Inflationary Cosmology – Robert Brandenberger
Abstract: The inflationary scenario has become the standard paradigm of early universe cosmology. In the first part of my lecture series, I will review this scenario, mention its shortcomings and introduce a couple of alternatives. The second part of my lectures will focus on the theory of cosmological perturbations, the main tool used to connect early universe models with cosmological observations. In the third part of my lectures I will introduce attempts to derive early universe cosmology from superstring theory.
1. Overview
1.1. Challenges for early universe cosmology
1.2. Inflation as a solution
1.3. Alternatives More specifically
2. Theory of Cosmological Perturbations
2.1. Classical perturbations
2.2. Quantum fluctuations
2.3. Application to inflation
2.4. Application to alternatives
3. Challenges to the Standard Paradigm
3.1. Trans-Planckian Censorship Criterion
3.2. Remarks on the Swampland Program
4. Superstring Cosmology
4.1. Overview of approaches
4.2. BFSS Matrix Cosmology
Cosmological Observables – Paolo Benincasa
Abstract: These lectures will go thorugh recent progress in formulating an approach to the early universe physics, based on first principles, how they constrain physical processes, and thus the mathematical structure of physical observables) and a novel formulation of the latter in terms of geometrical-combinatorial objects. We will go through their analysis and how to extract model independent physics out of them.
Lecture 1 : Observers and observables in expanding universes
- Introduction and motivation
- Observers in an expanding universe
- Observables in an expanding universe: wavefunctions, probability distributions, correlations
- An open question: What should we compute? - different answers for different issues.
- Bonus topic: physics problems as language problems
Lecture 2: The analytic structure of wavefunctions and correlators
- Perturbation theory and diagrammatics
- Singularities
- Unitarity
- Playing with graphs
Lecture 3: The combinatorial origin of cosmological observables
- From graphs to combinatorics
- A crash course on projective geometry and its extensions
- A first principle definition for wavefunctions and correlators
- Playing with combinatorics
Lecture 4: Extracting physics from combinatorics
- Compatibility conditions 1: Steinmann relations
- Compatibility conditions 2: Beyond Steinmann
- Unitarity revisited
Lecture 5: Late-time, infra-red effects and asymptotic behaviour.
- Massless and light states: infra-red effects
- The loop structure of cosmological observables
- A general analysis of the asymptotic behaviour of cosmological integrals
- Infra-red finite computables
- Wrapping up: a set of open problems
Topological Data Analysis for Cosmology – Matteo Biagetti
Abstract: The matter distribution in the late universe is organized into a cosmic web of clusters, filaments, walls, and voids. The statistics used to characterise it are the two-point correlation function and its Fourier transform: analytically predictable, well understood, optimal for a Gaussian field, and by a wide margin the most studied and most widely applied tools in the field. Gravitational evolution nevertheless drives the late-time field away from Gaussianity, and recovering the information that spreads into higher-order correlations has motivated a broad range of approaches — peak counts, wavelet scattering transforms, marked and field-level methods, learned summaries. Persistent homology is one of these, distinguished by the quantity it measures: the connectivity of the field, a topological rather than a correlational property, which the correlation hierarchy encodes only indirectly.
These lectures develop persistent homology from the ground up. The first part introduces topological data analysis in general — what it measures, where it has succeeded, and what it cannot do — and then builds the persistence diagram in full, together with the stability theorem that makes it usable on noisy data and the machinery that turns it into a feature a statistical or machine-learning method can consume. The second part treats the analysis pipeline stage by stage: every application is a chain of modelling choices, and the aim is to survey what is available at each stage and on what grounds one chooses. It closes with a review of what persistent homology has achieved in cosmology, and of what remains open.
No prior exposure to algebraic topology is assumed.
Part I — Foundations
1. Topological data analysis: an introduction. What it measures; where it has worked; what it cannot do; the state of the subject.
2. Why a topological summary in cosmology. Sufficient statistics and the model-agnostic disadvantage; connectivity versus correlation; when persistence is not the right tool.
3. From data to persistence diagrams. Complexes from point clouds and from fields; filtrations; homology, Betti numbers and the Euler characteristic; persistent homology, the diagram and the barcode.
4. Stability. Distances on diagrams; the stability theorem and its consequences.
5. From Diagrams to Features. Vectorizations; representative cycles; differentiable persistence.
Part II — Applications
6. The pipeline and its choices. The stages; what one might do with a diagram; fixed and learned as per-stage decisions.
7. Filtrations: where the data enters. Point clouds, fields on a grid, signals, and representation spaces; learned filtrations; what each construction ignores.
8. Vectorization and objective: where the task enters. Choose, optimise, or design; the Fisher objective; choosing in practice.
9. Cosmology: what has been achieved, and what has not. The classical baseline; persistence as a summary statistic; weak lensing and learned summaries; open problems.
Amplitude Methods for Gravitational Waves – Carlo Heissenberg
Abstract: The detection of gravitational waves emitted by binary systems has brought about a new era of precision measurements, putting the two-body problem in General Relativity in the spotlight. This has stimulated ground-breaking advancements on the theory side, one of which is due to an unexpected twist: the realization that scattering amplitudes, the bread-and-butter of particle-physics calculations, provide remarkably efficient tools to make predictions for gravitational waves. In these lectures, I will present an introduction to such amplitude-based methods applied to the gravitational two-body problem, covering in particular: eikonal exponentiation, in-in formalism, soft theorems, deflection angle, gravitational waveforms, losses of energy and angular momentum. This will also serve as an occasion to mention recent achievements and outline the challenges that lie ahead.
1. Introduction
1.1. Scales of the gravitational two-body problem
1.2. Analytical approximation methods, post-Minkowskian expansion, soft limit
2. Elastic dynamics
2.1. The 2->2 amplitude
2.2. Eikonal exponentiation and deflection angle at tree level
2.3. One-loop and (mention of the) two-loop correction
2.4. Comparison with the in-in formalism
3. Inelastic dynamics
3.1. Soft theorems and soft spectra
3.2. The 2->3 amplitude
3.3. Waveform
3.4. Losses of energy and angular momentum
