Light that set out towards here and was carried away

A photon of the microwave background reaching a detector today has been travelling for 13.8 billion years. It would be natural to think that the photon therefore started 13.8 billion light years away, and it would be natural also, having heard about the expansion, to correct that upward: the place where it started is now 45 billion light years away, a number quoted often enough to be familiar. Neither of those is the distance between the photon and this part of space when it set out. That distance was about 41 million light years, barely twenty times the separation of the Milky Way and Andromeda.

That number invites an obvious question. Light covers 41 million light years in 41 million years. The rest of the 13.8 billion years has to be spent on something, and the answer is that for the first four billion of them the photon, aimed straight at this place, was getting further away.

Nothing in that statement requires anything to break a speed limit. The horizons of this cosmology have been treated throughout as properties of integrals, and this is a property of the same integral, read at each moment instead of at the end. The behaviour follows directly from the distance that expands while the light crosses it, and once it is drawn, a number from a quite different part of observational cosmology turns out to be the same number.

A teardrop rather than a cone #

Textbooks draw the past light cone as a cone. In the coordinates where it is a cone, the expansion has been divided out of the distances. Put the expansion back, drawing each event at the proper distance it had from the observer’s location at the time it happened, and the shape changes. If $\chi(z)$ is the comoving distance light travels between redshift $z$ and today, the proper distance at the time is

$$D(z) = \frac{\chi(z)}{1+z}.$$

The numerator grows with $z$ because older light has come further. The denominator grows because everything was closer together then. At small redshift the first wins, and at large redshift the second does, so the function must rise to a maximum and fall back.

The result is a teardrop, and every photon reaching the observer at this moment lies on its one curve. Light from a galaxy at redshift 5 joined the curve near the bottom, at 4.3 billion light years, 1.2 billion years after the start. Light from redshift 1 joined it at 5.5 billion light years on the way back in. Light from last scattering has been on it from almost the very start. Those photons are not racing each other. They are all on the same path, in the same sense in which all the cars on a road at one instant are part of a single line of traffic, and a galaxy merely adds to that line at the moment the line passes it.

This resolves a confusion that the surprising size of the observable universe tends to leave behind. The number of 46 billion light years answers the question of how far away that material is now. The number of 41 million answers how far it was when the light left. The light-travel figure of 13.8 billion is neither, and no ruler laid at any moment would read it. The teardrop never gets wider than 5.85 billion light years. No light now arriving has ever been further from here than that.

What makes it turn #

The arithmetic that gives the widest point is short, and it identifies the surface the turning happens on. Differentiate the proper distance along the photon’s path with respect to cosmic time. The separation grows because the space between grows, at the Hubble rate times the distance, and shrinks because the photon is heading inward at $c$:

$$\frac{dD}{dt} = H(t)\,D - c.$$

So the distance is increasing whenever $D$ exceeds $c/H$, which is the radius of the Hubble sphere: the distance at which the recession equals the speed of light. Outside the sphere, light aimed inward loses ground. Inside it, the light gains. The light turns round exactly where it crosses the sphere, and it can cross because the sphere itself grows outward faster than the photon is being carried away.

A photon just released at last scattering, then, was receding from this place at sixty-five times the speed of light, while moving towards it at the speed of light through the gas around it. Locally everything is in order. An observer in that gas sees the photon pass at $c$ exactly, in the direction of the future Milky Way, and no measurement anyone could make there would hint at anything strange. Only the phrase “receding from here” has changed its meaning, because it now compares the rate of change of a distance that no one could measure at the time with the local speed of light. Tamara Davis and Charles Lineweaver surveyed how often this is misread in the literature, textbooks included, and the statement that nothing can recede faster than light turned up repeatedly.

The speed drops steadily as the photon moves, for two reasons. The Hubble rate falls as the universe thins out, and the photon is getting further inside its own teardrop relative to the sphere. There is nothing like a moment of decision. The net motion simply passes through zero at 4.05 billion years after the start, when the photon has been outrun by the Hubble sphere, and it then approaches at an increasing rate, reaching the full speed of light only at the end of its trip.

How long the journey spends going the wrong way #

The moment of turning is the same for every photon on the cone, since they all share a single path. A photon’s share of receding therefore depends on when it joined.

The curve flattens quickly, because the early universe passes so fast. A galaxy at redshift 20 has its light emitted only 180 million years after the start, so its share is nearly the same as that of the background. That makes for a slightly unsettling way to describe a deep-field image. For the earliest galaxies a telescope can reach, a good quarter of the time since the light left was spent moving away from the place the telescope now is.

One thing this does not mean is that the photon might have failed to arrive. The light in these figures is exactly the light that did arrive; the condition for reaching here is an integral over the whole future, and the question of whether a horizon exists is the question of whether that integral converges. Light emitted today from a galaxy at redshift 2 is outside the Hubble sphere and will recede for a while, and in a universe with a cosmological constant it will not recover. It was emitted too late, because the sphere stops growing fast enough to catch it. So being outside the sphere is necessary for losing ground and not sufficient for failing to arrive, and the distinction between those two surfaces, the Hubble sphere and the event horizon, is precisely the difference between “receding now” and “never”.

The same curve in a survey’s data #

Now consider a different question, one that was asked long before anyone drew a teardrop. How large does a galaxy look at redshift $z$? A rod of proper length $\ell$ subtends the angle it had at the moment the light left it, and the photons that define its two ends have been travelling along diverging paths ever since. In a flat universe the angle is $\ell/D_A$, with $D_A$ the angular-diameter distance:

$$D_A(z) = \frac{\chi(z)}{1+z}.$$

This is the same expression as before. The angular-diameter distance is usually presented as a construction, one of several ways of defining a distance in curved spacetime, and its odd behaviour of falling at large redshift is often presented as a curiosity. It is the teardrop. The angular-diameter distance to a galaxy is how far away that galaxy was, in proper distance, when the light that now reaches the observer left it, and the reason it decreases is that the light left when everything was close together.

So the galaxies that look smallest are exactly those at the waist of the teardrop, and a galaxy that looks larger than that is either close or very far away. An object at redshift 1.59 was farther from this place when its light left than any other object seen in the sky at any redshift. And this one coincidence ties together two facts that are usually stated in separate chapters: that the sky has a smallest apparent size, and that light from beyond a certain redshift first moved away. Each of them is the maximum of one function.

The same fact makes sense of what looks like an extraordinary number. The acoustic scale in the background is a ripple about 135 thousand parsecs across at the time, and it subtends six-tenths of a degree. A feature less than half a million light years across, seen across a universe billions of light years wide, covers an angle a little larger than the full Moon. It is large in the sky because it is 41 million light years away, in the only sense of distance that sets its apparent size. The patch of the sky that is the surface the background actually is is, in this sense, close by. It is a small shell close to where the observer’s own material then was, and the whole sky is an enormously magnified image of it.

What the turning point depends on #

The position of the widest point depends on how the expansion went, because it is where the Hubble radius catches up with the light.

For matter alone the answer takes a line. The comoving distance is $2(c/H_0)(1-\sqrt{a})$, so $D$ is proportional to $a(1-\sqrt{a})$, whose maximum is at $\sqrt{a} = 2/3$, or $a = 4/9$, which is redshift $5/4$. This classic result was used in the 1950s and 60s to propose the angular size of radio galaxies as a test of cosmology, with the expectation that the minimum could be located and would measure the matter density directly. It did not work out, because radio sources grow as they age, and a sample at redshift 2 is not a sample of the same objects as one at redshift 0.5. That is the fate of many standard rulers that are not fixed by physics, and the ruler that is has shown what the same geometric test looks like done properly. Its comoving length is set in the early universe, and it records the turnover directly.

The opposite limit is instructive too. In pure de Sitter space the Hubble radius is constant, and the light cone’s proper distance is $(c/H)(1-a)$, which only rises towards the Hubble radius and never exceeds it. All light is inside the sphere from the start. Nothing recedes. The whole phenomenon is therefore a product of early deceleration, when the Hubble radius was growing rapidly from small values while light released outside it waited to be overtaken. The cosmological constant’s presence moves the turning point a little further back, and if there were much more of it, the phenomenon would shrink, because the universe’s balance between matter and the cosmological term determines how much deceleration there was to do the overtaking.

The turn that a different coordinate removes #

The teardrop has a strong visual claim on the mind, and for that reason it needs a caution. Change the coordinates, and it goes away.

In the coordinates in which light travels in straight lines the photon approaches at a steady rate from the instant it leaves. It never recedes. It makes constant progress across a grid that does not move. Which description is correct? Both describe the same geometry, and neither carries more information. The word “receding” relates to a chosen way of cutting spacetime into moments: one slice for each value of cosmic time, with distance measured along the slice.

That choice is not arbitrary, as it is the one adapted to the matter in the universe, and it gives a physically meaningful answer: what a chain of comoving observers would add up, each measuring a short stretch of its own neighbourhood at the same cosmic time. It also makes the angular size physically real. The teardrop’s width is measured directly, as an apparent size on the sky. Still, the rate at which the distance changes has no local measurement, and an argument that something is really moving away at 65 $c$ is an argument about a convention. What is coordinate-free is the angular size, the redshift, the event of arrival, and the fact that the light takes 13.8 billion years. The proper distance is a way of organising those facts, and the teardrop is what they look like when organised that way.

The surprise inside an ordinary number #

There is a pleasing economy in where this ends up. A student of cosmology meets the angular-diameter turnover early, generally as a slightly mysterious feature of a formula. A student meets superluminal recession separately, usually as a warning about what not to say. And the Hubble sphere is a third topic, tied to the question of why inflation works. These three items are one fact, and it can be stated as follows: the maximum of $\chi/(1+z)$ is the point at which light crosses the Hubble sphere, and it is at redshift 1.59.

So a survey that measures the angular size of a standard ruler at redshifts on either side of 1.6 is doing something quite geometric. It is sampling the width of the observer’s past. Going further back in redshift means looking at things that were closer, and a large part of why the early universe is observable in the detail it is lies in that closeness: at a given physical size, high-redshift objects appear larger than they would have at intermediate redshift, though they are also far fainter, since surface brightness falls by the fourth power of one plus redshift. Distant objects being resolved is a gift of the teardrop’s narrowing, and their being faint is a penalty of the same expansion.

What the picture cannot show #

The figures assume a perfectly homogeneous flat universe, and light in the real one moves through lumps. The path is deflected, delayed and magnified along the way, and the relation between redshift and angular size is changed by lensing in individual directions, by amounts that average out over the sky but not along any particular line. The figures draw a single cone with a single width, whereas any actual sightline has its own slightly different width.

Spatial curvature has been set to zero, and with curvature the angular-diameter distance differs from the proper distance at emission by a sine or hyperbolic sine factor, breaking the exact identity. The measured curvature is small enough for this not to matter at the precision shown here.

The picture also implies a clarity about the very earliest moments that it does not have. At the lowest points of the teardrop, close to the bang, the light in question does not exist: the universe is opaque before redshift 1090, and the drawn curve below that represents only the path that light would have followed. Neutrinos and gravitational waves are the only messengers that could actually have travelled that part. The curve continues downward only because the geometry does.

Still open: the part of the teardrop nothing has yet been seen from #

The teardrop has a lower section, from redshift 1090 down to about redshift 15, in which almost nothing has been observed. Any object seen there would look large, being close, and very faint. The earliest galaxies now measured have reached into it only at its upper edge, and the redshifted 21-centimetre emission from neutral hydrogen, whose global trough is still unconfirmed, offers a way to map the rest of it.

Whether an actual standard ruler can be followed through the waist of the teardrop and down the far side with enough precision to see the angular turnover in a single population, and so read the expansion at the moment the light turned, depends on surveys of the acoustic scale reaching past redshift 2 at high completeness. Those already exist in part from the quasar forest. If they extend, the measured width of the observer’s past will have been traced from the inside out. If some departure from the expected shape appears, it would be the most direct sign yet that this universe’s early deceleration differed from the assumed one, since the width at every redshift is the integrated record of that deceleration.