MNRAS 000, 1?? (0000)

Compiled using MNRAS LATEX style le v3.0

Preprint 18 June 2021

Existence of Chern-Simons modied gravity based on
discovery of many galaxies without dark matter
1Kaito Takeda , Keiji Nakatsugawa

1,2, Kohkichi Konno

2,32,4, Toyoki Matsuyama, and Satoshi Tanda

1 Department of Applied Physics, Hokkaido University, Sapporo, 060-8628, Japan.
2 Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo, 060-8628, Japan.
3 Department of Natural and Physical Sciences, Tomakomai National College of Technology, Tomakomai, 059-1275, Japan.
4 Department of Physics, Nara University of Education, Nara, 630-8528, Japan.

18 June 2021

ABSTRACTThe existence of dark matter has been assumed to explain the at rotation curves of galaxies. Recently, however,many studies have reported galaxies without dark matter. Therefore, we need to reconsider dark matter theory.
Although modied Newtonian dynamics (MOND) (which is another candidate to explain at rotation curves without
dark matter) can describe spiral galaxies well, it is inapplicable to recent observations. Here, we focus on considering
the rotational motion of galaxies on the basis of Newtonian dynamics with a relativistic correction, and we nda new empirical formula for at rotation curves. The simple formula is derived from Chern-Simons (CS) modiedgravity. We may be able to understand the dynamics of all galaxies since the helicity C2 in CS theory can takedierent values. In this study, we calculate CS gravitational acceleration and compare it with MOND gravitationalacceleration. Furthermore, we compute the log variance ratio between CS data and MOND data. Consequently, wediscover that CS data ts the observed gravitational acceleration better than MOND data for a large radius whereat rotation curves appear. It also means that we can explain the at rotation curves of galaxies without dark matter.
Rotational dynamics regarded as MOND or dark matter phenomena may be interpreted as CS vortex phenomena.
Key words:

1Chern-Simons modied gravity  modied Newtonian dynamics  dark matter

1 INTRODUCTION

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There is a mystery regarding the at rotation curves of galax- 28
ies. Galaxies rotate as if they contain an invisible mass. To 29
explain this phenomenon, the existence of dark matter has 30
been assumed (Rubin et al. 1978). Although X-rays observed 31
in an elliptical galaxy oer its evidence, a black hole may 32
be an X-ray source in such a galaxy (Haaften et al. 2019). 33
In terms of gravitational lensing in a cluster of galaxies, we 34
cannot decide whether it is a proof of dark matter because 35
the amount of dark matter estimated by gravitational lensing 36
was frequently less than expected (Meneghetti et al. 2020). In 37
addition, many galaxies without dark matter have been dis- 38
covered (Dokkum et al. 2018; Gui et al. 2020; Montes et al. 392020). Thus, we need to reconsider dark matter theory (Lelli 40412014; Firmani et al. 2001).
Modied Newtonian dynamics (MOND) (Milgrom 1983; Mc- 42
Gaugh 2008; Smolin 2017) may provide another way to solve 43
this fundamental problem regarding the universe. Although 44
MOND is a phenomenological theory without dark matter 45(McGaugh & Lelli 2016; McGaugh 2008), it is inapplicable to 46
non-rotating elliptical galaxies. In contrast, we are interested 47
in Chern-Simons (CS) modied gravitational theory (Jackiw& Pi 2003; Konno et al. 2007, 2009; Alexander & Yunes 2009),
which deals with a vortex system including a singularity. In
this study, we discovered a new empirical formula which de-© 0000 The Authors

scribes at rotation curves. The formula corresponds with thehelicity C2 of the space-time structure in CS theory. Hereafter, we use CS∗ to indicate the empirical CS. In principle,CS∗ may be able to describe at rotation curves for any typeof galaxy from elliptical to spiral since C2 can take from zero
to an arbitrary value (Konno et al. 2008). We show that CS∗
is the most suitable theory for understanding galactic dynamics.
By using the new empirical formula, we calculated gravitational acceleration and tted observed data for spiral galaxies. Moreover, we computed the log variance ratio between
22CS∗ and MOND, k = δCS∗ /δMOND , for small values of gravitational acceleration from baryonic matter: gbar ≤ 10−10 m/s2(that is, r > 5 × 1021 m in the case of the Milky Way galaxy).
We show that CS∗ can be applied to the problem of at rotation curves. This is consistent with the discovery of galaxies
without dark matter.
This paper is organized as follows. In §2, we outline the limitations of dark matter and MOND. In §3, we describe therelationship between the new empirical formula and CS theory. Each theory is compared in §4, which comprises the discussion and conclusion.

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2 DARK MATTER AND MOND

Dark matter is one of the strong candidates for explaining
at rotation curves. In dark matter theory, spiral galaxies
contain several times more dark matter than luminous (baryonic) matter. However, it is currently unclear whether or not
this theory is applicable to elliptical galaxies (Romanowskyet al. 2003). For example, the deprivation of dark matter wasrecently observed in the elliptical galaxy NGC 7507 (Laneet al. 2015). In addition, many elliptical galaxies without darkmatter have been discovered (Dokkum et al. 2018; Gui et al.
2020). Further studies may solve this problem, the deprivation or the absence of dark matter (Bidin et al. 2012, 2015).
In contrast, MOND is a phenomenological theory that canalso describe the at rotation curves of spiral galaxies. Thismodel was rst proposed by Milgrom (Milgrom 1983), andMcGaugh developed the following gravitational acceleration
formula (McGaugh & Lelli 2016; McGaugh 2008) :gMOND =gbar√−1−egbar /g†

(1)

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where g† is a universal acceleration constant and gbar isgravitational acceleration from baryonic matter. Figure 1-(a)shows the results obtained with this equation and the observed gravitational acceleration of spiral galaxies. We theoretically consider MOND data (McGaugh & Lelli 2016). InFigure 1-(a), MOND provides a good description of at rotation curves. Equation (1) must also be applicable to (nonrotating) elliptical galaxies because it is a universal modication of Newtonian dynamics. This universality contradicts
recent discoveries of galaxies without dark matter.
Therefore, although dark matter theory and MOND are generally used to explain at rotation curves, they may not provide information about elliptical galaxies (except for certainrotating galaxies). Here, we describe at rotation curves from
Newtonian dynamics with a relativistic correction. In thenext section, we show that a new empirical formula is applicable to various galaxies because of a relationship with CStheory.

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3 NEW EMPIRICAL FORMULA

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In this study, we newly added a function to Newton's equation of motion. When we consider rotational galactic motion
based on this expansion, the next equation of rotational speed
vrot is assumed
vrot =88899091
9293r

CrotGM+r2(2)where M is the mass of a galaxy, G is the universal gravitational constant and r is the radius from the center of a galaxy.
The rst term indicates rotational speed derived from Newtonian mechanics. We assumed that Crot is a function of r.
We analyzed at rotation curves with this equation. First, we
assumed a standard Laurent expansion for Crot :Crot =∞X

an rnn=−∞MNRAS 000, 1?? (0000)(3)

bar in log scale andthe vertical axis represents gobs or gMOND in log scale. The redline shows gobs , which is the observed gravitational acceleration ofspiral galaxies. The blue line shows gMOND , which was calculatedwith Equation (1). (b) The horizontal axis represents gbar in log
scale and the vertical axis represents gobs or gCS∗ in log scale. The
red line shows gobs as in Figure 1-(a). The blue line shows gCS∗ ,which we calculated with Equation (4), (5) and (6). Each gureshows that MOND and CS∗ can describe at rotation curves sinceeach set of calculated data is comparable to observed data. In
particular, we focused on seven points from a long distance andconrmed the dierence between MOND and CS∗ in more detail.
In the insets, the horizontal axis represents gbar in log scale andthe vertical axis represents gMOND − gobs or gCS∗ − gobs , wheregMOND −gobs and gCS∗ −gobs are the dierence from observed data
represented by a thick black line. By computing the log variance
22ratio between CS∗ and MOND, k = δCS∗ /δMOND , based on theresult, we obtained k = 0.885. This indicates that CS∗ ts theobserved data better than MOND for a large radius.
Figure 1. (a) The horizontal axis represents g

Existence of Chern-Simons modied gravity without dark matter
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As explained below, we discovered that the following three 138 To verify the validity of our tting, we calculated the log
terms are eective for at rotation curves:
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variance dened by:
Crot = αr +96
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111112113114

β+γr(4)N

where α, β and γ are constant and expressed by an in Equation (3). We found α = 2.4 × 10−17 s−1 , β = −2.1 × 1026
m2 /s, γ = 3.5 × 105 m/s for the best tting. γ is the most 140eective term for Crot because r and r−1 work as correction 141
terms: these terms correct at a larger radius and a smaller 142
radius, respectively. In fact, the contribution of the r−1 term 143
is 1 to 2% of γ , thus the new empirical formula is just the 144
Laurent expansion around γ up to the rst order in r. If we 145
compare space-time and uid, we can treat αr as a forced 146
vortex, and βr−1 as a free vortex. These two terms are inter- 147
preted as the conservation of angular momentum. γ is linked 148
to a topological term. The most important γ term in Equa- 149tion (4) is derived from CS theory for the following reasons. 150
CS modied gravity in (3 + 1) dimensions was proposed by 151
Jackiw and Pi (Jackiw & Pi 2003). The eects of the CS 152
action in gravitational elds have been discussed in several 153
studies (Alexander & Yunes 2007a,b). A perturbative study 154
with small angular momentum provides the following velocity 155(Konno et al. 2008):
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vCS =r
2GM C2 rs C2 rs+ ++ 2 C1 + C2 + 2C2 log
r
2r2r



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

r − rs 159
rs 160(5) 161
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where rs = 2GMis the Schwarzschild radius, M is the mass 163
c2of a galaxy, G is the universal gravitational constant, c is the 164
speed of light, r is the radius of a galaxy and C1 , C2 are the 165
helicity in CS. At a large r, Equation (5) simply becomes 166
Equation (2) and Crot corresponds to the helicity C2 , which 167
is an important term in CS. This implies that our assumed 168
Equation (2) is strongly related to CS theory.
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Gravitational acceleration of CS∗ is calculated by the follow- 170
ing equation:
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(6) 174
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Figure 1-(b) shows the results of CS∗ gravitational accelera- 176tion with Equation (6). CS∗ can describe at rotation curves 177
as well as MOND. In particular, the results were comparable 178
to observed data when we added the r−1 term. This indi- 179
cates the importance of the r−1 correction term to explain 180
at rotation curves. CS∗ can be applied to various galaxies 181
since the helicity C2 in Equation (5) can take dierent values 182
including zero. We can understand the motion of galaxies by 183
considering CS gravity.
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4 DISCUSSION AND CONCLUSION

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We compared CS∗ and MOND by computing the dierences 190
between both theories and observed data. Those results are 191
shown in the insets in Figure 1. Gravitational acceleration 192
values based on CS∗ and MOND t well at a large radius. 193

2δth=1 X(log gobs − log gth )2
N n=1

(7)

where gth denotes gCS∗ or gMOND , and N is number of data.
Then, we calculated the log variance ratio between CS∗ and22−10MOND, k = δCS∗ /δMOND , for small values of gbar ≤ 10221m/s (that is, r > 5 × 10 m in the case of the Milky Way
22galaxy). As a result, we obtained k = δCS∗ /δMOND = 0.885.
The log variance ratio increases for a smaller radius. However, this deviation is not important here because at rotation curves appear for galaxies with larger radiuses. That is,
the result indicates that a CS∗ t can describe at rotation
curves better than a MOND t. Moreover, CS∗ has a decisively dierent feature, namely it can be applied to ellipticalgalaxies. In that situation, C2 in CS can take zero or a smallvalue. CS∗ may be a more general theory that solves the problem with MOND.
Figure 2 outlines the dierences between dark matter theory,MOND and CS∗ . Dark matter theory and MOND give us
information about the at rotation curves of spiral galaxies.
However, if they were the only theories that can be used toexplain at rotation curves, then paradoxically the dark matter deciency in elliptical galaxies favors dark matter theory.
This strange logic fails if we further consider CS∗ . In principle, CS∗ can describe not only spiral galaxies but also elliptical galaxies since the helicity C2 can take dierent values.
We may be able to classify galaxies based on Hubble classication with CS∗ . The r−1 term in C2 that we discovered may
be computed from a dierent form of CS, e.g. dynamical CS(Konno & Takahashi 2014; Yagi et al. 2012). Although it isdicult to nd an exact solution, we plan to work on thisproblem in a future study.
In spite of many eorts, there has as yet been no direct
laboratory detection of dark matter particles. On the otherhand, if we change our viewpoint and assume that CS
helicity exists in the universe, we can obtain a compatible
explanation for the at rotation curves of galaxies. In thisstudy, we conrmed that CS∗ can describe spiral galaxies
as well as MOND without dark matter. With a large radius(where at rotation curves appear) the log variance ratio
k = 0.885 implies that CS∗ ts observed gravitational
acceleration better than MOND. In addition, there is adenite dierence between CS∗ and MOND. CS, which isrelated to the new empirical formula, is a universal theory
and can be applied to all galaxies. This theory may provide
us some solutions for at rotation curves. CS structure
has the potential to solve the problems posed by many
astrophysical phenomena (Canizares et al. 2012; Barrientoset al. 2019) whereas dark matter or MOND is not required insome cases (Lane et al. 2009). Ultimately, it can be applied
to condensed matter physics and particle physics (Cisternaet al. 2019). We plan to use CS to research many problems
including exploring axial symmetry theory (Bahamondeet al. 2021).

MNRAS 000, 1?? (0000)

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∗ based 241

Figure 2. Classication of dark matter, MOND and CS

on Hubble classication. Arrows with solid lines indicate the re- 242
gion in which each model is applicable. The dashed lines indicate 243
the region where each theory may not be applicable. We do not 244
currently know why galaxies rotate so strangely (winding dilemma 245etc.). MOND can explain at rotation curves only for spiral galax- 246
ies and some rotating elliptical galaxies (Samurovi¢ & Vudragovi¢ 2472019), that is, this theory is not applicable to non-rotating galaxies. 248
Dark matter theory is a strong candidate for solving the problem of 249
at rotation curves. Recent observations for elliptical galaxies indi- 250
cate the existence of galaxies with less dark matter than expected. 251
On the other hand, we can understand all galaxies with CS∗ since 252
universal CS theory has the important helicity C2 . Each galaxy 253
may have dierent (including vanishing) values of C2 depending
on its type. Credit: Ville Koistinen.

194ACKNOWLEDGEMENTS

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We thank Koichi Ichimura and Masahito Sakoda for helpful
discussions. We also thank Yasuhiro Asano for stimulating
comments.

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