Concept

Dark energy — where it appears

Whatever has pressure negative enough to make the expansion accelerate, known only through its effect on distances and so far consistent with a constant. It contributes about seven tenths of the present energy budget, and every constraint on it comes from how distances depend on redshift rather than from any local measurement.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

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.

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

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.

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

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.

cosmology · Horizons

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.

cosmology · Expansion

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.

cosmology · Expansion

Three fits of the dark energy's equation of state, and the line they all cross. The equation of state w of the dark energy against redshift, for the three combined fits published with the second year of acoustic-scale data from the largest spectroscopic galaxy survey — the same galaxy and microwave-background data with three different supernova compilations — each written as w = w₀ + wₐ z/(1 + z). A cosmological constant is the horizontal line at −1. All three fits have w above −1 today (between −0.84 and −0.67) and below it in the past, and they cross it at redshift 0.35, 0.44, 0.41. Below −1 lies the phantom region, where a density grows as the universe expands; a single field with ordinary kinetic energy can be on either side of the line but cannot cross it. The preference for these fits over a constant is quoted at 2.8, 3.8 and 4.2 standard deviations, and the only thing that differs between the three is the supernovae.

A crossing that may belong to the fitting form

Allow the dark energy's equation of state to change and three independent fits say it has: above −1 today, below it before redshift half. The data hold w well at one redshift only, a line crossed from below is one no single field can cross, and a straight line in the scale factor fitted to a curved history crosses it anyway.

cosmology · Dark energy

Where solved fields land in the plane the surveys fit. The w₀–wₐ plane, with the cross at a cosmological constant. Each small point is the best two-parameter description, through the same mock survey as the earlier figures, of a scalar field solved from its own equation: four thawing potentials at seven values of w₀ from −0.97 to −0.62, and freezing trackers on inverse-power potentials with exponents from a quarter to two. The thawing fields all land on one line through the constant, wₐ ≈ −1.49(1 + w₀), with no point more than 0.08 from it; the freezing fields land on the other side, with wₐ positive. The three published fits (larger points, with their quoted errors) lie below the thawing line by −0.38, −0.59, −0.49 in wₐ: steeper than any thawing field with their present value would be. The diagonal is w₀ + wₐ = −1. The thawing line is steeper than it, so every thawing field is fitted as having crossed −1 — but early: these fits cross between redshift 1.7 and 3.3, where a line through the constant with this slope must, near z = 1/(1.49 − 1) = 2.0. The published fits cross near redshift 0.4.

Every thawing field falls on one line

A scalar field that is the dark energy sits still while the universe expands quickly, and begins to roll only when the expansion has slowed to the field's own mass. Solve the field instead of fitting a line to it, and four quite different potentials give one history and one line through the plane the surveys fit — a line the published fits do not sit on.

cosmology · Dark energy

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

A ratio in which the galaxies cancel

Matter falls in one gravitational potential and light is bent by the sum of two, and relativity says the two are equal. A survey that measures both the infall of galaxies and the lensing around them can divide one by the other so that the galaxies' bias and the fluctuations' amplitude both cancel — leaving a ratio that relativity predicts from the matter density alone, and that a gravity which bends light differently from the way it pulls matter cannot fake.

cosmology · Dark energy

Named alongside it

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

Equation of state Cosmological constant Phantom energy Quintessence Scale factor Coincidence problem Comoving distance Conformal time Cosmic acceleration Deceleration parameter Degeneracy direction Growth index

All concepts