Cosmic Inflation
A rigorous graduate-level course on cosmic inflation — from the problems of standard cosmology through inflaton field dynamics, slow-roll theory, primordial perturbations, inflationary models, reheating, and observational signatures — with full derivations, MathJax equations, and Python simulations.
Course Overview
Cosmic inflation is a period of exponential expansion in the very early universe ($t \sim 10^{-36}$ to $10^{-32}$ s), proposed to solve fundamental problems with the standard Big Bang model. During inflation, quantum fluctuations of the inflaton field are stretched to cosmological scales, seeding all the structure we observe today — galaxies, galaxy clusters, and the CMB anisotropies. This course derives every key result from first principles, following the tradition of Liddle & Lyth, Baumann, Mukhanov, and Weinberg.
What You Will Learn
- ● Why inflation is needed: horizon, flatness, and monopole problems
- ● Inflaton field dynamics: Klein-Gordon in expanding spacetime
- ● Slow-roll approximation and the number of e-folds
- ● Quantum origin of perturbations: scalar and tensor power spectra
- ● Inflationary models: chaotic, Starobinsky, natural, Higgs inflation
- ● Reheating and the transition to the hot Big Bang
- ● CMB observables: $n_s$, $r$, $f_\text{NL}$, B-modes
Central Equations
Friedmann: $H^2 = V(\phi)/(3M_P^2)$
Klein-Gordon: $\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0$
Slow-Roll: $\epsilon_V = \frac{M_P^2}{2}(V'/V)^2$, $\eta_V = M_P^2 V''/V$
Scalar Spectrum: $\mathcal{P}_\mathcal{R} = \frac{V}{24\pi^2 M_P^4 \epsilon_V}$
Spectral Index: $n_s - 1 = -6\epsilon_V + 2\eta_V$
Tensor-to-Scalar: $r = 16\epsilon_V$
e-folds: $N = \frac{1}{M_P^2}\int_{\phi_\text{end}}^{\phi} \frac{V}{V'}\,d\phi$
Consistency: $r = -8n_t$
1. Problems of Standard Cosmology
Horizon problem, flatness problem, magnetic monopole problem, and the need for a pre-Big-Bang epoch.
2. Inflaton Field Dynamics
Scalar field action, Klein-Gordon equation in FLRW, energy-momentum tensor, equation of state.
3. Slow-Roll Inflation
Slow-roll parameters, slow-roll approximation, Hubble slow-roll, number of e-folds, attractor behavior.
4. Primordial Perturbations
Quantum fluctuations, Mukhanov-Sasaki equation, scalar and tensor power spectra, spectral index, consistency relation.
5. Inflationary Models
Chaotic inflation, Starobinsky R², natural inflation, Higgs inflation, α-attractors, and the ns-r plane.
6. Reheating
Perturbative decay, parametric resonance, preheating, thermalization, and reheating temperature.
7. Observational Signatures
CMB angular power spectrum, B-mode polarization, non-Gaussianity, Planck/BICEP constraints.
8. Open Questions & Alternatives
Trans-Planckian problem, eternal inflation, measure problem, ekpyrotic/cyclic models, string gas cosmology.
Research Seminars
Advanced research talks on inflation, de Sitter space, and early universe physics.
An Observer in de Sitter Space, and Rereading Everett
Edward Witten
Combinatorics and Geometry of Fundamental Physics and Cosmology
Nima Arkani-Hamed
Constraints on Long-Range Forces in De Sitter Space
Callum Jones
Is String Theory Unique?
Clifford Cheung
Prerequisites & References
Prerequisites
- • General relativity (FLRW metric, Friedmann equations)
- • Quantum field theory (scalar fields, quantization)
- • Statistical mechanics (thermal equilibrium)
- • Cosmological perturbation theory (helpful but not required)
Recommended Texts
- • Baumann, Cosmology (Cambridge, 2022)
- • Liddle & Lyth, Cosmological Inflation and Large-Scale Structure
- • Mukhanov, Physical Foundations of Cosmology
- • Weinberg, Cosmology (Oxford, 2008)
- • Peter & Uzan, Primordial Cosmology