I have organized series of international coanference, MOGRA, named after "mogura (もぐら)" in Japanese
that means "mole" in English.
The focus of MOGRA series is recent progress in study of gravitational theory, specially modified
gravity theory, and its theoretical and phenomenological applications including:
Past MOGRAs are as follows:
I am now writing the lecture notes on F(R) gravity theory for beginners. Coming soon.
(Last update: September 5, 2025)
The study of gravitational theory began with Galileo, followed by Kepler's laws of celestial motion and Newton's law of universal gravitation, and Einstein established general relativity. In the ceaseless research on gravity, alternative theories to general relativity, known as modified gravity, have been investigated. F(R) gravity theory is a modified gravity theory that includes a simple extension of Einstein-Hilbert action and has been intensively applied to cosmology and astrophysics. This lecture note covers basics of the F(R) gravity theory and related research topics.
Part I summarizes fundamental but essential subjects including basics of GR.
Chapter 1. General Relativity
1.1 Einstein-Hilbert action
1.2 Einstein equation
1.3 Bianchi identity
1.4 Cosmological constant
Chapter 2. F(R) Gravity Theory
2.1 Basic concepts
2.2 Field equation
2.3 Conservation law
2.4 Scalaron field
Chapter 3. F(R) Gravity as Scalar-Tensor
3.1 Scalar-Tensor theory
3.2 In Jordan frame
3.3 In Einstein frame
3.4 Chameleon mechanism
Part II reviews several models of F(R) gravity theory and basic calculation methods.
Chapter 4. Basics
4.1 Modification for what?
4.2 R2 model for inflation
4.3 1/R model for DE
Chapter 5. Extensions
5.1 Rn model
5.2 R2-n model
5.3 ln R model
Chapter 6. DE Models
6.1 Viable DE models
6.2 Curvature singularity
6.3 R2 corrected DE model
Chapter 7. Advanced
7.1 R2 corrected ΛCDM
7.2 R2-n + Rn model
7.3 Other models
Part III is devoted to an overview of practical applications of F(R) gravity.
Chapter 8. Cosmology
8.1 FLRW spacetime
8.2 riedmann equation in GR
8.3 Modified Friedmann equations
8.4 Late-time Universe and DE
8.5 Topical reviews
Chapter 9. Compact Objects
9.1 SSS spacetime
9.2 TOV equation in GR
9.3 Modified TOV equation
9.4 BH solutions in GR
9.5 SSS BH in F(R) gravity
9.6 Topical reviews
Chapter 10. Gravitational Waves
10.1 Fluctuations of spacetime
10.2 GW in GR
10.3 GW in F(R) gravity
10.4 Scalar mode of GW
10.5 Topical reviews
Chapter 11. Particle Physics
11.1 Dilatonic coupling in Einstein frame
11.2 Coupling to scalar field
11.3 Coupling to fermion field
11.4 Coupling to vector field
11.5 Topical reviews
Part IV covers advanced topics, detailed calculations, and technical matters.
Chapter 12. Differential Geometry
12.1 Tensor and tensor density
12.2 Affine connection
12.3 Tangent space and vielbein
12.4 Metric-affine space
Chapter 13. Palatini and Cartan Formalism
13.1 Palatini formalism
13.2 Cartan formalism
13.3 Palatini F(R) gravity
13.4 Cartan F(R) gravity
Chapter 14. Conservation Law
14.1 Generalized Bianchi identity
14.2 Local and global conservation
14.3 Noether theorem
Chapter 15. Degrees of Freedom
15.1 Hamiltonian analysis
15.2 ADM formalism
15.3 DOF in GR
15.4 DOF in F(R) gravity
Chapter 16. Weyl Transformation
16.1 Transformation rules
16.2 Frame equivalence
16.3 Conformal transformation
Chapter 17. Screening Mechanism
17.1 Long-range fifth force
17.2 PPN formalism
17.3 Representative mechanisms
Chapter 18. Friedmann Equation
18.1 Single component
18.2 Two components
18.3 Einstein's static universe
18.4 Evolution of density parameters
Chapter 19. TOV Equation
19.1 Junction condition at surface
19.2 Boundary conditions
19.3 Models of EOS
19.4 M-R relation
Chapter 20. Black Hole Solutions
20.1 Uniqueness in GR
20.2 Uniqueness in Scalar-Tensor theory
20.3 Regular black holes
Chapter 21. Perturbations
21.1 Second-order perturbation
21.2 Linearized EH action
21.3 Spin-2 field theory
Chapter 22. Field Theory in Curved Spacetime
22.1 Covariant derivative
22.2 Spin connection
22.3 Scale Anomaly
22.4 Spin and force
Part V introduces other modified gravity theories related to the F(R) gravity.
Chapter A. Nordström Theory of Gravity
A.1 From Newtonian gravity to relativistic gravity
A.2 Relativistic scalar theory of gravity
A.3 Features of Nordström theory
Chapter B. TEGR, STEGR, and Extensions
B.1 GR as quadratic theory of connection
B.2 Teleparallel gravity
B.3 F(T) gravity
B.4 Symmetric teleparallel gravity
B.5 F(Q) gravity
Chapter C. Higher-Curvature Gravity
C.1 Curvature-squared correction
C.2 Critical gravity
C.3 F(Rμν) gravity
C.4 Eddington-inspired Born-Infeld gravity
Chapter D. Lovelock Theorem
D.1 Uniqueness of Einstein equation
D.2 Lovelock theory of gravity
D.3 Relation to higher-curvature gravity
Chapter E. Horndeski Theory
E.1 (under construction)
E.2 (under construction)
E.3 (under construction)
I upload short notes on matters of personal interest. Coming soon.