The Hubble Tension: Why Two Precise Measurements of Cosmic Expansion Disagree — Epoche C1
Two of the best-measured numbers in cosmology are both estimates of the present expansion rate of the universe, the Hubble constant $H_0$, and they disagree. The Cepheid-calibrated distance ladder gives $H_0 = 73.04 \pm 1.04$ km/s/Mpc (Riess et al. 2022). The Planck satellite's fit to the cosmic microwave background — the relic radiation of the hot early universe — gives $H_0 = 67.4 \pm 0.5$ km/s/Mpc (Planck Collaboration 2020). The difference is $5.64$ km/s/Mpc, or $8.4$ per cent of the Planck value. Because the two rest on disjoint instruments, disjoint objects and disjoint physical assumptions, their errors may be treated as independent and combined in quadrature: $\sqrt{1.04^2 + 0.50^2} \approx 1.15$ km/s/Mpc. The gap is therefore $5.64/1.15 \approx 4.9$ standard deviations, the "$5\sigma$ tension" of the recent literature; for a normal distribution the two-tailed probability of a departure that large is about $10^{-6}$. The usual response — that some experiment has underestimated its errors — is a coherent hypothesis, but it is not the only one, and this review argues that the more interesting possibility survives scrutiny: the two teams are not measuring the same quantity in the same way. 1. What the number means, and what it does not Before comparing the routes, the quantity itself needs pinning down, because one common gloss on it is slightly wrong. $H_0$ is defined by the Hubble–Lemaître law $v = H_0 d$: a galaxy at distance $d$ recedes at speed $v$. The distance unit is the megaparsec, $1\,\mathrm{Mpc} = 3.26$ million light-years $= 3.09\times10^{19}$ km, so a galaxy one megaparsec away recedes at about 70 km/s. Inverting, $1/H_0$ has the dimensions of time — the Hubble time. Taking one year as $3.16\times10^{7}$ s, the Planck value gives $1/H_0 \approx 14.5$ billion years and the local value $13.4$ billion. Those are not ages. The Hubble time would be the age only if the expansion rate had never changed; in a universe that decelerates while matter dominates and accelerates once dark energy does, the true age is an integral over the history, and for Planck's parameters it works out at about $0.95/H_0$, or 13.8 billion years. The point survives the correction, though: the age scales as $1/H_0$, so an $8.4$ per cent disagreement about $H_0$ is an $8.4$ per cent disagreement about the age of everything, roughly a billion years. That is not a rounding error, and it is why the discrepancy is worth taking apart. 2. The local route: climbing a ladder The first of the two measurements builds distances outward in three rungs, each calibrating the next, and this section walks up all three because the tension is often blamed on one of them. The organising formula throughout is the distance modulus. Flux falls as the inverse square of distance, and the astronomers' magnitude scale is logarithmic, defined so that five magnitudes correspond to a factor of one hundred in flux; hence $m - M = -2.5\log_{10}(F/F_{10\,\mathrm{pc}}) = 5\log_{10}(d/10\,\mathrm{pc})$, where the apparent magnitude $m$ is what the telescope records and the absolute magnitude $M$ is what the star would show at ten parsecs. Differentiating, a fractional distance error $\delta d/d$ costs $5\,\delta d/(d\ln 10) = 2.17\,\delta d/d$ magnitudes: one per cent in distance is $0.022$ magnitudes of photometry. The whole enterprise is a demand for photometry at the hundredth-of-a-magnitude level. Rung one is pure geometry. Three anchors give absolute distances without assuming anything about stars. Parallaxes from the Gaia satellite give trigonometric distances to Cepheids in our own Galaxy. Eclipsing binary stars in the Large Magellanic Cloud give distances from the ratio of their measured angular size to their physical size, the latter following from the orbital velocities. The third is the most elegant: water masers — clouds emitting coherent microwave radiation — orbit the black hole at the centre of the galaxy NGC 4258 in a disc seen almost edge-on. Their Doppler shifts give the orbital speed $v$; the slow drift of that shift over years measures the centripetal acceleration $a = v^2/r$, which yields the physical radius $r = v^2/a$; and radio interferometry measures the angular radius $\theta$. The distance is then simply $d = r/\theta$, with no astrophysics in it at all. Reid, Pesce and Riess (2019) obtained $7.576 \pm 0.082\,(\mathrm{stat}) \pm 0.076\,(\mathrm{sys})$ Mpc, an accuracy of 1.5 per cent. Rung two transfers that calibration to Cepheids and thence to supernovae. Cepheids are stars that pulsate radially, brightening and dimming with a period of days to weeks. Leavitt and Pickering (1912) found, from 25 variables in the Small Magellanic Cloud, that the period tracks the apparent brightness — decisive because all the stars in that cloud lie at essentially one distance, so differences in apparent brightness are differences in luminosity and nothing else. The link is now called the Leavitt law, and its physical basis is straightforward: the pulsation period of a star is set by its dynamical time, $P \sim (G\bar\rho)^{-1/2}$, where $\bar\rho$ is the mean density; Cepheids occupy a narrow strip of surface temperature, so their luminosity $L = 4\pi R^2\sigma T_{\mathrm{eff}}^4$ depends essentially on radius alone. A larger radius means both a lower mean density, hence a longer period, and a greater luminosity. Period therefore predicts luminosity, and a measured period plus a measured flux gives a distance. Riess et al. (2022) used Cepheids in 37 nearby galaxies that had also hosted a Type Ia supernova — the thermonuclear detonation of a white dwarf — to fix those supernovae's peak absolute magnitudes, 42 of them in all. Type Ia supernovae are not identical, but their peak luminosity correlates tightly with the width of the light curve and with colour, and applying that empirical correction leaves a scatter of roughly a tenth of a magnitude, a few per cent in distance per object. Rung three reads the expansion off the fa