TASI 2009.

https://arxiv.org/abs/0907.5424

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

**Abstract: **
In a series of five lectures I review inflationary cosmology.
I begin with a description of the initial conditions problems of the
Friedmann-Robertson-Walker (FRW) cosmology and then explain how
inflation, an early period of accelerated expansion, solves these
problems. Next, I describe how inflation transforms microscopic quantum
fluctuations into macroscopic seeds for cosmological structure
formation. I present in full detail the famous calculation for the
primordial spectra of scalar and tensor fluctuations. I then define the
inverse problem of extracting information on the inflationary era from
observations of cosmic microwave background fluctuations. The current
observational evidence for inflation and opportunities for future tests
of inflation are discussed. Finally, I review the challenge of relating
inflation to fundamental physics by giving an account of inflation in
string theory.

**Lecture 1: Classical Dynamics of Inflation**

The aim of this lecture is a first-principles introduction to the
classical dynamics of inflationary cosmology. After a brief review of
basic FRW cosmology we show that the conventional Big Bang theory leads
to an initial conditions problem: the universe as we know it can only
arise for very special and finely-tuned initial conditions. We then
explain how inflation (an early period of accelerated expansion) solves
this initial conditions problem and allows our universe to arise from
generic initial conditions. We describe the necessary conditions for
inflation and explain how inflation modifies the causal structure of
spacetime to solve the Big Bang puzzles. Finally, we end this lecture
with a discussion of the physical origin of the inflationary expansion.

**Lecture 2: Quantum Fluctuations during Inflation**

In this lecture we review the famous calculation of the primordial
fluctuation spectra generated by quantum fluctuations during
inflation. We present the calculation in full detail and try to avoid
`cheating' and approximations. After a brief review of fundamental
aspects of cosmological perturbation theory, we first give a qualitative
summary of the basic mechanism by which inflation converts microscopic
quantum fluctuations into macroscopic seeds for cosmological structure
formation. As a pedagogical introduction to quantum field theory in
curved spacetime we then review the quantization of the simple harmonic
oscillator. We emphasize that a unique vacuum state is chosen by
demanding that the vacuum is the minimum energy state. We then proceed
by giving the corresponding calculation for inflation. We calculate the
power spectra of both scalar and tensor fluctuations.

**Lecture 3: Contact with Observations**

In this lecture we describe the inverse problem of extracting
information on the inflationary perturbation spectra from observations
of the cosmic microwave background and the large-scale structure. We
define the precise relations between the gauge-invariant scalar and
tensor power spectra computed in the previous lecture and the observed
CMB anisotropies and galaxy power spectra. We give the transfer
functions that relate the primordial fluctuations to the late-time
observables. We then use these results to discuss the current
observational evidence for inflation. Finally, we indicate opportunities
for future tests of inflation.

**Lecture 4: Primordial Non-Gaussianity**

In this lecture we summarize key theoretical results in the study of
primordial non-Gaussianity. Most results are stated without proof, but
their significance for constraining the fundamental physical origin of
inflation is explained. After introducing the bispectrum as a basic
diagnostic of non-Gaussian statistics, we show that its momentum
dependence is a powerful probe of the inflationary action. Large
non-Gaussianity can only arise if inflaton interactions are significant
during inflation. In single-field slow-roll inflation non-Gaussianity is
therefore predicted to be unobservably small, while it can be
significant in models with multiple fields, higher-derivative
interactions or non-standard initial states. Finally, we end the lecture
with a discussion of the observational prospects for detecting or
constraining primordial non-Gaussianity.

**Lecture 5: Inflation in String Theory**

We end this lecture series with a discussion of a slightly more advanced
topic: inflation in string theory. We provide a pedagogical overview of
the subject based on a recent review article with Liam McAllister. The
central theme of the lecture is the sensitivity of inflation to
Planck-scale physics, which we argue provides both the primary
motivation and the central theoretical challenge for realizing inflation
in string theory. We illustrate these issues through two case studies of
inflationary scenarios in string theory: warped D-brane inflation and
axion monodromy inflation. Finally, we indicate opportunities for
future progress both theoretically and observationally.