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.
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. This structure carries cosmological information, but the standard statistical tools, the power spectrum and its higher-order generalizations, are not naturally suited to describing its connectivity, which is a topological rather than a correlational property. The power spectrum is optimal for a Gaussian field, whereas gravitational evolution and, potentially, primordial physics render the late-time universe non-Gaussian; higher n-point functions recover part of this information, but at increasing cost and decreasing signal-to-noise.
This course develops persistent homology as a complementary route. Rather than measuring correlations at fixed scales, persistent homology tracks the connectivity of a field across all scales at once, producing a stable, multi-scale summary of its topological structure. The first part of the course builds the mathematical foundations of the method from the ground up, at a level of generality that makes clear why it applies equally to a point cloud, a density field on a grid, or a signal evolving in time. The second part turns to applications, developing in detail a line of work in which persistent homology is used as a cosmological summary statistic, as a probe of primordial non-Gaussianity and, more broadly, as a competitive alternative to standard statistics for parameter inference. Two applications outside cosmology, in neuroscience and in the analysis of large language models, are presented briefly to indicate the generality of the framework.
No prior exposure to algebraic topology is assumed.
1. Motivation and Constructions from Data (1 hour)
1.1. Why topology in cosmology?
1.2. The cosmic web as a topological decomposition; when not to use TDA
1.3. Simplicial complexes; the Vietoris–Rips and Čech constructions; the Nerve theorem
2. Homology and Persistence (1 hour)
2.1. Cubical complexes and sublevel-set filtrations
2.2. Homology: components, loops, and voids
2.3. Persistent homology, the persistence diagram, and the barcode
3. Stability, Extensions, and the Bridge to Inference (1 hour)
3.1. The stability theorem and diagram distances
3.2. Extensions: zigzag, multi-parameter persistence, and Mapper
3.3. Vectorizations and differentiable persistence
4. Persistent Homology as a Cosmological Statistic (1 hour)
4.1. A simple construction on halo and galaxy catalogs for primordial non-Gaussianity
4.2. Realistic survey conditions and Fisher-forecasts
5. Learned Compression and Applications Beyond Cosmology (1 hour)
5.1. Learned neural compression for simulation-based inference
5.2. Applications beyond cosmology: time series, neuroscience, and large language models
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
