# Reassessing the Constraints from SH0ES Extragalactic Cepheid Amplitudes on
Systematic Blending Bias

Amir Sharon1, Doron Kushnir1, Wenlong Yuan2, Lucas Macri3 and Adam Riess2
1Department of Particle Physics & Astrophysics, Weizmann Institute of Science,
Rehovot 76100, Israel
2Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD
21218, USA
3NSF’s National Optical Infrared Astronomy Research Laboratory, Tucson, AZ
85726, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The SH0ES collaboration Hubble constant determination is in a
$\mathord{\sim}5\sigma$ difference with the Planck value, known as the Hubble
tension. The accuracy of the Hubble constant measured with extragalactic
Cepheids depends on robust stellar-crowding background estimation. Riess et
al. 2020 (R20) compared the light curves amplitudes of extragalactic and MW
Cepheids to constrain an unaccounted systematic blending bias,
$\gamma=-0.029\pm 0.037\,\rm{mag}$, which cannot explain the required,
$\gamma=0.24\pm 0.05\,\rm{mag}$, to resolve the Hubble tension. Further checks
by Riess et al. 2022 demonstrate that a possible blending is not likely
related to the size of the crowding correction. We repeat the R20 analysis,
with the following main differences: 1. We limit the extragalactic and MW
Cepheids comparison to periods $P\lesssim 50\,\rm{d}$, since the number of MW
Cepheids with longer periods is minimal; 2. We use publicly available data to
recalibrate amplitude ratios of MW Cepheids in standard passbands; 3. We
remeasure the amplitudes of Cepheids in NGC 5584 and NGC 4258 in two HST
filters (F555W and F350LP) to improve the empirical constraint on their
amplitude ratio $A^{555}/A^{350}$. We show that the filter transformations
introduce an $\mathord{\approx}0.04\,\rm{mag}$ uncertainty in determining
$\gamma$, not included by R20. While our final estimate, $\gamma=0.013\pm
0.057\,\rm{mag}$, is consistent with the value derived by R20, the error is
somewhat larger, and the best-fit value is shifted by
$\mathord{\approx}0.04\,\rm{mag}$. Although the obtained $\gamma$ for this
crowding test is consistent with zero, folding (in quadratures) it is
$\mathord{\approx}3.0\sigma$ away from aligning with Planck. Future
observations, especially with JWST, would allow better calibration of
$\gamma$.

###### keywords:

cosmological parameters – distance scale – stars: variables: Cepheids

††pubyear: 2021††pagerange: Reassessing the Constraints from SH0ES
Extragalactic Cepheid Amplitudes on Systematic Blending Bias–12

##  1 introduction

The latest determination of the Hubble constant by the SH0ES collaboration
(Riess et al., 2022, hereafter R22), $H_{0}=73.04\pm
1.04\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$, is in a $\mathord{\sim}5\sigma$
difference with the Planck value (Planck Collaboration et al., 2020),
$H_{0}=67.4\pm 0.5\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$, known as the Hubble
tension. The difference between the Cepheid- and Type Ia supernovae-based
SH0ES measurement and the cosmic microwave background temperature and
polarization anisotropies Planck measurement has led to numerous suggestions
for extensions of the standard $\Lambda$CDM cosmology model (see Di Valentino
et al., 2021, for a review). The SH0ES absolute distance scale is based on the
period-luminosity relation of Cepheids ($P-L$ relation; Leavitt & Pickering,
1912) measured in the HST F160W filter (similar to the NIR $H$ band). The
Cepheids reside in $37$ Type Ia supernovae host galaxies and other anchor
galaxies with an absolute distance measurement. The Hubble tension can be
expressed as $\mathord{\sim}0.1-0.2\,\textrm{mag}$ difference in the
magnitudes of SH0ES Cepheids (Riess, 2019; Efstathiou, 2020), in the sense
that the SH0ES Cepheids (in M31 and further away) are brighter than the
$\Lambda$CDM prediction.

The accuracy of the Hubble constant measured with extragalactic Cepheids
depends on robust photometry and background estimation in the presence of
stellar crowding. The SH0ES collaboration performs artificial Cepheid tests
and derives a crowding correction, $\Delta m_{H}$, which is added to the
photometry of each Cepheid (i.e., reducing the brightness of the Cepheid).
Riess et al. (2020, hereafter R20) pointed out that crowding by unresolved
sources at Cepheid sites reduces the fractional amplitudes of their light
curves. This is because the crowding adds a constant sky background flux that
compresses the relative flux amplitude variations of a Cepheid. R20 compared
the HST F160W amplitudes of over 200 Cepheid amplitudes in three hosts
(hereafter faraway galaxies) and in the anchor galaxy NGC 4258 to the observed
amplitudes in the Milky way (MW). This comparison allowed them to constrain a
possible systematic bias in the determination of the crowding correction,
$\gamma=-0.029\pm 0.037\,\rm{mag}$111Note a typo in R20, with reported
$\gamma=0.029\pm 0.037\,\rm{mag}$., which cannot explain the required
systematic error to resolve the Hubble tension. Note that the results of R20
suggests that both the calculated crowding correction, which estimated the
chance superposition of Cepheids on crowded backgrounds, is accurate and that
light from stars physically associated with Cepheids (with the prime
candidates being wide binaries and open clusters; Anderson & Riess, 2018) is
small. In other words, R20 constrained the total systematic blending bias to
be $\gamma=-0.029\pm 0.037\,\rm{mag}$.

In this paper, we repeat the analysis of R20 with a careful study of each step
required for the comparison of the extragalactic amplitudes to the MW
amplitudes. The main differences between our analysis and the analysis of R20
are:

  * •

We impose the period limit $\log P\equiv\log_{10}(P\,[\rm{d}])<1.72$ for the
comparison (the period range of the R20 extragalactic Cepheids is $1<\log
P<2$), since the amplitudes for longer period Cepheids cannot be reliably
determined for the MW (as the number of such MW Cepheids is minimal, see
Appendix A.3). We obtain similar results by removing the period cut but adding
increased uncertainties to the MW relations at long periods.

  * •

We use public available data to recalibrate amplitudes ratios of MW Cepheids
in standard bands along with their associate uncertainties. We show that a
calibration of the required filter transformations from Cepheid observations
introduces an $\mathord{\approx}0.04\,\rm{mag}$ uncertainty in the
determination of $\gamma$, not included by R20. We show that available
Cepheids templates are not accurate enough to reduce this error. Our
transformation between two HST filters (F555W and F350LP; $A^{555}/A^{350}$)
is different from the transformation used by R20. We show that the
transformation used by R20 did not optimally weight the data, and we calibrate
a new transformation based on updated amplitude measurements.

Our final estimate for a possible blending bias is $\gamma=0.013\pm
0.057\,\rm{mag}$. While the obtained $\gamma$ is consistent with the value
derived by R20, the error is somewhat larger, and the best fit value is
shifted by $\mathord{\approx}0.04\,\rm{mag}$. Although the obtained $\gamma$
is consistent with zero, folding (in quadratures) it is
$\mathord{\approx}3.0\sigma$ away from aligning with Planck (Section 5).

The method of R20 to compare the extragalactic amplitudes to the MW amplitudes
is described in Section 2. In Section 3 we calibrate the required HST filters
transformation and in Appendix C we calibrate the required ground-HST filter
transformations. In Section 4 we repeat the analysis of R20 using our methods.
We discuss some caveats of our analysis and the implications of our results in
Section 5.

We independently recalibrate MW Cepheids amplitude ratios by constructing a
galactic Cepheid catalog from publicly available photometry (Appendix A). We
employ Gaussian processes (GP) interpolations on the phase-folded light curves
to determine the mean magnitudes and amplitudes in different bands.

We follow the convention that a single Cepheid magnitude $x$ is the magnitude
of intensity mean, $x=\langle x\rangle$, and colors $(x-y)$ stand for $\langle
x\rangle-\langle y\rangle$. All fits in this paper includes global $2.7\sigma$
clipping. In order to decide on the optimal polynomial order for the fitting,
we normalized the errors to obtain a reduced $\chi^{2}$ of $1$, and we inspect
the difference $\Delta\chi^{2}$ obtained with a higher-by-one order
polynomial.

##  2 The method of R20

R20 compared between the amplitudes of extragalactic Cepheids and MW Cepheids
to constrain a possible systematic blending bias, $\gamma$. Specifically, R20
minimized

$\chi^{2}\left(\gamma\right)=\sum_{i=1}^{n}\left(\frac{A^{160}}{A^{350}}-\frac{A^{160,\rm{MW}}}{A^{350,\rm{MW}}}10^{-0.4\left(\Delta
m_{H}-\Delta m_{V}+\gamma\right)}\right)^{2}\sigma_{i}^{-2},$ (1)

where the summation is over all Cepheids in the sample (see R20 for a
derivation of Equation (1)). $A^{160}$ and $A^{350}$ are the observed
amplitudes222The amplitude is defined here as the magnitude difference between
the minimum and the maximum of the light curve. in F160W and the white filter
F350LP, respectively, $A^{160,\rm{MW}}/A^{350,\rm{MW}}$ is the observed MW
ratio between these amplitudes (that depends on the period of the Cepheid),
the factor $10^{-0.4(\Delta m_{H}-\Delta m_{V})}$ is the expected reduction in
the amplitude ratio because of crowding, which also depends on the (small)
crowding correction in the F350LP filter, $\Delta m_{V}$, and $\sigma_{i}$ is
the relevant error of the expression. The motivation to study the
$A^{160}/A^{350}$ amplitude ratio instead of the NIR amplitude is the
reduction in the observed scatter around the MW relation
($\mathord{\approx}0.05$, see Appendix B, compared with
$\mathord{\approx}0.1\,\rm{mag}$ for the NIR amplitude). The Cepheids in the
sample have $1<\log P<2$ with $A^{160}\sim 0.2\,\rm{mag}$ (measured with an
accuracy of $\mathord{\sim}0.1\rm{mag}$), compared with $A^{160,\rm{MW}}$ in
the range of $0.2-0.5\,\rm{mag}$ for the same period range. The crowding
corrections for the Cepheids in the sample are mostly $\Delta m_{H}\lesssim
0.6\,\rm{mag}$ with a smaller fraction of Cepheid found in regions with higher
surface brightness (up to $\Delta m_{H}\approx 2\,\rm{mag}$) than the limit
typically used to measure $H_{0}$.

The transformation of the MW relation, observed in $H$ and $V$ bands, to the
HST filters is performed in R20 with

$\frac{A^{160,\rm{MW}}}{A^{350,\rm{MW}}}=\frac{A^{H,\rm{MW}}}{A^{V,\rm{MW}}}\frac{A^{160}}{A^{H}}\frac{A^{V}}{A^{555}}\frac{A^{555}}{A^{350}},$
(2)

where $A^{555}$ is the amplitude in the F555W filter (similar to the $V$
band). The ratios $A^{160}/A^{H}$ and $A^{V}/A^{555}$ can be determined by
comparing ground-based observations to HST observations (see Riess et al.,
2021a, and references therein). The ratio $A^{555}/A^{350}$ can be determined
from HST observations of extragalactic Cepheids. In Section 3 and Appendix C
we independently calibrate each transformation in Equation (2). By a
minimization of Equation (1), R20 obtained $\gamma=-0.029\pm 0.037\,\rm{mag}$,
which cannot explain the required systematic error to resolve the Hubble
tension.

##  3 The $A^{555}/A^{350}$ ratio

In this section, we discuss the amplitude transformation $A^{555}/A^{350}$,
which is required for the comparison in Section 4 (see Equation (2)), and
significantly affects the estimation of $\gamma$. Unlike the situation with
ground filter amplitudes, where high signal to noise ratio photometric data is
available in a large number of epochs, the calibration of HST filter
amplitudes is less certain and involves some modeling. As we demonstrate
below, there are different calibrations of the $A^{555}/A^{350}$ ratio that
deviate significantly from each other. Before we discuss the actual
observations, we provide some intuition for the expected ratio and the
predictions of available templates.

Assume that the Cepheid at the time of maximum (minimum) light is a blackbody
with a temperature $T_{h}$ ($T_{l}$), with typical values $6000-7000\,\rm{K}$
($4400-5000\,\rm{K}$) (see, e.g., Figure 3 of Javanmardi et al., 2021,
hereafter J21). We can use the well observed relation $A^{I}/A^{V}\approx 0.6$
to determine a relation between $T_{h}$ and $T_{l}$ (regardless of the radii
of the Cepheid at extremum light)333Note that the filter F350LP contains a
significant overlap with the $V$ and $I$ bands, see Figure 2 of Hoffmann et
al. (2016).. From the relation between $T_{h}$ and $T_{l}$ we find
$A^{555}/A^{350}\approx 1.07-1.12$. In principle, well calibrated templates
can provide a more accurate estimate for this ratio. However, the results from
the Pejcha & Kochanek (2012, hereafter P12) templates, $A^{555}/A^{350}\approx
1.04$444Since no observations with HST filters were used to calibrate the P12
templates, the predictions of the P12 templates for these filters depends on
theoretical atmospheric models. We use the values $\beta=5.42,5.62,1.7$ for
the F350LP, F555W and F160W filters, kindly provided to us by Ondřej Pejcha
(see equation (3) of P12 for details). We discuss a method to improve the P12
templates predictions for HST filters in Section 5., and from the templates
used by J21, $A^{555}/A^{350}\approx 1.17$, deviate by more than $10\%$. This
deviation is probably related to the large wavelength range
($\mathord{\sim}0.3-1\,\mu\rm{m}$, see Figure 2 of Hoffmann et al., 2016,
hereafter H16), which the white filter F350LP spans, that is challenging to
describe accurately (see, e.g., the $\mathord{\sim}25\%$ deviation of the
$A^{H}/A^{V}$ between the P12 templates and observations in Appendix B), and
to the less precise prediction of the P12 templates for HST filters (see the
$\mathord{\sim}5\%$ deviation of the $A^{555}/A^{V}$ between the P12 templates
and our estimate in Appendix C).

In what follows, we discuss various analyses of the data from F555W and F350LP
observations, to estimate the $A^{555}/A^{350}$ transformation. In Section 3.1
we discuss an empirical calibration based on a sample of Cepheids in NGC 5584.
In Section 3.2 we discuss other, less robust, methods. We summerize our
findings in Section 3.3.

###  3.1 NGC 5584 empirical calibration

R20 used a transformation that was derived in H16 from a sample of Cepheids in
NGC 5584 with $A^{555}$ and $A^{350}$ determinations for each Cepheid:
$A^{555}/A^{350}=(0.308\pm 0.052)(\log P-1.5)+(1.024\pm 0.011)$555Note a typo
in Table 2 of H16. (with a scatter of $0.134$, red line in the top panel of
Figure 1). This transformation satisfies $A^{555}/A^{350}<1$ for $\log
P\lesssim 1.4$ and $A^{555}/A^{350}=0.87$ for $\log P=1$, which significantly
deviates from our expectation above. These amplitudes are publicly available
only for $199$ Cepheids above a period cut, and they are plotted in green
symbols. For the fit in H16, additional cuts were imposed on the
data666$1<A^{555}/A^{814}<2.2$, $0.55<A^{555}/A^{350}<1.4$,
$0.35<A^{814}/A^{350}<0.975$., and the data that passed these cuts are plotted
in blue symbols. We obtain a best fit of $A^{555}/A^{350}=(0.24\pm 0.07)(\log
P-1.5)+(1.05\pm 0.01)$ (with a scatter of $\mathord{\approx}0.13$ for $148$
Cepheids after clipping; the linear fit performs significantly better than a
constant ratio), which is similar to the H16 fit. The small inconsistency of
our fit and the H16 fit could be explained with the few additional Cepheids of
H16. While both the fit preformed here and the fit of H16 suggest a slope that
is significant by more than $3\sigma$, there are a few issues with this
procedure.

Figure 1: The $A^{555}/A^{350}$ transformation. Top panel: the
$A^{555}/A^{350}$ ratio for Cepheids in NGC 5584 with $A^{555}$ and $A^{350}$
determinations for each Cepheid as a function of period. Green symbols: all
$199$ Cepheids above a period cut. Blue symbols: only Cepheids that passed the
additional H16 cut. Black symbols: the recalibrated amplitudes that passed our
cuts. Black line: our best fit to the Cepheids that passed our cuts. Red line:
the H16 transformation. The H16 transformation satisfies $A^{555}/A^{350}<1$
for $\log P\lesssim 1.4$ and $A^{555}/A^{350}=0.87$ for $\log P=1$, while it
is expected that $A^{555}/A^{350}\gtrsim 1$. Second panel: The NGC 5584
analysis of J21 (black symbols). J21 found a very small $A^{555}/A^{350}$
scatter, $\mathord{\approx}0.015$, and they fitted the data with $0.073(\log
P-1.5)+1.167$ (blue dashed line). The obtained $A^{555}/A^{350}$ scatter is
significantly smaller than the scatter in any MW amplitude ratio, suggesting
that the J21 fit to the NGC 5584 Cepheids is artificially constrained by their
MW templates. We limit the data to $\log P<1.72$ and we obtain
$A^{555}/A^{350}=1.164\pm 0.003$ by fitting a constant value to the data (red
dashed line). Third panel: the full sample of H16, which includes $1325$
Cepheids with $A^{555}$ values (blue) and $1035$ Cepheids with $A^{350}$
values (black). We bin the data in the range $1.1\leq\log P\leq 2$ with a bin
size of $0.1$ and find the mean and scatter in each bin for each filter (red
and magenta symbols with error bars). Bottom panel: The means ratio as a
function of period (black symbols). Dashed black line: the best fit to the
binned data, $A^{555}/A^{350}=1.13\pm 0.01$. For reference, our best fit for
the NGC 5584 sample (solid black line), the H16 fit (solid red line), our fit
to the J21 data (dashed red line) and the P12 templates (blue line) are shown
as well. Except for the H16 fit, which is unreliable, all estimates suggest a
constant ratio for $A^{555}/A^{350}$ (or a very weak period dependence), with
the range of $1.04-1.16$. The best estimate that we have for this ratio is our
best fit for the NGC 5584 sample (solid black line), see text.

First, the errors of the amplitude ratios are dominated by the light-curve
fitting to a noisy photometric data in a small number of epochs (H16 did not
provide error bars), as the intrinsic scatter should be roughly bounded by the
scatter of the $A^{I}/A^{V}$ relation, $\mathord{\approx}0.06$ (see Appendix
B), while the obtained scatter is larger by a factor of $\mathord{\approx}2$.
The implication is that different error-bars should be assigned to each
Cepheid, as one cannot assume that the observed scatter is dominated by the
intrinsic scatter. The procedure of H16 is to assume a constant error bar for
all data, which is equivalent to assuming a good fit. As a result, one cannot
get an independent goodness-of-fit probability, making the significance of the
slope statistically meaningless. Moreover, the periods of the NGC 5584 sample
are $\log P>1.3$, and the extrapolation to shorter periods of the positive
slope fit amplifies the deviation between the fit and the expectations.
Indeed, $40$ light curves in NGC 4258 with $0.7<\log P<1.4$ (mean $0.95$)
measured by Yuan et al. (2022) in both F555W and F350LP yield a mean
$A^{555}/A^{350}$ value of $1.068\pm 0.022$, strengthening the case of a
constant ratio with period. A second issue with the determination of
$A^{555}/A^{350}$ from the NGC 5584 sample is the use of cuts. While the
motivation to use cuts in order to remove unreliable results (e..g, blending
with a nearby source that changes the amplitude ratio) is well justified, the
exact choice of the cuts affect the obtained $A^{555}/A^{350}$. For example,
in the case that no cuts are employed on the data, we obtain a best fit of
$A^{555}/A^{350}=1.09\pm 0.01$ (with a scatter of $\mathord{\approx}0.17$ for
$196$ Cepheids after clipping; we find no significant improvement for fitting
with a linear function). This transformation is significantly larger than the
H16 transformation for $\log P\lesssim 1.5$, see the black line that
represents a similar transformation. This large deviation is driven by the
tendency of the H16 cuts to remove observations with large value of
$A^{555}/A^{350}$, evident by comparing the green to the blue symbols in the
figure.

The amplitudes and amplitude ratios measured in H16 used a coarse grid
$(0.01)$ to identify the best fit amplitudes with no mapping of the $\chi^{2}$
space to measure individual uncertainties, thus errors were assumed constant.
Here we revise the fitting procedure (benefitting from faster CPUs than
available in 2015) to increase the grid sampling to $0.001$ and also to map
the space to measure individual amplitude for the Cepheids in NGC 5584777The
revised amplitudes are publicly available
https://drive.google.com/drive/folders/1pCWp0_QARVE6EzsDSI5bMOaKdvIRHV6D?usp=sharing..
Using the revised data and similar quality cuts in H16 (to reduce the impact
of blending)888$0.35<A^{814}/A^{555}<0.85$, $0.55<A^{555}/A^{350}<1.45$,
$0.3<A^{814}/A^{350}<0.9$. and the full period range yields a constant
amplitude ratio $A^{555}/A^{350}=1.074\pm 0.011$, and weak evidence of a trend
with $\log P$ ($0.052\pm 0.035$), in good agreement with the results of Yuan
et al. (2022). We further limit the data to $\log P<1.72$ and additionally
require a small crowding bias, $|\Delta m_{I}|<0.05\,\rm{mag}$, as derived by
J21 for the F814W filter999We thank B. Javanmardi for kindly providing us with
the crowding biases calculated in J21. (black symbols). We obtain
$A^{555}/A^{350}=1.09\pm 0.015$ by fitting a constant value to the data (black
line; we find $\chi^{2}_{\nu}\approx 0.90$ for $88$ Cepheids and a scatter of
$\mathord{\approx}0.15$; we find no significant improvement for fitting with a
linear function).

We performed simulations of the process of measuring amplitude ratios,
$A^{555}/A^{350}$, in a distant galaxy like NGC 5584. We randomly selected a
period from the observed range which defines the Yoachim et al. (2009) light
curve template in $3$ bands, F555W($V$), F814W($I$) and F350LP($V$). Using the
same light curve sampling and realistic noise as for NGC 5584, we produced
noisy light curves and fit them with the templates. We did the fitting two
ways. Method one (often used in past work) was to solve for the best fit
period, phase and three mean magnitudes and once found optimize these fits for
three amplitudes (PPM method). The second approach which is more
computationally intensive is to optimize all $8$ parameters simultaneously
(PPMA method). The results for recovering the amplitude ratio, shown in Figure
2, are quite similar for the two methods. The PPMA method has slightly larger
errors because all parameters are determined simultaneously. Further, we
performed this test two ways: 1) input amplitude was the same as the template
and 2) a randomized amplitude parameter (with amplitude ratios from H16 used
to scale the other bands). These two tests also yielded similar results.
Neither produces a bias in the period or mean magnitudes. The amplitudes are
measured with a mean precision of $\mathord{\sim}0.07-0.08$ per band (similar
to what we found in the real data) with no significant bias to the precision
of the test. There is a small bias in the fitted amplitude ratio,
$A^{555}/A^{350}$, where $\Delta=$output-input has a mean of
$\mathord{\sim}0.015$ (see Figure 2), which is significant given the precision
of the test with $10000$ fakes. The sense of this bias is a small overestimate
of the amplitude ratios from measured data. We have not corrected the
empirical estimate of the mean amplitude ratio for this bias but we make a
note of it here.

Figure 2: Fitted amplitude ratio, $A^{555}/A^{350}$, difference, where
$\Delta=$output-input as a function of $\log P$ obtained in simulations of the
process of measuring amplitude ratios in a distant galaxy like NGC 5584. PPM
method (left panels): solve for the best fit period, phase and three mean
magnitudes and once found optimize these fits for three amplitudes. PPMA
method (right panels): optimize all $8$ parameters simultaneously. Upper
panels: input amplitude was the same as the template. Lower panels: randomized
amplitude parameter (with amplitude ratios from H16 used to scale the other
bands). All methods show similar results. The PPMA method has slightly larger
errors because all parameters are determined simultaneously. Neither method
produces a bias in the period or mean magnitudes. The amplitudes are measured
with a mean precision of $\mathord{\sim}0.07-0.08$ per band (similar to what
we found in the real data) with no significant bias to the precision of the
test. There is a small bias in the fitted amplitude ratio of
$\mathord{\sim}0.015$, which is significant given the precision of the test
with $10000$ fakes. The sense of this bias is a small overestimate of the
amplitude ratios from measured data.

One issue with our estimate form above is related to the use of light curve
templates for the light curve fitting. We used the Yoachim et al. (2009) $V$
band templates for fitting both the F555W and the F350LP photometry, where the
amplitude in each band is allowed to change freely, providing an empirical
estimate for the amplitude ratio. However, while the use of the $V$ band
templates to estimate $A^{555}$ is justified due to the similarity of the $V$
band and F555W filters, it is not clear that the use of the $V$ band template
to estimate $A^{350}$ does not introduce any bias. The F350LP light curve
shape has not been measured accurately, and, as we demonstrated above, it is
challenging for available templates to accurately describe the behavior of
such a wide filter. We note that we found no change in the amplitude ratio by
substituting the $B$-band Yoachim et al. (2009) template for the $V$-band to
fit F350LP light curves, so this ratio does not appear particularly sensitive
to the shape of the template within reason. We discuss methods to improve this
situation in Section 5. We adopt $A^{555}/A^{350}=1.09\pm 0.015$ as our best
estimate at the moment, but keeping in mind the caveat with this method, we
describe other methods to estimate $A^{555}/A^{350}$ in the following Section.

###  3.2 Other methods

The same NGC 5584 observations were also analyzed by J21, in which an
independent light-curve modelling approach has been implemented (see details
below). They found a very small $A^{555}/A^{350}$ scatter,
$\mathord{\approx}0.015$, and they fit the data with $0.073(\log
P-1.5)+1.167$, see the second panel of Figure 1. We limit the data to $\log
P<1.72$ and we obtain $A^{555}/A^{350}=1.164\pm 0.003$ by fitting a constant
value to the data (dashed red line; we find a scatter of
$\mathord{\approx}0.03$ for $170$ Cepheids; we find no significant improvement
for fitting with a linear function). The $A^{555}/A^{350}$ values obtained in
this method are higher by $\mathord{\approx}7\%$ from the estimate in the
previous section, which is based on the SH0ES collaboration light-curve
modelling. As we explaine below, the J21 estimate is less reliable, since it
heavily relies on their MW templates, and the accuracy of their templates is
expected to be lower than $\mathord{\approx}10\%$.

The light-curve modelling approach of J21 for a given Cepheid includes a
simultaneous fit of all bands to their MW templates. These templates already
include some pre-determined $A^{555}/A^{350}$ amplitude ratio (we emphasize
that the F350LP light curve shape has never been measured for any MW Cepheid),
and their fitting process do not allow each light curve to independently
determine its own amplitude and thus to measure the amplitude ratio directly
from the data. This situation is evident from the fit results of J21. First,
their fitted line passes directly through the results of their MW templates,
and second, the obtained $A^{555}/A^{350}$ scatter is significantly smaller
than the scatter in any MW amplitude ratio (see Appendix B), suggesting that
the J21 fit to the NGC 5584 Cepheids is artificially constrained by their MW
templates. While the J21 MW templates are not publicly available, their
prediction for $A^{814}/A^{555}$ (see their equation 3), can be compared with
minimal manipulation to the measured $A^{I}/A^{V}$. We show in Appendix B that
they over-predict the observed ratio by $\gtrsim 10\%$. This result suggests
that the ability of J21 MW templates to predict $A^{555}/A^{350}$, which
include a challenging modelling of a wide filter, is limited by (at least)
$\mathord{\approx}10\%$.

Another estimate for $A^{555}/A^{350}$ can be obtained with the full sample of
H16, which includes $1325$ Cepheids with $A^{555}$ values and $1035$ Cepheids
with $A^{350}$ values. We emphasize that most of the data is obtained for
different Cepheids (except for the $199$ Cepheids in NGC 5584 with both
$A^{555}$ and $A^{350}$ values), which limit the robustness of the results
from this sample, as we explain below. The sample is plotted in the third
panel of Figure 1. We bin the data in the range $1.1\leq\log P\leq 2$ with a
bin size of $0.1$. We find the mean and scatter in each bin for each filter
(red and magenta symbols with error bars). We find that the means of $A^{555}$
are consistently larger than the means of $A^{350}$. The amplitude
distributions in each period bin are given in Appendix D, where it is evident
that the entire $A^{555}$ distribution is shifted from the $A^{350}$
distribution to higher amplitudes. We next plot the means ratio as a function
of the period (bottom panel of Figure 1), for which we can assign reliable
errors. We fit the data with $A^{555}/A^{350}=1.13\pm 0.01$ (dashed black
line; we find no significant improvement for fitting with a linear function).
This method can introduce a bias to the calibrated $A^{555}/A^{350}$ ratio,
since the distribution of amplitudes in each bin is determined by the
intrinsic amplitude distribution and by the observational error, which neither
is accurately constrained for F555W and for F350LP. While additional study is
required to calibrate this bias, we apply various cuts to the full H16 data
(ignoring M101 and NGC 4258 and/or using only Cepheids included in Riess et
al. (2016, hereafter R16)), and we do not find a significant effect on the
results.

The bottom panel of Figure 1 summarizes the different estimates. Except for
the H16 fit, which is unreliable, all estimates suggest a constant ratio for
$A^{555}/A^{350}$ (or a very weak period dependence), with the range of
$1.04-1.16$. The best estimate that we have for this ratio is based on the
updated measurements of the H16 amplitudes in NGC 5584,
$A^{555}/A^{350}=1.09\pm 0.015$, which is used as our preferred value in what
follows (hereafter empirical). As we explained above, this estimate is not
free from caveats. We also demonstrate the sensitivity of our results by
considering $A^{555}/A^{350}=1.15$, which represents the high-end range of
estimates (hereafter speculative). We emphasize that this high value is less
reliable than our preferred value, and it is only considered for the purpose
of demonstrating the sensitivity of our results to $A^{555}/A^{350}$ and to
motivate additional observations that will improve the accuracy of the
$A^{555}/A^{350}$ calibration, discussed in Section 5.

###  3.3 Summary

We conclude this section with the comparison in Figure 3 of our derived
$A^{160,\rm{MW}}/A^{350,\rm{MW}}$ (based on the empirical $A^{555}/A^{350}$ in
black and based on the speculative $A^{555}/A^{350}$ in green) to the relation
used by R20 (red line) and to the prediction of the P12 templates (blue line).
Our estimation uses the terms (see Equation (2))
$A^{H,\rm{MW}}/A^{V,\rm{MW}}=(0.20\pm 0.03)(\log P-1)+(0.30\pm 0.01)$ (see
Appendix B), $A^{160}/A^{H}$ and $A^{555}/A^{V}$ from Appendix C (that are
similar to the ratios of R20) and $A^{555}/A^{350}$ from this section. The
black and green shaded areas represent the (systematic) uncertainty of the
transformations, not considered by R20. As can be seen in the figure, our
derived relation is somewhat different from the relation used by R20, mostly
because of the different $A^{555}/A^{350}$ transformation, as discussed in
this section. Our derived relation agrees fairly well with the P12 templates
prediction, but this is a coincidence, as there are significant deviations in
some terms of Equation (2) that cancel out. Also presented in the figure is
the R20 extragalactic sample distribution of periods with $\log P$ bin widths
of $0.1$ (the last bin is between 1.6 and 1.72). Cepheids in NGC 4258 that
were measured with the F555W filter, not requiring the $A^{555}/A^{350}$
transformation to compare with the MW, are presented in red. Cepheids in the
faraway galaxies that were measured with the F350LP filter are presented in
black. The largest deviation between our results and R20 (at $\log P\lesssim
1.2$) is effecting only a small number of Cepheids.

Figure 3: Top panel: a comparison of our derived
$A^{160,\rm{MW}}/A^{350,\rm{MW}}$ (based on the empirical $A^{555}/A^{350}$ in
black and based on the speculative $A^{555}/A^{350}$ in green) to the relation
used by R20 (red line) and to the prediction of the P12 templates (blue line),
as a function of $\log P$. Our estimation uses the terms (see Equation (2))
$A^{H,\rm{MW}}/A^{V,\rm{MW}}=(0.20\pm 0.03)(\log P-1)+(0.30\pm 0.01)$ (see
Appendix B), $A^{160}/A^{H}$ and $A^{555}/A^{V}$ from Appendix C (that are
similar to the ratios of R20) and $A^{555}/A^{350}$ from this section. The
black and green shaded areas represent the (systematic) uncertainty of the
transformations, not considered by R20. Our derived relation is different from
the relation used by R20, mostly because of the different $A^{555}/A^{350}$
transformation. Our derived relation agrees fairly well with the P12 templates
prediction, but this is a coincidence, as there are significant deviations in
some terms of Equation (2) that cancel out. Bottom panel: the R20
extragalactic sample distribution of periods with $\log P$ bin widths of $0.1$
(the last bin is between 1.6 and 1.72). Red: Cepheids in NGC 4258 that were
measured with the F555W filter, not requiring the $A^{555}/A^{350}$
transformation to compare with the MW. Black: Cepheids in the faraway galaxies
that were measured with the F350LP filter. The largest deviation between our
results and R20 (at $\log P\lesssim 1.2$) is effecting only a small number of
Cepheids.

##  4 Constraining a blending bias

In this section, we repeat the analysis of R20 that compares the
$A^{160}/A^{350}$ amplitude ratios of extragalactic Cepheids to the
$A^{H}/A^{V}$ amplitude ratios of MW Cepheids to constrain a possible
systematic blending bias, $\gamma$, by minimizing Equation (1)101010We
hereafter assume that the errors are normally distributed.. The values of
$\log P$ (required for the MW relation), $A^{160}$, $A^{350}$, $\Delta m_{H}$,
$\Delta m_{V}$, and $\sigma_{i}$ are provided in Table 3 of R20111111The
provided $\sigma_{i}$ (and their properties, described below Equation (14) of
R20) were multiplyied by $A^{350}$, so one should divide the provided error by
$A^{350}$, which is typically smaller than 1, to be used in Equation (1).. The
MW relation is obtained by using Equation (2), where R20 used
$A^{160}/A^{H}=1.015$, $A^{555}/A^{V}=1.04$121212The relation
$A^{V}/A^{555}=1.04$ in R20 is a typo. and the H16 transformation for the
$A^{555}/A^{350}$ ratio. Note that the NGC 4258 Cepheid amplitudes were
measured with the F555W filter, so the transformation of H16 between $A^{350}$
and $A^{555}$ was used to derive the values in Table 3 of R20.

We first use the entire data set of R20 with the MW relation derived in R20
and we find $\gamma=-0.035\pm 0.037\,\rm{mag}$ (black line in Figure 4), in a
good agreement with the value $\gamma=-0.029\pm 0.037\,\rm{mag}$ obtained by
R20. We next limit the sample to Cepheids with $\log P<1.72$, as the MW
relation cannot be determined reliably for larger periods (see Section A.3).
We find a $\mathord{\approx}0.02\,\rm{mag}$ increase $\gamma=-0.016\pm
0.041\,\rm{mag}$ (red line). We next use $A^{160}/A^{H}$ and $A^{555}/A^{V}$
from Appendix C, and the expression for $A^{H,\rm{MW}}/A^{V,\rm{MW}}$ found in
Appendix B. These are similar to the ratios used by R20 and do not have a
large effect. We additionally use the empirical $A^{555}/A^{350}=1.09$ (see
Section 3) instead of the H16 relation, and we find a
$\mathord{\approx}0.03\,\rm{mag}$ increase, $\gamma=0.012\pm 0.041\,\rm{mag}$.
Not limiting the Cepheid periods to $\log P=1.72$ would lead to a smaller
change, as the H16 relation predicts $A^{555}/A^{350}>1.09$ for $\log P>1.72$.
Including the transformations uncertainty, see below, leads to our final
result , $\gamma=0.013\pm 0.057\,\rm{mag}$ (blue line). Using the speculative
$A^{555}/A^{350}=1.15$ (see Section 3) instead of the H16 relation, we find a
$\mathord{\approx}0.07\,\rm{mag}$ increase, $\gamma=0.054\pm 0.041\,\rm{mag}$.
Including the transformations uncertainty, see below, leads to our final
result in this case, $\gamma=0.055\pm 0.056\,\rm{mag}$ (green line).

Figure 4: Constrains on a possible systematic blending bias, $\gamma$ with
$\chi^{2}$ tests (Equation (1)). Black solid line: using the methods of R20.
Red line: limiting the sample to Cepheids with $\log P<1.72$, as the MW
relation cannot be determined reliably for larger periods (see Section A.3).
Blue line: Additionally using $A^{160}/A^{H}$ and $A^{555}/A^{V}$ from
Appendix C, the expression for $A^{H,\rm{MW}}/A^{V,\rm{MW}}$ found in Appendix
B, and the empirical $A^{555}/A^{350}=1.09$ (see Section 3) instead of the H16
relation. We find $\gamma=0.013\pm 0.057\,\rm{mag}$ (including the
transformations uncertainty). Green line: same as the blue line, but using the
speculative $A^{555}/A^{350}=1.15$ (see Section 3) instead of the H16
relation. We find $\gamma=0.055\pm 0.056\,\rm{mag}$ (including the
transformations uncertainty). Dashed brown lines: the distance (in $\sigma$s)
from the Planck results (see Section 5 for details). In order to remove the
Hubble tension, a value of $\gamma=0.24\pm 0.05\,\rm{mag}$ is required.

In order to calculate the contribution of the transformation uncertainties to
the total error (not included in the R20 analysis), we inspect the change of
$\gamma$ when the transformations are allowed to change within their
uncertainty values. We added in quadratures the contributions from the
uncertainty in $A^{555}/A^{V}$ (by changing $f$ between 0 and 1, see Appendix
C; $\delta\gamma\approx 0.012\,\rm{mag}$), the uncertainty in $A^{160}/A^{H}$
(by changing $f$ between 0 and 1, see Appendix C; $\delta\gamma\approx
0.008\,\rm{mag}$), the uncertainty in $A^{H,\rm{MW}}/A^{V,\rm{MW}}$ (by
scanning the uncertainty ellipse derived from the fit in Appendix B;
$\delta\gamma\approx 0.035\,\rm{mag}$), and the uncertainty in the empirical
$A^{555}/A^{350}$ ($\delta\gamma\approx 0.01\,\rm{mag}$). The total
transformation uncertainties ($\delta\gamma\approx 0.040\,\rm{mag}$) were
convoluted with $\exp(-\Delta\chi^{2}/2)$ found without these uncertainties to
increase the error in $\gamma$ from $\mathord{\approx}0.041\,\rm{mag}$ to the
values presented in Figure 4 (blue and green lines)131313Note that the best-
fit value slightly shifts because $\Delta\chi^{2}$ is not symmetric in
$\gamma$ around the best-fit value..

We claimed that comparing the extragalactic Cepheids to the MW Cepheids should
be limited to $\log P<1.72$. One could worry that we ignore too many
extragalactic Cepheids with this period cut, and that the period cut is too
abrupt. We repeat our analysis without any period cut, but in order to reflect
the more uncertain MW relation at long periods, we count for each
extragalactic Cepheid the number of MW Cepheids, $N_{i}$, within a $0.1$ $\log
P$ bin around its period, and add $\sigma_{\rm{MW}}/\sqrt{N_{i}}$ to its error
budget, where $\sigma_{\rm{MW}}\approx 0.04\,\rm{mag}$ is the intrinsic
scatter of $A^{H,\rm{MW}}/A^{V,\rm{MW}}$ (see Appendix B). We find in this
case (hereafter period weighting) an increase in $\gamma$ by only
$\mathord{\approx}0.01\,\rm{mag}$. We can also use a quadratic relation for
$A^{H,\rm{MW}}/A^{V,\rm{MW}}$ instead of the default linear relation (the
quadratic relation is preferred over the linear relation by less than
$2\sigma$, see Appendix B for details). We find in this case a small decrease
in $\gamma$ by $\mathord{\approx}0.01\,\rm{mag}$ for the $\log P<1.72$ limit
case and an additional small decrease by $\mathord{\approx}0.005\,\rm{mag}$
with period weighting.

A small fraction of the extragalactic Cepheids is found in regions with higher
surface brightness (up to $\Delta m_{H}\approx 2\,\rm{mag}$) than the limit
typically used to measure $H_{0}$. We repeat our analysis by limiting the
extragalactic Cepheids to small surface brightness ($\Delta
m_{H}<0.7\,\rm{mag}$). We find a small increases $\delta\gamma\lesssim
0.01\,\rm{mag}$.

While the obtained $\gamma$ is consistent with the value derived by R20, the
error is somewhat larger, and the best fit value is shifted by
$\mathord{\approx}0.04\,\rm{mag}$ (for the empirical $A^{555}/A^{350}$).

##  5 Discussion

In this paper, we repeated the analysis of R20 to constrain a systematic
blending bias, $\gamma$, through Cepheid amplitudes. The analysis compares MW
Cepheids to extragalactic Cepheids, so it requires an accurate determination
of Cepheid amplitudes in the MW and various filter transformations. The main
differences between our analysis and the analysis of R20 are:

  * •

We imposed the period limit $\log P<1.72$ for the comparison (the period range
of R20 extragalactic Cepheids is $1<\log P<2$), since the amplitudes for
longer period Cepheids cannot be reliably determined for the MW (as the number
of such MW Cepheids is minimal, see Appendix A.3). We obtained similar results
by using period weighting (i.e., removing the period cut but adding increased
uncertainties to the MW relations at long periods).

  * •

We used public available data to recalibrate amplitudes ratios of MW Cepheids
in standard bands along with their associate uncertainties. We showed that a
calibration of the required filter transformations from Cepheid observations
introduces an $\mathord{\approx}0.04\,\rm{mag}$ uncertainty in the
determination of $\gamma$, not included by R20. We showed that available
Cepheids templates are not accurate enough to reduce this error. The most
significant difference with R20 is obtained for the $A^{555}/A^{350}$
transformation. We showed that the H16 fit, used by R20, did not optimally
weight the data and that the best estimate that we have for this ratio is
based on updated measurements of the H16 amplitudes in NGC 5584,
$A^{555}/A^{350}=1.09\pm 0.015$, used as our preferred value (the empirical
ratio). This estimate is not free from caveats so we also demonstrate the
sensitivity of our results by considering $A^{555}/A^{350}=1.15$, which
represents the high-end range of (less robust) estimates (the speculative
ratio).

Our final estimates for a possible blending bias is $\gamma=0.013\pm
0.057\,\rm{mag}$ with the empirical $A^{555}/A^{350}$. While the obtained
$\gamma$ is consistent with the value derived by R20, the error is somewhat
larger, and the best fit value is shifted by
$\mathord{\approx}0.04\,\rm{mag}$.

We constructed a galactic Cepheid catalog from publicly available photometry
for the recalibration of the MW Cepheids amplitudes ratios (Appendix A). We
employed GP interpolations on the phase-folded light curves to determine the
mean magnitudes and amplitudes in different bands. The GP interpolations do
not depend on any presumed behavior and allowed us to assign reliable error
bars to our results. The catalog, as well as the light curves of all Cepheids
in the catalog, are publicly
available141414https://drive.google.com/drive/folders/1pCWp0_QARVE6EzsDSI5bMOaKdvIRHV6D?usp=sharing.

We next inspect the effect of our results on the significance of the Hubble
tension, by calculating $\partial H_{0}/\partial\gamma$ with the fitting
procedure of Mortsell et al. (2021) (which is similar to the procedure of R16;
a detailed description of the fitting process can be found in these papers)
and the early data set release of R22. We note that some improvements to the
fitting procedure and additional 18 hosts were introduced in R22, which are
not included in our analysis. However, the impact of these additions should
have a minor effect on $\partial H_{0}/\partial\gamma$. Note further that the
blending bias $\gamma$ deduced from Cepheid amplitudes is actually the
difference between the NIR blending bias and the white filter blending bias,
$\gamma_{160}-\gamma_{350}$, such that it is not straight forward to deduce
the blending bias in the Wesenheit index,
$\rm{F160}-0.386(\rm{F555}-\rm{F814})$, used for the $H_{0}$ calculation. In
what follows, we assume that the blending bias of the term
$0.386(\rm{F555}-\rm{F814})$ is small compared with $\gamma_{160}$ and we take
$\gamma_{350}=0$ to test the minimal effect of $\gamma$ on $H_{0}$
($\gamma_{160}>\gamma$ for any positive value of $\gamma_{350}$).

We first assume that all extragalactic Cepheids (beyond M31) are fainter by
some value $\gamma$. The change in $H_{0}$ for the usual choice of anchors
(MW, LMC and NGC 4258) is $\delta H_{0}/H_{0}\approx-0.32\gamma$ (with the
same change in $H_{0}$ error). Limiting the bias for Cepheids with $\log P>1$,
as the amplitudes observations are only available for such Cepheids, has a
small effect on the results. In what follows, we assume the latest
determination of the Hubble constant by the SH0ES collaboration,
$H_{0}=73.04\pm 1.04\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$ (R22) and the
derivative $\delta H_{0}/H_{0}\approx-0.32\gamma$. One can now calculate the
distance from the SH0ES result for any value of $\gamma$ (dashed brown lines
in Figure 4). In order to remove the Hubble tension, a value of
$\bar{\gamma}=0.24\pm 0.05\,\rm{mag}$ is required. Folding (in quadratures)
the $\gamma$ and $\bar{\gamma}$ uncertainty yields
$H_{0}=72.7\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$, which is
$\mathord{\approx}3.0\sigma$ away from Planck, with the empirical
$A^{555}/A^{350}$. A larger $\gamma$ is required to remove the tension with
other combinations of anchors. For example, we find $\delta
H_{0}/H_{0}\approx-0.23\gamma$ with just using the LMC and NGC 4258 anchors.
The determination of $H_{0}$ with only the NGC 4258 anchor is hardly affected
by $\gamma$ in this case, as almost all Cepheids (except M31 Cepheids) suffer
from the same blending bias. We study in detail various ways to determine
$H_{0}$ that are immune to blending biases in a companion paper (Kushnir &
Sharon, 2023).

R22 provided a few checks (see the comparison between fits 41 and 42 and the
checks in Appendix B of R22) demonstrating that a possible blending is not
likely related to the size of the crowding correction. A different scenario,
which is more difficult to test with the methods of R22, is of a possible
blending due to stars physically associated with Cepheids. Since the mass (and
the age) of Cepheids is correlated with their period, it is expected that
long-period Cepheids are more likely to be physically associated with stars.
For example, Anderson & Riess (2018) demonstrated by observing Cepheids in M31
that long period Cepheids have a higher chance of being in open clusters (see
their Figure 13). The available data, however, are limited to Cepheid ages
older than $\mathord{\sim}50\,\rm{Myr}$ (see also Breuval et al., 2023, with
similar age limitations). The ages of the long-period Cepheids, which dominate
the population in the faraway galaxies, are $\lesssim 20\,\rm{Myr}$ (see, e.g,
Table A1 of Anderson et al., 2016), probably shorter than the dispersing time
of open clusters. It is, therefore, reasonable to assume that a significant
fraction of long-period Cepheids reside in open clusters. Such an effect would
lead to an increased blending with the period. We, therefore, test for such a
period-dependency by modifying $\gamma$ in Equation (1) to
$\gamma_{0}+\gamma_{p}(\log P-1)$, and repeating the fits. We find
$\gamma_{0}=-0.28\pm 0.12\,\rm{mag}$, $\gamma_{p}=0.61\pm
0.25\,\rm{mag}\,\rm{dex}^{-1}$, with a change of $\Delta\chi^{2}\approx 5.2$
for the empirical $A^{555}/A^{350}$, indicating an insignificant (less than
$3\sigma$) evidence for a linear period dependency of $\gamma$.

While many assumptions are involved in our analysis, we demonstrated that the
R20 calibration of $\gamma=-0.029\pm 0.037\,\rm{mag}$ is not secured. As we
mentioned above, our results are sensitive to the $A^{555}/A^{350}$ ratio, and
the empirical ratio that we use is not free from caveats. We next consider the
impact of the speculative $A^{555}/A^{350}=1.15$, which represents the high-
end range of (less robust) estimates. In this case we find $\gamma=0.055\pm
0.056\,\rm{mag}$, which yields
$H_{0}=71.7\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1}$ that is
$\mathord{\approx}2.5\sigma$ away from Planck. We suggest below a few
directions for future studies in order to remove some of the assumptions made
in this work and to better constrain the blending effect.

We assumed that all Cepheids in NGC 4258 and the faraway galaxies suffer on
average from the same systematic blending bias, which we calibrated from a
smaller sample of Cepheids (and only in three faraway galaxies). Similar
information for more extragalactic Cepheids can be collected with future HST
observations. Better calibration of the $A^{555}/A^{V}$, $A^{160}/A^{H}$ and
$A^{555}/A^{350}$ transformations can be obtained by observations of Galactic
Cepheids in many epochs, either with HST or from the ground. Such observations
could also be useful to improve existing Cepheid templates (such as P12 or the
templates used by J21). For example, using the same approach of P12, but with
the additional (some of them already available) HST single epoch observations,
may significantly improve the accuracy of P12 templates (that is currently
estimated to be $\gtrsim 10\%$).

A different approach is to anchor the extragalactic Cepheid amplitudes to M31
Cepheids instead to the MW Cepheids. This has the advantage of observing the
Cepheids with the same instrument and filters, bypassing the need for filter
transformations and perhaps obtaining a larger number of long period Cepheids.
Finally, the possible underline open cluster population of extragalactic
Cepheids can be either examined with HST UV observations (Anderson et al.,
2021) or resolved with JWST (Anderson & Riess, 2018; Riess et al., 2021b; Yuan
et al., 2022).

## Acknowledgements

We thank Ondřej Pejcha, Dan Scolnic, Stefano Casertano, Eli Waxman, Boaz Katz,
and Eran Ofek for useful discussions. DK is supported by the Israel Atomic
Energy Commission – The Council for Higher Education – Pazi Foundation, by a
research grant from The Abramson Family Center for Young Scientists, by ISF
grant, and by the Minerva Stiftung. This research has made use of the
International Variable Star Index (VSX) database, operated at AAVSO,
Cambridge, Massachusetts, USA.

## Data availability

All data used in this study is either publicly available through other
publications or through the publicly available catalogs:
https://drive.google.com/drive/folders/1pCWp0_QARVE6EzsDSI5bMOaKdvIRHV6D?usp=sharing.

## References

  * Anderson et al. (2016) Anderson R. I., Saio H., Ekström S., Georgy C., Meynet G., 2016, A&A, 591, A8. doi:10.1051/0004-6361/201528031
  * Anderson & Riess (2018) Anderson R. I., Riess A. G., 2018, ApJ, 861, 36. doi:10.3847/1538-4357/aac5e2
  * Anderson et al. (2021) Anderson R. I., Casertano S., Riess A., Spetsieri Z., 2021, hst..prop, 16688
  * Barnes et al. (1997) Barnes T. G., Fernley J. A., Frueh M. L., Navas J. G., Moffett T. J., Skillen I., 1997, PASP, 109, 645. doi:10.1086/133927
  * Berdnikov (2008) Berdnikov L. N., 2008, yCat, II/285
  * Berdnikov et al. (2015) Berdnikov L. N., Kniazev A. Y., Sefako R., Dambis A. K., Kravtsov V. V., Zhuiko S. V., 2015, AstL, 41, 23. doi:10.1134/S1063773715020012
  * Berdnikov et al. (2019) Berdnikov Â. L. Â. N., Kniazev Â. A., Dambis Â. A., Kravtsov Â. V. Â. V., 2019, PZ, 39, 2
  * Berdnikov & Pastukhova (2020) Berdnikov L. N., Pastukhova E. N., 2020, AstL, 46, 235. doi:10.1134/S1063773720040027
  * Breuval et al. (2023) Breuval L., Riess A. G., Macri L. M., Li S., Yuan W., Casertano S., Konchady T., et al., 2023, arXiv, arXiv:2304.00037. doi:10.48550/arXiv.2304.00037
  * Chen et al. (2020) Chen X., Wang S., Deng L., de Grijs R., Yang M., Tian H., 2020, ApJS, 249, 18. doi:10.3847/1538-4365/ab9cae
  * Coulson & Caldwell (1985) Coulson I. M., Caldwell J. A. R., 1985, SAAOC, 9
  * Di Valentino et al. (2021) Di Valentino E., Mena O., Pan S., Visinelli L., Yang W., Melchiorri A., Mota D. F., et al., 2021, CQGra, 38, 153001. doi:10.1088/1361-6382/ac086d
  * Efstathiou (2020) Efstathiou G., 2020, arXiv, arXiv:2007.10716
  * Feast et al. (2008) Feast M. W., Laney C. D., Kinman T. D., van Leeuwen F., Whitelock P. A., 2008, MNRAS, 386, 2115. doi:10.1111/j.1365-2966.2008.13181.x
  * Fernie (1979) Fernie J. D., 1979, PASP, 91, 67. doi:10.1086/130443
  * Fernie et al. (1995) Fernie J. D., Evans N. R., Beattie B., Seager S., 1995, IBVS, 4148, 1
  * Follin & Knox (2018) Follin B., Knox L., 2018, MNRAS, 477, 4534. doi:10.1093/mnras/sty720
  * Groenewegen (2018) Groenewegen M. A. T., 2018, A&A, 619, A8. doi:10.1051/0004-6361/201833478
  * Groenewegen (2020) Groenewegen M. A. T., 2020, A&A, 635, A33. doi:10.1051/0004-6361/201937060
  * Henden (1996) Henden A. A., 1996, AJ, 111, 902. doi:10.1086/117837
  * Hertzsprung (1926) Hertzsprung E., 1926, BAN, 3, 115
  * Hoffmann et al. (2016) Hoffmann S. L., Macri L. M., Riess A. G., Yuan W., Casertano S., Foley R. J., Filippenko A. V., et al., 2016, ApJ, 830, 10. doi:10.3847/0004-637X/830/1/10
  * Javanmardi et al. (2021) Javanmardi B., Mérand A., Kervella P., Breuval L., Gallenne A., Nardetto N., Gieren W., et al., 2021, ApJ, 911, 12. doi:10.3847/1538-4357/abe7e5
  * Jayasinghe et al. (2020) Jayasinghe T., Stanek K. Z., Kochanek C. S., Shappee B. J., Holoien T. W.-S., Thompson T. A., Prieto J. L., et al., 2020, MNRAS, 491, 13. doi:10.1093/mnras/stz2711
  * Klagyivik & Szabados (2009) Klagyivik P., Szabados L., 2009, A&A, 504, 959. doi:10.1051/0004-6361/200811464
  * Koen et al. (2007) Koen C., Marang F., Kilkenny D., Jacobs C., 2007, MNRAS, 380, 1433. doi:10.1111/j.1365-2966.2007.12100.x
  * Kushnir & Sharon (2023) Kushnir D., Sharon A., 2022, in preparation
  * Laney & Stobie (1992) Laney C. D., Stobie R. S., 1992, A&AS, 93, 93
  * Leavitt & Pickering (1912) Leavitt H. S., Pickering E. C., 1912, HarCi, 173
  * Moffett & Barnes (1984) Moffett T. J., Barnes T. G., 1984, ApJS, 55, 389. doi:10.1086/190960
  * Monson & Pierce (2011) Monson A. J., Pierce M. J., 2011, ApJS, 193, 12. doi:10.1088/0067-0049/193/1/12
  * Mortsell et al. (2021) Mortsell E., Goobar A., Johansson J., Dhawan S., 2021, arXiv, arXiv:2105.11461
  * Ngeow (2012) Ngeow C.-C., 2012, ApJ, 747, 50. doi:10.1088/0004-637X/747/1/50
  * Pejcha & Kochanek (2012) Pejcha O., Kochanek C. S., 2012, ApJ, 748, 107. doi:10.1088/0004-637X/748/2/107
  * Pel (1976) Pel J. W., 1976, A&AS, 24, 413
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim N., Akrami Y., Ashdown M., Aumont J., Baccigalupi C., Ballardini M., et al., 2020, A&A, 641, A6. doi:10.1051/0004-6361/201833910
  * Reid, Pesce, & Riess (2019) Reid M. J., Pesce D. W., Riess A. G., 2019, ApJL, 886, L27. doi:10.3847/2041-8213/ab552d
  * Riess et al. (2016) Riess A. G., Macri L. M., Hoffmann S. L., Scolnic D., Casertano S., Filippenko A. V., Tucker B. E., et al., 2016, ApJ, 826, 56. doi:10.3847/0004-637X/826/1/56
  * Riess et al. (2018) Riess A. G., Casertano S., Yuan W., Macri L., Bucciarelli B., Lattanzi M. G., MacKenty J. W., et al., 2018, ApJ, 861, 126. doi:10.3847/1538-4357/aac82e
  * Riess (2019) Riess A. G., 2019, NatRP, 2, 10. doi:10.1038/s42254-019-0137-0
  * Riess et al. (2020) Riess A. G., Yuan W., Casertano S., Macri L. M., Scolnic D., 2020, ApJL, 896, L43. doi:10.3847/2041-8213/ab9900
  * Riess et al. (2021a) Riess A. G., Casertano S., Yuan W., Bowers J. B., Macri L., Zinn J. C., Scolnic D., 2021a, ApJL, 908, L6. doi:10.3847/2041-8213/abdbaf
  * Riess et al. (2021b) Riess A., Anderson R. I., Breuval L., Casertano S., Macri L. M., Scolnic D., Yuan W., 2021b, jwst.prop, 1685
  * Riess et al. (2022) Riess A. G., Yuan W., Macri L. M., Scolnic D., Brout D., Casertano S., Jones D. O., et al., 2022, ApJL, 934, L7. doi:10.3847/2041-8213/ac5c5b
  * Ripepi et al. (2019) Ripepi V., Molinaro R., Musella I., Marconi M., Leccia S., Eyer L., 2019, A&A, 625, A14. doi:10.1051/0004-6361/201834506
  * Rodrigo & Solano (2020) Rodrigo C., Solano E., 2020, sea..conf, 182
  * Samus’ et al. (2017) Samus’ N. N., Kazarovets E. V., Durlevich O. V., Kireeva N. N., Pastukhova E. N., 2017, ARep, 61, 80. doi:10.1134/S1063772917010085
  * Schechter et al. (1992) Schechter P. L., Avruch I. M., Caldwell J. A. R., Keane M. J., 1992, AJ, 104, 1930. doi:10.1086/116368
  * Soszyński et al. (2020) Soszyński I., Udalski A., Szymański M. K., Pietrukowicz P., Skowron J., Skowron D. M., Poleski R., et al., 2020, AcA, 70, 101. doi:10.32023/0001-5237/70.2.2
  * Szabados (1977) Szabados L., 1977, CoKon, 70, 1
  * Szabados (1980) Szabados L., 1980, CoKon, 76, 1
  * Pietrukowicz, Soszyński, & Udalski (2021) Pietrukowicz P., Soszyński I., Udalski A., 2021, AcA, 71, 205. doi:10.32023/0001-5237/71.3.2
  * Tammann, Sandage, & Reindl (2003) Tammann G. A., Sandage A., Reindl B., 2003, A&A, 404, 423. doi:10.1051/0004-6361:20030354
  * Turner (2016) Turner D. G., 2016, RMxAA, 52, 223
  * Udalski, Szymański, & Szymański (2015) Udalski A., Szymański M. K., Szymański G., 2015, AcA, 65, 1
  * Welch et al. (1984) Welch D. L., Wieland F., McAlary C. W., McGonegal R., Madore B. F., McLaren R. A., Neugebauer G., 1984, ApJS, 54, 547. doi:10.1086/190943
  * Yoachim et al. (2009) Yoachim P., McCommas L. P., Dalcanton J. J., Williams B. F., 2009, AJ, 137, 4697. doi:10.1088/0004-6256/137/6/4697
  * Yuan et al. (2022) Yuan W., Riess A. G., Casertano S., Macri L. M., 2022, arXiv, arXiv:2209.09101

##  Appendix A The construction of the MW catalog

In this appendix, we describe the construction of the MW catalog, which is
used to recalibrate MW Cepheids amplitude ratios. In Section A.1, we describe
the selection process of the Cepheids. In Section A.2, we present our method
to determine mean magnitudes and amplitudes from publicly available
photometry. In Section A.3, we discuss the content of our catalog and
determine the maximal period for which reliable results can be obtained.

###  A.1 The Cepheid selection process

We aim to construct a comprehensive list of secure classical galactic Cepheids
pulsating at the fundamental mode. We begin from the $1939$ fundamental mode
Cepheids in the list of Soszyński et al. (2020) (an updated version of the
catalog has been recently published; Pietrukowicz, Soszyński, & Udalski, 2021,
which is discussed in Section A.3). We remove $218$ Cepheids ($64$ Cepheids
with $\log P>1$) that do not have DCEP designation in the international
variable star index (VSX)151515https://www.aavso.org/vsx/index.php. We add ET-
Vul (Berdnikov & Pastukhova, 2020) and V0539-Nor to the list, with periods and
positions from VSX. We finally remove from the list Cepheids that are not
found in GCVS (Samus’ et al., 2017) or Cepheids that are identified as non-
fundamental mode Cepheids by Ripepi et al. (2019). Following this selection
process, we are left with a list of $1723$ Cepheids ($424$ with $\log P>1$).

We search the literature for high-quality, publicly available photometry of
the Cepheids in our list, emphasizing Cepheids with $\log P>1$. Since the
SH0ES Cepheids are observed with the F555W, F814W, and F160W filters, we look
for available photometry in the $V$, $I$ and $H$ bands, which are most similar
to the HST filters, respectively. Since optical (NIR) photometry sources
usually include observations in the $B$ band ($J$ and $K$ bands), we include
in our catalog values for the $BVIJHK$ bands. We use the following sources for
the optical photometry: Pel (1976), Szabados (1977), Szabados (1980), Moffett
& Barnes (1984), Coulson & Caldwell (1985), Henden (1996), Berdnikov (2008,
additional photometry is obtained from the Sternberg Astronomical Institute
database161616http://www.sai.msu.su/groups/cluster/CEP/PHE/, referred later on
as Bextr), Berdnikov et al. (2015), Berdnikov et al. (2019), the OGLE Atlas of
Variable Star Light Curves (Udalski, Szymański, & Szymański, 2015), and the
ASAS-SN Variable Stars Database (Jayasinghe et al., 2020). In the cases that
the $I$ band measurements are given in the Johnson system, we transform to the
Cousins system with the transformations given in Coulson & Caldwell (1985). We
use the following sources for the NIR photometry: Welch et al. (1984), Laney &
Stobie (1992), Schechter et al. (1992), Barnes et al. (1997), Feast et al.
(2008), and Monson & Pierce (2011). We transform the NIR photometry to the
Two-Micron All-Sky Survey photometry system, using the transformations in Koen
et al. (2007) and in Monson & Pierce (2011). In some cases, we use the
McMaster cepheid photometry and radial velocity data archive (Fernie et al.,
1995) to retrieve the photometry from the sources listed above.

###  A.2 Light curve parameters by Gaussian processes

We determine mean magnitudes and amplitudes from the retrieved photometry with
interpolation using Gaussian processes (GP). The advantage of this method over
template fitting methods is that it does not draw on any presumed behavior
and, for example, is not limited by intrinsic variations between light curves.
The method requires sufficient sampling of the light curve, and we, therefore,
require at least three epochs with the maximal phase difference between two
adjacent points $<0.5$. We used the built-in matlab functions fitrgp and
predict with a squared-exponential kernel for the covariance matrix. The
phase-folded light curve is duplicated to ensure continuity, and the
interpolation is performed over phases between $-0.5$ to $1.5$. The outcome of
this process is an estimated mean magnitude and an amplitude. The errors of
the obtained values are estimated by repeating the process many times with the
magnitude values in each phase randomly shifted according to the estimated
photometric error. We choose for the photometric error the maximum between the
provided observational errors (we apply a uniform error of $0.01\,\rm{mag}$ if
no errors are provided) and the noise standard deviation, as estimated by the
GP fit, which roughly corresponds to the scatter around the fit. In most cases
the GP-estimated photometric error is larger than the observational
photometric error, since the phase-folded light curve can have additional
errors due to uncertainties in the Cepheid period (or its drift over the
course of observations) or some other unknown source.

For Cepheids with $\log P>1$, we perform a consistency check of our results
with the P12 templates. The templates contain the radius and temperature phase
curves within the range $1\leq\log P\leq 2$, parametrized by a truncated
Fourier series, thus allowing the construction of light curves in any
photometric band. We fit three parameters for each Cepheid in a given band, by
minimising the deviation of the observed magnitudes, $m_{i}^{\text{obs}}$,
from the template-computed magnitudes at a given period, $m_{\log
P}^{\text{tmp}}$:

$\sum_{i}\left(m^{\text{obs}}_{i}(\phi_{i})-m_{\log
P}^{\text{tmp}}(\phi_{i}-\phi_{0},A^{2},\bar{m})\right)^{2}\sigma_{i}^{-2},$
(3)

where $\bar{m}$ is a constant offset magnitude, $A^{2}$ is the amplitude, and
$\phi_{0}$ is a constant phase offset. Note that $A^{2}$ and $\phi_{0}$ are
fitted to each band separately, which is different from the method of P12,
where a single value of $A^{2}$ and a single value of $\phi_{0}$ are used for
all bands. We find that the differences in $\phi_{0}$ between different bands
are negligible, but $A^{2}$ can change significantly between optical and NIR
bands, such that a single value of $A^{2}$ for all bands is inconsistent with
observations (see Appendix B). The advantage of template fitting over GP is
the reasonable fits that are obtained for light-curves with poor sampling.
However, the accuracy of the fits is limited by intrinsic variations of the
light curves (at the same period) and by small scale features that are not
captures by the templates. For example, the ”Hertzsprung Progression”
(Hertzsprung, 1926), seen as a ”bump” in the light curves of Cepheids with
periods $\mathord{\approx}10-20\,\rm{d}$, is not captured by the templates,
which leads in some cases to an underestimate of the inferred amplitudes by up
to $\mathord{\approx}0.1\,\text{mag}$. We, therefore, prefer to use the more
accurate GP-derived values, but we demand that they are within
$3\times\max(\sigma,0.01\,\rm{mag})$ from the templates-derived values. We
further demand that the derived amplitudes are different from zero by at least
$3\sigma$ (for any $\log P$).

As a final check, we visually inspect all fitted light curves. Usually, the
agreement between the GP-derived and the template-derived light curves is well
described by our conditions from above. In Figure 5 we provide two examples
for a good match between the two fits for light curves that are well sampled
(CT-Car in the $V$ band, upper left panel, and SV-Vul in the $H$ band, lower
left panel). In some cases, our conditions from above rejected the GP fit
because the template provides a poor fit to the data. Two such examples are
provided in Figure 5 (AD-Cam in the $V$ band, upper-middle panel, and XX-Cen
in the $H$ band, lower right panel). The template fits fail to capture the
behavior of the light curves, although they are well described by the GP fits.
In these cases, we keep the GP-derived values. In very few cases, our
conditions from above did not reject the GP fit, although it is significantly
different from the template in a phase region where no observations are
available. An example is provided in Figure 5 (OGLE-GD-CEP-0428 in the $V$
band, upper right panel). There are no observations between phases $0.2$ and
$0.5$, where the GP fit significantly deviates from the template. In these
cases, we reject the GP-derived values. Following these procedures, we obtain
a catalog that contains reliable mean magnitudes and amplitudes (with error
bars) for secure classical Cepheids, especially for $\log P>1$. Figures of the
obtained light curves (similar to Figure 5) for all Cepheids in our catalog
are publicly available.

Figure 5: Examples of five Cepheid light curves and fitting results. For each
Cepheid, the phase-folded observations are marked by blue circles, with error-
bars corresponding to the maximum between the GP-estimated noise and the
provided observational errors. The GP and template fits are indicated by the
red line and black line, respectively. The three examples in the upper panel
are in the $V$ band, while the two in the lower panels are in NIR bands. CT-
Car in the $V$ band and SV-Vul in the $H$ band (upper left and lower left
panels, respectively) are examples of a good match between the two fits, which
is the case for most of the sample. AD-Cam in the $V$ band and XX-Cen in the
$J$ band (upper-middle and lower right panels, respectively) illustrate two
cases where the template fits fail to capture the behavior of the light
curves, although they are well described by the GP fits. OGLE-GD-CEP-0428 in
the $V$ band (upper right panel) is a case where the GP fit significantly
deviates from the template in a phase region ($0.2-0.5$) where no observations
are available, and therefore the GP-derived values are rejected.

###  A.3 Properties of the catalog

Our final catalog includes $688$ Cepheids with at least one newly derived mean
magnitude or amplitude in some band. The number of newly determined mean
magnitudes and amplitudes from each source is given in Tables 1 and 2 for the
optical and the NIR bands, respectively.

Table 1: The number of newly determined values in the catalog from each source. Note that in some cases we reject the derived amplitude but not the derived mean magnitude. Since the data set of Berdnikov (2008) and Bextr are almost identical, in many cases, the choice of source between them is arbitrary (we choose the source with smaller errors for the derived values, but the estimation of the errors contain a random component, see main text). Source | $B$ | $A^{B}$ | $V$ | $A^{V}$ | $I$ | $A^{I}$
---|---|---|---|---|---|---
Pel (1976) | 1 | 1 | 0 | 0 | 0 | 0
Szabados (1977) | 1 | 1 | 0 | 0 | 0 | 0
Szabados (1980) | 1 | 1 | 1 | 1 | 0 | 0
Moffett & Barnes (1984) | 13 | 13 | 13 | 13 | 13 | 13
Coulson & Caldwell (1985) | 1 | 1 | 1 | 1 | 1 | 1
Henden (1996) | 0 | 0 | 0 | 0 | 12 | 12
Berdnikov (2008) | 124 | 124 | 146 | 145 | 124 | 123
Bextr | 144 | 144 | 173 | 173 | 121 | 119
Berdnikov et al. (2015) | 106 | 106 | 69 | 67 | 76 | 75
Berdnikov et al. (2019) | 51 | 51 | 49 | 49 | 50 | 50
Udalski, Szymański, & Szymański (2015) | 0 | 0 | 36 | 28 | 223 | 220
Jayasinghe et al. (2020) | 0 | 0 | 10 | 10 | 0 | 0
Table 2: Same as Table 1 for the NIR bands.  Source | $J$ | $A^{J}$ | $H$ | $A^{H}$ | $K$ | $A^{K}$
---|---|---|---|---|---|---
Welch et al. (1984) | 13 | 13 | 12 | 12 | 11 | 11
Laney & Stobie (1992) | 32 | 32 | 31 | 31 | 31 | 31
Schechter et al. (1992) | 14 | 15 | 13 | 13 | 14 | 14
Barnes et al. (1997) | 4 | 4 | 4 | 4 | 4 | 4
Feast et al. (2008) | 5 | 5 | 5 | 5 | 5 | 5
Monson & Pierce (2011) | 126 | 126 | 126 | 126 | 123 | 122

The distribution of periods in the catalog is shown in the upper left panel of
Figure 6. The catalog contains 356 (332) Cepheids with $\log P>1$ ($\log
P<1$). The vast majority of available extragalactic Cepheids for which
crowding corrections are significant (i.e., beyond M31) have $\log P>1$, see
the upper left panel of Figure 6. As a result, in what follows, we do not
consider the short-period ($\log P<1$) Cepheids, although we provide in our
catalog their derived values.

Figure 6: The distribution of periods in the catalog with $\log P$ bin widths
of $0.1$. Upper left panel: the entire catalog (black), which contains 356
(332) Cepheids with $\log P>1$ ($\log P<1$). We demonstrate with the H16
optical sample (blue) and the R22 early data release NIR sample (red) that the
vast majority of available extragalactic Cepheids for which crowding
corrections are significant (i.e., beyond M31) have $\log P>1$. Upper right
panel: Cepheids with both $A^{V}$ and $A^{H}$ (black). There are only two
Cepheids with $\log P>1.7$, such that one cannot reliably determine the
$A^{V}/A^{H}$ MW ratio in this period range. One of the two Cepheids is GY-Sge
with $\log P\approx 1.71$, so we set our default upper limit to be $\log
P=1.72$ to include the largest reasonable period range. Table 1 of R20
includes additional $10$ Cepheids (red) with $A^{H}$ values from unpublished
photometry. To keep our data homogeneous, we do not include the reported
$A^{H}$ values of these additional Cepheids. Since these Cepheids have $\log
P<1.4$, where we have numerous Cepheids, the impact of ignoring these Cepheids
is minimal. The R20 extragalactic sample is shown as well (in blue for NGC
4258 and in green for the faraway galaxies). All Cepheids in NGC4258 have
$\log P<1.7$, but there is a significant fraction of Cepheids in the faraway
galaxies with $\log P>1.7$ which cannot be reliably compared to the MW. Bottom
left panel: Cepheids with both HST (F555W, F814W, and F160W) and ground
observations ($V$, $I$, and $H$) are shown in black, red, and blue,
respectively. There are only 4 Cepheids with $\log P>1.5$, which limits the
reliability of the filter transformations in this period range. Bottom right
panel: Cepheids with sufficient data to determine $(B-V)_{0}$ (black) and
$(V-I)_{0}$ (red). There are only three such Cepheids with $\log P>1.7$,
limiting the intrinsic color’s reliability in this period range.

The number of Cepheids in our catalog with both $A^{V}$ and $A^{H}$ (and $\log
P>1$) is 77. The period distribution of these Cepheids is shown in the upper
right panel of Figure 6. As can be seen in the figure, there are only two
Cepheids with $\log P>1.7$, such that one cannot reliably determine the
$A^{V}/A^{H}$ MW ratio in this period range. One of the two Cepheids is GY-Sge
with $\log P\approx 1.71$, so we set our default upper limit to be $\log
P=1.72$ to include the largest reasonable period range. Table 1 of R20
includes additional $10$ Cepheids171717DR-Vel, KK-Cen, SS-CMa, XY-Car, SY-Nor,
SV-Vel, XX-Car, XZ-Car, YZ-Car, and V0340-Ara. (red histogram) with $A^{H}$
values from unpublished photometry, and are therefore not included in our
catalog. To keep our data homogeneous, we do not include the reported $A^{H}$
values of these additional Cepheids in what follows. Since these Cepheids have
$\log P<1.4$, where we have numerous Cepheids, the impact of ignoring these
Cepheids is minimal. The period distribution of the R20 extragalactic sample
is shown as well (in blue for NGC 4258 and in green for the faraway galaxies).
As can be seen, all Cepheids in NGC4258 have $\log P<1.7$, but there is a
significant fraction of Cepheids in the faraway galaxies with $\log P>1.7$
which cannot be reliably compared to the MW.

We supplement the catalog with HST observations in the F555W, F814W, and F160W
filters, as reported by Riess et al. (2018, 2021a). This data can be used to
determine various transformations between HST and ground filters. The bottom
left panel of Figure 6 shows the period distribution of Cepheids with both HST
and ground observations. As can be seen, there are only 4 Cepheids with $\log
P>1.5$, which limits the reliability of the filter transformations in this
period range. We finally provide selective extinction, $E(B-V)$, values that
can be used to calculate the intrinsic colors of Cepheids. The preferred
source for $E(B-V)$ values is Turner (2016), with additional values (in order
of preference) from Groenewegen (2020); Fernie et al. (1995); Ngeow (2012). We
multiply the estimates of Fernie et al. (1995) by $0.94$ (see discussion in
Tammann, Sandage, & Reindl, 2003; Groenewegen, 2018). The bottom right panel
of Figure 6 shows the period distribution of Cepheids with sufficient data to
determine various intrinsic colors. As can be seen, there are only three such
Cepheids with $\log P>1.7$, which limits the reliability of the intrinsic
color in this period range.

margin=1cm

Table 3: A few examples for entries in the catalog. For each Cepheid we provide position (RA and Dec in degrees), period (in d), source for the position and period (S20 (Soszyński et al., 2020) or VSX), and the fitting results (mean magnitude and amplitude in each band together with the source of photometry). The photometry sources initials are: P76 (Pel, 1976), S77 (Szabados, 1977), S80 (Szabados, 1980), M84 (Moffett & Barnes, 1984), C85 (Coulson & Caldwell, 1985), H96 (Henden, 1996), B08 (Berdnikov, 2008), Bextr (Photometry from the Sternberg Astronomical Institute database), B15 (Berdnikov et al., 2015), B19 (Berdnikov et al., 2019), OGLE (Udalski, Szymański, & Szymański, 2015), ASAS (Jayasinghe et al., 2020). Name | RA | DEC | Period | source | $B$ | $A^{B}$ | ref | $V$ | $A^{V}$ | ref | $I$ | $A^{I}$ | ref
---|---|---|---|---|---|---|---|---|---|---|---|---|---
FF-Aur | 73.82550 | 39.97975 | 2.121 | OF | $14.687\pm 0.006$ | $1.440\pm 0.027$ | B08 | $13.720\pm 0.004$ | $1.075\pm 0.019$ | Bextr | $12.455\pm 0.028$ | $0.501\pm 0.149$ | H96
BB-Gem | 98.64708 | 13.07911 | 2.308 | OF | $12.244\pm 0.015$ | $1.440\pm 0.055$ | B15 | $11.412\pm 0.011$ | $1.054\pm 0.041$ | B15 | $10.416\pm 0.007$ | $0.669\pm 0.029$ | B15
V2201-Cyg | 316.07011 | 49.74440 | 2.418 | OF | $13.722\pm 0.006$ | $0.751\pm 0.021$ | Bextr | $12.126\pm 0.002$ | $0.532\pm 0.009$ | Bextr | $99.900\pm 99.900$ | $99.900\pm 99.900$ |
CN-CMa | 107.39410 | -18.56329 | 2.546 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $13.664\pm 0.002$ | $0.750\pm 0.009$ | ASAS | $12.240\pm 0.018$ | $0.581\pm 0.042$ | B08
XZ-CMa | 105.10346 | -20.43169 | 2.558 | OF | $13.761\pm 0.006$ | $1.448\pm 0.022$ | B08 | $12.933\pm 0.004$ | $1.029\pm 0.014$ | Bextr | $11.904\pm 0.004$ | $0.637\pm 0.014$ | Bextr
V0620-Pup | 119.45788 | -29.38406 | 2.586 | OF | $12.967\pm 0.007$ | $0.778\pm 0.024$ | B19 | $11.959\pm 0.004$ | $0.541\pm 0.017$ | B19 | $10.730\pm 0.003$ | $0.325\pm 0.012$ | B19
IT-Lac | 332.32734 | 51.40539 | 2.632 | OF | $16.046\pm 0.014$ | $0.953\pm 0.054$ | B08 | $15.156\pm 0.012$ | $0.643\pm 0.059$ | B08 | $99.900\pm 99.900$ | $99.900\pm 99.900$ |
BW-Gem | 93.99954 | 23.74750 | 2.635 | OF | $12.991\pm 0.025$ | $1.123\pm 0.084$ | B15 | $11.975\pm 0.018$ | $0.802\pm 0.068$ | B15 | $10.711\pm 0.013$ | $0.499\pm 0.049$ | B15
V0539-Nor | 245.22592 | -53.55461 | 2.644 | VSX | $12.317\pm 0.004$ | $0.513\pm 0.016$ | B19 | $11.826\pm 0.003$ | $0.421\pm 0.015$ | B19 | $11.208\pm 0.003$ | $0.332\pm 0.010$ | B19
EW-Aur | 72.85342 | 38.18856 | 2.660 | OF | $14.595\pm 0.009$ | $1.107\pm 0.037$ | Bextr | $13.515\pm 0.006$ | $0.784\pm 0.029$ | Bextr | $99.900\pm 99.900$ | $99.900\pm 99.900$ |
. |  |  |  |  |  |  |  |  |  |  |  |  |
. |  |  |  |  |  |  |  |  |  |  |  |  |
V1467-Cyg | 301.00912 | 32.45075 | 48.677 | OF | $15.974\pm 0.005$ | $1.435\pm 0.022$ | Bextr | $13.485\pm 0.003$ | $1.006\pm 0.014$ | Bextr | $10.545\pm 0.010$ | $0.740\pm 0.059$ | Bextr
OGLE-GD-CEP-1499 | 288.47750 | 11.95300 | 49.142 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $15.154\pm 0.001$ | $0.643\pm 0.006$ | OGLE
CE-Pup | 123.53350 | -42.56817 | 49.322 | OF | $13.379\pm 0.003$ | $1.282\pm 0.012$ | B15 | $11.755\pm 0.003$ | $0.841\pm 0.011$ | B15 | $9.965\pm 0.003$ | $0.532\pm 0.011$ | Bextr
OGLE-GD-CEP-1505 | 288.60517 | 12.99211 | 50.604 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $15.794\pm 0.003$ | $0.538\pm 0.013$ | OGLE
V0708-Car | 153.90787 | -59.55131 | 51.414 | OF | $14.456\pm 0.002$ | $0.762\pm 0.008$ | B19 | $12.075\pm 0.002$ | $0.536\pm 0.008$ | B19 | $9.221\pm 0.002$ | $0.401\pm 0.008$ | B19
GY-Sge | 293.80679 | 19.20239 | 51.814 | OF | $12.445\pm 0.007$ | $1.009\pm 0.019$ | B08 | $10.154\pm 0.004$ | $0.646\pm 0.014$ | Bextr | $7.529\pm 0.011$ | $0.350\pm 0.024$ | B08
ET-Vul | 293.83375 | 26.43022 | 53.910 | VSX | $13.846\pm 0.005$ | $0.689\pm 0.014$ | B08 | $12.190\pm 0.003$ | $0.449\pm 0.008$ | Bextr | $99.900\pm 99.900$ | $99.900\pm 99.900$ |
II-Car | 162.20437 | -60.06306 | 64.836 | OF | $14.830\pm 0.006$ | $1.353\pm 0.028$ | B15 | $12.580\pm 0.003$ | $0.917\pm 0.015$ | B15 | $9.858\pm 0.004$ | $0.560\pm 0.014$ | Bextr
V1496-Aql | 283.74804 | -0.07678 | 65.731 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $10.185\pm 0.007$ | $99.900\pm 99.900$ | B15 | $7.743\pm 0.006$ | $0.345\pm 0.014$ | B15
S-Vul | 297.09921 | 27.28650 | 68.651 | OF | $10.851\pm 0.003$ | $0.912\pm 0.014$ | B08 | $8.965\pm 0.002$ | $0.552\pm 0.012$ | Bextr | $6.939\pm 0.008$ | $0.384\pm 0.020$ | Bextr
Table 4: Same as Table 3 for NIR photometry and $E(B-V)$ values. The photometry sources initials are: W84 (Welch et al., 1984), L92 (Laney & Stobie, 1992), S92 (Schechter et al., 1992), B97b (Barnes et al., 1997), F08 (Feast et al., 2008), M11 (Monson & Pierce, 2011). The $E(B-V)$ sources initials are: F95 (Fernie et al., 1995), N12 (Ngeow, 2012), T16 (Turner, 2016), G20 (Groenewegen, 2020). Name | Period | source | $J$ | $A^{J}$ | ref | $H$ | $A^{H}$ | ref | $K$ | $A^{K}$ | ref | $E(B-V)$ | ref
---|---|---|---|---|---|---|---|---|---|---|---|---|---
. |  |  |  |  |  |  |  |  |  |  |  |  |
V1467-Cyg | 48.677 | OF | $8.164\pm 0.003$ | $0.462\pm 0.015$ | M11 | $7.286\pm 0.014$ | $0.380\pm 0.050$ | M11 | $6.918\pm 0.005$ | $0.375\pm 0.013$ | M11 | $1.532\pm 99.900$ | F95
OGLE-GD-CEP-1499 | 49.142 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ |
CE-Pup | 49.322 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $0.740\pm 0.070$ | G20
OGLE-GD-CEP-1505 | 50.604 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ |
V0708-Car | 51.414 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ |
GY-Sge | 51.814 | OF | $5.540\pm 0.003$ | $0.273\pm 0.013$ | M11 | $4.836\pm 0.005$ | $0.237\pm 0.017$ | M11 | $4.520\pm 0.003$ | $0.239\pm 0.010$ | M11 | $1.147\pm 0.020$ | T16
ET-Vul | 53.910 | VSX | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ |
II-Car | 64.836 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $1.372\pm 99.900$ | F95
V1496-Aql | 65.731 | OF | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ | $99.900\pm 99.900$ |  | $99.900\pm 99.900$ |
S-Vul | 68.651 | OF | $5.412\pm 0.004$ | $0.234\pm 0.012$ | M11 | $4.816\pm 0.004$ | $0.225\pm 0.011$ | M11 | $4.551\pm 0.006$ | $0.224\pm 0.025$ | M11 | $0.999\pm 0.010$ | T16

All Cepheids in our final catalog are classified as fundamental mode Cepheids
in the updated catalog of Pietrukowicz, Soszyński, & Udalski (2021). There are
additional 333 Cepheids (57 with $\log P>1$) in the updated catalog of
Pietrukowicz, Soszyński, & Udalski (2021) that are not present in Soszyński et
al. (2020). The additional $\log P>1$ Cepheids are mostly from Chen et al.
(2020). We could not find NIR observations of the additional $\log P>1$
Cepheids, such that there is no available additional data that could modify
the main results of this paper.

The entire catalog is available online. A few examples for entries in the
catalog are given in Tables 3-4.

##  Appendix B The amplitude ratios of the MW Cepheids

In this appendix, we use our catalog to derive the amplitude ratios of the MW
Cepheids with $1<\log P<1.72$ in different bands. We present in Figure 7 the
ratios that are used to estimate the ground-HST filter transformations in
Appendix C. As can be seen in the top-left panel, we fit the ratio
$A^{V}/A^{B}$ with a quadratic function (black line). We find in this case
$\chi^{2}_{\nu}\approx 3.7$ for $123$ Cepheids after the removal of the
outliers GQ-Vul and SU-Cru, suggesting an intrinsic scatter of
$\mathord{\approx}0.018$. The results of the fit following the addition of the
calibrated intrinsic scatter are indicated in the figure. We find no
significant improvement for fitting with a third-order polynomial (but we do
find a significant improvement over fitting with a constant ratio or a linear
function). The templates of P12 (red line) reproduce the fitted function with
deviations $\lesssim 5\%$.

As can be seen in the bottom-left panel, we fit the ratio $A^{I}/A^{V}$ with a
constant ratio (black line). We find in this case $\chi^{2}_{\nu}\approx 4.7$
for $132$ Cepheids after the removal of the outliers OGLE-GD-CEP-0332 and SU-
Cru, suggesting an intrinsic scatter of $\mathord{\approx}0.031$. The results
of the fit following the addition of the calibrated intrinsic scatter are
indicated in the figure. We find no significant improvement for fitting with
higher-order polynomials. The templates of P12 (red line) reproduce the fitted
value with deviations $\lesssim 10\%$. The result of J21 for $A^{814}/A^{555}$
(dotted blue line, derived from their equation 3) is similar to our fit,
however, their results should be multiplied by
$(A^{I}/A^{814})(A^{555}/A^{V})$ for comparing to $A^{I}/A^{V}$. The factor
$(A^{I}/A^{814})(A^{555}/A^{V})$ is estimated in Appendix C (along with an
argument for this factor exceeding $1$) and the result of multiplying this
factor by the J21 result is plotted is solid blue line (the dashed blue lines
represent the estimated error of this factor). The obtained $A^{I}/A^{V}$
based on the J21 result over-predict our fitted value by $\gtrsim 10\%$. We
discuss in detail the J21 method in Section 3.

As can be seen in the top-right panel, we fit the ratio $A^{H}/A^{J}$ with a
quadratic function (black line). We find in this case $\chi^{2}_{\nu}\approx
3.7$ for $74$ Cepheids after the removal of the outlier AA-Gem, suggesting an
intrinsic scatter of $\mathord{\approx}0.067$. The results of the fit
following the addition of the calibrated intrinsic scatter are indicated in
the figure. We find no significant improvement for fitting with a third-order
polynomial (but we do find a significant improvement over fitting with a
constant ratio or a linear function). The P12 templates (red line) mostly
over-predict the fitted function with deviations smaller than
$\mathord{\approx}15\%$. As can be seen in the bottom-right panel, we fit the
ratio $A^{K}/A^{H}$ with a constant ratio (black line). We find a good fit in
this case, $\chi^{2}_{\nu}\approx 0.79$ for $73$ Cepheids after the removal of
the outliers RY-Cas and YZ-Aur, suggesting that the intrinsic scatter is
smaller than the observational error (the scatter of the observed ratios is
$\mathord{\approx}0.07$). The results of the fit are indicated in the figure.
We find no significant improvement for fitting with a linear function. The P12
templates (red line) slightly over-predict the fitted value with deviations
$\lesssim 5\%$.

Figure 7: Distributions of amplitude ratios as a function of the period. Top-
left panel: the ratio $A^{V}/A^{B}$. We fit the observations with a quadratic
function (black line). The templates of P12 (red line) reproduce the fitted
function with deviations $\lesssim 5\%$. Bottom-left panel: the ratio
$A^{I}/A^{V}$. We fit the observations with a constant ratio (black line). The
templates of P12 (red line) reproduce the fitted value with deviations
$\lesssim 10\%$. The result of J21 for $A^{814}/A^{555}$ (dotted blue line) is
similar to our fit. The obtained $A^{I}/A^{V}$ based in the J21 result (solid
blue line with estimated errors in dashed blue lines) over-predict our fitted
value by $\gtrsim 10\%$. Top-right panel: the ratio $A^{H}/A^{J}$. We fit the
observations with a quadratic function (black line). The P12 templates (red
line) over-predict the fitted function with deviations smaller than
$\mathord{\approx}20\%$. Bottom-right panel: the ratio $A^{K}/A^{H}$. We fit
the observations with a constant ratio (black line). The P12 templates (red
line) slightly over-predict the fitted value with deviations $\lesssim 5\%$.

We reproduced the known result that the $\gtrsim 0.1\,\rm{mag}$ scatter seen
in single-band amplitudes can in some cases be significantly reduced by
considering amplitude ratios between different bands (Klagyivik & Szabados,
2009, and references therein). This was the motivation of R20 to study the
ratio $A^{H}/A^{V}$ that is presented in Figure 8. We fit for this ratio a
linear function (solid black line). We find in this case
$\chi^{2}_{\nu}\approx 9.8$ for $75$ Cepheids after the removal of the outlier
V0340-Nor, suggesting an intrinsic scatter of $\mathord{\approx}0.037$. The
results of the fit following the addition of the calibrated intrinsic scatter
are indicated in the bottom of the figure181818the off-diagonal term in the
covariance matrix of the linear fit is $\mathord{\approx}-1.55\times 10^{-4}$,
which is required for the analysis in Sections 3-4.. We find a small
improvement for fitting with a quadratic function, $\Delta\chi^{2}\approx
2.9$, i.e. less than $2\sigma$191919Incuding the longest period Cepheid with
$A^{H}/A^{V}$ measurement, S-Vul with $\log P=1.84$, to the sample changes
$\Delta\chi^{2}$ to $\mathord{\approx}4.4$ between the quadratic and the
linear fit, indicating a larger but still insignificant
($\mathord{\approx}2\sigma$) improvement.. Nevertheless, we also use a
quadratic function (as used by R20) to check the sensitivity of our results.
We find for the quadratic fit (solid black line) $\chi^{2}_{\nu}\approx 8.3$
for $74$ Cepheids after the removal of the outliers V0340-Nor and HZ-Per,
suggesting an intrinsic scatter of $\mathord{\approx}0.034$. The results of
the fit following the addition of the calibrated intrinsic scatter are
indicated in the top of the figure. The P12 templates (red line) over-predict
the fitted functions by $\lesssim 25\%$202020The deviation of the P12
templates are probably related to the fact that the data of Monson & Pierce
(2011) were not included in the P12 fitting, while it dominates our $H$-band
catalog (O. Pejcha, private communication).. We also plot the data points from
Table 1 of R20212121Note that VZ-Pup has a double entry in Table 1 of R20 and
that the period of SV-Vul should be $\mathord{\approx}45\,\rm{d}$ and not as
stated there ($P=14.10\,\rm{d}$). and the best fit second-order polynomial
derived in R20 (blue)222222The best fit for the $A^{H}/A^{V}$ ratio is derived
from the best fit for the $A^{160}/A^{350}$ ratio, given in R20, multiply by
the filter transformation functions, as given in R20.. The best fit of R20 and
the quadratic fit derived here are similar.

Figure 8: The distribution of $A^{H}/A^{V}$ as a function of the period. The
observations (black symbols) are well fitted with a first-order and second-
order polynomials in $\log P$ (solid and dashed black lines, respectively).
The P12 templates (red line) over-predict the fitted functions by $\lesssim
25\%$. The data points from Table 1 of R20 and the best fit second-order
polynomial derived in R20 are plotted in blue. The best fit of R20 and the one
derived here are similar.

The deviations in $A^{H}/A^{V}$ of the P12 templates with a single value of
$A^{2}$ for all bands justify our procedure in Section A.2 to separately fit
$A^{2}$ in each band. The deviations also suggest that the accuracy of the
predicted amplitude in a wide filter, such as F350LP, is limited, see Section
3.

##  Appendix C Ground-HST amplitude transformations

In this appendix, we estimate the ground-to-HST amplitude ratios
$A^{555}/A^{V}$ and $A^{160}/A^{H}$, which are required for comparing the MW
amplitudes to the extragalactic amplitudes in Section 4 (see Equation (2)),
and the ratio $A^{814}/A^{I}$ (not required for our analysis). Since complete
light curves of the same Cepheids with both ground and HST filters are
unavailable, we are unable to directly calibrate the required ratios (see
Appendix B for a direct calibration of other bands). We are, therefore, forced
to make some approximations to estimate the required ratios. We suggest in
Section 5 future observations that will allow a more direct calibration.

The method of R20 to estimate $A^{Z}/A^{X}$, where $Z$ is an HST filter
(F555W, F814W or F160W) that is similar to a ground filter $X$ ($V$, $I$ or
$H$, respectively) is as follows. They first calibrate mean-magnitude
transformations in the form of

$Z=X+zp+b(X-Y),$ (4)

where $Y$ ($I$, $V$ or $J$, respectively) is a nearby filter, $zp$ is the zero
point, and $b$ is the slope of the color term. They next assume that the
transformation holds in each phase of the light curve and that the extremum
values of the $X$, $Y$ and $Z$ light curves are at the same phase. Then they
can derive the amplitude ratio $A^{Z}/A^{X}$ as

$\frac{A^{Z}}{A^{X}}=1+b-b\frac{A^{Y}}{A^{X}}.$ (5)

In reality, none of the assumptions from above hold, and the level at which
Equation (5) deviates from the actual ratio is difficult to estimate. Note
that Equation (5) depends only on $b$, while $b$ is highly degenerate with
$zp$. In other words, there is a range of $b$ values that is consistent with
the mean magnitude transformation (through degeneracy with $zp$) and
significantly changes the amplitude ratio transformation.

Here we choose to use a different method, which relies on the empirical
observation that for a given Cepheid the amplitude is a decreasing function of
the observed wavelength (Fernie, 1979; Klagyivik & Szabados, 2009). In Figure
9 we show $A^{Y}/A^{V}$ for the filters $Y=BVIJHK$, as calibrated in Appendix
B, as a function of $1/\lambda$, where $\lambda$ is the effective wavelength
of filter $Y$ (obtained from the SVO filter profile service; Rodrigo & Solano,
2020)232323http://svo2.cab.inta-csic.es/theory/fps/. We use
$\lambda=0.443,0.554,0789,1.235,1.662,2.159\,\mu\rm{m}$ for $Y=BVIJHK$ and
$\lambda=0.539,0.813,1.544\,\mu\rm{m}$ for $Z=$F555W,F814W,F160W.. The
motivation to use $1/\lambda$ is the linear relation that is obtained in the
optical and the UV bands (Fernie, 1979; Klagyivik & Szabados, 2009). As can be
seen in the figure, the function $A^{Y}/A^{V}$ is super-linear with
$1/\lambda$, such that estimating $A^{555}/A^{V}$ by interpolating between the
$V$ and $B$ bands is expected to overestimate the ratio, while extrapolating
with the $I$ and the $V$ bands is expected to underestimate the ratio. We can
therefore bound $A^{555}/A^{V}$ between these two estimates. A similar bound
can be obtained for $A^{160}/A^{H}$ ($A^{814}/A^{I}$) by considering
$A^{Y}/A^{H}$ ($A^{Y}/A^{I}$), interpolation with the $J$ ($J$) band, and
extrapolation with the $K$ ($V$) band. Because F555W, F814W and F160W are
close to the $V$, $I$ and $H$ band, respectively, our choice of using
$1/\lambda$ (instead of $\lambda$, for example) has a small effect on our
results.

Figure 9: $A^{Y}/A^{V}$ for the filters $Y=BVIJHK$ (calibrated in Appendix B)
as a function of $1/\lambda$, where $\lambda$ is the effective wavelength of
filter $Y$. Each solid line corresponds to a single value of $\log P$ within
the range $[1,1.7]$ with a spacing of $0.1$. The function $A^{Y}/A^{V}$ is
super-linear with $1/\lambda$, such that estimating $A^{555}/A^{V}$ by
interpolating between the $V$ and $B$ bands is expected to overestimate the
ratio, while extrapolating with the $I$ and the $V$ bands is expected to
underestimate the ratio. We can therefore bound $A^{555}/A^{V}$ between these
two estimates. A similar bound can be obtained for $A^{160}/A^{H}$
($A^{814}/A^{I}$) by considering $A^{Y}/A^{H}$ ($A^{Y}/A^{I}$), interpolation
with the $J$ ($J$) band, and extrapolation with the $K$ ($V$) band. Our choice
of using $1/\lambda$ (instead of $\lambda$, for example) has a small effect on
our results.

The results of the interpolations (extrapolations) with

$\left(\frac{A^{Z}}{A^{X}}\right)_{Y}=\frac{\left(\frac{A^{Y}}{A^{X}}-1\right)\left(\frac{\lambda_{X}}{\lambda_{Z}}-1\right)+\frac{\lambda_{X}}{\lambda_{Y}}-1}{\frac{\lambda_{X}}{\lambda_{Y}}-1}$
(6)

for $A^{555}/A^{V}$ are presented in the top panel of Figure 10. As can be
seen, we can bound $A^{555}/A^{V}$ (dark region) between
$(A^{555}/A^{V})_{B}\approx 1.05-1.06$ from $Y=B$ (green line) and between
$(A^{555}/A^{V})_{I}\approx 1.035$ from $Y=I$ (brown line). The ratio
$A^{555}/A^{V}=1.04$ used in R20, is within our bounded region. In what
follows we interpolate between the two estimates with
$A^{555}/A^{V}=f\left(A^{555}/A^{V}\right)_{B}+(1-f)\left(A^{555}/A^{V}\right)_{I}$,
where $f=0.5$ is our fiducial value (black line) and the error is estimated
with $f=0$ and $f=1$. The P12 templates (blue line) predict a value which is
larger from our estimate by $\mathord{\approx}5\%$. This deviation could be
related to less precise prediction of the P12 templates for HST filters (see
Section 3).

Figure 10: Top panel: $A^{555}/A^{V}$ as a function of the period. We can
bound $A^{555}/A^{V}$ (dark region) between $(A^{555}/A^{V})_{B}\approx
1.05-1.06$ from $Y=B$ (green line) and between $(A^{555}/A^{V})_{I}\approx
1.035$ from $Y=I$ (brown line). The ratio $A^{555}/A^{V}=1.04$ used in R20
(red line), is within our bounded region. We interpolate between the two
estimates with
$A^{555}/A^{V}=f\left(A^{555}/A^{V}\right)_{B}+(1-f)\left(A^{555}/A^{V}\right)_{I}$,
where $f=0.5$ is our fiducial value (black line) and the error is estimated
with $f=0$ and $f=1$. The P12 templates (blue line) predict a value which is
larger from our estimate by $\mathord{\approx}5\%$. This deviation could be
related to less precise prediction of the P12 templates for HST filters (see
Section 3). Bottom panel: $A^{160}/A^{H}$ as a function of the period. We can
bound $A^{160}/A^{H}$ (dark region) between $(A^{160}/A^{H})_{J}\approx
1.02-1.065$ from $Y=J$ (green line) and between $(A^{160}/A^{H})_{K}\approx
1.01$ from $Y=K$ (brown line). The ratio $A^{160}/A^{H}=1.015$ used in R20, is
within our bounded region. We interpolate between the two estimates with
$A^{160}/A^{H}=f\left(A^{160}/A^{H}\right)_{J}+(1-f)\left(A^{160}/A^{H}\right)_{K}$,
where $f=0.5$ is our fiducial value (black line) and the error is estimated
with $f=0$ and $f=1$. The P12 templates (blue line) predict a value which is
smaller from our estimate by $\mathord{\approx}1\%$.

As can be seen in the bottom panel of Figure 10, we can bound $A^{160}/A^{H}$
(dark region) between $(A^{160}/A^{H})_{J}\approx 1.02-1.065$ from $Y=J$
(green line) and between $(A^{160}/A^{H})_{K}\approx 1.01$ from $Y=K$ (brown
line). The ratio $A^{160}/A^{H}=1.015$ used in R20, is within our bounded
region. In what follows we interpolate between the two estimates with
$A^{160}/A^{H}=f\left(A^{160}/A^{H}\right)_{J}+(1-f)\left(A^{160}/A^{H}\right)_{K}$,
where $f=0.5$ is our fiducial value (black line) and the error is estimated
with $f=0$ and $f=1$. The P12 templates (blue line) predict a value which is
smaller from our estimate by $\mathord{\approx}1\%$.

We finally inspect the ratio $A^{160}/A^{I}$ (not required for our analysis)
in Figure 11. As can be seen in the figure, we can bound $A^{814}/A^{I}$ (dark
region) between $(A^{814}/A^{I})_{V}\approx 0.955$ from $Y=B$ (green line) and
between $(A^{555}/A^{V})_{I}\approx 0.965-0.98$ from $Y=J$ (brown line). The
ratio $A^{814}/A^{I}\approx 0.99$, derived by using the R20 method with the
relation F814W=$I+0.02-0.018(V-I)$ from R16 (their equation 11), over-predicts
our estimate by $\mathord{\approx}3\%$. This deviation could be related to the
problems with the R20 method discussed above. One can interpolate between the
two estimates with
$A^{814}/A^{I}=f\left(A^{814}/A^{I}\right)_{V}+(1-f)\left(A^{814}/A^{I}\right)_{J}$,
with $f=0.5$ for the fiducial value (black line) and the error can be
estimated with $f=0$ and $f=1$.

Figure 11: $A^{814}/A^{I}$ as a function of the period. We can bound
$A^{814}/A^{I}$ (dark region) between $(A^{814}/A^{I})_{V}\approx 0.955$ from
$Y=B$ (green line) and between $(A^{555}/A^{V})_{I}\approx 0.965-0.98$ from
$Y=J$ (brown line). The ratio $A^{814}/A^{I}\approx 0.99$, derived by using
the R20 method with the relation F814W=$I+0.02-0.018(V-I)$ from R16, over-
predicts our estimate by $\mathord{\approx}3\%$. This deviation could be
related to the problems with the R20 method discussed in the text. We
interpolate between the two estimates with
$A^{814}/A^{I}=f\left(A^{814}/A^{I}\right)_{V}+(1-f)\left(A^{814}/A^{I}\right)_{J}$,
with $f=0.5$ for the fiducial value (black line) and the error can be
estimated with $f=0$ and $f=1$.

##  Appendix D The H16 amplitude distributions of $A^{555}$ and $A^{350}$ in
different period bins

In this appendix, we supplement the analysis of Section 3 with the H16
amplitude distributions of $A^{555}$ and $A^{350}$ in each period bin
presented in Figure 1. The distributions are presented in Figure 12. As can be
seen in the figure, in each period bin the entire $A^{555}$ distribution is
shifted from the $A^{350}$ distribution to higher amplitudes.

Figure 12: The H16 amplitude distributions of $A^{555}$ (blue) and $A^{350}$
(black) in each period bin presented in Figure 1. In each period bin the
entire $A^{555}$ distribution is shifted from the $A^{350}$ distribution to
higher amplitudes. The mans of the distributions (dashed lines, corresponds to
the red and magenta symbols with error bars in the middle panel of Figure 1)
and the number of Cepheids in each period bins are indicated as well.