A clump that waits for the radiation to thin

There is a standard explanation of why the first structures took so long to form, and for the ordinary matter it is correct. Before recombination, protons and electrons were bound to the photons by scattering, and the photons had an enormous pressure. Any clump of that fluid that tried to collapse rang like a bell instead, and those are the oscillations whose frozen pattern fills the microwave background. The radiation held the baryons up.

The same sentence is often extended to dark matter, and for dark matter it is false. Dark matter does not scatter light. It feels the photons only through gravity, and on small scales the photons’ gravity averages to nothing, because a fluid that oscillates has no lasting clumps to pull with. So in the radiation era a clump of dark matter inside the horizon has nothing pushing back on it at all. It has self-gravity and no pressure, which is the recipe for collapse.

It does not collapse. It grows by about a factor of two or three over a span in which the universe expands by hundreds. The reason, worked out by Peter Mészáros in 1974, is one of the more instructive failures of intuition in cosmology, because nothing opposes the clump. It simply runs out of time, repeatedly, on every scale, for as long as radiation sets the expansion rate.

Nothing pushes, and the clump still stalls #

A small overdensity $\delta$ of pressureless matter in an expanding universe obeys an equation with two terms on the right that matter. The first is a friction, proportional to the Hubble rate $H$ times the rate of growth: as the universe expands, infalling matter is carried outward with it and its peculiar velocity decays. The second is the self-gravity of the clump, proportional to the density of the matter and to $\delta$:

$$\ddot\delta + 2H\dot\delta = 4\pi G\rho_m\,\delta .$$

When matter dominates, the two right-hand scales agree. The Hubble rate itself is set by the matter density, so $H^2\sim G\rho_m$, friction and gravity come out comparable, and the clump grows in proportion to the scale factor. That is the familiar result, and it is what every simulation of late structure formation begins from.

When radiation dominates, the balance breaks. The Hubble rate is set by a density much larger than the matter’s: $H^2\sim G\rho_r$. The friction therefore becomes very large compared with the gravitational term, and it is as if the clump were trying to fall through treacle. The treacle is not a substance. It is the universe expanding at a rate that the clump’s own weak gravity has no say in.

Rewritten in the variable $y=a/a_{\rm eq}$, the scale factor in units of its value at equality, the equation has an exact solution for the growing mode:

$$\delta \propto 1+\tfrac{3}{2}y .$$

That is the whole effect. For $y$ well below one the solution is a constant. From equality onward it is the familiar proportional growth. A clump does not grow between horizon entry and equality, apart from one slow logarithmic rise given to it at entry by the velocities the gas of the early universe already had.

The thing worth holding on to is what the equation does not contain. There is no pressure term and no term in which radiation acts on the matter. The stall comes from the left-hand side alone.

How long each wave has to wait #

Every comoving length starts outside the horizon, where no causal process acts on it, and gets inside as the horizon grows. So every Fourier mode of the density has an entry time. A long wave enters late and a short wave early, and the critical comparison is whether entry comes before or after equality.

A mode entering after equality has the easy history. It grows in proportion to the scale factor from the moment it enters until the present day, and nothing has held it back. A mode entering before equality reaches the inside of the horizon at $y_e$, sits there nearly frozen, and starts growing only near $y\approx 1$. The ratio of what it would have grown by to what it did grow by is therefore

$$\frac{\text{proportional growth from } y_e}{\text{actual growth}}\approx\frac{1+\tfrac32 y_e}{\tfrac32 y_e},$$

which is near one for late entry and rises as one over $y_e$ for early entry.

The entry condition turns $y_e$ into a wavenumber. During the radiation era the horizon grows as the scale factor, so a mode’s wavenumber $k$ is inversely proportional to the scale factor at which it entered: $y_e\propto 1/k$. In amplitude, a mode has fallen behind by a factor of about $k/k_{\rm eq}$. In power, which goes as amplitude squared, it has fallen behind by $(k/k_{\rm eq})^2$, with one logarithm clawed back from the kick at entry. A statement about one clump in one history has become a statement about the shape of the whole spectrum.

There is a good way to see this as a sum over the histories of waves instead of a dynamical claim. The universe has, at any moment, one length scale it is free to imprint: the horizon. It imprints exactly one on the spectrum during its whole history, because there is exactly one moment at which the expansion law changes. That moment is equality.

The turnover is the horizon at equality #

Inflation, or whatever came before, hands down a spectrum of nearly scale-free fluctuations, $P(k)\propto k^{n}$ with $n$ near 0.965. The value of that tilt, and what it may or may not say, is a separate question. What matters here is what the wait does to it.

Modes entering after equality keep it unchanged, and the spectrum rises as $k^n$. Modes entering before equality get multiplied by the square of the factor they lost, which falls as $k^{-4}$, so they follow $k^{n-4}$ apart from the logarithm.

This is one of the most remarkable objects in cosmology and it is easy to overlook how little is in it. The two power laws come from one inherited number. The bend comes from one moment. Everything in between, from the scale of the largest structures to the masses of dwarf galaxy haloes, is the transition between them. It is what comes out of a universe with a smooth radiation background, a pressureless component, and a change of expansion law.

The transfer function drawn here is the fit Bardeen, Bond, Kaiser and Szalay published in 1986, and it includes the wait, the logarithm and the slow change between the regimes; it ignores baryons, which suppress the spectrum further on small scales and print the acoustic feature that is used as a ruler. Those are real corrections, and they are drawn on top of a shape already fixed by the wait.

The bend as a weighing #

The moment of equality is set by a ratio. The radiation density today is known to better than a per cent, because the temperature of the microwave background is measured directly and the neutrinos follow from it by a calculation. The matter density is the unknown. Having more matter brings equality earlier, when the horizon was smaller, and a smaller horizon at equality moves the turnover to larger wavenumber. The position of the bend therefore weighs the matter, and the dependence is simple: $k_{\rm eq}\propto\Omega_m h^2$.

The observation behind this claim is the shape of galaxy clustering on very large scales, and a good deal of the argument for a low matter density in the 1990s came from it. Surveys measured more power on large scales than a universe at critical matter density should have had. In a universe with $\Omega_m=1$ equality comes late, the horizon at equality is smaller than measured clustering required, and the shape comes out wrong. This was among the first evidence that the matter density was well below critical, before the supernova results, and it did not rely on any measurement of distance.

It is an awkward measurement in practice, because the very largest scales are where a survey has the fewest independent modes, and the transition to homogeneity is where the signal is weakest and the systematic errors of calibration are largest. Galaxy surveys have measured the region of the turnover and only in recent years reached far enough beyond it to see the rise. The same shape appears in a different way in the suppression that a massive neutrino causes on small scales, which is a second bend written on top of the first, by a component that stayed fast for longer.

Two kinds of stall #

A clear way of seeing what is distinctive here is to compare it with the two other ways of slowing growth in cosmology, because they are easily conflated and they behave differently.

The baryons before recombination are stopped by pressure. They oscillate at the sound speed, they keep their amplitude up to a slow damping, and they retain a phase, which is why the microwave background has a pattern of peaks. That is a force stall. Its signature is a scale: the sound horizon, a preferred length that survives in the spectrum as a ripple.

Dark matter in the radiation era is stopped by friction, and it has no oscillation or phase. A mode that waited looks the same as one that did not, only smaller. That is why the small-scale spectrum is smooth, and it is why the turnover is broad. A wait prints a gradual bend, where a pressure prints a sequence of peaks.

The third is what dark energy does to late growth. As the expansion starts accelerating, the same friction term comes back and the same stall returns: growth slows and then stops altogether, and in a universe dominated by a cosmological constant all the structure that will ever bind has already bound. Mathematically this is the same effect as Mészáros’s. The term in the equation that does the work is the friction term, and the only difference is that what sets $H$ is no longer radiation but a density that does not dilute.

That is the surprising connection in the argument. The epoch before the first atoms, and the epoch that started about five billion years ago, are the same equation with two different things in the expansion rate. Structure formation is a window between two stalls, and the window lasts from redshift 3,400 to about redshift one.

Why small haloes and large ones are nearly the same age #

The most striking consequence of the bend lies in what it does to the formation of objects. A halo of mass $M$ forms from a patch of comoving size $R\propto M^{1/3}$. Whether that patch collapses depends on $\sigma(M)$, the variance of the density in spheres of that size, which is an integral of the spectrum.

Without the wait, the spectrum would keep rising with wavenumber and $\sigma$ would rise steeply toward small masses: the smallest objects would collapse first and enormously earlier than the larger ones. With the wait the small-scale spectrum falls nearly as $k^{-3}$, which is the slope at which every decade of wavenumber contributes the same to the variance. So $\sigma$ grows only as a logarithm toward small masses.

So three decades of mass below a billion solar masses gain less than a factor of two in variance. A typical halo collapses when the growth factor has reached about $1.686/\sigma$, and the growth factor is proportional to the scale factor during matter domination, so its collapse redshift is roughly proportional to $\sigma$. A million-solar-mass halo forms at only about twice the redshift of a billion-solar-mass halo.

This is why the universe seems to have assembled its small structures in a narrow range of time, why the transition from one generation of objects to the next is so rapid, and why the same picture has to be used for the dwarf haloes of a void and the much later clusters. The near-flatness has practical consequences: it makes the abundance of small objects sensitive to a threshold, since many masses are collapsing at nearly the same epoch. And it means that a modest change in the spectrum on small scales would have a large effect on when galaxies began.

How much the argument assumes #

The claim is exact only in a clean situation, and several things the figures draw could mislead.

The radiation is taken as smooth. Inside the horizon the photons oscillate and the neutrinos free-stream, so to good approximation neither has lasting clumps. That is what allows the equation to treat them as a uniform background, and it is accurate on scales well inside the horizon. It is not accurate at entry. The gravitational potential of the photon fluid decays at that moment, and the decay gives the dark matter a velocity kick. The kick is where the logarithm comes from, and the exact solution for a clump after entry is a logarithm that matches onto the growing solution.

Baryons are not in the picture. In the real universe about a sixth of the matter is in baryons, which do not fall in until after recombination. Dark matter alone is then the source of the clumps, so the growth just after equality is a little slower than $1+3y/2$ says, and the spectrum is smaller than the baryon-free fit on small scales by up to tens of per cent.

The bend is not a sharp feature. The turnover in the figures is broad, spanning more than a decade in wavenumber, because equality is not a moment but a gradual crossing. Reading a single scale off it, as the text has done for convenience, is a shorthand. The fit to data uses the whole shape, not one point.

Nothing here can show the clump’s actual history. Every curve is one solution of a linear equation. There are no objects in it, only a small quantity that is multiplied. Nonlinear collapse, the halo’s internal structure and everything that makes a real galaxy lie outside what a linear growth factor can say.

A test the universe has already performed #

There is a neat consistency check on all this, in what is measured at the two ends of the growth.

The microwave background gives the amplitude of the fluctuations at recombination, and the shape of its power spectrum reads the baryon and matter densities independently. The present-day clustering of galaxies, and the lensing of light by matter, give the amplitude today. Growth between the two is fixed by the wait, the proportional growth, and the late stall. There is no free parameter to adjust once the densities are known.

If dark matter had no such stall — if, for example, the radiation era had ended much earlier than the microwave temperature implies — the amplitude today would be enormously larger than measured, and clusters of galaxies would be far more common. The comparison works to within about ten per cent, and the remaining discrepancy, sometimes reported as a slight tension in the amplitude of clustering, is an active question about the small end of the same calculation rather than about the wait itself.

Run the same reasoning backward and the argument for dark matter becomes sharper than it sounds. Baryons alone, released from the photons at recombination with fluctuations of a few parts in $10^5$, could grow only by a factor of about 1,100 by the present, not enough to reach unity. Something must already have been clumped at recombination, and dark matter was able to do that only because, though it had been stalled until equality, it had been growing since. The window between equality and recombination, a factor of about three in scale, is small. But it is the head start on which galaxies exist.

A wait at the edge of the observable #

The bend also limits what can be read from the largest scales. Modes well below the turnover have kept their primordial form, and they are the ones that would test whether the primordial fluctuations were really Gaussian, or whether their spectrum has a feature at very long wavelengths. These modes entered the horizon late and are few. Their number is limited by the size of the observable volume and no improvement in instruments can raise it.

Modes well above the turnover have forgotten something different. Their amplitude is dominated by the factor they lost, which depends on $k$ almost entirely through the wait, so what comes through is mostly the expansion history at the time of entry. Any structure in the primordial spectrum at those wavenumbers has been scaled by the same function, and if the scaling is known it can be undone. But the transfer function has to be known to the same accuracy as the feature being searched for, and for very high wavenumbers the nonlinear collapse of structure has already mixed the modes together.

So the region of the spectrum that is both clean and linear is a limited range around and above the bend, and the wait helps to set that range too. The surveys now under way are trying to extend that range using tracers at high redshift, where the small scales have had less time to become nonlinear, and the turnover itself should be clearly resolved by the largest of them.

Still open: what the first small haloes looked like #

The argument gives the shape of the spectrum down to scales far below anything observed. It predicts dark-matter clumps of about an Earth mass if the particle is a typical weakly interacting one, with the cut-off set by how the particle stopped interacting and began to stream. Between the smallest measured scales, about a few million solar masses from the gaps in stellar streams and the distortions of strong lenses, and that predicted cut-off, the spectrum is entirely an extrapolation of the wait, over about twelve decades of mass.

That is a real exposure. Any new physics that made a difference in the radiation era, an early period of matter domination for instance, or a dark matter that kept interacting with radiation for longer, would change the small-scale slope in that range and nothing above it. Since haloes of those sizes hold no stars, the direct tests have to come from their gravity alone. Whether the wait really continues all the way down is a question that no galaxy can answer, because the waves it concerns are shorter than any galaxy. The bend is measured. The long straight run below it, on which most haloes in the universe depend, has yet to be checked.