Having developed the necessary mathematics in chapters 4 to 6, chapter 7 returns to physics Evidence for homogeneity and isotropy of the Universe at the largest cosmological scales is presented and Robertson-Walker metrics are introduced. Einstein’s equations are then used to derive the Friedmann equations, relating the cosmic scale factor to the pressure and density of matter in the Universe. The Hubble constant is discussed and an analytic form of the red-shift distance relation is derived, in terms of the matter density, the cosmological constant and the spatial curvature, and observational values of these three parameters are given. Some analytic solutions of the Friedmann equation are presented. The cosmic microwave background dominates the energy density in the early Universe and this leads to a description of the thermal history of the early Universe: the transition from matter dominated to radiation dominated dynamics and nucleosynthesis in the first 3 minutes. Finally the horizon problem and the inflationary Universe are described and the limits of applicability of Einstein's equations, when they might be expected to break down due to quantum effects, are discussed.

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.

We walk through the different epochs and eras of the universe, going forward in time from the Hot Big Bang. In the earliest universe, radiation (photons) dominated over matter. As the universe cools, electrons are able to recombine with protons, then helium and other light elements were formed in the first few minutes. Cosmic inflation is posited to overcome several problems, but investigations to probe and perhaps confirm inflation are ongoing.