This chapter introduces some of the basic tools of a cosmologist, including scale factor, redshift, and comoving distance. We start with the Hubble law, which is a key consequence of the expanding universe. Next, we cover the possible geometries of space (positively and negatively curved, and flat), and the associated Friedmann--Lemaître--Robertson--Walker metric that describes them. This leads us to define distance measures in cosmology, and introduce the Friedmann equation that describes the evolution of the universe given its contents. We end by discussing the role of critical density and curvature.

We describe gravitational lensing, which is a phenomenon of light from distant objects being deflected by mass that it encounters on its way to our telescopes. We start from mathematical foundations, introduce the lens equation, and derive the deflection angle of light rays in the simple case of a point-mass lens. We introduce the concepts of shear and convergence, and discuss lensing in more mathematical detail, including strong-lensing and weak-lensing regimes. We discuss weak gravitational lensing in sufficient detail to connect it to research in the field, and derive the formula for the weak-lensing convergence power spectrum. We end by discussing galaxy--galaxy lensing and the Bullet cluster, and explain how lensing points to evidence for the existence of dark matter.

This chapter describes how large-scale structure -- the distribution of galaxies on the sky -- can be used to probe the cosmological model. We start by defining the density perturbation and its most fundamental statistical property -- the correlation function. We briefly review the evolution of density perturbations in the standard cosmological model, emphasizing key results. We next discuss the growth of cosmic structure, and segue into talking about the power spectrum of density perturbations, its theoretical description, and its measurements. This leads us to discuss structure formation in the universe more generally, the role of numerical (N-body) simulations, and the mass function. We end by discussing how the statistical properties of dark-matter halos, which can easily be modeled in simulations, are related to those of galaxies that we typically observe.

This chapter reviews the Boltzmann equation, which is a starting point for some of the key results in cosmology. We introduce a general version of the Boltzmann equation, then study its implications in the simple scenario of a few interacting particles. We introduce the concept of a freezeout of particle species, and illustrate it using a simple example. We end the chapter by discussing baryogenesis (the process that generated the excess of baryons over antibaryons), and Sakharov conditions for successful baryogenesis to take place.

We study the evolution of particle species throughout the history of the universe. We introduce the phase-space distribution function, and review basic concepts in statistical mechanics as applied to early-universe cosmology, including thermal equilibrium, entropy, and chemical potential. We calculate the effective number of relativistic species, and show how it varies as a function of time.

We give a broad-brush overview of cosmology, including a timeline of events starting from the Big Bang until the present day. We introduce the three pillars of the Big Bang cosmological model, the concepts of homogeneity and isotropy, as well as parsec as a unit of distance. We also introduce natural units, and develop intuition on how to adopt and use them.

Here we review dark energy, the component that causes accelerated expansion of the universe. We start by reviewing the history of this fascinating discovery, describing in detail how type Ia supernovae were used to measure the expansion rate and find that the expansion is speeding up. We then outline modern evidence for the existence of dark energy, how dark energy is parametrically described, and what its phenomenological properties are. We review the cosmological-constant problem that encapsulates the tiny size of dark energy relative to expectations from particle physics. Next we introduce physical candidates for dark energy, including scalar fields and modified gravity. We end by explaining the controversial anthropic principle, and describe the possible future expansion histories of the universe dominated by dark energy.

Neutrinos have an important role in cosmology, and here we review them in some detail. We review the fascinating history of how neutrinos were first proposed then detected. We then mathematically describe neutrino oscillations. We describe decoupling of neutrinos from the thermal bath, point out the likely existence of the cosmic neutrino background, and discuss prospects for detecting it directly.