The 13 billion lightyears distance of the quasar mean 13 billion lightyears light-travel distance. In other words, the light took 13 billion years to reach us, independent of the distance the quasar is now away from us.
The current proper distance (a chain of rulers would measure) is called comoving distance; it is much larger than the light-travel distance for large redshifts.
The quasar is moving away from us in an accelerated fashion, and is now moving away faster than light. This means, light which is emitted now (in the sense of cosmic time) by the galaxy, the quasar may have evolved into in the meanwhile, will never reach us.
The light, the quasar emitted 13 billion years ago, left the quasar just in time to leave the regions of the universe which later have been receding faster than light from us, to eventually reach us.
As a thought experiment, imagine walking with 5 km/h (our playground speed of light) on a rubber band which is expanding with a constant rate of 1 km/h per meter actual length of the band. (This leads to an exponential acceleration of the distance between the two marks on the band). If you start closer than 5 meters away from your goal, e.g. from a start mark (our quasar) 4.50 m away, you'll finally reach the goal mark. The start mark will soon be further away from the goal mark than 5 meters, therefore receding with more than 5 km/h from the goal shortly after you left the start mark.
At the moment you arrive at the goal mark, the start mark will be much further away (comoving distance) than the distance you needed to walk (light travel distance). And you've been walking a longer distance than the (proper) distance between the marks was at the time you started walking.
Btw.: Acceleration is only felt as a force, when the velocity to the local rubber band (mataphoric space-time) is changed.
Example calculations with a protogalaxy of redshift $z=11.9$:
Based on the Cosmology Calculator on this website, the cosmological parameters $H_0 = 67.11$ km/s/Mpc,
$\Omega_{\Lambda} = 0.6825$ provided by the Planck project, and the scale factor $d(t) = d_0 / (1+z)$, setting $\Omega_M = 1- \Omega_{\Lambda} = 0.3175$, the
age of the universe is $13.820$ Gyr, and the comoving distance of the protogalaxy is $d_0 = 32.644$ Gly.
The age of the universe, we see the protogalaxy (at redshift 11.9), was 0.370 Gyr, light-travel distance has been 13.450 Gly, proper distance was 2.531 Gly.
After the protogalaxy has been emitting light 0.370 Gyr after the big bang, the light travelled towards us through space of redshift beginning with 11.9 shrinking to 0; the light arrived at us 13.820 Gyr after the big bang. The comoving distance (to us) of the space traversed by the light started with 32.644 Gly shrinking to 0. The remaining distance, the light needed to travel, started with 13.450 Gly shrinking to 0. The proper distance between the protogalaxy and us started with 2.531 Gly increasing to 32.644 Gly due to the expansion of spacetime.
Here some intermediate states described by
a couple of tuples, consisting of
- redshift $z$,
- according age $t$ of the universe (Gyr),
- comoving radial distance (at age $t$) of the emitted light, we can now detect from the protogalaxy (Gly),
- remaining light travel distance of that emitted light (Gly),
- proper distance of the protogalaxy at age $t$, according to $d(t) = d_0 / (1+z)$:
$$(11.9, 0.370, 32.644, 13.450, 2.531),$$
$$(11.0, 0.413, 32.115, 13.407, 2.720),$$
$$(10.0, 0.470, 31.453, 13.349, 2.968),$$
$$( 9.0, 0.543, 30.693, 13.277, 3.264),$$
$$( 8.0, 0.636, 29.811, 13.184, 3.627),$$
$$( 7.0, 0.759, 28.769, 13.061, 4.081),$$
$$( 6.0, 0.927, 27.511, 12.892, 4.663),$$
$$( 5.0, 1.168, 25.952, 12.651, 5.441),$$
$$( 4.0, 1.534, 23.952, 12.285, 6.529),$$
$$( 3.0, 2.139, 21.257, 11.680, 8.161),$$
$$( 2.0, 3.271, 17.362, 10.549, 10.881),$$
$$( 1.0, 5.845, 11.124, 7.974, 16.322),$$
$$( 0.0, 13.820, 0.0 , 0.0 , 32.644).$$
The Hubble parameter, meaning the expansion rate of space per fixed proper distance, is decreasing with time. This allowed the protogalaxy to recede almost with the speed of light, although it was just about 2.5 Gly away from us (proper distance) in the time, when it emitted the light we detect now.
Nevertheless distant objects in this space accelerate away from us, since their increasing distance is multiplied with the expansion rate of space.
