# On the time dependency of $a_{0}$

Antonino Del Popolo<EMAIL_ADDRESS>Dipartimento di Fisica e
Astronomia, University of Catania, Viale Andrea Doria 6, 95125 Catania, Italy
Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya str. 48,
119017 Moscow, Russia  Institute of Astronomy and National Astronomical
Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko Shosse Blvd., 1784
Sofia, Bulgaria  Man Ho Chan  Department of Science and Environmental Studies,
The Education University of Hong Kong, Tai Po, New Territories, Hong Kong,
China

###### Abstract

In this paper, we test one of the predictions of the Scale Invariant Vacuum
(SIV) theory on MOND. According to that theory, MOND’s acceleration $a_{0}$ is
not a universal quantity but depends on several parameters, including time. To
check this prediction, we compare the dependency of $a_{0}$ from redshift,
using the values of $a_{0}$ obtained in Marra  _et al._ (2020) by carrying a
Bayesian inference for 153 galaxies of the SPARC sample. Since this sample
does not contain galaxies at large redshift, we have estimated $a_{0}$ from
the data in Nestor Shachar  _et al._ (2023). The SPARC sample, for small
values of the redshift, in the redshift range $0.00032-0.032$ gives a
correlation with $z$, while the Nestor Shachar  _et al._ (2023) data, in the
higher redshift range $0.5-2.5$ gives an anti-correlation with the redshift
$z$. Both samples show a dependency of $a_{0}$ from $z$, although the
uncertainties involved are large, especially for the high-redshift galaxies.
The combined sample gives an overall correlation of $a_{0}$ with $z$. The
different behavior at low and high redshift can be related to a change of the
$a_{0}(z)$ with redshift, or to the lower precision with which the high-
redshift value of $a_{0}$ are known.

Dwarf galaxies; galaxy clusters; modified gravity; mass-temperature relation

###### pacs:

98.52.Wz, 98.65.Cw

##  I Introduction

In the past decades, the $\Lambda$CDM model showed to be a very good model in
predicting observations on cosmological scales Komatsu  _et al._ (2011a); Del
Popolo (2007, 2013), except some drawbacks, and on intermediate scales
((Spergel  _et al._ , 2003; Kowalski  _et al._ , 2008; Percival  _et al._ ,
2010; Komatsu  _et al._ , 2011b; Del Popolo, 2013, 2014). Some of the
drawbacks on cosmological scales are the cosmic coincidence problem (Velten
_et al._ , 2014), the cosmological constant problem (Weinberg, 1989;
Astashenok and del Popolo, 2012), the Hubble tension, namely a discrepancy in
$H_{0}$, the current value of the Hubble parameter, in which the value fitted
using the CMB is different from that in the local universe using supernovae
and stars (Di Valentino  _et al._ , 2021). The CMB also shows an unexpected
cold spot (Cruz  _et al._ , 2005, 2006, 2007), a quadrupole-octupole alignment
(Schwarz  _et al._ , 2004; Copi  _et al._ , 2006, 2007, 2010, 2015), a
hemispherical asymmetry (Eriksen  _et al._ , 2004; Hansen  _et al._ , 2004;
Jaffe  _et al._ , 2005; Hoftuft  _et al._ , 2009; Planck Collaboration  _et
al._ , 2014; Akrami  _et al._ , 2014), and another tension concerning the
growth rate of perturbations, $\sigma_{8}$, when using different measurements
(Macaulay  _et al._ , 2013; Raveri, 2015). Another problem of the model is
that the constituents of dark matter have never been detected, and there are
no clues on what “dark energy” could be. In the scale $1-10$ kpcs, the
$\Lambda$CDM model suffers from the so called “small scale problems, like the
“Cusp/Core” problem (Moore, 1994; Flores and Primack, 1994; Burkert, 1995; de
Blok  _et al._ , 2003; Swaters  _et al._ , 2003; Del Popolo, 2009)(Del Popolo
and Kroupa, 2009; Del Popolo, 2012; Del Popolo and Hiotelis, 2014; Navarro
_et al._ , 1996a, 1997, 2010) the “missing satellite problem” (Klypin  _et
al._ , 1999; Moore  _et al._ , 1999), etc. Apart drastic solutions to solve
the “small scale problems”, like modifying the particles constituting dark
matter ((Colín  _et al._ , 2000; Sommer-Larsen and Dolgov, 2001; Goodman,
2000; Peebles, 2000)), or modifying the power spectrum (e.g. (Zentner and
Bullock, 2003)), astrophysical solutions have been proposed based on the role
of baryons ((Navarro  _et al._ , 1996b; Gelato and Sommer-Larsen, 1999; Read
and Gilmore, 2005; Mashchenko  _et al._ , 2006; Governato  _et al._ , 2010;
El-Zant  _et al._ , 2001, 2004; Romano-Díaz  _et al._ , 2008; Del Popolo,
2009; Cole  _et al._ , 2011; Del Popolo, 2012; Saburova and Del Popolo, 2014;
Del Popolo and Hiotelis, 2014)). Another solution proposed to solve some of
the drawbacks of the $\Lambda$CDM model is modifying the theory of gravity
((Buchdahl, 1970; Starobinsky, 1980; Milgrom, 1983a, b; Ferraro, 2012)). In
some cases these theories do not even need dark matter and dark energy, like
for example the MOND theory (Milgrom, 1986). Another theory without the need
of the two quoted components is the Scale Invariant Vacuum (SIV) theory, based
on the hypothesis that the macroscopic space is scale invariant.

In (Maeder, 2023), the relations between SIV and MOND were studied. The author
found the deep-MOND limit equations are reobtained, and that contrarily to
MOND assumption, $a_{0}$ is not a universal constant but is time dependent. In
the present paper, we want to check the time dependency prediction of $a_{0}$.
For this aim, we will use the SPARC sample and Nestor Shachar  _et al._ (2023)
data, to calculate the values of $a_{0}$ and their relation with the redshift
$z$. The paper is organized as follows. Section II describes SIV, and its
connection with MOND. In Section III, we discuss the data and statistical
analysis. Section IV and V are devoted to discussion and conclusion,
respectively.

##  II $a_{0}$ time dependence

One of the central point in MOND is the existence of an universal
acceleration, namely $a_{0}$, that distinguishes between the Newtonian and
non-Newtonian behavior of a system. Nevertheless MOND theory claims that
$a_{0}$ is universal and constant with a value $a_{0}=1.2\times 10^{-8}$
cm/s2. Several studies have shown that this is not the case (Saburova and Del
Popolo, 2014; Rodrigues  _et al._ , 2018a, b, 2020; Marra  _et al._ , 2020;
Zhou  _et al._ , 2020; Del Popolo, 2023). However, none of the previous
studies has considered the possibility that $a_{0}$ is time-dependent. Based
on the claims in (Maeder and Gueorguiev, 2020; Maeder, 2023) that $a_{0}$
should be a time-dependent quantity, and the relation between SIV and MOND
theories (see the Appendix), in this paper, we would like to check this
possibility.

Figure 1:  Upper panel: The black dots with error bars represent the values of
$a_{0}$ (with $1\sigma$ standard deviation) against redshift $z$ obtained from
the SPARC data. The red solid line indicates the non-linear fit to the data
points. Bottom panel: same as the top panel but with a different ordinate axis
scale.

##  III Data and statistical analysis

As mentioned above, in the paper, we use the SPARC ($\it Spitzer$ Photometry
and Accurate Rotation Curves) dataset (Lelli  _et al._ , 2016) for analysis.
The dataset is constituted by 175 late-type galaxies with surface photometry
at 3.6 $\mu\rm m$, and with high quality rotation curves obtained from $\rm
HI/H\alpha$ studies. The gas mass is provided by 21 cm observations. The
morphologies present in SPARC span from S0 to Irr. Almost all SPARC galaxies
have a disc structure, with some having also bulges. The baryon component is
constituted by the disc, bulge, and gas components. The SPARC dataset, with
some quality cuts, was used by (McGaugh  _et al._ , 2016) to derive the radial
acceleration relation (RAR), and by (Marra  _et al._ , 2020) to perform a test
on MOND, based on (Rodrigues  _et al._ , 2018a, b, 2020). After the quoted
cuts described in (McGaugh  _et al._ , 2016; Marra  _et al._ , 2020), 153
galaxies are left. (Marra  _et al._ , 2020) carried a Bayesian inference for
the quoted 153 galaxies. They adopted for each galaxy the Gaussian likelihood

$\displaystyle\mathcal{L}(\theta)=|2\pi\Sigma|^{-1/2}e^{-\chi^{2}(\theta)/2}\,,$
(1)

where $\Sigma$ is the covariance matrix described in (Marra  _et al._ , 2020),
and $\chi^{2}=\chi^{2}(A_{0},Y_{b},Y_{d},D,I)$, with

$\chi^{2}=\sum_{i=1}^{N}\left(\frac{V_{M}(R_{i},A_{0},Y_{b},Y_{d},D)-V_{C,i}\frac{\sin
I_{0}}{\sin I}}{\sigma_{V,i}\frac{\sin I_{0}}{\sin I}}\right)^{2}\,.$ (2)

In Eq. (2), $N$ is the number of data points relative to the given galaxy
data, $V_{M}$ is the circular velocity of the model, $R_{i}$ represents the
galaxy radius at which the circular velocity $V_{C,i}$ was measured,
$\sigma_{V,i}$ is the corresponding error, and $I_{0}$ is the inclination,
given together with $V_{C,i}$ and $\sigma_{V,i}$ by SPARC. The model velocity
$V_{M}$ is obtained from MOND using the interpolating function

${\bf a}=\frac{{\bf a_{N}}}{1-e^{-\sqrt{a_{n}/a_{0}}}}$ (3)

the Newtonian velocity, $V_{N}$,

$V_{N}^{2}=\Upsilon_{b}|V_{b}|V_{b}+\Upsilon_{d}|V_{d}|V_{d}+|V_{\rm
gas}|V_{\rm gas}\,.$ (4)

Here, the subscripts $b$ and $d$ stand for bulge and disc respectively, and
$\Upsilon_{b}$ and $\Upsilon_{d}$ are the corresponding mass-to-light ratios,
with the distance correction given in Eq. 5 of (Marra  _et al._ , 2020). The
priors chosen, the quality cuts, and the determination of the global best
value are described in Sections 4.2, 4.3, and 4.4 of (Marra  _et al._ , 2020)
respectively.

Moreover, based on the data of the high-redshift galaxies in Nestor Shachar
_et al._ (2023), we can get a sample of high-redshift galaxies for analysis.
These samples can represent the average values of $a_{0}$ at high redshift
($z=0.5-2.5$). However, it is very difficult to obtain the value of $a_{0}$
for each of these high-redshift galaxies because the data of the galactic
rotation curve profiles are too few. Fortunately, there are some galaxies in
which the gravitational acceleration at the effective radius $R_{e}$ is
smaller than the characteristic acceleration scale $a_{0}$. That means the
position at $R_{e}$ for these galaxies are approximately located at the deep-
MOND regime. In this regime, we have (Sanders and McGaugh, 2002)

$V_{c}^{4}=GM_{B}a_{0},$ (5)

where $V_{c}$ is the rotation velocity at $R_{e}$ and $M_{B}$ is the total
baryonic mass within $R_{e}$. From Table 3 of Nestor Shachar  _et al._ (2023),
we find that there are 17 galaxies in which the gravitational acceleration is
smaller than the typical $a_{0}=1.2\times 10^{-8}$ cm/s2 at $R_{e}$. We plot
the values of $a_{0}$ as a function of $z$ in Fig. 2. We can see a very small
decreasing trend exists in the plot when $z$ is larger. Unfortunately, the
uncertainties of the high-redshift rotation curve data are very large.
Therefore, the result here can be used as a reference only.

Figure 2: Top panel: The black dots with error bars represent the values of
$a_{0}$ (with $1\sigma$ standard deviation) against redshift $z$ obtained from
the Nestor Shachar  _et al._ (2023) data. The red solid line indicates the
non-linear fit to the data points. Bottom panel: same as the top panel but
with a different ordinate axis scale.

Figure 3: The black dots with error bars represent the values of $a_{0}$ (with
$1\sigma$ standard deviation) against redshift $z$ obtained from all the data
we have. The red solid line indicates the non-linear fit to the data points.

##  IV Discussion

The MOND paradigm is successful in describing rotation curves, and other
astrophysical aspects. It is based on the idea that Newtonian mechanics is
valid only for higher values of gravitational acceleration, and for lower
values it must be modified. The threshold between the regimes in which
Newtonian mechanics, and MOND are to be applied is related to an universal
parameter of the model, the acceleration $a_{0}$. Several studies have shown
that in reality $a_{0}$ is not a universal constant (Saburova and Del Popolo,
2014; Rodrigues  _et al._ , 2018a, b, 2020; Zhou  _et al._ , 2020; Del Popolo,
2023) depending on several parameters. Prompted by the paper of (Maeder,
2023), showing that the SIV theory reduces to MOND prediction of the
acceleration, in certain limits, and that the acceleration $a_{0}$ of MOND
cannot be a universal constant but a parameter that changes with time, we
checked this possibility. Using the values of $a_{0}$, and their errors
obtained by (Marra  _et al._ , 2020), which are an improvement on that
obtained in (Rodrigues  _et al._ , 2018a) using Bayesian statistics, and the
redshift of the same galaxies, we studied the case of $a_{0}$ being a function
of redshift. Then, we performed a non-linear fitting using the Levenberg-
Marquardt algorithm, on the (Marra  _et al._ , 2020) data, in order to account
for errors in the dependent variable. The result is shown in Fig. 1. the top
and bottom panel represent the same fit considering different scale for the
box. The plot clearly shows a correlation between $a_{0}$, and redshift. The
Pearson’s r coefficient is 0.25661. The other estimated parameters of the fit
are written directly on the Fig. 1. It is interesting to note that the
redshift range is 0.00032-0.032, namely even for low redshift the correlation
is present. In order to see if the correlation continues to persist going
towards larger redshifts, we used the the data of the high-redshift galaxies
in Nestor Shachar  _et al._ (2023). These samples have much larger redshift
than SPARC, with redshift in the range $z=0.5-2.5$. While in the case of the
SPARC sample the values of $a_{0}$ were obtained from the rotation curves
using solid statistical methods, in the case of Nestor Shachar  _et al._
(2023) sample, we do not have the rotation curves of the galaxies, and we
cannot obtain $a_{0}$ with the methods applied to the SPARC sample. Since the
gravitational acceleration at the effective radius $R_{e}$, for some galaxies,
is smaller than $a_{0}$, and they are located in the deep-MOND regime, and in
this case we can get an approximate value of $a_{0}$ from the relation
$V_{c}^{4}=GM_{B}a_{0}$, where $M_{B}$ is the total baryonic mass within
$R_{e}$, and $V_{c}$ is the rotation velocity at $R_{e}$. Clearly this is just
an estimate that cannot be precise as the $a_{0}$ obtained from SPARC with
Bayesian statistics. We used 17 galaxies from Nestor Shachar  _et al._ (2023),
those having the gravitational acceleration smaller than the typical
$a_{0}=1.2\times 10^{-8}$ cm/s2 at $R_{e}$. Then, we fitted with the same
method used for the SPARC galaxies (Fig. 1), the $a_{0}$s estimated from
Nestor Shachar  _et al._ (2023) data. The result is shown in Fig. 2. The fit
shows a anti-correlation between $a_{0}$ and the redshift weaker than in the
case of the SPARC sample. The Pearson’s r coefficient is in this case -0.116,
and the other parameters of the fit are written on Fig. 2. As reported, Fig. 2
shows an anti-correlation of $a_{0}$ with $z$, while Fig. 1 a correlation
between the same quantities. The reason leading to this could be several. As
Fig. 2 shows, the error bars in $a_{0}$ are very large, and the fitting
algorithm is sensitive to the errors. Small changes in the errors could
produce a different behavior on the fit.

Note that the values of $a_{0}$ obtained in Fig. 1 and in Fig. 2 are following
different methods. Generally speaking, using Bayesian method to get the values
of $a_{0}$ would be more reliable. However, for the high-redshift galaxies,
the data points of the rotation curves are too few. The number of variables
involved in the Bayesian analysis would be larger than or almost the same as
the number of data points. In this situation, the results from the Bayesian
analysis would not be reliable. Therefore, we need to use another method (i.e.
the deep-MOND regime approximation) to get the values of $a_{0}$. Although the
methods used in the two samples are different, the values of $a_{0}$ found in
the two different samples originate from the same MOND framework.
Nevertheless, the values of $a_{0}$ obtained for the high-redshift galaxies
are not known with the precision of those of SPARC, and probably the result in
Fig. 2 is less reliable than that in Fig. 1. A last possibility is that the
$a_{0}$ dependence on redshift is non-linear: it increases reaches a maximum
and then decreases. In order to evaluate these possibilities, it would be
necessary to find other galaxies at high redshift for which the rotation
curves are known. In any case, we are more interested in the overall trend of
$a_{0}$ with redshift. For this reason, we plotted all the data on the same
plot in Fig. 3, and fitted again the data. The final result is that there is a
correlation between $a_{0}$, and redshift. The Pearson’s r coefficient is
0.26. Apart from the results using galaxies, some previous studies using the
data of galaxy clusters (HIFLUGCS and CLASH samples) have obtained larger
values of $a_{0}$ by analysing the radial acceleration relation (Chan and Del
Popolo, 2020; Tian  _et al._ , 2020). The redshift ranges of the data samples
HIFLUGCS and CLASH are $z\sim 0.01-0.2$ and $z\sim 0.2-0.7$ respectively,
which are in between the SPARC sample and the high-redshift galaxies used in
this study. The values of $a_{0}$ obtained are $a_{0}\sim 10^{-7}$ cm/s2,
which are larger than that shown in galaxies. Therefore, these show that the
value of $a_{0}$ might not be a constant. We anticipate that more data ranging
from galaxies to galaxy clusters with different redshifts can further verify
our proposal.

##  V Conclusion

According to the $\Lambda$CDM model, universe is dominated by dark matter, and
dark energy. Nevertheless, this dark matter component remains undetected from
many different experiments. Concerning dark energy, the situation is even
worse since we do not really know what dark energy is composed of. Moreover,
the $\Lambda$CDM model shows some drawbacks especially at small scales. In
order to solve those drawbacks in the $\Lambda$CDM model, several authors have
proposed to modify the theory of gravity ((Buchdahl, 1970; Starobinsky, 1980;
Milgrom, 1983a, b; Ferraro, 2012)). In some of these modified gravity
theories, dark matter and dark energy are not needed. One of these theories is
the Scale Invariant Vacuum (SIV) theory, based on the hypothesis that the
macroscopic space is scale invariant. One of the predictions of this theory is
that the quantity $a_{0}$ in MOND is time-dependent. In order to check this
prediction, we compared the dependency of $a_{0}$ from redshift, using two
groups of data. The first one, of higher quality, and low values of $z$, is
obtained in Marra  _et al._ (2020) by carrying a Bayesian inference for 153
galaxies of the SPARC sample. The second one, of lower quality but larger
redshift is obtained from the data in Nestor Shachar  _et al._ (2023). Both
sample shows a dependency $a_{0}$ from redshift, although the involved
uncertainties are large, especially for the high-redshift galaxies. The
$a_{0}$ obtained from the SPARC sample shows a correlation with $z$, at small
redshifts, while the $a_{0}$ obtained from the data of Nestor Shachar  _et
al._ (2023) shows an anti-correlation with redshift, at higher $z$. In other
terms, both samples show a dependency of $a_{0}$ from redshift. The difference
of behavior at small and larger redshift can be due to a change of the
behavior of $a_{0}$ with $z$, or to the lower quality of the high redshift
data. The combined data give a correlation of $a_{0}$ with $z$.

APPENDIX

Scale invariance was considered by Weyl, and Eddington (Eddington, 1923) in
order to account for gravitation, and electromagnetism by the space-time
geometry. It was initially abandoned because particle properties were
dependent on its past worldline. Considering the Weyl Integrable Geometry
(WIG), the problem is solved (Dirac, 1973; Canuto  _et al._ , 1977). By scale
invariance one means that through a transformation of the line element like

$ds^{{}^{\prime}}=\lambda(x^{\mu})ds,$ (6)

the basic equations do not change. $\lambda(x^{\mu})$ is the scale factor,
$ds^{{}^{\prime}}$ refers to general relativity (GR), while $ds$ refers to the
WIG space. (Bouvier and Maeder, 1978) studied the WIG geometrical properties,
and geodesics, while (Maeder and Bouvier, 1979) studied the weak field limit.
Differently from GR is present a small additional acceleration in the
direction of motion. Several studies, with positive results, have been
performed on the rotation of galaxies, the dynamics of clusters of galaxies,
on the growth of density fluctuation in the early universe, on inflation, on
lunar recession (see (Maeder and Gueorguiev, 2020; Maeder, 2023) for the
references).

In (Maeder and Gueorguiev, 2020; Maeder, 2023), the SIV basic theoretical
context, and the cosmological solutions are given. We are interested in the
weak field approximation, and the MOND approximation. The weak field
approximation was obtained by (Maeder and Bouvier, 1979), and can be
expressed, in spherical coordinates, as

$\frac{d^{2}{\bf r}}{dt^{2}}=-\frac{G_{t}M(t)}{r^{2}}\frac{{\bf
r}}{r}+\kappa(t)\frac{d{\bf r}}{dt}.$ (7)

The extra term is called the dynamical gravity. This term, which is
proportional to the velocity, favors outwards motion during the expansion, and
collapse during a contraction. In the case of a dust Universe, the
conservation law imposes $\varrho a^{3}\lambda=const.$. This means that the
inertial mass of a particle is not a constant, and moreover it depends on the
factor $\lambda$. The time dependency of mass can be expressed as
$M(t)=M(t_{0})(t/t_{0})$. In the case $\Omega_{\mathrm{m}}=0.3$, the mass at
the Big-Bang was,
$M(t_{\mathrm{in}})=\Omega_{\mathrm{m}}^{1/3}\,M(t_{0})=0.6694\;M(t_{0})$,
where is $t_{0}=1$ at present and $t_{\mathrm{in}}=\Omega^{1/3}_{\mathrm{m}}$
at the origin. The timescale $\tau$ in years is defined as $\tau_{0}=13.8$
Gyr, and $\tau_{\rm in}=0$ at the Big-Bang. $\tau$, and $t$ are related by

$\frac{\tau-\tau_{\mathrm{in}}}{\tau_{0}-\tau_{\mathrm{in}}}=\frac{t-t_{\mathrm{in}}}{t_{0}-t_{\mathrm{in}}}\,,$
(8)

and

$\tau\,=\,\tau_{0}\,\frac{t-\Omega^{1/3}_{\mathrm{m}}}{1-\Omega^{1/3}_{\mathrm{m}}}\,\quad\mathrm{and}\;\;t\,=\,\Omega^{1/3}_{\mathrm{m}}+\frac{\tau}{\tau_{0}}(1-\Omega^{1/3}_{\mathrm{m}})\,,$
(9)

At present time, $\tau_{0}$, the modified Newton’s equation is given by

$\frac{d^{2}\bf{r}}{d\tau^{2}}\,=\,-\frac{G\,M(\tau_{0})}{r^{2}}\,\frac{\bf{r}}{r}+\frac{\psi_{0}}{\tau_{0}}\frac{d\bf{r}}{d\tau}\,\,.$
(10)

where

$G=G_{t}(\frac{dt}{d\tau})^{2}$ (11)

and

$\psi(\tau)\,=\,\frac{t_{0}-t_{\mathrm{in}}}{t_{\mathrm{in}}+\frac{\tau}{\tau_{0}}(t_{0}-t_{\mathrm{in}})}\,;\;\mathrm{and}\;\psi_{0}=\psi(\tau_{0})\,=1-\Omega^{1/3}_{\mathrm{m}}\,.$
(12)

In the case of galaxies, having rotation periods of some hundred years,
$\lambda$, and $M$ can be considered constant at the level of 1%. In Eq. (10)
the dynamical gravity, namely the term proportional to $\kappa$ disappears.
Considering the transformations $r=\lambda\;r^{\prime}$ and
$t=\lambda\;t^{\prime}$, and applying to the Newton equation expressed in the
prime coordinates gives,

$\frac{d^{2}r^{\prime}}{dt^{\prime
2}}\,=\,-\frac{GM}{r^{\prime}{{}^{2}}}\,\equiv\,g^{\prime}_{N}.$ (13)

In the system $(r,t)$ the total acceleration $g$ is,

$g\,=\,\frac{d^{2}r}{dt^{2}}\,=\,\frac{1}{\lambda}\,\frac{d^{2}r^{\prime}}{dt^{\prime
2}}\,.$ (14)

The accelerations $g_{N}$ and $g^{\prime}_{N}$ are related by,

$g_{N}\,\equiv\,-\frac{G\,M}{r^{2}}\,=\,-\frac{1}{\lambda^{2}}\frac{G\,M}{r^{\prime
2}}\,\equiv\frac{1}{\lambda^{2}}\,g^{\prime}_{N}\,.$ (15)

Then the total acceleration is

$g\,=\,\frac{d^{2}r}{dt^{2}}\,=\,\frac{1}{\lambda}\,\frac{d^{2}r^{\prime}}{dt^{\prime
2}}\,=\,\frac{1}{\lambda}\,g^{\prime}_{N}\,=\,{\lambda}\,g_{N}\,,$ (16)

and finally

$g\,=\,\frac{d^{2}r}{dt^{2}}\,=\,{\lambda}\,g_{N}\,=\,\left(\frac{g^{\prime}_{N}}{g_{N}}\right)^{1/2}g_{N}\,=\left(g^{\prime}_{N}\,g_{N}\right)^{1/2}\,.$
(17)

In the deep-MOND regime limit we have

$g=\sqrt{a_{0}g_{N}}$ (18)

and then comparing, $g^{\prime}_{N}=a_{0}$. So, in the initial approximation
of constant $\lambda$, and $M$, the scale invariant theory leads us to the
deep-MOND limit, and moreover $a_{0}$ is time dependent like $\lambda$ (in the
general case). Another way to obtain the MOND behavior from Eq. (10) is to
consider low gravities. In this case, one can show that

$g\,=g_{\mathrm{N}}\left[1+\frac{\sqrt{2}\,\psi_{0}}{\xi}\,\left(\frac{g_{\mathrm{c}}}{g_{\mathrm{N}}}\right)^{1/2}\right]\,.$
(19)

where $g_{c}$ is the mean gravity at the edge of a sphere having the critical
density, $\xi=\tau_{0}H_{0}$, and $\psi_{0}$ depends on $\Omega_{m}$ (see Eq.
(21)). For regions at large distances from the galactic center the previous
equation reduces to

$g\,\simeq\,\frac{\sqrt{2}\,\psi_{0}}{\xi}\,\left(g_{\mathrm{c}}\,g_{\mathrm{N}}\right)^{1/2}\,$
(20)

where

$\frac{\sqrt{2}\,\psi_{0}}{\xi}=\frac{\sqrt{2}\,(1-\Omega^{1/3}_{\mathrm{m}})}{H(\tau_{0})\,\tau_{0}}=\frac{(1-\Omega_{\mathrm{m}})}{\sqrt{2}}\,,$
(21)

So, in the weak field limit, Eq. (10) leads to Eq. (20) which is equivalent to
the deep-MOND limit, namely Eq. (18).

In the case of SIV or $\Lambda$CDM models with $\Omega_{m}$ in the range
$0.2-0.3$, we have

$a_{0}=\frac{(1-\Omega_{\mathrm{m}})^{2}}{4}\,n\,c\,H_{0}\,$ (22)

or

$a_{0}=\frac{n\,c\,(1-\Omega_{\mathrm{m}})(1-\Omega^{1/3}_{\mathrm{m}})}{2\,\tau_{0}}$
(23)

$n$ depends on the cosmology, and is defined in (Maeder, 2023).

Summarizing, according to SIV, $a_{0}$ is not a universal constant, but
depends on time, $H_{0}$, and $\Omega_{m}$.

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Table 1:  (1) Galaxy name. (2) Best fit for $A_{0}=\log_{10}a_{0}$ ($\rm km/s^{2}$). (3–8) $A_{0}$ credible intervals ($1\sigma$, $3\sigma$, $5\sigma$). (9-10) Distance (Mpc), Error. (11) $Z=\log_{10}z$.  Galaxy |  $A_{0}$ best | $1\sigma_{-}$ | $1\sigma_{+}$ | $3\sigma_{-}$ | $3\sigma_{+}$ | $5\sigma_{-}$ | $5\sigma_{+}$ | D | $\rm D_{\rm Error}$ | Z
---|---|---|---|---|---|---|---|---|---|---
CamB | -14.645 | -0.614 | 0.462 | -5.328 | 0.727 | -5.328 | 1.287 | 3.36 | 0.26 | -3.1237
D512-2 | -13.241 | -0.416 | 0.463 | -1.388 | 1.963 | -3.752 | 4.179 | 15.2 | 4.56 | -2.4682
D564-8 | -13.486 | -0.147 | 0.164 | -0.424 | 0.585 | -0.71 | 1.212 | 8.79 | 0.28 | -2.7061
D631-7 | -13.14 | -0.075 | 0.08 | -0.214 | 0.259 | -0.337 | 0.462 | 7.72 | 0.18 | -2.7625
DDO064 | -13.03 | -0.317 | 0.363 | -0.935 | 1.609 | -1.679 | 2.535 | 6.80 | 2.04 | -2.8176
DDO154 | -13.007 | -0.066 | 0.07 | -0.187 | 0.222 | -0.298 | 0.377 | 4.04 | 20% | -3.0437
DDO161 | -11.576 | -0.438 | 0.641 | -0.996 | 1.505 | -1.401 | 2.143 | 7.50 | 2.25 | -2.775
DDO168 | -12.993 | -0.146 | 0.168 | -0.388 | 0.592 | -0.577 | 1.157 | 4.25 | 0.21 | -3.0217
DDO170 | -13.615 | -0.245 | 0.288 | -0.657 | 1.086 | -0.985 | 1.794 | 15.40 | 4.62 | -2.4626
ESO079-G014 | -12.785 | -0.242 | 0.284 | -0.654 | 1.137 | -1.008 | 1.776 | 28.70 | 7.17 | -2.1922
ESO116-G012 | -12.479 | -0.273 | 0.362 | -0.651 | 1.706 | -0.945 | 2.37 | 13.00 | 3.90 | -2.5361
ESO444-G084 | -12.322 | -0.308 | 0.37 | -0.807 | 1.457 | -1.153 | 2.589 | 4.83 | 0.48 | -2.9661
ESO563-G021 | -12.866 | -0.088 | 0.094 | -0.253 | 0.301 | -0.404 | 0.535 | 60.8 | 9.10 | -1.8662
F565-V2 | -12.742 | -0.258 | 0.306 | -0.687 | 1.149 | -1.011 | 2.185 | 51.8 | 10.00 | -1.9357
F568-3 | -13.065 | -0.381 | 0.468 | -1.09 | 2.097 | -2.011 | 3.371 | 82.40 | 8.24 | -1.7341
F568-V1 | -12.544 | -0.352 | 0.441 | -1.056 | 2.656 | -1.503 | 4.102 | 80.60 | 8.06 | -1.7437
F571-8 | -11.482 | -0.22 | 0.281 | -0.557 | 1.596 | -0.783 | 2.307 | 53.30 | 10.70 | -1.9233
F571-V1 | -13.074 | -0.531 | 0.674 | -1.317 | 4.444 | -6.797 | 6.756 | 80.10 | 8.00 | -1.7464
F574-1 | -13.03 | -0.196 | 0.22 | -0.572 | 0.841 | -0.975 | 1.564 | 96.80 | 9.68 | -1.6642
F583-1 | -12.998 | -0.254 | 0.293 | -0.763 | 1.384 | -1.067 | 2.574 | 35.40 | 8.85 | -2.1011
F583-4 | -13.091 | -0.321 | 0.364 | -0.937 | 1.353 | -1.806 | 2.401 | 53.30 | 10.70 | -1.9233
IC2574 | -13.099 | -0.067 | 0.08 | -0.177 | 0.32 | -0.26 | 0.56 | 3.91 | 0.20 | -3.0579
IC4202 | -12.607 | -0.452 | 2.562 | -0.612 | 4.467 | -1.012 | 4.686 | 100.40 | 10.00 | -1.6483
KK98-251 | -13.834 | -0.376 | 0.395 | -5.703 | 1.539 | -2.491 | 3.983 | 6.80 | 2.04 | -2.8176
NGC0024 | -12.862 | -0.095 | 0.097 | -0.277 | 0.297 | -0.459 | 0.515 | 7.30 | 0.36 | -2.7868
NGC0055 | -13.205 | -0.067 | 0.069 | -0.201 | 0.212 | -0.332 | 0.372 | 2.11 | 0.11 | -3.3258
NGC0100 | -12.603 | -0.282 | 0.368 | -0.702 | 1.706 | -1.027 | 2.507 | 13.50 | 4.05 | -2.5197
NGC0247 | -13.451 | -0.135 | 0.126 | -0.448 | 0.356 | -0.815 | 0.56 | 3.70 | 0.19 | -3.0819
NGC0289 | -13.489 | -0.165 | 0.17 | -0.49 | 0.535 | -0.773 | 0.891 | 20.80 | 5.20 | -2.332
NGC0300 | -12.493 | -0.351 | 0.47 | -0.8 | 2.343 | -1.057 | 3.43 | 2.08 | 0.10 | -3.332
NGC0801 | -13.546 | -0.059 | 0.061 | -0.175 | 0.191 | -0.283 | 0.342 | 80.70 | 8.07 | -1.7432
NGC0891 | -12.612 | -0.051 | 0.052 | -0.148 | 0.16 | -0.243 | 0.281 | 9.91 | 0.50 | -2.654
NGC1003 | -10.827 | -0.516 | 0.661 | -1.291 | 1.168 | -1.726 | 1.47 | 11.40 | 3.42 | -2.5932
NGC1090 | -13.181 | -0.14 | 0.151 | -0.4 | 0.491 | -0.629 | 0.784 | 37.00 | 9.25 | -2.0819
NGC2403 | -12.006 | -0.09 | 0.095 | -0.256 | 0.305 | -0.368 | 0.462 | 3.16 | 0.16 | -3.1504
NGC2683 | -13.241 | -0.096 | 0.096 | -0.287 | 0.295 | -0.489 | 0.496 | 9.81 | 0.49 | -2.6584
NGC2841 | -12.898 | -0.055 | 0.058 | -0.156 | 0.188 | -0.245 | 0.324 | 14.10 | 1.40 | -2.5009
NGC2903 | -12.317 | -0.104 | 0.106 | -0.304 | 0.317 | -0.437 | 0.462 | 6.60 | 1.98 | -2.8305
NGC2915 | -12.399 | -0.104 | 0.113 | -0.296 | 0.369 | -0.475 | 0.703 | 4.06 | 0.20 | -3.0415
NGC2955 | -12.903 | -0.134 | 0.139 | -0.39 | 0.433 | -0.633 | 0.727 | 97.9 | 9.80 | -1.6593
NGC2976 | -18.324 | -2.469 | 1.421 | -3.109 | 3.518 | -3.109 | 4.828 | 3.58 | 0.18 | -3.0962
NGC2998 | -13.369 | -0.1 | 0.105 | -0.29 | 0.338 | -0.471 | 0.613 | 68.1 | 10.2 | -1.8169
NGC3109 | -12.76 | -0.075 | 0.084 | -0.206 | 0.285 | -0.318 | 0.529 | 1.33 | 0.07 | -3.5262
NGC3198 | -12.92 | -0.094 | 0.101 | -0.264 | 0.334 | -0.392 | 0.511 | 13.8 | 1.40 | -2.5102
NGC3521 | -12.928 | -0.115 | 0.119 | -0.332 | 0.371 | -0.529 | 0.604 | 7.70 | 2.30 | -2.7636
NGC3726 | -13.133 | -0.149 | 0.159 | -0.428 | 0.526 | -0.682 | 0.984 | 18.00 | 2.50 | -2.3948
NGC3741 | -12.828 | -0.07 | 0.076 | -0.2 | 0.246 | -0.31 | 0.441 | 3.21 | 0.17 | -3.1436
NGC3769 | -12.98 | -0.141 | 0.153 | -0.402 | 0.519 | -0.626 | 1.052 | 18.00 | 2.50 | -2.3948
NGC3877 | -13.184 | -0.745 | 0.577 | -6.789 | 0.834 | -6.789 | 1.545 | 18.00 | 2.50 | -2.3948
NGC3893 | -12.737 | -0.139 | 0.148 | -0.402 | 0.486 | -0.639 | 0.908 | 18.00 | 2.50 | -2.3948
NGC3917 | -13.232 | -0.184 | 0.186 | -0.559 | 0.581 | -0.973 | 1.076 | 18.00 | 2.50 | -2.3948
NGC3949 | -12.935 | -0.32 | 0.292 | -6.868 | 0.889 | -7.053 | 1.382 | 18.00 | 2.50 | -2.3948
NGC3953 | -13.77 | -3.868 | 0.454 | -6.15 | 0.806 | -6.15 | 1.581 | 18.00 | 2.50 | -2.3948
NGC3972 | -12.766 | -0.19 | 0.19 | -0.6 | 0.608 | -1.073 | 1.077 | 18.00 | 2.50 | -2.3948
NGC3992 | -13.299 | -0.099 | 0.102 | -0.29 | 0.318 | -0.479 | 0.568 | 23.70 | 2.30 | -2.2753
NGC4010 | -12.84 | -0.168 | 0.175 | -0.501 | 0.58 | -0.839 | 1.128 | 18.00 | 2.50 | -2.3948
NGC4013 | -12.858 | -0.083 | 0.089 | -0.233 | 0.29 | -0.365 | 0.521 | 18.00 | 2.50 | -2.3948
NGC4051 | -13.571 | -5.087 | 0.431 | -6.366 | 0.933 | -6.366 | 1.766 | 18.00 | 2.50 | -2.3948
NGC4068 | -13.607 | -0.335 | 0.331 | -5.465 | 1.17 | -6.357 | 1.93 | 4.37 | 0.22 | -3.0096
NGC4085 | -12.829 | -0.212 | 0.212 | -0.682 | 0.682 | -1.417 | 1.396 | 18.00 | 2.50 | -2.3948
NGC4088 | -13.149 | -0.147 | 0.155 | -0.426 | 0.512 | -0.707 | 0.953 | 18.00 | 2.50 | -2.3948
NGC4100 | -13.182 | -0.094 | 0.099 | -0.273 | 0.315 | -0.445 | 0.573 | 18.00 | 2.50 | -2.3948
NGC4138 | -13.069 | -0.213 | 0.212 | -0.686 | 0.657 | -1.623 | 1.21 | 18.00 | 2.50 | -2.3948
NGC4157 | -12.978 | -0.121 | 0.13 | -0.343 | 0.433 | -0.543 | 0.779 | 18.00 | 2.50 | -2.3948
NGC4183 | -13.506 | -0.145 | 0.151 | -0.425 | 0.485 | -0.704 | 0.89 | 18.00 | 2.50 | -2.3948
NGC4217 | -12.419 | -0.181 | 0.208 | -0.496 | 0.815 | -0.745 | 1.504 | 18.00 | 2.50 | -2.3948
NGC4559 | -12.954 | -0.258 | 0.318 | -0.689 | 1.42 | -0.96 | 2.025 | 9.00 | 2.70 | -2.6958
NGC5005 | -12.783 | -0.28 | 0.246 | -6.503 | 0.704 | -7.205 | 1.001 | 16.90 | 1.50 | -2.4222
NGC5033 | -12.933 | -0.1 | 0.101 | -0.295 | 0.307 | -0.442 | 0.47 | 15.70 | 4.70 | -2.4542
NGC5055 | -12.821 | -0.098 | 0.098 | -0.29 | 0.293 | -0.426 | 0.444 | 9.90 | 0.30 | -2.6544
NGC5371 | -13.769 | -0.09 | 0.092 | -0.265 | 0.283 | -0.447 | 0.437 | 39.70 | 9.92 | -2.0513
NGC5585 | -12.21 | -0.37 | 0.552 | -0.764 | 1.895 | -1.156 | 2.194 | 7.06 | 2.12 | -2.8013
NGC5907 | -13.373 | -0.04 | 0.041 | -0.117 | 0.125 | -0.186 | 0.211 | 17.30 | 0.90 | -2.412
NGC5985 | -13.567 | -0.095 | 0.098 | -0.28 | 0.31 | -0.467 | 0.556 | 39.70 | 9.90 | -2.0513
NGC6015 | -13.536 | -0.094 | 0.098 | -0.272 | 0.306 | -0.443 | 0.498 | 17.00 | 5.10 | -2.4196
NGC6195 | -13.133 | -0.105 | 0.108 | -0.312 | 0.329 | -0.501 | 0.562 | 127.80 | 12.80 | -1.5435
NGC6503 | -12.827 | -0.043 | 0.045 | -0.126 | 0.142 | -0.201 | 0.239 | 6.26 | 0.31 | -2.8535
NGC6674 | -13.504 | -0.082 | 0.085 | -0.238 | 0.264 | -0.39 | 0.452 | 51.20 | 10.20 | -1.9408
NGC6789 | -12.214 | -0.284 | 0.314 | -0.841 | 1.141 | -1.492 | 2.393 | 3.52 | 0.18 | -3.1035
NGC6946 | -13.033 | -0.098 | 0.099 | -0.293 | 0.3 | -0.466 | 0.462 | 5.52 | 1.66 | -2.9081
NGC7331 | -12.785 | -0.063 | 0.066 | -0.18 | 0.211 | -0.275 | 0.353 | 14.70 | 1.50 | -2.4828
NGC7793 | -13.243 | -0.205 | 0.23 | -0.57 | 0.823 | -0.926 | 1.676 | 3.61 | 0.18 | -3.0926
NGC7814 | -12.572 | -0.058 | 0.058 | -0.18 | 0.171 | -0.307 | 0.284 | 14.40 | 0.66 | -2.4917
UGC00128 | -13.691 | -0.116 | 0.128 | -0.32 | 0.429 | -0.487 | 0.735 | 64.50 | 9.70 | -1.8405
UGC00191 | -13.65 | -0.241 | 0.259 | -0.707 | 0.887 | -1.159 | 1.378 | 17.10 | 5.10 | -2.4171
UGC00634 | -12.969 | -0.379 | 0.45 | -1.014 | 1.701 | -1.469 | 2.544 | 30.90 | 7.70 | -2.1601
UGC00731 | -13.257 | -0.32 | 0.369 | -0.98 | 1.834 | -2.166 | 2.757 | 12.50 | 3.75 | -2.5532
UGC00891 | -12.334 | -0.379 | 0.554 | -0.8 | 2.19 | -1.214 | 2.594 | 10.20 | 3.10 | -2.6415
UGC01281 | -12.981 | -0.083 | 0.08 | -0.263 | 0.233 | -0.476 | 0.384 | 5.27 | 0.24 | -2.9283
UGC02259 | -13.736 | -0.236 | 0.251 | -0.705 | 0.85 | -1.283 | 1.635 | 10.50 | 3.10 | -2.6289
UGC02487 | -13.349 | -0.074 | 0.075 | -0.22 | 0.229 | -0.357 | 0.377 | 69.10 | 10.40 | -1.8106
UGC02885 | -12.964 | -0.101 | 0.105 | -0.292 | 0.33 | -0.463 | 0.577 | 80.60 | 8.06 | -1.7437
UGC02916 | -13.551 | -0.156 | 0.153 | -0.515 | 0.465 | -1.21 | 0.758 | 65.40 | 9.80 | -1.8345
UGC02953 | -13.032 | -0.067 | 0.069 | -0.2 | 0.211 | -0.281 | 0.338 | 16.50 | 4.95 | -2.4326
UGC03205 | -13.201 | -0.085 | 0.088 | -0.245 | 0.275 | -0.39 | 0.461 | 50.00 | 10.00 | -1.9511
UGC03546 | -12.884 | -0.103 | 0.107 | -0.313 | 0.352 | -0.453 | 0.639 | 28.70 | 7.20 | -2.1922
UGC03580 | -10.529 | -0.525 | 0.646 | -1.473 | 1.102 | -2.141 | 1.434 | 20.70 | 5.20 | -2.3341
UGC04278 | -12.504 | -0.319 | 0.426 | -0.766 | 2.138 | -1.135 | 2.641 | 9.51 | 2.85 | -2.6719
UGC04325 | -13.656 | -5.926 | 0.625 | -6.163 | 1.357 | -6.163 | 3.269 | 9.60 | 2.88 | -2.6678
UGC04483 | -13.395 | -0.141 | 0.142 | -0.428 | 0.434 | -0.733 | 0.759 | 3.34 | 0.31 | -3.1263
UGC04499 | -13.231 | -0.297 | 0.344 | -0.872 | 1.609 | -1.371 | 3.036 | 12.50 | 3.75 | -2.5532
UGC05005 | -13.197 | -0.419 | 0.501 | -1.185 | 2.231 | -1.859 | 4.122 | 53.70 | 10.70 | -1.9201
UGC05253 | -12.707 | -0.085 | 0.086 | -0.255 | 0.259 | -0.398 | 0.411 | 22.90 | 5.72 | -2.2902
UGC05414 | -13.154 | -0.322 | 0.371 | -1.043 | 1.974 | -2.442 | 2.8 | 9.40 | 2.82 | -2.6769
UGC05716 | -13.273 | -0.285 | 0.343 | -0.732 | 1.225 | -1.11 | 1.652 | 21.30 | 5.30 | -2.3217
UGC05721 | -12.573 | -0.241 | 0.291 | -0.641 | 1.266 | -0.906 | 2.059 | 6.18 | 1.85 | -2.8591
UGC05750 | -13.355 | -0.285 | 0.313 | -0.861 | 1.124 | -2.09 | 2.168 | 58.70 | 11.70 | -1.8814
UGC05764 | -14.603 | -1.104 | 2.539 | -1.545 | 4.629 | -1.545 | 5.561 | 7.47 | 2.24 | -2.7768
UGC05829 | -13.033 | -0.602 | 0.717 | -2.34 | 4.449 | -6.898 | 5.976 | 8.64 | 2.59 | -2.7136
UGC05918 | -13.083 | -0.334 | 0.38 | -0.979 | 1.702 | -1.592 | 2.999 | 7.66 | 2.30 | -2.7658
UGC05986 | -12.436 | -0.24 | 0.306 | -0.605 | 1.531 | -0.851 | 2.004 | 8.63 | 2.59 | -2.7141
UGC06399 | -12.847 | -0.167 | 0.172 | -0.51 | 0.562 | -0.906 | 1.109 | 18.00 | 2.50 | -2.3948
UGC06446 | -13.18 | -0.266 | 0.301 | -0.76 | 1.177 | -1.187 | 1.956 | 12.00 | 3.60 | -2.5709
UGC06614 | -12.415 | -0.372 | 0.479 | -0.968 | 2.554 | -1.336 | 4.065 | 88.70 | 8.87 | -1.7022
UGC06667 | -12.561 | -0.136 | 0.149 | -0.388 | 0.516 | -0.608 | 1.042 | 18.00 | 2.50 | -2.3948
UGC06786 | -12.087 | -0.158 | 0.178 | -0.418 | 0.606 | -0.597 | 0.909 | 29.30 | 7.32 | -2.1832
UGC06787 | -13.387 | -0.048 | 0.05 | -0.139 | 0.156 | -0.218 | 0.277 | 21.30 | 5.32 | -2.3217
UGC06818 | -13.089 | -0.18 | 0.184 | -0.549 | 0.595 | -0.935 | 1.175 | 18.00 | 2.50 | -2.3948
UGC06917 | -12.882 | -0.166 | 0.172 | -0.491 | 0.567 | -0.819 | 1.112 | 18.00 | 2.50 | -2.3948
UGC06923 | -13.031 | -0.196 | 0.195 | -0.63 | 0.622 | -1.401 | 1.232 | 18.00 | 2.50 | -2.3948
UGC06930 | -13.382 | -0.356 | 0.37 | -1.354 | 1.399 | -4.689 | 3.052 | 18.00 | 2.50 | -2.3948
UGC06983 | -13.01 | -0.149 | 0.158 | -0.432 | 0.524 | -0.7 | 1.021 | 18.00 | 2.50 | -2.3948
UGC07089 | -13.308 | -0.189 | 0.189 | -0.604 | 0.609 | -1.175 | 1.143 | 18.00 | 2.50 | -2.3948
UGC07125 | -13.871 | -0.285 | 0.327 | -0.877 | 1.551 | -1.516 | 2.282 | 19.80 | 5.90 | -2.3534
UGC07151 | -13.265 | -0.131 | 0.12 | -0.437 | 0.337 | -0.837 | 0.524 | 6.87 | 0.34 | -2.8131
UGC07232 | -12.847 | -0.213 | 0.203 | -0.726 | 0.611 | -3.71 | 1.102 | 2.83 | 0.17 | -3.1983
UGC07261 | -13.236 | -0.719 | 0.839 | -6.698 | 3.382 | -6.698 | 6.573 | 13.10 | 3.93 | -2.5328
UGC07323 | -13.118 | -0.383 | 0.407 | -2.974 | 2.245 | -6.843 | 2.622 | 8.00 | 2.40 | -2.747
UGC07399 | -12.272 | -0.252 | 0.308 | -0.676 | 1.453 | -0.957 | 2.208 | 8.43 | 2.53 | -2.7242
UGC07524 | -13.326 | -0.142 | 0.143 | -0.436 | 0.438 | -0.765 | 0.751 | 4.74 | 0.24 | -2.9743
UGC07559 | -13.614 | -0.167 | 0.155 | -0.572 | 0.444 | -1.645 | 0.79 | 4.97 | 0.25 | -2.9537
UGC07577 | -14.597 | -3.045 | 0.474 | -5.333 | 0.813 | -5.333 | 1.459 | 2.59 | 0.13 | -3.2368
UGC07603 | -12.501 | -0.266 | 0.346 | -0.659 | 1.715 | -0.943 | 2.149 | 4.70 | 1.41 | -2.978
UGC07690 | -13.541 | -0.389 | 0.404 | -6.31 | 1.609 | -6.454 | 2.388 | 8.11 | 2.43 | -2.7411
UGC07866 | -13.476 | -0.269 | 0.265 | -0.935 | 0.861 | -6.501 | 1.448 | 4.57 | 0.23 | -2.9902
UGC08286 | -12.989 | -0.061 | 0.06 | -0.189 | 0.176 | -0.323 | 0.29 | 6.50 | 0.21 | -2.8372
UGC08490 | -12.963 | -0.119 | 0.128 | -0.34 | 0.413 | -0.54 | 0.735 | 4.65 | 0.53 | -2.9826
UGC08550 | -12.975 | -0.241 | 0.302 | -0.622 | 1.329 | -0.885 | 2.039 | 6.70 | 2.00 | -2.824
UGC08699 | -12.978 | -0.122 | 0.137 | -0.34 | 0.479 | -0.521 | 0.861 | 39.30 | 9.82 | -2.0557
UGC08837 | -13.599 | -0.129 | 0.12 | -0.439 | 0.35 | -0.909 | 0.588 | 7.21 | 0.36 | -2.7921
UGC09037 | -12.876 | -0.136 | 0.146 | -0.382 | 0.478 | -0.588 | 0.839 | 83.6 | 8.40 | -1.7279
UGC09133 | -13.146 | -0.065 | 0.065 | -0.192 | 0.197 | -0.318 | 0.297 | 57.10 | 11.40 | -1.8934
UGC09992 | -13.996 | -5.379 | 0.911 | -5.672 | 3.04 | -5.672 | 7.389 | 10.70 | 3.21 | -2.6207
UGC10310 | -13.29 | -0.772 | 0.746 | -6.666 | 1.808 | -6.666 | 4.33 | 15.20 | 4.60 | -2.4682
UGC11455 | -12.841 | -0.095 | 0.103 | -0.268 | 0.337 | -0.416 | 0.58 | 78.60 | 11.80 | -1.7547
UGC11557 | -13.209 | -1.105 | 1.227 | -6.676 | 4.01 | -6.676 | 6.456 | 24.20 | 6.05 | -2.2663
UGC11820 | -13.069 | -0.379 | 0.47 | -0.938 | 1.757 | -1.34 | 2.426 | 18.10 | 5.43 | -2.3924
UGC11914 | -12.274 | -0.128 | 0.129 | -0.38 | 0.384 | -0.61 | 0.575 | 16.90 | 5.10 | -2.4222
UGC12506 | -13.546 | -0.116 | 0.117 | -0.355 | 0.359 | -0.624 | 0.625 | 100.60 | 10.10 | -1.6475
UGC12632 | -13.674 | -0.35 | 0.361 | -2.922 | 1.682 | -6.297 | 2.197 | 9.77 | 2.93 | -2.6602
UGC12732 | -13.386 | -0.356 | 0.399 | -1.003 | 1.481 | -1.743 | 2.227 | 13.20 | 4.00 | -2.5295
UGCA442 | -12.786 | -0.133 | 0.165 | -0.329 | 0.629 | -0.462 | 1.305 | 4.35 | 0.22 | -3.0116
UGCA444 | -13.26 | -0.077 | 0.078 | -0.231 | 0.242 | -0.381 | 0.415 | 0.98 | 0.05 | -3.6588