Astrophysics and Cosmology

I. Gravitational waves: 1. Review of the properties of gravitational waves 2. Quadrupole radiation 2. TT-gauge, calculation of the gravitational luminosity of a source II. Gravitational Wave Astrophysics: 1. Rapidly rotating neutron stars, spin down gravitational 2. Relativistic binary pulsars 3. Coalescence of binary systems 4. Notes on gravitational wave detectors III. Structure of degenerate stars: 1. White dwarfs 2. Neutron stars 3. Equations of the stellar structure in general relativity 4. Stability criteria 5. Schwarzschild solution for constant density stars. 6. Buchdal's theorem. 7. Solution of Einstein's equations for an isolated body and stationary in the "far field limit". 8. Geodesic and Lense-Thirring precession of a gyroscope spin IV. Black holes: 1. Review of the Schwarzschild solution - Black hole ringdown 2. Coordinates of Kruskal 3. Kerr's solution: singularities, horizons, ergosphere. 4. Geodesics in the Kerr metric. 5. Penrose trial. V. Variational principles and Einstein equations. The course is divided in three parts. In the first part, devoted to the generation and to the detection of gravitational waves, it will be shown how to use the quadrupole formalism to compute the gravitational signal emitted by a source in the weak-field, slow-motion regime. It will be shown how to include the effect of radiation reaction when an astrophysical source emits gravitational waves; the consequences on the source evolution and on the emitted waveforms will be described. In particular we shall study binary systems formed by neutron stars or black holes close to coalescence, focussing on the main features of the signals which have been detected by LIGO and Virgo in recent years; in addition, the gravitational emission of rotating neutron stars will also be studied. In the second part of the course the final phases of the evolution of a star will be described, showing how the stellar fate can be different depending on the mass of the progenitor. Then, using the Newtonian theory of gravity, we will study the structure of a white dwarf and the concept of critical mass will be introduced. In order to describe the structure of a neutron star, General relativity is needed; therefore the first step will be to show how the equations of Thermodynamics have to be modified in General Relativity. Then it will be shown how the equations of stellar structure are derived from Einstein’s equations and how they can be solved. The third part of the course is devoted to the study of the Kerr solution of Einstein’s equations, which describes a rotating black hole, and of the complex phenomena occurring in its neighbourhood. In particular we shall study the structure of the spacetime of a rotating black hole, the curvature singularity, the presence of horizons and of an ergoregion. We shall study the geodesics of massive and massless bodies and the associated phenomena like, for instance, Penrose’s process to extract energy from a rotating black hole. Finally, as a completion of the study of General Relativity, we shall show how Einstein’s equations can be derived using a variational approach.

In order to profitably attend this course, it is essential to know Einstein’s theory of General Relativity. This matter is taught in the General Relativity course. We signal the following textbooks to fill the gap, if needed: V. Ferrari, L. Gualtieri, P. Pani, General Relativity and its Applications, CRC Press S. Carrol, Spacetime and Geometry, an Introduction to General Relativity, Addison & Wesley S. Weinberg, Gravitation and Cosmology, Wiley & Sons Einstein-Infeld - "Evolution of physics" Landau-Liftschitz “The Classical Theory of Fields” B. F. Schutz, A first course in general relativity, Cambridge University Press Poisson-Will "Gravity", Cambridge University Press

V. Ferrari, L. Gualtieri, P. Pani, General Relativity and its Applications, CRC Press M. Maggiore, Gravitational Waves Vol I and II, Oxford University Press I. Novikov, V. Frolov, Physics of Black Holes, Springer E. Poisson, C.M. Will, Gravity, Cambridge University Press Per la parte propedeutica di GR: S. Carroll, Spacetime and Geometry, An Introduction to General Relativity, Addison & Wesley S. Weinberg, Gravitation and Cosmology, Wiley & Sons A. Einstein, L. Infeld, Evolution of physics Landau-Liftschitz, The Classical Theory of Fields B. F. Schutz, A first course in general relativity, Cambridge University Press

This advanced course on gravity theory, is essentially devoted to the acquisition of a deeper knowledge of the the predictions of General Relativity, which will be illustrated mainly in lectures. In addition, there will be exercises, which will further develop some of the treated issues, and seminars to illustrate some observational aspects of the astrophysical phenomena where General Relativity plays a fundamental role. In the case of a continued COVID emergency, the course will be delivered online. All the relevant information can be found at the webpage of the course, http://www.roma1.infn.it/teongrav/onde.html

Attendance is not mandatory, but strongly suggested

The final grade will be based on the evaluation of : - the correctness and completeness in the exposition of the topics; - the clarity and rigour in the exposition; - the ability to analytically develop the theory.

V. Ferrari, L. Gualtieri, P. Pani, General Relativity and its Applications, CRC Press M. Maggiore, Gravitational Waves Vol I and II, Oxford University Press I. Novikov, V. Frolov, Physics of Black Holes, Springer E. Poisson, C.M. Will, Gravity, Cambridge University Press Per la parte propedeutica di GR: S. Carroll, Spacetime and Geometry, An Introduction to General Relativity, Addison & Wesley S. Weinberg, Gravitation and Cosmology, Wiley & Sons A. Einstein, L. Infeld, Evolution of physics Landau-Liftschitz, The Classical Theory of Fields B. F. Schutz, A first course in general relativity, Cambridge University Press

This advanced course on gravity theory, is essentially devoted to the acquisition of a deeper knowledge of the the predictions of General Relativity, which will be illustrated mainly in lectures. In addition, there will be exercises, which will further develop some of the treated issues, and seminars to illustrate some observational aspects of the astrophysical phenomena where General Relativity plays a fundamental role. In the case of a continued COVID emergency, the course will be delivered online. All the relevant information can be found at the webpage of the course, http://www.roma1.infn.it/teongrav/onde.html

I. Gravitational waves: 1. Review of the properties of gravitational waves 2. Quadrupole radiation 2. TT-gauge, calculation of the gravitational luminosity of a source II. Gravitational Wave Astrophysics: 1. Rapidly rotating neutron stars, spin down gravitational 2. Relativistic binary pulsars 3. Coalescence of binary systems 4. Notes on gravitational wave detectors III. Structure of degenerate stars: 1. White dwarfs 2. Neutron stars 3. Equations of the stellar structure in general relativity 4. Stability criteria 5. Schwarzschild solution for constant density stars. 6. Buchdal's theorem. 7. Solution of Einstein's equations for an isolated body and stationary in the "far field limit". 8. Geodesic and Lense-Thirring precession of a gyroscope spin IV. Black holes: 1. Review of the Schwarzschild solution - Black hole ringdown 2. Coordinates of Kruskal 3. Kerr's solution: singularities, horizons, ergosphere. 4. Geodesics in the Kerr metric. 5. Penrose trial. V. Variational principles and Einstein equations. The course is divided in three parts. In the first part, devoted to the generation and to the detection of gravitational waves, it will be shown how to use the quadrupole formalism to compute the gravitational signal emitted by a source in the weak-field, slow-motion regime. It will be shown how to include the effect of radiation reaction when an astrophysical source emits gravitational waves; the consequences on the source evolution and on the emitted waveforms will be described. In particular we shall study binary systems formed by neutron stars or black holes close to coalescence, focussing on the main features of the signals which have been detected by LIGO and Virgo in recent years; in addition, the gravitational emission of rotating neutron stars will also be studied. In the second part of the course the final phases of the evolution of a star will be described, showing how the stellar fate can be different depending on the mass of the progenitor. Then, using the Newtonian theory of gravity, we will study the structure of a white dwarf and the concept of critical mass will be introduced. In order to describe the structure of a neutron star, General relativity is needed; therefore the first step will be to show how the equations of Thermodynamics have to be modified in General Relativity. Then it will be shown how the equations of stellar structure are derived from Einstein’s equations and how they can be solved. The third part of the course is devoted to the study of the Kerr solution of Einstein’s equations, which describes a rotating black hole, and of the complex phenomena occurring in its neighbourhood. In particular we shall study the structure of the spacetime of a rotating black hole, the curvature singularity, the presence of horizons and of an ergoregion. We shall study the geodesics of massive and massless bodies and the associated phenomena like, for instance, Penrose’s process to extract energy from a rotating black hole. Finally, as a completion of the study of General Relativity, we shall show how Einstein’s equations can be derived using a variational approach.

In order to profitably attend this course, it is essential to know Einstein’s theory of General Relativity. This matter is taught in the General Relativity course. We signal the following textbooks to fill the gap, if needed: V. Ferrari, L. Gualtieri, P. Pani, General Relativity and its Applications, CRC Press S. Carrol, Spacetime and Geometry, an Introduction to General Relativity, Addison & Wesley S. Weinberg, Gravitation and Cosmology, Wiley & Sons Einstein-Infeld - "Evolution of physics" Landau-Liftschitz “The Classical Theory of Fields” B. F. Schutz, A first course in general relativity, Cambridge University Press Poisson-Will "Gravity", Cambridge University Press

V. Ferrari, L. Gualtieri, P. Pani, General Relativity and its Applications, CRC Press M. Maggiore, Gravitational Waves Vol I and II, Oxford University Press I. Novikov, V. Frolov, Physics of Black Holes, Springer E. Poisson, C.M. Will, Gravity, Cambridge University Press Per la parte propedeutica di GR: S. Carroll, Spacetime and Geometry, An Introduction to General Relativity, Addison & Wesley S. Weinberg, Gravitation and Cosmology, Wiley & Sons A. Einstein, L. Infeld, Evolution of physics Landau-Liftschitz, The Classical Theory of Fields B. F. Schutz, A first course in general relativity, Cambridge University Press

This advanced course on gravity theory, is essentially devoted to the acquisition of a deeper knowledge of the the predictions of General Relativity, which will be illustrated mainly in lectures. In addition, there will be exercises, which will further develop some of the treated issues, and seminars to illustrate some observational aspects of the astrophysical phenomena where General Relativity plays a fundamental role. In the case of a continued COVID emergency, the course will be delivered online. All the relevant information can be found at the webpage of the course, http://www.roma1.infn.it/teongrav/onde.html

Attendance is not mandatory, but strongly suggested

The final grade will be based on the evaluation of : - the correctness and completeness in the exposition of the topics; - the clarity and rigour in the exposition; - the ability to analytically develop the theory.

V. Ferrari, L. Gualtieri, P. Pani, General Relativity and its Applications, CRC Press M. Maggiore, Gravitational Waves Vol I and II, Oxford University Press I. Novikov, V. Frolov, Physics of Black Holes, Springer E. Poisson, C.M. Will, Gravity, Cambridge University Press Per la parte propedeutica di GR: S. Carroll, Spacetime and Geometry, An Introduction to General Relativity, Addison & Wesley S. Weinberg, Gravitation and Cosmology, Wiley & Sons A. Einstein, L. Infeld, Evolution of physics Landau-Liftschitz, The Classical Theory of Fields B. F. Schutz, A first course in general relativity, Cambridge University Press

This advanced course on gravity theory, is essentially devoted to the acquisition of a deeper knowledge of the the predictions of General Relativity, which will be illustrated mainly in lectures. In addition, there will be exercises, which will further develop some of the treated issues, and seminars to illustrate some observational aspects of the astrophysical phenomena where General Relativity plays a fundamental role. In the case of a continued COVID emergency, the course will be delivered online. All the relevant information can be found at the webpage of the course, http://www.roma1.infn.it/teongrav/onde.html

PART I (Gravitational waves, 16 hrs) 1. General information about the course. Primer on gravitational waves. Generation of gravitational waves: solution of the wave equation in the weak-field and low-velocity approximation. 2. Virial theorem. Quadrupole formula. Projection on the transverse-traceless (TT) gauge. Quadrupole moment of a harmonic oscillator. 3. Gravitational waves emitted by a harmonic oscillator. Gravitational waves emitted by a binary system in circular orbit. 4. Gravitational waves emitted by a triaxial ellipsoid rigidly rotating about one of its symmetry axes (model for a rotating star). Ellipsoid rigidly rotating about an axis that is not a symmetry axis: wobble angle and reference frames. 5. Gravitational waves emitted by an ellipsoid rigidly rotating about an axis that is not a symmetry axis. Stress-energy pseudotensor. 6. Gravitational wave energy flux. Gravitational-wave luminosity. Oblateness: the spin-down limit and the limit yielded by gravitational-wave detectors. 7. Gravitational-wave-driven evolution of a binary system: calculation of the separation, frequency, and period as a function of time for quasi-circular orbits. Gravitational wave emitted during the quasi-circular inspiral stage of a binary system. 8. The PSR 1913+16 plot. Gravitational-wave detectors (https://www.youtube.com/watch?v=0MGmQ9pPyA4). Analysis of GW150914, the first direct detection of gravitational waves: the inspiral, merger and ringdown epochs, inferring the chirp mass from the inspiral, arguing the objects are extremely compact, explaining the spin of the remnant black hole. Gravitational-wave catalog (https://waveview.cardiffgravity.org, https://catalog.cardiffgravity.org) PART II (Compact stars , 14 hrs) 9. Introduction to compact objects. Newtonian stellar structure equations and definition of equation of state. Stellar evolution. White dwarfs. Started the solution of the general case of equation of state for a fully degenerate fermionic gas. 10. Fully degenerate fermionic gas. Non-relativistic and ultra-relativistic limits: polytropic equations of state. Started the general case of question of state for a fully degenerate fermionic gas. 11. Completed the general case of question of state for a fully degenerate fermionic gas. The Chandrasekhar mass limit for white dwarfs from energy arguments. General-relativistic thermodynamics of perfect fluids: locally inertial comoving frame (LICF), baryon number conservation law. 12. First law of thermodynamics. Barotropic equation of state. Stress-energy tensor of a perfect fluid; isentropic fluid and relativistic Euler equation. 13. Tolman-Oppenheimer-Volkoff stellar structure equations. Boundary conditions to numerically integrate the Tolman-Oppenheimer-Volkoff equations; stellar radius. 14. Stellar gravitational mass and its interpretation. Buchdahl's theorem. Compact star equilibrium configurations and their stability. 15. The neutron star equation of state. The structure of neutron stars. Neutron star observations. Tidal deformability. PART III (Black Holes, 30 hrs) 16. Singularities in General Relativity, how to remove a coordinate singularity, extension of the manifold. An example: Rindlet's spacetime. Removing the r=2M singularity of the Schwarzschild metric. Maximal extension. Kruskal coordinates. Kruskal diagram. Eddington-Finkelstein coordinates and diagram. Eternal vs astrophysical BHs, white holes. 17. Non-coordinate basis. Fermi's coordinates, non-coordinate basis. The silhouette of a BH, BH shadows, light emitted by a static source near a BH, escape angle. 18. Basic aspects of black-hole perturbation theory; scalar toy model, spherical harmonic decomposition; ringdown & quasinormal modes (QNMs) of a Schwarzschild BH. Few words on gravitational perturbations. Numerical methods to find the QNMs. 19. Tidal deformability of a neutron star and tidal Love numbers. Impact on the waveform and implications for the constraints on the equation of state. Tidal Love numbers of a BH in GR. 20. Metric of an isolated and stationary source in the "far field" limit in the case of a weak gravitational field. Few words on the general case. Physical interpretation of M and J in the metric using the pseudotensor. Introduction to the Lense-Thirring effect and geodesic precession. Further kinematical test of gravity with the Gravity Probe B experiment. 21. The Kerr solution. Boyer Linquist coordinates. Main properties of the Kerr metric. 22. Horizons and ergoregion of the Kerr metric. Frame dragging. Static and stationary observers. 23. How to remove the horizon singularity in Kerr. Kerr coordinates and Kerr-Schild coordinates. Principal null geodesics in Kerr. Curvature singularity in Kerr. 24. The interior of a Kerr BH, Maximal extension of Kerr. Causality violation in the interior. Geodesics in Kerr. 25. Geodesics in the Kerr metric: integrability and Carter's constant. Equatorial geodeiscs. Equatorial Null geodesics, light ring of a Kerr BH. 26. Equatorial Timelike geodesics, Keplerian frequency. ISCO of a Kerr BH. 27. Energy/angular mom extraction from a Kerr BH: Penrose's process; Superradiance. 28. BH thermodynamics. Irriducible mass and area of a Kerr BH. BH Area theorem. 29. The laws of BH thermodynamics. BH entropy. Generalized second law. 30. Hawking evaporation and information loss paradox. Q&A session, examples of written test

The General Relativity course.

Ferrari, Gualtieri, Ferrari – General Relativity and its Applications: Black Holes, Compact Stars and Gravitational Waves, CRC Press

Optional.

The exam consists of an oral test, which covers the entire syllabus of the course. The oral test will start with a list of questions/problems to be answered in written form as if they were oral questions. No books or notes are allowed during the test, but the written questions will come with some relevant equations. Following this first part, there will be an individual oral test.

Blackboard lectures with occasional use of multimedia material.

PART I (Gravitational waves, 16 hrs) 1. General information about the course. Primer on gravitational waves. Generation of gravitational waves: solution of the wave equation in the weak-field and low-velocity approximation. 2. Virial theorem. Quadrupole formula. Projection on the transverse-traceless (TT) gauge. Quadrupole moment of a harmonic oscillator. 3. Gravitational waves emitted by a harmonic oscillator. Gravitational waves emitted by a binary system in circular orbit. 4. Gravitational waves emitted by a triaxial ellipsoid rigidly rotating about one of its symmetry axes (model for a rotating star). Ellipsoid rigidly rotating about an axis that is not a symmetry axis: wobble angle and reference frames. 5. Gravitational waves emitted by an ellipsoid rigidly rotating about an axis that is not a symmetry axis. Stress-energy pseudotensor. 6. Gravitational wave energy flux. Gravitational-wave luminosity. Oblateness: the spin-down limit and the limit yielded by gravitational-wave detectors. 7. Gravitational-wave-driven evolution of a binary system: calculation of the separation, frequency, and period as a function of time for quasi-circular orbits. Gravitational wave emitted during the quasi-circular inspiral stage of a binary system. 8. The PSR 1913+16 plot. Gravitational-wave detectors (https://www.youtube.com/watch?v=0MGmQ9pPyA4). Analysis of GW150914, the first direct detection of gravitational waves: the inspiral, merger and ringdown epochs, inferring the chirp mass from the inspiral, arguing the objects are extremely compact, explaining the spin of the remnant black hole. Gravitational-wave catalog (https://waveview.cardiffgravity.org, https://catalog.cardiffgravity.org) PART II (Compact stars , 14 hrs) 9. Introduction to compact objects. Newtonian stellar structure equations and definition of equation of state. Stellar evolution. White dwarfs. Started the solution of the general case of equation of state for a fully degenerate fermionic gas. 10. Fully degenerate fermionic gas. Non-relativistic and ultra-relativistic limits: polytropic equations of state. Started the general case of question of state for a fully degenerate fermionic gas. 11. Completed the general case of question of state for a fully degenerate fermionic gas. The Chandrasekhar mass limit for white dwarfs from energy arguments. General-relativistic thermodynamics of perfect fluids: locally inertial comoving frame (LICF), baryon number conservation law. 12. First law of thermodynamics. Barotropic equation of state. Stress-energy tensor of a perfect fluid; isentropic fluid and relativistic Euler equation. 13. Tolman-Oppenheimer-Volkoff stellar structure equations. Boundary conditions to numerically integrate the Tolman-Oppenheimer-Volkoff equations; stellar radius. 14. Stellar gravitational mass and its interpretation. Buchdahl's theorem. Compact star equilibrium configurations and their stability. 15. The neutron star equation of state. The structure of neutron stars. Neutron star observations. Tidal deformability. PART III (Black Holes, 30 hrs) 16. Singularities in General Relativity, how to remove a coordinate singularity, extension of the manifold. An example: Rindlet's spacetime. Removing the r=2M singularity of the Schwarzschild metric. Maximal extension. Kruskal coordinates. Kruskal diagram. Eddington-Finkelstein coordinates and diagram. Eternal vs astrophysical BHs, white holes. 17. Non-coordinate basis. Fermi's coordinates, non-coordinate basis. The silhouette of a BH, BH shadows, light emitted by a static source near a BH, escape angle. 18. Basic aspects of black-hole perturbation theory; scalar toy model, spherical harmonic decomposition; ringdown & quasinormal modes (QNMs) of a Schwarzschild BH. Few words on gravitational perturbations. Numerical methods to find the QNMs. 19. Tidal deformability of a neutron star and tidal Love numbers. Impact on the waveform and implications for the constraints on the equation of state. Tidal Love numbers of a BH in GR. 20. Metric of an isolated and stationary source in the "far field" limit in the case of a weak gravitational field. Few words on the general case. Physical interpretation of M and J in the metric using the pseudotensor. Introduction to the Lense-Thirring effect and geodesic precession. Further kinematical test of gravity with the Gravity Probe B experiment. 21. The Kerr solution. Boyer Linquist coordinates. Main properties of the Kerr metric. 22. Horizons and ergoregion of the Kerr metric. Frame dragging. Static and stationary observers. 23. How to remove the horizon singularity in Kerr. Kerr coordinates and Kerr-Schild coordinates. Principal null geodesics in Kerr. Curvature singularity in Kerr. 24. The interior of a Kerr BH, Maximal extension of Kerr. Causality violation in the interior. Geodesics in Kerr. 25. Geodesics in the Kerr metric: integrability and Carter's constant. Equatorial geodeiscs. Equatorial Null geodesics, light ring of a Kerr BH. 26. Equatorial Timelike geodesics, Keplerian frequency. ISCO of a Kerr BH. 27. Energy/angular mom extraction from a Kerr BH: Penrose's process; Superradiance. 28. BH thermodynamics. Irriducible mass and area of a Kerr BH. BH Area theorem. 29. The laws of BH thermodynamics. BH entropy. Generalized second law. 30. Hawking evaporation and information loss paradox. Q&A session, examples of written test

The General Relativity course.

Ferrari, Gualtieri, Ferrari – General Relativity and its Applications: Black Holes, Compact Stars and Gravitational Waves, CRC Press

Optional.

The exam consists of an oral test, which covers the entire syllabus of the course. The oral test will start with a list of questions/problems to be answered in written form as if they were oral questions. No books or notes are allowed during the test, but the written questions will come with some relevant equations. Following this first part, there will be an individual oral test.

Blackboard lectures with occasional use of multimedia material.