# Why does the cosmic comoving horizon expand with time?

Since the comoving distance is unchanged by cosmic expansion, why isn't the comoving horizon constant?

The comoving horizon (I'm used to calling it the 'particle horizon') is the furthest distance a particle can have travelled since the beginning of the Universe. The fastest a particle can travel is the speed of light, so the particle horizon is defined as $c\eta$ where $\eta$ is the conformal time:

$$\eta(a) = \int_0^a\frac{1}{a'H(a')}\frac{{\rm d}a'}{a'}$$

$a$ is the scale factor and $H$ is the Hubble parameter.

The particle horizon in comoving coordinates is always expanding because as time passes a particle moving at the speed of light always travels further. Physically how far it gets depends on the expansion history, but it must always keep moving, so in comoving coordinates the horizon expands.

The best paper to follow the different horizon descriptions I've seen is at https://arxiv.org/abs/astro-ph/0310808. It's got some very nice graphics and explanations. It was written to try to clarify the confusion about the different horizons.

It's a confusing thing, almost always, so the best thing to do is to keep a few things straight in your head.

You have the comoving distance right, if you mean the coordinate distance. So your question is a good question. It is always fixed. Unfortunately the comoving horizon is a little confusing, best to think of it as the particle horizon, i.e., the max physical distance that we can still receive light (signals or information) from. It is the comoving coordinate distance times the scale factor. I guess it's meant to mean the horizon for the further comoving object we can see - but that still means physical distances. So though comoving distance to it is fixed, the physical distance keeps on increasing due to the expansion. It is where something is now, from which we see light it emitted. The light we see is indeed old, not what it is emitting now. Best way to think of it is to think of it as the particle horizon. See the wiki article for different horizons at https://en.m.wikipedia.org/wiki/List_of_cosmological_horizons

EDITED SLIGHTLY TO ACCOUNT FOR @Kyle Oman's comment below. The issue is a bit tricky and easily confusing. By comoving coordinate distance (now inserting the word 'coordinate' in the answer to be exact and explicit) the OP meant just exactly that. It is what in the arXiv paper referenced, in Eqs 15 and 16, they call $\Xi$. It is unchanged for two objects (like galaxies or clusters) that are comoving, i.e., moving with the Hubble flow, or the expansion of the universe. In those coordinates they are not moving. If you know their value (which we don't in an atlas say, it has to be calculated, and later in the arxiv paper it shows how), then indeed you just multiply by scale factors. See Eq. 15 in the paper. Something to watch for, is the calculation of the coordinate distance from eg it's redshift z, or other parameters - see Eq. 24 and 27. The final point to make is that Figs. 1 and 3 shows the different horizons, and particularly the particle horizon.

• It sounds like you say the particle horizon = comoving distance * scale factor. It's a bit unclear what that means, and I don't think it's correct in any case. – Kyle Oman Feb 1 '17 at 18:24
• I'll check the exact correct statement and edit. – Bob Bee Feb 1 '17 at 22:28