# What is the theoretical limit for farthest we can see back in time and distance?

13.2 billion years ago the universe was rather small, having started only half a billion years ago. Today, with the help of Hubble Space Telescope, we are able to capture the light of galaxies emitted at that time.

The point at which Earth exists now must have been quite close to those galaxies back then. If so, why is it that it is only now, after 13.2 billion years later the light from those galaxies has reached us? Or in other words, are we sure that the light we are seeing from those galaxies indeed travelled for 13.2 billion years?

It looks like as if there was a race between our point running away from those galaxies (with the expansion of universe and space) and the light that was emitted at that time. And only now that light has reached and overtaken us. But if that is so, then wouldn't it put a limit on the oldest light we can see, no matter how powerful the telescope is (even it is more powerful than the James Webb Space Telescope)? This should be expected, because at the time just after the Big Bang, the light emitted by all objects must have already overtaken all other objects, including the location of earth. Therefore we will never see the light that old (close to the time of Big Bang) no matter how powerful the telescope. If this is so, what is the theoretical limit we can see far back in the past?

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Hi Ankur, and welcome to Physics Stack Exchange! Your title seems to be a bit different from what you're actually asking; could you look over your question and see if you could edit the title to more closely correspond to what you really want to ask? –  David Z Jun 8 '13 at 8:12
→ Ankur: may I suggest this title: How may we receive now images from 13 billion light years away within the Big Bang model? –  daniel Azuelos Jun 8 '13 at 13:08
It seems you are asking for the particle horizon (en.wikipedia.org/wiki/Observable_universe#Particle_horizon)? –  Johannes Jun 8 '13 at 14:51
@Johannes I think that is indeed the appropriate answer . It might also be worth noting (in contrast) the surface of last scattering. –  zhermes Jun 8 '13 at 18:34
@DavidZaslavsky I have changed the title. –  Ankur Jun 9 '13 at 19:35

It looks like as if there was a race between our point running away from those galaxies (with the expansion of universe and space) and the light that was emitted at that time. And only now that light has reached and overtaken us.

That's correct. A photon from a distant source has to overcome the expansion of the universe in order to reach us. I'll illustrate it with an example. The graph below shows the path of a photon in an expanding universe (based on the Standard Model of cosmology and the latest data).

The horizontal axis shows the distance to us, and the vertical axis is the cosmic time. Over time, our galaxy moves on the black vertical line, and we're currently located at the black dot: the current age of the universe is 13.8 billion years.

Suppose that we're now observing photons from a distant galaxy. The redshift of those photons allow us to calculate when they were emitted, what the distance of the galaxy was at that time, and what the present-day distance of the galaxy is. In the graph, the galaxy emitted the light when the universe was 2.5 billion years old. The galaxy was located at the purple dot, while our own galaxy was at the white dot, and the distance between both was 5.52 billion lightyears (white line).

Now, if the universe wouldn't be expanding, then the light would've only needed 5.52 billion years to reach us (moving on the dashed orange line). However, the universe does expand, and as a result the light followed the thick orange line, taking 11.3 billion years to reach us. So because of the expansion of the universe, the light needed about twice as much time to reach us. During that time, the expansion caused the source itself to recede from us, following the dotted purple line, and its current distance to us has increased to 19.89 billion lightyears (cyan line).

There's one more interesting point: the source galaxy is receding from us faster then the speed of light (yes, that's allowed in General Relativity). Because of that, the distance between us and the photons was initially increasing (the expansion was 'winning' the race). But gradually the photons moved through regions that were receding from us slower: the dark green line represents the so-called Hubble distance: the region of space that is receding from us at the speed of light. So when the photons crossed that line, their distance to us began to decrease. All photons that we observe today have been travelling on this teardrop-shaped curve, which is called our past light cone.

But if that is so, then wouldn't it put a limit on the oldest light we can see, no matter how powerful the telescope is?

Yes. The maximum distance of regions of space that we can observe is called the cosmic particle horizon, and is shown in the graph as the thick blue line. You can think of it as the path of a photon sent out from our location at $t=0$. If we zoom out the graph, it looks like this:

The current distance to the particle horizon is 46.2 billion lightyears, and everything inside it is called the observable universe. We cannot see anything beyond it.

This should be expected, because at the time just after the Big Bang, the light emitted by all objects must have already overtaken all other objects, including the location of earth.

No, in fact the expansion rate of the universe was very high in the beginning, so the photons from distant regions couldn't reach us. The expansion then slowed down, until it began to accelerate again when dark energy started to dominate (when the universe was around 7.7 billion years old).

If this is so, what is the theoretical limit we can see far back in the past?

In theory, all the way to the very early universe, when all particles were created (which, according to leading theories, was at the end of the inflation era). However, the early universe was so dense that it was opaque, so we cannot see photons from the first ~380,000 years (although in principle we could detect neutrinos from that era). When the universe was about 380,000 years old, the density was low enough for atoms to form, and photons could move freely. Those photons are the oldest light that we can see, forming the Cosmic Microwave Background.

For a more detailed and technical explanation, see this post.

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+1 Neat explanation –  Prathyush Jun 10 '13 at 6:34
What software(s) did you use to draw these graphs? Really appreciate the time it must have taken you to do this... 10x as long as putting the information as text only? But it makes it so much easier for duffers like me to understand it :) –  Eugene Seidel Jun 11 '13 at 7:20
@EugeneSeidel Thanks for the compliment! The graphs are made with matplotlib; I wrote a Python programme to do all the cosmological calculations. It takes a while to get it right, but I like the result :-) –  Pulsar Jun 11 '13 at 13:48

The light from formerly nearby galaxies needed this long time to get here because the Universe – and the distance in between the source galaxy and ours – was expanding as the light was travelling. So when the light got to the middle point, for example, the distance between both galaxies was already just a bit smaller than 1/2 of 13.7 billion light years.

Light from places that are even further than the observable Universe couldn't have gotten here at all because its attempt to overcome the expansion of the huge distance between this beyond-the-horizon galaxy and ours is as hopeless as attempts to surpass the speed of light.

If you actually want to compute how much time it takes for light to travel over a distance in an expanding Universe, you need the FRW geometry - metric tensor of the form $$ds^2 = -c^2 dt^2 + a(t)^2 (dX^2+dY^2+dZ^2)$$ I assume that the spatial sections at a given time are flat, in agreement with observations. The function $a(t)$ is an increasing function of time that quantifies how a unit distance at one time grows at another time – the overall scaling of $a(t)$ is irrelevant because it may be absorbed to the normalization of the coordinates $X,Y,Z$, too.

So the physical, proper distances at a given time $t$ are $a(t)\cdot \Delta Z$ rather than $\Delta Z$ itself. But $t$ measures directly the time from the Big Bang, according to a galaxy at rest.

To compute how far the light can get, it's useful to use another time coordinate $\tau$ so that $$ds^2 = (-c^2 d\tau^2+dX^2+dY^2+dZ^2) A(\tau)^2$$ We must have $A(\tau)=A(\tau(t))=a(t)$ to have the right coefficients in front of $dX^2+dY^2+dZ^2$. And $c\cdot d\tau \cdot A(\tau(t)) = c\cdot dt$ to match the time-related term which means $dt / d\tau = A(\tau(t))$ which allows you to integrate it and find the reparametrization from $t$ to $\tau$.

I don't want to be explicit about the form of these functions – they're a bit complicated as the expansion of the Universe has had different power-law stages etc. – but the point is that the light moves along trajectories with $c\cdot d\tau =\sqrt{dX^2+dY^2+dZ^2}$, i.e. along nice 45-degree curves in the $\tau,X,Y,Z$ coordinates if I set $c=1$ now.

The reason why light needed such a long time is that it actually needed a short time in the $\tau$ coordinate – just like you expect, the Universe was small, the distances were rather short half a billion years after the Big Bang, and $c\cdot \Delta \tau$ is directly equal to this short distance. But the problem is that $\Delta \tau$ isn't the actual proper time. The actual proper time $\Delta t$ is the integral $\int c\cdot A(\tau)\cdot d\tau$ and the factor $A(\tau)$ will be getting larger and larger as the Universe expands and indeed, it will be 13.2 billion years despite the proximity of the initial galaxies.

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are you saying that the rate of change of rate of expansion was non-uniform? I am sorry, I am not a technical guy and so missed most of your message. –  Ankur Jun 9 '13 at 20:01
I can't seem to wrap my head around this issue. With bigger and more powerful telescopes, we can see farther in the past. We can already see events and objects that took place 13.2 billion years ago. In future when we have James Webb Space Telescope, which is a much more powerful telescope than Hubble, how much farther we expect to see? If had a really huge telescope (even bigger than JWST, say 100,000 km mirror) would be able to see the birth of universe? Would we be able to see the entire universe of that time, including the point where we stand now? Would we see it in every direction? –  Ankur Jun 9 '13 at 20:15
Dear Ankur, in the telescopes, we never see "the point where we stand now". In each direction, the telescope sees objects that were (and probably still are) separated from us in the same direction. In the same direction, we may see objects that are arbitrarily far up to some maximum, and the further these observed objects are, the further in the past we see them. The maximally distant objects we may see with any telescope these days are those objects which we observe as they looked 13.7 billion years ago, right after the Big Bang. –  Luboš Motl Jun 10 '13 at 7:05
These most distant ones among the visible objects define the boundary of the "observable Universe" and the observable Universe is a ball surround the Solar System whose current radius happens to be 46 billion light years. This is greater than 13.7 billion because the distances that light traveled at various intervals of those previous 13.7 billion years were later expanding, and if one integrates (essentially sums) the distances from all intervals, he gets 46 and not 13.7 billion light years. But whatever telescope you use, you can't see behind the boundary of this observable universe yet. –  Luboš Motl Jun 10 '13 at 7:07
Thanks for taking the trouble to answer my question in detail. While I understand that we cannot see objects farther than 46 billion ly, I am wondering about what is the oldest object we can see. Hubble, with some 25 years old tech can show us objects and events that happened 13.2 billion years ago. So what we are seeing in the Hubble deep space images of galaxies as what was happening there just 600 million years after the Big Bang. There is every reason to believe that JWST will help us look at still older galaxies -- may be as they were just 50 million years ago (my guess). –  Ankur Jun 10 '13 at 8:34
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