Simulating Galactical Vorticity Using a Supersymmetric Operator

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Simulating galactical vorticity using a supersymmetric operator

Technical Report · March 2020 DOI: 10.13140/RG.2.2.34254.00323

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Sergio Manzetti 1,2∗ and Alexander Trounev 1,3 1. Fjordforsk A/S, Midtun, 6894 Vangsnes, Norway. Email: [email protected] 2.Uppsala University, BMC, Dept Mol. Cell Biol, Box 596, SE-75124 Uppsala, Sweden. 3. A & E Trounev IT Consulting, Toronto, Canada.

March 29, 2020

1 Abstract

1 Simulations of galactic vorticity require the development of complex systems of equations to generate models that account for the gravity, mass, kinetic, and electromagnetic eﬀects which govern cluster behaviour. In this study, a relationship is outlined through a series of steps, between the Einstein equations, the Ricci equations and the gravitational model of comoslogical bodies. This series of steps are then followed by the generation of a group of equations, including a supersymmetric equation which describes vorticity under the presence of an electromagnetic ﬁeld for quantized systems. The relationship between the Einstein equations and the equation for quantum vorticity is ultimately the described and we outline the model of galactic vorticity through a nonlinear

1Please cite as: Manzetti Sergio and Alexander Trounev (2020). Simulating galactical vorticity using a supersymmetric operator. Technical Reports 23, 21, pp. 1-32. Fjordforsk A/S - www.fjordforsk.no

1 cluster model, which has direct similarities to the Kosterlitz-Thouless model for vorticity in quasi-ordered systems. The resulting supersymmetric model for describing galactic vorticity we here show suggests that galactic vorticity broke symmetry already at T=0, and by the eﬀect of the galactic halo around the periphery, generates nonlinear vorticity through the shear property of the wavefunction. Most interestingly, this property of the wavefunction, which generates autonomous vorticity by the mere presence of a halo-boundary of the galactic cluster is independent of temperature. We describe by this that the nonlinear relationship between vortices in the cluster leads to a cumulative effect which generates an exchange interaction at the macroscopic level between clusters which is the source of irregular moment of inertia of galaxies, that has been attributed to dark matter. Our theory is thus that galactic rotations are both fueled by the mass of the galaxy by classical physics means as well as the nonlinear vortex interaction between clusters.

2 Introduction

Einstein’s general theory of relativity is widely used in cosmology and astrophysics, including for modeling the motion of stars in galaxies. The problem of many bodies in the general theory of relativity was considered in [1, 2, 3, 4, 3, 5, 6] and others. It was shown in [7] that the field equations for empty space are sufficient to describe the motion of matter represented in the form of point singularities. Consequently, the motion of material particles represented by the singularities of the gravitational field is one of the problems of the general theory of relativity. This approach uses only gravitational equations in empty space. As is known, to describe the motion of inert matter in the framework of the general relativity, Einstein and Infeld formulated the theory [8], where they declared that all attempts to represent matter by the energy- momentum tensor are unsatisfactory, and they want to free theory from the special choice of such a tensor. Therefore, theory [8] deals only with gravitational equations in empty space, and matter represented by singularities of the

2 gravitational field. Note that in the existing approaches to the description of black holes, the Schwarzschild metric [9] is widely used, which describes the singularity of the gravitational field, which is physically interpreted as the field of a particle of a given mass. In an alternative approach to the description of the motion of material bodies, the energy-momentum tensor is used [10, 4, 6] and [11, 12, 13, 14, 15, 16]. However, in this case, the theory is surrounded by a large number of hypotheses related to the indeterminacy of the concept of matter, especially in light of recent discoveries in astrophysics, indicating a significant contribution of dark matter and dark energy to the dynamics of the expanding Universe [16]. Einstein’s equations have the form [17, 18, 19, 20]: 1 8πG R − g R = g Λ + T (1) ik 2 ik ik c4 ik

Here Rik, gik,Tik is the Ricci tensor, the metric tensor and the energy-momentum tensor;Λ, G, c - Einstein’s cosmological constant, the gravitational constant and the speed of light, respectively. We put Λ = 0 suppose that due to the small in- ﬂuence of this parameter in the problem of many bodies associated with galaxy dynamics. Solutions of Einstein’s equations with axial symmetry were considered in [2, 3, 5], [18, 19, 20, 21, 22] and others (a review of publications is given, for example, in [5], [19, 20, 21]). The metric of such spaces, under certain assumptions, can be reduced to

ds2 = eµdt2 − e−µ(eν dρ2 + eν dz2) − e−µρ2dφ2 (2)

Here µ(ρ, z), ν(ρ, z) are functions that satisfy Einstein’s equations. Calculating the components of the Einstein tensor in the metric (2) and assuming for vacuum 1 Gik = Rik − 2 gikR = 0 we ﬁnd the ﬁeld equations: ∂2µ 1 ∂µ ∂2µ ω = + + = 0 (3) 1 ∂ρ2 ρ ∂ρ ∂z2

1 ω = ν − ρ(µ2 − µ2) = 0 (4) 2 ρ 2 ρ z

ω3 = νz − ρµρµz = 0 (5)

3 ∂2ν ∂2ν 1 ω = + + (µ2 + µ2) = 0 (6) 4 ∂ρ2 ∂z2 2 ρ z It can be veriﬁed that not all equations (3)-(6) are independent and that the following two relations are true [19]

∂ω ∂µ ∂ω ω = 2 + ρω + 3 (7) 4 ∂ρ 1 ∂ρ ∂z ∂µ ∂ω ∂ω ρω = 2 − 3 (8) 1 ∂z ∂z ∂ρ

Therefore, we can choose, for example, the equations (3) and (6) as a system of equations for determining two functions µ(ρ, z), ν(ρ, z), we have

∂2µ 1 ∂µ ∂2µ + + = 0 (9) ∂ρ2 ρ ∂ρ ∂z2

∂2ν ∂2ν 1 + + (µ2 + µ2) = 0, (10) ∂ρ2 ∂z2 2 ρ z

indicating that ω1 and ω2 and ω4 are linearly dependent, as also is ω1 and

ω2 and ω3. By solving the system of these linearly dependent equations (9) and (10), we can come to the determination of static gravity fields in the case of axial symmetry. Model (9), (10) has a unique expansion property in the theory of Ricci flows, which allows us to specify the motion in the system of N bodies. In this case, the gravitational potentials in the system of equations (9), (10) depend on time. To simulate a change in the metric during the movement of singularities, we use Ricci flows, which are described by the equation [23, 24, 25, 26] ∂g ik = D R (11) ∂t 0 ik

Here D0 is the diﬀusion coeﬃcient, which in the standard theory [23, 24, 25,

26] is assumed D0 = −2, however, in the metric (2), taking into account the signature, it should be put D0 = 2, then (9), (10) reduces to a system of parabolic equations:

∂µ ∂2µ 1 ∂µ ∂2µ eν−µ = + + (12) ∂t ∂ρ2 ρ ∂ρ ∂z2

4 ∂ν ∂2ν ∂2ν 1 eν−µ = + + (µ2 + µ2) (13) ∂t ∂ρ2 ∂z2 2 ρ z Comparing (12) (13) and (9), (10), we find that for steady flows under conditions µt = νt = 0, system (12)-(13) reduces to (9), (10). The justification for such a transition from the static system (9)- (10) to the parabolic system of equations (12)-(13) is similar to the theory developed in the following studies [23, 24, 25, 26] and others. For many practically important problems, such as the merging of black holes [1], it will be enough to know how the metric of the binary system changes (as spacetime is curved by two merging gravity sources) as the centers of gravity approach a given speed. The Ricci flow model (12)-(13) allows one to answer this and other questions related to a change in the metric. Consider the solutions of the system of equations (9), (10) of the form [5, 21]

N X mj µ = − (14) p 2 2 j=1 ρ + (z − Lj)

N 2 2 1 X mj ρ 1 X mjmk ν = − 2 2 2 − 2 fjk (15) 4 (ρ + (z − Lj) ) 2 (Lj − Lk) j=1 j6=k

2 Here f = √ ρ +(z−L√j )(z−Lk) − 1, m ,L are some constants. Note jk 2 2 2 2 j j ρ +(z−Lj ) ρ +(z−Lk) that expressions (14)-(15) differ in normalization from expressions given, for example, in [5] by virtue of the definition of metric (2). In the particular case, assuming in (14)-(15) that N = 2, we arrive at the expression of the potentials first obtained in [2]. In the particular case of three bodies, the potentials µ, ν were obtained in [3]. In the nonrelativistic case, the potential µ reduces to the expression µ = 2ϕ/c2, where ϕ is the gravitational potential in Newton’s theory of gravity. Since the point masses in metric (2) with potential (14) do not experience mutual displacement, although they should be attracted according to Newton’s theory, the author of [2] draws the erroneous conclusion that Einstein’s theory of relativity is not true. In reality, however, in Newton’s theory there is a static solution for two gravitating masses moving in circular orbits in a synodic coordinate system - a non-inertial reference frame rotating

5 synchronously with the period of revolution of bodies [27]. As is known, the synodic coordinate system is used in the formulation of the restricted three-body problem in classical mechanics [27, 28], which can be used in the formulation of a similar problem in the general relativity. The diﬀerence between these two problems lies in the presence of a potential ν that has no analogues in Newton’s theory, but plays a role analogous to the eﬀects of a non-inertial reference frame in classical mechanics. To extend the indicated analogy, we put in the metric

(2) z = R0φ1, here R0 is some constant (large radius of the torus), and φ1 is the cyclic coordinate. With such a replacement, metric (2) and equations (12)-(13), (9)-(10) will not change, but solution (14)-(15) has a diﬀerent physical meaning, since it describes the gravitational potentials of sources distributed around a circle. Such a distribution can be used to describe annular galaxies, planetary rings, and the asteroid belt. A further generalization of the theory is that as a coordinate z we can choose some function, such that the equation z(R, φ1) = 0 describes a plane curve, for example, a logarithmic spiral R = R0 exp kφ1. By setting the distribution of singularities on one or several curves, one can simulate the motion of particles in a spiral galaxy. Now we note that solutions similar to (14) can be obtained in another way using the wave function and the post-Newtonian metric [17]:

ds2 = eh(t,x,y,z)dt2 − e−h(t,x,y,z)(dx2 + dy2 + dz2) , (16) where h(t, x, y, z) is some function that is a solution of the Einstein equation:

2 1 2 3 −h 2 (17) ∇ h − 4 (∇h) + 4 (∂te ) = ρ

In the static case, in the linear approximation, from equation (17) we have Newton’s theory of gravity ∇2h = ρ (18)

In another case, when h = h(t), h(0) = 0, equation (17) describes the collapse (-) or expansion (+)

t 3 (∂ e−h)2 = ρ → h = − ln(1 ± √2 R ρ1/2dt) (19) 4 t 3 0

6 Since equation (17) is not quasilinear, we reduce it to the form of a quasilinear equation using the hypothesis of the dominant inﬂuence of ρ. Suppose that |ρ| >> max(|∇2h|, (∇h)2), then it is possible to bring (17) to the form of a parabolic equation

√ q √ 3 −h 2 1 2 1 2 1 2 (20) 2 ∂t(e ) = ρ − ∇ h + 4 (∇h) = ρ(1 − 2ρ ∇ h + 8ρ (∇h) + ...)

There are two possible forms of equation (20), depending on the sign of ρ. If ρ > 0, then keeping only the first terms of the expansion on the right-hand side of (10), we find the diffusion equation

√ −h 3 e√ ∂h 1 2 1 2 (21) 2 ρ ∂t = −1 + 2ρ ∇ h − 8ρ (∇h)

If ρ < 0, then setting ρ = −m2, we reduce equation (20) to the form of the nonlinear Schr¨odingerequation

√ 3 e−h ∂h 1 2 1 2 (22) −i 2 m ∂t = 1 + 2m2 ∇ h − 8m2 (∇h)

Note that equation (21) is similar to equation (13). We are interested in solving equations (12),(13) and (21),(22) with initial data in the form of quantum vorticity. For this, we devise a supersymmetric wave-equation which we developed recently over several works [29, 30, 31] when studying electron gas in quantized 2D systems. The Hamiltonian of the supersymmetric wave-equation is of particular appeal to model vorticity in a space-time continuum, as it generates vorticity spontaneously without need of auxiliary functions and has a particu- larly appealing spectrum of eigenvalues, where the range of trivial eigenvalues in its spectrum is positive inﬁnite, and the evolution of the spectrum is slowly crescent [30]. The algebraic properties of the operator of the supersymmetric wave-equation [30] suggest therefore that it is valid for both microscopic and quantized systems as well as for macroscopic continuous systems, yielding real eigenvalues towards inﬁnite levels of energy [30]. The equation, which reads:

1 ( /i∇ − (e/c)A~) · (− /i∇ − (e/c)A~)Ψ = EΨ (23) 2m ~ ~

7 was developed using supersymmetry rules on its factors [29], generating the inverse sign on the the two inherent factors as given in (23). The original form of this equation developed by Fang and Stiles [32] diﬀers from it as it does not have supersymmetric form of the factors, and therefore generates diﬀerent results. We consider a dimensionless nonlinear form of (23) describing the vorticity in a quantum system in the form

∂ψ 1 2 ∂ψ ∂ψ 2 ~ 2 (24) ∂t = 2 ∇ ψ − iΩ(x ∂y − y ∂x ) − β|ψ| ψ + (A) ψ

Here A~ is a dimensionless vector potential and β, Ω - parameters of the model describing the quasiparticle interaction and the angular velocity, respectively. Note that nonlinear Schr¨odinger equation (Gross-Pitaevskii model) follows from (24) when replacing t → it. Eq. (24) describes evolution of the wave function from some initial state ψ(x, y, z, 0) = ψ0(x, y, z) and up to stationary state with E = 0.

The nonlinear diffusion equation (21) is obviously related to equations (13) and (24), and equation (22) is related to the nonlinear Schrödingerequation. Hence, we describe a sequence of steps where we derive a diffusion equation (21) by the dominant influence of ρ in Einstein’s equation (17), which we here solve using a supersymmetric quantum vorticity equation (23) as initial data, as it has a valid spectrum for both non-relativistic as well as relativistic phenomena [30] and also is parental to the NLSE (non-linear Schrödingerequation), in its dimensionless form (24). from the system of equations with dependent variables, deriving from the Einstein field equations, We pose the question if vorticity has arisen by virtue of equations (23),(24), then how will this evolve by virtue of equations (12),(13), (21) or (22)? To simplify the problem, we put ρ = const in eq. (21) and m = const in eq. (22). But here other options are possible, which we will discuss below. As the initial state for equations (21) and (22), we take the analytical solution obtained in [31]

ir2 √ −iEmt+imφ+ √ m m 2 ψ = C e 2 r L (−i 2r ), (25) m m κ1

8 √ m 1 Here Lκ (x) is the generalized Laguerre polynomial, κ = 8 (2i 2mΩ − 4m + √ √ 2 2i 2E − i 2k − 4), and Cm is constant. Here we set the constant from the condition max|ψm| = 1 and generate a superposition of states with three diﬀer- ent m in the form

(26) ψ = Cm1 ψm1 + Cm2 ψm2 + Cm3 ψm3

Variants of the initial states for equation (21)-(22) are shown in Figure 1.

Figure 1.The modulus of the wave functions used as initial states

3 Numerical analysis and results

As main example we use the combination, including the third, eight and thir- teenth harmonics (middle in Figure 1), then the results of simulation are shown in Figures 2-5. Figure 2 demonstrates evolution of gravitational potentials µ, ν.

9 Figure 2.Evolution of the gravitational potential µ (top) and ν (bottom) with initial data (25),(26) with m1 = 3, m2 = 8, m3 = 13.

The potential distribution for t = 1 is shown in Fig. 3. In more detail, the evolution of the metric can be traced in Fig. 4 where the potentials are shown in the cross section y = 0. Finally, the evolution of metric (16) with potential (21) is shown in Fig. 5. Comparing the data in Figs. 2 and 5, we ﬁnd an obvious similarity. Therefore, we can calculate the potential h for the initial state with m1 = 1, m2 = 7, m3 = 13 (hexagon on Figure 1) assuming that µ looks similar to Fig. 6. We can also calculate the potentials for singularities distributed on the circle using equations (14),(15) - Fig. 7. And here we also ﬁnd similarities with the data in Fig. 2-6.

10 Figure 3.Distribution of the gravitational potential µ (left) and ν (right) at t = 1 computed with initial data (25),(26) with m1 = 3, m2 = 8, m3 = 13.

Figure 4.Evolution of the gravitational potential µ (left) and ν (right) at y = 0 computed with initial data (25),(26) with m1 = 3, m2 = 8, m3 = 13.

Figure 5.Evolution of the gravitational potential h computed with initial data

(25),(26) with m1 = 3, m2 = 8, m3 = 13.

11 Figure 6.Evolution of the gravitational potential h computed with initial data

(25),(26) with m1 = 1, m2 = 7, m3 = 13.

Figure 7.Distribution of the gravitational potential µ (top) and ν (bottom) computed for six (left) and ten (right) singularities with equations (14),(15).

In order to interpret the properties of the gravity potential described in the aforementioned images, consider the center to be situated at a mass center, such as the center of a galaxy. Allow then the concentric circles to be local centers of gravity circulating about the center. We here consider the gravity center as a rotating center, which by the number m, we can model according to the level

12 of energy of the system. It appears from the images that the higher the energy of the system, the more concentric mass centers arise in the periphery as also is observed by Feng and Gallo in [33]. This phenomenon can be associated with points in the gravity ﬁeld (singularities), as suggested by Einstein and Infeld [8]. These points form symmetrically about the center of gravity, and organize in a symmetric arrangement according to the level of m. Note that m is an oscillatory parameter, deﬁned in the Laguerre polynomials in (25), which is a solution to supersymmetric initial conditions (23).

Ricci gravity waves Note that along with the system of equations (12), (13), the equations (4) and (5) must also be satisﬁed. It is obvious, however, that these equations cannot be satisﬁed on arbitrary solutions of the system of equations (12), (13). It can be assumed that the discrepancy arising from this should be compensated by the Einstein equation (1) by some energy-momentum tensor

τik, which is interpreted as the energy of gravitational waves freely propagating outside the system. To describe the wave propagation process, we use the standard ﬁeld theory [17], which gives:

2 i i 16πG i (27) ∇ ψk − (ψk)tt = c4 τk

i Here ψk describe the perturbation of the Galilean metric during the passage of a gravitational wave. Using equations (12)-(13), we ﬁnd the components of the

Einstein tensor Gik in the metric (2) in Ricci ﬂows

2 1 2µ−ν ω2 ω3 −ρ −ν (28) G11 = 2 ∂te ,G22 = 2ρ ,G23 = G32 = 2ρ ,G44 = 2 ∂te

Using (28), we transform equation (27) to a form convenient for numerical integration:

2 i i ij ∇ ψk − (ψk)tt = 2g Gkj (29) The system of equations (29) makes it possible to determine the distribution of ﬁelds in the far zone, where the metric (2) tends to the Galilean metric [17]. However, we will use these equations directly in the region of collision between two singularities in the gravity ﬁeld, that may represent two black holes. In

13 order to visualize this event, we solve numerically the system of equations (12), (13) and (29) in a rectangular region ρ ≤ R, −L/2 ≤ z ≤ L/2. As the initial and boundary data for the system of equations (12)- (13), we used the solution in the form (14)-(15) with N = 2. We assume that the particles move towards each other at a constant speed u1, u2 up to a collision. For equations (29) we pose the homogeneous initial and boundary conditions provided that the gravitational wave formed in the region of particle collision does not reach the boundary of the region during the integration time. Figure 8 shows the results of modeling the propagation of gravitational waves generated during collision and particle fusion in Ricci ﬂows, performed according to (12), (13), (29) with the following parameters:

−2 m1 = m2 = 10 ,L1 = −L2 = 1,R = L = 10, u2 = −u1 = 0.5.

The data in Fig. 8 were obtained under the assumption that particles after a collision at time t = 1 form a new particle. The computational process stops at time t = 8. Note that the waves arise due to the nonlinearity of equations (12), (13). The nature of these waves is the same as that of waves arising from the merger of black holes [1, 34] - a metric perturbation caused by the merger of two ﬁeld singularities.

14 Figure 8. Evolution of the gravitational waves in 2D at diﬀerent time (above) and in 1D in diﬀerent directions (below).

1 Note that we used ψ1 = ψ1 to visualize gravitational waves in Fig. 8.

Logarithmic potential By processing data for 50 galaxies [35, 36], it was found that the best ﬁt with the entire data set is obtained when the dependence of the potential on the radial coordinate can be represented in the form

m 2 (30) φ(ρ, 0) = − ρ + a ln ρ + bρ + kρ + φ0

Here a, b, k, mφ0 are some constants characterizing the distribution of the gravitational potential in the inner region of the galaxy. Therefore, we can assume

15 that the gravitational potential in the inner region of the galaxy has the form

m 2 2 φ(ρ, z) = −√ + a ln ρ + b(ρ − 2z ) + kρ + φ0 (31) ρ2+z2

Expression (30) of gravitational potential, which is consistent with experimental data on the rotation speed of CO and neutral hydrogen in 50 galaxies [35, 36], is shown in Fig. 9.

Figure 9.Dependence of the gravitational potential (km2/s2) on the radial coordinate (kpc) in the galaxies NGC0000, NGC0253, NGC0660, NGC0891 according to the data of [35, 36] (points) and according to the model (30) (solid line): the ﬁeld contains a point source, a logarithmic singularity, and a quadratic potential describing the ﬂow in the cluster.

The first two terms on the right-hand side of (30) and (31) correspond to conditions in the galaxy, and the remaining three describe conditions in a cluster. m Term − ρ describes the field of a singular source (black hole), the second term describes the vortex field. Both terms are a solution of equations (9), (10) in the form µ = −√ m + a ln ρ, 2 2 ρ +z (32) m2ρ2 am a2 ν = − 2 2 − √ + ln ρ 4(ρ +z ) ρ2+z2 2

16 Thus, as can be seen from expressions (30)-(32), each galaxy contains a logarithmic potential, and therefore is a vortex. If we look at a cluster of galaxies, we will see many vortices. At this stage, we investigate how these vortices (galaxies) arise, by using a supersymmetric model to describe galaxy rotation.

3.1 Application of a supersymmetric model to describe galactical rotation

We want to apply our model [29, 31, 30] with which we described the occurrence of vortices in a magnetic field of a cluster, and attribute the vorticity behaviour to galactical spin. We assume that in a certain cylindrical region of space there √ exists a uniform magnetic field with vector potential A~ = (−y, x, 0)/ 2. There exists quantum matter with a density of |ψ|2 and ordinary matter with a density 2 of ρm = 1 − |ψ| . Quantum matter is described by the equation (24). At the 2 initial moment, the distribution of quantum matter is given as ψ2 = e−r . At the boundary of the region, a homogeneous Neumann condition is specified. Figure 10 shows the density distribution of ordinary matter during the decay of the initial state of quantum matter computed at β = 100, Ω = 1, 2. It was previously established that during the decay of the initial state, quantum vortices are formed that surround the giant vortex [29, 31, 30] - Fig. 1. Therefore, the minimum density of quantum matter and, accordingly, the maximum density of ordinary matter is observed in the central region and in quantum vortices - Fig. 10. If each quantum vortex is associated with a galaxy, then as a result of evolution there is a cluster of galaxies formed. Figure 11 shows the streamlines of the flow of quantum matter calculated for t = 4, β = 100, Ω = 1, 1.2, 1.5, 1.7. Note that the number of vortices increases with increasing Ω - Fig.10-11.

17 Figure 10. Distribution of the density of the ordinary matter at diﬀerent time computed at Ω = 1 (above) and Ω = 2 (below). Calculated using eqn. (24).

18 Figure 11. Streamlines of the ﬂow of quantum matter calculated for t = 4, β = 100, Ω = 1, 1.2, 1.5, 1.7.

3.2 Cluster expansion and vorticity

By observation of the evolution of quantum matter in Fig. 10 we see similarities to cluster expansion, where the matter tends to expand from the center of the system. We can describe the intensity of this expansion by the Ω factor, in (24), which we model in Figure 11 at higher and higher values. If we related Ω to the expansion factor of a galactic cluster, we can assume that model (24)

19 describes galactic expansion. To close the model, it is necessary to formulate the equation for the vector potential. We use the Ginzburg-Landau model with Coulomb calibration ∇.A = 0 to describe the dynamics of the electromagnetic ﬁeld:

∂A 1 σ = (ψ∗∇ψ − ψ∇ψ∗) − |ψ|2A + ∇2A (33) ∂t 2iκ where σ is the electrical conductivity of the normal state, κ is the Ginzburg- Landau constant. Using equation (33), we assume that matter exists in two states (macroscopic quantum state and macroscopic classical state). Then equation (24) must be modiﬁed taking into account two phases and temperature as follows

∂ 1 i ψ = ∇2ψ − (A.∇)ψ + (1 − T )ψ(1 − |ψ|2) + (γ − 1)(A)2Ψ, (34) ∂t κ2 κ

where T is temperature, γ - is a supersymmetric parameter. In a uniform magnetic field and at zero temperature, models (24) and (34) coincide. In a general case, we see that the cause of the rotation of the medium is the vector potential of a magnetic field. We assume that in the initial state before the formation of galaxies there was a strong magnetic field. After the decay of the initial state, a rotation of the medium arose. Figure 12 shows the formation of vortices during the decay of the uniform state ψ = 1, A = 0 calculated according to model (33) - (34) with parameters γ = 1.1, σ = 4, κ = 4,T = 0 ~ ∂ψ with the boundary conditions A = (−y, x, 0) and ∂n = 0.

20 Figure 12. Distribution of the density of the wave function at diﬀerent time computed using (33) and (34): vortices penetrate from the periphery into the interior of the region at T=0.

The temperature eﬀect can be traced in Figure 13, 14 where the formation of vortices during the decay of the uniform state is shown at T = 0.5, 0.85 consequently.

21 Figure 13. Distribution of the density of the wave function at diﬀerent time computed using (33) and (34) at T = 0.5

22 Figure 14. Distribution of the density of the wave function at diﬀerent time computed using (33) and (34) at T = 0.85

Figure 12 illustrates the dynamics of vortices in a macroscopic quantum state, for a cluster of galaxies, where T=0. The similarity between macroscopic quantum vorticity and microscopic quantum vorticity is evidently shown by the formation of galactic vortices from the boundaries, as for as in a case of microscopic quantum vortices in quantum ﬂuids where vorticity also arises from the boundary, as we have also found in other similar studies [37, 38]. It can be explain by the properties of the model (33) and (34) where all parameters are non-dimensional, and so can be applied to any scale of physics. The properties of galactic vorticity, which we here report as conserved and similar to quantum vorticity, by the rationale described above, following from the Einstein Equa-

23 tions, the Ricci equations and the supersymmetric equations, are rather similar to the effects described by the Kosterlitz–Thouless transition, where phase shift takes place via vortex pairs break the symmetry and form unpaired vortices at beyond a critical temperature. We see this from Figure 10-12, where a small change in temperature from T=0 to T=0.85 induces rapid changes in vorticity symmetry. As galactic matter can be described as evolving from different stages of density during its various stages, it is conceivable that this evolution pertains similarities to phase transitions observed for quantum liquids subjected to small temperature changes. During these very changes, the existing vortex symmetries go from ordered-states (governed by the gravitational potential such as shown in Fig 6) to quasi-ordered states (Fig 10, t=0-2.4, Fig 12, t=0-9) to disordered states (Fig 10 t=3.2,...t=4, Fig 12, t=13.5).By our model, it appears that vorticity is favored for galactic matter at critical temperature Tc = 0,and evolves from ordered states to semi-ordered states, even though the temperature remains at T=0. This is a principal difference from our model and the model of Kosterlitz–Thouless, where vorticity is stably ordered into a specific symmetry below and at Tc [39, 40]. We note also that our system is simulated without a magnetic field potential, however the boundary conditions described by A~ = (−y, x, 0) appear to impose vorticity of the system, in an autonomous manner (vorticity induction independent of magnetic field). This has to our knowledge only been observed in the quantum Hall effect, at the microscopic quantum level [41], where certain metals generate vorticity and thus acquire a superconductance without the influence of magnetic fields. This effect is thus propagated by the boundary conditions in of which the wavefunction can only assume vortex solutions with time without any external impulse, which, when we extrapolate to the galactic nuclei, a natural boundary with the Universal vacuum about the periphery creates the natural boundaries for galactic vorticity. Our result share similarities to existing studies on galactic vorticity and simulation [42, 43], where the so-called Hodge–Helmoltz modes decomposition for simulating galaxy requires specific boundary behaviour to generate galactic turbulence/vorticity. For the latter study, it emerges that the boundary of the

24 galaxy in simulations represents a physical halo boundary formed by the transition to intergalactic vacuum. We relate this physical halo [43] at the galactic cluster periphery to the boundary in our simulations that generate vorticity in a similar manner as observed in the quantum Hall state for microscopic vorticity.

4 Discussion

From the given property of vorticity arising from the boundary, we can assume that the individual mass of the galaxies in the galactic cluster modulates the strength of the magnetic field, the effect on vorticity and hence, with the boundary, the competitive forces that generate vorticity in our model - boundary properties (galactic radius, halo circumference) and the mass of the galaxy. With this in mind, we can further propose that each member of the cluster, here represented by vortices will contribute in its individual way tot he total magnetic field of the cluster and thus also add to the global vorticity generating potential for the entire cluster along with its total circumference. With this in mind, we thus propose that several cluster contribute an-harmonically to the oscillations of the Universe, and thus facilitate the non-linear evolution of Uni- versal behaviour, where both expansion and contraction result, as also recently proposed in an interesting study by Mirza [44]. Having thus a nonlinear behaviour of the Universe, by looking at the nonlinear properties of clusters (Fig. 10-12), we can propose that the contribution of dark matter may be minimal if not in-existing, as the an-harmonic oscillations of several clusters together generate a non-additivity effect which ultimately leads to dissipation and transfer of energy between clusters. Hence, the effects that are postulated to arise from dark matter [45, 46, 47] can be explained by energy exchange between clusters rather than arising from an invisible source. Existing models that disclaim the existence of dark matter are published [48, 49], however none of these do to our knowledge propose that dark matter is actually an exchange interaction between galactic clusters, and hence their vortices.

25 5 Conclusions

We examined a model for the formation of a cluster of galaxies. In this model, each galaxy is described as a vortex in a continuous medium. The state of the medium is described by the wave function and vector potential. We applied a supersymmetric wave equation describing the formation of a quantum vorticity around a giant vortex. The results show strong familiarity with both the Kosterlitz-Thouless model for phase transitions and also to have properties of the quantum Hall eﬀect. As we have a nondimensional model, we can model this at any scale of physics, and generate theoretical results for large macrocos- mic bodies, such as galactic clusters. We propose a theory on the inexistence of dark matter and note that the origins of anormal spatial physics for rotation of galaxies can arise from exchange interaction between large cosmological systems.

26 6 Acknowledgements and correspondence

The authors declare that they have no competing ﬁnancial interests. Correspondence and requests should be addressed to Sergio Manzetti, [email protected]).

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