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# A Step-by-Step Guide to Cosmology’s Best-Kept Secret

The views expressed are those of the author and are not necessarily those of Scientific American.

How far is each of these galaxies? Appearances may deceive.

In my previous post, I described the little-known and somewhat counterintuitive idea that objects in the distant universe appear larger and larger the farther they are, in a reversal of the usual rules of perspective. I called it the cosmic magnifying lens. As promised, I will now explain the physics behind it.

One possible way to discuss the reversal of perspective could be to show how it is just one among several optical effects of the curvature of spacetime. In fact, when in the late 1990s two separate teams of astronomers discovered that the universe’s expansion is now accelerating—the “dark energy” business that earned three astronomers this year’s Nobel Prize for Physics—they did so while studying anomalies in the optical effects of curvature.

But for this post, I want to avoid talking about curvature and instead I want to keep things as basic as possible. As you’ll see, the cosmic lensing follows from a few simple facts. The first one of those facts reminds us that all images we see in the sky are time-delayed.

Simple fact #1: Light propagates at finite speed.

We all know that light goes at finite speed, but it’s worth recalling what that means for astronomical observations: if we see a hypothetical galaxy that’s one million light-years away, we are detecting light that was emitted 1 million years ago. If we picture the path the light has covered during that time, and if we represent the arrow of time as pointing upwards, it will look something like this:

The diagonal line is the spacetime history of the light, and the two vertical lines represent the unchanging positions of the Milky Way and of the galaxy we are looking at; I have assumed that the galaxy did not move at all during that time, and in particular, that its distance from us has stayed constant. Of course, for most galaxies that is not true. Because the universe has been relentlessly expanding since the beginning of time, other galaxies have been moving away from us (with few exceptions in the Milky Way’s Local Group).

But while for galaxies that are relatively close to us (on a cosmic scale, 1 million light-years is nothing) you can safely neglect the expansion of the universe, for much more distant galaxies you must take expansion into account. One of the most important facts about cosmic expansion is the following, which is almost synonymous with saying that the universe started with a big bang.

Simple fact #2: The universe used to expand at a much faster pace than it does now.

Although it is true that the universe’s rate of expansion has picked up a bit in recent eons–that was the Nobel-worthy dark energy discovery–it’s still nowhere as fast as it used to be. The big bang theory (the actual scientific theory, not the equally awesome sit-com) says that the universe started out expanding at breakneck speed and spent most of its history slowing down, under the mutual gravitational pull of everything that it contains.

What does a slowing expansion look like? Imagine a galaxy—call it galaxy A—that is now 3 billion light-years from us, and picture its trajectory during the last 10 billion years. (That would be a sizable chunk of the universe’s history, which overall spans about 13.7 billion years.) At the beginning of that period, the galaxy would have been much closer, perhaps 1 billion light-years away (physically realistic numbers may be slightly different). If again we represent the arrow of time as pointing upwards, the galaxy’s path could look something like this:

(Again, the present is at the top and the past is at the bottom.) The line that represents the changing position of the galaxy, also known as its world line, leans more to the right at the beginning, and becomes closer to vertical as we approach the present, reflecting the fact that the speed at which the galaxy recedes has become much smaller.

(Once again: the numbers here are meant to give an idea rather than to be exctly accurate, and I am intentionally ignoring the fact that in recent times the rate of expansion has switched from slowing to accelerating.)

So far I have talked about a single galaxy, but what happens if we compare different galaxies that are at different distances?

Simple fact #3: Hubble’s Law.

The famous law discovered by Edwin Hubble says not only that distant galaxies recede from us, but also that the farther away they are, the faster they recede. (Their velocity is directly proportional to their distance, which is is consistent with a universe that expands at the same rate everywhere.)

Thus, we can imagine playing the same game not just with galaxy A but with a few equally-spaced galaxies, A, B, C, and D, currently at distances of tree, six, nine and twelve billion light-years, respectively. Ten billion years ago, those galaxies might have been one-fourth as far as they are now—namely, at distances of one, two, three and four billion light-years. If we pictured the galaxies’ trajectories in a decelerating universe, they would look something like this:

The beginning of the time period pictured here, 10 billion years ago, was a special time for galaxy D. It was the time when D emitted the light that we are now receiving from it. Now, for a galaxy so far away that its light took 10 billion years to reach us, Hubble’s law has a remarkable consequence.

Simple fact #4: Distant galaxies move faster than light.

Einstein’s special theory of relativity says that nothing can go faster than light, but that rule only applies to the velocities of objects that essentially pass by each other. It does not apply to objects that are far from each other: for those, space itself can expand between them and make them move apart faster than the speed of light.

In fact, faster-than-light, or superluminal, velocities are a necessary consequence of Hubble’s law: because galaxies recede at a speed that’s proportional to their distance, you can find galaxies that recede at arbitrarily high speeds: just look far enough.

Ten billion years ago, galaxy as far as galaxy D would have been moving away faster than the speed of light. That’s so fast that the light it emitted in our direction—which of course emanated from it at the speed of light—still wasn’t fast enough to make any progress getting closer to us. That light was dashing toward us as fast as the laws of physics allowed, and yet it was still moving away from us.

Of course, we are seeing that light now, which means that at some point in time between then and now it stopped receding and began to get closer instead. Correspondingly, its trajectory in spacetime would look something like this:

The light’s trajectory in spacetime first heads to the right, then reaches a maximum distance, turns around and starts coming toward us. (When it approaches our current location, the trajectory’s angle becomes 45 degrees, which is what we should expect: light covers one light-year in one year.)

Instad of following the path of light in spacetime from galaxy D to us, we could have just as easily taken the light that comes to us now and traced its path backwards. In fact, we could trace back the light we receive now from all four of the galaxies, not just from D. When the spacetime history of light crosses the history of a galaxy, that’s the time when the galaxy emitted its light.

Once we find those intersection points we can deduce, by counting vertical squares, how along ago the light was emitted for each of the galaxies. Thus (pretending that my hand-drawn sketch had accurate data) we could deduce that are seeing light from A that is a little less than three billion years old; about five billion years old in the case of B; and seven billion years old for C. For the case of D, we had already establish that its light is 10 billion years old.

Because A is the closest of the four galaxies, D the farthest, and B and C are in between, based on the ordinary rules of perspective you would expect their apparent sizes in the sky to be largest for A and smallest for D. But here’s where we get to our key point.

Simple fact #5: We don’t see objects where they are now.

It is clear that we see galaxies not as they are in the present but as they were at a different time in the past, depending how far they are: that is just a consequence of Simple fact #1.

But not only are we seeing those galaxies as they looked at different times; we are also seeing them where they were at those times. The galaxies appear just as large in the sky as if they had not moved since then. If you think about it, this makes sense: once the light has left a galaxy and gotten on its way, there is no reason why that light–or the images it will form in our eyes and in our telescopes–should be affected by the galaxy’s subsequent motion.

In the following figure, both dimensions represent space. The angle subtended by a galaxy in the observer’s visual field–which is the same as saying the galaxy’s apparent size–is what you would see at the time the light left the galaxy on its way to the observer. If the observer could somehow “see” the galaxy instantaneously where it is in the present, the galaxy would subtend a smaller angle.

Just as we estimated how long ago each galaxy emitted its light by counting vertical squares in the spacetime diagram, we can also estimate how far each galaxy was at that time (and thus how large it appears now in the sky) by counting horizontal squares. Here is the same spacetime illustration as before, but with distances highlighted for clarity.

We see that galaxy A was, at the time it emitted its light, only slightly closer than its current distance of 3 billion light-years. Galaxy B however was more than one billion light-years closer. And in case of galaxy D, the difference is most remarkable: its current distance of 12 billion light-years is fully three times farther than it was 10 billion years ago, the time when we see it. The discrepancy between current distance and apparent size is very small for galaxies that are not too far away such as A, but very large for the most remote galaxies.

But the figure demonstrates something even stranger when instead of comparing each galaxy’s actual size with its apparent size, we compare the apparent sizes of the various galaxies with one another. Assume that all of our four galaxies are identical. In the case of galaxy C and galaxy D, we are comparing an image of galaxy D from ten billion years ago with an image of galaxy C from about eight billion years ago. But even though D is and always has been farther away than C, D will look larger. That’s because at the respective times of emission, galaxy D was closer than galaxy C.

So that’s where reverse perspective comes from. The foreground object looks smaller than a background object of the same size. The following video (by London-based video artist Jeremy Mooney-Somers) portrays reverse perspective quite spectacularly. At about 00:22, you’ll see a squadron of teapots rotating in space; the teapots are all the same size, but the ones farther away look larger than those closer to the observer.

In the universe, reverse perspective applies only to very distant objects; ordinary perspective is valid not only at the scales of our experience but also for billions of light years around us. The point where the observable universe transitions from direct perspective to reverse is precisely where the trajectory of light bends backwards. In my approximate diagrams, it is at about 8 billion years; the actual value is more like 10 billion years in the past. Although that’s a long time ago, the universe is 13.7 billion years old, which means that reverse perspective applies to a full 3.7 billion years of its history.

(Side note for astronomy nerds: in matter-dominated models the transition takes place at a redshift of exactly 1.25, while in our dark-energy-dominated universe it is at a redshift of about 1.65, according to Roger Blandford, director of the Kavli Institute for Particle Astrophysics and Cosmology at Stanford.)

What if we were to look at the most distant past? The big bang theory predicts that, as you play the movie backwards, you’ll see all of the content of the universe converge toward a single point. That includes the trajectory of light that we traced backward from us. It also includes the world lines of all galaxies, no matter how far they are now.

Of course, if you rewind the movie of the history of a galaxy, you will see the stages of its formation in reverse order. Because galaxies and the stars they are made of coalesced from a cloud of hydrogen and helium gas, going backwards you will see the galaxy evaporate into a cloud. Going even farther, you will see the clouds merging, and the whole universe fill uniformly with gas.

Eventually, when you get to about 13.7 billion years ago, and more precisely to the epoch just 400,000 years or so after the big bang, the trajectory of light will come to a stop. In even earlier epochs, the content of the universe was too hot to be a gas, and instead it was a plasma, which is opaque to light. Thus, that is as far as we can see. We have reached the edge of the observable universe.

What does reverse perspective look like at that point? The world line of light cannot quite go all the way back to the big bang, but the plasma we can now see was, back then, on the order of 10 million light-years away. (Away from what? From the plasma that later evolved into the Milky Way, and into us.) Thus, the farthest observable object in the universe was, at the time when we see it, at a distance so small that in the current universe it would put it in the Local Group. And the cosmic magnifying lens has now reached its limit. Its magnification factor is now more than 1,000X.

If a galaxy had existed back then at the edge of the observable universe, that galaxy by now would have receded so much that it would be 40 billion light-years away. The light we receive from such a galaxy would be so redshifted (that’s the stretching of light waves due to the expansion of the space in which the light waves travel) to be far outside of the visible spectrum. Moreover, its image would be exceedingly faint.

Still, it is fun to imagine what this impossible galaxy might look like in the night sky if we could see it with the naked eye. It could appear almost as large as the moon. Hundreds of other, seemingly smaller galaxies, looking like tiny little dots, would show up in front of it, looking hundreds of times smaller than the big disk behind them.

I end this post by pointing out a riddle that was implicit in my diagrams and that has surely occurred to the most attentive readers: how far would you say is galaxy D?

Read the prequel to this post: The Cosmic Magnifying Lens

• Poetry of the Universe, by Robert Osserman (RIP 1926-2011). Anchor Books, 1995.
• An Introduction to the Science of Cosmology, by D. J. Raine and E. G. Thomas. Institute of Physics, 2001

Hubble deep field image courtesy of NASA.

About the Author: Davide Castelvecchi is a freelance science writer based in Rome and a contributing editor for Scientific American magazine. Follow on Twitter @dcastelvecchi.

The views expressed are those of the author and are not necessarily those of Scientific American.

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1. 1. Torbjörn Larsson, OM 2:46 pm 11/26/2011

The observed distance to D would be measured by the redshift of the emitted light, I take it. Seems it experience traveling through spacetime that stretches a factor ~ 2 meanwhile (eyeballing the intersections of galaxy pathways. If redshift is defined as z in relative wavelength change = 1+z, it would be z ~ 1.

[Hmm. Is that significant? z ~ 1-2 is where the parametrized curves of Kellerman minimizes: "The data suggested that the angular size bottomed out at a redshift of between 1 and 2".]

Ned Wright’s redshift calculator gives (with space flat) a comoving radial distance of ~ 11 Gly to where D is now and an age at redshift of ~ 6 Gyr; not too shoddy. The light travel time would then be ~ 8 Gyr.

So D looks to be 8 Gly away, when using redshift for estimate?

2. 2. Hephaestus 8:54 pm 11/26/2011

This is a very timely posting, as I have spent the last week wondering if it would be possible to detect blurring of very distant galaxies because of super-luminal expansion.

It appears intuitive that we would see light from such a galaxy in a continuum of locations until expansion slows below light speed. In your diagram showing the path of light from “D Then” to “Milky Way Now”, I would expect that we would see light from D at all positions along its location curve until D had progressed far enough that its light curve on your graph would never have a negative slope.

I recognize that “intuitive” and “cosmology” don’t belong in the same sentence, but I wonder if you could explain where I have gone off the rails on this.

3. 3. @dcastelvecchi 6:19 pm 11/27/2011

Torbjorn: I think your calculation is correct but it doesn’t match my data because my data was not physically realistic
In my diagrams, every vertical square is one billion years, and the travel time of light from D is 10 squares, so 10 billion light years.
But mine was actually a trick question to make people ponder a fact that I find very interesting: the distances conventionally quoted by astronomers, which are based on light travel times, are just that — conventions. They do not match either the comoving distance of the object at the present time nor its comoving distance at the time of emission!

4. 4. @dcastelvecchi 6:28 pm 11/27/2011

Hephaestus: In spacetime, objects are represented as a continuum of events, but that’s not how a physical observer sees them. In this respect, spacetime diagrams are a bit misleading in that they give you a god’s eyeview of the history of the universe that is not realistic for any actual observer. To properly answer your question I’d need to add a diagram or two, which I can’t do here in the comments. But you gave me a good idea for a follow-up post!

5. 5. CherryBombSim 11:38 pm 11/27/2011

My compliments on a very lucid explanation, with excellent diagrams. I learned something today.

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