Modern cosmology s bonometto

MODERN COSMOLOGY
Studies in High Energy Physics, Cosmology and Gravitation
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MODERN COSMOLOGY
Edited by
Silvio Bonometto
Department of Physics,
University of Milan—Bicocca, Milan
Vittorio Gorini and Ugo Moschella

Department of Chemical, Mathematical and Physical Sciences,
University of Insubria at Como
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Contents
Preface xiii
1 The physics of the early universe (an overview)
Silvio Bonometto 1
1.1 The physics of the early universe: an overview 1
1.1.1 The middle-age cosmology 2
1.1.2 Inflationary theories 4
1.1.3 Links between cosmology and particle physics 6
1.1.4 Basic questions and tentative answers 7
2 An introduction to the physics of cosmology
John A Peacock 9
2.1 Aspects of general relativity 9
2.1.1 The equivalence principle 11
2.1.2 Applications of gravitational time dilation 12
2.2 The energy–momentum tensor 13
2.2.1 Relativistic fluid mechanics 14
2.3 The field equations 16
2.3.1 Newtonian limit 16
2.3.2 Pressure as a source of gravity 17
2.3.3 Energy density of the vacuum 17
2.4 The Friedmann models 19
2.4.1 Cosmological coordinates 19
2.4.2 The redshift 21
2.4.3 Dynamics of the expansion 22
2.4.4 Solutions to the Friedmann equation 24
2.4.5 Horizons 27
2.4.6 Observations in cosmology 27
2.4.7 The meaning of an expanding universe 29

2.5 Inflationary cosmology 32
2.5.1 Inflation field dynamics 34
2.5.2 Ending inflation 36
2.5.3 Relic fluctuations from inflation 38
vi
Contents
2.5.4 Gravity waves and tilt 40
2.5.5 Evidence for vacuum energy at late times 42
2.5.6 Cosmic coincidence 43
2.6 Dynamics of structure formation 47
2.6.1 Linear perturbations 47
2.6.2 Dynamical effects of radiation 50
2.6.3 The peculiar velocity field 53
2.6.4 Transfer functions 54
2.6.5 The spherical model 57
2.7 Quantifying large-scale structure 58
2.7.1 Fourier analysis of density fluctuations 59
2.7.2 The CDM model 61
2.7.3 Karhunen–Lo`eve and all that 63
2.7.4 Projection on the sky 68
2.7.5 Nonlinear clustering: a problem for CDM? 72
2.7.6 Real-space and redshift-space clustering 74
2.7.7 The state of the art in LSS 76
2.7.8 Galaxy formation and biased clustering 81
2.8 Cosmic background fluctuations 86
2.8.1 The hot big bang and the microwave background 86
2.8.2 Mechanisms for primary fluctuations 88
2.8.3 The temperature power spectrum 90
2.8.4 Large-scale fluctuations and CMB power spectrum 93
2.8.5 Predictions of CMB anisotropies 95

2.8.6 Geometrical degeneracy 97
2.8.7 Small-scale data and outlook 100
References 104
3 Cosmological models
George F R Ellis 108
3.1 Introduction 108
3.1.1 Spacetime 109
3.1.2 Field equations 109
3.1.3 Matter description 110
3.1.4 Cosmology 111
3.2 1 + 3 covariant description: variables 112
3.2.1 Average 4-velocity of matter 112
3.2.2 Kinematic quantities 113
3.2.3 Matter tensor 114
3.2.4 Electromagnetic field 114
3.2.5 Weyl tensor 115
3.3 1 + 3 Covariant description: equations 115
3.3.1 Energy–momentum conservation equations 115
3.3.2 Ricci identities 116
Contents
vii
3.3.3 Bianchi identities 119
3.3.4 Implications 120
3.3.5 Shear-free dust 120
3.4 Tetrad description 121
3.4.1 General tetrad formalism 121
3.4.2 Tetrad formalism in cosmology 123
3.4.3 Complete set 124
3.5 Models and symmetries 124
3.5.1 Symmetries of cosmologies 124

3.5.2 Classification of cosmological symmetries 127
3.6 Friedmann–Lemaˆıtre models 130
3.6.1 Phase planes and evolutionary paths 131
3.6.2 Spatial topology 131
3.6.3 Growth of inhomogeneity 132
3.7 Bianchi universes (s = 3) 132
3.7.1 Constructing Bianchi universes 133
3.7.2 Dynamical systems approach 134
3.7.3 Isotropization properties 138
3.8 Observations and horizons 139
3.8.1 Observational variables and relations: FL models 139
3.8.2 Particle horizons and visual horizons 141
3.8.3 Small universes 141
3.8.4 Observations in anisotropic and inhomogeneous models 142
3.8.5 Proof of almost-FL geometry 143
3.8.6 Importance of consistency checks 146
3.9 Explaining homogeneity and structure 146
3.9.1 Showing initial conditions are irrelevant 147
3.9.2 The explanation of initial conditions 150
3.9.3 The irremovable problem 153
3.10 Conclusion 154
References 154
4 Inflationary cosmology and creation of matter in the universe
Andrei D Linde 159
4.1 Introduction 159
4.2 Brief history of inflation 160
4.2.1 Chaotic inflation 161
4.3 Quantum fluctuations in the inflationary universe 164
4.4 Quantum fluctuations and density perturbations 168
4.5 From the big bang theory to the theory of eternal inflation 169

4.6 (P)reheating after inflation 172
4.7 Conclusions 183
References 183
viii
Contents
5 Dark matter and particle physics
Antonio Masiero and Silvia Pascoli 186
5.1 Introduction 186
5.2 The SM of particle physics 188
5.2.1 The Higgs mechanism and vector boson masses 189
5.2.2 Fermion masses 191
5.2.3 Successes and difficulties of the SM 192
5.3 The dark matter problem: experimental evidence 192
5.4 Lepton number violation and neutrinos as HDM candidates 194
5.4.1 Experimental limits on neutrino masses 194
5.4.2 Neutrino masses in the SM and beyond 195
5.4.3 Thermal history of neutrinos 196
5.4.4 HDM and structure formation 197
5.5 Low-energy SUSY and DM 198
5.5.1 Neutralinos as the LSP in SUSY models 198
5.5.2 Neutralinos in the minimal supersymmetric SM 199
5.5.3 Thermal history of neutralinos and 
CDM
200
5.5.4 CDM models and structure formation 202
5.6 Warm dark matter 203
5.6.1 Thermal history of light gravitinos and WDM models 203
5.7 Dark energy, CDM and xCDM or QCDM 204
5.7.1 CDM models 205
5.7.2 Scalar field cosmology and quintessence 206

References 207
6 Supergravity and cosmology
Renata Kallosh 211
6.1 M/string theory and supergravity 211
6.2 Superconformal symmetry, supergravity and cosmology 212
6.3 Gravitino production after inflation 215
6.4 Super-Higgs effect in cosmology 216
6.5 M
P
→∞limit 217
References 218
7 The cosmic microwave background
Arthur Kosowsky 219
7.1 A brief historical perspective 220
7.2 Physics of temperature fluctuations 222
7.2.1 Causes of temperature fluctuations 223
7.2.2 A formal description 224
7.2.3 Tight coupling 226
7.2.4 Free-streaming 227
7.2.5 Diffusion damping 227
7.2.6 The resulting power spectrum 228
7.3 Physics of polarization fluctuations 229
Contents
ix
7.3.1 Stokes parameters 230
7.3.2 Thomson scattering and the quadrupolar source 231
7.3.3 Harmonic expansions and power spectra 232
7.4 Acoustic oscillations 234
7.4.1 An oscillator equation 235
7.4.2 Initial conditions 236

7.4.3 Coherent oscillations 237
7.4.4 The effect of baryons 238
7.5 Cosmological models and constraints 239
7.5.1 A space of models 239
7.5.2 Physical quantities 241
7.5.3 Power spectrum degeneracies 242
7.5.4 Idealized experiments 244
7.5.5 Current constraints and upcoming experiments 247
7.6 Model-independent cosmological constraints 251
7.6.1 Flatness 252
7.6.2 Coherent acoustic oscillations 254
7.6.3 Adiabatic primordial perturbations 254
7.6.4 Gaussian primordial perturbations 255
7.6.5 Tensor or vector perturbations 255
7.6.6 Reionization redshift 257
7.6.7 Magnetic fields 257
7.6.8 The topology of the universe 257
7.7 Finale: testing inflationary cosmology 258
References 261
8 Dark matter search with innovative techniques
Andrea Giuliani 264
8.1 CDM direct detection 264
8.1.1 Status of the DM problem 264
8.1.2 Neutralinos 265
8.1.3 The galactic halo 266
8.1.4 Strategies for WIMP direct detection 267
8.2 Phonon-mediated particle detection 271
8.2.1 Basic principles 272
8.2.2 The energy absorber 272
8.2.3 Phonon sensors 273

8.3 Innovative techniques based on phonon-mediated devices 273
8.3.1 Basic principles of double readout detectors 273
8.3.2 CDMS, EDELWEISS and CRESST experiments 274
8.3.3 Discussion of the CDMS results 276
8.4 Other innovative techniques 279
References 280
x
Contents
9 Signature for signals from the dark universe
The DAMA Collaboration 282
9.1 Introduction 282
9.2 The highly radiopure ∼100 kg NaI(Tl) set-up 285
9.3 Investigation of the WIMP annual modulation signature 286
9.3.1 Results of the model-independent approach 286
9.3.2 Main points on the investigation of possible systematics
in the new DAMA/NaI-3 and 4 running periods 287
9.3.3 Results of a model-dependent analysis 290
9.4 DAMA annual modulation result versus CDMS exclusion plot 292
9.5 Conclusion 294
References 295
10 Neutrino oscillations: a phenomenological overview
GianLuigi Fogli 296
10.1 Introduction 296
10.2 Three-neutrino mixing and oscillations 297
10.3 Analysis of the atmospheric data 298
10.4 Analysis of the solar data 302
10.4.1 Total rates and expectations 302
10.4.2 Two-flavour oscillations in vacuum 305
10.4.3 Two-flavour oscillations in matter 305
10.4.4 Three-flavour oscillations in matter 308

10.5 Conclusions 309
References 311
11 Highlights in modern observational cosmology
Piero Rosati 312
11.1 Synopsis 312
11.2 The cosmological framework 312
11.2.1 Friedmann cosmological background 313
11.2.2 Observables in cosmology 314
11.2.3 Applications 317
11.3 Galaxy surveys 321
11.3.1 Overview 321
11.3.2 Survey strategies and selection methods 322
11.3.3 Galaxy counts and evolution 325
11.3.4 Colour selection techniques 328
11.3.5 Star formation history in the universe 331
11.4 Cluster surveys 334
11.4.1 Clusters as cosmological probes 334
11.4.2 Cluster search methods 337
11.4.3 Determining 
m
and 

339
References 342
Contents
xi
12 Clustering in the universe: from highly nonlinear structures to
homogeneity
Luigi Guzzo 344
12.1 Introduction 344

12.2 The clustering of galaxies 344
12.3 Our distorted view of the galaxy distribution 347
12.4 Is the universe fractal? 353
12.4.1 Scaling laws 353
12.4.2 Observational evidences 355
12.4.3 Scaling in Fourier space 357
12.5 Do we really see homogeneity?
Variance on ∼1000h
−1
Mpc scales 359
12.5.1 The REFLEX cluster survey 359
12.5.2 ‘Peaks and valleys’ in the power spectrum 361
12.6 Conclusions 363
References 364
13 The debate on galaxy space distribution: an overview
Marco Montuori and Luciano Pietronero 367
13.1 Introduction 367
13.2 The standard approach of clustering correlation 367
13.3 Criticisms of the standard approach 368
13.4 Mass–length relation and conditional density 369
13.5 Homogeneous and fractal structure 369
13.6 ξ(r) for a fractal structure 369
13.7 Galaxy surveys 370
13.7.1 Angular samples 371
13.7.2 Redshift samples 371
13.8 (r) analysis 372
13.9 Interpretation of standard results 374
References 376
14 Gravitational lensing
Philippe Jetzer 378

14.1 Introduction 378
14.1.1 Historical remarks 379
14.2 Lens equation 381
14.2.1 Point-like lenses 381
14.2.2 Thin lens approximation 383
14.2.3 Lens equation 384
14.2.4 Remarks on the lens equation 386
14.3 Simple lens models 390
14.3.1 Axially symmetric lenses 390
14.3.2 Schwarzschild lens 393
14.3.3 Singular isothermal sphere 395
xii
Contents
14.3.4 Generalization of the singular isothermal sphere 396
14.3.5 Extended source 397
14.3.6 Two point-mass lens 398
14.4 Galactic microlensing 398
14.4.1 Introduction 398
14.5 The lens equation in cosmology 406
14.5.1 Hubble constant from time delays 409
14.6 Galaxy clusters as lenses 409
14.6.1 Weak lensing 413
14.6.2 Comparison with results from x-ray observations 416
References 417
15 Numerical simulations in cosmology
Anatoly Klypin 420
15.1 Synopsis 420
15.2 Methods 421
15.2.1 Introduction 421
15.2.2 Equations of evolution of fluctuations in an expanding

universe 423
15.2.3 Initial conditions 425
15.2.4 Codes 429
15.2.5 Effects of resolution 433
15.2.6 Halo identification 437
15.3 Spatial and velocity biases 439
15.3.1 Introduction 439
15.3.2 Oh, bias, bias 440
15.3.3 Spatial bias 442
15.3.4 Velocity bias 447
15.3.5 Conclusions 451
15.4 Dark matter halos 451
15.4.1 Introduction 451
15.4.2 Dark matter halos: the NFW and the Moore et al profiles 454
15.4.3 Properties of dark matter halos 457
15.4.4 Halo profiles: convergence study 462
References 471
Index 474
Preface
Cosmology is a new science, but cosmological questions are as old as mankind.
Turning philosophical and metaphysical problems into problems that physics can
treat and hopefully solve has been an achievement of the 20th century. The main
contributions have come from the discovery of galaxies and the invention of a
relativistic theory of gravitation. At the edge of the new millennium, in the spring
of 2000, SIGRAV—Societ`a Italiana di Relativit`a e Gravitazione (Italian Society
of Relativity and Gravitation) and the University of Insubria sponsored a doctoral
school on ‘Relativistic Cosmology: Theory and Observation’, which took place
at the Centre for Scientific Culture ‘Alessandro Volta’, located in the beautiful
environment of Villa Olmo in Como, Italy. This book brings together the reports
of the courses held by a number of outstanding scientists currently working in

various research fields in cosmology. Topics covered range over several different
aspects of modern cosmology from observational matters to advanced theoretical
speculations.
The main financial support for the school came from the University of
Insubria at Como–Varese. Other contributors were the Department of Chemical,
Physical and Mathematical Sciences of the same University, the National Institute
of Nuclear Physics and the Physics Departments of the Universities of Milan,
Turin, Rome La Sapienza and Rome Tor Vergata.
We are grateful to all the members of the scientific organizing committee and
to the scientific coordinator of the Centro Volta, Professor Giulio Casati, for their
invaluable help in the organization. We also acknowledge the essential support
of the secretarial conference staff of the Centro Volta, in particular of Chiara
Stefanetti.
S Bonometto, V Gorini and U Moschella
23 January 2001
xiii
Chapter 1
The physics of the early universe (an
overview)
Silvio Bonometto
Department of Physics, University of Milan–Bicocca, Milan, Italy
1.1 The physics of the early universe: an overview
Modern cosmology has a precise birthdate, Hubble’s discovery of Cepheids and
ordinary stars in Nebulae. The nature of nebulae had been disputed for centuries.
As early as 1755, in his General History of Nature and Theory of the Sky,
Immanuel Kant suggested that nebulae could be galaxies. The main objection
to this hypothesis has been supernovae. Today we know that, close to its peak, a
supernova can exceed the luminosity of its host galaxy. But, while this remained
unknown, single stars as luminous as whole nebulae were a severe objection to
the claim that nebulae were made of as many as hundreds of billions stars. For

instance, in 1893, the British astronomer Mary Clark reported the observation of
two stellar bursts in a single nebula, one 25 years after the other. She wrote that:
The light of the nebula has been practically cancelled by the bursts, which. . .
should have been of an order of magnitude so large, that even our imagination
refuses in conceiving it. Clark was not alone in having problems conceiving the
energetics of supernovae.
After the recognition that most nebulae were galaxies, Hubble also claimed
that they receded from one another, as fragments of a huge explosion. Such an
expansive trend, currently named the Hubble flow, has been confirmed by the
whole present data-set. Although there are no doubts that Hubble’s intuition was
great, the point is that his data-set did not show that much. At the distances where
he pretended to see an expansive trend, the ‘Hubble flow’ is still dominated by
peculiar motions of individual galaxies. Discovering the true nature of nebulae
was, however, essential. It is the galactic scale which sets the boundary above
which dynamical evolution is mostly due to pure gravity. Dissipative forces, of
course, still play an essential role above such a scale. But even the huge x-ray
1
2
The physics of the early universe (an overview)
emission from galaxy clusters, now the principal tool for their detection, bears
limited dynamical effects.
Galaxies, therefore, are the inhabitants of a super-world whose rules are
set by relativistic gravitation. Their average distances are gradually increasing,
within the Hubble flow. The Friedmann equations tell us the ensuing rate of
matter density decrease and how such a rate varies with density itself. No doubts,
then, that the early universe must have been very dense. The cosmic clock, telling
us how long ago density was above a given level, is set by the Hubble constant
H = 100h km s
−1
Mpc

−1
.Hereh conveys our residual ignorance, but it is likely
that 0.6 < h < 0.8, while almost no one suggests that h lies outside the interval
0.5–0.9. (One can appreciate how far from reality Hubble was, considering that
he had estimated that h  5.)
A realistic measure of h came shortly before the discovery of the cosmic
background radiation (CBR). The Friedmann equations could then also determine
how temperature varies with time and it was soon clear that, besides being dense,
the early universe was hot. This defined the early environment and, until the
1980s, modern cosmologists essentially used known physics within the frame
of such exceptional environments. In a sense, this extended Newton’s claim
that the same gravity laws hold on Earth and in the skies. On the basis of
spectroscopical analysis it had already become clear that such a claim could be
extended beyond gravity to the laws governing all physical phenomena, thereby
leading cosmologists to extend these laws back in time, besides far in space.
1.1.1 The middle-age cosmology
This program, essentially based on the use of general relativity, led to great results.
It was shown that, during its early stages, the universe had been homogeneous
and isotropic, apart from tiny fluctuations, seeds of the present inhomogeneities.
Cosmic times (t) can be associated with redshifts (z), which relate the scale factor
a(t) to the present scale factor a
0
, through the relation
1 + z = a
0
/a(t).
The redshift z also tells us the temperature of the background radiation, which is
T
0
(1 + z) (T

0
 2.73 K is today’s temperature).
On average, linearity held for z > 30–100. For z > 1000, the high-energy
tail of the black body (BB) distribution contained enough photons, with an energy
exceeding B
H
= 13.6 eV, to keep all baryonic matter ionized. Roughly above the
same redshift, the radiation density exceeds the baryon density. This occurs above
the so-called equivalence redshift z
eq
= 2.5 × 10
4

b
h
2
.Here
b
is the ratio
between the present density of baryon matter and the present critical density ρ
cr
,
setting the boundary between parabolic and hyperbolic models. It can be shown
that ρ
cr
= 3H
2
0
/8π G.
The relativistic theory of fluctuation growth, developed by Lifshitz, also

showed that, in their linear stages, inhomogeneities would grow proportionally
The physics of the early universe: an overview
3
to (1 + z)
−1
, if the content of the universe were assumed to be a single fluid.
This moderate growth rate tells us that the actual inhomogeneities could not arise
from purely statistical fluctuations. When the Lifshitz result was generalized to
any kind of matter contents, it also became clear that fluctuations compatible with
observed anisotropies in the CBR were too small to turn into galaxies, unless
another material component existed, already fully decoupled from radiation at
z  1000, besides baryons.
Various hypotheses were then put forward, on the nature of such dark matter,
whose density, today, is 
c
ρ
cr
. (The world is then characterized by an overall
matter density parameter 
m
= 
c
+
b
.) But, as far as cosmology is concerned,
only the redshift z
d
when the quanta of dark matter become non-relativistic
matters. Let M
d

be the mass scale entering the horizon at z
d
and let us also recall
that the mass scale entering the horizon at z
eq
= 2.5 × 10
4

m
h
2
is ∼10
16
M

.
Early fluctuations, over scales <M
d
, are fully erased by free-streaming, at the
horizon entry. If one wants to preserve a fluctuation spectrum extending to quite
small scales, it is therefore important for z
d
to be large.
As far as cosmology is concerned, the nature of dark matter can therefore
be classified according to the minimal size of fluctuations able to survive. If
fluctuations are preserved down to scales well below the galactic scale (M
g
∼
10
8

–10
12
M

),wesaythatdarkmatteriscold. If dark matter particles are too
fast, and become non-relativistic only at late times, so that M
d
> M
g
, we say that
dark matter is hot. In principle, in the latter case galaxies could also form, because
of the fragmentation of greater structures in their nonlinear collapse, which, in
general, is not spherically symmetric. But such top–down scenarios were soon
shown not to fit observational data. This is why cold dark matter (CDM) became
a basic ingredient of all cosmological models.
This argument is quite independent from the assumption that 
m
has to
approach unity, in order for the geometry of spatial world sections to be flat.
However, once we accept that CDM exists, the temptation to imagine that 
m
= 1
is great. There is another class of arguments which prevents 
b
from approaching
unity by itself alone. These are related to the early formation of light elements,
like
2
H,
4

He,
7
Li. The study of big-bang nucleosynthesis (BBNS) has shown
that, in order to obtain the observed abundances of light nuclides, we ought
to have 
b
h
2
 0.02. BBNS occurred when the temperature of the universe
was between 900 and 60 keV (ν decoupling and the opening of the deuterium
bottleneck, respectively). At even larger temperatures, strongly interacting matter
had to be in the quark–hadron plasma form. Going backwards in time we reach
T
ew
, when the weak and electromagnetic interactions separated. To go still further
backwards, we need to speculate on physical theories, as experimental data are
lacking. The physics of cosmology, therefore, starts from hydrodynamics and
reaches advanced particle physics. In this book, a review of the physics of
cosmology is provided in the contribution by John Peacock.
All these ages, starting from the quark–hadron transition, through the
era when lepton pairs were abundant, then through BBNS, to arrive at the
4
The physics of the early universe (an overview)
moment when matter became denser than radiation and finally to matter–radiation
decoupling and fluctuation growth, are the so-called middle ages of the world.
Their study, until the 1980s, was the main duty of cosmologists. Not all problems,
of course, were solved then. Moreover, as fresh data flowed in, theoretical
questions evolved. In his contribution Piero Rosati reviews the present status
of observational cosmology, in relation to the most recent data.
The world we observe today is the result of fluctuation growth through linear

and nonlinear stages. The initial simplicity of the model has been heavily polluted
by nonlinear and dissipative physics. Tracing back the initial conditions from data
requires both a theoretical and a numerical effort. In his contribution Anatoly
Klypin presents such numerical techniques, the role of which is becoming more
and more important. Using recent parallel computing programs, it is now possible
to try to reproduce the events leading to the shaping of the universe.
The point, however, is that, once this self-consistent scenario became clear,
cosmology was ready for another leap. Since the 1980s, it has become a new
paradigm within which very high-energy physics could be tested.
1.1.2 Inflationary theories
The world we observe is extremely complex and inhomogeneous. The level of
inhomogeneity gradually decreases when we go to greater scales (on this subject,
see the contribution by Luigi Guzzo; another less shared point of view is exposed
by Marco Montuori and Luciano Pietronero). But only the observations of CBR
show a ‘substance’ close to homogeneity. In spite of this, the driving scheme
of the cosmological quest had been that the present complexity came from an
initial simplicity and much effort has been spent in developing a framework
able to show that this is what truly occurred. When this desire for unity was
fulfilled, cosmologists realized that it had taken them to a deadlock: the conditions
from which the observed world had evidently arisen, which so nicely fulfilled
their intimate expectations, were so exceptional as to require an exceptional
explanation.
This is the starting point of the next chapter of cosmological research, which
started in the 1980s and was made possible by the great achievements of previous
cosmological research. The new quest took two alternative directions. The most
satisfactory possibility occurred if, starting from generic metric conditions, their
eventual evolution necessarily created the exceptional ‘initial conditions’ needed
to give a start to the observed world. An alternative, weaker requirement, was
that, starting from a generic metric, its eventual evolution necessarily created
somewhere the exceptional ‘initial conditions’ needed to give a start to the

observed world.
The basic paradigm for implementing one of such requirement is set by
inflationary theories. The paradoxes such theories are called to justify can be
listed as follows:
(i) Homogeneity and isotropy: apart from tiny fluctuations, whose distribution
The physics of the early universe: an overview
5
is itself isotropic, the conditions holding in the universe, at z > 1000, are
substantially identical anywhere we can observe them. The domain our
observations reach has a size ∼ct
0
(c, the speed of light; t
0
, the present
cosmic time). This is the size of the regions causally connected today.
At z ∼ 10
3
, the domain causally connected was smaller, just because the
cosmic time was ∼10
4.5
times smaller than t
0
. Let us take a sphere whose
radius is ∼ct
0
. Its surface includes ∼1000 regions which were then causally
disconnected one from another. In spite of that, temperature, fluctuation
spectrum, baryon content, etc, were equal anywhere. What made them so?
(ii) Flatness: According to observations, the present matter density parameter


m
cannot deviate from unity by more than a factor 10. (Recent observations
on the CBR have reduced such a possible discrepancy further.) But, in order
for 
m
∼ 0.1 today, we need to fine-tune the initial conditions, at the Planck
time, by 1:10
60
. To avoid such tuning we can only assume that the spatial
section of the metric is Euclidean. Then it remains as such forever.
(iii) Fluctuation spectrum: Let us assume that it reads:
P(k) = Ak
n
.
Here k = 2π/L and L are comoving length scales. This spectral shape,
apparently depending on A and n only (spectral amplitude and spectral
index, respectively), tries to minimize the scale dependence. But a fully
scale-independent spectrum is obtained only if n = 1. It can then be shown
that fluctuations on any scale have an identical amplitude when they enter the
horizon. This fully scale-independent spectrum, first introduced by Harrison
and Zel’dovich, approaches all features of the observed large-scale structure
(LSS). How could such fluctuations arise and why did they have such a
spectrum?
Apart from these basic requirements, there are a few other requests such as
the absence of topological monsters that we shall not discuss here.
The scheme of inflationary theories amounts then to seeking a theory
of fundamental interactions which eliminates these paradoxes. The essential
ingredient in achieving such an aim is to prescribe a long period of cosmic
expansion dominated by a false vacuum, rather than by any kind of substance.
Early periods of vacuum dominance are indeed expected, within most elementary

particle theories, and this sets the bridge between fundamental interaction theories
and cosmological requirements.
In this book, inflationary theories and their framework are discussed in detail
by Andrei Linde and George Ellis, and therefore we refrain from treating them
further in this introduction. Let us rather outline what is the overall resulting
scheme. One assumes that, around the Planck time, the universe emerges from
quantum gravity in a chaotic status. Hence, anisotropies, inhomogeneities,
discontinuities, etc, were dominant then.
However, such a variety of initial conditions has nothing to do with the
present observed variety. The universe is indeed anisotropic, inhomogeneous,
6
The physics of the early universe (an overview)
discontinuous, etc, today; and more and more so, as we go to smaller and smaller
scales. But such secondary chaos has nothing to do with the primeval chaos. It is
akindofmoderate chaos that we have reached after passing through intermediate
highly symmetric conditions. The sequence complex → simple → complex had
to run, so that today’s world could arise.
1.1.3 Links between cosmology and particle physics
There are, therefore, at least two fields where the connections between particle
physics and cosmology have grown strong. As we have just outlined, explaining
why and how an inflationary era arose and runs is certainly a duty that
cosmologists and particle physicists have to fulfill together.
In a sense, however, this is a more speculative domain, compared with the
one opened by the need for a dark component. The first idea on the nature of
dark matter was that neutrinos had mass. A neutrino background, similar to the
CBR, must exist, if the universe ever had a temperature above ∼1MeV.Sucha
background would be made by ∼100 neutrinos/cm
3
, for each neutrino flavour.
It is then sufficient to assume that neutrinos have a mass ∼10–100 eV, to reach


m
∼ 1.
Such an appealing picture, which needs no hypothetical new quanta, but
refers to surely existing particles only, was, however, shown not to hold.
Neutrinos could be hot dark matter, as they become non-relativistic around z
eq
.
As we have already stated, the top–down scenario, where structures on galactic
scales form thanks to greater structure fragmentation, is widely contradicted by
observations.
This does not mean that massive neutrinos may not have a role in shaping the
present condition of the universe. Models with a mix of cold and hot dark matter
were considered quite appealing until a couple of years ago. Their importance,
today, has somehow faded, owing to recent data on dark energy. Recent data on
the neutrino mass spectrum are reviewed by Gianluigi Fogli in his contribution.
Alternative ideas on the nature of dark matter then came from
supersymmetries. The lightest neutral supersymmetric partner of existing bosons
is likely to be stable. In current literature this particle is often called the
neutralino. There are quite a few parameters, concerning supersymmetries, which
are not deducible from known data and, after all, supersymmetries themselves
have not yet been shown to be viable. However, well within observationally
acceptable values, it is possible for neutralinos to have mass and abundance such
as to yield 
m
∼ 1.
In their contribution Antonio Masiero and Silvia Pascoli focus on the
interface between particle physics and cosmology, discussing in detail the nature
of CDM. Andrea Giuliani’s paper deals with current work aiming at detecting
dark matter quanta in laboratories and the contribution by Rita Bernabei et al

relates possible evidence for the detection of neutralinos. Various hypotheses
were considered, about dark matter setting. Its distribution may differ from visible
The physics of the early universe: an overview
7
matter, on various scales. By definition, its main interaction, in the present epoch,
occurs via gravity and gravitational lensing is the basic way to trace its presence.
In his contribution Philippe Jetzer reviews the basic pattern to detect dark matter,
over different scales, using the relativistic bending of light rays.
1.1.4 Basic questions and tentative answers
There can be little doubt that the last century has witnessed a change of the context
within which the very word ‘cosmology’ is used. Man has always asked basic
questions, concerning the origin of the world and the nature of things. The only
answers to such questions, for ages, came from metaphysics or religious beliefs.
During the last century, instead, a large number of such questions could be put
into a scientific form and quite a significant number could be answered.
As an example, it is now clear that the universe is evolutionary. At the
beginning of modern cosmology, models claiming a steady state (SS) of the
universe had been put forward. They have been completely falsified, although
it is now clear that the stationary expansion regime, introduced by SS models,
is not so different from the inflationary expansion regime, needed to make big-
bang models self-consistent. Furthermore, if recent measures of the deceleration
parameter are confirmed, we seem to be living today in a phase of accelerated
expansion, quite similar to inflation. It ought to be emphasized that the strength of
the data, supporting this kind of expansion, is currently balanced by the theoretical
prejudices of wise researchers. In fact, an accelerated expansion requires a
desperate fine-tuning of the vacuum energy, which seems to spoil all the beauty
of the inflationary paradigm.
Since Hubble’s hazardous conclusion that the universe was expanding, the
century which has just closed has seen a number of results, initially supported
more by their elegance than by data. The Galilean scheme of experimental

science is not being forgotten, but one must always remember that such a
scheme is far from requiring pure experimental activity. The basic pattern to
physical knowledge is set by the intricate network of observations, experiments
and predictions that the researcher has to base on data, but goes well beyond
them. With the growing complication of current research, the theoretical phase
of scientific thought is acquiring greater and greater weight. During such a stage,
the lead is taken by the same criteria which drove mathematical research to its
extraordinary achievements.
Besides Hubble’s findings, within the cosmological context, we may quote
Peebles’ discovery of the correlation length r
0
, based on angular data, which have
recently been shown to allow quite different interpretations. Outside cosmology,
the main example is given by gauge theories, which are now the basic ingredient
of the standard model of fundamental interactions, and were deepened, from 1954
to the early 1970s, only because they were too beautiful not to be true. At least
two other fields of research in fundamental physics are now driven by similar
criteria—supersymmetries and string theories (see the paper by Renata Kallosh).
8
The physics of the early universe (an overview)
While supersymmetries can soon be confirmed, either by the discovery of
neutralinos by passive detectors or at CERN’s new accelerator, string theories
might only find confirmation if signals arriving from the Planck era can be
observed. This might be possible if future analyses of CBR anisotropies and
polarization show the presence of tensor modes. In this book a review of current
procedures for CBR analysis is provided by Arthur Kosowsky.
Also within the cosmological domain, leading criteria linked to aesthetical
categories are now being pursued. However, in this field, the concept of beauty
is often directly connected with ideological prejudices. Questions such as ‘can
the universe tunnel from nothing’ have been asked and replied within precise

physical contexts. It is, however, clear that the ideological charge of such research
is dominant. Moreover, when theoretical results, in this field, are quoted by the
media, the distinction between valid speculations and scientific acquisitions often
fully fades.
But the main question, for physicists, is different. For at least two centuries,
basic mathematics has developed without making reference to experimental
reality. The criterion driving mathematicians to new acquisitions was the
mathematical beauty. Only a tiny part of such mathematical developments then
found a role in physics. Tensor calculus was developed well before Einstein
found a role for it in special and general relativity. Hilbert spaces found a role
in quantum mechanics. Lie groups found a role in gauge theories. But there
are plenty of other chapters of beautiful advanced mathematics which are, as yet,
unexplored by physicists and may remain so forever.
There is, however, no question about that. Mathematics is an intellectual
construction and its advancement is based on intellectual criteria. The problem
arises when physicists begin to use similar criteria to put order in the physical
world. Let us emphasize that this is not new in the history of research. The
Pythagorean school, in ancient Greece, centered its teaching on mathematical
beauty. They also found important physical results, e.g. in acoustics, starting
from their criterion that the world should be a reflection of mathematical purity.
In the ancient world, the views of Pythagoreans were then taken up by the whole
Platonic school, in opposition to the Aristoteleans who thought that the world was
ugly and complicated, so that attempting a quantitative description was in vain.
Even though we now believe that the final word has to be provided by the
experimental data, there is no doubt that theoretical developments, often long and
articulate, are grounded on mathematical beauty. This is true for any field of
physics, of course, but the impact of such criteria in the quest for the origin is
intellectually disturbing. What seems implicit in all this is that the human mind,
for some obscure reason, although in a confused form, owns in itself the basic
categories enabling it to distinguish the truth and to assert what is adherent to

physical reality.
It is not our intention to take a stand on such points. However, we believe that
they should be very present in the mind of all readers, when considering recent
developments in basic physics and modern cosmology.
Chapter 2
An introduction to the physics of cosmology
John A Peacock
Institute for Astronomy, University of Edinburgh, United
Kingdom
In asking me to write on ‘The Physics of Cosmology’, the editors of this book
have placed no restrictions on the material, since the wonderful thing about
modern cosmology is that it draws on just about every branch of physics. In
practice, this chapter attempts to set the scene for some of the later more
specialized topics by discussing the following subjects:
(1) some cosmological aspects of general relativity,
(2) basics of the Friedmann models,
(3) quantum fields and physics of the vacuum and
(4) dynamics of cosmological perturbations.
2.1 Aspects of general relativity
The aim of general relativity is to write down laws of physics that are valid
descriptions of nature as seen from any viewpoint. Special relativity shares the
same philosophy, but is restricted to inertial frames. The mathematical tool for
the job is the 4-vector; this allows us to write equations that are valid for all
observers because the quantities on either side of the equation will transform in
the same way. We ensure that this is so by constructing physical 4-vectors out of
the fundamental interval
dx
µ
= (c dt, dx, dy, dz)µ= 0, 1, 2, 3,
using relativistic invariants such as the the rest mass m and proper time dτ.

For example, defining the 4-momentum P
µ
= m dx
µ
/dτ allows an
immediate relativistic generalization of conservation of mass and momentum,
9
10
An introduction to the physics of cosmology
since the equation P
µ
= 0 reduces to these laws for an observer who sees a
set of slowly-moving particles.
None of this seems to depend on whether or not observers move at constant
velocity. We have in fact already dealt with the main principle of general relativity,
which states that the only valid physical laws are those that equate two quantities
that transform in the same way under any arbitrary change of coordinates. We
may distinguish equations that are covariant—i.e. relate two tensors of the same
rank—and invariants, where contraction of a tensor yields a number that is the
same for all observers:
P
µ
= 0covariant
P
µ
P
µ
= m
2
c

2
invariant.
The constancy of the speed of light is an example of this: with dx
µ
=
(c dt, −dx, −dy, −dz),wehavedx
µ
dx
µ
= 0.
Before getting too pleased with ourselves, we should ask how we are going
to construct general analogues of 4-vectors. We want general 4-vectors V
µ
to
transform like dx
µ
under the adoption of a new set of coordinates x
µ
:
V
µ
=
∂x
µ
∂x
ν
V
ν
.
This relation applies for 4-velocity U

µ
= dx
µ
/τ , but fails when we try to
differentiate this equation to form the 4-acceleration A
µ
= dU
µ
/dτ :
A
µ
=
∂x
µ
∂x
ν
A
ν
+
∂
2
x
µ
∂τ∂x
ν
U
ν
.
The second term on the right-hand side is zero only when the transformation
coefficients are constants. This is so for the Lorentz transformation, but not in

general.
The need is therefore to be able to remove the effects of such local coordinate
transformations from the laws of physics. Technically, we say that physics should
be invariant under Lorentz group symmetry.
One difficulty with this programme is that general relativity makes no
distinction between coordinate transformations associated with the motion of
the observer and a simple change of variable. For example, we might decide
that henceforth we will write down coordinates in the order (x , y, z, ct) rather
than (ct, x, y, z). General relativity can cope with these changes automatically.
Indeed, this flexibility of the theory is something of a problem: it can sometimes
be hard to see when some feature of a problem is ‘real’, or just an artifact of the
coordinates adopted. People attempt to distinguish this second type of coordinate
change by distinguishing between ‘active’ and ‘passive’ Lorentz transformations;
a more common term for the latter class is gauge transformations.
Aspects of general relativity
11
2.1.1 The equivalence principle
The problem of how to generalize the laboratory laws of special relativity is solved
by using the equivalence principle, in which the physics in the vicinity of freely
falling observers is assumed to be equivalent to special relativity. We can in fact
obtain the full equations of general relativity in this way, in an approach pioneered
by Weinberg (1972). In what follows, Greek indices run from 0 to 3 (spacetime),
Roman from 1 to 3 (spatial). The summation convention on repeated indices of
either type is assumed.
Consider freely falling observers, who erect a special-relativity coordinate
frame ξ
µ
in their neighbourhood. The equation of motion for nearby particles is
simple:
d

2
ξ
µ
dτ
2
= 0; ξ
µ
= (ct, x, y, z),
i.e. they have zero acceleration, and we have Minkowski spacetime
c
2
dτ
2
= η
αβ
dξ
α
dξ
β
,
where η
αβ
is just a diagonal matrix η
αβ
= diag(1, −1, −1, −1). Now suppose
the observers make a transformation to some other set of coordinates x
µ
.What
results is the perfectly general relation
dξ

µ
=
∂ξ
µ
∂x
ν
dx
ν
,
which, on substitution, leads to the two principal equations of dynamics in general
relativity:
d
2
x
µ
dτ
2
+ 
µ
αβ
dx
α
dτ
dx
β
dτ
= 0
c
2
dτ

2
= g
αβ
dx
α
dx
β
.
At this stage, the new quantities appearing in these equations are defined only in
terms of our transformation coefficients:

µ
αβ
=
∂x
µ
∂ξ
ν
∂
2
ξ
ν
∂x
α
∂x
β
g
µν
=
∂ξ

α
∂x
µ
∂ξ
β
∂x
ν
η
αβ
.
This tremendously neat argument effectively uses the equivalence principle
to prove what is often merely assumed as a starting point in discussions of
relativity: that spacetime is governed by Riemannian geometry. There is a metric
tensor, and the gravitational force is to be interpreted as arising from non-zero
derivatives of this tensor.