Observing the Big Bang
Matthias Bartelmann
Institut für Theoretische Astrophysik
Universität Heidelberg
Contents
1
The cosmological standard model
1
1.1
Friedmann models . . . . . . . . . . . . . . . . . . .
1
1.1.1
Introduction . . . . . . . . . . . . . . . . . . .
1
1.1.2
The metric . . . . . . . . . . . . . . . . . . .
1
1.1.3
Redshift and expansion . . . . . . . . . . . . .
3
1.1.4
Age, distances and horizons . . . . . . . . . .
4
1.1.5
The radiation-dominated phase . . . . . . . . .
5
Structures . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1
Structure growth . . . . . . . . . . . . . . . .
6
1.2.2
The power spectrum . . . . . . . . . . . . . .
7
1.2.3
Non-linear evolution . . . . . . . . . . . . . .
8
1.2
2
3
The age of the Universe
10
2.1
Nuclear cosmo-chronology . . . . . . . . . . . . . . .
10
2.1.1
The age of the Earth . . . . . . . . . . . . . .
10
2.1.2
The age of the Galaxy . . . . . . . . . . . . .
12
2.2
Stellar ages . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
Cooling of white dwarfs . . . . . . . . . . . . . . . .
16
2.4
Summary . . . . . . . . . . . . . . . . . . . . . . . .
17
The Hubble Constant
18
3.1
Hubble constant from Hubble’s law . . . . . . . . . .
18
3.1.1
Hubble’s law . . . . . . . . . . . . . . . . . .
18
3.1.2
The distance ladder . . . . . . . . . . . . . . .
19
i
CONTENTS
3.1.3
4
The HST Key Project . . . . . . . . . . . . . .
23
3.2
Gravitational Lensing . . . . . . . . . . . . . . . . . .
23
3.3
The Sunyaev-Zel’dovich effect . . . . . . . . . . . . .
25
3.4
Summary . . . . . . . . . . . . . . . . . . . . . . . .
27
Big-Bang Nucleosynthesis
28
4.1
The origin and abundance of Helium-4 . . . . . . . . .
28
4.1.1
Elementary considerations . . . . . . . . . . .
28
4.1.2
The Gamow criterion . . . . . . . . . . . . . .
30
4.1.3
Elements produced . . . . . . . . . . . . . . .
31
4.1.4
Deuterium fusion . . . . . . . . . . . . . . . .
32
4.1.5
Expected abundances and abundance trends . .
33
Observed element abundances . . . . . . . . . . . . .
34
4.2.1
Principles . . . . . . . . . . . . . . . . . . . .
34
4.2.2
Evolutionary corrections . . . . . . . . . . . .
35
4.2.3
Specific results . . . . . . . . . . . . . . . . .
35
4.2.4
Summary of results . . . . . . . . . . . . . . .
37
4.2
5
The Matter Density in the Universe
39
5.1
Mass in galaxies . . . . . . . . . . . . . . . . . . . . .
39
5.1.1
Stars
. . . . . . . . . . . . . . . . . . . . . .
39
5.1.2
Galaxies . . . . . . . . . . . . . . . . . . . . .
40
5.1.3
The galaxy population . . . . . . . . . . . . .
41
Mass in galaxy clusters . . . . . . . . . . . . . . . . .
42
5.2.1
Kinematic masses . . . . . . . . . . . . . . . .
42
5.2.2
Mass in the hot intracluster gas . . . . . . . . .
43
5.2.3
Alternative cluster mass estimates . . . . . . .
45
5.3
Mass density from cluster evolution . . . . . . . . . .
46
5.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . .
48
5.2
6
ii
The Cosmic Microwave Background
49
CONTENTS
6.1
6.2
7
The isotropic CMB . . . . . . . . . . . . . . . . . . .
49
6.1.1
Thermal history of the Universe . . . . . . . .
49
6.1.2
Mean properties of the CMB . . . . . . . . . .
51
6.1.3
Decoupling of the CMB . . . . . . . . . . . .
52
Structures in the CMB . . . . . . . . . . . . . . . . .
54
6.2.1
The dipole . . . . . . . . . . . . . . . . . . .
54
6.2.2
Expected amplitude of CMB fluctuations . . .
54
6.2.3
Expected CMB fluctuations . . . . . . . . . .
55
6.2.4
CMB polarisation . . . . . . . . . . . . . . . .
58
6.2.5
The CMB power spectrum . . . . . . . . . . .
58
6.2.6
Microwave foregrounds . . . . . . . . . . . .
59
6.2.7
Measurements of the CMB . . . . . . . . . . .
60
Cosmic Structures
67
7.1
Quantifying structures . . . . . . . . . . . . . . . . .
67
7.1.1
Introduction . . . . . . . . . . . . . . . . . . .
67
7.1.2
Power spectra and correlation functions . . . .
68
7.1.3
Measuring the correlation function . . . . . . .
69
7.1.4
Measuring the power spectrum . . . . . . . . .
70
7.1.5
Biasing . . . . . . . . . . . . . . . . . . . . .
73
7.1.6
Redshift-space distortions . . . . . . . . . . .
74
7.1.7
Baryonic acoustic oscillations . . . . . . . . .
75
Measurements and results . . . . . . . . . . . . . . . .
75
7.2.1
75
7.2
8
iii
The power spectrum . . . . . . . . . . . . . .
Cosmological Weak Lensing
79
8.1
Cosmological light deflection . . . . . . . . . . . . . .
79
8.1.1
Deflection angle, convergence and shear . . . .
79
8.1.2
Power spectra . . . . . . . . . . . . . . . . . .
82
8.1.3
Correlation functions . . . . . . . . . . . . . .
83
Cosmic-shear measurements . . . . . . . . . . . . . .
85
8.2
CONTENTS
9
iv
8.2.1
Typical scales and requirements . . . . . . . .
85
8.2.2
Ellipticity measurements . . . . . . . . . . . .
87
8.2.3
Results . . . . . . . . . . . . . . . . . . . . .
89
Supernovae of Type Ia
91
9.1
Standard candles and distances . . . . . . . . . . . . .
91
9.1.1
The principle . . . . . . . . . . . . . . . . . .
91
9.1.2
Requirements and degeneracies . . . . . . . .
92
Supernovae . . . . . . . . . . . . . . . . . . . . . . .
95
9.2.1
Types and classification . . . . . . . . . . . .
95
9.2.2
Observations . . . . . . . . . . . . . . . . . .
97
9.2.3
Potential problems . . . . . . . . . . . . . . . 100
9.2
10 The Normalisation of the Power Spectrum
102
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 102
10.2 Fluctuations in the CMB . . . . . . . . . . . . . . . . 103
10.2.1 The large-scale fluctuation amplitude . . . . . 103
10.2.2 Translation to σ8 . . . . . . . . . . . . . . . . 105
10.3 Cosmological weak lensing . . . . . . . . . . . . . . . 107
10.4 Galaxy clusters . . . . . . . . . . . . . . . . . . . . . 108
10.4.1 The mass function . . . . . . . . . . . . . . . 108
10.4.2 What is a cluster’s mass? . . . . . . . . . . . . 110
11 Inflation and Dark Energy
113
11.1 Cosmological inflation . . . . . . . . . . . . . . . . . 113
11.1.1 Motivation . . . . . . . . . . . . . . . . . . . 113
11.1.2 The idea of inflation . . . . . . . . . . . . . . 115
11.1.3 Slow roll, structure formation, and observational constraints . . . . . . . . . . . . . . . . 117
11.2 Dark energy . . . . . . . . . . . . . . . . . . . . . . . 119
11.2.1 Motivation . . . . . . . . . . . . . . . . . . . 119
11.2.2 Observational constraints? . . . . . . . . . . . 120
CONTENTS
12 Appendix
v
123
12.1 Cosmological parameters . . . . . . . . . . . . . . . . 123
12.2 Cosmic time, lookback time and redshift . . . . . . . . 124
12.3 Linear growth factor . . . . . . . . . . . . . . . . . . 125
12.4 Distances . . . . . . . . . . . . . . . . . . . . . . . . 126
12.5 Density and Hubble parameters . . . . . . . . . . . . . 127
12.6 The CDM power spectrum . . . . . . . . . . . . . . . 128
Chapter 1
The cosmological standard
model
1.1
1.1.1
Friedmann models
Introduction
• For a few years now, cosmology has a 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 cosmologies 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).
• Rather, after a brief summary of the cosmological model, the lecture will discuss what we know from observations about the evolution of the Universe and its contents, and what we conclude
about the origin and the future of the Universe and the structures
it contains.
1.1.2
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
1
CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
2
small scales. Cosmic magnetic fields can perhaps reach coherence 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.
• 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. Cosmological 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
2
2
2
2
2
2 2
2
ds = −c dt + a (t)
+ r dθ + sin θdφ
, (1.1)
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 fourdimensional 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
h
i
ds2 = −c2 dt2 + a2 (t) dw2 + fk2 (w) dθ2 + sin2 θdφ2 , (1.2)
CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
where the radial function fk (w) is given by



sin(w)
(k = 1)




fk (w) = 
w
(k = 0) ;




sinh(w) (k = −1)
3
(1.3)
sometimes one or the other form of the metric is more convenient;
1.1.3
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 λ and λ0 are the emitted
and observed wavelengths, respectively, they are given by
λ0 a0 1
=
= ,
λ
a
a
(1.4)
where a is the scale factor at the time of emission; the relative
wavelength change is the redshift,
z≡
λ0 − λ 1
= −1,
λ
a
and thus
1+z=
1
,
a
a=
1
;
1+z
(1.5)
(1.6)
• when inserted into Einstein’s field equations, two ordinary differential equations for the scale factor a(t) result; when combined,
they can be brought into the form
#
"
Ωr,0
1 − Ωm,0 − Ωr,0 − ΩΛ
2
2 Ωm,0
H = H0
+ 4 + ΩΛ +
a3
a
a2
≡ H 2 E 2 (a) ;
(1.7)
this is Friedmann’s equation, in which the relative expansion rate
ȧ/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
ρcr,0 ≡
3H02
;
8πG
(1.8)
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;
CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
4
• a Robertson-Walker metric whose scale factor satisfies Friedmann’s equation is called a Friedmann-Lemaı̂tre-RobertsonWalker 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 ;
• since the critical density evolves in time, so do the density parameters; their evolution is given by
Ωm (a) =
Ωm,0
a + Ωm,0 (1 − a) + ΩΛ,0 (a3 − a)
(1.9)
for the matter-density parameter and
ΩΛ (a) =
ΩΛ,0 a3
a + Ωm,0 (1 − a) + ΩΛ,0 (a3 − a)
(1.10)
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?
1.1.4
Age, distances and horizons
• since Friedmann’s equation gives the relative expansion rate ȧ/a,
we can use it to infer the age of the Universe,
Z 1
Z 1
Z t
Z 1
da
1
da
da
0
=
=
, (1.11)
t=
dt =
ȧ
H0 0 aE(a)
0 aH(a)
0
0
which illustrates that the age scale is the inverse Hubble constant
H0−1 ; a simple example is given by the Einstein-de Sitter model,
which (unrealistically, as we shall see later) assumes Ωm,0 = 1,
Ωr,0 = 0 and ΩΛ = 0; then, E(a) = a−3/2 and
Z 1
√
1
2
t=
ada =
;
(1.12)
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
Z
c
da
dDprop = cdt ⇒ Dprop =
,
(1.13)
H0
aE(a)
CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
5
where the integral has to be evaluated between the scale factors
of emission and observation of the light signal;
• 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
Z
Z
c
c
cdt
da
da
=
=
;
(1.14)
dDcom = dw =
2
a
H0
aȧ 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)] ,
(1.15)
where fK (w) is given by (1.3);
• the luminosity distance Dlum is analogously defined to reproduce
the Euclidean relation between the luminosity of an object and its
observed flux; this gives
Dlum (a) =
Dang (a)
fK [w(a)]
fK [Dcom (a)]
=
=
,
2
a
a
a
(1.16)
• 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
,
(1.17)
D(z) =
H0
which will turn out to be very important later;
• 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;
1.1.5
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 1. THE COSMOLOGICAL STANDARD MODEL
6
• equation (1.7) 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 acquired 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 ;
1.2
1.2.1
Structures
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 density ρ̄,
δ≡
ρ − ρ̄
;
ρ̄
(1.18)
• linear perturbation theory shows that the density contrast δ is described by the second-order differential equation
δ̈ + 2H δ̇ − 4πGρ̄δ = 0
(1.19)
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 (4πGρ̄)−1/2 ,
and a damping term 2H δ̇ which shows that the cosmic expansion
slows down the gravitational instability;
• equation (1.19) 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 1. THE COSMOLOGICAL STANDARD MODEL
7
formula
"
!
!#−1
5a
1
1
4/7
D+ (a) = Ωm Ωm − ΩΛ + 1 + Ωm 1 + ΩΛ
,
2
2
70
(1.20)
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
;
(1.21)
req =
p
H0 2Ωm,0
1.2.2
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. (1.19) 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 δ̂(~k) in Fourier space is called
the power spectrum
D
E
δ̂(~k)δ̂∗ (~k0 ) ≡ (2π)3 Pδ (k) δD (~k − ~k0 ) ,
(1.22)
CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
8
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 k0 )
k
Pδ (k) ∝ 
,
(1.23)

k−3 (k k0 )
where k0 = 2πreq is the wave number of the horizon size at the end
of radiation domination; the steep decline for structures smaller
than req reflects the suppression of structure growth during matter
domination;
1.2.3
Non-linear evolution
• as the density contrast approaches unity, its evolution becomes
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 nonlinear evolution proceeds, it is remarkable for two applications;
first, it allows a computation of the shapes of collapsing darkmatter 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;
• 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
The linear CDM power spectrum
with its characteristic shape (red),
and the deformation by non-linear
evolution at the small-scale end
(green).
CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
9
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 2
The age of the Universe
2.1
2.1.1
Nuclear cosmo-chronology
The age of the Earth
• How old is 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. As we shall see, this sets
interesting constraints on its expansion history.
• nuclear cosmo-chronology compares the measured abundance of
certain radioactive isotopes with their initial abundance, which is
eliminiated by comparing abundances in different probes;
• to give a specific example, consider the two uranium isotopes 235 U
and 238 U; they both decay into stable lead isotopes, 235 U → 207 Pb
through the actinium series and 238 U → 206 Pb 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 Ṅ = −λN, where N is the
number of decaying nuclei in a closed sample and λ is the decay
rate, integration gives
N(t) = N0 e−λt
10
(2.1)
CHAPTER 2. THE AGE OF THE UNIVERSE
11
for the remaining number of initially N0 radioactive nuclei, or
N̄ = N0 1 − e−λt = N(t) eλt − 1
(2.2)
for the number of nuclei of the stable decay product;
• thus, the present abundance of 207 Pb nuclei is its primordial abundance N207,0 plus the amount produced,
N207 = N207,0 + N235 eλ235 t − 1 ,
(2.3)
where N235 is the abundance of 235 U nuclei today; a similar equation with 235 replaced by 238 and 207 replaced by 206 holds for
the decay of 238 U to 206 Pb; the decay constants for the two uranium isotopes are measured as
λ235 = (1.015 Gyr)−1 ,
λ238 = (6.45 Gyr)−1 ;
(2.4)
• 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 fractionation 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;
• the unstable lead isotope 204 Pb has no long-lived parents and is
therefore a measure for the primordial lead abundance; thus, the
abundance ratios between 207 Pb and 208 Pb to 204 Pb calibrate the
abundances in different samples;
• suppose we have two independent samples a and b, in which the
abundance ratios
R206 ≡
N206
N204
and R207 ≡
N207
N204
(2.5)
are measured; according to (2.2), they are
N238
N204
N235
= R207,0 +
N204
R206 = R206,0 +
eλ238 te − 1 ,
R207
eλ235 te − 1 ;
(2.6)
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
Ra207 − Rb207 N235 eλ235 te − 1
=
;
(2.7)
Ra206 − Rb206 N238 eλ238 te − 1
CHAPTER 2. THE AGE OF THE UNIVERSE
12
once the lead abundance ratios have been measured in the two
samples, and the present uranium isotope ratio
N235
= 0.00725
N238
(2.8)
is known, the age of the Earth te is the only unknown in (2.7); this
method yields
te = 4.6 ± 0.1 Gyr ;
(2.9)
2.1.2
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; again, we can assume that the galaxy formed quickly compared to its age tg ;
• 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
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 235 U/238 U is, we can conclude from its present value (2.8) 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;
• the differential equation we have to solve now is
Ṅ = −λN + p ,
(2.10)
where p is the constant production rate; we solve it by variation
of constants, starting from the ansatz
N = C(t)e−λt
(2.11)
which solves (2.10) if
p λt
e +D
(2.12)
λ
with a constant D; thus, the abundance of a radioactive element
with decay constant λ is
p
N = De−λt +
(2.13)
λ
C=
CHAPTER 2. THE AGE OF THE UNIVERSE
before tg − te , and
N = N0 e−λ[t−(tg −te )]
13
(2.14)
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
(2.15)
N(0) = D + = f Np ,
λ
and thus
p λ(tg −te )
N(tg − te ) = e−λ(tg −te ) f Np +
e
−1
(2.16)
λ
when the Earth formed, and
p λ(tg −te )
N(tg ) = e−λtg f Np +
e
−1
(2.17)
λ
• the production rate must be
p=
(1 − f )Np
,
tg − te
which gives the present abundance
"
#
(1 − f ) λ(tg −te )
−λtg
N = Np e
f+
e
−1
λ(tg − te )
(2.18)
(2.19)
in terms of the produced abundance Np ;
• supernova theory says that the produced abundance ratio of the
isotopes 235 U and 238 U is
N235,p
= 1.4 ± 0.2 ;
N238,p
(2.20)
taking the ratio of (2.19) for the present abundances of 235 U and
238
U, inserting the decay constants from (2.4), the abundance ratios from (2.8) and (2.20), and the age of the Earth te from (2.9)
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




tg = 
(2.21)
8.0 ± 0.6 Gyr f = 0.5




12 ± 2 Gyr
f =0
• 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 ;
(2.22)
CHAPTER 2. THE AGE OF THE UNIVERSE
2.2
14
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 colour;
• 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
GMρ
dM
dP
=−
,
= 4πr2 ρ ,
(2.23)
dr
r
dr
which simply state hydrostatic equilibrium and mass conservation, and
3Lκρ
dL
dT
=
,
= 4πr2 ρ ,
(2.24)
dr
4πr2 acT 3
dr
which describe energy transport and production; κ is the opacity
of the stellar material, is the energy production rate per mass,
and
2π5 k4
erg
a=
= 5.67 × 10−5
(2.25)
3
2
15h c
cm2 K4
is the Stefan-Boltzmann constant;
• assuming κ is independent of temperature, the energy-transport
equation, mass conservation, hydrostatic equilibrium and the
equation-of-state for an ideal gas yield the scaling relations
L'
RT 4
,
ρ
M ' ρR3 ,
P'
ρM
,
R
P ' ρT ;
(2.26)
the second pair of equations gives T ' M/R, which yields L ' M 3
when inserted into the first pair;
• the total lifetime τ of a star on the main sequence must scale as
Lτ ' M because the total energy radiated, Lτ, must be a fraction
of the total rest-mass energy; together with the earlier result, we
find
M
' M −2 ' L−2/3 ;
(2.27)
τ'
L
• according to the Stefan-Boltzmann law, the star’s luminosity must
be
M3
L ' R2 T 4 ⇒ R2 ' 4 ,
(2.28)
T
but we also know from above that T ' M/R; thus
M 3 R4 R4
'
⇒ R ' M 1/2 , T ' M 1/2 ,
(2.29)
4
M
M
and the lifetime τ on the main sequence turns out to scale as τ '
T −1 ;
R2 '
CHAPTER 2. THE AGE OF THE UNIVERSE
15
• thus, 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 ) ' (τ−3/2 , τ−1 );
• old, coeval stellar populations exist, they are the globular clusters which surround the centre of the Galaxy in an approximately
spherical halo; therefore, the main-sequence turn-off points 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 diagrams of globular clusters and assigning the age of the best-fitting
stellar-evolution model to the cluster;
• two difficulties are typically met: first, the simulated stellarevolution tracks depend on the assumed metallicity of the stellar
material, which changes the opacity and thus the energy transport
through the stars; second, the light from the clusters is reddened
and attenuated by interstellar absorption;
• reddening 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 using
other well-defined features of the diagram like the red giant or
horizontal branches;
• since observations cannot tell the luminosity of the turn-off point
on the main sequence, but only its apparent brightness, age determinations from globular clusters require that the cluster distances
be known; there are several ways for estimating cluster distances;
one uses the period-luminosity relation of certain classes of variable stars, such as the Cepheids that will play a central role in the
next chapter; another 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;
• 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 improvements in stellar-evolution theory have lowered the globularcluster ages, while recently determined cosmological parameters
Colour-magnitude diagram of a
globular cluster. The turn-off point
in the main sequence is clearly visible, but not very well defined.
CHAPTER 2. THE AGE OF THE UNIVERSE
16
now yield a higher age for the Universe as assumed before; now,
globular-cluster ages imply
t & 13 Gyr
(2.30)
for the age of the Universe, allowing for a Gyr between the Big
Bang and the formation of the globular clusters;
2.3
Cooling of white dwarfs
• a final method for cosmic age determinations is based on the cooling of white dwarfs; they are the end products in the evolution
of low-mass stars; since stars with lower mass have longer lifetimes, there is a lower limit to the mass of white dwarfs set by
the stellar mass for which the lifetime equals the age of the Universe; similarly, white dwarfs cannot be more massive than the
Chandrasekhar mass limit of ≈ 1.4 M ; thus, the first generation
of white dwarfs corresponds to stars with Chandrasekhar mass
which cooled passively after their nuclear fuel was exhausted;
• suppose we study white dwarfs with increasing mass; white
dwarfs just above the lower mass limit have high luminosities,
but are rare because they are just now forming; white dwarfs with
higher masses are more abundant, but could form earlier and are
thus typically cooler and less luminous; a limit is reached with
white dwarfs of Chandrasekhar mass, which are typically the oldest, coolest, and therefore least luminous white dwarfs observable; this implies that the white-dwarf luminosity function should
have a low-luminosity cut-off related to the largest possible cooling times, and thus for the age of the Galaxy and the Universe;
• white dwarfs have a simple internal constitution because they are
stabilised by the pressure of a degenerate electron gas; the degeneracy implies that the mean free electron path is very large, and
therefore the cores are isothermal; moreover, their core pressure
does not depend on temperature;
• however, cooling models are difficult to construct because of the
complicated opacity of their atmospheres; a simple model based
on Kramer’s opacity
κ ∝ ρT −3.5
(2.31)
implies that the luminosity of a white dwarf scales with time as
L ' Mt−7/5 ,
(2.32)
which enables us to translate the luminosity cutoff to an age estimate;
Luminosity distribution of white
dwarfs.
CHAPTER 2. THE AGE OF THE UNIVERSE
17
• this methods yields
tg ≈ 9.3 ± 2 Gyr
(2.33)
for the age of the Galactic disk; if we assume that massive spiral disks form at redshifts below z . 3, the implied age of the
Universe is approximately
t0 ' 11 ± 1.4 Gyr ;
2.4
(2.34)
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
Friedmann-Lemaı̂tre 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 (1.12) that
t0 =
2
& 11 Gyr
3H0
⇒
H0 . 2 × 10−18 s ,
(2.35)
which reads
H0 . 61 km s−1 Mpc−1
(2.36)
in conventional units;
• as we shall see in the next chapter, the Hubble constant is measured to be larger than this, which can immediately be interpreted
as an indication that we are not living in an Einstein-de Sitter universe;
Constraints on the cosmic age
have meaningful implications
on the cosmological parameters, in particular on the cosmic
density parameter.
The three
curves for each cosmological
model are obtained assuming
H0 = (64, 72, 80) km s−1 Mpc−1 .
Chapter 3
The Hubble Constant
3.1
3.1.1
Hubble constant from Hubble’s law
Hubble’s law
• 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 ,
(3.1)
and determined the constant of proportionality as H0
570 km s−1 Mpc−1 ;
≈
• we had seen in (1.17) before that all distance measures in a
Friedmann-Lemaı̂tre universe follow the linear relation
D=
cz
H0
(3.2)
to first order in z 1; since cz = v is the velocity according to
the linearised relation for the Doppler shift,
1+z=
c+v
v
≈1+ ,
c−v
c
(3.3)
(3.2) is exactly the relation that Hubble found;
• there is little doubt that (3.1) is the result that Hubble wanted
to find because he wanted his measurements to support the
Friedmann-Lemaı̂tre cosmology; he even left out data points from
the analysis that did not support his conclusion;
• the problem with any measurement of the Hubble constant from
(3.1) is that, while (3.2) holds for an idealised, homogeneous and
isotropic universe, real galaxies have peculiar motions on top of
18
The relation between recession velocity and distance originally published by Hubble and Humason in
1931. Note the value of the Hubble
constant!
CHAPTER 3. THE HUBBLE CONSTANT
19
their Hubble velocity which are caused by the attraction 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 is attracted by the Local Group of
galaxies and approaches the Milky Way;
• 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; requiring that the typical peculiar velocities of order 300 . . . 600 km s−1
be less than 10% of the Hubble velocity, galaxies with redshifts
z & 10 ×
300 . . . 600 km s−1
≈ 0.01 . . . 0.02
c
(3.4)
must be observed; this is already so distant that it is hard or impossible to apply direct distant estimators;
• 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;
3.1.2
The distance ladder
• 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 stave
calibrates the next;
• the only direct distance measurement that can be applied here
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 10−5 00 are
thus necessary to measure distances of order 10 kpc;
• in this way, aided by the astrometric satellite Hipparcos, it was
possible to determine the distance to the Large Magellanic Cloud
as DLMC = (50.1 ± 3) kpc;
• the next important step in the distance ladder is formed by the
Cepheids; these are stars in late evolutionary stages which undergo periodic variability; the underlying instability is driven by
the temperature dependence of the atmospheric opacity in these
CHAPTER 3. THE HUBBLE CONSTANT
20
stars, which is caused by the transition between singly and dubly
ionised Helium;
• the cosmologically important aspect of the Cepheids is that their
variability period τ and their luminosity L are related,
L ∝ τ1.3 ,
(3.5)
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=
;
(3.6)
S
at the relevant distances, any distinction between differently defined distance measures is irrelevant;
• it is of crucial importance here that the calibration of the periodluminosity relation depends on the metallicity of the Cepheids,
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;
• measuring the periods of Cepheids 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
DM 31
= 15.28 ± 0.75 ,
(3.7)
DLMC
while Cepheids in the member galaxies of the Virgo cluster yield
DVirgo
= 316 ± 25 ;
DLMC
(3.8)
direct distance measurements to Virgo with Cepheids give
DVirgo
= 314 ± 16 ,
DLMC
(3.9)
in perfect agreement with the relative measurement;
• 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 . 30 Mpc;
Some examples
lightcurves
for
Cepheid
In some Cepheids, overtones of the
pulsation are excited rather than the
fundamental mode.
CHAPTER 3. THE HUBBLE CONSTANT
21
• 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
−0.85
Re ∝ σ1.4
;
v Ie
(3.10)
since the luminosity is evidently
L ∝ Ie R2e ,
(3.11)
the fundamental-plane relation implies
−0.7
L ∝ σ2.8
;
v Ie
(3.12)
such a relation follows directly from the virial theorem if the
mass-to-light ratio in elliptical galaxies increases gently with
mass,
M
∝ M 0.2 ;
(3.13)
L
• knowing that any elliptical galaxy can be expected to fall on the
fundamental plane, we still do not know where on the fundamental plane it will be; this can be found with the help of the empirical
Faber-Jackson relation,
L ∝ σαv ,
(3.14)
with the exponent α = 3 . . . 4 depending on the filter band of the
observation; thus, if it is possible to measure the effective surface
brightness Ie and the velocity dispersion σv of an elliptical galaxy,
the intersection between the fundamental plane and the FaberJackson relation gives the luminosity, which can be compared to
the flux to find the distance;
• a relation similar to (3.14), the Tully-Fisher relation, holds for
spiral galaxies if the velocity dispersion σv is replaced by the rotational velocity vrot ; 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;
• 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
√ Poisson 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;
CHAPTER 3. THE HUBBLE CONSTANT
22
• planetary nebulae, which are late stages in the evolution of massive stars, have a luminosity function with a steep upper cut-off;
moreover, their spectra are dominated by sharp nebular 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;
• an important class of distance indicators are supernovae of type
Ia; they occur in binary systems in which one of the components
is a white dwarf accreting 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 ;
(3.15)
when this correction is applied, type-Ia supernova are turned into
precise standard candles with a dispersion of only 6%;
• although they are not standard (or standardisable) candles, corecollapse 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 = aR2 T 4 ,
(3.16)
where a is again the Stefan-Boltzmann constant from (2.25); 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 Magellanic Cloud, the Baade-Wesselink method yields a distance of
DLMC = (44 . . . 50) kpc ,
which agrees with the direct distance measurements;
(3.17)
CHAPTER 3. THE HUBBLE CONSTANT
3.1.3
23
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 galaxies1 for testing and calibration;
• double-blind photometry was applied to the identified distance indicators; since Cepheids tend to lie in star-forming regions and are
thus attenuated by dust, and since their period-luminosity relation
depends on metallicity, respective corrections had to be carefully
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
H0 = (72 ± 8) km s−1 Mpc−1 ,
Hubble laws as measured by the
Hubble Key Project in different
wave bands (top to bottom) and in
different stages of correction (left to
right).
(3.18)
where the error is the square root of the systematic and statistical
errors summed in quadrature;
3.2
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
(3.19)
n=1− 2 ,
c
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 additionally 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,
2
1
(3.20)
τ = ~θ − ~β − ψ(~θ) ,
2
1
see http://www.ipac.caltech.edu/H0kp/H0KeyProj.html
Probability distributions for H0 obtained with different measurement
techniques applied in the Hubble
Key Project, and the combined distribution.
CHAPTER 3. THE HUBBLE CONSTANT
24
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,
~ θ τ = 0;
i.e. ∇
• the projected lensing potential ψ is related to the surface-mass
density Σ of the deflector by
~ 2 ψ = 2 Σ ≡ 2κ ,
∇
Σcr
(3.21)
where the critical surface-mass density
Σcr ≡
c2 Ds
4πG Dd Dds
(3.22)
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 (3.20) is related to the true
physical time delay t by
t∝
τ
,
H0
(3.23)
where the proportionality constant is a dimension-less combination of the distances Dd,s,ds with the Hubble radius cH0−1 and the
deflector redshift 1 + zd ; (3.23) shows that the true time delay is
proportional to the Hubble time, as it can intuitively be expected;
• 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) ,
(3.24)
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
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 3. THE HUBBLE CONSTANT
25
delays of the images; applying this technique to five different lens
systems2 , Kochanek (2002) found
H0 = (73 ± 8) km s−1
(3.25)
assuming that the lensing galaxies have radially constant massto-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;
3.3
The Sunyaev-Zel’dovich effect
• another method should finally be mentioned because it is physically 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
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 ,
√
√
(3.26)
jX ∝ ne ni T = CX n2e T ,
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 ,
(3.27)
where the integral extends along the line-of-sight; the energy flux
seen by the observer from this surface-area element is
R
dA dl jX
dE
dS =
=
,
(3.28)
4πD2lum
4πD2lum
these are: PG 1115 + 80, SBS 1520 + 530, B 1600 + 434, PKS 1830 − 211 and
HE 2149 − 2745
2
Values for the Hubble constant
obtained with alternative methods
(gravitational lensing, GL, and the
thermal Sunyaev-Zel’dovich effect,
SZ) not depending on the distance
ladder.
CHAPTER 3. THE HUBBLE CONSTANT
26
• by definition of the angular-diameter distance, the surface-area
element dA spans the solid angle element dΩ = dA/D2ang , so the
X-ray flux per unit solid angle, or the X-ray surface brightness, is
D2ang
dS
I=
=
dΩ 4πD2lum
Z
1
dl jX =
4π(1 + z)4
Z
dl jX ,
(3.29)
where we have used the remarkable Etherington relation between
the angular-diameter and luminosity distances,
Dlum = (1 + z)2 Dang ,
(3.30)
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
σT ne ,
(3.31)
y=
dl
me c 2
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
Z
Z
1
kT
1
dAdl y = 2
dV
σT ne , (3.32)
Y=
dΩ y = 2
Dang
Dang
me c2
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 (3.29) and the integrated Compton-y parameter (3.32), 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 distance, which is proportional to the Hubble length cH0−1 and thus
to the inverse Hubble constant;
CHAPTER 3. THE HUBBLE CONSTANT
27
• 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;
3.4
Summary
• if we accept the result of the Hubble Key Project for now,
H0 = (72 ± 8) km s−1 Mpc−1 ,
(3.33)
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
H0 = (2.3 ± 0.3) × 10−18 s ,
(3.34)
which implies the Hubble time, i.e. the inverse of the Hubble constant
1
= (13.6 ± 1.5) Gyr
(3.35)
H0
and the Hubble radius
c
= (1.3 ± 0.1) × 1028 cm = (4.1 ± 0.5) Gpc ;
H0
(3.36)
the critical density of the Universe turns out to be
ρcr =
3H02
= (9.65 ± 2.1) × 1030 g cm−3 ;
8πG
(3.37)
• the uncertainty in H0 is conventionally expressed in terms of
the dimension-less parameter h ≡ H0 /100 km s−1 Mpc−1 ; since
lengths in the Universe are typically measured with respect to
the Hubble length, they are often given in units of h−1 Mpc; similarly, luminosities are typically obtained by multiplying fluxes
with squared luminosity distances and are thus often given in
units of h−2 L ; we avoid this notation in the following and insert
h = 0.72 where needed;
Chapter 4
Big-Bang Nucleosynthesis
4.1
4.1.1
The origin and abundance of Helium-4
Elementary considerations
• 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 4 He nucleus, the amount of energy released corresponds to
0.7% of the masses involved, or
∆E = ∆mc2 = 0.007 (2mp + 2mn )c2 ≈ 0.028 mp c2
≈ 26 MeV ≈ 4.2 × 10−5 erg ;
(4.1)
• suppose a galaxy such as ours, the Milky Way, shines with a luminosity of L ≈ 1010 M ≈ 3.8 × 1043 erg s−1 for a good fraction
of the age of the Universe, say for τ = 5 × 109 yr ≈ 1.5 × 1017 s;
then, it releases a total energy of
Etot ≈ Lτ ≈ 5.7 × 1060 erg ;
(4.2)
• the number of 4 He nuclei required to produce this energy is
E
5.7 × 1060
≈ 1.4 × 1065 ,
∆N =
≈
−5
∆E 4.2 × 10
(4.3)
which amounts to a Helium-4 mass of
MHe ≈ 4mp ∆N ≈ 9.3 × 1041 g ;
28
(4.4)
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
29
• 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
Sun; then, the mass in hydrogen was MH ≈ 1010 M ≈ 2 × 1043 g
initially, and the final Helium-4 abundance by mass expected
from the energy production amounts to
9.3 × 1041
≈ 5% ,
(4.5)
2 × 1043
which is much less than the Helium-4 abundance actually observed; this discrepancy is exacerbated by the fact that 4 He is
destroyed in later stages of the evolution of massive stars;
Y∗ ≈
• we thus see that the amount of 4 He 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 4 He which is now observed
must have existed already before the galaxies formed;
• nuclear fusion of 4 He and similar light nuclei in the early Universe is possible only if the Universe was hot enough for a sufficiently 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 T 0 ∼ 3 K as we shall see later, this corresponds to times
when the scale factor of the Universe was
3
≈ 2.5 × 10−10 ;
(4.6)
anuc ∼
10
1.2 × 10
• 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
−2
expansion function can be simplified to read E(a) = Ω1/2
r,0 a , and
we find for the cosmic time according to (1.11)
Z a
a2
1
a0 da0 =
≈ 4.3 × 1019 a2 s , (4.7)
t(a) = 1/2
1/2
Ωr,0 H0 0
2Ωr,0 H0
where we have inserted the Hubble constant from (3.18) and the
radiation-density parameter today Ωr,0 ≈ 2.5 × 10−5 , which will
be justified later;
• inserting anuc from (4.6) into (4.7) 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;
• it is instructive for later purposes to establish a relation between
time and temperature based on (4.7); using T = T 0 /a, we substitute a = T 0 /T to obtain
T −2
T 2
0
t = 4.3 × 1019
s ≈ 1.6
s;
(4.8)
T
MeV
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
4.1.2
30
The Gamow criterion
• a crucially important step in the fusion of 4 He is the fusion of
deuterium 2 H or d,
p+n→d+γ
(4.9)
because the direct fusion of 4 He from two neutrons and two protons is extremely unlikely;
• if too much deuterium is produced, all neutrons are locked up in it
and no 4 He is produced, and if too little deuterium is produced, no
4
He is produced either 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
nB hσvit ≈ 1 ,
(4.10)
where np is the baryon number density, hσvi is the velocityaveraged cross section for the reaction (4.9), and t is the available
time for the fusion, which we have seen in (4.8) to be set by the
present temperature of the cosmic radiation background, T 0 , 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 T 0 of the
cosmic radiation background; already in the early 1940’s, Gamow
was able to predict T 0 ≈ 5 K!
• summarising, we have arrived at two remarkable arguments so
far; first, the observation that the 4 He 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 4 He 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 4. BIG-BANG NUCLEOSYNTHESIS
4.1.3
31
Elements produced
• the fusion of deuterium (4.9) is the crucial first step; since the photodissociation 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 (4.8) shows, this happens t ≈ 270 s after the Big
Bang;
• from there, Helium-3 and tritium (3 H or t) can be built, which
can both be converted to 4 He; these reactions are now fast, immediately converting the newly formed d; in detail, these reactions
are
d+p
d+d
d+d
3
He + n
→ 3 He + γ ,
→ 3 He + n ,
→ t + p , and
→ t+p,
(4.11)
followed by
3
He + d →
t+d →
He + p and
4
He + n ;
4
(4.12)
• fusion reactions with neutrons are irrelevant because free neutrons are immediately locked up in deuterons once deuterium fusion begins, and passed on to t, 3 He and 4 He in the further fusion
steps;
• since there are no stable elements with atomic weight A = 5, addition of protons to 4 He is unimportant; fusion with d is unimportant because its abundance is very low due to the efficient followup reactions; we can therefore proceed only by fusing 4 He with t
and 3 He to build up elements with A = 7,
t + 4 He →
3
He + 4 He →
7
Be + e− →
Li + γ ,
Be + γ ,
7
Li + νe ;
7
7
followed by
(4.13)
some 7 Li is destroyed by
7
Li + p → 2 4 He ;
(4.14)
the fusion of two 4 He nuclei leads to 8 Be, which is unstable; further fusion of 8 Be in the reaction
8
Be + 4 He → 12 C + γ
(4.15)
Nuclear fusion reactions responsible for primordial nucleosynthesis
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
32
is virtually impossible because the low density of the reaction
partners essentially excludes that a 8 Be nucleus meets a 4 He nucleus during its lifetime;
• thus, while the reaction (4.15) is possible and extremely important
in stars, it is suppressed below any importance in the early Universe; this shows that the absence of stable elements with A = 8
prohibits any primordial element fusion beyond 7 Li;
4.1.4
Deuterium fusion
• 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 + ν̄e
(4.16)
until the interaction rate falls below the expansion rate of the Universe;
• while equilibrium is maintained, the abundances nn and n p are
controlled by the Boltzmann factor
!3/2
Q
Q
mn
nn
=
exp −
≈ exp −
,
(4.17)
np
mp
kT
kT
where Q = 1.3 MeV is the energy equivalent of the mass difference 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
Q
nn mn
nn
≈
= 1 + exp
≈ 0.17 ;
Xn (0) ≡
nn mn + n p m p nn + n p
kT n
(4.18)
detailed calculations show that this value is kept until tn ≈ 20 s
after the Big Bang, when T n ≈ 3.3 × 109 K;
• afterwards, the free neutrons undergo β decay with a half life of
τn = 886.7 ± 1.9 s, thus
!
t − tn
Xn = Xn (0) exp −
≈ Xn (0)e−t/τn ;
(4.19)
τn
when d fusion finally sets in at td ≈ 270 s after the Big Bang, the
neutron abundance has dropped to
Xn (td ) ≈ Xn (0)e−td /τn ≈ 0.125 ;
(4.20)
Light-element abundances as a
function of cosmic time during primordial nucleosynthesis
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
33
now, essentially all these neutrons are collected into 4 He because
the abundances of the other elements can be neglected to first
order; this yields a 4 He abundance by mass of
Y ≈ 2Xn (td ) = 0.25
(4.21)
because the neutrons are locked up in pairs to form 4 He nuclei;
• the Big-Bang model thus allows the prediction that 4 He 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;
4.1.5
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,
η=
nB
= 1010 η10 ,
nγ
η10 ≡ 273ΩB h2 ;
(4.22)
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 at a fiducial η parameter of η10 = 5 are
0.4% for 4 He, 15% for d and 3 He, and 42% for 7 Li;
• the 4 He abundance depends only very weakly on the η because
the largest fraction of free neutrons is swept up into 4 He without
strong sensitivity to the detailed conditions;
• the principal effects determining the abundances of d, 3 He and
7
Li are the following: with increasing η, they can more easily be
burned to 4 He, and so their abundances drop as η increases; at low
Dependence of the 4 He, d, and 7 Li
abundances on the parameter η
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
34
η, an increase in the proton density causes 7 Li to be destroyed by
the reaction (4.14), while the precursor nucleus 7 Be is more easily
produced if the baryon density increases further; this creates a
characteristic “valley” of the predicted 7 Li abundance near η ≈
(2 . . . 3) × 10−10 ;
4.2
4.2.1
Observed element abundances
Principles
• of course, the main problem with any comparison between lightelement 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 can be observed in cool, neutral hydrogen gas (HI regions) via resonant UV absorption from the ground state, or in
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
3
He+ 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
4
He in stars is virtually impossible to correct precisely; rather,
4
He 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 7 Li abundance;
Deuterium signature in the wing of
a damped (saturated) hydrogen absorption line in a QSO spectrum
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
4.2.2
35
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;
• inferences from Helium-3 are different because 3 He is produced
from deuterium in stars during the pre-main sequence evolution;
it is burnt to 4 He during the later phases of stellar evolution in
stellar cores, but conserved in stellar exteriours; observations indicate that a net destruction of 3 He must happen, possibly due
to extra mixing in stellar interiours; for these uncertainties, 3 He
commonly excluded from primordial abundance measurements;
4.2.3
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 velocityshifted hydrogen lines, H abundances from saturated H lines need
to be corrected by comparison with higher-order lines, and highresolution spectroscopy is required for accurate continuum subtraction;
• at high redshift, a deuterium abundance of
nD
= 3.4 × 10−5
nH
(4.23)
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
36
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 (4.24) may be due to lack of spectral resolution;
the d abundance in the local interstellar medium is typically lower
nd
∼ (1 . . . 1.5) × 10−5
nH
(4.24)
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, 4 He 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 the oldest stars
in the Galaxy, which are halo (Pop-II) stars 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 7 Li may
have been produced by cosmic-ray spallation on the interstellar
medium;
7
• in the limit of low stellar metalicity, the observed Li abundance
turns towards the Spite plateau, which is asymptotically independent of metalicity,
A(7 Li) = 12 + log(nLi /nH ) = 2.2 ± 0.1 ,
(4.25)
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 7 Li abundance, but a corrections are probably moderate; a possible increase of 7 Li with the iron abundance indicates low production of
7
Li, 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(7 Li) ≤ 2.3 ;
(4.26)
The Spite plateau in the 7 Li abundance as a function of the metalicity
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
37
• in absence of depletion, this value falls into the valley expected in
the primordial 7 Li at the boundary between destruction by protons
and production from 8 Be; however, if 7 Li was in fact depleted, its
primordial abundance was higher than the value (4.26), and then
two values for η10 are possible;
4.2.4
Summary of results
• through the relation η10 = 273 ΩB h2 , 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;
• the 7 Li abundance predicted from this value of η is then A(7 Li)p =
2.1 . . . 2.8 which is fully consistent with the observed value
A(7 Li) = 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(7 Li) = 1.9 . . . 2.7 and
YP = 0.225 . . . 0.241, which are also compatible with the observations;
• we thus find that Big-Bang nucleosynthesis alone implies
ΩB h2 = 0.019 ± 0.0024
or ΩB = 0.037 ± 0.009
(4.27)
at 95% confidence level if conclusions are predominantly based
on the deuterium abundance in high-redshift absorption systems;
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
Predicted primordial element abundances as a function of η, overlaid with the measurements (boxes).
The η parameter compatible with all
measurements is marked by the vertical bar.
CHAPTER 4. BIG-BANG NUCLEOSYNTHESIS
38
a higher abundance of 4 He; in this way, the 4 He abundance was
found to limit the number of lepton families to three;
Chapter 5
The Matter Density in the
Universe
5.1
5.1.1
Mass in galaxies
Stars
• given the luminosity of a stellar population, what is its mass? if
all stars were like the Sun, the answer would be trivial, but this
is not the case; we shall focus the discussion here on stars which
fall on the main sequence of the colour-magnitude diagram;
• stars are formed with an initial mass distribution, called the “initial mass function”, which has the Salpeter form
dN
∝ M −1.35 ;
(5.1)
d ln M
expressing the mass M in solar units, m ≡ M/M , and normalising the mass distribution to unity in the mass range m0 ≤ m < ∞
yields
m 2.35 dm
dN
0
= 1.35
;
(5.2)
dm
m
m0
• the accepted lower mass limit for a star is m0 = 0.08 because nuclear fusion cannot set in objects of lower mass; however, we are
interested in stars that are able to produce visible or near-infrared
light so that we can translate the respective measured luminosities
into mass;
• for a simple estimate, assume that stars have approximate Planck
spectra, for which Wien’s law holds, relating the wavelength λmax
of the peak in the Planck curve to the temperature T ,
!
K
λmax = 0.2898
cm ;
(5.3)
T
39
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
40
to give an example, the effective solar temperature T = 5780 K
implies λmax = 5.0 × 10−5 cm;
• stars releasing the majority of their energy in the optical and nearinfrared regime should have λmax . 1µm = 10−4 cm and thus
T & 2900 K ≈ 0.5 T ; we saw in (2.29) that the temperature
scales as T ∼ M 1/2 ; thus, T & 0.5 T implies m0 ≈ 0.25;
• also, we saw following (2.26) that the luminosity scales as L ∼
M 3 ; with l ≡ L/L , the mean mass-to-light ratio of the visible
stellar population is thus expected to be
Z ∞
m Z ∞ dN m 1.35
dN dm
=
dm =
=
≈ 6.4 ; (5.4)
2
l
3.35 m20
m0 dm l
m0 dm m
this shows that an average stellar population visible in the optical
and near-infrared spectral ranges is expected to require ≈ 6.4 solar
masses for one solar luminosity; in order to produce, say, 1010 L ,
a galaxy thus needs to have a mass of at least ≈ 6.4×1010 M ; this
is a crude estimate, of course, but it illustrates the central aspect
of mass estimates from stellar luminosities;
5.1.2
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;
• in a spherically-symmetric mass distribution, test particles on circular orbits have orbital velocities of
v2rot (r) =
GM(r)
;
r
(5.5)
flat rotation curves thus imply that M(r) ∝ r; based on the continuity equation dM = 4πr2 ρdr, this requires that the density falls
off as ρ(r) ∝ r−2 ; 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;
• 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 density ρ̄;
After a quick rise, stellar velocities
in spiral galaxies remain approximately constant with radius. (The
galaxy shown is NGC 3198.)
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
41
• let R be the radius enclosing this overdensity, and M the mass
enclosed, then
3M
M
=
= 200ρ̄
V
4πR3
⇒
M 800πρ̄R2
=
;
R
3
at the same time, (5.5) needs to be satisfied, hence
!1/2
3v2rot
800πρ̄R2 v2rot
=
⇒ R=
;
3
G
800πGρ̄
inserting a typical numbers yields
vrot
;
R = 290 kpc
200 km s−1
(5.6)
(5.7)
(5.8)
with (5.5), this implies
M=
3
Rv2rot
vrot
;
= 2.7 × 1012 M
G
200 km s−1
(5.9)
• typical luminosities of spiral galaxies are given by the FaberJackson relation,
3...4
vrot
L = L∗
,
(5.10)
220 km s−1
with L∗ ≈ 2.4 × 1010 L ; thus, the mass-to-light ratio of a massive
spiral galaxy is found to be
m
≈ 150
l
(5.11)
in solar units, where it is assumed that the galaxy extends out to
the virial radius of ≈ 290 kpc with the same density profile r−2 ;
evidently, this exceeds the stellar mass-to-light ratio by far;
• evidently, the mass-to-light ratio of galaxies depends on the limiting radius assumed; values of m/l ≈ 30 are often quoted, which
are typically based on the outermost radius to which rotation
curves can be measured;
5.1.3
The galaxy population
• galaxy luminosities are observed to be distributed according to
the Schechter function,
!−α
!
dN Φ∗ L
L
=
exp −
,
(5.12)
dL
L∗ L∗
L∗
where the normalising factor is Φ∗ ≈ 3.7 × 10−3 Mpc−3 , the scale
luminosity is L∗ ≈ 2 × 1010 L and the power-law exponent is
α ≈ 1;
Far beyond the stars, flat rotation
curves are inferred from the motion
of neutral-hydrogen clouds (blue;
the galaxy shown is NGC 2915).
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
42
• irrespective of what physical processes this distribution originates
from, it turns out to characterise mixed galaxy populations very
well, even in galaxy clusters;
• the luminosity density in galaxies is easily found to be
Z ∞
Z ∞
dN
dL = Φ∗ L∗
l1−α e−l dl
Lg =
L
dL
0
0
L
= Γ(2 − α)Φ∗ L∗ ≈ Φ∗ L∗ ≈ 7.4 × 107
; (5.13)
Mpc3
the average mass-to-light ratio (5.11) then allows converting this
number into a mass density,
m
M
Lg ≈ 1.1 × 10−4
≈ 7.5 × 10−31 g cm−3 (5.14)
hρg i =
l
Mpc3
and thus, with ρcr = 9.65 × 1030 g cm−3 ,
Ωg ≈ 0.08 ;
(5.15)
of course, estimates based on the more conservative mass-to-light
ratio m/l ≈ 30 find values which are lower by a factor of ∼ 5; in
summary, this shows that the total mass expected to be contained
in the dark-matter halos hosting galaxies contributes of order 8%
to the critical density in the Universe;
5.2
5.2.1
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
themselves contain a total amount of mass
m
2
Mg . 10 L∗
≈ 3 × 1014 M ;
(5.16)
l
the mass in stars is of course considerably less; with the mean
stellar mass-to-light ratio of m/l ≈ 6.4 from (5.4), the same luminosity implies
M∗ . 1.3 × 1013 M ;
(5.17)
the stellar mass of the Coma cluster, for instance, is inferred to
be M∗,Coma ≈ 4.5 × 1013 M ;
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
43
• 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 ;
• if these galaxies were not gravitationally bound to the clusters,
the clusters would disperse within . 1 Gyr; since they exist over
cosmological time scales, clusters need to be (at least in some
sense) gravitationally stable;
• assuming virial equilibrium, the potential energy of the cluster
galaxies should be minus two times the kinetic energy, or
GM
≈ 3σ2v ,
R
(5.18)
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;
Galaxies move so fast in galaxy
clusters (here the Coma cluster) that
much more than the visible mass is
needed to keep them gravitationally
bound.
• we combine (5.18) with (5.6) to find
9σ2v
R=
800πGρ̄
!1/2
≈ 2.5 Mpc ,
(5.19)
and, with (5.18),
M ≈ 2 × 1015 M ;
(5.20)
hence, the mass required to keep cluster galaxies bound despite
their high velocities exceeds the mass in galaxies by about an
order of magnitude, even if the entire virial mass of the galactic
halos is accounted for; the stellar mass apparently contributes just
about one per cent to the total cluster mass;
5.2.2
Mass in 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
√
(5.21)
jX = Z 2 gff CX n2 T
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
CX = 2.68 × 10−24
in cgs units, if T is measured in keV;
(5.22)
Galaxy clusters are the most luminous emitters of diffuse X-ray radiation. The figure shows the X-ray
emission of the Coma cluster observed with the Rosat satellite.
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
44
• a common simple, axisymmetric model for the gas-density distribution in clusters is
r
n0
, x≡ ,
(5.23)
n(x) =
2
3β/2
(1 + x )
rc
where rc is the core radius;
• the line-of-sight projection of the X-ray emissivity yields the Xray surface brightness as a function of the projected radius ρ,
√
√
Z ∞
πΓ(3β − 1/2) Z 2 gff CX T n20
S X (ρ) =
jX dz =
, (5.24)
Γ(3β)
(1 + ρ2 )3β−1/2
−∞
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 (5.23), 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 ∞
√ 2 πΓ(3β − 3/2)
3
2
3 2
,
LX = 4πrc
jX x dx = 4πrc Z gff CX T n0
4Γ(3β)
0
(5.25)
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
n(x)x2 dx ,
(5.26)
MX (R) = 4πrc
0
which is a complicated function for general β; for β = 2/3, which
is a commonly measured value,
!
R
R
3
− arctan
,
(5.27)
MX (R) = 4πrc n0
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
hypothetic cluster with LX = 1045 erg s−1 , a temperature of kT =
10 keV and a core radius of rc = 250 kpc = 7.75 × 1023 cm;
• then, (5.25) yields the central ion density
n0 = 5 × 10−3 cm−3
(5.28)
and thus the central gas mass density
ρ0 = mp n0 = 8.5 × 10−27 g cm−3 ;
(5.29)
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
45
• based on the virial radius (5.19) and on the mass (5.27), we find
the total gas mass
MX = 1.0 × 1014 M ;
(5.30)
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
Ωdm,0 ≈ Ωb,0
M
≈ 10Ωb,0 ≈ 0.4 ;
M∗ + MX
(5.31)
more precise estimates based on detailed investigations of individual clusters yield
Ωdm,0 ≈ 0.3 ;
(5.32)
5.2.3
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
GM(r)
1 dp
=− 2 ,
ρ dr
r
(5.33)
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
mp ρ dr
r
(5.34)
or, solving for the mass
M(r) = −
rkT d ln ρ
;
Gmp d ln r
(5.35)
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
46
• for the β model introduced in (5.23), the logarithmic density slope
is
d ln ρ d ln n
r2
=
= −3β
,
(5.36)
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,
M(r) =
3βrkT r2
;
Gmp 1 + r2
(5.37)
• 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
M(R) ≈ 1.1 × 1015 M ,
(5.38)
in reassuring agreement with the mass estimate (5.20) from
galaxy kinematics;
• a third, completely independent way of measuring cluster masses
is provided by gravitational lensing; without going into any detail on the theory of light deflection, we mention here that it can
generate image distortions from which the projected lensing mass
distribution can be reconstructed; mass estimates obtained in this
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 mechanical 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;
5.3
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
≈ 9 × 108 Mpc3 ; of the 2712 clusters in the catalog, 818 fall into
(the poorest) richness class 0; excluding those, there are 1894
Strong gravitational lensing in
galaxy clusters can cause strong
distortions of background galaxies
into arcs (shown is the large arc in
the cluster Abell 370). They allow
independent cluster-mass estimates.
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
47
clusters with richness class ≥ 1 in that volume, which yields an
estimate for the spatial cluster density of
nc ≈ 2 × 10−6 Mpc−3 ;
(5.39)
• 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
!
δc
1
,
(5.40)
pc (z) = erfc √
2
2σ(z)
where
σ(z) = σ0 D+ (z)
(5.41)
because the linear growth of the density contrast is determined by
the growth factor, a fitting formula for which was given in (1.20);
• now, the present-day standard deviation σ0 must be chosen such
as to reproduce the observed number density of clusters given in
(5.39); the measured probability for finding a cluster is approximated by
Mnc
≈ 3 × 10−3 Ω−1
(5.42)
p0c =
m0 ;
ρc Ωm
the standard deviation σ in (5.40) must now be chosen such that
this number is reproduced, which yields



0.61 Ωm0 = 1.0
σ0 ≈ 
;
(5.43)

0.72 Ωm0 = 0.3
Cluster probability as a function of
σ for two different values of Ωm0 .
• equations (5.40) and (5.41) can now be used to estimate how the
cluster abundance should change with redshift; simple evaluation
reveals that the cluster abundance is expected to drop very rapidly
with increasing redshift if Ωm0 is high, and much more slowly if
Ωm0 is low;
Evolution of the cluster abundance,
depending on the density parameter
Ωm0 .
• qualitatively, this behaviour is easily understood; if, in a lowdensity universe, cluster do not form early, they cannot form at
all because the rapid expansion due to the low matter density prevents them from growing late in the cosmic evolution;
• 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 ,
(5.44)
in good agreement with the preceding determinations;
CHAPTER 5. THE MATTER DENSITY IN THE UNIVERSE
5.4
48
Conclusions
• What does it all mean? The preceding discussion should have
demonstrated that the matter density in the Universe is considerably less than its critical value, approximately one third of it.
Since only a small fraction of this matter is visible, we call the
invisible 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. (4.27). In the framework of the Friedmann-Lemaı̂tre 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 (1.6), the
scale factor at this time was a ≈ 10−10 , and thus the matter density was of order ρm ≈ 1030 ρcr ≈ 10 g cm−3 . The horizon size is
rH ≈ ct ≈ 1.8 × 1012 cm, thus the mass enclosed by the horizon
is ≈ 3 × 104 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
M
70
,
(5.45)
τbh ≈ 4 × 10 s
M
which is shorter than the Hubble time (3.35) if
M . 4 × 1015 g .
(5.46)
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
order 103...4 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 6
The Cosmic Microwave
Background
6.1
6.1.1
The isotropic CMB
Thermal history of the Universe
• How does the Universe evolve thermally? We have seen earlier
that the abundance of 4 He 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 absolutely 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
49
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
50
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
particles; thus, the collision time scale was τcoll ∝ a3 ;
• according to Friedmann’s equation, the expansion rate in the very
early Universe was determined by the radiation density, and thus
proportional to ∝ ȧ/a ∝ a−2 , and the expansion time scale was
τexp ∝ a2 ;
• equilibrium could be maintained as long as the collision time
scale was sufficiently shorter than the expansion time scale,
τcoll τexp ,
(6.1)
which is easily achieved in the early Universe when a 1; thus,
even though the expansion rate was very high in the early Universe, 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 6. THE COSMIC MICROWAVE BACKGROUND
6.1.2
51
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 4 He; 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 fitting the latest measurement of the CMB spectrum by the WMAP
satellite is
T 0 = 2.726 K ,
(6.2)
which implies a mean number density of CMB photons of
nCMB = 405 cm−3
(6.3)
and an energy density in the CMB of
uCMB = 4.2 × 10−13 erg cm−3 ,
(6.4)
which corresponds to a mass density of
ρCMB = 4.7 × 10−34 g cm−3 ;
(6.5)
• the density parameter of the CMB radiation is thus
Ωr0 = 4.8 × 10−5 ,
(6.6)
which shows that the scale factor at matter-radiation equality was
aeq =
Ωm0
≈ 6200 ;
Ωr0
(6.7)
• the number density of baryons in the Universe is approximately
nB ≈
ΩB0 ρcr
≈ 2.3 × 10−7 cm−3 ,
mp
(6.8)
confirming that the photon-to-baryon ratio is extremely high,
1
405
≈
≈ 1.8 × 109 ;
−7
η 2.3 × 10
(6.9)
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
6.1.3
52
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 neglect 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
(6.10)
can thermodynamically be described by minimising the free energy
 Np N NH 
 Zp Ze e ZH 
 ,
(6.11)
F = −kT ln Z = −kT ln 
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 grandcanonical 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,
∂Ne
(6.12)
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
me c 2
0.26 me c2
−χ/kT
= √
e
≈
e−χ/kT ,
1 − x 4 2ζ(3)η kT
η
kT
(6.13)
where χ is the ionisation energy of hydrogen, χ = 13.6 eV, and ζ
is the Riemann Zeta function;
• notice that Saha’s equation contains the inverse of the η parameter (6.9), which is a huge number due to the high photon-tobaryon ratio in the Universe; this counteracts the exponential
which would otherwise guarantee that recombination happens
Once recombination sets in, the
ionisation fraction x drops very
quickly.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
53
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; putting x ≈ 0.5 in (6.13) yields a recombination temperature of
kT rec ≈ 0.3 eV ,
T rec ≈ 3500 K
(6.14)
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
Z a
√
1
2a3/2
da0
0 da0 =
≈
a
t =
√
√
0
0
H0 Ωm0 0
3H0 Ωm0
0 a H(a )
≈ 360, 000 yr ;
(6.15)
• 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
∆T ≈
∆z ≈
− 1 ∆T ≈
≈ 75 ; (6.16)
z
dT zrec
dT T 0
T0
rec
or a time interval of
∆a
∆z
∆t ≈
≈
≈ 35, 000 yr ;
√
aH H0 Ωm0 (1 + z)5/2
(6.17)
• 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 fact that the temperature of the Universe dropped by ≈ 200 K
while the CMB was released leads to another remarkable realisation: How can the CMB have a Planck spectrum with a single
temperature if it was released from a plasma with a fairly broad
range of temperatures? In a Friedmann-Lemaı̂tre model universe, this is easy to understand: Photons released from highertemperature plasma were released somewhat earlier and were
subsequently redshifted by a somewhat larger amount. The range
of temperatures is thus precisely compensated by the redshift,
which confirms the expectation that T ∝ a−1 in FriedmannLemaı̂tre models. Thus, the fact that the CMB has a Planck spectrum with a single temperature indirectly confirms that we are
living in a Friedmann-Lemaı̂tre universe.
The FIRAS instrument on-board the
COBE satellite confirmed that the
CMB has the most perfect Planck
spectrum ever measured.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
6.2
6.2.1
54
Structures in the CMB
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⊕ ;
• Lorentz transformation shows that, to lowest order in v⊕ /c, the
Earth’s motion imprints a dipolar intensity pattern on the CMB
with an amplitude of
v⊕
∆T
=
;
(6.18)
T0
c
the dipole’s amplitude has been measured to be ≈ 1.24 mK, from
which the Earth’s velocity is inferred to be
v⊕ ≈ 371 km s−1 ;
(6.19)
• 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;
6.2.2
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
δ=
4T 3 δT
δρ
−1
−1≈
ρ
T4
⇒
δT
1+δ
≈
;
T
4
(6.20)
• 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
knowing the scale factor of recombination, we can thus infer that
relative temperature fluctuations of order
!
δT
1
1
1
≈
1+
≈
≈ 10−3
(6.21)
T
4
D+ (arec )
4arec
The Earth’s motion with respect to
the CMB rest frame imprints a dipolar temperature pattern on the CMB
with milli-Kelvin ampitude.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
55
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.
6.2.3
Expected CMB fluctuations
• before we come to the results of CMB observations and their significance 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;
• 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
1 δΦ
δT
=
,
(6.22)
T
3 c2
which is called the Sachs-Wolfe effect after the people who first
described it;
• let us briefly look into the expected statistics of the Sachs-Wolfe
effect; we introduced the power spectrum of the density fluctuations in (1.22) as the variance of the density contrast in Fourier
space; Fourier-transforming Poisson’s equation, we see that
δ̂
,
(6.23)
k2
and thus the power spectrum of the temperature fluctuations due
to the Sachs-Wolfe effect is determined by


hδ̂δ̂∗ i Pδ 
k−3 k k0
PT ∝ PΦ ∝ 4 ∝ 4 ∝ 
(6.24)

k−7 k k
k
k
δΦ̂ ∝ −
0
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
56
according to (1.23); 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;
• 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 baryonphoton 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
= √ ≈ 0.58 c ;
(6.25)
cs ≈
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
(6.26)
rs ≈ 360, 000 yr × √ ≈ 63 kpc ;
3
larger-scale density fluctuations could not oscillate;
• we saw in (1.15) that the angular-diameter distance from today to
scale factor a 1 is
Z
ca 1 dx
Dang (a) =
(6.27)
H0 a x2 E(x)
if we assume for simplicity that the universe is spatially flat; then,
the denominator in the integrand is
s
p
p
1 − Ωm0 3
x2 E(x) = Ωm0 x + (1 − Ωm0 )x4 = Ωm0 x 1 +
x
Ωm0
(6.28)
• inserting, we find that we can approximate the angular-diameter
distance for a 1 by
!
2carec
1 − Ωm0
Dang (arec ) ≈
1−
≈ 7.3 Mpc ,
(6.29)
√
6Ωm0
H0 Ωm0
and the sound horizon sets an angular scale of
θs =
2rs
≈ 1◦ ,
Dang (arec )
to which we shall shortly return;
(6.30)
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
57
• inserting the time directly from (6.15), the sound speed from
(6.25) and the angular-diameter distance from (6.29) reveals a
weak dependence of θs on Ωm0 even for a flat Universe,
!
2a1/2
1 − Ωm0
rec
;
(6.31)
θs ≈ √ 1 +
6Ωm0
3 3
• 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
λ≈
1
;
xnB σT
(6.32)
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 ;
(6.33)
the number of scatterings per unit time is
dN ≈ xnB σT cdt ,
and thus the diffusion scale is given by
Z
Z
cdt
2
2
λD ≈
λ dN ≈
;
xnB σT
(6.34)
(6.35)
• 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
λ2D ≈
2c∆t
;
nB σT
(6.36)
• around recombination, the baryon number density is
nB ≈
ΩB0 ρcr
≈ 500 cm−3 ,
−3
mp arec
(6.37)
and we find
λ ≈ 2 kpc
and λD ≈ 4.5 kpc
(6.38)
from Eqs. (6.32) and (6.36), respectively; λD thus corresponds to
an angular scale of θD ≈ 50 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;
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
6.2.4
58
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 polarised radiation must reflect the same physical effects as
the unpolarised radiation, and the two must be cross-correlated;
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;
6.2.5
The anisotropy of Thomson scattering causes the CMB to be partially
linearly polarised.
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
δT
alm =
d2 θ
Ylm (~θ) ,
(6.39)
T
and from them the power spectrum
Cl ≡
l
1 X
|alm |2 ,
2l + 1 m=−l
(6.40)
which is equivalent to the matter power spectrum (1.22) 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 Earth’s surface, and its
lowest-order multipoles: dipole,
quadrupole and octupole (left
column below the map), and the
multipoles with l = 4 . . . 7 (right
column).
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
59
• 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;
6.2.6
Microwave foregrounds
• 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;
The CMB power spectrum is characterised by three physical effects:
the Sachs-Wolfe effect, acoustic oscillations, and Silk damping.
• 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;
• 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
by the bulk motion of the cluster as a whole, which causes 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;
Galaxy clusters appear as characteristic point-like sources on the CMB.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
60
• diffuse CMB foregrounds are mainly caused by our Galaxy itself;
there are three main components: synchrotron emission, emission
from warm dust, and bremsstrahlung;
• synchrotron radiation is emitted by relativistic electrons in the
Galaxy’s magnetic field; it is highly linearly polarised and has
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
warmer than the CMB itself; it has a Planck spectrum which is
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 prominent; among them are line emission from CO molecules embedded in cool gas clouds;
• 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;
6.2.7
Measurements of the CMB
• Wien’s law (5.3) 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
∆θ ≈ 1.44
λ
D
(6.41)
relating the angular resolution ∆θ to the ratio between wavelength
and telescope diameter D, we find that mirrors with
D . 1.44
λmax
≈ 75 cm
θD
(6.42)
Relativistic electrons gyrating in the
Galaxy’s magnetic field emit synchrotron radiation.
Warm dust in the galaxy also adds
to the microwave foregrounds.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
61
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;
• 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 crosscorrelation 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 Microwave 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 three years of operation have been published, and cosmological parameters have been obtained fitting
theoretically expected to the measured temperature-fluctuation
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 6. THE COSMIC MICROWAVE BACKGROUND
62
power spectrum and the temperature-polarisation power spectrum; results are given in the following table:
CMB temperature
total density
matter density
baryon density
cosmological constant
decoupling redshift
age of the Universe
age at decoupling
power-spectrum normalisation
T CMB
Ωtot
Ωm0
Ωb0
ΩΛ0
zdec
t0
tdec
σ8
2.728 ± 0.004 K
1.01+0.02
−0.01
0.25+0.01
−0.03
0.045+0.001
−0.002
0.72 ± 0.04
1089 ± 1
13.7 ± 0.2 Gyr
379+8
−7 kyr
0.74+0.05
−0.06
• 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
spectrum to be H0 = 73 ± 3 km s−1 Mpc−1 , 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 2008; it will observe the
microwave sky in ten frequency bands between 30 and 857 GHz
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;
The COBE (top) and WMAP satellites.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
63
Figure 6.1: The European Planck satellite, to be launched in 2008 (left),
and the expected error bars on the temperature and polarisation power
spectra (right).
Figure 6.2: Left: Comparison between the WMAP temperature maps
obtained after one (top) and three years of measurement. Right: Decomposition of the WMAP 3-year temperature map into low-order multipoles.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
64
Figure 6.3: Power spectrum of CMB temperature fluctuations as measured from the 3-year data of WMAP and several additional groundbased experiments.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
65
Figure 6.4: Top: Polarisation map obtained by WMAP. Bottom: The
temperature-fluctuation power spectrum (top) and the temperaturepolarisation cross-power spectrum determined from the WMAP 3-year
data.
CHAPTER 6. THE COSMIC MICROWAVE BACKGROUND
66
Figure 6.5: 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 7
Cosmic Structures
7.1
7.1.1
Quantifying structures
Introduction
• We have seen before that there is a very specific prediction for
the power spectrum of density fluctuations in the Universe, characterised by (1.23). 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.
• Since the location k0 of the maximum in the power spectrum is determined by the horizon radius at matter-radiation equality (1.21),
√
√
2π 2 2πH0 Ωm0
,
(7.1)
=
k0 =
req
c
a3/2
eq
and the scale factor at equality is
aeq =
Ωr0
,
Ωm0
(7.2)
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 .
67
CHAPTER 7. COSMIC STRUCTURES
68
• 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.
7.1.2
Power spectra and correlation functions
• The definition (1.22) 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 (twopoint) correlation function
ξ(x) ≡ δ(~y)δ(~x + ~y) ,
(7.3)
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
d3 k ~ −i~k~x
δ̂(k)e
δ(~x) =
(2π)3
(7.4)
of the density contrast into (7.3), using the definition (1.22) of the
power spectrum and taking into account that the Fourier transform
δ̂ must obey δ̂(−~k) = δ̂∗ (~k) because δ is real, it is straightforward
to show that the correlation function ξ is the Fourier transform of
the power spectrum,
Z
d3 k
~
P(k)e−ik~x .
(7.5)
ξ(x) =
3
(2π)
• Assuming isotropy, the integral over all relative orientations between ~x and ~k can be carried out, yielding
Z ∞
1
sin kx 2
ξ(x) = 2
P(k)
k dk ,
(7.6)
2π 0
kx
whose inverse transform is
P(k) = 4π
Z
0
∞
ξ(x)
sin kx 2
x dx .
kx
(7.7)
This indicates one way to determine the power spectrum via measuring the correlation function ξ(x).
CHAPTER 7. COSMIC STRUCTURES
7.1.3
69
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
dP = hn(~x1 )n(~x2 )idV1 dV2 ,
(7.8)
where n is the number density of the galaxies as a function of
position.
• If we introduce a density contrast for the galaxies in analogy to
the density contrast for the matter,
n
(7.9)
δn ≡ − 1 ,
n̄
and assume for now that δn = δ, we find from (7.8) with n =
n̄(1 + δ)
The correlation function quantifies
the probability to find a galaxy in
the small volume dV2 if there is a
galaxy in the small volume dV1 , a
distance r = |~r2 − ~r1 | away.
dP = n̄2 h(1 + δ1 )(1 + δ2 )idV1 dV2 = n̄2 [1 + ξ(x)]dV1 dV2 , (7.10)
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.
• Thus, the correlation function can be measured by counting
galaxy pairs and comparing the result to the Poisson expectation,
i.e. to the pair counts expected in a random point distribution.
Symbolically,
hDDi
,
(7.11)
1 + ξ1 =
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
,
hDRi
hDDihRRi
=
,
hDRi2
h(D − R)2 i
,
= 1+
hRRi2
1 + ξ2 =
1 + ξ3
1 + ξ4
(7.12)
which are all equivalent in the ideal situation of an infinitely extended point distribution. For finite point sets, ξ3 and ξ4 are superiour to ξ1 and ξ2 due to their better noise properties.
Correlations between points can be
determined by counting pairs.
CHAPTER 7. COSMIC STRUCTURES
70
• 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
hξi = 0 ,
hξ2 i =
1
,
Np
(7.13)
where Np is the number of pairs found. Thus, the Poisson error
on the correlation function is
∆ξ
1
= p .
1+ξ
Np
(7.14)
• This is a lower limit to the actual error, however, because the
galaxies are in fact correlated. It turns out that the result (7.14)
should be multiplied with 1 + 4πn̄J3 , 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 (7.7) to find P(k), but
this is difficult in reality because of the inevitable sample limitations. Consider (7.6) 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 (7.6) 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 largescale correlation function is propagated to the power spectrum
even on small scales.
• A further problem is the uncertainty in the mean galaxy number
density n̄. Since 1 + ξ ∝ n̄−1 according to (7.10), the uncertainty
in ξ due to an uncertainty in n̄ is
∆n̄
∆ξ
≈ ∆ξ =
,
1+ξ
n̄
(7.15)
showing that ξ cannot be measured with an accuracy better than
the relative accuracy of the mean galaxy density.
7.1.4
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 7. COSMIC STRUCTURES
71
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,
n(~x) =
N
X
δD (~x − ~xi ) .
(7.16)
i=1
• The Fourier transform of the density contrast is then
N
1 X i~k~xi
~
e .
δ̂(k) =
N i=1
(7.17)
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
hδ̂(~k)δ̂∗ (~k)i =
N
1 X i~k~xi −i~k~xi
1
e e
.
=
2
N i=1
N
(7.18)
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,
P(k) =
1X ~ 2 1
|δ̂(k)| − ,
m
N
(7.19)
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) ,
(7.20)
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 δ̂(~k),
Z
b
ˆ
f δ = f ∗ δ̂ ≡
fˆ(~k0 )δ̂(~k0 − ~k)d3 k0 .
(7.21)
CHAPTER 7. COSMIC STRUCTURES
72
If the Fourier phases of fˆ and δ̂ 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,
Pobs = Ptrue ∗ | fˆ(~k)|2 .
(7.22)
• This convolution typically has two effects; first, it smoothes 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
R
.
(7.23)
P(k) → P(k) R
f 2 d3 x d3 x
• If the Poisson error dominates in the survey, the different modes
δ̂(~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
wi (k) =
1
,
1 + n̄i P(k)
(7.24)
where n̄i 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. (7.17) and (7.18) become
P i~k~xi
wi e
~
δ̂(k) = P
,
wi
w2i
~
h|δ̂(k)| i = P 2 .
( wi )
P
2
(7.25)
• 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 7. COSMIC STRUCTURES
73
density. Since, by definition, the mean of the density contrast δ0
within the survey vanishes, we must have
Z
0
δ =δ−
f (~x)δ(~x)d3 x ,
(7.26)
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 (7.22). 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
P0obs (k) = Pobs (k) − Pobs (0) ∗ | fˆ(~k)|2 .
7.1.5
(7.27)
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
b ~k) = b(k)δ̂(~k) ,
δn(
(7.28)
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.
Pgal = b2gal (k) P(k) .
(7.29)
It constitutes a major uncertainty in the determination of the matter power spectrum from the galaxy power spectrum.
CHAPTER 7. COSMIC STRUCTURES
7.1.6
74
Redshift-space distortions
• Of course, for the estimate (7.17) 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 ,
(7.30)
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,
vHubble + vpec
v
=
= Dtrue + ∆D ,
(7.31)
D=
H0
H0
giving rise to a distance error δD = vpec /H0 , the so-called redshiftspace distortion.
• Fortunately, the redshift-space distortions have a peculiar pattern through which they can be corrected. Consider a matter
overdensity such as a galaxy cluster, containing galaxies moving
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-ofsight, 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
2
Pz (k) = P(k) 1 + βµ2 ,
(7.32)
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
β≡
f (Ωm )
,
b
(7.33)
Peculiar velocities give rise to
redshift-space distortions, whose
characteristic shape constrains the
bias.
CHAPTER 7. COSMIC STRUCTURES
75
and f (Ωm ) is the logarithmic derivative of the growth factor
D+ (a),
d ln D+ (a)
f (Ωm ) ≡
≈ Ω0.6
(7.34)
m .
d ln a
• 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.
7.1.7
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 (6.26), rs ≈ 63 kpc. By now, this was
increased by the cosmic expansion to 1280 × 63 kpc ≈ 80.6 Mpc,
or k0 ≈ 0.078 Mpc−1 .
• This must be compared to the horizon size at matter-radiation
equality (1.21). With aeq ≈ 6200 from (6.7), we find req ≈
11.0 kpc, which was stretched by now to 6200 × 11.0 kpc ≈
68.3 Mpc, or ks ≈ 0.092 Mpc−1 .
• 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.
7.2
7.2.1
Measurements and results
The power spectrum
• Spectacularly successful measurements of the power spectrum
became recently possible with the two largest galaxy surveys to
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 apples with oranges, estimating the window function of the survey,
Top: The telescope dedicated to
the Sloan Digital Sky Survey. Bottom: The two-degree field camera
in the prime focus of the AngloAustralian Telescope.
CHAPTER 7. COSMIC STRUCTURES
76
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:
Ωm0
Ωb0 /Ωm0
0.233 ± 0.022
0.185 ± 0.046
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.
• Based on 205, 443 galaxies, the power spectrum inferred from the
SDSS also confirms the CDM shape. The estimate for the matter density parameter is somewhat higher than from the 2dFGRS,
Ωm0 = 0.30 ± 0.03, but both estimates already overlap within 3-σ
error bars.
Top: Geometry of the 2dFGRS survey volume. Middle: Galaxy distribution therein. Bottom: The area
covered by the 2dFGRS on the sky.
CHAPTER 7. COSMIC STRUCTURES
77
Figure 7.1: 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 7. COSMIC STRUCTURES
78
Figure 7.2: The galaxy power spectrum obtained from the SDSS (bottom), and the ratio between the power spectra of red and blue galaxies
(top).
Chapter 8
Cosmological Weak Lensing
8.1
8.1.1
Cosmological light deflection
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.
• Matter inhomogeneities deflect light. Working out this effect in
the limit that the Newtonian gravitational potential is small, Φ c2 leads to the deflection angle
Z w
2
fk (w − w0 ) ~
~
~ (θ) = 2
α
dw0
∇⊥ Φ[ fk (w0 )~θ] .
(8.1)
c 0
fk (w)
It is determined by the weighted integral over the gradient of the
Newtonian gravitational potential Φ perpendicular to the line-ofsight into direction θ on the observer’s sky, and the weight is given
by the comoving angular-diameter distance fk (w) defined in (1.3).
The integral extends along the comoving radial distance w0 along
the line-of-sight to the distance w of the source.
• Equation (8.1) can be intuitively understood. Light is deflected
due to the pull of the dimension-less Newtonian gravitational
~ ⊥ Φ/c2 perpendicular to the otherwise unperturbed line-offield ∇
sight, and the effect is weighted by the ratio between the angulardiameter distances from the deflecting potential to the source,
fk (w − w0 ), 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
79
Density inhomogeneities along the
way deflect light rays.
CHAPTER 8. COSMOLOGICAL WEAK LENSING
80
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,
Z w
2
fk (w − w0 ) fk (w0 ) ∂2 Φ
∂αi
= 2
dw0
[ fk (w0 )~θ] .
(8.2)
∂θ j c 0
fk (w)
∂xi ∂x j
The additional factor fk (w0 ) 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
fk (w − w0 ) fk (w0 )
(8.3)
W(w0 , w) ≡
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 (8.2) and its trace,
Z w
2
∂αi
∂2 Φ
= 2
tr
dw0 W(w0 , w) 2 [ fk (w0 )~θ] ,
(8.4)
∂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
(8.4). Thus, we find
Z w
2
∂αi
=
dw0 W(w0 , w) ∆Φ .
(8.5)
tr
∂θ j c2 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 deflection is caused by density perturbations, and that we need the
Laplacian in terms of comoving rather than physical coordinates.
Thus,
1
∆Φ = 4πGρ̄δ ,
(8.6)
a2
where δ is the density contrast and
ρ̄ = ρ̄0 a−3 = ρcr Ωm0 =
is the mean matter density.
3H02
Ωm0 a−3
8πG
(8.7)
CHAPTER 8. COSMOLOGICAL WEAK LENSING
81
• Thus, Poisson’s equation reads
δ
3 2
H0 Ωm0 ,
2
a
∆Φ =
(8.8)
and (8.5) becomes
∂αi 3H02 Ωm0
tr
=
∂θ j
c2
Z
w
dw0 W(w0 , w)
0
δ
≡ 2κ ,
a
(8.9)
where we have introduced the (effective) convergence κ.
• The trace-free part of the matrix (8.2) is
∂αi ∂αi
∂αi 1
− δi j tr
=
− δi j κ ≡
∂θ j 2
∂θ j ∂θ j
γ1 γ2
γ2 −γ1
!
,
(8.10)
which defines the so-called shear components γi . Specifically,
!
Z w
1
∂2 Φ ∂2 Φ
0
0
γ1 = 2
− 2 ,
dw W(w , w)
c 0
∂x12
∂x2
!
Z w
2
∂2 Φ
.
(8.11)
γ2 = 2
dw0 W(w0 , w)
c 0
∂x1 ∂x2
• Combining the results, we can write the matrix of deflectionangle derivatives as
!
∂αi
κ + γ1
γ2
=
.
(8.12)
γ2
κ − γ1
∂θ j
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 ,
(8.13)
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
a=
r
,
1−κ−γ
b=
r
,
1−κ+γ
(8.14)
where γ ≡ (γ12 + γ22 )1/2 . The ellipticity of the observed image of a
circular source thus provides an estimate for the shear,
≡
a−b
γ
=
≈γ.
a+b 1−κ
(8.15)
The gravitational tidal field (shear)
of large-scale structures distorts the
images of background galaxies (exaggerated).
CHAPTER 8. COSMOLOGICAL WEAK LENSING
8.1.2
82
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 densityperturbation 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
~
g(θ) =
dw0 q(w0 )δ[ fk (w0 )~θ] .
(8.16)
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
2
0
l
0 q (w )
,
(8.17)
Pg (l) =
dw 2 0 P
fk (w0 )
fk (w )
0
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 (8.9) is clearly of the form (8.16) with the weight function
H02 W(w0 , w)
3
0
q(w ) = Ωm0 2
,
(8.18)
2
c
a
thus the power spectrum of the convergence is, according to Limber’s equation,
!
Z w
9Ω2m0 H04
l
0
2
0
dw W̄ (w , w) P
Pκ (l) =
,
(8.19)
4 c4 0
fk (w0 )
with a new weight function
W̄(w0 , w) ≡
W(w0 , w)
.
a fk (w0 )
(8.20)
CHAPTER 8. COSMOLOGICAL WEAK LENSING
83
• 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) .
(8.21)
Thus, the statistics of the image distortions caused by cosmological weak lensing contains integral information on the power spectrum of the matter fluctuations.
• Since the shear is defined on the two-dimensional sphere (the observer’s sky), its power spectrum is related to its correlation function ξγ through the two-dimensional Fourier transform
Z ∞
Z
ldl
d2 l
~~l
iφ
Pγ (l)e =
Pγ (l)J0 (lφ) ,
(8.22)
ξγ (φ) =
2
(2π)
2π
0
where Jν is the ordinary Bessel function of order ν.
8.1.3
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 .
(8.23)
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
Pκ (l) J0 (lφ) + J4 (lφ)
ξ++ (φ) = hγ+ (θ)γ+ (θ + φ)i =
2 0 2π
(8.24)
The power spectrum of the weaklensing convergence κ for three different source redshifts.
CHAPTER 8. COSMOLOGICAL WEAK LENSING
84
and
1
ξ×× (φ) = hγ× (θ)γ× (θ + φ)i =
2
Z
∞
ldl
Pκ (l) J0 (lφ) − J4 (lφ) ,
2π
0
(8.25)
while the cross-correlation between the tangential and cross components must vanish,
ξ+× (φ) = 0 .
(8.26)
• This suggests to define the correlation functions ξ± = ξ++ ± ξ×× ,
which are related to the power spectrum through
Z ∞
ldl
Pκ (l)J0 (lφ) ,
ξ+ =
2π
0
Z ∞
ldl
ξ− =
Pκ (l)J4 (lφ) .
(8.27)
2π
0
The convergence (or shear) correlation function for three different
source redshifts.
• 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 2
dϑ ~
γ(ϑ) ,
(8.28)
γ̄(θ) ≡
2
0 πθ
which is related to the power spectrum by
"
#2
Z ∞
2J1 (lθ)
ldl
2
Pκ (l)
h|γ̄(θ)| i =
.
2π
lθ
0
(8.29)
• The principle of all these measures for cosmic shear is the same:
They are integrals of the weak-lensing power spectrum times socalled filter functions which describe the detailed response of the
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 weaklensing effect. The power ∆κ in the weak-lensing quantities such
as the cosmic shear is given by the power spectrum Pκ (l) found in
(8.19), times the volume in l-space,
∆κ (l) = l2 Pκ (l) .
(8.30)
• Assuming a cosmological model with Ωm0 = 0.3 and ΩΛ0 = 0.7,
the CDM power spectrum and a reasonable source redshift distribution, ∆κ (l)1/2 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.
The power of cosmological weak
lensing as a function of angular
scale.
CHAPTER 8. COSMOLOGICAL WEAK LENSING
8.2
Cosmic-shear measurements
8.2.1
Typical scales and requirements
85
• How can cosmic gravitational lensing effects be measured? As
shown in (8.15), 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
(8.31)
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 152 π = 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. Assuming their shapes are intrinsically independent, the intrinsic ellipticities will average out, and the shear will remain,
hi ≈ h (s) i + hγi ≈ hγi .
(8.32)
• 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 galax√
ies, 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) )2 i1/2 = √
(8.33)
N
CHAPTER 8. COSMOLOGICAL WEAK LENSING
86
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
Np =
1
2πn2 Aθδθ ,
2
(8.34)
and thus the Poisson noise due to the intrinsic ellipticities will be
noise ≈
2σ
,
√
n πAθδθ
(8.35)
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
ξ ≈ l2 Pκ (l)δ ln l ≈ l2 Pκ (l)
δθ
δl
≈ l2 Pκ (l) ,
l
θ
(8.36)
where we have used in the last step that θ = 2π/l.
• Thus, the signal-to-noise ratio turns out to be
p
√
√
S
ξ
lnδθ πAPκ n π3 APκ δθ
≈
≈
=
.
N noise
2σ
σ
θ
(8.37)
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 (8.37) numerically, we have to take into account that
l2 Pκ (l) must be a dimension-less number, which implies that the
power spectrum Pκ must have the dimension steradian. Therefore,
either the survey area A and the number density n in (8.37) 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.
The estimated signal-to-noise ratio
of weak-lensing measurements for
a hypothetical survey on an area of
one square degree.
CHAPTER 8. COSMOLOGICAL WEAK LENSING
87
• Matters are more complicated in reality, but the orders-ofmagnitude are well represented by this rough estimate. Bearing 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. Storing only one 4-byte number per pixel (i.e. the photon count), this
amounts to 4.6 × 1011 /240 = 0.8 TBytes.
8.2.2
Ellipticity measurements
• The determination of image ellipticities is straightforward in principle, but difficult in practice. Usually, the surface-brightness
quadrupole
R
I(~x)xi x j d2 x
(8.38)
Qi j = R
I(~x)d2 x
The sobering appearance of real
data.
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 (8.38) can be
found.
• Even if the surface-brightness quadrupole of the image on the
detector can be accurately determined, the image appears affected
by imperfections of the telescope optics and by the turbulence in
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
∆θ ≈ 1.44
λ
D
(8.39)
The compatibility of the lower data
points signals the almost complete
absence of systematic effects in the
data show above.
CHAPTER 8. COSMOLOGICAL WEAK LENSING
88
as mentioned in (6.41) 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
is slightly astigmatic, and this anisotropy may, and will in general, 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
mosaic are not exactly in the same plane, or if the telescope is
slightly out of focus, systematic image deformations may result
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 polynomial 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 gravitational 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.
The point-spread function of the
Canada-France-Hawaii telescope.
Illustration of systematic image distortions in the CFHTLS and their
correction.
E- and B-mode distortion patterns.
CHAPTER 8. COSMOLOGICAL WEAK LENSING
8.2.3
89
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
below an arc minute to several degrees. The best correlation functions could be shown to be at most negligibly contaminated by
B-modes.
The first published measurements
of the cosmic-shear correlation
function.
• 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 (8.19) shows, the weak-lensing power spectrum Pκ (l) depends
on the product of a factor Ω2m0 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 W̄(w0 , w), but this is not very important. By and large, therefore, the cosmic-shear correlation function measures the product AΩ2m0 , 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
spectrum A is conventionally described by a parameter σ28 which
will be defined and described in more detail later. Weak lensing
thus measures the product σ8 Ωm0 , and current measurements find
σ8 Ωm0 ≈ 0.2.
• Weak gravitational lensing is a fairly new field of cosmological
research. Within a few years, it has considerably matured and returned cosmologically interesting constraints. Considerable potential is expected from weak lensing in wide-area surveys in
particular when combined with photometric redshift information.
We shall return to this issue later.
The CFHT dome (top) and the
Mega-Prime Camera in its prime focus (bottom).
CHAPTER 8. COSMOLOGICAL WEAK LENSING
90
Figure 8.1: Recent constraints in the Ωm0 − σ8 plane obtained from
weak-lensing measurements. The Universe is assumed spatially flat
here.
Chapter 9
Supernovae of Type Ia
9.1
9.1.1
Standard candles and distances
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 (1.16), the distance can be inferred from the measured
flux, f , through
s
Dlum =
L
.
4π f
(9.1)
• Besides the redshift z, the luminosity distance will depend on the
cosmological parameters,
Dlum = Dlum (z; Ωm0 , ΩΛ0 , H0 , . . .) ,
(9.2)
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 (1.17), all distance measures tend to
D≈
cz
H0
(9.3)
at low redshift and lose their sensitivity to all cosmological parameters except H0 .
91
CHAPTER 9. SUPERNOVAE OF TYPE IA
92
• 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 (9.3) 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,
czi
L = 4π fi
H0
!2
.
(9.4)
Since all cosmological distance measures are proportional to the
Hubble length c/H0 , the dependences on H0 on both sides of (9.1)
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 (9.3) 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 (9.1) 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.
9.1.2
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 9. SUPERNOVAE OF TYPE IA
93
• 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,
∆Dlum ≈
dDlum
∆Ωm0 ,
dΩm0
(9.5)
about a fiducial model, such as a ΛCDM model with Ωm0 = 0.3.
• At a fiducial redshift of z ≈ 0.8, we find numerically
d ln Dlum
≈ −0.5 ,
dΩm0
(9.6)
which shows that a relative distance accuracy of
∆Dlum
≈ −0.5∆Ωm0
Dlum
(9.7)
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. Assuming Poisson statistics for simplicity and distance measurements to N supernovae, the combined accuracy is
2 ∆Dlum
|∆Ωm0 | ≈ √
.
N Dlum
(9.8)
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 (9.1) yields the distance uncertainty
1/2

!
!2

 dDlum 2 2
dDlum
2
∆L +
∆ f  ,
(9.9)
Dlum = 
dL
df
or the relative uncertainty

!2
!2 1/2
∆Dlum 1  ∆L
∆ f 
 .
= 
+
Dlum
2 L
f
(9.10)
Even if the flux could be measured precisely, the intrinsic luminosity scatter currently forbids distance determinations to better
than 10%.
Logarithmic derivative of the luminosity distance with respect to Ωm0 .
CHAPTER 9. SUPERNOVAE OF TYPE IA
94
• 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
photon fluxes of fγ ≈ 85 s−1 at peak and fγ ≈ 7 s−1 35 days afterwards. These fluxes are typically distributed over a few CCD
pixels.
• 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
√
N
N
S
≈ √ =
,
(9.11)
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 certainly 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
∂Dlum ∂Dlum
~g ≡
,
,
(9.12)
∂Ωm0 ∂ΩΛ0
at a fiducial ΛCDM model with Ωm0 = 0.3. When normalised to
The luminosity distance in a universe with Ωm0 = 0.3 and ΩΛ0 =
0.7 with Hubble constant h = 0.72.
CHAPTER 9. SUPERNOVAE OF TYPE IA
unit length, it turns out to point into the direction
!
−0.76
~g =
.
0.65
95
(9.13)
• 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
(9.14)
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.
9.2
9.2.1
Supernovae
Types and classification
• Supernovae are “eruptively variable” stars. A sudden rise in
brightness is followed by a gentle decline. They are unique events
which at peak brightness reach luminosities comparable to those
of an entire galaxy, or 1010 . . . 1011 L . They reach their maxima
within days and fade within several months.
Supernova 1994d in its host galaxy.
• 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, unlike type-Ib/c supernovae, which are distinguished by the prominence 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.
• Except for type-Ia, supernovae arise due to the collapse of a massive stellar core, followed by a thermonuclear explosion which
disrupts the star by driving a shock wave through it. Corecollapse 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.
Lightcurves of supernovae of different types.
CHAPTER 9. SUPERNOVAE OF TYPE IA
96
• 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 Chandrasekhar 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 56 Ni, which
later decays through 56 Co into the stable 56 Fe. 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
of its temperature. The mass accreted from the companion star increases the pressure. Once it exceeds the Fermi pressure, inverse
beta decay sets in,
p + e− → n + νe + γ ,
Early (top) and late spectra of different supernova types.
(9.15)
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 unnoticed because they can leave the supernova essentially without
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 56 Ni. 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.
Typically, ∼ 0.5 M of 56 Ni is produced in theoretical models.
• The peak brightness is reached when the deflagration front
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
Supernova classification.
Type-Ia supernovae occur when
white dwarfs are driven over the
Chandrasekhar mass limit by mass
flowing from a companion star.
CHAPTER 9. SUPERNOVAE OF TYPE IA
97
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 56 Co into 56 Fe 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.
9.2.2
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 pointspread 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 typeIa 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 9. SUPERNOVAE OF TYPE IA
98
• 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 56 Ni 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)
their use as standardisable candles for cosmology.
• The remaining relative uncertainty is now typically between
10 . . . 15% for individual supernovae. Since, as we have seen following (9.7), 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 (9.8).
• 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 (lunations).
• 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-
Lightcurves of type-Ia supernovae
before (top) and after correction.
CHAPTER 9. SUPERNOVAE OF TYPE IA
99
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, socalled “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.
• After applying the corrections for absorption and duration, each
supernova yields an estimate for the luminosity distance to its redshift. Together, the supernovae in the observed sample constrain
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.
Distances to type-Ia supernovae (in
logarithmic units) as a function of
their redshift, as measured by the
Supernova Legacy Survey.
• 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 .
(9.16)
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
parameter constraints from supernovae differ from those from the
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
Cosmological parameter constraints
derived from the same data.
CHAPTER 9. SUPERNOVAE OF TYPE IA
100
constant is much more important. As (9.14) shows, the luminosity distance constrains the difference between the two parameters,
ΩΛ0 = 1.17 Ωm0 + P ,
(9.17)
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
(9.18)
from the SNLS first-year result (9.16), illustrating that the survey
has constrained the density parameters to follow the relation
ΩΛ0 ≈ 1.17 Ωm0 + 0.43 .
(9.19)
• The relative acceleration of the universe, ä/a, is given by the
equation
!
Ωm0
ä
2
= H0 ΩΛ0 − 3
(9.20)
a
2a
if matter is pressure-less, which follows directly from Einstein’s
field equations. Thus, the expansion of the universe accelerates
today (a = 1) if ä = H02 (ΩΛ0 − Ωm0 /2) > 0, or ΩΛ0 > Ωm0 /2.
Given the measurement (9.19), the conclusion seems inevitable
that the Universe’s expansion does indeed accelerate today.
• If the Universe is indeed spatially flat, then the transition between
decelerated and accelerated expansion happened at
1 − 0.263 ≈
0.263
2a3
⇒
a = 0.56 ,
(9.21)
or at redshift z ≈ 0.78. Luminosity distances to supernovae at
larger redshifts should show this transition, and in fact they do.
9.2.3
Potential problems
• 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. Depending on the line-of-sight, they are either magnified or demagnified. Since, due to nonlinear structures, high magnifications
Above redshift z ≈ 1, the cosmic acceleration seems to turn into deceleration.
CHAPTER 9. SUPERNOVAE OF TYPE IA
101
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
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. Moreover, recent supernova simulations reveal good physical reasons
why they should in fact be reliable, standardisable candles.
Chapter 10
The Normalisation of the Power
Spectrum
10.1
Introduction
• We saw in Chapter 7 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 darkmatter 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−1 Mpc 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
102
CHAPTER 10. THE NORMALISATION OF THE POWER SPECTRUM103
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 ∞ 3
dk
2
Pδ (k)WR2 (k) ,
(10.1)
σR =
3
0 (2π)
where WR (k) is a window function selecting the k modes contributing 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
−k .
(10.2)
WR (k) = Θ(kR − k) = Θ
R
This is a step function dropping to zero for k > 2π/R.
• Inserting this into (10.1), we find
Z 2π/R 2
k dk
2
σR =
Pδ (k) .
2π2
0
(10.3)
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 .
10.2
Fluctuations in the CMB
10.2.1
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
Φ
δT
≡τ= 2
T
3c
(10.4)
in terms of the Newtonian potential fluctuations Φ; see also
Eq. (6.22).
CHAPTER 10. THE NORMALISATION OF THE POWER SPECTRUM104
• The three-dimensional temperature-fluctuation power spectrum is
then
1
(10.5)
Pτ (k) = 4 PΦ (k) .
9c
The Poisson equation in its form (8.8) implies that the power
spectra of potential- and density fluctuations are related through
!
9H04 2 D+ (a) 2 Pδ (k)
PΦ (k) =
Ω
,
(10.6)
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 threedimensional 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 (6.29) that
D≈
1
2ca
∝
√
√
H0 Ωm0 H0 Ωm0
(10.7)
to first order in Ωm0 . Thus, the angular wave number under which
the mode appears is
2π
≈ Dk .
(10.8)
l≈
θ
• Expressing now the power spectrum (10.5) in terms of the angular wave number l yields
!2
!
H 4
D+ (a) 1 D4
l
0
2
Ωm0
Pτ (l) ∝
Pδ
,
(10.9)
c
a
D2 l 4
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,



(k < k0 )
kn
Pδ (k) = A 
.
(10.10)

n−4
k
else
Inserting its long-wave limit, Pδ (k) = Akn , into (10.9) yields
!2
H 4
D+ (a) 1 D 4−n
0
2
Ωm0
.
(10.11)
Pτ (l) ∝ A
c
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 angulardiameter distance, the growth factor and the power-spectrum exponent n.
CHAPTER 10. THE NORMALISATION OF THE POWER SPECTRUM105
• 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
−1−n/2 −2−n D+ (a)
A ∝ Ωm0 h
Pτ (l) .
(10.12)
a
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.
10.2.2
Translation to σ8
• Regarding σ8 , we are not done yet. Inserting the model power
spectrum (10.10) into the definition (10.3) gives


 kn−1 kn−1

 kn+3 
0
8

A
−
(n
,
1)

0
 ,
n−1
n−1
+
(10.13)
σ28 = 2 

ln k8
2π n + 3 
(n
=
1)
k0
where k8 = 2π/(8 h−1 Mpc).
• Since n ≈ 1, the second term is close to logarithmic and thus
weakly dependent on the cosmological parameters in k0 . Then,
we see by combining (10.13) with (10.12) that
D+ (a)
.
(10.14)
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.
2+n/2
σ8 ∝ Ω1+n/4
m0 h
• 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.
The translation of the CMB temperature fluctuations depends on cosmological parameters, e.g. on Ωm0 .
CHAPTER 10. THE NORMALISATION OF THE POWER SPECTRUM106
• The optical depth for Thomson scattering is
Z
τ=
dx ne σT ,
(10.15)
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 Thomson 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 fluctuation 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 largescale 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 10. THE NORMALISATION OF THE POWER SPECTRUM107
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.
10.3
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 (8.19), the
cosmic-shear power spectrum is proportional to Ω2m0 times the
amplitude A of the dark-matter power spectrum, which leads to
the approximate degeneracy Ωm0 σ8 ≈ 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 W̄(w0 , w) shown in (8.20), 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 analytically, non-linear growth can only be quantified by means of large
numerical simulations and recipes derived from them. Insufficient 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 10. THE NORMALISATION OF THE POWER SPECTRUM108
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 matterdensity parameter, is perhaps the most promising method for
precisely determining σ8 . Table 10.1 lists values of σ8 derived
from some cosmic-shear measurements under the assumption of
Ωm0 = 0.3 in a spatially-flat universe.
σ8
0.86+0.09
−0.13
0.71+0.12
−0.16
0.72 ± 0.09
0.97 ± 0.13
1.02 ± 0.16
0.83 ± 0.07
0.68 ± 0.13
0.85 ± 0.06
data
RCS
CTIO
Combo-17
Keck-II
HST/STIS
Virmos-Descart
GEMS
CFHTLS
reference
Hoekstra et al. 2002
Jarvis et al. 2003
Brown et al. 2003
Bacon et al. 2003
Rhodes et al. 2004
van Waerbeke et al. 2005
Heymans et al. 2005
Hoekstra et al. 2006
Table 10.1: Values for σ8 derived from cosmic-shear measurements under the assumption of a spatially-flat universe with Ωm0 = 0.3.
10.4
Galaxy clusters
10.4.1
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
density ρ̄ into the game.
CHAPTER 10. THE NORMALISATION OF THE POWER SPECTRUM109
• The standard deviation σR is related to the power spectrum. For
convenience, we introduce an effective slope
n=
d ln P(k)
d ln k
(10.16)
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
∗
,
(10.17)
σR =
M
with
1
n
α≡
1+
.
(10.18)
2
3
• In terms of the dimensionless mass m ≡ M/M∗ , the PressSchechter mass function can then be written in the form
r
!
δ2c
2 ρ̄δc
2α
α−2
N(m, a)dm =
αm exp − 2 m dm .
π M∗2 D+ (a)
2D+ (a)
(10.19)
• 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
!
δ2c
−5/3
2/3
N(m, a)dm ∝ m
exp − 2 m
dm ,
(10.20)
2D+ (a)
with an amplitude characterised by M∗ , the mean dark-matter
density ρ̄, and the growth factor D+ (a).
• This opens a way to constrain cosmological parameters as well
as σ8 with galaxy clusters: if the abundance and evolution of the
cluster mass function can be measured, they can be determined
from the mass scale of the exponential cut-off and the amplitude
of the power-law end. Today, the non-linear mass scale is a few
times 1013 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.
The X-ray flux (top) or luminosity
functions of galaxy clusters can be
converted to a mass function if it is
possible to measure cluster masses
sufficiently accurately.
CHAPTER 10. THE NORMALISATION OF THE POWER SPECTRUM110
10.4.2
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 (10.20),
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 immdiately 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 (5.37)
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. Being 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 10. THE NORMALISATION OF THE POWER SPECTRUM111
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 T X or its X-ray luminosity LX should be related
to the depth of the gravitational-potential well they are embedded 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
1.02 ± 0.07
0.77 ± 0.07
0.75 ± 0.16
0.79+0.06
−0.07
0.77+0.05
−0.04
0.69 ± 0.03
0.78 ± 0.17
0.67+0.04
−0.05
data
M-T relation
M-T relation
lensing masses
luminosity function
temperature function
lensing masses
optical richness
lensing masses
reference
Pierpaoli et al. 2001
Seljak 2002
Smith et al. 2003
Pierpaoli et al. 2003
Pierpaoli et al. 2003
Allen et al. 2003
Eke et al. 2006
Dahle 2006
Table 10.2: Values of σ8 derived from the galaxy-cluster population
based on different observational data.
• 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
Several recent determinations of σ8 .
CHAPTER 10. THE NORMALISATION OF THE POWER SPECTRUM112
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 therefrom are given in Tab. 10.2.
Chapter 11
Inflation and Dark Energy
11.1
Cosmological inflation
11.1.1
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 FriedmannLemaı̂tre-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. (1.2)]. Therefore,
Z t
Z t
Z t 0
da
dt
=c
.
(11.1)
w(t) =
dw = c
0 aȧ
0
0 a
• Between the Big Bang and the recombination time trec , the integrand in (11.1) can be approximated by the expansion rate for a
matter-dominated universe, or
ȧ2
= H02 Ωm0 a−3
a2
(11.2)
according to (1.7). Thus,
p
aȧ = H0 Ωm0 a ,
113
(11.3)
CHAPTER 11. INFLATION AND DARK ENERGY
the comoving particle horizon becomes
√
Z arec
2c arec
c
da
w(trec ) =
,
√
√
√ =
a H0 Ωm0
H0 Ωm0 0
114
(11.4)
and the physical particle horizon at the time of recombination is
rrec = arec wrec .
• On the other hand, we have seen in (6.29) that the angulardiameter distance to the CMB is
Dang (arec ) ≈
2carec
,
√
H0 Ωm0
(11.5)
which implies that the angular size of the particle horizon is
√
rrec
θrec =
≈ arec ≈ 5◦ .
(11.6)
Dang (arec )
• The physical meaning of the particle horizon is that no event between the Big Bang and recombination can exert any influence
on a given particle if it is more than the horizon length away.
Our simple calculation thus shows that we can understand how
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 all plausible how
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
H 2 (a) =
Λ Kc2
8πG
ρ+ − 2
3 " 3
a
#
Kc2
= H (a) Ωtotal (a) − 2 2 ,
a H
2
(11.7)
from which we conclude
|Ωtotal (a) − 1| =
Kc2
.
a2 H 2
(11.8)
Illustration of the causality problem: The particle horizon at CMB
decoupling corresponds to a circle
of ∼ 5◦ radius.
CHAPTER 11. INFLATION AND DARK ENERGY
115
• According to (11.3), we have, in the matter-dominated era,
√
ada
dt =
⇒ a ∝ t2/3 ,
(11.9)
√
H0 Ωm0
hence
a2 H 2 = ȧ2 ∝ t−2/3
(11.10)
|Ωtotal (a) − 1| ∝ t2/3 .
(11.11)
or, from (11.8),
• 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
≈ 10−5 ,
(11.12)
|Ωtotal (arec ) − 1| . 1%
10
1.4 × 10
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?
11.1.2
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,
√
H ∝ a−3/2 and thus rH ∝ a, while H ∝ a−2 and rH ∝ a during the
radiation-dominated era. Therefore, the comoving Hubble radius
typically grows with time. Since we can write (11.8) as
|Ωtotal (a) − 1| = KrH2 ,
(11.13)
this is equivalent to the flatness problem.
• This motivates the idea that at least the flatness problem would
be solved if the comoving Hubble radius could, at least for some
sufficiently long period, shrink with time. If that could be arranged, any deviation of Ωtotal (a) from unity would be driven towards zero.
• Conveniently, such an arrangement would also remove or at least
alleviate the causality problem. Since the Hubble length characterises 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
The universe can be driven into flatness (top) if the comoving Hubble radius can shrink for sufficiently
long time (middle). This can also
solve the causality problem (bottom).
CHAPTER 11. INFLATION AND DARK ENERGY
116
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 arranged? Obviously, we require
!
c
c
ȧ2
ȧ2
d c
=−
(ȧH + aḢ) = −
+ ä −
<0,
dt aH
(aH)2
(aH)2 a
a
(11.14)
which is possible if and only if ä > 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
.
(11.15)
3
In other words, cosmic acceleration is possible if the dominant
ingredient of the cosmic fluid has sufficiently negative pressure.
ρc2 + 3p < 0
⇒
p<−
• When applied to a cosmic sub-volume V = a3 , the first law of
thermodynamics
dE + pdV = 0
⇒
d(ρc2 a3 ) + pda3 = 0
(11.16)
because any heat current would violate isotropy. We thus obtain
the equation
(ρ̇a3 + 3ρa2 ȧ)c2 + 3pa2 ȧ = 0 ,
(11.17)
which implies the density evolution
p
ȧ ρ̇ = −3 ρ + 2 .
a
c
(11.18)
• The cosmological constant must have ρ̇ = 0 and therefore p =
−ρ/c2 . It has a suitable equation-of-state for cosmic acceleration.
This endorses the earlier conclusion that type-Ia supernoave indicate the need for a cosmological constant and thus for cosmic
acceleration.
• If we bring Friedmann’s equation (1.7) into the form
h
i
a2 H 2 = H02 Ωm0 a−2 + Ωm0 a−1 − ΩK + ΩΛ0 a2 ,
(11.19)
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, (11.19) becomes
p
p
ȧ = H0 ΩΛ0 a ⇒ a ∝ exp H0 ΩΛ0 t ,
(11.20)
and the universe enters into exponential expansion.
CHAPTER 11. INFLATION AND DARK ENERGY
11.1.3
117
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
pφ = wρφ c
2
with
w≡
1 2
φ̇
2
1 2
φ̇
2
−V
+V
.
(11.21)
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, φ̇ = 0, we have w = −1 or p = −ρc2 ,
in agreement with the conclusion from (11.18).
• In other words, a suitably strongly self-interacting scalar field has
exactly the properties we need. Inflation, i.e. accelerated expansion, broadly requires φ̇2 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
!
V0
1 V 00
1
1 and η ≡
1.
(11.22)
≡
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.
• Estimates show that inflation needs to expand the Universe by
∼ 50 . . . 60 e-foldings (i.e. by a factor of e50...60 ) for solving the
causality and flatness problems.
• 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 .
The slow-roll conditions mean that
the potential must be sufficiently flat
for inflation to set in, and gently
curved for it to last long enough.
CHAPTER 11. INFLATION AND DARK ENERGY
118
• Once approaches unity, the kinetic term φ̇2 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.
• These vacuum oscillations cause the spontaneous creation and
annihilation of particle-antiparticle pairs. Once inflation sets in,
vacuum fluctuation modes are quickly driven out of the horizon
and loose causal connection. Then, they cannot decay any more
and “freeze in”. Thus, inflation introduces the breath-taking notion 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,
Pδ (k) ∝ kn ,
(11.23)
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 .
(11.24)
The latest WMAP measurements do in fact show that
n = 0.951+0.015
−0.019 .
(11.25)
Inflation may expand quantum fluctuations to cosmological scales. It
is possible – and likely! – that the
large-scale structure in the universe
originated from inevitable quantum
fluctuations in the very early universe.
CHAPTER 11. INFLATION AND DARK ENERGY
119
When combined with the galaxy power spectrum obtained from
the 2dFGRS, this result changes very little,
+0.014
n = 0.948−0.018
.
(11.26)
The completely scale-invariant spectrum, n = 1, is thus excluded
at more than 3σ.
• The measured deviation of n from unity also restricts the number N of e-foldings completed by inflation. Under fairly general
assumptions,
N = 54 ± 7
(11.27)
based on the WMAP data.
• 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 .
(11.28)
• 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. Together 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.
11.2
Dark energy
11.2.1
Motivation
• The CMB shows us that the Universe is at least nearly spatially
flat. Constraints from kinematics, from cluster evolution and from
the CMB 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 constituent 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 tell us that it behaves at least very
similar to a cosmological constant.
CHAPTER 11. INFLATION AND DARK ENERGY
120
• 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.
• However, this is deeply dissatisfactory 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.
11.2.2
Observational constraints?
• If the dark energy is indeed dynamical and provided by a selfinteracting scalar field, how can we find out more about it? Reviewing 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
CHAPTER 11. INFLATION AND DARK ENERGY
121
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.
• Let us therefore assume that the dark energy is a suitably selfinteracting, homogeneous scalar field. Then, its pressure can be
described by
p = w(a)ρc2 ,
(11.29)
where the equation-of-state parameter w is some function of a.
According to (11.15), 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 selfinteraction potential V(φ) through (11.21) or from a simple adhoc parameterisation. Then, (11.18) implies
ȧ
ρ̇
= −3(1 + w) ,
ρ
a
(11.30)
or
(
ρ(a) = ρ0 exp −3
a
Z
)
da0
[1 + w(a )] 0 ≡ ρ0 f (a) .
a
0
1
(11.31)
• If w = const., this simplifies to
ρ(a) = ρ0 exp [−3(1 + w) ln a] = ρ0 a−3(1+w) .
(11.32)
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 (1.7) by
ΩDE0 f (a), and the expansion function E(a) turns into
h
i1/2
E(a) = Ωr0 a−4 + Ωm0 a−3 + ΩDE0 f (a) + ΩK0 a−2
,
(11.33)
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
CHAPTER 11. INFLATION AND DARK ENERGY
122
growth factor D+ (a) satisfying (1.19), this implies that cosmological observables measure integrals over the integrated equationof-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.
• 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)
,
dw
d ln D+ (z)
,
dw
(11.34)
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. Moreover,
the largest hope is put on so-called tomographic measurements,
which trace the evolution of structures throughout cosmic history.
An example is given by weak gravitational lensing: Since its geometrical sensitivity peaks approximately half-way between the
sources and the observer, sources at higher redshift also probe
more distant, and thus less evolved, cosmic structures. If lensing effects can be measured for sub-samples of sources in different redshift shells, the growth factor can be probed differentially.
First examples for this technique have been published. They give
rise to the expectation that clarifying the nature of the dark energy
may indeed be feasible in the near future.
Logarithmic derivatives of the
angular-diameter distance and the
growth factor with respect to the
equation-of-state parameter.
Chapter 12
Appendix
12.1
Cosmological parameters
Parameter
WMAP
100Ωb0
4.307+0.139
−0.176
Ωm0 0.2446+0.0139
−0.0183
ΩΛ0
0.758+0.035
−0.058
σ8
0.744+0.050
−0.060
ns
0.951+0.015
−0.019
WMAP and 2dFGRS
2.223+0.127
−0.160
0.2434+0.0087
−0.0120
0.739+0.026
−0.029
0.737+0.033
−0.045
0.948+0.014
−0.018
Table 12.1: 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.
123
CHAPTER 12. APPENDIX
12.2
124
Cosmic time, lookback time and redshift
Figure 12.1: 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 12. APPENDIX
12.3
125
Linear growth factor
Figure 12.2: The linear growth factor as a function of redshift, plotted
linearly (left) and logarithmically (right).
CHAPTER 12. APPENDIX
12.4
126
Distances
Figure 12.3: 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 12. APPENDIX
12.5
127
Density and Hubble parameters
Figure 12.4: The density parameters Ωm and ΩΛ (left) and the Hubble
parameter as functions of redshift (right).
CHAPTER 12. APPENDIX
12.6
128
The CDM power spectrum
Figure 12.5: Linearly and nonlinearly evolved CDM power spectra today.
Index
4
He
and power spectrum, 68
measurement, 69
two-point, 68
cosmic expansion
adiabatic expansion of the Uniacceleration, 100
verse, 49
cosmological redshift, 3
age
critical density, 3
of globular clusters, 16
of the Earth, 12
dark energy, 119
of the Galaxy, 13
and self-interacting scalar
of the Universe, 13
field, 120
age of the universe, 4
dynamical, 120
angular-diameter distance, 5
dark matter, 48
and appearance of the CMB,
bias factor, 73
55
Big Bang, 5
density parameter
black holes
from 2dFGRS power specevaporation timescale, 48
trum, 76
from cluster evolution, 47
causality problem, 114
from type-Ia supernovae, 99
Cepheids
in galaxies, 42
period-luminosity relation, 20
in galaxy clusters, 45
populations, 20
density
parameters, 3
CMB
distance measurement
acoustic oscillations, 56
with parallaxes, 19
COBE satellite, 51
distance measurements
common detectors, 61
with Cepheids, 19
dipole, 54
distances
Galactic foregrounds, 60
from
Baade-Wesselink
linear polarisation, 58
method,
22
mean properties, 51
from planetary-nebula lumiPlanck satellite, 62
nosity function, 22
power spectrum, 58
from
surface-brightness flucrecombination temperature,
tations, 21
53
from type-Ia supernovae, 22
recombination time, 53
temperature prediction, 30
ellipticity measurements, 87
WMAP satellite, 51, 61
equations
comoving distance, 5
of stellar structure, 14
correlation function
Etherington relation, 26
and excess probability, 69
abundance by mass, 33
2dFGRS, 75
129
INDEX
Faber-Jackson relation, 21, 41
flatness problem, 115
Friedmann’s equation, 3
fundamental plane, 21
galaxy clusters, 42
gas mass, 45
gravitational-lensing mass, 46
number density, 47
stellar mass, 42
virial mass, 43
X-ray mass, 45
galaxy rotation curves, 40
Gamow criterion, 30
Gaussian random field, 7
filamentary structures, 8
globular clusters, 15
gravitational field
index of refraction, 23
gravitational lensing
E and B modes, 88
convergence, 81
current results, 89
deflection angle, 79
distortion matrix, 80
magnification, 81
of type-Ia supernovae, 100
shear, 81
typical numbers, 84
grey dust, 99, 100
growth factor, 6
Hertzsprung-Russell diagram, 14
horizon, 5
at matter-radiation equality, 7
particle, 113
problem, 114
sound, 56
Hubble constant, 3
inflation
and negative pressure, 116
and spectral index, 118
equation of state, 117
exponential expansion, 116
Gaussian random field, 118
inflaton field, 118
reheating, 118
130
shrinking comoving horizon,
115
structure formation from
quantum
flyuctuations,
118
tensor perturbations, 119
lensing potential, 24
Limber’s equation, 82
luminosity distance, 5
mass function
Press-Schechter, 109
Salpeter, 39
mass-to-light ratio
of galaxies, 41
of stars, 40
metric
Friedmann-Lemaı̂treRobertson-Walker form,
4
Robertson-Walker form, 3
mode coupling, 8
nuclear cosmo-chronology, 10
nucleosynthesis
3
He detection, 34
7
Li detection, 34
deuterium detection, 34
of 4 He, 28
peculiar
motion, 18
velocity, 19
point-spread function, 88
Poisson’s equation, 81
polarisation
due to Galactic foreground,
106
power spectrum, 7
amplitude, 102
and Sachs-Wolfe effect, 104
CDM shape, 102
cold dark matter, 8
normalisation with the CMB,
105
of weak gravitational lensing,
82
INDEX
shot-noise contribution, 71
window function, 71
proper distance, 4
radiation-dominated era, 6
redshift-space distortions, 74
spatial pattern, 74
reionisation, 105
from Gunn-Peterson effect,
107
optical depth, 106
secondary polarisation, 106
Sachs-Wolfe effect, 55
Saha’s equation, 52
Schechter luminosity function, 41
SDSS, 75
seeing, 87
Silk damping, 57
slow-roll conditions, 117
spherical-collapse model, 47
Spite plateau, 36
standard candles, 91
distance uncertainty, 93
sensitivity to cosmological parameters, 93
Sunyaev-Zel’dovich effect, 26
as a CMB foreground, 59
supernovae
classification, 95
type-Ia photon counts, 94
symmetry assumptions, 2
isotropy, 2
thermal equilibrium, 50
time delay, 23
measurement, 24
Tully-Fisher relation, 21
type-Ia supernovae, 96
degeneracy direction in parameter plane, 95
empirical calibration, 97
legacy survey, 98
silicon lines, 96
spectroscopic identification,
94
white dwarf
131
luminosity function, 16
Wien’s law, 39
Zel’dovich approximation, 8