# The Pantheon+ Analysis: Dependence of Cosmological Constraints on
Photometric-Zeropoint Uncertainties of Supernova Surveys

Sasha Brownsberger Department of Physics, Harvard University
17 Oxford Street
Cambridge, MA 02138, USA<EMAIL_ADDRESS>Dillon Brout
Department of Physics, Harvard University
17 Oxford Street
Cambridge, MA 02138, USA  Center for Astrophysics, Harvard & Smithsonian
60 Garden Street
Cambridge, MA 02138, USA  Daniel Scolnic Department of Physics, Duke
University
Science Drive
Durham, NC 27710, USA  Christopher W. Stubbs Department of Physics, Harvard
University
17 Oxford Street
Cambridge, MA 02138, USA  Center for Astrophysics, Harvard & Smithsonian
60 Garden Street
Cambridge, MA 02138, USA  Adam G. Riess Department of Physics and Astronomy,
Johns Hopkins University
3400 Charles Street
Baltimore, MD 21218, USA  Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218, USA

###### Abstract

Type Ia supernovae (SNe Ia) measurements of the Hubble constant, H0, the
cosmological mass density, $\Omega_{M}$, and the dark energy equation-of-state
parameter, $w$, rely on numerous SNe surveys using distinct photometric
systems across three decades of observation. Here, we determine the
sensitivities of the upcoming SH0ES+Pantheon+ constraints on H0, $\Omega_{M}$,
and $w$ to unknown systematics in the relative photometric zeropoint
calibration between the 17 surveys that comprise the Pantheon+ supernovae data
set. Varying the zeropoints of these surveys simultaneously with the
cosmological parameters, we determine that the SH0ES+Pantheon+ measurement of
H0 is robust against inter-survey photometric miscalibration, but that the
measurements of $\Omega_{M}$ and $w$ are not. Specifically, we find that
miscalibrated inter-survey systematics could represent a source of uncertainty
in the measured value of H0 that is no larger than $0.2$ km s-1 Mpc-1. This
modest increase in H0 uncertainty could not account for the $7$ km s-1 Mpc-1
“Hubble Tension” between the SH0ES measurement of H0 and the Planck
$\Lambda$CDM-based inference of H0. However, we find that the SH0ES+Pantheon+
best-fit values of $\Omega_{M}$ and $w$ respectively slip, to first order, by
$0.04$ and $-0.17$ per $25$ mmag of inter-survey calibration uncertainty,
underscoring the vital role that cross-calibration plays in accurately
measuring these parameters. Because the Pantheon+ compendium contains many
surveys that share low-$z$ Hubble Flow and Cepheid-paired SNe, the
SH0ES+Pantheon+ joint constraint of H0 is robust against inter-survey
photometric calibration errors, and such errors do not represent an impediment
to jointly using SH0ES+Pantheon+ to measure H0 to 1% accuracy.

Type Ia supernovae(1728) — Cosmology(343) — Flux calibration(544) — Hubble
constant(758) — Dark energy(351) Astronomy data analysis(1858)

##  1 Introduction

Since the first determination by Perlmutter et al. (1999); Riess et al. (1998)
that dark energy (DE) is causing the Universe to expand at an accelerating
rate, observations of type Ia supernovae (SNe Ia) have been integral in the
formulation of the canonical $\Lambda$CDM cosmological model. Predictions
based on the $\Lambda$CDM cosmology, which is dominated at our moment in
cosmic history by a cosmologically-constant dark energy density ($\Lambda$)
and by nonrelativistic, collisionless (‘cold’) dark matter (CDM), have
accurately matched a diversity of cosmological observables, such as the
formation of cosmic structure (Eisenstein et al., 2005; DES Collaboration et
al., 2021), the rate of the Universe’s expansion (Perlmutter et al., 1999;
Riess et al., 1998; Scolnic et al., 2018), and the Cosmic Microwave Background
(CMB) (Bennet et al., 1993; Jones et al., 2006; Hinshaw et al., 2013; Planck
Collaboration et al., 2020).

However, local measurements of the contemporary rate of cosmic expansion (the
Hubble constant, H0) using SNe Ia (Riess et al., 2021) are in approximately
4.5$\sigma$ tension with the $\Lambda$CDM-based inferences of H0 derived from
observations of the CMB (Planck Collaboration et al., 2020). This
inconsistency, dubbed the Hubble Tension (HT), is the source of much debate
and has motivated (1) additional methods to produce independent measurements
of $H_{0}$ and (2) investigations into sources of systematic uncertainty that
these measurements face. Comprehensive reviews on the matter, such as those of
Verde et al. (2019); Di Valentino et al. (2021), show a general tension
between near-Universe measurements of H0 and early-Universe $\Lambda$CDM-based
inferences of H0, with the largest inter-data set tension being that between
SH0ES and Planck, owing in part to their relatively small measurement
uncertainties.

Here we focus on a critical aspect of the SH0ES H0 measurement: SN Ia cross-
survey calibration. Low-redshift ($z<0.01$) SNe Ia are critical for measuring
H0, as only in such nearby galaxies can the SNe luminosities be tied to
collocated Cepheid (or other intermediate standard candle) calibrators.
However, because SNe Ia occur approximately once every century in a typical
galaxy, observations of low-redshift ($z<0.01$) SNe are volume limited, with
the historical frequency of detection being less than one per year. Owing to
this fundamental infrequency of occurrence, recent analyses (Scolnic et al.,
2018; Riess et al., 2021), in order to obtain a statistically meaningful
sample, rely on a heterogeneous catalogue of low-redshift SNe observations
spanning three decades and multiple photometric systems usually in the $g$,
$r$, $i$ bands. Though imperfect and idiosyncratic, this painstakingly
accumulated catalogue of low-redshift SNe Ia is both critical for SNe Ia-based
measurements of H0 and irreplaceable by future surveys. In contrast, high-
redshift supernovae are much more numerous and considerably fainter, and are
best observed by large survey telescopes operating in the $r$, $i$ and $z$
bands with greater sensitivities and wider fields of view. Thus, in order to
fully utilize all SN Ia distance measurements, contemporary SN Ia analyses
combine separate datasets acquired across three decades of observations on
many distinct photometric systems.

The calibration of each photometric system used in SN Ia cosmology is
performed in each respective data release using heterogeneous methodologies.
Surveys such as Pan-STARRS (PS1), Foundation, the Supernova Legacy Survey
(SNLS), the Sloan Digital Sky Survey (SDSS), and the Dark Energy Survey (DES)
utilize multiple Calspec standards to define their photometry on the AB system
and tie nightly photometry to stellar catalogs in each survey’s natural
system. Some surveys utilize sophisticated forward modeling approaches (Stubbs
& Tonry, 2006) to infer top of the atmosphere magnitudes of their nightly
photometry standards (e.g. UberCal as part of the PS1 analysis: Schlafly et
al. 2012; Scolnic et al. 2018, and the Forward Global Calibration Method used
in the DES analysis: Burke et al. 2018). Conversely, some low-$z$ samples are
tied to the Vega system while others are tied to the AB system using only
BD$+17^{\circ}$4708 to tie magnitudes from each filter to the AB system.
Furthermore, the low-$z$ surveys’ nightly photometric zeropoints are
determined differently from those of the larger high redshift surveys. For the
low-$z$ surveys this is done either by transforming Landolt (Landolt & Uomoto,
2007) and Smith standards (Smith et al., 2002) onto the respective natural
system or by transforming the nightly photometry onto the system of Landolt
standards.

Additionally, each respective SNe sample has historically had its own
proprietary photometric pipeline. Some use sophisticated forward modeling
photometry (Holtzman et al., 2008; Brout et al., 2019a; Burke et al., 2018),
some use difference imaging (Rest et al., 2014; Foley et al., 2018), and some
use difference imaging for SNe that are near their host galaxies while
foregoing it when host-galaxy contamination appears minimal (Riess et al.,
1999). Furthermore, subtle photometric issues have gradually been identified.
For example, Kessler et al. (2015) and Brout et al. (2019a) report the host
brightness anomaly, which appears to be worsened by photometric analysis and
astrometric uncertainty, and Scolnic et al. (2018) identify photometric non-
linearities. These issues have been addressed in some contemporary surveys,
but retroactive correction of some older surveys is impossible since the
original data are no longer accessible.

These systematic uncertainties in the flux calibration of the SNe Ia can be
distilled into the following aspects: systematic uncertainty in the
calibration of standard star observations to the AB primary standard(s),
systematic uncertainties in the set of primary standard(s) themselves used by
each survey, systematic uncertainties in the transfer of zeropoints between
the local standards near the SN on the night of observations to the primary
standards, systematic uncertainties in the measurements of the local standards
themselves, and finally systematic uncertainties arising from each photometric
pipeline. The analysis processes applied across the numerous SNe Ia surveys
are heterogeneous.

Stitching many distinct surveys into a unified data set requires cross-
calibration, and considerable effort has already been invested to place SNe Ia
on the same the flux scale for a self-consistent and homogeneous Hubble
diagram. For example, in the SuperCal calibration method (Scolnic et al.,
2015), the uniformity of the Pan-STARRS photometric system across a large
portion of the sky is used to tie every observed field to background
catalogues of reference calibration stars for all 10 surveys used in the
original Pantheon Analysis (Scolnic et al., 2018). Likewise, SuperCal-
Fragilistic (Brout et al. in prep.) perform incremental improvements on
SuperCal and allow for covariance between surveys including Pan-STARRS, and
most notably extend the cross-calibration to include all 17 surveys analyzed
in the Pantheon+ analysis (Brout in prep.). However, there remain subtleties
and assumptions made during the cross-calibration process and it is entirely
possible that residual inter-survey photometric zeropoint differences remain.
As an alternate to SuperCal, Currie et al. (2020) propose the CROSS-CALIB
Baysian-based approach to inter-survey calibration, tying tertiary stars
across surveys to Pan-STARRS and SDSS objects.

This exhaustive cross-calibration effort has made possible the compilation of
17 heterogeneous SNe Ia surveys into the single Pantheon+ SNe compendium,
which will provide the strongest near-universe constraints of cosmological
parameters to-date (Brout in prep.). However, it is possible that some
undiagnosed inter-survey photometric zeropoint errors remain in Pantheon+. By
introducing floating, color-independent (gray) flux offsets for each survey,
we, in this paper, assess the effect that such residual systematics can have
on the on the SH0ES+Pantheon+ joint constraints on H0, on the matter fraction,
$\Omega_{M}$ and on the dark energy equation-of-state parameter, $w$. The
SH0ES+Pantheon+ joint constraint is obtained by combining an updated version
of the SH0ES data set of low-$z$ SNe and calibration Cepheids (Riess et al.,
2018) with the updated Pantheon+ version of the Pantheon compendium of
low-$z$, mid-$z$, and high-$z$ SNe (Scolnic et al., 2018).

Our paper is organized as follows. In Section 2.1, we summarize the Pantheon+
SNe Ia compendium. In Section 2.2, we show how we incorporate inter-survey
zeropoint offsets into the Pantheon+ SNe Ia data. In Section 2.3, we explain
the MCMC that we run with these inter-survey zeropoints as free parameters. In
Section 3.1, we show how the inclusion of conservative but realistic inter-
survey zeropoint offsets expands the MCMC posterior distributions and thus the
Pantheon+ uncertainties of H0, $\Omega_{M}$ and $w$. In Section 3.2, we
demonstrate that the increase in these posterior uncertainties have clear
maximum values that they cannot exceed, no matter how poorly the inter-survey
photometric calibration is known. We explain why these maxima exist in
Appendix B and discuss their implications for HT and for SNe Ia measurements
of $\Omega_{M}$ and $w$ in Section 3.3. We conclude our analysis in Section 4,
where we also discuss its immediate implications and possible future
applications. In Section 4.1, we compare the relative statistical power of
adding more SNe to the Pantheon+ data set vs improving our confidence in the
existing inter-survey calibrations, and in Section 4.2 we note the
vulnerability that a measurement of H0 that relies on a single survey of
Hubble Flow (HF) SNe would have to zeropoint inter-survey calibration errors.

##  2 Data & Methodology

In this section, we review the upcoming Pantheon+ SNe Ia compendium and
explain the methodology that we used to determine the effect that
miscalibrated inter-survey zeropoints could have on measurements of
cosmological parameters. Our analysis is agnostic as to the source of these
miscalibrations, assumes that the offsets are gray, and assumes that the
offsets between surveys are uncorrelated.

Figure 1:  The distribution of Pantheon+ SNe Ia in redshift, $z$, and distance
modulus residual, $\mu-\mu_{\textrm{model}}$, space, where
$\mu_{\textrm{model}}$ is calculated using the best-fit SH0ES+Pantheon+
$\Lambda$CDM cosmology (black line). The horizontal extents of the rectangles
span the redshift ranges of the surveys, and their vertical extents define the
$\pm 1\sigma$ bounds of the the surveys’ mean $\mu-\mu_{\textrm{model}}$. The
Pantheon+ data set (Scolnic et al, in prep.; Brout et al in prep.) is the most
expansive collection of spectroscopically-matched SNe Ia ever compiled,
containing 1800 SNe gathered from 17 distinct surveys spanning three decades
of observations. To place these SNe on this single plot, the Pantheon+
analysis team undertook an exhaustive cross-calibration effort. To assess what
effect gray errors in this inter-survey cross-calibration could have on the
SH0ES+Pantheon+ cosmological parameter constraints, we introduce, for each
survey, a zeropoint offset, $\Delta\mu_{S}$, as a free parameter that
vertically shifts all SNe in that survey. Those surveys (CFA1, for example)
that are most excursive from the best-fit Pantheon+ $\Lambda$CDM cosmological
fit have the largest best-fit $\Delta\mu_{S}$ values (see Table 1).

###  2.1 Pantheon+ Data Sample

We utilize the upcoming Pantheon+ (Scolnic et al. in prep.; Brout et al. in
prep.) compendium of publicly available, spectroscopically classified,
photometric light curves of SNe Ia that have been cross-calibrated in Brout et
al. (in prep.) with redshifts reported by Carr et al (in prep.) and Peterson
et al (in prep.). The SNe range in redshift from very low-$z$ ($z<0.01$) used
for Cepheid-SN Ia luminosity calibration, to low-$z$ ($0.02<z<0.15$) for
filling out the Hubble flow (HF), to high-$z$ ($z>0.15$) usable for
measurements of dark energy. The low-$z$ SNe used here are from the Carnegie
Supvernova Project (CSP, Stritzinger et al. (2010)), the Center for
Astrophysics SNe Ia data sets (CfA1-4, Riess et al. (1999); Jha et al. (2006);
Hicken et al. (2009a, b, 2012)), the Lick Observatory Supernova Search (LOSS,
(Ganeshalingam et al., 2010; Stahl et al., 2019)), the Swift Optical Archive
(SWIFT, Brown et al. (2014)), the All Sky Automated Survey for SuperNovae
(ASASSN, Chen et al. (2020)), and the recently released 180 low-$z$ SNe from
the Foundation sample (Foley et al., 2018). At high-z, we include SNe from the
first data release of the Pan-STARRS Medium-Deep Field observations (PS1MD,
Rest et al. (2014); Scolnic et al. (2018)), SDSS (Sako et al. (2011)), SNLS
(Betoule et al. (2014)), the DES 3-year sample (Brout et al., 2019a, b; Smith
et al., 2020), and the Hubble Space Telescope (GOODS: Riess et al. 2004, 2007,
SCP: Suzuki et al. 2012, HST: Riess et al. (2018)). The distribution of SNe Ia
in redshift and best-fit $\Lambda$CDM residual space can be found in Figure 1.

The full derivation of distances will be detailed in Brout et al. (in prep.).
In short, the light-curve fit parameters of each SN ($m_{B}$,$c$,$x_{1}$) are
determined using the SALT2 model (Guy et al., 2007) as re-trained on the
Supercal-Fragilistic system in Brout et al (in prep.). Distances are inferred
following the Tripp estimator (Tripp, 1998). The distance modulus ($\mu$) to
each candidate SN Ia is obtained by:

$\mu=m_{B}+\alpha x_{1}-\beta c-M+\delta_{\rm bias},$ (1)

where $m_{B}$ is the $B-$band peak-brightness based off of the light-curve
amplitude (log${}_{10}(x_{0})$), where $M$ is the absolute magnitude of a SN
Ia with $x_{1}=c=0$, and where $\delta_{\rm bias}$ is the bias-correction term
that corrects for selection effects. $\alpha_{\rm SALT2}$ and $\beta_{\rm
SALT2}$ are the correlation coefficients that standardize the SNe Ia and are
determined following Marriner et al. (2011); Popovic et al. (2021).

Typical selection cuts are applied on the observed data sample: we require
fitted color uncertainty $<0.05$, fitted stretch uncertainty $<1$, fitted
light-curve peak date uncertainty $<2$, light-curve fit probability (from
SNANA) $>0.01$, and we apply Chauvenaut’s criterion to the distance modulus
residuals, relative to the best-fit cosmological model, at 3.5$\sigma$. All
SNe data are processed on the same SALT2 light curve fitter.

###  2.2 The Effects of Inter-Survey Zeropoints on SNe Distance Moduli

We seek to determine the effect that gray inter-survey zeropoint calibration
errors could have on the SH0ES+Pantheon+ constraints of cosmological
parameters. In this Section, we detail how we introduce model agnostic inter-
survey zeropoint offsets, representing hypothetical gray miscalibrations, into
our analysis.

For HF SNe Ia $i$ in survey $S$, we adjust its initially determined distance
modulus by a new free fit parameter, $\Delta\mu_{S}$, that applies to all SNe
in survey $S$. All SNe in survey $S$ are shifted by the same $\Delta\mu_{S}$,
of which we have a total of 17, one for each Pantheon+ survey. As we derive in
Appendix A, this shifts the measured distance modulus of HF SN $i$ from the
initial value of $\mu_{\textrm{meas, init},i}$ to a shifted value of
$\mu_{\textrm{meas, shift},i}$ according to:

$\displaystyle\mu_{\textrm{meas, shift},i}=\mu_{\textrm{meas,
init},i}-\Delta\mu_{L}-\Delta\mu_{S}\ ,$ (2)

where $\Delta\mu_{L}$ is the distance modulus correction that results from our
redetermination of the fundamental SNe Ia luminosity, and is the same for
every SNe, regardless of survey. As we describe in Appendix A3, we calculate
$\Delta\mu_{L}$ from all $\Delta\mu_{S}$’s of the Cepheid-paired SNe Ia:

$\displaystyle\Delta\mu_{L}=-\frac{\sum_{i=1}^{N_{Ceph}}\Delta\mu_{S,i}\sigma_{\mu,i}^{-2}}{\sum_{i=1}^{N_{Ceph}}\sigma_{\mu,i}^{-2}}\
,$ (3)

where $\sigma_{\mu,i}$ is the reported uncertainty in the distance modulus of
SN $i$ and the sum is taken only over Cepheid-paired SNe. Including
$\Delta\mu_{L}$ is necessary because adjusting the photometric zeropoint of a
Cepheid-paired SN adjusts our inference of that SN’s luminosity based on the
Cepheid-determined distance to the host galaxy.

As we detail in Appendix B, because $\Delta\mu_{L}$ is the negative average
inter-survey zeropoint offset of all Cepheid paired SNe,
$\Delta\mu_{L}+\Delta\mu_{S}$ only shifts the zeropoint of survey $S$ relative
to the average shift of surveys with Cepheid-paired SNe. Applying inter-survey
zeropoint offsets (the $\Delta\mu_{S}$’s) thus cannot shift all SNe distance
moduli collectively, but only relative to each other.

Figure 2:  The triangle plot showing the posterior distributions of all varied
parameters ($\Delta$H0, $\Delta\Omega_{M}$, $\Delta w$, and one
$\Delta\mu_{S}$ for each survey) with 25 mmag-wide Gaussian priors on the
$\Delta\mu_{S}$ variables. The $\Delta\mu_{S}$’s are given in units of mmag,
and $\Delta H_{0}$ is given in units of km s-1 Mpc-1 (we omit units in the
Figure to provide more space for the labels). The $\Delta$’s on the
cosmological parameters indicate how much that parameter has shifted from the
best-fit value measured when all inter-survey zeropoint offsets are set to 0.
For the inter-survey zeropoint offsets ($\Delta\mu_{S}$’s), and blue cross
hairs mark $0$, corresponding to total certainty in the Pantheon+ inter-survey
photometric calibrations. We list the covariances of H0, $\Omega_{M}$, and $w$
with all $\Delta\mu_{S}$’s in Table 1. Circular posteriors indicate weak
covariance, while elongated posteriors indicate strong covariance. The
covariance between a cosmological parameter and an inter-survey zeropoint
offset indicates how much that parameter shifts as the zeropoint of the
survey, relative to all other surveys, is varied. The covariance between two
inter-survey zeropoint offsets indicates how tightly two zeropoints move
together. The tightest covariances are between the largest surveys in the
Pantheon+ compendium: DES, SDSS, PS1MD, and SNLS. This tight correlation is
unsurprising - separating the zeropoints of two large surveys bifurcates the
resulting Hubble diagram, worsening the cosmological fit.

###  2.3 Sampling the Space of Cosmological Parameters and Inter-Survey
Zeropoint Offsets

To assess what information a data set of SNe Ia measurements provides about
cosmological parameters, we compare, for HF SN $i$ the shifted measured
distance modulus, $\mu_{\textrm{meas, shift},i}$, and the theoretically
predicted distance modulus, $\mu_{\textrm{model, }i}$. As we discuss in
Section 2.2, $\mu_{\textrm{meas, shift},i}$ shifts when we introduce the
inter-survey zeropoint offsets, $\Delta\mu_{S}$. In contrast,
$\mu_{\textrm{model, }i}$ is determined entirely by our choice of cosmological
parameters and the SN’s redshift, $z_{i}$:

$\displaystyle\mu_{\textrm{model, }i}(z_{i};H_{0},\Omega_{M},w)\
=25+5\log_{10}\Big{(}{\frac{c(1+z_{i})}{1\textrm{Mpc}\times
H_{0}}\int_{0}^{z_{i}}\frac{dz^{\prime}}{H(z^{\prime},\Omega_{M},w)/H_{0}}}\Big{)},$
(4)

where we assume a flat cosmology and where $c$ is the speed of light.

We minimize the $\chi^{2}$ statistic for each of the $N_{HF}$ Hubble flow
Pantheon+ supernovae:

$\displaystyle\chi^{2}=\sum_{i=1}^{N_{HF}}\frac{\Delta\mu_{i}^{2}(z_{i};H0,\Omega_{M},w,L,f_{i})}{\sigma_{\mu,i}^{2}},$
(5)

where $\sigma_{\mu,i}^{2}$ is the variance of $\mu_{\textrm{meas, }i}$ and
where we define the predicted vs measured distance modulus residual as

$\displaystyle\Delta$
$\displaystyle\mu_{i}(z;H0,\Omega_{M},w)=\mu_{\textrm{meas,
}i}-\mu_{\textrm{model, i}}(z;H_{0},\Omega_{M},w).$ (6)

For supernova $i$ in a survey with zeropoint offset $\Delta\mu_{S}$ with
redshift $z_{i}$ and measured distance modulus $\mu_{\textrm{meas, init, }i}$,
the distance modulus residual for a given choice of cosmological parameters
and inter-survey offsets is:

$\displaystyle\Delta$
$\displaystyle\mu_{i}(z_{i};\bm{H_{0}},\bm{\Omega_{M}},\bm{w},\bm{\Delta\mu_{S}})=\mu_{\textrm{meas,
init},i}-\bm{\Delta\mu_{L}}-\bm{\Delta\mu_{S}}-25-5\log_{10}\Big{(}{\frac{c(1+z_{i})}{1\textrm{Mpc}\times\bm{H_{0}}}\int_{0}^{z_{i}}\frac{dz^{\prime}}{H(z^{\prime},\bm{\Omega_{M}},\bm{w},\bm{H_{0}})/\bm{H_{0}}}}\Big{)},$
(7)

where the luminosity offset, $\Delta\mu_{L}$, is calculated from all Cepheid
paired SNe and the corresponding values of $\Delta\mu_{S}$ using Equation 3.
For emphasis in Equation 7, we write in boldface the parameters that we vary
in the MCMC minimization (note $\Delta\mu_{L}$ is not directly varied, but is
a weighted sum of several varied $\Delta\mu_{S}$’s).

To determine the best-fit cosmological parameters (H0, $\Omega_{M}$ and $w$)
and inter-survey zeropoint offsets (the 17 $\Delta\mu_{S}$’s), we use the
emcee MCMC sampling Python library described by Foreman-Mackey et al. (2013)
to minimize $\chi^{2}$ (Equation 5). We use flat priors to sample the
cosmological parameters so that our best-fit posterior values are unbiased.
However, we use Gaussian priors when sampling $\Delta\mu_{S}$ to reflect the
confidence that the Pantheon+ analysis team has in the accuracy of their
inter-survey calibrations (inter-survey zeropoint errors are more likely small
than large). We center the Gaussian prior for Survey $S$ at $\Delta\mu_{S}=0$,
and use its width, $\sigma_{P,S}$, to represent our confidence in the
Pantheon+ zeropoint calibration of survey $S$. A prior width of
$\sigma_{P,S}<1$ mmag represents effectively complete confidence in the
Pantheon+ calibration of survey $S$, and a prior width of $\sigma_{P,S}>100$
mmag represents effectively no knowledge of survey $S$’s zeropoint. We
estimate that a prior width of $\sigma_{P,S}\simeq 25$ mmag on all
$\Delta\mu_{S}$ is a reasonable and conservative representation of the
uncertainty of the Pantheon+ cross-calibration (Brout et al. in prep.).

By running multiple MCMCs with different $\sigma_{P,S}$, we assess the extent
to which the conclusions of the standard SH0ES+Pantheon+ analysis
(corresponding to setting the $\sigma_{P,S}=0$) would degrade if the Pantheon+
photometric zeropoints were moderately ($\sigma_{P,S}\leq 25$ mmag), severally
($25$ mmag $<\sigma_{P,S}\leq 50$ mmag), or egregiously ($\sigma_{P,S}>50$
mmag) miscalibrated .

##  3 Results

In this Section, we discuss the results of the analysis described in Section
2.

###  3.1 Updated SH0ES+Pantheon+ Cosmological Parameter Posterior
Uncertainties

$\Delta\mu_{S}$ |  $<\Delta\mu_{S}>$(mmag) | $\frac{dH_{0}}{d\Delta\mu_{S}}(\frac{\textrm{km}}{\textrm{s Mpc 25 mmag }})$ | $\frac{d\Omega_{M}}{d\Delta\mu_{S}}(\frac{1}{\textrm{25 mmag}})$ | $\frac{dw}{d\Delta\mu_{S}}(\frac{1}{\textrm{25 mmag}})$ | $N_{\textrm{cal}}$ | $N_{\textrm{HF}}$ | $N_{\textrm{cal}}/N_{\textrm{HF}}$
---|---|---|---|---|---|---|---
$\Delta\mu_{SWIFT}$ | $-1\pm 20$ | $-0.056\pm 0.00054$ | $0.000916\pm 0.00014$ | $-0.00745\pm 0.00037$ | 14 | 45 | 0.311
$\Delta\mu_{ASASSN}$ | $3\pm 21$ | $-0.0153\pm 0.0011$ | $-0.00408\pm 0.00031$ | $0.0068\pm 0.00095$ | 4 | 11 | 0.364
$\Delta\mu_{CFA1}$ | $-27\pm 21$ | $0.00715\pm 0.00052$ | $-0.00024\pm 0.00015$ | $-0.00479\pm 0.00039$ | 2 | 12 | 0.167
$\Delta\mu_{CFA2}$ | $-19\pm 20$ | $-0.0197\pm 0.00038$ | $0.00125\pm 0.00016$ | $-0.0123\pm 0.00059$ | 4 | 20 | 0.2
$\Delta\mu_{LOWZ}$ | $-11\pm 18$ | $-0.0273\pm 0.00038$ | $0.0038\pm 8.9e-05$ | $-0.0207\pm 0.00046$ | 8 | 46 | 0.174
$\Delta\mu_{KAITM}$ | $0\pm 14$ | $-0.0417\pm 0.00075$ | $0.00211\pm 0.00017$ | $-0.0186\pm 0.00061$ | 11 | 93 | 0.118
$\Delta\mu_{CFA4p2}$ | $16\pm 22$ | $0.00415\pm 0.00043$ | $0.00224\pm 0.00014$ | $-0.0121\pm 0.0004$ | 1 | 11 | 0.0909
$\Delta\mu_{KAIT}$ | $10\pm 17$ | $0.0028\pm 0.0012$ | $0.000806\pm 0.0002$ | $-0.0102\pm 0.00085$ | 7 | 42 | 0.167
$\Delta\mu_{CFA3S}$ | $12\pm 18$ | $-0.0162\pm 0.0014$ | $0.00113\pm 0.00017$ | $-0.00817\pm 0.00062$ | 4 | 31 | 0.129
$\Delta\mu_{CSP}$ | $9\pm 13$ | $0.0372\pm 0.0015$ | $0.00689\pm 0.00017$ | $-0.037\pm 0.0007$ | 10 | 78 | 0.128
$\Delta\mu_{CFA3K}$ | $-8\pm 15$ | $0.0343\pm 0.00072$ | $0.0032\pm 0.0002$ | $-0.0237\pm 0.00079$ | 4 | 54 | 0.0741
$\Delta\mu_{CFA4p1}$ | $6\pm 17$ | $0.0294\pm 0.00055$ | $0.00347\pm 0.00017$ | $-0.021\pm 0.00062$ | 1 | 35 | 0.0286
$\Delta\mu_{PS1MD}$ | $16\pm 10$ | $0.0724\pm 0.00095$ | $-0.0362\pm 0.00051$ | $0.153\pm 0.0016$ | 4 | 442 | 0.00905
$\Delta\mu_{SDSS}$ | $-28\pm 11$ | $0.0172\pm 0.001$ | $-0.0479\pm 0.00039$ | $0.194\pm 0.00094$ | 0 | 346 | 0.0
$\Delta\mu_{DES}$ | $-3\pm 13$ | $-0.0176\pm 0.0015$ | $-0.0295\pm 0.00028$ | $0.148\pm 0.00089$ | 0 | 203 | 0.0
$\Delta\mu_{SNLS}$ | $-7\pm 13$ | $0.0207\pm 0.00097$ | $0.00981\pm 0.0002$ | $0.0152\pm 0.00081$ | 0 | 229 | 0.0
$\Delta\mu_{HST}$ | $24\pm 22$ | $0.00694\pm 0.0011$ | $0.0125\pm 0.00024$ | $-0.0374\pm 0.00096$ | 0 | 18 | 0.0
$\Delta\mu_{SNAP}$ | $0\pm 24$ | $0.0126\pm 0.00039$ | $0.00502\pm 0.00022$ | $-0.0178\pm 0.0009$ | 0 | 4 | 0.0
$\Delta\mu_{CANDELS}$ | $10\pm 24$ | $0.0213\pm 0.00053$ | $0.00994\pm 0.00013$ | $-0.0324\pm 0.00048$ | 0 | 6 | 0.0
Table 1: When the 17 inter-survey offsets, $\Delta\mu_{S}$, are varied with
conservative 25 mmag wide Gaussian priors, we list in this Table their median
values (column 2), the slopes of linear fits to the posterior distributions
shown in Figure 2 between each $\Delta\mu_{S}$ and the three fitted
cosmological parameters, H0 (column 3), $\Omega_{M}$ (column 4), and $w$
(column 5), and the number of calibrator SNe ($N_{\textrm{cal}}$, column 6),
the number of Hubble Flow (HF) SNe ($N_{\textrm{HF}}$, column 7), and the
calibrator-to-HF SNe ratio (column 8) in each survey. Each posterior slope
(columns 3, 4, and 5) are given as the rate of the parameter change per 25
mmag offset shift. We estimate the uncertainty of each fit via statistical
bootstrapping. The effect that miscalibration of a survey’s zeropoint would
have on the best-fit value of a cosmological parameter increases with the
magnitude of the corresponding slope. For example, a miscalibration of 25 mmag
in the SWIFT zeropoint would shift the best-fit value of H0 by about -0.056 km
s-1 Mpc-1, while a miscalibration of the same size in the SNAP zeropoint would
shift H0 by only 0.013 km s-1 Mpc-1.

We first consider the example of a single MCMC run with a uniform, realistic,
and conservative prior width of $\sigma_{P,S}=25$ mmag for all Pantheon+
surveys. This choice of $\sigma_{P,S}$ allows the MCMC to accommodate
conservative but possible inter-survey photometric miscalibrations while
preventing them from expanding unphysically. We show the full posterior
triangle plot of this MCMC in Figure 2. In that Figure, the single-parameter
histogram for each $\Delta\mu_{S}$ shows how the relative photometric
zeropoint of survey $S$ departs from its Pantheon+ reported value
($\Delta\mu_{S}=0$ by definition) when our MCMC allows $\Delta\mu_{S}$ to
float.

The covariance elements of Figure 2 indicate how tightly two fit parameters
are correlated: circular posteriors indicate little correlation and elongated
contours indicate strong correlation. We find the tightest correlations
between the zeropoint offsets of DES, SDSS, PS1MD, and SNLS. These are the
largest surveys within Pantheon+, and the tight correlations in their offsets
are expected: if the zeropoints of these large surveys shift separately, they
cannot be easily matched by a single Hubble curve and the overall goodness of
fit declines.

For the data shown in Figure 2, we list in Table 1 the medians of inter-survey
zeropoint offsets and their posterior covariances with H0, $\Omega_{M}$ and
$w$. We find that the strength of the dependence of the best-fit cosmological
parameters on the inter-survey offsets varies, survey to survey. The first-
order contribution of a floating survey offset on the shift in the posterior
value of a cosmological parameter can be approximated by multiplying the
derivative of the parameter with respect to that offset by the median value of
that offset. For example, we estimate that the the floating zeropoints of SDSS
and CFA1 shift the best-fit value of $w$ respectively by:

$\displaystyle<\Delta w>_{SDSS}$
$\displaystyle\simeq\frac{dw}{d\Delta\mu_{SDSS}}\Delta\mu_{SDSS}\simeq\frac{0.194\pm
0.001}{25\textrm{ mmag}}(-28\pm 11)\textrm{ mmag}\simeq-0.22\pm 0.09,$ (8)

and

$\displaystyle<\Delta w>_{CFA1}$
$\displaystyle\simeq\frac{dw}{d\Delta\mu_{CFA1}}\Delta\mu_{CFA1}\simeq\frac{-0.0048\pm
0.0004}{25\textrm{ mmag}}(-27\pm 21)\textrm{ mmag}\simeq 0.05\pm 0.04.$ (9)

In Figure 3, we show the posterior contours of H0, $\Omega_{M}$, and $w$ when
all $\Delta\mu_{S}$ are fixed to 0 (filled contours), reflecting the default
analysis paradigm, and when all $\Delta\mu_{S}$ are varied with 25 mmag-wide
Gaussian priors centered at 0 (open contours, same as the lower right contours
in Figure 2). As expected, the posterior distributions of these cosmological
parameters broaden when we introduce additional degrees of freedom in the form
of the $\Delta\mu_{S}$’s.

The contour shifts in Figure 3 and the nonzero derivatives of the cosmological
parameters with respect to the inter-survey offsets in Table 1 both indicate
that shifting some of the floating $\Delta\mu_{S}$ values change the posterior
Pantheon+ cosmic constraints. The shift in the H0 and $\Omega_{M}$ centroids
are modest, less than $1\sigma$ relative to their respective posterior
centroids when the $\Delta\mu_{S}$ offsets are fixed. However, the central
value of $w$ shifts appreciably, by about $-0.15$ from its null value, which
exceeds the null $w$ posterior width of $0.13$. Examining columns 2 and 5 of
Table 1, this shift does not appear to be the result of a single survey
offset. Rather, the majority of survey offsets impose a negative shift in the
posterior value of $w$, and the net effect of these modest shifts is the
appreciable shift in the $w$ central value shown in Figure 3.

Figure 3:  The posterior $1\sigma$, $2\sigma$, and $3\sigma$ contours of the
$\Delta$H0, $\Delta\Omega_{M}$, and $\Delta w$ cosmological parameters when
the inter-survey zeropoint offsets, $\Delta\mu_{S}$, are fixed to 0 (filled
contours), or allowed to vary with Gaussian priors of widths of
$\sigma_{P,S}=25$ mmag centered at 0 (unfilled contours). To maintain the
integrity of the SH0ES and Pantheon+ blind, we subtract the median of the
posterior distribution of each cosmological parameter when the inter-survey
zeropoint offsets are fixed to 0. The $\Delta$’s on each parameter indicate
that the listed value is measured relative to the base-case ($\Delta\mu=0$)
median values. The inclusion of the $\Delta\mu_{S}$ as free fit parameters
both expand and shift the posterior contours, indicating that they adjust the
best-fit cosmological parameter values and increase their reported
uncertainties. The shifts in H0 are modest, not appreciably relieving HT.
However, the shift in the uncertainties and especially the centroids of
$\Omega_{M}$ and $w$ are substantial. Overall, our results suggest that
SH0ES+Pantheon+ measurements of H0 are robust against even large inter-survey
calibration errors, but that the accuracy of the SH0ES+Pantheon+ reported
values of $\Omega_{M}$ and $w$ are heavily reliant on accurate inter-survey
photometric calibration.

###  3.2 The Dependence of the Cosmological Parameter Constraints on Inter-
Survey Zeropoint Offset Uncertainties

To assess what impact systematic errors in the cross-survey calibration could
have on the SH0ES+Pantheon+ constraints of cosmological parameters, we run
several MCMC analyses with inter-survey zeropoint Gaussian priors,
$\sigma_{P,S}$, of varying widths. These varying prior widths represent
varying degrees of certainty on the accuracy of the Pantheon+ cross-survey
calibration. Priors of $\sigma_{P,S}\leq 10$ mmag represent high confidence in
the Pantheon+ inter-survey photometric calibration, priors of
$10<\sigma_{P,S}\leq 25$ represent reasonable confidence, priors of
$25<\sigma_{P,S}\leq 50$ mmag represent extremely conservative but not
outlandish calibration uncertainties, and priors of $\sigma_{P,S}>50$ mmag
represent unrealistically large uncertainties in the Pantheon+ cross-
calibration.

Figure 4:  The dependencies of the SH0ES+Pantheon+ H0 uncertainty (left) and
of the SH0ES+Pantheon+ joint $\Omega_{M}$ and $w$ uncertainty (right) on the
widths of the inter-survey calibration offsets’ MCMC Gaussian priors,
$\sigma_{P,S}$. Larger $\sigma_{P,S}$ correspond to less confidence in the
reported Pantheon+ relative photometric zeropoints. The dashed black line
indicates a conservative estimate of the worst reasonable Pantheon+ inter-
survey calibration error of $25$ mmag. A $\Delta\mu_{S}$ Gaussian prior width
of 0 mags is equivalent to running the MCMC with no inter-survey offsets
($\sigma_{P,S}$ fixed to 0 for each survey $S$). The posterior widths increase
approximately linearly as the $\sigma_{P,S}$ expands from 0 to 40 mmag,
increase more slowly for priors between $40<\sigma_{P,S}\leq 200$ mmag, and
largely cease to change at all for $\sigma_{P,S}>200$ mmag. As we explain in
Appendix B, this plateauing effect occurs because inter-survey zeropoint
offsets can source only relative differences in distance moduli between
surveys, and these relative differences can only grow so large before they
became inconsistent with a single Hubble curve. We estimate the realistic
dependence of the SH0ES+Pantheon+ measurements on the accuracy of the inter-
survey photometric calibrations by fitting a line to the $\sigma_{P,S}\leq 40$
mmag data points, shown as a dashed red line. The slopes, which we write in
the plots, estimate the rate at which the SH0ES+Pantheon+ parameter
measurements slip per $25$ mmag of inter-survey calibration error. The right
vertical axis of the H0 plot (left) shows the HT between the SH0ES measurement
of H0 (Riess et al., 2021) and the Planck inference of H0 (Planck
Collaboration et al., 2020) when the additional statistical uncertainty
introduced by the $\Delta\mu_{S}$ offsets (left vertical axis) is added in
quadrature with the statistical uncertainties reported by SH0ES and Planck.
The change in HT is marginal, reducing the number of statistical $\sigma$
separating SH0ES and Planck measurements of H0 from $4.14\sigma$ to
$4.11\sigma$. gray inter-survey calibration errors cannot be the source of HT,
as they reduce the statistical discrepancy between SH0ES and Planck only
negligibly.  Figure 5:  Same as Figure 5, except the change in the
SH0ES+Pantheon+ $H_{0}$, $\Omega_{M}$, and $w$ centroids, indicated with
$<\cdot>$, as we increase $\sigma_{P,S}$. The slopes of the $\sigma_{P,S}\leq
40$ fitted lines estimate the realistic dependencies of the SH0ES+Pantheon+
best-fit cosmological parameter values on the accuracy of the inter-survey
photometric zeropoint calibrations. The slippage in $<H_{0}>$ of 0.13 km s-1
Mpc-1 $(25$ mmag$)^{-1}$ is negligible, underscoring the robustness of the
SH0ES+Pantheon+ measurement of H0 to cross-survey calibration errors. However,
the $<\Omega_{M}>$ and $<w>$ slopes of 0.04 $(25$ mmag$)^{-1}$ and -0.17 $(25$
mmag$)^{-1}$ respectively are quite substantial, representing meaningful
changes in the class of cosmology measured by SH0ES+Pantheon+. We do note that
contemporary measurements of $\Omega_{M}$ and $w$ typically leverage
orthogonal cosmic probes, such as SNe Ia and the CMB (see, for example,
Scolnic et al. (2018)). The slip in the joint CMB+SH0ES+Pantheon+ measurements
of $\Omega_{M}$ and $w$ as $\sigma_{P,S}$ increases would be less than the
slopes we measure, as the CMB would provide additional stabilization. Our
results do not indicate that the SH0ES+Pantheon+ measurements of cosmological
parameters are inaccurate, but rather that their reliability is highly
dependent on the reliability of the Pantheon+ inter-survey calibrations,
underscoring the importance of the painstaking cross-calibration work of the
Pantheon+ analysis team.

The cosmological parameters’ (H0, $\Omega_{M}$, $w$) posterior uncertainties
expand when the inter-survey zeropoint offsets ($\Delta\mu_{S}$) are allowed
to vary with Gaussian priors of widths specified by $\sigma_{P,S}$. In Figures
4 and 5, we respectively show the dependence of the posterior widths and
centroids of the H0, $\Omega_{M}$, and $w$ posterior widths on $\sigma_{P,S}$.
For $\sigma_{P,s}<40$ mmag, the cosmological parameter posteriors expand and
shift rapidly. As the priors broaden from 40 mmag to about 200 mmag, allowing
for progressively larger zeropoint calibration errors, the parameter posterior
uncertainties and centroids continue to shift, but at a slower pace. Past this
200 mmag threshold the expansions of the MCMC posteriors plateau, remaining
approximately constant even as the widths of the priors on $\Delta\mu_{S}$
increase. As we further detail in Appendix B, the combination of many distinct
surveys of overlapping redshift ranges spanning both Cepheid calibrators and
HF SNe prevents the individual survey zeropoints from drifting apart while
remaining consistent with a single set of cosmological parameters. This
effective ‘self-calibration’ is the source of the plateauing effect in Figure
4, and places a limit on how large an impact inter-survey calibration errors
could have on the SH0ES+Pantheon+ measurement of cosmological parameters, even
if the true inter-survey zeropoints are effectively unknown
($\sigma_{P,S}>100$ mmag in our analysis).

Generically, additional surveys introduce additional, independent measurements
of H0 and therefore more resilience against inter-survey calibration errors.
However, because the calibration methods of several surveys are correlated,
not every additional survey introduces a totally independent H0 probe. There
are, therefore, diminishing returns as additional surveys are introduced
unless those new surveys are wholly independent from the surveys that came
before.

Even the most conservative estimates of the Pantheon+ inter-survey photometric
calibration uncertainty is less than $50$ mmag, or within the $\sigma_{P,S}$
range at which the cosmological parameters posteriors depend approximately
linearly on $\sigma_{P,S}$. To estimate the realistic dependence on the
SH0ES+Pantheon+ measurements of cosmological parameters on the accuracy of the
compendium’s inter-survey calibrations, we fit lines to the dependencies of
the H0, $\Omega_{M}$, and $w$ posterior widths and centroids on the inter-
survey calibration priors, $\sigma_{P,S}$ (red dashed lines in Figures 4 and
5). The slopes of these lines, shown in Figures 4 and 5, estimate the
sensitivities of the SH0ES+Pantheon+ cosmological parameters constraints to
errors in inter-survey zeropoint calibrations.

###  3.3 Implications for HT and SNe Ia Measurements of Cosmological
Parameters

The linear change of the SH0ES+Pantheon+ H0 uncertainty and centroid on
realistic ($\sigma_{P,S}<40$ mmag) inter-survey zeropoint calibration errors
are respectively 0.06 km s-1 Mpc-1 (25 mmag)-1 and 0.13 km s-1 Mpc-1 (25
mmag)-1. Even the inclusion of effectively free-floating ($\sigma_{P,S}>200$)
inter-survey offsets increases the SH0ES+Pantheon+ posterior uncertainty of H0
by 0.15 km s-1 Mpc-1 and shifts the H0 posterior centroid by 0.37 km s-1
Mpc-1. This change in H0 posterior width and centroid are both small,
indicating that the SH0ES+Pantheon+ measurement of H0 is quite robust against
even egregious inter-survey miscalibrations.

Using the most updated SH0ES measurement (Riess et al., 2021) of
H${}_{0}=73.2\pm 1.3$ km s-1 Mpc-1 as a proxy (the SH0ES best-fit value
remains blinded, pending the publication of Brout et al. in prep.), this
increase in the H0 uncertainty negligibly enlarges the H0 uncertainty from
$1.30$ km s-1 Mpc-1 to $\sqrt{1.30^{2}+0.16^{2}}\simeq 1.31$ km s-1 Mpc-1.
This would correspond to a reduction in HT between the Cepheid+SNe Ia distance
ladder measurement and the Planck $\Lambda$CDM-based prediction of H0
($67.27\pm 0.6$ km s-1 Mpc-1) from $4.14\sigma$ to $4.11\sigma$, and would not
relieve HT. In principle, this calculation could also include the centroid
shift in H0 (see Figure 5) which would (because the centroid shift in H0 is
positive) actually marginally increase HT. We opt to take the more
conservative approach, only including the increase in the H0 posterior
uncertainty. Inter-survey photometric calibration errors in Pantheon+ cannot
appreciably relieve HT.

Because the SH0ES+Pantheon+ joint data set contains many low-$z$ surveys that
include both Cepheid-paired and HF SNe Ia, the data set’s effective ‘self-
calibration’ makes the SH0ES+Pantheon+ constraint on H0 quite robust against
inter-survey systematic errors. As we detail in Appendix B, calibration errors
in surveys with Cepheid-paired SNe do not strongly shift the best-fit value of
H0 because the shift is compensated for by the correction to the inferred SNe
Ia absolute luminosity. Further, photometric offsets in surveys with only HF
SNe can grow only so large before the surveys’ distance modulus residuals
split at overlapping redshifts in the Hubble diagram, degrading the resulting
goodness of fit.

However, because Pantheon+ has fewer high-redshift ($z>0.4$) SNe Ia and SNe Ia
surveys, where changes in $\Omega_{M}$ and $w$ are most significant, the self-
calibration that makes the SH0ES+Pantheon+ measurements of H0 so robust
against inter-survey photometric miscalibration does not meaningfully
stabilize SH0ES+Pantheon+ measurements of $\Omega_{M}$ nor of $w$. SNe Ia
measurements of $\Omega_{M}$ and $w$ are somewhat degenerate, and we therefore
report in Figure 5 the area of the 1$\sigma$ contour in the $\Omega_{M}\times
w$ posterior space as the joint uncertainty in these two parameters. As we
show in the right panel of Figure 4, effectively free-floating inter-survey
offsets ($\sigma_{P,S}>200$) increases the $\Omega_{M}\times w$ joint
posterior uncertainty by 150%. In the center and right panels of Figure 5, the
centroids of $\Omega_{M}$ and $w$ respectively shift by as much as 0.12 and
-0.5 if the inter-survey zeropoint calibration is effectively unconstrained.
Focusing on the more realistic ($\sigma_{P,S}\leq 25$) inter-survey
calibration errors, an increase in inter-survey calibration uncertainty of 25
mmag produces slips of, respectively, 0.04 and -0.17 in the SH0ES+Pantheon+
best-fit values of $\Omega_{M}$ and $w$ (center and right panels of Figure 5)
and of 50% in their joint posterior uncertainty (right panel of Figure 4).
These best-fit shift and posterior uncertainty increases are large,
substantially larger than the corresponding changes in H0. They would, if
realized, represent dramatic changes in the class of cosmology measured by
SH0ES+Pantheon+. Reliable SNe Ia compendium-based measurements of $\Omega_{M}$
and $w$ therefore rely heavily on accurate inter-survey photometric
calibration.

Notably, gray offsets are sufficient for H0 measurements, which rely on the
comparison of ultra low-$z$ ($0.01<z$) to low-$z$ ($0.1<z$) SNe at roughly the
same colors. However, our analysis does not encompass possible chromatic
inter-survey systematics that could subtly effect SH0ES+Pantheon+ measurements
of $\Omega_{M}$ and $w$.

##  4 Discussion and Conclusions

###  4.1 Statistical Power of More Surveys vs Improved Inter-Survey
Calibrations

Ongoing efforts to leverage SNe Ia to gain ever more accurate measurements of
the Universe’s recent expansion could pursue one of two broad tracks: improved
calibration of existing data sets or acquisition of new observations,
introducing more data at the cost of additional inter-survey systematics. With
the hope of providing insight into how limited scientific resources could most
efficiently be used, we use artificial SNe data to study how measurements of
cosmological parameters improve with additional data vs with improved
knowledge of inter-survey systematics.

We produce artificial low-$z$, mid-$z$, and high-$z$ surveys by randomly
drawing 300 SNe logarithmically distributed over redshift ranges of $0<z<0.1$,
$0.1<z<0.4$, and $0.4<z<1$, respectively, with associated $\mu$ values
determined from the best-fit SH0ES+Pantheon+ cosmology, subject to the median
$\mu$ uncertainty of the real Pantheon+ SNe Ia in this redshift range. By
repeating the analysis discussed Section 2.3 on the Pantheon+ data with the
inclusion of each of these artificial surveys, we determine how the
SH0ES+Pantheon+ constraints of H0, $\Omega_{M}$, and $w$ depend on the
redshift distribution of SNe added to the Pantheon+ compendium and on the
uncertainties of the Pantheon+ inter-survey photometric zeropoints. We list
these results in Table 2.

Relative H0 uncertainty when: |  LightCyan Just Pantheon+ |  \+ Low-$z$ Survey |  \+ Mid-$z$ Survey |  \+ High-$z$ Survey
---|---|---|---|---
$\sigma_{P,S}=5$ mmag |  LightCyan 95.5% | 90% | 97.8% | 92.9%
LightCyan $\sigma_{P,S}=10$ mmag |  DarkCyan 100% | 94.6% | 100% | 94.1%
$\sigma_{P,S}=25$ mmag |  LightCyan 107% | 101% | 107% | 361%
$\sigma_{P,S}=50$ mmag |  LightCyan 112% | 105% | 105% | 103%
Relative $\Omega_{M}\times w$ area when: |  LightCyan Just Pantheon+ | \+ Low-$z$ Survey | \+ Mid-$z$ Survey | \+ High-$z$ Survey
$\sigma_{P,S}=5$ mmag |  LightCyan 87.4% | 91.7% | 85% | 53.8%
LightCyan $\sigma_{P,S}=10$ mmag |  DarkCyan 100% | 106% | 99.3% | 63.3%
$\sigma_{P,S}=25$ mmag |  LightCyan 148% | 133% | 137% | 501%
$\sigma_{P,S}=50$ mmag |  LightCyan 165% | 167% | 156% | 114%
Relative $w$ uncertainty reduction when: |  LightCyan Just Pantheon+ | \+ Low-$z$ Survey | \+ Mid-$z$ Survey | \+ High-$z$ Survey
$\sigma_{P,S}=5$ mmag |  LightCyan 91.2% | 93.7% | 83.9% | 64.4%
LightCyan $\sigma_{P,S}=10$ mmag |  DarkCyan 100% | 102% | 97.8% | 75.6%
$\sigma_{P,S}=25$ mmag |  LightCyan 131% | 110% | 124% | 1640%
$\sigma_{P,S}=50$ mmag |  LightCyan 134% | 128% | 136% | 95.7%
Table 2: The changes in the SH0ES+Pantheon+ cosmological parameter posterior
constraints for various inter-survey zeropoint offset prior widths
($\sigma_{P,S}$, different rows) and with the inclusion of low-$z$
($0.01<z<0.1$), mid-$z$ ($0.1<z<0.4$), and high-$z$ ($0.4<z<1.0$) artificial
surveys (different columns). Each survey contains 300 ‘new’ SNe, randomly
distributed logarithmically in the corresponding redshift range. We express
these constraints as percentages of the ‘gold-standard’ scenario, with an
impressive $\Delta\mu_{S}$ prior of 10 mmag and with the Pantheon+ compendium
as it presently exists (the darkly shaded cyan cells). Smaller percentage
values indicate tighter constraints. As expected, the posterior uncertainties
decline as we add surveys and increase as the inter-survey offset priors grow.
H0 is better constrained by the addition of low-$z$ SNe, while $\Omega_{M}$
and $w$ are better constrained with the addition of high-$z$ SNe. The
constraining power of a well-chosen survey of 300 SNe is comparable to an
improvement in the inter-survey zeropoint cross-calibration confidence from
$25$ mmag to $10$ mmag. For example, adding a low-$z$ survey of 300 SNe would
improve the SH0ES+Pantheon+ constraint on H0 by about 6% (from 100% to 94.6%
in this Table) if the inter-survey zeropoints are calibrated to an accuracy of
10 mmag. Similarly, improving the inter-survey zeropoint calibration
uncertainty from 25 mmag to 10 mmag would improve the constraining power of
the existing SH0ES+Pantheon+ data by about 7% (from 107% to 100% in this
Table). Accurate inter-survey cross-calibration is essential if surveys of SNe
Ia are to be fully leveraged.

Examining Table 2, we find, as expected, that the uncertainties in the best-
fit cosmological parameter values decrease both as we add SNe of an
appropriate redshift range, and as we decrease the uncertainty (i.e. Gaussian-
prior width) of the inter-survey zeropoint offsets. The posterior uncertainty
in H0 would be improved by about $6-7$% either with the inclusion of a low-$z$
survey of 300 SNe or with an improvement in the Pantheon+ inter-survey
zeropoint calibration from 25 mmag to 10 mmag.

These results underscore the importance of good inter-survey cross-
calibration. With a compendium of nearly 2000 SNe Ia, a modest improvement in
inter-survey calibration (from 25 mmag to 10 mmag uncertainties, for example)
provides as much additional cosmological insight as an entirely new,
independent survey of 300 additional SNe Ia.

###  4.2 Assessing the Impact of Single Hubble Flow Survey Analysis

The Carnegie Chicago Hubble Program (CCHP) measures H0 using only HF SNe Ia
observed as part of the Carnegie Supernova Project (CSP) low to intermediate
redshift ($z<0.1$) survey (Krisciunas et al., 2017; Phillips et al., 2019)
anchored to a very low-$z$ SNe observed in a variety of surveys tied to tip of
the red giant branch (TRGB) measured galactic distances.

Figure 6:  The dependence of the Hubble Constant, H0, posterior uncertainty on
the width of the $\Delta\mu_{S}$ Gaussian priors when we use only HF SNe Ia
detected by the Carnegie Supernova Project (CSP). In contrast to the
plateauing effect in Figure 4, the posterior uncertainty on the measured value
of H0 increases without bound as we increase the width of the $\Delta\mu_{S}$
priors. Because we use HF SNe from only CSP in this analysis, a single cross-
calibration error between this HF survey and the other surveys with Cepheid-
paired SNe shifts all HF $\mu$ values, and thus the inferred value of H0,
without a statistical penalty, leading to a posterior H0 uncertainty that
increases effectively linearly with $\sigma_{P,S}$ with a slope of $0.77$ km
s-1 Mpc-1 per every 25 mmag of Pantheon+ calibration uncertainty. Focusing on
the conservative $\sigma_{P,S}=25$ mmag estimation of the Pantheon+ cross-
calibration uncertainty (dashed black line), the H0 posterior uncertainty
increases by $0.77$ km s-1 Mpc-1, or a full 1% of the best-fit H0 value,
potentially challenging the goal of measuring H0 to better than 1% accuracy.
This underscores the importance of accurate inter-survey photometric
calibration, as the additional H0 uncertainty depends linearly on the accuracy
of the inter-survey zeropoint calibration.

CCHP was initiated in part to sidestep systematic errors in distance ladder
measurements of cosmological parameters that could arise from cross-
calibrating heterogeneous observations of HF SNe Ia. However, because
observations of TRGB-paired SNe Ia are volume limited (direct observations of
TRGB is only possible in nearby host galaxies with an observed SNe Ia, of
which there are a volume-limited number), CCHP is still forced to use numerous
optical systems to establish the normalized standard candle luminosity of SNe
Ia. Cross-calibration errors, therefore, are still present in CCHP.

In this work, we have thus far demonstrated that the inclusion of multiple
surveys with HF SNe makes the resulting measurement of H0 inherently robust
against large inter-survey calibration errors. In Figure 4, the increase in
the posterior H0 uncertainty has a clear maximum value, no matter how large
the inter-survey zeropoints might be.

To approximate the analysis of CCHP, we repeat the analysis of Section 3.2
using all Cepheid-paired SNe Ia but only CSP HF SNe Ia. Because the CSP SNe Ia
in the Pantheon+ compendium are all below redshift 0.08, they do not provide
strong constraints on $\Omega_{M}$ nor on $w$. We therefore fix those
cosmological parameters to $\Omega_{M}=0.3$ and $w=-1$, varying only $H_{0}$
and the relevant inter-survey zeropoint offsets, $\Delta\mu_{S}$, in the MCMC.

We present the dependence of the H0 posterior uncertainty on the Gaussian
prior width, $\sigma_{P,S}$, of all $\Delta\mu_{S}$ in Figure 6. Notably, and
in contrast to the results of Figure 4, Figure 6 shows no plateauing in the
uncertainty on $H_{0}$. Thus, where the SH0ES+Pantheon+ constraint on H0
cannot grow arbitrarily large, even if the Pantheon+ inter-survey zeropoints
are grossly miscalibrated, the accuracy of the CCHP constraint on H0 is
entirely dependent on the accuracy of the photometric calibration between CSP
and the remaining TRGB calibrator surveys. Taking the conservative but not
outlandish estimation of $\sigma_{P,S}\simeq 25$ mmag used throughout this
work, the additional H0 uncertainty that CCHP might experience as a result of
inter-survey photometric miscalibration photometric could be as large as
$0.77$ km s-1 Mpc-1.

###  4.3 Conclusion

In this paper, we assessed the extent to which miscalibrated zeropoint offsets
between the surveys of the Pantheon+ SN Ia compendium, $\Delta\mu_{S}$, could
degrade constraints on measurements of cosmological parameters. We determined
that (effectively) freely floating photometric zeropoint offsets,
$\Delta\mu_{S}$, could contribute an additional statistical uncertainty of
$0.15$ km s-1 Mpc-1 to SH0ES+Pantheon+ measurements of H0, but could increase
the SH0ES+Pantheon+ joint uncertainty on $\Omega_{M}$ and $w$ by as much as
150%. Those floating $\Delta\mu_{S}$ could shift the best-fit values of H0,
$\Omega_{M}$, and $w$ by $0.35$km s-1 Mpc-1, 0.12, and 0.5, respectively.
Notably, we determined that the systematic uncertainty contributions of the
$\Delta\mu_{S}$ could grow no larger than these maximum values, as larger
inter-survey zeropoint offset uncertainties do not improve the correspondence
between the cosmological fit and the Pantheon+ data. We explained the reasons
for this plateauing in Appendix B.

Focusing specifically on the Hubble Tension (HT) between SH0ES SNe Ia distance
ladder measurements of H0 and Planck $\Lambda$CDM-based inferences H0, we
found that uncertainties in the inter-survey zeropoint offsets could reduce
this tension by only $0.03\sigma$. Gray miscalibrations between Pantheon+
surveys cannot account for HT, at least not alone.

Our findings suggest that SNe Ia compendium-based distance ladder measurements
of cosmological parameters benefit from an effective self-calibration of gray
systematics between surveys, capping the inter-survey systematic error in
compendium-based measurements of cosmological parameters. However, this self-
calibration occurs only when there are numerous surveys that span both the
middle and Hubble Flow (HF) rungs of the cosmic distance ladder. Programs that
leverage only one HF survey, such as CCHP, are more susceptible to undetected
inter-survey systematic than those that include many HF surveys.

We focused exclusively on gray inter-survey calibration errors. Although
chromatic cross-survey photometric calibration errors, owing to distinct
filter systems, chromatic telescope throughputs, etc., are a matter of
legitimate concern, the SNe Ia processing pipeline does not presently lend
itself to varying the chromatic inter-survey photometry. Such chromatic
differences are embedded in the SNe Ia lightcurve fitting algorithms, making
the prospect of varying them significantly more complicated than the fixed,
post-lightcurve fitting $\mu$ offsets that we considered here. A future, more
in-depth analysis could take up the task of measuring the effects of chromatic
inter-survey calibration errors by varying the SNe light curve parameters as
part of the larger MCMC sampling loop.

Accurate inter-survey calibrations in compendia of SNe Ia are still extremely
important. SH0ES+Pantheon+ measurements of $\Omega_{M}$ and $w$ are reliable
only if the inter-survey zeropoints are well-calibrated. Further, the
additional cosmic constraining power achieved when additional surveys are
added to the Pantheon+ compendium is comparable to the greater accuracy
achieved by improving the inter-survey zeropoint calibration (see Section
4.1). As more surveys of SNe Ia are gathered, effort should be taken to
integrate them into the Pantheon+ (or a future larger) compendium, and the
relative survey flux scales should be calibrated with utmost accuracy.

The creation of a well-calibrated SNe Ia compendium is a monumental
undertaking, and our results underscore how valuable such compendia are in
measuring the recent expansion history of the Universe. Even if a SNe
compendium suffers from undiagnosed inter-survey calibration errors, it can
still provide meaningful cosmic constraints, especially on H0. We particularly
demonstrate that the SH0ES+Pantheon+ measurement of H0 is degraded by $<1$%
even for unrealisticaly larger inter-survey calibration errors, and thus that
cross-survey calibration uncertainty does not represent a substantial
impediment to the longstanding objective of measuring H0 to within 1%. The HT
between Cepheid + SNe Ia distance ladder measurements of H0 and Planck
$\Lambda$CDM inferences of H0 is not ascribable to gray inter-survey
miscalibration, and is not likely to diminish with additional surveys of even
hundreds of spectroscopically matched SNe Ia.

However, we do find that the SH0ES+Pantheon+ measurements of $\Omega_{M}$ and
$w$ are highly sensitive to the accuracy of inter-survey cross-calibration.
Our results emphatically do not demonstrate that the SH0ES+Pantheon+ cosmology
is inaccurate, but rather underscores the immense importance of the
painstaking cross-survey calibration performed by the Pantheon+ analysis team.
Only a well orchestrated, all sky campaign to observe thousands of
spectroscopically-matched SNe on a single photometric system, such as the
ambitious SNe Ia observing program of the Vera Rubin Observatory, (Ivezić et
al., 2019), could use SNe Ia to measure $\Omega_{M}$ and $w$ free of possible
cross-survey calibration errors.

##  5 Acknowledgements

We are grateful to the US Department of Energy for their support of our
assessments of the cosmic constraints offered by SNe Ia, under DOE grant DE-
SC0007881, and to Harvard University for its support of our program. Dillon
Brout acknowledges support for this work was provided by NASA through the NASA
Hubble Fellowship grant HST-HF2-51430.001 awarded by the Space Telescope
Science Institute, which is operated by Association of Universities for
Research in Astronomy, Inc., for NASA, under contract NAS5-26555. This work
was completed in part with resources provided by the Harvard University FAS
Research Computing and University of Chicago Midway Research Computing Center.
SN distance determination pipeline managed by PIPPIN (Hinton & Brout, 2020).
D.S. is supported by DOE grant DE-SC0010007 and the David and Lucile Packard
Foundation. D.S. is supported in part by the National Aeronautics and Space
Administration (NASA) under Contract No. NNG17PX03C issued through the Roman
Science Investigation Teams Programme.

##  Appendix A Effects of Inter-Survey Zeropoint Offsets on SNe Ia Distance
Moduli

Fundamentally, adjusting distance moduli represents an adjustment of the
supernova flux from an initial value, $f_{init}$, to a shifted value,
$f_{shift}$:

$\displaystyle f_{shift}=f_{init}10^{-\Delta\mu_{S}/2.5}\ .$ (A1)

By adjusting the distance moduli in Survey $S$, we conjecture that the fluxes
of the SNe in that survey are greater than initially reported (if
$\Delta\mu_{S}>0$) or less than initially reported (if $\Delta\mu_{S}<0$).

Adjusting the fluxes of Cepheid-paired SNe also changes our measurement of the
normalized SN Ia absolute luminosity from an initial value, $L_{init}$, to a
shifted value, $L_{shift}$. As we will see, the logarithmic ratio of this
adjusted luminosity introduces an additional distance modulus offset to the
distance moduli of the Hubble Flow SNe. For Cepheid-paired supernova $i$, this
luminosity-sourced distance modulus offset is:

$\displaystyle\Delta\mu_{L,i}=2.5\log_{10}{\frac{L_{shift,i}}{L_{init,i}}}=2.5\log_{10}{\frac{L_{ceph}f_{shift,i}/f_{ceph}}{L_{ceph}f_{init,i}/f_{ceph}}}=2.5\log_{10}{\frac{f_{shift,i}}{f_{init,i}}}=2.5\log_{10}(10^{-\Delta\mu_{S}/2.5})=-\Delta\mu_{S}\
,$ (A2)

where we used the fact that the ratio of the initial and shifted fluxes is the
exponential of the appropriate distance modulus residual survey offset
(Equation A1) and where $L_{ceph}$ and $f_{ceph}$ are respectively the
absolute luminosity and measured flux of the Cepheid star to which SNe $i$ is
tied. The best-fit inferred value of this luminosity distance modulus offset,
$\Delta\mu_{L}$, is the weighted mean of all the individual luminosity
distance modulus offsets:

$\displaystyle\Delta\mu_{L}=\frac{\sum_{i=1}^{N_{Ceph}}\Delta\mu_{L,i}\sigma_{\mu,i}^{-2}}{\sum_{i=1}^{N_{Ceph}}\sigma_{\mu,i}^{-2}}=-\frac{\sum_{i=1}^{N_{Ceph}}\Delta\mu_{S,i}\sigma_{\mu,i}^{-2}}{\sum_{i=1}^{N_{Ceph}}\sigma_{\mu,i}^{-2}}\
,$ (A3)

where $\sigma_{\mu,i}$ is the reported uncertainty in the distance modulus of
SN $i$, $\Delta\mu_{S,i}$ is the distance modulus offset applied to the survey
containing SN $i$, and the sum is taken only over Cepheid-paired SNe.

The initially measured distance modulus for Hubble Flow supernova $i$,
$\mu_{init,i}$, is derived from $f_{init,i}$ and $L_{init,i}$:

$\displaystyle\mu_{\textrm{meas,
init},i}=25+5\log_{10}{\Big{(}\sqrt{\frac{L_{\mathrm{init}}}{4\pi
f_{\mathrm{init},i}}}\mathrm{Mpc}^{-1}\Big{)}}\ .$ (A4)

Applying inter-survey distance modulus offsets, $\Delta\mu_{S}$, to the entire
reported set of Pantheon+ distance moduli (both Cepheid and Hubble Flow)
adjusts the measured Hubble Flow distance moduli in two ways: by changing the
measured flux and by changing the measured value of the SN Ia luminosity. The
shifted distance modulus of Hubble Flow supernova $i$, $\mu_{\textrm{meas,
shift, }i}$, in survey $S$ is thus:

$\displaystyle\mu_{\textrm{meas, shift,
}i}=25+5\log_{10}{\Big{(}\sqrt{\frac{L_{\mathrm{shift}}}{4\pi
f_{\mathrm{shift},i}}}\mathrm{Mpc}^{-1}\Big{)}}$ (A5)
$\displaystyle=25+5\log_{10}{\Big{(}\sqrt{\frac{L_{\mathrm{init}}}{4\pi
f_{\mathrm{init},i}}}\mathrm{Mpc}^{-1}\Big{)}}\ \ \
+2.5\log_{10}{\Big{(}\frac{L_{\mathrm{shift}}}{L_{\mathrm{init}}}\Big{)}}+2.5\log_{10}{\Big{(}\frac{f_{\mathrm{init},i}}{f_{\mathrm{shift},i}}\Big{)}}=\mu_{\textrm{meas,
init}}+\Delta\mu_{L}+\Delta\mu_{S}\ .$

As we discuss further in Section B, because $\Delta\mu_{L}$ is the average
inter-survey zeropoint offset of all Cepheid paired SNe,
$\Delta\mu_{L}+\Delta\mu_{S}$ only shifts the zeropoint of survey $S$ relative
to the average shift of surveys with Cepheid-paired SNe.

##  Appendix B Self-Correcting Effect of Cepheid-paired SNe Data Sets in
Determining Cosmological Parameters

Figure 7:  A simple artificial SNe Ia data set with two surveys: one of which
(Survey A, blue) includes Cepheid paired and Hubble flow SNe Ia, while the
other (Survey B, pink) has only Hubble flow SNe Ia. From top to bottom, we
introduce the following inter-survey zeropoint offsets to these data: no
offset, an offset of 1 magnitude to survey B, an offset of 1 magnitude to
survey A, and a 1 magnitude offset to both surveys. The values of the offsets
for Surveys A and B are indicated with dashed lines of the appropriate color.
Because only Survey A includes Cepheid-paired SNe, the luminosity offset,
$\Delta\mu_{L}$ indicated with a black dashed line, is exactly opposite the
applied distance modulus offset of Survey A (see Equation 3). According to
Equation 7, all surveys are shifted by $\Delta\mu_{L}$. Thus, applying a
zeropoint offset to Survey A does not shift Survey A, but effectively
introduces the opposite shift to Survey B. Inter-survey offsets can only
separate Surveys A and B, but cannot shift the surveys collectively.
Therefore, the relative survey offsets can grow only so large before they
become incompatible with a single cosmic fit. This effect is why, in Figure
LABEL:fig:H0PostOnGaussPrior, the posterior uncertainties of H0, $\Omega_{M}$,
and $w$ plateau at some threshold $\Delta\mu_{S}$ prior width.

In Figure 4, the widths of the H0 and $\Omega_{M}\times w$ posterior
distributions do not increase with $\Delta\mu_{S}$ when $\Delta\mu_{S}>200$
mmag. This occurs because, as we show in Equation 7, the updated SN Ia
luminosity introduces an additional distance modulus offset, $\Delta\mu_{L}$.
This luminosity offset is opposite in sign and equal in magnitude to the
weighted average of the inter-survey zeropoints of all Cepheid paired SNe
(Equation 3). Therefore, any $\Delta\mu_{S}$ offset shared between surveys has
no effect on the $\chi^{2}$ statistic because $\Delta\mu_{L}$ exactly cancels
it. Differential $\Delta\mu_{S}$ offsets are not cancelled and pull the SNe
distance moduli vertically apart in the Hubble Diagram. As these differential
offsets become larger, the distance moduli residuals become less compatible
with a single set of cosmological parameters.

To illustrate this phenomenon, we consider a simplified SNe compendium that
consists of only two surveys: Survey A, which contains both Hubble Flow and
Cepheid paired SNe Ia, and survey B which contains only Hubble Flow SNe Ia. As
we show in Figure 7, applying an inter-survey offset to only one survey pulls
the $\Delta\mu$ values of each survey vertically apart, while applying the
same offset to both surveys produces no net change in $\Delta\mu$. Thus, the
best-fit value of H0 cannot change without limit, even for arbitrarily large
inter-survey offsets.

This cancellation of $\Delta\mu_{L}$ with the average inter-survey zeropoint
offsets of Cepheid-paired SNe is perfect only because we assume (incorrectly)
that the distance moduli of the Cepheid-calibrators are known exactly. A more
complete analysis of this data would include additional degrees of freedom to
reflect the uncertainty of HST measurements of Cepheid fluxes. Adjustments to
the Cepheid distance moduli would add an additional term to Equation 3,
partially decoupling $\Delta\mu_{L}$ from the average $\Delta\mu_{S}$ of
Cepheid-paired SNe.

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