type-IIL and type-IIP. Type-IIb supernova spectra are dominated by helium instead. Lightcurves of supernovae of diﬀer- • Except for type-Ia, supernovae arise due to the collapse of a mas- ent types. sive stellar core, followed by a thermonuclear explosion which disrupts the star by driving a shock wave through it. Core- collapse supernovae of type-I (i.e. types Ib/c) arise from stars with masses between 8 ... 30 M , those of type-II from more massive stars. CHAPTER 12. SUPERNOVAE OF TYPE IA 130

• Type-Ia supernovae, which we are dealing with here, arise when a white dwarf is driven over the Chandrasekhar mass limit by mass overﬂowing from a companion star. In a binary system, the more massive star evolves faster and can reach its white-dwarf stage before its companion leaves the main sequence and becomes a red giant. When this happens, and the stars are close enough, matter will ﬂow from the expanding red giant on the white dwarf.

• Electron degeneracy pressure can stabilise white dwarfs up to the Chandrasekhar mass limit of ∼ 1.4 M . When the white dwarf is driven over that limit, it collapses, starts a thermonuclear runaway and explodes. Since this type of explosion involves an approximately ﬁxed amount of mass, it is physically plausible that the explosion releases a ﬁxed amount of energy. Thus, the Chan- drasekhar mass limit is the main responsible for type-Ia supernovae to be approximate standard candles.

• The thermonuclear runaway in type-Ia supernovae converts the carbon and oxygen in the core of the white dwarf into 56Ni, which later decays through 56Co into the stable 56Fe. According to detailed numerical explosion models, the nuclear fusion is started at random points near the centre of the white dwarf.

• Since the core material is degenerate, its pressure is independent Early (top) and late spectra of dif- of its temperature. The mass accreted from the companion star in- ferent supernova types. creases the pressure. Once it exceeds the Fermi pressure, inverse beta decay sets in,

− p + e → n + νe + γ , (12.15)

which suddenly removes the degenerate electrons. Under the high pressure, the temperature rises dramatically and ignites the fusion. The neutrinos carry away much of the explosion energy un- noticed because they can leave the supernova essentially without Supernova classiﬁcation. further interaction.

• The presence of silicon lines in the type-Ia spectra indicates that not all of the white dwarf’s material is converted into 56Ni. This shows that there is no explosion, but a deflagration, in which the flame front propagates at velocities below the sound speed. The deflagration can burn the material fast enough if it is turbulent, because the turbulence dramatically increases the surface of the flame front and thus the amount of material burnt per unit time. Type-Ia supernovae occur when 56 Typically, ∼ 0.5 M of Ni is produced in theoretical models. white dwarfs are driven over the Chandrasekhar mass limit by mass • The peak brightness is reached when the deflagration front flowing from a companion star. reaches the former white dwarf’s surface and drives it as a rapidly expanding envelope into the surrounding space. The γ photons released in the nuclear fusion processes are redshifted by scattering CHAPTER 12. SUPERNOVAE OF TYPE IA 131

with the expanding material and ﬁnally leave the explosion site as X-ray, UV, optical and infrared photons.

• Once the thermonuclear fusion has ended, additional energy is released by the β decay of 56Co into 56Fe with a half life of 77.12 days. The exponential nature of the radioactive decay causes the typical exponential decline phase in supernova light curves.

• Since the supernova light has to propagate through the expanding envelope before we can see it, the opacity of the envelope and thus its metallicity are important for the appearance of the supernova.

12.2.2 Observations

• Since supernovae are transient phenomena, they can only be detected by suﬃciently frequent monitoring of selected areas in the sky. Typically, ﬁelds are selected by their accessibility for the telescope to be used and the least degree of absorption by the Galaxy. Since a type-Ia supernova event lasts for about a month, monitoring is required every few days.

• Supernovae are then detected by diﬀerential photometry, in which the average of all preceding images is subtracted from the last image taken. Since the seeing varies, the images appear convolved with point-spread functions of variable width even if they are taken with identical optics, thus the objects on them appear more or less blurred. Before they can be meaningfully subtracted, they therefore have to be convolved with the same eﬀective point- spread function. This causes several complications in the later analysis procedure, in particular with the photometry.

• Of course, this detection procedure returns many variable stars and supernovae of other types, which are not standard candles and have to be removed from the sample. Pre-selection of type- Ia candidates is done by colour and the light-curve shape, but the identiﬁcation of type-Ia supernovae requires spectroscopy in order to identify the decisive silicon lines at 6347 Å and 6371 Å. Since these lines move out of the optical spectrum for redshifts z & 0.5, near-infrared observations are crucially important for the high-redshift supernovae relevant for cosmology.

• Nearby supernovae, which we have seen need to be observed for calibration, show that type-Ia supernovae are not standard candles but show a substantial scatter in luminosity. It turned out that there is an empirical relation between the duration of the supernova event and its peak brightness in that brighter supernovae last longer. CHAPTER 12. SUPERNOVAE OF TYPE IA 132

• This relation between the light-curve shape and the brightness can be used to standardise type-Ia supernovae. It was seen as a major problem for their cosmological interpretation that the origin for this relation was unknown, and that its application to high-redshift supernovae was based on the untested assumption that the relation found and calibrated with local supernovae would also hold there. Recent simulations indicate that the relation is an opacity effect: brighter supernovae produce more 56Ni and thus have a higher metallicity, which causes the envelope to be more opaque, the energy transport through it to be slower, and therefore the supernova to last longer. • Thus, before a type-Ia supernova can be used as a standard candle, its duration must be determined, which requires the light-curve to be observed over sufficiently long time. It has to be taken into account here that the cosmic expansion leads to a time dilation, due to which supernovae at redshift z appear longer by a factor of (1 + z). We note in passing that the confirmation of this time dilation effect indirectly confirms the cosmic expansion. After the standardisation, the scatter in the peak brightnesses of nearby supernovae is substantially reduced. This encourages (and justifies) Lightcurves of type-Ia supernovae their use as standardisable candles for cosmology. before (top) and after correction. • The remaining relative uncertainty is now typically between 10 ... 15% for individual supernovae. Since, as we have seen following (12.7), we require relative distance uncertainties at the per cent level, of order a hundred distant supernovae are required before meaningful cosmological constraints can be placed, which justifies the remark after (12.8). • An example for the several currently ongoing supernova surveys is the Supernova Legacy Survey (SNLS) in the framework of the Canada-France-Hawaii Legacy Survey (CFHTLS), which is carried out with the 4-m Canada-France-Hawaii telescope on Mauna Kea. It monitors four fields of one square degree each five times during the 18 days of dark time between two full moons (luna- tions). • Differential photometry is performed to find out variables, and candidate type-Ia supernovae are selected by light-curve fitting after removing known variable stars. Spectroscopy on the largest telescopes (mostly ESO’s VLT, but also the Keck and Gemini telescopes) is then needed to identify type-Ia supernovae. To give a few characteristic numbers, the SNLS has taken 142 spectra of type-Ia candidates during its first year of operation, of which 91 were identified as type-Ia supernovae. • The light curves of these objects are observed in several different filter bands. This is important to correct for interstellar absorp- CHAPTER 12. SUPERNOVAE OF TYPE IA 133

tion. Any dimming by intervening material makes supernovae appear fainter, and thus more distant, and will bias the cosmological results towards faster expansion. Since the intrinsic colours of type-Ia supernovae are characteristic, any deviation between the observed and the intrinsic colours signals interstellar absorption which is corrected by adapting the amount of absorption such that the observed is transformed back into the intrinsic colour. • This correction procedure is expected to work well unless there is material on the way which absorbs equally at all wavelengths, so- called “grey dust”. This could happen if the absorbing dust grains are large compared to the wavelength. Currently, it is quite difficult to concusively rule out grey dust, although it is implausible based on the interstellar absorption observed in the Galaxy. Distances to type-Ia supernovae (in • After applying the corrections for absorption and duration, each logarithmic units) as a function of supernova yields an estimate for the luminosity distance to its red- their redshift, as measured by the shift. Together, the supernovae in the observed sample constrain Supernova Legacy Survey. the evolution of the luminosity distance with redshift, which is then fit varying the cosmological parameters except for H0, i.e. typically Ωm0 and ΩΛ0. This yields an “allowed” region in the Ωm0-ΩΛ0 plane compatible with the measurements which is degenerate in the direction calculated before. • More information or further assumptions are necessary to break the degeneracy. The most common assumption, justified by the CMB measurements, is that the Universe is spatially flat. Based upon it, the SNLS data yield a matter density parameter of

Ωm0 = 0.263 ± 0.037 . (12.16) This is a remarkable result. First of all, it conﬁrms the other independent measurements we have already discussed, which were based on kinematics, cluster evolution and the CMB. Second, it shows that, in the assumed spatially ﬂat universe, the dominant contribution to the total energy density must come from something else than matter, possibly the cosmological constant.

• It is important for the later discussion to realise in what way the Cosmological parameter constraints parameter constraints from supernovae differ from those from the derived from the same data. CMB. The fluctuations in the latter show that the Universe is at least nearly spatially flat, and the density parameters in dark and baryonic matter are near 0.25 and 0.045, respectively. The rest must be the cosmological constant, or the dark energy. Arising early in the cosmic history, the CMB itself is almost insensitive to the cosmological constant, and thus it can only constrain it indirectly. • Type-Ia supernovae, however, measure the angular-diameter distance during the late cosmic evolution, when the cosmological CHAPTER 12. SUPERNOVAE OF TYPE IA 134

constant is much more important. As (12.14) shows, the luminosity distance constrains the diﬀerence between the two parameters,

ΩΛ0 = 1.17 Ωm0 + P , (12.17)

where the degenerate parameter P is determined by the measurement. Assuming ΩΛ0 = 1 − Ωm0 as in a spatially-ﬂat universe yields P = 1 − 2.17 Ωm0 ≈ 0.43 (12.18) from the SNLS ﬁrst-year result (12.16), illustrating that the survey has constrained the density parameters to follow the relation

ΩΛ0 ≈ 1.17 Ωm0 + 0.43 . (12.19)

• The relative acceleration of the universe,a ¨/a, is given by the equation ! a¨ Ω = H2 Ω − m0 (12.20) a 0 Λ0 2a3 if matter is pressure-less, which follows directly from Einstein’s ﬁeld equations. Thus, the expansion of the universe accelerates 2 today (a = 1) ifa ¨ = H0(ΩΛ0 − Ωm0/2) > 0, or ΩΛ0 > Ωm0/2. Given the measurement (12.19), the conclusion seems inevitable that the Universe’s expansion does indeed accelerate today. Above redshift z ≈ 1, the cosmic ac- • If the Universe is indeed spatially ﬂat, then the transition between celeration seems to turn into decel- decelerated and accelerated expansion happened at eration. 0.263 1 − 0.263 ≈ ⇒ a = 0.56 , (12.21) 2a3 or at redshift z ≈ 0.78. Luminosity distances to supernovae at larger redshifts should show this transition, and in fact they do.

12.2.3 Potential problems

• The main observational issues is dust in the SN host galaxy. If one is careful enough (an observational issue), then one can estimate the reddening of the supernova accurately. Yet, there is a terri- ble conceptual issue that is currently one of the major stumbling- blocks of the ﬁeld - we do not know with certainty what the relationship between reddening and the extinction (the amount of light lost - this aﬀects the inferred absolute magnitude).

• There are two ways around this issue - trying to better estimate the attenuation curve shape (the relationship between reddening and extinction, this is not straightforward and requires excellent photometry over a long wavelength range; see Wood-Vasey et al. CHAPTER 12. SUPERNOVAE OF TYPE IA 135

2007), and going to systems that one a priori expects to be free of dust (elliptical galaxies; this involves a considerable reduction in sample size; see, e.g., Sullivan et al. 2003, MNRAS, 340, 1057). Both approaches have been taken. • The problem with possible grey dust has already been mentioned: While the typical colours of type-Ia supernovae allow the detection and correction of the reddening coming with typical interstellar absorption, grey dust would leave no trace in the coulours and remain undetectable. However, grey dust would re-emit the absorbed radiation in the infrared and add to the infrared background, which is quite well constrained. It thus seems that grey dust is not an important contaminant, if it exists. • Gravitational lensing is inevitable for distant supernovae. De- pending on the line-of-sight, they are either magnified or demag- nified. Since, due to nonlinear structures, high magnifications can occasionally happen, the magnification distribution must be skewed towards demagnification to keep the mean of zero magnification. Thus, the most probable magnification experienced by supernova is below unity. In other words, lensing may lead to a slight demagnification if lines-of-sight towards type-Ia supernovae are random with respect to the matter distribution. In any case, the rms cosmic magnification adds to the intrinsic scatter of the supernova luminosities. It may become significant for redshifts z & 1. • It is a difficult and debated question whether supernovae at high redshifts are intrinsically the same as at low redshifts where they are calibrated. Should there be undetected systematic differences, cosmological inferences could be wrong. In particular, it may be natural to assume that metallicties at high redshifts are lower than at low redshifts. Since supernovae last longer if their atmospheres are more opaque, lower metallicity may imply shorter supernova events, leading to underestimated luminosities and overestimated distances. Simulations of type-Ia supernovae, however, seem to show that such an effect is probably not significant. • It was also speculated that distant supernovae may be intrinsically bluer than nearby ones due to their lower metallicity. Should this be so, the extinction correction, which is derived from reddening, would be underestimated, causing intrinsic luminosities to be under- and luminosity distances to be overestimated. Thus, this effect would lead to an underestimate of the expansion rate and counteract the cosmological constant. There is currently no indication of such a colour effect. • Supernovae of types Ib/c may be mistaken for those of type Ia if the identification of the characteristic silicon lines fails for CHAPTER 12. SUPERNOVAE OF TYPE IA 136

some reason. Since they are typically fainter than type-Ia supernovae, they would contaminate the sample and bias results towards higher luminosity distances, and thus towards a higher cosmological constant. It seems, however, that the possible contamination by non-type-Ia supernovae is so small that it has no noticeable eﬀect.

• Several more potential problems exist. It has been argued for a while that, if the evidence for a cosmological constant was based exclusively on type-Ia supernovae, it would probably not be considered entirely convincing. However, since the supernova observations come to conclusions compatible with virtually all independent cosmological measurements, they add substantially to the persuasiveness of the cosmological standard model. More- over, recent supernova simulations reveal good physical reasons why they should in fact be reliable, standardisable candles. Chapter 13

Galaxies in a cosmological context

In the preceding, we have explored in some depth the observational ba- sis of our cosmological paradigm, and introduced the state-of-the-art techniques that are used to critically test this paradigm and estimate its free parameters. Constraints have come from diverse physical systems — from the behavior of cooling degenerate matter in a white dwarf, through to nucleosynthesis and the abundances of light elements, the clustering of galaxies on large scales, and the super-horizon scale ﬂat- ness of the cosmic microwave background, to name just a few observables — and have converged towards one cosmological paradigm. This paradigm involves three challenging, as yet poorly-understood, ingredients: dark matter, dark energy, and inﬂation. This paradigm, despite these three huge and basic ill-constrained ingredients, has proved remarkably successful at explaining the evolution of the scale factor of the Universe and the evolution of structure in the linear and somewhat non-linear regime. In this section, we focus on what cosmology predicts for the properties of galaxies (i.e., some of the smallest cosmologically-relevant scales). We will see that overall there is a qualitative match between the predictions of theory and observations, but some puzzling (and likely related) discrepancies remain — these represent the frontier of the study of galaxy formation in a cosmological context.

137 CHAPTER 13. GALAXIES IN A COSMOLOGICAL CONTEXT138

13.1 Predictions for galaxy formation in a cosmological context

13.1.1 Evolution of the dark matter framework

The distribution of dark matter at • Recall that in the early universe dark matter perturbations could relatively late times; collapsed into collapse, damped by the strong radiation pressure that attempted ﬁlaments and halos; the concentra- to counteract collapse (in the radiation-dominated era). The bary- tions of dark matter contain galax- onic signatures of this collapse are imprinted in the cosmic mi- ies crowave background.

• After matter domination and recombination, the pressure support that counteracted the gravity forces of the dark matter overdensities were dramatically reduced, and the dark matter overdensities collapsed uninhibited.

• The ﬁrst dark matter halos to form are relatively small, and rapidly grow (primarily) through merging into larger and larger dark matter halos. Small dark matter halos continue to form, act- ing as the ’raw material’ for further growth.

• Most of the mass growth in dark matter halos comes from (relatively rare) mergers between two comparably-sized halos - there are many more frequent accretions of small dark matter halos that do not contribute much to the mass budget but are by far the most frequent occurrences.

13.1.2 Evolution of the baryonic content of dark matter halos

• The baryonic content of these dark matter halos acts a little differently. Baryons in very small halos cannot cool after reionization (the temperature of the intergalactic medium is too high to cool into small halos; e.g., Bullock et al. 2000). Baryons in small 12 8 halos (10 > M/M > 10 ) can cool effectively (although the detailed mechanism is still being debated; e.g., Keres et al. 2005). Baryons in larger halos tend to collide and shock, developing a hot envelope of gas (for large halos — galaxy clusters — this emits X-rays and was discussed earlier as an important diagnos- Top: a major merger of three galax- tic of cluster properties and as a cosmological probe). ies. Botton: an accretion of a • The baryons that effectively cool collapse into a dense baryonic low-mass galaxy onto a larger one ’nugget’ in the center of its dark matter halo; this is what one (Martinez-Delgado et al. 2008; commonly refers to when one talks about galaxies. NGC 5907). CHAPTER 13. GALAXIES IN A COSMOLOGICAL CONTEXT139

• When dark halos merge, the galaxies within them also merge (although typically later than common deﬁnitions of halo merger would deﬁne as a ’merger time’).

• In the case of the (rare) merger of two near equal-mass objects, the galaxies are typically of similar size and merge reasonably rapidly (after 1-2Gyr) and the result is a spectacular major merger. In the case of (much more common) accretion of smaller halos, the galaxies that are being accreted are usually very low-mass and one has to look hard to ﬁnd evidence of that accretion.

• Thus, cosmological theory predicts that i) there should be a low but detectable level of major galaxy merging (mergers between A montage of ongoing galaxy merg- two comparably-sized galaxies), and ii) there should be frequent ers with very high star formation but hard-to-detect accretion of small galaxies onto larger ones; rates (indicating lots of gas has been furthermore theory predicts the rate of such events. dumped into the centers of these galaxies; Borne et al. 1999).

13.2 Major galaxy merging

• Major galaxy merging clearly happens (see ﬁgure). Most of the most intensely star-forming galaxies in the local Universe are ongoing galaxy mergers - the argument here is that the merging leads to orbit crossing in the gas, and the gas dissipates its orbital angular momentum and heads to the center, leading to a dense lump of gas and lots of star formation.

• Galaxy merging plays a role in another key phenomenology in the local Universe: the diﬀerence between elliptical and spiral galaxies. Spiral galaxies have a dominant disk of stars (with some gas), supported by rotational motions. Elliptical galaxies have instead a dominant spheroid of stars supported primarily by random motions (there may be some rotational support, but rarely is rotation the dominant source of support). It has long been argued (since Toomre & Toomre 1972) that mergers could randomize stellar motions in disk galaxies, leading to an elliptical merger remnant. There has been a lot of work on this issue over the subsequent years, and while a lot of details have been ﬁlled in the bottom line remains the same.

• The rate of galaxy merging can be estimated using a number of ways. The incidence of morphologically-messed up things (see ﬁgure) can be used to count mergers; then some estimate of the timescale over which such ’messed-up’ morphologies exist in Top: a spiral galaxy, Bottom: an el- typical interactions can be used to infer a rate from the number liptical galaxy of such systems. There are a number of practical diﬃculties to such an approach; in particular, there is no clear consensus on CHAPTER 13. GALAXIES IN A COSMOLOGICAL CONTEXT140

what a ’messed-up’ morphology is, and no automated classiﬁca- tion system has been developed that allows reliable counting of such systems.

• One can instead count the number of close pairs of galaxies as a proxy for merger rate. The argument here is that close pairs of galaxies (especially massive galaxies) are likely to be pre-merger systems. One can count these much more reliably and quantitatively than merger remnants (a key advantage). There have been a number of discussions of how this translation between pair counts and mergers happens in practice (Patton et al. 2000; Masjedi et al. 2005; Bell et al. 2006). The average number of galaxies in • A powerful approach is to use the two point correlation func- pairs/triplets/etc, per unit galaxy for tion of galaxies. One measures the projected correlation function luminous (left) and massive (right) w(rp): Z ∞ galaxies. In black are observa- 2 2 1/2 tional points estimated by Bell et al. w(rp) = ξ([rp + π ] )dπ, (13.1) −∞ (2006), in grey are literature values where rp is the transverse distance between two galaxies, π is (data points) and model predictions their line-of-sight separation (Davis & Peebles 1983), and ξ is from a galaxy formation model in a the real-space correlation function. In practice, what one actually cosmological context (Somerville et does to estimate w(rp) is w(rp) = ∆(DD/RR − 1), where ∆ is the al. 2008). path length being integrated over (one cannot integrate over an inﬁnite path length), DD is the histogram of separations between real galaxies (it’s a histogram, as a function of separation) and RR is the histogram of separations of mock catalog galaxies.

• The projected correlation function is clearly related to the real- space correlation function via the Abel integral (Davis & Peebles 1983). In the special case when the real-space and projected correlation functions can be adequately ﬁt with power laws, the parameters of the two ﬁts are intimately related. If the real-space −γ γ 1−γ correlation function ξ(r) = (r/r0) , then w(rp) = Cr0rp , where √ Γ([γ−1]/2) C = π Γ(γ/2) . • Given the real-space correlation function parameters and the number density n of galaxies, one can then infer the average number of galaxies in pairs/triplets/etc, per unit galaxy: 4πn P(r < r ) ∼ rγr3−γ, (13.2) f 3 − γ 0 f

where P(r < r f ) is the average number of galaxies in pairs/triplets/etc, per unit galaxy (it’s like a probability, but not required to be less than 1) with real separations of r < r f and r0 and γ are the parameters of the power-law real-space correlation function (Masjedi et al. 2005; Bell et al. 2006). CHAPTER 13. GALAXIES IN A COSMOLOGICAL CONTEXT141

• The results of such an exercise are shown in the ﬁgure. The fraction of galaxies in close pairs (data points) agrees reasonably well with the expectations of a galaxy formation model in a cosmological context (grey line); the predicted rate of mergers in ΛCDM is consistent with our best estimates of the rate of galaxy merging. • A further consistency check can be carried out. Recall that it was argued that elliptical galaxies were the result of galaxy mergers. In that case, one would expect that the integral of the merger rate would agree with the observed number density of elliptical galaxies. One needs to make a number of assumptions (all mergers produce ellipticals, all close pairs merge, a merger timescale that The rate of growth of the elliptical ∼ πr/v is equal to the typical orbital timescale 2 , and that the only galaxy population (black points). way to make an elliptical galaxy is through merging). The in- The line is the integral of the galaxy tegral of the merger rate for galaxies with masses in excess of merger rate. 2.5 × 1010 solar masses (grey line) is indeed rather similar to the build-up of the elliptical galaxy population (here, the observed comoving number density of non star-forming galaxies above a mass of 5×1010 solar masses, shown as black data points, a decent proxy for the elliptical galaxy population) lending further weight to the notion that we understand the evolution of the merger rate reasonably well.

13.3 The accretion of small galaxies through minor mergers

• Minor mergers (or accretions), defined as the merger of a substantially smaller galaxy by a much larger one, are much more common. Unfortunately, because the smaller merger partner is very faint, these are much more challenging to observe. Again, there are a number of possible strategies for assessing their rate - the counting of faint tidal tails around external galaxies, the frequency of disturbed large galaxies (assuming that their distur- bances are likely from minor merging), etc. • It is clear that such mergers are reasonably frequent, in qualitative agreement with galaxy formation theory. In the systems where it has been possible to observe them deeply (either through star counts or ultra-deep observations of the integrated light from Top: A map of the stellar content of these galaxies), tidal tails are rather frequent. the diffuse stars around M31; significant inhomogeneities and streams • Quantitative agreement is harder to demonstrate. The most quan- can be seen, a signature of the dis- titative analysis to date has taken place for the Milky Way galaxy truption of dwarf galaxies by M31 (Bell et al. 2008). Here, inhomogeneities in the diffuse stellar in the last several Gyr (Ibata et al. envelope around the Milky Way (usually called the stellar halo) 2002, 2005; Ferguson et al. 2003). were quantified using the RMS of the star counts in different bins Bottom: A stellar stream around NGC 4013; Martinez-Delgado et al. (2008). CHAPTER 13. GALAXIES IN A COSMOLOGICAL CONTEXT142

of magnitude and position on the sky, compared to the expectation from a smooth distribution of the stars in the stellar halo (solid points).

• The same quantity was derived for simulations of stellar halo formation in a cosmological context (Bullock & Johnston 2005), by simulating the distribution of the same type of stars used in the observations, imposing the observational window function and uncertainties, and using the same analysis technique. The results are shown by the grey lines - the models clearly show a lot of model-to-model scatter (these are all drawn from simulations of the formation of a Milky Way-sized halo, so they indicate the expected degree of stochastic variation in stellar properties), but they are similar to the observed degree of structure. While it is clear that much work remains to be done, the properties of the stellar halo of the Milky Way are quantitatively consistent with being built up through minor mergers and accretions, as expected in ΛCDM. The run of RMS/total number of stars for the Milky Way (black line) 13.4 Challenges to ΛCDM on small scales and models of stellar halo growth in a ΛCDM context (grey lines) as a function of Heliocentric distance. • We have seen that there is a qualitative (and tentative quantitative) agreement between observations and theory on one of the most important predictions of the ΛCDM paradigm — the frequency of galaxy merging (linked to and driven by the merging of dark matter halos). Here, we examine some (potentially related discrepancies between observations and our current understanding of how galaxies should form in ΛCDM. All of these problems apply to the smallest scales — ∼kpc scales — and are complicated by the importance of baryonic physics on these scales. Yet, these discrepancies may be critical, as all go in the sense that one ob- serves less dark matter on kpc scales than is predicted, potentially indicating a lack of CDM on these scales.

13.4.1 Rotation curve problem

• This is perhaps the most serious problem affecting ΛCDM. From the mid-1990s onwards, it has become clear that ΛCDM predicts a nearly ’universal’ density profile for dark matter halos that has a relatively steep cusp ∼ 1/r in the central parts (mass goes as radius squared), yielding a predicted√ rotation√ curve of the form (M ∝ v2r so v2 ∝ M/r so) v ∝ M/r ∝ r. A rotation curve of an LSB galaxy, • With the discovery that low surface brightness galaxies (galaxies taken from Kuzio de Naray et al. with low surface brightness, thus low stellar density, thus small 2008. The left-hand side gives an isothermal halo with a constant- density core, the right-hand side shows a ρ ∝ 1/r halo. The different rows make different assumptions about the contribution of stars to the rotation curve; the third row is the most realistic assumption. CHAPTER 13. GALAXIES IN A COSMOLOGICAL CONTEXT143

amounts of mass density in stars) were dominated by dark matter even in their central parts (e.g., de Blok et al. 1996), the rotation curves of these galaxies has been studied in detail. Such galaxies have v ∝ r (i.e., constant density core). Extensive work has demonstrated√ that such rotation curves cannot be reconciled with v ∝ r rotation curves by any combination of plausible observational eﬀects (e.g., de Blok et al. 2008; Kuzio de Naray et al. 2009); instead, the density proﬁle of the dark halo is likely to have a constant density core in the inner parts.

• This is an interesting problem. It could have ’boring’ solutions — it is thought that it might be possible, with strong ∼ 20km/s non- circular motions (the galaxy’s baryons would be sloshing around in the center of the dark matter halo) or intense enough stellar feedback (the argument is that intense star formation early in the galaxy’s history would eject a large fraction of the baryonic content of the galaxy, dragging along much of the dark matter with it). The ’exciting’ solution would be that the actual dark matter distribution has a constant-density core — this would necessitate a ’warm’ dark matter particle (with a mass in the ∼ 1 − 10keV range that would give it a velocity dispersion high enough to form constant ∼kpc-scale density cores) or a suppression of the power spectrum at ∼kpc scales. Either solution would have important implication on the dark matter content of the least massive galaxies.

• There are constraints on warm dark matter — lyman alpha forest structure sets constraints on a WDM candidate particle (sterile neutrino) mass that it is >10keV (Seljak et al. 2006); such a mass may make it harder to produce low-density cores in LSB galaxies (see Dalcanton & Hogan 2001).

13.4.2 The substructure problem

• This problem is a simple and challenging problem, and is related The circular velocity distribution in many ways to the other problems. The issue here is that the for a simulated galaxy cluster and Local Group halo (lines), and the number of dark matter halos per unit mass goes as NDM(M) ∝ M−2, whereas the number of galaxies per unit mass goes roughly observed inferred circular velocity −1.2 distributions for the Virgo Cluster as Ngal(M) ∝ M . (agrees with the predictions) and the • The ﬁgure is taken from Moore et al. (1999), and shows the nor- Local Group (does not agree with malized circular velocity distribution of subhalos in a simulated predictions). cluster (dashed line) and Local Group (solid line); the x-axis is vc,subhalo/vc,group. One can see that the distribution of halo masses is scale-free; in contrast the galaxy mass distributions are not scale- free. Clusters of galaxies (e.g., Virgo) have reasonably steep mass functions, in agreement with the subhalo mass function. Groups CHAPTER 13. GALAXIES IN A COSMOLOGICAL CONTEXT144

such as the Local Group appear to be deﬁcient in low-mass galaxies.

• There are two possible explanations for this. It has been argued that the gas in the lowest-mass halos will not be able to cool. Pre- vious to reionization gas in these halos can cool, but after reionization if the potential well is too small and the thermal pressure is too high then the gas cannot collapse into the potential well, and so the lowest mass halos will not accrete gas or form stars. Fur- thermore, because the gas is so weakly-bound to these low-mass halos it will be easy to remove through supernova-driven winds or ram-presure stripping (e.g., Bullock et al. 2000). It is also possible that there are many fewer low-mass galaxies because there are many fewer low-mass dark halos, either because of a trun- cated power spectrum, or because the dark matter has a velocity dispersion that suppresses small-scale structure (i.e., Warm Dark Matter).

13.4.3 The bulgeless galaxy problem

• This may be a rather more mundane problem, although it could be solved with the suppression of small-scale power in CDM (or the introduction of WDM). Hydrodynamical simulations of galaxy formation in a cosmological context are at a relatively early stage. Yet, all simulations agree (and indeed, simpler semi- analytic models that simply track the expected gas accretion his- tories of galaxies also predict) that there is predicted to be an essentially universal period of intense star formation and merging, in which a reasonably substantial stellar bulge would be created.

10 • Observationally, many disk galaxies with masses < 10 M lack a bulge component (e.g., M33); apparently, while our current galaxy formation models in a CDM context cannot avoid bulge formation, real galaxies manage to avoid bulge formation just fine. It is possible that this reflects a reasonably serious misun- derstanding of how gas gets into galaxies at redshifts higher than one; it is also possible that it signposts very efficient feedback in the earliest star formation events. Again, a reduction of small- scale power in the power spectrum or the introduction of WDM could potentially help. Top: Simulated edge-on galaxy stellar light images of a Milky-Way and dwarf galaxy from Governato et al. (2006). Bottom: M33. Chapter 14

Appendix

14.1 Cosmological parameters

Parameter WMAP WMAP and 2dFGRS +0.139 +0.127 100Ωb0 4.307−0.176 2.223−0.160 +0.0139 +0.0087 Ωm0 0.2446−0.0183 0.2434−0.0120 +0.035 +0.026 ΩΛ0 0.758−0.058 0.739−0.029 +0.050 +0.033 σ8 0.744−0.060 0.737−0.045 +0.015 +0.014 ns 0.951−0.019 0.948−0.018

Table 14.1: The main cosmological parameters as obtained from the WMAP three-year data alone and together with the galaxy power spectrum obtained from the 2dFGRS data, plus the Hubble constant as measured by the Hubble Key Project, h = 0.72 ± 0.08.

145 CHAPTER 14. APPENDIX 146

14.2 Cosmic time, lookback time and redshift

Figure 14.1: Cosmic time and lookback time as functions of redshift, plotted linearly (top left) and logarithmically (top right). The plot below shows the redshift as a function of cosmic time. CHAPTER 14. APPENDIX 147

14.3 Linear growth factor

Figure 14.2: The linear growth factor as a function of redshift, plotted linearly (left) and logarithmically (right). CHAPTER 14. APPENDIX 148

14.4 Distances

Figure 14.3: Linear plot of the angular-diameter distance as a function of redshift (top left), and plotted logarithmically together with the luminosity distance (top right). The plot below shows the angular-diameter and luminosity distances as functions of lookback time. CHAPTER 14. APPENDIX 149

14.5 Density and Hubble parameters

Figure 14.4: The density parameters Ωm and ΩΛ (left) and the Hubble parameter as functions of redshift (right). CHAPTER 14. APPENDIX 150

14.6 The CDM power spectrum

Figure 14.5: Linearly and nonlinearly evolved CDM power spectra today.