# Learning to Segment Medical Images from Few-Shot Sparse Labels: Suplementary
Material

Pedro H. T. Gama Hugo Oliveira Jefersson A. dos Santos

(2021)

##  1 Additional results

Table 1: Jaccard Score for Montgomery Lungs segmentation.

From Scratch From JSRT From OpenIST From Shenzhen WeaSeL Points Points Points
Points Points Shot 1 5 10 20 1 5 10 20 1 5 10 20 1 5 10 20 1 5 10 20 1 28.64
31.26 35.99 39.91 15.71 29.78 43.19 49.71 88.42 84.54 84.82 85.89 82.26 78.84
79.49 84.08 43.23 63.28 77.49 78.40 5 28.93 33.34 46.75 60.65 28.92 50.63
63.43 71.06 83.96 86.24 88.79 89.16 80.31 84.09 86.79 89.14 59.67 80.21 81.55
85.26 10 30.52 48.87 66.53 76.56 38.16 64.46 73.88 78.73 87.54 88.69 89.51
90.15 84.38 86.54 89.19 90.97 69.64 78.92 84.54 87.05 20 39.15 63.05 77.65
82.16 52.73 73.82 79.81 83.29 89.48 90.15 90.02 91.33 86.07 89.46 89.07 91.20
74.17 85.01 87.80 89.02 Grid Grid Grid Grid Grid Shot 8 12 16 20 8 12 16 20 8
12 16 20 8 12 16 20 8 12 16 20 1 46.92 36.34 37.81 40.40 62.90 59.17 56.89
49.50 91.50 90.61 92.08 92.45 92.89 91.07 91.23 92.54 87.81 83.72 82.68 78.03
5 50.22 58.43 54.46 48.80 82.28 77.55 75.42 71.00 93.46 92.06 91.64 91.91
93.58 91.18 91.46 92.32 91.74 89.86 88.43 86.12 10 70.54 79.54 77.58 59.96
85.30 83.05 80.90 76.87 92.90 92.54 91.63 92.16 92.94 92.24 92.09 92.62 92.05
91.14 89.93 87.71 20 88.70 87.11 85.47 83.13 88.99 87.27 85.41 83.62 93.40
93.24 92.84 92.78 93.23 92.59 92.12 92.04 92.89 92.14 91.54 90.36

Table 2: Jaccard Score for Shenzhen Lungs segmentation.

From Scratch From JSRT From OpenIST From Montgomery WeaSeL Points Points
Points Points Points Shot 1 5 10 20 1 5 10 20 1 5 10 20 1 5 10 20 1 5 10 20 1
32.26 35.75 38.07 43.05 27.11 38.09 47.46 58.88 83.42 81.27 86.57 86.03 81.99
79.33 84.82 84.37 53.40 59.86 71.58 74.80 5 36.66 47.53 54.35 58.12 51.78
65.35 67.90 69.09 87.35 87.31 88.46 85.61 85.90 86.96 85.42 83.70 68.91 82.97
81.78 81.26 10 40.57 55.62 65.17 71.85 58.36 69.33 72.92 75.01 87.74 85.51
87.51 85.48 86.24 85.29 86.26 84.14 69.82 80.57 82.58 83.00 20 48.14 65.88
73.50 77.62 61.15 74.01 77.04 78.09 86.31 84.70 85.74 84.94 83.03 84.82 84.84
83.10 71.44 80.71 82.39 82.78 Grid Grid Grid Grid Grid Shot 8 12 16 20 8 12 16
20 8 12 16 20 8 12 16 20 8 12 16 20 1 44.67 54.18 40.81 40.35 69.78 63.92
59.97 52.63 84.05 81.50 83.45 82.83 81.79 80.50 81.07 81.65 79.90 76.85 75.89
78.63 5 52.53 50.58 52.81 49.43 75.19 71.59 71.60 71.26 87.24 85.85 87.21
85.73 86.40 84.06 85.74 86.14 82.88 83.59 83.18 79.72 10 73.60 76.38 69.71
55.20 80.92 78.97 78.04 75.16 86.99 86.71 87.01 83.39 86.12 84.64 85.63 82.96
85.53 84.09 82.85 80.97 20 81.76 81.77 80.97 76.54 83.56 82.37 81.43 79.00
86.86 85.94 86.40 84.55 85.91 85.26 84.76 82.88 86.93 86.33 85.73 84.11

(a)

(b)

Figure 1: Jaccard results for lung segmentation in two target datasets:
Montgomery (a) and Shenzhen (b). Solid lines indicate the performance of the
sparse labeling (Points on the top and Grid on the bottom), while the dashed
line presents the performance of dense labeling for the target using the same
methods and baselines.

As mentioned in the paper, the results for Montgomery and Shenzhen lungs
segmentation task, had results similar to the OpenIST dataset, as can be seen
in tables 1 and 2, and in Figure 1. So the same conclusions from the paper
holds for these datasets: From Scratch being the worst baseline, Fine Tuning
from similar domains providing the best results, while WeaSeL reaching
comparable results to the Fine-Tuning baselines with more data samples.

Table 3: Jaccard Score for JSRT Clavicles segmentation.

From Scratch From JSRT Hearts From JSRT Lungs WeaSeL Points Points Points
Points Shot 1 5 10 20 1 5 10 20 1 5 10 20 1 5 10 20 1 4.36 4.37 4.90 5.30 5.32
6.85 7.73 14.55 7.33 7.31 7.85 9.13 7.52 7.65 10.67 12.04 5 4.23 4.64 5.40
13.32 8.98 16.20 15.41 23.60 7.65 10.50 11.58 17.38 7.44 10.77 15.23 22.04 10
4.53 7.50 14.21 22.80 9.49 18.54 22.66 28.73 7.44 11.21 13.81 21.38 8.33 17.38
24.62 29.63 20 5.83 14.75 23.94 31.48 10.63 21.17 26.06 31.62 9.17 13.78 17.12
26.95 11.55 23.58 33.27 39.58 Grid Grid Grid Grid Shot 8 12 16 20 8 12 16 20 8
12 16 20 8 12 16 20 1 3.05 3.27 2.87 2.89 3.99 .80 9.89 9.33 19.43 20.47 14.38
12.31 14.74 7.17 11.76 11.00 5 2.93 2.54 3.05 2.71 4.66 .96 .07 1.92 18.37
23.71 25.29 13.20 30.35 21.06 22.76 6.03 10 3.09 1.97 3.40 3.09 32.55 27.86
25.46 18.09 31.59 28.15 30.12 23.83 48.88 36.44 35.56 24.74 20 11.02 7.54 5.22
3.58 47.00 36.19 27.93 25.06 49.58 39.37 37.10 31.78 59.18 48.44 45.42 36.89

Similar to the heart segmentation in JSRT, the clavicles segmentation followed
the same tendency, as showed in table 3. Although, having lower scores in
general, we can see that the methods hold the same order of performance.
WeaSeL proving to be a reliable method when the target task is absent in
source tasks, or when they highly differ from sources. The overall low scores,
can be explained by the use of small resized images, that can deform and blend
multiple objects which is problematic for small structures like clavicles.

##  2 Visual Examples

In this section we show some examples of segmentation produced by our method
WeaSeL.

Below, in Table 4 is the relation of task and examples. In each example, the
lines are the number of shots in the experiment, and the columns are the
parameters $n$ for points and $s$ for grids. These parameters, were the ones
used to simulate the sparse labels of the support set of the few-shot task.

Table 4: Relation of examples and tasks Dataset | Class | Figures
---|---|---
JSRT | Lungs | Figures 2 and 3
JSRT | Clavicles | Figures 4 and 5
JSRT | Heart | Figures 6 and 7
Montgomery | Lungs | Figures 8 and 9
OpenIST | Lungs | Figures 10 and 11
Shenzhen | Lungs | Figures 12 and 13

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 2: Example 1, Visual Examples of Segmentation from JSRT Lungs task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 3: Example 2, Visual Examples of Segmentation from JSRT Lungs task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 4: Example 3, Visual Examples of Segmentation from JSRT Clavicles task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 5: Example 4, Visual Examples of Segmentation from JSRT Clavicles task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 6: Example 5, Visual Examples of Segmentation from JSRT Heart task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 7: Example 6, Visual Examples of Segmentation from JSRT Heart task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 8: Example 7, Visual Examples of Segmentation from Montgomery Lungs
task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 9: Example 8, Visual Examples of Segmentation from Montgomery Lungs
task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 10: Example 9, Visual Examples of Segmentation from OpenIST Lungs task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 11: Example 10, Visual Examples of Segmentation from OpenIST Lungs task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 12: Example 11, Visual Examples of Segmentation from Shenzhen Lungs
task

| Image | Ground Truth
---|---|---
|  |
| Points | Grid
Shots | 1 | 5 | 10 | 20 | 8 | 12 | 16 | 20
1 |  |  |  |  |  |  |  |
5 |  |  |  |  |  |  |  |
10 |  |  |  |  |  |  |  |
20 |  |  |  |  |  |  |  |

Figure 13: Example 12, Visual Examples of Segmentation from Shenzhen Lungs
task


# Dark sector interaction and the supernova absolute magnitude tension

Rafael C. Nunes<EMAIL_ADDRESS>Divisão de Astrofísica, Instituto
Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758, São José dos
Campos, 12227-010, São Paulo, Brazil  Eleonora Di Valentino  eleonora.di-
<EMAIL_ADDRESS>Institute for Particle Physics Phenomenology,
Department of Physics, Durham University, Durham DH1 3LE, UK.

###### Abstract

It has been intensively discussed if modifications in the dynamics of the
Universe at late times is able or not to solve the $H_{0}$ tension. On the
other hand, it has also been argued that the $H_{0}$ tension is actually a
tension on the supernova absolute magnitude $M_{B}$. In this work, we robustly
constraint $M_{B}$ using Pantheon Supernovae Ia (SN) sample, Baryon Acoustic
Oscillations (BAO), and Big Bang Nucleosynthesis (BBN) data, and assess the
$M_{B}$ tension by comparing three theoretical models, namely the standard
$\Lambda$CDM, the $w$CDM and a non-gravitational interaction (IDE) between
dark energy (DE) and dark matter (DM). We find that the IDE model can solve
the $M_{B}$ tension with a coupling different from zero at 95% CL, confirming
the results obtained using a $H_{0}$ prior.

##  I Introduction

The standard $\Lambda$-Cold Dark Matter ($\Lambda$CDM) scenario provides a
wonderful fit to astronomical observations carried out over the past two
decades. On the other hand, recently some tensions and anomalies became
statistically significant while analyzing different data sets, placing the
$\Lambda$CDM cosmology in a crossroad. The most discussed and statistically
significant tension in the literature is in the estimation of the Hubble
constant, $H_{0}$, between the Cosmic Microwave Background (CMB) and the
direct local distance ladder measurements. Assuming the $\Lambda$CDM scenario,
Planck-CMB data analysis provides $H_{0}=67.4\pm 0.5$ km s-1Mpc-1 Aghanim  _et
al._ (2020), which is in $4.4\sigma$ tension with the local measurement
$H_{0}=74.03\pm 1.42$ km s-1Mpc-1 Riess  _et al._ (2019) by the SH0ES team, or
$4.2\sigma$ considering the latest one $H_{0}=73.2\pm 1.3$ km s-1Mpc-1 Riess
_et al._ (2021). Additionally, many other late time measurements are in
agreement with a higher value for the Hubble constant (see the discussion in
Verde  _et al._ (2019); Riess (2019); Di Valentino (2020) and in tension with
the Planck-CMB estimate, as for example the Megamaser Cosmology Project Pesce
_et al._ (2020) that gives $H_{0}=73.9\pm 3.0{\rm\,km\,s^{-1}\,Mpc^{-1}}$, or
using the Surface Brightness Fluctuations Blakeslee  _et al._ (2021) that find
$H_{0}=73.3\pm 0.7\pm 2.4{\rm\,km\,s^{-1}\,Mpc^{-1}}$. The lower value of
$H_{0}$ inferred from the Planck-CMB data is instead in very good agreement
with Baryon Acoustic Oscillations (BAO) + Big Bang Nucleosynthesis (BBN)
constraints Alam  _et al._ (2021), and other CMB experiments like ACTPol-DR4
Aiola  _et al._ (2020) or SPT-3G Dutcher  _et al._ (2021). Finally, there are
measurements in the middle that can not discriminate between the two values,
such as the Tip of the Red Giant Branch Freedman (2021) or the astrophysical
model dependent measurements based on the strong lensing effect Birrer  _et
al._ (2020). Motivated by these observational discrepancies, unlikely to
disappear completely by introducing multiple and unrelated systematic errors,
it has been widely discussed in the literature whether new physics beyond the
standard cosmological model can solve the $H_{0}$ tension (see Refs. Di
Valentino  _et al._ (2020a, 2021a); Perivolaropoulos and Skara (2021) and
reference therein for a review).

The SH0ES team measures the absolute peak magnitude, $M_{B}$, of Type Ia
supernovae (SN), assumed to be standard candles, by calibrating the distances
of SN host galaxies to local geometric distance anchors via the Cepheid period
luminosity relation. The magnitude $M_{B}$ is then converted into a value of
$H_{0}$ via the magnitude-redshift relation of the Pantheon SN sample Scolnic
_et al._ (2018). Therefore, it has been argued that the $H_{0}$ tension is
actually a tension on the supernova absolute magnitude $M_{B}$ Efstathiou
(2021); Camarena and Marra (2021), because the SH0ES $H_{0}$ measurement comes
directly from $M_{B}$ estimates. The CMB constraint on the sound horizon to
the SN absolute magnitude $M_{B}$ using the parametric-free inverse distance
ladder predicts $M_{B}=-19.401\pm 0.027$ mag Camarena and Marra (2020), while
the the SN measurements from SH0ES corresponds to $M_{B}=-19.244\pm 0.037$ mag
Camarena and Marra (2021). These measurements are at $3.4\sigma$ tension.
Thus, as argued in Efstathiou (2021); Camarena and Marra (2021), rather than
explaining the $H_{0}$ tension one should instead focus on the supernova
absolute magnitude tension, because this is what the Cepheid calibrations are
designed to measure. For this reason, it has been shown that modifications of
the expansion history at late times can not help with the $H_{0}$ and/or
$M_{B}$ tension, motivating discussions and investigations in this direction
Marra and Perivolaropoulos (2021); Alestas  _et al._ (2021); Alestas and
Perivolaropoulos (2021); Adil  _et al._ (2021); Zhou  _et al._ (2021); Felice
_et al._ (2020); De Felice  _et al._ (2021). On the other hand, other
possibilities, including early time solutions that introduce new physics prior
to recombination, also has been considered as good alternatives to solve the
$H_{0}$ tension (see Poulin  _et al._ (2019); Mörtsell and Dhawan (2018);
Niedermann and Sloth (2021); Ye and Piao (2020); Vagnozzi (2021); Chudaykin
_et al._ (2020); D’Amico  _et al._ (2021); Seto and Toda (2021); Blinov  _et
al._ (2020); Hryczuk and Jodłowski (2020); Di Valentino  _et al._ (2018);
Braglia  _et al._ (2020); Lin  _et al._ (2020); Smith  _et al._ (2021); Murgia
_et al._ (2021); Keeley  _et al._ (2020); Velten  _et al._ (2021); Krishnan
_et al._ (2021); Vagnozzi  _et al._ (2021); de la Macorra  _et al._ (2021) and
reference therein), even if they are not completely successful Arendse  _et
al._ (2020); Lin  _et al._ (2021).

In particular, alternative scenarios involving a non-gravitational interaction
between the two main dark species of our Universe, namely, dark matter (DM)
and dark energy (DE), have been intensively studied as a possibility to
resolve the current cosmological $H_{0}$ tension (see for example Kumar and
Nunes (2016, 2017); Di Valentino  _et al._ (2017); Yang  _et al._ (2019a,
2018, b); Kumar  _et al._ (2019); Pan  _et al._ (2019a, b); Di Valentino  _et
al._ (2020b, c); Di Valentino and Mena (2020); Di Valentino  _et al._ (2021b);
Gómez-Valent  _et al._ (2020); Yang  _et al._ (2020a); Lucca and Hooper
(2020); Martinelli  _et al._ (2019); Yang  _et al._ (2020b); Yao and Meng
(2020a); Pan  _et al._ (2020); Di Valentino (2020); Yao and Meng (2020b);
Amirhashchi and Yadav (2020); Yang  _et al._ (2021a); Gao  _et al._ (2021);
Lucca (2021a); Kumar (2021); Yang  _et al._ (2021b); Lucca (2021b); Halder and
Pandey (2021) for a short list and reference therein). This class of models
has been listed as an example of ”not working” solution, i.e. able to solve
the $H_{0}$ tension, but unable to alleviate the $M_{B}$ tension. Therefore,
we aim with this paper to check if modifications to the late time evolution
involving a non-gravitational coupling different from zero in the dark sector,
can bring the $M_{B}$ measured from distant supernovae of the Pantheon sample
and calibrated by BAO + BBN into agreement with the value obtained calibrating
the local SN with the Cepheid measurements. We are interested in the
background only analysis, so we don’t include CMB data in this paper. But,
without loss of generality, using the mentioned data, we will get good
accuracy in our constraints. Our main conclusion is that an Interacting DE-DM
model (IDE) with a dark coupling different from zero is able to solve the
$H_{0}-M_{B}$ tension, and it is in agreement with the Pantheon SN sample.

This paper is structured as follows. In Section II we introduce the
Interacting DE-DM model considered. In Section III, we present the data sets
and methodology used in this work. In Section IV, we discuss the main results
of our analysis. In Section V, we outline our final considerations and
perspectives.

##  II Theoretical framework

In this Section we introduce the Interacting DE-DM model studied here. In a
homogeneous and isotropic Universe, the dark interaction is quantified as

$\displaystyle\nabla_{\mu}T_{i}^{\mu\nu}=Q_{i}^{\nu}\,,\quad\sum\limits_{\mathrm{i}}{Q_{i}^{\mu}}=0~{},$
(1)

where the index $i$ runs over DM and DE. The four-vector $Q_{i}^{\mu}$ governs
the interaction. We can assume that $Q_{i}^{\mu}$ is given by

$\displaystyle Q_{i}^{\mu}=(Q_{i}+\delta
Q_{i})u^{\mu}+a^{-1}(0,\partial^{\mu}f_{i}),$ (2)

where $u^{\mu}$ is the velocity four-vector and $Q_{i}$ is the background
energy transfer. Let us note that from now on we shall use the notation
$Q_{i}\equiv Q$. The symbol $f_{i}$ refers to the momentum transfer potential.
In the Friedmann-Lemaître-Robertson-Walker (FLRW) background, one can write
down the conservation equations of the DM and DE densities as

$\displaystyle\dot{\rho}_{c}+3H\rho_{c}=Q~{},$ (3)
$\displaystyle\dot{\rho}_{x}+3H(1+w)\rho_{x}=-Q~{},$ (4)

where $H=\dot{a}/a$ is the expansion rate of the Universe and $w$ is the
equation of state (EoS) of DE.

In the present work, we consider a very well known parametric form of the
interaction function $Q$, namely, $Q=\mathcal{H}\xi\rho_{x}$, where $\xi$ is
the coupling parameter between the dark components. From the sign of $\xi$,
one can identify the direction of the energy flow between the dark sectors.
The condition $\xi<0$ corresponds to the energy flow from DM to DE, and
$\xi>0$ represents the opposite scenario. The functional form
$Q=\mathcal{H}\xi\rho_{x}$ can avoid the instabilities in the perturbations at
early times on the dark sector species, although it will not be necessary to
consider the evolution of the perturbations in this work. It should be noted
that the inclusion of the global factor $\mathcal{H}$ into the interaction
function $Q$ was motivated to quantify a possible global interaction through
cosmic history. Although, as already argued by the authors Väliviita  _et al._
(2008), the interaction in the dark sectors should depend on the local
quantities too. Nevertheless, as the present interaction function is widely
studied in the literature, we aim to revisit this model.

On the other hand, the choice of the interaction function is not unique and it
is very difficult to provide a specific functional form since the
nature/properties of both dark components is completely unknown at present.
From the observational perspective, we do not find any strong signal which
could reveal the nature of the dark components, and hence, we believe that the
nature of the interaction function will probably remain unknown for the next
decade(s). We can only approximate the interaction function $Q$ through
theoretical arguments and the consistency with observational data. However, it
has been argued by many investigators that the interaction between the dark
components may appear from some effective field theory Gleyzes  _et al._
(2015), disformal coupling de Bruck and Morrice (2015), axion monodromies
D’Amico  _et al._ (2016), varying dark matter mass and fundamental constants
Marsh (2017) and Horndeski theories Kase and Tsujikawa (2020). Thus, one can
see that some viable formalism can be given for the interacting dark energy
theory. Motivated to check the strength of this dark coupling and its
observational viability, let us assume in the present work the simple
parametric function $Q=\mathcal{H}\xi\rho_{x}$ to quantify such dark
interaction.

Table 1: Constraints at 95% CL for the four models under consideration in this work for the Pantheon + BAO + BBN joint analysis. The parameter $H_{\rm 0}$ is measured in units of km s-1 Mpc-1 and $M_{B}$ in units of mag. The last column represents the degree of tension with $M_{B}$ from SH0ES. Model | $M_{B}$ | $H_{0}$ | $\Omega_{m}$ | $w$ | $\xi$ | Tension
---|---|---|---|---|---|---
$\Lambda$CDM | $-19.393^{+0.042}_{-0.040}$ | $68.5^{+1.2}_{-1.1}$ | $0.302^{+0.025}_{-0.025}$ | -1 | 0 | 3.4$\sigma$
$w$CDM | $-19.376^{+0.095}_{-0.11}$ | $69.2^{+3.3}_{-3.7}$ | $0.306^{+0.032}_{-0.034}$ | $-1.02^{+0.11}_{-0.11}$ | 0 | 2.1$\sigma$
IDE | $-19.385^{+0.094}_{-0.089}$ | $69.1^{+3.1}_{-2.8}$ | $0.274^{+0.044}_{-0.050}$ | -0.999 | $>-0.35$ | 2.4$\sigma$
Table 2: Constraints at 95% CL for the four models under consideration in this work for the Pantheon + BAO + BBN + $M_{B}$ joint analysis. The parameter $H_{\rm 0}$ is measured in units of km s-1 Mpc-1 and $M_{B}$ in units of mag. The last column represents the degree of tension with $M_{B}$ from SH0ES. Model | $M_{B}$ | $H_{0}$ | $\Omega_{m}$ | $w$ | $\xi$ | Tension
---|---|---|---|---|---|---
$\Lambda$CDM | $-19.340^{+0.047}_{-0.046}$ | $70.0^{+1.4}_{-1.4}$ | $0.314^{+0.024}_{-0.025}$ | -1 | 0 | 2.1$\sigma$
$w$CDM | $-19.285^{+0.049}_{-0.060}$ | $72.4^{+1.9}_{-2.2}$ | $0.323^{+0.028}_{-0.025}$ | $-1.10^{+0.09}_{-0.10}$ | 0 | 1.0$\sigma$
IDE | $-19.288^{+0.064}_{-0.063}$ | $72.3^{+2.4}_{-2.3}$ | $0.256^{+0.061}_{-0.065}$ | -0.999 | $-0.31^{+0.27}_{-0.28}$ | 0.9$\sigma$

Figure 1: Two-dimensional joint posterior distributions in the $M_{B}$-$H_{0}$
plane, with the corresponding 68% CL and 95% CL contours, obtained for
Pantheon + BAO + BBN and Pantheon + BAO + BBN + $M_{B}$ joint analyses within
the $\Lambda$CDM (top left panel), $w$CDM (top right panel), and IDE (bottom
panel) models. The vertical grey band denotes the $68\%$ CL interval of the SN
absolute magnitude $M_{B}$ based on the SH0ES measurements. The horizontal
yellow band denotes the $68\%$ CL interval of the $H_{0}$ parameter based on
the SH0ES measurements.

##  III Datasets and Methodology

In this Section, we present the data sets and methodology used to obtain the
observational constraints on the model parameters by performing Bayesian Monte
Carlo Markov Chain (MCMC) analysis. In order to constrain the parameters, we
use the following data sets:

  * •

Type Ia Supernovae distance moduli measurements from the Pantheon sample
Scolnic  _et al._ (2018). These measurements constrain the uncalibrated
luminosity distance $H_{0}d_{L}(z)$, or in other words the slope of the late-
time expansion rate (which in turn constrains $\Omega_{m}$). We refer to this
dataset as Pantheon. For a SN at redshift $z$, the theoretical apparent
magnitude $m_{B}$ is given by

$\displaystyle m_{B}=5\log_{10}\Big{[}\frac{d_{L}(z)}{1Mpc}\Big{]}+25+M_{B},$
(5)

where $M_{B}$ is the absolute magnitude. The distance modulus is then written
as $\mu(z)=m_{B}-M_{B}$.

  * •

Baryon Acoustic Oscillations (BAO) distance and expansion rate measurements
from the 6dFGS Beutler  _et al._ (2011), SDSS-DR7 MGS Ross  _et al._ (2015),
BOSS DR12 Alam  _et al._ (2017) galaxy surveys, as well as from eBOSS DR14
Lyman-$\alpha$ (Ly$\alpha$) absorption de Sainte Agathe  _et al._ (2019) and
Ly$\alpha$-quasars cross-correlation Blomqvist  _et al._ (2019). These consist
of isotropic BAO measurements of $D_{V}(z)/r_{d}$ (with $D_{V}(z)$ and $r_{d}$
the spherically averaged volume distance, and sound horizon at baryon drag
respectively) for 6dFGS and MGS, and anisotropic BAO measurements of
$D_{M}(z)/r_{d}$ and $D_{H}(z)/r_{d}$ (with $D_{M}(z)$ the comoving angular
diameter distance and $D_{H}(z)=c/H(z)$ the Hubble distance)for BOSS DR12,
eBOSS DR14 Ly$\alpha$, and eBOSS DR14 Ly$\alpha$-quasars cross-correlation.

  * •

The state-of-the-art assumptions on Big Bang Nucleosynthesis (BBN). The BBN
data consist of measurements of the primordial abundances of helium, $Y_{P}$,
from Aver  _et al._ (2015), and the deuterium measurement,
$y_{DP}=10^{5}n_{D}/n_{H}$, obtained in Cooke  _et al._ (2018). This BBN
likelihood is sensitive to the constraints of the physical baryon density
$\omega_{b}\equiv\Omega_{b}h^{2}$ and the effective number of neutrino species
$N_{\rm eff}$. Let us fix $N_{\rm eff}=3.046$ in this present work.

  * •

A gaussian prior on $M_{B}=-19.244\pm 0.037$ mag Camarena and Marra (2021),
corresponding to the SN measurements from SH0ES.

We first consider the standard $\Lambda$CDM model, spanned by the following
parameters: the Hubble constant $H_{0}$, the physical baryon density
$\omega_{b}\equiv\Omega_{b}h^{2}$, the physical cold dark matter density
$\omega_{c}\equiv\Omega_{c}h^{2}$, and the absolute peak magnitude, $M_{B}$ of
SN. The matter density parameter today $\Omega_{m}$ is given by
$\Omega_{m}=(\omega_{b}+\omega_{c})/h^{2}$. Then, we consider two 1-parameter
extensions of the previous model. Firstly, we introduce a dark energy equation
of state free to vary $w$ and we call this model $w$CDM. Secondly, we consider
a dark coupling between DM and DE with a single free parameter $\xi$,and we
refer to this extended scenario as the IDE model.

It should be noticed here that in order to avoid the gravitational
instabilities present when $w=-1$ Valiviita  _et al._ (2008); He  _et al._
(2009), we fix the dark energy equation of state to $w=-0.999$, as previously
done in Salvatelli  _et al._ (2013); Di Valentino  _et al._ (2020b, d),
without affecting the results, as shown by using simulated data in Di
Valentino and Mena (2020). Moreover, in order to avoid early-time
instabilities Valiviita  _et al._ (2008); Gavela  _et al._ (2009); Clemson
_et al._ (2012); Gavela  _et al._ (2010); He  _et al._ (2009); Jackson  _et
al._ (2009), we also assume $\xi<0$.

We use the Metropolis - Hastings mode in CLASS+MontePython code Blas  _et al._
(2011); Audren  _et al._ (2013); Brinckmann and Lesgourgues (2019) to derive
the constraints on cosmological parameters from the data sets described above
in this section III, ensuring a Gelman - Rubin convergence criterion of
$R-1<10^{-3}$ Gelman and Rubin (1992).

##  IV Results

In this work, we consider two different data combinations, namely, Pantheon +
BAO + BBN and Pantheon + BAO + BBN +$M_{B}$ applied on three different
cosmological models, namely, $\Lambda$CDM, $w$CDM, and IDE models.

The first joint analysis, shown in Table 1, is chosen to break any possible
degeneracy in the $H_{0}-\Omega_{m}$ plane, because we know that BAO + BBN can
constrain these parameters very well. Also, several authors have pointed out
the ability of combined BAO and BBN data to probe the background cosmological
history, independently of CMB data. As a second step for the analysis, we will
add a $M_{B}$ gaussian prior corresponding to the SN measurements from SH0ES,
and we will analyse Pantheon + BAO + BBN + $M_{B}$, showing the results in
Table 2. In this work, we will analyze our baseline regardless of Planck-CMB
data and then discuss the potential of the model of solving and/or alleviating
the $M_{B}$ tension.

Fig 1 show the 2D joint posterior distributions at 68% CL and 95% CL in the
$M_{B}$-$H_{0}$ plane, obtained from Pantheon + BAO + BBN and Pantheon + BAO +
BBN + $M_{B}$ for the three scenarios considered in this work. In particular
we compare $\Lambda$CDM in the top left panel of Fig 1, the $w$CDM framework
in the top right panel of Fig 1, and the IDE scenario in the bottom panel of
Fig 1.

Figure 2: Left panel: Best fit values for the $\Lambda$CDM, $w$CDM and IDE
models obtained by Pantheon + BAO + BBN + $M_{B}$ compared against the
magnitude-redshift relation of the Pantheon SN sample in the redshift range
$0<z<1$. Right panel: $100\times(1-m_{B}^{i}/m_{B}^{\Lambda CDM})$, where $i$
run on the $w$CDM and IDE models from the difference of the best fit values
between $\Lambda$CDM and $w$CDM or IDE of the left plot

.

Once we have obtained constraints on the above cosmological parameters, we can
quantify the level of concordance or discordance (if any) on $M_{B}$ between
different data-set $i$ and $j$ using a simple 1D tension metric as follows:

$T_{M_{B}}\equiv\frac{M_{B,i}-M_{B,j}}{\sqrt{\sigma^{2}_{M_{B,i}}+\sigma^{2}_{M_{B,j}}}}\,,$
(6)

where the value of $T_{M_{B}}$ can directly be interpreted as the level of
tension in equivalent Gaussian $\sigma$.

The first thing to notice is that, taking as a reference the SN measurements
from SH0ES team obtained by Ref. Camarena and Marra (2021), $M_{B}=-19.244\pm
0.037$, our constraints on $M_{B}$ are in tension above $3\sigma$ for the
$\Lambda$CDM model.

On the contrary, both the $w$CDM and IDE models can reduce the $M_{B}$ tension
below $3\sigma$. This shows that a late time modification such as a phantom
dark energy equation of state can alleviated the $M_{B}$ tension as suggested
when a $H_{0}$ prior is used.

Regarding the IDE scenario instead, the constraining power of the dataset
combinations we are using is not enough to rule out the value of the coupling
found from the works in the literature to alleviate the Hubble tension Kumar
and Nunes (2016, 2017); Di Valentino  _et al._ (2017); Yang  _et al._ (2019a,
2018, b); Kumar  _et al._ (2019); Pan  _et al._ (2019a, b); Di Valentino  _et
al._ (2020b, c); Di Valentino and Mena (2020); Di Valentino  _et al._ (2021b);
Gómez-Valent  _et al._ (2020); Yang  _et al._ (2020a); Lucca and Hooper
(2020); Martinelli  _et al._ (2019); Yang  _et al._ (2020b); Yao and Meng
(2020a); Pan  _et al._ (2020); Di Valentino (2020); Yao and Meng (2020b);
Amirhashchi and Yadav (2020); Yang  _et al._ (2021a); Gao  _et al._ (2021);
Lucca (2021a); Kumar (2021). Actually, the dark coupling estimate obtained in
Table 1 has just a lower limit and it is in agreement with both zero and the
large coupling needed to solve the $H_{0}$ tension, within one standard
deviation, because of the large error bars.

Considering the Pantheon + BAO + BBN combination (Table 1), we can see that
the introduction of a non null interaction between DM and DE shifts slightly
both $H_{0}$ towards higher values and $M_{B}$ towards lower values, relaxing
the error bars and reducing the tension with SH0ES below $3\sigma$ (see also
Fig. 1, bottom panel).

If we now add a gaussian prior $M_{B}$ to the previous dataset combination, we
obtain the results in Table 2. Here we can see that, even if in disagreement
with the $\Lambda$CDM model at more than $3\sigma$, the addition of the
$M_{B}$ prior can shift its value reducing the tension below $3\sigma$. On the
contrary, for the $w$CDM model, the combination with the gaussian $M_{B}$
prior solves the tensions with the SH0ES measurement thanks to the evidence
for a phantom dark energy at more than 95% CL. Moreover, for the IDE model the
$M_{B}$ prior can be included safely, restoring the complete agreement with
the SH0ES data at the price of a negative coupling $\xi=-0.31^{+0.27}_{-0.28}$
at 95% CL. This result confirms the claim we find in the literature Kumar and
Nunes (2016, 2017); Di Valentino  _et al._ (2017); Yang  _et al._ (2019a,
2018, b); Kumar  _et al._ (2019); Pan  _et al._ (2019a, b); Di Valentino  _et
al._ (2020b, c); Di Valentino and Mena (2020); Di Valentino  _et al._ (2021b);
Gómez-Valent  _et al._ (2020); Yang  _et al._ (2020a); Lucca and Hooper
(2020); Martinelli  _et al._ (2019); Yang  _et al._ (2020b); Yao and Meng
(2020a); Pan  _et al._ (2020); Di Valentino (2020); Yao and Meng (2020b);
Amirhashchi and Yadav (2020); Yang  _et al._ (2021a); Gao  _et al._ (2021);
Lucca (2021a); Kumar (2021), i.e. that a flux of energy from the DM to the DE
can solve the Hubble tension, and show that the use of a gaussian prior on
$H_{0}$ or $M_{B}$ is equivalent for this class of models.

To conclude the analysis, we plot the magnitude-redshift relation for the
Pantheon sample in the redshift range $0<z<1$, together with the best fits
obtained for Pantheon + BAO + BBN + $M_{B}$ for a $\Lambda$CDM model (blue
line), the IDE model (orange line) and the $w$CDM scenario (green line) in the
left panel of Fig 2. We show instead the $100\times(1-m_{B}^{i}/m_{B}^{\Lambda
CDM})$, where $i$ run on the $w$CDM and IDE models from the difference of the
best fit values between $\Lambda$CDM and $w$CDM scenarios (blue line), or
$\Lambda$CDM and IDE models (orange line) in the right panel of Fig 2, i.e.
the percentage difference between the best fits of the models with respect to
the $\Lambda$CDM case. We find that in the $z$ range adopted in SH0ES SN data
analysis, the difference between the IDE scenario and the $\Lambda$CDM one is
$<$ 1.4%, decreasing up to $<0.6\%$ at high $z$. For the $w$CDM model, we note
practically the same difference at very low $z$ ($<$ 1.35%), and $<0.5\%$ at
high $z$.

To better quantify the agreement (and/or disagreement) between the models and
the Pantheon SN sample, we perform a statistical comparison of the IDE model
with the $\Lambda$CDM scenario by using the well-known Akaike information
criterion (AIC) Akaike (1974).

The AIC is defined through the relation

$\text{AIC}\equiv-2\ln\mathcal{L}_{\rm max}+2N,$ (7)

where $\mathcal{L}_{\rm max}$ is the maximum likelihood function of the model,
and $N$ is the total number of free parameters in the model. For the
statistical comparison, the AIC difference between the model under study and
the reference model is calculated for the joint analysis Pantheon + BAO + BBN
+ $M_{B}$. This difference in AIC values can be interpreted as the evidence in
favour of the model under study over the reference model. It has been argued
in Tan and Biswas (2011) that one model can be preferred with respect to
another if the AIC difference between the two models is greater than a
threshold value $\Delta_{\rm threshold}$. As a rule of thumb, $\Delta_{\rm
threshold}=5$ can be considered the minimum value to assert a strong support
in favour of the model with a smaller AIC value, regardless of the properties
of the models under comparison Liddle (2007).

The result we obtain by computing $AIC^{\Lambda
CDM}-AIC^{IDE}=\Delta\text{AIC}=1.54$, which shows that the IDE model can best
fit the joint analysis than $\Lambda$CDM model, but the $\Lambda$CDM framework
cannot be rule out with high statistical significance. Actually, also a simple
$\chi^{2}$ comparison points in the same direction. We find for the fit of
Pantheon + BAO + BBN + $M_{B}$ a $\chi^{2}_{min}=1046.74$ for the IDE model,
and $\chi^{2}_{min}=1050.28$ for the $\Lambda$CDM model, considering the same
joint analysis, corresponding to a $\Delta\chi^{2}_{min}=-3.57$ for one
additional degree of freedom. This shows that the IDE model can fit these data
better than the $\Lambda$CDM scenario.

A similar result is obtained for the $w$CDM models, that gives $AIC^{\Lambda
CDM}-AIC^{wCDM}=\Delta\text{AIC}=1.52$, improving the fit of the joint
analysis with respect to the standard $\Lambda$CDM, even if not in a
significant way. Moreover, the $\chi^{2}_{min}=1046.76$ for the $w$CDM model,
with a $\Delta\chi^{2}_{min}=-3.52$ and only one more degree of freedom.

Therefore, even if the $\Lambda$CDM model fits slightly better the SN data,
the best fit for the IDE case, which can solve the Hubble tension with a 95%
CL of evidence for a dark coupling different from zero, cannot be excluded by
the Pantheon + BAO + BBN + $M_{B}$ combination of data. In other words, for
this class of Interacting models, solving the $H_{0}$ or the $M_{B}$ tension
is equivalent, and using a gaussian prior from SH0ES on the Hubble constant
does not bias the results. Actually, the constraints obtained are completely
in agreement with the magnitude-redshift relation of the Pantheon SN sample.

##  V Final Remarks

It has been intensively discussed in Efstathiou (2021); Camarena and Marra
(2021) if modifications in the dynamics of the Universe at late times is able
or not to solve the $H_{0}$ tension once BAO and Pantheon data are taken into
account. On the other hand, it has also been argued that the $H_{0}$ tension
is actually a tension on the supernova absolute magnitude $M_{B}$.

We investigate this issue analysing three different models, namely the
$\Lambda$CDM scenario and two possible late times solutions to the Hubble
tension, i.e. a $w$CDM model and Interacting DE-DM model. We analyse these
scenarios making use of two possible dataset combinations, that are Pantheon +
BAO + BBN and Pantheon + BAO + BBN + $M_{B}$. We find that the IDE model can
alleviate the $M_{B}$ tension, as well as the $H_{0}$ disagreement, and a
coupling different from zero at more than 95% CL is actually preferred by the
Pantheon + BAO + BBN + $M_{B}$ dataset combination. The same thing it is true
also for a phantom solution with $w<-1$, because it can solve the $M_{B}$
tension, as does for the $H_{0}$ tension.

As a final step we investigate if the best fit from an interaction between the
DM and the DE for the Pantheon + BAO + BBN + $M_{B}$ data, than can solve the
Hubble tension, is in agreement with the Pantheon data and we find that this
model is preferred against the $\Lambda$CDM case. This result allows us to
conclude that applying a SH0ES $H_{0}$ prior, instead of a $M_{B}$ prior, does
not bias the results for an IDE late time solution. On the contrary, if we
repeat the same analysis for a phantom DE equation of state, we do not see an
indication for this solution from Pantheon data only, that prefers in any case
a cosmological constant.

###### Acknowledgements.

We thank George Efstathiou, Sunny Vagnozzi, Valerio Marra, Stefano Gariazzo
and Olga Mena for their useful comments on the draft. R.C.N. acknowledges
financial support from the Fundação de Amparo à Pesquisa do Estado de São
Paulo (FAPESP, São Paulo Research Foundation) under the project No.
2018/18036-5. EDV acknowledges the support of the Addison-Wheeler Fellowship
awarded by the Institute of Advanced Study at Durham University.

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