Recently, ultra diffuse galaxy (UDG) of Dragonfly 44 in the Coma Cluster was
observed and observations of the rotational speed suggest that its mass is
almost same as the mass of the Milky Way. On the other hand, interestingly, the
galaxy emits only 1 \% of the light emitted by the Milky Way. Then, astronomers
reported that Dragonfly 44 may be made almost entirely of dark matter. In this
study we try to show that the dark matter that constitutes Dragonfly 44 can
form the wormhole or not. Two possible dark matter profiles are used, namely,
ultra diffuse galaxy King's model and generalized Navarro-Frenk-White (NFW)
dark matter profile. We have shown that King's model dark matter profile does
not manage to provide wormhole whereas generalized Navarro-Frenk-White (NFW)
dark matter profile is managed to find wormholes.Comment: 8 pages, two columns. Accepted for publication in Canadian Journal of
  Physic

Halilsoy, Mustafa

Islam, Sayeedul

Rahaman, Farook

Övgün, Ali

English

arXiv

Recently, ultra diffuse galaxy (UDG) of Dragonfly 44 in the Coma Cluster was observed and observations of the rotational speed suggest that its mass is almost same as the mass of the Milky Way. On the other hand, interestingly, the galaxy emits only 1 % of the light emitted by the Milky Way. Then, astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. In this study we try to show that the dark matter that constitutes  Dragonfly 44 can form the wormhole or not. Two possible dark matter profiles are used, namely, ultra diffuse galaxy King's model and generalized Navarro-Frenk-White (NFW) dark matter profile. We have shown that King's model dark matter profile does not manage to provide wormhole whereas generalized Navarro-Frenk-White (NFW) dark matter profile is managed to find wormholes.The presentation of the authors' names and (or) special characters in the title of the pdf file of the accepted manuscript may differ slightly from what is displayed on the item page. The information in the pdf file of the accepted manuscript reflects the original submission by the author

TSpace (University of Toronto)

For Review Only      Formation of Wormholes by Dark Matter in the Galaxy Dragonfly 44   Journal: Canadian Journal of Physics Manuscript ID cjp-2017-0864.R1 Manuscript Type: Article Date Submitted by the Author: 29-Dec-2017 Complete List of Authors: Islam, Sayeedul; Jadavpur University, Department of Mathematics Rahaman, Farook; Jadavpur University Övgün, Ali; Instituto de F\'{\i}sica, Pontificia Universidad Cat\'olica de Valpara\'{\i}so, Casilla 4950, Valpara\'{\i}so, Chile; Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey; TH Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland Halilsoy, Mustafa; Eastern Mediterranean University, Physics Department, Arts and Science Faculty Keyword: Wormholes;, Dark matter;, Ultra Diffuse Galaxy;, Dragonfly 44;, NFWs Dark Matter profile Is the invited manuscript for consideration in a Special Issue? : Not applicable (regular submission)     https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review OnlyFormation of Wormholes by Dark Matter in the Galaxy Dragonfly 44Sayeedul Islam∗ and Farook Rahaman†Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, IndiaAli Övgün‡Instituto de F́ısica, Pontificia Universidad Católica de Valparáıso, Casilla 4950, Valparáıso, ChilePhysics Department, Arts and Sciences Faculty, Eastern Mediterranean University,Famagusta, North Cyprus via Mersin 10, Turkey andTH Division, Physics Department, CERN, CH-1211 Geneva 23, SwitzerlandMustafa Halilsoy§Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University,Famagusta, North Cyprus via Mersin 10, Turkey(Dated: December 29, 2017)Recently, ultra diffuse galaxy (UDG) of Dragonfly 44 in the Coma Cluster was observed andobservations of the rotational speed suggest that its mass is almost same as the mass of the MilkyWay. On the other hand, interestingly, the galaxy emits only 1 % of the light emitted by the MilkyWay. Then, astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. Inthis study we try to show that the dark matter that constitutes Dragonfly 44 can form the wormholeor not. Two possible dark matter profiles are used, namely, ultra diffuse galaxy King’s model andgeneralized Navarro-Frenk-White (NFW) dark matter profile. We have shown that King’s modeldark matter profile does not manage to provide wormhole whereas generalized Navarro-Frenk-White(NFW) dark matter profile is managed to find wormholes.Keywords: Wormholes; Dark matter; Ultra Diffuse Galaxy; Dragonfly 44; NFWs Dark Matter profileI. INTRODUCTIONToday, one of the challenging questions in a theoreti-cal physics is the question of the existence of traversablewormholes [1–21]. Other mysteries in physics is darkmatter (DM) [22–24]. Like black holes there is anothermiraculous objects of our universe which are wormholes.After the prediction of Einstein-Rosen bridge in 1935 [1],with a lot of theoretical evidence researchers have pro-posed the existence of wormholes in space-time whichacts like a shortcut path to travel between any two widelyseparated or infinite region of the universe or between an-other universes in multi universe model. Structurally itlooks like a tunnel (called its throat) with two mouths(most likely spheroidal). Here most interesting thing isexotic matter needed to open its throat (violates nullenergy conditions [2–4]. The curiosity about wormholesphysics has vigorously been increased since publicationof Morris and Thorne’s research article where they pro-posed the prospect of the existence of traversable worm-holes, as a solution of Einstein’s field equations, that doesnot contain event horizon and traveler can easily move inboth the regions in space-time through its straight stretchthroat [2, 3]. By the existence of these hypothetical ob-jects one can realize time machines or shortcut among∗Electronic address: sayeedul.jumath@gmail.com†Electronic address: rahaman@associates.iucaa.in‡Electronic address: ali.ovgun@pucv.cl§Electronic address: mustafa.halilsoy@emu.edu.trfaraway places of space.Maximum mass of the universe is made up by the mostmysterious substance which do not interact with the elec-tromagnetic force, i.e. can’t absorb, reflect or emit lightare dark matter whose nature and composition remainoverall question mark [22, 23]. Researchers hypothesizethe existence of dark matter only from the gravitationaleffect. Observation of the spiral galaxy rotation curvesis an enthralling experimental evidence for the existenceof dark matter. The dark matter candidate particles aregenerally classified into cold, warm and hot categories.Interaction of a scalar field whose energy density is darkenergy may be caused of the dark matter particle mass.Over several decades it is openly accepted that almostevery galaxy contain a large amount of non luminousmatter forming massive dark matter halos around thegalaxy based on different lines of evidence like flat ro-tation curves of spiral galaxies [24] and strong lensingsystem [25].The model under consideration predicts that the ultradiffuse galaxies (UDGs) space distribution for 90 globularclusters observed around Dragonfly 44 [27] has the spacedensity:a. King’s model like [28, 29]:ρ(r) ∝ κ(rr02+ λ)η, (1)where η , κ, r0 and λ are parameters . We assume thatthe averaged relative speed of the galaxies in the Comacluster is approximately equal to the velocity dispersionin the cluster v ∼= 1000 km/s. This density profile isPage 1 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review Only2shown in fig 1.b. Generalized Navarro-Frenk-White (NFW) profile [26]:ρ(r) ∝1rγ[1 +(rr0)]3−γ , (2)where γ and r0 are parameters .FIG. 1: Plot ρ versus r ( κ = 1,λ = 1 and r0 = 1 ).This paper is organized as follows. Sec. II we intro-duces the traversable wormholes and their basic equa-tions. In Sec. III and IV, we study the possibility ofwormholes in the galaxy of Dragonfly 44 by using theUDGs dark matter profile [27–29] and NFWs dark mat-ter profile [26],respectively. Finally, in Sec. V, we presentour remarks.II. WORMHOLES FORMULATIONThe traversable wormhole space-time is given by fol-lowing [2, 3]ds2 = −e2f(r)dt2+(1−b(r)r)−1dr2+r2(dθ2+sin2 θ dφ2).(3)It is noted that the b(r) and f(r) stands for the spa-tial shape function, and for the redshift function, respec-tively. The range of the radial coordinate is from +∞to b(r0) = r0 where r0, is the minimum value of the r(throat of the wormhole).The Einstein field equations are given byGµν = 8πTµν (4)where T µν and Gµν are the stress-energy tensors and theEinstein tensor, respectively. Then one can calculate thenon-zero Einstein tensors:Gtt =b′r2, (5)Grr =−br3+ 2(1−br)f ′r, (6)Gθθ =(1−br)[f ′′ + f ′2+f ′r−(f ′ +1r){b′r − b2r(r − b)}],(7)Gφφ = Gθθ (8)where a prime is ddr .DM is generally defined in the form of generalanisotropic energy-momentum tensorT µν = (ρ+ pr)uµuν + prgµν + (pt − pr)ηµην , (9)where uµuµ = −12ηµηµ = −1. Note that pt stands forthe transverse pressure, pr is the radial pressure and ρ isthe energy density. A possible set of uµ and ηµ are givenby uµ = (e2f(r), 0, 0, 0) and ηµ = (0, 0, 1r ,1r sin θ ). Thenthe stress-energy tensors T µν are calculated as followsT tt = −ρ, (10)T rr = pr, (11)T θθ = Tφφ = pt. (12)III. WORMHOLES WITH THE UDG KING’SMODEL DMOne can find the tangential velocity from the flat rota-tion curve for the circular stable geodesic motion in theequatorial plane as [21]vφ =√rf ′, (13)which is responsible to fit the flat rotational curve for theDM. Rahaman et. al observe the rotational curve profilein the DM region as follows [19, 20]vφ = αr exp(−k1r) + β[1− exp(−k2r)] (14)where α, β, k1, and k2 are constant positive parameters.Using the Eqns.(13) and (14), the redshift function isobtained as followsf(r) = −α2r2k1e(2k1r)−α24k21e(2k1r)−2αβk1e(k1r)+2αβe(−k1r−k2r)k1 + k2+ β2 ln(r) + 2β2Ei(1, k2r)−β2Ei(1, 2k2r) +D. (15)Page 2 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review Only3where Ei and D are the exponential integral andintegration constant, respectively. At large scales, itbecomes e2f(r) = B0r(4vφ).Now we discuss two cases for different values of theparameter η by using UDGs king’s dark matter profilefor the stability of wormholes in the galaxy Dragonfly44.A. The case IFor the UDGs King’s DM density profile [27–29]ρ(r) = κ[(rr0)2+ λ]η, (16)we assume in this case κ = 1, λ = 1 and η = −3/2. Afterone uses the Eq.s (13) and (14) and the UDG’s darkmatter density profile under the Einstein field equations,the shape function is calculated (8π = 1) asb(r) =rr208(r2 + r0)+r3/2o8tan−1(r√r0)−r30r4(r2 + r0)2+C(17)Note that C is the integration constant and it is chosenasC = r0−r208(1 + r0)+r3/2o8tan−1(√r0)+r204(1 + r0)2(18)to satisfy the condition of b(r0) = r0, then it is checkedthe flare-out condition (b′ < 1), where r0 is the radius ofthroat.b′ = ρr2 =r30r4(r2 + r0)3(19)which is satisfied (see fig.(2)).Furthermore from the Eq.(10) the second derivative ofb with respect to r is calculated asb′′ = −2r30r(2r2 − r0)(r2 + r0)4. (20)The radial of pressure is showed as follows by substi-tuting f(r) and b(r) into the solution (Eqns. 5- 12):pr =−br3+ 2(1−br)f ′r, (21)f ′ =α2r(ek1 r)2− 2αβ e−r(k1+k2) + 2αβek1 r+β2e−2 k2 rr− 2β2e−k2 rr+β2r. (22)FIG. 2: The figure is shown for b(r)/r versus r.FIG. 3: The figure is shown for b′(r) versus r.It is showed in the figure (4) that the null energy condi-tion (ρ + pr < 0) is violated so that one of the essentialcondition for a wormhole is satisfied. However, the mostimportant criterion for wormhole, namely, b(r)r < 1 isviolated (see fig.2). So in this case wormhole does notexists.B. The case IIIn this section we follow the same method using in caseI and the shape function is calculated .Page 3 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review Only4FIG. 4: Null energy condition for the case I.FIG. 5: The figure is shown for b(r)/r versus r.It is checked the flare-out condition (b′ < 1) is satisfiedand plotted in Fig. (6).It is also showed in the Fig. (7) that the null energycondition (ρ + pr < 0) is violated as needed to hold awormhole. However, as in case I, b(r)r < 1 is violated(seefig.5), so in this case wormhole does not exists. ThusKing’s model dark matter profile that constitutes Drag-onfly 44 does not manage to provide wormhole. Nowwe will perform a study whether generalized NFW darkmatter profile that constitutes Dragonfly 44 does manageto provide wormhole or not.FIG. 6: The figure is shown for b′(r) versus r.FIG. 7: Null energy condition for the case II.IV. WORMHOLES WITH THE NFWS DMIn this section we describe the dark matter halo dis-tribution using the following generalized Navarro-Frenk-White (NFW) profile for the existence of wormholes inthe galaxy Dragonfly 44.ρ(r) =1rγ[1 +(rr0)]3−γ (23)In this expression, γ is the inner slope of the profile(γ = 1 corresponds to the case of a standard NFWPage 4 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review Only5profile), and r0 is the scale radius and taking variationconstant as unity.Here we have discussed three different cases for theseveral values of the parameter γA. case IIn this case we assume r0 = 10, ρ0 = 0.05 and γ = 3.After using NFW dark matter density profile under theEinstein field equations, the shape function b(r) is calcu-lated (8π = 1) asb(r) = 8πρ0 [ln(r) + C] (24)Note that C is the integration constant and it is choosenasC =r08πρ0− lnr0 (25)FIG. 8: The figure is shown for b(r) versus r.Here fig.(8) represents the shape function b(r) and wesee that the null energy condition (ρ+pr < 0) is violated( see fig.10). Also we have checked the most importantflare-out condition (b(r) − r < 0, after the throat radiusi.e. b′ < 1) which is satisfied and is plotted in the fig.(9).Thus the generalized NFW dark matter profile that con-stitutes Dragonfly 44 can manage to provide wormhole.FIG. 9: The figure is shown for b(r)− r versus r.FIG. 10: Null energy condition for the case I.B. case IIIn this case we choose r0 = 10, ρ0 = 0.05 and γ = 4After using NFW dark matter density profile under theEinstein field equations, the shape function b(r) is calcu-lated (8π = 1) asb(r) = 8πρ0[−1r+ln(r)r0+ C](26)Page 5 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review Only6Note that C is the integration constant and it is choosenasC =r08πρ0+1r0−1r0lnr0 (27)In this section we follow the same method used inFIG. 11: The figure is shown for b(r) versus r.FIG. 12: The figure is shown for b(r)− r versus r.case I and the shape function b(r) is plotted in thefig. (11). To form a wormhole, the essential criteriaregarding flare-out condition (b(r) − r < 0, after thethroat radius i.e. b′ < 1) and violation of null energycondition (ρ+pr < 0) are satisfied ( see fig.12 and fig.13 ).FIG. 13: Null energy condition for the case II.C. case IIIIn this case we assume r0 = 10, ρ0 = 0.05 and γ = 5After using NFW dark matter density profile under theEinstein field equations, the shape function b(r) is calcu-lated (8π = 1) asb(r) = 8πρ0[−2rr0+ln(r)r20−12r2+ C](28)Note that C is the integration constant and it is choosenasC =r08πρ0+2r20−1r20lnr0 +12r20(29)Here we use the same NFW dark matter density profilelike above cases and similarly the shape function b(r) iscalculated and plotted in the fig. (14).In the fig. (15) and fig. (16), it is clearly shown thatthe flare-out condition ((b(r) − r < 0, after the throatradius i.e. b′ < 1)) is satisfied and null energy condition(ρ+ pr < 0) is violated to hold a wormhole open.So we can claim that the generalized NFW dark matterprofile that constitutes Dragonfly 44 provides wormhole.Page 6 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review Only7FIG. 14: The figure is shown for b(r) versus r.FIG. 15: The figure is shown for b(r)− r versus r.V. CONCLUSIONThe presence of stable traversable wormholes is a note-worthy issue in theoretical physics. There is doubtlesslythat wormholes is the most interesting objects in uni-verse. This work is persuaded primarily by Ref.s [19–21]. In this paper, we use the UDG systems in the ComaCluster which is known as Dragonfly 44 that astronomersreported that Dragonfly 44 may be made almost entirelyof dark matter. Moreover, there is another possibility: inthis study we show that this dark matter can form thewormhole, and it affects the observations. All the nor-FIG. 16: Null energy condition for the case III.mal matter might be passed through this wormhole. Forthis purpose we firstly use the UDG profile and try toconstruct wormhole solution, then we repeat our calcu-lations for the Navarro-Frenk-White (NFW) profile. Wehave shown that only Navarro-Frenk-White (NFW) pro-file provide wormhole solutions. Thus we able to findthe solutions of the wormhole in the Dragonfly 44 galaxyso that the dark matter‘s halos around the Dragonfly 44galaxy is suitable to harbor wormholes.AcknowledgmentsThis work was supported by the Chilean FONDECYTGrant No. 3170035 (AÖ). AÖ is grateful to the CERNtheory (CERN-TH) division for hospitality where partof this work was done. FR would like to thank the au-thorities of the Inter-University Centre for Astronomyand Astrophysics, Pune, India for providing research fa-cilities. FR and SI are also grateful to DST-SERB andDST-INSPIRE, Govt. of India , for financial supportrespectively.[1] A. Einstein and N. Rosen, Physical Review. 48: 73(1935).[2] M.S. Moris and K.S. Thorne, Am. J. Phys. 56, 395(1988).Page 7 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of PhysicsFor Review Only8[3] M. Morris, K. S. Thorne and U. Yurtsever,Phys.Rev.Lett. 61, 1446 (1988).[4] M. Visser, Lorentzian Wormholes: From Einstein toHawking, (Springer, Berlin, 1997).[5] M. Halilsoy, A. Ovgun, S. Habib Mazharimousavi, Eur.Phys. J. C 74, 2796 (2014).[6] F. Rahaman, M. Kalam and S. Chakraborty, Gen. Rel.Grav. 38, 1687 (2006).[7] M. G. Richarte, I. G. Salako, J. P. Morais Graa,H. Moradpour and A. Övgün, Phys. Rev. D 96, no. 8,084022 (2017).[8] S. Capozziello and M. Francaviglia, Gen. Rel. Grav. 40,357 (2008)[9] M. S. R. Delgaty and R. B. Mann, Int. J. Mod. Phys. D4, 231 (1995)[10] G. P. Perry and R. B. Mann, Gen. Rel. Grav. 24, 305(1992).[11] M. Cataldo and F. Orellana, Phys. Rev. D 96, no. 6,064022 (2017)[12] M. Cataldo, L. Liempi and P. Rodriguez, Phys. Lett. B757, 130 (2016)[13] M. Cataldo, P. Labrana, S. del Campo, J. Crisostomoand P. Salgado, Phys. Rev. D 78, 104006 (2008)[14] K. Jusufi, A. Övgün and A. Banerjee, Phys. Rev. D 96,no. 8, 084036 (2017)[15] I. Sakalli and A. Ovgun, Eur. Phys. J. Plus 130, no. 6,110 (2015).[16] I. Sakalli and A. Ovgun, Astrophys. Space Sci. 359, no.1, 32 (2015).[17] A. Ovgun, Eur. Phys. J. Plus 131, 389 (2016).[18] S. Kar, S. Lahiri, S. SenGupta, Phys. Lett. B. 750, 319-324 (2015).[19] F. Rahaman, P.K.F. Kuhfittig, S. Ray, N. Islam, Eur.Phys. J. C (2014) 74:2750.[20] F. Rahaman, P. Salucci, P.K.F. Kuhfittig, S. Ray, M.Rahaman, Ann. Phys. 350, 561 (2014).[21] A. Ovgun, M. Halilsoy, Astrophys Space Sci (2016)361:214.[22] V. Sahni, Lect. Notes Phys. 653, 141 (2004)[23] J. L. Feng, Ann. Rev. Astron. Astrophys. 48, 495 (2010)[24] T. S. V. Albada, R. Sancisi, M. Petrou and R. J. Tayler,Phil. Trans. Roy. Soc. Lond. A 320, no. 1556, 447 (1986).[25] C. R. Keeton, C. S. Kochanek and E. E. Falco, Astro-phys. J. 509, 561 (1998).[26] J. F. Navarro, C. S. Frenk and S. D. M. White, Astro-phys. J. 462, 563 (1996).[27] van Dokkum et al., Astrophys.J. 828, no.1, L6 (2016).[28] I. R. King, ApJL 174, L123 (1972).[29] A. N. Baushev, arXiv:1608.04356.Page 8 of 8https://mc06.manuscriptcentral.com/cjp-pubsCanadian Journal of Physics

Canadian Journal of Physics

Formation of Wormholes by Dark Matter in the Galaxy Dragonfly 44

Sayeedul Islam

Farook Rahaman

Ali Övgün

Mustafa Halilsoy

Crossref

Formation of wormholes by dark matter in the galaxy Dragonfly 44

Recently, the ultra-diffuse galaxy (UDG) Dragonfly 44 in the Coma Cluster was observed and observations of the rotational speed suggest that its mass is almost the same as the mass of the Milky Way. On the other hand, interestingly, the galaxy emits only 1% of the light emitted by the Milky Way. Thus, astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. In this study we try to determine whether or not the dark matter that constitutes Dragonfly 44 can form a wormhole. Two possible dark matter profiles are used, namely, UDG King’s model and generalized Navarro–Frenk–White (NFW) dark matter profile. We have shown that King’s model dark matter profile does not manage to provide a wormhole whereas the generalized NFW dark matter profile manages to find wormholes.Recently, ultra diffuse galaxy (UDG) of Dragonfly 44 in the Coma Cluster was observed and observations of the rotational speed suggest that its mass is almost same as the mass of the Milky Way. On the other hand, interestingly, the galaxy emits only 1 \% of the light emitted by the Milky Way. Then, astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. In this study we try to show that the dark matter that constitutes Dragonfly 44 can form the wormhole or not. Two possible dark matter profiles are used, namely, ultra diffuse galaxy King's model and generalized Navarro-Frenk-White (NFW) dark matter profile. We have shown that King's model dark matter profile does not manage to provide wormhole whereas generalized Navarro-Frenk-White (NFW) dark matter profile is managed to find wormholes

Ovgün, Alï

CERN Document Server

Recently, ultra diffuse galaxy (UDG) of Dragonfly 44 in the Coma Cluster was observed and observations of the rotational speed suggest that its mass is almost same as the mass of the Milky Way. On the other hand, interestingly, the galaxy emits only 1 % of the light emitted by the Milky Way. Then, astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. In this study we try to show that the dark matter that constitutes  Dragonfly 44 can form the wormhole or not. Two possible dark matter profiles are used, namely, ultra diffuse galaxy King's model and generalized Navarro-Frenk-White (NFW) dark matter profile. We have shown that King's model dark matter profile does not manage to provide wormhole whereas generalized Navarro-Frenk-White (NFW) dark matter profile is managed to find wormholes.The accepted manuscript in pdf format is listed with the files at the bottom of this page. The presentation of the authors' names and (or) special characters in the title of the manuscript may differ slightly between what is listed on this page and what is listed in the pdf file of the accepted manuscript; that in the pdf file of the accepted manuscript is what was submitted by the author

Formation of Wormholes by Dark Matter in the Galaxy Dragonfly 44

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