How the Universe Works: Introduction to Modern Cosmology

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Praktyka kliniczna Konie Galaktyka Dietz Olof, Huskamp Bernhard Year: 2015 Language: polish File: PDF, 162.73 MB

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Praktyka kliniczna Konie Galaktyka Dietz Olof, Huskamp Bernhard Year: 2015 Language: polish File: DJVU, 171.88 MB

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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

HOW THE UNIVERSE WORKS
Introduction to Modern Cosmology
Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.

ISBN 978-981-3234-94-9
For any available supplementary material, please visit
http://www.worldscientific.com/worldscibooks/10.1142/10847#t=suppl
Desk Editor: Christopher Teo
Typeset by Stallion Press
Email: enquiries@stallionpress.com
Printed in Singapore

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Dedicated to the centennial of cosmology

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Preface

This book is about the history and the current state of the art in cosmology —
the science about the Universe as a whole. It mostly aims to explain the
main ideas of modern cosmology: the expanding Universe, its creation in
a Big Bang, its evolution, characteristics and so on. We tried to answer
some frequently asked questions about cosmology. We also discuss the two
mysteries of the modern science, directly related to cosmology — dark
matter and dark energy.
This book is based on our earlier book “The Introduction to Modern
Cosmology” published in Russian in 2013, which was well received by
a much broader audience than we anticipated. We significantly reworked
it based on this fact and the numerous feedbacks. We tried to make this
book more suitable for a general audience by expanding the explanations
and cutting some of the more complicated material. We nearly doubled the
number of illustrations, added a whole new Section explaining the basics
of General Relativity, and updated the text to reflect the latest advances in
cosmology. The book was significantly restructured as well. We also tried
to explain all scientific terms we use. In case we failed to do so, or you
feel that our explanations are insufficient, we kindly ask you to look it up
on Wikipedia, which should provide quite enough information to proceed
with the reading.
This book is different from most popular science books. The golden
rule of writing popular science books is: each formula in the text cuts
the number of potential readers in half. Still, we took the risks and used
equations, but only where they were indispensable. We tried to reduce the
vii

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How the Universe Works: Introduction to Modern Cosmology

number of formulas to the minimum and keep them simple. They should
be understandable to anyone who majored in mathematics or physics in
college. We labelled such parts “Advanced material” and they should be
treated as such. Skipping them should not impair the understanding of the
material, but there are a few references to these parts in the main text,
so we advise to at least try reading them. Each such part begins with a
brief summary. They provide a sort of a simple textbook on cosmology for
those unfamiliar with the mathematical formalism of General Relativity
who would like to understand where the laws of cosmology come from.
These “Advanced material” parts are marked with an asterisk and a picture
of a thoughtful octopus:

The rest of the book is aimed at the general audience, although some
level of prior knowledge of mathematics and physics is expected. For those
lacking even basic knowledge of astronomy we strongly recommend to read
some popular books on astronomy first. Our suggestion for the first book on
astronomy is “The Universe: From Flat Earth to Quasar” by Isaac Asimov
[1966], which is, however, somewhat outdated. Other reading suggestions
are listed in the end of the Summary.
We avoided cutting corners in explanations and tried justifying various
assumptions or estimations made in cosmology. We did not hide problems
faced by modern cosmology as well as issues the community has no
consensus about. We did not try to pass hypotheses for established theories,
which is not uncommon in scholarly articles. In some sense, this book stands
between a popular science book and a textbook, acting as a sort of a bridge
across the great chasm, separating popular science from true science.
We would like to thank many people who assisted us in preparation
of this book. Special thanks go to Igor Zhuk and Gayane Terzyan who
prepared most of the original artwork, and to Natalia Atamas who provided
a lot of valuable advices on improving the book’s structure.
Serge Parnovsky
Aleksei Parnowski

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page ix

Contents

Preface

vii

List of Table

xiii

List of Figures

xv

Chapter 1. The Laws of the Universe
1.1 Roots of Cosmology . . . . . . . . . . . . .
1.2 Principles of General Relativity . . . . .
1.2.1 Perihelion precession . . . . . . .
1.2.2 Deviation of light . . . . . . . . . .
1.2.3 Gravitational redshift . . . . . . .
1.2.4 Other effects and tests . . . . . . .
1.2.5 Chosen frame . . . . . . . . . . . .
1.2.6 Gravity, inertia, and tidal forces
1.2.7 Lunar tides . . . . . . . . . . . . . .
1.2.8 Space, time, and space-time . . .
1.2.9 Curved space-time . . . . . . . . .
1.3 How Much Does Light Weigh? . . . . . .
1.3.1 Baryonic matter . . . . . . . . . . .
1.3.2 Radiation . . . . . . . . . . . . . . .
1.3.3 Dark energy and antigravity . . .

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How the Universe Works: Introduction to Modern Cosmology

Chapter 2. The Expanding Universe
2.1 Einstein’s Static Universe . . . . . . . . . . . . . . . . . . . .
2.2 Expansion and Redshift . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Other galaxies and their recession . . . . . . . . . .
2.2.2 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Hubble’s Law∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Friedmann Models . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Arrow of time . . . . . . . . . . . . . . . . . . . . . . .
2.5 Geometry of the Universe . . . . . . . . . . . . . . . . . . . .
2.6 Scale Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Deceleration parameter . . . . . . . . . . . . . . . . .
2.7 Non-Relativistic Friedmann Solutions∗ . . . . . . . . . . .
2.7.1 Cosmological evolution without cosmological
constant∗ . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Study of solutions∗ . . . . . . . . . . . . . . . . . . . .
2.7.3 Deceleration parameter∗ . . . . . . . . . . . . . . . .
2.7.4 Matter with nonzero pressure in the expanding
Universe∗ . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Modern Modification of the Model . . . . . . . . . . . . . .
2.8.1 Cosmological constant strikes back . . . . . . . . .
2.8.2 Standard cosmological model . . . . . . . . . . . . .
2.9 Distances in Astronomy . . . . . . . . . . . . . . . . . . . . .
Chapter 3.

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Early Universe

3.1 The Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Cosmic Microwave Background: an Echo of the Big Bang
3.2.1 CMB discovery . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 CMB anisotropy . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Bringing cosmology to space . . . . . . . . . . . . . . .
3.2.4 Ground studies of the CMB . . . . . . . . . . . . . . . .
3.2.5 CMB fluctuations spectrum . . . . . . . . . . . . . . . .
3.2.6 Conservation of energy . . . . . . . . . . . . . . . . . . .
3.2.7 Speculations . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Evolution of the Early Universe∗ . . . . . . . . . . . . . . . . .
3.4 Cosmological Horizon . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

3.5 Distance to the Cosmological Horizon∗
3.6 Inflation of the Universe . . . . . . . . . .
3.6.1 Inflation models . . . . . . . . . . .
3.7 Multiverse and the Anthropic Principle
3.7.1 Pulsating Universe . . . . . . . . .
3.8 The Matter in Making . . . . . . . . . . . .
3.8.1 Big Bang nucleosynthesis . . . .
3.8.2 Stellar nucleosynthesis . . . . . .
3.8.3 The antimatter problem . . . . . .
Chapter 4.

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Dark Energy

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Black Holes and Other Exotics

6.1 Black Holes . . . . . . . . . . . . . . . . . .
6.1.1 Schwarzschild black holes . . . .
6.1.2 Reissner–Nordström black hole
6.1.3 Kerr black hole . . . . . . . . . . .

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5.1 Cosmological Evidence for Dark Matter and Dark Energy
5.1.1 Type Ia supernovae . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Baryonic acoustic oscillations . . . . . . . . . . . . . .
5.1.3 CMB spectrum . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Time to Big Rip∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Other Kinds of Matter . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6.

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Dark Matter

4.1 Revolution Comes . . . . . . . . . . . . . . . .
4.2 Evidence for Dark Matter . . . . . . . . . . .
4.2.1 Virial mass . . . . . . . . . . . . . . . .
4.2.2 Galactic rotation curves . . . . . . . .
4.2.3 Mass-to-luminosity ratio . . . . . . .
4.2.4 Galactic mergers . . . . . . . . . . . .
4.2.5 Cosmic flows . . . . . . . . . . . . . . .
4.2.6 Growth rate of density fluctuations
4.2.7 Gravitational lensing . . . . . . . . . .
4.2.8 Alternative models . . . . . . . . . . .
4.3 What Makes up Dark Matter? . . . . . . . .
Chapter 5.

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How the Universe Works: Introduction to Modern Cosmology

6.2 Naked Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Wormholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Summary

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Appendix A. Cosmological Evolution with a
Cosmological Constant∗
Solution∗

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A.1 De Sitter
.............................
∗
A.2 CDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Flat CDM Model∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bibliography

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Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index

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∗Advanced

material

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page xiii

List of Table

Table 3.1

Isotopes created in Big Bang nucleosynthesis. . . . . .

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List of Figures

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4
Figure 1.5
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4

Forces acting upon a body resting at the Earth
surface in the Newtonian (a) and relativistic
(b) frames. Here N is the floor reaction force
(a.k.a. pressure force) and W is the gravitational force
(a.k.a. weight) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Forces acting upon an orbiting spacecraft in the
Newtonian (a) and relativistic (b) frames. Here
W is the gravitational force and Fcf is the centrifugal
force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Forces acting on balls in a free falling elevator in the
Newtonian (a) and relativistic (b) frames. The scale is
largely exaggerated . . . . . . . . . . . . . . . . . . . . . . .
Lunar tides on the Earth in the Newtonian (a) and
relativistic (b) frames . . . . . . . . . . . . . . . . . . . . . .
A light cone in Minkowski (flat) space-time . . . . . .
To the explanation of the Hubble’s law . . . . . . . . . .
Relative scale factor (left) and Hubble constant (right)
vs. time for three Friedmann models . . . . . . . . . . .
A portion of Fig. 2.2 illustrating the evolution of the
scale factor of the closed universe . . . . . . . . . . . . .
In a static closed Universe the ratio of a circle’s length
to its radius is less than 2π and decreases when the
radius increases. A circle’s length first increases and
then decreases as its radius increases . . . . . . . . . . .
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Figure 2.5

In a closed Universe the sum of angles in a triangle is
greater than 180 degrees. Thick lines show a triangle
on a sphere, crosshatched area shows a flat triangle for
comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.6 In an open Universe the sum of angles in a triangle is
less than 180 degrees. Thick lines show a triangle on a
pseudosphere, dashed lines show a flat triangle for
comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.7 Possible results of kicking a soccer ball upwards . . .
Figure 2.8 Three Friedmann models. Relative scale factor u (left)
and Hubble constant H (right) vs. time since the Big
Bang t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.9 CDM model. Relative scale factor u (left) and
Hubble constant H (right) vs. time since the
Big Bang t . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.10 CDM model. Redshift z vs. time since the
Big Bang t . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.11 Flat CDM model. Density parameters of matter m
and cosmological constant  vs. time since the Big
Bang t. Their sum is fixed to unity . . . . . . . . . . . . .
Figure 3.1 A map of CMB temperature fluctuations. Image:
ESA/Planck . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.2 Power spectrum of temperature fluctuations of CMB
radiation. Dots represent experimental data, and a line
is a fitted theoretical curve . . . . . . . . . . . . . . . . . .
Figure 3.3 The cosmological horizon . . . . . . . . . . . . . . . . . . .
Figure 3.4 A size of a causally linked region in the recombination
epoch in the case without inflation (not to scale) . . .
Figure 3.5 Vacuum energy in a scalar field. The Universe rests in
the global minimum . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.6 Vacuum energy in a scalar field. The Universe rests in
the local minimum. It can tunnel into either of the
global minima . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.7 Vacuum energy in a scalar field. The Universe
unstably rests in a local maximum and classically
slides into either of the global minima . . . . . . . . . .

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List of Figures

Figure 4.1
Figure 4.2

Figure 4.3
Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7
Figure 5.1
Figure 5.2

Figure 5.3

Modern estimation of the Universe’s contents . . . . .
A galaxy’s rotation velocity can be estimated by
measurements in the atomic hydrogen line (H i). A
vertical line on each spectrum indicates a position of a
certain spectral line shifted according to the galaxy’s
radial velocity. . . . . . . . . . . . . . . . . . . . . . . . . . .
A rotation curve of the M33 galaxy . . . . . . . . . . . .
Luminous parts of each galaxy — disk and bulge
(white) and stellar halo (dark grey) — are surrounded
by a much larger dark halo (light grey) . . . . . . . . . .
“Bullet” cluster. Green lines show equal density
levels, red and blue shading show colder and hotter
X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formation of the large-scale structure in the Universe.
Simulations were performed at the National Center
for Supercomputer Applications by Andrey Kravtsov
(The University of Chicago) and Anatoly Klypin (New
Mexico State University). Visualizations by Andrey
Kravtsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Strong gravitational lensing . . . . . . . . . . . . . . . . .
Angular distance can be tricky. All three elements are
of the same length . . . . . . . . . . . . . . . . . . . . . . . .
Correlation function vs. distance between SDSS
galaxies (dots) and theoretical dependencies for
different matter densities (curves). A peak corresponds
to the BAO scale . . . . . . . . . . . . . . . . . . . . . . . . .
Constraints on density parameters of matter and
cosmological constant obtained with three described
methods (marked with white text): CMB fluctuations
spectrum (orange), supernovae bursts (blue), and
baryonic acoustic oscillations (green). Regions
with qualitatively different scenarios of Universe’s
expansion are marked with black text: a line
corresponding to flat Universe with open Universe
below and closed Universe above, as well as the
region which excludes the Big Bang. Originally

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published as Figure 5 in [Suzuki et al., 2012]. ©AAS.
Reproduced with permission . . . . . . . . . . . . . . . . .
Figure 5.4 Constraints on the dark energy’s equation of state.
Designations are the same as in Figure 5.3. Originally
published as Figure 6 in [Suzuki et al., 2012]. ©AAS.
Reproduced with permission . . . . . . . . . . . . . . . . .
Figure 5.5 Possible options for the future of the Universe
depending on the dark energy’s equation of state
(scale is exaggerated) . . . . . . . . . . . . . . . . . . . . . .
Figure 5.6 Calculation of the time before the hypothetical Big
Rip. The horizontal axis shows the value of w. The
vertical axis represents the time before the Big Rip.
The curve tends to infinity at w → −1 . . . . . . . . . .
Figure 6.1 An affordable black hole simulator . . . . . . . . . . . . .
Figure 6.2 Light cones in the vicinity of a Schwarzschild
black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 6.3 Light cones in the vicinity of a Kerr black hole
(equatorial plane section). Dots indicate points
of origin and small circles indicate light cone in
the future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure A.1 Relative Hubble constant vs. relative scale factor . . .
Figure A.2 Deceleration parameter vs. relative scale factor . . . .
Figure A.3 Dependence of density parameters of matter and
cosmological constant on the relative scale factor
in CDM model . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 3.1. A map of CMB temperature fluctuations. Image: ESA/Planck.

Figure 4.5. “Bullet” cluster. Green lines show equal density levels, red and blue shading
show colder and hotter X-rays.

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xx

Figure 4.6. Formation of the large-scale structure in the Universe. Simulations were
performed at the National Center for Supercomputer Applications by Andrey Kravtsov (The
University of Chicago) and Anatoly Klypin (New Mexico State University). Visualizations
by Andrey Kravtsov.

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Figure 5.3. Constraints on density parameters of matter and cosmological constant
obtained with three described methods (marked with white text): CMB fluctuations spectrum
(orange), supernovae bursts (blue), and baryonic acoustic oscillations (green). Regions with
qualitatively different scenarios of Universe’s expansion are marked with black text: a line
corresponding to flat Universe with open Universe below and closed Universe above, as
well as the region which excludes the Big Bang. Originally published as Figure 5 in [Suzuki
et al., 2012]. ©AAS. Reproduced with permission.

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xxii

Figure 5.4. Constraints on the dark energy’s equation of state. Designations are the same as
in Figure 5.3. Originally published as Figure 6 in [Suzuki et al., 2012]. ©AAS. Reproduced
with permission.

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Chapter 1

The Laws of the Universe
1.1 Roots of Cosmology
This book is about cosmology — a science about the structure and evolution
of the Universe in its entirety, its past and its future. Cosmology is a very
young science; it just celebrated its centennial. Its creation is associated
with the publication of Albert Einstein’s 1917 paper “Cosmological
Considerations on the General Theory of Relativity” [Einstein et al., 1952].
It was for the first time when physical laws were applied to the whole
Universe. Specifically, this article applied the equations of the General
Theory of Relativity, formulated by Einstein shortly before.
Technically, nothing prevented this science to appear some 250 years
earlier, right after the discovery of the Law of Universal Gravitation by Sir
Isaac Newton. The physicists of 17th –19th centuries speculated about an
infinite Universe filled with stars with planetary systems. Such a Universe
existed eternally, and all it took to predict its future state was the knowledge
of the laws of mechanics and the current position of all objects. However,
the gravitational force in Newtonian mechanics has one peculiarity: it is
always the attracting force that never becomes a repulsion force. Therefore,
individual stars in an infinite Universe were bound to gather at some
point due to attraction. This problem was evaded using a simple, but
incorrect, idea. Since the Universe is infinite, each particle is attracted to
an infinite number of other particles. If particles fill the Universe with
constant density, the resulting force would become zero and the gravitation
can be neglected when considering the dynamics of the Universe as a
whole.

1

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This idea is as productive as an attempt to set a pencil standing on its
tip. The reason for failure in both cases is the instability of the equilibrium.
If we somehow managed to make a pencil stand on its tip, any deviation from
this position will induce a momentum in the same direction, magnifying
the initial deviation and ultimately ruining the equilibrium. In engineering
this effect is known as positive feedback.
A closer analogy is an upturned glass of water. Many of you know a
classical demonstration when a glass of water covered with a firm sheet
is turned upside down and the sheet is held in place by the atmospheric
pressure force, which is equivalent to the weight of 10.3 metres of water.
But few ponder why the sheet is needed for that demonstration. The answer
is Rayleigh–Taylor instability: when a denser liquid (water) is put on top of
a rarer liquid (aira ), any deviation from a flat boundary will grow over time
exponentially, ruining the boundary very quickly. This is popularly known
as spilling of liquid. This is why a firm sheet is needed: it does not apply
any forces, but prevents the Rayleigh–Taylor instability from developing.
Note that the mutual attraction of the stars that fill the infinite Universe
not only leads to an increase of its density irregularities, but also to
the accelerated contraction of the whole Universe, that is, to decreasing
distances between the stars.
Naturally, it was known by that time that deviations from a homogeneous distribution of matter density led to their growth, but this mechanism
was then considered only on the spatial scale of the Solar System.According
to Laplace hypothesis, the planets in the Solar System were formed from
a primordial nebula made of gas and dust due to mutual gravitational
attraction. Such a treatment was not extended to larger scales. In the big
picture of that time, the growth of matter density inhomogeneities led
to the formation of planets which did not fall on the Sun only because
they orbited it. On scales larger than distances between nearest stars,
the Universe was considered as something homogeneous and a belief
was held that the attraction force of any body to different stars is fully
compensated.
The only problem which marred this glorious picture was the so-called
Olbers paradox, formulated in 1823 by a German amateur astronomer
a In hydrodynamics gases are often called liquids as well.

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Heinrich Olbers, a medic by trade. Its point was that in an infinite static
Universe instead of a night sky we would see a burning sphere as bright as
the Sun. It can be explained in the following way. Let us divide the Universe
into geocentric nested spherical layers with constant thickness. The number
of stars in each layer will grow as a square of distance, and the flux from
each single star will decrease as a square of distance. Thus, the flux from
each layer will be the same. Since the number of layers if infinite, the total
flux will be also infinite.
However, if we take into account that stars can cover each other, we
obtain that the luminosity will be finite, because no matter which direction
we look at, our line of sight will inevitably reach some star. Although,
anyone who looked up in the sky at night knows that it looks quite different.
A simple way to resolve the Olbers paradox was to write it off for the
absorption of light by interstellar dust. However, this solution sounds
credible only for those who studied physics poorly, as after long enough
exposure this dust would be heated to the temperature of surrounding stars
and become luminous.
The progress in astronomy led to a new model of the Universe proposed
by William Herschel in the end of 18th century. In this model the stars
did not fill the whole Universe, but formed a single lens-shaped cluster
called Galaxy. Why did they not fall on the centre of the Galaxy? The
answer was the same as to why the planets did not fall on the Sun — they
orbited it. Likewise, individual stars in the Galaxy orbited its centre. In 1783
Herschel discovered Sun’s movement around the centre of the Galaxy. This
model with minor corrections was generally accepted till the beginning of
the 20th century. The idea of the Galaxy solved the Olbers paradox, since
the matter now filled a finite volume in the Universe. Nevertheless, the
discovery of other galaxies revived the Olbers paradox.
So, cosmology, which could potentially emerge in late 17th century,
appeared only in the beginning of the 20th century. Usually new sciences are
created in the simplest formulation and then evolve towards more complex
models and calculations and use modern physical theories. For example,
condensed matter physics relied on classical mechanics for centuries and
successfully switched to quantum mechanics much later.
The cosmology is curious because it was created in its most complex
form — relativistic cosmology. Only several decades later cosmologists

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reached a surprising conclusion that they could consider a much simpler
non-relativistic cosmology. This is possible because a uniform Universe
evolves identically in every point of space, and to study it in its entirety
it is sufficient to consider a small volume, e.g. 1 cm3 . And when studying
1 cm3 one can ignore space-time curvature and other complicated problems
of General Relativity. But this is true only in the case of a homogeneous
and isotropic Universe. In such a world there are no chosen places and
preferred directions, every point is not better or worse than any other, and
each direction is not better or worse than any other; this is known as the
Copernicus principle.
Not every result of relativistic cosmology can be obtained in this way,
but the main ones can be obtained quite easily. To derive, understand, and
analyze these results it is sufficient to know physics at college level. For
this reason, when we simply can not resist the urge to write some formulae
in this book, we shall limit ourselves to non-relativistic cosmology.
We marked the parts containing mathematics “Advanced material”.
They can be skipped without much loss to the understanding.
Question: What is the principal difference between cosmology and other
areas of physics?
Answer: Cosmology studies a unique object, only one copy of which
exists, which is changing in time, and containing us as a part. Thus, we can
achieve neither repeatability nor reproducibility, and we can forget about
active experiments. As a result, it is very difficult to check cosmological
theories for falsifiability, which is required of any scientific theory.
A similar situation is encountered in some other scientific disciplines,
such as history and evolutionary biology.

1.2 Principles of General Relativity
The emergence of cosmology as a science was preceded by the creation of
the General Theory of Relativity (GTR), finally formulated by Einstein in
1916. This theory is one of the pinnacles of modern physics. Since its ideas
and terminology are widely used in cosmology, we decided to describe the
basics of GTR, which are simple enough to understand and can be explained
without the use of its very sophisticated mathematical formalism. We start
with the three classical GTR effects.

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1.2.1 Perihelion precession
The first effect was discovered by the astronomers long before the
emergence of General Relativity. It is the perihelionb precessionc of
Mercury, which is manifested as the rotation of Mercury’s orbit as a
whole around the Sun with very small angular velocity — less than
6 arcseconds per year. This was not the first deviation from the laws of
celestial mechanics since their discovery by Johann Kepler. Earlier in the
middle of 19th century a similar behaviour of Uranus’orbit was successfully
explained by interference from a then-unknown planet, later called Neptune.
One of Neptune’s predictors, Urbain Le Verrier, carried over the same
approach to Mercury’s orbit, assuming the existence of a new planet Vulcan,
which should have been located very close to the Sun and was hidden by his
light. This hypothetical planet’s transits over solar disk were reported by
both professional and amateur astronomers for a few decades afterwards,
but these were later dismissed with the improvement of telescopes. Now
we know for certain that Vulcan does not exist, and this was almost certain
100 years ago. Thus, perturbations of Mercury’s orbit should have been
explained in a different way.
General Relativity not only explained the perihelion precession of
Mercury but also provided an accurate quantitative agreement with the
observed precession rate. With further improvement of the observational
accuracy, a similar perihelion precession of Venus was discovered, which,
together with other effects described below, heavily supported GTR. As a
result, the International Astronomical Union (IAU) — the supreme world
authority on astronomy — issued a resolution on the mandatory inclusion of
the General Relativity effects in precise orbit calculation of celestial bodies
in the Solar System. An even more impressive manifestation of this effect
is displayed in binary systems with pulsars,d where two massive bodies

b Perihelion is the point of the heliocentric orbit (orbit around the Sun), closest to the Sun.
c Perihelion precession is a rather slow rotation of the heliocentric orbit in the orbital

plane.
dA pulsar is a highly magnetized, rotating neutron star that emits a beam of electromagnetic
radiation. This radiation can be observed only when the beam is pointing towards Earth and
thus is registered as a series of pulses.

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rotate at a small distance with a period of a few days. GTR describes their
motion up to 0.01 per cent accuracy. The discovery of such systems brought
a 1993 Nobel Prize in Physics to Russell Alan Hulse and Joseph Hooton
Taylor, Jr.

1.2.2 Deviation of light
The second effect is the bending of light rays in the gravitational field of
massive objects. The bending itself was not a humbling sensation at the time
and could be explained within the framework of the Newtonian mechanics.
But the General Relativity predicted the angle of deviation to be twice as
large compared to the Newtonian prediction. The nature of this coefficient
will be discussed slightly later, in Subsection 1.3.2.
At that time the effect was purely hypothetical but this discrepancy
prompted astronomers to measure its value. To do so, it was necessary
to measure the position of a star, whose light travelled near the Sun and
deviated in his gravitational field, changing the star’s apparent position.
With modern accuracy using a very-long base radio interferometer (VLBI)
this effect can be measured even in the perpendicular direction to the Sun,
but in the beginning of the 20th century, it could be measured only in a very
small area around the Sun.
This was done by the expedition of Sir Arthur Eddington which
measured star positions during the total solar eclipse of 1919. The
total solar eclipse was required because at that time astronomers could
perform observations only in visible light and the sunshine would make
it impossible to observe stars. Eddington and his collaborators performed
observations from Brazil and from the west coast of Africa. Comparing
the photographs of the sky near the eclipsed Sun and of the same area far
from the Sun, they measured the deviation angle, which appeared to be in
favour of Einstein’s prediction. These observations were plagued by poor
accuracy, which was substantially improved only with the invention of radio
telescopes.
This effect is the basis for the so-called gravitational lensing, which
produces several images of the same object. It is actively studied today and
even used as an exotic tool for observing extremely distant objects. We
discuss this in Subsection 4.2.7.

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1.2.3 Gravitational redshift
The third effect is called gravitational redshifte and describes the difference
in the rate of time at different gravity potentials.f Simply speaking, time
runs faster on the top floor of the building than in the basement. A source
transmits, say, 1000 signals per second. They propagate to the receiver, but
for the receiver a second has a different duration, so during that second it
receives not 1000, but, e.g. 999 signals. In other words, the frequency at
the receiver is shifted with respect to the frequency of the source.
Astronomers observed gravitational redshift in white dwarf stars, in
particular, in Sirius B, which packs roughly the mass of the Sun in roughly
the Earth’s volume. As a result, the gravitational potential at its surface far
surpasses the top values available in the Solar System.
This effect was also demonstrated in laboratory conditions by Robert
Pound and Glen Rebka in 1959. They built their experiment around a
fundamental idea of quantum mechanics that for an atom to be excited
from the ground stateg it should absorb a photon with exactly the same
energy or wavelength an excited atom emits when transiting to a ground
state. If something (in our case, gravitational redshift) changes the energy
or wavelength of the photon while it travels from one atom to another even
the smallest bit, it will not get absorbed. However, it can still be absorbed
if the receiving atom moves so that the change in the wavelength due to
Doppler effecth compensates for the change in the wavelength due to the
gravitational redshift.
Pound and Rebka put one iron plate in the basement, and attached
another one to a loudspeaker cone on the roof and measured the phase of
e Redshift means an increase of wavelength. The opposite effect is called blueshift. The

names come from the fact that red light has longer waves than blue light, although both
terms are applied to any frequency band of electromagnetic radiation, not necessarily visible
light.
fA gravitational potential is a potential energy per mass unit in a gravitational field.
g Ground state is a state of an atom with minimum energy. Any state other than the ground
one is called excited.
h Doppler effect is a shift of frequency of periodic signals caused by the movement of either
the source or the recipient or both and the finiteness of the signal propagation velocity.
Contrary to a popular belief, it applies not only to waves but to any periodic signals. When
the source and the recipient move towards each other, the frequency is increased, and when
they move away from each other, it is decreased.

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the speaker at which the gamma-ray flux produced by excited iron atoms
in the basement was mostly absorbed. This gave them the change in the
photon energy due to the difference in gravitational potential on the roof
and in the basement. Their results supported the GTR predictions within
10 per cent error margin.
Further support to this effect was given by the Gravity Probe A
experiment in 1976, which was a rocket carrying a hydrogen maser used
as an extremely stable frequency generator. An identical maser rested on
the ground. This experiment confirmed the gravitational redshift within a
0.01 per cent error margin. Today gravitational redshift is routinely taken
into account when precise time measurements are required, for example
in the GPS and other navigation satellites. It is also taken into account by
astronomers in the Terrestrial Time, the Geocentric Coordinate Time and
the Barycentric Coordinate Time introduced by the IAU in 1991, which
represent proper time at sea level, at the centre of the Earth, and at the
barycentrei of the Solar System respectively.
1.2.4 Other effects and tests
The Gravity Probe A experiment also confirmed another important General
Relativity effect: the equivalence principle, which states that the object
behaves the same regardless of whether it is uniformly accelerating or
placed in a uniform gravitational field.
Since then, all the predictions of General Relativity were experimentally confirmed. One of the most widely known predictions was the
existence of black holes (see Section 6.1) — massive compact objects,
from which nothing can escape including light. While they were indirectly
observed (e.g. by tracing proper motions of nearby stars) for quite some
time, the first direct observation of a visible outburst coming from a socalled accretion discj surrounding a black hole was performed in June 2015
[Kimura et al., 2016].

iA barycentre is a common centre of mass of a gravitationally bound system.
jAn accretion disc is an area around a star, a black hole, or another massive object, filled with

orbiting falling matter. Due to gravitation this matter heats up and emits radiation. Accretion
discs of black holes radiate X-rays.

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The latest confirmed prediction was the discovery of gravitational
waves by the Advanced LIGO detector in September 2015 [Abbot et al.,
2016a], later confirmed in December 2015 [Abbot et al., 2016b]. For this
discovery, Kip Thorne, Rainer Weiss, and Barry Barish were awarded the
2017 Nobel Prize in Physics. Naturally, experimental tests of the GTR
continue with higher and higher precision.
Let us tell about other principles of General Relativity as well.
1.2.5 Chosen frame
The Newtonian mechanics is built around the idea of the inertial frame.
The first Newton’s law holds true only in such frames. Inertial frame is
tied to a body, which is not affected by the rest of the Universe. Is this
even possible? Any body can be affected by mechanical forces, such as
a tension force from an attached string,k and by four fundamental forces:
electromagnetic, weak,l strong,m and gravitational. Electromagnetic, weak,
and strong forces act only on some of the particles, which have non-zero
charges of respective type. The gravitational force, on the other hand, is
universal; it acts upon each and every body in the Universe. Even massless
particles such as photons are affected by gravitational attraction. Therefore,
it is not clear how it is possible to provide an inertial reference system where
a gravitational field is present.
General Relativity also has chosen frames, but, in contrast to Newtonian
mechanics, these frames must be unaffected by all forces except gravitational ones. In such frames all physical laws hold true, including the laws
of Special Relativity. To accelerate in such a frame, a body must be affected
by an external force other than gravity. In other words, these are the frames,
where an observer falls freely. We shall refer to them as relativistic frames.
Let us illustrate this by two simple examples.
k Strictly speaking, such forces are a complex combination of electromagnetic and strong
forces. However, explaining this would take us into the depths of quantum mechanics, so
we kindly ask readers to take it as a given.
l Weak interaction is one of four fundamental interactions (besides electromagnetic,
gravitational, and strong), which works at subatomic scales and is responsible for e.g.
radioactive decay. Currently it is considered together with the electromagnetic interaction
as a part of a more general electroweak interaction.
m Strong interaction is responsible for keeping atomic nuclei together. It acts upon composite
subatomic particles called hadrons. It works on very short scales about 10−15 m.

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A person sleeps in his or her bed. To be extremely specific, the bed
is static with respect to the ground, i.e. its geographical coordinates are
constant. From the Newtonian perspective, this person can be considered
to be at rest in an approximately inertial frame. It is not truly inertial because
this person rotates (together with the bed) around the centre of the Earth,
around the Sun (together with the Earth), around the centre of the Milky Way
(together with the Solar System), falls towards the Virgo cluster (together
with the Milky Way galaxy), towards the Great Attractor (together with the
Virgo cluster) and so on.n But let us not be too picky and call this system
inertial. The person is acted upon by two main forces (and a multitude
of minor ones): the Earth’s gravitational attraction (popularly known as
weight) and the pressure from the bed caused by elastic forces. These forces
compensate each other, causing the person to stay at rest.
Let us consider the same situation from the point of view of General
Relativity. In this case the chosen frame is quite different: it is a frame of a
free-falling observer. A person sleeping in a bed is prevented from staying
at rest in this frame by the pressure from the bed.
A second case is an astronaut orbiting the Earth. From the Newtonian
point of view, his frame is not chosen in any way because he or she is acted
upon by the gravity and follows a curved path. This can be described in two
ways. In the Earth frame the gravitational force acts as a centripetal force,
causing his trajectory to bend. In the spacecraft (non-inertial) frame the
gravitational force is compensated by a so-called centrifugal force, which
makes the astronaut to feel zero gravity.
This situation is much simpler in the framework of the General
Relativity. The astronaut is acted upon only by the gravitational force, and
thus rests in a chosen frame. This is due to the fact that an orbiting spacecraft
falls freely towards the Earth, but constantly misses it due to the tangentialo
velocity. This is the primary principle which makes spaceflight possible.
For this reason, the astronaut experiences weightlessness. However, if he or
she is affected by some non-gravitational force (shoved with a stick, pulled

n Sadly, it is quite difficult to find oneself some genuine rest nowadays.
o Tangential direction is the direction in the rotation plane perpendicular to the radius. When

applied to orbital motion, it is either the direction of flight (prograde) or the opposite one
(retrograde).

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Figure 1.1. Forces acting upon a body resting at the Earth surface in the Newtonian (a) and
relativistic (b) frames. Here N is the floor reaction force (a.k.a. pressure force) and W is the
gravitational force (a.k.a. weight).

Figure 1.2. Forces acting upon an orbiting spacecraft in the Newtonian (a) and relativistic (b) frames. Here W is the gravitational force and Fcf is the centrifugal force.

by a string, given a strong enough magnet etc.), this astronaut would move
according to the second Newton’s law.
We illustrate the forces involved in both cases in Figures 1.1 and 1.2.
1.2.6 Gravity, inertia, and tidal forces
It is easy to live in a world with uniform gravity. It can be effortlessly
faked with acceleration, for example in a rocket or even in an elevator.
No experiments inside a closed rocket or elevator can distinguish between
gravity and inertia. However, this happens only in fiction books by Terry
Pratchett and the like, but not in real life. In real world, the gravitational
field can be considered uniform only on very small scales, such as your
house. The problem is that at larger scales the Earth’s gravity field is much
more similar to that of a point mass than a uniform field.
This is expressed in two effects: the reduction of the free fall acceleration with altitude, and the difference in the directions of the gravitational

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force in two different points on the surface. For example, in two antipodal
points on the surface, say, in the UK and in Australia, the directions of the
gravitational forces are nearly opposite. At smaller distances, like between
China and Japan, the gravitational forces are directed at much smaller but
considerable angle with respect to each other. The second effect could be
faked with the help of an inflating sphere, but it would be nearly impossible
to simulate different free fall accelerations near the floor and the ceiling.
In General Relativity the term gravity is understood not as the attraction
towards some massive body (this is provided by the chosen frame’s
motion), but as small differences in the direction and the magnitude of
the gravitational field at different nearby locations called tidal forces. The
name originates from the long-known fact that these forces cause oceanic
tides on the Earth.
To explain this let us consider a free-falling elevator — an example
coined up by Einstein himself due to unavailability of rockets at that time —
with seven almost weightless balls, which are initially immobile relative to
the elevator and to each other. One of the balls is placed at the centre of mass,
another one — near the ceiling right above the first one, the third one —
near the floor right beneath them, and other four — near the walls at the
height of the first one, as shown in Figure 1.3. We assume that the elevator’s

Figure 1.3. Forces acting on balls in a free falling elevator in the Newtonian (a) and
relativistic (b) frames. The scale is largely exaggerated.

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walls have negligible weight and their only functions are to protect the balls
from the ram air and to maintain the rigidity of the construction.
Each of the balls falls freely together with the elevator, but due to the
difference in initial positions their movement will be slightly different. This
is clearly seen in the reference frame tied to the elevator. In this frame ball 1
is static; ball 2, which is always located in the area with slightly smaller
free fall acceleration, drifts upwards; similarly, ball 3 drifts downwards;
and balls 4 to 7 have a small component of the gravitational force directed
towards the centre,p and drift towards ball 1. Note that the scale of the tidal
forces in Figure 1.3 is largely exaggerated.
If we now add mutual gravitational attraction between the balls in the
elevator, we get a model of tidal forces on the Earth. Actually, tides on the
Earth are caused by the Moon and by the Sun, but for clarity we consider
only lunar tides. Thus, the tide will be high in the vertical direction and low
in the horizontal direction.
Tidal force can be quite strong and plays an important role in astronomy.
For example, in 1992 a Shoemaker–Levy 9 comet was torn apart by tidal
forces in Jupiter’s gravity field. Another example is Magellanic Clouds —
two satellites of our Milky Way galaxy, which are deformed by its tidal
forces. Especially strong tidal forces are encountered in the vicinity of
compact objects, such as neutron stars and black holes (see Section 6.1).
From Newtonian point of view, the falling elevator is a non-inertial
reference frame with uniform field of inertial forces precisely compensating
the gravitational force at the centre of mass. At all other locations, however,
this balance is broken, and thus the field of tidal forces is created, see
Figure 1.3. These forces replace gravity in an orbiting spacecraftq ; for this
reason, space scientists and engineers use the term microgravity instead of
weightlessness.
1.2.7 Lunar tides
To explain lunar tides on the Earth within Newtonian framework, first we
consider the gravitational field of the static Moon. Considering her a point
mass, her gravitational field lines, i.e. directions of the free-fall acceleration,
p For the same reason a plummet hung near the wall would be slightly inclined compared to
the plummet hung at the centre.
q In a real spacecraft there are many other much stronger forces.

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Figure 1.4. Lunar tides on the Earth in the Newtonian (a) and relativistic (b) frames.

are radial. Thus, the total gravitational force acting on any particle on the
Earth is a sum of an attraction towards the rest of the Earth and towards
the Moon (we do not consider other celestial bodies to avoid confusion).
To switch to the terrestrial frame, we should subtract the gravitational
acceleration of the Earth’s centre of mass from the accelerations acting
on each point, see Figure 1.4. As a result, we get a familiar picture from
the relativistic case: high tides in the direction towards and from the Moon,
and low tides in perpendicular directions.
Thus, from the Newtonian point view, tidal force is simply a difference
in free-fall accelerations between an arbitrary point and some reference
point, e.g. the centre of the Earth. From the relativistic point of view, the
tidal force is what distinguishes gravity from inertia in, for example, an
accelerating rocket. By switching to a free-falling frame, you negate gravity
at only one point, usually at centre of mass, but at any other point there is
a non-zero remainder — the tidal force. In General Relativity tidal force is
a manifestation of space-time curvature.
1.2.8 Space, time, and space-time
What does space-time mean? Let us begin with space. Our space is threedimensional. This means that we can move forward or backward, sideward
towards each side, upward or downward, i.e. change our location in three
spatial coordinates. Every physical process takes place in these three
coordinates and in time. In General Relativity, time is considered the fourth
coordinate in addition to the three spatial ones. Together they form the
four-dimensional space-time. However, time has one important difference
compared to space: we can deliberately choose how to move in space, but

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we cannot affect our motion in time. We always move in time from the
past to the future and at a rate beyond our control, unless we move with
the speed close to the speed of light. If we move this fast, we can make
things more complicated, but still have to move along the time axis towards
the future. For this reason, even when we combine spatial and temporal
dimensions into space-time, we do not make them equal and still treat time
in a special way.
What good is the concept of space-time? When we consider the
trajectory of a body in space, it gives us no idea on the velocity, acceleration
and other kinematical properties of its motion, except for the fact that
the body was present at each point of the trajectory. When we switch
to the space-time, the trajectory of a body tells us not only about its
location, but also when the body was at each location and for how long.
This gives us a complete description of its kinematics throughout the
considered time frame. Such trajectory in space-time is called a world line of
a body.
Any world line has one fundamental limitation: it can not exhibit
velocities greater than the speed of light. According to Special Relativity,
only massless particles can (and must) travel at speed of light. As of today,
only two such particles are known: photon and gluon, which are gauge
bosonsr of electromagnetic and strong forces, respectively. Of the two,
only photons were directly observed, because gluons are confined within
hadronss and never exist in free form. Some theoreticians speculate that a
class of particles called tachyons could exist, which always move faster than
the speed of light; however all attempts to detect them were unsuccessful.
To illustrate space-time, cosmologists use a concept of a light cone. It
shows world lines of photons either emitted from or observed in a given

rA gauge boson is a bosonic elementary particle, which acts as a carrier of any of the
fundamental forces. A boson is an elementary particle with angular momentum equal to an
integer number of reduced Planck’s constants. Any number of such particles can be in the
same quantum state at the same time. The only other possible situation is when a particle’s
angular momentum is equal to a half-integer number of reduced Planck’s constants, i.e.
1/2, 3/2, and so on. Such particles are called fermions and no more than one such particle
can have any particular quantum state at any time.
sA hadron is a massive composite elementary particle participating in strong interaction.
Hadrons include baryons and mesons, see footnotes on p. 97.

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Figure 1.5. A light cone in Minkowski (flat) space-time.

point at a given time. These two distinct cases are called, respectively,
future and past light cones. To keep this plot two-dimensional, two spatial
dimensions are usually discarded, making it look like the one shown in
Figure 1.5, which addresses the simplest case of a flat space-time, called
the Minkowski space-time, which has no gravity at all, space is not curved,
and Special Relativity laws can be used. The absence of gravity also makes
any reference frame, which moves without acceleration, both inertial and
chosen from the relativistic point of view (frame of free falling observer).
The region within the future light cone is called absolute future, the region
within the past light cone is called absolute past, and the region on the
outside of the light cones is called external region.
For any given point in absolute future it is possible to find such an
inertial reference frame, in which this point is located at the same place as
the origin, but occurs later. A similar transformation is possible for a point
in absolute past, but with the opposite sequence of events. It is then said
that the interval — a four-dimensional analogy of distance — between the
observer and either of these points is timelike. For any given point in the
external region, i.e. outside the light cone, it is possible to find such an
inertial reference frame, in which both events happen at the same time but

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at different locations. Such interval is called spacelike. Finally, if a point is
precisely on the edge of the light cone, neither transformation is possible,
but there is a photon, which visits both points consequently, and the interval
is called lightlike.
The concept of a light cone is directly related to the causality principle,
which plays a pivotal role in physics. The idea of the causality principle
is that any event can influence the events in the future, but not in the past.
Special Relativity adds to that that information can not propagate faster
than light in vacuum. Combining these two ideas, we get a very important
interpretation of the light cone: an event can influence only events upwards
inside the cone, i.e. its absolute future, and can be influenced by events
downwards inside the cone, i.e. in its absolute past. The events outside of
the cone are totally independent of the event in its origin, although they
could be caused by the same reason. This also means that the body’s world
line is confined within all light cones originating from its every point.
An important thing to remember is that the speed of light in vacuum
is always the same regardless of the motion of its source and observer. For
this reason, the light cone does not depend on the velocity of the body.
1.2.9 Curved space-time
The more experienced of the readers must be expecting some sort of catch
with all this space-time business at this point, as it sounded way too simple
so far. Guess what — their doubts are right. For this space-time appears to
be curved in all cases but the simplest case of Minkowski flat space-time.
This brings up the difficulty of analyzing the dynamics of space-time and
its contents in a fairly complicated gravitational field — sometimes referred
to as geometrodynamics — quite a bit. The local shape of the space-time is
defined by the so-called metric, which binds the interval between two very
close points with their four-dimensional coordinates. All properties of the
space-time can be recovered from its metric alone.
All space-times we deal with in this book are described by
their metrics and all these metrics are named after their discoverers.
They include Minkowski metric (flat space time), Friedmann–Lemaître–
Robertson–Walker metric (expanding Universe filled with dust-like matter,
Section 2.4), de Sitter metric (very rapidly expanding Universe without
matter but with cosmological constant, Section A.1), and Schwarzschild,

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Reissner–Nordström, Kerr, and Kerr–Newman metrics (different kinds of
black holes, Section 6.1), whose formal description, however, exceeds the
scope of this book.
The light cone in curved space-time could be much more complex
than in flat space-time. For example, gravitational lensing produces several
images of the same object. This means that the photons emitted by this
object travel to us along different trajectories (also taking different time
to arrive). And for exotic objects such as black holes the situation is even
weirder. That’s why light cones are often plotted in complicated cases to
illustrate geometrical properties of the space-time.
Curved space-time can be easily understood if we drop out one of the
spatial dimensions. Then the space-time can be visualised as an elastic
film with different objects lying on top of it. These objects will deform the
film making it curved and affecting the movement of other objects. The
change in the height of this film represents gravitational potential, the pitch
of its surface represents free-fall acceleration, and its local curvature is
connected with tidal force. This very appropriate analogy was invented by
Einstein. There are quite a lot of videos on the Internet demonstrating this
analogy.
Summarizing this section, General Relativity does not only provide a
quantitative correction to Newtonian physics; it also predicts some totally
new effects and objects, such as gravitational waves or black holes.
Question: Why are tides caused by the small Moon are stronger than tides
caused by the massive Sun?
Answer: A formula for tidal force can be found in textbooks; it states
that they fall off as the distance cubed. However, instead of using this
formula directly, we demonstrate how this dependence on distance can be
obtained using a simple analogy.
Consider two point unit masses: one in the centre of the Earth and
another on the Earth’s surface. By definition, the tidal force at the second
point is the difference of the forces acting on these points. It can depend
only on three parameters: on the distance between these points, on the
distance to the Moon, and on the angle between the direction to the Moon
and the line between these points.
By an electrostatic analogy, let us replace these points with unit charges,
and the Moon — with an external point-like charge, whose value is chosen

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so that the forces acting on the points are identical to the gravitational
case.t Let us invert the sign of the central charge (this is why we needed
to switch to electric field, since there is no such thing as a negative
mass). Now, it is affected by the force of the same strength but opposite
in direction. The tidal force, which equals the difference of the forces
acting on these points, is then equal to the sum of the forces acting on
the second point and the first point with inverse sign. These two points
have opposite unit charges and can be treated as a dipole. Due to the third
Newton’s law, the force exerted upon this dipole by the Moon equals the
force exerted upon the Moon by the dipole. The dipole field falls off as
an inverse cube of the distance, and so does the tidal force. Switching
back to gravity, we finally obtain that tidal forces fall off as the third
power of the distance and are directly proportional to the mass of the
gravitating body.
Now let us do some simple math. The Sun weighs 2.0 · 1030 kg and is
located 1.5 · 108 km away. The Moon weighs 7.3 · 1022 kg and is located
3.8 · 105 km away. Thus, the Sun is 2.7 · 107 times heavier and 395 times
farther than the Moon. If we take the distance ratio to the third power, we
get 6.2 · 107 , which is 2.2 times larger than the mass ratio. Thus, lunar
tides are 2.2 times stronger than solar tides.
However, if we are interested in the Newtonian gravitational force, we
should take the ratio of distances squared, which is 176 times smaller than
the mass ratio, and the Sun wins this fight easily. If we are interested in the
gravitational potential, which is inversely proportional to the distance, a
far stronger contribution will be made by the Virgo cluster, located some
54 million light-years away.

1.3 How Much Does Light Weigh?
General Relativity provided a mathematical formalism for cosmology.
However, in cosmological problems one should take into account the
properties of the medium, which fills the Universe. Since Einstein’s
lifetime our notions of the contents of the Universe essentially changed.
100 years ago physicists knew only about ordinary matter, which makes
up stars, planets and other familiar objects, such as our bodies, and about
electromagnetic radiation. Today ordinary matter is called baryonic matter,

t This is possible because both forces fall off as the distance squared.

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and is believed to account for about 5 per cent of the contents of the Universe,
and the electromagnetic radiation accounts for much less than 1 per cent.
The other 95 per cent are comprised of two or three additional types of
matter. The two types, which are genuinely new, are dark matter and dark
energy, which we discuss in Chapters 4 and 5 respectively. The third one,
which is only arguably a new type of matter, is the neutrinos.u These types
of matter differ with respect to each other in their equations of state, which
is the relation between their mass density ρ and pressure p. Mass density
is connected with the energy density ε through a simple relation ε = ρc2 ,
which is obtained by applying the well-known relation E = mc2 to a unit
volume. Although the equation of state can have any form, we consider only
the simplest possible form p = wε = wρc2 , where w is a dimensionless
constant.
Note that the energy density includes the rest energy, which is very
high due to the c2 factor. How high actually? Let us reformulate this
question: if we considered ordinary air, and would like to make w = 1, what
pressure should it have? At standard conditions air has the mass density of
1.23 kg/m3 . Multiplied by the speed of light squared this translates into
the energy density about 1017 J/m3 , which corresponds to the pressure of
1017 Pa. So, we would have to compress the air to 1012 atmospheresv to
make its equation of state w = 1. Such pressure is encountered within the
Solar System only at the centre of the Sun, but the mass density there is also
much higher: about 1.6 · 105 kg/m3 . Thus, we can safely assume w = 0 for
all baryonic matter. This kind of matter is called cold matter or dust-like
matter in cosmology.
From the point of view of General Relativity, the equation of state of
matter defines how it is involved in the gravitational interaction. This is
different from Newton’s gravitation, where the pressure does not affect the
force of the gravitational interaction. Let us determine how different types
of matter interact gravitationally.
uA neutrino is a very light elementary particle, which participates only in gravitational and

weak interactions. There are three known flavours of neutrinos: electron neutrino (νe ), muon
neutrino (νμ ), and tau neutrino (ντ ), each having a corresponding antineutrino.
v This value is highly underestimated because compressed air has higher density as well.
The equation of state w = 1 will be achieved when air pressure increases 1012 more times
than its density. This is impossible to achieve in gaseous state.

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1.3.1 Baryonic matter
For the baryonic matter, this was done in late 18th century by Henry
Cavendish. The results of his experiment were published in 1798 in
Philosophical Transactions of the Royal Society of London, the top
scientific journal of that time, and are considered a major milestone in
physics. The goal of the experiment was to determine the mean density of
the Earth, which directly translates into an estimation of the gravitational
constant. Cavendish measured the force of the gravitational interaction
between two pairs of lead balls, with changeable masses of the balls and
adjustable distance between them. His experimental setup was built around
a novel contraption of that time — the torsion scales. The same idea was
used a few years later by Charles-Augustin de Coulomb to measure the
force of electromagnetic interaction. However, Cavendish faced a much
tougher challenge due to the much weaker force of the gravitational
interaction. He managed to measure forces as small as 10−7 N, which was an
unparalleled feat at that time. His estimation of the gravitational constant
differs from the modern one by some mere 1 per cent and its accuracy
was improved only a century later. He also experimentally confirmed the
Newton’s Gravitation Law.
But this confirmation works only for ordinary baryonic matter in an
environment familiar to us, i.e. when the gravitational fields are weak and
the velocities are much less than the speed of light in vacuum. Physicists
have studied the GTR laws and calculated the amount of force of the
gravitational interaction in the case of a weak gravitational field (the gravity
field at the Sun’s surface is still considered weak). It slightly differs from
Newton’s Gravitation law. The difference is that we should replace body
mass m with an expression m + 3pV /c2 , or, in other words, introduce an
additional factor of 1 + 3w for each of the gravitating bodies. This wellknown replacement is obtained as the Newtonian limit of General Relativity.
The real Cavendish experiment used lead balls (which, as we already
learned, have w = 0) as attracting masses, but there is no reason why
we cannot run a similar imaginary experiment with other types of matter.
First, let us replace one of the balls with a vessel filled with air at standard
conditions. The vessel is weightless and used only to confine the gas. Thus
we get an attraction force, which is stronger by approximately three parts
in 1012 , compared to the lead ball of equal mass due to non-zero — yet very

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small — value of w. If we used some matter from the solar core instead
(please don’t try this at home), we would get an increase in attraction by
approximately seven parts in a million.
1.3.2 Radiation
The greatest increase in attraction is achieved with a bottle of light or other
electromagnetic radiation, which has w = 1/3, the highest possible value
of w. The idea of radiation pressure was introduced by Johannes Kepler back
in 1619. The relation between energy density and pressure was determined
by James Clerk Maxwell in 1862 and confirmed by Pyotr Lebedev’s
1899 experiment on measuring light pressure, thus presenting the final
experimental proof of Maxwell equations. Its attraction force would be
twice as strong as for baryonic matter: 1 + 3 · 1/3 = 2 — that’s where a
factor of 2 comes from in the formula for the deviation of light.
1.3.3 Dark energy and antigravity
Who said pressure must be positive? By definition, pressure is the force per
unit surface area. Positive pressure means that this force acts outwards, and
negative pressure means that it acts inwards. Note that the situation when the
external pressure compresses the vessel does not count as negative pressure.
Negative pressure occurs under normal conditions, for example, due to
Bernoulli’s law when the liquid flows through a tube at high velocity (this
is the primary reason why aircraft can fly). However, all these situations
produce small negative pressures, which vanish next to the ρc2 term. In
modern cosmology we deal with media with negative pressures comparable
to ρc2 .
When a gas with positive pressure expands in a cylinder with piston, it
performs work and loses energy. Due to the First Law of Thermodynamics,
its density drops. When we put a medium with negative pressure in a
cylinder with piston, it gains energy when expanding, and its density can
change both directions.
In the next Chapter we introduce the so-called cosmological constant
proposed by Einstein. It is equivalent to a medium with some energy density
and pressure, which retain their values regardless of the cosmological
expansion. The pressure is negative and corresponds to the equation of state

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w = −1. This is precisely the type of the medium we are talking about:
it has negative pressure and thus gains energy when expanding. The total
balance keeps energy density the same. But we are not sure that we deal with
cosmological constant and not some medium with similar manifestations,
in which the balance is broken. If it gained slightly less energy its density
would decrease, and if it gained more energy its density would increase.
This generalised cosmological constant is called dark energy and will be
discussed in Chapter 5.
Let us put this dark energy thing into the Cavendish machine (in our
imagination, naturally). What we get is the antigravity, i.e. gravitational
repulsion. This is not some kind of trick due to buoyancy force in air;
it works in pure vacuum. The reason is that the sum ρc2 + 3p becomes
negative (the equation of state is such that w < −1/3), and so does the
gravitational force. For the cosmological constant w = −1, and antigravity
is very prominent.
According to modern views, dark energy with an equation of state very
close to that of the cosmological constant constitutes the majority of the
Universe’s contents — about 69 per cent. It is the gravitational repulsion
or antigravity caused by the dark energy that provides the accelerated
expansion of the Universe.
So far we considered the gravitational interaction between baryonic
matter represented by lead balls and various exotic types of matter. But there
is no reason why we could not consider the interaction between two exotic
types of matter. According to the third Newton’s law, the force exerted on
the exotic matter by the baryonic matter exactly equals the force exerted on
the baryonic matter by the exotic matter. Using this fact and the equations of
state given above, it is possible to calculate gravitational forces between any
two types of matter. We leave these calculations to readers as an exercise.

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Chapter 2

The Expanding Universe
2.1 Einstein’s Static Universe
In his 1917 paper Einstein considered the Universe homogeneously filled
with matter. The first result he obtained was that the force of attraction
would bring all the matter together. Unlike the older Herschel model where
all the matter was confined within a finite volume, Einstein considered
an infinite Universe uniformly filled with matter. The contraction of such a
Universe as a whole did not mean that it changed its size, but that its density
increased in every point of space. This can be understood by considering
any finite part of the Universe, which would shrink and become denser. He
believed, as any other scientist of that time, that the Universe was static,
i.e. did not change with time.
For this reason he needed some kind of a repulsion force to compensate
the gravity. Einstein introduced this force artificially by adding in his GTR
equations an additional summand containing the so-called cosmological
constant. Einstein denoted it λ; nowadays it is customary to denote it ,
so the corresponding summand in Einstein equation is also called lambda
term. The cosmological constant was proposed by Einstein without any
empirical foundations, merely as a consequence of a hypothesis that the
Universe should be static. The cosmological constant provided, in addition
to the Newtonian gravitational attraction, some sort of a repulsion force
between any two objects in the Universe. At a certain matter density these
two forces would compensate each other, making the Universe static.
It was very soon realized that Einstein’s static Universe is unstable. If
some region becomes slightly denser for some reason, it will attract more
matter, becoming even denser and so on. Similarly, regions with lower
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density become more rarefied. This is one of the manifestations of the
gravitational instability. Furthermore, the whole static Universe itself is
also unstable. If it becomes slightly larger, the forces of repulsion will
outperform the forces of attraction and the Universe will expand infinitely.
If it becomes slightly smaller, the attraction will dominate and the Universe
will collapse into a point. This circumstance prompted Einstein to abandon
his solution.
2.2 Expansion and Redshift
2.2.1 Other galaxies and their recession
Initially, cosmology was considered a purely theoretical science because
its conclusions could not be verified. This rapidly changed after Edwin
Hubble discovered in 1923 that the Andromeda nebula was located beyond
our Galaxy and soon was classified as a separate galaxy. Thus, it turned out
that our Galaxy was just one of many galaxies.a By 1929 many galaxies
were known and for some of them radial velocitiesb and distances were
measured. These data surprised the astronomers because they yielded that
most galaxies move away from us with very high velocities. Based on
the data about 24 galaxies, Hubble in 1929 obtained his famous lawc
that the galaxies’ velocity v is proportional to the distance r to them. It
is mathematically expressed with the formula
v = Hr.

(2.1)

The coefficient H is called the Hubble constant.
a Please note that the word “galaxy” is capitalized in mid-sentence only when referring to

our Galaxy — the Milky Way.
bA radial velocity of an object is the projection of its velocity vector on the line of sight —
a line connecting this object to the observer.
c The authorship of the Hubble’s law is disputed by some historians (http://arxiv.
org/abs/1104.3031), who attribute it to Georges Lemaître, who allegedly published this result
in French in the 1927 issue of the local journal Annals of the Scientific Society of Brussels,
and then published an English translation in the 1931 issue of Monthly Notices of Royal
Astronomical Society, which lacked two pages containing this result for an unknown reason.
Some of them went as far as blamed Hubble for plagiarism (http://arxiv.org/abs/1106.3928),
although this conclusion is disputed by other historians (http://arxiv.org/abs/1107.0442). We
provide the links to the papers expressing different points of view on the subject, leaving it
to readers to make conclusions by themselves. The original Hubble’s article is available at
(http://www.pnas.org/content/15/3/168.full.pdf+html).

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According to GTR, the value of H changes with time, but very slowly —
the time scale of its change is comparable to the age of the Universe which
is now estimated as 13.8 · 109 years. This often leads to confusion because
a variable value H is historically referred to as Hubble constant. Its current
value H0 is called Hubble parameter. This quantity is usually measured in
kilometres per second per megaparsecd (designation (km/s)/Mpc).
The Hubble parameter is one of the most important cosmological
parameters. It is necessary to measure distances to remote objects (more on
that later in Section 2.9), it is directly related to the age of the Universe, and
it is used to calculate many other cosmological parameters, such as matter
density. Thus, an improvement of its accuracy also improves the accuracy
of cosmological parameters and thus leads to better understanding of the
Universe.
The latest estimations of the Hubble parameter are provided by the
Planck spacecraft (2013): H0 = (67.80 ± 0.77)(km/s)/Mpc, Sloan
Digital Sky Survey (2016): H0 = (67.6 ± 0.7)(km/s)/Mpc, and Hubble
Space Telescope (2016): H0 = (73.00 ± 1.75)(km/s)/Mpc. These three
estimations are different because they are provided by different methods
and are completely independent of each other. Some of them could be biased
due to some unaccounted systematic errors.
2.2.2 Expansion
Galaxy recession and the Hubble’s Law mean that the Universe is
expanding. How to understand this? How could one visualise the expansion
of the Universe in a uniform world without a fixed centre? Consider a
two-dimensional Universe made of an elastic film, with galaxies lying
on it. This film stretches itself, increasing the distance between galaxies,
since they are fixed with respect to a local patch of the film. This is
the expansion of the Universe. Note that the galaxies themselves do not
expand with the film because mutual gravitational attraction is dominant at
galactic scale. In other words, non-Hubble motions, i.e. local small-scale
d Parsec (parallax-second, designation pc) is a widely used astronomical length unit defined

as a distance from which the Earth’s orbit has the angular size 1 second of arc. Parsec is
expressed through the astronomical unit (mean distance between the Sun and the Earth,
designation AU) and is connected to other length units through the following relations:
1 pc ≈ 206.26 · 103 AU ≈ 3.26156 ly ≈ 30.857 · 1015 m. Here ly means light year.

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motions overpower the global Hubble expansion on the cosmological scale.
As a result, nearby galaxies do not move apart, but move in a common
gravitational well.
Our well contains the Local Group galaxies, which include the Milky
Way, Andromeda, Triangle, both Magellanic Clouds, and about a hundred
dwarf galaxies. These deviations from the general expansion are the result
of deviations from homogeneity of the Universe. Naturally they cannot
be described in the frame of homogeneous cosmology and require special
treatment. At larger scale the Universe is quite homogeneous and we can
use the Hubble law.
The expansion means the increasing of the spatial scale of the Universe
with time. We write “spatial scale” and not “size of the Universe” because
the latter term is ill-defined if the Universe is infinite. To quantify the
spatial scale, we introduce a value a called scale factor. We will define
it in Section 2.6. In a truly homogeneous Universe there are no non-Hubble
motions and all distances between any individual points, tied to the matter
filling this Universe, are proportional to each other and grow at the same
rate. In other words, if we measure the distance between two remote objects,
the scale factor describes how this distance changed over time.
In terms of the scale factor the Hubble constant is its relative rate of
change: H = a−1 da/dt, i.e. its absolute rate of change divided by its value,
all measured at the same epoch. Thus, all that matters is not the exact value
of the scale factor a, but a ratio of its values at different epochs, which we
shall refer to as a relative scale factor u. For practical reasons, a scale factor
at the present epoch is used as a denominator of this ratio: u = a/a0 , where
a0 is the current value of the scale factor.
So all we need to calculate the time-depending Hubble constant is
a ratio of the distances between any far enough objects (so they are not
gravitationally bound) at all given epochs to its value at the present epoch.
Thus it can be obtained without an exact definition of scale factor. We will
return to a further discussion of the concept of scale factor in Section 2.6.
2.2.3 Redshift
The expansion of the Universe is manifested in the so-called redshift of
the radiation spectrum. Every distant astronomical object like a galaxy or
a quasar has its spectrum of radiation shifted. This shift is usually towards

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increasing wavelengths — hence the name redshift — and characterises its
radial velocity.
It is possible to explain it in two ways, all describing the same effect.
Note that only one of them should be used to avoid taking this effect into
account multiple times.
The first approach connects the redshift with the expansion of space
itself (and the light waves) during the propagation of light. As a result, the
observed wavelength λobs is longer than the emitted wavelength λemit . Their
ratio equals the relative scale factor at the epoch the light was emitted.
The quantitative characteristic of redshift is given by value z, also
called redshift, which equals z = 1/u − 1. It equals zero for local objects
and tends to infinity for very distant objects. Redshift is commonly used
to determine the distance to the most remote objects. At extremely large
distances, astronomers prefer using redshift to the expansion velocity.
The second approach is treating the redshift as a result of the Doppler
effect considering that the emitting object moves away at the expansion
velocity. For velocities less than the speed of light c, its value is given by
√
the formula for the Doppler effect: 1 + z = 1/u = (1 + v/c)/(1 − v/c).
At small redshifts z  1 this can be approximated as v = cz.
The expansion velocity is converted into distance using the Hubble’s
law. Due to the uncertainty in the value of the Hubble’s parameter,
the distance is often expressed through its dimensionless value h =
H0 /(100(km/s)/Mpc). So the expansion velocity of 10000 km/s translates
to the distance of 100 h−1 Mpc, sometimes shortened to 100 Mpc/h. The
reason for doing so is that the accuracy of the redshift measurement is much
higher than that of Hubble’s parameter, and the distances written this way
stay current after an update to the Hubble’s parameter value.
We emphasize once again that these are not two different effects, but
two different explanations of the same effect. They should not be combined
together.
Note that non-Hubble motions also cause a redshift, but only in the
sense of the Doppler effect. No stretching of space occurs when a galaxy
falls towards a density enhancement. For example, nearby galaxies are
sometimes blueshifted, which means that they move towards us. This effect
has nothing to do with the expansion of the Universe and is caused by
the mutual attraction of nearby galaxies, for example of the Milky Way

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and Andromeda. Therefore, astronomers do not use radial velocities to
determine distance to nearby objects, but for faraway objects this is the
main method of distance measurement.
2.3 Hubble’s Law∗

In this Section we derive Hubble’s law assuming
isotropic and homogeneous Universe.
Consider three points 1, 2 and 3 somewhere in the Universe, which
form a triangle, as shown in Figure 2.1. The lengths of the triangle’s sides
are r21 , r31 and r32 . The length r31 depends not only on the length of the
other two sides, but also on the angle between them. By changing the angle,
we can get any value for r31 in the range from |r21 − r32 | to r21 + r32 .
Due to the cosmological expansion the points recess from each other.
Consider the motion of particles in two other points observed from a certain
point, say point 1. Each of the points 2 and 3 can move only radially
away from the point 1, otherwise the Universe would be anisotropic. In
an isotropic Universe there are no preferred directions except for the radial
direction. The velocities of all particles at a distance r from the observer
should be the same regardless of the direction; otherwise the Universe
also would be anisotropic. Thus, the isotropy condition fixes the rate of
expansion in the form v = f(r)r , where f(r) is some still unknown
function.

Figure 2.1. To the explanation of the Hubble’s law.

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The velocity of point 2 relative to the observer at point 1 equals v21 =
f(r21 )r21 . Point 3 moves at the speed of v32 = f(r32 )r32 relative to the
point 2. Adding these, we obtain that point 3 moves relative to the point 1 at
the speed v31 = v21 +v32 = f(r21 )r21 +f(r32 )r32 . On the other hand, it also
equals v31 = f(r31 )r31 = f(r31 )(r21 + r32 ). This gives us the condition
(f(r21 ) − f(r31 ))r21 = (f(r31 ) − f(r32 ))r32 .

(2.2)

Because vectors r21 and r32 may have different directions, it is possible
only if
f(r21 ) = f(r31 ) = f(r32 ) = const.

(2.3)

Thus, the function f reduces to a constant, which we call the Hubble
constant H. Thus, in a homogeneous and isotropic universe at any given
time the only possible law of expansion is the Hubble’s law (2.1).
Note that when deriving the Hubble’s law, we ignored relativistic
effects. In relativistic case we no longer can simply add velocities, and
formulae become trickier.
Let us derive them. According to Special Relativity if a body moves
with the velocity v relative to the fixed observer and a second body moves
in the same direction with the speed w relative to the first one, then the
velocity of the second body relative to the fixed observer is
V =

v+w
1 + vw/c2

(2.4)

This equation can be rewritten in the form
tanh−1

V
v
w
= tanh−1 + tanh−1 .
c
c
c

(2.5)

The inverse hyperbolic tangent can be reduced to the natural logarithm as


1+x
1
−1
tanh x = ln
.
(2.6)
2
1−x
If the Hubble constant H was really constant, then we would have to use
the function c · tanh−1 (v/c) instead of v in the Hubble law (2.1) and get


Hr
c
c+v
−1 v
v = c tanh , Hr = c tanh
= ln
.
(2.7)
c
c
2
c−v

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However, for velocities much less than the speed of light the Hubble law in
the form (2.1) works well.
Note that the velocity v of recession of galaxies and other objects in
space cannot reach the speed of light c and there is no contradiction with
Special Relativity. The distance r = c/H is thus not the border of the
observable part of the Universe, as many erroneously believe. In Section 3.4
we discuss the cosmological horizon, which is a real border of visible part
of the Universe, but it exists for a completely different reason.
In any case the time dependence of the Hubble constant leads to the fact
that scientists use the Hubble law in its simple form (2.1) only for objects
located not too far away with v  c. More distant objects are characterised
by their redshifts z. Their light was emitted long ago when the value of the
Hubble constant differed from the current Hubble parameter.
Question: Why do galaxies move away from us? Are we in the centre of
the Universe?
Answer: Any point in the Universe is as good as any other. In Figure 2.1,
point 3 moves away from point 1 at the velocity v31 = Hr31 , and point 2
moves away from point 1 at the velocity v21 = Hr21 . If we switch to the
frame of point 2, the radius vector of point 3 would be r32 = r31 − r21 ,
and the velocity would be v32 = v31 − v21 = Hr31 − Hr21 = Hr32 . So,
the Hubble’s law holds regardless of the location of the origin.
Question: How could one estimate the age of the Universe from Hubble’s
law?
Answer: Any two galaxies separated by the distance r move away from
each other with the velocity v = Hr. Let us find when the distance
between them was zero. Dividing the distance by the velocity we get
T = r/v = H −1 . Inverting the Hubble parameter H0 = 68(km/s)/Mpc
we get the estimated age of the Universe 14.4 · 109 yr. Note that this is
a rough estimation, because the Hubble constant changes with time and
the expansion velocity changes with distance.
Question: Why the value H, which changes with time, is called Hubble’s
constant?
Answer: Inverting the result of the previous problem, we estimate the
Hubble’s constant as H = T −1 , where T is age of the Universe. Thus, a
relative change of H in 1 year equals 1 yr/13.8 · 109 yr = 7.25 · 10−11 .
Thus, in 75 years since the discovery of the Hubble’s law this quantity

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changed approximately by 5 parts in 109 . Like before, this estimation is
correct only to an order of magnitude. The reason is the influence of the
so-called dark energy, which we will explain later.

2.4 Friedmann Models
The Hubble’s law is so important for cosmology because it was its first
serious observational confirmation, as this dependence followed from
theoretical predictions made shortly before its discovery.
In 1922 a Russian physicist Alexander Friedmann derived the solution
of Einstein equation, which described the whole Universe without a
cosmological constant. The peculiarity of this solution was that the Universe
was dynamic, i.e. it first expanded and then, depending on its initial density,
either continued expanding eternally, or compressed. In 1927 the same
solution was also derived by a Belgian priest Georges Lemaître, and in
1935 — by an American physicist Howard P. Robertson and a British
mathematician Arthur G. Walker. In English literature this solution is often
called FLRW solution after their initials (in older books it is sometimes
cited as FRW unfairly omitting Lemaître). We refer to it as Friedmann’s
solution recognizing his priority. This solution is extremely important
because it lies at the foundation of all modern cosmological theories, which
combine it with an idea of a cosmological constant. Although it is known
today that the original Friedmann’s solution does not describe the real
Universe, it is very important to learn its properties in order to understand modern cosmological models. For this reason, we give its detailed
description.
Friedmann and his successors applied Einstein’s equation to a homogeneous and isotropic Universe. Let us remind that in such a world there
are no chosen places and preferred directions, every point is as good as any
other, and each direction is neither better nor worse than any other. These
requirements are sufficient in order to obtain, for example, the Hubble’s
law as the most general solution, like we did in Section 2.3.
Friedmann’s solution describes three physically different situations,
distinguished by the value of the ratio of the Universe’s matter density
to the so-called critical density, which depends on the Hubble’s constant.
The exact formula for critical density (2.11) will be given a bit later in
the advanced section. This ratio is called matter density parameter and is

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denoted m . When this parameter is less than unity, the Universe has infinite
volume and expands forever with the Hubble’s constant tending to a positive
value. This situation is called an open model or an open Universe. When
the density parameter is equal to unity, the Universe is also infinite and also
expands forever, but the Hubble’s constant tends to zero. This situation is
called a flat model or a flat Universe, and differs from the open model by its
asymptotic behaviour in the future: an open Universe expands faster than
a flat Universe.
A totally different situation arises when the density parameter is greater
than unity. In this case, which is called a closed model or a closed Universe,
the Universe has finite volume, yet has no edges. To understand how this
is possible, imagine a globe or any other sphere, whose area is finite, yet
there is no “edge of the world”. The closed Universe looks exactly the same
but in 3 dimensions.e At some point a closed Universe stops expanding and
starts contracting, which is accompanied by the reversal of the sign of the
Hubble’s constant, and some time after that it collapses into a singularity
called Big Crunch.
Note that although both the matter density and the critical density vary
over time, their ratio m cannot cross the value equal to unity. In other
words, the type of the model — open, flat, or closed — is fixed and cannot
change.
We illustrate these models in Figure 2.2 by plotting how their key
parameters (scale factor and the Hubble constant) change with time.
Immediately after the Big Bang (to be described in Section 3.1) the Hubble
constant was infinitely large and immediately before the Big Crunch in the
closed model it will tend to minus infinity. Naturally, it vanishes when the
closed universe achieves its maximal size.
From the fact that the Universe is expanding now, we conclude that it is
either open or flat, or closed but on the expansion phase. In any case, its scale
grew monotonically so far. The plot of the scale factor from the Big Bang
to the current epoch looks like the curve in Figure 2.2. The wavelength of a
photon emitted at some epoch increased 1/u times, where u is the relative
scale factor at that epoch. Therefore, its redshift equals z = 1/u − 1. The
e The ability to imagine space with more than 3 dimensions is extremely useful when studying
Relativity.

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Figure 2.2. Relative scale factor (left) and Hubble constant (right) vs. time for three
Friedmann models.

older is the light emitted by the object, the further is the object, and the
greater is its redshift. This is the reason why redshift is a good proxy for
distance.
2.4.1 Arrow of time
One philosophical issue to consider with respect to Friedmann models is
the so-called arrow of time — a term introduced by Eddington. Time is
different from spatial coordinates because any object including ourselves
must move along the time axis in a pre-defined direction regardless of
its will. The arrow of time characterizes this direction: from the past to
the future. We can distinguish between past and future using the causality
principle, which states that any cause must precede its consequence. The
direction of the arrow of time due to the causality principle is called a
causal arrow of time. There are also other arrows of time, which we do not
describe here. Although it is a philosophical concept, it is very important for
physics. For example, General Relativity has solutions with closed timelike
trajectories, which allow any object to return to the original location is
space-time visiting its past en-route. Such solutions are marked as nonphysical due to the violation of the causality principle. Thus, it is believed
that the arrow of time is always present in real world.
How does the arrow of time manifest itself? The laws of mechanics
are symmetric to the reversal of time until a dissipative force such as
friction appears. Thermodynamic processes are split into reversible, which

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are insensitive to the arrow of time, and irreversible, where the arrow of
time is the direction in which entropy increases. From the perspective of the
elementary particle physics, the Universe should be CPT-invariant.f Since
CP symmetry is broken, there is also no symmetry to the reversal of time.
In electrodynamics the situation is a bit trickier: Maxwell’s equations are
symmetric to the reversal of time, and the asymmetr