# How has light from 13 billion years ago not passed us by?

*How is it possible that the light emitted billions of years ago is still reaching us today? How can objects be 13 billion years away by some few hundred thousand years after the big bang?*

To answer this let us construct an analogy to the universe.

To begin, consider a sheet of paper, one that we can stretch and compress (i.e. one that can grow and shrink). Let us also draw a grid on the paper (so that it is a sheet of stretchable graph paper).

Now let’s imagine this sheet represents the Universe today, and so let’s draw some galaxies on the sheet. Their coordinates can be given by their location on the grid, or we could use a ruler to specify distances to them. (The coordinates on the grid will be the so called comoving coordinates, since the grid will expand with the paper, while the ruler will give physical distances since the ruler won't expand).

Now since the universe is expanding, we should be stretching the sheet of paper as time moves forward. When we speak of an expanding universe this is exactly what we mean, the very space-time on which our universe is "drawn" is stretching, dragging each "drawing" with it. We would see that while galaxies remain at their locations on the grid, they do seem to get further apart since the grid itself is becoming larger (each cell of our stretchable graph paper would expand). This is to say comoving coordinates are fixed but physical separations change.

So lets us now instead run time backwards and compress the paper. Each cell shrinks and things appear to move closer and closer. If we do this for a while the paper shrinks to a point. Even though we know there is a grid on it that we can use to identify each point on the paper, it looks like the grid has become a point too (from our vantage point outside the paper) and each grid point is overlapping with the others. Thus, our space-time has become singular. Notice that each point on the grid has the big bang occurring there so to speak as all points are singular at this time. Note also we could have done this for an infinite sheet of paper, that is started with an infinite one with a nice well defined grid and then let it contract until all points on the grid were once again overlapping and thus we had an infinite singularity. We believe the universe to be infinite today, which means it was infinite in its formation so this infinite singularity is the picture to have in mind for the Big Bang. So now that we have the big bang setup, we can run time forward to address the question.

Let us assume we can place a clock at each point on the grid, and of course set them all going from time=0 at the instant of the big bang. Run time forward a bit so that the sheet can expand to some non-singular state, and then let’s pause expansion to look at some simple things.

Consider an ant walking on the page to be a photon (a piece of light). Suppose that we choose a drawing on the paper to represent us, and suppose then that each other drawing has an ant that starts walking towards us at time =0. Suppose further that each ant can move 10 cm per second. If the sheet doesn't expand an ant 10 cm away reaches us in 1 second, an ant 20 cm away arrives after two seconds etc. Thus, only after three seconds do we know about a galaxy 30 cm away, as the first light (the ant) from that galaxy takes that long to reach us. The ant that arrives at this time tells us about how the galaxy it started from looked when it set out (i.e. 3 seconds ago). Note that in spite of the ant coming at t=3 to tell us about the galaxy at t=0, the galaxy it started from is actually also sitting at t=3 since all the clocks are ticking. Thus, all parts of the Universe are the same age, however we see things farther out as younger since the ants are more and more delayed in reaching us to tell us about them. Thus, we can see things at various ages simply due to the fact that it takes time for the ants (light) to reach us.

But then if this is the case, and the universe is small early on, that is nearly singular, how can we see light coming from billions of light years away?

The answer to this lies in understanding the first "trick" of the universe. Very shortly after the big bang the sheet grows incredibly fast (this rapid is expansion is what we call inflation). In a matter of a fraction of a second the Universe expands to a size comparable to its size today. So from this small point like clump of paper, we get a full sheet of paper almost immediately. Now as an organism living on the paper, we would see things in a circle around us, and the size of the circle would be determined by the maximum distance from which light will have reached us, as discussed in terms of ants above. The size of this circle is c*t where c is the speed of light and t is the time since the big bang. Initially at the big bang the circle is size zero, at the time just before inflation the circle has some small but non zero size. Now because inflation expands space so rapidly, grid points that were in the circle just before inflation move outside the circle after, i.e. the circle only grows at a rate c*t, but the expansion makes the space between objects grow much faster. So some things we could see before inflation get pushed past the circle of sight.

In terms of ants, recall the ant coming from 30 cm away? Let’s now do inflation so that we expand the sheet to 10 times the size in no time at all, then the 30 cm away galaxy is now 300 cm away, so that an ant emitted just after inflation will be 30 seconds away, and so not lie in our present 30 cm (3 second) radius circle of vision. (Note in reality there are no galaxies or even stars around at the time of inflation, but let’s pretend for the sake of our ant discussion that there are. Also note that really since light is continuously emitted the light from this galaxy would keep coming for a while seeming, like it is frozen in a 3 second old state, until the after inflation light can reach us normally).

Then after inflation ends, the circle continues to grow and eventually might recapture some things (I say might because there is still a slower expansion that occurs, which, for sufficiently distant objects results in an overall rapid recession from us). Each time an object enters our circle of vision for the first time we see its earliest light, and so the object looks young, even though the object itself is in actuality the same age as our region of space.

The point of discussing inflation is mainly the fact that it sets up objects at vast distances from us, so that as our circle of vision grows we continue to add new things. Indeed, during inflation the expansion is so rapid that things would seem to move away faster than the speed of light, but this is allowed since there is no real motion of the objects through space time, i.e. our drawings don't move at all on the sheet but end up at greater and greater distances simply because we are inserting more space into the space-time between drawings. Thus, we can almost instantly set up objects billions of light years from us.

There is another point to mention here, if the Universe is finite, then ultimately at some point our circle of light will stop adding new things, i.e. once it encompasses the whole of the Universe. However, it is held that the universe is likely infinite, so that as time goes on we will continue to see farther and farther. In either case however, expansion can intervene and make it so we don’t actually see any new things, since once we see out to a certain distance there will be no light that reaches us, as it is constantly dragged back by expansion.

That is consider now a post inflation universe where ants (light) move at 10 cm per second and space expands at a rate of 1 cm per second for every 10 cm of space that exists (that is space expands 10% per second). For simplicity, let’s further suppose that in each second first light moves then we apply expansion. That is, let’s consider a galaxy 20 cm away emitting an ant. In the first second the ant travels 10 cm and so is now 10 cm away from us, that 10 cm then expands 1 cm so the ant is 11 cm away at the end of 1 second. In the next second the ant travels 10 cm again, and then the 1 cm left expands by ten percent again and so the ant is 1.1 cm away after 2 seconds. Some time in the next second then it will arrive. Thus, with expansion ants arrive later than expected as the distance they must travel also expands. Now, the statement about not seeing new things should be clear, we can imagine being 110 cm away to start. An ant from this distance will travel 10 cm in a second then be 100 cm away but that 100 cm will expand by 10 cm so that at the end of the second the ant is again 110cm away, it makes no progress towards us!

So the key ideas:

The Big Bang occurs at all points in space, so relics associated with it like the Cosmic Microwave Background are emitted from all points in space and can be seen in all directions.

The Universe is thought to be infinite, so as time goes on we can see farther and farther away, and thanks to its infinite size and inflation, there are already objects at all distances in the universe for us to see there. These objects long ago emitted light that reflects their earliest state and since they are great distances away (and the journey is lengthened by expansion), that light reaches us only today.

Things that start emitting light from too far away won’t ever be seen as expansion prevents photons from ever reaching us.

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