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.

Which of the four-parameter family of Friedman–Robertson–Walker (FRW) cosmological models best fits our universe and why? This chapter addresses these two central questions for observation and theory in cosmology. Of the four parameters that define an FRW model, only two are determined by observations so far: the Hubble constant; and the ratio of energy density in radiation to the critical density. To determine the others, the spacetime geometry of the universe must be measured on large scales through a study of how matter moves through it. We describe two illustrative ways of doing that – one based on observations of distant supernovae, and the other on observations of the cosmic background radiation. Remarkably, the best cosmological parameter values are consistent with the universe being spatially flat – right on the borderline between positive and negative spatial 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.