If faster than light travel is impossible, how is it that light emitted from matter so close together in the time soon after the Big Bang is only now just reaching us? I would assume that there would be a "limit" to how far back we can see, but exactly how long after the Big Bang are we able to observe? I'm sure there's an easy explanation, but it has been bugging me for a while... I am aware of how this works, but I am curious as to how long after the Big Bang are we able to see in images such as the Hubble Deep Field? If light travels at a constant speed no matter the conditions the observer is experiencing, then why would there be a longer stretch of time for light to travel through as the universe expanded? Please let me know if I should elaborate more on my question because I do feel like my writing is a little bit hard to understand...
Early in the universe the expansion rate was much greater than it is today, which is a way of saying that spacetime was strongly curved. You really need general relativity to properly work out what happens, but a good way to think about it is that by the time a light ray gets from A to where B was, the expansion of the universe has carried B even further away from A. It is true that nothing can move faster than light locally, but there is no such restriction on the expansion of space itself. Every observer still sees light rays moving past them at the speed of light, but distant points can be carried apart so fast no message can pass between them. Or a message sent early on just reaches us today.
The surface beyond which no information can reach us is called the cosmological horizon. In practice we only "see" back to the surface of last scattering (the place the cosmic microwave background comes from). This is about 300,000 yr after the big bang. Before that the universe is filled with a plasma that is optically opaque. But we can infer what is going on earlier through a number of other lines of evidence (such as big bang nucleosynthesis for example). The Hubble ultra deep field comes from a later time when stars exist and galaxies are forming. Have a look here for a timeline of the universe. That page says the Hubble ultra deep field goes back 13 billion years.
Does the expansion of the universe soon after the Big Bang affect the amount of time that light takes to reach us?
The time light has had to travel is simply by the age of the universe (or slightly less, because the very early universe was opaque). The age of the universe is 14 billion years, so that's how long the most ancient light has had to travel.
What cosmological expansion has affected is the distance between us and the point of emission of that light. You would think that a point from which that light was emitted would be 14 billion light years away from us, but actually it's about 46 billion light years away, because the universe has been expanding while the light was on its way.
I'll expand a bit (no pun intended) on the answers given by Michael Brown and Ben Crowell. The expansion rate of the early universe was indeed much greater than it is today. The expansion decelerated first, and only started to accelerate again when the universe was 7.7 billion years old, due to the effect of dark energy. Furthermore, distant regions of space are receding from us faster than the speed of light: the Hubble law $$ v = HD $$ simply states that the recession velocity of a distant galaxy is proportional to its proper distance $D$. If $D$ is large enough, then $v$ exceeds the speed of light. The distance $D_H$ for which $v=c$ is called the Hubble distance, and is currently about 14.5 billion lightyears. However, this distance is much smaller than the radius of the observable universe, which is about 46.2 billion lightyears. Have a look at this graph, which displays the expansion of the universe:
The horizontal axis shows the distance to us (in Gigalightyears), and the vertical axis is the cosmic time (left), and the corresponding scale factor (right). We are located at the cross-section of the thick black lines; our universe is currently 13.8 billion years old.
The blue curve is the size of the observable universe, and the red curve is our event horizon. The yellow lines are the paths of photons; in particular, the orange lines are the paths of the photons that we observe today: our current past light cone. The black dotted lines are regions of space (clusters of galaxies) that are moving away from as the universe expands. Finally, the green curves are lines of constant recession velocity: from dark-green to light-green we have $v=c$ (the Hubble distance), $v=2c$, $v=3c$ and $v=4c$.
As you can see, the edge of the observable is currently receding from us at more than 3 times the speed of light.
When a photon is emitted in our direction from a region of space, it has to overcome the expansion of space in able to reach us: while it is travelling towards us, the amount of space through which it has to travel increases. It's like swimming upstream or running on a treadmill, and the result is that it takes photons much longer to reach us (and if they're too far away, they can't reach us at all).
Let's zoom in on the graph:
As I mentioned before, the orange lines are the paths of the photons that we observe today. Notice the characteristic teardrop shape: photons emitted in the early universe were emitted from regions that were receding from us faster than the speed of light. As a result, those photons were at first also moving away from us ($v-c>0)$. However, those photons gradually moved through regions that were receding at lower velocities, until they eventually crossed the Hubble distance (the dark-green curve), the region that recedes at $v=c$. This happened when the universe was about 4.1 billion years old. After that, the photons were fast enough to overtake the expansion of space, and their distance to us decreased, until they finally reach us today.
See also this post of mine, for a more detailed and technical answer.
The universe expands at a constant rate of 74 km per/MPC (roughly 3 million years) The rate of expansion today is much less than it was during the early years near the start of the big bang. Since faster than light travel is not possible; "expansion" of the early universe is simply caused by early inflation.
If a star Is 13.82 billion light years away It takes 13.82 billion years for us to see the Image so 13.82 billion + 13.82 billion = 27.64 billion years old because while that Image was flying through space, time was still moving forward also If I set up a camcorder telescope 1 light year away and look at earth then I can see the past on earth the closest thing to time travelling.