# Radial Tully-Fisher relation and the local variance of Hubble parameter

Balakrishna S. Haridasu,1,2,3 Paolo Salucci,1,2 Gauri Sharma,1,2,3,4,5,6
1SISSA-International School for Advanced Studies, Via Bonomea 265, 34136
Trieste, Italy
2INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
3IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151
Trieste, Italy
4Department of Physics and Astronomy, University of the Western Cape, Cape
Town 7535, South Africa
5University of Strasbourg, CNRS UMR 7550, Observatoire astronomique de
Strasbourg, F-67000 Strasbourg, France
6Department of Physics and Astronomy, University of the Western Cape, Cape
Town 7535, South Africa sharidas<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Utilizing the well-established Radial Tully-Fisher (RTF) relation observed in
a ‘large’ (843) sample of local galaxies, we report the maximum allowed
variance in the Hubble parameter, $H_{0}$. We estimate the total intrinsic
scatter in the magnitude of the RTF relation(s) implementing a cosmological
model-independent cosmographic expansion. We find that the maximum allowed
local variation in our baseline analysis, using 4 RTF relations in the galaxy
sample is $\Delta H_{0}/H_{0}\lesssim 3\%$ at a $95\%$ C.L. significance.
Which is implied form a constraint of $\Delta
H_{0}/H_{0}=0.54^{+1.32}_{-1.37}\%$ estimated at $D_{\rm L}\sim 10\,[{\rm
Mpc}]$. Using only one ‘best-constrained’ radial bin we report a conservative
$95\%$ C.L. limit of $\Delta H_{0}/H_{0}\lesssim 4\%$. Through our estimate of
maximum variation, we propose a novel method to validate several late-
time/local modifications put forth to alleviate the $H_{0}$ tension. We find
that within the range of the current galaxy sample redshift distribution
$10\,[{\rm Mpc}]\leq D_{\rm L}\leq 140\,[{\rm Mpc}]$, it is highly unlikely to
obtain a variation of $\Delta H_{0}/H_{0}\sim 9\%$, necessary to alleviate the
$H_{0}$-tension. However, we also elaborate on the possible alternative
inferences when the innermost radial bin is included in the analysis.
Alongside the primary analysis of fitting the individual RTF relations
independently, we propose and perform a joint analysis of the RTF relations
useful to create a pseudo-standardizable sample of galaxies. We also test for
the spatial variation of $H_{0}$, finding that the current samples’ galaxies
distributed only in the southern hemisphere support the null hypothesis of
isotropy, within the allowed noise levels.

###### keywords:

(cosmology:) dark matter – galaxies – cosmological parameters

††pubyear: 2021††pagerange: Radial Tully-Fisher relation and the local
variance of Hubble parameter–13

##  1 Introduction

To determine the value of the “Hubble Parameter": $H(z)$ at different
redshifts and in particular at present ($H_{0}$) has become one of the most
important and telling cosmological measurements. The well-established and
increasingly prominent $H_{0}$-tension (Verde et al., 2019; Di Valentino et
al., 2021) has paved the way to speculate several modifications to the
concordance model of cosmology. At present, the significance of this
discrepancy between the local model-independent Cepheid calibration-based
supernovae yield $H_{0}=73.04\pm 1.04\,\text{km/s Mpc${}^{-1}$}$ (Riess et
al., 2021) and the Cosmic Microwave Background (CMB) based model-dependent
($\Lambda$CDM ) indirect estimate $H_{0}=67.66\pm 0.49\,\text{km/s
Mpc${}^{-1}$}$ from the improved PR4 analysis in Tristram et al. (2023) is
about $\sim 5\sigma$. In addition, the latter CMB estimate is corroborated by
the Baryon Acoustic Oscillation data (Alam et al., 2021; du Mas des Bourboux
et al., 2020; Zhao et al., 2019), providing $H_{0}=67.81\pm 0.38\,\text{km/s
Mpc${}^{-1}$}$ which only increases the significance of the said tension. A
more recent claim for a $8.2\sigma$ tension was presented in Riess et al.
(2024), addressing the cepheid crowding and their power-luminosity relation
using JWST observations. Several reviews and discussions now provide a very
good overview of the state of the tension (Abdalla et al., 2022; Riess, 2019;
Perivolaropoulos & Skara, 2022; Schöneberg et al., 2021; Shah et al., 2021; Hu
& Wang, 2023; Efstathiou, 2020; Freedman, 2021; Knox & Millea, 2020; Verde et
al., 2023; Akarsu et al., 2024). Also, other local calibration techniques and
measurements either provide a similar disagreement with the CMB estimate or at
least do not yield immediate resolutions to explore (Pesce et al., 2020; de
Jaeger et al., 2020; Schombert et al., 2020; Chen et al., 2019; Bonvin et al.,
2017; Yuan et al., 2019; Blakeslee et al., 2021; Freedman et al., 2019; Wong
et al., 2020; Jee et al., 2019; Shajib et al., 2020; Huang et al., 2019).

To resolve longstanding Hubble tension, several approaches have been proposed
and explored ranging from modifications to early-universe physics (Poulin et
al., 2021; Karwal & Kamionkowski, 2016; Niedermann & Sloth, 2021a, b; Zhao et
al., 2017; de la Macorra et al., 2022; Jedamzik et al., 2021; Kreisch et al.,
2020) to late-time/intermediate redshift physics (Nygaard et al., 2023; Vattis
et al., 2019; Akarsu et al., 2020; Blinov et al., 2020; Anchordoqui, 2021;
Haridasu & Viel, 2020; Tutusaus et al., 2019; Solà et al., 2017; Tutusaus et
al., 2023; Khosravi et al., 2019; Adil et al., 2024; Akarsu et al., 2023; Lapi
et al., 2023), local Universe (Keenan et al., 2013; Whitbourn & Shanks, 2014;
Shanks et al., 2019; Hoscheit & Barger, 2017; Colgáin et al., 2018; Luković et
al., 2019; Cai et al., 2020; Alestas et al., 2021; Kenworthy et al., 2019),
leading to extended discussions (Lee et al., 2023; Addison et al., 2018;
Poulin, 2020; Haridasu et al., 2021; Vagnozzi, 2023; Bernal et al., 2021;
Krishnan et al., 2021; Vagnozzi, 2020; Gómez-Valent et al., 2024; Cao & Ratra,
2023; Gómez-Valent et al., 2024). Alongside modifications to the physics of
the Universe, possibilities that tension can arise due to systematics have
been explored many times (e.g., Mortsell et al., 2022, 2021). Several
techniques to study the possible resolutions of the Hubble tension under the
least possible cosmological assumptions have also driven the need for model-
independent techniques that have substantial significance in recent times, for
example, (Liao et al., 2020; Pandey et al., 2020; Haridasu et al., 2018;
Gómez-Valent, 2018; Lemos et al., 2019; Lyu et al., 2020; Liu et al., 2023; Qi
et al., 2023; Du et al., 2023; Li & Liao, 2024, refernces therin).

In this context, the Tully-Fisher (Kourkchi et al., 2020) and the Baryionic
Tully-Fisher (Kourkchi et al., 2022) relations, that have the advantage of
possessing a clear physical justification (see Salucci et al. (1993); Salucci
(2019)), have been utilized to estimate the local expansion rate, which has
been consistent and at times providing even larger values of $H_{0}$ with
respect to the SH0ES estimate using SNe. These methods essentially highlight
the necessity of alternate local estimations of $H_{0}$ aiding immensely the
discussion on $H_{0}$-tension and providing robust support to the Chepehid-
SNe-based local determination of the same. For a recent discussion see also
Tully (2023). Along these lines, we intend to introduce the possibility of
assessing the same using the Radial Tully-Fisher (RTF) relation for the first
time. The RTF relation has been well established in Yegorova & Salucci (2007),
following the discovery of a strong global relationship between the rotation
velocities and the absolute magnitudes ($M_{\rm I}$) of the $\sim 800$ nearby
galaxies (Persic et al., 1996; Salucci, 2019, and references therein). The RTF
relation indicates that there exist independent radial Tully-Fisher-like
relations at different galactocentric radii within spiral galaxies. In the
current work, we take advantage of the RTF relation to obtain limits in the
allowed variation of the Hubble rate within the local Universe $z<0.035$, more
precisely at the edge of the ‘Hubble-Flow’.

Several local ($z<0.1$) and ultra-local ($z<0.01$) (Alestas et al. (2021);
Marra & Perivolaropoulos (2021); Desmond & Sakstein (2020); Desmond et al.
(2019); Benevento et al. (2020); Banerjee et al. (2023)) physical resolutions
and possible variations in the standardization of SNe (Alestas et al. (2022);
Benisty (2023); Camarena & Marra (2023); Ruchika et al. (2023);
Perivolaropoulos & Skara (2023)), have also been suggested to alleviate the
$H_{0}$-tension, essentially relying on the fast transitions of physics in
addition to the local void Keenan et al. (2013); Hoscheit & Barger (2017);
Whitbourn & Shanks (2014) and extremely local ($z<0.015$) sharp transition of
the dark energy equation of state Camarena & Marra (2019) etc. We anticipate
the ability of the RTF relation to constrain proposed local modifications to
the cosmological scenario by estimating the allowed variance in the local
estimation of the Hubble parameter. We begin by reanalyzing the RTF relations
presented in Persic et al. (1996) (here onwards PSS95) and Yegorova & Salucci
(2007) (here onwards YS07) using improved Bayesian techniques. Given the
nature of the empirical relations, we also propose a methodology improving the
RTF relations by introducing a joint analysis to constrain the relations
simultaneously, modeling a covariance among the individual relations.

The organization of the paper is as follows. After a brief introduction to the
Radial Tully-Fisher relation in Section 2, in Section 2 we present the data.
The cosmographic methodology and fitting of the RTF relations is described in
Section 3. Finally, in Section 4, we present the results with an extended
discussion.

##  2 Radial Tully-Fisher relation

Adopting $R_{\rm opt}$, the radius encompassing 83% of the total light of a
Spiral Galaxy, as the reference size of its stellar disk, the radial Tully-
Fisher relation is a family of TF-like relations observed in disk systems
between the galaxy absolute Magnitude in a certain $j$ band (e.g. $M_{\rm I}$,
the infrared band) and the rotational velocity $V(R/R_{\rm opt})$ measured at
fixed normalized radii $R/R_{\rm opt}$. These relations have been well
established in YS07 with the help of the large sample of galaxies with good
quality RCs from PSS95 and that of two additional samples with 86 and 81 high-
quality RCs Courteau (1997); Vogt et al. (2004). In Fontaine et al. (2018) the
RTF relationships have been established for a sample of 36 Dwarf Spirals and
Irregulars.

The RTF relations, for a given magnitude $M_{j}$, are represented by a class
of linear models,

$M_{j}=a_{n}\times\log_{10}(V_{n})+b_{n}$ (1)

where the subscript $n$ tags the radial bins in which the RTF is established.
These bins are centered at the radii $R_{n}$, defined in terms of fractions of
the optical radii. For the PSS95 sample, that we use in this work:
$R_{n}\equiv n/5R_{\rm opt}$ and the bin width is $1/5R_{\rm opt}$.
Noticeably, the RTF has emerged also by adopting a smaller bin size, e.g.
$1/15R_{\rm opt}$, as in YS07 for the Courteau (1997); Vogt et al. (2004)
samples of high-resolution RCs.

For the galaxies of the current Sample, $V_{n}\equiv V(R_{n})$ is the average
value of the velocity data in each $n^{\rm th}$ bin, $a_{n}$ and $b_{n}$ are
the slope and the intercept of the linear RTF relations found for the data
belonging to the $n^{\rm{th}}$ bin111Since the surface luminosity density of
the stellar disk $I(R)$ is very similar in all spirals and it is represented
by the well-known Freeman profile: $I(R)\propto e^{-R/R_{\rm D}}$, the
lengthscale $R_{\rm opt}=3.2R_{\rm D}$ describes the distribution of luminous
matter for every spiral galaxy in the same consistent way.. Let us notice that
in the constant $b_{n}$ we can incorporate the term
$M_{j}(\infty)-M_{j}(R_{n})$ which at a fixed $n$ is equal in all galaxies, so
that the L.H.S of Equation 1 can be interpreted as the aperture $j$-magnitude
at the radius $R_{n}$, a quantity related to the mass in stars inside such
radius.

It is worth demonstrating here that the RTF relationship with the features
described above has a strong physical background as the originating TF one: it
is a direct consequence of the fact that spiral galaxies are rotationally
supported; their circular velocities, at a radius $R$ balance the
gravitational attraction of the galaxy mass inside this radius $R$.
Remarkably, in Spirals the mass distribution has universal features (see
Salucci (2019)) and it includes 1) a Freeman stellar disk of mass $M_{\rm D}$
and length-scale $R_{\rm D}(M_{\rm I})$, whose contribution to the circular
velocity $V(R)$ can be written as $V_{\rm d}(R/R_{\rm D};M_{\rm D})$ with the
disk mass $M_{\rm D}$ as a free parameter of the mass model. 2) a cored dark
matter halo whose contribution to $V(R)$ can be written as $V_{\rm
h}(R;\rho_{0},r_{0})$ (Salucci (2008)), with the central density $\rho_{0}$
and the core radius $r_{0}$ also as free parameters of the mass model. The
resulting circular velocity model:

$V_{\rm model}(R)=\left[V^{2}_{\rm d}(R/R_{\rm D};M_{\rm D})+V^{2}_{\rm
h}(R;\rho_{0},r_{0})\right]^{1/2}$ (2)

successfully fits the circular velocities $V(R)$ of the entire family of
Spiral galaxies (PSS95; Salucci & Burkert (2000); Karukes & Salucci (2017);
Salucci (2019)) provided that the above three free structural parameters
become specific functions of the galaxy’s infrared magnitude $M_{\rm I}$

$M_{D}(M_{I});\ \ \ r_{0}(M_{I});\ \ \ \rho_{0}(M_{I}),$ (3)

shown in Figure 1 and given in PSS95 (see also Salucci & Burkert (2000)). By
inserting the Equation 3 into Equation 2, one obtains that the latter becomes
equivalent to the Equation 1 with the values of parameters given by Table 1
(see YS07 for the more details).

Figure 1: The parameters of the mass model of Spiral galaxies plotted as
functions of the infrared Magnitude $M_{\rm I}$. $\tilde{\rho}_{0}(M_{\rm
I})=\rho_{0}(M_{\rm I})/(1.2\times 10^{-25}\rm{g/cm^{3}})$ (blue),
$\tilde{r}_{0}(M_{\rm I})=r_{0}(M_{\rm I})/(54\,{\rm kpc})$ (orange),
$\tilde{M}_{\rm D}(M_{\rm I})=M_{\rm D}(M_{\rm I})/(4\times 10^{11}M_{\odot})$
(green).

A further prediction of the physical origin of the RTF relationship is that in
the innermost bin (i.e. for $n=1$) the relation should show a scatter sensibly
larger than in the outer bins (i.e. for $2\leq n\leq 5$). This is due to the
presence, in a good fraction of the objects of the sample, of a central
stellar bulge that provides an important contribution to the mass enclosed in
the innermost radius. The mass of this spheroid has a trend with that of the
stellar disk, so that, the RTF continues to exist also for $n=1$, but with a
scatter larger than those at farther radii, not affected by the central bulge
mass (see YS07). Let us also point out that, according to the above velocity
model and directly supported by RCs data, we have

$V_{\rm model}(n/5\,R_{\rm opt},-23.)\simeq 10^{2.5}\ \rm{km/s}$ (4)

for $1\leq n\leq 5$. Thus the RTF relationship, that will be used here as a
distance indicator, reflects the condition of the centrifugal equilibrium of a
stellar disk embedded in a dark halo.

Data: In the current work, we utilize the same dataset that has been used to
work out the ‘Universal rotational curve’ in PSS95 and later analyzed in
Yegorova & Salucci (2007) to work out the Radial Tully Fisher. The dataset
provides the magnitude of the galaxies in the ‘I’ band against the binned
rotational velocity in each of the optical bins. In Figure 2 we show the
redshift distribution of the galaxies which range between $0.005\leq z\leq
0.035$, incidentally centered around $z\sim 0.015$, similar to the lower limit
of $z>0.023$ usually taken to estimate the local value of $H_{0}$ in SH0ES
analysis Riess et al. (2016, 2018b), allowing the SNe to be in the ‘Hubble
flow’. Notice that the peculiar motions of the galaxies in our sample, assumed
to be $200\,\rm{km/s}$, affect the determination of their redshift with an
error, on average, of about $\pm 0.08$ mag, i.e. a value smaller than the
intrinsic scatter of the RTF relations, that amount to $0.12-0.25$ mag;
therefore, systematics that may arise in having not detailed these motions is
small and likely washed out by the random uncertainty of the RTF relations.
Moreover, in this work, the RTF distance indicator is used for tasks, whose
complement does not require a good knowledge of the peculiar motions of
galaxies.

In Figure 8 we show the RTFs for the different optical radii, alongside the
best-fit linear relations, elaborated in Section 4. The distribution of these
galaxies in the sky is shown in the Figure 3, which are a subset of those
presented in the southern sky survey Mathewson et al. (1992). Here we show the
distribution of the galaxies in Figure 3, in the J2000 system in contrast to
the B1950 as was originally presented in Mathewson et al. (1992).

In summary, the Radial TF has been established in the process of
investigating, in an innovative way (YS07), the mass distribution of Spirals:
for the region inside their optical radii essentially providing evidence for:
i) the presence of a massive Dark Component ii) the decrease of the DM/total
matter fraction with increasing galaxy luminosity iii) a very shallow halo
density profile iv) the presence of a central bulge component. The role of the
RTF thus far was therefore primarily intended to study dark matter properties
in galaxies (e.g. Salucci (2019)). However, given the recent Hubble Tension,
this tight and physically motivated Tully-Fisher-like relationship, even using
the ‘original’ dataset can be a very efficient distance-indicator. The
improvement, with respect to the TF is obvious, that the latter uses one
circular velocity at a reference radius per galaxy, while the RTF exploits the
full RC rotation curve inside the same optical radius of each galaxy.

Figure 2: Redshift distribution of the PSS95 sample consisting of 489 galaxies
below $z<0.015$ and 384 galaxies from $z>0.015$.

##  3 Method

We adopt the linear models shown in Equation 1 to perform regression to obtain
the posteriors for each of the radial bins defined above in terms of the
optical radius. In detail, firstly we perform a simple linear regression for
each of the sub-samples split into the 6 bins with centers located at each of
the first six $0.2$ multiples of $R_{\rm opt}$, wherein all the data points
are assumed to be independent and no correlation is assumed in the analysis.
For consistency, after performing the initial regression we exclude the data
points that are more than $3\sigma$ confidence level away from the posterior
regressed line, to exclude the outliers. However, we also note that this
exclusion of the outliers does not significantly affect the final inferences
drawn from the analysis. The analysis requires 3 parameters for each of the
RTF relations taking into account the slope, intercept, and intrinsic scatter
$\\{a_{n},b_{n},\sigma_{n}\\}$, amounting to a total of 18 total parameters to
fit the 6 optical bins considered in the current dataset. Note that throughout
the analysis we assume and fix a fiducial value of $H_{0}=70\,\text{km/s
Mpc${}^{-1}$}$ and $q_{0}=-0.55$. Varying the value of $H_{0}$ within the
current analysis with a simple cosmographic background, presented in the next
section, only amounts to an overall shift of all RTF relations with no
implication for the shape of the RTF relations. This remains a valid
assumption given no significant empirical indications to go beyond the linear
relations.

###  3.1 Cosmography

To utilize the given galaxy samples to estimate the maximum allowed variance
in the Hubble parameter we implement a simple cosmographic approach to model
the luminosity distance as,

$D_{\rm L}(z)=\frac{c}{H_{0}}\times\left[z+\frac{1}{2}(1-q_{0})z^{2}\right],$
(5)

where $H_{0}$ is the current expansion rate and $q_{0}$ is the deceleration
parameter. The distance modulus can now be written as the difference between
the apparent magnitude and the absolute magnitude,

$m_{\rm I}-M_{\rm I}=25+5\log_{10}\left(D_{\rm L}(z)\left[{\rm
Mpc}\right]\right),$ (6)

wherein we utilise the recession velocity ($V_{\rm rec}$) of each of the
galaxies to obtain the redshift $z=V_{\rm rec}/c$. Note that the above
equation can further be approximated, taking only the first-order term in
Equation 5 into account,

$m_{\rm I}-M_{\rm I}=25+5\log_{10}\left(V_{\rm rec}/H_{0}\right).$ (7)

Finally, given the assumed cosmography to obtain the absolute magnitude of the
data points and the posteriors of linear regression models obtained through
the fitting relation Equation 1 we construct the residuals of the absolute
magnitude as a function of the redshift. The redshift of the galaxies is
consistently obtained utilizing the recession velocity of the galaxies, as
described earlier. Also, one could equivalently present the same in terms of
the luminosity distance of the galaxies instead of redshift, which we present
as the final result (elaborated in Section 4).

###  3.2 Joint analysis

The methodology described so far follows YS07 and constrains each of the RTF
relations individually. While we utilize the same as a first step, in this
work we also introduce a method to perform joint analysis and simultaneously
constrain the RTF relations. As can be seen in the Figure 8 and also in Figure
2. of YS07, all the individual RTF relations at each optical bin converge at a
‘pivot’ that remains to be a fixed point depending only on the value of
$H_{0}$ assumed in the conversion of the observed apparent magnitudes to the
absolute magnitude $M_{\rm I}$ through Equation 5, and the distance modulus
expression. As shown in Section 2 we remind that the mass model of spiral
galaxies at all radii is observationally unrelated to the RTF, implying the
existence of this pivot quantity.

This, in turn, modifies the individual RFT description in Equation 1 with a
pivot as,

$M_{j}-M_{\rm
I}^{\rm{Pivot}}=a_{n}\times(\log_{10}(V_{n})-\log_{10}(V^{\rm{Pivot}}))$ (8)

where the pivot is given by $\\{M_{\rm
I}^{\rm{Pivot}},\log_{10}(V^{\rm{Pivot}})\\}$ and the corresponding slopes
$a_{n}$ of the individual relations. In contrast to the total of 18 parameters
describing the 6 independent relations, within the joint analysis setup, we
have 14 parameters: 12 describing the slopes and intrinsic scatter of the six
relations, and two fixing the pivot.

Finally, the likelihood of the individual linear regression analysis within
each of the radial bins $n$ can be written as,

$-2\log(\mathcal{L})=\sum^{N_{\rm gal}^{n}}\left[\frac{(M_{\rm I}^{\rm
obs}-M_{\rm I}^{\rm theo}(V_{n}))^{2}}{\sigma_{n}^{2}}\right],$ (9)

where the $M_{\rm I}^{\rm obs}$ is constructed using Equation 6, while the
$M_{\rm I}^{\rm theo}(V_{n})$ assume the form in Equation 1, with the free
parameters $\Theta\equiv\\{a_{n},b_{n}\\}$, and $\sigma_{n}$ is a free
parameter assessing the intrinsic scatter of the data. Here $N_{\rm gal}^{n}$
is the number of galaxies with rotational curve velocities in $n^{\rm th}$
radial bin. Similarly, the likelihood for the joint analysis is written as,

$-2\log{(\mathcal{L})}=\sum^{n}\sum^{N_{\rm gal}^{n}}\left[\frac{(M_{\rm
I}^{\rm obs}-M_{\rm I}^{\rm theo}(V_{n}))^{2}}{\sigma_{n}^{2}}\right],$ (10)

wherein a summation over the radial bins $n$ is included and the $M_{\rm
I}^{\rm theo}(V_{n})$ is given by Equation 8, with free parameters
$\Theta\equiv\\{M_{\rm
I}^{\rm{Pivot}},\log_{10}(V^{\rm{Pivot}}),\mathbf{a_{n}}\\}$ amounting to a
total of 14 parameters as described earlier. Note that one could include an
additional parameter $\tau$ as a penalty term $\log(\tau+\sigma_{n}^{2})$ to
the total likelihood. This would describe a degeneracy/covariance between the
6 RTF relations, to be more conservative and validate the utility of the
magnitude used in RTF relation at each optical bin. In other words, the
parameter $\tau$ takes into account the variability between the individual RTF
relations, enforcing a covariance amongst them. However, note that this does
not curtail the use of individual RTF relations to asses the Hubble variance,
but aids an opportunity to construct a ‘standardizable’ sample of galaxies
that follow a scaling relation, and can be utilized for additional distance
ladder analysis, which we intend to perform as an independent full-fledged
analysis. As we shall elaborate later in the Section 4.3, we do not find
strong correlations between individual radial bins, and therefore we leave
this assessment to a future analysis.

To perform the fully Bayesian analysis we utilize the publicly available
emcee222http://dfm.io/emcee/current/ package (Foreman-Mackey et al., 2013),
implementing an affine-invariant ensemble sampler. We analyse the generated
MCMC samples using corner and/or GetDist 333https://getdist.readthedocs.io/
Lewis (2019) packages.

Figure 3: Distribution of the galaxies utilized in the current analysis. The
vertical color bar shows the redshift distribution of the galaxies.

###  3.3 Isotropy of the local universe

Given the availability of the galaxy positions Mathewson et al. (1992), we
also estimate the isotropy and the subsequent variance in the sky of the
current sample. Note that the current sample only covers the southern sky and
is not an isotropic survey allowing us to assess the overall isotropy being
biased on large scales. Therefore, we estimate the overall noise level that
would be expected in an isotropic universe, utilizing the bootstrapping
methodology presented in Soltis et al. (2021). In this method, the positions
of the galaxies are permuted amongst themselves essentially rearranging under
the assumption that in an isotropic universe, the galaxies could have been
observed in any of the given positions within the sky coverage of the survey.
In Soltis et al. (2021), the residuals of the supernovae magnitudes within the
MCMC fitting are utilized as the indicators for the noise levels. Similarly,
we utilize

$r_{\rm{i}}=\frac{M_{\rm{I,i}}^{\rm{theory}}-M_{\rm I,i}^{\rm
data}}{\sigma_{M_{\rm I,i}^{\rm data}}}$ (11)

as the indicators for the same, where $i$ iterates over the number of
galaxies. The assumption of these residuals as an indicator is valid as we do
not intend to estimate the actual isotropy itself but rather the noise level
present within the distribution of the galaxies and contrast against the
observed positions of the galaxies. Using these residuals in the sky we
estimate the clustering of the galaxies by estimating the angular power
spectrum $C_{\rm l}$, the range of which depends on the assumption of the
NSIDE444The NSIDE fixes the resolution of the maps.  of the constructed maps.
In this analysis, we present our results for NSIDE =16, while we validate that
a different assumption does not change our inference substantially. For each
of the galaxies, we take the total residual obtained from all the available
optical bins and then an average value of these residuals for all the galaxies
falling within each pixel of the map. We perform 1000 bootstrap steps for
every set of the model parameters within the MCMC chain that are iterated 1000
times, yielding a total of a million steps taking into account uncertainty in
the RTF relations. It is important to note that the linear regression of the
RTF relations described in the earlier sections is performed assuming no
uncertainty ($\sigma_{M_{\rm I,i}^{\rm data}}=0$)555The anticipated intrinsic
scatter of the RTF relations is much larger than error on the magnitudes of
the galaxies and the inclusion of $\sigma_{M_{\rm I,i}^{\rm data}}$ error
would fall completely within the prior. For this reason, we remain with
fitting eh RTF relations to estimate the maximum scatter. on the $M_{\rm
I,i}^{\rm data}$. However, in estimating the noise level associated with the
statistical isotropy, we assume the reasonable conservative value of
$\sigma_{M_{\rm I,i}^{\rm data}}=0.075$ (Mathewson et al., 1992) for the
photometric measurements errors.

##  4 Results

Table 1: We show the posteriors, including the $68\%$ C.L. limits for the linear regression parameters, performed for each of the radial bins shown in Figure 8. The first column is the radial bins and the next three columns present eh results obtained by fitting the RTFs individually. In columns 4 and 5 we show the results obtained in the joint analysis. In the last column, we show the number of data points utilized in the regression in each bin. All constraints reported here are obtained assuming $H_{0}=70\,\text{km/s Mpc${}^{-1}$}$.  $R_{n}\,[R_{\rm opt}]$ |  |  |  | Joint | $N_{\rm D}$
---|---|---|---|---|---
$b_{n}$ | $a_{n}$ | $\sigma_{n}^{\rm int}$ | $a_{n}$ | $\sigma_{n}^{\rm int}$
0.2 | $-11.98\pm 0.11$ | $-4.67\pm 0.06$ | $0.402\pm 0.011$ | $-4.62\pm 0.04$ | $0.370\pm 0.012$ | 749
0.4 | $-8.18\pm 0.07$ | $-6.15\pm 0.03$ | $0.191\pm 0.005$ | $-6.16\pm 0.03$ | $0.108\pm 0.010$ | 793
0.6 | $-5.78\pm 0.06$ | $-7.11\pm 0.03$ | $0.148\pm 0.004$ | $-7.11\pm 0.02$ | $0.013\pm 0.009$ | 799
0.8 | $-4.21\pm 0.09$ | $-7.72\pm 0.04$ | $0.173\pm 0.005$ | $-7.76\pm 0.03$ | $0.068\pm 0.016$ | 663
1.0 | $-3.02\pm 0.15$ | $-8.20\pm 0.07$ | $0.214\pm 0.007$ | $-8.18\pm 0.04$ | $0.145\pm 0.012$ | 454
1.2 | $-2.06\pm 0.26$ | $-8.61\pm 0.12$ | $0.262\pm 0.013$ | $-8.44\pm 0.06$ | $0.157\pm 0.004$ | 231

We begin by re-analyzing the PSS95 sample containing 6 different radial bins
($R_{\rm opt}$). The results obtained for the fitting of the individual RTFs
are summarised in Table 1 for completeness, and are consistent in comparison
to those presented in YS07 (see Table A1. therein). In the current Bayesian
formalism the mean values of the scatter are mildly larger along with
associated uncertainty. Following this to assess the redshift variation, as a
preliminary step, we split the galaxy sample into two redshift bins performing
the linear regression in each of the radial optical bins. In Figure 4 we show
the intrinsic scatter as obtained for each of the radial bins when the
regression is performed with only samples $z\leq 0.015$ and $z\geq 0.015$ are
considered independently. We find a very good consistency between the
intrinsic scatter estimated for the galaxies with the redshift cut which
asserts that a redshift-dependent analysis would not be biased due to the
variation in the galaxy dataset over redshift. This allows us to estimate the
variance of the Hubble parameter as a function of redshift and, consequently
luminosity distance. For the innermost optical bin $0.2\,R_{\rm opt}$, we find
the intrinsic scatter is mildly larger for $z<0.015$, however consistent
within $2\sigma$. Given the large intrinsic scatter in this bin, mostly due to
the random presence of a not-negligible compact bulge component in our
galaxies, it is difficult to estimate the rotational velocity very accurately
along the lines of the bulge-free Equation 2. Therefore, we exclude this bin
in our baseline analysis when estimating the overall variance 666 The effect
of the bulge in RTF is discussed in YS07.. We also exclude the outermost bin,
owing to the low sample density having only $\sim 230$ data points, and the
fact that the accuracy in measuring the rotational velocity from the
$H_{\alpha}$ line is low at the outskirts of the spiral galaxies.
Nevertheless, we retain the advantage of utilizing 4 independent radial bins
to estimate the overall scatter. From here onwards we remain with 4 optical
bins as our ‘baseline’ dataset to present our main results to evaluate the
cosmological variance of the Hubble parameter. However, we do comment on the
implications of utilizing these two bins as they present interesting scenarios
in assessing the Hubble variance. The $0.6\,R_{\rm opt}$ bin having the least
scatter777Also for the RTFs emerging in the Vogt et al. (2004); Courteau
(1997) samples. and being best constrained RTF, is a conservative estimate to
which we compare the joint constraint in our analysis.

Figure 4:  The intrinsic scatter and the $1\sigma$ error, within each of the
radial bins, however, split at redshift $z\sim 0.015$ to assess the variation
in the different redshift bins.  Figure 5: The variation in the Hubble
parameter utilizing all 5 radial optical bins simultaneously. The data points
have been slightly shifted in the x-axis for better visualization.

Having the Tully-Fisher relation fitted in each of the bins, and establishing
that the dataset is suitable for the assessment of redshift variation, we now
evaluate the residual of the absolute magnitude, w.r.t the fitted linear
regression. We conservatively anticipate the intrinsic scatter and hence the
absolute residual, $\Delta M_{\rm I}$ to account for cosmological variation
which can be converted to the variance in the expansion rate as given by
Equation 12, also considering the uncertainty of the RTF relations888We
validate that including the uncertainty of the RTF relation itself makes very
little difference to our final estimates, since the intrinsic scatter much
larger in comparison.. Note that we also exclude the galaxies that are more
that more than $3\sigma$ away from the residuals deeming them to be
outliers999We find only about 5-10 galaxies per radial bin to satisfy the
outlier condition, which however could be an important assessment as the lower
$D_{\rm L}<20\,[{\rm Mpc}]$ and $D_{\rm L}>120\,[{\rm Mpc}]$ distance bins
with a low number of galaxies can get mildly affected. We also validate that
this step does not make a significant difference to our final inference. .
Following Equation 7, we now translate the average dispersion in the residual
of RTF relation as the maximum possible variance in the value of $H_{0}$ as,

$\displaystyle\frac{\Delta H_{0}}{H_{0}}(z)$ $\displaystyle=$
$\displaystyle\frac{\ln{10}}{5}\overline{\Delta M}(z).$ (12)

Which is now recast as the redshift evolution of the allowed fractional
variation in the Hubble parameter as shown in Figure 5, by binning the
residuals appropriately. In this figure, we show the mean of the residuals for
the best-fit RTF relations and the uncertainity on the same, estimated as the
standard error. We present the residuals in all the radial bins while
assessing the corresponding cosmological distance ($D_{\rm L}$) using Equation
7, assuming that the redshift is given by the recession velocity ($V_{\rm
rec}$) as $z=V_{\rm rec}/c$. We then compute the average variation of the
$\Delta M_{\rm I}$ in each of the redshift/distance bins split accordingly.
Needless to say, the difference in the number of data points (galaxies) in
each bin is reflected in the uncertainity estimated as standard error
$\sigma_{\rm\overline{\Delta H_{0}/H_{0}}}=\sigma_{\rm{\Delta
H_{0}/H_{0}}}/\sqrt{n_{\rm{i}}}$, where $n_{\rm{i}}$ is the number of data
points (galaxies) in each redshift bin. As we have already mentioned, each of
the RTF relations fitted in the 6 optical bins can be considered independent
relations with no correlations whatsoever. Therefore, in the joint analysis of
all the radial bins $R_{n}[R_{\rm opt}]$ considered, we follow the same
procedure considering each velocity measurement in each radial bin as an
independent data point. In Figure 5 we show the variation in the residuals of
absolute magnitudes of the galaxies, as a function of the luminosity distance,
wherein we have binned the luminosity distance into linear bins of size
$\Delta D_{\rm L}=20\,[{\rm Mpc}]$.

As can be seen in Figure 5, we find that there is no discernible redshift
evolution of the Hubble variance, always consistent within $\sim 5\%$,
suggesting no variation in the current redshift range. We find that the joint
constraint to sub percent precision always being consistent with $\Delta
H_{0}/H_{0}=0$ within the $\sigma_{\rm\overline{\Delta H_{0}/H_{0}}}$. For the
$0.4\,R_{\rm opt}$ we find the maximum variation to be $\sim 10\%$ comparing
the data point at $\sim 120\,{\rm Mpc}$ with the innermost data point at $\sim
17\,{\rm Mpc}$. Similarly, the $1.0\,R_{\rm opt}$ also shows a mild increase
in the variance in the innermost distance bin. However, this trend is not
immediately corroborated in the $0.6\,R_{\rm opt}$ and $0.8\,R_{\rm opt}$
radial bins, which are the better-constrained RTF relations showing no
statistically significant trend. This is also reflected in the joint
constraint shown as black data points in Figure 5. Moreover, there seems to be
a mild decrease in the values of $\Delta H_{0}/H_{0}$, especially around the
$D_{\rm L}=100\,[{\rm Mpc}]$ range, however, completely consistent within the
$\sim 2\sigma$ C.L. for $\Delta H_{0}/H_{0}=0$.

We contrast our results for the uncertainty in the evolution of the Hubble
parameter against the constraints obtained from the Supernovae datasets in the
local universe. Although the current galaxy dataset utilized in this work only
extends up to $D_{\rm L}\sim 140[{\rm Mpc}]$, we confirm that within this
range such an under-density is not suggested by utilizing the RTF relation.
This is in agreement with the earlier analysis in Kenworthy et al. (2019);
Luković et al. (2019); Camarena & Marra (2019) using SNe datasets. An under
density of size $\sim 300\,[{\rm Mpc}]$ Hoscheit & Barger (2017) and a similar
less significant local hole of $\sim 150\,[{\rm Mpc}]$ Shanks et al. (2019);
Whitbourn & Shanks (2014) were proposed as possibilities to alleviate the
Hubble tension by modifying the current expansion rate measured by the local
supernovae. To comment on which one would need to extend the current sample to
a higher redshift range Stone et al. (2022) (left for a future investigation).
Within the redshift range of the current dataset, we find that local $z<0.01$
sharp phantom transitions as proposed in Alestas et al. (2021) are less
likely. Similarly, sharp transitions in the gravitational constant ($G_{\rm
N}$) local/ultra-local universe Marra & Perivolaropoulos (2021); Alestas et
al. (2022); Benisty (2023); Camarena & Marra (2023); Perivolaropoulos & Skara
(2023) and local modifications Ruchika et al. (2023) have also been suggested
as a possibility for the discrepancy in the locally measured $H_{0}$ and the
value inferred from the CMB assuming model, which our results so far do not
immediately suggest. In other words, we notice that in the current linear
binning of distance, the innermost constraint obtained at $D_{\rm L}\sim
17\,[{\rm Mpc}]$ is completely consistent with no evolution of the $\Delta
H_{0}/H_{0}$. This constraint is also consistent with the second bin centered
at $D_{\rm L}\sim 30\,[{\rm Mpc}]$. As can be seen in Figure 5, our
conservative inference using only the radial bin $0.6\,R_{\rm opt}$ is
completely consistent with the joint constraint of the 4 considered so far.
However, please refer also to the Sections 4.1 and A for mild yet alternate
perspectives mostly arising due to different binning schemes and the inclusion
of inner and outermost radial bins.

###  4.1 Inclusion of the innermost bin and alternate binning

In this section, we present the results as obtained when the innermost
$0.2\,R_{\rm opt}$ and outer bins $1.2\,R_{\rm opt}$ are also included in the
analysis. We also change the binning scheme of the distances to test for the
validity of including the innermost bin while accommodating variations also on
the closet and the farther-distance bins. In Figure 6, the black data points
show the joint constraint taking into account all the $R_{\rm opt}$ bins and
increasing the number of distance bins in contrast to the earlier results in
Figure 5. As one can immediately notice the the joint constraints show a trend
of increasing $\Delta H_{0}/H_{0}$ for lower distances. This change is
completely driven by the $0.2\,R_{\rm opt}$ bin alone, while the outermost
bin, having only $231$ data points, makes minimal difference to joint
constraints.

We show results for finer binning in distances with $\Delta D_{\rm L}\sim
5\,[{\rm Mpc}]$, at the expense of precision in each bin to evaluate the
overall shape of $\Delta H_{0}/H_{0}(z)$. We find that the $0.2\,R_{\rm opt}$
strongly influences the joint constraint, showing significant evidence for a
possible cosmological signature for an increasing value of the Hubble
parameter in the ‘ultra-local’ ($D_{\rm L}\lesssim 30\,[{\rm Mpc}]$) regime.
Needless to say, to make this inference is heavily reliant on the binning
scheme, as the closer distance bins tend to have very few galaxies providing
the jump we notice therein. For instance, within the joint analysis of all the
radial bins (black data points in Figure 6), the first bin centered at $D_{\rm
L}\sim 14.4\,[{\rm Mpc}]$ and the last bin at $D_{\rm L}\sim 133.2\,[{\rm
Mpc}]$ contain merely 5 and 3, galaxies, respectively. In this case the
difference in the bins is $\Delta H_{0}/H_{0}\sim 0.1$, which is about $\sim
10\%$ variation in the value of absolute $H_{0}$. Note that this level of
variation difference could play a significant role in explaining the
$H_{0}$-tension which is $\sim 8\%$ difference between the local and the CMB
estimates. We also show and validate that the $0.6\,R_{\rm opt}$ radial bin
shows no such behavior using the finer bins, with mildly increasing scatter of
the mean at larger distances.

The initial consideration to leave this bin out of the joint analysis is
because the central bulge in the spiral galaxies does not allow one to measure
the rotational velocity accurately. Therefore, the validity of the redshift-
dependent behavior of $\Delta H_{0}/H_{0}$ we find in Figure 6, is subject to
the confidence in the rotational velocities measured in the innermost regions
of the spiral galaxies. In YS07, the authors provide a relationship (see Eq.14
therein) between the slopes of the individual RTFs and the optical radii,
indicating the validity of the RTFs also in the bulge of the galaxies. This is
once again an empirical relation that is observed and fitted in the current
galaxy sample and it is also shown that this expectation is consistent when
considering the two other galaxy samples they have tested. However, in a
conservative approach, we tend to remain with our 4-bin analysis or
equivalently 5 bins including the outermost radial bin, not claiming
significant evidence for a redshift evolution of $\Delta H_{0}/H_{0}$. This
variation has to be validated in larger galaxy samples with better estimations
of the central rotational velocities and would be untimely to claim a possible
detection of local variation in $H_{0}$ using the current dataset alone. We
leave this for a future investigation using the PROBES (Arora et al., 2021)
dataset. Howbeit, it is interesting to note that the innermost bin carries
significant statistics being able to sway the joint analysis using all 6 bins.

Figure 6: Same as Figure 5 with different binning schemes and inclusion of the
innermost radial $0.2\,R_{\rm opt}$ bin. Black data points show the results
using the same binning as in Figure 5. Figure 7: Top: We show the $1\sigma$
and $2\sigma$ C.L. of the fit to the $0.6R_{\rm opt}$ bin. Center: Similar to
the top panel, for the baseline Joint analysis with the 4 radial bins and for
comparison, we also show the results from all the 6 radial bins in the Bottom
panel.

###  4.2 Smoothing the binned variance

While we have so far presented our results by binning the galaxies in
distance, we now implement a simple smoothing of the scatter using the locally
weighted scatter regression of the scatter (LOWESS 101010Locally weighted
scatter plot smoothing (LOWESS) (Cleveland & Devlin, 1988; Cleveland, 1979).
We utilize the publicly available statsmodels package Seabold & Perktold
(2010) for this purpose. A simple example of how to perform the same can be
found here. Please see also Cook & Weisberg (1999); Fox (2008, 2024); Andersen
(2009) for more details.) obtained in the $\Delta H_{0}/H_{0}\,\rm{vs.}$
$D_{\rm L}$ plane. This technique has been utilized in various contexts in
Montiel et al. (2014); Escamilla-Rivera et al. (2021); Bernardo & Levi Said
(2021); Fernández-Hernández et al. (2019). We implement this essentially to
represent the variation in $\Delta H_{0}/H_{0}$ as a smooth function of
$D_{\rm L}$ and simultaneously obtain the uncertainty on the estimated
variance. Similar to the binning schemes where we have to assume the size of
the $\Delta D_{\rm L}$, here we need to assume a fraction ($f\in\\{0,1\\}$) of
the data that will be considered to obtain the locally weighted least squares
fit. Higher values of this fraction will consider a larger number of data
points eventually providing a smoother reconstruction of the scatter plot.
This technique while being ’non-parametric’ also allows one to obtain the
uncertainty region by simply bootstrapping on the scatter points. We utilize
this method to present our final constraints on the $\Delta H_{0}/H_{0}$ as
our main result in terms of uncertainty and the upper limits.

We show the results of the LOWESS smoothing in Figure 7 for three different
scenarios: using only the central $0.6\,R_{\rm opt}$ bin alone, our baseline
analysis with 4 radial bins and all 6 radial bins included. Here we have
utilized $f=0.2$, while we have tested the results also with the larger
fractions. We show a comparison of the constraints and $95\%$ upper limits on
$\Delta H_{0}/H_{0}$ in Table 2, for $f=0.2$ and $f=0.5$ fractions. Using
lower fractions ($f<0.1$) tends to provide very few galaxies for each locally
weighted regression, essentially mimicking the scatter with very large mean
squared errors. Note that traditionally the fraction of data to be utilized is
optimized for the least possible mean square error through cross-validation
techniques (Montiel et al., 2014, see references therin). In our case, this
typically corresponds to the larger values of $f$, as can be seen in the last
column of Table 2. Therefore we remain with the conservative $f=0.2$ fraction
of data to present the limits expected on $\Delta H_{0}/H_{0}$.

As shown in Table 2, using only the $0.6\,R_{\rm opt}$ bin we find $\Delta
H_{0}/H_{0}\,[\%]=-0.11^{+2.06}_{-1.96}$ with a $95\%$ C.L. upper limit of
$\Delta H_{0}/H_{0}<3.9\%$. We interpret this limit as a conservative upper
limit on the allowed local variation on $H_{0}$. Note that this limit is
estimated at a distance $D_{\rm L}\sim 10\,[{\rm Mpc}]$. Similarly, we find
the upper limit at $95\%$ C.L. limits is $\Delta H_{0}/H_{0}\lesssim 3\%$
which we infer as our best estimate. Needless to say, these limits are tighter
when we utilize $f=0.5$, as can be seen in the last two columns of Table 2. We
also find that similar features are reconstructed in both these cases,
validating the use of 4 bins as our baseline dataset. We notice a mild dip at
$D_{\rm L}\sim 100\,[{\rm Mpc}]$ and a rise at $D_{\rm L}\sim 30\,[{\rm
Mpc}]$, albeit with a low significance of $\sim 1\sigma$. Finally, with the
inclusion of the innermost radial bin, the redshift dependence of $\Delta
H_{0}/H_{0}$ is evident as depicted in the bottom panel of Figure 7. This also
translates to a detection of $\Delta H_{0}/H_{0}>0$ at C.L. of $\sim
1\sigma(f=0.2)$ and $\sim 2\sigma(f=0.5)$.

Bins | $\Delta H_{0}/H_{0}[\%]$
---|---
$f=0.2$ | $f=0.5$
$68\%$ C.L. |  $<95\%$ C.L. |  $68\%$ C.L. |  $<95\%$ C.L.
$0.6R_{\rm opt}$ | $-0.11^{+2.06}_{-1.96}$ | $3.90$ | $0.35^{+1.04}_{-1.18}$ | $2.64$
4 bins | $0.54^{+1.32}_{-1.37}$ | $2.98$ | $0.34^{+0.63}_{-0.74}$ | $1.74$
6 bins | $2.08^{+1.65}_{-1.39}$ | $5.17$ | $1.45^{+0.72}_{-0.69}$ | $2.79$
Table 2: We show the $95\%$ C.L. upper limits on the $\Delta H_{0}/H_{0}[\%]$,
for the two cases of the fraction of data utilized in the analysis. The
$f=0.2$ column corresponds to the uncertainty regions presented in Figure 7.

###  4.3 Joint analysis

Figure 8: Scatter corresponding to the RTF relations in the 6 bins of $R_{\rm
opt}$ taken from PSS95 and the linear best-fits obtained from our joint
analysis assuming $H_{0}=70\,\text{km/s Mpc${}^{-1}$}$.

As anticipated in Section 3, we perform the joint analysis of the RTFs
considering the pivot point as parameters of the fit and individual slopes and
intrinsic scatters for each of the RTFs as free parameters. To validate this
assumption, we first perform a rolling regression111111We use PyMC Wiecki ;
Salvatier et al. (2016) for this purpose. analysis where no binning is
considered but all the data points from all the bins are simultaneously fitted
assuming a gradually changing slope. Wherein we recover that all the
regression lines pass through the pivot point, without an explicit assumption
of the same. In Figure 8, we show the RTF relations fitted against data in
this approach. As can be seen in Figure 8 and also in Table 1 we find a very
good agreement with the scenario when performing individual fits, especially
contrasting the slope of the RTF relations. We obtain the following
constraints for the pivot parameters,

$\displaystyle M_{\rm I}^{\rm{Pivot}}$ $\displaystyle=$
$\displaystyle-23.830\pm 0.066,$ $\displaystyle\log_{10}(V^{\rm{Pivot}})$
$\displaystyle=$ $\displaystyle 2.526\pm 0.009.$

The estimates of the intrinsic scatter and their uncertainty, are at times
lower than those obtained in the individual fits. This is clearly because a
fraction of the uncertainty is now attributed to the estimation of the pivot,
and reduction in the overall degrees of freedom, in the joint analysis. In
Figure 12, we show the correlations between the posteriors of the slopes and
the pivot point. As expected there exists a strong anti-correlation between
the parameters $\log_{10}(V^{\rm{Pivot}})$ and $M_{\rm I}^{\rm{Pivot}}$. The
slopes of the individual RTF relations are correlated with the coordinates of
the pivot point, ranging from anti-correlations for the inner-most $0.2R_{\rm
opt}$ RTF to positive correlation for the outer-most $1.2R_{\rm opt}$ RTF. In
this context, it is interesting to note that the slope of $0.6R_{\rm opt}$ RTF
is almost completely uncorrelated to the pivot point and shows only mild
correlations with the slopes of the other RTFs. This also validates the
existence of a very well-constrained individual RTF relation in this optical
bin.

We show the complete contour plot of all the parameters of the MCMC analysis
in Figure 13, for brevity in the main text. The analysis so far has been
performed assuming $H_{0}=70\,\text{km/s Mpc${}^{-1}$}$, which is necessary to
estimate the distances to galaxies. In Figure 11 we show the comparison the
posteriors when the $H_{0}$ is assumed differently, being either
$67\,\text{km/s Mpc${}^{-1}$}$ Aghanim et al. (2020) or $73\,\text{km/s
Mpc${}^{-1}$}$ Riess et al. (2021). As anticipated the magnitude of the pivot
point is strongly affected by the assumption, while the velocity remains
completely unchanged. Also indicating that not assuming a particular value of
$H_{0}$ mainly affects the overall scale of the RTF relations and not
necessarily the shape of individual relations. However, the redshift
dependence of the data can be affected by the assumption of underlying
cosmology and tentatively this in turn can allow one to use the current
dataset to constrain the value of $H_{0}$. We intend to present this in the
second installment of this pilot analysis, utilizing also the necessary local
distance calibrators.

###  4.4 Variance on the sky

As yet we have only estimated the variation of the Hubble parameter as a
function of redshift, however now we turn to the anisotropy in the estimate of
the $H_{0}$ on the sky. For this purpose, we utilize the method presented in
Soltis et al. (2021), wherein the angular clustering of the Supernovae sample
Riess et al. (2018a); Scolnic et al. (2018) was estimated and a percent level
spatial variation in the Hubble constant was reported. Following this
methodology, we compute the spatial clustering of the galaxies in the current
PSS95 sample, which is a collection of galaxies in the southern hemisphere as
shown in Figure 3. Note however, that in contrast to the SNe datasets that
extend all the to $z\lesssim 2.3$, our dataset of local galaxies is restricted
to $z\leq 0.03$, allowing us to test explicitly for the local anisotropy.

In Figure 9, we show the power spectrum of the spatial clustering of the
galaxies. The black data points and the associated uncertainties represent the
clustering of the galaxies varying on the parameters of the RTF relations as
fitted in the MCMC analysis Figure 13. The blue-shaded region shows the $68\%$
C.L. uncertainty on the noise level associated with the statistical variation
expected in an isotropic universe, given the observed positions of the current
galaxies. As can be seen the angular power spectrum of distribution of the
galaxies in the current catalog is perfectly consistent within the $\lesssim
1\sigma$ level, with the isotropic expectation, indicating no signal for
anisotropy. The current sample only occupies the southern hemisphere, which is
reflected in the data and the noise level as large values of the angular power
spectrum for $l=1,2$ and is completely consistent with what is expected. Also,
note that the uncertainty in the data shown as the error bars in Figure 9,
only takes into account the variation of parameters of the RTFs and a fixed
$H_{0}=70\,\text{km/s Mpc${}^{-1}$}$. We do not anticipate that the values of
$H_{0}$ (also $q_{0}$) or so any variation in cosmology would affect distances
to all the galaxies almost equivalently unless neglecting the redshift
dependence in the current redshift range. Therefore, a full assessment can be
warranted when $H_{0}$ is utilized as a free parameter alongside the local
distance calibrators, to obtain more quantitative limits on the level of
isotropy. This however is not expected to change the inference here that the
current galaxy sample is completely consistent with the null hypothesis of
isotropy.

Figure 9: Power spectrum of the spatial clustering of the galaxies. The blue
line shows the mean of the noise level, while the shaded region shows the
$68\%$ C.L. uncertainty on the noise level.

##  5 Conclusions and Outlook

We have presented the usefulness of the astrophysical scaling relation, in
particular, the Radial Tully-Fisher Yegorova & Salucci (2007), to assess the
variation in the Hubble parameter in the local universe. While similar studies
have been performed using the TF relation earlier, here we use RTF for the
very first time. This empirical relation exploits the information of the full
RC rather than one single reference value as done using the TF relation. From
a physical point of view, in contrast to the TF relation, the RTF takes into
account the presence of DM in galaxies and the variation in them of the
stellar mass-to-light ratios. The primary results of our analysis are
summarised as follows:

  * •

We constrain the maximum possible variance in the Hubble parameter as a
function of redshift in the range of $0.005<z<0.035$ to be $\Delta
H_{0}/H_{0}\lesssim 3\%$ at $95\%$ C.L., showing no significant redshift
dependence.

  * •

Conservatively, using only the $0.6R_{\rm opt}$ radial bin we find the
consistent with our baseline analysis using 4 radial bins with $\Delta
H_{0}/H_{0}\lesssim 4\%$ at $95\%$, estimated at $D_{\rm L}\sim 10\,[{\rm
Mpc}]$.

  * •

These constraints allow us to conservatively conclude that any local solutions
to alleviate the $H_{0}$-tension are not supported within the redshift range
of the current galaxy sample.

  * •

We introduce a joint analysis of the ‘independent’ RTF relations, reducing the
number of parameters while assessing the possible correlation between them.
This also provides us with a possible pseudo-standardization of the RTFs.

  * •

While occupying only the southern hemisphere, spatial clustering of the
current galaxies shows no deviation from the null hypothesis of isotropy.

Constraining the value of the present expansion rate ($H_{0}$) is one of the
most crucial aspects of the current cosmological crisis. Exploring different
independent methods and possible synergies in these datasets will yield a
better understanding of changing the current cosmological paradigm in a more
consistent direction. Needless to say, it is important to validate the
analysis with newer datasets. In this context, we intend to extend the current
analysis to much larger samples of galaxies, like PROBES (Stone et al., 2022)
and MANGA (Arora et al., 2021), also extending to farther redshifts (Sharma et
al., 2021; Sharma et al., 2022) to provide a comprehensive understanding of
the evolution of the local and late universe. This is necessary also to
validate the possible alternative inference when the innermost radial bin
$0.2\,R_{\rm opt}$ is included in the joint analysis, which indicates a
variation of $\Delta H_{0}/H_{0}\sim 10\%$, which is sufficiently apt to
explain the $H_{0}$-tension.

On the other hand, as a second installment to this pilot study, we intend to
perform a joint analysis utilizing the local distance estimators (Riess et
al., 2021) to calibrate the RTFs and constrain the value $H_{0}$. To this end,
we have introduced here the joint analysis of the RTFs that will be a
preliminary step in this direction, allowing a pseudo-standardization of the
empirical Radial Tully-Fisher relations.

## Acknowledgements

The authors are grateful to Leandros Perivolaropoulos, Stephane Courteau, and
Anto I. Lonappan for insightful discussions. BSH is supported by the INFN
INDARK grant.

## Data availability

The data products utilized in the manuscript are all publicly available and
appropriately referenced. And code utilized to perform the analysis here can
be made available on reasonable request soon after acceptance of the paper.

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##  Appendix A Alternate binning of the distance

In the main text, we have presented our primary results using a bin size of
$\Delta D_{\rm L}=20\,[{\rm Mpc}]$, using linear binning of the magnitude
residuals. In this section, we briefly report the mild differences we notice
when different binning schemes are utilized. In Figure 10 we show the $\Delta
H_{0}/H_{0}$ using a logarithmic binning scheme. We notice that the joint
constraint shows a mild jump in the mean value of $\Delta H_{0}/H_{0}$ towards
lower redshifts in the closest distance bin, although with no strong
significance. And while being completely consistent with $\Delta
H_{0}/H_{0}=0$ for distances larger than $20\,[{\rm Mpc}]$. Note that this is
particularly interesting in the context of some of the ultra late-time
resolutions to $H_{0}$-tension, as we mentioned in the main text. The closet
distance bins centered at $\sim 15\,[{\rm Mpc}]$ and $\sim 23\,[{\rm Mpc}]$
contain about 10, 100 galaxies in each, respectively. The most local distance
bin, with very few galaxies, seems to consistently suggest a mild jump in
$H_{0}$, even for our conservative $0.6R_{\rm opt}$ bin. This as we have
already mentioned in the main text is more evident when the innermost
$0.2\,R_{\rm opt}$ bin is included in the joint analysis.

Figure 10: Same as Figure 5 for the logarithmic binning of distance as
described in Appendix A.

##  Appendix B Contour plots of the Joint analysis

For the sake of brevity in the main text, we show the contour plots of the
joint analysis here. As elaborated in the main text, we can see that the
slopes of the individual RTFs are correlated to a certain extent in the joint
analysis, while the intrinsic scatters of the same have negligible
correlation. Also, one can notice that the intrinsic scatter of the bin
$\sigma_{n}(0.6\,\rm{Ropt})$ almost shows lower intrinsic scatter, validating
that this radial bin is well constrained. Similarly, also the slope
$a_{n}(0.6\,\rm{Ropt})$ can be seen to have negligible correlation with the
rest of the RTFs slopes.

Figure 11: We show the $68\%$ and $95\%$ C.L. posteriors for the parameters
$\log_{10}(V^{\rm{Pivot}})$ and $M_{\rm I}^{\rm{Pivot}}$, when the likelihood
analysis is performed assuming $H_{0}=67\,\text{km/s Mpc${}^{-1}$}$ (blue) and
$H_{0}=73\,\text{km/s Mpc${}^{-1}$}$ (red). The $1\sigma$ parameter
constraints for the case of $H_{0}=73\,\text{km/s Mpc${}^{-1}$}$ are shown on
top of the 1D posteriors.  Figure 12: Correlation between the posteriors of
the estimated slopes for the individual RTF relations and the pivot point.
Figure 13: Contour plots showing the $68\%$ and $95\%$ C.L. for all the
parameters in the joint analysis of the RTFs. We assume the
$H_{0}=70\,\text{km/s Mpc${}^{-1}$}$.