Astronomy & Celestial Mechanics
the sky, computed
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Dark energy — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · cosmology
  2. 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.

    part 2 · cosmology
  3. 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.

    part 3 · cosmology

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