Here we discuss inflation, a period of accelerated expansion that occurred in the very early history of the universe. We motivate inflation by describing the flatness and horizon problems, then explain how inflation resolves them. We describe the early history of inflationary ideas, then move on to modern work where we outline the standard scalar-field model for inflation, and define the slow-roll parameters that phenomenologically describe the dynamics of inflation. We briefly outline how inflation leads to the generation of density fluctuations in the universe; we mathematically describe the spectrum of these fluctuations, and confront it with modern observations. We end by discussing more speculative ideas in this area, including eternal inflation and multiverse.

In this chapter we study the primordial process in which nuclei of atoms formed -- the Big Bang nucleosynthesis, or BBN for short. We show that BBN principally leads to the synthesis of hydrogen and helium, along with a trace amount of a few more of the lightest elements. We describe the basic thermodynamical and nuclear-physics conditions in the early universe, and explain how they determine the primordial origin and abundance of nuclei in the universe. Finally, we illustrate how the lightest-element abundances are in spectacularly good agreement with observations, making BBN one of the pillars of the hot Big Bang cosmological model.

In this chapter we review dark matter, whose physical nature remains unknown 90 years after astronomer Fritz Zwicky found the first evidence for it. We go over the historical evidence for dark matter, than review modern indications for the existence of this component. We then review particle candidates for dark matter, outlining the properties that dark matter of each particle candidate would have. Next we go over the direct, indirect, and laboratory searches for dark matter. We end by discussing a dramatically different alternative (to particles) for the nature of dark matter -- that of a modified theory of gravity (MOND).

This chapter lays out the physics of the cosmic microwave background radiation (CMB). We start by describing the discovery of the CMB, the blackbody property of the radiation, and its basic properties like mean temperature and dipole. We then discuss the physics of the epoch of recombination when the CMB was generated, and derive key properties of the CMB anisotropy starting from basic principles. We continue to mathematically describe the CMB anisotropy, and outline ways in which it is measured from CMB maps and compared to theory and inflationary predictions. Along the way, we emphasize the statistical properties of the angular power spectrum of the CMB, and how they are used to confront measurements and theory. We end by discussing cutting-edge topics in CMB research, such as CMB polarization, Sunyaev--Zeldovich effect, and primordial non-Gaussianity.

We complete the basic equations of cosmology by introducing the second Friedmann and continuity equations. We next introduce the equation of state parameter, and describe the evolution of the universe in the simplest cosmological models: matter-only, radiation-only, and lambda-only. We introduce the concept of a cosmological horizon, and explain how to calculate the age of the universe in a given model. Along the way, we establish the fiducial cosmological model -- a set of cosmological parameters that we will use for all our results in the remainder of the book. We end by introducing two observationally accessible distance measures -- luminosity distance and angular-diameter distance.

This chapter reviews statistics and data-analysis tools. Starting from basic statistical concepts such as mean, variance, and the Gaussian distribution, we introduce the principal tools required for data analysis. We discuss both Bayesian and frequentist statistical approaches, with emphasis on the former. This leads us to describe how to calculate the goodness of fit of data to theory, and how to constrain the parameters of a model. Finally, we introduce and explain, both intuitively and mathematically, two important statistical tools: Markov chain Monte Carlo (MCMC) and the Fisher information matrix.

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.