11institutetext: Richard I. Anderson 22institutetext: Institute of Physics,
École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny,
Chemin Pegasi 51, 1290 Versoix, Switzerland, 22email<EMAIL_ADDRESS>
# On Cepheid Distances in the $H_{0}$ measurement

Richard I. Anderson

###### Abstract

Classical Cepheids were the first stellar standard candles and have played a
crucial role for astronomical distance measurements ever since the discovery
of the Leavitt law (period-luminosity relation). Enormous improvements in
distance accuracy have been achieved since Hertzsprung’s first application of
Leavitt’s law to measure the distance to the Small Magellanic Cloud in 1913,
notably in very recent years thanks to a large data set of highly accurate
space astrometry from the ESA mission Gaia. Complemented by homogeneous space
photometry, Cepheids enable the most accurate distance estimates to galaxies
hosting type-Ia supernovae up to approximately 70 Mpc distant. Here, I review
the history of Cepheid distance measurements, open questions on the side of
stellar astrophysics, and recent studies seeking to quantify and mitigate
systematics with a view to further improve the accuracy on the Hubble
constant. For example, the recently launched James Webb Space Telescope will
enhance precision due to 4x lower sensitivity to source blending in crowded
regions and greater sensitivity in dust-insensitive infrared bands. Future
30m-class telescopes could in principle further improve Cepheid distance
measurements towards the Hubble flow, if technical challenges related to a
continuously evolving instrument can be overcome.

##  1 Cepheids: From variable stars to cosmic yardsticks

The variability of classical Cepheids (henceforth for simplicity: Cepheids,
cf. Sec. 2) was first discovered by visual obserations in the 18th century by
John Goodricke and Edward Piggott (Goodricke1786, ). Friedrich Argelander
began systematically surveying stars on both hemispheres in the 1850s, and
Charles Pickering pushed for systematic observations of variable stars towards
the end of the 19th century. Pickering hired female assistants (the so-called
Harvard “Computers”) to process the observations collected (sobel2016glass, ).
The photographic plates were astrometrically aligned and variable stars
identified by blinking observations of the same sky area observed at different
epochs. Henrietta Leavitt thus conducted an unprecedented census of 1777
variable stars in the Small Magellanic Cloud (Leavitt1908, ).

In 1912, Pickering reported on Leavitt’s groundbreaking discovery based on 25
classical Cepheids, that the apparent magnitude $m$ of 25 SMC Cepheids
correlated well with the logarithm of their period of variability $P$, both
near minimum and maximum light (Leavitt1912, ). Since all SMC stars were
assumed to reside at a common distance, it was clear that the observed
relation between brightness and period must be an intrinsic property of the
stars. This period-luminosity relation (PLR) is nowadays referred to as
Leavitt’s law (LL).

Just one year later, Ejnar Hertzsprung (Hertzsprung1913, ) made the first
attempt to measure distance using Leavitt’s discovery. He estimated the
absolute magnitude of Cepheids at a period near $6.6$ d using secular parallax
and, in turn, determined the distance modulus of the SMC Cepheids by comparing
their apparent magnitude at the same period to that of the MW Cepheids.
Hertzsprung’s work laid the foundation for the distance ladder in use today.
Despite concluding on an incorrect distance ($2000$ lightyears), he
convincingly demonstrated that the SMC is external to the MW based on a
comparison of the scale height of stars in the Solar Neighborhood. Thus,
Cepheid distances already informed the (extragalactic) distance scale before
their variability mechanism was understood Shapley1914 ; Eddington1917 ;
Wesselink1946 , and well before the Great Debate of 1920, Hubble’s works on
M33 (Hubble1926, ), as well as the subsequent discovery of the Hubble-Lemaître
law (Lemaitre1927, ; Hubble1929, ).

###  1.1 The Hubble constant measured using the distance ladder

The relation between Hubble’s constant, $H_{0}$, and $D_{L}$ is given by the
Friedmann equation, developed here to second order using the Robertson-Walker
metric:

$D_{L}=\frac{cz}{H_{0}}\left[1+\frac{1}{2}(1-q_{0})z-O(z^{2})+\ldots\right]\
,$ (1)

where $q_{0}\approx-0.55$ is the deceleration parameter, whose observed
negative value implies the Universe’s accelerated expansion (riess1998, ;
Perlmutter1999, ). Clearly, the wish to measure $H_{0}$ accruately is
intimately tied to measuring accurate distances.

Eq. 1 requires the measured redshifts to be purely cosmological, that is, any
motions due to graviational attraction (large scale flows, peculiar motions,
etc.) must be subtracted or negligible, so that $H_{0}$ is best measured in
the Hubble flow at distances of $90-600$ Mpc ($0.01<z<0.0233$) (Riess2016, ).
Such distances are accessible thanks to the extremely luminous type-Ia
supernovae (SNeIa, cf. Vincenzi this volume) that enable $\sim 3\%$ relative
distance estimates. However, since SNeIa are very rare, no SNIa has as yet
been observed in a galaxy whose distance is known by geometrical means. SNeIa
thus require external calibration by other means, such as Cepheids, via the
_distance ladder (DL)_.

Figure 1: The extragalactic distance ladder uses a two-rung approach for
measuring $H_{0}$. Geometric distances calibrate stellar standard candles for
determining luminosity distances, which in turn calibrate the luminosity of
SNeIa that trace the Hubble flow. Credit: NASA, ESA, A. Feild (STScI), and A.
Riess (STScI/JHU)

The modern DL consists of three rungs as illustrated in Fig. 1. On the first
rung, Cepheids are calibrated as standard candles (SCs) using distances
measured by geometric methods. On the second rung, Cepheids are used to
determine distances to galaxies (SN-hosts) where at least one SNIa has been
observed, out about $80$ Mpc (Riess2022H0, ). On the last rung, SNeIa map
cosmic expansion in the Hubble flow.

Seen the other way around, the DL connects the SNeIa distance-redshift
relation to an angular scale provided by geometry using Cepheids as the
intermediary. By further connecting the Hubble diagram to the early Universe’s
angular scale (the cosmological sound horizon) via the cosmic expansion
history ($H(z)$) using baryon acoustic oscillations, the DL enables a crucial
cosmological end-to-end test involving two angular scales at opposite ends of
the cosmos—trigonometric parallaxes and the quantum fluctuations of the early
Universe (Riess2020nature, ).

$H_{0}$ measurements using a DL have a rich history (cf. Tully, this volume),
involving much controversy and disagreements. The HST Key Project to measure
the Hubble constant was designed to resolved the ongoing disputes and for the
first time achieved a $10\%$ measurement of $H_{0}$ (Freedman2001, ).

In the late 2000’s, the SH0ES (Supernova $H_{0}$ for the Equation of State)
project was launched to measure $H_{0}$ in order to determine the nature of
Dark Energy using a differential DL built on the principle of data homogeneity
(Riess2009a, ; Riess2009, ). In its most recent form, the SH0ES DL is
implemented as a least squares problem that minimizes (Riess2022H0, ):

$\chi^{2}=(y-Lq)^{T}C^{-1}(y-Lq)\ ,$ (2)

with the data vector $y$, the design matrix $L$ that encodes the relevant
equations, such as the LL, distance moduli, etc., the covariance matrix $C$,
and the best-fit parameter vector $q$. Covariance among observables can be
directly considered by quantifying the corresponding elements of $C$, and very
significant amount of covariance information is already being considered,
notably among background corrections of Cepheids in SN host galaxies
(Scolnic2022, ; Brout2022, ; Riess2022H0, ). Using 3445 degrees of freedom, 5
best fit parameters are determined by the SH0ES DL (see Sec. 1.2 for
definitions): the LL slope $\beta$ ($b_{W}$), metallicity effect $\gamma$
($Z_{W}$), the fiducial absolute magnitude of a 10-day Cepheid in the HST NIR
Wesenheit magnitude $M_{H,1}^{W}$, the absolute $B-$band magnitude of SNeIa,
$M_{B}^{0}$, and Hubble’s constant $5\log_{10}{H_{0}}$. Alternative symbols
used in literature are listed in parenthesis.

The matrix formalism is both simple and accurate, and allows straightforward
investigation of analysis variants, notably to assess the systematic effects
of modeling choices or dataset selection. Nevertheless, the results from the
matrix inversion are cross-checked with computationally intensive Markov chain
Monte Carlo (MCMC) simulations that allow inspection of marginalized posterior
distributions and correlations among fit parameters. TRGB distances have also
been included in this procedure and can help to further improve precision.
Including the latest results based on cluster Cepheids, this DL yields
$H_{0}=73.15\pm 0.97\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (Riess2022clusters, ).

The specific construction of the SH0ES DL, with the simultaneous fit to all
observables resembles a tightly controlled _experiment_ that seeks to maximize
the benefits of purely differential flux comparisons to leverage statistical
precision while clearly quantifying the systematic uncertainties involved in
calibrating the absolute luminosity scale. In this spirit, the SH0ES DL
emphasized the use of homogeneous and very well calibrated Hubble Space
Telescope (HST) photometry. The greatly improved spatial resolution and
enhanced sensitivity of the James Webb Space Telecope (JWST) is increasingly
replacing the near-infrared (NIR) photometry observed using HST WFC3/IR
(Riess2023crowd, ). Careful cross-calibration between HST and JWST NIR
photometry is thus required to ensure direct compatibility. As a result of its
specific construction, any modifications to the experimental setup must
consider consequences across the entire DL in order to ensure apples-to-apples
comparisons.

###  1.2 On standard candles, calibration, and standardization

Astronomical standard candles (SCs) are objects that fulfill at least two
conditions: a) their intrinsic brightness (absolute magnitude $M$) can be
calibrated in a given photometric band, or combinations of several bands, and
b) they can be faithfully recognized in different observational contexts,
notably in different galaxies. SCs are crucial to astronomy because they allow
to determine _luminosity distances_ , $D_{L}$ (cf. Eq. 2), by exploiting the
fact that the observed flux $F$ decreases with distance squared, that is,
$F\propto L/D_{L}^{2}$. In magnitude space, this becomes the distance modulus
equation $\mu=m-M-A=5\log_{10}{D_{L}}+25$, with $D_{L}$ in Mpc and extinction
$A$. As noted by Hertzsprung, using standard candles for distance
determination presumes that the absolute magnitude established in one context,
e.g., in the Milky Way, $M_{\mathrm{cal}}$, is representative of the absolute
magnitude of sources seen in other contexts, such as SN host galaxies,
$M_{0}$. Assuming this correspondence, distance is determined by comparing the
observed (apparent) magnitude to the calibrated absolute magnitude,
$\mu=m_{\mathrm{obs}}-M_{\mathrm{cal}}-A$.

_Calibrating_ standard candles to establish $M_{\mathrm{cal}}$ requires
observing them at known distance, $M_{\mathrm{cal}}=m-\mu-A$. Galaxies where
this is possible are frequently referred to as _anchors_ , and three anchors
are of particular importance for Cepheids: The Milky Way (MW), the Large
Magellanic Cloud (LMC), and the water megamaser-host galaxy NGC 4258. Despite
an available geometric distance estimate of the SMC, complications related to
its elongated line-of-sight orientation have thus limited its usefulness for
measuring $H_{0}$ at the current levels of precision. Interestingly, the
distance measurements used for all three anchors are systematically very
different. In the MW, trigonometric parallaxes of SCs, notably from the
European Space Agency Gaia mission (cf. Lindegren, this volume, and references
therein) (Prusti2016gaia, ; Brown2021gaiaedr3, ; GaiaDR3summary, ), set the
current gold standard. In particular, parallaxes of open star clusters that
host Cepheids have provided the most accurate Cepheid calibration to date
($0.9\%$ in distance) (CruzReyes2023, ) thanks to better understood
systematics of the fainter, non-variable cluster stars, and the statistical
gain of averaging over many cluster members. The typical uncertainty of such
cluster parallaxes is $7\,\mu$arcsec, three times better than for individual
field Cepheids, so that one cluster Cepheid carries as much weight as 9 field
Cepheids (Riess2022clusters, ).

The geometric distance measurements of the LMC (detached eclipsing binaries)
and NGC4258 (Keplerian disk model of observed megamaser features observed
using VLBI) rely on very different systematics, including instrumentation used
and modeling assumptions, rendering them truly independent checks of the
calibration based on parallax. The close agreement among the Cepheid
calibrations based on these anchors is therefore a strong attestation to the
accuracy of the anchor distances as well as the consistency among Cepheid
observations spanning an enormous contrast range from approximately $8$th to
$28$th magnitude (8 dex in flux).

_Standardization_ refers to the process of mitigating differences that disturb
the equivalence between $M_{\mathrm{cal}}$ and $M_{0}$. All stellar standard
candles (including Cepheids, SNeIa, and the TRGB method) require
standardization to provide distances to better than a few percent.

As Leavitt showed more than a century ago, Cepheids differ very significantly
in luminosity depending on their pulsation period (by about 3 mag/dex of
period), which ranges from a few days to more than 100 days. Cepheid distances
are thus obtained by comparing _fiducial_ magnitudes. The most commonly
adopted form of the Leavitt law of Cepheids is

$M(P)=\alpha+\beta\cdot\log{P/P_{0}}+\gamma\cdot\mathrm{[Fe/H]}\ ,$ (3)

with $\alpha$ the fiducial absolute magnitude of a Solar metallicity Cepheid
at the pivot period $P_{0}$ (also often referred to as the zero-point),
$\beta$ the LL slope, and $\gamma$ the metallicity term that allows to correct
the impact of chemical composition using iron (also: oxygen) abundances, cf.
below. Importantly, $\alpha$, $\beta$, and $\gamma$ generally depend on the
photometric passband (or combinations thereof) used.

The fiducial absolute Cepheid magnitude in the SH0ES DL, $M_{H,1}^{W}$,
represents a $10$ d Cepheid at the center of the instability strip in a
specific combination of photometric passbands (cf. Sec. 2.2). The commonly
used pivot period of $10$ d is motivated by literature reports of LL non-
linearity (Bhardwaj2016, ) and the period-amplitude diagram of Cepheids
(Klagyivik2009, ), and corresponds to an intermediate period among Cepheids in
anchor galaxies, although Cepheids at greater distance tend to have
increasingly longer periods. $\beta$ and $\gamma$ are determined in the global
DL fit using all Cepheids in anchor and SN host galaxies simultaneously.

At present, there is no clearly demonstrated need to consider LL non-linearity
when using Cepheids to measure distances, and allowing for two different
slopes near the pivot period does not significantly improve the fit. Moreover,
the astrophysical motivation for using non-linear LLs is limited, since
stellar models of single chemical composition yield log-linear LLs across a
large range of periods (e.g. Anderson2016rot, ; DeSomma2021PA, ). However,
stellar evolution models do predict a small dependence of chemical composition
on LL slope (Anderson2016rot, ). Since the effect is rather small, it has not
yet been confidently detected (Ripepi2020, ). LL slope changes due to
metallicity are furthermore particularly small in the infrared, rendering the
SH0ES Wesenheit formalism very insensitive to them.

It seems that observed LL non-linearity arises in a context-specific way from
an uneven sampling of the period-color relation, which is caused by the finite
width of the classical instability strip. Multi-phase PL relations (Ngeow2012,
) compare LLs at different pulsation periods and have demonstrated broken PL
relations. However, Cepheid magnitudes used for measuring distances correspond
to the average flux of Cepheids, and cannot be directly compared to PL
relations observed at specific phases. The coincidence of the reported LL
breaks near $10$ d strongly suggest an origin in the Hertzsprung1926
progression.

Two further considerations are in order concerning LL non-linearity. Firstly,
$H_{0}$ bias due to LL non-linearity arises only in case of systematic
differences among the distributions of Cepheid periods in anchor and SN-host
galaxies, and this effect is of order $\lesssim 0.15\%$ (the author & Bastian
Lengen, priv. comm.). Secondly, variants of the SH0ES DL that include a break
period of $10\,d$ are presented alongside the baseline analysis. These
variants also indicate a very insignificant effect on $H_{0}$, while not being
preferred according to the change in $\chi^{2}$.

Aside from pulsation period, the most common parameter used for
standardization considers metallicity differences (Kennicutt1998, ; Rizzi2007,
; Freedman2011, ; Breuval2022, ). For Cepheids, most studies assume a global
offset to the LL using a linear dependence on chemical composition usually
quantified via oxygen or iron abundances relative to Solar composition, that
is, $\Delta M_{\mathrm{met}}=M_{0}-M_{\mathrm{cal}}=\gamma\cdot\Delta[O/H]$.
Determining $\gamma$ is challenging because the effect is rather small
($\sim-0.25$ mag/dex) (Riess2022H0, ; Breuval2022, ), and metallicity
differences among most Cepheids used in the DL are typically less than
$0.5-0.6$ dex. Oxygen abundances estimated in SN-host galaxies fall within a
rather narrow range that is fully contained within the abundance ranges of
Cepheids in the Milky Way, LMC, and the SMC (Riess2022H0, ). As a result, the
uncertainty in the metallicity effect does not significantly affect $H_{0}$.
However, metallicity can be a limiting factor for determining distances to
individual Cepheids, e.g., to map Galactic structure, or for determining
Gaia’s parallax offsets (Molinaro2023, ).

Standardization is best done using quantities than can be directly observed in
the specific sources of interest, such as individual elemental abundances
derived from Cepheid spectra. Such comparisons are presently limited to MW and
LMC Cepheids, although future 30m class telescopes will significantly increase
the ability to observe individual stellar spectra in other nearby galaxies.
However, the majority of SN-host galaxies will remain off-limits to individual
star spectroscopy. At such distances, equivalent information based on the
closest proxys is typically used, such as oxygen abundances of H II regions
derived by the strong line method (Zaritsky1994, ; Bresolin2011, ), or
abundances of blue supergiants (Kudritzki2008, ; Bresolin2022, ). Ensuring the
compatibility of proxy information used for standardization is thus crucial to
obtaining unbiased distance estimates. For the realm where this has been
feasible, such agreement has been shown, for example among HII regions in the
Milky Way and Cepheid abundances determined by optical spectra (Riess2022H0,
).

#### A sidenote on TRGB standardization

As all SCs, the TRGB method (Lee1993, ) is also subject to standardization.
However, metallicity corrections are complicated by the statistical nature of
the measurement ($m_{\mathrm{TRGB}}$ is measured based on a sample, not for
each star individually) and the diverse population of giant stars composed of
different ages and chemical composition. As a result, there is less agreement
over how population differences should be treated, with some arguing for
color-based metallicity corrections (Rizzi2007, ) and others assuming that
differences among the bluest (interpreted to be the oldest) stars will be
negligible (Freedman2021, ).

Ubiquitous small amplitude pulsations observed in luminous red gaints near the
tip (Ita2002, ; Eyer2002, ) provide useful addition information that may
resolve this deadlock. Anderson2023  recently showed that all stars near the
TRGB feature are variable, and that the variabilty of red giants provides
crucial information about the composition of the diverse red giant population.
Comparison between the Large and Small Magellanic Clouds (Koblischke &
Anderson, in prep.) reveals that the PL sequences of small amplitude red
giants in the OGLE (Optical Gravitational Lensing Experiment) catalog of
variable stars (OSARGs) are systematically shifted to shorter period in the
SMC. This is indicative of a metallicity effect that causes lower-metallicity
stars to be more compact, increasing the average density of stars, and thus
reducing their periods, since $P\propto 1/\sqrt{\rho}$. Thus, OSARG-like
variability can be used to standardize metallicity differences for the TRGB
method, provided such low levels of variability (amplitudes of order
$0.01-0.04$ mag) can be detected.

Another form of standardization is required for TRGB distances due to
methodological choices. The CATs (Comparative Analysis of TRGBs) team Wu2023 ;
Li2023 ; Scolnic2023  demonstrated a significant tip-contrast relation (TCR)
for TRGB measurements, whereby fields observed in the same (very distant)
galaxy differed in TRGB magnitude according to the ratio of the number of
stars that are fainter and brighter ($R=N_{+}/N_{-}$) than the TRGB feature.
Standardization to a fiducial value of $R=4$ was thus proposed to ensure
consistency. Since TCR standardization does not depend on distance, the effect
mainly increases the scatter of TRGB distances to SN host galaxies, rather
than biasing $H_{0}$. Meanwhile, Anderson2023  showed that the TCR is a
methodological artifact introduced by weighting the edge detection response
(Sobel filter output) in an effort to assign statistical significance to it
(Madore2009, ) and that unweighted edge detection responses do not exhibit a
significant TCR. Moreover, simulations showed that unweighted EDRs yield less
biased TRGB magnitudes than weighted EDRs (Anderson2023, ). Since $R$ is
defined _after_ determining $m_{\mathrm{TRGB}}$, TCR corrections are slightly
circular, and unweighted EDRs should be the preferred choice of measuring
$m_{\mathrm{TRGB}}$. Additional details on the TRGB method are presented in
Beaton (this volume), and Beaton2018review  provides a detailed account of
other methodological aspects of TRGB measurements that must be considered to
obtain consistent results.

##  2 Cepheids and where to find them – observational considerations

###  2.1 On the usefulness of Cepheids in the DL

Classical Cepheids afford the best precision for measuring distances thanks to
the fortuitous combination of a) high luminosity, b) highly regular
variability, c) comparatively straightforward identification thanks to large
(optical) amplitudes, d) and relatively high prevalence in star forming
galaxies. Of course, Cepheids also come with their own observational
challenges and are not suitable for every context. For example, Cepheids in
highly inclined galaxies suffer from enhanced crowding and reddening issues.
In such cases, the Tip of the Red Giant Branch (TRGB) method can offer a
suitable alternative (Lee1993, ).

An important advantage of Cepheids is that they are observed specifically and
individually. This differs notably from the detection of the TRGB feature,
which is ascribed to an astrophysically extremely short-lived phenomenon and
which is always measured using an inhomogeneous and contaminated (e.g., by AGB
stars) sample of stars. Galaxies where multiple types of SCs can be observed
thus provide cross-checks essential for understanding systematic uncertainties
while allowing to cross-calibrate multiple SC types, e.g., to increase the
number of SN host galaxies in the DL. The following reviews observational
aspects of Cepheids and any known issues involving their calibration or use as
SCs.

###  2.2 Cepheid measurements used in the DL

#### Cepheid types, variability, and classification

This chapter uses the term “Cepheid” synonymous with classical (type-I)
Cepheids that belong to the young population I, and whose prototype is
$\delta$ Cephei. However, it is important to note that other Cepheid types
exist, notably the type-II (subgroups W Vir, BL Her, and RV Tau) and anomalous
Cepheids that belong to the old population II and exhibit relatively similar
light variability, at least in terms of timescales and amplitudes.
Distinguishing these groups is crucial to ensure distance accuracy and indeed
led to a major revision of the distance scale by Walter Baade (Baade1956, ).

Classical Cepheids are, first and foremost, identified by their variability.
Additional context, such as Galactic latitude for MW stars, or the position
within a galaxy, usefully separates between young and old stars. However,
misclassification between type-I and type-II Cepheids is relatively common if
parallax information is not reliable or available. For stars at a common
distance, PL sequences are extremely powerful at distinguishing among
different types, notably in the Magellanic Clouds, where many PL sequences
have been identified (Soszynski2015, ). Since classical Cepheids are easily
identified by their high luminosity, large number, and period range, there is
little room for confusion among Cepheid types in SN host galaxies.

Where available, multi-band time-series data are helpful to establish
temperature variations that distinguish chromatic pulations from usually
achromatic eclipse-related phenomena. Spectroscopy can provide certainty by
allowing to measure effective temperature variations, asymmetric line shapes
(a telltale of pulsations), RV variations, as well as surface gravity and
abundances.

Gaia’s magnitude limited survey has provided the largest homogeneous sample of
Cepheids classified solely based on (multi-band) light curves, containing 15
006 Cepheids of all sub-types, including approximately 12 554 classical
Cepheids. These Cepheids belong to galaxies within the Local Group ($<1$ Mpc),
including the MW, the Magellanic Clouds, M31, and M33, and nearby dwarfs. In
the MW, there are approximately $3500$ classical Cepheids (Pietrukowicz2021, ;
GDR3Cepheids, ), whereas nearly $10\,000$ Cepheids are nowadays known in the
Magellanic System (Soszynski2015, ; GDR3Cepheids, ).

The prime targets for $H_{0}$ are Cepheids that pulsate in the fundamental
radial mode (FM Cepheids). These are the most numerous, most regular in terms
of their variability, and the longest-period Cepheids. Their detection is
further aided by saw-tooth shaped light curves that exhibit a characteristic
dependence on pulsation period. This so-called Hertzsprung progression
(Hertzsprung1926, ) provides important evidence of the similarity of nearby
and distant Cepheids. FM Cepheids have their highest amplitudes between
approximately $5-7$ and $15-30$ d, and amplitudes decrease for the longest-
period Cepheids (Klagyivik2009, ).

Shorter-period Cepheids that pulsate in the first overtone (FO Cepheids)
feature sinusoidal light curves of lower amplitudes, and higher overtones
feature progressively lower amplitudes that can become challenging to detect
and distinguish from other types of variability. Approximate relations between
FO and FM pulsation periods (MACHO, ) are occasionally used to combine samples
of FO and FM Cepheids. In these cases, FO Cepheid periods are
“fundamentalized” using period ratios of FO and FM Cepheids in well observed
samples, such as the LMC. However, the metallicity dependence of
$P_{\mathrm{FO}}/P_{\mathrm{FM}}$ (Kovtyukh2016, ) complicates this process
and adds systematic uncertainty that is difficult to quantify (e.g.
Csoernyei2023M51, ). As a result, FO Cepheids are not usually considered in
the modern DL.

Besides being more luminous, FM Cepheids also have the advantage of being more
regular in their variability than FO Cepheids (Evans2015MOST, ; Anderson2014,
; Smolec2016, ; Sueveges2018a, ; Sueveges2018b, ). However, the very long
period Cepheids ($P\gtrsim 25$ d) exhibit period fluctuations that can be
stochastic or (semi-)periodic in nature (Turner2005, ), notably causing issues
with de-phasing to mean magnitudes due to variable pulsation ephemerides
(Csoernyei2022periods, ). However, even very fast period changes of $100$ s/yr
($\dot{P}/P\sim 4\cdot 10^{-5}$ at $30$ d, $\Delta\log{P}\sim 6\cdot 10^{-7}$)
are irrelevant for determining distances using LLs.

#### Reddening-free Wesenheit magnitudes

Near-infrared (NIR) photometry significantly improves the use of Cepheids as
standard candles due to the intrinsically reduced sensitivity to extinction,
lower amplitudes, and an intrinsically reduced width of the instability strip,
which results in lower dispersion. Until recently, the main drawback of NIR
photometry for the DL was the reduced spatial resolution of HST’s WFC3/IR
channel compared to WFC3/UVIS. However, NIRCAM on board the JWST now provides
$4\times$ better spatial resolution than WFC3/IR, with a larger aperture,
resulting in much improved Cepheid photometry at large distances
(Riess2023crowd, ). Unfortunately, the reduced amplitudes of Cepheids in the
NIR compared to the optical render Cepheid discovery using JWST not applicable
to large samples.

Issues related to dust extinction are commonly mitigated by computing
Wesenheit111The German word “Wesenheit” relates to the abstract innate nature
(or essence) of an entity, cf. https://www.dwds.de/wb/Wesenheit. magnitudes
(VanDenBergh1975, ; Madore1982, ), $m^{W}$, which are reddening-free by
construction for a given reddening law (SH0ES uses $R_{V}=3.3$ from
Fitzpatrick1999, ; Schlafly2011, ). The SH0ES near-IR Wesenheit formalism
combines $H-$band (F160W) magnitudes with optical colors,
$m_{H}^{W}$$=\mathrm{F160W}-R^{W}\cdot(\mathrm{F555W}-\mathrm{F814W})$, where
$R^{W}=A_{H}/(A_{V}-A_{I})=0.386$ set by the reddening law. For Cepheids
observed by Gaia, $R^{W}_{G}=A_{G}/(A_{G_{BP}}-A_{G_{RP}})\approx 1.91$.
Importantly, $R^{W}$ depends on intrinsic source color when using wide-band
photometry due to the shape of the spectral energy distribution incident on
the photometric passband (Anderson2022, ). The SH0ES approach is thus
particularly robust against reddening issues thanks to the combination of
intrinsically extinction-insensitive NIR photometry with a rather small color-
dependent term. For example, a $10\%$ change in reddening law (from
$R_{V}=3.3$ to $3.0$) would result in a $0.01-0.05$ change in $R_{H}$
(depending on reddening law), which multiplied by a typical color excess of
$E(V-I)\approx 0.6$ would merely change $m_{H}^{W}$ by $\sim 0.03*0.6=0.02$
mag, that is, much less than the dispersion in most SN host galaxies, cf. Fig.
22 in (Riess2022H0, ).

Wesenheit magnitudes have the desirable side-effect of narrowing LL dispersion
due to the finite width of the instability strip, since $R^{W}$ happens to be
close to the slope of the period-color relation. Of course, Wesenheit
magnitudes require knowledge of the reddening law, which can vary among
sightlines, and this can be an issue for specific sightlines, such as Cepheids
in the SMC, which has been reported to have an abnormal reddening law
(Gordon2003, ). However, given the thousands of sightlines among Cepheids in
anchor galaxies (MW, LMC, NGC4258) and the thousands of sightlines to Cepheids
in SN-host galaxies it is highly unlikely for reddening-law related bias to
significantly affect $H_{0}$, as was also found empirically (Moertsell2022, ;
Riess2022H0, ).

#### Cepheids in the Milky Way

MW Cepheids are particularly important for the DL due to the ability to
measure trigonometric parallax (Feast1997, ). However, MW Cepheids are subject
to several challenges that arise from their location inside the MW’s disk,
where dust extinction is high and a strong function of distance, and where the
LL cannot be recovered using only apparent magnitudes. Observationally, the
brightness of Cepheids can be an asset in that high-resolution spectra can be
observed (daSilva2022, ; Ripepi2022, ; Trentin2023, ). However, the same
brightness can easily saturate photometric observations of Cepheids, notably
from space, using large apertures, and in the infrared. Another practical
issue is the distribution of MW Cepheids across a very large sky area,
requiring MW Cepheids to be targeted individually and causing large
observational overheads.

Drift scanning observations with HST have allowed the collection of photometry
in the SH0ES Wesenheit system for a legacy sample of 75 MW FM Cepheids that
feature low extinction and are sufficiently nearby ($\sim 1-5$ kpc) for high-
quality Gaia parallaxes (Riess2018phot, ; Riess2021phot, ). HST drift scan
observations have further provided parallaxes of 8 Cepheids, with a typical
accuarcy of $40-50\,\mu$as (Riess2014, ; Casertano2016, ; Riess2018plx, ).

Hundreds of high-quality parallaxes of MW Cepheids have been published by Gaia
(Prusti2016gaia, ; Brown2021gaiaedr3, ; GaiaDR3summary, ; GaiaDR3astrometry, ;
Lindegren2021, ), an order of magnitude increase over ESA’s predecessor
Hipparcos (Hipparcos, ; VanLeeuwen2007, ). Clearly, Gaia has been a _game
changer_ for DL calibration and has ushered in a new era for SCs. However,
Gaia (E)DR3 parallaxes are subject to unfortuante systematics at the level of
$10-20\,\mu$as (a $3-6\%$ error at $3$ kpc) that correlate strongly with sky
position (ecliptic latitude), magnitude, and color, cf. Lindegren, this
volume. Lindegren2021  (L21) determined parallax offset corrections based on
these parameters using bright physical pairs, LMC stars, and quasars. However,
several follow-up studies Zinn2021 ; Riess2022H0 ; Molinaro2023  have shown
that stars brighter than $G\lesssim 11$ mag require additional correction at
levels similar to the L21 corrections (Groenewegen2023Budapest, ). Very
recently, Khan2023plx  showed that the L21 corrections require significant
further corrections at bright ($G\lesssim 11$ mag) magnitudes and that these
residual corrections depend significantly on sky position, likely because the
bright physical pairs insufficiently sample the full sky. As a result, all SC
calibrations in this magnitude range must simultaneously solve for a sample
parallax offset, which requires careful analysis and generally comes at the
cost of precision, and this notably includes Cepheids that are typically
brighter than $11$ mag.

The best parallax accuracy for MW Cepheids is afforded by the fainter non-
variable member (main sequence) stars of open star clusters that host Cepheids
(Riess2022clusters, ; CruzReyes2023, ). The numerous member stars improve
statistical precision while Gaia’s parallax sysetmatics are best understood in
the magnitude probed by the members stars (Lindegren2021, ; MaizApellaniz2022,
). This fortuitous combination yields a typical accuracy of $7\,\mu$arcsec for
cluster Cepheids, affording them the same weight as $\sim 9$ field Cepheids
for determining $M_{H,1}^{W}$. Combining cluster and field Cepheids thus has
allowed to calibrate $M_{H,1}^{W}$, and an analogous magnitude based on Gaia
photometry, to better than $0.020$ mag ($0.9\%$ in distance), while
simultaneously solving the sample’s overall parallax offset to $-13\pm
5\,\mu$as ($-19\pm 3\,\mu$as for the larger sample based on Gaia photometry)
(CruzReyes2023, ).

#### Cepheids in the Magellanic Clouds

The Magellanic Clouds host very large Cepheid populations that have provided
the most important proving grounds for testing stellar models thanks to the
ability to observe the Cepheids in great detail at a virtually constant
distance.

LMC Cepheids are particularly valuable due to the extremely accurate distance
determined based on 20 detached eclipsing binaries (cf. Pietrzynski & Graczyk,
this volume; Pietrzynski2019 ). The intrinsic depth of the SMC combined with
the relatively few (8) detached eclipsing binaries that define its distance
(Graczyk2020, ) has thus far limited the use of the SMC as an anchor galaxy.
Photometric observations using HST’s DASH mode have enabled efficient
photometric observations of a significant sample of long-period Cepheids,
despite the large angular separation among LMC Cepheids (Riess2019, ). While
ground-based photometry of LMC Cepheids abounds from several surveys (MACHO, ;
Soszynski2019, ; Macri2015LMC, ; Ripepi2022VMC, ), HST’s photometry provides
unmatched photometric consistency and avoids issues related to Earth’s
atmosphere, notably in the NIR.

Cepheids in the Magellanic Clouds usefully constrain the slope of the
metallicity term, $\gamma$ Romaniello2022 , since they are less metallic than
most Cepheids in the Solar neighborhood, are not heavily extincted, and not
subject to Gaia parallax offsets. However, the internal depth of both galaxies
must be considered for best results, and correcting for the LMC plane geometry
(Ripepi2022VMC, ) reduces the scatter of the observed Cepheid LL. This effect
is even more pronounced in the SMC (Scowcroft2016, ), which also allows to
extend the metallicity lever to the lowest values (Breuval2022, ).

#### Cepheids in NGC 4258

The spiral galaxy NGC 4258 (Messier 106) is a crucial DL anchor thanks to a
water megamaser that orbits its supermassive black hole (Humphreys2013, ), cf.
Braatz & Pesce this volume. Modeling the observed maser emission as a warped
Keplerian disk has allowed to determine the distance to this galaxy to within
$1.5\%$ (Reid2019, ). Importantly for the DL, this measurement shares
virtually no common systematics with Gaia parallaxes, or the modeling of
binary star orbits (apart from Kepler’s laws) that underlies the LMC distance.
The largest population of $669$ Cepheids in NGC 4258 has been observed in the
SH0ES Wesenheit system by Yuan2022 . However, long-period Cepheids in NGC 4258
are sufficiently close to detect them using ground-based, 8m-class telescopes
Shapee+2011 ; Fausnaugh+2015 .

At a distance of $7.6$ Mpc, Cepheids in NGC 4258 are observed under conditions
that resemble the conditions of other galaxies much more than Cepheids in the
MW, or the LMC, notably with regards to the process by which the photometry is
collected (cf. Sec. 2.3), the typical flux range, the blending of sources, and
the metallicity corrections based on HII regions. This makes NGC 4258 the only
anchor galaxy where Cepheids, as well as other SCs, such as the TRGB and
J-region AGB methods (Zgirski2021, ; Madore2022, ), can be calibrated in the
same way as they are used for determining distances. Thus, NGC 4258 allows
very direct apples-to-apples comparisons for determining distances to
individual galaxies, such as M51 (Csoernyei2023M51, ). Additionally, the
consistent absolute LL calibrations determined using MW parallaxes, the LMC
distance, and NGC 4258 underline the accuracy of the measurements in all three
anchor galaxies.

#### Cepheids in SN-host galaxies

The SH0ES DL uses 2150 Cepheids in 37 SN-host galaxies to calibrate the SNIa
luminosity zero-point, and these galaxies are located across a wide range of
distances, from $6.7$ (M101) to $80$ Mpc (NGC 105), with an average distance
of approximately $27$ Mpc. This dataset represents a major time investment,
totaling approximately $1050$ orbits, or the equivalent of $\sim 66$ full days
of continuous HST observations. In the nearest galaxies, several fields can be
monitored to map Cepheids across galaxies, whereas a single frame (WFC3/NIR
footprint is $2.7\times 2.7$ arcmin2) centered on the galactic bulge captures
the majority of the Cepheids in distant galaxies.

Cepheids in SN-hosts are discovered at optical wavelengths where their
amplitudes are largest, $\sim 1$ mag. $V-$ (F555W) and $I-$band (F814W)
observations are de-phased to estimate mean magnitudes, whereas $H-$band
(F160W) observations can usually be used at random phase thanks to the reduced
amplitudes. The combination of $H-$band magnitudes, and $V-I$ colors is then
used to compute the Wesenheit magnitudes. Further information on Cepheid
photometry beyond the Local Group and Wesenheit magnitudes is provided in
Sect. 2.3.

Bona fide Cepheids used for the DL are discovered based on their light
variations and subsequently vetted using several quality criteria, cf.
Yuan2022 . These cuts include amplitudes, amplitude ratios, dispersion of
Cepheid observations around template light curve fits, minimum period of $5$
d, and average color to ensure a matching temperature range. Finally, the
sample is cleaned by $3\sigma$ clipping in all three bands. Impressively,
binning the light curves of many Cepheids in period ranges allows to
illustrate the Hertzsprung progression, which adds further confidence
concerning the equivalence of the distant and nearby Cepheids.

The constant angular resolution of the photometric system corresponds to
increasingly larger physical scales, the more distant the SN host galaxy. This
creates a fairly clean correlation between the degree to which Cepheids are
blended with other sources and the distance at which they are observed.
Variations in surface source density, e.g., due to inclination or the region
of the galaxy observed, further impact the amount of blending. At large source
densities, the term _crowding_ refers to heavy blending in fields with many
overlapping sources.

_Crowding corrections_ are used to remove background light contributions due
to unresolved sources (Ferrarese2000, ). The background due to blended sources
is measured using artificial star tests that locally insert and remeasure the
flux of sources in the vicinity of the target star, and with similar
brightness (Riess2011, ; Riess2016, ). This is a significant effect in many SN
host galaxies (mean of $\sim 0.4$ mag, up to nearly $1\,$mag at the largest
distances), although corrections as small as a few hundredths of a magnitude
are also estimated in low density regions of nearby galaxies. The accuracy of
crowding corrections has been extensively tested (Riess2022H0, ) and recently
been validated using JWST’s higher spatial resolution (Riess2023crowd, ).

_Stellar association bias_ arises because wide binaries and open star clusters
add flux to Cepheids in NGC 4258 and SN-host galaxies, while being resolved
(and hence not contributing) to Cepheid flux measurements used to calibrate
the LL in the LMC and MW (Anderson2018, ). Given a typical physical scale of
clusters being approximately $4$ pc, the artificial star tests used for
crowding corrections cannot correct for this effect, leading to a small,
albeit one-sided bias for the DL because the effect occurs primarily in the
SN-host galaxies. Using M31 as a SN-host analog, Anderson2018  estimated an
effect of $\sim 0.007$ mag per SN-host Cepheid, and a correction of this size
has been included in the SH0ES analysis. A similar estimate has been found in
M33 (Breuval2023, ), and additional work based on M101 and NGC 4258 is ongoing
(Spetsieri et al., Lengen et al., both in prep.). While a better
quantification of this bias will further improve $H_{0}$ accuracy, it is
important to note that stellar association bias cannot resolve the $H_{0}$
tension, since both the requirements of Cepheid discovery (notably
sufficiently large amplitudes and characteristic light curve shapes) and the
quality cuts applied to Cepheid candidates (notably based on color and LL
$\sigma$ clipping) provide very strong protection against significant bias. In
essence, Cepheids blended by very luminous clusters are either not discovered
due to diminished amplitudes, or rejected due to their blue color.

Metallicity corrections in SN-host galaxies can unfortunately not (yet) be
measured directly from Cepheid spectra. Instead, oxygen abundances of Cepheids
in SN-host galaxies are estimated based on their galactocentric location and
metallicity gradients estimated using HII regions (Bresolin2011, ). Care must
be taken to ensure the consistency of the derived oxygen abundances, notably
with regards to the Solar oxygen abundance problem. In that case, a high
degree of agreement between Cepheid oxygen abudances and HII region abundances
has been shown (Riess2022H0, , Appendix C).

_Relativistic effects_ can bias $H_{0}$ measurements based on the DL because
SN-host galaxies are systematically redshifted relative to the anchor galaxies
due to cosmic expansion. For example, time dilation leads to systematically
overestimated Cepheid periods at greater distances, and thus, to overestimated
luminosities and underestimated distances. Correcting the dilated observed
Cepheid periods therefore has the effect of _increasing_ $H_{0}$, by
approximately $0.2\%$ (Anderson2019rlb, ). This effect is already accounted
for in the SH0ES DL. Two additional relativistic effects have been considered
(Anderson2022, ): $K-$corrections and the redshifting of the reddening law on
Wesenheit magnitudes. While both effects are very small when working with NIR
Wesenheit magnitudes, single-band IR observations (e.g., of TRGB stars
observed with JWST) can exceed $1\%$ distance bias at $100$ Mpc and must
therefore be considered as the use of stellar SCs is pushed to ever greater
distances.

###  2.3 Cepheid Photometry in SN-hosts and NGC 4258

The quality of the observed photometry is paramount for accurate luminosity
estimates and $H_{0}$. As such, a few words on the measurement process are in
order.

Point spread function (PSF) photometry remains the state-of-the-art method for
measuring Cepheid light curves, mostly using DAOPHOT and DOLPHOT (DAOPHOT, ;
DOLPHOT, ). PSF photometry treats each star as a point source that can be
distinguished from neighboring sources and background, and it is used for busy
and inhomogeneous scenes, where the simpler aperture photometry approach is
not suitable. An important benefit of PSF photometry over aperature photometry
is its ability to distinguish resolved and unresolved sources, and PSF fitting
produces quantitative information that can be used to study the shape of the
light’s distribution over the detector. However, the key limitation of PSF
photometry is the need to know how light is dispersed by the optical system
either a priori, or by measuring the PSF directly from isolated sources. The
details of how temporal variations in PSF, e.g., due to telescope breathing or
focus changes, affects photometry are challenges where observer experience
frequently outperforms automated procedures.

In many ways, measuring photometry _well_ is an _art_ that requires a
significant time investment to master. Given the high hurdle for participating
and the prevalence of two methods commonly adopted as black boxes, there is
certaintly room for more independent cross-checks. A notable recent study
presented new photometric measurements based on the raw HST frames of NGC 5584
(Javanmardi2021, ). Comparison with the photometry by the SH0ES team revealed
very good agreement. While further cross-checks would be welcome, it is worth
noting that $H_{0}$ bias would primarily arise due to inconsistency among
photometric measurements between near and far galaxies, rather than due to
errors involving the absolute flux scale of all galaxies.

###  2.4 Baade-Wesselink distances

The large scale radial pulsations of Cepheids enable distance determination
using the ratio of the observed angular ($\Delta\Theta$) and linear radius
variations ($\Delta R$), since $D\propto\Delta R/\Delta\Theta$. Such distances
are commonly referred to as Baade-Wesselink distances (Baade1926, ;
Wesselink1946, ), and their development was intricately linked with
establishing pulsations as the cause of Cepheid pulsations (Eddington1917, ;
Lindemann1918, ; Tiercy1927, ). $\Delta\Theta$ is observable for a relatively
small number of MW Cepheids using very long baseline optical interferometry
(Mourard1997, ; Kervella1999, ; Lane2000, ) as well as surface brightness
color relations (SBCRs) (Kervella2004, ), which require only photometric
measurements and are thus applicable to longer distances. The SpectroPhoto-
Interferometry of Pulsating Stars (SPIPS) algorithm performs simultaneous
modeling of interferometry, multi-band photometry, and radial velocity
observations to determine distance using the same concepts (Merand2015SPIPS, ;
Trahin2021, ). $\Delta R=p\cdot\int{v_{r}d\phi}$ is the product of the
integral of the line of sight (radial) velocity variations measured using
optical spectra and the projection factor $p$, which translates the observed
line of sight velocity to the pulsational velocity as measured from the center
of the star. The requirement for optical spectra limits the use of BW
distances to relatively nearby objects, such as Cepheids in the MW and the
Magellanic Clouds (Storm2011pfac, ; Storm2011met, ; Gieren2018, ). However,
the projection factor $p$ has been a limiting factor to he precision
attainable for BW distances due to its complex task of simultaneously
correcting for geometric effects, limb darkening, and dynamical aspects of
stellar atmospheres (Nardetto2004, ; Nardetto2007, ; Anderson2014, ;
Anderson2016c2c, ; Anderson2016vlti, ). For example, despite the availability
of Gaia parallaxes, it has not been possible to determine whether or how $p$
may depend on Cepheid properties, such as pulsation period $P$
(Breitfelder2016, ; Trahin2021, ). While BW distances thus provide a
potentially interesting alternative for measuring individual distances to
Cepheids, they do not currently afford the accuracy required to inform
$H_{0}$.

###  2.5 On the impact of Cepheid multiplicity on $H_{0}$

The majority of Cepheids reside in multiple systems (Evans2015crav, ; Moe2017,
; Kervella2019pm, ; Kervella2019pm2, ), with estimates of the multiplicity
fraction ranging from $60\%$ to more than $100\%$ (many Cepheids occur in
triple or even quadruple systems). Cepheid companions can affect distances
either astrometrically or photometrically.

Astrometric issues falsify the parallax measurements of Cepheids and can arise
either due to real or apparent orbital motion. The former is caused by the
movement around the common center of gravity, whereas apparent orbital motion
is a photometric effect in disguise and caused by the shifting photocenter in
a binary system containing stars of similar magnitude. Timescales can be used
to separate these two effects, with true orbital motion generally being much
longer than the pulsation period, and apparent motion occuring in phase with
the Cepheid’s variability. Poor astrometry thus obviously can affect LL
calibration, albeit on a star-by-star basis, increasing scatter rather than
biasing LL calibration. Thankfully, radial velocity monitoring can very
sensitively estimate whether orbital motion can significantly affect parallax
measurements (Anderson2016rvs, ). Gaia’s astrometric quality flags, such as
RUWE, are also commonly used to identify high quality astrometric solutions.

Photometric distance bias can in principle arise if the calibrated LL does not
correspond to the observed LL due to differences in the nature of the
companion stars, or the prevalence of multiple star systems. However, these
concerns are essentially limited to individual Cepheids (Gallenne2018, ),
which may stand out significantly from observed LLs. For example, abnormally
luminous and red LL outliers in the LMC have been successfully identified as
high-value double-lined spectroscopic binaries (SB2s) (Pilecki2021, ;
Pilecki2022, ). A similar population of SB2 Cepheids in the MW has not yet
been detected, which prompts the interesting question of how much the
companion star population of Cepheids may differ from galaxy to galaxy.
Reasonable assumptions on the distribution of wide binary separations and
contrasts have allowed to constrain a possible DL bias to $<0.004\%$
(Anderson2018, ).

###  2.6 Cepheids in (open) star clusters

In the Milky Way, approximately 10% of Cepheids are associated with
(gravitationally bound to) open clusters (CruzReyes2023, ). This estimate is
based on Gaia (E)DR3 data and considers the 2 kpc surrounding the Sun, where a
rather complete census of both Cepheids and star clusters is currently
feasible. Different galaxies present different clustered Cepheid fractions,
with the LMC having the highest known fraction. In M31, the fraction of
Cepheids in star clusters is much less, approximately 2.5% (Senchyna2015, ;
Anderson2018, ), and M33 has a clustered Cepheid fraction of $3.3\%$
(Breuval2023, ). HST UV photometry of M101 shows that Cepheid clustering is
highly variable across the two fields in which Cepheids have been observed and
that it correlates strongly with star formation (Spetsieri et al. in prep.).

Dynamical simulations (Dinnbier2022, ) explain the low fraction of Cepheids
that occur in clusters as a consequence of cluster dissociation over time.
Cluster dispersal is driven by the violent gas expulsion due to massive star
winds at very early times and the dynamical interactions over longer
timescales, and mostly occurs well before intermediate-mass stars reach the
Cepheid stage. Despite sensitivity to initial conditions, the observed trend
that younger Cepheids cluster more frequently strongly supports the paradigm
that a majority, if not all, stars are born in clusters
(Dinnbier2022formation, ). Cepheid clustering is thus both dependent on global
and local properties of galaxies, as is observed.

#### Circumstellar environments

Circumstellar environments (CSEs) of Cepheids have been detected using mid-IR
photometry (Marengo2010, ; Barmby2011, ) and HI imaging (Matthews2012, ),
near- and mid-IR interferometry (Kervella2006CSE, ; Merand2006CSE, ;
Gallenne2013CSE, ), as IR excess among spectral energy distributions
(Schmidt2015, ; Groenewegen2020MWSED, ; Groenewegen2023SED, ), and using
Spitzer spectroscopy (Hocde2020, ). Based on the IR excesses, it appears that
$H-$band is just short of the region where IR excess becomes most noticeable,
and that only a fraction of Cepheids exhibit significant mid-IR excess. At
present, it is unclear whether and how CSEs can bias Cepheid distances.
However, the following preliminary conclusions appear reasonably secure: a)
CSEs can lead to DL bias only if the nature of CSEs differs systematically
among anchor and SN-host galaxies — in this regard, CSE distance bias behaves
like photometric bias due to companions; b) CSEs may lead individual Cepheids
to be outliers from LLs; c) long-wavelength observations, such as JWST
NIRCAM/F277W and F444W are likely to be more sensitive to CSEs than $H-$band
and shorter wavelength observations; d) if chemical composition significantly
affects the presence of CSEs, then metallicity corrections in use today likely
already correct the majority of the effect induced by CSEs.

###  2.7 Stellar evolution models

Cepheids pose many interesting challenges to stellar evolution theory,
starting from the occurrence of blue loops during core He burning, to observed
rates of period change, to predicting the correct instability strip
boundaries, to predicting the right luminosity (mass) range where Cepheids
occur, to the mass-luminosity relation and its dependence on metallicity,
convection, and rotation, and the ability to predict Cepheid variability,
among others. Stellar models thus benefit from a large number of observational
constraints that allow to understand the physical nature of these important
stars and to rigorously test model predictions along several axes at once.
With thousands of Cepheids in dozens of galaxies, the DL offers unprecedented
means for comparisons, albeit at a reduced level of detail at greater
distances. However, given their systematic uncertainties, stellar models have
limited predictive power for Cepheid distances.

Figure 2: The slope of the metallicity effect on Wesenheit magnitudes
determined by different methods (bottom x-axis) and the impact of metallicity
on $H_{0}$ (top x-axis). Circles show results based on different galaxies,
squares results based on galactic abundance gradients, diamonds results based
on MW Cepheids observed with Gaia, downward triangles results based on BW-type
distances, and asterisks results based on stellar evolution models. Figure
courtesy of Louise Breuval.

Areas where stellar models _can_ usefully inform the DL include, among others,
a) the linearity of the LL (cf. above); b) the dispersion of the LL observed
in different passbands; c) differential comparisons of Cepheid luminosity
under otherwise identical conditions as required, e.g., to determine the
effects of chemical composition on the LL. For example, predictions based on
Geneva stellar evolution models (Anderson2016rot, ) predict a negative sign
for the metallicity term, $\gamma$, both in the individual $V-$ and $H-$band
PL-relations and in the optical $W_{\mathrm{VI}}$ and near-IR Wesenheit
systems $W_{\mathrm{H,VI}}$ used in the SH0ES analysis. Conversely, a widely
used set of Cepheid models (Marconi2005, ; Fiorentino2013, ; DeSomma2021PA, ;
DeSomma2022, ) predicts a positive sign for PL-relations in the individual
passbands ($\gamma_{\lambda}>0$ in those cases) and negatively signed period-
Wesenheit relations, that is, when combining a single photometric passband
with a color ($\gamma_{W}<0$). Recent empirical work by different teams has
shown the metallicity effect to have negative sign across several individual
photometric bands as well as in the Wesenheit formulations (Gieren2018, ;
Wielgorski2017, ; Breuval2022, ; Ripepi2022, ; Trentin2022, ; Bhardwaj2023, ),
cf. Fig. 2. Hence, it appears that the disagreement between the two sets of
models is due to different predicted locations of the instability strip
boundaries, as well as their dependence on metallicity.

##  3 Conclusions

Classical Cepheids provide the base calibration for the $H_{0}$ measurement
using the extragalactic DL. They benefit from more than $110$ years of study
for measuring distances. Recent progress in quantifying systematics has
centered on sub-percent effects for $H_{0}$, which ended up sharpening the
accuracy of the measurement and the significance of the Hubble tension. In the
process, corrections that increase and decrease the value of $H_{0}$ have been
identified. Gaia has revolutionized the calibration of the LL in the MW based
on trigonometric parallaxes, and JWST stands to significantly improve $H_{0}$
precision by better sensitivity and reduced LL dispersion in SN-host galaxies
due to lower background (enabled by the $4\times$ better spatial resolution
than HST’s WFC3/IR).

Further improvements of the distance ladder and its experimental setup will
yield significant improvements in $H_{0}$ accuracy, hopefully to deliver the
desired $1\%$ direct measurement. Further improvements to Gaia’s parallax
systematics are paramount to achieving this level of accuracy. However,
despite very significant efforts to identify systematics that could bias
$H_{0}$ sufficiently to explain the $H_{0}$ constant tension, no such biases
have been identified. For this reason, the author is cautiously optimistic
that the problem with the $H_{0}$ constant will eventually enable new
interesting insights into the physics governing our Universe.

###### Acknowledgements.

RIA is funded by the SNSF through an Eccellenza Professorial Fellowship, grant
number PCEFP2_194638 and acknowledges support from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 947660).

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