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The Universe expands according to the Hubble's law: velocity is proportional to distance. There must be some distance, therefore, at which the velocity reaches the speed of light. This defines the horizon. The Doppler shift is so huge that the length of waves becomes infinitely long: We can not see beyond that horizon. Is this correct?

On the other hand, when we look in the sky we look back in time. Since the Universe has finite age we can not see beyond some distance.

So these are two horizons. Are they related to each other?

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+1 good question –  Mathew Foscarini Oct 17 '12 at 22:50

3 Answers 3

It is the same horizon or at least very, very, very close. The light that is (almost) infinitely red-shifted is the light from the big bang. This is the "Cosmic Background Radiation".

Think of it this way, space expands uniformly, so there is a fixed distance away that always moves away from us at the speed of light(assuming constant accelleration for simplicity). Things beyond we will never see. This is the first of your horizons.

But when will we see the light just inside this horizon? In the beginning the light is just crawling towards us because the space is almost moving away at the same speed, but eventually it starts moving faster towards us. The light will stay longer and longer near the horizon the closer it is to it, because the net initial speed towards us drops lower and lower. Take a point a little closer to the horizon and the light might hang out at the horizon for another billion years.

So my answer is that the second horizon is moving asymptotically towards the first one, never quite reaching it, but already very, very close.

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The CMB was emitted some 300,000 years after the Big Bang, and its redshift has the perfectly finite value of 1100. The only reason we can't see any earlier is because the universe was opaque. –  Chris White Oct 31 '12 at 23:14

Because the universe is approaching a DeSitter-like expansion, eventually light emitted more than about 10 billion light years from us, will never reach us.

The distance of light we are currently receiving is no further than the surface of the last scattering of the cosmic microwave background, and is about 30 billion light years.

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Short answer: They are different.

Long answer: There are a number of subtleties involving horizons in cosmology. I refer anyone who is interested in the details to a paper by Davis and Lineweaver, which I have referenced many times on this site. Here I'll only refer to Figure 1 of that paper, shown below. All three frames are the same, with the lower two being nonlinear but convenient rescalings of the "normal" one on top. (If you are familiar with spacetime diagrams, the bottom frame is conformally related to the top, so light travels on straight lines, which are at $45^\circ$ in the bottom pane. In any case, the rescalings never alter the structure of which regions encompass/border which others.)

horizon diagram

(Diagrams correspond to concordance cosmology of $\Omega_\mathrm{m} = 0.3$, $\Omega_\Lambda = 0.7$, and $H_0 = 70\ (\mathrm{km}/\mathrm{s})/\mathrm{Mpc})$.)

There are four different surfaces of interest. The simplest one to define is the Hubble sphere, which is where the recession velocity given by Hubble's Law is equal to the speed of light. However, the Hubble sphere doesn't mean much at all, despite its enigmatic definition. In general relativity, there's nothing too unusual about proper distances growing faster than $c$.

Our past light cone is also marked. This demarcates the region of the universe capable of influencing us today. That is, events within and on the light cone, and only those events, can originate a photon that we receive right now. Note that you can see beyond the Hubble sphere - there are events outside the Hubble sphere still within our past light cone. At the same time, there are things within the Hubble sphere that have not yet come into view. Thus neither the light cone nor the Hubble sphere is fully contained within the other.

You might also ask what locations in spacetime will we be able to receive information from, given that we wait long enough. This limit of the past light cone as we move its apex toward future infinity is the event horizon. This includes all of the light cone and its interior, since anything seen today can certainly be seen some time between today and future infinity. It also includes all of the Hubble sphere and its interior, since if something is not receding faster than light right now, photons it emits will eventually get here.

Finally, there's the particle horizon. Be careful, as different authors use slightly different definitions. Here, think of it as a function of time. At a particular time, the particle horizon is the distance - at that time - to the furthest object that can be seen. Note that in our universe, with its accelerated expansion, an object can right now be outside our event horizon - thus not capable of ever influencing us any more - while still inside our particle horizon, since we can see an older version of it with light that was emitted back when it was still inside our event horizon.

In summary, "what we can see" and "what is presently receding faster than light" are two different sets of things. Also, I would advise against trying to apply special relativistic Doppler effects to GR. The main purpose of the paper is in displaying all the ways this goes wrong. Have a look at Appendix B to see a list of examples were scientists got confused on this point.

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