MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 16 October 2018 (MN L A TEX style file v2.2) H Revisited
George Efstathiou
Kavli Institute for Cosmology and Institute of Astronomy, Madingley Road, Cambridge, CB3 OHA.
16 October 2018
ABSTRACT
I reanalyse the Riess et al. (2011, hereafter R11) Cepheid data using the revisedgeometric maser distance to NGC 4258 of Humphreys et al. (2013, hereafter H13). Iexplore different outlier rejection criteria designed to give a reduced χ of unity andcompare the results with the R11 rejection algorithm, which produces a reduced χ that is substantially less than unity and, in some cases, leads to underestimates of theerrors on parameters. I show that there are sub-luminous low metallicity Cepheids inthe R11 sample that skew the global fits of the period-luminosity relation. This hasa small but non-negligible impact on the global fits using NGC 4258 as a distancescale anchor, but adds a poorly constrained source of systematic error when using theLarge Magellanic Cloud (LMC) as an anchor. I also show that the small Milky Way(MW) Cepheid sample with accurate parallax measurements leads to a distance toNGC 4258 that is in tension with the maser distance. I conclude that H based on theNGC 4258 maser distance is H = 70 . ± . − Mpc − , compatible within 1 σ withthe recent determination from Planck for the base six-parameter ΛCDM cosmology. Ifthe H-band period-luminosity relation is assumed to be independent of metallicity andthe three distance anchors are combined, I find H = 72 . ± . − Mpc − , whichdiffers by 1 . σ from the Planck value. The differences between the
Planck results andthese estimates of H are not large enough to provide compelling evidence for newphysics at this stage. Key words : cosmology: distance scale, cosmological parameters.
The recent
Planck observations of the cosmic microwave background (CMB) lead to a Hubble constant of H = 67 . ± . − Mpc − for the base six-parameter ΛCDM model (Planck Collaboration 2013, herafter P13). This value is intension, at about the 2 . σ level, with the direct measurement of H = 73 . ± . − Mpc − reported by R11. If thesenumbers are taken at face value, they suggest evidence for new physics at about the 2 . σ level (for example, exotic physics inthe neutrino or dark energy sectors as discussed in P13; see also Wyman et al. et al. et al. H measurements to intense scrutiny.Direct astrophysical measurements of the Hubble constant have a checkered history (see, for example, the reviews byTammann, Sandage and Reindl 2008; Freedman and Madore 2010). The Hubble Space Telescope (HST) Key Project led to asignificant improvement in the control of systematic errors leading to ‘final’ estimate of H = 72 ± − Mpc − (Freedman et al. H : the Supernovae and H for the Equation of State (SH0ES) programme of R11 (with earlier results reported in Riess et al. Carnegie Hubble Program of Freedman et al. (2012). In addition, other programmes are underway using geometricalmethods, for example the Megamaser Cosmology Project (MCP) (Reid et al. et al. et al. et al. et al. H measurement from these data has the smallest errorand has been used widely in combination with CMB measurements for cosmological parameter analysis ( e.g. Hinshaw et al. et al. et al. ∼ χ values of their fits, c (cid:13) a r X i v : . [ a s t r o - ph . C O ] F e b G. Efstathiou
Figure 1.
Period-luminosity relation for the LMC Cepheids. The line shows the best fit of equation (4a). The vertical dotted lines showthe range of periods used in the fit of equation (4b). and the variations of some of the parameter values with different distance anchors, particularly the metallicity dependence ofthe period-luminosity relation.The layout of this paper is as follows. Section 2 reviews the near-IR period-luminosity (P-L) relation of a sample of LMCCepheids. Section 3 describes a reanalysis of the R11 sample using the maser distance to NGC 4258 as an anchor. Section4 investigates the use of the LMC and MW Cepheids as anchors. Section 5 investigates combinations of the three distanceanchors and presents some internal consistency checks. The conclusions are summarized in Section 6.
I will start with the LMC Cepheids which I will use as a reference since the slope of the P-L relation is tightly constrainedfrom this sample. I use the 53 LMC Cepheids with H-band magnitudes listed in Persson et al. (2004) and V, I magnitudeslisted in Sebo et al. (2002). Figure 1 shows the Wesenheit magnitudes m W = m H − . V − I ) , (1)plotted against period P (in units of days). The line shows a least-squares fit to m PW = A + b W (log P − , (2) i.e. minimising χ = (cid:88) i ( m W,i − m PW ) ( σ e,i + σ ) , (3)where σ e,i is the error on m W,i and σ int is the ‘internal’ scatter that gives a reduced χ (denoted ˆ χ in this paper) of unity.Since σ int depends on the parameters A and b W , the minimisation is performed iteratively until convergence. The best fitparameters are A = 12 . ± . , b W = − . ± . , σ int = 0 . , log P < . , (4a) A = 12 . ± . , b W = − . ± . , σ int = 0 . , . < log P < . . (4b)The upper period limit is imposed because there is evidence that the P-L relation departs from a power law for periods > ∼ et al. , 2004; Freedman et al. et al. et al. et al. (2004). Evidently, a single power law is anextremely good fit to the LMC Cepheids at least to periods of 60 days, and the slope of the P-L relation is determined tohigh accuracy. c (cid:13) , 000–000 Revisited H mags W magsGalaxy N fit N rej σ (mag) ˆ χ σ (mag) ˆ χ N4536 69 20 0 .
32 1 .
25 0 .
33 1 . .
41 1 .
25 0 .
41 1 . .
32 0 .
90 0 .
32 0 . .
32 0 .
79 0 .
33 0 . .
39 0 .
73 0 .
39 0 . .
31 0 .
95 0 .
32 0 . .
32 1 .
01 0 .
32 1 . .
34 0 .
77 0 .
36 0 . .
36 1 .
01 0 .
37 0 . Table 1.
H-band rejection. N fit and N rej lists the number of Cepheids accepted and rejected by the R11 outlier rejection algorithm. σ is the standard deviation of the magnitude residuals about the best fit P-L relation with slope constrained to b = − . χ lists thereduced χ for each fit (with no additional contribution from ‘intrinsic’ scatter). Results are given for fits to H-band and Wesenheit P-Lrelations. As discussed by R11, there are several reasons to expect outliers in the P-L relation. These include variables misidentified asclassical Cepheids, blended images, errors in crowding corrections and possible aliasing of the periods.The R11 rejection algorithm works as follows: • The H-band only P-L relations are fitted galaxy-by-galaxy to a power law with slope fixed at b H = − . >
205 days are excluded. • Cepheids are rejected if they deviate from the best-fit relation by ≥ .
75 mag, or by more than 2.5 times the magnitudeerror. • The fitting and rejection is repeated iteratively 6 times.Once the outliers have been removed, R11 proceed to global fits using all of their galaxies, now adding a 0 .
21 mag. errorin quadrature to the magnitude errors listed in their Table 2 (Riess, private communication). One of the consequences ofadding this additional error term is that the ˆ χ values of the R11 global fits are always less than unity (with typical valuesof ˆ χ ∼ .
65) so R11 rescale their covariance matrices by 1 / ˆ χ to compute errors on parameters.There are several aspects about this rejection algorithm that are worrisome: • The rejection algorithm is applied galaxy-by-galaxy before the global fit to the entire sample. • A large fraction of the data are rejected (about 20% of the total sample). • The imposition of an absolute cut of 0 .
75 mag will accept points with large magnitude errors and small residuals, i.e. pointsthat just happen to lie close to the best-fit P-L relation for each galaxy. • As a consequence, ˆ χ for the global fits is guaranteed to be less than unity if an additional ‘intrinsic’ error of 0 .
21 mag isadded to the magnitude errors.The last three points are evident from the entries in Table 1. N fit is the number of Cepheids accepted by R11 and N rej is the number rejected. σ is the standard deviation of the magnitude residuals of the accepted Cepheids around the best fitP-L relation with slope constrained to b = − .
1. ˆ χ gives the reduced χ computed using the magnitude errors listed in R11.Although the dispersions exceed 0 . χ for many galaxies are already less than unity. I also list results forfits to the Wesenheit magnitudes (1). These numbers are similar to those for the H-band fits, so adding colour informationproduces very little change to the scatter. Since the values of ˆ χ in Table 1 are already low, ˆ χ for the global fits will besubstantially less than unity if an additional 0 .
21 mag. is added in quadrature to the R11 magnitude errors. Table 1 showsthat there is simply no room for additional scatter (irrespective of colour corrections). The choice of adding a 0 .
21 magnitudeerror to the Cepheids accepted by the R11 rejection algorithm is not supported by the data.Instead of rejecting Cepheids on a galaxy-by-galaxy basis, I reject outliers from the global fit. As in R11, we write theP-L relation for galaxy i as m PW,i = ( µ ,i − µ , ) + p W + Z W ∆log (O / H) + b W log P, (5)and minimise, χ = (cid:88) ij ( m W,ij − m PW,i ) ( σ e,ij + σ ) , (6)(where j is the index of the Cepheid belonging to galaxy i and σ e,ij is the magnitude error listed in Table 2 of R11) with c (cid:13)000
21 magnitudeerror to the Cepheids accepted by the R11 rejection algorithm is not supported by the data.Instead of rejecting Cepheids on a galaxy-by-galaxy basis, I reject outliers from the global fit. As in R11, we write theP-L relation for galaxy i as m PW,i = ( µ ,i − µ , ) + p W + Z W ∆log (O / H) + b W log P, (5)and minimise, χ = (cid:88) ij ( m W,ij − m PW,i ) ( σ e,ij + σ ) , (6)(where j is the index of the Cepheid belonging to galaxy i and σ e,ij is the magnitude error listed in Table 2 of R11) with c (cid:13)000 , 000–000 G. Efstathiou
Global fits: NGC 4258 anchorT=2.5 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W .
00 71.1 (3.2) (2.4) 26.30 (0.17) -3.05 (0.13) -0.45 (0.15) 0.32 N N2 484 1 .
00 70.3 (3.2) (2.4) 26.35 (0.17) -3.08 (0.13) -0.31 (0.13) 0.32 W N3 481 1 .
00 69.7 (3.1) (2.3) 26.38 (0.16) -3.10 (0.12) -0.006 (0.020) 0.32 S N4 482 1 .
00 69.0 (3.0) (2.1) 26.49 (0.11) -3.18 (0.08) -0.006 (0.020) 0.32 S YT=2.25 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W .
00 70.6 (3.0) (2.1) 26.59 (0.15) -3.24 (0.11) -0.53 (0.13) 0.21 N N6 459 1 .
00 70.3 (3.0) (2.1) 26.59 (0.15) -3.23 (0.11) -0.40 (0.11) 0.22 W N7 447 1 .
00 70.8 (3.0) (2.0) 26.61 (0.14) -3.22 (0.10) -0.007 (0.020) 0.18 S N8 447 1 .
00 70.8 (2.9) (2.0) 26.61 (0.10) -3.23 (0.07) -0.007 (0.020) 0.18 S YR11 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W .
64 72.3 (2.8) (1.8) 26.43 (0.13) -3.10 (0.09) -0.33 (0.11) 0 .
21 N N10 390 0 .
64 72.1 (2.8) (1.8) 26.44 (0.13) -3.11 (0.10) -0.25 (0.10) 0 .
21 W N11 390 0 .
65 71.2 (2.8) (1.8) 26.48 (0.13) -3.14 (0.09) -0.007 (0.016) 0 .
21 S N12 390 0 .
65 70.8 (2.7) (1.7) 26.56 (0.09) -3.19 (0.06) -0.007 (0.016) 0 .
21 S Y
Table 2. N fit gives the number of Cepheids accepted by the outlier rejection criteria. The numbers in brackets give the 1 σ errors onthe parameters computed from the diagonals of the inverse covariance matrix. For H the first number in brackets gives the total erroron the Hubble constant, including the megamaser distance error and the SNe magnitude errors. The second number in brackets liststhe error in H from the P-L relation only. The column labelled σ int lists the internal scatter. The last two columns indicate the priorapplied to metallicity dependence Z W (N: no prior; W: weak prior; S: strong prior) and to the P-L slope b W (N: no prior; Y: prior) assummarized in (8a) - (8c). Fits 1-4 list results for T = 2 . T = 2 .
25 outlier rejection and fits 9-12 for theR11 rejection algorithm. respect to the parameters of the global fit. I use only Cepheids with periods
P <
60 days in these fits. (See the Appendix Afor remarks on the effects of extending the period range).I set σ int = 0 .
30 initially and reject Cepheids with absolute magnitude residuals relative to the global fit that are greaterthan T (cid:112) ( σ e,ij + σ ) for a chosen threshold T . I then recompute σ to give ˆ χ = 1 and repeat until the fits and rejectionconditions converge. The algorithm is statistically self consistent , in the sense that the solutions coverge with ˆ χ = 1 for apositive value of σ , as long as T is chosen to be greater than T = 2 .
1. Below I will show results for T = 2 . T = 2 . σ int is expected to be non-zero since there will be scatter in the P-L relation from the finite width of theinstability strip. The analysis of the LMC Cepheids in Section 2 suggests that at H-band the internal scatter of the P-Lrelation is about 0 . . σ int ≈ .
14 mag. Systematic under-estimation of magnitudeerrors ( e.g. crowding corrections) or contamination by outliers will result in higher values of σ int .Results of the global fits, propagated through to values of H , are listed in Table 2. Here I have used the new NGC4258 maser distance of 7 . ± .
23 Mpc (H13) which improves on the maser distance of Herrnstein et al. (1999) and is higherthan the distance adopted by R11 of 7 . ± .
22 Mpc (see Riess et al. , 2012). This change alone revises H downwards byapproximately 3 km s − Mpc − . The SNe magnitudes and errors as listed in Table 3 of R11. Figure 2 shows the magnituderesiduals with respect to global fit number 5 for Cepheids in NGC 4258 (left) and for the SNe host galaxies (right). The resultsof Table 2 are in agreement with the conclusion of R11, namely that the primary sensitivity to outliers comes from a smallnumber of highly deviant points that are easy to identify and that are rejected by all three rejection conditions. However,the effects of applying different rejection criteria is non-negligible. Comparing pairs of fits with the different outlier rejectionconditions explored in in Table 2 shows differences in H of ∼ . − Mpc − . An estimate of the error associated withoutlier rejection should therefore be folded into the total error on H . (In fact, R11 add 0 . − in quadrature to their finalerror estimate on H to account for systematic errors such as sensitivity to outlier rejection.) For a Gaussian distribution, the application of a threshold rejection will bias ˆ χ low by factors of 0 .
921 and 0 .
856 for T = 2 . T = 2 .
25 respectively. These biases are neglected in this analysis. c (cid:13) , 000–000 Revisited Figure 2.
P-L magnitude residuals relative to global fit 5 of Table 2: Residuals for NCG4258 Cepheids are shown on the left and the SNehost galaxy Cepheids are shown on the right. Filled (red) dots show residuals for Cepheids that are accepted by both the R11 and the T = 2 .
25 rejection criterion. Filled (blue) squares are rejected by both algorithms. Open (red) squares are rejected by R11 but acceptedby the T = 2 .
25 rejection criterion and the single open (blue) circle is rejected by the T = 2 .
25 criterion but accepted by R11. The errorsshow the H-band magnitude errors as listed in R11. The dotted lines show offsets that would produce changes of ± − Mpc − inthe value of H (increasing H is shown by the direction of the arrow in each plot). More worryingly, fits 1, 5 and 9 all show a strong metallicity dependence, apparently at the 3 − σ significance level. Themetallicity dependence of the P-L relation has been controversial for many years. At optical wavelengths, there is evidencefor a metallicity dependence of ∼ − .
25 mag . dex − (Kennicutt et al. et al. et al. et al. et al. et al. Z H = 0 . ± .
02 mag . dex − , (7)for the metallicity dependence at H-band. An updated version of this analysis is presented in Freedman et al. (2011) in whichthe metallicity dependence of LMC Cepheids at H-band appears to be even weaker than in equation (7). Equation (7) clearlyconflicts with the results of Table 2 .Yet as can be seen from Figure 3, the R11 data contain sub-luminous low metallicity Cepheids. In these plots, we show themagnitude residual with respect to fit 5, but setting Z w = 0 ( i.e. neglecting the metallicity dependence of the P-L relation).The right hand panel of Figure 3 shows low metallicity Cepheids, almost all of which lie below the mean relation. Some ofthese Cepheids are rejected by R11 but accepted by the T = 2 .
25 rejection (open red symbols). This is mainly because theseCepheids fail the R11 0 .
75 mag. cut which is applied to the H-band magnitudes before fitting for a metallicity dependence ofthe P-L relation. This difference in rejection explains why the metallicity dependence is stronger in the T = 2 . T = 2 . / H) > .
6. Eliminating the small number of starswith 12 + log(O / H) < . Z W , based on [O/H] metallicities, are discussed in Section 5.Another way of reducing the sensitivity to outliers and possible systematics in the data is to impose priors on theparameters of the P-L relation. I have explored the following priors: (cid:104) Z w (cid:105) = 0 , σ Z w = 0 . , weak metallicity prior , (8a) The constraint of equation (7) is derived from a small sample of 21 LMC Cepheids. Since the metallicity dependence is a small effect,it is possible that subtle effects such as a correlation between effective temperature and metallicity, or between metallicty and age leadto biases in the inferred metallicity dependence of the P-L relation. Equation (7) is consistent with a weak metallicity dependence ofthe P-L relation at near-infrared wavelengths, however, both the variation of the metallicity dependence with waveband reported byFreedman and Madore (2011; see also Romaniello et al. , 2008) and their error estimates should be treated with caution.c (cid:13)000
6. Eliminating the small number of starswith 12 + log(O / H) < . Z W , based on [O/H] metallicities, are discussed in Section 5.Another way of reducing the sensitivity to outliers and possible systematics in the data is to impose priors on theparameters of the P-L relation. I have explored the following priors: (cid:104) Z w (cid:105) = 0 , σ Z w = 0 . , weak metallicity prior , (8a) The constraint of equation (7) is derived from a small sample of 21 LMC Cepheids. Since the metallicity dependence is a small effect,it is possible that subtle effects such as a correlation between effective temperature and metallicity, or between metallicty and age leadto biases in the inferred metallicity dependence of the P-L relation. Equation (7) is consistent with a weak metallicity dependence ofthe P-L relation at near-infrared wavelengths, however, both the variation of the metallicity dependence with waveband reported byFreedman and Madore (2011; see also Romaniello et al. , 2008) and their error estimates should be treated with caution.c (cid:13)000 , 000–000
G. Efstathiou
Figure 3.
P-L relations for the data used in fit 5. The symbols and colour coding of the points are as in Figure 3. In these plots, theF160W magnitudes have not been corrected for a metallicity dependence. The figure to the left shows high metallicity Cepheids and thefigure to the right shows low metallicity Cepheids. The lines show the best fit P-L relation for the entire sample. (cid:104) Z w (cid:105) = 0 , σ Z w = 0 . , strong metallicity prior , (8b) (cid:104) b (cid:105) = − . , σ b = 0 . , P − L slope prior . (8c)The first of these imposes a weak prior on the metallicity dependence of the P-L relation. The second imposes a strong prior,effectively eliminating a metallicity dependence of the P-L relation. The last condition is motivated by constraints on theslope of the LMC P-L relation discussed in Section 2.The results of applying these priors are listed in Table 2. For the R11 rejection, applying these priors drives H downwardsby ∼ . − Mpc − . The results for the T = 2 .
25 rejection are, however, extremely stable to the imposition of the priors.To determine a final ‘best-estimate’ of H using NGC 4258 as an anchor, I have averaged the H values of fits 3, 7, 11,and added the scatter between these estimates in quadrature with the largest of the error estimates. I have also added a1 km s − Mpc − error to account for systematics associated with the SNe magnitudes and light curve fitting. This gives H = 70 . ± . − Mpc − , NGC 4258 . (9)This value is lower than the value H = 72 ± − Mpc − quoted by H13 using the revised maser distance to NGC4258. This difference is caused mainly by the use of difference outlier rejection criteria and the imposition of a metallicityprior. Note also that (9) is within 1 σ of the Planck value of H for the base ΛCDM model. Since the mean metallicities of NGC 4258 and SNe hosts are similar (12 + log (O / H) ≈ . H if NGC 4258 is used as a distance anchor. However, ifwe use the LMC as an anchor (for which we assume 12 + log (O / H) = 8 . Z W ∼ − . − lead to a substantial reduction in the value of H compared tofits in which Z W is constrained to be zero (see Table 3).For the LMC distance, I use the new eclipsing binary distance of 49 . ± .
13 kpc determined by Pietrzy´nski et al. (2013).I minimise the sum of the χ of equations (3) and (6) (which are denoted χ and χ respectively) with respect to theparameters of the P-L relation applying the rejection criteria described in Section 3.1 to the R11 Cepheids. The LMC Cepheidsample is as described in Section 2 and σ int , LMC is kept fixed at 0 . T = 2 . T = 2 .
25 rejection criteria, σ int , WFC3 is adjusted to maintain ˆ χ = 1. In using the LMC (or MW) Cepheids as a distance anchor, the only role ofthe NGC 4258 Cepheids in determining H is to influence the slope and metallicity dependence of the global fit to the P-Lrelation. I have chosen to retain the NGC 4258 Cepheids (as did R11), though the results are very similar if these Cepheidsare excluded.Table 3 lists the results for the three rejection criteria. If no priors are included, the values of H show some sensitivityto the rejection algorithm. This sensitivity is caused by the low metallicity Cepheids in the R11 sample, which pull Z W towards negative values (less strongly for the R11 rejection algorithm). Applying the strong metallicity prior, we find verylittle sensitivity to the rejection algorithm. Furthermore, imposing the slope prior of (8c) has very little effect on the solutionsbecause the slopes of the global fits are well constrained by the LMC Cepheids. Averaging the results for fits 15, 19 and 23,we find: c (cid:13) , 000–000 Revisited Global fits: LMC anchorT=2.5 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W
13 479 1 .
00 68.6 (3.5) (2.2) 15.59 (0.08) -3.23 (0.05) -0.47 (0.14) 0.32 N N14 478 1 .
00 69.7 (3.4) (2.1) 15.64 (0.08) -3.23 (0.05) -0.35 (0.13) 0.32 W N15 481 1 .
00 73.5 (2.9) (1.9) 15.76 (0.06) -3.21 (0.05) -0.005 (0.020) 0.32 S N16 480 1 .
00 73.5 (2.8) (1.9) 15.77 (0.06) -3.22 (0.05) -0.006 (0.020) 0.32 S YT=2.25 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W
17 458 1 .
00 67.4 (3.2) (2.0) 15.57 (0.08) -3.23 (0.05) -0.53 (0.13) 0.21 N N18 459 1 .
00 68.7 (3.2) (2.0) 15.62 (0.08) -3.23 (0.05) -0.40 (0.11) 0.22 W N19 447 1 .
00 73.3 (2.8) (1.8) 15.78 (0.06) -3.23 (0.05) -0.007 (0.020) 0.18 S N20 447 1 .
00 73.3 (2.8) (1.8) 15.78 (0.05) -3.23 (0.05) -0.007 (0.020) 0.18 S YR11 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W
21 390 0 .
64 70.7 (3.1) (1.8) 15.63 (0.07) -3.21 (0.04) -0.31 (0.11) 0 .
21 N N22 390 0 .
64 71.3 (3.0) (1.8) 15.64 (0.06) -3.21 (0.04) -0.24 (0.10) 0 .
21 W N23 390 0 .
65 73.5 (2.7) (1.6) 15.76 (0.05) -3.21 (0.04) -0.006 (0.016) 0 .
21 S N24 390 0 .
65 73.4 (2.6) (1.5) 15.77 (0.05) -3.22 (0.04) -0.006 (0.016) 0 .
21 S YGlobal fits: MW Cepheids anchorT=2.5 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
25 486 1 .
00 76.5 (4.0) (3.7) -5.89 (0.05) -3.22 (0.11) -0.47 (0.15) 0.30 N N26 484 1 .
00 77.7 (4.1) (3.8) -5.88 (0.05) -3.15 (0.10) -0.31 (0.13) 0.32 W N27 482 1 .
00 76.5 (3.9) (3.6) -5.89 (0.05) -3.17 (0.11) -0.006 (0.020) 0.32 S N28 482 1 .
00 75.7 (3.4) (3.1) -5.89 (0.05) -3.20 (0.07) -0.006 (0.020) 0.32 S YT=2.25 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
29 458 1 .
00 75.2 (3.8) (3.5) -5.90 (0.05) -3.26 (0.10) -0.53 (0.13) 0.21 N N30 459 1 .
00 74.8 (3.7) (3.5) -5.90 (0.05) -3.26 (0.10) -0.40 (0.11) 0.22 W N31 459 1 .
00 73.6 (3.6) (3.3) -5.90 (0.05) -3.26 (0.10) -0.009 (0.020) 0.23 S N32 447 1 .
00 74.7 (3.2) (2.9) -5.90 (0.05) -3.24 (0.07) -0.007 (0.020) 0.18 S YR11 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
33 390 0 .
64 77.9 (3.3) (2.9) -5.88 (0.04) -3.15 (0.08) -0.32 (0.11) 0 .
21 N N34 390 0 .
64 77.4 (3.3) (2.9) -5.88 (0.04) -3.16 (0.08) -0.24 (0.10) 0 .
21 W N35 390 0 .
65 76.0 (3.2) (2.8) -5.89 (0.04) -3.18 (0.08) -0.006 (0.016) 0 .
21 S N36 390 0 .
65 75.4 (2.8) (2.4) -5.89 (0.04) -3.21 (0.06) -0.006 (0.016) 0 .
21 S Y
Table 3.
As in Table 2, the numbers in brackets give the 1 σ errors on the parameters computed from the diagonals of the inversecovariance matrix. For H the first number in brackets adds the errors arising from the distance anchors and SNe magnitudes. Thesecond number in brackets lists the error in H from the P-L relation alone. N fit gives the number of R11 Cepheids retained in the fits.The remaining columns are as defined in Table 3. H = 73 . ± . − Mpc − , LMC , strong metallicity prior . (10)More generally, the solutions of Table 3 show a strong sensitivity to the metallicity dependence of the P-L relation: H ≈ (73 . Z w ) km s − Mpc − . (11)An accurate determination of H using the LMC as an anchor therefore requires a precise determination of Z w . As discussedin Section 3.2, the strong metallicity dependence at H-band wavelengths derived from the R11 data conflict with the weakmetallicity dependence of equation (7). Applying the strong metallicity prior (10), H is in tension, at about the 1 . σ , levelwith the Planck base ΛCDM value for H . This tension can be relieved if the metallicity dependence of the P-L relation issomewhat stronger than implied by equation (7). c (cid:13)000
As in Table 2, the numbers in brackets give the 1 σ errors on the parameters computed from the diagonals of the inversecovariance matrix. For H the first number in brackets adds the errors arising from the distance anchors and SNe magnitudes. Thesecond number in brackets lists the error in H from the P-L relation alone. N fit gives the number of R11 Cepheids retained in the fits.The remaining columns are as defined in Table 3. H = 73 . ± . − Mpc − , LMC , strong metallicity prior . (10)More generally, the solutions of Table 3 show a strong sensitivity to the metallicity dependence of the P-L relation: H ≈ (73 . Z w ) km s − Mpc − . (11)An accurate determination of H using the LMC as an anchor therefore requires a precise determination of Z w . As discussedin Section 3.2, the strong metallicity dependence at H-band wavelengths derived from the R11 data conflict with the weakmetallicity dependence of equation (7). Applying the strong metallicity prior (10), H is in tension, at about the 1 . σ , levelwith the Planck base ΛCDM value for H . This tension can be relieved if the metallicity dependence of the P-L relation issomewhat stronger than implied by equation (7). c (cid:13)000 , 000–000 G. Efstathiou
Figure 4.
Period-luminosity relation for 13 MW Cepheids with parallax measurements (from van Leeuwen et al.
I use the sample of 13 MW Cepheids with parallax measurements and photometry as listed in van Leeuwen et al. (2007)(eliminating Polaris). The Wesenheit P-L relation (HVI photometry) for these Cepheids is shown in Figure 5. A fit to thesegives M W = − . ± . , b W = − . ± . , ˆ χ = 0 . , (12)where M W replaces A in equation (1). Note that the slope is not well constrained because of the dearth of Cepheids withperiods greater than 10 days in this small sample. The slope is, however, compatible with the slope determined from the LMCCepheids. The reduced χ of this sample is smaller than unity, but because of the small sample size this is not statisticallysignificant and leaves room for a significant ‘internal dispersion’ (which cannot be well constrained from ˆ χ ). In the fits below,I adopt an internal dispersion of σ int = 0 .
10, consistent with the internal dispersion of the LMC sample. The parametersof the MW Cepheid fits discussed below are insensitive to this value, though adopting σ int = 0 .
10 has the effect of slightlydownweighting the MW Cepheids compared to the LMC and/or NGC 4258 when combining distance anchors.The global fits using the MW parallax distances are summarized in Table 3. The metallicity dependences of these fits areskewed by the low metallicity outliers in the R11 sample. These raise H and introduce a sensitivity to the outlier rejectionalgorithm. Imposing the strong metallicity and P-L slope priors reduces the sensitivity to these outliers. Averaging the resultsof fits 28, 32 and 36 gives H = 75 . ± . − Mpc − , MW , strong metallicity and slope priors . (13)This result is in tension at about the 2 . σ , level with the Planck base ΛCDM value for H . Because of the small size of theMW Cepheid sample, the instability strip is not well sampled. In addition, the lack of overlap between the periods of theMW Cepheids and the R11 sample leads to a sensitivity of H to the slope prior and to the choice of period range for theR11 Cepheids (see Appendix A). Coupled with possible systematic errors associated with matching ground-based and HSTphotometry, I consider the MW Cepheids to be the least reliable of the three distance anchors. Whereas R11 found similar values of H using NGC 4258 and the MW Cepheids as distance anchors, with H for the LMClying low, we find reasonable agreement between the H values for the LMC and MW Cepheids (equations 10 and 13) with H for NGC 4258 lying low (equation 9). There are two reasons why our results differ from R11. Firstly, the revised megamaserdistance to NGC 4258 of H13 lowers H by about 3 km s − Mpc − . Secondly, we have argued that the strong metallicitydependence of the global fits is caused by sub-luminous outliers in the P-L relation, and may be unphysical. Imposing a strongmetallicity prior centred around Z W = 0 raises the LMC solutions for H substantially. A further difference between R11 c (cid:13) , 000–000 Revisited Global fits: NGC 4258+LMC anchors
T=2.5 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
37 477 1 .
00 69.3 (2.4) -6.10 (0.06) -3.22 (0.05) -0.45 (0.13) 0.30 N N38 478 1 .
00 69.8 (2.4) -6.07 (0.06) -3.22 (0.05) -0.35 (0.11) 0.30 W N39 477 1 .
00 71.4 (2.4) -5.99 (0.05) -3.23 (0.05) -0.009 (0.020) 0.30 S N40 477 1 .
00 71.5 (2.4) -5.99 (0.05) -3.24 (0.05) -0.009 (0.020) 0.30 S YT=2.25 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
41 458 1 .
00 69.0 (2.3) -6.10 (0.06) -3.23 (0.05) -0.48 (0.12) 0.21 N N42 459 1 .
00 69.4 (2.3) -6.07 (0.06) -3.23 (0.05) -0.38 (0.11) 0.22 W N43 448 1 .
00 71.8 (2.3) -5.97 (0.05) -3.24 (0.05) -0.009 (0.020) 0.18 S N44 448 1 .
00 71.9 (2.3) -5.97 (0.05) -3.23 (0.04) -0.009 (0.020) 0.18 S YR11 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
45 390 0 .
64 71.0 (1.9) -6.06 (0.05) -3.21 (0.04) -0.29 (0.10) 0 .
21 N N46 390 0 .
64 71.2 (1.9) -6.04 (0.05) -3.21 (0.04) -0.24 (0.09) 0 .
21 W N47 390 0 .
66 72.1 (1.9) -5.98 (0.04) -3.22 (0.04) -0.008 (0.016) 0 .
21 S N48 390 0 .
65 72.1 (1.9) -5.98 (0.04) -3.22 (0.04) -0.008 (0.016) 0 .
21 S Y
Global fits: NGC 4258+MW Cepheid anchors
T=2.5 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
49 476 1 .
00 72.2 (2.9) -5.96 (0.05) -3.31 (0.09) -0.51 (0.14) 0.29 N N50 478 1 .
00 71.7 (2.9) -5.96 (0.05) -3.32 (0.09) -0.35 (0.13) 0.30 W N51 477 1 .
00 70.9 (2.8) -5.96 (0.05) -3.32 (0.09) -0.007 (0.020) 0.30 S N52 477 1 .
00 71.6 (2.6) -5.96 (0.05) -3.28 (0.07) -0.007 (0.020) 0.30 S YT=2.25 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
53 456 1 .
00 72.1 (2.8) -5.95 (0.06) -3.32 (0.09) -0.51 (0.12) 0.21 N N54 455 1 .
00 71.4 (2.7) -5.95 (0.05) -3.34 (0.09) -0.38 (0.11) 0.21 W N55 447 1 .
00 71.5 (2.7) -5.94 (0.05) -3.30 (0.08) -0.007 (0.020) 0.18 S N56 448 1 .
00 72.2 (2.5) -5.95 (0.05) -3.27 (0.05) -0.007 (0.020) 0.19 S YR11 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
57 390 0 .
65 73.8 (2.3) -5.95 (0.04) -3.23 (0.07) -0.30 (0.11) 0 .
21 N N58 390 0 .
64 73.5 (2.3) -5.95 (0.04) -3.24 (0.07) -0.23 (0.07) 0 .
21 W N59 390 0 .
66 72.6 (2.3) -5.95 (0.04) -3.25 (0.07) -0.006 (0.016) 0 .
21 S N60 390 0 .
66 72.7 (2.1) -5.95 (0.04) -3.24 (0.05) -0.006 (0.016) 0 .
21 S Y
Global fits: LMC+MW Cepheid anchors
T=2.5 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
61 479 1 .
00 72.4 (2.5) -5.98 (0.05) -3.25 (0.05) -0.30 (0.12) 0.31 N N62 479 1 .
00 72.6 (2.5) -5.96 (0.05) -3.25 (0.05) -0.24 (0.11) 0.31 W N63 480 1 .
00 74.1 (2.5) -5.92 (0.05) -3.23 (0.05) -0.005 (0.020) 0.32 S N64 477 1 .
00 74.1 (2.5) -5.92 (0.05) -3.23 (0.04) -0.005 (0.020) 0.32 S YT=2.25 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
65 459 1 .
00 71.6 (2.4) -5.99 (0.05) -3.26 (0.05) -0.36 (0.11) 0.21 N N66 455 1 .
00 71.9 (2.4) -5.97 (0.05) -3.26 (0.05) -0.28 (0.10) 0.21 W N67 448 1 .
00 73.6 (2.4) -5.92 (0.05) -3.24 (0.05) -0.007 (0.020) 0.18 S N68 448 1 .
00 73.7 (2.4) -5.92 (0.05) -3.24 (0.04) -0.007 (0.020) 0.18 S YR11 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
69 390 0 .
64 73.4 (2.0) -5.96 (0.04) -3.23 (0.04) -0.20 (0.09) 0 .
21 N N70 390 0 .
65 73.6 (2.0) -5.95 (0.04) -3.23 (0.04) -0.16 (0.09) 0 .
21 W N71 390 0 .
65 74.1 (2.0) -5.92 (0.04) -3.23 (0.04) -0.005 (0.016) 0 .
21 S N72 390 0 .
65 74.0 (2.0) -5.92 (0.04) -3.23 (0.04) -0.006 (0.016) 0 .
21 S Y
Table 4.
Solutions combining distance anchors. The columns are as defined in Tables 2 and 3.c (cid:13)000
Solutions combining distance anchors. The columns are as defined in Tables 2 and 3.c (cid:13)000 , 000–000 G. Efstathiou
Global fits: NGC 4258+LMC+MW Cepheids anchor
T=2.5 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
73 479 1 .
00 71.5 (2.2) -6.00 (0.04) -3.26 (0.05) -0.33 (0.12) 0.30 N N74 479 1 .
00 71.6 (2.3) -5.99 (0.04) -3.26 (0.05) -0.27 (0.11) 0.31 W N75 477 1 .
00 72.3 (2.3) -5.95 (0.04) -3.26 (0.05) -0.008 (0.020) 0.30 S N76 477 1 .
00 72.4 (2.2) -5.95 (0.04) -3.25 (0.04) -0.008 (0.020) 0.30 S YT=2.25 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
77 455 1 .
00 71.3 (2.2) -5.99 (0.04) -3.26 (0.05) -0.34 (0.11) 0.21 N N78 455 1 .
00 71.4 (2.2) -5.98 (0.04) -3.26 (0.05) -0.29 (0.10) 0.21 W N79 447 1 .
00 72.4 (2.2) -5.94 (0.04) -3.25 (0.05) -0.007 (0.020) 0.18 S N80 447 1 .
00 72.5 (2.2) -5.94 (0.04) -3.25 (0.04) -0.007 (0.020) 0.18 S YR11 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W
81 390 0 .
64 72.6 (1.8) -5.98 (0.03) -3.24 (0.04) -0.22 (0.09) 0 .
21 N N82 390 0 .
65 72.6 (1.8) -5.97 (0.03) -3.24 (0.04) -0.18 (0.08) 0 .
21 W N83 390 0 .
66 72.9 (1.8) -5.95 (0.03) -3.24 (0.04) -0.006 (0.016) 0 .
21 S N84 390 0 .
66 72.9 (1.8) -5.95 (0.03) -3.24 (0.04) -0.006 (0.016) 0 .
21 S Y
Table 4. (contd.)
Solutions combining distance anchors. results and the results presented here is that our errors on H are larger. This is most noticeable for the MW solutions inTable 3.To account for correlated errors between ground-based and HST photometry and correlated errors between the SNemagnitudes, I minimise: χ = (cid:88) ij,j =1 − ( m W,ij − m PW,i ) ( σ e,ij + σ ) + ( m W,i − m PW )( C LMC+MW ) − ij ( m w,j − m PW ) + ( m V,i − m PV )( C SNe ) − ij ( m V,j − m PV )+ ( µ , − µ M , ) σ µ , + ( µ , LMC − µ M , LMC ) σ µ , LMC . (14)The first term is summed over the R11 Cepheids as in equation (6). The second term is summed over the LMC and MWCepheids, where the covariance matrix C LMC , MW is C LMC , MW ij = ( σ i + σ ) δ ij + σ , (15)where σ i is the magnitude error, σ int is the internal scatter, and σ cal is the calibration error between the ground based andWFC3 photometry (assumed to be σ cal = 0 .
04 mag. as in R11). The third term is summed over the SNe magnitudes m V,i and the covariance matrix C SNe is C SNe ij = ( σ i ) δ ij + σ a V , (16)where σ i is the SNe magnitude error and σ a V is the error in 5 a V , where a V is the intercept of the SNe Ia magnitude-redshiftrelation ( σ a V = 0 . µ , and µ , LMC are the distance moduli of NGC 4258 and the LMC, µ M , and µ M , LMC are their measured values (from the maser and eclipsing binary distances respectively) with errors σ µ , and σ µ , LMC . The theoretically predicted magnitudes areSNe hosts : m Pij = µ ,i + M W + b W (log P ij −
1) + Z W ∆log(O / H) ij , (17a)NGC 4258 : m Pj = µ , + M W + b W (log P j −
1) + Z W ∆log(O / H) j , (17b)LMC : m Pj = µ ,LMC + M W + b W (log P j −
1) + Z W ∆log(O / H) j , (17c)MW : m Pj = M W + b W (log P j −
1) + Z W ∆log(O / H) j , (17d)SNe : m PV i = µ ,i + 5log H − − a V . (17e)Results for joint distance anchor fits are given in Table 4. There are several points worth noting: • As in the analyses presented in previous Sections, the low metallicity Cepheid outliers in the R11 data lead to a sensitivityto the outlier rejection criterion. • With no metallicity or slope priors, the combined solutions using NGC 4258 + LMC anchors are discrepant at > ∼ σ withthe MW anchor solutions given in Table 3. (Compare, for example, the results for fits 33 and 45.) • Applying a strong metallicity prior, the solutions for H for the combined NGC 4258 and LMC fits become insensitive tothe outlier rejection criteria (and are consistent to within ∼ . − Mpc − ). c (cid:13) , 000–000 Revisited Figure 5.
Period-luminosity fit used to determine the distance modulus to NGC 4258 using the MW Cepheids as a distance anchor.The MW Cepheids are shown by the (green) filled stars. The rest of the points (red and blue) show the R11 Cepheids. As in Figure 2,filled (red) circles are accepted by both the R11 and T = 2 .
25 rejection criteria while filled (blue) sqaures are rejected by both criteria.Open (red) squares are rejected by R11 but accepted by the T = 2 .
25 criterion. The line shows the best fit P-L relation.. • Since the slope of the P-L relation is well constrained by the LMC Cepheids, adding a slope prior has almost no effect onsolutions that include the LMC Cepheids.If we accept each of these distance anchors at face value, and average over the outlier rejection criteria as in the previousSection using the solutions with strong metallicity and no slope priors for (18a) and (18d) and strong metallicity and slopepriors for (18b) and (18c), then we find: H = 71 . ± . − Mpc − , NGC 4258 + LMC , (18a) H = 72 . ± . − Mpc − , NGC 4258 + MW , (18b) H = 73 . ± . − Mpc − , LMC + MW , (18c) H = 72 . ± . − Mpc − , NGC 4258 + LMC + MW . (18d)These values differ by 1 . σ , 1 . σ , 2 . σ and 1 . σ respectively from the Planck value of H for the base ΛCDM model. Evidently,using the LMC and especially the MW Cepheids as distance anchors pulls H to higher values than those derived using themegamaser distance. Note that when R11 combine all three distance anchors, they conservatively adopt the largest error fromany pair of distance anchors. Adopting the same approach would increase the error in (18d) to 2 . − Mpc − . Before accepting (18a) - (18d) it is worth investigating internal consistency tests of the three distance anchors. We first usethe MW Cepheids to compute the distance modulus to the LMC assuming no metallicity dependence of the P-L relation. Theresult is µ ,LMC = 18 . ± . , (19)in good agreement with the Pietrzy´nski et al. (2013) eclipsing binary distance modulus of 18 . ± . Z W = 0 . ± .
16 mag . dex − , (20)consistent with a weak metallicity dependence of the P-L relation at H-band.Next, we use the LMC Cepheids to determine a distance modulus to NGC 4258, assuming the Pietrzy´nski et al. (2013)eclipsing binary distance. This solution is based on the likelihood of equation (14). I impose the strong metallicity prior andaverage over the three outlier rejection criteria, adding the scatter to the final error estimate. The result is µ , = 29 . ± . , (21)which is within 1 . σ of the H13 megamaser distance modulus of 29 . ± . c (cid:13)000
16 mag . dex − , (20)consistent with a weak metallicity dependence of the P-L relation at H-band.Next, we use the LMC Cepheids to determine a distance modulus to NGC 4258, assuming the Pietrzy´nski et al. (2013)eclipsing binary distance. This solution is based on the likelihood of equation (14). I impose the strong metallicity prior andaverage over the three outlier rejection criteria, adding the scatter to the final error estimate. The result is µ , = 29 . ± . , (21)which is within 1 . σ of the H13 megamaser distance modulus of 29 . ± . c (cid:13)000 , 000–000 G. Efstathiou
Figure 6.
The direct estimates (red) of H (together with 1 σ error bars) for the NGC 4258 distance anchor (equation 9) and for all threedistance anchors (equation 18d). The remaining (blue) points show the constraints from P13 for the base ΛCDM cosmology and someextended models combining CMB data with data from baryon acoustic oscillation surveys. The extensions are as follows: m ν , the massof a single neutrino species; m ν + Ω k , allowing a massive neutrino species and spatial curvature; N eff , allowing additional relativisticneutrino-like particles; N eff + m sterile , adding a massive sterile neutrino and additional relativistic particles; N eff + m ν , allowing a massiveneutrino and additional relativistic particles; w , dark energy with a constant equation of state w = p/ρ ; w + w a , dark energy with atime varying equation of state. I give the 1 σ upper limit on m ν and the 1 σ range for N eff . See P13 for further details on these extendedmodels. Z W = − . ± .
26 mag . dex − , (22)consistent with zero but with even lower precision than the estimate of (20).Finally, I use the MW Cepheids to compute a distance modulus to NGC 4258. Since the MW Cepheids have similarmetallicities to the mean of the Cepheids in NGC 4258, uncertainties in the metallicity dependence of the P-L relation do notintroduce a significant source of error into the distance modulus. Nevertheless, I impose the strong metallicity prior on thesolutions and average over the three outlier criteria. This gives µ , = 29 . ± . , (23)which is about 1 . σ lower than the H13 distance modulus. Figure 5 shows the T = 2 .
25 fit to the MW and NGC 4258Cepheids.The differences between the H values for the three distance anchors are reflected by these differences in the distancemoduli of NGC 4258. The MW and LMC Cepheids (assuming zero metallicity dependence) give a shorter distance to NGC4258 than the H13 revised megamaser distance. It is possible that the true distance to NGC 4258 is substantially lower thanthe H13 central value. However, it is also possible that the tension is caused by a more subtle effect, for example, a residual ∼ . The SH0ES project was cleverly designed to minimise the impact of metallicity, crowding, and photometric calibration biaseswhen comparing Cepheids measured in SNe hosts with those in NGC 4258. These methodological reasons argue that higher c (cid:13) , 000–000 Revisited weight should be placed on the NGC 4258 anchor than either the LMC or MW anchors when using the R11 data. However, thevalue of H derived using NGC 4258 as an anchor relies on the fidelity of the geometric maser distance. Despite the extensiveVLBI campaign described by H13, systematic errors contribute significantly to the total error in the megamaser distance.It may therefore be dangerous to place very high weight on the NGC 4258 distance without cross-checks with independentdistance anchors.However, in addition to the methodological issues that drove the design of the SH0ES project, there are significantproblems in using the LMC and MW distance anchors. Although there are several accurate and consistent eclipsing binarydistance estimates to the LMC (Fitzpatrick et al. et al. et al. H derived usingthe LMC as a distance anchor is extremely sensitive to any metallicity dependence of the P-L relation (equation 11). I showthat the R11 sample contains sub-luminous low metallicity Cepheids, pointing either to a stronger than expected metallicitydependence of the near-IR P-L relation (in conflict with the Freedman and Madore 2011 analysis), a possible misidentificationof these objects as classical Cepheids, or to some unidentified systematic error in their magnitudes. The presence of thesesub-luminous Cepheids causes some sensitivity to the rejection criteria used to identify outliers from the mean P-L relation.However, if a strong metallicity prior is imposed, the global fits and derived values of H become insensitive to the outlierrejection criteria. (It is also worth noting that the strong metallicity prior also affects the value of H derived using NGC 4258as a distance anchor: we find H = 70 . ± . − Mpc − compared to the value H = 72 . ± . − Mpc − quoted byH13.) One would have greater confidence in using the LMC anchor if there were stronger observational constraints on Z W .The sample of MW Cepheids with parallax measurements is small and contains only one star that overlaps with theperiod range sampled by Cepheids in the SNe host galaxies ( cf Figures 2 and 4). Use of the MW Cepheids as an anchor istherefore susceptible to sample biases and small number statistics. The distance modulus to NGC 4258 derived from the MWCepheids is lower by about 1 . σ compared to the H13 megamaser distance modulus. As a consequence, H derived using theMW Cepheids as a distance anchor is higher than that derived from the megamaser distance and is discrepant by about 2 . σ with the Planck base ΛCDM value. This is the largest discrepancy reported in this paper with the Planck determination of H . Observations with the GAIA satellite will increase the number of Galactic Cepheids with accurate parallaxes into themany thousands. It will be interesting to see whether the tensions with the megamaser distance and with the Planck baseΛCDM cosmology persist.The value of H derived here from the megamaser distance is within 1 σ of the Planck base ΛCDM value of H . Althoughthere are some tensions between the three distance anchors, none are sufficiently compelling to justify excluding either theMW or LMC anchors from a joint fit. Imposing the strong metallicity prior, the combination of all three distance anchorsraises H to 72 . ± . − Mpc − , which is within 1 . σ of the Planck base ΛCDM value.Figure 6 compares these two estimates of H with the P13 results from the Planck +WP+highL+BAO likelihood forthe base ΛCDM cosmology and some extended ΛCDM models. I show the combination of CMB and BAO data since H is poorly constrained for some of these extended models using CMB temperature data alone. (For reference, for this datacombination H = 67 . ± .
77 km s − Mpc − in the base ΛCDM model.) The combination of CMB and BAO data is certainlynot prejudiced against new physics, yet the H values for the extended ΛCDM models shown in this figure all lie within 1 σ of the best fit value for the base ΛCDM model. For example, in the models exploring new physics in the neutrino sector, thecentral value of H never exceeds 69 . − Mpc − . If the true value of H lies closer to, say, H = 74 km s − Mpc − , thedark energy sector, which is poorly constrained by the combination of CMB and BAO data, seems a more promising place tosearch for new physics.In summary, the discrepancies between the Planck results and the direct H measurements shown in Figure 5 are notlarge enough to provide compelling evidence for new physics beyond the base ΛCDM cosmology. Acknowledgements:
I am particularly grateful to Adam Riess, who has answered patiently my many questions on theanalysis of the SH0ES data. I also thank Rob Kennicutt, Wendy Freedman and Brian Schmidt for valuable discussions andcorrespondence and the referee for a helpful report.
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APPENDIX A: EXTENDING THE PERIOD RANGE OF THE GLOBAL FITS
Throughout this paper, I imposed an upper period limit of 60 days on the R11 Cepheid sample. As noted in Section 2, thereis evidence from analyses of LMC Cepheids that the P-L relation flattens for Cepheids with periods >
60 days (Persson etal. , 2004; Freedman et al. et al. <
205 days in theiranalysis. The purpose of this Appendix is to show how the results of Sections 3 and 4 change if the Cepheid period range isextended. Table A1 is the equivalent of Tables 2 and 3, but using R11 Cepheids with periods <
205 days. The main changeis that the slopes of the P-L relation in many of the fits become substantially flatter than the LMC slopes of equation (4).As a consequence, the global fits in Table A1 become more sensitive to the imposition of an LMC slope prior, whereas thefits in Tables 2 and 3 are insensitive to the slope prior. This is particularly true for fits A25-A36 using the MW Cepheids asan anchor. Without any slope prior, H is about 80 km s − Mpc − and is inconsistent with H determined using the maserdistance to NGC 4258 (fits A1-A12). This provides further evidence that the MW Cepheids are the least reliable of thethree distance anchors. R11 adopted a slope prior of − . ± . H usingeither NGC 4258 or the LMC Cepheids as a distance anchor. However, adopting an upper period of limit of 60 days, as in themain body of this paper, all of the P-L slopes are consistent with the LMC P-L relation. The global fits are then insensitiveto the imposiition of an LMC slope prior (even for the MW Cepheids). c (cid:13) , 000–000 Revisited Global fits: NGC 4258 anchor
T=2.5 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W A1 552 1 .
00 71.5 (3.1) (2.2) 26.08 (0.12) -2.88 (0.08) -0.33 (0.14) 0.30 N NA2 550 1 .
00 71.6 (3.1) (2.2) 26.09 (0.12) -2.88 (0.08) -0.21 (0.12) 0.29 W NA3 551 1 .
00 71.0 (3.0) (2.1) 26.12 (0.12) -2.91 (0.08) -0.005 (0.020) 0.30 S NA4 551 1 .
00 69.2 (2.9) (2.0) 26.33 (0.09) -3.06 (0.06) -0.004 (0.020) 0.31 S YT=2.25 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W A5 523 1 .
00 72.0 (2.9) (2.0) 26.21 (0.10) -2.95 (0.07) -0.46 (0.12) 0.21 N NA6 524 1 .
00 71.7 (2.9) (1.9) 26.22 (0.10) -2.96 (0.07) -0.32 (0.11) 0.21 W NA7 524 1 .
00 71.3 (2.9) (1.9) 26.20 (0.10) -2.95 (0.07) -0.006 (0.020) 0.21 S NA8 519 1 .
00 70.9 (2.9) (1.9) 26.38 (0.09) -3.06 (0.06) -0.006 (0.020) 0.20 S YR11 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W A9 448 0 .
65 72.3 (2.8) (1.8) 26.30 (0.10) -3.01 (0.07) -0.26 (0.10) 0 .
21 N NA10 448 0 .
65 72.1 (2.8) (1.7) 26.31 (0.10) -3.01 (0.07) -0.21 (0.07) 0 .
21 W NA11 448 0 .
66 71.4 (2.7) (1.7) 26.35 (0.07) -3.04 (0.07) -0.006 (0.016) 0 .
21 S NA12 448 0 .
66 70.7 (2.7) (1.6) 26.45 (0.07) -3.11 (0.05) -0.006 (0.016) 0 .
21 S Y
Global fits: LMC anchor
T=2.5 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W A13 553 1 .
00 71.3 (3.6) (2.2) 15.54 (0.08) -3.11 (0.04) -0.27 (0.14) 0.31 N NA14 552 1 .
00 72.1 (3.4) (2.1) 15.57 (0.07) -3.11 (0.04) -0.20 (0.12) 0.31 W NA15 551 1 .
00 74.2 (2.9) (1.9) 15.65 (0.05) -3.11 (0.04) -0.003 (0.020) 0.31 S NA16 551 1 .
00 73.9 (2.9) (1.8) 15.67 (0.05) -3.13 (0.04) -0.003 (0.020) 0.31 S YT=2.25 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W A17 523 1 .
00 69.9 (3.2) (2.0) 15.49 (0.07) -3.11 (0.04) -0.38 (0.12) 0.21 N NA18 523 1 .
00 70.5 (3.2) (2.0) 15.52 (0.07) -3.11 (0.04) -0.31 (0.11) 0.21 W NA19 519 1 .
00 74.2 (2.8) (1.8) 15.65 (0.05) -3.11 (0.04) -0.006 (0.020) 0.20 S NA20 518 1 .
00 73.9 (2.8) (1.7) 15.68 (0.05) -3.14 (0.04) -0.006 (0.020) 0.20 S YR11 Rejection PriorsFit N fit ˆ χ H p W b W Z w σ int Z w b W A21 448 0 .
65 72.1 (3.1) (1.7) 15.58 (0.06) -3.13 (0.03) -0.22 (0.10) 0 .
21 N NA22 448 0 .
65 72.5 (3.0) (1.7) 15.60 (0.06) -3.13 (0.03) -0.18 (0.09) 0 .
21 W NA23 448 0 .
66 74.2 (2.7) (1.7) 15.68 (0.04) -3.14 (0.03) -0.005 (0.016) 0 .
21 S NA24 448 0 .
66 73.9 (2.6) (1.5) 15.70 (0.04) -3.15 (0.03) -0.005 (0.016) 0 .
21 S Y
Global fits: MW Cepheids anchor
T=2.5 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W A25 552 1 .
00 82.0 (3.9) (3.6) -5.85 (0.05) -2.96 (0.08) -0.31 (0.13) 0.31 N NA26 549 1 .
00 82.1 (3.9) (3.5) -5.85 (0.05) -2.96 (0.08) -0.20 (0.12) 0.29 W NA27 551 1 .
00 80.2 (3.7) (3.4) -5.85 (0.05) -3.00 (0.08) -0.004 (0.020) 0.31 S NA28 551 1 .
00 77.7 (3.4) (3.0) -5.87 (0.05) -3.08 (0.06) -0.006 (0.020) 0.30 S YT=2.25 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W A29 522 1 .
00 81.2 (3.6) (3.3) -5.86 (0.05) -2.99 (0.06) -0.45 (0.12) 0.20 N NA30 521 1 .
00 80.9 (3.6) (3.3) -5.86 (0.05) -2.99 (0.07) -0.36 (0.11) 0.20 W NA31 514 1 .
00 81.2 (3.5) (3.2) -5.85 (0.06) -2.96 (0.06) -0.007 (0.020) 0.18 S NA32 519 1 .
00 77.8 (3.3) (2.9) -5.87 (0.05) -3.08 (0.05) -0.006 (0.020) 0.20 S YR11 Rejection PriorsFit N fit ˆ χ H M W b W Z w σ int Z w b W A33 448 0 .
66 79.9 (3.2) (2.8) -5.87 (0.04) -3.05 (0.06) -0.25 (0.11) 0 .
21 N NA34 448 0 .
66 79.5 (3.2) (2.8) -5.87 (0.04) -3.06 (0.06) -0.20 (0.10) 0 .
21 W NA35 448 0 .
67 78.2 (3.1) (2.7) -5.87 (0.04) -3.08 (0.06) -0.006 (0.016) 0 .
21 S NA36 448 0 .
67 76.6 (2.8) (2.4) -5.87 (0.04) -3.14 (0.05) -0.005 (0.016) 0 .
21 S Y
Table A1.
Fits as in Tables 2 and 3, but using all Cepheids in the R11 sample with periods less than 205 days. The columns are asdefined in Table 2.c (cid:13)000