Scale factor (cosmology)

Lore & Background

The scale factor is a geometrical scaling factor that relates proper distances between objects at different cosmic times. For example, the proper distance between two galaxy clusters at time t is given by d(t) = a(t) d₀, where d₀ is the distance at the reference time t₀. The scale factor is independent of location and direction, and its evolution is determined by the equations of general relativity, specifically the Friedmann equations for a locally isotropic, locally homogeneous universe. In the early stages of the Big Bang, most energy was in the form of radiation, leading to a radiation-dominated era. As the universe expanded and cooled, matter became dominant, and recent results suggest a transition to a dark-energy-dominated era about 4 billion years ago. The effective energy densities of radiation and matter scale differently with the scale factor, driving these transitions. The Hubble parameter H(t) is defined as the time derivative of the scale factor divided by the scale factor itself: H(t) ≡ ȧ(t)/a(t). This parameter varies with time, and its current value is the Hubble constant H₀. From the relation d(t) = d₀ a(t), one derives Hubble's law: ȧ(t) = H(t) d(t).

Reader's Guide

The scale factor is fundamental to modern cosmology because it provides a simple mathematical framework for describing the expansion of the universe. By setting a(t₀)=1 at the present time, all distances at earlier epochs are scaled relative to today. This allows cosmologists to model how the universe evolved from a hot, dense state through radiation-dominated and matter-dominated eras to the current dark-energy-dominated epoch. The scale factor's evolution is governed by the Friedmann equations, which incorporate the energy densities of radiation, matter, and dark energy. The Hubble parameter, derived from the scale factor, directly gives the expansion rate and leads to Hubble's law, which relates the recession velocity of distant galaxies to their distance. Current evidence indicates that the expansion is accelerating, implying a positive second derivative of the scale factor. Understanding the scale factor is essential for interpreting observations of cosmic microwave background radiation, galaxy surveys, and supernovae, as it links theoretical models to measurable quantities like redshift and proper distances.

Did You Know?

• The scale factor is dimensionless and conventionally set to 1.0 at the present time t₀.
• The proper distance between objects at time t is given by d(t) = a(t) d₀, where d₀ is the distance at the reference time.
• The Hubble parameter H(t) is defined as ȧ(t)/a(t) and varies with time, not space.
• The universe transitioned from a radiation-dominated era to a matter-dominated era, and about 4 billion years ago entered a dark-energy-dominated era.