Contents -- Notations -- Chapter 1. Modern ideas of gravitation and cosmology - a brief essay -- Einstein after Einstein -- The technological breakthrough -- To quantize or not? -- The zoo of theories -- Gravitation and the Universe -- Part I Gravitation -- Chapter 2. Fundamentals of general relativity -- 2.1 Special relativity.Minkowski geometry -- 2.1.1 Geometry -- 2.1.2 Coordinate transformations -- 2.1.3 Kinematic effects -- 2.1.4 Elements of relativistic point mechanics -- 2.2 Riemannian space-time. Coordinate systems and reference frames -- 2.2.1 Covariance, maps and atlases -- 2.2.2 Reference frames and relativity -- 2.2.3 Reference frames and chronometric invariants -- 2.2.4 Covariance and relativity -- 2.3 Riemannian space-time. Curvature -- 2.4 The gravitational field action and dynamic equations -- 2.4.1 The Einstein equations -- 2.4.2 Geodesic equations -- 2.4.3 The correspondence principle -- 2.5 Macroscopic matter and nongravitational fields in GR -- 2.5.1 Perfect fluid -- 2.5.2 Scalar fields -- 2.5.3 The electromagnetic field -- 2.6 The most symmetric spaces -- 2.6.1 Isometry groups and killing vectors -- 2.6.2 Isotropic cosmology. The dS and AdS spaces -- Chapter 3. Spherically symmetric space-times. Black holes -- 3.1 Spherically symmetric gravitational fields -- 3.1.1 A regular centre and asymptotic flatness -- 3.2 The Reissner-Nordstrom-(anti-)de Sitter solution -- 3.2.1 Solution of the Einstein equations -- 3.2.2 Special cases -- The (anti-)de Sitter metric -- The Schwarzschild metric and the Newton law -- The Reissner-Nordstrom metric -- Metrics with a nonzero cosmological constant -- 3.3 Horizons and geodesics in static, spherically symmetric space-times -- 3.3.1 The general form of geodesic equations -- 3.3.2 Horizons, geodesics and the quasiglobal coordinate -- 3.3.3 Transitions to Lemaıtre reference frames.

3.3.4 Horizons, R- and T-regions -- 3.4 Schwarzschild black holes. Geodesics and a global description -- 3.4.1 R- and T-regions -- 3.4.2 Geodesics in the R-region -- 3.4.3 Particle capture by a black hole -- 3.4.4 A global description: The Kruskal metric -- 3.4.5 From Kruskal to Carter-Penrose diagram for the Schwarzschild metric -- 3.5 The global causal structure of space-times with horizons -- 3.5.1 Crossing the horizon in the general case -- 3.5.2 Construction of Carter-Penrose diagrams -- 3.6 A black hole as a result of gravitational collapse -- 3.6.1 Internal and external regions. Birkhoff's theorem -- 3.6.2 Gravitational collapse of a spherical dust cloud -- Chapter 4. Black holes under more general conditions -- 4.1 Black holes andmassless scalar fields -- 4.1.1 The general STT and the Wagoner transformations -- On phantom fields -- 4.1.2 Minimally coupled scalar fields -- 4.1.3 Conformally coupled scalar field -- Solutions with nonconformal coupling -- 4.1.4 Anomalous (phantom) fields. The anti-Fisher solution -- 4.1.5 Cold black holes in the anti-Fisher solution -- 4.1.6 Vacuum and electrovacuum in Brans-Dicke theory -- 4.1.7 Summary for massless scalar fields -- 4.2 Scalar fields with arbitrary potentials. No-go theorems -- 4.2.1 What is the use of no-go theorems? -- 4.2.2 Basic equations -- 4.2.3 Global structure theorems -- 4.2.4 No-hair theorem -- 4.2.5 Two expressions for the mass and the properties of particle-like solutions -- 4.3 Rotating black holes -- 4.4 Black hole thermodynamics -- 4.4.1 Four laws of BH thermodynamics -- 4.4.2 Black hole evaporation -- 4.5 Regular black holes and black universes -- 4.5.1 Different kinds of regular black holes -- 4.5.2 Black universes with a minimally coupled scalar field -- 4.5.3 A black universe in a brane world -- Reasons for neglecting -- Brane gravity with a scalar field.

4.5.4 A black universe with a trapped ghost -- Chapter 5. Wormholes -- 5.1 The notion of a wormhole -- 5.2 A wormhole as a time machine -- 5.3 Wormholes as solutions to gravitational field equations -- 5.3.1 Spherically symmetric wormholes. General properties -- 5.3.1.1 Wormholes with scalar fields -- Wormholes in the anti-Fisher solution -- Wormholes with nonminimally coupled scalar fields -- 5.3.2 Wormhole construction by solving the trace of the Einstein equations -- 5.3.3 Alternative gravity and vacuum as wormhole supporters -- 5.4 Observational effects. Wormhole astrophysics -- Chapter 6. Stability of spherically symmetric configurations -- 6.1 Preliminaries -- 6.2 Perturbation equations -- 6.2.1 General form of the field equations -- 6.2.2 Gauge δβ ≡ 0 -- 6.2.3 Gauge-invariant perturbations -- 6.2.4 Regularized potential near a throat -- 6.2.5 Regular perturbations near a throat -- 6.3 Instabilities of the Fisher and anti-Fisher solutions -- 6.3.1 The static solutions -- 6.3.2 Perturbations: The Fisher solution -- 6.3.3 Perturbations: The anti-Fisher solution -- Branch A -- Branch B -- 6.4 Extensions and related problems -- Part II Cosmology -- Chapter 7. Stages of the Universe's evolution -- 7.1 The cosmological principle and the Einstein equations -- Some solutions for the scale factor -- 7.2 De Sitter space -- Is there real expansion in de Sitter space? -- 7.3 Inflation -- 7.4 Post-inflationary stages -- 7.4.1 Post-inflationary reheating of the Universe -- 7.4.2 The radiation-dominated stage -- 7.4.3 The matter-dominated stage -- 7.4.4 The modern stage of accelerated expansion (secondary inflation) -- 7.4.5 Future of the Universe: Is a Big Rip expected? -- 7.5 The scale factor in the general case -- 7.6 Why do we need an inflationary period? -- 7.6.1 The flatness problem -- 7.6.2 The initial size of the Universe.

7.6.3 Causal connections at inflation and after it -- 7.7 Basic properties of expanding space -- 7.7.1 The redshift -- 7.7.2 The luminosity distance -- 7.7.3 The velocity of particles in FRW space-time -- Chapter 8. Field dynamics in the inflationary period -- 8.1 Quadratic inflation -- 8.2 Quantum fluctuations during inflation -- A detailed consideration -- Homogeneous fluctuations -- 8.3 Hybrid inflation -- 8.4 Influence of massive fields on the process of inflation -- 8.5 Suppression of vacuum decay by virtual particles -- Chapter 9. The large-scale structure -- 9.1 The cosmic microwave background -- Classical evolution of quantum fluctuations -- 9.2 The development of density fluctuations -- 9.2.1 Density fluctuations in Minkowski space -- 9.2.2 Density perturbations in the expanding Universe -- 9.3 The baryonic asymmetry of the Universe -- 9.3.1 Baryogenesis -- 9.3.2 Large-scale fluctuations of the baryonic charge -- 9.4 Massive primordial black holes -- 9.4.1 Field fluctuations near an extremum of the potential -- 9.4.2 A specific example -- 9.4.3 Suppressed intermediate-mass black hole formation -- 9.4.4 PBH mass spectra and the scalar field dynamics -- 9.4.5 Discussion -- Part III Extra Dimensions -- Chapter 10. Multidimensional gravity -- 10.1 Compact extra dimensions. A brief review -- 10.1.1 A Kaluza-Klein model with a single extra dimension -- How to extract gauge symmetry from extra dimensions -- Generalization: G55 const -- The geometric meaning of gµν -- Vielbeins -- 10.1.2 Kaluza-Klein models. The general case -- Killing vectors -- A non-Abelian gauge group as an isometry group of the extra space -- 10.2 Multidimensional gravity with higher-order derivatives. Basic equations -- 10.2.1 F(R)-theory -- 10.2.2 Slow-change approximation. The Einstein frame -- Some estimates.

10.2.3 The first generalization: A more general form of the Lagrangian -- 10.2.4 The second generalization: Several extra factor spaces -- 10.2.5 Slow-change approximation. Reduction to d0 dimensions -- 10.3 Extra dimensions and low-energy physics -- 10.3.1 Self-stabilization of an extra space -- 10.3.2 On the influence of the number of extra dimensions on low-energy physics -- 10.3.3 Extra dimensions and inflation -- A single extra factor space -- 10.3.4 Two factor spaces: Inflation and modern acceleration -- Inflation -- Matter-dominated stage -- Modern stage -- Discussion -- 10.3.5 Rapid particle creation in the post-inflationary period -- The field behavior near K = 0 -- Multiple factor spaces and a spatially varying size of the extra dimensions -- 10.3.6 Conclusions -- 10.4 The origin of gauge symmetries and fundamental constants -- 10.4.1 Why is the extra space symmetric? -- Entropy of a compact space -- Decay of excitations of a compact space -- The transition rate to a symmetric state -- 10.4.2 Fundamental constants and the properties of an extra space -- Chapter 11. The emergence of physical laws -- 11.1 Fine tuning of the universe parameters -- The electron and the properties of the Universe -- The carbon level -- Slow reactions in stars -- Stellar lifetime -- Passage through a needle's eye -- It is good time to ask a question: what should an Ultimate Theory look like? -- Choosing particles and interactions -- 11.2 Fine tuning mechanisms -- 11.2.1 Cascade birth of universes in multidimensional spaces -- 11.2.2 Simultaneous formation of space-time and the parameters of the theory -- 11.2.3 Reduction cascades -- 11.2.4 A step of the cascade in detail -- Quantum formation of space -- The classical evolution stage -- 11.3 Quadratic gravity as an explicit example -- 11.3.1 Formation of low-energy physics -- 11.3.2 Numerical computations.

11.4 Discussion.