# Bulk Flow Motion Detection in the Local Universe with Pantheon$+$ Type Ia
Supernovae

Maria Lopes  Observatório Nacional, Rua General José Cristino 77, São
Cristóvão, 20921-400 Rio de Janeiro, RJ, Brasil 0000-0001-9181-5675 Armando
Bernui  Observatório Nacional, Rua General José Cristino 77, São Cristóvão,
20921-400 Rio de Janeiro, RJ, Brasil 0000-0003-3034-0762 Camila Franco
Observatório Nacional, Rua General José Cristino 77, São Cristóvão, 20921-400
Rio de Janeiro, RJ, Brasil 0000-0002-6320-425X Felipe Avila  Observatório
Nacional, Rua General José Cristino 77, São Cristóvão, 20921-400 Rio de
Janeiro, RJ, Brasil 0000-0002-0562-2541

###### Abstract

The bulk flow in the Local Universe is a collective phenomenon due to the
peculiar motions of matter structures, which, instead of moving in random
directions, appears to follow an approximate dipole velocity flow. We apply a
directional analysis to investigate, through the Hubble-Lemaître diagram, the
angular dependence of the Hubble constant $H_{0}$ of a sample of Type Ia
Supernovae from the Pantheon+ catalog in the Local Universe ($0.015\leq z\leq
0.06$). We perform a directional analysis that reveals a statistically
significant dipole variation of $H_{0}$, at more than $99.9\%$ confidence
level, showing that matter structures follow a dipole bulk flow motion towards
$(l,b)=(326.^{\circ}1\pm 11.^{\circ}2,27.^{\circ}8\pm 11.^{\circ}2)$, close to
the Shapley supercluster
$(l_{\scalebox{0.6}{Shapley}},b_{\scalebox{0.6}{Shapley}})=(311.^{\circ}5,32.^{\circ}3)$,
with velocity $132.14\pm 109.3$ km s-1 at the effective distance $102.83\pm
10.2$ Mpc. Interestingly, the antipodal direction of this dipole points close
to the Dipole Repeller structure. Our analyses confirm that the gravitational
dipole system Shapley-Dipole Repeller explains well the observed bulk flow
velocity field in the Local Universe. Furthermore, we performed robustness
tests that support our results. Additionally, our approach provides a
measurement of the Hubble constant $H_{0}=70.39\pm 1.4$ km s-1 Mpc-1, at the
effective distance $102.8$ Mpc, $z\simeq 0.025$. Note that this value was
obtained using the first order approximation of the Hubble law because our
methodology is model-independent. If one assumes, for instance, cosmography at
second order with the $\Lambda$CDM value $q_{0}=-0.55$, which is a model-
dependent hypothesis, then $H_{0}=72.6\pm 1.5$ km s-1 Mpc-1, but our results:
bulk flow velocity, dipole direction and its statistical significance remain
the same.

(Received 19-Mar, 2024; Accepted 21-Mar, 2024; to appear in ApJ)

Keywords: Observational cosmology, Large-scale structure of the universe, Type
Ia supernovae

## 1 Introduction

Our Local Universe is manifestly inhomogeneous, plenty of matter structures,
like galaxies assembled in small groups or in galaxy clusters, and cosmic
voids, with various underdense matter contents and sizes, around us (Rubin,
1951; de Vaucouleurs, 1953; Gregory and Thompson, 1978; de Lapparent et al.,
1986; Tully and Fisher, 1987; Coles, 1996; Courtois et al., 2013; Tully et
al., 2019; Avila et al., 2019; Franco et al., 2024). Peculiar velocities of
cosmic objects are the result of gravitational fields, which, in turn, are
originated by the surrounding matter distribution (Peebles, 1980; Kaiser,
1987). Therefore, peculiar motions are one of the best tracers of matter
density fluctuations in the universe.

Near the Local Group of galaxies, to which the Milky Way belongs, a large
underdensity termed the Dipole Repeller (DR) was recently discovered (Hoffman
et al., 2017). Another prominent feature in our cosmic neighborhood is the
Shapley supercluster (Raychaudhury, 1989; Scaramella et al., 1989), the
largest cluster of galaxies of the Local Universe.

The system Shapley-DR acts, approximately, as a gravitational dipole system
producing a bulk flow motion of matter in the Local Universe (Hoffman et al.,
2017). In this scenario the DR acts as it were a repulsive body causing the
evacuation of matter in its surroundings; on the other side, the Shapley
supercluster acts as the dominant matter attractor, pulling matter structures
from every side. One important motivation for studying this gravitational
dipole system, and the features of the induced bulk flow, is due to the
dynamical effects produced as large peculiar velocities in our nearby galaxies
(Tully et al., 2019; Peterson et al., 2022), information needed to a better
calibration of the standard candles for precise measurements of $H_{0}$ at low
redshifts (Scolnic et al., 2018).

Measurements resulting from some earlier studies already suggested a large
bulk flow (Rubin et al., 1976; Lynden-Bell et al., 1988). More recently,
reports in the literature (see, e.g., Hong et al. (2014); Kalbouneh et al.
(2023); Mc Conville and Ó Colgáin (2023); Perivolaropoulos (2023)) have
studied the magnitude and direction of this bulk flow velocity with diverse
approaches and investigating various cosmic tracers, where these studies
confirm the dipole nature of the bulk flow, but point out some differences in
the velocity magnitude and/or direction (Turnbull et al., 2012; Scrimgeour et
al., 2016; Qin et al., 2018; Avila et al., 2023; Watkins et al., 2023;
Whitford et al., 2023).

In this work, we perform an accurate directional analysis of the Type Ia
Supernovae (SNe Ia) data from Pantheon$+$ (Brout et al., 2022) to measure the
bulk flow velocity, both in intensity and in direction. Our analyses differ
from other studies in various aspects. First, our methodology follows previous
studies looking for preferred directions on the sky using several cosmic
tracers (Bernui et al., 2008; Marques et al., 2018; Kester et al., 2024);
second, to avoid systematics from different observables, we use only SNe Ia
data from Pantheon+; this choice is a challenge due to the possibility of
having few SNe in some directions of the sky; third, we measure $H_{0}$ and
its uncertainty in a set of $N$ directions, which covers the celestial sphere,
performing a best fit procedure using the Pantheon+ catalog and its covariance
matrix, without cosmological model assumptions; fourth, we perform consistency
and robustness tests to support our analyses and results.

Our directional analysis confirms an approximate dipole behaviour of the
Hubble constant, with a dipole direction close to the Shapley-DR direction.
The statistical significance is obtained by comparison with a set of simulated
maps, produced by randomizing the angular positions of the SNe sample in
study, and repeating our directional analysis procedure. For consistency, our
analyses scan the celestial sphere considering two angular resolutions, but
our final results and conclusions are drawn from the best angular resolution,
i.e., scanning the sky with $N=192$ ($N_{\text{side}}=4$) spherical caps.

The outline of this work is the following. In Section 2 we select the SNe
data, subsample of the Pantheon+ SNe Ia catalog, for our directional analysis
in the Local Universe. In Section 3 we describe in detail the directional
analysis approach, including the measurements and uncertainties involved. In
Section 4, we present the $H_{0}$-maps, calculate their dipole component, and
determine its statistical significance. In Section 5 we present our
conclusions and final remarks. We leave for the Appendix our consistency and
robustness tests that support our analyses.

## 2 Data: The Pantheon+ SNe Ia Catalog

The recently released catalog of SNe Ia Pantheon+ (Brout et al., 2022),
successor of the original Pantheon sample (Scolnic et al., 2018), compiles a
larger number of SNe Ia events, with more precise data, and it includes events
located in host galaxies whose distances were obtained using Cepheids. The
Pantheon+ catalog contains $1701$ type Ia SNe, with redshifts $0.001\leq z\leq
2.26$, and their angular distribution is shown in the top panel of Figure 1
(in galactic coordinates, where the Milky Way is in the center of the figure.

Figure 1: Distribution of SNe Ia on the celestial sphere with a Mollweide
projection in galactic coordinates. Left panel: the Pantheon$+$ sample,
showing $1701$ SNe Ia. Right panel: the sample of 501 SNe Ia selected for our
directional analysis, within the redshift interval $0.015\leq
z_{\scalebox{0.6}{CMB}}\leq 0.06$.

In addition to encompassing several redshift reference frames, such as the
cosmic microwave background (CMB), and Heliocentric, the Pantheon+ catalog
offers a comprehensive collection of precise data. This includes the light
curves, covariance matrix, distance modulus, bolometric magnitude, and their
associated uncertainties (Scolnic et al., 2022). In our analyses, we shall use
the following information from SN Ia data: the sky angular position, the
redshift, the distance modulus, the peculiar velocities, all with their
respective uncertainties except the last one, and the covariance matrix.
Regarding the peculiar velocities’ uncertainties, it is worth mentioning that
they are undetermined (in the catalog appears the value $250$ km s-1 for all
the SNe listed there).

Our objective in this work is to detect the bulk flow motion in the Local
Universe investigating the latest SNe Ia data from Pantheon+. The choice of
the redshift interval for analyses was motivated by the following: (i) the
gravitational dipole in analysis is limited by the region spatially close to
the Shapley supercluster, where $z_{\text{Shapley}}\simeq 0.05$; for this, we
consider data with $z_{\scalebox{0.5}{CMB}}\leq 0.06$; (ii) galaxies very
close to us are not expected to follow the Hubble flow, as overdense and
underdense regions in our neighborhood affect their motion, making it dominant
over the Hubble expansion, then we adopt a conservative low-redshift limit
$z_{\scalebox{0.5}{CMB}}\geq 0.015$ (Riess et al., 2016; Peterson et al.,
2022). In this case, our analyses are restricted to the Local Universe;
therefore one can use the Hubble-Lemaître (HL) law to find $H_{0}$ (Visser,
2004). Notice that, for consistency, in Appendix A we consider other four
redshift intervals, close to $z_{\scalebox{0.6}{CMB}}\in[0.015,0.06]$, where
we apply our directional analysis to study the SNe samples contained in these
intervals; as observed there, we basically obtain the same results.

According to these selection criteria, the sample chosen for analyses has
$501$ SNe Ia with redshifts $0.015\leavevmode\nobreak\
\leq\leavevmode\nobreak\ z_{\scalebox{0.6}{CMB}}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 0.06$, i.e., measured in the CMB reference frame. The
angular distribution of this sample is shown in galactic coordinates in the
lower panel of Figure 1, and its redshift distribution is given in Figure 2.

Figure 2: Frequency histogram of the 501 SNe Ia sample selected for our
directional analysis, in different redshift intervals within the range
$0.015\leq z_{\scalebox{0.6}{CMB}}\leq 0.06$; the bin size is $\Delta
z=0.002$.

## 3 Methodology

In this section, we present our methodology to perform the directional
analysis of the set of 501 SNe Ia selected in the previous section from the
Pantheon+ catalog. Our study consists in determining the value of $H_{0}$ in
different sky directions. For this we define a set of $N$ directions, chosen
according to the HEALpix scheme, that uniformly scan the celestial sphere with
spherical caps of angular radius $\gamma$ centered at these $N$ directions.
Our directional analysis considers two angular resolutions to scan the
celestial sphere: with $N=48$ ($N_{\text{side}}=2$) and $N=192$
($N_{\text{side}}=4$) spherical caps. Then, we perform the best fit analysis
in the HL diagram for those SNe Ia inside the $i$th spherical cap, centered at
direction $i$, obtaining the best fit value $H_{0}^{b\text{-}\\!f}\rightarrow
H_{0}^{i}$ at the $i$th direction. We repeat this procedure for the $N$
directions, obtaining the data set $\\{H_{0}^{i}\\}$, for $i=1,2,\cdots,N$.
The details of this analysis will be discussed in Section 3.2.1.

###  3.1 Finding $H_{0}$ with the Hubble-Lemaître Law

In the CMB reference frame, the measured velocity of any cosmic object is
(Courteau and van den Bergh, 1999; Baumann, 2022)

$\vec{\text{v}}=H_{0}\,\vec{r}+\vec{\text{v}}_{pec}\,,$ (1)

where peculiar motions contribute to deviate cosmic objects from the Hubble
flow (i.e., the universe expansion). Considering the radial component of this
equation, one obtains

$c\,z_{\scalebox{0.6}{CMB}}=H_{0}\,D+\text{v}_{pec}^{r}\,,$ (2)

where $D$ is the radial physical distance to the cosmic object, and
$\text{v}_{pec}^{r}$ is the radial projection of its peculiar velocity (to
simplify the notation, from now on, we write:
$\text{v}_{pec}=\text{v}_{pec}^{r}$).

The way we adopt to minimize the impact of the peculiar velocities is to
choose for analysis SNe with $z_{\scalebox{0.6}{CMB}}\geq 0.015$, because in
such a case, the peculiar velocities are a small fraction of the radial
recession velocity. In fact, as observed in the plots shown in Figure 3 the
peculiar velocities of our sample in analysis are, on average, lower than 5%
of the recession velocity at redshifts $z_{\scalebox{0.6}{CMB}}\geq 0.015$.
The impact of the SNe peculiar velocities will be considered later in our
analyses, in Section 4.2.1.

Figure 3: Left panel: peculiar velocities histogram of the 501 SNe Ia selected
for our bulk flow analyses; this histogram shows that mean $-81.87$, median
$-97.44$, and standard deviation $157.90$, all in units of km s-1. Right
panel: fraction, in percentage, of the peculiar velocities with respect to the
recession velocity for the sample in analysis.

####  3.1.1 Luminosity distance

To estimate the value of the Hubble constant, $H_{0}$, through a best fit
analysis, we use the SNe Ia luminosity distance and the redshift of each SN.
In the Local Universe, luminosity distance is a good approximation for
estimating proper distances to objects with small redshifts, $z<0.2$ (Visser,
2004). In effect, to perform the best fit for $H_{0}$, we use the first-order
expansion of the luminosity distance derived for the HL law, resulting from
the series expansion for the physical distance $D(z)=D_{L}(z)$,

$D_{L}(z_{\scalebox{0.5}{CMB}})=\frac{c\,z_{\scalebox{0.5}{CMB}}}{H_{0}},$ (3)

where $z_{\scalebox{0.5}{CMB}}$ is the SN redshift in the CMB frame.

To calculate the SNe Ia luminosity distances, we use the distance modulus
$\mu$, where the fiducial magnitude was determined from SNe host distances
(Brout et al., 2022). The distance modulus is defined as the difference
between the apparent magnitude $m_{B}$ and the absolute magnitude $M_{B}$
(Sparke and Gallagher, 2007)

$\mu\equiv m_{B}-M_{B}=5\log_{10}(D_{L}/\text{Mpc})+25\,,$ (4)

where $D_{L}$ is given in Mpc. Then, we calculate the luminosity distance as

$D_{L}=10^{\frac{\mu-25}{5}}\,.$ (5)

####  3.1.2 Uncertainties associated to $D_{L}$ and $\delta H_{0}$

To determine the $1\sigma$ uncertainty of a function $X$ that depends on two
or more variables $u,v,...$, $X=f(u,v,...)$, we calculate it by quadrature

$\sigma_{{}_{X}}=\sqrt{\left(\frac{\partial X}{\partial
u}\right)^{2}(\sigma_{u})^{2}+\left(\frac{\partial X}{\partial
v}\right)^{2}(\sigma_{v})^{2}+\cdots}\,\,,$ (6)

where the terms $\frac{\partial X}{\partial u}$, $\frac{\partial X}{\partial
v},...$, are the partial derivatives of $X$ with respect to $u,v,...$, and
$\sigma_{u},\sigma_{v},...$, are the uncertainties associated with the
measured quantities $u,v,...$, respectively. According to this, the errors
associated to measurements of the luminosity distance, $D_{L}$, are

$\displaystyle\sigma_{{}_{D_{L}}}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\ln(10)\,D_{L}\,\sigma_{\mu}\,,$ (7)

where $\sigma_{\mu}$ is directly obtained from the Pantheon+ catalog. To
estimate the error of $H_{0}$ we take into account the uncertainties from the
best fit procedure, which in turn uses the covariance matrix of the Pantheon+
sample, and also consider the statistical error due to the number of SNe Ia in
each spherical cap. Therefore, the total error in the measurement of $H_{0}$
is

$\sigma_{H_{0}}=\sqrt{\sigma_{b\text{-}\\!f}^{2}+\sigma_{st}^{2}+\sigma_{dp}^{2}}\,\,,$
(8)

where $\sigma_{b\text{-}\\!f}$ is the uncertainty in the best fit, and
$\sigma_{st}$ corresponds to the statistical error calculated as

$\sigma_{st}^{i}=\frac{1}{\sqrt{n_{i}}}\,,$ (9)

where $n_{i}$ is the number of SNe Ia observed in the spherical cap $i$, and
$\sigma_{dp}$ corresponds to the uncertainty due to the dipole nature of the
$H_{0}$-map (see Section 4.1) uncertainty to be calculated using a Monte Carlo
simulation in Appendix B.

###  3.2 The SNe $H_{0}$-maps

To investigate in what direction our sample of 501 SNe Ia, with redshifts
$0.015\leq z\leq 0.06$, show a larger value of $H_{0}$, or otherwise does not
show a preferred direction, we perform our directional analysis. For this, we
shall scan the celestial sphere with $N$ spherical caps, of radius $\gamma$,
and consider two cases: $N=48$ and $N=192$ caps.

Let
$\Omega_{i}^{\gamma}\equiv\Omega(\theta_{i},\phi_{i},\gamma)\in\mathcal{S}^{2}$
be a spherical cap on the celestial sphere, of $\gamma$ degrees of aperture,
with vertex at the $i$th pixel with coordinates $(\theta_{i},\phi_{i})$, that
is, $(\theta^{i}_{\text{cap}},\phi^{i}_{\text{cap}})=(\theta_{i},\phi_{i})$,
for $i=1,2,\cdots,N_{\text{pix}}$ (for details of our directional analysis see
Bernui et al. (2008)). Both, the number of spherical caps $N$ and the angular
coordinates of their centers
$\\{(\theta^{i}_{\text{cap}},\phi^{i}_{\text{cap}})\\},\,i=1,2,\cdots,N$, are
defined using the Healpy HEALPix111https://healpy.readthedocs.io/en/latest/
pixelization algorithm (Górski et al., 2005).

The pixelization parameter $N_{\text{side}}$ provides the grid resolution that
defines the quantity of pixels, $N_{\text{pix}}$, in which the sphere is
pixelized

$N=N_{\text{pix}}=12\,N_{\text{side}}^{2}\,,$ (10)

where $N_{\text{side}}=2^{k}$, and $k$ is an integer number sometimes called
order. This means that $N_{\text{side}}=2$ produces $N=48$ cap centres, and
$N_{\text{side}}=4$ produces $N=192$ cap centers, which correspond to the
angular resolutions we will use in our directional analysis.

As observed in the lower plot of Figure 1, the SNe are not uniformly
distributed on the sky, this is because the Pantheon+ catalog is a compilation
of SNe that are short-lived random events in the sky, not long-lived cosmic
tracers that can be observed at any time with an astronomical survey. This
implies that the number of SNe is not equal in all caps, a fact that we take
into account in our error analysis, and in the choice of $\gamma$, the radius
of the spherical caps.

####  3.2.1 From the $H_{0}$-caps to the $H_{0}$-map

Our directional analysis consists in calculating $H_{0}$ in each one of the
$N$ caps that scan the celestial sphere, computation that is done performing
the best fit of the HL diagram for the SNe observed in each cap, and assigning
the $H_{0}$ value found to the centre of the cap. For example, suppose that in
the $i$-th spherical cap, of radius $\gamma$, we observe $n_{i}$ SNe, then we
perform the best fit of the HL diagram with these $n_{i}$ events finding the
value $H_{0}^{i}$. In Appendix C, we investigate the correlation between the
number of SNe and the best fit $H_{0}$, where we conclude that the number of
SNe observed in each of the spherical caps has a negligible impact in the
dipolar behaviour of $H_{0}$. The set of $N$ values: $\\{H_{0}^{i}\\}$, for
$i=1,\cdots,N$, are then assembled together into a full-sky map, hereafter the
$H_{0}$-map; attributing a color scale to these set of real numbers
$\\{H_{0}^{i}\\}$ we obtain a colored $H_{0}$-map, as seen in the left panels
of Figure 4. The next step is to calculate the dipole component of the
$H_{0}$-map, and establish its statistical significance with the help of
simulated isotropic maps (see Section 4.3).

The radius of the spherical cap, $\gamma$, is a challenge: it cannot be small
because the number of SNe observed in that caps would be so small that the
best fit is dominated by statistical noise, and cannot be large that one loses
precision222Because including SNe that do not contribute to the direction of
interest causes the decrease of the effect we want to measure. in determining
the directions where $H_{0}$ is maximum or minimum. As a lower limit for the
number $n_{i}$ we consider the number of the SNe analyzed in 1998 (Riess et
al., 1998; Perlmutter et al., 1999), that is $n_{i}\sim 40$ SNe. After various
tests, we adopt $\gamma=60^{\circ}$. However, we also performed robustness
tests with other cap radius to verify that the value $\gamma=60^{\circ}$ is
not biasing our results. As shown in Appendix D, we redo our directional
analysis considering caps with $\gamma=65^{\circ}$ and $\gamma=70^{\circ}$
obtaining completely similar results.

## 4 Analyses and Results

In this section, we shall present our analyses and corresponding results. We
start calculating the $H_{0}$-maps, for the two angular resolutions, with
$N=48$ and $N=192$ spherical caps. Then, we calculate the dipole components of
these maps and determine their statistical significance.

###  4.1 Dipole structure and the angular power spectrum

With all the data in hand, we then calculate the $H_{0}$-maps, scanning the
celestial sphere with spherical caps of radius $\gamma=60^{\circ}$. Our
results are displayed in Figure 4. The next step is to calculate the dipole
component $C_{1}$ of the corresponding angular power spectra. Clearly, the
dipole term $C_{1}$ is the larger one, $C_{1}^{48}=1.52$ for the $H_{0}$-map48
and $C_{1}^{192}=1.66$ for the $H_{0}$-map192, indicating that the
$H_{0}$-maps show a net dipolar behavior. However, one needs to evaluate the
statistical significance of this result, and this is done by comparison with
$H_{0}$-maps obtained by redoing our directional analysis procedure but
randomizing the choice of the SNe. The production of these sets of isotropic
$H_{0}^{\text{Iso}}$-maps considering $48$ and $192$ caps, and the statistical
analyses to determine the statistical significance of the dipole components of
the $H_{0}$-map48 and $H_{0}$-map192 are described in Section 4.3.

We also calculate the direction of the dipole term finding, in galactic
coordinates, $(l,b)=(327.^{\circ}08\pm 22.^{\circ}5,28.^{\circ}23\pm
22.^{\circ}5)$, for the $H_{0}$-map with $N=48$ caps, and
$(l,b)=(326.^{\circ}06\pm 11.^{\circ}2,27.^{\circ}79\pm 11.^{\circ}2)$, for
the $H_{0}$-map with $N=192$ caps. The direction and bulk flow velocity of the
dipoles, in the directional analysis with 48 and 192 spherical caps, are
summarized in Table 1 (see also Figures 4, 5, and 6).

Regarding the dipole interpretation of our $H_{0}$-maps, we observe in Figure
4 that this is a good approximation, but not an exact result. In fact, the
color of each pixel in our $H_{0}$-maps represents the dominance of inflows or
outflows, phenomena that increases or decreases the recession speed of matter
structures, respectively, relative to the Hubble flow. That is, the diversity
of colors in the pixels of a $H_{0}$-map reflects the dominance of one
phenomenon or the other, or even the balance between both, effects that are
quantified by the best fit $H_{0}^{i}$ from the HL diagram of those SNe in the
$i$th cap: reddish (bluish) pixels represent $H_{0}^{i}$ values above (below)
the mean, and greenish for values close to the mean.

Remarkably, the prevalence of each of these phenomena is determined by the
large-scale distribution of clustered matter and large voids, and the Local
Universe is plenty of them (Courtois et al., 2013; Hoffman et al., 2017; Tully
et al., 2023; Dias et al., 2023). Large matter structures besides the Shapley
supercluster, like the South Pole Wall (SPW, Pomarède et al., 2020), a huge
structure at $z\simeq 0.04$, produces an inflow that contributes to distort a
pure dipolar pattern. On the other hand, the presence of large underdense
regions besides the DR (Franco et al., 2024), like the Perseus-Pisces void
($z\simeq 0.027$), is producing outflows that also leave their peculiar
signature in our $H_{0}$-maps. All these complex gravitational interactions
produce inflows and outflows that affect the dynamics of the SN hosts,
phenomena that are captured by our directional analysis and revealed in our
$H_{0}$-maps. The observed dipole structure of the $H_{0}$-maps is just the
consequence of a dominant gravitational dipole system, i.e., Shapley-DR
(Hoffman et al., 2017).

| $l(^{\circ})$ | $b(^{\circ})$ |  $V_{\scalebox{0.7}{BF}}$ (km s-1) |  $\delta H_{0}$ (km s-1Mpc-1)
---|---|---|---|---
$H_{0}$-map48 | $327.08\pm 22.5$ | $28.23\pm 22.5$ | $127.69\pm 110.97$ | $2.44\pm 1.88$
$H_{0}$-map192 | $326.06\pm 11.2$ | $27.79\pm 11.2$ | $132.14\pm 109.30$ | $2.57\pm 1.87$

Table 1: Bulk flow direction and velocity, for directional analysis with 48
and 192 spherical caps of radius $\gamma=60^{\circ}$.

Figure 4:  First row: the $H_{0}$-map48 with resolution $N_{\text{side}}=2$
(48 spherical caps, left panel), and its dipole component map (right panel).
Second row: the $H_{0}$-map192 with resolution $N_{\text{side}}=4$ (192
spherical caps, left panel), and its dipole component map (right panel)
showing the position of the Shapley supercluster as a multiplication sign X
(in fuchsia color) to evidence that the dipole direction obtained in our
analyses is very close to this matter structure. Note that all these maps are
in units km s-1 Mpc-1.

###  4.2 Bulk flow velocity

Given a data set of low-redshift SNe Ia, $z\ll 1$, one can perform a best fit
analysis of the linear HL law and write

$\langle\frac{c\,z_{\scalebox{0.6}{CMB}}}{r}\rangle=H_{0}^{b\text{-}\\!f}\,,$
(11)

where the angular parentheses mean the procedure done to obtain the best fit
value $H_{0}^{b\text{-}\\!f}$. Clearly, according to the left panels of Figure
4, the $H_{0}$-maps, the value one obtains in the best fit analysis,
$H_{0}^{b\text{-}\\!f}$, depends on the direction in study. Analyzing these
maps one determines the positive ($+$) and negative ($-$) dipole directions of
the $H_{0}$-maps (right panels of Figure 4). Consider the best fit value of
the $H_{0}$-map obtained in the positive (negative) dipole direction, that is,
$H_{0}^{+}$ ($H_{0}^{-}$). Then, the directional analysis of the SNe data in
these directions, with effective distance $R$, states

$\langle\frac{c\,z_{\scalebox{0.6}{CMB}}}{r}\rangle_{\pm}=H_{0}^{\pm}\simeq
H_{0}\pm\frac{\text{V}_{\scalebox{0.6}{BF}}(R)}{R}\,,$ (12)

where we interpret the excess (defect) in $H_{0}$ as being the effect of the
bulk flow motion, with velocity $\text{V}_{\scalebox{0.6}{BF}}(R)$ at the
effective distance $R$, on the universe expansion rate measured by the SNe. As
a consequence of this, one has

$H_{0}^{+}-H_{0}^{-}\equiv\delta
H_{0}(R)\simeq\frac{2\,\text{V}_{\scalebox{0.6}{BF}}(R)}{R}\,.$ (13)

Therefore,

$\text{V}_{\scalebox{0.6}{BF}}(R)\simeq\frac{1}{2}R\,\,\delta H_{0}(R)\,.$
(14)

In what follows we will show how to obtain the effective distance $R$.

####  4.2.1 Effective distance $R$ and peculiar velocities

To calculate the effective distance $R$ of a data sample, necessary to compute
the bulk flow velocity, we use the thermal noise model of Turnbull et al.
(2012),

$R=\frac{\sum_{i=1}^{N_{\scalebox{0.5}{SNe}}}{D_{L}}_{i}/\sigma_{i}^{2}}{\sum_{i=1}^{N_{\scalebox{0.5}{SNe}}}1/\sigma_{i}^{2}}\,,$
(15)

where ${D_{L}}_{i}$ is the luminosity distance of the $i$-th SN, $\sigma_{i}$
is the peculiar velocity uncertainty of the $i$-th SN, and
$N_{\scalebox{0.7}{SNe}}$ is the number of SNe in our sample. Unfortunately,
the Pantheon$+$ catalog does not provide uncertainties of the SNe peculiar
velocities measurements333The column of this quantity exists, but the
information displayed there is $250$ km s-1 for all the SNe in the Pantheon+
catalog.. Therefore, for the calculation of the effective distance, $R$, we
need to estimate the set
$\\{\sigma_{i}\\}=\\{{\sigma_{\text{v}_{pec}}}_{i}\\}$.

From equation (2), the peculiar velocity in the radial direction is given by

$\text{v}_{pec}=c\,z_{\scalebox{0.6}{CMB}}-H_{0}\,D_{L}\,.$ (16)

To calculate the uncertainty of $\text{v}_{pec}$, $\sigma_{\text{v}_{pec}}$,
we use equation (6) multiplied by a factor444The factor $1/\sqrt{2}$ is an
average of all the possible directions of the peculiar velocity, relative to
radial direction; see, e.g., Brout et al. (2022). $1/\sqrt{2}$,
obtaining555The uncertainty in the redshift measurements,
$\sigma_{z_{\scalebox{0.5}{CMB}}}$, is negligible.

$\sigma_{\text{v}_{\text{pec}}}=\frac{1}{\sqrt{2}}\sqrt{D_{L}^{2}\sigma_{H_{0}}^{2}+H_{0}^{2}\sigma_{D_{L}}^{2}}\,.$
(17)

We then compute $\sigma_{\text{v}_{\text{pec}}}$ using the best fit values of
$H_{0}$ and $\sigma_{H_{0}}$, together with $D_{L}$ and $\sigma_{D_{L}}$, for
each SN located in the spherical caps corresponding to the positive and
negative dipole direction of the $H_{0}$-map. Performing error propagation,
the uncertainty of $R$, $\sigma_{R}$, is calculated from the uncertainties in
the luminosity distance, ${\sigma_{D_{L}}}_{i}$, and in the peculiar velocity,
$\sigma_{i}$, that is,

$\sigma_{R}=\frac{\sum_{i=1}^{N_{\scalebox{0.5}{SNe}}}{\sigma_{D_{L}}}_{i}/\sigma_{i}^{2}}{\sum_{i=1}^{N_{\scalebox{0.5}{SNe}}}1/\sigma_{i}^{2}}\,.$
(18)

We then calculate the effective distance using equation (15), and its
associated uncertainty using equation (18), obtaining $R^{48}=104.67\pm 10.31$
Mpc and $R^{192}=102.83\pm 10.24$ Mpc, for $N=48$ and $N=192$ spherical caps,
respectively. For consistency, we have also estimated the effective distance
using Monte Carlo simulations, as presented in Appendix E, obtaining values
fully consistent with these ones.

####  4.2.2 Bulk flow velocity with SNe Ia

Our measurements of the directional variation of the Hubble constant were done
in Section 4.1, where we have obtained a net dipolar structure in the
$H_{0}$-maps, as seen in Figure 4, with maximum and minimum values
$H_{0}^{max}=71.91\pm 1.42$ and $H_{0}^{min}=68.88\pm 1.32$ for the
$H_{0}^{48}$-map, and $H_{0}^{max}=72.80\pm 1.26$ and $H_{0}^{min}=68.88\pm
1.17$ for the $H_{0}^{192}$-map, respectively (all $H_{0}$ values are in units
of km s-1Mpc-1). The differences among diverse directional measurements of
$H_{0}$, observed in the $H_{0}$-maps shown in the left panels of Figure 4,
are interpreted as being originated in the combination of motions of the SNe
host galaxies due to the Hubble flow (i.e., the universe expansion) and due to
peculiar motions relative to the matter structures, including the bulk flow
motion. Clearly, in a universe without peculiar motions, the $H_{0}$-maps
should be mono-colour, indicative of the isotropy of $H_{0}$, that is, the
same value $H_{0}$ in any direction (except statistical fluctuations
originated in the best fit procedure of the HL diagrams). According to
equation (14), to obtain the magnitude of the bulk flow velocity, we need
$\delta H_{0}$. We calculate $\delta H_{0}\equiv H_{0}^{+}-H_{0}^{-}$,
considering the values in the $H_{0}$-maps (left panels of Figure 4)
corresponding to the $+/-$ dipole directions shown in the right panels of
Figure 4, obtaining $H_{0}^{+/48}=71.78\pm 1.34$ and $H_{0}^{-/48}=69.34\pm
1.32$, and $H_{0}^{+/192}=72.31\pm 1.34$ and $H_{0}^{-/192}=69.74\pm 1.30$,
where all $H_{0}$ values are in units of km s-1 Mpc-1. The effective
distances, as calculated before, are $R^{48}=104.67\pm 10.31$ Mpc and
$R^{192}=102.83\pm 10.24$ Mpc, then

$V_{{\scalebox{0.6}{BF}}_{48}}=\frac{1}{2}(104.67\pm 10.31)(2.44\pm 1.88)\,,$
(19) $V_{{\scalebox{0.6}{BF}}_{192}}=\frac{1}{2}(102.83\pm 10.24)(2.57\pm
1.87)\,,$ (20)

finding $V_{{\scalebox{0.6}{BF}}_{48}}=127.81\pm 110.97$ km s-1 and
$V_{{\scalebox{0.6}{BF}}_{192}}=132.14\pm 109.30$ km s-1, respectively. Our
results are robust, and in good agreement with the values reported in the
literature (see Figure 5).

Figure 5:  This plot compares several bulk flow velocity measurements with the
value expected in the $\Lambda$CDM model (continuous line plus $1\sigma$ and
$2\sigma$ limits): A23 (Avila et al., 2023); H14 (Hong et al., 2014); Q18 and
Q′18 (Qin et al., 2018); N11 (Nusser and Davis, 2011); W09 (Watkins et al.,
2008); M13 (Ma and Scott, 2012); T12 (Turnbull et al., 2012); S16 (Scrimgeour
et al., 2016); C11 (Colin et al., 2011); D11 (Dai et al., 2011); W23 and W′23
(Watkins et al., 2023); WA23 and WA′23 (Whitford et al., 2023), and the result
of our analyses. All effective distances, $R$, are given in units of Mpc
$h^{-1}$; in this plot, we used $h=0.7$ for comparison purposes. Figure 6:
Mollweide projection in galactic coordinates of the structures and directions
characterizing flows in the Local Universe: the Dipole Repeller, Shapley
supercluster, CMB dipole, and the direction of the bulk flow we found for
$H_{0}$-map192.

In Figure 6 we can observe some interesting directions on the celestial
sphere, including the dipole bulk flow motion found in our analyses. Our
results, summarized in Table 1, confirm that the gravitational system Shapley-
DR explains well the dipole nature of the bulk flow motion, and that matter
structures in the Local Universe appear to move in the direction of the
Shapley supercluster with a velocity of $132.14\pm 109.30$ km s-1. 666 The
performance of our approach depends on having a large number of SNe per square
degree uniformly distributed on the celestial sphere, which is not the present
case; in consequence, SNe unevenly distributed on large spherical caps
contribute only partially to point out the true bulk flow velocity: the
potential contribution $\vec{\text{v}}_{pec}$ reduces to its component in the
radial direction $\text{v}_{pec}^{r}=\vec{\text{v}}_{pec}\cdot\widehat{r}$,
which can be a small quantity because depends on the angle between
$\vec{\text{v}}_{pec}$ and the unit vector $\widehat{r}$, angle that can be
large, and its cosine small, in large spherical caps. As a consequence, our
measurement is underestimated.

###  4.3 Isotropic maps and statistical confidence analysis

We want to verify that the dipole signature found in our $H_{0}$-map analyses
is not a biased result, perhaps due to the methodology of analysis, or some
artifact in the data. Our statistical confidence test considers the comparison
with isotropic $H_{0}$-maps, as follows.

Suppose that the number of SNe in the $i$th spherical cap is $n_{i}$, for
$i=1,\cdots,N$. These $n_{i}$ SNe were selected in the $i$th cap because their
angular separation with respect to the center of the cap is less or equal
$\gamma$ degrees. In this test, instead, for the $i$th cap we randomly choose
$n_{i}$ SNe from any part of the sky, where $i=1,\cdots,N$; then, we redo the
best fit procedure in the $N$ HL diagrams obtaining an isotropic $H_{0}$-map,
termed $H_{0}^{\text{Iso}}$-map. We repeat this process $1000$ times,
obtaining 1000 $H_{0}^{\text{Iso}}$-maps. For illustrative purposes, in Figure
7 we show two of these $H_{0}^{\text{Iso}}$-maps, both for $N=48$ and $192$
spherical caps, i.e., $H_{0}^{\text{Iso-48}}$-maps and
$H_{0}^{\text{Iso-192}}$-maps, respectively. We also calculate the angular
coordinates of the dipole components from these $H_{0}^{\text{Iso}}$-maps, and
show them collectively as histograms in Figure 8, again for both angular
resolutions, that is, for $N=48$ and $N=192$ spherical caps.

Furthermore, we calculate the corresponding dipole component of the
$H_{0}^{\text{Iso}}$-maps in both angular resolutions,
$\\{C_{1}^{\text{Iso-48/192}}\\}$, useful information to calculate the
statistical significance of the dipole components $C_{1}^{48}$ and
$C_{1}^{192}$, from the $H_{0}$-map48 and $H_{0}$-map192 (shown in Figure 4),
respectively. The distribution of the dipole components from the
$H_{0}^{\text{Iso-48}}$-maps, $\\{C_{1}^{\text{Iso-48}}\\}$, and from the
$H_{0}^{\text{Iso-192}}$-maps, $\\{C_{1}^{\text{Iso-192}}\\}$, as well as the
values $C_{1}^{48}=1.52$ and $C_{1}^{192}=1.66$, are displayed in Figure 9;
from these distributions, we obtain the means and standard deviations:
$C_{1}^{\text{Iso-48}}=0.08$, $C_{1}^{\text{Iso-192}}=0.02$ and
$\sigma_{\text{Iso-48}}=0.07,\,\sigma_{\text{Iso-192}}=0.02$.

In Appendix B we perform Monte Carlo simulations to calculate the
uncertainties in $C_{1}^{48}$ and $C_{1}^{192}$, i.e.,
$\sigma_{C_{1}^{\text{Ran-48}}}=0.67$ and
$\sigma_{C_{1}^{\text{Ran-192}}}=0.35$, respectively. With these data one can
finally calculate the statistical significance of the dipole components found
in our directional analysis; for instance, for the dipole case $C_{1}^{192}$
one computes the following:
$|C_{1}^{192}-C_{1}^{\text{Iso-192}}|/\sqrt{\sigma_{C_{1}^{\text{Ran-192}}}^{2}+\sigma_{\text{Iso-192}}^{2}}$,
finding that this dipole is significant at more than $99.9\%$ confidence
level.

Figure 7: Illustrative examples of simulated isotropic maps,
$H_{0}^{\text{Iso-48}}$-maps, for $N=48$ caps (first row), and
$H_{0}^{\text{Iso-192}}$-maps for $N=192$ caps (second row).

Figure 8: Histogram of the distribution of galactic coordinates $l$ and $b$,
in degrees, for the dipole directions in the generated random
$H_{0}^{\text{Iso}}$-maps. The first and second panels analyze the random maps
$H_{0}^{\text{Iso-48}}$-map (48 caps), while the third and fourth panels
analyze the random maps $H_{0}^{\text{Iso-192}}$-map (192 caps). In both
cases, our results confirm that the isotropic $H_{0}^{\text{Iso-48/192}}$-maps
do not exhibit a preferred direction, as expected.

Figure 9: Histograms of the dipole components, $C_{1}$, obtained from $1000$
randomized maps, plus the vertical dashed line draw for comparison from the
$H_{0}$-map48 (left panel) and $H_{0}$-map192 (right panel), with $C_{1}=1.52$
and $C_{1}=1.66$ from the original maps, respectively. As observed, in both
cases, our directional analysis show that the dipole components of the
$H_{0}$-maps displayed in Figure 4 are statistically significant, at more than
$99.9\%$ confidence level.

###  4.4 Measuring the Hubble constant

Although our directional approach is not intended to perform a $H_{0}$
measurement, even so, one can obtain this important information from the
monopole components of our $H_{0}$-maps48/192.777$H_{0}=\sqrt{C_{0}/4\pi}$,
where $C_{0}$ is the monopole term of the $H_{0}$-maps. This allows us to
calculate the Hubble constant at the effective distance $R^{192}=102.83\pm
10.24$ Mpc, that is, $z\simeq 0.025$, obtaining $H_{0}=70.39\pm 1.40$ km s-1
Mpc-1 using the $H_{0}$-map192 and considering spherical caps with
$\gamma=60^{\circ}$. For completeness, we also calculate for other angular
resolutions and spherical cap sizes used in our scrutiny; the results are
summarized in Table 2.

$\gamma\,\backslash\,H_{0}$-map |  $H_{0}$ [ km s-1 Mpc-1]
---|---
$\gamma=60^{\circ}\,\backslash\,H_{0}$-map48 | $70.35\pm 1.50$
$\gamma=60^{\circ}\,\backslash\,H_{0}$-map192 | $70.39\pm 1.40$
$\gamma=65^{\circ}\,\backslash\,H_{0}$-map48 | $70.42\pm 1.45$
$\gamma=65^{\circ}\,\backslash\,H_{0}$-map192 | $70.39\pm 1.27$
$\gamma=70^{\circ}\,\backslash\,H_{0}$-map48 | $70.43\pm 1.38$
$\gamma=70^{\circ}\,\backslash\,H_{0}$-map192 | $70.41\pm 1.18$

Table 2: $H_{0}$ measurements for different spherical cap sizes.

In Appendix F we discuss the influence that overdensity and underdensity
matter structures can produce on $H_{0}$ measurements.

## 5 Conclusions

Cosmic matter structures grow over time due to gravitational interaction. This
explains why our cosmic neighborhood, $z\simeq 0$, exhibits amazingly large
underdense and overdense matter structures (Rubin, 1951; de Vaucouleurs, 1953;
Courtois et al., 2013; Hoffman et al., 2017; Tully et al., 2019; Colgáin,
2019; de Carvalho et al., 2020; Avila et al., 2021, 2022; Marques et al.,
2024). In this scenario, it was recently proposed that the DR and the Shapley
supercluster structures dominate the galaxy velocity field in the Local
Universe, acting like a dipolar gravitational system. This scenario deserves
investigation because in the Local Universe the effect of peculiar motions
caused by the distribution of matter competes with the expansion of the
universe making the analysis of the HL diagram difficult, affecting the
measurement of the Hubble constant.

Motivated by this proposal, we investigated the subsample of 501 SNe Ia of the
Pantheon+ catalog in the Local Universe, with redshifts $0.015\leq z\leq
0.06$, aiming to measure both the direction and magnitude of the bulk flow
velocity. Our directional analysis scan the celestial sphere considering two
angular resolutions: with $N=48$ and $N=192$ spherical caps, although our
conclusions are drawn from the best angular resolution case, i.e., for $N=192$
caps. We found a statistically significant dipole variation of the Hubble
constant, at more than 99.9% confidence level, with $H_{0}$ values in the
$+/-$ dipole directions: $H_{0}^{+/192}=72.31\pm 1.34$ km s-1 Mpc-1 and
$H_{0}^{-/192}=69.74\pm 1.30$ km s-1 Mpc-1, for the $H_{0}^{192}$-map. The
Table 1, and Figures 4 and 5, summarize our results.

Our studies confirm that matter structures in the Local Universe are indeed
following a dipolar bulk flow motion toward $(l,b)=(326.^{\circ}1\pm
11.^{\circ}2,27.^{\circ}8\pm 11.^{\circ}2)$, a direction close to the Shapley
supercluster
($l_{\scalebox{0.7}{Shapley}},b_{\scalebox{0.7}{Shapley}})=(311.53^{\circ},32.31^{\circ}$),
with the velocity of $132.14\pm 109.30$ km s-1 at the effective distance
$102.83\pm 10.24$ Mpc = $71.98\pm 7.17$ Mpc $h^{-1}$, using $h=0.7$ to
illustrate the measurements’ comparison shown in Figure 5.

Not less interesting, the antipodal direction of the dipolar bulk flow found
points close to the DR structure, as illustrated in Figure 6.

As shown in Figure 5, recent studies of the CosmicFlows-4 data (CF4, Tully et
al., 2023) at scales greater than $100$ Mpc $h^{-1}$ have reported bulk flow
velocities that are in great tension with the velocities expected in the
standard cosmological model, like the measurements of Watkins et al. (2023)
who found $395\pm 29$ km s-1 at 150 Mpc $h^{-1}$ and $427\pm 37$ km s-1 at 200
Mpc $h^{-1}$, and the measurement of Whitford et al. (2023) who found $428\pm
108$ km s-1 at 173 Mpc $h^{-1}$, values with very low probability of
occurrence: $0.015\%$, $0.00015\%$, and $0.11\%$, respectively (these data are
also displayed in Figure 5). As suggested by Watkins et al. (2023), these
results may be an indication that the matter rest frame of the universe is not
that determined from the dipole in the CMB radiation (see, e.g., Migkas et
al., 2021).

However, several analyses performed at moderate scales $\lesssim 100$ Mpc
$h^{-1}$ estimate the bulk flow velocities finding that they are statistically
consistent with the $\Lambda$CDM prediction, as well as the measurement of
Whitford et al. (2023) at $49$ Mpc $h^{-1}$ analysing the CF4 dataset, and
also our measurement at the effective distance of $72$ Mpc $h^{-1}$.

Concluding, our directional analysis shows that the bulk flow velocity field
in the Local Universe is well explained by the gravitational dipole system
Shapley-DR (Hoffman et al., 2017). In addition, we are confident that our
results are robust, and the methodology adopted is unbiased due to the various
consistency tests done, which we present in Appendix.

## Acknowledgments

M.L., C.F., and A.B. acknowledge to CAPES and CNPq for their corresponding
fellowships. F.A. thanks CNPq and FAPERJ, Processo SEI 260003/014913/2023, for
financial support.

##  Appendix A Consistency test for other redshift intervals

We have investigated the consistency of our results studying the SNe present
in other redshift intervals, close to the original interval analyzed in this
work. Thus, we performed our directional analysis for four SNe samples with
redshift intervals as follows: $z\in[0.01,0.06]$, $z\in[0.01,0.055]$,
$z\in[0.01,0.065]$, and $z\in[0.015,0.065]$; the number of SNe Ia in these
intervals was 565, 552, 576, and 512, respectively. The $H_{0}$-maps192
obtained in each case are displayed in the left column of Figure 10. To
illustrate how different these maps are with respect to the original
$H_{0}$-map192, analyzed for the SNe with $z\in[0.015,0.06]$ and displayed in
the left panel, second row, of Figure 4, we calculate the map difference
$D^{i}\equiv$ [$H_{0}$-map${}^{192/\text{orig}}$ \- $H_{0}$-map192/i], where
the index $i$ refers to each one of the four samples mentioned above. The maps
$\\{D^{i}\\}$, displayed in the right column of Figure 10, show that the
differences with respect to $H_{0}$-map${}^{192/\text{orig}}$ are very tiny
for the first two cases, and differences $\lesssim 2\%$ for the last two
cases, implying that our results regarding the interval analyzed in this work,
$z\in[0.015,0.06]$, are robust.

Figure 10: Left column: the $H_{0}$-maps192 for different redshift intervals.
Right column: the maps of the difference between the original $H_{0}$-map192
(see Figure 4) and the corresponding map to the left.

##  Appendix B Uncertainty Due to the Dipole $\delta H_{0}$: Monte Carlo
Simulations

The dipolar nature of $\delta H_{0}(l,b)$ contributes to the uncertainty in
the measurement of $H_{0}^{b\text{-}\\!f}$ (see Section 3.1.2). To calculate
this uncertainty, $\sigma_{dp}$, due to the directional dependence observed in
$H_{0}^{b\text{-}\\!f}\\!(l,b)$, we perform Monte Carlo simulations.

We first generate $1000$ random samples of SNe, where their angular positions
$(l,b)$ were preserved, but their luminosity distances were estimated randomly
from a Gaussian distribution, considering, as the mean and the standard
deviation of such distribution, the luminosity distance of each SN $D_{L}$,
and its uncertainty $\sigma_{D_{L}}$, respectively. We then performed the same
process of directional analysis considering both angular resolutions, that is,
with $N=48$ and $N=192$ caps. Then, performing the $H_{0}$ best fit procedure
in the HL diagrams, we obtain $1000$ random $H_{0}$-maps${}^{\text{Ran-48}}$
and $1000$ random $H_{0}$-maps${}^{\text{Ran-192}}$. 888Not to be confused
with the isotropic $H_{0}^{\text{Iso-48/192}}$-maps, see Section 4.3.

For each one of these maps, we calculate the amplitudes $\\{\delta
H_{0}\\}^{\text{Ran-48/192}}$, the dipole components
$\\{C_{1}\\}^{\text{Ran-48/192}}$, and the dipole directions
$\\{(l,b)\\}^{\text{Ran-48/192}}$. The corresponding histograms of the
distributions obtained are shown in Figure 11 (the first and second rows
correspond to $N=48$, and the third and fourth rows correspond to $N=192$).
The standard deviation of the distributions $\\{\delta
H_{0}\\}^{\text{Ran-48/192}}$ provides a measure of the uncertainty due to the
dipole nature of the $H_{0}$-map, $\delta H_{0}$, where we obtain
$\sigma_{dp}^{48}=1.26$ km s-1 Mpc-1 and $\sigma_{dp}^{192}=1.24$ km s-1
Mpc-1, respectively.

Another quantity that we analyze in Figure 11 are the set of dipole components
$\\{C_{1}\\}^{\text{Ran-48/192}}$ of the $H_{0}^{\text{Ran-48/192}}$-maps,
where their standard deviations provide a measurement of the uncertainties in
the dipole components $C_{1}^{48}$ and $C_{1}^{192}$ of our $H_{0}$-maps
(shown in Figure 4), obtaining $\sigma_{C_{1}^{\text{Ran-48}}}=0.67$ and
$\sigma_{C_{1}^{\text{Ran-192}}}=0.35$, respectively. These quantities are
used to estimate the statistical significance of the dipole nature of our
results (see Section 4.3).

Figure 11:  These plots show the results of our Monte Carlo simulations
performed to produce $H_{0}$-maps${}^{\text{Ran-48/192}}$ from which one can
estimate the uncertainties of quantities used in our approach. In the first
row we display the amplitudes $\\{\delta H_{0}\\}^{\text{Ran-48}}$ (left
panel) and the dipole component $\\{C_{1}\\}^{\text{Ran-48}}$ (right panel)
calculated from random $H_{0}$-maps${}^{\text{Ran-48}}$, where their standard
deviations provide the uncertainties $\sigma_{dp}^{48}=1.26$ km s-1 Mpc-1, and
$\sigma_{C_{1}^{\text{Ran-48}}}=0.67$, respectively (see Appendix B for
details). The third row show similar calculations, but for the $N=192$ case,
namely, $\sigma_{dp}^{192}=1.24$ km s-1 Mpc-1, and
$\sigma_{C_{1}^{\text{Ran-192}}}=0.35$. The second and fourth rows provide the
distributions of the dipole directions of the
$H_{0}$-maps${}^{\text{Ran-48/192}}$, showing mean and standard deviations as
expected due to the procedure used in these Monte Carlo simulations, that is,
$(l,b)^{\text{Ran-48}}=(326.^{\circ}58\pm 14.^{\circ}33,27.^{\circ}33\pm
12.^{\circ}76)$ and $(l,b)^{\text{Ran-192}}=(325.^{\circ}68\pm
6.^{\circ}76,27.^{\circ}17\pm 6.^{\circ}1)$. The vertical dashed lines in the
$\\{C_{1}\\}^{\text{Ran-48/192}}$ plots (right panels in the first and third
row) represent the dipole values $C_{1}^{48}=1.52$ and $C_{1}^{192}=1.66$ of
the $H_{0}$-maps displayed in Figure 4 (see Section 4.3 and Figure 9).

##  Appendix C Studying the correlation between the number of SNe and the best
fit $H_{0}$

An interesting question regards the possible bias of the number of SNe
observed in different directions could have an impact in the best fit analysis
of the HL diagram to obtain $H_{0}$.

For this, we perform a correlation analysis of the $H_{0}$-map with the
number-of-SNe-map (that is, the map where the color in each pixel represents
the number of SNe used to construct our $H_{0}$-map, see Figure 12), using the
linear Pearson correlation coefficient $\cal{P}$. Our analyses show (the bars
mean absolute value): $|{\cal P}|=0.234$, for the case with $N=48$ spherical
caps, and $|{\cal P}|=0.198$, for the case with $N=192$ spherical caps.
According to the literature, for values of the Pearson coefficient in the
interval, $|{\cal P}|\in[0.0,0.199]$ means that the correlation between the
pairs of maps with the same angular resolution is very weak.

Notice that, for consistency, our analyses scan the celestial sphere
considering two angular resolutions: with $N=48$ and with $N=192$ spherical
caps. However, our conclusions are drawn from the best angular resolution
case, i.e., scanning the celestial sphere with $N=192$ spherical caps. Due to
this, we are led to conclude that the number of SNe observed in the 192
spherical caps has a negligible impact in the –statistically
significant–dipolar behavior that our directional analysis found in the
$H_{0}$-map.

Figure 12: Number of SNe maps, for 48 (left) and 192 (right) spherical caps
analyses. These maps show the number of SNe analyzed in each spherical cap, as
in the directional analysis procedure, by assigning a color to the
corresponding number of SNe inside the $60^{\circ}$ cap. These maps are useful
to study the possible correlation with the maps shown in Figure 4; our
results, discussed in Appendix C, confirm that the correlation is very weak.

##  Appendix D Robustness for Other Spherical Cap Sizes:
$\gamma=65^{\circ},\,70^{\circ}$

In this appendix, we perform robustness tests to verify our directional
analyses results considering spherical caps with radius different from
$\gamma=60^{\circ}$, the case considered along this work, namely
$\gamma=65^{\circ}$ and $\gamma=70^{\circ}$. Moreover, we make these tests for
both angular resolutions, with 48 and 192 spherical caps.

Observing Table 3, and the $H_{0}$-maps displayed in Figure 13, we obtain an
excellent agreement with the results found along the text considering
spherical caps with $\gamma=60^{\circ}$. Therefore, we conclude that our
results, summarized in Table 1, are robust considering directional analysis
done with spherical caps of different sizes used to scan the celestial sphere.

| $l(^{\circ})$ | $b(^{\circ})$
---|---|---
$H_{0}$-map48 / $\gamma=65^{\circ}$ | $323.87\pm 22.5$ | $28.04\pm 22.5$
$H_{0}$-map192 / $\gamma=65^{\circ}$ | $328.25\pm 11.2$ | $27.42\pm 11.2$
$H_{0}$-map48 / $\gamma=70^{\circ}$ | $323.87\pm 22.5$ | $28.04\pm 22.5$
$H_{0}$-map192 / $\gamma=70^{\circ}$ | $326.63\pm 11.2$ | $29.77\pm 11.2$

Table 3: Robustness tests to verify our directional analysis results
considering different spherical cap sizes.

Figure 13:  Robustness tests to confirm our directional analysis results
considering different spherical cap sizes, namely: $\gamma=65^{\circ}$ and
$\gamma=70^{\circ}$, tests done for both angular resolutions, with 48 and 192
spherical caps. The plots at the top correspond to the cases with 48 caps for
$\gamma=65^{\circ}$ (left) and $\gamma=70^{\circ}$ (right). The plots at the
bottom correspond to the cases with 192 caps for $\gamma=65^{\circ}$ (left)
and $\gamma=70^{\circ}$ (right). As observed, comparing the dipole directions
of these $H_{0}$-maps, displayed in Table 3, with those in Table 1 both
results are in excellent agreement.

##  Appendix E Consistency in the Calculation of the Effective Distance

The calculation of the effective distance is an important step in our
methodology of finding the velocity of the bulk flow. Equation (15) is based
on the thermal noise model of Turnbull et al. (2012), that needs the
information of the peculiar velocities’ uncertainties, which are individually
unknown in the Pantheon+ catalog, and just appears as value $250$ km s-1 for
all SNe.

In Section 4.2.1, we have calculated $\sigma_{v_{pec}}$ using Equation (17).
However, one can adopt the Monte Carlo method to find $\sigma_{v_{pec}}$ and
the effective distance $R$. For this, we simulate Gaussian distributions for
each SN located within the spherical caps corresponding to both the $+/-$
dipole directions (see Table 1), considering the value $\text{v}_{pec}^{i}$
(data in the Pantheon+ catalog) as the mean of the distribution and $250$ km
s-1 as its standard deviation. For each distribution, one for each SN in the
analysis, we randomly choose a value that we consider the uncertainty
$\sigma_{v_{pec}}$ of such SN. The set of values $\\{\sigma_{v_{pec}}\\}$
obtained in this form is termed one realization. With this set of
uncertainties, we calculate the value of the effective distance $R$ using
Equation (15) for each realization.

We repeat this procedure to produce a set of $N_{\scalebox{0.8}{realiz}}=5000$
realizations999We verified that for a greater number of realizations the final
value of the effective distance $R$ remains practically the same., for each
angular resolution, that is, for $48$ and $192$ spherical caps. The set of
effective distances obtained $\\{R_{j}\\},j=1,\cdots,5000$ produces two
distributions for $48$ and $192$ caps, respectively, that we plot as
histograms in Figure 14. The median of these distributions provides the value
of the effective distance $R$. Therefore, we obtain $R^{48}=118.33\pm 33.71$
Mpc, and $R^{192}=117.57\pm 33.60$ Mpc, results that are in good agreement
with those found in Section 4.2.1, with larger uncertainties than previous
ones.

Figure 14: Histograms of the distributions obtained using the Monte Carlo
approach (see the text for details) to calculate the effective distance $R$,
in units of Mpc, for $H_{0}$-map48 (left panel) and $H_{0}$-map192 (right
panel). The median values of these distributions, that provide our
measurements of $R$, are: $R^{48}=118.33\pm 33.71$ Mpc, and $R^{192}=117.57\pm
33.60$ Mpc, respectively.

##  Appendix F The Hubble Tension: Are Overdensities and Underdensities
Biasing the $H_{0}$ Measurements?

Recent works (Giani et al., 2024; Mazurenko et al., 2024; Kenworthy et al.,
2019) have investigated the possibility that overdensities (i.e., large galaxy
clusters) and underdensities (i.e., large voids) affect the $H_{0}$
measurements in the Local Universe, because such structures produce large
inflows and outflows, respectively. In fact, this biasing effect exists, being
important quantify the impact of such phenomena in the measurements of
$H_{0}$. According to Giani et al. (2024), the large supercluster Laniakea,
which hosts the Milky Way, produces a negative average expansion of $\sim-1.1$
km s-1 Mpc-1, inducing a variation in the Hubble constant $\Delta H_{0}\approx
0.5$ km s-1 Mpc-1. Therefore, the inflows produced by Laniakea can increase
the Hubble tension when applying this correction to the SNe Ia dataset. On the
other hand, Mazurenko et al. (2024) noticed that a large supervoid could be
generating outflows on scales $100-250$ Mpc $h^{-1}$ inducing a change in the
bulk flow velocities at such scales, then decreasing the tension due to the
measurements reported by Watkins et al. (2023) and Whitford et al. (2023);
data that we commented at the end of Section 5. Lastly, Kenworthy et al.
(2019) affirm that overdensities and underdensities can change the Hubble
constant in $2.2\%$, not affecting the Hubble tension.

In this context, our directional analysis can didactically explain these
results, and also provide a quantification of this biasing effect. The color
of each pixel in our $H_{0}$-maps (see, e.g., the left column in Figure 4) is
the manifestation of the dominance of inflows or outflows, phenomena that
increases or decreases the recession speed of matter structures with respect
to the Hubble flow. In fact, the prevalence of these phenomena in the Local
Universe is determined by the large-scale distribution of clustered matter and
large voids, which can vary substantially along different directions as
recently noticed by Franco et al. (2024). However, this diversity is captured
by our methodology, which provides, for the $i$th spherical cap in analysis,
the best fit $H_{0}^{i}$ from the HL diagram of those SNe in the $i$th cap;
since the $i$th cap has its vertex at the position of pixel $i$, it will have
the color corresponding to the dominant effect in that direction: reddish
(bluish) for values above (below) the mean, and greenish for values close to
the mean. The relative differences of the pixel values of the $H_{0}$-map with
respect to the $H_{0}$ value shown in Table 2 are $\lesssim 3\%$ in all cases,
i.e., considering different angular resolutions and cap sizes, in perfect
agreement with the outcomes of Kenworthy et al. (2019).

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