Astronomy & Celestial Mechanics
the sky, computed
Concept

Growth rate — where it appears

The logarithmic rate at which density fluctuations grow with the scale factor, written f. In general relativity it is close to the matter density parameter to the power 0.55, a relation many modified theories of gravity violate — which is what makes measuring it a test.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

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.

cosmology · Large-scale structure 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 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.

cosmology · Large-scale structure 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.

cosmology · Sunyaev zeldovich A void: a deficit of matter, and the outflow it drives. The density contrast of a typical cosmic void against distance from its centre in units of its radius (solid), from the profile fitted to stacked voids in simulations, with a central density 15 per cent of the mean and a compensating overdense ridge just outside the edge. The dashed curve is the excess expansion rate the deficit drives, v/(Hr) = −(f/3)Δ(r), for a growth rate f = 0.75: the centre expands 21 per cent faster than the universe around it, and the outflow speed, which is this rate times the radius, peaks near 0.7 void radii. Beyond the ridge the flow reverses: matter at the ridge is pulled back towards the denser wall. A void is a small open universe expanding inside a flat one, and every galaxy in it is moving outward.

An emptiness that expands faster

A cosmic void is not a hole in the universe but a region with a fifth of the average density, and gravity makes it behave like a small open universe: it expands faster than its surroundings, every galaxy in it streams outward, and in a redshift survey it is stretched along the line of sight — the opposite of what the same gravity does around a cluster. The stretch measures how fast structure grows, and the average void's roundness measures the geometry of the universe.

cosmology · Large-scale structure A velocity error that grows with distance. The uncertainty in one galaxy's peculiar velocity, v = cz − H₀d, against its distance in Mpc/h, when the distance comes from an indicator with a fractional error of 5 per cent, 20 per cent, 25 per cent (type Ia supernovae, 5%; Tully–Fisher, 20%; fundamental plane, 25%). The redshift is exact to a few kilometres a second, so the whole error is the distance error times H₀, and it grows in proportion to the distance. The horizontal line is the rms velocity of a galaxy along one direction in linear theory for the model drawn, 302 km/s. Each galaxy's own velocity is measured to better than its own size only inside 60 Mpc/h for 5 per cent, 15 Mpc/h for 20 per cent, 12 Mpc/h for 25 per cent; beyond that a single velocity is mostly noise, and a velocity survey is useful only as an average over many galaxies.

Velocities that see past the edge of the map

A redshift survey infers the motions of galaxies from the distortions of its own map. A peculiar-velocity survey measures them directly, one galaxy at a time, by subtracting a distance from a redshift — and every one of those measurements is dominated by its own error beyond a few tens of megaparsecs. What makes the survey worth doing is that a velocity answers for structure far larger than the volume it is measured in.

cosmology · Large-scale structure How much each change moves the growth rate. The percentage difference in fσ₈ from growth in relativity inside a cosmological constant's expansion, against redshift, for three alternatives started from the same early amplitude. Changing the expansion to the evolving fit with DES-SN5YR and keeping relativity moves it by −1.2 per cent at redshift 0.5, and by at most 2.1 per cent anywhere below redshift 2. Keeping the constant's expansion and changing gravity to γ = 0.68 moves it by −8.6 per cent; doing both, −10.3. The shaded band is ±8 per cent, about the error of one of the better redshift-space measurements so far. The difference between a constant and the evolving fit is several times smaller than any single measurement; the difference between two laws of gravity is comparable to one. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is gravity.

The growth that tells a substance from a law

In relativity an expansion history leaves no freedom in how fast structure grows inside it, so every fitted history predicts its own growth. The evolving dark energy the surveys prefer changes that growth by one or two per cent; a change in the law of gravity, with the expansion held fixed, changes it by nine. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is there at all.

cosmology · Dark energy How deep a hole the Hubble tension needs, and how deep holes are. The typical departure of the matter density from the mean, averaged over a sphere — one standard deviation, σ, from the linear power spectrum with σ₈ = 0.81 — against the sphere's radius, both on logarithmic axes. It falls from 0.55 at 20 Mpc to 0.022 at 300 Mpc. A local expansion rate 8.3 per cent above the global one, the size of the Hubble tension, needs the sphere the supernovae sample to be underdense by 0.47 (dashed), with a growth rate of 0.53. Across the radii the supernovae span (shaded, 100–600 Mpc) that is 4 to 67 standard deviations.

A hole too deep for the sky to have made

If the Milky Way sat inside a vast underdense region, everything around it would be expanding faster than the universe as a whole, and the locally measured Hubble constant would come out high — which is what Cepheids and supernovae find. The idea needs no new physics. It needs a void about half empty across the six hundred megaparsecs the supernovae sample, and the same cosmology that predicts the microwave background prices such a void at dozens of standard deviations.

cosmology · Hubble constant

Named alongside it

The objects these essays reach for when they reach for this one.

Peculiar velocityModified gravityRedshift-space distortionCorrelation functionGalaxy biasSigma eightCosmic varianceCosmic voidFingers of GodGrowth indexKaiser effectPower spectrum

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