Why “distance” is not what it seems in the expanding Universe

Take a look at any object, and what you see will be dependent on how far away it is from you. In reality, this object will have specific physical properties: size, color, brightness, etc., and will be located a particular distance from you. Typically, your way of detecting this object will be based in electromagnetic radiation, also known as light. The light that you see is the light arriving at your eyes right now, but that light would have been emitted by this object a certain amount of time in the past: the time it takes that light to travel from that object to your eyes.

In addition, the object will appear smaller and less bright the farther it is away from you. Light spreads out in a sphere once it leaves the source that generated it, and hence the farther it is from you, the fainter it appears. In addition, all physical objects take up a particular physical size, so the farther away they are, the smaller — in terms of angular size — they appear to you.

But these relationships, which are straightforward in our everyday experience, are wildly counterintuitive in the expanding Universe. Light-travel time, brightness, and angular size all defy our expectations. In a very real way, distances are not as they appear in the expanding Universe. Here’s how to make sense of them.

We often visualize space as a 3D grid, even though this is a frame-dependent oversimplification when we consider the concept of spacetime. In reality, spacetime is curved by the presence of matter-and-energy, and distances are not fixed but rather can evolve as the Universe expands or contracts. Prior to Einstein, space and time were thought to be fixed and absolute for everyone; today we know this cannot be true.

The way we normally conceive of things is the same way that Euclid and Newton did: as though the world was defined by some sort of fixed three-dimensional space, where you could define all objects by their position. In classical physics, as well as in Euclidean geometry, all observers everywhere can agree on the definition of what position and location mean: both in an absolute sense as well as in a relative sense. If that were the case for our actual, physical Universe, then everything would be simple and straightforward when it came to measuring the cosmic distances to various objects.

• An object’s angular size would simply be determined by its physical size at a physical distance. If you knew how big it actually was and could measure its angular size, you can determine how far away it is.
• An object’s apparent brightness would simply be determined by its intrinsic brightness at a physical distance. If you knew how luminous it was and could measure how bright it appeared, you could determine its distance.
• And if you looked back at an object and knew how long had elapsed from when the light was emitted until you observed it, you could know how far away it was. Just multiply the light-travel time by the speed of light, and there you have its distance.

The brightness distance relationship, and how the flux from a light source falls off as one over the distance squared. A satellite that’s twice as far away from Earth as another will appear only one quarter as bright, but the light-travel time will be doubled and the amount of data throughput will also be quartered. Gravitation, light, sound, and electromagnetism all fall off as the inverse distance squared.

It seems so simple and straightforward that it couldn’t possibly be wrong. Unless, that is, one of the underlying assumptions behind this reasoning were false, and in this case, it is.

The falsehood?

Assuming that our three-dimensional space is fixed, and that observers at different locations in space and/or time would agree on what “positions” actually are, is the problem. Space, among many other properties in the Universe, is relative to the observer. In this case, “relative” means:

• what you perceive changes as your velocity changes,
• what you perceive changes as time passes,
• what you perceive changes dependent on the distribution of masses in the nearby Universe,
• and what you perceive changes, on a global scale, as the Universe itself expands.

Although all of these components play a role, the largest role is played by the expanding Universe.

The ‘raisin bread’ model of the expanding Universe, where relative distances increase as the space (dough) expands. The farther away any two raisins are from one another, the greater the observed redshift will be by the time the light is received. The redshift-distance relation predicted by the expanding Universe is borne out in observations and has been consistent with what’s been known since the 1920s.

Perhaps the best analogy for the expanding Universe is a loaf of leavening raisin bread. Imagine you have a ball of dough with raisins distributed all throughout it. Imagine the ball is in the zero-gravity environment of space, so that no direction or dimension is preferred over any other. And furthermore, I want you to imagine that the dough itself is transparent, so that you could see “through it” to the individual raisins inside.

Each raisin is like a galaxy or other luminous object that’s gravitationally bound together, but that isn’t gravitationally bound up as part of a larger structure. The dough is like the fabric of space: it expands over time.

If you were to imagine that you, yourself, were located on one of the raisins, what would you see? At each moment, you’d see the raisins as they were relative to you: as the light from them that traveled through the “expanding dough” was just arriving. Only, because of that expansion, the light you saw wouldn’t have precisely the same properties as the light that was emitted.

Light may be emitted at a particular wavelength, but the expansion of the Universe will stretch it as it travels. Light emitted in the ultraviolet will be shifted all the way into the infrared when considering a galaxy whose light arrives from 13.4 billion years ago. The more the expansion of the Universe accelerates, the greater the light from distant objects will be redshifted and the fainter it will appear.

The first and most obvious thing that occurs is that the light gets affected by the expansion of the Universe. As the fabric of space “stretches” while the light journeys through it, the light itself also stretches: to longer and longer wavelengths. Longer wavelengths mean redder colors, cooler temperatures, and lower energies, so it really is true that the Universe cools as it expands. It also means that if we were to take three snapshots of the Universe:

• at the moment the light was emitted from any particular “raisin” in this raisin bread,
• at the moment where the distance between the emitting “raisin” and the observer “raisin” equaled the light-travel time (multiplied by the speed of light) that the light wave will traverse over its entire journey,
• and at the moment the light arrives at the observer “raisin” in the raisin bread,

we would obtain three different answers to the question of “How far away are these two raisins from one another?” The light was emitted back in the distant past, and was emitted farther back in time the farther away the emitting raisin is located. And although the light takes a journey (that we can measure in years) through the Universe, by the time that light arrives, the expansion of the Universe has pushed that object so that it’s much farther away than either the light-travel time or the initial separation distance would indicate.

The relative importance of different energy components in the Universe at various times in the past. Note that when dark energy reaches a number near 100% in the future, the energy density of the Universe (and, therefore, the expansion rate) will remain constant arbitrarily far ahead in time. Owing to dark energy, distant galaxies are already speeding up in their apparent recession speed from us. Way off the scale of this diagram, to the left, is when the inflationary epoch ended and the hot Big Bang began. Dark energy’s energy density is ~123 orders of magnitude lower than the theoretical expectation.

Another thing that happens, although it isn’t necessarily obvious from the raisin bread analogy, is that the expansion rate itself changes as the Universe evolves. At any particular moment in time, there are only three things that determine how quickly the Universe expands:

1. the sum of all the matter-and-energy, in all its various forms, at the particular location in space you’re occupying,
2. the initial expansion rate as it was at the start of the hot Big Bang,
3. and what the curvature of space is, right now, at the particular location in space that you’re occupying.

If the initial expansion rate was the same everywhere, and the curvature of space is negligible everywhere, then the biggest factor in the evolution of light that travels through the expanding Universe is how the various types of energy change, relative to one another, in terms of density. Fortunately for all of us, the last few decades in cosmology have been revolutionary for teaching us both what our Universe is made out of, including in what ratios at all sorts of different times, and how its expansion rate has changes over its history.

While matter and radiation become less dense as the Universe expands owing to its increasing volume, dark energy is a form of energy inherent to space itself. As new space gets created in the expanding Universe, the dark energy density remains constant.

What we’ve learned is that, today, our Universe is made out of:

• 68% dark energy, which is a form of energy inherent to the fabric of space itself,
• 27% dark matter, which is massive, clumps and gravitates, but always moves slow compared to the speed of light,
• 4.9% normal matter, which is massive, clumps and gravitates, interacts collisionally with both itself and radiation, and moves fast compared to the speed of light at very early times,
• 0.1% neutrinos, which are massive, clump and gravitate at cosmically late times, but which move close to the speed of light for at least several million years,
• and 0.01% photons, which are massless and always behave as radiation.

If you behave like matter, your density drops as the Universe expands, since density is mass over volume and the volume increases. If you behave like radiation, your density drops more quickly than matter’s density drops, since not only does the volume increase, but the wavelength of your radiation gets stretched by the expanding Universe. And if you behave like dark energy, your energy density remains constant, since new space gets created when the Universe expands, preventing the density from decreasing.

When you factor in everything we know about the expanding Universe, we can immediately learn some valuable lessons about what distance actually is, and how it differs from our Euclidean/Newtonian prejudices.

This simplified animation shows how light redshifts and how distances between unbound objects change over time in the expanding Universe. Note that the objects start off closer than the amount of time it takes light to travel between them, the light redshifts due to the expansion of space, and the two galaxies wind up much farther apart than the light-travel path taken by the photon exchanged between them.

Lookback time. Have you ever wondered why we say things like, “we see this object as it was 13.4 billion years ago, and it’s 32 billion light-years away”? That’s because of how distances evolve in the expanding Universe. When the light was first emitted, the Universe was much smaller, and that object was not even 1 billion light-years away from us. As that light traveled to us, the Universe expanded; initially, the light got farther away, but as the expansion rate slowed, it caught up to us. Finally, today, after a journey of 13.4 billion years, the light arrives. But the object itself is now even farther away: 32 billion light-years.

As we look to farther and farther distances, the difference between “lookback time” in years and “actual distance, today” in light-years gets progressively greater, as the expanding Universe has a more pronounced effect. For example:

• Light arriving from 100 million years ago corresponds to an object presently 101 million light-years away.
• Light arriving from 1 billion years ago corresponds to an object presently 1.036 billion light-years away.
• Light arriving from 5 billion years ago corresponds to an object presently 6.087 billion light-years away.
• Light arriving from 10 billion years ago corresponds to an object presently 16.03 billion light-years away.
• Light arriving from 13.78 billion years ago corresponds to an object presently 41.6 billion light-years away.

The important takeaway is that the distance to an object is not the light-travel time multiplied by the speed of light; it’s greater in the expanding Universe.

Two of the most successful methods for measuring great cosmic distances are based on either their apparent brightness (left) or their apparent angular size (right), both of which are directly observable. If we can understand the intrinsic physical properties of these objects, we can use them as either standard candles (left) or standard rulers (right) to determine how the Universe has expanded, and therefore what it’s made of, over its cosmic history. The geometry of how bright or how large an object appears is not trivial in the expanding Universe.

Luminosity distance. The farther away an object is, the fainter it is; that’s exactly what you’d expect. But the farther away you go, as you measure just how faint it gets, you’ll progressively begin to discover that things get fainter faster than you expected. This is because the light is affected by the curvature of space, by the cosmological redshift, and by time dilation of cumulatively intervening matter. If the Universe were static, an object that was twice as distant would appear to be just one-quarter the brightness, but in reality, it’s even less bright than that, largely due to the “redshift” factor.

Travel the Universe with astrophysicist Ethan Siegel. Subscribers will get the newsletter every Saturday. All aboard!

One of the most surprising finds in all of human history was the discovery of dark energy, which was revealed by the study of ultra-distant supernovae. The fact that the expansion of the Universe, at very late times, is dominated by a form of energy (dark energy) that doesn’t get less dense as the Universe expands is why objects get fainter, more severely, the longer it takes their light to arrive at our eyes. This isn’t necessarily intuitive, but it’s not the biggest surprise in the Universe.

Arguably, that designation goes to the last type of distance measure we could use: one based on angular size.

This three-panel image shows the simulated view of the same astronomical target, NGC 3603, as seen with Hubble (at left), the Very Large Telescope with adaptive optics (at center), and the currently under-construction European Extremely Large Telescope (at right). The increase in sharpness is a reflection of the increased resolution that comes about from having a larger primary mirror. This angular resolution is also distance-dependent, but in an unintuitive way.

Angular diameter distance. You’d think this would be the most straightforward: know how big an object is, intrinsically, then measure how large it appears on the sky, and you can calculate how far away it is. Only, that’s no good, because for every angular size that you’d see, there are actually two distances that the object could be at: one closer to you and one farther away.

Why is this the case? Because, way back when the Universe was younger, the objects that are now very distant were once much nearer, and hence, they appeared to take up a much larger angle on the sky. Even though the Universe has expanded since then, it once expanded faster than light could travel, and only after the expansion rate drops does the light begin to “catch up” to us.

In other words, as you look farther away, objects appear smaller and smaller until a critical point: a minimum size that objects will appear in our Universe, which occurs for objects that are somewhere around 15 billion light-years away. Beyond that, they start to appear larger again; if something comes from us close by or very far away, they will appear to be the same angular size on the sky.

A simulated view of the same part of the sky, with the same observing time, with both Hubble (L) and the initial architecture of LUVOIR (R). The difference is breathtaking, and represents what civilization-scale science can deliver: resolution of just a few hundred light-years-per-pixel of every object in the Universe. Many of the galaxies suspected to be there, but beyond the reach of Hubble, will finally be revealed.

This is not a bug with the expanding Universe; it is a feature. It means that if we can build a telescope to resolve objects that are a certain size across — a thousand light-years, or a hundred, or ten, or one — that are presently 15 billion light-years away, that telescope would be able to see every object in the Universe at that resolution or better. If you want to build a large space telescope, like the National Academies just rated as their highest priority for NASA astrophysics, this is what it gets you: superior resolution all across the observable Universe.

In the expanding Universe, distant objects are fainter than they appear. They’re farther away than they appear. But in terms of angular size, once you get beyond a certain distance, they actually appear larger than they truly are. We can exploit this, if we’re clever enough and willing to invest our resources in it, to probe more distant objects in the Universe to better accuracy and resolution than ones that are closer to us. When it comes to seeing across the great cosmic expanse, there are precious few ways that nature gives us a helping hand. By making ultra-distant galaxies appear larger than they actually are, the expanding Universe highlights one undoubtable way that the cosmos, at least in terms of discovery potential, has been kind to us.