Dark Matter as a Non-Relativistic Bose-Einstein Condensate with Massive Gravitons
ssymmetry SS Article
Dark Matter as a Non-Relativistic Bose–EinsteinCondensate with Massive Gravitons
Emma Kun , Zoltán Keresztes * , Saurya Das and László Á. Gergely Institute of Physics, University of Szeged, Dóm tér 9, Szeged H-6720, Hungary;[email protected] (E.K.); [email protected] (L.Á.G.) Theoretical Physics Group and Quantum Alberta, Department of Physics and Astronomy, University ofLethbridge, 4401 University Drive, Lethbridge, AB T1K 3M4, Canada; [email protected] * Correspondence: [email protected]: 15 September 2018; Accepted: 15 October 2018; Published: 17 October 2018
Abstract:
We confront a non-relativistic Bose–Einstein Condensate (BEC) model of light bosonsinteracting gravitationally either through a Newtonian or a Yukawa potential with the observedrotational curves of 12 dwarf galaxies. The baryonic component is modelled as an axisymmetricexponential disk and its characteristics are derived from the surface luminosity profile of the galaxies.The purely baryonic fit is unsatisfactory, hence a dark matter component is clearly needed. Therotational curves of five galaxies could be explained with high confidence level by the BEC model.For these galaxies, we derive: (i) upper limits for the allowed graviton mass; and (ii) constraints ona velocity-type and a density-type quantity characterizing the BEC, both being expressed in termsof the BEC particle mass, scattering length and chemical potential. The upper limit for the gravitonmass is of the order of 10 − eV/c , three orders of magnitude stronger than the limit derived fromrecent gravitational wave detections. Keywords: dark matter; galactic rotation curve
1. Introduction
The universe is homogeneous and isotropic at scales greater than about 300 Mpc. It is also spatiallyflat and expanding at an accelerating rate, following the laws of general relativity. The spatial flatnessand accelerated expansion are most easily explained by assuming that the universe is almost entirelyfilled with just three constituents, namely visible matter, Dark Matter (DM) and Dark Energy (DE),with densities ρ vis , ρ DM and ρ DE , respectively, such that ρ vis + ρ DM + ρ DE = ρ crit ≡ H /8 π G ≈ − kg / m (where H is the current value of the Hubble parameter and G the Newton’s constant),the so-called critical density, and ρ vis / ρ crit = ρ DM / ρ crit = ρ DM / ρ crit = Symmetry , a r X i v : . [ a s t r o - ph . C O ] M a y ymmetry , , 520 2 of 13 Therefore, what exactly are DM and DE remain as two of the most important open questions intheoretical physics and cosmology.Given that DM pervades all universe, has mass and energy, gravitates and is cold (as otherwise itwould not clump near galaxy centers), it was examined recently whether a Bose–Einstein condensate(BEC) of gravitons, axions or a Higgs type scalar can account for the DM content of our universe [11,12].While this proposal is not new, and in fact BEC and superfluids as DM have been considered by variousauthors [13–34], the novelty of the new proposal was twofold: (i) for the first time, it computed thequantum potential associated with the BEC; and (ii) it showed that this potential can in principleaccount for the DE content of our universe as well. It was also argued in the above papers that, if theBEC is accounting for DE gravitons, then their mass would be tightly restricted to about 10 − eV/c .Any higher, and the corresponding Yukawa potential would be such that gravity would be shorterranged than the current Hubble radius, about 10 m, thereby contradicting cosmological observations.Any lower and unitarity in a quantum field theory with gravitons would be lost [35].In this paper, we discuss the possibility of a BEC formed by scalar particles, interactinggravitationally through either the Newton or Yukawa potential. Such a BEC, interacting only throughmassless gravitons has been previously tested as a viable DM candidate by confronting with galacticrotation curves [30,36].In this paper, we solve the time-dependent Scrödinger equation for the macroscopic wavefunctionof a spherically symmetric BEC, where in place of the potential we plug-in a sum of the externalgravitational potential and local density of the condensate, proportional to the absolute square ofthe wavefunction itself, times the self-interaction strength. The resultant non-linear Schrödingerequation is known as the Gross–Pitaevskii equation. For the self-interaction, we assume a two-body δ -function type interaction (the Thomas–Fermi approximation), while we assume that the externalpotential being massive-gravitational in nature, satisfying the Poisson equation with a mass term.The BEC-forming bosons could be ultra-light, raising the question of why we use the non-relativisticSchrödinger equation. This is because, once in the condensate, they are in their ground states withlittle or no velocity, and hence non-relativistic for all practical purposes. Solving these coupled set ofequations, we obtain the density function, the potential outside the condensate and also the velocityprofiles of the rotational curves. We then compare these analytical results with observational curves for12 dwarf galaxies and show that they agree with a high degree of confidence for five of them. For theremaining galaxies, no definitive conclusion can be drawn with a high confidence level. Nevertheless,our work provides the necessary groundwork and motivation to study the problem further to providestrong evidence for or against our model.This paper is organized as follows. In the next section, we set the stage by summarizing thecoupled differential equations that govern the BEC wavefunction and gravitational potential and findthe BEC density profiles. In Section 3, we construct the corresponding analytical rotation curves. InSection 4, we compare these and the rotational curves due to baryonic matter with the observationalcurves for galaxies. In Section 5, we find most probable bounds on the graviton mass, as well as derivelimits for a velocity-type and a density-type quantity characterizing the BEC.
2. Self-Gravitating, Spherically Symmetric Bec Distribution in the Thomas-Fermi Approximation
A non-relativistic Bose–Einstein condensate in the mean-field approximation is characterized bythe wave function ψ ( r , t ) obeying i ¯ h ∂∂ t ψ ( r , t ) = (cid:34) − ¯ h m ∆ + mV ext ( r ) + λρ ( r , t ) (cid:35) ψ ( r , t ) , (1) ymmetry , , 520 3 of 13 known as the Gross–Pitaevskii equation [37–39]. Here, ¯ h is the reduced Planck constant, r is theposition vector; t is the time; ∆ is the Laplacian; m is the boson mass; ρ ( r , t ) = | ψ ( r , t ) | (2)is the probability density; the parameter λ > V sel f = λδ (cid:0) r − r (cid:48) (cid:1) ; (3)and finally V ext ( r ) is an external potential. For a stationary state, ψ ( r , t ) = (cid:113) ρ ( r ) exp (cid:18) i µ ¯ h t (cid:19) (4)where µ is a chemical potential energy [40,41]. When µ is constant, Equation (1) reduces to presentworks [22], [30] mV ext + V Q + λρ = µ , (5)where V Q is the quantum correction potential energy: V Q = − ¯ h m ∆ √ ρ √ ρ . (6)We mention that Equation (5) is valid in the domain where ρ ( r ) (cid:54) = V Q has significant contribution only close to the BEC boundary [21],therefore it can be neglected in comparison to the self-interaction term λρ . This Thomas–Fermiapproximation becomes increasingly accurate with an increasing number of particles [42].We assume V ext ( r ) to be the gravitational potential created by the condensate. In the case ofmassive gravitons, it is described by the Yukawa-potential in the non-relativistic limit: V ext = U Y ( r ) = − (cid:90) G ρ BEC ( r (cid:48) ) | r − r (cid:48) | e − | r − r (cid:48) | Rg d r (cid:48) , (7)with ρ BEC = m ρ , gravitational constant G , and characteristic range of the force R g carried by thegravitons with mass m g . The relation between R g and m g is R g = ¯ h / (cid:0) m g c (cid:1) , where c is the speed oflight and ¯ h is the reduced Planck constant. The Yukawa potential obeys the following equation: ∆ U Y − U Y R g = π G ρ BEC . (8)Contrary to Equation (5), Equation (8) is also valid in the domain where ρ ( r ) =
0. In the masslessgraviton limit, we recover Newtonian gravity, in particular Equations (7) and (8) reduce to theNewtonian potential and Poisson equation.
The Laplacian of Equation (5) using Equation (8) gives ∆ ρ BEC + π Gm λ ρ BEC = − m λ R g U Y . (9) ymmetry , , 520 4 of 13 For a spherical symmetric matter distribution, Equations (8) and (9) become d ( rU Y ) dr − R g ( rU Y ) = π G ( r ρ BEC ) , (10) d ( r ρ BEC ) dr + R ∗ ( r ρ BEC ) = − m λ R g ( rU Y ) . (11)where we introduced the notation 1 R ∗ = π Gm λ . (12)This system gives the following fourth order, homogeneous, linear differential equation for r ρ BEC : d ( r ρ BEC ) dr + Λ d ( r ρ BEC ) dr = Λ = (cid:115) R ∗ − R g . (14)In the case of massless gravitons, π R ∗ gives the radius of the BEC halo [30]. To have a real Λ , R g > R ∗ should hold, constraining the graviton mass from above. Typical dark matter halos have π R ∗ of the order of 1 kpc which gives the following upper bound for the graviton mass: m g c < × − eV.Then, the general solution of Equation (13) is r ρ BEC = A sin ( Λ r ) + B cos ( Λ r ) + C r + D . (15)with integration constants A , B , C and D . This is why we impose the reality of Λ . For the imaginarycase the general solution would contain runaway hyperbolic functions. This is also the solution ofthe system in Equations (10)–(11). Requiring ρ BEC to be bounded, we have D = − B . Then, the coredensity of the condensate is 0 < ρ ( c ) ≡ ρ BEC ( r = ) = A Λ + C , (16)and the solution can be written as ρ BEC ( r ) = (cid:16) ρ ( c ) − C (cid:17) sin ( Λ r ) Λ r + B cos ( Λ r ) − r + C . (17)Substituting ρ BEC ( r ) in Equation (11), the gravitational potential is − m λ R g ( rU Y ) = (cid:16) ρ ( c ) − C (cid:17) sin ( Λ r ) Λ R g + B R g cos ( Λ r ) − B R ∗ + C R ∗ r . (18)Being related to the mass density by Equation (5) gives B = C = − m µλ R g Λ . (19)The BEC mass distribution ends at some radial distance R BEC (above which we set ρ BEC to zero),allowing to express C in terms of ρ ( c ) , R BEC and Λ as C = ρ ( c ) sin ( Λ R BEC ) Λ R BEC (cid:18) sin ( Λ R BEC ) Λ R BEC − (cid:19) − . (20) ymmetry , , 520 5 of 13 Finally, we consider the massless graviton limiting case m g →
0. Then, R g → ∞ implies Λ = √ π Gm / λ = R ∗ and C = ρ BEC ( r ) coincides with Equation (40) [22]. The potential U is determined up to an arbitrary constant A , i.e. U out = U outY + A . (21)Here, U outY satisfies Equation (8) with ρ BEC =
0. The solution for U outY is U outY = B e − rRg r + C e rRg r . (22)Since an exponentially growing gravitational potential is non-physical, C = U out = A + B e − rRg r . (23)The constants A and B are determined from the junction conditions: the potential is bothcontinuous and continuously differentiable at r = R BEC : A = π G ρ ( c ) + R BEC R g R ∗ R g − sin ( Λ R BEC ) Λ R BEC (cid:20) Λ R g sin ( Λ R BEC ) R ∗ sin ( Λ R BEC ) Λ R BEC − cos ( Λ R BEC ) R g (cid:35) , (24) B = π G ρ ( c ) R BEC + R g R ∗ − sin ( Λ R BEC ) Λ R BEC (cid:20) cos ( Λ R BEC ) − sin ( Λ R BEC ) Λ R BEC (cid:21) e RBECRg . (25)In the next section, we see that the continuous differentiability of the gravitational potentialcoincides with the continuity of the rotation curves.
3. Rotation Curves in Case of Massive Gravitons
Newton’s equation of motions give the velocity squared of stars in circular orbit in the plane ofthe galaxy as v ( R ) = R ∂ U ∂ R . (26)Here, R is the radial coordinate in the galaxy’s plane and U is the gravitational potential. In thecase of massive gravitons, U is given by U = U Y + A , where U Y satisfies the Yukawa-equation withthe relevant mass density and A is a constant.The contribution of the condensate to the circular velocity is v BEC ( R ) = π G ρ ( c ) R ∗ − sin ( Λ R BEC ) Λ R BEC (cid:20) sin ( Λ R ) Λ R − cos ( Λ R ) (cid:21) (27)for r ≤ R BEC and v BEC ( R ) = − B (cid:18) R + R g (cid:19) e − RRg (28)for r ≥ R BEC . ymmetry , , 520 6 of 13 In the relevant situations, the stars orbit inside the halo and their rotation curves are determinedby the parameters: ρ ( c ) R ∗ , R BEC and Λ . In the limit m g →
0, the v of the BEC with massless gravitonsis recovered, given as Böhmer proposed [22] v BEC ( R ) = π G ρ ( c ) R ∗ (cid:20) sin ( R − ∗ R ) R − ∗ R − cos ( R − ∗ R ) (cid:21) (29)for r ≤ R BEC and v BEC ( R ) = G ρ ( c ) R ∗ R (30)for r ≥ R BEC .
4. Best-Fit Rotational Curves
The baryonic rotational curves are derived from the distribution of the luminous matter, given bythe surface brightness S = F / ∆Ω (radiative flux F per solid angle ∆Ω measured in radian squaredof the image) of the galaxy. The observed S depends on the redshift as 1/ ( + z ) , on the orientationof the galaxy rotational axis with respect to the line of sight of the observer, but independent fromthe curvature index of Friedmann universe. Since we investigate dwarf galaxies at small redshift( z < z -dependence of S is negligible. Instead of S given in units of solar luminosity L (cid:12) per square kiloparsec ( L (cid:12) / kpc ), the quantity µ given in units of mag / arcsec can be employed,defined through S ( R ) = × × ( M (cid:12) − µ ( R )) , (31)where R is the distance measured the center of the galaxy in the galaxy plane and M (cid:12) is the absolutebrightness of the Sun in units of mag . The absolute magnitude gives the luminosity of an object, on alogarithmic scale. It is defined to be equal to the apparent magnitude appearing from a distance of10 parsecs. The bolometric absolute magnitude of a celestial object M (cid:63) , which takes into account theelectromagnetic radiation on all wavelengths, is defined as M (cid:63) − M (cid:12) = − ( L (cid:63) / L (cid:12) ) , where L (cid:63) and L (cid:12) are the luminosity of the object and of the Sun, respectively.The brightness profile of the galaxies µ ( R ) was derived in some works [43–45] from isophotal fits,employing the orientation parameters of the galaxies (center, inclination angle and ellipticity). Thisanalysis leads to µ ( R ) which would be seen if the galaxy rotational axis was parallel to the line-of-sight.We used this µ ( R ) to generate S ( R ) .The surface photometry of the dwarf galaxies are consistent with modeling their baryoniccomponent as an axisymmetric exponential disk with surface brightness [46]: S ( R ) = S exp [ − R / b ] (32)where b is the scale length of the exponential disk, and S is the central surface brightness. To convertthis to mass density profiles, we fitted the mass-to-light ratio ( Υ = M / L ) of the galaxies.In Newtonian gravity, the rotational velocity squared of an exponential disk emerges as Freemanproposed [46]: v ( R ) = π GS Υ b (cid:18) Rb (cid:19) ( I K − I K ) , (33) ymmetry , , 520 7 of 13 with I and K the modified Bessel functions, evaluated at R /2 b . In Yukawa gravity, a more cumbersomeexpression has been given in the work of De Araujo and Miranda [47] as v ( R ) = π GS Υ R × (cid:34) (cid:90) ∞ b / λ √ x − b / λ ( + x ) J (cid:18) Rb (cid:112) x − b / λ (cid:19) dx − (cid:90) b / λ √ b / λ − x ( + x ) I (cid:18) Rb (cid:112) b / λ − x (cid:19) dx (cid:35) , (34)where λ = h / m g / c = π R g is the Compton wavelength. For b / λ (cid:28)
1, the Newtonian limitis recovered.
We chose 12 late-type dwarf galaxies from the Westerbork HI survey of spiral and irregulargalaxies [43–45] to test rotation curve models. The selection criterion was that these disk-like galaxieshave the longest R -band surface photometry profiles and rotation curves. For the absolute R -magnitudeof the Sun, M (cid:12) , R = m [48] was adopted. Then, we fitted Equation (32) to the surface luminosityprofile of the galaxies, calculated with Equation (31) from µ ( R ) . The best-fit parameters describing thephotometric profile of the dwarf galaxies are given in Table 1.We derived the pure baryonic rotational curves by fitting the square root of Equation (33) tothe observed rotational curves allowing for variable M / L . The pure baryonic model leads to best-fitmodel-rotation curves above 5 σ significance level for all galaxies (the χ -s are presented in the firstgroup of columns in Table 1), hence a dark matter component is clearly required.Then, we fitted theoretical rotation curves with contributions of baryonic matter and BEC-typedark matter with massless gravitons to the observed rotational curves in Newtonian gravity. Themodel–rotational velocity of the galaxies in this case is given by the square root of the sum of velocitysquares given by Equations (29) and (33) with free parameters Υ , ρ ( c ) and R ∗ . The best-fit parametersare given in the second group of columns of Table 1. Adding the contribution of a BEC-type darkmatter component with zero-mass gravitons to rotational velocity significantly improves the χ for allgalaxies, as well as results in smaller values of M/L. The fits are within 1 σ significance level in fivecases (UGC3851, UGC6446, UGC7125, UGC7278, and UGC12060), between 1 σ and 2 σ in three cases(UGC3711, UGC4499, and UGC7603), between 2 σ and 3 σ in one case (UGC8490), between 3 σ and 4 σ in one case (UGC5986) and above 5 σ in two cases (UGC1281 and UGC5721). We note that the bumpycharacteristic of the BEC model results in the limitation of the model in some cases, the decreasingbranch of the theoretical rotation curve of the BEC component being unable to follow the observedplateau of the galaxies (UGC5721, UGC5986, and UGC8490). The theoretical rotation curves composedof a baryonic component plus BEC-type dark matter component with massless gravitons are presentedon Figure 1. ymmetry , , 520 8 of 13 H kpc L v r o t I k m s M UGC12060 H kpc L v r o t I k m s M UGC7278 H kpc L v r o t I k m s M UGC6446 H kpc L v r o t I k m s M UGC3851 H kpc L v r o t I k m s M UGC7125 H kpc L v r o t I k m s M UGC3711 H kpc L v r o t I k m s M UGC4499 H kpc L v r o t I k m s M UGC7603 H kpc L v r o t I k m s M UGC8490 H kpc L v r o t I k m s M UGC5986 H kpc L v r o t I k m s M UGC1281 H kpc L v r o t I k m s M UGC5721
Figure 1.
Theoretical rotational curves of the dwarf galaxy sample. The dots with error-bars denotearchive rotational velocity curves. The model rotation curves are denoted as follows: pure baryonicin Newtonian gravitation with dotted line, baryonic + BEC with massless gravitons in Newtoniangravitation with dashed line, and baryonic + BEC with the upper limit on m g in Yukawa gravitationwith continuous line. ymmetry , , 520 9 of 13 Table 1.
Parameters describing the theoretical rotational curve models of the 12 dwarf galaxies. Best-fit parameters of the pure baryonic model in the first groupof columns: central surface brightness S , scale parameter b , M / L ratio Υ , along with the χ of the fit. This model results in best-fit model-rotation curves above5 σ significance level for all galaxies. Best-fit parameters of the baryonic matter + BEC with massless gravitons appear in the second group of columns: M / L ratio Υ , characteristic density ρ ( c ) , distance parameter R ∗ , along with the χ of the fit and the respective significance levels. Constraints on the parameter m / λ are alsoderived. In five cases, the fits χ are within 1 σ and marked as boldface. The fits are between 1 σ and 2 σ in three cases, between 2 σ and 3 σ in one case, between 3 σ and4 σ in one case and above 5 σ in two cases. Best-fit parameters of the baryonic matter + BEC with massive gravitons are given in the third group of columns only for thewell-fitting galaxies: the range for R BEC and the upper limit on m g are those for which the fit remains within 1 σ . Corresponding constraints on the parameter m / µ arealso derived. Pure Baryonic Baryonic + BEC with m g = m g > ID S b Υ χ Υ ρ ( c ) R ∗ m λ χ sign. lev. R BEC m g m µ sign. lev.10 L (cid:12) kpc kpc M (cid:12) kpc kpc − kgs m kpc − eVc − s m UGC12060 0.7 0.90 11.23 155 5.50 ± ± ± ± σ = ÷ < < σ = ± ± ± ± σ = ÷ < < σ = ± ± ± ± σ = ÷ < < σ = ± ± ± ± σ = ÷ < < σ = ± ± ± ± σ = ÷ < < σ = σ = σ = σ = σ = σ = σ = σ = ymmetry , , 520 10 of 13 We attempted to distinguish among galaxies to be included in well-fitting or less well-fittingclasses based on their baryonic matter distribution. Several factors affect the goodness of the fits, asfollows. The best-fit falls outside the 1 σ significance level in the case of seven galaxies. Among thesegalaxies, UGC8490 and UGC5721 have ( a ) steeply rising rotational curve due to their centralizedbaryonic matter distribution ( b < v max > kms − ) with ( a ) long, approximately constantheight observed plateau. Joint fulfilment of these criteria does not occur for the well-fitting galaxies,as b (cid:38) σ significancelevel have ( b ) slowly rising rotational curve due to their less centralized baryonic matter distribution( b > v max < kms − ) with ( b ) short, variable height observed plateau, holding relativelysmall number of observational points ( N ≤
15, a small N lowers the 1 σ significance level). Thewell-fitting galaxies do not belong to this group, as either they hold more observational points, orhave a longer, approximately constant height observed plateau. We expect that for the galaxies notfalling in the classes with baryonic and observational characteristics summarized by either properties( a )–( a ) or ( b )–( b ) the BEC dark matter model represents a good fit. Finally, we note the galaxyUGC3711 represents a special case due to the lack of sufficient observational data. Although the shapeof its rotational curve is very similar to that of the best-fitting galaxy, UGC12060, it is based on just sixobservational points, lowering the 1 σ level. Its points also have smaller error bars, which increases the χ . This results in the best-fit rotational curve of UGC3711 falling outside out the 1 σ significance level.Finally, we fitted the theoretical rotational curves given by both a baryonic component and anon-relativistic BEC component with massive gravitons, employing Yukawa gravity. The parameters Υ , ρ ( c ) and R ∗ were kept from the best-fit galaxy models composed of baryonic matter + BEC withmassless gravitons. The model–rotational velocity of the galaxies arises as the square root of the sumof velocity squares given by Equations (27) and (34) with free parameters R BEC and R g . Adding massto the gravitons in the BEC model leads to similar performances of the fits.
5. Discussion and Concluding Remarks
We estimated the upper limit on the graviton mass, employing first the theoretical condition ofthe existence of the constant Λ , then analyzing the modelfit results of those five dwarf galaxies forwhich the fit of the BEC model with massive gravitons to data was within 1 σ significance level.Keeping the best-fit parameters ρ ( c ) , R ∗ , we varied the value of R BEC and R g and calculated the χ between model and data. The upper limit on the graviton mass m g has been estimated from thevalues of R g , for which χ = σ has been reached. The results are given in Table 1. We plotted thetheoretical rotation curves given by a baryonic plus a non-relativistic BEC component with massivegravitons with limiting mass in Figure 1. As shown in Table 2, the fit with the rotation curve data hasimproved the limit on the graviton mass in all cases. Table 2.
Constraints for both the upper limit for the mass of the graviton (first from the existence of Λ ,second from the rotation curves) and for the velocity-type and density-type BEC parameters (related tothe mass of the BEC particle, scattering length and chemical potential) in the case of the five well-fittinggalaxies. ID m g ( Λ ∈ IR ) m g ¯ v BEC ¯ ρ BEC − eVc − eVc ms M (cid:12) kpc UGC12060 < < < < < < < < < < Comparing the theoretical rotation curves derived in our model with the observational ones,we found the upper limit to the graviton mass to be of the order of 10 − eV/c . We also note that ymmetry , , 520 11 of 13 the constraint on the graviton mass imposed from the dispersion relations tested by the first threeobservations of gravitational waves, 7.7 × − eV/c [49], is still weaker than the present one.For the BEC, we could derive two accompanying limits: (i) first m / λ has been constrained fromthe corresponding values of R ∗ arising from the fit with the massless gravity model; and then (ii) m / µ has been constrained from the constraints derived for the graviton mass and our previous fits throughEquations (19) and (20). These are related to the bosonic mass, chemical potential and scattering length,but only two combinations of them, a velocity-type quantity¯ v BEC = (cid:114) µ m (35)and a density-type quantity ¯ ρ BEC = m λ ¯ v BEC (36)were restricted, both characterizing the BEC. Their values are also given in Table 2 for the set of fivewell-fitting galaxies.If the BEC consists of massive gravitons with the limiting masses m = m g determined in Table2, the chemical potential µ and the constant characterizing the interparticle interaction λ can bedetermined as presented in Table 3. Table 3.
Constraints on µ and λ assuming m = m g in case of the five well fitting galaxies. ID µ ( m = m g ) λ ( m = m g ) − m s kg − m s kg UGC12060 < < < < < < < < < < With this, we established observational constraints for both the upper limit for the mass of thegraviton and for the BEC.
Author Contributions:
Conceptualization, L.Á.G. and S.D.; Data curation, E.K.; Formal analysis, L.Á.G., E.K. andZ.K.; Funding acquisition, L.Á.G., Z.K. and S.D.; Investigation, E.K.; Methodology, L.Á.G., E.K. and Z.K.; Software,E.K.; Supervision, L.Á.G. and Z.K.; Validation, L.Á.G., Z.K. and S.D.; Visualization, E.K.; Writing—original draft,L.Á.G., E.K., Z.K. and S.D.;Writing—review & editing, L.Á.G.
Funding:
This work was supported by the Hungarian National Research Development and Innovation Office(NKFIH) in the form of the grant 123996 and by the Natural Sciences and Engineering Research Council of Canadaand based upon work from the COST action CA15117 (CANTATA), supported by COST (European Cooperationin Science and Technology). The work of Z.K. was also supported by the János Bolyai Research Scholarship of theHungarian Academy of Sciences and by the UNKP-18-4 New National Excellence Program of the Ministry ofHuman Capacities.
Acknowledgments:
This work was supported by the Hungarian National Research Development and InnovationOffice (NKFIH) in the form of the grant 123996 and by the Natural Sciences and Engineering Research Councilof Canada and based upon work from the COST action CA15117 (CANTATA), supported by COST (EuropeanCooperation in Science and Technology). The work of Z.K. was also supported by the János Bolyai ResearchScholarship of the Hungarian Academy of Sciences and by the UNKP-18-4 New National Excellence Program ofthe Ministry of Human Capacities.
Conflicts of Interest:
The founding sponsors had no role in the design of the study; in the collection, analyses, orinterpretation of data; in the writing of the manuscript, or in the decision to publish the results. ymmetry , , 520 12 of 13 References
1. Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.;Goobar, A.; Groom, D.E.; Hook, I.M.; Kim, A.G.; Kim, M.Y.; Lee, J.C.; Nunes, N.J.; Pain, R.; Pennypacker,C.R.; Quimby, R.; Lidman, C.; Ellis, R.S.; Irwin, M.; McMahon, R.G.; Ruiz-Lapuente, P.; Walton, N.; Schaefer,B.; Boyle, B.J.; Filippenko, A.V.; Matheson, T.; Fruchter, A.S.; Panagia, N.; Newberg, H.J.M.; Couch, W.J.;Project, T.S.C. et al. Measurements of Ω and Λ from 42 high-redshift supernovae. Astrophys. J. , , 565–586.2. Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiatti, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan,C.J.; Jha, S.; Kirshner, R.P.; Leibundgut, B.; Phillips, M.M.; Reiss, D.; Schmidt, B.P.; Schommer, R.A.; Smith,R.C.; Spyromilio, J.; Stubbs, C.; Suntzeff, N.B.; Tonry, J. et al. Observational evidence from supernovae foran accelerating universe and a cosmological constant. Astrophys. J. , , 1009–1038.3. Young, B.L. A survey of dark matter and related topics in cosmology. Front. Phys. , , 121201.4. Plehn, T. Yet another introduction to Dark Matter. arXiv , arXiv:1705.01987.5. Copeland, E.J.; Sami, M.; Tsujikawa, S. Dynamics of Dark Energy. Int. J. Mod. Phys. , , 1753–1935.6. Frieman, J.A.; Turner, M.S.; Huterer, D. Dark Energy and the accelerating universe. ARAA , , 385–432.7. Gergely, L. Á.; Tsujikawa, S. Effective field theory of modified gravity with two scalar fields: Dark energyand dark matter. Phys. Rev. Lett. , , 064059.8. Gergely, L.Á. Friedmann branes with variable tension. Phys. Rev. Lett. , , 0084006.9. Kleidis, K.; Spyrou, N.K. Polytropic dark matter flows illuminate dark energy and accelerated expansion. A&A , , 23.10. Kleidis, K.; Spyrou, N. Dark Energy: The Shadowy Reflection of Dark Matter? Entropy , , 94,11. Das, S.; Bhaduri, R.K. Dark matter and dark energy from a Bose-Einstein condensate. Class. Quantum Gravity , , 105003.12. Das, S.; Bhaduri, R.K. Bose-Einstein condensate in cosmology. arXiv , arXiv:1808.10505.13. Hu, W.; Barkana, R.; Gruzinov, A. Fuzzy cold dark matter: the wave properties of ultralight particles. Phys.Rev. Lett. , , 1158–1161,14. Ureña-López, L.A. Bose-Einstein condensation of relativistic Scalar Field Dark Matter. J. Cosmol. Astropart.Phys. , , 014.15. Sinha, K.P.; Sivaram, C.; Sudarshan, E.C.G. Aether as a superfluid state of particle-antiparticle pairs. Found.Phys. , , 65–70.16. Sinha, K.P.; Sivaram, C.; Sudarshan, E.C.G. The superfluid vacuum state, time-varying cosmological constant,and nonsingular cosmological models. Found. Phys. , , 717–726.17. Bohua Li, M.A. Cosmology with Bose-Einstein-Condensed Scalar Field Dark Matter. Master of Arts, TheUniversity of Texas at Austin, Austin, Texas, USA, 2013.18. Morikawa, M. 22nd Texas Symp. on Relativistic Astrophysics at Stanford University , .19. Fukuyama, T.; Morikawa, M. The relativistic gross-pitaevskii equation and cosmological bose-einsteincondensation quantum structure in the universe. Prog. Theor. Phys. , , 1047–1068.20. Moffat, J.W. Spectrum of cosmic microwave fluctuations and the formation of galaxies in a modified gravitytheory. arXiv , arXiv:astro-ph/0602607.21. Wang, X.Z. Cold Bose stars: Self-gravitating Bose-Einstein condensates. Phys. Rev. D , , 124009.22. Böhmer, C.G.; Harko, T. Can dark matter be a Bose Einstein condensate? J. Cosmol. Astropart. Phys. , , 025.23. Harko, T.; Mocanu, G. Cosmological evolution of finite temperature Bose-Einstein condensate dark matter. Phys. Rev. D , , 084012.24. Sikivie, P. Dark Matter Axions. Int. J. Mod. Phys. A , , 554–563.25. Dvali, G.; Gomez, C. Black Hole’s Quantum N-Portrait. arXiv , arXiv:1112.3359.26. Chavanis, P.H. Growth of perturbations in an expanding universe with Bose-Einstein condensate darkmatter. A&A , , A127.27. Kain, B.; Ling, H.Y. Cosmological inhomogeneities with Bose-Einstein condensate dark matter. Phys. Rev. D , , 023527.28. Suárez, A.; Robles, V.H.; Matos, T. A review on the scalar field/Bose-Einstein condensate dark matter model. Accel. Cosmic Expans. , ,
38, 107-142. ymmetry , , 520 13 of 13
29. Ebadi, Z.; Mirza, B.; Mohammadzadeh, H. Infinite statistics condensate as a model of dark matter.
J. Cosmol.Astropart. Phys. , , 057.30. Dwornik, M.; Keresztes, Z.; Gergely, L.A. Rotation curves in Bose-Einstein Condensate Dark Matter Halos, in Recent Development in Dark Matter Research (N. Kinjo and A. Nakajima, eds.), Nova Science Publishers, ,pp. 195–219.31. Bettoni, D.; Colombo, M.; Liberati, S. Dark matter as a Bose-Einstein Condensate: The relativistic non-minimallycoupled case.
J. Cosmol. Astropart. Phys. , , 004.32. Gielen, S. Quantum cosmology of (loop) quantum gravity condensates: An example. Classi. Quantum Gravity , , 155009.33. Schive, H.Y.; Chiueh, T.; Broadhurst, T. Cosmic structure as the quantum interference of a coherent dark wave. Nat.Phys. , , 496–499.34. Davidson, S. Axions: Bose Einstein condensate or classical field? Astropart. Phys. , , 101–107.35. Ali, A.F.; Das, S. Stringent theoretical and experimental bounds on graviton mass. Int. J. Mod. Phys. D , , 1644001.36. Dwornik, M.; Keresztes, Z.; Kun, E.; Gergely, L.A. Bose-Einstein condensate Dark Matter halos confronted withfalactic rotation curves. Adv. High Energy Phys. , , 14.37. Gross, E.P. Structure of a quantized vortex in boson systems. Nuovo Cim. , , 454.38. Gross, E.P. Hydrodynamics of a superfluid condensate. J. Math. Phys. , , 195.39. Pitaevskii, L.P. Vortex lines in an imperfect bose gas. Sov. Phys. JETP , , 451.40. Rogel-Salazar, J. The Gross-Pitaevskii equation and Bose-Einstein condensates. Eur. J. Phys. , , 247.41. Giorgini, S.; Pitaevskii, L.P.; Stringari, S. Thermodynamics of a trapped Bose-condensed gas. J. Low Temp. Phys. , , 309.42. Lieb, E.H.; Seiringer, R.; Yngvason, J. Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional. arXiv , arXiv:math-ph/9908027.43. Swaters, R.A. Dark Matter in Late-type Dwarf Galaxies. Ph.D., University of Groningen, Location of University,Groningen, Netherlands, 1999.44. Swaters, R.A.; Balcells, M. The Westerbork HI survey of spiral and irregular galaxies. II. R-band surface photometryof late-type dwarf galaxies. A&A , , 863–878,45. Swaters, R.A.; Sancisi, R.; van Albada, T.S.; van der Hulst, J.M. The rotation curves shapes of late-type dwarf galaxies. A&A , , 871–892,46. Freeman, K.C. On the Disks of Spiral and S0 Galaxies. Astrophys. J. , , 811. doi:10.1086/150474.47. De Araujo, J.C.N.; Miranda, O.D. A solution for galactic disks with Yukawian gravitational potential. Gen. Relativ.Gravit. , , 777–784.48. Binney, J.; Merrifield, M. Galactic Astronomy, Princeton University Press, 1998.49. Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari,R.X.; Adya, V.B.; et al. GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift0.2.
Phys. Rev. Lett. , , 221101.c (cid:13)(cid:13)