.70 | $\mathbf{.858~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$1.3910-201$})}$ | $.183~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$2.0210-8$})$ | $.191~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$2.1910-11$})$ | $.160~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$7.5110-6$})$ | $.173~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$9.4710-10$})$ | $.149~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$1.3110-6$})$ | $.158~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$8.2110-6$})$
.80 | $\mathbf{.887~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$3.4810-145$})}$ | $.271~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$6.7010-11$})$ | $.032~{}($1.0210-1$)$ | $.203~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$2.1710-6$})$ | $.181~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$2.2210-8$})$ | $.209~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$4.3010-9$})$ | $.218~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$4.3010-7$})$
.90 | $\mathbf{.887~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$3.4810-145$})}$ | $.352~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$2.7110-9$})$ | $.033~{}($1.5710-1$)$ | $.320~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$1.8910-8$})$ | $.194~{}($1.4710-4$)$ | $.179~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$2.4210-5$})$ | $.253~{}($8.0510-5$)$
$\hat{\tau}_{\mathtt{RD}}$ | .10 | $-.272~{}($1.2910-1$)$ | $-.215~{}($4.6910-1$)$ | $\mathbf{.059~{}($7.3810-1$)}$ | $-.190~{}($9.4010-2$)$ | $-.123~{}($4.6810-1$)$ | $-.312~{}($1.2210-2$)$ | $-.034~{}($3.3610-1$)$
.20 | $-.558~{}($1.2210-2$)$ | $-.013~{}($7.7010-1$)$ | $\mathbf{.109~{}($5.3710-1$)}$ | $-.012~{}($9.3410-1$)$ | $-.162~{}($1.1910-1$)$ | $-.161~{}($1.4310-1$)$ | $.042~{}($6.5210-1$)$
.30 | $.073~{}($6.7110-1$)$ | $-.058~{}($5.4810-1$)$ | $-.067~{}($6.3810-1$)$ | $.103~{}($1.9010-1$)$ | $\mathbf{.726~{}($3.6310-4$)}$ | $-.473~{}($8.9510-3$)$ | $-.273~{}($4.3510-3$)$
.40 | $\mathbf{.322~{}($1.0310-1$)}$ | $.027~{}($7.6910-1$)$ | $-.022~{}($8.4410-1$)$ | $.092~{}($1.0610-1$)$ | $-.153~{}($2.1310-1$)$ | $-.069~{}($5.4210-1$)$ | $.055~{}($5.1510-1$)$
.50 | $\mathbf{.354~{}($1.3810-1$)}$ | $.058~{}($6.2010-1$)$ | $.144~{}($2.3810-1$)$ | $.010~{}($8.8310-1$)$ | $.157~{}($1.1610-1$)$ | $.125~{}($1.2510-1$)$ | $-.018~{}($8.0610-1$)$
.60 | $\mathbf{.465~{}($2.1210-1$)}$ | $.093~{}($4.6310-1$)$ | $.041~{}($7.6710-1$)$ | $.014~{}($8.4710-1$)$ | $.090~{}($2.7210-1$)$ | $.155~{}($3.7010-2$)$ | $-.171~{}($1.6010-1$)$
.70 | $.357~{}($3.8210-1$)$ | $-.180~{}($1.3510-1$)$ | $\mathbf{.638~{}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$5.7710-5$})}$ | $-.064~{}($4.2910-1$)$ | $-.044~{}($6.1510-1$)$ | $.073~{}($3.5210-1$)$ | $-.114~{}($5.6410-1$)$
.80 | $-$ | $.137~{}($3.6010-1$)$ | $.072~{}($6.9310-1$)$ | $-.152~{}($3.2110-1$)$ | $.176~{}($3.3310-2$)$ | $.077~{}($3.2810-1$)$ | $\mathbf{.278~{}($1.3610-1$)}$
.90 | $-$ | $.065~{}($7.9110-1$)$ | $-.222~{}($1.9410-1$)$ | $-.023~{}($9.0110-1$)$ | $.032~{}($8.4410-1$)$ | $\mathbf{.167~{}($1.0610-1$)}$ | $.008~{}($9.5410-1$)$
Accuracy | .00 | $\mathbf{.871}$ | $\mathbf{.871}$ | $\mathbf{.871}$ | $\mathbf{.871}$ | $\mathbf{.871}$ | $\mathbf{.871}$ | $\mathbf{.871}$
.10 | $.879$ | $.880$ | $.864$ | $.883$ | $\mathbf{.884}$ | $.883$ | $.877$
.20 | $\mathbf{.892}$ | $.889$ | $.850$ | $.889$ | $.891$ | $.891$ | $.877$
.30 | $\mathbf{.897}$ | $.893$ | $.838$ | $.893$ | $.891$ | $.891$ | $.877$
.40 | $.877$ | $.894$ | $.836$ | $.890$ | $.900$ | $\mathbf{.902}$ | $.886$
.50 | $.842$ | $.892$ | $.814$ | $.886$ | $.892$ | $\mathbf{.896}$ | $.887$
.60 | $.789$ | $.890$ | $.783$ | $.885$ | $\mathbf{.897}$ | $.885$ | $\mathbf{.897}$
.70 | $.726$ | $\mathbf{.900}$ | $.730$ | $.887$ | $.887$ | $.871$ | $.892$
.80 | $.632$ | $\mathbf{.899}$ | $.687$ | $.884$ | $.869$ | $.870$ | $.891$
.90 | $.632$ | $\mathbf{.885}$ | $.685$ | $.876$ | $.852$ | $.850$ | $.872$
1.00 | $.512$ | $.842$ | $.682$ | $.838$ | $.838$ | $.832$ | $\mathbf{.848}$

###  D.3 How to validate estimates under Scenario 2

When considering Scenario 2, we require Assumption 1 to hold. Here, we provide
a few sanity checks that can be implemented to validate the estimates (see
Appendix C for a detailed description).

####  D.3.1 Placebo Tests

##### Placebo cutoff test setup.

We consider the same setup presented in Section 2, except for the following:

  1. 1.

we estimate two different cutoff values $\overline{\kappa}_{c,L}$ and
$\overline{\kappa}_{c,H}$

     * •

$\overline{\kappa}_{c,L}$ is obtained by considering the 75-th percentile on
the instances with a reject score below $\overline{\kappa}_{c}$,

     * •

$\overline{\kappa}_{c,H}$ is estimated by considering the 25-th percentile of
the reject scores above $\overline{\kappa}_{c}$;

  2. 2.

we run a local kernel polynomial regression to estimate
$\hat{\tau}_{\mathtt{RD}}$, substituting the true cutoff value
$\overline{\kappa}_{c}$ with both $\overline{\kappa}_{c,L}$ and
$\overline{\kappa}_{c,H}$.

##### Placebo outcome test setup.

We consider the same setup presented in Section 2, except for the following:

  1. 1.

we sample a placebo outcome $\tilde{T}$, such that
$\tilde{T}\sim\mathtt{Bernoulli}(p)$ and $p=.5$;

  2. 2.

we run the same local kernel polynomial regression used to estimate
$\hat{\tau}_{\mathtt{RD}}$ substituting the target variable $T$ with
$\tilde{T}$.

##### Results

Figure 6: Estimated $\hat{\tau}_{\mathtt{RD}}$ for the best baselines over the
five datasets. The first row provides the estimates of
$\hat{\tau}_{\mathtt{RD}}$. The second row provides the estimates for the
lower cutoff placebo test. The third row provides the estimates for the higher
cutoff placebo test. The fourth row provides the estimates for the placebo
outcome test.

Figure 6 provides the results for the placebo tests above. The first row shows
the original estimates for $\hat{\tau}_{\mathtt{RD}}$ under Scenario 2 for the
best baselines.

Regarding synth, all the coefficients of the placebo tests are not
significant, with the sole exception of the highest coverage $c=.90$ when
considering the highest cutoff (third row of Figure 6).

When considering cifar10h, hatespeech, and xray-airspace, all the coefficients
for all the placebo tests are not significant. This suggests that our
estimated $\hat{\tau}_{\mathtt{RD}}$ should be robust.

On the other hand, when looking at galaxyzoo, we see statistically significant
estimates for the fake cutoff placebo estimates. This means we should take
with a grain of salt our estimated $\hat{\tau}_{\mathtt{RD}}$, as assumptions
could be violated.

We can exploit other tests, such as the one detailed in the next subsection,
to understand why the placebo tests fail.

####  D.3.2 Density Estimation

##### Density estimation test setup.

We consider the same setup presented in Section 2, and we validate
$\hat{\tau}_{\mathtt{RD}}$ estimates by estimating the empirical density of
reject scores. This is done separately for instances below the cutoff and
above the cutoff, using the R package rddensity [101] with default parameters.
This allows us to: $(i)$ statistically assess the continuity of estimated
reject score densities around the cutoff using a permutation test (null
hypothesis stands for continuity at the cutoff); $(ii)$ visualize whether
there is a discontinuity in the estimated reject score densities around the
cutoffs.

##### Results

Figures 7-11 provide the density estimation plots and the permutation tests
p-values (high values mean we can’t falsify the assumption) for the best
baseline on all the datasets. If the running variable is not manipulable, we
would expect to see no difference in the estimated densities from both sides
of the cutoff.

We can see that for the majority of cutoffs, the estimated densities are
close, thus not falsifying Assumption 1. The only exception is galaxyzoo
(Figure 9), where most tests reject the null hypothesis. The main reason for
this behavior is that the reject score density peaks around one value, with
little variation in the reject scores. Accordingly, small changes in the
reject score imply abrupt changes in predictive accuracy. This sheds light on
a potential criticality of the deferring system, i.e., the reject score
estimation. Moreover, this behavior also explains why most placebo cutoff
tests failed for galaxyzoo.

Regarding synth, the null hypothesis is rejected only for $c=.10$ and $c=.90$.
Once again, this is because the reject scores peak at $-1$ and $1$,
potentially harming the quality of the estimates. We see similar trends for
cifar10h (Figure 8), where the p-values are significant only at $c=.10$, and
xray-airspace, where values are significant for $c>.70$ and around the
accuracy maximizing cutoff value, i.e., .$\overline{\kappa}_{.50}$ and
$\overline{\kappa}$.

Regarding hatespeech, we see no statistically significant p-values, suggesting
the validity of $\hat{\tau}_{\mathtt{RD}}$ across all cutoff values (see
Figure 10).

(a) $\overline{\kappa}_{.10}$.

(b) $\overline{\kappa}_{.20}$.

(c) $\overline{\kappa}_{.30}$.

(d) $\overline{\kappa}_{.40}$.

(e) $\overline{\kappa}_{.50}$.

(f) $\overline{\kappa}_{.60}$.

(g) $\overline{\kappa}_{.70}$.

(h) $\overline{\kappa}_{.80}$.

(i) $\overline{\kappa}_{.90}$.

Figure 7:  synth estimated best baseline (ASM) reject scores densities at the
left and right of cutoff $\overline{\kappa}_{c}$. All the plots are zoomed
around the cutoff values.

(a) $\overline{\kappa}_{.10}$.

(b) $\overline{\kappa}_{.20}$.

(c) $\overline{\kappa}_{.30}$.

(d) $\overline{\kappa}_{.40}$.

(e) $\overline{\kappa}_{.50}$.

(f) $\overline{\kappa}_{.60}$.

(g) $\overline{\kappa}_{.70}$.

(h) $\overline{\kappa}_{.80}$.

(i) $\overline{\kappa}_{.90}$.

Figure 8:  cifar10h estimated best baseline (CC) reject scores densities at
the left and right of cutoff $\overline{\kappa}_{c}$. All the plots are zoomed
around the cutoff values.

(a) $\overline{\kappa}_{.10}$.

(b) $\overline{\kappa}_{.20}$.

(c) $\overline{\kappa}_{.30}$.

(d) $\overline{\kappa}_{.40}$.

(e) $\overline{\kappa}_{.50}$.

(f) $\overline{\kappa}_{.60}$.

(g) $\overline{\kappa}_{.70}$.

(h) $\overline{\kappa}_{.80}$.

(i) $\overline{\kappa}_{.90}$.

Figure 9:  galaxyzoo estimated reject scores densities at the left and right
of cutoff $\overline{\kappa}_{c}$ for the best baseline CC. All the plots are
zoomed around the cutoff values.

(a) $\overline{\kappa}_{.10}$.

(b) $\overline{\kappa}_{.20}$.

(c) $\overline{\kappa}_{.30}$.

(d) $\overline{\kappa}_{.40}$.

(e) $\overline{\kappa}_{.50}$.

(f) $\overline{\kappa}_{.60}$.

(g) $\overline{\kappa}_{.70}$.

(h) $\overline{\kappa}_{.80}$.

(i) $\overline{\kappa}_{.90}$.

Figure 10:  hatespeech estimated best baseline (RS) reject scores densities at
the left and right of cutoff $\overline{\kappa}_{c}$. All the plots are zoomed
around the cutoff values.

(a) $\overline{\kappa}_{.10}$.

(b) $\overline{\kappa}_{.20}$.

(c) $\overline{\kappa}_{.30}$.

(d) $\overline{\kappa}_{.40}$.

(e) $\overline{\kappa}_{.50}$.

(f) $\overline{\kappa}_{.60}$.

(g) $\overline{\kappa}_{.70}$.

(h) $\overline{\kappa}_{.80}$.

(i) $\overline{\kappa}_{.90}$.

Figure 11:  xray-airspace estimated best baseline (RS) reject scores densities
at the left and right of cutoff $\overline{\kappa}_{c}$. All the plots are
zoomed around the cutoff values.


# A Preliminary Calibration of the JAGB Method Using Gaia EDR3

Abigail J. Lee  Department of Astronomy & Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637  Kavli Institute for Cosmological
Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
Wendy L. Freedman  Department of Astronomy & Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL 60637  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL 60637  Barry F. Madore  Observatories of the Carnegie Institution for
Science 813 Santa Barbara Street, Pasadena, CA 91101  Department of Astronomy
& Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL
60637  Kayla A. Owens  Department of Astronomy & Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL 60637  In Sung Jang  Department
of Astronomy & Astrophysics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL 60637  Kavli Institute for Cosmological Physics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL 60637 Abigail J. Lee
<EMAIL_ADDRESS>
###### Abstract

The recently-developed J-region Asymptotic Giant Branch (JAGB) method has
extraordinary potential as an extragalactic standard candle, capable of
calibrating the absolute magnitudes of locally-accessible Type Ia supernovae,
thereby leading to an independent determination of the Hubble constant. Using
Gaia Early Data Release 3 (EDR3) parallaxes, we calibrate the zeropoint of the
JAGB method, based on the mean luminosity of a color-selected subset of
carbon-rich AGB stars. We identify Galactic carbon stars from the literature
and use their near-infrared photometry and Gaia EDR3 parallaxes to measure
their absolute $J$-band magnitudes. Based on these Milky Way parallaxes we
determine the zeropoint of the JAGB method to be $M_{J}=-6.14\pm 0.05$ (stat)
$\pm$ 0.11 (sys) mag. This Galactic calibration serves as a consistency check
on the JAGB zeropoint, agreeing well with previously-published, independent
JAGB calibrations based on geometric, Detached-Eclipsing Binary (DEB)
distances to the LMC and SMC. However, the JAGB stars used in this study
suffer from the high parallax uncertainties that afflict the bright and red
stars in EDR3, so we are not able to attain the higher precision of previous
calibrations, and ultimately will rely on future improved DR4 and DR5
releases.

Parallax (1197), Distance indicators (394), Stellar distance (1595),
Observational cosmology (1146), Asymptotic giant branch stars (2100), Carbon
stars (199), Milky Way Galaxy (1054), Hubble constant (758)

††software: Astropy (Astropy Collaboration et al., 2013, 2018), TOPCAT
(Taylor, 2005), NumPy (van der Walt et al., 2011), Matplotlib (Hunter, 2007),
Pandas (McKinney, 2010), HessDiagram (https://github.com/rwk506/HessDiagram/)

##  1 Introduction

Measuring the Hubble constant ($H_{0}$) precisely and accurately has remained
a challenging endeavor for observational cosmologists. $H_{0}$ parameterizes
the current expansion rate of the universe, and together with measurements of
the cosmic microwave background radiation, it provides a powerful constraint
on the standard model of cosmology (e.g., Freedman & Madore, 2010; Planck
Collaboration et al., 2020). Because of its significance, the value of $H_{0}$
has undergone intense scrutiny and re-determination over the past few decades.
Currently, early universe measurements of $H_{0}$ (forward modeled to the
present day) as inferred from the measurements of the power spectra of
temperature and polarization anisotropies of the Cosmic Microwave Background
(CMB) appear to differ by $4-6\sigma$ from measurements directly measured by
late-time universe distance ladders (Verde et al., 2019). Even more
perplexingly, the two most accurate and precise late-time universe values of
$H_{0}$ measured by the SHoES Group via the Leavitt Law (Riess et al., 2016;
Riess et al., 2019, 2021) and by the Carnegie-Chicago Hubble Program (CCHP)
via the Tip of the Red Giant Branch (TRGB) method (Freedman et al., 2019,
2020; Freedman, 2021) differ by about 2-$\sigma$. Clearly, the local universe
tension should be reconciled before concluding unambiguously that new physics
is required to bridge the gap between early- and late-universe measurements of
$H_{0}$.

Recently, a new independent distance indicator was proposed as an alternative
Type Ia supernovae calibrator to Cepheids and the TRGB in order to act as a
‘tie-breaker,’ especially for more distant galaxies where the agreement
between the two standard methods begins to break down, and where hidden
systematic uncertainties may be revealed. This new method, the J-region
Asymptotic Giant Branch (JAGB) method, was introduced by Madore & Freedman
(2020), further applied and tested by Freedman & Madore (2020), and also
independently developed by Ripoche et al. (2020).

JAGB stars are thermally-pulsating asymptotic giant branch (TP-AGB) stars that
are undergoing dredge-up events: instabilities in the two energy-producing
shells in these stars induce episodes of upper-envelope convection that
eventually penetrate deep into the helium-fusing shell that can result in
freshly-produced carbon being brought to the surface. The carbon then gives
these stars a much redder appearance than their bluer oxygen-rich counterparts
(Habing & Olofsson, 2003). In order to distinguish this region of C-rich stars
from the O-rich AGB stars to the blue and the extreme carbon stars to the red,
Nikolaev & Weinberg (2000) defined color limits of $1.40<(J-K_{s})<2.00$ mag
for this class of stars they labeled as ‘Region J’ in a $K$ versus $(J-K_{s})$
color-magnitude diagram. The JAGB stars are also bounded in luminosity: JAGB
stars have an upper magnitude limit as more luminous (and thus more massive)
stars undergo “Hot-Bottom Burning,” where as the carbon is being transported
to the surface of the star during dredge-ups, the star is so hot the carbon is
converted to nitrogen before it can reach the atmosphere. There also exists a
lower magnitude limit because the dredge-up events for fainter (and thus less
massive) stars are ineffective in transporting carbon to the surface.
Therefore, JAGB stars have well defined color and magnitude limits and can be
easily photometrically identified.

Over 15 years ago, the mean absolute I-band magnitude of carbon stars was
shown to be remarkably stable from galaxy to galaxy and was therefore proposed
as a standard candle by Battinelli & Demers (2005). Recently, Lee et al.
(2021) compared distances determined by the JAGB method, Cepheid Leavitt Law,
and TRGB in the local group galaxy Wolf-Lundmark-Melotte, finding agreement at
the 3% level amongst the three methods. Lee et al. (2021) also found that the
JAGB method had comparable precision to the Cepheid Leavitt law and TRGB,
indicating enormous potential for its own future, independent $H_{0}$
measurement.

Madore & Freedman (2020) were the first to calibrate the JAGB method in the
LMC and SMC via Detached-Eclipsing Binary (DEB) distances (Pietrzyński et al.,
2019; Graczyk et al., 2014). In this paper, we provide an independent
consistency check on that calibration using the Gaia EDR3 parallaxes. Gaia is
a European Space Agency satellite that recently provided accurate astrometry
for over 1.8 billion nearby sources in their Early Data Release 3 (EDR3) (Gaia
Collaboration et al., 2016, 2021). However, limits in the accuracy of the Gaia
parallaxes are still present, especially for bright and red sources, which we
discuss in Section 4. Hence, our Gaia calibration remains preliminary only,
and we expect that it will be superseded by future Gaia releases.

The outline of this paper is as follows: in Section 2, we discuss the
photometry of our sample of Galactic JAGB stars. In Section 3, we give an
overview of the Gaia distance measurements used in this study. In Section 4,
we present an absolute calibration of the JAGB Method zeropoint via Gaia
parallaxes, and discuss the limitations of the Gaia parallaxes for bright and
red stars. In Section 5, we compare our absolute calibration with previous
calibrations. Finally, in Section 6, we present a summary of this paper and
the future direction of the JAGB method.

##  2 Photometry

In this section, we describe the Milky Way carbon stars used in this study. We
identified two robust catalogs of Galactic carbon stars from Whitelock et al.
(2006) and Chen & Yang (2012).

In Appendix A, we also describe our attempts to identify JAGB stars solely on
the basis of their color from the 2MASS Point-Source Catalog (Skrutskie et
al., 2006). Unfortunately, stars in the 2MASS catalog with $m_{J}<5.5$ mag, a
magnitude limit that many Galactic JAGB stars are brighter than, have large
photometric uncertainties on the order of $0.25$ mag (Skrutskie et al., 2006).
Gaia also reported potential large systematic parallax uncertainties for both
bright and red stars, of which JAGB stars are both (Lindegren et al., 2021).
In conclusion, we found the large photometric and parallax uncertainties of
bright JAGB stars made it difficult to discern them solely on the basis of
color.

###  2.1 The Milky Way Carbon Star Sample

Whitelock et al. (2006) identified 239111We chose to exclude the 18 stars in
their list which were labeled as peculiar or unlikely bona fide carbon stars.
Galactic carbon-rich variable stars from the IRAS Point Source Catalogue
(Beichman et al., 1988), the Aaronson et al. (1989) carbon star catalog, and
their own observations of C stars, for which they measured $JHKL$ photometry,
defined on the South African Astronomical Observatory (SAAO) magnitude system
(Carter, 1990), which we converted to the 2MASS magnitude system using
transformations from Koen et al. (2007). These carbon stars were selected on
the basis of their color, $K$-band variability, carbon star SiC emission
feature at 11.2 $\mu$m, 25-to-12 $\mu$m flux ratio, and 12 $\mu$m flux. The
goal of their study was to identify Galactic carbon stars with accurate
distances and radial velocities in order to gain insight into the ages and
masses of the local carbon star population.222We removed 32 stars from the
Whitelock et al. (2006) catalog that were missing $J$-band photometry.

Chen & Yang (2012) compiled a sample of 972333There were 2 objects, IRAS
03291+4116 and IRAS 20204+2914, for which there were duplicate entries.
However, neither star resided in the JAGB star color range so they ended up
being inconsequential to the final calculation. Galactic carbon stars from the
literature, with 2MASS $JHK_{s}$ photometry. These stars were selected by
their color, carbon star SiC emission feature at 11.2 $\mu$m, flux ratios, and
HCN + $\textrm{C}_{2}\textrm{H}_{2}$ feature in absorption spectra at 3.1
$\mu$m. The goal of this study was to investigate carbon-rich AGB stars in the
late stages of evolution.

Both Chen & Yang (2012) and Whitelock et al. (2006) applied Galactic
interstellar extinction corrections to their photometry, using the Drimmel et
al. (2003) three-dimensional Galactic extinction model, and following the
reddening law from Glass (1999).

##  3 Gaia EDR3 Distances

To obtain distance estimates for each source, we chose not to perform a simple
cone search between our carbon star catalog and the Gaia EDR3 catalog, as
generally the most probable match is often not necessarily the nearest
neighbor. Instead, the Gaia team has already matched the Gaia EDR3 catalog
with the 2MASS catalog, calculating the most probable neighbor based on
positional and source density properties (Marrese et al., 2017, 2019). This
catalog is publicly available444Find the documentation at
https://gea.esac.esa.int/archive/documentation/GEDR3/Gaia_archive/chap_datamodel/sec_dm_crossmatches/ssec_dm_tmass_psc_xsc_best_neighbour.html
and can be queried at https://gaia.aip.de/query/. To find each carbon star’s
corresponding 2MASS identifier, we located its Simbad entry using its
R.A./Dec. coordinates and name. Every star in the Whitelock et al. (2006)
catalog had a corresponding 2MASS identifier. 14/972 stars in the Chen & Yang
(2012) catalog lacked 2MASS identifiers, and were removed.

Using the tmass_psc_xsc_best_neighbour catalog, we then determined the unique
Gaia EDR3 source id for each star from its corresponding 2MASS identifier.
There were several sources in both catalogs that lacked Gaia counterparts: two
in the Whitelock et al. (2006) catalog and 85 in the Chen & Yang (2012)
catalog. This may occur because as carbon stars are moderately dusty, they may
be too faint in the optical for Gaia to detect. This is one reason why it was
beneficial to use this procedure instead of a simple cone search; while a cone
search will always result in a nearest neighbor, utilizing the Gaia team’s
best-neighbor match catalog revealed whether a given source had a genuine Gaia
counterpart.

We then combined the two catalogs, finding 108 sources in common between them.
This brought the total number of unique sources in our list to 970. If a
source overlapped in both lists, we averaged its flux.

Then, using each star’s Gaia EDR3 source id, we acquired a distance to each
star from the Bailer-Jones et al. (2021) catalog555Publicly available at
https://dc.zah.uni-heidelberg.de/gedr3dist/q/cone/form of stellar distances,
which was derived from Gaia EDR3 information. 30/970 stars did not have
Bailer-Jones et al. (2021) distances available, and were removed. The catalog
gives two types of distance estimates for a given source: the geometric and
photogeometric distance. The geometric distances were calculated from the
star’s parallax, parallax uncertainty, and Galactic latitude and longitude.
The photogeometric distances were calculated from the aforementioned
parameters, as well as the star’s apparent $G$-band magnitude and $BP-RP$
color. Bailer-Jones et al. (2021) noted their model for calculating
photogeometric distances failed for stars at the bluest and reddest ends of
the color range [ ($BP-RP$) $\lesssim 0$ or ($BP-RP$) $\gtrsim 4$ mag,
respectively; see Figures 4 and 5 of their paper]. As carbon stars are
extremely red stars, we found a fraction (7%) of our carbon star sample had
($BP-RP>4$) mag, and thus lacked photogeometric distances. Therefore, we opted
to use the geometric distances.

A common way of selecting sources with reliable astrometry from Gaia is to
only use parallaxes much larger than their estimated uncertainties (Fabricius
et al., 2021). We eliminated sources with $\sigma_{\pi}/\pi>0.20$, where $\pi$
is the measured parallax by Gaia and $\sigma_{\pi}$ is the uncertainty on the
parallax. The Gaia team also recommends cutting on the parameter ruwe, the
primary indicator for the quality of a given astrometric solution (Fabricius
et al., 2021). We used a cut of ruwe$<2.0$, determined by Maíz Apellániz et
al. (2021) to be a likely safe cut for determining high-quality parallaxes.
After making these two data cuts, 585 carbon stars remained in our final
catalog. We have made this catalog publicly available and show a sample of it
in Table 1. Finally, we calculated the absolute $J$-band magnitude $M_{J}$ for
each star, using its apparent $J$-band magnitude and distance from the Bailer-
Jones et al. (2021) catalog.

Table 1: Carbon star catalog

Gaia EDR3 source_id | 2MASS ID |  $J$ [mag] |  $(J-K_{s})$ [mag] | Parallax [mas] | DistanceaaGeometric distance from Bailer-Jones et al. (2021) [pc]
---|---|---|---|---|---
5118511817421484544 | $02291531-2605559$ | 4.074 | 2.536 | 1.807 | 540.0
3314149224049447040 | $04312193+1739103$ | 6.337 | 2.715 | 0.671 | 1454.6
2901319097862449536 | $05434305-3223287$ | 9.402 | 3.573 | 0.571 | 1573.3
3400017749285334784 | $05453941+2041420$ | 2.031 | 1.705 | 1.501 | 649.7
3340248194120411776 | $05453669+1216152$ | 6.277 | 2.261 | 0.318 | 2728.1
3373737664738495744 | $06114782+1908200$ | 8.225 | 3.342 | 0.577 | 1536.4
3325351357754529152 | $06215800+0720576$ | 4.413 | 2.068 | 0.662 | 1442.4

Note. — Table 1 is published in its entirety in the machine-readable format. A
portion is shown here for guidance regarding its form and content.

##  4 Absolute Calibration of the JAGB Zero point

Figure 1: Color Magnitude Diagram (left panel) and GLOESS smoothed $J$-band
luminosity function (right panel) for our sample of Milky Way carbon stars.
The smoothed luminosity function is only shown for illustrative purposes and
was not used in any calculations. The JAGB sample of stars between
$1.40<(J-K_{s})<2.00$ mag is clearly defined. The extreme carbon stars, which
cannot be used as standard candles, are excluded with our red color limit of
$(J-K_{s})=2.0$ mag.

We present our final $J$ vs. $(J-K_{s})$ color magnitude diagram and $J$-band
luminosity function for our sample of 585 carbon stars in Figure 1. While this
is a carbon star sample, not all carbon stars are JAGB stars. Extreme carbon
stars with $(J-K_{s})>2.0$ mag do not have a constant J magnitude, but
decrease in magnitude with increasing $(J-K_{s})$ color. These extreme carbon
stars are heavily affected by dust obscuration and strong winds (Boyer et al.,
2011), and have been excluded from our JAGB star sample after making the color
cuts of $1.40<(J-K_{s})<2.00$ mag following Nikolaev & Weinberg (2000). Based
on the 153 remaining JAGB stars, we measured the median absolute $J$-band
magnitude for the JAGB stars to be $M_{J}=-6.14$ mag. We chose not to utilize
the mean as has been used in previous papers (Madore & Freedman, 2020;
Freedman & Madore, 2020; Lee et al., 2021), as there were three outliers $>-4$
mag and $<-8$ mag that significantly biased the mean $M_{J}$ to $-6.03$ mag
(eliminating these three outliers gave a mean $M_{J}=-6.15$ mag). We plan to
explore the merits of using the mean, the median, and other methods for
determining the JAGB magnitude in a future program aimed at obtaining high-
precision $JHK$ photometry for several nearby galaxies. Finally, we discuss
the statistical and systematic uncertainties associated with this measurement
in Section 4.1 below.

###  4.1 Gaia Parallax Uncertainties for Bright and Red Stars

Weinberg & Nikolaev (2001) were the first to utilize carbon stars as a
distance indicator in the broad-band near-infrared filters. For their ‘Region
J’ stars, they measured the scatter to be $\pm 0.38$ mag about their $K_{s}$
vs. ($J-K_{s}$) relation, using single-epoch data. In comparison, the observed
scatter on our JAGB star sample’s median $M_{J}$ (excluding the three outliers
$>-4$ mag and $<-8$ mag) was measured to be $\pm 0.60$ mag, yielding an error
on the mean of $0.60/\sqrt{150}=0.05$ mag, which we adopt as a statistical
uncertainty on our measurement.

We note that the scatter in the observed luminosity function must include
contributions from the intrinsic variability of the JAGB stars ($\sim 0.2$
mag), the variance due to extinction, parallax uncertainties, and photometric
uncertainties (Weinberg & Nikolaev, 2001). The Whitelock et al. (2006)
$J$-band photometry has a precision better than $\pm 0.03$ mag and the median
photometric uncertainty of the Chen & Yang (2012) $J$-band photometry is $\pm
0.026$ mag, indicating that the photometric uncertainties contribute little to
the observed scatter.

Next, we estimated the uncertainty introduced from the extinction corrections
applied to our photometry. Whitelock et al. (2006) provided $A_{V}$ values
determined by their models for 144 C-rich Miras. We found that 17 of these
Miras were also JAGB stars and passed our data cleaning cuts (ruwe>2.0 and
$\sigma_{\pi}/\pi<0.20$), with a median extinction value in the $J$ band of
$A_{J}=0.16$ mag.666Assuming $A_{J}=0.272A_{V}$ (Schlegel et al., 1998). The
Carnegie-Chicago Hubble Program procedure is to adopt half of the reddening
value as its systematic uncertainty, which in this case would then be 0.08 mag
(e.g., Jang et al. 2021). However, even adopting the full median extinction
value of $A_{J}=0.16$ mag as an uncertainty would still not be enough to
explain the large observed scatter in the absolute $J$-band luminosity
function.

Therefore, the large scatter may be explained, in part, by the fact that the
astrometric precision of the Gaia parallaxes in EDR3 has strong color and
magnitude dependencies, relying heavily on accurate PSF fitting and
calibration (Rowell et al., 2021). While we found the median parallax
uncertainty of our sample of JAGB stars to be only $\pm 0.027$ mas, recent
publications have found the Gaia EDR3 parallax uncertainties to be
underestimated for brighter stars (El-Badry et al., 2021). For stars brighter
than $G=13$ mag, the instrument switches its main CCD sampling scheme to
mitigate saturation effects. However, this secondary mode is more sensitive to
PSF modeling deficiencies, leading to larger uncertainties (Rowell et al.,
2021). These uncertainties are relevant for not only the bright JAGB stars, as
all of the JAGB stars used in this study were brighter than $G=13$ mag, but
other local distance indicators777For example, the Cepheid sample of Riess et
al. (2021) recently used to calibrate the Leavitt Law using Gaia EDR3 is
brighter than $G=11.2$ mag. as well. Fortunately, as detailed by Rowell et al.
(2021), improvements in the PSF modeling, particularly for bright stars
($G<13$ mag), are already being implemented for DR4. Advancements in the PSF
modeling for Gaia DR4 will benefit the astrometry for several distance
indicators including the JAGB stars and Cepheids.

In addition to the large statistical uncertainties, the Gaia parallaxes for
the JAGB stars also likely suffer from systematic offsets. Although in theory,
Gaia should be able to measure absolute parallaxes: that is, for sufficiently
distant sources such as quasars, the measured parallax by Gaia should be zero.
However, the accuracy of Gaia’s astrometry has been shown to be subject to
various instrumental effects which cause a global systematic offset in the
parallaxes with respect to the International Celestial Reference System
(Lindegren et al., 2018), referred to as the parallax zeropoint offset. For
their Early Data Release 3 in December 2020, Gaia reported a median zeropoint
offset of $-17$ $\mu$as, with systematic variations on the order of $10$
$\mu$as, defined by over 1.6 million background quasars (Lindegren et al.,
2021). However, this parallax zeropoint offset was derived from quasars in the
magnitude and color range $G>16$ mag and $(GP-RP)<1.6$ mag (Lindegren et al.,
2021), respectively, a range within which none of our bright and red JAGB
stars fall. Lindegren et al. (2021) speculated that sources outside of the
region of parameter space well populated by quasars could have additional
systematic uncertainties ranging from “a few $\mu$as” to “several tens of
$\mu$as.”

Several recent publications have attempted to also quantify additional
parallax offsets in the Gaia EDR3 data. Using first-ascent red giant branch
stars with asteroseismic parallaxes, Zinn (2021) derived an offset of $-15\pm
3~{}\mu$as. Huang et al. (2021) found a global offset of $-9.8\pm 1.0~{}\mu$as
for the $<10.8$ mag LAMOST primary red clump (PRC) stellar sample. Stassun &
Torres (2021) compared the EDR3 parallaxes with distances derived from
eclipsing binaries and found a maximum additional offset of $+33$ $\mu$as.
Using globular clusters, Maíz Apellániz et al. (2021), measured an offset of
$+6.9\pm 2.2~{}\mu$as for stars in the range $9.3<G<11$ mag. Finally, Riess et
al. (2021) derived a $-14\pm 6~{}\mu$as offset by comparing the EDR3
parallaxes with HST photometrically-predicted parallaxes. Thus, due to the
lack of consensus on a parallax offset value, we conservatively adopt $\pm 30$
$\mu$as as the systematic uncertainty on the parallax measurements. We
propagated this uncertainty on the parallax estimates into an error on
$M_{J}$, which amounted to $\pm 0.11$ mag. We thus adopt $\pm 0.11$ mag as a
systematic uncertainty on the absolute calibration.

We look forward to the Gaia DR4, which promises better astrometry and self-
calibration of the global parallax zeropoint, especially for brighter and
redder stars (Lindegren et al., 2021). In conclusion, we report our final
value for the JAGB zeropoint to be $M_{J}=-6.14\pm 0.05$ (stat) $\pm$ $0.11$
(sys) mag.

##  5 Comparison with Previous Calibrations

Table 2: Previous JAGB Calibrations  Study |  $M_{J}$ [mag] | Anchor
---|---|---
Madore & Freedman (2020) |  $-6.22\pm 0.01$ (stat) $\pm$ 0.03 (sys) | LMC DEB (Pietrzyński et al., 2019)
Madore & Freedman (2020) |  $-6.18\pm 0.01$ (stat) $\pm$ 0.05 (sys) | SMC DEB (Graczyk et al., 2014)
Ripoche et al. (2020) | $-6.28\pm 0.004$ | LMC DEB (Pietrzyński et al., 2019)
Ripoche et al. (2020) | $-6.16\pm 0.015$ | SMC Leavitt Law (Scowcroft et al., 2016)
Ripoche et al. (2020) | $-5.60\pm 0.026$ |  Gaia DR2 Galactic Parallaxes
This Work |  $-6.14\pm 0.05$ (stat) $\pm$ 0.11 (sys) |  Gaia EDR3 Galactic Parallaxes

Previous JAGB calibrations are compiled in Table 2. This is the first study to
date that has utilized the Gaia EDR3 parallaxes as the geometric anchor for
the JAGB method. Two previous studies, Madore & Freedman (2020) and Ripoche et
al. (2020) both anchored the JAGB method to the LMC and SMC. Ripoche et al.
(2020) also measured a JAGB zeropoint based on Gaia DR2 parallaxes.

For the LMC-based calibrations, Madore & Freedman (2020) found $M_{J}=-6.22$
$\pm$ 0.01 (stat) $\pm$ 0.03 (sys) mag, based on a DEB distance to the LMC
(Pietrzyński et al., 2019). Using the same LMC DEB distance, Ripoche et al.
(2020) measured $M_{J}=-6.28\pm 0.004$ mag.

For the SMC-based calibrations, Madore & Freedman (2020) found $-6.18\pm 0.01$
(stat) $\pm$ 0.05 (sys) mag based on the SMC DEB distance from Graczyk et al.
(2014). Ripoche et al. (2020) found $-6.16$ $\pm$ $0.015$ mag based on the
Scowcroft et al. (2016) Leavitt Law SMC distance.

Finally, Ripoche et al. (2020) measured an $M_{J}=-5.60\pm 0.026$ mag based on
Galactic carbon stars. This calibration was measured using Gaia DR2
parallaxes. They attributed their considerably fainter calibration to the high
parallax uncertainties from Gaia DR2.

##  6 Summary and Conclusions

The JAGB method is a geometrically calibrated and simply-applied stellar
distance indicator that capitalizes on the fact that JAGB stars are easily
identified in the near infrared. We have identified a sample of 153 JAGB stars
from Milky Way carbon star catalogs, for which we derived distances using Gaia
EDR3 parallaxes. We present an absolute calibration of the JAGB method,
$M_{J}=-6.14\pm 0.05$ (stat) $\pm$ 0.11 (sys) mag for this Galactic sample.

Our Milky Way JAGB calibration is completely independent of, and agrees well
with the Madore & Freedman (2020) LMC and SMC DEB-based calibrations of
$M_{J}(LMC)=-6.22\pm 0.01$ (stat) $\pm$ 0.03 (sys) mag and
$M_{J}(SMC)=-6.18\pm 0.01$ (stat) $\pm$ 0.05 (sys) mag. Averaging these three
calibrations yields an updated JAGB zero-point of $M_{J}=-6.18\pm 0.02$ (stat)
$\pm$ 0.04 (sys) mag. Moreover, no statistically significant trend with
increasing host-galaxy metallicity is found here, with the SMC ($M_{J}=-6.18$
mag, [Fe/H]$\sim-0.8$ dex), LMC ($M_{J}=-6.22$ mag, [Fe/H]$\sim-0.5$ dex), and
Milky Way ($M_{J}=-6.14$ mag, [Fe/H]$\sim 0$ dex) already spanning a large
range in present-day metallicity. We also note that our calibration is also in
excellent agreement with the independent Ripoche et al. (2020) SMC calibration
of $-6.16\pm 0.015$ mag, and consistent to within $1.2\sigma$ of their LMC
calibration of $-6.28\pm 0.004$ mag.

We have demonstrated that the JAGB method’s absolute J-band calibration is
consistent across three independent zero-point calibrators. With this
increasing accuracy, the stage is being set for the JAGB method to be more
widely applied as a primary extragalactic distance indicator. We look forward
to the release of Gaia DR4 in 2022, which will bring with it several
improvements in its astrometry.

A. J. L. would like to thank Andrew Madigan for providing helpful comments on
code used in this work. We thank Rachel Wagner-Kaiser for making her Hess
Diagram Python code publicly available on Github. We thank the Observatories
of the Carnegie Institution for Science and the University of Chicago for
their support of our long-term research into the calibration and determination
of the expansion rate of the Universe. Finally, we thank the anonymous referee
for their constructive suggestions. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research made use of
the cross-match service and SIMBAD database (Wenger et al., 2000), provided by
and operated at CDS, Strasbourg. This research has made use of NASA’s
Astrophysics Data System Bibliographic Services. Gaia

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##  Appendix A Identifying JAGB stars with 2MASS photometry

In addition to identifying JAGB stars from spectroscopically selected carbon
star catalogs in the literature, we also attempted to photometrically select
JAGB stars solely on the basis of their color. This approach follows that of
Madore & Freedman (2020), who photometrically selected JAGB stars without
using carbon star catalogs for their LMC and SMC DEB-based calibrations.

###  A.1 2MASS Sample

To build a Galactic near-infrared color magnitude diagram, $JHK_{s}$ data for
Galactic stars were obtained from the 2MASS All-Sky Point Source Catalog X
Gaia EDR3 best neighbor cross-match catalog (tmass_psc_xsc_best_neighbour)
using a Galactic latitude constraint of |l|>30. We chose to limit our catalog
to sources with high Galactic latitude in order to avoid high line-of-sight
extinction contamination from Galactic dust and source crowding (Skrutskie et
al., 2006). We also applied more stringent data cuts (ruwe<1.4 and
$\sigma_{\pi}/\pi<0.10$) than in Section 3 to obtain the highest-quality
parallaxes.

###  A.2 Issues in the Astrometry and Photometry

In Figure A1, we show in a color magnitude Hess diagram that the JAGB stars at
$1.4<(J-K_{s})<2.0$ mag were heavily contaminated by bluer O-rich AGB stars.
In this section, we explore several potential explanations for why the
photometry and astrometry of the 2MASS sample was too inaccurate to
distinguish between the O-rich AGB stars and JAGB stars using the standard
color cuts. First, as discussed in Section 4.1, bright ($G<13$ mag) and red
stars in the Gaia catalog suffer from high parallax errors resulting from
their parallax zeropoint offset measurement and uncertainties in the PSF-
fitting routines. Second, reddening could also be a source of error. In the
future, we will explore applying the Wesenheit function (Madore, 1982) as a
means of correcting the reddening. Third, Skrutskie et al. (2006) reported
large photometric uncertainties on the order of $\sim 0.25$ mag for bright
stars with $m_{J}<5.5$ mag. Removing stars with $m_{J}<5.5$ mag or $G<13$ mag
was not a viable solution because it would have removed many of the brighter
JAGB stars and introduced a systematic bias in $M_{J}$.

Attempts to clean our photometry with cuts on distance ($d<1$ kpc) and
fractional parallax ($\sigma_{\pi}/\pi<0.05$), shown in Figure A2 also proved
unsuccessful. In both cases, a large fraction of the O-rich AGB stars remained
post-cleaning, continuing to make it impossible to distinguish between the
O-rich AGB stars and the JAGB stars. Therefore, we found that photometrically
selecting a sample of Galactic JAGB stars was not a viable option at this
time, and for the time being, chose to select JAGB stars from Galactic carbon
star catalogs instead.

Figure A1: $J$ vs. ($J-K_{s}$) color magnitude Hess diagram for the full
2MASS-selected sample. The JAGB stars are obscured by O-rich AGB stars likely
because of large errors from reddening, Gaia parallax uncertainties, and 2MASS
photometric uncertainties.  Figure A2: Cleaned $J$ vs. ($J-K_{s}$) color
magnitude Hess diagrams for the 2MASS-selected sample, filtered by distance
(left) and fractional parallax uncertainty (right). We determined that
attempts to clean the 2MASS photometry using aggressive cuts on distance and
fractional parallax uncertainty were still ineffective in isolating the JAGB
stars, particularly at the blue end.