Gravitational Waves theory & observations

Course goals

The first half of this course will have different content for students in the experimental physics and in the theoretical physics track. The second half, which deals with observations, will be joint between the two tracks.
At the end of the course, students will have the following knowledge and skills.
Experimental physics track:

1. The student has a working knowledge of differential geometry, and understands the structure of the Einstein field equations.
2. The student is able to discuss some of the basic solutions to the Einstein equations: relativistic stars, black holes, homogeneous and isotropic cosmologies.
3. The student is able to explain how the linearized Einstein equations follow from the full field equations, and how this leads to gravitational waves.
4. The student is able to derive, to leading order, the gravitational wave signal emitted by the inspiral of binary neutron stars and black holes.

Theoretical physics track:

1. The student is able to discuss relativistic compact objects, particularly neutron stars and their interior structure, through the Tolman-Oppenheimer-Volkoff equations.
2. The student has a grasp of the post-Newtonian formalism, particularly in the context of the inspiral of compact binary objects, and of how the properties of black holes and neutron stars get imprinted on the gravitational wave signals that are emitted.
3. The student has an understanding of black hole perturbation theory.
4. The student is able to discuss the mathematics and numerics behind the state-of-the-art understanding of the inspiral-merger-ringdown process of compact binary coalescence as a whole.

Both tracks:

5. The student is able to discuss how interferometers like LIGO and Virgo are used to detect gravitational waves, with an emphasis on the data analysis techniques that allow us to find weak signals in detector noise.
6. The student has a grasp of how Bayesian inference techniques are used to extract information about the source from a gravitational wave signal, through parameter estimation and model selection.
7. The student knows how Bayesian data analysis methodology enables us to use gravitational waves to probe the strong-field dynamics of gravity, the basic structure of neutron stars, and the evolution and contents of the Universe.

The student is able to discuss how future gravitational wave observatories on the ground (Einstein Telescope) and in space (LISA) will open up new avenues in gravitational physics.

Content

First half of the course: For the experimental physics track, the first half of the course involves an introduction to general relativity. The basics of differential geometry are explained, and the Einstein field equations are motivated. Some exact solutions of the Einstein equations are discussed: the structure of relativistic stars (e.g. neutron stars) and black holes, as well as cosmological solutions. Next we show how from the full Einstein equations, the linearized equations are derived, which yield a description of gravitational waves in the limit of weak gravitational fields. With this in hand, we study the gravitational wave signal emitted by binary neutron star or black holes as they spiral towards each other, highlighting how the different nature of the objects involved affects the shape of the signal. For the theoretical physics track, prior knowledge of general relativity is required. Here we will focus on the importance of theoretical models for using gravitational waves from binary systems of compact objects to probe black holes, nonlinear gravity, ultra-dense matter in neutron stars, and physics beyond the standard model. We will start with an introduction to gravitational waves and their properties, and discuss the structure of relativistic compact objects as well as their gravitational wave signals in binary systems. We will study the link between asymptotic gravitational waves and source properties by using various approximation methods, including black hole perturbation theory, post-Newtonian methods, and effective approaches for combining available information on different aspects of dynamical spacetimes. Second half of the course: The second half of the course, which is joint between the two tracks, focuses on gravitational wave observations and what can be learned from them. The response of interferometric detectors to gravitational waves is derived, and it is explained how signals are filtered out of detector noise. Bayesian parameter estimation is introduced, and it is shown how this can be used to efficiently measure e.g. the masses, spins, and other properties of compact objects from the gravitational wave signal that was received. This enables gravitational wave astronomy. It is shown how gravitational waves can be used to probe the interior structure of neutron stars, test the strong-field dynamics of gravity, and study the structure and expansion of the Universe at large scales. We end with an outlook on the future prospects of this vibrant field of research.