Inspection of Table 13 of R05 and Table 1 of S06 reveals part of the problem; the
distances of the four calibrators are generally discrepant in the two analyses, with S06
having the higher value. In the best case, SN 1994ae in NGC 3370, the discrepancy is only
0.08 mag^{8}.
In the worst case (SN 1990N in NGC 4639) R05 quotes a distance modulus ^{0} = 31.74, whereas the
value obtained by S06 is 32.20, corresponding to a 20 – 25% difference in the inferred distance and hence in
H_{0}. The quoted ^{0} is formed by a combination of observations at two optical bands, V and I, and is
normally defined as 2.52(m – M)_{I} – 1.52(m – M)_{V}, although the coefficients differ slightly between
different authors. The purpose of the combination is to eliminate differential effects due to
reddening, which it does exactly provided that the reddening law is known. This law has one
parameter, R, known as the “ratio of total to selective extinction”, and defined as the number of
magnitudes of extinction at V corresponding to one magnitude of difference between B and
V.

We can investigate what is going on if we go back to the original photometry. This is given in [126] and has
been corrected for various effects in the WFPC2 photometry discovered since the original work of Holtzman
et al. [57]^{9}.
If we follow R05, we proceed by using the Cepheid P-L relation for the LMC given by [156]. We then apply
a global metallicity correction, to account for the fact that the LMC is less metallic than NGC 4639
by about 0.6 dex [130], and we arrive at the R05 value for ^{0}. Alternatively, we can use
the P-L relations given by S06 and derived from earlier work by Sandage et al. [133]. These
authors derived relations both for the LMC and for the Galaxy. Like the P-L relations in [156],
the LMC relations are based on the OGLE observations in [164], with the addition of further
Cepheids at long periods, and the Galactic relations are based on earlier work on Galactic
cepheids [48, 44, 7, 9, 73]. Following S06, we derive ^{0} values separately for each Cepheid for the
Galaxy and LMC. We then assume that the difference between the LMC and Galactic P-L relations is
entirely due to metallicity, and use the measured NGC 4639 metallicity to interpolate and find
the corrected ^{0} value for each Cepheid. This interpolation gives us much larger metallicity
corrections than in R05. We then finally average the ^{0} values to recover S06’s longer distance
modulus.

The major difference is not specifically in the P-L relation assumed for the LMC, because the relation in [156] used by R05 is virtually identical to the P-L relation for long-period Cepheids used by S06. The difference lies in the correction for metallicity. R05 use a global correction

from [130] ([O/H] is the metallicity of the observed Cepheids minus the metallicity of the LMC), whereas S06’s correction by interpolation between LMC and Galactic P-L relations is [126] Which is correct? Both methods are perfectly defensible given the assumptions that are made. The S06
crucially depends on the Galactic P-L relations being correct, and in addition depends on the hypothesis
that the difference between Galactic and LMC P-L relations is dominated by metallicity effects (although it
is actually quite hard to think of other effects that could have anything like the same systematic effect as
the composition of the stars involved). The S06 Galactic relations are based on Tammann et al. [150]
(TSR03) who in turn derive them from two other methods. The first is the calibration of Cepheids in open
clusters to which the distance can be measured independently (see Section 2.1), as applied
in [40]. The second is a compilation in [48] including earlier measurements and compilations (see
e.g. [43]) of stellar angular diameters by lunar occultation and other methods. Knowing the
angular diameters and temperatures of the stars, distances can be determined [171, 6] essentially
from Stefan’s law. These two methods are found to agree in [150], but this agreement and the
consequent steep P-L relations for Galactic Cepheids, are crucial to the S06 case. Macri et
al. [92] explicitly consider this assumption using new ACS observations of Cepheids in the
galaxy NGC 4258 which have a range of metallicities [179]. They find that, if they assume the
P-L relations of TSR03 whose slope varies with metallicity, the resulting ^{0} determined
from each Cepheid individually varies with period, suggesting that the TSR03 P-L relation
overcorrects at long period and hence that the P-L assumptions of R05 are more plausible.
It is probably fair to say that more data is needed in this area before final judgements are
made.

The R05 method relies only on the very well-determined OGLE photometry of the LMC Cepheids and not on the Galactic measurements, but does rely on a global metallicity correction which is vulnerable to period-dependent metallicity effects. This is especially true since the bright Cepheids typically observed in relatively distant galaxies which host SNe Ia are weighted towards long periods, for which S06 claim that the metallicity correction is much larger.

Although the period-dependent metallicity correction is a major effect, there are a number of other
differences which affect H_{0} by a few percent each.

The calibration of the type Ia supernova distance scale, and hence H_{0}, is affected by the selection of
galaxies used which contain both Cepheids and historical supernovae. Riess et al. [124] make the case for
the exclusion of a number of older supernovae with measurements on photographic plates. Their exclusion,
leaving four calibrators with data judged to be of high quality, has the effect of shrinking the
average distances, and hence raising H_{0}, by a few percent. Freedman et al. [45] included six
galaxies including SN 1937C, excluded in [124], but obtained approximately the same value for
H_{0}.

There is a selection bias in Cepheid variable studies in that faint Cepheids are harder to see. Combined with the correlation between luminosity and period, this means that only the brighter short-period Cepheids are seen, and therefore that the P-L relation in distant galaxies is made artificially shallow [132] resulting in underestimates of distances. Neglect of this bias can give differences of several percent in the answer, and detailed simulations of it have been carried out by Teerikorpi and collaborators (see e.g. [152, 111, 112, 113]). Most authors correct explicitly for this problem – for example, Freedman et al. [45] calculate the correction analytically and find a maximum bias of about 3%. Teerikorpi and Paturel suggest that a residual bias may still be present, essentially because the amplitude of variation introduces an additional scatter in brightness at a given period, in addition to the scatter in intrinsic luminosity. How big this bias is is hard to quantify, although it can in principle be eliminated by using only long-period Cepheids at the cost of increases in the random error.

Further possible effects include differences in SNe Ia luminosities as a function of environment. Wang
et al. [170] used a sample of 109 supernovae to determine a possible effect of metallicity on SNe Ia
luminosity, in the sense that supernovae closer to the centre of the galaxy (and hence of higher metallicity)
are brighter. They include colour information using the indicator C_{12} (B – V)_{12} days, the
B – V colour at 12 days after maximum,as a means of reducing scatter in the relation between
peak luminosity and m_{15} which forms the traditional standard candle. Their value of H_{0} is,
however, quite close to the Key Project value, as they use the four galaxies of [124] to tie the
supernova and Cepheid scales together. This closeness indicates that the SNe Ia environment
dependence is probably a small effect compared with the systematics associated with Cepheid
metallicity.

In summary, local distance measures have converged to within 15%, a vast improvement on the factor of 2 uncertainty which prevailed until the late 1980s. Further improvements are possible, but involve the understanding of some non-trivial systematics and in particular require general agreement on the physics of metallicity effects on Cepheid P-L relations.

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