# What's the furthest distance that something could travel and eventually come back to Earth?

Imagine u shoot a photon into the sky at a mirror far away in space, and you want the photon to bounce off the mirror and eventually come back to you. Considering the cosmological constant, what's the furthest that mirror could be (at the moment you shot the photon) and still have the photon get to it, bounce back and eventually return?

I know the mirror would start moving away, and the photon would have to catch up with it. I think it would have to be around 15 BLY away by the time the photon reached it (I'm not sure), but I don't know how far it would have to be at the start to be 15 BLY away by then.

## 1 Answer

There's an easy way to work out the distance to the mirror at the present time, if it moves with the Hubble flow: imagine that instead of being emitted by us, the light is emitted by the mirror image of the Earth and simply passes through the mirror to us. The farthest comoving distance that the mirror Earth could be is the same as the farthest present-day distance from which light emitted in the present day can ever reach us; assuming ΛCDM is correct, that's around 16.5 billion light years. The comoving distance to the mirror (and the metric distance to it in the present day) is exactly half that.

Working out the metric distance to the mirror at the time of reflection is trickier. If (nonstandard terminology warning) $$D(a)$$ is the comoving distance light can travel from scale factor $$a$$ to $$∞$$, then the scale factor at which the light hits the mirror is $$D^{-1}\left(\frac12 D(1)\right)$$. If all energy was dark energy and the expansion was exactly exponential then the solution would be $$2$$ exactly. Using central parameters from here I get $$a\approx 2.09$$, so the mirror distance is a bit higher, around 17 billion light years.

For what it's worth there's an exact expression for this function in a flat perfect dust+dark energy universe:

$$D(a) = \frac{c}{a \sqrt{Ω_Λ} H_0} \; {}_2F_1\left(\frac13,\frac12;\frac43;-\frac{Ω_m}{a^3Ω_Λ}\right)$$

where $${}_2F_1$$ is the hypergeometric function.

• thank you so much! i was basically wondering about this cuz i wanted a hard limit on how much of the universe we could ever keep gravitationally bound to us (like by going out and gathering matter, maybe using stellar engines or starlifting or something), and ur way of thinking about it like just one long trip that passes thru the mirror is genius! (if obvious in hindsight lol) 🙂 – Aleon Ethulyn Oct 27 '20 at 1:53