Field

Cosmology

One object, seen once, from inside — and every number in it the output of a model.

The shift that is not a Doppler shift

Every galaxy beyond the Local Group has its lines shifted to the red, by an amount proportional to its distance. Read as a velocity that looks like a confession that everything is fleeing from here; read as a change of scale it says the opposite, because a uniform expansion produces exactly the same law measured from any galaxy in it.

Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround.

One number that is an age, a size and a density

The slope of the velocity–distance relation has units of inverse time, so it can be read as an age, multiplied by c to give a length, or squared to give a density. All three readings are natural, all three are quoted, and not one of them is the quantity it appears to be.

The same constant, measured twice, five sigma apart

The distance ladder gives an expansion rate of about 73 kilometres per second per megaparsec. The microwave background gives 67.4. Both quote errors near one per cent, both have been rebuilt from scratch by rival teams, and the gap between them has grown as the measurements have improved.

Four distances to the same galaxy

Inside the Local Group the word "distance" has one meaning. Past a redshift of about a tenth it has four, they disagree by factors of a hundred by the time the light is old, and one of them stops increasing and starts coming back.

Distance modulus against redshift, for three universes. The distance modulus μ = 5 log₁₀(D_L/10 pc) against redshift for three universes, all with H₀ = 67.36 km/s/Mpc, with 60 model supernovae drawn from the ΛCDM curve with 0.15 magnitudes of scatter. The point of the figure is how little difference there is: across two decades of redshift the three curves stay within a few tenths of a magnitude, and at z = 0.5 the accelerating and decelerating cases differ by 0.387 mag. A cosmology is not read off this plot. It is read off the residual, which is the next figure.

An expansion that was supposed to be slowing

Gravity is attractive, so an expanding universe full of matter must be decelerating, and the only question was by how much. Two teams set out to measure the deceleration and both found a quarter of a magnitude of extra faintness at redshift half — which is the wrong sign.

The most perfect blackbody ever measured

The whole sky glows at 2.7255 kelvin, and its spectrum matches the Planck curve to fifty parts in a million. Nothing else in astronomy is thermal to a part in twenty thousand, and a spectrum that exact is not a description of the radiation — it is a constraint on the history of everything that could have disturbed it.

The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has.

A standing wave frozen at one instant

The temperature of the microwave background varies across the sky by one part in a hundred thousand, and the sizes of the patches are not random. There is a preferred angular scale near one degree, and it is a sound wave that stopped ringing four hundred thousand years after the beginning.

$Four abundances, one free parameter. The abundances big-bang nucleosynthesis predicts, against the one number it is free to choose: η₁₀, the ratio of baryons to photons in units of 10⁻¹⁰. Four curves spanning nine decades, from a helium mass fraction of about a quarter down to a lithium abundance of one atom in ten billion, and they are not four independent predictions — they all come out of the same reaction network run at the same density. The horizontal bands are what is measured in the sky, each at its published one sigma. The measurement that matters is deuterium, because its curve is the steep one: inverting the drawn curve at D/H = 2.527e-5 gives η₁₀ = 6.11, and the ends of the observed interval give 6.06 to 6.15. The vertical band is what the microwave background gives, 6.13 ± 0.04, from the height of the second acoustic peak relative to the first. Those two agree to 0.4 per cent, and they have nothing whatever in common: one is a nuclear-reaction network run in the first three minutes and read off a quasar absorption line, the other is a fluid oscillation at four hundred thousand years read off a sky map. Lithium is the exception and it is not a small one — the network predicts 4.70e-10 at the microwave background's density and the halo stars show 1.60e-10, a factor of 2.9 too much, which is unresolved.$

Four abundances and one free parameter

In the first three minutes the universe ran a nuclear reaction network with exactly one adjustable number in it. That number predicts four abundances spanning nine orders of magnitude, three of them are observed and match, and the fourth is wrong by a factor of three and has been for twenty-five years.

$The same universe, four times, as a fraction of itself. Each bar is the fractional contribution of the four components to the total density at one epoch, computed from the Planck 2018 parameters by scaling each component from today: radiation as (1+z)⁴, both kinds of matter as (1+z)³, and Λ as a constant. The familiar figure — five per cent baryons, twenty-six dark matter, sixty-nine dark energy — is the top bar and only the top bar. At recombination the same universe is three-quarters dark matter and Λ is one part in ten million; before matter–radiation equality it is mostly radiation. A pie chart of the contents of the universe is therefore a statement about a moment, and the moment is the one it happens to be drawn in.$

A budget whose familiar part is five per cent

Five per cent ordinary matter, twenty-six per cent dark matter, sixty-nine per cent dark energy. The figures are quoted everywhere and each one comes from a different measurement, the denominator they are fractions of is itself built out of the expansion rate, and the whole chart is a statement about one instant that was a different chart at every earlier time.

A bump at a hundred megaparsecs. The two-point correlation function of galaxies, multiplied by the square of the separation so that the interesting part is not buried under the power law. The smooth dashed curve is the broad-band clustering — the ordinary fact that galaxies are near other galaxies, which carries no cosmological information and is treated as a nuisance term in the real analysis. The bump on top of it is the whole measurement, and its position here is not fitted: it is the comoving sound horizon at the drag epoch, integrated in this file from ∫c_s da/a²H with c_s = c/√(3(1+R)), which comes out at 146.9 Mpc — 99.0 in the h⁻¹ Mpc a survey works in. The excess says that a galaxy is very slightly more likely to have a companion at that separation than at 90 or 115, by about one part in two hundred, and the reason is that a pressure wave in the photon–baryon fluid ran outward from every overdensity for four hundred thousand years and stopped where it was when the photons let go. The points are drawn with the error a survey of a million galaxies achieves; a single pair of galaxies at 100 Mpc means nothing, which is why the measurement waited for the surveys.

The same ruler measured twice, ten billion years apart

The sound wave that produced the acoustic peaks in the microwave background also left a faint excess in how galaxies are spaced, at a separation of about a hundred megaparsecs. It is the only cosmological distance indicator whose length is set by physics rather than by a chain of calibrations.

Every shell contributes the same. Concentric shells of equal thickness around an observer, with stars scattered uniformly in volume. The number of stars in a shell grows as its radius squared and the flux from each falls as the radius squared, so the two cancel exactly and every shell delivers the same total light — the drawn counts are 6, 15, 31, 55, 88, in proportion to r³ − r₀³, and the drawn sizes fall as 1/r. That cancellation is the paradox, and it is why no amount of dust helps: dust absorbs the light and then re-radiates it, and in a universe old enough for the sum to converge it would come to the same temperature as the stars. The sum diverges linearly with radius, so something has to stop it — and the only two candidates are that the shells eventually overlap, or that there are no shells beyond a certain distance because there has not been time for their light to arrive.

Why the sky is dark

In an infinite universe of stars every sight line ends on a stellar surface, so the whole sky should be as bright as the Sun's disc. It is not, and the usual answer — that the expansion redshifts the light away — accounts for a factor of six out of a hundred trillion.

Three horizons, and a light cone that bulges. Cosmic time upwards, comoving distance sideways, for the Planck 2018 cosmology. Galaxies sit still in these coordinates, so their worldlines are the vertical grey lines: comoving distance is defined to take the expansion out. The solid inner curve is the past light cone — the set of events whose light reaches here and now — and it reaches out to the particle horizon at 46.1 billion light years, which is the diagram's central number and the one that sounds impossible. The universe is 13.80 billion years old and light has travelled for 13.80 billion years, and yet the material that emitted the oldest light is now 46 billion light years away. Nothing has outrun light: the comoving distance covered is ∫c dt/a, and dividing by a scale factor that was small early on makes the integral three times ct. The dashed curve is the Hubble sphere, where the recession speed equals c, at 14.5 Gly today — and the light cone lies outside it for most of its length, which is exactly why a galaxy can be observed while receding faster than light. The outer dot-dash curve is the event horizon, at 16.7 Gly: a signal sent from here today never reaches anything beyond it.

A horizon three times larger than the age allows

The universe is 13.8 billion years old and light travels one light year a year, so the observable universe should be 13.8 billion light years across. It is 46.1, nothing has outrun light, and the discrepancy is a piece of arithmetic rather than a paradox.

One length, leaving and returning. The comoving Hubble radius c/aH against the scale factor, both logarithmic. Everything to the right of the kink is exact and is the same cosmology as every other figure here: after inflation the comoving Hubble radius grows, as a in the radiation era and as √a in the matter era, then turns over once Λ takes hold. Everything to the left is a schematic — the energy scale of inflation is unmeasured, so neither the height of the plateau nor the 62 e-folds drawn is a number anybody has — but the shape is not negotiable: in any accelerated expansion H is nearly constant, so c/aH falls as 1/a. The consequence is the mechanism. The solid horizontal line is a fixed comoving length of 209 Mpc: it starts inside the Hubble radius, is carried outside during inflation, sits frozen while nothing can act across it, and re-enters at z = 1090, which is last scattering — it is the scale the first acoustic peak is made of. The dashed line above it is the comoving size of the whole observable universe, 14148 Mpc, and the figure's quiet second finding is that it is still outside the Hubble radius: the very largest angular scales in the microwave background have never re-entered, which is why they show the primordial spectrum with no acoustic processing on it at all. One crossing does two jobs: it makes the sky uniform, because everything now visible was once in causal contact, and it makes it not quite uniform, because a quantum fluctuation stretched beyond the horizon has nothing left that can smooth it out.

Two coincidences with one mechanism

Opposite sides of the microwave sky were never in causal contact and have the same temperature to one part in a hundred thousand. The total density sits on the knife edge of flatness, which is an unstable equilibrium. Two fine-tunings of quite different kinds, and one epoch of accelerated expansion removes both.

A slice 1,000 megaparsecs across. 872 galaxies in a wedge 1000 comoving megaparsecs deep, generated as a Thomas process — Poisson centres at 0.00012 per square megaparsec, 13 galaxies each on average, scattered about their centre with a Gaussian of 22 Mpc. That is not a simulation of how structure formed and does not pretend to be; it is the simplest clustered process whose correlation function is known exactly, which is what lets the next figure check a measurement against something rather than against itself. What it does reproduce is the impression: at this scale the distribution is obviously not uniform, there are groups and there are gaps, and the eye finds patterns in it readily — including several that are not there, because the eye finds patterns in a Poisson field too. The question the next two figures ask is the only one that settles it: at what separation does the clustering stop, and is there a scale above which a box of this universe looks like any other box?

Homogeneous above a hundred megaparsecs

Every cosmological calculation in this field assumes the universe is the same everywhere. Looked at on any scale a person can picture, it plainly is not — it is stars in galaxies in groups in clusters in filaments around voids. The assumption is not a claim about appearance; it is a claim about a statistic, and the statistic has a scale attached.

$One ionising photon per atom, and where that happens. The number of photons energetic enough to ionise hydrogen, per baryon, against temperature — with temperature falling to the right, so the universe ages to the right. The quantity is the fraction of a Planck distribution above 13.6 eV, integrated numerically, divided by the 6.13·10⁻¹⁰ baryons there are per photon. It crosses one at 5,844 K, which is z = 2143, and it falls by 19 orders of magnitude across the plot because it is the tail of an exponential. That crossing is the answer to why recombination waits until three thousand kelvin. At hydrogen's own ionisation temperature of 157,803 K there are two billion ionising photons per atom and the gas has no chance; the temperature has to fall by a factor of forty before the supply runs out, and it runs out suddenly because an exponential tail does. The number that sets the factor of forty is not an energy at all — it is the photon-to-baryon ratio, which is to say the entropy per baryon, which is to say a number fixed long before any of this and measured today from the second acoustic peak and from deuterium alike.$

The surface the background actually is

Hydrogen ionises at 157,800 kelvin and the universe became transparent at 3,000. The factor of fifty between them is not an error, and it is not about energy — it is about there being two billion photons for every atom, so the far tail of the distribution can keep the gas ionised long after the typical photon has become useless.

$A neutral fraction of 2.4·10⁻⁶ is already opaque. The Gunn–Peterson optical depth against the neutral fraction of the intergalactic medium, at z = 3, 5, 6.3, for Ω_b = 0.0493 and h = 0.674. Note the range of the vertical axis. A fully neutral medium at z = 6.3 gives τ = 4.1·10⁵, which is not absorption but extinction of everything; the medium reaches τ = 1 — the point at which it stops transmitting most of the light — at neutral fractions of 6.1·10⁻⁶ at z = 3, 3.3·10⁻⁶ at z = 5, 2.4·10⁻⁶ at z = 6.3. That is why the argument runs from the flux that survives rather than from the flux that does not. A spectrum showing any transmission at all between Lyman α and Lyman β is a measurement that the medium is ionised to better than one part in 164,727, and no fit to any absorption line is needed to establish it.$

A trough that proves the forest survived

A uniform neutral medium at redshift six would absorb Lyman alpha with an optical depth of four hundred thousand. So the existence of any transmitted light in a quasar's spectrum is a measurement — of a neutral fraction below one part in ten thousand — made from a detection rather than from an absorption.

Beyond redshift 1.87, what a galaxy does today will never be seen. For a galaxy at each redshift, the cosmic time of the last event on it that will ever be visible from here — not the last that has arrived, but the last that ever arrives, integrated to infinite future time. The horizontal line is the present, 13.8 billion years. A galaxy below redshift 1.87 has its curve above that line: its entire future will be seen from here, arriving ever more slowly and ever more redshifted, so it never quite disappears. A galaxy above redshift 1.87 has its curve below the line, and that is the whole content of an event horizon: what such a galaxy is doing today will never be seen, ever, by anybody here. Only a finite slice of its history is coming, and when the last of that light arrives the object stops changing. The redshift at which the curve crosses is 1.87, the comoving distance there is 16.7 billion light years against a particle horizon of 46.1, and the ratio of the volumes says that 95 per cent of the galaxies now observable are already beyond reach. Superluminal recession is not what does this. Everything past about redshift 1.5 has always been receding faster than light and is seen perfectly well, because the Hubble sphere grows to meet the photon; what closes the horizon is that with a cosmological constant the comoving Hubble radius stops growing and begins to shrink, so a photon that has not already been overtaken never will be.

The galaxies that are already out of reach

With a cosmological constant the comoving distance a photon can ever cover converges, so there is a redshift beyond which light leaving today never arrives. It is 1.87, and about ninety-five per cent of the galaxies now visible are past it — which superluminal recession has nothing to do with.

w = −0.9 is 34 millimagnitudes, and one supernova scatters by 120. Above: how much the distance modulus moves when the dark energy is not a constant. Each curve is a universe with the same Ωₘ = 0.315 and a different equation of state w, drawn as a difference from w = −1 in magnitudes. At redshift a half, w = −0.9 is worth 34 millimagnitudes — the shaded band is the 0.12-magnitude intrinsic scatter of a single standardised type Ia supernova, and the signal is a fifth of it. Nothing about one object can see this; the measurement is the mean of 1500, whose error on the mean is 3.1 millimagnitudes, and even that only works because the shape of the curve in redshift is different from every systematic anybody has thought of. Below: why the supernovae are not enough on their own. Each locus is the set of (Ωₘ, w) that a measurement cannot tell apart from the fiducial model — computed, not sketched: the supernova curve is the ridge of the same sum of squares a fit would minimise over 0.02–1 in redshift, and the acoustic-scale curve is the exact set of models with the same comoving distance to last scattering, which is what fixes the angle the microwave background's first peak subtends. They cross at 36 degrees. Neither is a measurement of w and the pair is, which is why the constraint on the equation of state is a picture of two loci crossing rather than a number read off a curve — and why −1.03 ± 0.03 is a statement about how well they cross rather than about how well anything was measured.

The number that would say whether it is a constant

Whether dark energy is a cosmological constant is the question of whether w is exactly −1, and w = −0.9 changes a distance modulus by thirty-four millimagnitudes at redshift a half — a fifth of the scatter of a single supernova, along a degeneracy only the acoustic scale can cut across.

A sphere reconstructed as a spheroid, and two distortions 0.15 apart. Left: the acoustic scale in the plane of separation across the line of sight against separation along it, one quadrant of it. In the cosmology that actually holds, the sound horizon is a sphere of 99.0 h⁻¹ Mpc and its locus here is a quarter circle. That is the whole content of the Alcock–Paczyński test: nothing about the early universe distinguishes the radial direction from the transverse one, so any departure from a circle is a statement about the observer's arithmetic rather than about the ruler. Converting angles into transverse separations needs the transverse comoving distance and converting redshift intervals into radial ones needs H(z), so assuming distances 1.1 times too large and rates 0.94 times too small returns an ellipse of axis ratio 0.855 — and the ellipticity measures that distance times the expansion rate over c, in which the sound horizon has cancelled. A ruler of unknown length still measures a shape. The third curve is the difficulty: peculiar velocities also distort the same correlation function along the same axis, squashing it by 1/(1+β) = 0.704 for β = 0.42, and a squashing is a squashing. Separating a geometric distortion from a dynamical one is the entire art of the measurement, and it is done by using the fact that they have different dependences on scale — the velocities act on the broad-band shape and the ruler is a feature. Right and below: the two numbers the same feature gives at each redshift. Across the line of sight, the transverse distance divided by the sound horizon; along it, c divided by the expansion rate times the sound horizon. Two functions of the expansion history, from one bump in one correlation function, and their agreement with a single model is one of the sharper consistency tests in the subject.

A ruler measured along and across

The sound horizon is a sphere, and a sphere in a redshift survey is measured twice over — across the line of sight it gives an angle, along it a redshift interval. Two different functions of the cosmology out of one feature, and their ratio is a measurement with no ruler in it at all.

Fingers 3.1 times long, a large scale squashed to 0.92, and a test of gravity. Left: a 360-megaparsec slice of a clustered universe, as the galaxies actually sit. Right: the same galaxies as a redshift survey records them, with the line of sight up the page. Nothing has moved sideways, because an angle is an angle; every displacement is along the line of sight, because that coordinate came from a redshift and a redshift is the expansion plus whatever the galaxy is doing on its own. Two effects, opposite in sign and separated by scale. Inside a cluster the motions are virial and random, 720 kilometres a second of them, which at H₀ = 67.36 is 11 megaparsecs of smearing on an object a few across: the clusters become fingers 3.1 times longer than they are wide, all pointing at the observer, which is the one structure in cosmology that is definitely not real. On the scale of a supercluster the motions are coherent — everything is falling in, so the far side is approaching and the near side receding — and the structure is compressed rather than stretched, to 0.92 of its true extent here, measured on cluster centroids so the fingers have already averaged away. Below: why that compression is worth having. Its amplitude is the rate at which structure is currently growing, and the growth rate is Ωₘ(z) raised to a power that general relativity fixes at about 0.55. A theory of gravity that differs from general relativity on cosmological scales while matching every solar-system test changes that exponent and nothing else, and the two curves drawn — γ = 0.55 and γ = 0.68 — differ by only 4 per cent at redshift a half, against error bars of 12 per cent on the points beside them. The worst systematic in a redshift survey is the measurement — and it is a hard one, because a quarter of a change in the exponent that governs how gravity assembles structure moves the observable by less than the width of the curve it is drawn on. What the picture cannot show is the degeneracy that limits it: what is measured is fσ₈, a product, and separating the growth rate from the amplitude of clustering needs something else entirely.

A map that is not of positions

One axis of every redshift survey is not a distance but a velocity, and the difference is not noise. Inside a cluster it smears the galaxies into a finger pointing at the observer; on the scale of a supercluster it compresses the structure — and the amount of that compression is a test of gravity.

$One cluster moved to z = 1.8: one signal unchanged, the other 57 times fainter. The same cluster of galaxies placed at a series of redshifts, with two ways of detecting it compared. The Sunyaev–Zel'dovich decrement is flat, because it is a fraction of the microwave background and the microwave background has the same surface brightness at every redshift a cluster can sit at — moving the cluster further away shrinks it on the sky but does not make its shadow shallower. The X-ray surface brightness of the identical object falls as (1+z)⁻⁴, the dimming every surface brightness suffers in an expanding universe, and by z = 1.8 it is 57 times below where it started. This is why the cluster surveys that reach the early universe are millimetre surveys: an X-ray telescope's cluster catalogue thins out with distance and a Sunyaev–Zel'dovich catalogue is limited only by how large the cluster looks and by how faint a fractional distortion the instrument can measure. What the effect cannot supply on its own is a distance — the signal that does not know how far away the cluster is also cannot say — so every one of these clusters still needs a redshift measured the ordinary way, from a spectrum of a galaxy inside it.$

A shadow that does not get fainter with distance

Every other way of finding a cluster of galaxies gets harder the further away the cluster is. One does not. A cluster's hot gas scatters about one microwave background photon in a hundred to a higher frequency, and because the result is a fraction of a background rather than a flux from a source, the same cluster is exactly as detectable at ten billion light years as at one.

A map stretched by the thing it measures

A redshift survey plots galaxies at distances derived from their redshifts, and a galaxy's redshift contains its own motion as well as the expansion. So the map is systematically distorted — squashed on large scales, drawn out into radial spikes on small ones — and both distortions are caused by the gravity the survey exists to measure.

$Bubbles that meet at z = 5.3, and a scattering depth of 0.047. The fraction of the volume of the universe filled by ionised bubbles, integrated from redshift 20 down to 4.5. The equation has two terms and no others: photons escaping from young galaxies open new volume, at a rate taken from the measured cosmic star formation history with an escape fraction of 0.2; recombinations inside the bubbles close it again, on a timescale that is one over the density times the recombination coefficient times a clumping factor of 3. Early on the density is high and recombination wins almost everything; the curve is nearly flat. As the universe expands the recombination time lengthens as the cube of one plus the redshift while the star formation rate is still rising, the balance tips, and the filling factor runs to one in under half a billion years. It reaches unity at redshift 5.31, which is overlap — the moment the bubbles meet and the last neutral walls between them disappear. The same integration gives an electron-scattering optical depth of 0.0465 for the microwave background, against the 0.054 that is measured, and that agreement is the check: the two observations constrain the same history from opposite ends, one fixing when it finished and the other how long it took.$

It ends when the walls meet

Reionisation is usually described as the universe becoming transparent, which makes it sound like a change of state. It is not. Each young galaxy opens a bubble of ionised gas around itself and recombination closes it again, and what ends the epoch is geometric — the bubbles meet, and the last neutral walls between them disappear.

$The damping tail measures a thickness: ℓ_D = 1400 for a shell 80 deep in redshift. The suppression of small-scale structure in the microwave background, against multipole, on logarithmic axes. Every other feature of the power spectrum measures the last-scattering surface as a surface — its distance, the sound horizon written on it, the ruler it provides. This one measures how thick it is. Recombination takes time: the ionised fraction falls over a range of redshift rather than all at once, and while it is falling the photons are still scattering, so each one random-walks. A photon taking N steps of a mean free path λ diffuses √N λ, which is much further than a single step and much less than the whole interval, and any temperature fluctuation smaller than that distance is mixed away before it can be frozen in. What is left is a Gaussian cut-off, drawn here for three shell thicknesses. A thicker shell means more steps and a longer walk, so it damps at a smaller multipole: Δz = 40 gives ℓ_D = 1980, Δz = 80 gives ℓ_D = 1400, Δz = 160 gives ℓ_D = 990. The observed cut-off is near ℓ = 1400, corresponding to a diffusion length of about 0.03 proper megaparsecs at the time — a scale the reader should compare with the sound horizon, some hundred and fifty comoving megaparsecs, which is what the peaks measure. The tail is therefore a genuine probe of the inside of the transition rather than of its position, and because the damping depends on the free-electron density it is also one of the cleanest constraints on anything that changes it: extra relativistic species, a varying fine-structure constant, or an early energy injection all move ℓ_D while leaving the peak positions nearly alone. The normalisation here is set once by the observed value, so what the figure asserts is the scaling with thickness and the shape of the cut-off, not the absolute number.$

A blur that measures a depth

Every other feature of the microwave background measures the last-scattering surface as a surface — its distance, the ruler written on it, the geometry between. The damping of its small-scale structure measures how thick it is, because a photon random-walking through a finite transition smears away anything smaller than its walk.

A torque that stops when the patch lets go. Left, the mechanism: a protogalactic patch drawn as an ellipsoid of axis ratios 1:0.72:0.5, with the principal axes of the surrounding tidal field drawn across it at 30 degrees to its own. The torque is proportional to the difference of the patch's principal moments times the sine of twice that angle, so it is exactly zero when the two sets of axes agree — checked at both alignments — and largest at forty-five degrees. A spherical patch takes no torque whatever the field around it does, which is why the spin of every galaxy begins as a statement about its shape. Right, the angular momentum against time in units of the turnaround time: in linear theory the torque acts on a patch still expanding with the universe and the angular momentum grows as the first power of time, measured off the drawn curve as t^1.000. At turnaround the patch detaches from the expansion, its quadrupole shrinks, and the torque switches off — so a galaxy's spin is fixed before it has collapsed at all, by neighbours it will never interact with again. What the figure cannot show is the sign: the same mechanism gives no preferred direction, and the observed near-absence of alignment between neighbouring galaxies' spins is the check on that.

Spin acquired before there was anything to spin

Every galaxy turns, and nothing in a smooth expanding universe turns. The rotation was applied while the material was still a mildly overdense patch spread across megaparsecs — torqued by the tidal field of its neighbours, growing steadily with time, and switching off the moment the patch stopped expanding.

Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived.

The seed that cannot be remembered

A galactic dynamo affords something like thirty e-foldings over the age of a galaxy, which multiplies any seed field above a ten-thousandth of a billionth of a microgauss up to the microgauss actually observed. The field a galaxy has today therefore says nothing about the field it started with, and the only place a seed survives unamplified is the emptiness between clusters.

$A dipole 187 times the signal, and its own harmonics under it. The amplitude of each harmonic of the observer's own motion imprinted on the microwave sky, against the anisotropies of the sky itself. Moving at 369.8 kilometres a second through a blackbody field makes it hotter ahead and cooler behind by a fraction β = v/c, giving a dipole of 3.36 millikelvin — 187 times the 18 microkelvin anisotropies. Each further harmonic is smaller by another factor of β, so the kinematic quadrupole is 4.15 microkelvin, which is comparable to the real quadrupole and has to be subtracted separately. The dipole is not a nuisance in one respect: it is the measurement of the solar system's motion with respect to the radiation, and it is the most precisely known velocity in astronomy.$

A dipole a hundred times the signal

The largest structure in the microwave sky is the observer. Moving through a blackbody radiation field makes it hotter ahead and cooler behind by three and a third millikelvin — nearly two hundred times the anisotropies that all of cosmology is read from — and removing it is the first operation on any map.

$A 5 per cent continuum error, and an optical depth wrong by 1.1. The mean transmitted flux of the Lyman-alpha forest against redshift, with a 5 per cent uncertainty in the quasar continuum drawn as a band. The continuum is not observed: at these redshifts every part of the spectrum blueward of the emission line is absorbed, so the level has to be extrapolated from the red side across a region where the quasar's own spectrum has structure. A fractional error in that level is a fractional error in the flux, and since the optical depth is minus the logarithm of the flux, the resulting error in the optical depth is the fractional error divided by the flux — which grows without bound as the forest goes black. At z = 2 it is 0.06; at z = 6.2 it is 1.1. That is why measurements of when reionisation ended are quoted as limits rather than values above about redshift six.$

A forest with no continuum left

Measuring how much neutral hydrogen sits between here and a distant quasar means measuring the fraction of its light that survives, which means knowing how much light there was. At high redshift nothing survives at the wavelengths that would show it, so the level is extrapolated across the region being measured — and the optical depth is the logarithm of a number divided by a guess.

One curve the temperature fixes, and one line the polarisation adds. The plane of the optical depth to reionisation against the amplitude of the primordial fluctuations. The temperature power spectrum of the microwave background measures the product of the amplitude and the exponential of minus twice the optical depth, so it constrains a curve rather than a point: more electrons scattering the photons out is indistinguishable from fewer fluctuations to begin with, and the two trade along the drawn locus across a factor of 1.22 in amplitude. What breaks it is the polarisation at the largest angular scales, where rescattered photons regenerate a signal whose amplitude is proportional to the optical depth itself rather than to its exponential. That constraint is nearly vertical here, it comes from a handful of multipoles at the very largest scales, and it is the single hardest measurement the microwave background has demanded — because at those scales the Galaxy's own polarised emission is larger than the signal.

An amplitude and a depth that arrive multiplied

The microwave background's temperature fluctuations are the primordial ones damped by everything that scattered them since. The damping is uniform, so a smaller starting amplitude and more scattering produce identical maps — and separating them requires a signal from a handful of the largest angular scales, where the Galaxy's own emission is larger than what is being measured.

Take 8 per cent off the horizon and the tension is gone. The Hubble constant against the sound horizon at recombination, along the locus the microwave background's measured angular scale fixes. What is measured is an angle — the angular size of the horizon, to a part in three thousand — and an angle is a length divided by a distance, so extracting an expansion rate requires the length. That length is computed from the physics of the first four hundred thousand years: the baryon density, the radiation density, the number of relativistic species and the recombination history. Change any of those and the locus is unchanged while the point on it moves. The horizontal band is the late-universe measurement from the distance ladder, and it meets the locus at 136 megaparsecs — 8 per cent shorter than the standard model gives. That is the arithmetic behind every proposal to resolve the disagreement by changing the early universe rather than the late one.

A constant that is an angle divided by a length

The microwave background does not measure an expansion rate. It measures one angle — the apparent size of the sound horizon at recombination — to a part in three thousand, and converting that angle into a rate requires the horizon's physical length, which is computed from a model of the first four hundred thousand years rather than observed.

$Residuals of 112 per cent nearby and 2.2 far out. Deviations from a pure Hubble flow, in per cent, against distance. Each galaxy carries a peculiar velocity of a few hundred kilometres a second — part a coherent bulk flow shared with its neighbours and part a random dispersion — and that velocity is added to its recession. Since the recession grows with distance and the peculiar velocity does not, the fractional error falls as one over the distance: it is 112 per cent at 5 megaparsecs and 2.2 at 250. The practical consequence is a lower cut-off on any Hubble-constant measurement: below about 40 megaparsecs the motions dominate, and the coherent part does not average away over a sample because neighbouring galaxies share it. Choosing that cut-off is one of the analysis decisions a local expansion rate depends on.$

A residual that is somebody else's velocity

A redshift is not a distance until the galaxy's own motion has been removed, and galaxies move at a few hundred kilometres a second. Nearby that is comparable to the expansion itself, so the local Hubble diagram's scatter is motions rather than measurement — and the motions are shared between neighbours, so they do not average away.

$The helium abundance as a count of neutrino species. The primordial helium mass fraction against the number of light neutrino species, integrated at a deuterium-bottleneck temperature of 0.0855 MeV and a neutron lifetime of 877.75 s. The curve rises at 0.01348 in Y_p per species near the standard model, and it is not quite a line: freeze-out temperature goes as the sixth root of g_*, so a species added at N_eff = 4.5 buys 25 per cent less helium than one added at 2 — a slope ratio of 0.749 against the 0.769 that scaling requires. The reading below is therefore taken off the drawn curve rather than off a slope. The horizontal band is the measurement — ⁴He (Aver 2015), Y_p = 0.2449 ± 0.004, taken from recombination lines in metal-poor dwarf galaxies and extrapolated to zero metallicity. Where the band crosses the line is the answer: N_eff = 2.84, with the ends of the observed interval giving 2.56 to 3.13. The standard model has three, and 3.046 rather than 3 because the neutrinos are not quite decoupled when the electron–positron pairs annihilate and take a sliver of the heat. The strength of this is not its precision, which is a third of a species and worse than the microwave background's; it is that the two constraints come from utterly different epochs, and that this one is a laboratory result about particle content obtained from an emission line in a galaxy.$

A particle count taken from a dwarf galaxy

Nearly every neutron that survives the first three minutes ends inside a helium nucleus, so the primordial helium abundance is not chemistry — it is the reading of a race between the weak interaction and the expansion. The expansion rate carries the square root of the number of relativistic species, which is why an emission line in a metal-poor galaxy counts neutrinos.

A length in centimetres, measured against an angle

A cluster's hot gas offers two line integrals of the same electrons — one linear in density, one quadratic. Two equations in two unknowns give back the path length in centimetres, and a length divided by the angle it subtends is a distance with no rung of any ladder beneath it.

The residue that failed to burn

Deuterium's abundance is not a measure of what the first three minutes made. It is a measure of what escaped being used — a two-body destruction rate losing a race to a one-body expansion, which is why its curve against the baryon density is steep and helium's is flat.

A factor of three, and the flatness that prices every cure

The oldest stars in the Galaxy show a third of the lithium the first three minutes should have left. Three kinds of resolution have been offered, and the datum that rules on all three is not how much lithium is missing but how uniformly it is missing.

$A bound on the baryon density from an abundance nobody can extrapolate. Primordial deuterium against the baryon density, with the upper bounds that the solar system's own deuterium and helium-3 place on it. The inequality is D_p ≤ D_obs + ³He_obs/g₃, and it holds for any star-formation history whatever: a deuteron entering a star becomes ³He, so the pair can only be moved from one member to the other and then destroyed, never increased. With pre-solar values of D/H = 2.0e-5 and ³He/H = 1.5e-5, the bounds are 3.50e-5 at g₃ = 1, 5.00e-5 at g₃ = 0.5, 8.00e-5 at g₃ = 0.25 — and because deuterium falls with density, each upper bound on the abundance is a lower bound on η: η₁₀ > 4.98, η₁₀ > 3.99, η₁₀ > 2.97. The weakest of them, at a survival fraction of 0.25, still excludes 52 per cent of the density range below the answer. That was the state of the measurement for most of the 1980s, and the shape of it is the thing worth carrying: an abundance too poorly understood to be extrapolated at all still constrained the quantity, because its direction of change under processing was known even though its magnitude was not. The microwave background's 6.13 sits above every bound drawn, which is not a coincidence and is not evidence — a bound that excluded the answer would have been an error, and a bound that admits it is only a bound.$

An abundance with no direction to correct in

Every primordial abundance is measured today and extrapolated backwards, and the extrapolation works because processing moves each species one way. Helium-3 is made by small stars and destroyed by large ones, so the sign of its correction is not merely uncertain — it is unknown.

A lumpy universe makes more of both, and one of them was already too much. The deuterium–lithium plane, with the curve a homogeneous universe traces as its baryon density varies and the points a two-zone universe reaches at a fixed mean density of η₁₀ = 6.13. The dense zone occupies 15 per cent of the volume, and the contrast between the zones runs from 1 — which is the homogeneous case — to 100. Every mixture lies up and to the right of its own homogeneous point, and that is not a modelling choice: an abundance is measured per baryon, so what a telescope averages is η times the abundance — and for both deuterium and lithium that product is a convex function of the density, whose average therefore exceeds its value at the average. The excess is large. At a contrast of 100 the mixture gives D/H = 1.13e-4 against 2.51e-5 smooth, and ⁷Li/H = 1.77e-8 against 4.70e-10, a factor of 37.7. The extra deuterium is exactly what the proposal was for: it lets the mean baryon density be raised while the observed D/H is still matched, which in the 1980s was the one way to make the baryons account for all the matter that dynamics required. The extra lithium is what it costs. The measured abundance was already a factor of 2.9 below the homogeneous prediction, and every step toward the lumpy universe that fixes the density makes that discrepancy worse — which is why the answer to "the baryons are lumpy" turned out to be that the extra matter is not baryons.

The universe that was lumpy at one second

If the baryons were unevenly spread when the network fired, each region ran its own nucleosynthesis and what is observed is an average. For a decade that was the one way to make ordinary matter account for all the matter — and the reason it fails is a theorem about convex curves.

The clock on which light travels in straight lines

Cosmic time makes light cones bulge and horizons curve. There is another time coordinate on which a photon's worldline is a forty-five degree line at every epoch, and on it the horizon problem stops being a piece of arithmetic and becomes a question about whether two triangles overlap.

$Whether an event horizon exists at all, against one number. The comoving event horizon today — the distance a signal sent now will ever cover — against the equation of state of the dark energy, for a flat universe with the measured matter density. The curve runs away at w = −1/3 and does not exist above it: that is where the expansion stops accelerating, and in a universe that does not accelerate the integral ∫da/a²E diverges and every galaxy is eventually reachable, however far away. Below −1/3 the horizon is finite and shrinks as w falls, because a more negative equation of state makes the dark energy density grow with time rather than stay constant. At the cosmological constant's w = −1 the horizon is 16.7 billion light years against a particle horizon of 46.1, so 4.7 per cent of the volume now observable is still reachable. The band is the measured −1.03 ± 0.03. What the figure is for is the asymmetry in what the measurement still allows: two sigma toward zero puts the horizon at 17.5 Gly and two sigma the other way at 14.7, and the shape of the curve means that the closer the true value sits to −1/3 the more violently the answer moves. The reachable fraction is not a robust number in the way the particle horizon is.$

Whether there is a horizon at all

An event horizon exists precisely when the expansion accelerates, and its size is not a smooth function of how much. The integral that defines it runs away as the equation of state approaches minus a third, so two sigma either way on a measured number are two very different futures.

Two horizons, one formula, and a factor of two. Horizon temperature against horizon radius, on logarithmic axes, for the two kinds of horizon this collection has. The upper line is a de Sitter horizon at T = ħc/2πk_BR and the lower is a Schwarzschild horizon at T = ħc/4πk_BR — the same expression with the same constants, differing by exactly two, and both falling as one over the radius so that a bigger horizon is a colder one. The cosmological horizon today has a radius of 4,451 megaparsecs and a temperature of 2.65e-30 K, which is thirty orders of magnitude below the microwave background and will never be measured by anything. A solar-mass black hole sits at 6.2e-8 K, and a black hole as cold as the sky would weigh 2.32e+22 solar masses — of the order of the mass inside the observable universe, which is not a coincidence, since both numbers are c³/GH up to factors of order one. The factor of two between the two lines is the one place the analogy is not exact, and it is not a convention: it comes from the periodicity of the Euclidean time coordinate, which is 8πGM/c³ for a black hole and 2π/H for de Sitter space. The entropies, by contrast, agree exactly.

Two horizons that differ only in who is inside

A black hole's horizon and the cosmological one share an entropy formula exactly and differ in temperature by precisely a factor of two. The factor of two is the whole of the difference, and what it encodes is which side of the surface the observer stands on.

How much a galaxy's redshift changes in 10 years, and which way. The change in a galaxy's apparent recession velocity over 10 years of the observer's time, c ż/(1 + z), against the galaxy's redshift, for three universes with the same present expansion rate of 67.36 km/s/Mpc. The drift is (1 + z) H₀ − H(z): positive if the expansion rate at the galaxy's epoch was less than (1 + z) times today's, which is to say if the expansion has been accelerating since. In the empty universe the expansion rate is exactly (1 + z) H₀ at every epoch and nothing drifts at all. In the matter-only universe the expansion has only ever slowed, and every redshift falls: −8.6 cm/s over 10 years at z = 1 and −25.5 at z = 4. In ΛCDM the drift is positive nearby, largest at z = 0.63 where it reaches 2.51 cm/s, changes sign at z = 1.91, and is −5.5 cm/s at z = 4. The whole signal is a few centimetres per second in a decade, against the thirty kilometres per second of the Earth's own orbital motion that has to be removed from every spectrum first.

A redshift that changes while it is watched

A galaxy's redshift is a ratio of two sizes of the universe, and the second one is still growing while the light is being collected. So every redshift drifts, by a few centimetres per second in a decade, and the direction of the drift says whether the expansion has been speeding up since the light left — the one test of that question that needs no distance and no model of any source.

The expansion rate and the acceleration, and which of them reaches inside an orbit. Two quantities per unit distance through cosmic time, both in units of today's H₀². The square of the expansion rate, H², falls steeply from the big bang and is 1.00 today by definition. The acceleration of the expansion, ä/a, is the sum of a matter-and-radiation part, −Ωₘ/(2a³) − Ωᵣ/a⁴, drawn dashed, and the cosmological constant's part, ΩΛ = 0.685, which is the same at every time. The sum was negative — the expansion decelerating — until the universe was 7.7 Gyr old, at a = 0.614 or redshift 0.63, and is 0.527 today. The equation of motion of anything orbiting inside a bound system carries ä/a and never H: the rate at which distant galaxies recede does not appear in it at all. Of ä/a, the matter part is the mean density of the universe, which inside a galaxy or a planetary system is already counted in the mass that is doing the holding, many million times over. What is left is the constant: a fixed outward acceleration per unit distance, ΩΛ H₀², that does not grow with time and does not care how fast the universe is expanding.

An orbit feels the acceleration and never the rate

If space expands, it is natural to ask why the Earth's orbit does not. The answer is not that gravity resists the stretching. It is that the expansion rate never appears in the equation of motion of a bound orbit at all — only the acceleration does, and of that only the cosmological constant's part survives, as a fixed outward push that moves the Earth's orbit once, by twelve picometres, and never again.

Two galaxies that stopped moving apart, and the mass that did it. The separation of two galaxies on a radial orbit that began together at the big bang, against cosmic time, solved so that after 13.797 Gyr they are 770 kpc apart and approaching at 110 km/s — the present separation and approach speed of the Milky Way and the Andromeda galaxy. The curve is a cycloid, r = A(1 − cos θ) and t = B(θ − sin θ), and only one cycloid passes through that point with that slope. It rose to 1037 kpc, turned round when the universe was 8.5 Gyr old, and on this purely radial orbit the two meet 3.3 Gyr from now. Its period fixes the mass: A³/(GB²) = 4.2·10¹² solar masses. The dashed curve is the same calculation with the cosmological constant's outward push included, integrated rather than solved; to arrive at the same place at the same speed against that push it needs 4.76·10¹² solar masses, 13 per cent more. Far more than the stars of the two galaxies, it is the timing argument's measurement of the Local Group's dark matter.

The age of the universe weighs the Local Group

The Andromeda galaxy is approaching the Milky Way, and in an expanding universe that means the two once moved apart, stopped and turned round. One radial orbit passes through their present separation with their present speed after exactly the age of the universe, and its period fixes the mass that turned them — four trillion suns, twenty-five times what their stars can account for.

The Sun's speed counted in quasars comes out twice too large

The microwave background is warmer in one direction by a part in a thousand, and that dipole is read as the Sun's motion through it at 370 kilometres per second. The same motion must crowd the counts of distant galaxies and quasars towards the same direction by an amount that can be calculated exactly. When a million quasars were counted, the direction agreed and the size came out more than twice too large — as though the Sun were moving at 800 kilometres per second relative to the matter.

m²φ²: where the observed scales left, and where inflation ends. The m²φ² potential, drawn as a shape with its height divided out, against the field in reduced Planck masses. The field rolls downhill towards zero and inflation ends where the slow-roll parameter ε reaches one, at φ = 1.41. The shaded band is the stretch of field the scales now seen on the sky left the Hubble radius from: 60 e-folds before the end at φ = 15.56 and 50 before it at φ = 14.21. Nothing about the sky depends on the rest of the curve. Across that band the two numbers the tilt is made of are ε = 9.01e-3 and η = 0.0090, which give a spectral index of 0.9640 and a tensor-to-scalar ratio of 0.1441 at 55 e-folds. The height is not in either: it is fixed separately by the amplitude of the fluctuations, which puts the potential at (2.0 × 10¹⁶ GeV)⁴ there — and multiplying the whole curve by any constant leaves the band, the tilt and the ratio exactly where they are, because every slow-roll quantity is a ratio of the potential to its own derivatives. The field travels 13.49 Planck masses from the middle of the band to the end.

The tilt knows the slope and not the height

The measured spectral index, 0.965, is quoted as the strongest evidence for inflation, and it is a statement about two dimensionless numbers — how steeply the potential fell and how sharply that slope was changing, over the few e-folds the sky can see. The height of the potential is not in it at all, which is why potentials that look nothing alike reproduce it.

6 potentials against the tilt and the tensor bound. Predictions in the plane of spectral index and tensor-to-scalar ratio, each drawn as a short track from 50 e-folds to 60, the range usually allowed for the pivot scale to have left before the end of inflation. The vertical band is the measured index, 0.9649 ± 0.0042 with its two-sigma extent, and the shaded region above 0.036 is excluded at 95 per cent by the B-mode polarisation limit — drawn as two independent limits, which the published constraint is not quite: the real likelihood is a correlated contour, and it is somewhat tighter than this box in the corner where the tilt is high. λφ⁴: nₛ 0.9412, r 0.3137 at 50 e-folds; 0.9508, 0.2623 at 60 — outside; m²φ²: nₛ 0.9604, r 0.1584 at 50 e-folds; 0.9669, 0.1322 at 60 — outside; linear φ: nₛ 0.9701, r 0.0796 at 50 e-folds; 0.9751, 0.0664 at 60 — outside; φ^⅔: nₛ 0.9734, r 0.0532 at 50 e-folds; 0.9778, 0.0443 at 60 — outside; natural, f = 7: nₛ 0.9569, r 0.0906 at 50 e-folds; 0.9628, 0.0670 at 60 — outside; Starobinsky: nₛ 0.9616, r 0.0042 at 50 e-folds; 0.9678, 0.0030 at 60 — inside. The simplest potential of all, a mass term, is excluded over its whole range of e-folds, and not by the tilt, which it matches — by the tensors.

A ratio that is an energy and a distance

The tensor-to-scalar ratio is the one inflationary observable that measures the height of the potential rather than its shape, and the quantity it fixes is an energy — a ratio of 0.01 means inflation happened at 10¹⁶ GeV. It fixes a second thing as well, how far the field travelled, and near the present bound that distance is several Planck masses, which is where the theory stops being able to vouch for itself.

Starobinsky: e-folds before the end, against how reheating went. N, the number of e-folds between the pivot scale leaving the Hubble radius and the end of inflation, for the Starobinsky potential, against the temperature at which reheating finished, for 3 equations of state during it. All the lines meet on the right at instant reheating, 2.6 × 10¹⁵ GeV, where N = 55.6. The left edge is 5 MeV, below which nucleosynthesis would not have happened. w = 0, oscillating field: 42.0 at 5 MeV; w = ⅓, like radiation: 55.6 at 5 MeV; w = 1, kination: 69.0 at 5 MeV. The reason is how far the universe stretches while the energy density falls: an oscillating field dilutes like matter, as a⁻³, so for the same fall in density it expands further than radiation would, more of the growth of today's scales happens after inflation, and fewer e-folds of inflation are needed to put them where they are. A stiff epoch with w = 1 dilutes as a⁻⁶, stretches less, and needs more. A radiation-like epoch changes nothing. None of this epoch has been observed; the lines are the arithmetic of energy and entropy, and the spread between them is how much an unobserved history moves a quantity the spectral index depends on.

The epoch nobody saw moves the tilt

Between the end of inflation and the hot universe that made the light elements lies an interval nothing has observed, in which the energy of the inflaton became radiation. How long that took changes how many e-folds before the end the observed scales left — by as many as fourteen — and that moves every model's predicted spectral index by more than the measurement's uncertainty. A potential is never tested by the tilt alone; a potential and a reheating history are tested together.

One sky with and without a local non-Gaussianity of fNL·σ = 0.3. The same scale-invariant random field drawn twice, from one seed: on the left as it is, Gaussian, and on the right after the local transformation Φ → Φ + fNL(Φ² − ⟨Φ²⟩) with fNL·σ = 0.3. Solid contours are one and two standard deviations above the mean, dashed ones below, and each map is measured against its own mean and spread. The transformation adds to every value in proportion to its square, so peaks are pushed up and troughs are pulled back towards the mean: the area above +2σ goes from 1.8 to 4.7 per cent of the map and the area below −2σ from 2.4 to 0.0, and the skewness of the values rises from −0.073 to 1.484. This is exaggerated by a factor of about 2200. The primordial potential varies by about 3 × 10⁻⁵, so even fNL = 5 — the size of the current uncertainty — makes fNL·σ about 10⁻⁴.

A test that can only fail one way

Single-field inflation predicts a local non-Gaussianity of 0.015, a skewness in the primordial potential of a few parts in a million. The measurement is −0.9 ± 5.1. A detection at the level of one would eliminate every model with a single clock at once; a null result at any reachable precision confirms nothing, because the prediction lies below anything the sky has enough independent modes to measure.

$A delay that counts the gas nothing can see. The mean dispersion measure of a radio pulse against the redshift it comes from, for several shares of the baryons residing in diffuse ionised gas between galaxies. A pulse is delayed by free electrons in proportion to the column it crosses, and that column is an integral over the expansion history of a density the baryon budget fixes — so the only unknown in the whole expression is the share itself. At z = 1 the relation gives 1085 pc cm⁻³ if every baryon is out there and 543 at 50 per cent. The measurement runs the other way: a burst with a known host redshift and a measured delay returns the fraction, and the answer came out consistent with nucleosynthesis. A pulse a millisecond long weighs the half of the ordinary matter that no survey could find, and it does so because the thing that delays it is the thing that does not shine.$

Half the ordinary matter was missing, and a millisecond found it

Nucleosynthesis fixes how many baryons there are to better than a per cent. Every survey of where they are came up about half short for two decades — and what closed the gap was the delay a radio pulse picks up crossing gas too thin and too hot for any telescope to have seen.

$A mass measured by the structure it prevents. The fractional suppression of the small-scale matter power spectrum against the sum of the neutrino masses. The relic density follows from the thermal history with no astrophysics in it — the neutrinos’ share of the critical density, times h², is the mass sum divided by 93.14 eV exactly — and the suppression is about eight times the neutrinos' share of the matter, because a particle moving at a large fraction of the speed of light streams out of an overdensity while it is still growing and takes its own gravity with it. The two mass orderings put a floor under the sum at 58 meV and 100 meV; the cosmological upper bound is near 0.12 eV, which is 7.2 per cent of suppression. The floor and the ceiling are within a factor of two of each other, so this is a measurement about to happen or a model about to break, and the quantity it returns is a sum rather than any individual mass.$

A mass measured by what it stopped from forming

Neutrinos were relativistic in the early universe and are not now, so they are the one entry in the cosmic budget that changes category. What cosmology measures is not their density but the hole they leave — they stream out of a growing clump and take their gravity with them, and the missing structure bounds a particle mass more tightly than any laboratory has.

$Structure stops forming above about 14 times the observed Λ. The fraction of matter that ever collapses into a bound object, against the cosmological constant in units of the observed one, holding the primordial fluctuation amplitude fixed. Growth of structure stops once Λ dominates the expansion, so the linear growth factor approaches a finite limit rather than rising for ever, and a larger Λ freezes it earlier and smaller. The asymptotic growth factor at the observed Λ is 1.110; at a hundredth of it, 5.152; at a hundred times, 0.239. The collapsed fraction falls from 0.86 to 0.40 to 0.000 across the same range, and passes a tenth of its present value at 14 times the observed constant. The observed value sits a factor of 14 below the largest one that leaves anything at all — which is the whole of the anthropic argument, and that factor is what has to be compared against the 10¹²⁰ by which the naive theoretical estimate misses.$

A coincidence that is a factor of fourteen

The cosmological constant is famously wrong by a hundred and twenty orders of magnitude, and famously coincidental in sitting near the matter density now. Computing how large it could be and still leave any structure at all turns the second complaint into a number — and the number is fourteen, not a hundred and twenty.

The darkness has a number in it

The night sky is not black. It carries about sixty nanowatts per square metre per steradian, and that number is the sum of every photon every star has ever emitted, redshifted and added up over thirteen billion years.

A background weighed by what it stops

The faintest light in the universe cannot be photographed from inside the Solar System, because the zodiacal foreground is a hundred times brighter. It can be weighed instead, by the bite it takes out of a blazar at a trillion electronvolts.

Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one.

Two skies where the paradox comes out right

Olbers argued that every sight line should end on a source and the sky should blaze. In neutrinos and in gravitational waves it does — the walls are at one second and at no time at all — and both of those skies have now been detected.

A cluster moving at +500 km/s, and the frequency where only the motion is left. The two distortions one cluster imprints on the microwave background, against observing frequency, scaled to the largest excursion drawn. The thermal effect is from the random motion of electrons at 8 keV, with a central Compton parameter of 10⁻⁴; the kinematic effect is from the bulk motion of the same gas at 500 km/s along the line of sight, positive meaning receding, through an optical depth of 0.00639, which is the Compton parameter divided by kT/mₑc². A bulk velocity shifts every scattered photon by a common Doppler factor, and a blackbody shifted by a common factor is a blackbody at another temperature — so the kinematic distortion has exactly the shape of a temperature change, ΔT/T = −τv/c, which here is −29.0 µK at every frequency. In intensity that shape is the derivative of the Planck spectrum, and its largest value falls at 217.5 GHz, the same frequency at which the thermal distortion crosses zero, 217.5 GHz: both conditions reduce to x coth(x/2) = 4. At 150 GHz the thermal decrement is −260 µK, so the motion is 11.2 per cent of it there and all of the signal at the null. What the kinematic spectrum cannot be told apart from is the primary microwave background itself, which is also a temperature change with this shape — so the frequency that isolates the velocity from the gas is no help at all against the sky behind it.

A velocity that has the colour of the sky

A cluster moving through the microwave background shifts the light it scatters by a common Doppler factor, which leaves a spectrum shaped exactly like a change of temperature. That shape is loudest precisely where the hot gas falls silent — and it is the one shape the background itself already has.

At 15 keV the decrement is 9 per cent shallower and the null has moved to 224.4 GHz. The thermal Sunyaev–Zel'dovich distortion at one fixed Compton parameter, 10⁻⁴, computed with the relativistic kinetic equation expanded to second order in kTₑ/mₑc² for gas at 5, 10, 15 keV, against the non-relativistic shape that is the same for every temperature. All are scaled to the non-relativistic curve's largest excursion. Heating the gas at fixed y does two things to the spectrum. The decrement becomes shallower — by 9.3 per cent at its deepest point for 15 keV — and the increment becomes lower and broader, by 17.3 per cent at its peak, because fast electrons scatter photons over a wider spread of frequencies than slow ones and some of the boost is carried to frequencies above the drawn range. The crossing moves up, from 217.5 GHz to 224.4 GHz. A cluster's temperature is therefore written into the shape of its distortion and not only its amplitude, which is what a thermometer needs; and a Compton parameter read off one frequency with the non-relativistic shape is biased low by the drawn amount, which is what a mass estimate does not need. The expansion is good to well under a per cent below 15 keV; above 20 it has to be replaced by the exact integral.

A null that moves with the temperature

The frequency at which a cluster's hot gas vanishes from the microwave sky was derived for slow electrons. The electrons in a massive cluster move at a quarter of the speed of light, the null drifts half a gigahertz per keV, and what is left at the old frequency reads as a velocity as large as the ones being sought.

1020 clusters or 372, and the survey cannot say which parameter moved. The number of clusters per unit redshift a survey finds above an integrated Compton signal of 8·10⁻⁵ arcmin² over 6 per cent of the sky, computed from the Sheth–Tormen halo count grown by the linear growth factor, the comoving volume in each redshift slice, and the calibrated relation between signal and mass. Each curve is one pair of assumptions: the amplitude of structure, σ₈, and the hydrostatic mass bias, 1 − b, which says how far the X-ray masses the relation was calibrated on fall below the true masses. With σ₈ = 0.811 and 1 − b = 0.8 the survey finds 1020; σ₈ = 0.811 with 1 − b = 0.6 gives 372; σ₈ = 0.75 with 1 − b = 0.8 gives 548. The counts fall at low redshift because there is little volume, and at high redshift because massive halos have not yet formed, and the peak sits near z = 0.24. Lowering 1 − b pushes the threshold onto more massive and rarer halos; lowering σ₈ makes every halo rarer. The two lower curves differ in total by 47 per cent and in normalised shape by at most 2 per cent of the peak. A survey that assumed 1 − b = 0.8 would read the 372 clusters of the curve with 1 − b = 0.6 as σ₈ = 0.716, with a redshift distribution that differs from it by at most 12 per cent of the peak — which is the only handle the survey has on the difference, and it is smaller than the counting noise in any redshift bin holding fewer than about 67 clusters. A total count cannot distinguish a universe with less structure from a survey that has misjudged its masses, and it is that degeneracy, not the counting, that has been argued about since the first large catalogue.

Too few clusters, or a scale that reads light

A catalogue selected on the microwave shadow is, past redshift one half, very nearly a catalogue of everything above a fixed mass — so its count by redshift is the growth of structure read almost directly. Almost, because the mass behind the threshold comes from a calibration, and a scale that reads twenty per cent light is indistinguishable from a universe with less in it.

The ladders in this field

16 anchors · one idea each

Expansion Hubble constant Distance ladder Dark energy Microwave background Nucleosynthesis Density parameters Baryon acoustic oscillations Olbers's paradox Horizons Inflation Large-scale structure Reionisation Sunyaev zeldovich Tidal torque theory Primordial fields

All fields · All essays