# A simple direct empirical observation of systematic bias of the redshift as
a distance indicator

Lior<EMAIL_ADDRESS>
Kansas State University
1701 Platt St
Manhattan, KS 66506, USA

###### Abstract

Recent puzzling observations such as the $H_{0}$ tension, large-scale
anisotropies, and massive disk galaxies at high redshifts have been
challenging the standard cosmological model. While one possible explanation is
that the standard model is incomplete, other theories are based on the
contention that the redshift model as a distance indicator might be biased.
These theories can explain the recent observations, but they are challenged by
the absence of a direct empirical reproducible observation that the redshift
model can indeed be inconsistent. Here I describe a simple experiment that
shows that the spectra of galaxies depend on their rotational velocity
relative to the rotational velocity of the Milky Way. Moreover, it shows that
the redshift of galaxies that rotate in the opposite direction relative to the
Milky Way is significantly smaller compared to the redshift of galaxies that
rotate in the same direction relative to the Milky Way ($p<0.006$). Three
different datasets are used independently, each one was prepared in a
different manner, all of them show similar redshift bias. A fourth dataset of
galaxies from the Southern Galactic pole was also analyzed, and shows similar
results. All four datasets are publicly available. While a maximum average z
difference of $\sim$0.012 observed with galaxies of relatively low redshift
(z$<$0.25) is not extreme, the bias is consistent, and can potentially lead to
explanations to puzzling observations such as the $H_{0}$ tension.

##  1 Introduction

Recent observations have shown unexplained tensions and anomalies at
cosmological scales. For instance, the $H_{0}$ determined by the Cosmic
Microwave Background (CMB) radiation is different from the $H_{0}$ determined
by using Ia supernovae and the redshift of their host galaxies (Wu and
Huterer,, 2017; Mörtsell and Dhawan,, 2018; Bolejko,, 2018; Davis et al.,,
2019; Pandey et al.,, 2020; Camarena and Marra,, 2020; Di Valentino et al.,,
2021; Riess et al.,, 2022). The relatively new JWST provides unprecedented
imaging power, showing massive disk galaxies with mature stellar populations
at unexpectedly high redshifts that can be as high as 15 (Whitler et al.,,
2023). In fact, the presence of large disk galaxies at unexpectedly high
redshifts was reported also before JWST saw first light (Neeleman et al.,,
2020). The existence of these galaxies is unexpected, as pre-JWST analyses
predicted that such galaxies do not exist according to $\Lambda$CDM cosmology
(Cowley et al.,, 2018), and therefore were not expected to be observed.

These unexpected observations challenge our understanding of the Universe. If
the common distance indicators are complete and fully accurate, the standard
cosmological theories are incomplete. Similarly, if the standard cosmological
model is complete, it is not possible that the distance indicators as
currently used are fully accurate. Therefore, assuming that all distance
indicators are reliable and consistent, explaining these observations might
reinforce certain modifications of some of the foundations of cosmology. But
in addition to theories that shift from the standard cosmological model, other
theories are based on the contention that the redshift as a distance indicator
at cosmological scales might be biased (Crawford,, 1999; Pletcher,, 2023;
Gupta,, 2023; Lee,, 2023; Seshavatharam and Lakshminarayana, 2023b, ;
Seshavatharam and Lakshminarayana, 2023a, ; Lovyagin et al.,, 2022). While the
assumption that the redshift might be biased or inconsistent can explain these
observations without modifying the standard model, there is currently no clear
reproducible empirical evidence that the redshift might indeed be biased.

The redshift of a luminous moving object is determined by the linear component
of the Doppler shift effect. But because galaxies have rotational velocity in
addition to their linear velocity, their redshift can also be affected by the
rotational velocity, as the rotational velocity of a luminous object also
leads to a Doppler shift effect (Marrucci,, 2013; Lavery et al.,, 2014; Liu et
al.,, 2019).

Since the rotational velocity of a galaxy is far smaller than its linear
velocity relative to Earth, the rotational velocity component of the Doppler
shift is often ignored when determining the distance of a galaxy. But while
the Doppler shift effect driven by the rotational velocity of the galaxy is
expected to be subtle, that has not yet been tested. It should also be
mentioned that the physics of galaxy rotation is one of the most mysterious
observations, and its nature cannot be explained unless making assumptions
such as dark matter (Zwicky,, 1937; Oort,, 1940; Rubin,, 1983; El-Neaj et
al.,, 2020), modified Newtonian dynamics (MOND) (Milgrom,, 1983, 2007; De Blok
and McGaugh,, 1998; Sanders,, 1998; Sanders and McGaugh,, 2002; Swaters et
al.,, 2010; Sanders,, 2012; Iocco et al.,, 2015; Díaz-Saldaña et al.,, 2018;
Falcon,, 2021), or other theories (Sanders,, 1990; Capozziello and De
Laurentis,, 2012; Chadwick et al.,, 2013; Farnes,, 2018; Rivera,, 2020;
Nagao,, 2020; Blake,, 2021; Gomel and Zimmerman,, 2021; Skordis and Złośnik,,
2021; Larin,, 2022). But despite over a century of research, there is still no
single clear proven explanation to the physics of galaxy rotation (Sanders,,
1990; Mannheim,, 2006; Kroupa,, 2012; Kroupa et al.,, 2012; Kroupa,, 2015;
Arun et al.,, 2017; Akerib et al.,, 2017; Bertone and Tait,, 2018; Aprile et
al.,, 2018; Skordis and Złośnik,, 2019; Sivaram et al.,, 2020; Hofmeister and
Criss,, 2020; Byrd and Howard,, 2021; Haslbauer et al., 2022a, ; Haslbauer et
al., 2022b, ), and that phenomenon is still not fully understood.

The purpose of the simple experiment described in this paper is to test the
impact of the rotational velocity component of galaxies on the Doppler shift
effect, and consequently on the redshift as a distance indicator. Section 2
describes the data used in the experiment, Section 3 provides the results of
the analysis, Section 4 compares the results shown in Section 3 to several
other datasets collected by different methods and different telescope systems,
and Section 5 discusses possible explanations in the light of recently
observed anisotropies and current cosmological theories.

##  2 Data

The experiment is based on one primary dataset, and two additional independent
datasets of galaxies from the Northern Galactic pole to which the results are
compared. Each of these datasets was prepared in a different manner. A fourth
dataset of galaxies from the Southern Galactic pole is also used for
comparison, as will be discussed in Section 4.3.

The primary dataset contains SDSS DR8 galaxies with spectra sorted by their
direction of rotation, as explained and used in (Shamir, 2020b, ). Instead of
using galaxies in the entire SDSS footprint, this experiment is focused on
galaxies that rotate in the same direction relative to the Milky Way, and
galaxies that rotate in the opposite direction relative to the Milky Way.
Therefore, only galaxies that are close to the Galactic pole are used, and the
field is limited to the $20\times 20$ degrees centered at the Northern
Galactic pole. Obviously, because the Milky Way rotates around its own pole,
galaxies that are close to the galactic pole rotate either in the same
direction relative to the galactic pole, or in the opposite direction relative
to the galactic pole. The analysis included objects with spectra in SDSS DR8
that have r magnitude of less than 19 and Petrosian radius of at least 5.5”.
The redshift of the galaxies in that initial set was limited to z$<$0.3, and
the redshift error smaller than $10^{-4}$. That selection eliminated the
possible effect of bad redshift values, that in some cases can be very high
and skew the dataset. The initial set of galaxies that meet these criteria in
that field contained 52,328.

The process by which the galaxies were sorted by their direction of rotation
is explained in detail in (Shamir, 2020b, ), and is similar to the process of
annotating galaxies imaged by other telescopes (Shamir,, 2016; Shamir, 2020a,
; Shamir, 2022b, ; Shamir, 2022f, ; Shamir, 2022c, ; Mcadam et al.,, 2023;
Shamir and McAdam,, 2022). In summary, the annotation is done by using the
Ganalyzer algorithm (Shamir,, 2011), where each galaxy image is transformed
into its radial intensity plot such that the value of the pixel at Cartesian
coordinates $(\theta,r)$ in the radial intensity plot is the median value of
the 5$\times$5 pixels at coordinates $(O_{x}+\sin(\theta)\cdot
r,O_{y}-\cos(\theta)\cdot r)$ in the original galaxy image, where r is the
radial distance measured in percentage of the galaxy radius, $\theta$ is the
polar angle in degrees relative to the galaxy center, and $(O_{x},O_{y})$ is
the coordinates of the galaxy center. A peak detection algorithm is then
applied to the rows in the radial intensity plot, and the direction of the
peaks determines the direction of the curves of the galaxy arms.

Figure 1 displays examples of the original galaxy images, their radial
intensity plots, and the detected peaks. The direction of the curves of the
arms is determined by the sign of the slope. Based on previous experiments
(Shamir, 2020b, ; Shamir, 2020a, ; Shamir, 2022b, ; Shamir, 2022f, ; Shamir,
2022c, ; Mcadam et al.,, 2023; Shamir and McAdam,, 2022), to avoid incorrect
annotations, a direction was determined only for galaxies that had at least 30
peaks are identified in the radial intensity plot. If less than 30 peaks are
identified the galaxy is not used, as its direction of rotation cannot be
identified. The algorithm is described with experimental results in (Shamir,,
2011), as well as (Shamir, 2020a, ; Shamir, 2022b, ; Shamir, 2022f, ; Shamir,
2022c, ; Mcadam et al.,, 2023).

Figure 1: Examples of original galaxy images (left), the radial intensity plot
transformations (center), and the peaks detected in the the radial intensity
plot lines (right).

The primary advantage of the algorithm is that its simple “mechanical” nature
makes it fully symmetric. Experiments of mirroring the galaxy images lead to
identical inverse results compared to when using the original images (Shamir,,
2016; Shamir, 2020a, ; Shamir, 2022b, ; Shamir, 2022f, ; Shamir, 2022c, ;
Mcadam et al.,, 2023).

After applying the algorithm to the galaxy images, the final dataset included
1,642 galaxies with identifiable directions of rotation, such that 817
galaxies rotate clockwise, and 825 galaxies rotate counterclockwise. Applying
the algorithm to the mirrored images led to an identical inverse dataset.
Testing a random subset of 200 galaxies showed that all galaxies were
annotated correctly. Figure 2 shows the redshift distribution of the galaxies.
The dataset is available at
https://people.cs.ksu.edu/~lshamir/data/zdifference/. In addition to this
dataset, two other previous public datasets were used, as will be described in
Section 4.

Figure 2: The redshift distribution of the galaxies in the dataset.

##  3 Results

Table 1 shows the redshift differences in the 20$\times$20 degree field
centered at the Northern Galactic pole, as well as the smaller 10$\times$10
degree field. The errors are the standard error of the mean, computed by
$\frac{\sigma_{z}}{\sqrt{N}}$, where $N$ is the number of galaxies. The mean
redshift of the galaxies in the dataset described in Section 2 that rotate in
the same direction relative to the Milky Way (MW) is 0.09545$\pm$0.0017, while
the mean redshift of the galaxies in the same field that rotate in the
opposite direction relative to the Milky Way (OMW) is 0.08895$\pm$0.0016. That
shows a $\Delta z$ of $\sim$0.0065$\pm$0.0023 between galaxies that rotate in
the same direction relative to the Milky Way and galaxies that rotate in the
opposite direction relative to the Milky Way.

By applying a simple Student t-test, the two-tailed probability that the two
means are different by mere chance is ($p\simeq$0.0058). To verify the
statistical significance, a simulation test was also performed. The simulation
randomly selected a set of 817 galaxies, and another set of 825 galaxies
regardless of their spin direction. That was repeated 100,000 times, and in
each run the mean redshift of the galaxies in the first set were compared to
the mean redshift of the galaxies in the second set. In 614 of the runs the
difference between the mean redshifts of the two sets was greater than 0.0016.
That provide a probability of 0.0061, which is similar to the two-tailed
t-test probability. The similarity between the probability of the t-test and
the probability of the simulation is not surprising, but provides consistency
and ensures that the t-test P-values are not driven by a certain unusual
distribution of the redshift values.

If the observed difference in redshift is driven by the rotational velocity of
the observed galaxy relative to the rotational velocity of the Milky Way, the
difference should increase when the observed galaxies are closer to the
Galactic pole. As Table 1 shows, $\Delta z$ indeed increases in the
10$\times$10 field. Despite the lower number of galaxies the difference is
still statistically significant.

Table 1: The mean redshift difference of galaxies in 20$\times$20 degree field centered at the Galactic pole and the 10$\times$10 degree field centered at the Galactic pole. The galaxies are separated into galaxies that rotate in the same direction relative to the Milky Way (clockwise to an Earth-base observer) and galaxies that rotate in the opposite direction relative to the Milky Way (OMW). The P values are the two-tailed P values determined by the standard Student t-test. The errors are the standard error of the mean. Field (o) | # MW | # OMW | $Z_{mw}$ | $Z_{omw}$ |  $\Delta$z | t-test P
---|---|---|---|---|---|---
10$\times$10 | 204 | 202 | 0.0996$\pm$0.0036 | 0.08774$\pm$0.0036 | 0.01185$\pm$0.005 | 0.02
20$\times$20 | 817 | 825 | 0.09545$\pm$0.0017 | 0.08895$\pm$0.0016 | 0.0065$\pm$0.0023 | 0.0058

If the redshift difference peaks at the Northern Galactic pole, it is expected
that galaxies that are on or close to the galactic pole would show higher
redshift difference, while when using galaxies that are more distant from the
Galactic pole the redshift difference $\Delta z$ would decrease. Figure 3
shows the change in $\Delta z$ when the size of the field centered at the
Galactic pole changes.

Figure 3: The $\Delta z$ when the size of the field changes. The analysis was
done such that the larger field contains also the galaxies of the smaller
field inside it (blue), and also when the galaxies in the smaller field are
excluded so that the two fields are orthogonal, and do not have overlapping
galaxies (green).

As the figure shows, the $\Delta z$ decreases as the field gets larger. That
can be explained by the fact that when the field gets larger, it includes more
galaxies that are more distant from the Galactic pole. While it does not fully
prove a link to the rotational velocity of these galaxies relative to the
Milky Way, that observation is in agreement with the contention that the
redshift difference is linked to the rotational velocity of the galaxies
relative to the rotational velocity of the Milky Way. The figure includes two
graphs. The first shows all galaxies inside the field. For instance, when the
field size is 20$\times 20$ degrees it also includes the galaxies inside the
10$\times$10 degree field centered at the Galactic pole. The other analysis
excludes overlapping galaxies, so that a galaxy can only be used in one field.
That is, the galaxies in the 20$\times$20 degree field centered at the
Galactic pole excludes the galaxies in the 10$\times$10 degree field. That
provides analysis with independent sets of galaxies that do not overlap. The
analysis shows a small $\Delta z$ of $\sim$0.002 also when the dataset
includes galaxies that are more distant from the Galactic pole, but it is
smaller compared to the $\Delta z$ observed when the set of galaxies is
limited to galaxies closer to the Galactic pole.

To test a possible change in different redshift ranges, the galaxies in the
20$\times$20 field centered at the Galactic pole were separated into galaxies
with redshift lower than 0.1 and redshift greater than 0.1. Using 0.1 for
separating the dataset into lower redshift and higher redshift galaxies
provided two sets, one with 1,012 galaxies with z$<$0.1, and another dataset
of 630 galaxies with z$>$0.1. Table 2 shows the differences in redshift when
the galaxies are separated to the two different redshift ranges. The $\Delta
z$ of the galaxies where $z<0.1$ is 0.0028$\pm$0.0013, while when $z>0.1$ the
$\Delta z$ is 0.0053$\pm$0.0028. The t-test probability that the difference
occurs by chance is P$\simeq$0.51, and therefore not statistically
significant. While not statistically significant, the dataset used here might
not be sufficiently large to provide statistically significant conclusions
regarding the $\Delta z$ in the different redshift ranges, if such indeed
exist.

Table 2: The mean redshift difference of galaxies with z$<$0.1 and z$>$0.1 that rotate in the same direction as the Milky Way or in the opposite direction relative to the Milky Way. z range | # MW | # OMW | $Z_{mw}$ | $Z_{omw}$ |  $\Delta$z
---|---|---|---|---|---
z$<$0.1 | 491 | 521 | 0.0629$\pm$0.001 | 0.0601$\pm$0.001 | 0.0028$\pm$0.0013
z$>$0.1 | 326 | 304 | 0.1441$\pm$0.002 | 0.1388$\pm$0.002 | 0.0053$\pm$0.0028

Table 3 shows the differences between the flux on the different filters, taken
from the specObjAll table in SDSS DR8. The spectrum flux difference shows a
consistent difference of $\sim$10% across the different filters. Unlike the
redshift, the differences in the flux of the specific filters are not
statistically significant, and therefore a definite conclusion about the flux
differences cannot be made.

Table 3: Flux in different filter galaxies that rotate in the same direction relative to the Milky Way and galaxies that rotate in the opposite direction relative to the Milky Way. The t-test P values are the two-tailed P value. Band | MW | OMW | $\Delta$ | t-test P
---|---|---|---|---
spectroFlux_g |  25.969$\pm$0.8669 |  28.554$\pm$1.0918 | -2.585 | 0.063
spectroFlux_r |  53.2433$\pm$1.765 |  58.6214$\pm$2.3422 | -5.378 | 0.066
spectroFlux_i |  77.4189$\pm$2.513 |  85.0868$\pm$3.407 | -7.667 | 0.067

##  4 Comparison to other datasets

The annotation algorithm used to sort the galaxies by their direction of
rotation as discussed in Section 2 is simple and symmetric, and there is no
known bias that can prefer the redshift of a certain set of galaxies as
annotated by the algorithm. Also, experimenting with the same images when the
images were mirrored leads to inverse results, as also shown in detail in
(Shamir,, 2016; Shamir, 2020a, ; Shamir,, 2021; Shamir, 2022b, ; Shamir,
2022f, ; Shamir, 2022c, ; Mcadam et al.,, 2023). To further test for a
possible impact of unknown or unexpected biases in the annotation process, two
additional annotation methods were used to test whether these algorithms
provide different results.

###  4.1 Comparison to annotations by Galaxy Zoo

The first annotation method that was used is the crowdsourcing-based Galaxy
Zoo 1 (Lintott et al.,, 2008). In Galaxy Zoo, anonymous volunteers used a web-
based interface to sort galaxy images by their direction of rotation. After
several years of work by over 100,000 volunteers, a relatively large set of
over $8\cdot 10^{5}$ galaxies were annotated. One of the downsides of Galaxy
Zoo was that in the vast majority of the cases the volunteers who annotated
the galaxies made conflicting annotations, and the disagreement between the
annotators makes it difficult to use the majority of the galaxies. Another
substantial downside is that the annotations were subjected to the bias of the
human perception, which is very difficult to model and fully understand,
challenging the reliability of the annotations as a tool for primary science.
Despite these known weaknesses, there is no known human perceptual bias that
would associate galaxies with lower redshift to a certain direction of
rotation. Therefore, although Galaxy Zoo might not necessarily be considered a
complete tool when used as the sole dataset, comparing to Galaxy Zoo can
provide an indication of whether a different annotation method leads to
different results shown in Section 3.

Because the annotations of the volunteers often disagree with each other,
Galaxy Zoo defined the “superclean” criterion as galaxies that 95% of the
human annotators agree on the annotation. That is, if 95% of the annotations
or more are for a galaxy that rotates clockwise, the annotation is considered
“superclean”. While these annotations are normally correct, only 1.7% of the
galaxies annotated by Galaxy Zoo 1 meet that criterion. Out of the 667,944
galaxies in the specZoo table in SDSS DR8, just 324 galaxies meet that
criterion and are also inside the 20$\times$20 degree field centered at the
Northern Galactic pole.

The mean z of the Galaxy Zoo 1 galaxies that rotate in the same direction
relative to the Milky Way in that field is 0.073834$\pm$0.0041. The mean z of
the galaxies that rotate in the opposite direction relative to the Milky Way
is 0.068292$\pm$0.00348. That shows a $\Delta z$ of 0.00554, which is similar
in both direction and magnitude to the $\Delta z$ of 0.0065 observe with the
dataset described in Section 2. The one-tailed P value of that difference to
occur by mere chance is 0.15. That is not statistically significant, and that
can be attributed to the small size of the dataset, but the similar $\Delta z$
in both direction and magnitude shows consistency between the annotation
methods. From the 324 galaxies annotated by Galaxy Zoo, 263 were also included
in the dataset described in Section 2. The value of the comparison is
therefore not by analyzing a new dataset, but by using a different annotation
method that is independent of the method used in Section 2, and therefore not
subjected to the same possible unknown or unexpected biases in that method if
such exist. An analysis of a completely different dataset acquired by a
different telescope system and does not have any overlap with the SDSS dataset
will be described in Section 4.3.

###  4.2 Comparison to annotations by SpArcFiRe

Another dataset that is used is the dataset of SDSS galaxies annotated by the
SpArcFiRe (Scalable Automated Detection of Spiral Galaxy Arm) algorithm (Davis
and Hayes,, 2014; Hayes et al.,, 2017). SpArcFiRe is implemented by an open
source software 222https://github.com/waynebhayes/SpArcFiRe, and the method is
described in detail in (Davis and Hayes,, 2014). In summary, the algorithm
first identifies arm segments in the galaxy image, and then fits these
segments to logarithmic spiral arcs to determine the direction of rotation
based on the curves of the arms. One of the advantages of SpArcFiRe is that it
is not based on data-driven machine learning or deep learning approaches that
are difficult to analyze, and is therefore not subjected to the complex biases
that are often very difficult to notice (Dhar and Shamir,, 2022). The downside
of SpArcFiRe is that it has an annotation error of about 15% (Mcadam et al.,,
2023). More importantly, since SpArcFiRe is a relatively sophisticated
algorithm, it is more difficult to ensure that it is completely symmetric, and
in some seldom cases a mirrored galaxy image is not annotated as rotating to
the opposite direction compared to the original image. That characteristic of
the algorithm is discussed in the appendix of (Hayes et al.,, 2017). That
weakness of the algorithm can be addressed by repeating the analysis twice,
such that in the first experiment the original images are used, and in the
second experiment the mirrored images are used. Then, the results of the two
experiments can be compared. While that practice might not be ideal, it can be
used to compare the results to the results shown in Section 3.

The dataset used here is the dataset of spiral galaxies annotated SpArcFiRe
used in (Mcadam et al.,, 2023), which is a reproduction of the experiment
described in (Hayes et al.,, 2017). As explained in (Hayes et al.,, 2017), the
set of galaxies is the same set of galaxies selected for annotation in
(Lintott et al.,, 2008), although the manual annotations are not used in any
form in the analysis. The dataset is available at
https://people.cs.ksu.edu/~lshamir/data/sparcfire. More details about the
dataset are available in (Mcadam et al.,, 2023).

The dataset was prepared with the original images, and then again with the
mirrored galaxy images. The dataset prepared with the original images contains
138,940 galaxies, and the dataset prepared with the mirrored images contains
139,852 galaxies. That difference is expected due to the fact that the
SpArcFiRe method is not fully symmetric, as explained in the appendix of
(Hayes et al.,, 2017). All of the galaxies used in the experiment have
spectra, and therefore can be used to compare the reshift. As before, galaxies
with redshift greater than 0.3 or redshift error greater than $10^{-4}$ were
ignored. Table 4 shows the mean redshift in the 10$\times$10 field centered at
the Northern Galactic pole and in the 20$\times$20 field, for both the
original images and the mirrored images.

Table 4: The mean redshift of galaxies annotated by the SpArcFiRe algorithm. The table shows results when using the original images, as well as the results when the algorithm is applied to the mirrored images, leading to inverse $\Delta z$. The t-test P values are the one-tailed P value. Field (o) | # MW | # OMW | $Z_{mw}$ | $Z_{omw}$ |  $\Delta$z | t-test P
---|---|---|---|---|---|---
Original 10$\times$10 | 710 | 732 | 0.07197$\pm$0.0015 | 0.06234$\pm$0.0014 | 0.00963$\pm$0.002 |  $<$0.0001
Mirrored 10$\times$10 | 728 | 709 | 0.06375$\pm$0.0014 | 0.07191$\pm$0.0014 | -0.00816$\pm$0.002 |  $<$0.0001
Original 20$\times$20 | 2903 | 2976 | 0.07285$\pm$0.0007 | 0.071164$\pm$0.0007 | 0.001686$\pm$0.0009 | 0.04
Mirrored 20$\times$20 | 3003 | 2914 | 0.07113$\pm$0.0007 | 0.07271$\pm$0.0007 | -0.00158$\pm$0.0009 | 0.05

As the table shows, both the original images and the mirrored images show
consistent results. These results are also consistent with the results shown
in Section 3. The $\Delta z$ is lower than the $\Delta z$ observed with the
dataset used in Section 3, and that could be due to the certain error rate of
the annotations made by the SpArcFiRe algorithm, which is expected to weaken
the signal as also shown quantitatively in Section 7.1 in (McAdam and Shamir,,
2023). To test for the effect of the annotation error, the dataset described
in Section 2 was used such that 15% of the galaxies were assigned
intentionally with the wrong spin direction. That reduced the $\Delta z$ from
0.0065 as shown in Section 3 to 0.0032$\pm$0.0023, which is closer to the
$\Delta z$ value shown in Table 4, and within 1$\sigma$ statistical
fluctuation from it. That also shows that the magnitude of the $\Delta z$
depends on the accuracy of the annotation, which is another indication to the
link between the spin direction of the galaxies and the $\Delta z$.

###  4.3 Comparison to galaxies from the Southern Galactic pole

The data used in the experiments described above was all taken from the
Northern hemisphere, and the galaxies it contains are around the Northern
Galactic pole. To verify the observed redshift difference, it is also required
to test if it exists in the Southern Galactic pole as well. If the observed
difference in redshift is also observed in the Southern Galactic pole, it can
provide an indication that it indeed could be related to the Galactic pole.
Since the three experiments above all used data collected by SDSS, using a
different telescope can show that the difference is not driven by some unknown
or unexpected anomaly in a specific telescope system. Also, a dataset of
galaxies from the Southern Galactic pole will have no overlap with the three
datasets of SDSS galaxies used above.

The set of galaxies used for the analysis are galaxies imaged by DECam used in
(Shamir,, 2021) that had spectroscopic redshift through the Set of
Identifications Measurements and Bibliography for Astronomical Data (SIMBAD)
database (Wenger et al.,, 2000). As explained in (Shamir,, 2021), DECam galaxy
images were acquired through the API of the DESI Legacy Survey server. The
initial set of galaxies contained all objects in the South bricks of DESI
Legacy Survey Data Release 8 that had g magnitude of less than 19.5 and
identified as de Vaucouleurs $r^{1/4}$ profiles (“DEV”), exponential disks
(“EXP”), or round exponential galaxies (‘’REX”).

The galaxy images were then annotation by the Ganalyzer algorithm as described
in Section 2, and also in (Shamir,, 2021). Unlike the other methods used in
Sections 4.1 and 4.2, Ganalyzer provides a high level of accuracy in the
annotations. Also, DESI Legacy Survey does not have datasets of galaxies
annotated through as used in Section 4.1.

The entire dataset of annotated galaxies contains $\sim 8.07\cdot 10^{6}$
galaxies, but because only galaxies with spectra in the 20$\times$20 field
centered at the Galactic pole can be used, the dataset used here is reduced to
3,383 galaxies. Figure 4 shows the redshift distribution of the galaxies. The
dataset of annotated galaxies is available at
https://people.cs.ksu.edu/~lshamir/data/zdifference/.

Figure 4: The redshift distribution of the galaxies in the dataset of galaxies
near the Southern Galactic pole.

Table 5 shows the mean redshift of the galaxies that rotate in the same
direction relative to the Milky Way and in the opposite direction relative to
the Milky Way. Due to the perspective of the observer galaxies that are close
to the Southern Galactic pole that rotate in the same direction relative to
the Milky Way seem to rotate in the opposite direction compared to galaxies in
the Northern Galactic pole that rotate in the same direction.

Table 5: The mean redshift of galaxies in 20$\times$20 field and the 10$\times$10 field centered at the Southern Galactic pole. The P values are the one-tailed Student t-test P values. Field (o) | # OMW | # MW | $Z_{omw}$ | $Z_{mw}$ |  $\Delta$z | t-test p
---|---|---|---|---|---|---
10$\times$10 | 414 | 376 | 0.1270$\pm$0.0025 | 0.1352$\pm$0.0027 | -0.0082$\pm$0.0036 | 0.018
20$\times$20 | 1702 | 1681 | 0.1273$\pm$0.0014 | 0.1317$\pm$0.0013 | -0.0044$\pm$0.0018 | 0.008

As the table shows, the redshift differences are statistically significant in
both fields, and increases when the galaxies are closer to the Galactic pole.
These results are in good agreement with the results shown with galaxies
located around the Northern Galactic pole. The table also shows that the mean
redshift is higher compared to the mean redshift observed with SDSS. That
difference can be expected due to the superior imaging power of DECam compared
to SDSS, allowing DECam image galaxies at deeper redshifts. Unlike SDSS, where
all redshifts are collected by the same instrument, the SIMBAD database
collects redshifts from several different available sources, each source might
use a different instrument. Still, it is reasonable to believe that the
redshifts are distributed uniformly across the instruments that collected
them, with no link between the rotation of direction and the specific
instrument that provided the reddshift.

When combining the data from the Southern Galactic pole with the data from the
Northern Galactic pole described in Section 2, the mean z for galaxies that
rotate in the same direction relative to the Milky Way is 0.11985$\pm$0.001,
while galaxies that rotate in the opposite direction relative to the Milky Way
have mean z of 0.11476$\pm$0.001. That provides a $\Delta z$ of
0.0051$\pm$0.0015, and the two-tailed Student t-test probability to have such
difference by chance is $p\simeq$0.0003.

##  5 Possible explanations and future experiments

Recent puzzling observations such as the $H_{0}$ tension and large disk
galaxies at high redshifts have been challenging cosmology. Explaining such
observations require to assume that either the standard cosmological models
are incomplete, or that the redshift as a distance model is incomplete. This
study shows first direct observational evidence of bias in the redshift as a
distance indicator.

While the link between the redshift of the galaxy and its direction of
rotation relative to the Milky Way is consistent, it is definitely difficult
to provide an immediate trivial explanation to that link. The redshift of
galaxies has a linear velocity component, but also a rotational velocity
component. Since the rotational component is expected to be very small
compared to the linear component, most models ignore the rotational component,
and use the redshift base on the linear velocity component alone. The analysis
shown here provides evidence that the rotational component of the redshift
might not be negligible. Although the analysis does not provide a
straightforward explanation to the anomaly, it should be reminded that the
physics of galaxy rotation is still not fully understood. Explanations such as
dark matter or MOND are the common explanations to the physics of galaxy
rotation, but despite several decades of research there is no proven
explanation to that physics. Due to the mysterious nature of the physics of
galaxy rotation, a link between the galaxy rotation anomalies and the
observation reported in this paper is one of the possible directions to
explaining the observation.

Besides the theoretical aspects of the observation reported in this paper,
redshifts are used in practice as the most common way to determine distances
at cosmological scales. A possible systematic bias might therefore have impact
on other studies that rely on redshift as a distance metrics. Although the
bias is relatively small, it might have impact on experiments that use
populations of galaxies to study the large-scale structure of the Universe.
Examples of such studies that can potentially be affected by subtle redshift
biases will be discussed later in this section.

The analysis done here is limited to galaxies at relatively low redshift of
$z<0.3$. When the redshift of the galaxies is higher the velocity is also
higher, and therefore the rotational velocity of the galaxies compared to the
linear velocity is smaller. That can lead to a smaller change in the redshift
relative to the linear velocity of the galaxy. To test that, an experiment can
be done by imaging a relatively deep field at the galactic pole with space-
based instruments such as HST or JWST. The spectra of galaxies in that field
can provide information of the effect of the rotational velocity to galaxies
at higher redshifts.

Also, deeper and larger datasets of clear galaxies with spectra such as the
data provided by the Dark Energy Spectroscopic Instrument (DESI) will allow a
more detailed profiling of the observed anomaly. While the observations do not
explain directly the existence of early massive disk galaxies as revealed by
JWST, they demonstrate that the current redshift model might be incomplete,
and might need to be expanded to also include the rotational velocity of the
galaxy to better estimate its distance. In that case, the existence of such
galaxies can be explained without the need to modify the standard cosmological
models.

While the bias can also be attributed to the algorithm that selects
spectroscopic targets, it is difficult to think of how that algorithm could be
affected by the direction of rotation relative to the Milky Way. Also, if the
target selection algorithm has such unknown and complex biases, these biases
are expected to be consistent throughout the sky, and is not expected to
decrease when the angular distance of the galaxy from the Galactic pole gets
larger, or flip when analyzing galaxies from the opposite side of the Galactic
pole. The fact that two different telescopes show similar results further
reduces the possibility that the results are driven by an unknown anomaly in
the selection algorithm of the spectroscopic surveys, or another unexpected
anomaly in the telescope system.

Another possible explanation to the observation is an unexpected anomaly in
the geometry of the Universe or its large-scale structure. While a certain
alignment in galaxy direction of rotation is expected (d’Assignies D et al.,,
2022; Kraljic et al.,, 2020), if the observation reported in Section 3 reflect
the real distribution of galaxies in the Universe, the scale of that structure
that covers two hemispheres is far larger than any known supercluster of
filament. If the redshifts represent the accurate distances of the galaxies,
and is not affected by their rotational velocity, the galaxies form a
cosmological-scale structure driven by the alignment in the direction of
rotation of the galaxies, and peaks around the Galactic pole. That explanation
assumes no anomaly in the physics of galaxy rotation, but it is aligned with
cosmological models that shift from the standard model (Aluri et al.,, 2023).
As discussed also in (Shamir, 2022b, ; Shamir, 2022a, ; Shamir, 2022d, ;
Shamir, 2022c, ; Shamir, 2022e, ; Shamir,, 2023), the observation of such
large-scale structure that forms a cosmological-scale axis is aligned with
alternative theories such as dipole cosmology (Ebrahimian et al.,, 2023;
Krishnan et al., 2022b, ; Allahyari et al.,, 2023; Krishnan et al., 2023b, ;
Krishnan et al., 2023a, ), or theories that assume a rotating universe
(Gödel,, 1949; Ozsváth and Schücking,, 1962; Gödel,, 2000; Chechin,, 2016;
Campanelli,, 2021) such as Black Hole Cosmology (Pathria,, 1972; Stuckey,,
1994; Easson and Brandenberger,, 2001; Seshavatharam,, 2010; Popławski,, 2010;
Christillin,, 2014; Dymnikova,, 2019; Chakrabarty et al.,, 2020; Popławski,,
2021; Seshavatharam and Lakshminarayana,, 2022; Gaztanaga, 2022a, ; Gaztanaga,
2022b, ) which is also linked to holographic universe (Susskind,, 1995; Bak
and Rey,, 2000; Bousso,, 2002; Myung,, 2005; Hu and Ling,, 2006; Rinaldi et
al.,, 2022). The presence of a cosmological-scale axis agrees also with the
contention of ellipsoidal universe (Campanelli et al.,, 2006, 2007; Gruppuso,,
2007; Campanelli et al.,, 2011; Cea,, 2014). In that case, the alignment of
such hypothetical axis with the Galactic pole is a coincidence. The
observation described in this paper can also be related to the theory of
stationary Universe, which explains multiple anomalies but is challenged by
the luminosity-distance relationship (Sanejouand,, 2022).

The results shown here might also provide an indication that the $H_{0}$
tension can be explained by the slight differences in the redshift. While
$H_{0}$ anisotropy has been reported in the past (Javanmardi et al.,, 2015;
Krishnan et al., 2022a, ; Cowell et al.,, 2022; McConville and Colgain,, 2023;
Aluri et al.,, 2023), its nature is still unclear. But these studies are
normally based on a limited number of galaxies with redshift that also host Ia
supernovae. A higher number of galaxies that rotate in a certain direction can
lead to a slight difference in the $H_{0}$, and therefore to $H_{0}$
anisotropy. These reports can be linked to the observed anisotropy in the
brightness of Ia supernova (Perivolaropoulos,, 2023). Explaining the $H_{0}$
tension might require new physics, which is not necessarily limited to physics
that applies to the early Universe alone (Vagnozzi,, 2023).

Differences in the redshift that are based on the rotational velocity of the
galaxies relative to the Milky Way can explain the $H_{0}$ anisotropy, and
potentially also the $H_{0}$ tension. If the rotational velocity of Ia
supernovae and their host galaxies relative to the Milky Way affect their
estimated distance, when the rotational velocity relative to the Milky Way is
normalized, the $H_{0}$ tension is expected to be resolved. As demonstrated in
(McAdam and Shamir,, 2023), when limiting the SH0ES collection (Khetan et
al.,, 2021) of Ia supernovae to galaxies that rotate in the same direction as
the Milky Way, the computed $H_{0}$ drops to $\sim$69.05 $km/sMpc^{-1}$, which
is closer to the $H_{0}$ determined by the CMB, and within statistical error
from it. When using just galaxies that rotate in the opposite direction
relative to the Milky Way, the $H_{0}$ does not drop, but instead increases to
$\sim$74.2 $km/sMpc^{-1}$ to further increase the $H_{0}$ tension (McAdam and
Shamir,, 2023). While the analysis was done with the relatively small
collection of SH0ES, it suggests that the distance indicators might respond to
the rotational velocity of the observed objects compared to the rotational
velocity of the Milky Way. This explanation agrees with the contention that
the explaining the $H_{0}$ tension might require new physics, which is not
necessarily limited to physics that applies to the early Universe alone
(Vagnozzi,, 2023).

The observed $\Delta z$ between galaxies with opposite rotational velocities
as shown here is between around 0.0065 to 0.012. If that difference is due to
the rotational velocity, that difference corresponds to velocity of between
roughly 2,000 to 3,600 km$\cdot$s-1. That is about 5 to 8 times the rotational
velocity of the Milky Way compared to the observed galaxies, which is $2\cdot
220=\sim 440$ km$\cdot$s-1, assuming that the observed galaxies have the same
rotational velocity as the Milky Way. That velocity difference is in good
agreement with the velocity difference predicted in (Shamir, 2020a, ) by using
analysis of the photometric differences between galaxies rotating with or
against the rotational velocity of the Milky Way. That analysis was based on
the expected and observed differences in the total flux of galaxies that
rotate in the same direction relative to the Milky Way and the flux of
galaxies that rotate in the opposite direction. Based on the expected flux
difference due to the Doppler shift driven by the rotational velocity as shown
in (Loeb and Gaudi,, 2003), it was predicted that light emitted from the
observed galaxies agrees with rotational velocity that is 5-10 times faster
than the rotational velocity of the Milky Way (Shamir, 2020a, ; McAdam and
Shamir,, 2023). These predictions are close to the results of comparing the
redshift as done here.

The observed redshift difference, if indeed driven by the rotational velocity
of the Milky Way and the observed galaxies, corresponds to a much higher
rotational velocity than the $\sim$220 km$\cdot$s-1 rotational velocity of the
Milky Way. Such consistent bias in the redshift can be linked to other
unexplained anomalies that challenge cosmology, and related to distances at
cosmological scale. The physics of galaxy rotation is one of the most puzzling
and provocative phenomena in nature, and despite over a century of research it
is still not fully understood (Opik,, 1922; Babcock,, 1939; Oort,, 1940; Rubin
and Ford Jr,, 1970; Rubin et al.,, 1978, 1980, 1985; Sanders,, 1990; Sofue and
Rubin,, 2001; Mannheim,, 2006; Kroupa,, 2012; Kroupa et al.,, 2012; Kroupa,,
2015; Arun et al.,, 2017; Akerib et al.,, 2017; Bertone and Tait,, 2018;
Aprile et al.,, 2018; Skordis and Złośnik,, 2019; Sivaram et al.,, 2020;
Hofmeister and Criss,, 2020; Byrd and Howard,, 2021; Haslbauer et al., 2022b,
). Due to the unexplained tensions in cosmology, the unknown physics of galaxy
rotation should be considered as a factor that can be associated with these
tensions and explain them.

##  6 Conclusion

Recent puzzling observations such as the $H_{0}$ tension and large disk
galaxies at high redshifts have been challenging cosmology. Explaining such
observations require to assume that either the standard cosmological models
are incomplete, or that the redshift as a distance model is incomplete. This
study shows first direct observational evidence of bias in the redshift as a
distance indicator. The bias is relatively small, and was observed in galaxies
with relatively low redshift. Studies that are based on a relatively small
number of galaxies such the anisotropy of $H_{0}$ might be affected by the
distribution of galaxies that rotate in opposite direction. If the direction
of rotation of the galaxies are not normalized, a slightly higher prevalence
of galaxies that rotate in a certain direction can lead to small but
consistent bias.

There is no immediate proven physical explanation to the difference between
the redshifts of galaxies that rotate with or against the direction of
rotation of the Milky Way. While a certain difference is expected, the
magnitude of the difference is expected to be small given the rotational
velocity of the Milky Way. But as discussed above, the physics of galaxy
rotation is not fully understood. Namely, the rotational velocity of galaxies
cannot be explained without assumptions such as dark matter or MOND. It is
therefore possible that the mysterious physics of galaxy rotation affects the
light emitted by galaxies in a way that is not proportionate to their rotation
velocity.

Future experiments will include using larger redshift survey such as DESI to
profile the redshift differences using a far higher number of galaxies with
redshift. Other experiments can be based on space-based instruments pointing
at the galactic poles to study the redshift difference at higher redshifts and
the earlier Universe.

## Data availability

All data used in this paper are available publicly. The coordinates, redshift,
and direction of rotation of galaxies of the datasets of galaxies from the
Northern and Southern Galactic pole are available at
https://people.cs.ksu.edu/~lshamir/data/zdifference/. The information of
galaxies annotated by SPARCFIRE are available at
https://people.cs.ksu.edu/~lshamir/data/sparcfire/. Galaxy Zoo data are
available through SDSS CAS http://casjobs.sdss.org/CasJobs/default.aspx.

## Acknowledgments

I would like to thank the three anonymous reviewers and the associate editor
for the helpful comments. This study was supported in part by NSF grant
2148878.

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