Sounds Discordant:
Classical Distance Ladder & $\Lambda$CDM-based Determinations of the Cosmological Sound Horizon

Classical distance ladder (CDL) approaches using Cepheids and supernovae (Riess et al., 2018b, hereafter R18) find higher Hubble constant estimates than those derived from cosmic microwave background (CMB) data, that assume the standard cosmological model, $\Lambda$CDM (Planck Collaboration VI, 2018). The statistical significance of this discrepancy has grown over time with fairly steady progress on the distance ladder (Riess et al., 2009, 2011, 2016) and with a sudden jump in precision of the $\Lambda$CDM prediction with the first release of the Planck cosmology data in 2013 (Planck Collaboration XVI, 2014). Comparing the R18 value of $H_{0}=73.52\pm 1.62$ km/s/Mpc with the value inferred from Planck CMB temperature and polarization power spectra plus CMB lensing, assuming $\Lambda$CDM, $H_{0}=67.27\pm 0.60$ km/s/Mpc, there is a 3.6$\sigma$ discrepancy (Planck Collaboration VI, 2018).

Along with the reduction of statistical errors in the Cepheids plus supernovae determination of $H_{0}$, other supporting evidence in favor of a high value of $H_{0}$ has been growing as well. A number of independent analyses of the data have served to largely confirm the conclusions of R18 (Efstathiou et al., 2014; Cardona et al., 2017; Zhang et al., 2017; Feeney et al., 2018a; Follin & Knox, 2018). Despite claims that the CDL $H_{0}$ departs significantly from the cosmic mean $H_{0}$ due to a local void, Wu & Huterer (2017) have shown that within $\Lambda$CDM the sample variance in the R18 measurement is only 0.3 km/sec/Mpc, or 0.2$\sigma$. In addition, Birrer et al. (2018) have provided the latest inference of $H_{0}$ from the H0LiCOW (Suyu et al., 2017) collaboration’s use of strong-lensing time delays: $H_{0}=72.5^{+2.1}_{-2.3}$ km/s/Mpc. Combining the result of this completely independent probe of the distance-redshift relation at low redshift, with that from R18, results in a 4.1$\sigma$ discrepancy with the Planck result quoted above. Finally, the tip of the red giant branch distance method shows consistency with Cepheid distances to a handful of supernova host galaxies (Jang & Lee, 2017; Hatt et al., 2018a, b).

Recently, Shanks et al. (2018a) claimed that corrections applied to the Gaia data (Lindegren et al., 2018) in the R18 analysis have introduced significant systematic error, a claim that has sparked a debate (Riess et al., 2018a; Shanks et al., 2018b). We simply point out here that the Riess et al. (2016) result of $H_{0}=(73.24\pm 1.74)$ km/s/Mpc makes no use of Gaia data and is thus immune to this controversy, while nearly as discrepant from the $\Lambda$CDM plus CMB-inferred values.

The case against systematic errors in CMB data as the source of this discrepancy is very strong. First of all, the result from Planck is robust to choice of frequency channels (Planck Collaboration et al., 2016), arguing against foreground modeling, or any channel-specific systematic errors as a source of bias in the $H_{0}$ inference. Further, the consistency of Planck measurements with WMAP on large angular scales (Planck Collaboration et al., 2014) and the 2500 deg² SPT-SZ measurements on small angular scales (Hou et al., 2018; Aylor et al., 2017) also argues against any significant systematic errors on all angular scales relevant for determination of cosmological parameters from Planck data.

Additionally, the conclusion of a low $H_{0}$ from CMB data and the assumption of $\Lambda$CDM can be reached without any use of Planck data. The inverse distance ladder results (Percival et al., 2010; Heavens et al., 2014; Aubourg et al., 2015; Cuesta et al., 2015; Bernal et al., 2016a; DES Collaboration et al., 2017; Verde et al., 2017b; Lemos et al., 2018; Joudaki et al., 2018; Feeney et al., 2018b) also show that the combination of measurement of the baryon acoustic oscillation (BAO) feature in galaxy surveys, Type Ia supernova observations, and CMB data with or without Planck (e.g., WMAP9, Bennett et al. 2013) lead to low (Planck-like) values of $H_{0}$. Finally, Addison et al. (2018) point out that assuming the $\Lambda$CDM model, using BAO data, and light element abundance measurements as constraints on the baryon-to-photo ratio, one infers a Planck-like value of $H_{0}$; i.e., without use of any CMB anisotropy data. The above results indicate that systematic errors in CMB data, and Planck CMB data in particular, are not the major driver of the discrepancies in inferences of $H_{0}$.

Recently, some works in the literature have proposed solutions both pre- and post-recombination to address the $H_{0}$ discrepancy. For example, Karwal & Kamionkowski (2016); Evslin et al. (2018); Poulin et al. (2018) propose the existence of an early dark energy which reduces the size of the sound horizon subsequently increasing the CMB inferred value of $H_{0}$; Lin et al. (2018) propose a modified gravity solution at the time of recombination and Chiang & Slosar (2018) alter the duration of the recombination to reduce the tension between CMB and CDL results. On the other hand, Di Valentino et al. (2016, 2017); Joudaki et al. (2017) use an extended parameter space and point out that an interacting phantom-like dark energy with equation of state $w_{\rm DE}<-1$ can also reduce the tension in $H_{0}$ measurements.

In this paper, to further explore what can, and what cannot, explain this discrepancy, we follow Bernal et al. (2016b) and use the sound horizon $r_{\rm s}$, rather than $H_{0}$, as the point of comparison between CDL and $\Lambda$CDM-based estimates. With this approach, the BAO data are on the CDL side: Cepheids calibrate Type Ia supernovae, which are then used to determine distances to redshifts for which BAO measurements of the angular size of the sound horizon exist, thereby revealing the size of the sound horizon. We also note that strong-lensing time delays can be used to calibrate the BAO measurements, so we convert the H0LiCOW result into a sound horizon constraint. For a more comprehensive use of data sensitive to distances and the expansion rate at low redshifs ($z\lesssim 1$), see Bernal & Peacock (e.g. 2018), which includes use of “cosmic clocks.”

We find use of the sound horizon as a point of comparison, to be a particularly useful way of examining the data for several reasons: first, there is added insensitivity with this method to extreme changes in the $z<0.1$ cosmology since one does not need to extrapolate to $z=0$, circumventing issues with peculiar velocities brought up by  Shanks et al. (2018a) ; second, the $\Lambda$CDM predictions for the sound horizon are more robust than those for $H_{0}$; third, as with the inverse distance ladder, this approach clarifies that reconciliation can not be delivered by altering cosmology at $z<1$. Fourth, it serves to clarify that the reconciliation of distance ladder, BAO, and CMB observations via a cosmological solution is likely to include a change to the cosmological model in the two decades of scale factor evolution prior to recombination; and finally, $\sigma(r_{\rm s})/r_{\rm s}$ from CMB data, assuming $\Lambda$CDM is four times smaller than the $\sigma(H_{0})/H_{0}$ from the same data and assumed model.

Since Bernal et al. (2016b), we have new supernova data available (Scolnic et al., 2018), an updated supernova absolute luminosity calibration (R18), and results from the final data release from Planck (Planck Collaboration VI, 2018). We examine the tension in $r_{\rm s}$ given these most recent data. As mentioned above, strong-lensing time delays can also be used to calibrate the distance to the BAO redshifts. So we also use the latest H0LiCOW results (Birrer et al., 2018) to produce a constraint on $r_{\rm s}$.

We find that the CDL-inferred sound horizons are in 2-3$\sigma$ tension with $\Lambda$CDM-determined ones¹¹1The inverse distance ladder results in a more significant tension because the BAO error, in this case, gets added to the $r_{\rm s}$ error which is fractionally smaller than the CDL error on $H_{0}$. . While a statistical fluke could explain this discrepancy, we believe there is sufficient evidence to motivate the exploration of cosmological solutions. In Section 4, we argue that if there is to be a cosmological solution to the discrepancies in $r_{\rm s}$ values, it is likely to be significantly different from $\Lambda$CDM in the two decades of scale factor growth prior to recombination. Such model changes are likely to lead to predictions testable with future CMB data. We examine the $r_{\rm s}$ predictions in the case of a two-parameter extension of the standard cosmological model, and forecasts for $r_{\rm s}$ errors in these extended model spaces given the survey to come from SPT-3G, a third-generation camera outfitted on the South Pole Telescope (Benson et al., 2014; Bender et al., 2018).

We present here the empirical, CDL, approach to determining the sound horizon scale, $r_{\rm s}$, and then the more cosmological-model dependent approach. Although the former also requires modeling, we demonstrate with a parametric Spline model for the history of the expansion rate that the results are at most only mildly dependent on cosmological model assumptions. We also describe how we use the recent H0LiCOW results (Birrer et al., 2018) to determine $r_{\rm s}$.

Before presenting the constraints on $r_{\rm s}$ from different methods described in §2, in Figures 1 and 2 we display the BAO and SNe data as they constrain $D_{A}(z)$ and $H(z)$. For these figures, we assume the fiducial values $r_{\rm s}=138.09$ Mpc and $M=-19.26$, which are the best-fit values for the Spline model parameter space described in §3.1.2 given the BAO, SNe, and Cepheid data. We choose the best-fit values from the spline model, as opposed to the $\Lambda$CDM model, primarily for specificity, and secondarily in order to have less model dependence in the resulting distance estimates.

Examining the residuals from these fits in Fig. 1 and Fig. 2 we see no obvious problems for either the $\Lambda$CDM model or the Spline model. For the $\Lambda$CDM model, we find for the best fit $\chi^{2}_{\rm SNe}=39.3$ and $\chi^{2}_{\rm BAO}$ = 3.5, summing to $\chi^{2}_{\rm tot}$ = 42.8 for 43 degrees of freedom (40 SNe data points, 6 BAO data points, and 3 parameters, not counting $H_{0}$). For the Spline model, we find the best fit $\chi^{2}_{\rm SNe}=38.0$ and $\chi^{2}_{\rm BAO}$ = 3.4, with $\chi^{2}_{\rm tot}$ = 41.4 for 40 degrees of freedom (6 parameters, once again not counting $H_{0}$).

Now we turn to results reporting the inferred value of $r_{\rm s}$ using the CDL approach from the $\Lambda$CDM and the Spline model in §3.1. This is followed by the results obtained using CMB data for the $\Lambda$CDM model in §3.2. Next, we discuss the $2$ to $3\sigma$ tension in the value of $r_{\rm s}$ obtained from these two methods in §3.3. In §3.4, we look at a couple model extensions and forecast the expected constraints on $r_{\rm s}$ that can be obtained by combining Planck results with SPT-3G (Benson et al., 2014), a stage-3 CMB temperature and polarization survey. Finally, in §4,we argue that if the origin of the discrepancies is cosmological, the cosmological solution must make its important changes at times prior to recombination.

The inverse distance ladder papers we cited earlier, and also Poulin et al. (2018), indicate that the combination of BAO and supernova data make a cosmological solution unlikely with changes restricted to $z\lesssim 1$. Here we go further and argue that any viable cosmological solution to sound horizon discrepancies is likely to differ significantly from the standard cosmological model in the two decades of scale factor expansion immediately prior to recombination. Changes that are only important earlier cannot reduce the sound horizon significantly. This is because in the standard cosmological model, near the best-fit location in parameter space given Planck data, greater than 95% of $r_{\rm s}$ is generated in the final two decades of scale factor growth prior to recombination.

What about changes after recombination? These would have to make a fractional change in $r_{\rm s}$ of $\delta r_{\rm s}/r_{\rm s}=x$ where $x\simeq-0.07$ to bring the model $r_{\rm s}$ values in line with the CDL values. If the changes were only important after recombination, then our $r_{\rm s}$ calculation is unchanged so we still have $\delta r_{\rm s}/r_{\rm s}\simeq-0.25\delta\omega_{\rm m}/\omega_{\rm m}$ so we need $\delta\omega_{\rm m}/\omega_{\rm m}=-4x$. To preserve $\theta_{\rm s}$ (which we would need to do to stay at high likelihood given the CMB data; e.g., Pan et al. 2016) we would also need to change the angular-diameter distance to last-scattering by $\delta D/D=x$. However, another important length scale for interpretation of CMB data, the comoving size of the horizon at matter-radiation equality, $r_{\rm EQ}=c/(a_{\rm EQ}H_{\rm EQ})$, responds much more rapidly to changes in $\omega_{\rm m}$. We find, assuming $\Lambda$CDM, as is appropriate here, $\delta r_{\rm EQ}/r_{\rm EQ}=-3\delta\omega_{\rm m}/\omega_{\rm m}=12x$ and therefore the change in distance required to keep $\theta_{\rm EQ}=r_{\rm EQ}/D$ from changing would be 12 times greater than required to keep $\theta_{\rm s}$ fixed. We can not make changes to the late-time cosmology, and therefore $D$, that keep both of these angular scales fixed. To make this work, the changes in the post-recombination cosmology would have to introduce new anisotropies that would confuse our inference of $\theta_{\rm EQ}$ and/or $\theta_{\rm s}$. The consistency of the $\Lambda$CDM results for $r_{\rm s}$ (which depend primarily on $\omega_{\rm m}$, which is inferred from $\theta_{\rm EQ}$; see, e.g., Section 4 of Planck Collaboration et al. (2017) ) across angular scale argue against this possibility. We find it to be highly unlikely that whatever confuses our interpretation of the $l<800$ TT data (perhaps ISW effects) would also similarly confuse our interpretation of the $l>800$ TT data, as well as our interpretation of other data selections such as TE+lowE.

Our claim in this section, that any viable cosmological solution is likely to include significant changes from $\Lambda$CDM in the epoch immediately prior to recombination, is an interesting one, as this is an epoch that we will probe better with improved measurements of CMB polarization (and also temperature on small angular scales). It has this exciting implication: viable cosmological solutions are likely to make predictions that are testable by so-called stage 3 CMB experiments, as well as CMB-S4.

Soon after we posted this paper on the arXiv (and prior to publication) Poulin et al. (2018) appeared on the arXiv. This paper presents a cosmological solution reconciling CMB, BAO, and Cepheid-calibrated supernova data. The solution is consistent with our analysis here: namely it has an early dark energy component contributing significantly in the scale factor window we have just described. It also leads to predictions that appear to be testable by future measurements of CMB polarization.

Following Bernal et al. (2016b) we have compared, using more recent data, empirical CDL determination of $r_{\rm s}$ with its inference assuming the $\Lambda$CDM model and given a variety of CMB datasets. Casting the tension between the CDL and $\Lambda$CDM + CMB datasets in terms of $r_{\rm s}$, as opposed to $H_{0}$, weakens the statistical significance, but helps to clarify the space of possible cosmologies that could reconcile these datasets. As the inverse distance ladder analyses have pointed out, modifying the shape of $D_{A}(z)$ at $z<1$ can at most be a sub-dominant part of the solution.

Because SNe cover the range of redshifts of the BOSS galaxy BAO data, our CDL inferences of $r_{\rm s}$ are highly model-independent. For the Spline model, which we prefer for this purpose over $\Lambda$CDM due to its modest cosmological model assumptions⁴⁴4There is an implicit assumption of zero mean curvature. As discussed above, we expect that if we relaxed this assumption our results would only shift a small amount, as was the case for a similar analysis (Verde et al., 2017b)., from the Cepheid, SNe, and BAO datasets, we find $r_{\rm s}=137.7\pm 3.6$ Mpc.

This result is 2.6$\sigma$ lower than the result from Planck TT+TE+EE+lowE (which we have referred to simply as Planck). We calculated the statistical significance of the difference between the CDL result and the $\Lambda$CDM + CMB data results for a variety of CMB datasets and found they ranged from 2.1 to 3.0$\sigma$. Perhaps of particular interest, the combination of the highest precision non-Planck data, WMAP9 + SPT-SZ + ACT, gives an $r_{\rm s}$ that is 3.0$\sigma$ discrepant from the above CDL result. It is clear that the sound horizon differences cannot be explained by an unknown systematic error in the Planck data.

Expanding the model space to $\Lambda$CDM + $N_{\rm eff}$ does not reduce the tension of the CDL $r_{\rm s}$ with the Planck $r_{\rm s}$. Although the error bar for the Planck-determined $r_{\rm s}$ increases considerably, the CDL error remains larger, and the central value for the Planck-determined $r_{\rm s}$ shifts to a slightly higher value. Expanding further to $\Lambda$CDM + $N_{\rm eff}+Y_{\rm P}$ only reduces the tension from 2.6$\sigma$ to 2.3$\sigma$. The CMB data have no significant preference for these extensions.

While the CMB data show no preference for those particular extensions, we point out here that there are hints/weak evidence of inconsistencies of the CMB data with the $\Lambda$CDM model. Parameter constraints derived from different angular scales, such as the Planck temperature power spectra at $l<800$ compared to $l>800$, are uncomfortably different, with a statistical significance that varies between 1.5 and 2.9$\sigma$ depending on details of the analysis and how the question of consistency is posed (Addison et al., 2016; Planck Collaboration et al., 2017; Kable et al., 2018). Driven by small angular scales better measured by the South Pole Telescope, there is a 2.1$\sigma$ tension between SPT-SZ determination of cosmological parameters and those from Planck (Aylor et al., 2017). It is possible that these are hints relevant to the sound horizon discrepancy, but current data are not yet clear on the matter, and no model has been discovered, to our knowledge, that both addresses the sound horizon discrepancy and improves CMB internal consistency.

We argued that viable cosmological model solutions are likely to include important changes from $\Lambda$CDM in the two decades of scale factor growth prior to recombination. This statement is interesting because it has an exciting implication: significant changes in this time period are likely to lead to consequences observable with near-future precision observations of CMB polarization.

We produced forecasts for one such model adjustment: allowing for $N_{\rm eff}$ to be a free parameter, which directly alters pre-recombination dynamics. We found a three-fold improvement in the constraints on $r_{\rm s}$ when combining Planck with the SPT-3G (Benson et al., 2014) dataset. Whether or not the solution to the discrepancy is cosmological, we can expect future observations of the CMB from SPT-3G and other future CMB surveys like AdvACT (Henderson et al., 2016); Simons Observatory (The Simons Observatory Collaboration et al., 2018), CMB-S4 (CMB-S4 Collaboration et al., 2016), and PICO (Young et al., 2018), to reveal further clues via their sensitivity to the acoustic dynamics of the plasma.