# Einstein’s Geometrical versus Feynman’s Quantum-Field Approaches to Gravity Physics: Testing by Modern Multimessenger Astronomy

## Abstract

**:**

Contents | ||||

1 | Introduction | 3 | ||

1.1 | Key Discoveries of Modern Multimessenger Astronomy.......................................................... | 4 | ||

1.1.1 | Gravitational Waves.................................................................................... | 4 | ||

1.1.2 | Imaging of Black Holes Candidates: Relativistic Jets and Disks......................................... | 5 | ||

1.1.3 | Tensions between Local and Global Cosmological Parameters.............................................. | 6 | ||

1.1.4 | Conceptual Problems of the Gravity Physics............................................................. | 7 | ||

1.2 | The Quest for Unification of of the Gravity with Other Fundamental Forces................................... | 8 | ||

1.2.1 | Future Unified Theory.................................................................................. | 8 | ||

1.2.2 | Quantum Electrodynamics as the Paradigmatic Theory..................................................... | 9 | ||

1.3 | Einstein’s Geometrical and Feynman’s Quantum-Field Gravitation Physics.................................... | 11 | ||

1.3.1 | Two Ways in Gravity Physics............................................................................ | 11 | ||

1.3.2 | Special Features of the Geometrical Approach........................................................... | 13 | ||

1.3.3 | Problem of the Gravitational Field Energy-Momentum in GRT.............................................. | 14 | ||

1.3.4 | Attempts to Resolve the Gravitational Energy-Momentum Problem in Geometrical Approach.................. | 15 | ||

1.3.5 | Special Features of the Feynman Approach............................................................... | 15 | ||

1.3.6 | Why Is QFGT Principally Different from GRT?............................................................ | 16 | ||

1.3.7 | Conceptual Tensions between Quantum Mechanics and General Relativity................................... | 18 | ||

1.3.8 | Astrophysical Tests of the Nature of Gravitational Interaction......................................... | 19 | ||

2 | Einstein’s Geometrical Gravitation Theory | 20 | ||

2.1 | Basic Principles of GRT..................................................................................... | 20 | ||

2.1.1 | The Principle of Geometrization........................................................................ | 20 | ||

2.1.2 | The Principle of Least Action.......................................................................... | 20 | ||

2.2 | Basic Equations of General Relativity....................................................................... | 21 | ||

2.2.1 | Einstein’s Field Equations............................................................................ | 21 | ||

2.2.2 | The Equation of Motion of Test Particles............................................................... | 21 | ||

2.3 | The Weak Field Approximation................................................................................ | 21 | ||

2.3.1 | The Metric Tensor...................................................................................... | 22 | ||

2.3.2 | The Field Equations.................................................................................... | 22 | ||

2.3.3 | The Equation of Motion in the Weak Field............................................................... | 23 | ||

2.4 | Major Predictions for Experiments/Observations.............................................................. | 23 | ||

2.4.1 | The Classical Relativistic Gravity Effects in the Weak Field........................................... | 23 | ||

2.4.2 | Strong Gravity Effects in GRT: Schwarzschild Metric.................................................... | 24 | ||

2.4.3 | Tolman-Oppenheimer-Volkoff Equation.................................................................... | 24 | ||

2.5 | Modifications of GRT to Aviod Field Energy Problem.......................................................... | 24 | ||

2.5.1 | Geometrical Approach without Black Holes?.............................................................. | 25 | ||

2.5.2 | The Energy-Momentum of the Space Curvature?............................................................ | 25 | ||

2.5.3 | Non-Localizability of the Gravitation Field Energy in GRT.............................................. | 26 | ||

2.5.4 | The Physical Sense of the Space/Vacuum Creation in the Expanding Universe.............................. | 28 | ||

2.5.5 | Conclusions............................................................................................ | 28 | ||

3 | Feynman’s Quantum-Field Approach to Gravitation Theory | 28 | ||

3.1 | Initial Principles.......................................................................................... | 29 | ||

3.1.1 | The Unity of the Fundamental Interactions.............................................................. | 29 | ||

3.1.2 | The Principle of Consistent Iterations................................................................. | 30 | ||

3.1.3 | The Principle of Stationary Action..................................................................... | 30 | ||

3.1.4 | Lagrangian for the Gravitational Field................................................................. | 30 | ||

3.1.5 | Lagrangian for Matter.................................................................................. | 31 | ||

3.1.6 | The Principle of Universality and Lagrangian for Interaction........................................... | 31 | ||

3.2 | Basic Equations of the Quantum-Field Gravity Theory......................................................... | 32 | ||

3.2.1 | Field Equations........................................................................................ | 32 | ||

3.2.2 | Remarkable Features of the Field Equations............................................................. | 33 | ||

3.2.3 | Scalar and Traceless Tensor Are Dynamical Fields in QFGT............................................... | 34 | ||

3.2.4 | The Energy-Momentum Tensor of the Gravity Field........................................................ | 36 | ||

3.2.5 | The Retarded Potentials................................................................................ | 37 | ||

3.3 | Equations of Motion for Test Particles...................................................................... | 37 | ||

3.3.1 | Derivation from Stationary Action Principle............................................................ | 38 | ||

3.3.2 | Static Spherically Symmetric Weak Field................................................................ | 39 | ||

3.3.3 | The Role of the Scalar Part of the Field............................................................... | 40 | ||

3.4 | Poincaré Force and Poincaré Acceleration in PN Approximation................................................ | 40 | ||

3.4.1 | The EMT Source in PN-Approximation..................................................................... | 40 | ||

3.4.2 | Relativistic Physical Sense of the Potential Energy.................................................... | 41 | ||

3.4.3 | The PN Correction Due to the Energy of the Gravity Field............................................... | 41 | ||

3.4.4 | Post-Newtonian Equations of Motion..................................................................... | 41 | ||

3.4.5 | Lagrange Function in Post-Newtonian Approximation...................................................... | 43 | ||

3.5 | Quantum Nature of the Gravity Force......................................................................... | 43 | ||

3.5.1 | Propagators for Spin-2 and Spin-0 Massive Fields....................................................... | 44 | ||

3.5.2 | Composite Structure of the Quantum Newtonian Gravity Force............................................. | 46 | ||

4 | Relativistic Gravity Experiments/Observations in QFGT | 47 | ||

4.1 | Classical Relativistic Gravity Effects...................................................................... | 47 | ||

4.1.1 | Universality of Free Fall.............................................................................. | 48 | ||

4.1.2 | Light in the Gravity Field............................................................................. | 48 | ||

4.1.3 | The Time Delay of Light Signals........................................................................ | 49 | ||

4.1.4 | Atom in Gravity Field and Gravitational Frequency Shift................................................ | 49 | ||

4.1.5 | The Pericenter Shift and Positive Energy Density of Gravity Field...................................... | 50 | ||

4.1.6 | The Lense-Thirring Effect.............................................................................. | 50 | ||

4.1.7 | The Relativistic Precession of a Gyroscope............................................................. | 50 | ||

4.1.8 | The Quadrupole Gravitational Radiation................................................................. | 51 | ||

4.2 | New QFGT Predictions Different from GRT..................................................................... | 52 | ||

4.2.1 | The Quantum Nature of the Gravity Force................................................................ | 52 | ||

4.2.2 | Translational Motion of Rotating Test Body............................................................. | 52 | ||

4.2.3 | Testing the Equivalence and Effacing Principles........................................................ | 54 | ||

4.2.4 | Scalar and Tensor Gravitational Radiation.............................................................. | 56 | ||

4.2.5 | The Binary NS System with Pulsar PSR1913+16............................................................ | 57 | ||

4.2.6 | Detection of GW Signals by Advanced LIGO-Virgo Antennas................................................ | 58 | ||

4.2.7 | The Riddle of Core Collapse Supernova Explosion........................................................ | 60 | ||

4.2.8 | Self-Gravitating Gas Configurations.................................................................... | 62 | ||

4.2.9 | Relativistic Compact Objects Instead of Black Holes.................................................... | 63 | ||

5 | Cosmology in GRT and QFGT | 66 | ||

5.1 | General Principles of Cosmology............................................................................. | 66 | ||

5.1.1 | Practical Cosmology.................................................................................... | 66 | ||

5.1.2 | Empirical and Theoretical Laws......................................................................... | 67 | ||

5.1.3 | Global Inertial Rest Frame Relative to Isotropic CMB................................................... | 67 | ||

5.1.4 | Gravitation Theory as the Basis of Cosmological Models................................................. | 68 | ||

5.2 | Friedmann’s Homogeneous Model as the Basis of the SCM...................................................... | 68 | ||

5.2.1 | Initial Assumptions: General Relativity, Homogeneity, Expanding Space.................................. | 68 | ||

5.2.2 | Friedmann’s Equations for Dark Energy and Matter...................................................... | 69 | ||

5.2.3 | Observational and Conceptual Puzzles of the SCM........................................................ | 71 | ||

5.3 | Possible Fractal Cosmological Model in the Frame of QFGT.................................................... | 73 | ||

5.3.1 | Initial Assumptions of the FGF Model................................................................... | 73 | ||

5.3.2 | Universal Cosmological Solution and Global Gravitational Redshift...................................... | 74 | ||

5.3.3 | The Structure and Evolution of the Field-Gravity Fractal Universe...................................... | 77 | ||

5.3.4 | Crucial Cosmological Tests of the Fractal Model........................................................ | 78 | ||

6 | Conclusions | 80 | ||

References | 81 |

## 1. Introduction

#### 1.1. Key Discoveries of Modern Multimessenger Astronomy

- detection of the gravitational waves from coalescent relativistic compact objects by LIGO-Virgo antennas;
- imaging of the supermassive black hole candidates M87* and SgrA* by Event Horizon Telescope; and,
- establishing tensions between Local and Global cosmological parameters.

#### 1.1.1. Gravitational Waves

#### 1.1.2. Imaging of Black Holes Candidates: Relativistic Jets and Disks

#### 1.1.3. Tensions between Local and Global Cosmological Parameters

#### 1.1.4. Conceptual Problems of the Gravity Physics

#### 1.2. The Quest for Unification of of the Gravity with Other Fundamental Forces

#### 1.2.1. Future Unified Theory

#### 1.2.2. Quantum Electrodynamics as the Paradigmatic Theory

- the inertial reference frames;
- the flat Minkowski space-time;
- the relativistic vector field ${A}^{i}(t,\overrightarrow{x})$;
- the Least (Stationary) Action Principle;
- the conservation of charges;
- the gauge invariance principle;
- the localizable positive energy density (${T}_{\left(em\right)}^{00}$) of the field.

**definition of energy density of the field must exist**within the conceptual bases of the principle of stationary action, ${j}^{\phantom{\rule{0.166667em}{0ex}}i}$-four-current, ${A}^{i}$-four-potential, and ${F}^{ik}$-electromagnetic field tensor

**fixed sources ${j}^{\phantom{\rule{0.166667em}{0ex}}i}$**we get field equations with conserved sources

- ${T}_{\left(em\right)}^{ik}={T}_{\left(em\right)}^{ki}$ -symmetry condition;
- ${T}_{\left(em\right)}^{00}=({E}^{2}+{H}^{2})/8\pi >0$ - localizable field energy density, positive for both static and wave field, corresponding to the positive photon energy ${E}_{photon}=h\nu $;
- ${T}_{\left(em\right)}={\eta}_{ik}{T}_{\left(em\right)}^{ik}=0$ -trace of the EMT is zero; for mass-less particles (photons);
- the EMT from action S is defined not uniquely; and,
- the EMT is gauge invariant.

**fixed four-potential**gives the four-equations of motion for charged particle:

**concept of force, work produced by force, positive energy density of the field, and its localization**.

#### 1.3. Einstein’s Geometrical and Feynman’s Quantum-Field Gravitation Physics

#### 1.3.1. Two Ways in Gravity Physics

**reducible**symmetric second rang tensor ${\psi}^{ik}$, which can be presented as a direct sum of three irreducible representations of the Poincare group: four-tensor (traceless), four-vector, and four-scalar (5 + 4 + 1 = 10 components). Gauge invariance (and corresponding EMT conservation) only excludes four components (four-vector) and, hence, leaves direct sum of

**two irreducible**representations:

**spin-2 and spin-0**parts (i.e., six independent components). The irreducible spin-2 representation

**${\varphi}^{ik}={\psi}^{ik}-(1/4){\eta}_{ik}{\psi}^{ik}$**describes

**the attractive force**and Feynman (as many other authors of the spin-2 derivations of GRT equations) tried to construct gravitation theory based on the spin-2 field only, so excluding spin-0 field.

**$\psi ={\eta}_{ik}{\psi}^{ik}$**), which is the second irreducible part of the gauged total reducible symmetric tensor potential ${\psi}^{ik}$, was found and developed by Sokolov and Baryshev [19,20,65,124,125,126,127,128,129]. Intriguingly this irreducible

**intrinsic 4-scalar**field corresponds to

**the repulsive dynamical field**, which in the sum with the pure spin-2 field gives the Newtonian gravity force and also all classical relativistic gravity effects. As a result, a consistent field gravity theory (FGT) has been developed, where the central role belongs to the inertial frames, Minkowski space and localizable positive energy of the gravitational field,

**including its scalar part**. Although many important questions are waiting for further work.

#### 1.3.2. Special Features of the Geometrical Approach

- the non-inertial reference frames;
- the equivalence principle and geometrization of gravity;
- the curved Riemannian space-time with metric ${g}_{ik}$;
- the geodesic motion of matter and light;
- the general covariance; and,
- the geometrical extension of Stationary Action Principle.

**experimental/observational proof**of the GRT strong gravity effects, which are still hypothetical models for the observed astrophysical phenomena (see discussion in Section 4 and Section 5).

#### 1.3.3. Problem of the Gravitational Field Energy-Momentum in GRT

**“prior geometry” of the Minkowski space**in the field theories has the advantage of

**guarantee the tensor character of the energy-momentum, its localization and its conservation**for the fields. However, in GRT, there is no global Minkowski space, so there is no EMT of the gravitation field and its conservation.

#### 1.3.4. Attempts to Resolve the Gravitational Energy-Momentum Problem in Geometrical Approach

**geometrization principle**) and they predict some differences with GRT only in the case of the strong gravity field, which is not directly observed yet. However, as we shall show below, the results of the consistent Poincare–Feynman field approach has led to predictions that differ from GRT, even in the weak field conditions, which, in principle, can be tested by experiments in the Earth laboratories and by observations while using terrestrial and space observatories.

#### 1.3.5. Special Features of the Feynman Approach

- the inertial reference frames;
- the flat Minkowski space-time wih metric ${\eta}^{ik}$;
- the reducible symmetric tensor potentials ${\psi}^{ik}\left({x}^{m}\right)$ with trace $\psi \left({x}^{m}\right)={\psi}^{ik}{\eta}_{ik}$;
- the universality of gravitational interaction;
- the Stationary Action Principle (Lagrangian formalism);
- the conservation law of energy-momentum tensor;
- the gauge invariance principle;
- the localizable positive energy density of the gravitational field;
- the gravitational field energy quanta as mediators of the gravity force; and
- the uncertainty principle and other quantum postulates.

#### 1.3.6. Why Is QFGT Principally Different from GRT?

**scalar-tensor**field gravitation theory. Note that the

**intrinsic**scalar part of the symmetric tensor field (trace $\psi \left({x}^{m}\right)={\psi}^{ik}{\eta}_{ik}$) is an observable part of the classical gravity experiments (see Section 3). The most radical difference of QFGT from GRT is that the field approach works with the two parts of the gravity physics - the traceless spin-2

**attraction**field and the intrinsic scalar spin-0

**repulsion**field (the trace of the tensor potential). This fact demonstrates the principal incompatibility of FGT and GRT, though there is common region of applicability of geometrical and field approaches (coincident predictions for classical relativistic gravity effects in the weak field regime).

**rigorous identities must be fulfilled**for the metric tensor of any Riemannian space:

**the convention**that indices are rased and lowered by the flat metric ${\eta}^{ik}$. However strictly speaking in the frame of the geometrical approach such procedure is “illegal”, because it violates the general covariance principle (${\eta}^{ik}$ and ${h}^{ik}$ are not tensors of the initial curved space). For the rasing and lowering indices, one should use the sum (Equation (16)), which must obey the strict identities (Equation (17)). As Schuts 2009 [104] (p. 199) emphasized: “Thus ${\eta}^{im}{\eta}^{nk}{h}_{mn}={h}^{ik}$—is not the deviation of ${g}^{ik}$ from flatness”.

#### 1.3.7. Conceptual Tensions between Quantum Mechanics and General Relativity

- general relativity having black hole solutions violates the simple topological structure of the Minkowski space of the quantum field theory;
- general relativity has lost the energy-momentum tensor of the gravity field together with the conservation laws, while in the Standard Model the EMT and its conservation is the direct consequences of the global symmetry of the Minkowski space.

#### 1.3.8. Astrophysical Tests of the Nature of Gravitational Interaction

## 2. Einstein’s Geometrical Gravitation Theory

#### 2.1. Basic Principles of GRT

#### 2.1.1. The Principle of Geometrization

#### 2.1.2. The Principle of Least Action

**no interaction Lagrangian**), because gravitation is not a matter in GRT (while other fields contain the interaction part ${S}_{\left(\mathrm{int}\right)}$, see Equations (1) and (39)). So the GRT action is:

#### 2.2. Basic Equations of General Relativity

#### 2.2.1. Einstein’s Field Equations

**gravitation is not a material field**in general relativity (as was discussed in Introduction).

#### 2.2.2. The Equation of Motion of Test Particles

**continuity equation**also gives the equations of motion for a considered matter. 5 It implies the geodesic equation of motion for a test particle:

#### 2.3. The Weak Field Approximation

#### 2.3.1. The Metric Tensor

#### 2.3.2. The Field Equations

#### 2.3.3. The Equation of Motion in the Weak Field

#### 2.4. Major Predictions for Experiments/Observations

#### 2.4.1. The Classical Relativistic Gravity Effects in the Weak Field

- universality of free fall for non-rotating bodies,
- the deflection of light by massive bodies,
- gravitational frequency-shift,
- the time delay of light signals,
- the perihelion shift of a planet,
- the Lense–Thirring effect,
- the geodetic precession of a gyroscope, and
- the emission and detection of the quadrupole gravitational waves.

#### 2.4.2. Strong Gravity Effects in GRT: Schwarzschild Metric

#### 2.4.3. Tolman-Oppenheimer-Volkoff Equation

#### 2.5. Modifications of GRT to Aviod Field Energy Problem

- the physical sense of the energy-momentum of the space curvature,
- the physical sense of the black hole horizon and singularity, and
- the physical sense of the space creation in the expanding Universe.

#### 2.5.1. Geometrical Approach without Black Holes?

#### 2.5.2. The Energy-Momentum of the Space Curvature?

#### 2.5.3. Non-Localizability of the Gravitation Field Energy in GRT

#### Attempts to Overcome the Energy Problem by Using Simultaneously Minkowski and Riemannian Spaces

#### Absence of the Required EMT Physical Properties in the Metric Gravity Theories

- symmetry, ${T}^{ik}={T}^{ki}$;
- positive localizable energy density, ${T}^{00}>0$; and,
- zero trace for massless fields, $T=0$.

#### 2.5.4. The Physical Sense of the Space/Vacuum Creation in the Expanding Universe

#### 2.5.5. Conclusions

## 3. Feynman’s Quantum-Field Approach to Gravitation Theory

#### 3.1. Initial Principles

#### 3.1.1. The Unity of the Fundamental Interactions

- the inertial reference frames and Minkowski space with metric ${\eta}^{ik}$;
- the
**reducible**symmetric second rank tensor potential ${\psi}^{ik}\left({x}^{m}\right)$; - two irreducible parts which correspond to spin-2 and spin-0 (trace) fields;
- the Lagrangian formalism and Stationary Action principle;
- the principle of consistent iterations;
- the universality of gravitational interaction;
- the conservation law of the energy-momentum;
- the gauge invariance of the linear field equations;
- the positive localizable energy density and zero trace of the gravity field EMT;
- the quanta of the field energy as the mediators of the gravity force; and,
- the uncertainty principle and other quantum postulates.

#### 3.1.2. The Principle of Consistent Iterations

#### 3.1.3. The Principle of Stationary Action

#### 3.1.4. Lagrangian for the Gravitational Field

#### 3.1.5. Lagrangian for Matter

#### 3.1.6. The Principle of Universality and Lagrangian for Interaction

#### 3.2. Basic Equations of the Quantum-Field Gravity Theory

#### 3.2.1. Field Equations

#### 3.2.2. Remarkable Features of the Field Equations

**tensor part of the field corresponds to attraction**, while the

**scalar part gives repulsion**. This result is caused by the fact that in the Lagrangian (40) the tensor and scalar parts have opposite signs, which does not mean negative energy of the scalar field, but it reflects the opposite signs of the pure tensor and pure scalar forces.

#### 3.2.3. Scalar and Traceless Tensor Are Dynamical Fields in QFGT

**four gauge conditions (56)**exclude four independent components (dofs) of the symmetric potential which corresponds to deleting the irreducible four-vector field ($3+1=4$ components. Hence, the initial reducible symmetric tensor potential will only contain two irreducible parts corresponding to spin-2 tensor field (5 dofs) and spin-0 scalar field (1 dof):

**four restrictions from the conservation laws (52)**of the energy-momentum tensor, which delete four source components corresponding to particles with spin-1 and spin-0’, we get

#### 3.2.4. The Energy-Momentum Tensor of the Gravity Field

**attraction force**has opposite sign relative to the spin-0

**repulsion force**, the canonical symmetric gravity EMT (70) of the field ${\psi}^{ik}={\psi}_{2}^{ik}+{\psi}_{0}^{ik}$ may be presented as the

**difference between to positive EMTs**of the spin-2 and spin-0 fields:

**two independent parts**(in linear approximation) that correspond to two independent particles with spin 2 (${\varphi}^{ik}$) and spin 0 ($\psi $). Thus, we have two following free field Lagrangians

#### 3.2.5. The Retarded Potentials

#### 3.3. Equations of Motion for Test Particles

#### 3.3.1. Derivation from Stationary Action Principle

**fixed gravitational potential**${\psi}^{ik}$. The free particle Lagrangian is

#### 3.3.2. Static Spherically Symmetric Weak Field

#### 3.3.3. The Role of the Scalar Part of the Field

#### 3.4. Poincaré Force and Poincaré Acceleration in PN Approximation

#### 3.4.1. The EMT Source in PN-Approximation

#### 3.4.2. Relativistic Physical Sense of the Potential Energy

#### 3.4.3. The PN Correction Due to the Energy of the Gravity Field

#### 3.4.4. Post-Newtonian Equations of Motion

#### 3.4.5. Lagrange Function in Post-Newtonian Approximation

#### 3.5. Quantum Nature of the Gravity Force

#### 3.5.1. Propagators for Spin-2 and Spin-0 Massive Fields

#### 3.5.2. Composite Structure of the Quantum Newtonian Gravity Force

## 4. Relativistic Gravity Experiments/Observations in QFGT

#### 4.1. Classical Relativistic Gravity Effects

- universality of free fall for non-rotating bodies,
- the deflection of light by massive bodies,
- gravitational frequency-shift,
- the time delay of light signals,
- the perihelion shift of a planet,
- the Lense–Thirring effect,
- the geodetic precession of a gyroscope, and
- the quadrupole gravitational radiation.

#### 4.1.1. Universality of Free Fall

#### 4.1.2. Light in the Gravity Field

#### 4.1.3. The Time Delay of Light Signals

#### 4.1.4. Atom in Gravity Field and Gravitational Frequency Shift

#### 4.1.5. The Pericenter Shift and Positive Energy Density of Gravity Field

#### 4.1.6. The Lense-Thirring Effect

#### 4.1.7. The Relativistic Precession of a Gyroscope

#### 4.1.8. The Quadrupole Gravitational Radiation

#### 4.2. New QFGT Predictions Different from GRT

#### 4.2.1. The Quantum Nature of the Gravity Force

#### 4.2.2. Translational Motion of Rotating Test Body

#### 4.2.3. Testing the Equivalence and Effacing Principles

- does not depend on the rest mass ${m}_{0}$ of the test body, and
- does depend on its velocity $\overrightarrow{V}$ (both on value and direction) and on the value of the gravitational potential ${\phi}_{N}$ at the location of the body.

#### 4.2.4. Scalar and Tensor Gravitational Radiation

#### 4.2.5. The Binary NS System with Pulsar PSR1913+16

**larger**than the predicted value of the energy radiated by pure tensor gravitational waves.

#### 4.2.6. Detection of GW Signals by Advanced LIGO-Virgo Antennas

#### 4.2.7. The Riddle of Core Collapse Supernova Explosion

#### 4.2.8. Self-Gravitating Gas Configurations

#### 4.2.9. Relativistic Compact Objects Instead of Black Holes

## 5. Cosmology in GRT and QFGT

#### 5.1. General Principles of Cosmology

#### 5.1.1. Practical Cosmology

#### 5.1.2. Empirical and Theoretical Laws

- experimentally measured empirical laws, and
- logically inferred theoretical laws.

- the cosmological redshift-distance law $cz=Hr$,
- the thermal law of isotropic cosmic background radiation ${B}_{\nu}\left(T\right)$, and
- the power-law correlation of galaxy clustering. $\Gamma \left(r\right)\sim {r}^{-\gamma}$.

#### 5.1.3. Global Inertial Rest Frame Relative to Isotropic CMB

**global inertial rest frame**(GIR). Very important that relative to this GIR reference frame it is possible to measure both the velocity and acceleration of any body in the Universe. If a two bodies are at rest relative to CMBR (i.e., they see isotropic global background radiation), then they are at rest relative to each other (for non-expanding space-vacuum). This fact delivers new relativistic and quantum physical situation in the 21st century cosmology and, in general, physics too.

#### 5.1.4. Gravitation Theory as the Basis of Cosmological Models

#### 5.2. Friedmann’s Homogeneous Model as the Basis of the SCM

#### 5.2.1. Initial Assumptions: General Relativity, Homogeneity, Expanding Space

- General relativity can be applied to the whole Universe (${g}_{ik}$; $\phantom{\rule{0.222222em}{0ex}}{\Re}_{iklm}$; $\phantom{\rule{0.222222em}{0ex}}{T}_{(\mathrm{m}+\mathrm{de})}^{ik}$).
- Homogeneity and isotropy of matter distribution in the expanding Universe ($\rho =\rho \left(t\right)$; ${g}^{ik}={g}^{ik}\left(t\right)$).
- Laboratory physics works in the expanding space.
- Inflation in the early universe is needed for explanation of the flatness, isotropy and initial conditions of large scale structure formation.

#### 5.2.2. Friedmann’s Equations for Dark Energy and Matter

- Cosmological redshift $(1+z)={\lambda}_{0}/\left({\lambda}_{1}\right)={S}_{0}/{S}_{1}\phantom{\rule{0.166667em}{0ex}},$ is the Lemaitre effect and the linear expansion velocity-distance relation ${V}_{exp}=H\times r\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$ is the consequence of the space expansion $r\left(t\right)=S\left(t\right)\times \chi \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$ of the homogeneous Universe.
- Cosmic microwave background radiation is the result of the photon gas cooling in the expanding space and the CMBR temperature is $T\left(z\right)={T}_{0}(1+z)$.
- Small anisotropy $\Delta T/T\left(\theta \right)$ of the CMBR is determined by the initial spectrum of density fluctuations which are the source of the large scale structure of the Universe.
- The physics of the expanding Universe is described by the LCDM model which predicts the following matter budget at present epoch: 70% of unobservable in lab dark energy, 25% unknown nonbaryonic cold dark matter, 5% ordinary matter. Visible galaxies contribution is less than 0.5%.

#### 5.2.3. Observational and Conceptual Puzzles of the SCM

**Absurd Universe.**The visible matter of the Universe, the part which we can actually observe, is a surprisingly small (about 0.5%) piece of the predicted matter content and this looks like an “Absurd Universe” [246]. What is more, about 95% of the cosmological matter density, which determine the dynamics of the whole Universe has unknown physical nature. Turner [229] emphasized that modern SCM predicts with high precision the values for dark energy and nonbaryonic cold dark matter, but “we have to make sense to all this”.**The cosmological constant problem.**One of the most serious problem of the LCDM model is that the observed value of the cosmological constant Λ is about 120 orders of magnitude smaller than the expectation from the physical vacuum (as discussed by Weinberg [247] and Clifton et al. [7]). In fact, the critical density of the $\Omega =1$ universe is ${\varrho}_{\mathrm{crit}}=0.853\times {10}^{-29}$ g/cm^{3}, while the Planck vacuum has ${\varrho}_{\mathrm{vac}}\approx {10}^{+94}$ g/cm^{3}.**The cold dark matter crisis on galactic and subgalactic scales.**There are number of problems with predicting behavior of baryonic and nonbaryonic matter within galaxies. It was discussed by Kroupa [54] that there are discrepancies between observed and predicted galaxy density profiles (the cusp problem), small number of observed satellites galaxies (missing satellites problem), and observed tight correlation between dark matter and baryons in galaxies, which is not expected within LCDM galaxy formation theory.**The LCDM crisis at super-large scales.**The most recent observational facts which contradict the LCDM picture of the large scale structure formation, come from: the 2MASS, 2dF, and SDSS galaxy redshift surveys (Sylos Labini [55]), problems with observations of baryon acoustic oscillations (Sylos Labini et al. [245]), existence of structures with sizes ∼ 400 Mpc/h in the local Universe (Gott et al. [248], Tully et al. [51], and Pomarede et al. [48]) and ∼1000 Mpc/h structures in the spatial distribution of distant galaxies, quasars, and gamma-ray bursts (Nabokov & Baryshev [249], Clowes et al. [56], Einasto et al. [250], and Horvath et al. [57]), existence of old galaxies, and supermassive black holes in quasars at redshift up to $\phantom{\rule{0.166667em}{0ex}}z\sim 7\phantom{\rule{0.166667em}{0ex}}$ (Dolgov [60], Yang et al. [61], alternative interpretation of the shape of the CMBR correlation function (Lopez-Corredoira & Gabrielli [251]), lack of CMBR power at angular scales larger 60 degrees and correlation of CMBR quadrupole with ecliptic plain (Copi et al. [252]).

**Vacuum energy paradox:**in the framework of the Einstein’s geometrical gravity theory (GRT) there is the paradox of too small value of the Lambda term, considered as the physical vacuum [247].**1st Harrison’s paradox (“energy-momentum non-conservation”):**physics of space expansion contains such puzzling phenomena as continuous creation of vacuum and violation of energy-momentum conservation for matter in any comoving volume, including photon gas of cosmic background radiation [8,76,77,78,80].**2nd Harrison’s paradox (“motion without motion”):**a galaxy cosmological velocity is conceptually different from the galaxy peculiar velocity, in particular, the cosmological redshift in expanding space is not the Doppler effect, but the Lemaitre effect is applicable to a receding galaxy, which can have velocity larger than the velocity of light (so cosmological redshift is a new physical phenomenon, which also includes the global gravitational cosmological redshift) [8,76,77,79,80].**Hubble-deVaucouleurs’ paradox (“Hubble law is not a consequence of homogeneity”):**in the expanding space the linear Hubble law is the fundamental consequence of the assumed homogeneity, however modern observations reveal existence of strongly inhomogeneous (power-law correlated) large-scale galaxy distribution at interval of scales $1\xf7100$ Mpc, where the linear Hubble law is firmly established, i.e., just inside inhomogeneous spatial galaxy distribution of the Local Universe [8,77,262,263,264].

#### 5.3. Possible Fractal Cosmological Model in the Frame of QFGT

#### 5.3.1. Initial Assumptions of the FGF Model

- the gravitational interaction is described by the Poincare covariant Feynman’s quantum-field gravitation theory in Minkowski spacetime; and,
- the total baryonic matter distribution (visible and dark) in the Local Universe is described by the stochastic fractal density law with critical fractal dimension ${D}_{crit}=2$.