# What is the farthest light from Earth can reach, given accelerating expansion?

At what current distance is the nearest point Q at which the photon sent from Earth right now never reaches it?

There is the concept of a Hubble volume, which "is a spherical region of the observable universe surrounding an observer beyond which objects recede from that observer at a rate greater than the speed of light due to the expansion of the Universe.", according to Wikipedia. This is a radius of 14.4 billion light years.

However due to the accelerating expansion of the universe, if I were to send a photon at a target 14.3 billion light years away, I assume that by the time it naively should have arrived, the target will have accelerated to beyond light speed, and the photon would never arrive.

Like the Hubble volume of universe which can causally affect me, there has to be a similar smaller volume which I can causally affect, due to this accelerating expansion. Since we have estimates for the behavior of expansion, this should be not too hard to calculate, but I seem unable to find any such estimates? Surprising, since it seems like an interesting property to me.

The maximum comoving distance that a light signal emitted today $$(t_0)$$ can travel until the 'end' of time at $$t=\infty$$ is given by the current proper distance to the Cosmic Event Horizon:

$$D_{eh} (t_0) = \int_{t_0}^\infty \frac{cdt}{a(t)}\simeq 16 \;Gly$$

in the $$\Lambda CDM$$ model, with the normalization $$a(t_0)=1$$. Consider a galaxy G currently located at a proper distance $$D$$ from us.

• If $$D\le 16$$ Gly, a light signal sent today from Earth will be able to reach G at some time in the future.
• If $$D>16$$ Gly, no light signal sent today from Earth can ever reach G.

This means that light sent today from Earth can reach objects that are currently receding from us faster than the speed of light c (the Hubble radius is $$D_H= 14.4$$ Gly at the present time).

The above seems impossible. Suppose that we sent a light signal right now towards a galaxy G at $$D=16$$ Gly. Since $$D>D_H$$, the galaxy G is at present receding from us faster than c, and therefore the proper distance from a photon to G is actually increasing. So, how could light reach something that is receding faster than light?

ANSWER: From the perspective of an observer in G, the photons sent by us are iniatially moving away from G, because these photons are initiatilly moving through a region of space outside the Hubble sphere centered on G. But the Hubble distance is not constant; it is increasing and will continue to increase in the future to a maximum of about 17.6 Gly. If $$D\le 16$$ Gly, at some moment in the future, the Hubble sphere centered on G will grow to emcompass the photons sent by us. Once that happens, the photons will find themselves in a region of space that is receding from G with a velocity $$v_{rec}, and therefore these photons will begin to approach the galaxy G and will eventually reach it.

If the target galaxy is more than 16 Gly from us, the Hubble sphere centered on G will grow but not enough to emcompass the photons sent by us. In this case, these photons will forever remain beyond the Hubble sphere centered on G and therefore they will never be able to approach G.