Dark Matter Is a Manifestation of the Vacuum Bose-Einstein Condensate

Dark matter is a manifestation of the vacuum Bose-Einstein condensate

Valeriy I. Sbitnev∗ St. Petersburg B. P. Konstantinov Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, Leningrad district, 188300, Russia; Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720, USA (Dated: September 3, 2018)

The vorticity equation stemming from the modified Navier-Stokes equation gives a solution for a flat profile of the orbital speed of spiral galaxies. Solutions disclose existence of the Gaussian vortex clouds, the coherent vortices with infinite life-time, what can be a manifestation of the dark matter. The solutions also disclose what we might call a breathing of the galaxies - due to an exchange of the vortex energy with zero-point fluctuations in the vacuum. Keywords: Quantum vacuum; Bose-Einstein condensate; vortex; axion; graviton; flat orbital speed; spiral galaxy; dark matter;

I. INTRODUCTION cent) is at perpetual movement. Marco Fedi[2] proposes to name this ocean as ”the superfluid The belief in the existence of an aether [1] - quantum space”, metric of which is ”flat”. It a world medium filling all of space, which is the means, it corresponds to the Euclidean geome- building material for all kinds of substances and try [3]. This space contains enormous amount movements manifesting themselves as force fields of virtual particle-antiparticle pairs, which arise - has accompanied natural science from the most and annihilate again and again. In effect, it is ancient times up to now. The building material the Bose-Einstein mega-condensate [4–6]. we currently admit is ordinary (baryonic) mat- We can account for dark matter in terms of a ter, but this material accounts only for 5 per- more fundamental building material if we treat cent of all matter/energy in the observed Uni- dark matter as a dark fluid [2, 7, 8], whose mo- verse. Factually, this matter moves through a tion mimics a Madelung quantum fluid, governed vast ocean of the dark energy (∼68 %) and some- by the hydrodynamic Euler equations [9]. The thing else, Fig. 1. The scientific community be- motion of that fluid could be described by the Gross-Pitaevskii equation [9, 10]. Baryonic matter, 5% From this perspective, the abundance of dark matter is due to the vacuum energy density baryonic object which fills a non-trivial space-time structure [11]. It originates from the quantum vacuum, which is ≥ /2 everywhere densely filled by the virtual particle- antiparticle pairs. With this assumption on vacuum particle-antiparticle fluctuations board, dark matter can be explained by gravita- Dark energy and something else, tional polarization of the quantum vacuum [12]. arXiv:1601.04536v2 [physics.gen-ph] 24 Jul 2016 68% + 27% = 95% The dark matter was originally invoked to cor- rect observation with theory when it comes to FIG. 1: Baryon particles move over a vast ocean of the dark the flat profiles we observe in the orbital speeds energy and something else named the dark matter. of spiral galaxies [13]. Observations show that the orbital speeds of the spiral arms of galaxies lieves that unaccounted (invisible) substances is stay almost constant as distance increases from the dark matter (∼27 %). the galactic cores [14]. This vast ocean of the dark essence (∼95 per- We adopt here the following facts: (i) based on observations of the Planck Observatory [15] it is determined that the parameter of curvature ∗Electronic address: [email protected] very close to zero. In other words, the universe 2 is flat with high precision ΩK = 0.0008 ± 0.004; fundamental element that creates a geometrody- (ii) the speed of rotation of the external regions namic picture of the quantum world both in the of galaxies is no more than 300 km/s; (iii) the non-relativistic domain [18, 19], and in the rela- speeds of galactic and extragalactic particles can tivistic domain [6, 17, 21–23]. It is proportional 3 3 reach 10 km/s and maybe up to 2 · 10 km/s, to the internal pressure P = P1 + P2 divided by at least [16]. In other words, the speeds are non- the density distribution ρ of the matter in the relativistic. In accordance with these observa- space. The pressures P1 and P2 follows from the tions, we may apply further the non-relativistic Fick’s law, J~ = −mD∇ρ, applied to distribution equations in the Euclidean geometry. of the particle-antiparticle condensate inhabiting The aim of this article is to explain the flat the vacuum [17, 19]: profile of orbital speeds in spiral galaxies as a natural consequence of a superfluid vacuum [17].  h¯2  P = D∇J~ = − ∇2ρ, Sec. II employs the modified Navier-Stokes equa-  1 4m ~!2 2 2 (2) tion to describe the motion of a special super-  ρ J h¯ (∇ρ)  P = = . fluid medium - the physical vacuum [18, 19]. The  2 2m ρ 8m ρ vorticity equation stemming from this modified Navier-Stokes equation [20] gives a solution de- Here D =h/ ¯ 2m is the diffusion coefficient [24], scribing the flat profile of orbital speeds. This andh ¯ and m are the reduced Planck constant is followed by concluding remarks and numerical and the particle mass, respectively. estimations Sec. III. So, the quantum potential reads

P h¯2 ∇2ρ h¯2 (∇ρ)2 II. MODIFIED NAVIER-STOKES EQUATION Q = = − + . (3) AND MOTION OF SUPERFLUID BEC ρ 4m ρ 8m ρ2

The modified Navier-Stokes equation is [19] The quantum potential looks as the ratio of the pressure to the density distribution of the mat- ∂~v F~ ter that is proportional to the energy density. m + (~v · ∇)~v = − ∇Q + mν(t) ∇ 2~v. ∂ t N As a consequence, the internal pressure opposes (1) the gravitational attraction. This means that This equation describes a velocity field under the the osmotic pressure pushes the baryonic matter influence of different forces acting on this special apart. This may be a cause of the formation of deformable medium. The difference between this voids in the distribution of galaxies [25]. equation and the familiar Navier-Stokes equa- The presence of the quantum potential in the tion is in two last terms, which describe internal Navier-Stokes equation allows us to describe mo- forces arising in the medium. The first force, tion of this medium through the complex-valued √ ∇Q = ∇(P/ρ), is the gradient from the internal wave function Ψ = ρ exp{iS/h¯} to a solution pressure P divided by the density distribution of the non-linear Schrödingerequation, or more of the matter, ρ. The quantum potential Q is a specifically, a Gross-Pitaevskii-like equation [19]

∂Ψ 1 d ln(ρ) ih¯ = (−ih¯∇ + m~v )2Ψ + U(~r)Ψ + mν(t) Ψ − CΨ. (4) ∂ t 2m R d t | {z } (a)

Here we write ~v = ~vS + ~vR , where ~vS = ∇S/m is the irrotational velocity (S is the action), and 3

~vR is the rotational (solenoidal) velocity. The the viscosity coeﬃcient ν is a function of time,

term m~vR represents the angular momentum of we can assume that (a) time-averaged, the vis- a swirling motion. An extra term enclosed by cosity coefficient vanishes and (b) its variance is the curly bracket (a) fluctuates about zero due to not zero. So the viscosity coefficient is a func- the fluctuating about zero parameter ν(t). The tion fluctuating about zero. We suppose that potential energy U(~r) comes from the expession such fluctuations represent an exchange of en- ∇U = −F~ /N, and C is the integration constant. ergy between the existing baryonic matter and zero-point fluctuations of this special superfluid medium - the physical vacuum, Fig. 1. We A. Vorticity equation and solutions for orbital speeds of spiral galaxies can imagine this special medium as the abso- lute black body at ultra-low temperature with The modified Navier-Stokes equation describes which the baryonic matter is in thermodynamic also a motion of the vortex against background equilibrium [10, 26]. of the superfluid BEC medium. The second in- Let us apply the curl operator to the Navier- y Stokes equation. We come to the equation for z the vorticity ~ω = [∇ × ~v] [27, 28]

∂ ~ω + (~v · ∇)~ω = ν( t )∇2~ω. (5) ∂ t

This vector is directed along the rotation axis. In order to simplify this task, let us move to x the coordinate system where the rotation occurs in the plane (x, y) and the z-axis lies along the rotation axis Fig. 2. Under this transformation the equation for the vorticity takes a particularly simple form: FIG. 2: A simulation of a rotating spiral galaxy: the orbital velocity ~v lies in the plane (x, y). The vorticity vector ~ω is ∂ ω ∂ 2ω 1 ∂ ω ! oriented perpendicular to this plane. Orientations of these = ν( t ) + . (6) vectors ~ω and ~v obey the right hand rule. ∂ t ∂ r2 r ∂ r ternal force in Eq. (1) is dissipative due to a ﬂuc- A general solution of this equation has the fol- tuating presence of a viscosity of the medium. If lowing view [17]

Γ ( r2 ) ω(r, t) = exp − , (7) 4Σ(ν, t, σ) 4Σ(ν, t, σ)

r 1 Z Γ ( r2 )! v(r, t) = ω(r0, t)r0dr0 = 1 − exp − . (8) r 2r 4Σ(ν, t, σ) 0

The ﬁrst function is the vorticity and the second lies on z-axis and the orbital velocity lies in the is the orbital speed. We do not mark the arrows (x, y) plane. The denominator Σ(ν, t, σ) in these above the letters ω and v, since the vorticity 4

formulas has the following view 0.1 (a) t 0.09 Z (b) 2

Σ(ν, t, σ) = ν(τ)dτ + σ . (9) σ/ν 0.08 . (c) 0 0.07

Here σ is an arbitrary constant such that the 0.06 denominator is always positive. 0 5 10 15t 20 The extra parameter σ ensures the existence of FIG. 3: Family of oscillations of the orbital speed v(r, t) for the Gaussian coherent vortex cloud [29] having different values of σ: (a) σ = 2; (b) σ = 3; (c) σ = 6. The factor σ/ν reduces the speed to dimensionless representation. permanent the vorticity and the angular speed Γ ( r2 ) substance, possesses the self-similarity, what can ωc(r) = 2 exp − 2 , (10) 4σ 4σ be seen from the above evaluations. Let us as- Γ ( r2 )! sume in this key that the fluctuating viscosity v (r) = 1 − exp − (11) c 2r 4σ2 has the following presentation [17] with the circulation Γ and the average radius c2 σ initially existing. Here the subscript c comes νn(t) = cos{Ωnt}. (14) Ωn from the Gaussian coherent vortex cloud. Factu- 2 ally, the coherent vortex represnts localized con- The kinetic viscosity coefficient c /Ωn has di- centration of energy and vorticity with a life- mension [length2/time]. Here c is the speed of time tending to infinity [27]. It does not interact light and Ωn is the angular frequency of a vac- with any form of matter and exists in itself in- uum oscillation.The viscosity obeys to the 1/f- −1 finitely in time. We can express a hypothesis law (the flicker-noise law) when Ωn ∼ n tends that these Gaussian coherent vortex clouds hav- to zero at n going to infinity. From the above ing a long-time memory can be a manifestation formula it follows that the viscosity goes to in- of the dark matter. finity as Ωn tends to zero, see Fig. 4. In this For the sake of demonstration let us set 10 eiΩt + e−iΩt 10 ν(t) = ν = ν cos(Ωt). (12) c 2 8 In this case 10 c2 2

Σ(ν, t, σ) = (ν/Ω) sin(Ωt) + σ . (13) { b 6 10 Fig. 3 shows family of oscillations of the orbital

speed v(r, t) for different values of the parame- { a 4 ter σ. Variations of the speeds are shown in a 10 region of their maximal changes, at r = 2σ. For bringing the speeds to an equal level, that are 100 -19 -17 -15 -13 -11 calculated at different values of σ, they are mul- 10 10 10 10 [1/s]10 tiplied by the factor σ/ν, leading the speeds to FIG. 4: The dimensionless coefficient c2/Γ Ω vs. Ω: (a) Ω dimensionless representation. One can see that ranges from 10−11 s−1 to 4 · 10−13 s−1 for the distance r from at increasing σ oscillations of the speed diminish. the galactic core changing form 0 to 20 kpc, Γ ≈ 3.2 · 1025 2 −11 −1 −15 −1 So, at very large σ the oscillations vanish. m /s; (b) Ω ranges from 10 s to 5 · 10 s for the distance r changing form 0 to 40 kpc, Γ ≈ 2.75 · 1025 m2/s. The arrow points to the limiting edge of the visible universe. B. Hierarchy of the parameters ν and σ and the problem of the flat orbital speed figure intervals of changing Ωn are marked by −11 the curly brackets: (a) Ωn ranges from 10 to Observe first that the Gaussian vortex clouds, 4·10−13 1/s when n goes from 1 to 25. In this case which manifest themselves like an unknown dark the flat profile of the orbital speed is observed 5

from about 10 to 20 kpc which has been shown Here the coefficient σ0 has to be more than 1. −11 −15 in [17]; (b) Ωn ranges from 10 to 5 · 10 1/s The fact that σ0 exceeds 1, prevents possible loss when n goes from 1 to 2000. The flat profile ob- of stability of the vortex (galaxy in our case), 2 served in this case is from about 10 to 40 kpc. So, since sin(Ωnt) + σ0 is always positive. the lower the frequency Ω is taken, the larger n Since Ω ∼ n−1, the coefficient Σ ∼ c2/Ω2 radius of the flat profile of the orbital speeds can n n n tends to infinity as Ω goes to zero with growth be achieved. n of n. From here it follows, that the expression De Broglie wavelength, λ = c/Ω , in these n n 1−exp{−r2/4Σ } reaches 1 the more slowly with cases ranges from about 20 kpc to 2 Mpc and n increasing r, the larger is Σ . A set of the co- covers the galactic scales. We can evaluate n efficients Σ for n = 1, 2, ··· can give output to masses, m =h ¯Ω /c2, for the axion-like parti- n n the flat profile. cles [30] which range from about 10−62 kg to 5·10−66 kg. They are in the range shown in [31]. In statistical mechanics, as in quantum mechanics, one computes the canonical partition function for all energetic states of a system be- C. Solution for a flat profile of the orbital speed ing in equilibrium with its environment. So, we A rich gallery of the galaxy rotation curves should compute an analogous sum of possible or- showing output on a flat profile is presented bital speeds of the galaxy for all entangled modes in [14]. Let us try to compute the flat profile. on which the baryonic matter is allowed to exchange fluctuations with the physical vacuum and the zero-point fluctuating background. Our statistical sum in that case becomes

V N ( 2 )! Γ X r 200 V (r, t) = 1 − exp − . (16) 2r n=1 4Σn( t )

km/s The orbital speed V (r, t) versus r and t is shown 5 100 10 in Fig. 5. Here for evaluated calculations we used 6 . 25 2 2.0810 Γ = 3·10 m /s and the angular frequency Ωn 6 . −11 −1 −13 −1 4.0610 ranges from 10 s to 1.667·10 s as n 6 ly . 6.0410 runs from 1 to 60. The angular frequencies are 6 r .10 30 extremely small. While the wavelengths, λn = 8.02 20 7 10 10 c/Ωn, are in the range from 0.97 to 58.3 kpc. 0 kpc tc These oscillating modes cover areas from the galactic core up to the sizes of the galaxy itself. FIG. 5: Orbital speed V is a function of the radius r from the galactic center (in kiloparsec) and time t (in light years). Fig. 5 shows also that the orbital speed experi- ences small fluctuations in time, resembling the To see the formation of the flat profile we need breathing of the galaxy. This breathing is caused to perform computations of sets collected from by the exchange of vortex energy with the phys- modes (14), n = 1, 2, ··· ,N, at different N. Let ical vacuum on the ultra-low frequencies Ωn. In us substitute the expression (14) into the inte- other words, the galaxy breathes in response to gral (9). After computing it we get the following an energy exchange with the vacuum. It should view of the denominator Σ(ν, t, σ): be noted here, that the trembling of galaxy with the 1/f spectrum most likely emulates music of 2 c 2 the heaven spheres (Fractal Memory [32]), than Σn(t) = 2 (sin(Ωnt) + σ0). (15) Ωn the breathing. 6

III. CONCLUSION ation that comes from the model of the holo- graphic screen to be the boundary of the visible The flat profile can be achieved by an exchange Universe [31]. This evaluation is in agreement of the orbital energy of the baryonic matter with with the graviton mass given in [35]. Ultra-light the quantum vacuum zero-point energy. This en- dark matter particles producing out of vacuum ergy exchange is represented through a change has been predicted in the work of [8]. of sign in the viscosity of the special superfluid The frequency spectrum obeys the 1/f-law medium, the physical vacuum. In this time- (the flicker-noise law). As the frequency goes averaged function the viscosity vanishes on av- to zero the viscosity coefficient (14) approaches erage, but its variance is not zero. infinity. We may imagine that on these limit- The exchange takes place on the ultra-low fre- ing frequencies the universe can have positive quencies. Recalculating the wavelengths of these viscosity during one stage of its evolution and fluctuations we find that they cover sizes rang- negative viscosity during other. When the vis- ing from the galactic core up to the radius of the cosity is negative, the physical vacuum returns galaxy and more. the energy to the baryonic matter. This means, Let us take a look at an extreme point sug- that the baryonic matter is created out of noth- gested by arrow c in Fig. 4. We note that ing and we observe the expansion of the universe. −18 the frequency Ωc is 2.2 · 10 1/s. We find The most likely candidates for creating baryonic 26 c/Ωc ≈ 1.36·10 m, which is almost close to the matter may be heavy stars and black holes jets. Compton wavelength evaluated for the visible On the other hand, when the viscosity is posi- universe in [33]. This corresponds to the radius tive baryonic matter loses energy. This means of the Hubble sphere rHS = c/H0, which is about that baryonic matter gradually disappears and 4.4 · 103 Mpc, or about 14 billions light years the universe shrinks towards zero. Theoretically, (the Hubble constant H0 = 67.8 km/(s·Mpc)). only a perfect vacuum remains. One can observe This corresponds with the radius of the observ- that such cyclic variations of the baryonic matter able universe. and the emtpy space find the correlations with We could continue this summation (16) up to the Baum-Frampton model [36–38]. According −18 the point Ωc = 2.2 · 10 1/s. This allows us to to this model the Universe periodically comes affirm that the observable universe rotates about back to ”empty space” and returns the bary- some center with an orbital speed, which has onic matter. This research confirms also that a flat profile through enormous distances. Ex- a spatially flat Universe exists in time without cepting a central region where the orbital speed any singularities such as the Big Bang. After grows from zero to the maximal value corre- some time, about 38 billion years [8], the solu- sponding to the profile level. This rotation pos- tion becomes unstable and characterizes the in- sibly takes place around the richest SuperClus- verse process of absorption of dark particles us- ter in the Sloan Great Wall, SCl 126, and es- ing the vacuum in the compression mode of the pecially around its core, resembling a very rich Universe. Gurzadyan and Penrose [39] also have filament [34]. described in 2010 a scenario of cyclic repeating On these cosmological scales we can evaluate of evolution between a series of big bangs, each a mass of a graviton, with a wavelength that is of which creates a new the Universe history. commensurable with the radius of the universe stated above. An extremal mass of the axion-like particle for the observable universe is Acknowledgments h¯Ω c −69 The author thanks Mike Cavedon, Thad mg = 2 ≈ 2.6 · 10 kg. (17) c Roberts, and Denise Puglia for valuable remarks This value finds a good agreement with evalu- and propositions. 7

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