If we send a photon from Earth at this moment, what is the most distant object that it could reach?

This question is partly inspired by this video which states we cannot ever travel beyond our local group.

I'm trying to understand why that would be. The recession velocities listed in this Wikipedia article are all in the order of hundreds of km/s, and light travels a thousand times faster. So that would seem reachable, for a photon at least.


The most distant object that light we emit today can reach in the distant future is at the event horizon

$$eH(t) = a(t)\cdot \int_{t}^{t_{max}} \frac{c\cdot \text{d}t'}{a(t')}$$

which is now approximately 17 billion lightyears away, see the future light cone in comoving coordinates which converges to this distance:

comoving grid

If the light was emitted at the big bang you can use the particle horizon

$$pH(t) = \int_{0}^{t} \frac{\text{d}t'}{a(t')}$$

as the future light cone of t=0, this light has travelled some 46 billion lightyears up to now and will in the distant future converge to a comoving distance of 63 billion lightyears where the farthest galaxies it will ever reach are located today.

The detailed calculation with the cosmological parameters from Planck 2013 can be found here from In[24] to Out[26]. For a detailed explanation of the spacetime diagram see here at page 3.


The metric for the de Sitter spacetime, which approximates the observable universe in stationary coordinates is $$ ds^2~=~-\left(1~-~\frac{r^2\Lambda}{3}\right)dt^2~+~\left(1~-~\frac{r^2\Lambda}{3}\right)^{-1}dr^2~+~r^2d\Omega^2 $$ The important term is $$ \left(1~-~\frac{r^2\Lambda}{3}\right), $$ that looks a bit like the Schwarzschild factor. This cosmological constant is related to the Hubble factor H by $$ H^2~=~\frac{\Lambda}{3c^2}, $$ where galaxies are moving outwards at $v = Hd$, for $d = r$ the distance of the galaxy. In observing outwards the galaxies with red shift $z < 1$ occur for this factor positive. For $z = 1$ the galaxies are moving outwards at $c = Hd$, which happens at the cosmological horizon distance $d~\simeq~1.3\times 10^{10}$ light years, and for $z > 1$ we have galaxies moving away faster than light. These galaxies are not moving by standard special relativity type of velocities, but because space itself if expanding and frame dragging them outwards relative to our rest frame.

A galaxy that is further out than $r = \sqrt{3/\Lambda}$ on the Hubble frame we can never send a signal to, nor can a signal from that galaxy reach us. The Hubble frame is a spatial surface where local galaxies are approximately at rest with respect to each other, and more distant galaxies are receeding away in an isotropic distribution of velocities that depend on $v = Hd$. This is even for galaxies we currently see with $z > 1$. The reason we can see them now is the photons left the galaxy in the past when it was much closer and within this cosmological horizon distance of $r = \sqrt{3/\Lambda}$.

This is one reason the CMB is quoted to be $45$ billion light years away, but represents the universe at $3.8\times 10^5$ years after the initial event of the big bang. This frame dragging has stretched space considerably so the CMB with $z~\simeq~1100$ is much further away. If we could do neutrino astronomy and detect gravitons from the earliest moments we could push this much further.


Well first of all,

Considering dark energy and other factors, what is the most distant object light could reach?

If you are talking about an OBJECT like stars, galaxies etc... the farthest object we can "see" is located 13.39 bilions light-years (Galaxy GNz-11, you can search that) The 11 on the name indicates its redshift z=11.


Now, if you're saying:

If we send a photon from Earth at this moment, what is the most distant object that it could reach?

Let me tell you, the photon will reach Alpha-Centauri in 4 years, in 26,000 years it would reach the center of the Milky Way.

And of course, if you give it time it will get away from our local group indeed... it even might reach at some point GNz11 (or some other galaxies biolions of light-years from here.)

The distance light can reach depends on the grouwth of the ratio of the observable universe. So if you send a signal right now, the distance it can travel will be biigger as time passes. If you could stop the universe expansion that signal might travel around 14 bilion light years.

  • $\begingroup$ No worries, i take mine back too, and we can move on! $\endgroup$ – IamZack May 20 '16 at 4:06

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