# Main Page

## Contents |

# Dynamics of the Universe in Problems

**The total number of problems on this website.**

### About

There are thousands of websites with names that contain the term “cosmology”. Many of them are devoted to discussion of fundamental questions: whether there is life on Mars, what was there when there was nothing and the like. Our aim is much more modest. We present here an online living version of our book of problems on cosmology.

### Why problems, which and for whom?

The only way to rise above the “popular” level in any science is to master its alphabet, that is, to learn to solve problems, even if most simple at the beginning. To our best knowledge, there are no problem books on cosmology yet, that would include its spectacular recent achievements. Of course, most of excellent modern textbooks on cosmology include problems. However, a reader, exhausted by high theory, may often be thwarted by the lack of time and strength to solve them.

You can familiarize oneself with the modern portrayal of the dynamics of the Universe, including the latest achievements of cosmology, by solving problems from the start. The perseverant will traverse the thorny path of cosmology’s evolution from the traditional Big Bang model to the presently universally accepted Standard Cosmological Model. It is based on the revolutionary discovery of accelerated expansion of the Universe, made at the very end of the last century. The attempts to explain such an unexpected observation led to fundamental reconstruction of the view on the evolution of the Universe that was formed in the XX century.

One only has to be acquainted with the basics of Special and General Relativities and the theory of elementary particles in order to start solving problems. We want to show that even this minimal knowledge is sufficient to be able to solve a very wide range of cosmological problems, and moreover, to understand the essence of difficulties that are encountered by modern cosmology.

No generation of Homo sapiens could withstand the temptation to claim that the true understanding of the nature of the Universe is (almost?) within its grasp. The current generation is not an exception, of course, naturally assuming it has better reason to believe so than ever before. We hope to give our reader the means to evaluate objectively whether this is true or not by himself.

### Second life online

We published the first edition of the problem book a couple of years ago, in Russian. However, a printed book, even not taking into account the language, has its obvious drawbacks in itself. The audience is limited by the number of copies printed. The inevitable misprints and errors cannot be corrected – the errors in classical problem books on physics are known to be sometimes corrected in the fifteenth edition. But the main problem that almost immediately manifested itself was that the content of the book with pretentious title “Modern Cosmology” was getting out of date by leaps and bounds. New sections and subsections demanded to be created and populated with problems. The online living problem book is the solution we have come to.

At the moment we have about 1500 problems, almost all of them with solutions, and we hope to bring them online by the fall of 2012. We tried to make references to the authors of problems whenever possible and would be happy to add the references that might still be missing. Some problems have already acquired the status of folklore and their authorship is hard to determine. We welcome all comments, suggestions and corrections to the formulations of problems and their solutions, but most valuable to us would be new problems suggested by the community.

### Good luck

Finally, we hope that after getting to grip with the problems our reader not only preserves his interest to cosmology, but tries to make the next step: read original papers. If this transition is overcome seamlessly, we will have achieved our goal.

Welcome!

# CONTENTS

1.1 Astronomy "before the Common Era" 1.2 Quantities large and small 1.3 Geometric warm-up 1.4 Astrophysical warm-up 1.5 Planck scales and fundamental constants $c$, $G$, $\hbar$ 1.6 Gravity 1.7 Forest for the trees 1.8 Life on Mars? 1.8.1 Fine-tuning of the Universe 1.9 Thermo warm-up

2. **Dynamics of the Expanding Universe**

2.1 Homogeneous and isotropic Universe, Hubble's Law 2.2 Equations of General Relativity 2.3 Friedman-Lemaître-Robertson-Walker (FLRW) metric 2.4 Expanding Universe: ordinarity, difficulties and paradoxes 2.4.1 Warm-up 2.4.2 The tethered galaxy problem 2.4.3 Cosmological redshift 2.5 Friedman equations 2.6 Newtonian cosmology 2.7 Energy balance in an expanding Universe 2.8 Cosmography 2.9 Light and distances

3. **Dynamics of the Universe in the Big Bang Model**

3.1 General questions 3.2 Solutions of Friedman equations in the Big Bang model 3.3 Polytropic equation of state 3.4 The role of curvature in the dynamics of the Universe 3.5 The Milne Universe 3.6 Cosmological horizons 3.7 Energy conditions and the Raychaudhuri equation 3.6.1 Energy conditions 3.6.2 Raychaudhuri equation 3.6.2 Sudden future singularities 3.8 Influence of cosmological expansion on local systems 3.9 Dynamics of the Universe in terms of redshift and conformal time

4. **Black Holes**

4.1 Technical warm-up 4.1.1 Uniformly accelerated observer, Rindler metric 4.1.2 Metric in curved spacetime 4.2 Schwarzschild black hole 4.2.1 Simple problems 4.2.2 Symmetries and integrals of motion 4.2.3 Radial motion 4.2.4 Blackness of black holes 4.2.5 Orbital motion, effective potential 4.2.6 Miscellaneous problems 4.2.7 Different coordinates, maximal extension 4.3 Kerr black hole 4.3.1 General axially symmetric metric 4.3.2 Limiting cases 4.3.3 Horizons and singularity 4.3.4 Stationary limit 4.3.5 Ergosphere and the Penrose process 4.3.6 Integrals of motion 4.3.7 The laws of mechanics of black holes 4.3.8 Particles' motion in the equatorial plane 4.4 Particles' motion in general black hole spacetimes 4.4.1 Frames, time intervals and distances 4.4.2 Fiducial observers 4.4.3 Collision of particles: general relationships 4.5 Astrophysical black holes 4.5.1 Preliminary 4.5.2 Quantum effects

5. **Cosmic Microwave Background (CMB)**

5.1 Thermodynamics of Black-Body Radiation 5.2 Time Evolution of CMB 5.3 Statistical properties of CMB 5.4 Primary anisotropy of CMB 5.5 CMB interaction with other components 5.6 Extras

6.1 Thermodynamical Properties of Elementary Particles 6.2 Thermodynamics of Non-Relativistic Gas 6.3 Entropy of Expanding Universe 6.4 Connection between Temperature and Redshift 6.5 Peculiarities of Thermodynamics in Early Universe 6.6 The Saha equation 6.7 Primary Nucleosynthesis 6.8 Extras

7. **Perturbation theory in cosmology**

7.1 Non relativistic small perturbation theory 7.2 Introduction to relativistic theory of small perturbations 7.2.1 Perturbations on flat background 7.2.2 Metrics perturbations, coordinates transforms and perturbed energy--momentum tensor 7.3 Expansion of cosmological perturbations in helicities 7.3.1 Scalar perturbations in conformal Newtonian gauge 7.3.2 Evolution of vector and tensor perturbations 7.4 CMB anisotropy 7.5 Initial perturbations in the Universe 7.5.1 Fluctuations power spectrum: non-relativistic approach

8. **Inflation**

8.1 Problems of the Hot Universe (the Big Bang Model) 8.1.1 The Horizon Problem 8.1.2 The Flatness Problem 8.1.3 The Entropy Problem 8.1.4 The Primary Inhomogeneities Problem 8.2 Cosmological Inflation: The Canonic Theory 8.2.1 Scalar Field In Cosmology 8.2.2 Inflationary Introduction 8.2.3 Solution of the Hot Big Bang Theory Problems 8.3 Different Models of Inflation 8.3.1 Chaotic Inflation (The Inflation with Power Law Potential) 8.3.2 The Novel Inflation (the Inflation Near Minimum of the Potntial) 8.3.3 Inflation with Exponential Potential (the Power Law Inflation) 8.3.4 The Hybrid Inflation 8.3.5 Other Types of Inflation

9. **Dark Energy**

9.1 The Cosmological Constant 9.2 Geometry and Destiny 9.3 Time-dependent Cosmological Constant 9.4 Static Einstein's Universe 9.5 Dynamical Forms of Dark Energy 9.5.1 The Quintessence 9.5.2 The K-essence 9.5.3 Phantom Energy 9.5.4 Disintegration of Bound Structures 9.5.5 Big Rip, Pseudo Rip, Little Rip 9.5.6 The Statefinder 9.5.7 Crossing the Phantom Divide 9.6 Lost and Found 9.7 Single Scalar Cosmology

10. **Dark Matter**

10.1 Observational Evidence of the Dark matter Existence 10.2 ANONIMOUS 10.3 Dark Matter Halo 10.4 Candidates for Dark Matter Particles 10.4.1 Standard Model Particles as Dark Matter Candidates 10.4.2 Supersymmetric Candidate Particles 10.5 Dark Matter Detection 10.6 The Dark Matter in the Solar System 10.7 The Dark Stars

11. **Interactions in the Dark Sector**

11.1 Physical mechanism of energy exchange 11.2 Phenomenology of interacting models 11.3 Simple linear models 11.4 Cosmological models with a change of the direction of energy transfer 11.5 Non-linear interaction in the dark sector 11.6 Phase space structure of models with interaction 11.7 Peculiarity of dynamics of scalar field coupled to dark matter 11.7.1 Interacting quintessence model 11.7.2 Interacting phantom 11.8 Realization of interaction in the dark sector 11.8.1 Vacuum decay into cold dark matter 11.8.2 Time-dependent cosmological "constant" 11.9 Statefinder parameters for interacting dark energy and cold dark matter 11.10 Interacting holographic dark energy 11.11 Transient acceleration

12. **Standard Cosmological Model**

12.1 Characteristic Parameters and Scales 12.2 Evolution of Universe

13. **Weak field limit and gravitational waves**

13.1 Motivation and symmetries / Introduction 13.2 Linearized Einstein equations 13.3 Gauge transformations and degrees of freedom 13.4 Wave equation 13.5 Transverse traceless gauge 13.6 Gravitational waves and matter 13.7 Energy of gravitational waves 13.8 Binary systems 13.9 Gravitational Waves: scale of the phenomenon 13.10 Generation and detection of gravitational waves

14.1 Point gravitational lenses 14.2 Microlensing and weak lensing

15.1 Information, Entropy and Holographic Screen

16. **Horizons**

16.1 Simple English 16.2 Simple Math 16.3 Composite Models 16.4 Causal Structure 16.5 Conformal diagrams: stationary black holes 16.5.1 Schwarzschild-Kruskal black hole solution 16.5.2 Other spherically symmetric black holes 16.6 Hubble sphere 16.7 Proper horizons 16.8 Inflation 16.9 Holography