We now examine the breakdown of uncertainties contributing to the statistical error, which include the multiple statistical and systematic uncertainties in SN Ia parameters making upCη as constructed in
Sections 3.4 and 4.5.3.
To visually assess the impact on confidence contours we compare results from MultiNest with dif- ferent covariance matrix inputs. For an example global fit (with Cepheid fitT = 2.5, NGC 4258 anchor, no priors, default SN cuts) we test each systematic, and compare their results from MultiNest. The full expression for the covariance matrix Cη for observed SN Ia quantities is described in Section 3.4.
As entries of Cη in MultiNest we try the following: only the statistical contribution Cstat (described in Section 3.4), each single systematic term added toCstat, and all systematics added i.e.Cstat+Csys (the default for all global fits). The confidence contours with statistical uncertainties only and with all systematics are easily distinguishable in Figure 4.11, but the contours with individual systematics are not. Thus for clarity we only show in Figure 4.11 the systematic term from the uncertainty in host mass correction (Chost in Equation 3.16, described in Section 3.4.3), in addition to Cstat and Cstat+Csys. The difference between the contours is slight, indicating that the uncertainties in the parameters only increase slightly when covariance matrices for different systematics are added to the statistical termCstat. Following the method in JLA (Betoule et al., 2014, section 6.2), we quantify the relative contribu-
§4.7 Uncertainties 91
Figure 4.11 Constraints on parameters H, MB, and MW from an example global MultiNest fit (with
Cepheid fit T = 2.5, NGC 4258 anchor, no priors, default SN cuts) with partial and full contributions to the full SN Ia covariance matrix. Confidence contours are shown with the statistical contribution Cstatonly (turquoise filled), with one systematic term (the host mass correction) added i.e.Cstat+Chost (orange solid line), and with all SN Ia systematics, i.e. Cstat+Csys (red dashed). The inclusion of systematic terms only increases the widths of contours marginally relative to the Cstat-only (turquoise) contours, reflecting that the statistical contribution dominates the uncertainty in the parameters.
Table 4.11 Relative contributions to the uncertainty in H0 (i.e. the variance) from individual statistical and systematic sources uncertainties, calculated as described in Betoule et al. (2014, Section 6.2).
Source of Relative contribution Described in uncertainty toσ2(H 0) (%) Section Statistical: Lightcurves 62.1 3.4.1 SALT2 training 1.2 3.4.1 Total statistical 63.3 Systematic: Malmquist bias 13.7 3.4.2 Host galaxy 13.0 3.4.3 Lightcurve model 6.8 3.4.7 Calibration 3.1 4.5.3 Peculiar velocities 0.04 3.4.4
Milky Way extinction 0.03 3.4.5
tions, replacing the parameters {Ωm, w, α, β, MB,∆M} with our parameters {H0, MB, α, β} (the only
parameters in Θ which can be influenced by the low-z SN Ia covariance matrices). The breakdown of relative contributions to the variance inH0 from each term (the purely statistical termCstat, and each systematic) are reported in Table 4.11. We emphasise that each of these numbers represents a proportion of the uncertainty (terms (i) and (ii) in Section 4.7 combined) from the systematic or statistical term in question alone, rather than reflecting an uncertainty inH0.
From Table 4.11 and Figure 4.11 it is clear that Cstat is the largest component of Cη. Even
though the contributions to Cη from SN systematics are included in the statistical uncertainty, all of
these systematics together are smaller than the SN statistical uncertainties: the relative contributions to the variance in H0 are dominated by Cstat,diag (Table 4.11), and the contours with and without systematic covariance matrices added toCstatin Figure 4.11 are similar to each other. Of the systematic terms, the most significant in their impact on H0 are from uncertainties in the Malmquist bias correction (including the selection function) and the host mass correction (Sections 3.4.2 and 3.4.3), followed by the uncertainty in lightcurve model. While JLA had found the photometric calibration (Section 3.1.1), especially from low-zSNe, to be the dominant uncertainty for Ωmandw, its effect onH0 is decidedly smaller. It is interesting to note that despite conservative estimates of both the uncertain- ties in Milky Way extinction and peculiar velocity correction, their effects on the error inH0are negligible. We comment on these results, particularly the increased error, in the next chapter.
Chapter 5
Implications for the Hubble constant
In this chapter, we reflect on the findings in Chapter 4, and their place in the literature. We then review other efforts to elucidate the current tension inH0in the literature, including several other reanalyses of R11 and R16. Finally, we look toward future prospects for the Hubble constant.
5.1
Reflections on Chapter 4
First, we examine the results in Chapter 4, which had been published in Z17. As we will describe below, our reanalysis of R11 in Z17 confirmed the central value, a strong affirmation of the R11 result (noting we use the same data – just the analysis techniques differ), because our analysis used a different set of techniques, and was blinded. However, our uncertainty is substantially larger than in R11, and it is worthwhile to understand this discrepancy because it is central to the notion that there is tension between the CMB-related values of Planck, and SN Ia and Cepheid distance ladder derived values from the local Universe.