The
scale factor,
cosmic scale factor or sometimes the
Robertson-Walker scale factor parameter of the
Friedmann equations is a function of time which represents the
relative expansion of the
universe. It relates the proper distance (which can change over time, unlike the
comoving distance which is constant) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting
FLRW universe at any arbitrary time
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=3854)
to their distance at some reference time
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=1162)
. The formula for this is:
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=1507)
where
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=1593)
is the proper distance at epoch
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=3854)
,
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=2751)
is the distance at the reference time
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=1162)
and
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=4131)
is the scale factor. Thus, by definition,
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=173)
.