What is the scale factor?

The scale factor a(t) describes how distances in the universe have changed over time. By convention a = 1 today; at the Big Bang a = 0.

When would I use a scale factor other than 1?

To find the physical distance at an earlier epoch. For instance, at redshift z the scale factor was a = 1 / (1 + z).

Not directly. We infer it from redshift and a cosmological model. What we observe is the luminosity distance or angular-diameter distance.

Convert comoving distance to the physical (proper) distance at a given epoch using d_proper = a(t) * d_comoving, where a is the scale factor.

Comoving Distance

Mpc

Scale Factor a(t)

Proper Distance

428.0000 Mpc

Proper Distance (Mly)1,395.9648 Mly

Proper Distance vs Comoving Distance

The proper distance is the physical distance between two objects at a particular moment in cosmic time.

d_proper = a(t) * d_comoving

*a(t)* is the cosmic scale factor (a = 1 today)

*d_comoving* is the comoving distance

At the present epoch a = 1, so proper distance equals comoving distance. In the past (a < 1) the physical distance was smaller.

A galaxy with comoving distance 428 Mpc observed at the present epoch (a = 1).