A balance that is a saddle

When Einstein applied general relativity to the universe as a whole in 1917, there was no reason to think that it was doing anything. The stars moved slowly, and the galaxies were not yet known to be galaxies. A universe of matter under gravity alone cannot stand still, so he added one term to his field equations, with one constant in it, which on large scales acts as a repulsion growing in proportion to distance. A universe with the right density and that term had a size, a uniform curvature, and no reason to change.

The story that follows is usually told as a joke with a punchline. Hubble found the expansion, Einstein is said to have called the term his greatest blunder, and the constant returned seventy years later for other reasons. That account skips the more interesting event, which happened in between. In 1930, a year after Hubble, Arthur Eddington pointed out that the static solution was unstable, so that even without any observation it could not have been what it was meant to be. The argument takes one figure to see. It also explains why the universe that exists now, which has been accelerating for some billions of years, passed through the same balance without so much as a pause.

Two forces with opposite slopes #

Take a closed universe of pressureless matter, whose size is the scale factor $a$. Its acceleration comes from two terms:

$$\frac{\ddot a}{a} = -\frac{4\pi G}{3}\rho + \frac{\Lambda}{3}.$$

Matter thins as the universe grows, $\rho\propto a^{-3}$, so its attraction on the scale factor falls as $a^{-2}$, an inverse-square law, which is what Newton’s sphere of matter would have given. The constant’s repulsion on the scale factor grows in proportion to $a$. Set the two equal and there is one size, $a_E$, at which they cancel. Choosing the time unit $1/\sqrt{4\pi G\rho_E}$ and the size unit $a_E$, the equation simplifies to

$$\ddot a = \frac{1}{3}\left(a - \frac{1}{a^2}\right).$$

Everything is in the slope. A stable balance would need the net force to oppose a displacement, as a marble in a bowl has. Here, a universe larger than the balance feels more repulsion and less attraction, and a smaller one feels less repulsion and more attraction. The displacement is reinforced in both directions. Neither term is a mistake. Their functional forms force the crossing to have the wrong sign, and no value of the constant can change that, because the constant only moves the crossing.

Linearising around $a=1$ makes the point in one line. With $a=1+\varepsilon$,

$$\ddot\varepsilon = \tfrac13\left(\varepsilon+2\varepsilon\right)=\varepsilon .$$

There is no oscillation, no restoring term, no free parameter. A displacement grows as $e^{t}$ in units of $1/\sqrt{4\pi G\rho}$.

How fast a balance is lost #

That time unit is worth pricing, because its size is what makes the instability physically decisive. For a density like the mean matter density of the present universe, about $2.6\times10^{-27}$ kilograms per cubic metre, the unit is about 21 billion years. For a static universe at the density Einstein was thinking of, where his universe needed a larger density than this, the time is correspondingly shorter. In any case it is a time comparable to the age of whatever could be observed in it.

The logarithm is the part that defeats any rescue. A universe that starts one part in a thousand out reaches half as big again after seven units. To make it last ten units more, the starting error would need to shrink by four orders of magnitude; to last a hundred, by more than forty. The stars and galaxies make density fluctuations of order one everywhere, in any case. A region even slightly overdense relative to the balance has a different local value and starts falling. There is no way to set up a real, lumpy universe to sit here at all. This is what Eddington saw, and it is why he and Lemaître immediately took the static solution not as a state of the universe but as one point on the map of its possible histories.

The instability has another face in the physics of galaxies, and the comparison is worth noting. A cloud held up by its own pressure, and the Jeans argument for when it falls, have an exponential on the same timescale, $1/\sqrt{4\pi G\rho}$. Einstein’s balance is a cloud with no pressure and a uniform repulsion instead, and its fall is the same fall. Exactly the same number governs the rate at which a star forms from gas and the rate at which a static cosmos would have come apart.

A map of every closed history #

The second integral of the equation of motion is the Friedmann equation, and it is an energy conservation law for the scale factor. In these units, for a closed universe,

$$\dot a^2 = \frac{2}{3a} + \frac{1}{3}a^2 - 1,$$

with a constant on the right for any universe not exactly balanced. Plotting $\dot a$ against $a$ gives the map on which all such universes live, and it is a remarkably clear one.

The static universe sits at a crossing point. In the language of dynamics it is a saddle: two directions lead into it, and two lead away. The thick curves are the separatrices. One approaches the static point for an infinite time, starting from a big bang. This is the Eddington–Lemaître model, a universe that begins as small as can be, expands more and more slowly, and never quite arrives. The other leaves the static point after an infinite time, and is the model Eddington himself favoured for philosophical reasons: a universe with an infinitely long, nearly static past and a slow, gentle start to expansion.

Every other curve belongs to one of four families, distinguished by which side of the separatrices it lies on. That is a complete classification with no calculation in it. Before 1930 it was not clear that there might be universes that bounce, and it had not been fully appreciated that recollapse was available at all with a positive constant. The bouncing family on the right is real as a solution, though a bounce needs the universe to have been extremely large and contracting, and it needs a very particular kind of matter content. It has been checked and largely excluded by observation, because the largest measured redshift limits how small the universe can have been at the bounce.

A pause bought with a logarithm #

Set the constant slightly above the static value and a new kind of history appears. A universe starting from a big bang with positive curvature rises toward the balance size, slows down enormously, and at last moves on. Georges Lemaître wrote this down in 1927 and 1931. It was soon called the hesitating or loitering universe, and for some decades it was the only way anybody had of giving a universe a long age with a large expansion rate today.

The duration is a pure logarithm of the fine tuning. A pause of any length is available, and the price is exponential in the length: each additional 2.3 time units needs a factor of ten closer tuning. It is the same factor as the failure of stability above, seen from the other direction, because a pause is simply a trajectory that passes close to the saddle, and the time it spends near the saddle is the time it takes to come back out along the unstable direction.

This mechanism was useful, and it played a concrete role in the history of the age problem. In the 1930s, Hubble’s constant was thought to be about 500 kilometres per second per megaparsec, which gave a universe younger than the Earth’s rocks. A hesitating universe of any desired age could fix that. As the distance scale was corrected over the following decades the need faded.

A pile of redshifts #

The loitering model came back in the late 1960s for a different reason, and this episode makes the most concrete test of the whole picture. When quasars were first catalogued their redshifts seemed to crowd near two. Very few were known beyond that value, and those that were found there were many. Several authors (Petrosian, Salpeter and Szekeres in 1967, and Shklovsky in the same year) noticed the implication. If the universe had paused at a scale factor corresponding to redshift two, objects at a steady rate per unit time would collect in a narrow interval of redshift, and the pile would be the pause itself.

The pile did not survive. Quasars at redshifts well beyond 2 began to be found in the early 1970s, and with better selection the excess at two turned out to be mostly a result of how they were picked out from colours and from the emission lines falling in particular filters, with a real peak in quasar activity near that redshift behind it. A real rise and fall of quasar activity, which comes from the growth of black holes and has nothing to do with the expansion, accounts for what remained.

The idea was worth trying and it was worth rejecting. A pause would have been an extraordinary discovery, and it makes a very specific prediction: a cut-off in the number of objects at redshifts beyond the pause, and a light-travel distance dominated by a single epoch. Neither was found. The modern measurements of the expansion rate over the same range of redshift, read from standard candles and from the acoustic ruler, show no such feature, and they rule out a hesitation of any noticeable size since about redshift five.

The crossing that happened #

What survives the whole story is the balance condition itself, which is simple enough to state without a static universe at all. The scale factor’s acceleration vanishes when

$$\rho_m = 2\rho_\Lambda,$$

and that is Einstein’s condition, with no reference to curvature. The curvature is needed for a static solution only to make the speed zero as well as the acceleration. A flat universe of matter and a constant satisfies the condition at exactly one instant, and it passes straight through because it is moving at that moment.

That is the connection across a century which makes the instability worth knowing about. The real universe reached Einstein’s balance about six billion years ago. The geometry of the saddle says that a universe coming up to that density ratio with zero speed would hover, and every actual universe with non-zero speed passes through, which the measured one has done. The slope of the net force at the crossing is the same as the one Eddington found, with the same sign: the repulsion wins by more and more from then on. So the same instability that ruled out a static universe is what makes the acceleration run away now, and its exponential rate, in the limit of pure constant, is what sets the event horizon.

The closeness of that moment to the present is a coincidence that has a separate literature, and its real size is smaller than it is usually said to be. It does not change the shape of the argument, since a crossing must happen sometime in any universe with both components.

Why it was worth Einstein’s trouble, and what the blunder was #

Reconstructing the reasoning, there are at least three different mistakes that could be attributed to the 1917 paper, and it is useful to separate them.

The term itself was not a mistake. It is the only other term allowed by the same requirements of symmetry and second-order equations that give the rest of the theory. Physicists had it ready for use when it was needed in the 1990s, and all the measurements since are consistent with it. If anything, it was the omission of the term that would have been arbitrary.

The motive was reasonable given the observations available, which were radial velocities of nearby stars. Slipher’s redshifts of spiral nebulae were already being measured, but what the spirals were was not known. A universe that was static at the largest scale was what everyone assumed.

The real error was the failure to check the stability of the solution, which takes one line. Einstein found a state of balance and treated it as a state of the world. A second-year student of mechanics, handed that force law, would be asked what happens if the system is displaced. De Sitter’s and Friedmann’s alternatives soon showed that the equations had dynamic solutions, and for some years Einstein argued against them, at first incorrectly. It was thirteen years before the sign was looked at and that argument was closed, by someone else. The attribution of the “blunder” remark itself rests only on Gamow’s recollection, and it is doubted by historians; what is firmly on record is that Einstein dropped the term in 1931.

What the map leaves out #

The figures take a strict idealisation, and the classification they suggest is less clean than it appears.

The matter is pressureless throughout. A real early universe was radiation-dominated, and the same analysis with radiation has a different exponent, and a balance at a different ratio. For the histories drawn here that hardly matters except very close to the big bang.

The universe is uniform. Lumps are not a small correction to a saddle, because a saddle amplifies small displacements exponentially, and no region of a real universe has the mean density. The figures describe an average that has no reason to obey its own equation exactly. Whether the average of an inhomogeneous universe follows the Friedmann equation at all is a subtle question, and although most estimates find the difference is small, it has not been possible to prove it in general.

Positive curvature has been assumed where it is needed. The real universe is measured to be flat to within a part in a few hundred. The static solution then does not exist as an equilibrium of speed, and the map becomes the single family with zero curvature. The curves on the figures are not candidates for the real universe. They remain the correct way to see what the possibilities were, and the flat universe’s own trajectory is the limit of them.

The constant is assumed to be constant. If dark energy changes with time, the force law’s right-hand side acquires a term with a different slope, and a stable balance is not impossible in principle: a component whose repulsion falls with size faster than matter’s attraction can restore. No known field does this with the density and pressure measured today.

The same equation, inside a cluster #

The balance is not only a matter of whole universes, and that gives the argument a test on a scale that can be observed. Every bound region in an expanding universe obeys the same two terms. A sphere of mass $M$ feels an inward pull that falls as the inverse square of its radius and, if the constant is real, an outward push that grows in proportion to its radius. They cancel at one radius,

$$r_{\rm eq} = \left(\frac{3GM}{\Lambda c^2}\right)^{1/3}.$$

For the Local Group that is about a megaparsec and a half, and for a rich cluster of galaxies about ten megaparsecs. Inside that radius matter can stay bound. Outside it, the push wins for ever. This is Einstein’s balance applied to one object, and it has the same wrong-signed slope: a shell a little beyond the radius is accelerated away and one a little inside falls back. Each massive object thus has its own small static sphere round it, unstable in exactly the same way, and the flow of galaxies round the Local Group passes through it.

This fixes something about the very distant future that is otherwise hard to state. As the expansion accelerates, each bound object is left alone inside its own balance radius, and everything else recedes beyond the horizon that a constant creates. Structure stops forming, for the same reason that it stopped forming in a cruder way when the static model was perturbed: the term that grows with distance has no competitor at large distance. The largest bound structures that will ever exist have already started to be defined, and the saddle marks each of their edges.

It also offers a modest and rarely used test of the constant. The radius at which the local flow turns round should lie inside the balance radius for any given mass. Measured turnaround radii of nearby groups are close to the limit but below it, which is consistent with the constant and with a good deal else. The test has no great power, but it is one of the few places where the repulsion can be compared with a measured mass on a scale smaller than the universe as a whole.

Still open: whether the saddle was ever approached #

The final question concerns the far past. Some universes on this map have long histories before the present expansion. One proposal that has attracted serious attention is an emergent universe, which starts as the Einstein static state itself, at a very early time and a very high density, from which it slowly leaves on the unstable direction. It requires no singularity at the beginning, since an infinite past near the saddle has no first moment, and some authors have proposed that an inflationary epoch could begin naturally in this way.

The objections are the ones Eddington’s calculation would predict. A state balanced on a saddle has to have got there, and has to survive perturbations of every kind for an infinite time; quantum fluctuations in the scale factor do not politely avoid the unstable direction, and analyses of tunnelling suggest the state decays in a finite time. No observation that is presently possible would distinguish such a past, because the inflation that comes after it would have stretched any such evidence away. Whether an instability that took one line to find in 1930 can be made into a beginning, or whether it merely says again that nothing can begin there, has not been settled.