# How far away are objects whose light will never reach us, because of the expansion of the universe?

I thought I had read this question on Stack Exchange before, but now I cannot find it... In fact, I thought I had posted this question before, somewhere, on Stack Exchange...

In other words, if an object is currently 62 billion light years away, or more, the light it is emitting right now will never reach Earth...

Is there a term for this 'line' or 'sphere'?

• This link seems to be useful here. It points to a pdf. bit.ly/36WmZpR – Alchimista Oct 12 '20 at 10:58

You're completely correct!

The farthest we can see (in principle, not in practice) is called the particle horizon. Currently, the distance to the particle horizon is $$d_\mathrm{P} \simeq 46\,\mathrm{Glyr}$$, but as time goes on, light from more and more distant regions will reach us.

If the Universe contains only "regular stuff" such as normal matter, dark matter, and radiation, there would be no limit to how far we could see, if we just wait long enough (and invent increasingly powerful telescopes, but that's a technical detail).

However, it seems our Universe contains — and in fact is dominated by — dark energy, which has the unfortunately effect of accelerating the expansion. This implies that, eventually, space expands faster than light can keep up with (note that I don't say "expands faster than light", since this is always, and has always been, the case for sufficiently distant region; faster-than-light expansion is in itself no hindrance for light to reach us, se this answer, or this).

For the amount of dark energy observed in our Universe, and assuming that its so-called equation of state does not evolve, we can calculate that the particle horizon will never expand beyond the regions of the Universe that are currently 62 Glyr away (in the future, the distance to those regions will increase without bounds because of the expansions).

There are several different quantities of this sort that you can define, and the definitions are fairly confusing. Hopefully the following diagram will make things clearer.

                   Z                      <- future infinity
/ \
/   \
/     \
D   C    B   A   B    C   D         <- now
.   .   /   / \   \   .   .
.   .  /   /   \   \  .   .
.   . /   /     \   \ .   .
.   ./   /       \   \.   .
.   /   /         \   \   .
.  /.  /           \  .\  .
. / . /             \ . \ .
./  ./               \.  \.
~~~~~~d~~~c~~~~~~~~a~~~~~~~~c~~~d~~~~~~   <- last scattering (universe opaque below this)


The horizontal axis on this diagram is "comoving distance", with respect to which objects that move with the Hubble flow are at rest. The vertical axis is "conformal time", with respect to which light travels along diagonal lines of a constant slope (in combination with the comoving distance).

A is our current location; Z is the ultimate location in the far future of the same matter, supposing it doesn't deviate much from the Hubble flow.

Light emitted in the current cosmological era from closer than B will reach us at some point in the future. Light emitted in the current era from farther than B will never reach us (if we don't deviate too far from the Hubble flow). The distance from A to B is around 16 Gly (billion light years). This seems like the closest match to the quantity you were asking about in your question. I'm not sure that it has a name.

Light emitted at the last scattering time from c is just reaching us now. This is the cosmic microwave background radiation. The distance in the present era from us to the presumed current location of the matter than emitted that light (if it hasn't deviated too much from the Hubble flow) is the distance from A to C. That distance is around 46 Gly. This is what's normally called the radius of the observable universe. Note that the distance from a to c is around 1100 times smaller, or 42 Mly.

Light emitted at the last scattering time from closer than d will eventually reach us. Light emitted at the last scattering time from farther than d never will (if we don't deviate from the Hubble flow). The present-day distance from A to D, which is the extrapolated current location of that matter supposing it doesn't deviate from the Hubble flow, is around 62 Gly. Again, the distance from a to d is around 1100 times smaller. (And the distance at future infinity is ∞ times larger.)

So your statement that "if an object is currently 62 billion light years away, or more, the light it is emitting right now will never reach Earth" is incorrect, but it would be correct if you replaced 62 by 16, or if you replaced "is emitting right now" by "emitted at the last scattering time, supposing it didn't deviate too much from the Hubble flow". Also, you should probably replace "Earth" by "whatever matter is left at Earth's location in the distant future, where 'location' is defined by the Hubble flow".