The age of the universe weighs the Local Group

Almost every galaxy’s spectrum is shifted to the red, and the shift grows with distance because the universe is expanding. The Andromeda galaxy’s spectrum is shifted to the blue. Measured from the Sun it is approaching at about 300 kilometres per second, and once the Sun’s own orbit round the centre of the Milky Way is taken out, the two galaxies are closing on each other at about 110 kilometres per second, 770 kiloparsecs apart. At that distance the expansion would carry a galaxy away at about 50 kilometres per second. Andromeda is not in the expansion. It is falling in.

That single fact carries a remarkable amount of information, because of what it implies about the past. At the big bang the matter that became the two galaxies was together and moving apart with everything else. Now it is approaching. At some moment in between, the two must have stopped separating and turned round, and whatever turned them was their mutual gravity. The time available for all of that is the age of the universe, and that is enough to weigh them. The argument was made in 1959 by Franz Kahn and Lodewijk Woltjer, who found that the pair had to be far more massive than their visible stars — among the first dynamical arguments that most of the mass in the universe does not shine.

One orbit, with the age of the universe as its clock #

Treat the two galaxies as point masses on a straight line through each other. A bound orbit under an inverse-square force is an ellipse, and an orbit with no angular momentum at all is the most eccentric ellipse the force allows: a straight segment, travelled out and back, with the two bodies at its ends meeting at the other. Its separation and time are written in terms of a single angle $\theta$:

$$r = A\,(1-\cos\theta), \qquad t = B\,(\theta-\sin\theta), \qquad A^3 = GM\,B^2.$$

The separation starts at zero when $\theta = 0$, reaches its maximum $2A$ at $\theta = \pi$, and returns to zero at $\theta = 2\pi$. The curve traced in the plane of time and separation is a cycloid — the path of a point on the rim of a rolling wheel.

Three numbers are known: the separation now, 770 kiloparsecs; the rate at which it is shrinking, 110 kilometres per second; and the time since it was zero, the age of the universe. The ratio of the velocity to the separation divided by the time, $v t / r$, depends only on $\theta$, so it fixes where on the cycloid the pair is. Then the separation fixes $A$, the age fixes $B$, and $A^3/B^2$ is $GM$. There is exactly one cycloid through that point with that slope, and its mass is 4.2·10¹² solar masses.

An orbit is ordinarily specified by five geometric elements and one clock reading — where the body was at some known moment. The timing argument supplies the clock reading from cosmology: the moment the separation was zero is the beginning of the universe. Everything else follows from the two quantities a telescope measures.

The history the cycloid describes is worth reading off the figure. The two galaxies-to-be separated to 1,037 kiloparsecs, a third further apart than they are now. They stopped when the universe was 8.5 billion years old, and they have been falling together for 5.3 billion years since. On this idealised orbit they will meet 3.3 billion years from now.

Measured in time rather than in angle, the pair is four fifths of the way through its orbit: 13.8 billion years of a 17.1-billion-year round trip. The value of $vt/r$ that puts it there is about $-2$. A pair still riding the expansion would have a value near $+1$, because the product of the Hubble constant and the age is close to one, and a pair at the moment of turnaround would have exactly zero. Where a pair sits on that scale is a direct reading of how far its own gravity has taken it from the expansion it started in.

Why “began together at the big bang” is not an extra assumption #

Starting the orbit at zero separation at the moment the universe began sounds like a choice made for convenience. It is not, and the cycloid itself shows why. Near its start, where $\theta$ is small, the two expressions become $r \approx A\theta^2/2$ and $t \approx B\theta^3/6$, and eliminating $\theta$ gives $r \propto t^{2/3}$ — exactly the law by which a universe of matter expands. Early on, the pair’s separation grew at the same rate as every other separation in the universe. A region that is going to become bound does not start out bound; it starts out expanding with everything else, very slightly denser than average, and the extra density decelerates it a little more than its surroundings until, billions of years later, it stops.

So the timing argument is not claiming that Andromeda and the Milky Way were once touching. It is claiming that the matter now in them was, at early times, moving apart with the expansion, and that its departure from the expansion has been due to its own gravity since. Both parts of that are what cosmology says about every galaxy, and neither was adjusted to make the argument work.

Twenty-five times the stars #

The Milky Way contains about sixty billion solar masses of stars and Andromeda about a hundred billion. The timing mass is twenty-five times their sum. Kahn and Woltjer suggested the missing mass was hot gas between the galaxies; it is now understood as the dark matter halos that surround both, the same halos whose presence is inferred from rotation curves that stay flat far beyond the visible discs.

What makes the timing argument valuable is that it is independent of those. A rotation curve measures the mass inside the radius of the last gas cloud whose velocity can be measured, and says nothing about what lies beyond. A velocity dispersion measures a mass through the virial theorem, which assumes the system has settled into a steady state — and the Local Group has not settled; its two dominant galaxies are still falling together for the first time. Galaxy clusters can be weighed three independent ways because they are old and relaxed. The Local Group can be weighed one way, and the way uses precisely the fact that it is young.

The same curve as a closed universe #

The cycloid has appeared before in cosmology, and not by coincidence.

A closed universe made only of matter expands from a big bang, slows, stops, and collapses, and its scale factor follows the same cycloid, with the same parameter, as the radial Kepler orbit. The two computations share nothing in the figure except their units, and they agree to the accuracy of the integration.

They agree because they are the same equation. A spherical shell of matter in an expanding universe responds only to the mass inside it, and the equation for its radius is the Friedmann equation with that mass and the shell’s own energy. A region whose density is high enough for the shell’s energy to be negative is, to the shell, a closed universe: it expands, reaches a maximum, and falls back. The Local Group is a small closed universe that turned round 5.3 billion years ago, inside a large open one that never will. The same picture, applied to every overdense region, is the spherical collapse model on which the calculation of how galaxies and clusters form is built.

What the push adds #

The universe the Local Group sits in is not only matter. Its cosmological constant exerts a small outward push inside every bound system, negligible for the Solar System and not negligible here, because the Local Group reaches 57 per cent of the radius at which the push would balance its gravity.

The dashed curve in the first figure is the same problem with the push included. There is no closed form, so the orbit is integrated, and its mass is adjusted until the pair arrives at 770 kiloparsecs at 110 kilometres per second after 13.8 billion years. It needs 4.76·10¹² solar masses, 13 per cent more than the pure Kepler orbit. The reason is direct: for the whole history of the universe the push has been opposing the infall, so a pair that arrives at today’s speed against it must have been pulled harder.

Thirteen per cent is well inside the uncertainties that follow, but it is not zero, and its sign is certain. A timing mass computed without the cosmological constant is an underestimate.

A younger universe would need a heavier group #

The age enters the argument as the length of the clock, and its effect is easy to isolate.

With less time available, the pair has had to separate less far, turn round sooner and fall back faster to reach today’s speed, and all of that takes a stronger pull: a universe younger by 28 per cent requires a group heavier by 34 per cent. The push matters less, 8 per cent rather than 13, because it has had less time to act.

For most of the argument’s history this was its weakest point. Through the second half of the twentieth century the age of the universe was uncertain by several billion years, and the Hubble constant from which it was estimated was argued over by a factor of two. The age is now known to a few tens of millions of years, and the clock is no longer the problem.

How the answer depends on what is measured #

The approach speed matters more than the age over the ranges now allowed. A faster approach requires more mass, for the same reason a shorter clock does: the pair has had to be pulled harder to reach it. The speed itself is a difference of two measured velocities — Andromeda’s velocity along the line of sight, which is easy, and the Sun’s velocity round the Galaxy, which has to be removed and is the larger source of error.

With the approach speed known to a few kilometres per second and the age to a fraction of a per cent, the formal uncertainty of the timing mass from these two quantities is small — a few per cent at the present age. That precision is illusory, and the reason is in what the calculation assumed rather than in what it measured.

The orbit is not radial #

Andromeda moves sideways too. The argument assumes all of the relative velocity is along the line joining the galaxies, and a sideways velocity adds angular momentum, so the orbit is an ellipse rather than a line, and the mass required to produce today’s approach changes. Sideways motion is measured from the drift of Andromeda’s stars across the sky over years, a few millionths of a degree, and the measurements disagree: estimates from the Hubble Space Telescope put it near twenty kilometres per second, and later ones from the Gaia satellite at fifty to eighty. The larger values raise the timing mass appreciably.

The Milky Way is not where it would be. The Large Magellanic Cloud, the Milky Way’s largest satellite, is massive enough to have pulled the Milky Way’s inner regions off their course by tens of kilometres per second, which alters the relative velocity the argument uses. Analyses that include its pull find a lower timing mass.

The galaxies are not points, and they are not alone. Their halos are hundreds of kiloparsecs across and may already overlap; the Triangulum galaxy and dozens of dwarfs add mass and tides of their own; and the matter of the group did not start as two points at the big bang but as a region that collapsed. Simulations that form groups like this one and apply the timing argument to them find it recovers the true mass to within a factor of about two — good enough to have established that the group’s mass is mostly dark, not good enough to measure it to better than that.

A second clock, at the edge of the group #

The same cycloid answers a different question, and its answer is a check on the first. Galaxies just outside the Local Group are not in the pure expansion either: the group’s mass has slowed them. At some distance from the group’s centre a galaxy is at exactly the turnaround point of its own radial orbit today, neither receding nor approaching, and outside that distance galaxies recede with speeds that climb towards the Hubble law. The surface at which the recession speed is zero is the zero-velocity surface, and a shell that is turning round today is at $\theta = \pi$ on its cycloid, so its radius $R_0$ is $2A$ and the age is $\pi B$. The mass inside it is then

$$M = \frac{\pi^2 R_0^3}{8\,G\,t_0^2},$$

which for a radius of one megaparsec and an age of 13.8 billion years is about 1.4·10¹² solar masses.

The radius is measured from the distances and velocities of the dwarf galaxies that surround the group, with each distance taken from the brightness of the dwarf’s most luminous red giants — one link in a chain in which every distance is calibrated by the one before — and it comes out close to a megaparsec. The mass it implies is lower than the timing mass by a factor of two or more. The two methods use the same physics and different parts of the group, and they should agree. That they do not is part of why the timing mass is quoted as uncertain by a factor of about two, and of the corrections that follow, one lowers the timing mass towards the edge’s value and another raises it further away.

Where the orbit ends #

On the radial orbit the two galaxies meet 3.3 billion years from now. With a sideways velocity they do not collide head-on; they pass, lose orbital energy by dragging on each other’s halos, and return. Whether and when they eventually merge depends on the same sideways velocity that limits the mass, and recent analyses incorporating the other galaxies of the group put a merger within the next ten billion years at roughly even odds. If it happens, the encounter will throw out bridges and tails of stars of the kind seen in interacting galaxies elsewhere, and the merged galaxy will be an elliptical.

Meanwhile the rest of the universe will have continued to expand. By the time the two have merged, every galaxy outside the Local Group will be receding faster, and the nearest of them will be on its way to being out of reach for good. The closed universe will have collapsed inside the open one.

What the argument shows #

The timing argument is a measurement of mass made with a clock rather than a scale. It uses the age of the universe as the time since the pair’s orbit began, and the one fact about that orbit that cosmology guarantees — that it began at zero separation — to turn two telescope measurements into the mass of a group of galaxies. That it lands at twenty-five times the stars, and that simulations confirm it lands within a factor of two of the truth, is how the Local Group’s dark matter was known to exist a decade before rotation curves showed it in individual galaxies.

Still open: whether the matter and the light agree about which way is at rest #

Andromeda’s approach speed is measured after removing the Sun’s motion round the Galaxy, and the Galaxy’s own motion is measured against the microwave background: the whole Local Group is moving through it at about 600 kilometres per second, a velocity that gravity from distant matter has built up over the history of the universe. That motion should put a pattern into the counts of distant galaxies as well as into the temperature of the background — more sources ahead than behind — and whether the two patterns agree is a test of the assumption the whole of cosmology rests on.