After a series of observational and theoretical breakthroughs in the 1960s, the Steady State theory was discarded, whereas the Big Bang cosmological paradigm remained viable. This model is described by the Friedmann equations with a Robertson-Walker metric. The metric describes the dynamic spacetime intervals and the Friedmann equations describe the expansion dynamics. The latter are derived from Einstein’s field equations of General Relativity assuming an isotropic and homogeneous medium, conservation of energy density, and an equation of state known as the “continuity equation.” Friedmann’s equations are conveniently written in terms of a time-dependent scale factor, the Hubble constant, and four present-epoch cosmological parameters. Today, we live in an era known as precision cosmology, in which the Hubble constant and cosmological parameters are measured with 1% or better uncertainties. In this chapter, we present an abridged derivation of the Friedmann equations and discuss the cosmological parameters and their temporal evolution in detail. The Robertson-Walker metric is then rewritten in terms of radial and transverse components suitable for convenient practical application.

Hubble’s law gives us the simple and obvious interpretation that we currently live in an expanding universe. The inverse of Hubble’s constant defines the "Hubble time" which effectively marks the time in the past since the expansion began. more realistically, one would expect the universe expansion to be slowed by the persistent inward pull of gravity from its matter. We consider how various theoretical models for the universe connect with the observable redshift that indicates its expansion.

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.

Hubble’s law gives us the simple and obvious interpretation that we currently live in an expanding universe. The inverse of Hubble’s constant defines the “Hubble time,” which effectively marks the time in the past since the expansion began. More realistically, one would expect the universe expansion to be slowed by the persistent inward pull of gravity from its matter. We consider how various theoretical models for the universe connect with the observable redshift that indicates its expansion.