Parallax
Parallax is both the oldest astrometric technique, and the easiest to understand. Hold out your thumb at arm’s length, relative to some fixed point on the wall, and alternately close one eye and then the other. Relative to the fixed wall, the apparent position of your thumb will change, and this is how our depth perception works: the smaller the change in position, the longer your arm must be. The same principle works with stellar distance, where now our “binoculars” correspond to the Earth’s position on opposite sides of its orbit. The change in a fixed star’s position \(2\varpi = \theta _1 - \theta _2\) arcseconds, due to the change in the Earth’s position by 2 a.u. over 6 months leads to the distance \(d = 1/\varpi \) parsecs.Footnote 5 Our measurement of position must be very precise. Although the nearest star, Proxima Centauri, has a parallex of approximately 0.77 arcseconds, modern measurements target a remarkable 10 \(\,\mu \)as, which is the size of a thumbnail on the Moon as seen from Earth.
Modern parallax measurements began with the satellite Hipparcos (the name alludes to the ancient Greek astronomer Hipparchus, who measured the distance to the Moon). Launched by the European Space Agency in 1989, it measured the parallaxes of 100,000 stars at an accuracy of up 0.5 milliarcsecond (mas), the fixed background frame now being extragalactic sources such as quasars. Although undoubtedly impressive, at 1000 light years an error of 0.5 mas would still be a distance error of 15%. A further drawback is Cepheids, an important part of the distance ladder we discuss shortly, are relatively rare stars and only a handful are located in our neighbourhood of the Milky Way.
Moving on to the present day, our two best current sources of parallax are Gaia and the Fine Guidance Sensor/Wide Field Cameras (FGS/WFC3) aboard the Hubble Space Telescope (HST). Gaia was launched in 2013 and the mission goal is to measure over a billion stars (including 9,000 Cepheids and half a million reference quasars), both in our galaxy and satellites like the LMC. The mission-expectation precision is 7\(\,\mu \)as at \(m=12\), rising to 26\(\,\mu \)as at \(m=20\) (Gilmore 2018). Gaia does this by slowly scanning the sky with two telescopes set at relative angles of \({106.5}^{\circ }\), to make a one-dimensional measurement of the time and position of each star that slowly drifts across the CCD. Up to 70 measurements will be made for each star, which allows the additional calculation of proper motions, and even small changes in position caused by the gravitational tug of planets orbiting the star. The HST operates on similar principles in “spatial scanning” mode; although it cannot survey like Gaia, when focused on nearby Cepheids its errors appear competitive (Riess et al. 2018a).
Gaia’s high precision is dependent on a very stable mechanical structure of the spacecraft. A variation in the angle between the two fields (which might be caused by thermal expansion) could cause spatial variations in apparent parallax, or if synchronous with the scan period even a fixed systematic offset. Indeed, such a variation has been inferred from the interim Data Release 2 (DR2) (Gaia Collaboration 2018). The average of the quasar parallaxes in it is \(-29\,\mu \) as (negative parallaxes can happen when position measurement error is larger than the parallax), and there were indications this “zero point” may vary with stellar colour, luminosity and position on the sky (Arenou et al. 2018). Riess et al. (2018b) compared HST Cepheid parallaxes to Gaia, simultaneously solving for the Gaia zero point and Cepheid calibration. They found a difference between them \(46\pm 15 \,\mu \) as, with Gaia parallaxes again appearing too low. As the typical parallax of a Milky Way Cepheid is \(400\,\mu \) as, this is very material to distance estimates. Breuval et al. (2020) creatively replaced DR2 Cepheid parallaxes with those of resolved bound binary companions (where available), which being dimmer are closer to the ideal magnitude range for Gaia.
During the preparation of this review, Gaia Early Data Release 3 was made available, and has already by used by Riess et al. (2021) to revisit Gaia parallaxes. Gaia EDR3 is not intended to be the final word, but indeed Cepheid zero points seem now to be reduced below \(10\,\mu \) as. A calibration of Cepheids using 75 Gaia parallaxes only gives \(H_0 = 73.0 \pm 1.4 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\), which is slightly lower, but consistent with their previous result based on HST parallaxes.
Before moving on to discuss alternative calibrations, we will first digress into a discussion of parallax bias. The potential for bias occurs in any astrophysical observation, and is often the subject of lengthy analysis in \(H_0\) papers. As it is most easily understood in the context of parallax, it is helpful to discuss it here.
Parallax bias is usually referred to as Lutz–Kelker–Hanson (LKH) bias (Lutz and Kelker 1973; Hanson 1979). This is a summation of three quite different effects: non-linearity of the desired variable (distance) with respect to the observed variable (parallax), population bias (have we observed the object we intended to, or did we confuse it with something else?), and selection bias (our surveys are normally magnitude limited, so we will only “select” objects for which \(m<m_0\)).
To explain non-linear bias, imagine we have a symmetric error in our parallax measurement, such as might be caused by an instrumental point spread function. To be concrete, suppose the likelihood of measuring \(150\,\mu \) as is the same as measuring \(50\,\mu \) as when \(100\,\mu \) as is the true value. If we average the distance, we will obtain \({\bar{d}} = 6666.67 + 20000 = 13,333\) which is biased with respect to the true distance of 10,000 parsecs. In mathematical terms, because \(d = 1/\varpi \), then \({\mathbb {E}}[d] \ne 1/{\mathbb {E}}[\varpi ]\).
Population bias arises in parallax when we consider a broad survey of stars at a given distance r from our position. Assuming a roughly constant spatial density of similar stars, there are more stars in the shell \((r, r+\varDelta r)\) than there are in the shell \((r - \varDelta r, r)\) for some finite \(\varDelta r\). Hence, there are more (further) stars whose parallaxes may be over-estimated to place them at r than (closer) stars whose parallaxes may be over-estimated. Taken to extremes, there are huge numbers of stars with effective parallaxes of zero, waiting for their small but finite chance to “crowd in” to a given measured parallax. This will bias observed parallaxes too low. Note that if we were certain of our identification of the star (as we would be for a Milky Way Cepheid close to us), we need not consider population bias: it would stand out from the crowd (Fig. 5).
Conversely, suppose we were observing close to our magnitude limitations. Now the opposite bias would occur: we cannot see the further stars, so they cannot crowd in. But the same number of closer stars are available to crowd out, so our observed parallaxes will now be biased too high. This is the well-known Malmquist bias (Malmquist 1922), and is a major issue for surveys as naturally we will try to see as far as we can!
To deal with bias then involves modelling of the instrumental error, the selection function, the population scatter and so forth. In modern surveys, this is normally done by constructing simulated catalogs with known physical parameters and some assumptions, and putting those catalogs through the same analysis pipeline as the real data to see what biases emerge. For example, Riess et al. (2018a) compute distance modulus biases of between 0.03 and 0.12 mag for MW Cepheids using a model for galactic stellar distributions. If working in a Bayesian framework, a posterior distribution for the distance may be derived using the method of Schönrich et al. (2019). An alternative is to work directly with the parallaxes, instead converting Cepheid magnitudes to predicted parallaxes as done by Riess et al. (2018b). As the magnitudes are measured considerably more accurately than the parallaxes, bias corrections to the magnitude to parallax conversion are not necessary. To check the predictions of LKH bias, Oudmaijer et al. (1998) compared ground-based to Hipparcos parallaxes, finding a bias towards brighter magnitudes up to relative error \(\approx 30\%\), and dimmer magnitudes for larger error when Malmquist bias is dominant, as one would expect from the discussion above.
We end our digression on biases here and move on to discuss other geometrical distances.
Detached eclipsing binaries
Imagine we had a star for which we knew the surface radiant flux density J, and the physical size R. The stellar luminosity would straightforwardly follow as \(L = 4\pi R^2 J\), and we would have a standard candle.
Cool, stable, helium-burning giants (that is, red clump) are sufficiently bright to be seen outside the Milky Way. For these “late-type” stars, just such an empirical relationship can be established for the surface brightness \(S_V\) defined as
$$\begin{aligned} S_V \equiv V + 5 \log (\phi ) \;\;\text{ mag } , \end{aligned}$$
(27)
where \(\phi \) is the stellar angular diameter, and V the visual band magnitude. This relationship has been calibrated by angular diameters obtained from optical interferometry of nearby stars, and is given by
$$\begin{aligned} S_V = (1.330 \pm 0.017) \times ((V-K) - 2.405) +(5.869 \pm 0.003) \;\;\text{ mag } , \end{aligned}$$
(28)
where \(V-K\) is the colour difference between magnitudes in the V and \(2.2 \;\,\mu \)m near-infrared K band (Pietrzyński et al. 2019). The scatter is just 0.018 mag. Rearranging the definition of surface brightness and with \(\phi = 2R / d\) it then follows that
$$\begin{aligned} d = 9.2984 \times \left( \frac{R}{R_{\odot }}\right) (10^{0.2(V - S_V)} \;\text{ mas}) \;\; \text{ parsecs }, \end{aligned}$$
(29)
where R and \(\phi \) have been converted to solar radii and milli–arcseconds, respectively. The pre-factor is purely geometric.
But how can we know the radius of distant stars? Eclipsing binaries allow just that. If the stars are well separated enough to spectroscopically resolve each one,Footnote 6 but close enough to eclipse each other, we can obtain their individual surface brightnesses, colours, radial velocities, eclipse depths and shapes, and the orbital period. With this data, the radius (and other physical parameters such as mass, eccentricity and inclination of the orbital plane) of each star can be solved for. Such an alignment of the stars is of course rare, but sufficient numbers do exist! By painstakingly observing 20 systems in the LMC over more than 20 years (covering many eclipses) Pietrzyński et al. (2019) determine the stellar radii to 0.8% accuracy. Crowding can be easily spotted in the light curve and removed. They derive \(\,\mu _{\mathrm{{LMC Centre}}} = 18.477\pm 0.004\; \text{(stat) } \pm 0.026\; \text{(sys) }\), where the main contribution to the systemic error budget is the \(S_V\) relation above.
This is the most accurate measurement of the distance to the LMC to date. As we shall see shortly when we discuss standard candles, this result has become key to many recent \(H_0\) results: every standard candle can be found in the LMC, and the low error budget allows for a very accurate calibration.
Looking forwards, although DEBs have been found in M31 they are “early-type” stars with hot atmospheres, for which a reliable surface brightness to colour relation has not been established. The hope is future 30 m-class telescopes will have sufficient spectroscopic resolution to extend this to late-types in M31 and other local group galaxies (Beaton et al. 2019).
Masers
Maser emission occurs when thermal collisions in warm gas in an accretion disk around the central black hole of a galaxy drive a population inversion of molecular energy levels. Such systems are rare: the disk must be “just right”, not too hot, and not too cold, and have suitable local molecular abundance. The Type 2 Seyfert galaxy NGC 4258 at a distance of 7.5 Mpc is just such a system. Isolated bright spots of 22.235 GHz maser emission (from a hyperfine transition of \({\mathrm {H}}_{2}{\mathrm {O}}\)) can be seen in three regions on the sky stretching 20 mas (\(\sim 0.1\) ly) long, with the overall shape of a warped line. After subtracting the overall system redshift, the spots on one side are blue-shifted by \(\sim 1000 \,\hbox {kms}^{-1}\), the other side is red-shifted by the same amount, and the spots in the middle are low velocity (Argon et al. 2008). If the spots are observed for long enough, their accelerations and proper motion can be obtained from the steady drift of their Doppler velocities and positions. The central spots show the lowest l.o.s. velocity and highest acceleration, and the outer spots having the highest velocity and lowest l.o.s. acceleration.
The key assumption these observations support is that all the spots form an orbital system with shared parameters, as it is anticipated that disk viscosity will have reduced the orbits to close to circular. The disk shape is then modelled (including such parameters as inclination, warp, residual eccentricity and so forth), fitted and the orbital parameters of the spots derived. The outer dot velocities v show a Keplerian behaviour with radius: \(v^2 \propto GM/r\), and as the acceleration is then \({\dot{v}} \propto GM/r^2\), the physical radius of the system can be determined. The angular diameter distance is then
$$\begin{aligned} d_A = \frac{v^2}{{\dot{v}} \times \theta } , \end{aligned}$$
(30)
where \(\theta \) is the angular impact parameter. Humphreys et al. (2013) have observed NGC 4258 for 10 years at 6 monthly intervals using VLBI interferometry, which resolves the relative position of the maser spots to \(<3\;\,\mu \)as accuracy (even the tectonic drift of the ground radio telescopes must be corrected for). The Doppler shifts are measured to within an accuracy of \(1 \;\hbox {kms}^{-1}\), and the central spots accelerate by \(\sim 10\;\hbox {kms}^{-1} \mathrm {yr}^{-1}\). The proper motions \({\dot{\theta }} \;\,\mu \text{ as } \text{ yr}^{-1}\) provide additional information as \(d_A = v / {\dot{\theta }} \; \text{ Mpc }\). This pattern fits an accretion disk orbiting a \(10^7 \,M_{\odot }\) black hole.
Reid et al. (2019) derive the distance as \(d=7.576 \pm 0.082 \text{(stat) } \pm 0.076 \text{(sys) }\) Mpc, an accuracy of 1.4%, competitive with the LMC distance error. The main statistical contribution to the error budget is the positional error of the \(\sim 300\) spots.
Only 8 megamaser galaxies have been detected, with the furthest being NGC 6264 at 141 Mpc, and it is unlikely more will now be found at usable distances. We illustrate the data for UGC 3789 from Reid et al. (2013) in Fig. 6a, b. The Megamaser Cosmology project (Pesce et al. 2020) have used six of these galaxies to find \(H_0 = 73.9 \pm 3.0 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\), independently of distance ladders.
Although it is too close to determine \(H_0\) directly (as its peculiar motion could be a large fraction of its redshift), NGC 4258 has become a key calibrator of the distance ladder, owing to its low error budget. Its particular usefulness is that, unlike the LMC or SMC, it is a fairly typical barred spiral galaxy, similar in morphology and environmental conditions (metallicity, star-formation rate and so forth) to the ones in which Cepheids and Type Ia supernovae are seen at greater distances. Using Cepheid and SN Ia data from Riess et al. (2016), Reid et al. (2019) find \(H_0 = 72.0 \pm 1.9 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) using solely NGC 4258 as the geometric calibrator.
Cepheids
Having discussed calibrators, we can now talk about the “engine room” of distance ladders. Cepheids form two classes, but it is the younger, population I, classical Cepheids which are of interest. These are yellow bright giants and supergiants with masses 4–20 \(M_{\odot }\) and their brightness cycles over a regular period, between days and months, by around 1 magnitude. They are bright, up to 100,000 \(L_{\odot }\), and they can be seen out to 30 Mpc with the HST. Although Milky Way Cepheids had been observed and catalogued since the 18th century, it was first discovered by Henrietta Swan Levitt in 1908 that there was linear relation between the logarithm of their oscillation period and absolute magnitudes, the Leavitt period-luminosity law (we use the term P-L law for Miras). She had been observing Cepheids in the SMC and LMC, using the Harvard College Observatory telescope, and decided to rank them in order of magnitude. As stars in the LMC will have roughly the same distance from the Earth, the Leavitt law was immediately clear from their apparent magnitudes. In fact, one could say modern extragalactic astrometry began with her groundbreaking discovery.
That stars can pulsate is not so surprising; after all, a star is in local equilibrium, and would be expected to oscillate about the equilibrium given any perturbation. However, something must drive the oscillation otherwise it would dissipate. For Cepheids, the driver is heat-trapping by an opaque layer of doubly-ionised He surrounding a He-burning core. The trapped heat increases pressure, which expands the star, cooling the ionised He to the point where it can recombine and, therefore, becomes transparent. As the radiation escapes, the core cools and re-contracts, which in turn releases gravitational energy into the He layer. The He heats, re-ionises and the cycle repeats, with period proportional to the energy released. The Cepheid population lies in an instability strip in the horizontal branch of the Hertzsprung–Russell diagram; the cool (red) edge of the population is thought to be due to the onset of convection in the He layer, and the hot (blue) edge by the He ionisation layer being too far into the atmosphere for pulsations to occur. They, therefore, form a well-defined population.
A straightforward understanding of the origin of the Leavitt law can be found in thermodynamic and dynamic arguments. The luminosity of a Cepheid will depend on its surface area via the Stefan–Boltzmann law \(L = 4\pi R^2 \sigma T^4 \), which expressed in bolometric magnitudes is
$$\begin{aligned} M = -5 \log _{10} R - 10 \log _{10} T + \text{ const. } \;\;. \end{aligned}$$
(31)
The stellar radius can be mapped to the period by writing an equation of motion for the He layer as
$$\begin{aligned} m \ddot{r} = 4 \pi r^2 p - \frac{G M m}{r^2}, \end{aligned}$$
(32)
where r denotes the radial position of the layer, p is the pressure and m the mass of the layer. For adiabatic expansion, it is then straightforward to solve for the period P and we find \(P \sqrt{{\bar{\rho }}} = \text{ const. }\) where \({\bar{\rho }}\) is the mean density (for further details see Cox 1960). With \({\bar{\rho }} \propto \; R^{-3}\) and temperature mapped to colour \(B-V\), we obtain the Leavitt law as
$$\begin{aligned} M = \alpha \log _{10} P + \beta (B-V) + \gamma , \end{aligned}$$
(33)
where P is the period in days, \(\alpha \), \(\beta \) are empirically calibrated from the Cepheid data, and for the zero-point \(\gamma \) we need a distance measurement.
It is preferable to use so-called Wesenheit magnitudes, which are constructed to be reddening-free, and have a reduced colour dependence (intrinsically redder Cepheids are fainter). Let’s see how this works in practice. Riess et al. (2019) define an NIR Wesenheit magnitude \(m_H^W \equiv m_{\mathrm{F160W}} - 0.386(m_{\mathrm{F555W}} - m_{\mathrm{F814W}})\) using HST filter magnitudes, where the numerical constant is derived from a reddening law. They observed 70 LMC Cepheids with long periods, and after setting \(\beta = 0\), they derive \(\alpha = -3.26\) with intrinsic scatter 0.075 mag. By comparison, the scatter using solely optical F555W magnitudes is 0.312 mag. After subtracting the DEB distance modulus (Pietrzyński et al. 2019), we obtain \(\gamma = -2.579\). The formal error in the Cepheid sample mean is 0.0092 mag, equivalent to 0.42% in distance (Fig. 7).
We would now like to feel that we can deduce the distance to any galaxy we can find Cepheids in, by applying this law to convert periods to absolute magnitudes, and comparing to the apparent magnitudes we observe. As is usual, though, things are not that simple! There are three principle objections:
-
Are Cepheids in the LMC the same as the ones in distant galaxies? The LMC has a lower metallicity than a typical spiral galaxy, so can we expect the Cepheids found there to have the same brightness? Unfortunately, it is hard to determine the metallicity of a Cepheid from its colour alone, and studies on the effects of metallicity are variable (Ripepi et al. 2020). One might try extending the Leavitt law to add a metallicity term \(\kappa \) [Fe/H]. Riess et al. (2019) have estimated Cepheid metallicity in the LMC based on optical spectra of nearby HII regions, and find \(\kappa = -0.17 \pm 0.06\), which is consistent with an earlier estimate from Freedman and Madore (2010) that LMC Cepheids are 0.05 mag dimmer than galactic Cepheids, and later results (Gieren et al. 2018; Breuval et al. 2021).
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Is the Leavitt law linear? This matters for \(H_0\), because the Cepheids seen in distant galaxies have longer periods than those used to calibrate the Leavitt law: they are brighter and more easily observed at distance. So, \(\log {P} \sim 1.5\) for Cepheids in SN Ia host galaxies, whereas the average for the LMC is \(\log {P} \sim 0.8\). If the Leavitt law is curved, a bias would be introduced. This can be dealt with by either introducing a separate calibration for short and long period Cepheids, or selecting only long period nearby Cepheids for calibration.
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Can we obtain clean astrometry of very distant Cepheids? We want to observe Cepheids as far away as we can, to maximise the overlap with the SN Ia observations. But pushing the limits of resolution brings the risk of crowding, whereby the Cepheid photometry is blended with nearby dimmer, redder and cooler stars. Indeed, this seems to be the main cause of the increased scatter in residuals for distant galaxies. There are some ways to deal with crowding: Riess et al. (2016) (hereafter R16) add random Cepheids to images of the same galaxy and put them through the same analysis pipeline, to see how well input parameters are recovered. Figure 8 shows an example of this. Cepheids which are outliers in colour, indicative of high blending, may be discarded (removing the colour cut lowers \(H_0\) by \(1.1 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\)). Another way to test for a crowding bias is to look for compression of the relative flux difference from peak to trough (a more blended Cepheid will be more compressed) as is done in Riess et al. (2020). Using an optical Wesenheit magnitude, in which stars may be less crowded than the NIR one, reduces \(H_0\) by \(1.7 \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (R16), although the argument can be made this is due to higher metallicity effects in the optical (Riess et al. 2019). As with metallicity, the matter of crowding continues to debated.
Systematic error analysis is provided in Sect. 4 and Table 8 of R16, where the effect of some different analytical choices such as breaks in the Leavitt law, different assumed values for the deceleration parameter \(q_0\), and methods of outlier rejection are shown to be \(\pm 0.7\%\). This may not cover all potential systematics. In a recent talk, Efstathiou (2020) re-examined the Leavitt laws of galaxies presented in R16. Calibrating each galaxy individually, the slopes of SN Ia host galaxies are generally shallower than M31 or the LMC, which should not be the case if Cepheids are a single population. A change in the slope will alter the zero-point, as would a change in the distance calibration. It is then noted that forcing the slope of the Leavitt law to \(-3.3\) (the M31 value), in combination with using only NGC4258 as the anchor, lowers the R16 \(H_0\) value to \(70.3 \pm 1.8 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (Equation 4.2b). Additionally, there appears to be tension between the relative magnitudes of Cepheids in the LMC and NGC4258, and the distance modulus implied by Masers and DEBs; calibrating with solely the LMC gives a higher \(H_0\) value by \(4.4 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). It is a feature of \(\chi ^2\) fits (as used in R16) that the joint solution will be drawn to the data with the lowest error, which in this case is the LMC value. But if two subsets of the data are in tension, it is uncertain that the one with the lowest \(\chi ^2\) would have the least systematics.
Such analyses do not show one value is preferable over another, nor can they show where any discrepancy may lie—it might be metallicity effects, crowded Cepheid photometry, the NGC4258 distance, or some other systematic. Re-analyses of the results of R16 by various authors (Feeney et al. 2018; Cardona et al. 2017; Zhang et al. 2017; Follin and Knox 2018; Dhawan et al. 2018) use the same Cepheid photometric reduction data so are not independent as such.
Research has accelerated to close down these potential issues. We have already mentioned replacing LMC and NGC4258 Cepheids with Milky Way Cepheids in the section on parallax. In a recent paper, Riess et al. (2020) show crowding effects can be detected as a light curve amplitude compression, and that their correction method has been robust. Finally, Javanmardi et al. (2021) fully re-derive the Cepheid periods and luminosities from the original HST imaging for NGC 5584 (which is face-on to the line of sight), intentionally using different analytical choices, finding no systematic difference in the light curve parameters.
All this said, it would be clearly preferable if we had some other candles to check against Cepheids. We now turn to two possibilities, Miras and the Tip of the Red Giant Branch.
Miras
Miras are variable stars that have reached the tip of the Asymptotic Giant Branch (AGB), comprising an inert C-O core, and a He-burning shell inside a H-burning shell. Their mass is 0.8–8 \(M_{\odot }\), although they are typically at the lower end of this range. Their large size of \(\sim 1\) a.u. means they are actually brighter by 2–3 mag than Cepheids in the NIR. That they follow a P-L law was first established in 1928 (Gerasimovic 1928), but being tricky to categorise and observe, were not extensively studied until the demand for cross-checks on Cepheids has brought renewed interest in them.
Miras form two distinct populations with O-rich and C-rich spectra, with the O-rich ones exhibiting somewhat less scatter than the C-rich (Feast 2004), but more than Cepheids. Miras have very long periods \(90<P<3000\) days and their light curves have many peaks of variable amplitude superimposed (see for example Fig. 3 in Yuan et al. 2017), so investment in observation time is needed to reliably determine P. The P-L law curves upwards at around \(P \sim 400\) days, where it seems likely the extra luminosity has been due to episodes of Hot-Bottom-Burning, so an extra quadratic term is required. Miras have high mass loss rates, and the surrounding dust cloud means Wesenheit magnitudes will be less reliable at subtracting reddening compared to Cepheids (because a standard reddening law is assumed in constructing them). Lastly, a size of 1 a.u. would mean their angular diameter is comparable to their parallax, and in addition the photocentre moves around the star, making parallax measurements challenging.
So why bother with them? Their advantages versus Cepheids is that they are (a) more numerous, and, therefore, easier to find in halos where there is less crowding (b) older, so can be found in all types of galaxies including SN Ia hosts with no Cepheids (c) \(2-3\) magnitudes brighter than Cepheids in the near infrared. Given imaginative observation strategies and careful population analysis, the issues above can be addressed, and modern studies now exist for Miras in the LMC (Yuan et al. 2017), NGC 4258 (Huang et al. 2018), and the SN Ia host NGC 1559 (Huang et al. 2020).
In the first of the above, Yuan et al. (2017) establish a P-L curve using 600 Miras in the LMC. They have sparse JHK-band observations from the LMC Near Infrared Synoptic Survey (LMCNISS, Macri et al. 2015), which on their own would not be enough to establish the period, but using much denser Optical Gravitational Lens Experiment (OGLE) I-band observations (Szymański et al. 2011), they are able to establish a regression rule for the relationship between passbands, and use OGLE to “fill in” the light curves. The classification of Miras into O-rich or C-rich is also obtained from OGLE. The reference magnitude is defined as the median of maxima and minima of the fitted light curves. The period is obtained by fitting to a sum of sine functions, progressively adding harmonics if supported by a Bayesian Information Criterion.Footnote 7 They fit a lawFootnote 8
$$\begin{aligned} M = a_0 + a_1 (\log _{10} P - 2.3) + a_2 (\log _{10} P - 2.3)^2 . \end{aligned}$$
(34)
The K-band for O-rich Miras shows the lowest scatter of 0.12 mag, with \(a_0 =-6.90 \pm 0.01\), \(a_1 = -3.77 \pm 0.08\), \(a_2 = -2.23 \pm 0.2\). The zero-point is obtained using the DEB distance to the LMC given by Pietrzyński et al. (2013). By comparison, the \(m_H^W\) scatter for LMC Cepheids obtained by SH0ES was 0.075 mag (Riess et al. 2019) (Fig. 9).
Huang et al. (2018) observed NGC4258 with the HST WFC in 12 epochs over 10 months. Mira candidates were identified by their V and I amplitude variation, and it was possible to use LMC data to show that using this method contamination between O-rich and C-rich could be kept to a minimum. Their “Gold” sample size was 161 Miras, and fitting Eq. (34) for apparent magnitudes gave \(a_0 = 23.24 \pm 0.01\) for F160W with scatter 0.12. Adjusting the ground-based photometry of LMCNISS to F160W equivalent, the authors calculate the relative distance modulus between the LMC and NGC4258 of \(\varDelta \,\mu = 10.95 \pm 0.01 \text{(stat) } \pm 0.06 \text{(sys) }\). This is consistent with the Cepheid value of \(\varDelta \,\mu = 10.92 \pm 0.02\) (Riess et al. 2016) and the Maser-DEB value of \(\varDelta \,\mu = 10.92 \pm 0.041\) (Reid et al. 2019; Pietrzyński et al. 2019). Similar methods are used by Huang et al. (2020) for observations of NGC 1559, with the addition of a low period cut to deal with potential incompleteness bias of fainter Miras. Using the LMC DEB and NGC 4258 distances as calibrators, they obtain \(H_0 = 73.3 \pm 4.0 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). So Miras are consistent with Cepheids, but due to their larger error budgets, they are also consistent with Planck results.
Tip of the Red Giant branch
Red Giant branch stars are stars of mass \(\sim 0.5 {-} 2.0 M_{\odot }\) that are at a late stage in their evolution, having moved off the main sequence towards lower temperatures and higher luminosities. The core of the star is degenerate He, and as He ash rains down from the H-burning shell surrounding it, it contracts and heats up. Since the temperature range for He fusion ignition is rather narrow, and the degenerate matter means there is a strong link between core mass and temperature, the core in effect forms a standard candle inside the star, just prior to ignition. As soon as the helium flash ignites, the star moves over the course of 1 My or so (almost instantaneously in stellar evolution terms) to the red clump at a higher temperature and somewhat lower luminosity. Therefore, there is a well-defined edge in a colour-magnitude diagram that can be used to identify the tip of the red giant population just before the flash (TRGB). The TRGB is then not one single star, but a statistical average for a suitably large population.
The TRGB offers many advantages compared to other standard candles. Although red giants are fainter than Cepheids in the optical, in the K-band they are \(\sim 1.6\) magnitudes brighter. Stars at or near the TRGB are relatively common, and can be observed readily in uncrowded and dust-free galactic halos. They are also abundant in the solar neighbourhood, meaning great numbers are available for calibration by parallax (by contrast, Cepheids with good parallaxes in Gaia DR3 will number in the hundreds). Sufficient numbers to resolve the tip can be seen in globular clusters such as \(\omega \) Centuari, where well-formed colour-magnitude diagrams can resolve metallicity and extinction effects, and an absolute magnitude calibration can be made either from Gaia DR3 parallaxes (Soltis et al. 2021) or DEB distances (Cerny et al. 2021).
Like Miras, they are found in all galactic types. Observation is efficient: one does not need to revisit fields to determine periods. The James Webb Space Telescope is capable of observing red giants in the near IR at distances of \(\sim 50 \,\hbox {Mpc}\), which is comparable to Cepheid distances obtainable from the HST.
It is also beneficial that the modelling of late-stage stellar evolution is quite well understood. Empirical calibrations of absolute magnitude and residual dependence on metallicity, mass and age can be checked against the results of stellar codes. For example, the dependency on metallicity is somewhat complex: the presence of metals dims optical passbands by absorption in stellar atmospheres, and brightens the NIR. Serenelli et al. (2017) express the TRGB as a curve with linear and quadratic terms in colour. The authors find an I-band median \(M_I^{\mathrm{TRGB}} = -4.07\) with a variation with colour of \(\pm 0.08\), and a downward slope in the colour-magnitude diagram as metallicity increases. This median value is consistent with empirical calibrations.
Fitting the TRGB is equivalent to finding the mode of the gradient of the (noisy) stellar distribution in the colour-magnitude diagram, and techniques are borrowed from image processing to do this. The tip has a background of AGB stars, and the edge is sensitive to the distribution of them close to it. If the contamination is large or variable, there is a risk the mode can shift. It is, therefore, important to have a dense population of stars, and enough fields to test the robustness of the fitting process. For example, Hoyt et al. (2021) bin data by separation from the galactic centre and fit the tip separately for each bin.
The Carnegie-Chicago Hubble program (CCHP) have determined TRGB distances for 10 SN Ia host galaxies that also have Cepheid distances, using HST I-band imaging (Freedman et al. 2019). They calibrate the zero point from the TRGB of the LMC and SMC using the DEB distance, finding \(M_I = -4.049 \pm 0.022 \;\text{(stat) } \pm 0.039 \;\text{(sys) }\), which is consistent with the theoretical value above (Fig. 10).
We show in Fig. 11 their comparison of TRGB and Cepheid distances, covering a range of distances from 7 Mpc to almost 20 Mpc. By expanding their sample to non-SN Ia host galaxies, the authors show the TRGB has a lower scatter versus their Hubble diagram than do Cepheids, by a factor of 1.4. Calibrating the Pantheon SN Ia sample on TRGB distances alone gives \(H_0 = 70.4 \pm 1.4 \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\), with the difference to SH0ES results due to the TRGB distances to the SN Ia host galaxies being further than Cepheid distances.
Currently, this result remains the subject of debate. Yuan et al. (2019) find \(M_{I} = -3.99\), which gives \(H_0 = 72.4 \pm 2.0 \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). They attribute their result to a different methodology to determine the LMC extinction. Freedman et al. 2019 determine the extinction applicable to the TRGB by comparison of LMC and SMC photometry in VIJHK bands, and using the lower SMC extinction as a reference point. Yuan et al. 2019 adopt the standard reddening law of Fitzpatrick 1999). They also revise the blending corrections of the older, lower resolution photometry of the SMC. Conversely, a calibration of the TRGB in the outskirts of NGC 4258 to the maser distance by Jang et al. (2021) gives a value fully consistent with Freedman et al. (2019). In a followup paper, Madore and Freedman (2020) present an upgraded methodology in which stellar photometry is itself used to separate metallicity and extinction effects (by exploiting their different sensitivities in J, H and K-band magnitudes), confirming their earlier results. Other results (Soltis et al. 2021; Reid et al. 2019; Capozzi and Raffelt 2020; Cerny et al. 2021—see for example Table 3 in Blakeslee et al. 2021) cluster evenly around these two values. We also note the difference between SN Ia zero points (Fig. 6 of Freedman et al. 2019) of the TRGB vs Cepheids appears to increase with distance, and is particularly large for one galaxy, N4308.
More recently, an updated value of \(H_0 = 69.8 \pm 0.6 \;\text{(stat) } \pm 1.6 \;\text{(sys) } \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) was published by Freedman (2021). New and updated values for \(M_I\) based on the anchors of the LMC, SMC, NGC 4258 and galactic globular clusters are derived. Additionally, potential zero-point systematic problems with the direct application of Gaia EDR3 data to globular clusters (as done by Soltis et al. 2021) are discussed. Each anchor is consistent with each other (although the closeness of the agreement suggests the errors may be over-estimated), and the paper brings the TRGB method on par with Cepheids in terms of number of anchors used.
The TRGB continues to attract attention because the CCHP result is very interesting: it is a late universe distance ladder that is in reduced \(2\sigma \) tension with the Planck value of \(H_0 = 67.4 \pm 0.5 \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). While it cannot by itself fully resolve the Hubble tension, in combination with some other systematic—perhaps in the calibration of SN Ia (see next section)—it may offer a resolution of it that does not involve new physics.
What is also clear is that quality of a distance ladder is less a matter of quantity of objects, but rather the accuracy of calibration and control of systematics. The TRGB seems very promising, however. Once the calibration is agreed among the community, they offer the tantalising prospect of either confirming the \(H_0\) tension through two independent data sets, or reducing it to a lower statistical significance.
Type Ia supernovae
It is hard to overstate the impact of Type Ia supernovae in cosmology. At \(M \sim -19\), they are both bright enough to be seen well into the Hubble flow (the furthest to date is SN Wilson at \(z = 1.914\)), and have a standardisable luminosity. The mainstream view is that they originate from the accretion of material from a binary companion onto a carbon-oxygen white dwarf. When the white dwarf reaches the maximum mass that can be sustained by degeneracy pressure, the Chandrasekhar mass \(M_{\mathrm{ch}} = 1.4 M_{\odot }\), a runaway fusion detonation occurs destroying the white dwarf. Approximately \(\sim 0.6 M_{\odot }\) fuses into heavier elements during the first few seconds of the explosion, and the observed light curve, which peaks at \(\sim 20\) days and lasts for \(\sim 60\) days is powered by radioactive decay of \(^{56}\)Ni to \(^{56}\)Co and then to \(^{56}\)Fe. Hence, a SN Ia is a “standard bomb”, primed to explode when it reaches critical mass.
However, there is surprisingly little observational evidence to confirm the accretion theory. Both the lack of H lines and computer modelling suggest the donor star should survive the explosion, but a search of the site of the nearby SN2011fe both pre- and post- explosion have revealed no evidence of a companion. This may support an alternative “double degenerate” theory that some, or all, SN Ia originate from the merger of two white dwarves (for a review, see Maoz et al. 2014). But if that is the case, why are their luminosities so uniform? Whether SN Ia have one or two types of progenitors has important implications for the Hubble constant, which we return to shortly.
Individual SN Ia luminosities can vary by up to a factor of 2, but they are empirically standardisable by the Tripp estimator (Tripp and Branch 1999):
$$\begin{aligned} \,\mu = m_b - M_{\mathrm{fid}} + \alpha x - \beta c + \varDelta _M + \varDelta _B, \end{aligned}$$
(35)
where \(\,\mu \) is the distance modulus, \(m_b\) ia the apparent AB magnitude, and \(M_{\mathrm{fid}}\) is the absolute magnitude of a typical SN Ia. The parameter x is the “stretch” of the light curve, which is a dimensionless measure of how long the bright peak of the light curve lasts: longer duration SN Ia are brighter. c is a measure of colour: redder SN Ia are dimmer. \(\varDelta _M\) is a correction for the environment of the SN Ia, and \(\varDelta _B\) a statistical bias correction for selection effects, analogous to the LKH bias of parallax we described earlier. \(M_{\mathrm{fid}}\) can be calibrated by finding SN Ia in galaxies for which there are Cepheid or TRGB luminosity distances, or in the inverse distance ladder approach by combination with angular diameter distances from baryon acoustic oscillations, or the CMB.
SN Ia are rare events: approximately 1 per galaxy per century, but by scanning a reasonably sized patch of sky, supernova surveys can detect hundreds per year.Footnote 9 Hence, supernovae catalogs are not uniformly distributed on the sky. However, SN Ia close enough to calibrate their magnitudes are few: there are just 19 SN Ia in galaxies with Cepheid distances from HST observations, although more are expected soon from new cycles (see for example HST proposal 16,198).
To maximise statistics and sky coverage, it is common to use SN Ia from multiple surveys in cosmological analyses. Scolnic et al. (2018) have compiled the Pantheon sample of 1,048 SN Ia from CfA, CSP, PS1, SDSS, SNLS and HST surveys in a uniform light curve calibration, representing almost 40 years of observational data. The sample divides into 180 mostly older Low-z (\(z < 0.1\)) SN Ia which are uniformly distributed on the sky, and 868 recent High-z SN Ia concentrated in the survey fields, notably the thin “Stripe 82” along the celestial equator. The sets have a small overlap at \(z \sim 0.1\), and cross-checks on their spectral and other parameters show they are likely to be from the same underlying population (that is, High-z SN Ia are not intrinsically different to Low-z). However, the mean values of light curve parameters like stretch and colour do drift with z, possibly due to selection effects or changes in the host galaxy properties at higher redshift. Large numbers of Type Ia supernova have also been found, or are in the process of being surveyed, by the Dark Energy Survey (Abbott et al. 2019), All-Sky Automated Survey for Supernovae (Holoien et al. 2018), Foundation Survey (Jones et al. 2019), and Zwicky Transient Facility (Yao et al. 2019).
The process of fitting a SN Ia lightcurve is straightforward. ugriz photometry is corrected for the background sky obtained from imaging the site after the nova has faded (a bonus of SN Ia being transient), and the light curve is fitted to a template, outputting the parameters x, c and \(m_b\). However, the bias correction \(\varDelta _B\) is challenging and subtle: it depends on the astrophysical modelling of sources of scatter and contamination by mis-classified supernovae, and the selection function of survey. For example, more distant SN Ia may be preferentially targeted for spectroscopic confirmation when their host galaxy is faint, so if their luminosities do depend on the galactic environment, this could bias the High-z population relative the to Low-z. The selection function of early Low-z surveys are not easy to model, and their biases can be up to 0.06 magnitudes. This could account for a somewhat elevated scatter of Pantheon residuals around \(z \sim 0.1\) in the Hubble diagram where the two sets join (Fig. 12). After fitting \(\alpha \) and \(\beta \), the residual intrinsic scatter of the SN Ia population is \(\sigma _{\mathrm{int}} \sim 0.1\) magnitudes, which compares favourably with other standard candles.
We return now to the question of whether SN Ia have additional environmental correlations that affect their luminosity (that is, what the nature of the \(\varDelta _M\) term in Tripp estimator is), and why this may matter for \(H_0\). It is generally accepted that SN Ia in galaxies with stellar mass \(M_{*} > 10^{10} M_{\odot }\) are intrinsically brighter by \(\varDelta _{M,*} \sim 0.04 - 0.06\) than those in lower mass galaxies, and that the transition is sharp, rather than a gradual evolution (see Smith et al. 2020 and references therein). It has, therefore, become common to use a step-function for \(\varDelta _{M,*}\) in fits for \(H_0\). Riess et al. (2016) calculate the effect is to lower \(H_0\) by 0.7%, as the mass of the calibrator galaxies is slightly lower than the mass of the Hubble flow set. The key point is then this: given that Cepheids are young, bright stars formed in active galaxies, is \(\varDelta _{M,*}\) sufficiently discriminating to ensure the 19 supernovae in galaxies with Cepheid distances are representative of the thousands in the Hubble flow?
Host galaxy mass itself cannot matter to an individual SN Ia, so it must be a proxy for some other environmental condition. Metallicity does vary with galactic mass, but the step-like feature suggests a connection with star formation: galaxies with mass below the threshold tend to be active, whereas those above are mostly passive. One hypothesis is that there is a prompt/bright SN Ia type that is continuously renewed by star formation, and an delayed/dimmer type originating from stars formed when the galaxy was young (Rigault et al. 2015, 2013; Maoz et al. 2014). But any link between this and SN Ia formed by accretion onto or mergers of white dwarfs is speculative. Also, with estimates of “global” properties being effectively light-weighted, they are biased to galactic centres rather than the outskirts where most supernovae are seen.
If some SN Ia are associated with sites of star-formation, we can expect a better proxy to be local specific star formation rates (that is, star formation normalised by local stellar mass or LsSFR for short), on the grounds that the younger SN Ia population will not have had enough time to disperse from their birth region. Because the pixel resolution of typical SN Ia surveys can resolve a 2 kpc aperture only out to \(z \sim 0.1\), studies concentrate on the Low-z sample. Star formation can be estimated from photometry, but is best tracked by \(\hbox {H}_{\alpha }\) lines from ionised gas or UV imaging, when available.
Rigault et al. (2020) have examined 141 SN Ia from the Nearby Supernova Factory sample (which has high resolution spectroscopy) and find a significant correlation between LsSFR and \(\alpha \), the stretch slope, and no correlation for \(\beta \). This suggests there is indeed an intrinsic difference between SN Ia originating in active areas versus passive ones. The size of the LsSFR step is \(\varDelta _{M,LsSFR} = 0.163 \pm 0.029\) mag, and including it in the Tripp luminosity estimator eliminates the need for a further global host mass step. The size of the step is interesting as it is significantly larger than \(\varDelta _{M,*}\), and because most SN Ia in the Cepheid calibrators are in regions of high LsSFR, whereas in the rest of the Low-z sample the fraction is about a half. Hence, a potential bias to \(H_0\) could be as much as 3%. In contrary results, Jones et al. (2018) find a local stellar mass step the same size as \(\varDelta _{M,*}\) and little effect on \(H_0\), and Riess et al. (2016) found just \(0.1 \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) difference in their \(H_0\) calculation after restricting to SN Ia found in spiral galaxies. In a recent paper, Brout and Scolnic (2021) argue the true factor is host galaxy extinction (correlated to star formation and hence mass), and present evidence of a distribution separated by colour into blue “clean” SN Ia with lower scatter than their red “dusty” counterparts. The host mass step is absent in the blue sample, and they suggest to use solely these in cosmological analyses to avoid bias (another possibility is to use near-infrared photometry; Dhawan et al. 2018).
In summary, while there is strong evidence SN Ia depend on local specific star formation rates, there is not yet a consensus on the cause, whether they constitute one or two populations, and the level of bias (if any) to \(H_0\). They are numerous and, therefore, high relative precision beyond \(z>0.1\), and have been dubbed the “guard rails” of the Hubble diagram. However, their absolute accuracy in distance ladders is limited by the small numbers available to calibrate their luminosities, and some uncertainty about the underlying astrophysics. Work is underway to address both of these issues: for example, the Foundation and Zwicky Transient Facility surveys (Jones et al. 2019; Yao et al. 2019) are targeting a new high-quality Low-z set with local spectroscopy to test environmental effects or extensions to the Tripp estimator. New HST observation cycles aim to extend the number of Cepheid calibrators up from 19, and the JWST will access a greater volume, perhaps increasing the calibration set beyond a hundred for both TRGBs and Cepheids. This would make a 1% calibration from at least two independent sources achievable.
Time delay of lensed quasars
Quasars are bright and can be variable on short timescales. Some are seen to be gravitationally lensed by a foreground galaxy, and each image will have a time delay caused by the extra path length and the gravitational time dilation of general relativity:
$$\begin{aligned} \varDelta t_i = D_{\varDelta t}\left[ \frac{1}{2} (\mathbf {\theta _i} - \mathbf {\beta })^2 - \psi (\mathbf {\theta _i}) \right] , \end{aligned}$$
(36)
where \(\mathbf {\theta _i}\) is the sky position of the image, \(\mathbf {\beta }\) the (unknown) source position, and \(\psi (\mathbf {\theta _i})\) the potential of the lensing galaxy integrated along the line of sight. The relevant quantity for \(H_0\) in the above equation is
$$\begin{aligned} D_{\varDelta t} \equiv (1+z_l) \frac{D_l D_s}{D_{ls}} \propto H_0 ^{-1} , \end{aligned}$$
(37)
where \(D_l, D_s, D_{ls}\) is the angular diameter distance to the lens, source, and between the lens and the source, respectively, as specified by Equations (11, 14)—in the case of \(D_{ls}\) the integration would be done between the redshifts of the lens and source. Then, if we have a system with multiple images, the relative time delay \(\varDelta t_{ij} = \varDelta t_i - \varDelta t_j\) of the quasar variability between the images scales with \(H_0^{-1}\). The proportionality follows from Eqs. (11, 14) with a weak dependence on other cosmological parameters. Acquiring the time delays \(\varDelta t_{i}\) requires extensive observations: relative delays range from weeks to a few months. Errors are typically \(\sim 6\%\), although these are reducing over time. A typical lensing system may have \(z_s \sim 1.5\) and \(z_l \sim 0.5\), so we are probing intermediate cosmological distances.
Each single lens system may provide an independent estimate of \(H_0\). But to do so, we need an idea of the lensing potential \(\psi (\mathbf {\theta _i})\) above. There is also a tricky degeneracy to deal with, known as the mass sheet transformation. For our purposes, it is sufficient to understand that the same observed time delays may be produced by simultaneously scaling the line-of-sight surface density \(\varSigma (\mathbf {\theta })\) of the lensing system by a constant factor \(\varSigma (\mathbf {\theta })\rightarrow \epsilon \varSigma (\mathbf {\theta })\) (the “mass-sheet”) and \(H_0 \rightarrow \epsilon H_0\). For a clear account of the details, see Falco et al. (1985).
The ingredients for modelling are then this: (a) a range of plausible mass profiles for the lensing galaxy with associated nuisance parameters to be marginalised over (b) a way to deal with the mass sheet degeneracy, through independent constraints on the mass distribution of the lensing galaxy and along the line-of-sight.
This is a challenging, time-consuming and model dependent process for each system! Some other choices must be made, such as which of the galaxies along the line of sight will be modelled and which will be absorbed into the line of sight mass average (Chen et al. 2019). Galactic stellar velocity dispersions provide information on the lensing potential, but as a function of which galactic mass profile is assumed, and with an anistropy nuisance parameter. Also, the accuracy of velocity dispersions at such distances are \(\sim 10\%\). If there are visible background galaxies, they may also be lensed and the shape distortions provide extra information on the mass profile. The line of sight mass may be estimated from the density of foreground galaxies, by searching in computer simulations of large scale structure for similar lines of sight, and calculating the line of sight density from them (lenses are associated with over-densities and, therefore, focusing is more likely). But a major advantage is that this modelling can be done blind to final value of \(H_0\), eliminating the risk of confirmation bias on the part of the experimenter.
The H0LiCOW team have analysed six such systems so far (Wong et al. 2020), and we show their results below. The main contributions to the errors are uncertainty in the time delay, and velocity dispersion of the lens. H0LiCOW use flat priors of \(\varOmega _{M} \in (0.05, 0.5)\) and \(H_0 \in (0,150) \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\), to avoid any dependence on other cosmological datasets (although they do assume a flat universe). The results are:
|
Lens name
|
\(D_{\varDelta t}\) (Mpc)
|
\(D_l\) (Mpc)
|
\(H_0\;(\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1})\)
|
|---|
|
PG1115+080
|
\(1470^{+137}_{-127}\)
|
\(697^{+186}_{-144}\)
|
\(81.1^{+8.0}_{-7.1}\)
|
|
RXJ1131-1231
|
\(2096^{+98}_{-83}\)
|
\(804^{+141}_{-112}\)
|
\(78.2^{+3.4}_{-3.4}\)
|
|
HE0435-1223
|
\(2707^{+183}_{-168}\)
|
-
|
\(71.7^{+4.8}_{-4.5}\)
|
|
WFI2033-4723
|
\(4784^{+399}_{-248}\)
|
-
|
\(71.6^{+3.8}_{-4.9}\)
|
|
B1608+656
|
\(5156^{+296}_{-236}\)
|
\(1228^{+177}_{-151}\)
|
\(71.0^{+2.9}_{-3.3}\)
|
|
SDSS1206+4332
|
\(5769^{+589}_{-471}\)
|
\(1805^{+555}_{-398}\)
|
\(68.9^{+5.4}_{-5.1}\)
|
Their combined result is \(H_0 = 73.3 \pm 1.8 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\), a final error budget of 2.4%. However, a key assumption in the combination is independence, and there are many modelling features in common between the lenses, in particular the specific mass profiles used. There is also an apparent drift in \(H_0\) values with lensing distance, although the statistical significance is not strong enough to say this is not just random chance.
A re-analysis has been undertaken by the TD Cosmo team (Birrer et al. 2020). By treating the mass sheet degeneracy as a fully unknown parameter at the population level, their analysis reduces reliance on specific mass profiles. For same set of systems, they find \(H_0 = 74.5^{+5.6}_{-\,6.1} \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). The central value is consistent with H0LiCOW, but the wider confidence intervals reflect the greater freedom allowed on lens mass distribution, combined with the small number of time-delay lenses. To re-narrow the uncertainty range, the paper investigates what effect introducing a set of non-time delay lensing systems from the SDSS survey may have. The purpose of this larger set is to use their imaging to constrain the mass-profile of lensings galaxy, on the assumption they are drawn from the same parent population as the H0LiCOW sample. TD Cosmo then find \(H_0 = 67.4^{+4.1}_{-\,3.2} \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). However, the H0LiCOW sample is selected as those systems capable of producing a clear time-delay signal, which may not have the same characteristics as those selected as having clean shear images. So the assumption they share the same mass profile as each other may not be valid.
It is, therefore, premature to claim lensed quasars are consistent with either SH0ES or Planck. More data is needed: upcoming surveys by the Vera Rubin Observatory, Euclid and Nancy Grace Roman telescope will result in several hundred new time delay systems being discovered (so improvements in analysis efficiency will certainly be needed!). Adaptive optics can provide spatially resolved velocity dispersions to improve the mass models. TD Cosmo estimate 200 extra systems will be needed to constrain \(H_0\) within \(1.2\%\), which may be within reach in the next five years.
Gravitational waves
The gravitational wave signal emitted by the merger of two compact objects can be used as a self-calibrating standard candle. There are now operational detectors at LIGO Hanford and LIGO Livingston in the USA, Virgo in Italy, and KAGRA in Japan, with a further planned LIGO India (Abbott et al. 2020). The detectors measure the strain amplitude of a gravitational wave using laser interferometry to detect the miniscule changes in the length of perpendicular beams as a wave passes by. The purpose of the two sites in the USA is to filter out local seismic vibrations. The wave amplitude is related to the chirp mass \({\mathcal {M}}\) which is in turn derivable from the waveform calculated for a merger. A simplified form of the relevant equations are
$$\begin{aligned} {\mathcal {M}}&= \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}} \end{aligned}$$
(38)
$$\begin{aligned}&= \frac{1}{G} \left[ \frac{5}{96} \pi ^{-8/3}f^{-11/3}{\dot{f}} \right] ^{3/5} \end{aligned}$$
(39)
$$\begin{aligned} {\mathcal {M}}_z&= (1+z_{\mathrm{obs}}) {\mathcal {M}} \end{aligned}$$
(40)
$$\begin{aligned} h(t)&= \frac{{\mathcal {M}}_{z}^{5/3} f(t)^{2/3}}{d_L} F(\theta , i) \cos {\varPhi (t)} \;, \end{aligned}$$
(41)
where f is the frequency, \(m_1\) and \(m_2\) the merging masses, \(\varPhi (t)\) the phase, and h(t) the measured dimensionless strain of the strongest harmonic (Abbott et al. 2016). The rest-frame chirp mass is redshifted by \(z_{\mathrm{obs}}\), and F is a function of the angle between the sky position of the source and detector arms, and the inclination i between the binary orbital plane and line of sight (Arun et al. 2009). Note that as we are measuring amplitude rather than energy flux, \(h(t) \propto d_L^{-1}\).
In a loose sense, every wave cycle is a measurement of the chirp mass \({\mathcal {M}}\) as it sets \(f, {\dot{f}}\), although in practice the full waveform is fitted. The relative precision is floored by sensitivity of the strain measurement, currently around \(5\%\). If the redshift and angles were known, \(d_L\) and hence \(H_0\) would be determined to the same precision.
When a binary neutron star (BNS) system merges, there is an accompanying burst of light from matter outside the combined event horizon. For this reason, it is known as a “bright siren”. If the flash can be observed, the host galaxy is identified and one can use its redshift in Eq. (41). The event GW170817 was just such a BNS merger (Abbott et al. 2017b). The gravitational wave was measured in Hanford and Livingston, which was enough to locate the sky position to 28 \(\text{ deg}^2\) (see next paragraph for how). Given the search region, an optical counterpart was found in NGC 4993 at a distance of \(\sim 40\) Mpc.Footnote 10 Around 3000 cycles of the wave resolved the chirp mass in the detector frame as \({\mathcal {M}} = 1.197M_{\odot }\) to accuracy of 1 part in \(10^3\), consistent with a BNS merger. The main remaining uncertainty is then the inclination angle i. Using a flat prior for \(\cos {i}\), Abbott et al. (2017a) obtain \(H_0 = 70.0^{+12}_{-8} \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). Hotokezaka et al. (2019) improved this to \(H_0 = 70.3^{+5.3}_{-\,5.0} \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) by constraining i with observations of the interaction of the ejecta with the ISM.
For black hole mergers, no optical counterpart is generated, and these are called “dark sirens”. However, it is still possible to constrain \(H_0\) using them if a probable redshift can be estimated. To do this, the relative amplitudes and time delay between detectors located around the globe are used to approximately determine the sky position (Soares-Santos et al. 2019). For example, if 3 detectors observe the wave with perfect accuracy, the two independent time delays and three measured amplitudes will in turn determine the two sky position angles, two polarisation amplitudes, and one phase lag between polarisations. Hence, simultaneous detections of an event are essential in narrowing the size of the region on the sky where the source is located. A reasonable prior for \(H_0\) will constrain the redshift range, and hence determine a localisation volume. Given a suitably complete galaxy catalog with sufficient sky coverage (the best available currently being SDSS and DES), galaxies within this volume can be averaged over with a suitable weighting to determine a value for \(H_0\) (Schutz 1986). Although each individual event is not very accurate, there are many more black hole mergers than neutron star mergers, so the errors are competitive in aggregate. It is also straightforward to mix dark and bright sirens to produce a combined result (Palmese et al. 2020) (Fig. 13).
As each event’s inclination angle is independent, averaging over N events will improve the errors relative to a single event by a factor of \(N^{-1/2}\). The main limiting factor is then detector calibration and sensitivity, which determines the event rates. Work is underway at the LIGO sites to improve this, using quantum engineering techniques such as squeezed light.Footnote 11 Chen et al. (2018) estimate a 5yr observing run by the upgraded LIGO, Virgo, KAGRA and LIGO India detectors will be enough to measure \(H_0\) to \(1\%\) precision by 2030.
In summary, the appeal of gravitation waves is that if one believes General Relativity correct, they are self-calibrating systems independent of the distance ladder and with minimal astrophysics input. Event rates and improvements in detector calibration are sufficient to converge to a \(\sim 2\%\) precision within the next decade.
Baryon acoustic oscillations
Baryon acoustic oscillations (BAO) are the relic of density fluctuations propagating in the pre-recombination universe. The baryons in an initial localised seed of overdensity (such as would originate from inflation) are subject to radiation pressure and expand outwards at the speed of sound \(c_s\), leaving the dark matter behind. At recombination, temperatures have dropped enough so that protons and electrons can combine to neutral hydrogen and the Thomson scattering rate quickly drops below the Hubble expansion rate. At the drag redshift \(z_{\mathrm{drag}}\) shortly after, baryons are released from radiation pressure and the baryon overdensity shell is “frozen in” at a characteristic distance relative to the central dark matter overdensity. This distance
$$\begin{aligned} r_d = \int _{z_{\mathrm{drag}}}^{\infty }\frac{c_s(z)}{H(z)} \mathrm{d}z, \end{aligned}$$
(42)
in comoving coordinates is what is referred to as the sound horizon. As the universe continues to evolve, both the baryon shell and the central dark matter overdensity attract further gravitational infall of matter, forming galaxies. Thus, the sound horizon becomes visible as a characteristic (statistical) physical separation between galaxies. The sound speed \(c_s = 1/ \sqrt{3(1+R)}\) depends on the baryon-to-photon ratio with \(R = 3 \rho _b/4 \rho _\gamma \). \(z_{\mathrm{drag}}\) has weak cosmological dependence, and Planck data gives \(z_{\mathrm{drag}} = 1059 \pm 0.3\). The key point is that given a cosmological model, \(r_d\) is calculable and its scale is large enough that the evolution of structure from \(z_{\mathrm{drag}}\) to \(z \sim 1\) is nearly linear. Thus we can compare observations of galaxy clustering in the late universe to \(r_d\). Armed with data for \(r_d\) we can use Eq. (42) above to solve for \(H_0\) in our model.
The theory of measuring BAO was largely worked out by the mid 1990s (Feldman et al. 1994). One assumes galaxies are a Poisson sample of the relative matter density field \(1+\delta (\mathbf {r})\) and so
$$\begin{aligned} P({\text{ Volume element }} \delta V {\text{ contains a galaxy}}) = \delta V {\bar{n}} (1+\delta (\mathbf {r})), \end{aligned}$$
(43)
where \({\bar{n}}\) is the expected mean space density of galaxies. It is possible to work either with the two-point correlation function \(\xi (\mathbf {r}) \equiv \langle \delta (\mathbf {r}^{'}) \delta (\mathbf {r}^{'}+\mathbf {r}) \rangle \) or equivalently the power spectrum
$$\begin{aligned} P(\mathbf {k}) \equiv \int d^3 r \xi (\mathbf {r}) \exp {(i \mathbf {k}\cdot \mathbf {r})}. \end{aligned}$$
(44)
It might seem preferable to use the power spectrum, as we are looking for a characteristic wavelength, and the modes of the power spectrum are independent if the density field is Gaussian. Also, the galaxy survey footprint in real space becomes an easily understood point spread function in Fourier space. In practice though, because of issues with binning such as cut-offs in Fourier space, most research papers calculate both the correlation function and power spectrum. Adjusting \({\bar{n}}\) above can correct for bias introduced by edge effects of angular and redshift cuts of the survey.
Expanding the power spectrum in spherical harmonics, the monopole determines \(d_A^2 H^{-1} (z)\), and the quadropole determines \(d_A H (z)\) (Padmanabhan and White 2008) in a given redshift bin centred at z. With \(\varDelta \theta \) and \(\varDelta z\), the density peak separation transverse to and along the line of sight, we can solve for \(d_A\) and H. We start with a mock galaxy catalog constructed in a fiducial cosmology seeded with density perturbations. This is matched to our real data with stretches \(\alpha _{\perp }, \alpha _{\parallel }\):
$$\begin{aligned} \alpha _{\perp }&= \frac{d_M(z) r_{d,\mathrm fid}}{d_{M,\mathrm fid}(z) r_d}&\alpha _{\parallel } = \frac{H_{\mathrm{fid}}(z)r_{d,\mathrm fid}}{H(z) r_d} \end{aligned}$$
(45)
$$\begin{aligned} \varDelta \theta&= r_d / d_{M}(z) \end{aligned}$$
(46)
$$\begin{aligned} \varDelta z&= H(z) r_d, \end{aligned}$$
(47)
where \(d_M = (1+z) d_A\) is the comoving angular distance. Here \(\alpha = \alpha _{\perp }^{2/3} \alpha _{\parallel }^{1/3}\) is the monopole term and \(\epsilon = \alpha _{\perp }/\alpha _{\parallel }\) is the quadropole.
There is the suggestion of circularity in this: a random catalog constructed in \({\varLambda} {\text{CDM}}\) (with small scale power suppressed) is then “tweaked” to compare to the real sky to justify \({\varLambda} {\text{CDM}}\). However, this method is essentially perturbative. If the real cosmology is far from the fiducial one, a bias might be introduced, but if it is close it should work well.
A technique called reconstruction is used to sharpen the BAO signal (Eisenstein et al. 2007; Anderson et al. 2012), and we describe it briefly here as we will mention it in Sect. 4, when we discuss modified gravity. The observed positions of galaxies will be somewhat moved from their original positions “on” the sound horizon, due to their subsequent infall towards overdensities. This has the effect of blurring the peak of the correlation function, and reduces precision. Reconstruction reverses the infall in the following way: (a) take a volume and smooth out small scale structure \(< 20\) Mpc (b) embed it into a larger scale random structure to remove edge effects (c) estimate the displacement \(\mathbf {q}\) of each galaxy by the continuity equation \(\nabla \cdot \mathbf {q} = -\delta _{\mathrm{gal}}/b_{\mathrm{gal}}\) where \(\delta _{\mathrm{gal}}\) is the fractional galaxy overdensity and \(b_{\mathrm{gal}}\) the galaxy bias (d) shift the galaxies by \(-\mathbf {q}\) plus an additional shift for redshift space distortions (e) do the same to the mock catalog.
Historically, the first galaxy surveys to report a BAO detection were the Two Degree Field survey (Cole et al. 2005), and the Sloan Digital Sky Survey (Eisenstein et al. 2005), followed by WiggleZ (Blake et al. 2011, 2012) and others. We discuss here results from Baryon Oscillation Spectroscopic Survey (BOSS) (Anderson et al. 2012). BOSS has spectroscopically surveyed 1.5 million luminous galaxies over 10,000 \(\hbox {deg}^2\) between \(0.2< z <0.7\), with angular separations greater than 62\(^{\prime \prime }\) (which is 0.5 Mpc at \(z \sim 0.7\)). Alam et al. (2017) analyse the BAO signal in Data Release 12. The galaxies are divided into three bins each of redshift and \(\cos {\theta }\). From a fiducial cosmology with \(h = 0.676\), they find \(\alpha = 1.042 \pm 0.016\) after reconstruction. Their result is summarised as a set of three values for \(d_m (z_i)\) and \(H(z_i) \times r_d/ r_{d,\mathrm fid}\) for \(z_i =0.38, 0.51, 0.61\), with precision \(\sim 2.5\%\) (Fig. 14).
The construction of mock galaxy catalogs is a key part of this analysis. An important parameter is \(b_{\mathrm{gal}}\) which is the (assumed linear) ratio of the galaxy overdensity to the matter overdensity. Constructing a mock galaxy catalog by brute force of N-body simulations is prohibitively expensive. Instead, construction starts with a fast, approximate gravity solver for the matter density field seeded with initial fluctuations, and resolution down to halo size. Galaxies are inserted into the matter halos using the bias ratio, while preserving the two- and three- point correlation functions. BOSS assumed \(b_{\mathrm{gal}}\) is constant, but more sophisticated models exist. For example, bias tends to be higher for red galaxies (see for example Zehavi et al. 2011). Attention also needs to be paid to what types of galaxies are being counted: massive luminous galaxies of the type measured by BOSS reside in high density nodes of the cosmic web, whereas emission line galaxies are strung out along the filaments. For further details, see Kitaura et al. (2016). BOSS DR12 uses two independent mock catalogs to mitigate against biases introduced by simulation effects.
BAO in isolation do not constrain \(H_0\), but rather the combination of H(z) and the sound horizon \(r_d\) described above. To obtain \(H_0\), we must either fit a cosmological model with constraints on its density parameters supplied by other datasets (for example SN Ia, CMB, or even the full shape of the matter-power spectrum), or we must supply a prior for \(r_d\) (which is often obtained from the CMB, but see also below). We describe three independent results below.
Alam et al. (2017) used \({\varLambda} {\text{CDM}}\) calibrated to a combination of the Planck power spectrum and the JLA sample of SN Ia. They find \(H_0 = 67.6 \pm 0.5 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). This value is consistent with the Planck result (Planck Collaboration 2020), but is not independent of it.
Lemos et al. (2019) use a parametric formula for H(z), in conjuction with a WMAP-derived prior for \(r_d\) and the Pantheon SN Ia sample, to derive \(H_0 = 67.9 \pm 1.0 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). This result is now independent of Planck, and their parametric H(z) allows for deviation from \({\varLambda} {\text{CDM}}\) by separating early and late universe expansion histories.
Addison et al. (2018) use data (almost) independent of CMB in \({\varLambda} {\text{CDM}}\). They determine \(\varOmega _m = 0.292 \pm 0.02\) by combining galaxy and Ly\(\alpha \) BAO.Footnote 12\(\varOmega _r\) is derived from the CMB temperature as measured by COBE/FIRAS and numerous balloon experiments (Fixsen 2009). Setting \(\varOmega _k = 0\) determines \(\varOmega _{\varLambda } = 1 - \varOmega _m - \varOmega _r\). To get the sound horizon, we now just need the baryon density \(\varOmega _b\) (see Eq. (42)). Without using the CMB power spectrum, the best available way to get it is from primordial deuterium abundances. In Big Bang nucleosynthesis (BBN), deuterium is burned to create \(^{4}\)He, with some uncertainty due to the reaction rate of \(d(p,\gamma )^3\)He. The reaction rate increases with physical baryon density so [D/H] decreases with \(\varOmega _b h^2\). The authors use the primordial [D/H] abundance measured from metal-poor damped Ly-\(\alpha \) systems, which is \(2.547 \pm 0.033 \times 10^{-5}\). Putting this all together, they derive \(H_0 = 66.98 \pm 1.18 \; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\). The only CMB data that has been used is the temperature, and the result corroborates the Planck value.Footnote 13 Repeating the analysis with galaxy data from the Dark Energy Survey (DES) Year 1 produces a consistent result (Abbott et al. 2018).
These results are consistent with more recent ones. DES Y3 survey data has been used to derive a set of three correlation measures (referred to as “3x2pt”). These measure the angular correlation of galaxy positions with each other, with the tangential shear of their shapes caused by weak lensing, and the cross-correlation in the two components of their ellipticities also due to weak lensing. Combining this with Planck, SN Ia data, BAO and redshift space distortions, the DES Collaboration (Abbott et al. 2021) find \(H_0 = 68.0^{+0.4}_{-\,0.3} \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) when fitting to the \({\varLambda} {\text{CDM}}\) model. The eBOSS survey, the culmination of more than 20 years of survey work at the Apache Point Observatory, find \(H_0 = 68.19 \pm 0.37 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) when fitting to a \({\varLambda} {\text{CDM}}\) model using their data plus the above additional probes to also fit to \({\varLambda} {\text{CDM}}\) (Alam et al. 2021). These seem to be the tightest constraints on \(H_0\) yet published.
We also note that the logic explained above to derive \(H_0\) can be reversed: BAO can be combined with a \(H_0\) prior from nearby universe results, instead treating the sound horizon \(r_d\) as a free parameter (Bernal et al. 2016). The \(r_d\) values so inferred are in tension with the CMB, and we return to this point in Sect. 4.2.
In summary, BAO serve as a standard ruler in the late universe which are almost independent of astrophysical assumptions. They may be used in extended models beyond \({\varLambda} {\text{CDM}}\), albeit with the caveat they have been derived perturbatively against a \({\varLambda} {\text{CDM}}\) background simulation. The two main data sets of BOSS and DES are soon to be joined by others. The Dark Energy Spectroscopic Instrument is operational, the ESA survey satellite Euclid is scheduled for launch in 2022 which is also when science operations on the Vera Rubin Observatory LSST in Chile commence. All of these surveys are complementary and will increase the precision and range of BAO measurements. Combined with deuterium abundances, BAO are an important corroboration of the Planck and WMAP results for \(H_0\).
Cosmic microwave background
The CMB power spectrum carries the imprint of the same acoustic oscillations we described for BAO, sampled on the surface of last photon scattering. The Planck 2018 value of \(H_0 = 67.4 \pm 0.5 \; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (Planck Collaboration 2020) is notable for its precision, and is consistent with previous results such as Boomerang \(64 \pm 10 \; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (Percival et al. 2002) and WMAP \(70.0 \pm 2.2 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (Hinshaw et al. 2013). More recently, the Atacama Cosmology Telescope (ACT) finds \(67.9 \pm 1.5 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (Aiola et al. 2020). There is, however, a moderate tension between Planck and the South Pole Telescope (SPT) result of \(71.6 \pm 2.0 \;\,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\), and the origin of the difference is not yet clear (Henning et al. 2018; Handley and Lemos 2021).
Two questions are immediately raised: how is \(H_0\) derived from a power spectrum sampled on the two-dimensional sky of the early universe, and why is the Planck value so precise?
To answer the former, we need to explain how the CMB power spectrum determines cosmological parameters. The spectrum is created by primordial fluctuations that evolve as a function of their scale and the sound speed, until recombination. Whilst this radiation is highly isotropic, we can obtain cosmological results from measurements of small anisotropies in both the temperature and the polarization of CMB photons. The former of these provides the largest amount of cosmological information, and can in fact measure the Hubble constant at high enough accuracy to be in tension with direct measurements without relying on polarization. These temperature anisotropies (\(\varDelta T \equiv T(\mathbf {n}) - T_0\)) can be expanded in a basis of spherical harmonics:
$$\begin{aligned} \varDelta T (\mathbf {n}) = \sum _{\ell =0}^{\infty } \sum _{m = - \ell }^{\ell } a_{\ell m} Y^{m}_{\ell } (\mathbf {n}) . \end{aligned}$$
(48)
The power spectrum of the coefficients is commonly expressed as \({\mathcal {D}}_{\ell } = \ell (\ell +1) C_{\ell }/ 2\pi \), where
$$\begin{aligned} C_{\ell } \equiv \frac{1}{2\ell +1} \sum _{m} a_{\ell m} a_{\ell \, -m} \end{aligned}$$
(49)
is the estimator of the power spectrum (we have only one CMB, so averaging over m is a way to estimate the true “universe average” we want). The spectrum is shown in Fig. 16 and contains all the statistical information of the temperature anisotropies, as long as these are Gaussian (no evidence supporting non-Gaussianity in the CMB has been detected at the time of writing). This power spectrum has three main features:
-
Large-scale plateau at scales (\(\ell < 100\)) that are not affected by post-recombination physics, and, therefore, reflect the primordial power spectrum. As noted above, we have just one “sampling” of the random seeds, so the amount of information we can extract from low multipoles is limited by cosmic variance \(\propto (2\ell +1)^{-1/2}\).
-
Acoustic oscillations which depend on the sound horizon size
$$\begin{aligned} r_s(z_\star ) = \frac{1}{H_0 \varOmega _m^{1/2}} \int _{0}^{a_\star } \frac{c_s}{\sqrt{a+a_{\mathrm{eq}}}}\mathrm{d}a \end{aligned}$$
(50)
in flat \({\varLambda} {\text{CDM}}\) where \(a_{\mathrm{eq}} = \varOmega _r / \varOmega _m\), \(\varOmega _b h^2\) determines \(c_s\) and both determine \(a_\star = a(z_\star )\), the scale factor at the surface of last scattering. These create the peaks in the spectrum.
-
Silk damping which is photon diffusion from hotter regions to colder ones, suppressing small scale power (Silk 1968). The mean free path of a photon is \(\lambda = 1/(\sigma _T n_e)\) where \(n_e\) is the number density of charged particles and \(\sigma _T\) the Thomson cross-section, so photons can diffuse a length \(r_{\mathrm{Silk}} = \sqrt{\lambda /H}\). As the sound horizon \(r_s \propto 1/H\), then \(r_{\mathrm{Silk}}/r_s \propto \sqrt{H/n_e}\). Silk damping creates the downward slope in the spectrum, so this is sensitive to \(n_e \propto \varOmega _b\).
With all of this, how does the CMB constrain the Hubble parameter? First, we can measure the acoustic scale length \(r_s\), which serves as a standard ruler. As described in Eq. 50, this can be calculated as a function of \(\varOmega _r\), \(\varOmega _m h^2\) and \(\varOmega _b h^2\): The radiation density \(\varOmega _r\) is determined by the temperature of the CMB. \(\varOmega _m h^2\) can be estimated from the effect of the integrated Sachs–Wolfe effect (Sachs and Wolfe 1967)—the redshift of CMB photons through wells or hills of gravitational potential during eras of non-matter domination—in the low multipoles of the power spectrum. Finally, the relative amplitudes of the acoustic peaks in the CMB serve to measure the ratio \(\varOmega _b/\varOmega _m\). Once we have computed the acoustic scale, we need a measurement of its comoving angular diameter distance \(\theta _s\), which is given by the distance between the acoustic peaks.Footnote 14 In a flat \({\varLambda} {\text{CDM}}\) model, this provides a measurement of the matter density \(\varOmega _m\), which combined with the measurement of \(\varOmega _m h^2\) gives an estimate on the Hubble parameter. This is illustrated graphically in Fig. 15.
Planck measured full-sky temperature T and polarisation E fluctuations in 9 frequencies, with angular resolution up to 5 arcmins (\(\ell \simeq 2000\)) and sensitivity \(2 \times 10^{-6}\,\mu \)K. Before calculating the TT, TE and EE power spectra, foreground effects must be calculated and removed, and much of the work done by the Planck collaboration between successive data releases has been to develop the foreground model, and reduce the uncertainties it may introduce. Foreground effects include:
-
Peculiar velocity of the Earth-Sun system induces a dipole which is easily subtracted. The annual motion of the Earth is actually helpful to calibrate the detector response.
-
Galactic synchrotron and free-free emission caused by cosmic rays interacting with the magnetic field and ISM of the MW. This has a non-thermal frequency spectrum (in effect its colour is different to the CMB).
-
Galactic dust emission whose contribution to the spectrum is also subtracted by estimating its mapped “colour” difference to the CMB.
-
Point sources of microwave emission are masked. Sky masking introduces some correlation between multipoles.
-
Sunyaev–Zel’dovich (SZ) effect where CMB photons interact with hot intracluster gas and are Doppler-shifted (the kinetic SZ effect) by bulk motion or up-scattered (the thermal SZ effect) (Sunyaev and Zel’dovich 1972). The former effect does not alter the black-body spectrum, but the latter does. The kinetic SZ effect is sensitive to epoch of re-ionisation at \(z \sim 8\) (Planck Collaboration 2014b). The thermal SZ effect may be exploited to produce a cluster map from which information on inhomogeneity can be extracted (Planck Collaboration 2014a). Modelling of intracluster gas predicts a spectral template (which peaks around \(\ell \simeq 2000\)) to subtract.
-
Weak lensing of the CMB by matter. Weak lensing peaks between \(1<z<2\), and may be used as a probe of the distribution of foreground matter. The effect increases as we move to smaller scales, particularly \(\ell > 2000\) (Zaldarriaga and Seljak 1998). There are two signals of lensing: (a) its non-gaussianity, which can be measured by a four-point correlation function (b) a smoothing effect on the peaks and troughs of the spectrum, both of which can be calculated as a function of \(\varOmega _M\). However, there is more smoothing apparent in the spectrum than is predicted by the correlation function so a phenomenological parameter \(\varOmega _{m,\mathrm spectrum} = A_{\mathrm{lens}} \varOmega _{m,\mathrm correlation}\) has been introduced to capture their relative amplitude. \(A_{\mathrm{lens}} \simeq 1.2 \pm 0.1\) is seen in both Planck and South Polar Telescope (SPT) data (Bianchini et al. 2020), but appears to be absent in the Atacama Cosmology Telescope (ACT) data (Aiola et al. 2020).
The latter two are perhaps better characterised as secondary anisotropies, as they have been used to derive cosmological information in their own right. The secondary anisotropies then provide a consistency check of certain parameters derived from the primary anisotropies, if the modelling is correct. We examine \(A_{\mathrm{lens}}\) in greater depth below.
Cross-checks on the effectiveness of foreground modelling are done by switching frequency bands, using alternative astrophysical models, and numerical simulations of parameter recovery from random CMB backgrounds (Planck Collaboration et al. 2016, 2020). The spectrum is then compared to one computed in a standard code such as CMBFAST (Seljak and Zaldarriaga 1996), CAMB (Lewis et al. 2000) or CLASS (Blas et al. 2011) and a posterior for each cosmological parameter is computed. In each check, no serious discrepancy with the main results was found. Although each \(C_{\ell }\) individually has \(\simeq 7\%\) noise, there are 2000 of them and the error is reduced further by the lensing signal. The binned spectrum and residuals is shown in Fig. 16. We now consider two questions regarding the data.
Is Planck self-consistent? A feature which is (barely) visible by eye in Fig. 16 is an oscillating residual pattern lining up to the spectral peaks, indicating the Planck spectrum is slightly tilted versus theory (see Fig. 24 of Planck Collaboration (2020) for an enlarged view). As each multipole is nearly independent, we can split the spectrum and run the analysis on each half. Addison et al. (2016) find that \(A_{\mathrm{lens}}\) is equivalent to a drift of cosmological parameters with different scales in the CMB. They fix \(A_{\mathrm{lens}} =1\) and compare \(H_0\) for \(\ell <1000\) and \(\ell >1000\) finding \(H_0 = 69.7 \pm 1.7 \; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) and \(H_0 = 64.1 \pm 1.7 \; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\), respectively. Allowing \(A_{\mathrm{lens}} = 1.3\), or a tilt parameter restores concordance.
We caution that \(A_{\mathrm{lens}}\) is not a physical parameter connected to lensing; it is a “fudge factor” that resolves internal inconsistency in CMB data. It does not resolve the \(H_0\) tension, but does resolve matter power amplitude \(\sigma _8\) tension with galaxy survey data such as the Dark Energy Survey and Kilo-Degree Survey. Could it be a foreground effect? Its value is relatively stable for various band channels and sky masks, which argues that it is not. The Planck Collaboration followed up on this curiosity (Planck Collaboration 2020). They note a dip in the power spectrum for \(l<30\) pulls low-\(\ell \) \(H_0\) higher; this may account for the lower-resolution WMAP measurement being somewhat higher than Planck. The dip between \(1420 \le \ell \le 1480\) mimics lensing but may be an unaccounted foreground effect. Potential explanations such as negative curvature density \(\varOmega _k <0\) and modified gravity are investigated but no convincing evidence is found in favour of these models when BAO data are added to the fit. A re-analysis of the Planck data using a different foreground subtraction methodology (Efstathiou and Gratton 2019) found that \(A_{\mathrm{lens}}\) decreased from 1.26 to 1.06 as the data were broadened from the temperature-only power spectrum to include polarisation and lensing data. Further, Planck data passes other consistency tests with the baseline \({\varLambda} {\text{CDM}}\) model where A-type parameters may be introduced, such as the expected magnitude of the lensing correlation function and the Sachs–Wolfe effect. Hence, the preferred explanation of the Planck Collaboration is that \(A_{\mathrm{lens}}\) is a statistical fluke. Nevertheless, while \(A_{\mathrm{lens}}\) does seem to vary over the sky, it is perhaps of concern that it seems to be larger for full-sky Planck data is and is a roughly \(2\sigma \) effect.
Is Planck consistent with other current CMB measurements? The SPT covers a \(6\%\) patch of the southern sky at a resolution 6\(\times \) that of Planck, finding \(H_0 = 71.3 \pm 2.1\; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (Henning et al. 2018). SPT is limited to \(\ell >650\) due its low sky coverage and atmospheric effects and has greater noise, but when the results are compared on a like-for-like basis (“Planck-in-patch” with SPT for \(650< \ell < 1800\)), they are consistent with Planck. The analyses are independent and so argue against systematic errors (in this patch and multipole range) in either experiment. However, including higher SPT multipoles in the 150 GHz spectrum causes \(H_0\) to drift higher: \(H_0 \simeq 74 \pm 3\; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) for \(650< \ell < 3000\) (Aylor et al. 2017). While this is the opposite direction to Planck data, this may be due to the \(l<30\) effect in Planck noted above. SPT confirm the Planck result of a greater lensing effect than predicted, and also note a spectral tilt versus the best-fit \({\varLambda} {\text{CDM}}\). However, the trend is not apparent in 143 GHz data, so it is not at all clear if this is a physical effect, rather than merely chance or systematics. The ACT Data Release 4 derives cosmological parameters from a \(6,000 \mathrm deg ^2\) patch of the southern sky up to \(\ell \sim 4000\), finding \(H_0 = 67.9 \pm 1.5\; \,\hbox {kms}^{-1} \,\hbox {Mpc}^{-1}\) (Aiola et al. 2020), however, ACT appears to be in \(2.6\sigma \) tension with Planck Handley and Lemos (2021); this appears to be caused by differing physical baryon densities.
In summary, Planck 2018 results for \(H_0\) are consistent with previous and current independent CMB data. The high precision for \(H_0\) is a consequence of Planck’s high resolution, lack of atmospheric interference, and full sky coverage. Checks have been performed on the codes, the methods used to remove foregrounds, and beam effects. Internal inconsistencies hint at new physics or foreground effects at small angular scales, manifesting either as a tilt or a smoothing of the spectrum, and non-physical parameters like \(A_{\mathrm{lens}}\) have been introduced to capture them. However, after their introduction the results for \(H_0\) are not materially affected, and remain in tension with late universe results. Nevertheless, the persistence of these effects across differing sky patches, multipole ranges and experiments make them less likely to be a statistical fluke. It is hoped next generation CMB experiments with resolution up to \(\ell =5000\) will shed further light on this.
Other methods
Promising methods that we did not have space to discuss are Surface Brightness Fluctuations (Tonry and Schneider 1988; Blakeslee et al. 2021; Khetan et al. 2021), Cosmic Chronometers (Jimenez and Loeb 2002; Moresco et al. 2018), the Tully–Fisher relation (Tully and Fisher 1977; Kourkchi et al. 2020; Schombert et al. 2020), Type II supernovae (de Jaeger et al. 2020), HII galaxies (Terlevich and Davies 1981; Arenas et al. 2018), and Galaxy Parallax (Croft 2021). Quasars (Risaliti and Lusso 2019) and GRBs (Schaefer 2007) offer the prospect of extending the Hubble diagram up to \(z \sim 5\), further testing \({\varLambda} {\text{CDM}}\). However, despite improvements in characterising their intrinsic luminosity from observables (for example X-ray and UV flux in the case of quasars), it remains challenging at present to regard them as standard candles. We refer the reader to the above for the latest work in these fields.