Is flat rotation curve a sign of cosmic expansion?

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1 MNRAS 433, (2013) Advance Access publication 2013 June 11 doi: /mnas/stt847 Is flat otation cuve a sign of cosmic expansion? F. Daabi Depatment of Physics, Azabaijan Shahid Madani Univesity, Tabiz , Ian Accepted 2013 May 13. Received 2013 May 3; in oiginal fom 2013 Mach 18 ABSTRACT Fou altenative poposals as the possible solutions fo the otation cuve poblem ae intoduced on the basis of assumption that the cosmic expansion is engaged with the Galactic dynamics ove the halo. The fist one poposes a modification of equivalence pinciple in an acceleating Univese. The second one poposes a modification of the Mach pinciple in an expanding univese. The thid one poposes a dynamics of vaiable-mass system fo the halo in an expanding univese, and the fouth one poposes the eplacement of physical adius with comoving one ove the halo, in an expanding univese. Key wods: Galaxy: halo Galaxy: kinematics and dynamics lage-scale stuctue of Univese. 1 INTRODUCTION Obsevations on the otation cuves of galaxies have tun out that they ae not otating in the same manne as the Sola system, accoding to the classical Newtonian dynamics. Galactic otation cuves illustating the velocity of otation vesus the distance fom the galactic cente cannot be explained by the luminescence matte. This suggests that eithe a lage amount of the mass of galaxies is contained in the dak galactic halo, o Newtonian dynamics does not apply univesally. The dak matte poposal is mostly efeed to Zwicky (1957) who found that the motion of the galaxies of the clustes induced by the gavitational field of the cluste can be explained by the assumption of dak matte in addition to the total visible matte content of the obseved galaxies. Late, it tuned out that dak matte is not the specific popety of clustes, athe it can be found as well in single galaxies to explain thei flat otation cuves. The second poposal esulted in the modification of Newtonian dynamics (MOND) (Milgom 1983). The modification poposed by Milgom was the following ( ) a F = mμ a, (1) a 0 { 1 if x 1 μ(x) = x if x 1, (2) whee a 0 = ms 2 is a poposed new constant. At the galactic scale outside the cental bulge a a 0,sowehavethe modified dynamics ( ) a 2 F = m. (3) a 0 f.daabi@azauniv.edu Using this new law of dynamics fo the gavitational foce, we obtain ( ) a 2 =, (4) 2 a 0 whee M is the total mass of the bulge. This esults in v 2 = a 0. (5) Altenative solutions in this diection has also been poposed by assuming the gavitational attaction foce to be 1/ beyond some galactic scale (Sandes 2003), o assuming the dak matte as the manifestation of the Mach pinciple showing the effective gavitational mass of astophysical bodies to be dependent (Bozeszkowski & Tede 1998). Recently, some authos have addessed the otation cuve poblem by esoting to modified theoies of gavity (Lin et al. 2013; Stabile & Capozziello 2013). In what follows, we will intoduce fou altenative poposals fo the otation cuve poblem without esoting to the dak matte, MOND o modified gavity. These poposals ae based on the possibility that the cosmic expansion o acceleation is next to the galactic stuctues and the dynamics of the oute (halo) stuctue of the galaxies is engaged with the cosmological dynamics. The velocity pofile obtained in the fist poposal is like that of suggested initially by Gumille (Gumille 2010) and discussed late by Lin et al. (2013). 2 MODIFIED EQUIVALENCE PRINCIPLE DUE TO COSMIC ACCELERATION Recent cosmological obsevations obtained by SNe Ia (Pelmutte et al. 1999), Wilkinson Micowave Anisotopy Pobe (Bennett et al 2003), Sloan Digital Sky Suvey (Tegmak et al. 2004) and X-ay (Allen et al. 2004) indicate that ou univese is globally expeiencing an acceleated expansion. This acceleation is imposed on empty space all ove the univese, so that fo an obseve in a local C 2013 The Autho Published by Oxfod Univesity Pess on behalf of the Royal Astonomical Society Downloaded fom

2 1730 F. Daabi est fame, the space within the fame is isotopically expanded adially outwads in an acceleating way, namely we have an isotopic acceleation of space aound each point in this est fame. Let us now evisit the equivalence pinciple in the pesence of this space acceleation. We know that in a static space, the equivalence pinciple states that a local fame ested in the vicinity of a gavitational field with gavitational stength g is indistinguishable fom an acceleated fame (in empty static space) having an acceleation a out equal to the gavitational stength g. The equivalence pinciple is also the main eason by which, fo example, a massive body otates aound a gavitational mass M ove a constant adius as v 2 =, (6) 2 whee the centifugal and the gavitational acceleations play the ole of a out and g, espectively. Howeve, the stoy is diffeent in a fame in which the backgound space is not static. Conside a fee local fame 1 in such an expanding space. If we put a body of mass m in this fame, then it expeiences a adially outwad and isotopic cosmic acceleation in all diections, so the oveall acceleation imposed on the body vanishes. Now, let us impose an upwad diectional acceleation on this fame by an oute agent. It destoys the isotopy of space fom the obseve s point of view and a local acceleation is induced in this fame. This is simila to the spontaneous symmety beaking phenomena whee the system afte symmety beaking takes a gound state. Fo example, the inteactions between the atoms in a feomagnet is invaiant unde otation. Howeve, when the tempeatue eaches the citical tempeatue the otational symmety disappeas and a gound state appeas in which all the spins ae aligned, and this is clealy not otationally invaiant. In ou case, befoe the oute agent of acceleation is applied on the fame, the space within the fame has isotopic symmety so that a massive body theein expeiences a same acceleation in all adial diections, and so the esultant foce imposed on this body vanishes. The symmety disappeas when a pefeed diection is singled out by an oute agent which imposes an acceleation in a pefeed upwad diection on the fame. Theefoe, a local acceleation (as a singled out gound state) emeges within the fame which is clealy not isotopically invaiant. It is easonable to suppose that the diection of the induced acceleation within the fame is in the same diection of the upwad acceleation, imposed by the oute agent, which beaks down the isotopy. Hence, if the oute upwad acceleation of the fame is a out = g, the body inside this local fame beas an upwad induced acceleation a c, despite the downwad acceleation g. Note that, the induced acceleation a c depends on the value of the cosmic acceleation, hence we call it induced cosmic acceleation a c. Now, in ode to validate the equivalence pinciple, we assume a local fame 2 in the vicinity of a gavitational field, with stength g, in a static backgound space and put a body of same mass m in this local fame. In this fame, the body falls down towads the gavitational cente by an acceleation g. But, in the pevious paagaph we ealized that the body of mass m in fame 1 (unde the oute acceleation a out = g) does not fall with the acceleation g,atheit falls with an acceleation as the esultant combination of g and a c. It is clea that the discepancy between the obsevations in fames 1 and 2 is due to the fact that the backgound space is not static within fame 1. Once we take a static space within fame 1, both masses in fames 1 and 2 fall with the same acceleation g, so the discepancy is emoved and the equivalence pinciple is eestablished. Howeve, it is still possible to eestablish the equivalence pinciple between two fames 1 and 2, even if the space within fame 1 is not static. To this end, the oute acceleation imposed on fame 1 should fist compensate the induced cosmic acceleation a c (acceleation of the backgound space itself) within this fame and then povide the exta acceleation equal to g (by which the body falls down in fame 2) as a out = a c + g, (7) which gives a esultant acceleation in fame 1, as ā = a out a c = g. (8) Note that, the value of a c may depend on the cosmic acceleation and the value of a out imposed on fame 1. This is what we mean by modified equivalence pinciple. Accoding to this equation, both masses in fames 1 and 2 now fall downwads with the same acceleation g. Now, suppose a galaxy with a dens mass in the bulge and a diluted mass in the halo. We assume that, at the pesent state of low-acceleating phase, the cosmic acceleation is just able to affect the halo and not the tight bound gavitational stuctue of the bulge. Howeve, when the univese entes the phantom phase, not only the bulge but also all the gavitational stuctues, the electomagnetic stuctues and even the stongly coupled stuctues will be ton apat by the cosmic acceleation. Let us now conside a typical galaxy with a bulge of adius R 0 and a body of mass m located at a adius R 0 in the halo within a local fame otating aound the cente of galaxy. This body expeiences a eal gavitational foce F = GmM/ 2,whee M is the total mass of the bulge and is the distance fom the cente of the galaxy. Howeve, it expeiences a centifugal foce as well, due to the otation. Now, let us assume that the cosmic acceleation is next to the galaxy and so affects the halo. Using the above modified equivalence pinciple, we may conside the otating fame hee as: (1) fame 1 concening the centifugal acceleation as a out, and (2) fame 2 concening the gavitational acceleation as g. Accoding to the above discussion, the centifugal foce as the agent of oute acceleation beaks the space isotopy within fame 1 and esults in an upwad induced cosmic acceleation a i inside this local fame. Theefoe, using the equivalence of inetial and gavitational masses m i = m g, we may ewite the modified equivalence pinciple equation (7) as v 2 m i = m ia c + Gm gm. (9) 2 Note that the induced cosmic foce m i a c which is upwads fom the obseve s point of view in the otating fame consideed hee, is actually tying to pull out adially the body of inetial mass m i fom the halo s gavitational configuation towads the cosmological void backgound. One may intepet the induced foce m i a c in the ight-hand side of equation (9) as the eaction foce which the gavitational system of the galaxy imposes against the cosmic foce attempting to pull out the body of mass m fom halo. Equation (9) esults in the following expession fo the obital velocity within the halo v = a c +. (10) Note that since the bulge is not affected by the cosmic acceleation, the velocity pofile within the bulge is Kepleian () v =. (11) Downloaded fom

3 Rotation cuve and cosmic expansion 1731 Howeve, if we assume that the bulge is also affected by the cosmic acceleation, then we have v = a c + (), (12) fo > 0. Now, we may ask the inteesting questions: (1) What is the intepetation of the oute agent? (2) Why the induced cosmic acceleation is simply upwads? To answe the fist question, we note that in pinciple the oute agent is just meaningful in the famewok of the equivalence pinciple in geneal elativity. In a thought expeiment, when the oute agent acceleates a est fame by a out, a same amount of pue gavitational acceleation g = a out is effectively poduced inside the fame, exeting on a test body. In the famewok of ou genealized equivalence pinciple, the oute agent acceleates a est fame by a out and a pue gavitational acceleation g plus an induced cosmic acceleation a c, oiginating fom the existence of cosmic acceleation in space and beakdown of space isotopy, is effectively poduced inside the fame, exeting on the test body. In fact, the oute agent does not eally exist in these thought expeiments, but helps us to ealize the equivalence between the inetial and gavitational foces: Wheeve thee is a gavitational foce, we may look fo the coesponding equivalent inetial foce. In this pape, we ae looking fo the oigin of a gavitational foce which poduces a flat otation cuve within the halo. Hence, we esot to the (genealized) equivalence pinciple to find the oigin of the exta gavitational foce which causes deviation fom the Kepleian cuve. We find that the oigin of this exta gavitational foce on a test body within the halo may be nested in the equal inetial foce exeting by an induced cosmic acceleation a c on this test body in a local fame (within the halo) which is being acceleated by the thought oute agent with a out.inothewods, the oute agent hee is nothing but a thought agent acceleating (adially outwads) a est fame within the halo to poduce a locally equivalent gavitational foce (adially inwads) inside the fame, exeting on the test body to keep it on a local obit ove a flat otation cuve within the halo. To answe the second question, we note that upwad acceleation is just a elative concept. A local est fame, which is otating on an obit aound the cente of galaxy, is also a locally fee-falling fame along a adial diection towads the cente of galaxy. Hence, in this fame upwads and downwads means adially outwads and adially inwads, espectively. By symmetic consideations, it is easonable to suppose that the diection of the induced cosmic acceleation exeting on the test body in this local fame is in the same adial diection as that of the acceleation imposed by the thought oute agent which poduces this induced cosmic acceleation. The eason fo an upwad, instead of downwad, acceleation is simply because accoding to ou assumption the space within the halo is adially acceleating the bulge outwads, so fo a test body (o a est obseve) in the local fame within the halo, the adially outwad acceleation means exactly an upwad acceleation. It is appealing to connect the cosmic acceleation hee with the acceleation constant intoduced by Milgom in his theoy of MOND. Accoding to Milgom (2003), the acceleation constant, which maks the bounday between the validity egions of MOND and Newtonian dynamics, tuns out to have a value that matches acceleations appeaing in the context of cosmology, namely the ecently discoveed acceleation of the univese which is of the same ode of magnitude as ch 0 and c /3wheec, H 0 and ae the velocity of light, Hubble constant and the emeging value of the cosmological constant, espectively. Simila to Milgom, we take this statement as a hint that Milgom s acceleation is somehow connected with the cuent acceleating phase of the univese. Motivated by the above idea, we assume that the cosmic acceleation a c is calculable fom a deepe theoy so that its estimate value at the edge of bulge coincides with the Milgom s acceleation constant a ms 2.Thisissimilatotheconstant g, namely the fee-fall acceleation nea Eath s suface, as it appeas, say, in Galilean mechanics. The deepe theoy in this case is Newtonian univesal gavity, which tells us that g is calculable fom the mass and adius of the Eath (Milgom 2003). Now, putting the value of a 0 in equation (10) gives the following fomula fo the obital velocity at R 0 v (13) Applying this fomula to a typical galaxy indicates that (i) the maximum value of velocity at R 0 is a little geate than that of Kepleian otation cuve due to the fist tem including a 0, (ii) the velocity deceases to a local minimum at a distance R 0 /a 0, (iii) the velocity vey steadily climbs fo R 0. Equation (13) esults in the otation cuve pofile of the halo in good ageement with that of dwaf galaxies, o lage (spial) galaxies, see e.g. Rhee et al. (2004) and Sofue & Rubin (2001), espectively. Howeve, since thee is no empiical hint that the otation cuves of galaxies ise linealy at lage distances, see e.g. de Blok & McGaugh (1997), so we may assume the induced cosmic acceleation as a function of distance ove the halo, and ewite equation (10) as v = a c () +. (14) This is simila to the fact that the gavitational acceleation g at the suface of Eath with adius 0 is almost constant, while in pinciple it is -dependent fo lage adii > 0. Hence, we may assume that the induced cosmic acceleation at the edge of bulge with adius R 0 is almost constant but it is -dependent fo R 0 in the halo. Actually, in equation (8) we have aleady assumed the dependence of a c on a out in the modified equivalence pinciple. Since a out = v 2 /, then we may expect a c = a c (). Actually, allowing fo such an -dependent function one can fit any given otation cuve. In fact, such an -dependence may be well justified fo the induced cosmic acceleation consideed hee, as follows. The induced cosmic acceleation in the void and distant spaces between the galaxies is almost vanishing because in these egions the space is acceleating adially in all diections in an isotopic way, so the esultant acceleation imposed on a test body becomes zeo at each abitay point in the void and distant spaces between the galaxies. Fo this eason, vey fa away fom the whole gavitational stuctue of a typical galaxy, the induced cosmic acceleation is also zeo fo a test body at any point. This popety limits the scope of the induced cosmic acceleation to the galactic halo as follows. Inside the bulge, because of the stong and tight gavitational bound between the gavitational ingedients, thee is no engagement of the cosmic acceleation and galactic dynamics. So, thee is no oom fo the induced cosmic acceleation inside the bulge. As is discussed above, well beyond the halo, the induced cosmic acceleation is also vanishing due to the full isotopy of space. Theefoe, it is just left fo the halo to exhibit the engagement of the cosmic acceleation and the galactic dynamics. This means that the induced cosmic Downloaded fom

4 1732 F. Daabi acceleation is non-vanishing and has local popety just within the halo. Fo a given mass of a test body at the edge of bulge, namely at the beginning of the halo, thee is a consideable centipetal acceleation adially diected towads the cente of galaxy, and this singled out adial diection easily distubs the isotopy of space at this egion. Hence, the esultant induced cosmic acceleation in this egion has a athe consideable non-vanishing value along the adial diection. It is theefoe plausible that fo a given mass at moe distant egions along the adial diection within the halo, whee the centipetal acceleation becomes smalle, the distubance on the isotopy of space should be smalle as well and so the esultant induced cosmic acceleation in this egion becomes smalle. In othe wods, the moe we get close to the gavitational cente within the halo, the moe centipetal acceleation we have, the moe isotopy of space is distubed and the moe esultant induced cosmic acceleation we have. On the othe hand, the moe we go away fom the gavitational cente within the halo, the less centipetal acceleation we have, the less isotopy of space is distubed and the less esultant induced cosmic acceleation we have. Hence, the induced cosmic acceleation becomes small and smalle at lage and lage distances within the halo, espectively, and vanishes in the fa void space beyond the galaxy, as we fist expected. Theefoe, the fomula (14) with a local -dependent and halodependent induced cosmic acceleation is well justified, and can pedict the flat otation cuves by consideing the chaacteistic featues of each halo in each galaxy. This may justify the vaiety of flat otation cuves coesponding to diffeent types of galaxies. A vey simple function fo the induced cosmic acceleation may be poposed as a c = R 0 a 0, (15) whee R 0 is the chaacteistic size of the bulge. Putting this into equation (14) gives v = R 0 a 0 +, (16) which poduces an almost flat otation cuve fo > R 0. Since the cosmic acceleation cannot affect the inne bulge < R 0 (with tight gavitational stuctue), thee is no induced cosmic acceleation in the bulge (a c = 0) and so the obital velocity within < R 0 and at R 0 is obtained, espectively, as v =, (17) and v 0. (18) R 0 Using equation (18), we may eplace fo R 0 in equation (16) and obtain v v0 2 a 0, (19) whee / is almost ignoed fo > R 0. This fomula coincides numeically with that of Milgom, namely (5), because numeically we have v v 0. Theefoe, anothe esult of the poposal discussed in this section is that the Milgom s constant acceleation is nothing but the induced -dependent cosmic acceleation at = R 0, just like g which is nothing but the -dependent gavitational acceleation at the suface of Eath. The fomulae (13) and (14) ae in close ageement with those suggested initially by Gumille (2010) and discussed aftewads by Lin et al. (2013) in fitting to the otation cuve data of halo fo some galaxies. The majo diffeence lies in the physical oigins of the exta acceleations appeaed in the fomulas hee and those suggested by Gumille. The effective potential in the Gumille s modified gavity includes the Newtonian potential and a Rindle tem, so the exta acceleation suggested by Gumille has its oigin in Rindle tem. Howeve, the induced acceleation in this wok has its oigin in the cosmic acceleation. 3 MODIFIED MACH PRINCIPLE AND COSMIC INERTIAL MASS Accoding to the Mach pinciple (Mach 1960), the distant mass distibution of the static univese has been consideed as being esponsible fo geneating the local inetial popeties of the close mateial bodies. One may modify the Mach pinciple in an expanding univese so that the local inetial popeties of the mateial bodies ae affected by the expansion of univese. In this line of thought, we assume that the inetial mass of a body is constant within the bulge of a typical galaxy whee a dense distibution of matte is pesent. This is in accodance with the spiit of the Mach pinciple in that as long as a lage and almost constant mass configuation exists aound a body, it has a constant inetial mass. In fact, accoding to Mach, the coodinates in space ae defined by the pesence of matte configuations. In othe wods, the space does not exist without the pesence of matte. Theefoe, a deep connection exists between the inetial popeties of matte and the coodinates in space. In this egad, one may think that within the dense pat of the galaxy, namely the bulge, whee all the mass configuation is otating as a igid body, the inetial popeties of each individual body, namely the inetial mass, is constant. This is because the coodinates of bodies (configuations) in this almost igid body is constant. So, thei inetial popeties (mass) which ae induced by this constant spatial configuation becomes constant too. In fact, the matte configuation in the bulge is constucted by the gavitational inteaction which tightly bounds togethe all the bodies inside. These tightly bound (dense) bodies define the tightly bound (dense) spatial coodinates within the space of bulge, and these dense coodinates detemine the tight inetial popety o constant inetial mass of bodies in the bulge. Howeve, the situation changes fo bodies beyond the bulge in the halo whee, accoding to ou assumption, they ae affected by the expansion of univese. In fact, we assume that the inetial mass of bodies beyond the bulge is diluted because thei physical configuation is subject to the expansion of univese. In othe wods, the spatial coodinate beyond the bulge, > R 0, is subject to the expansion of univese so that unlike the dense spatial comoving coodinates within the bulge, < R 0,thecomoving coodinate beyond the bulge is not dense, athe it is diluted due to the expansion of univese. Theefoe, accoding to Mach, the inetial popety o mass of bodies beyond the bulge may be diluted too, in compaison to the tight inetial popety o constant mass of bodies within the bulge. Then, one may suggest the following definitions fo the inetial mass within and beyond the bulge, espectively, as { mi = C, R 0, (20) m i = C, > R 0, whee C and C ae constants. We call the fist one as inetial mass vesus gavitational inteaction within the bulge, and the second one Downloaded fom

5 Rotation cuve and cosmic expansion 1733 as inetial mass vesus cosmological expansion beyond the bulge. Equation (20) is what we mean by the modified Mach pinciple. Now, using equation (20), we may wite the equation of motion fo a body at > R 0 as follows: m i v 2 = Gm gm 2 v = R 0, (21) whee C = m g R 0 is taken based on the dimensional consideations to obtain the flat otation cuve. This fomula again gives a otation cuve in ageement with obsevations. Note that it seems at fist glance that the equivalence pinciple is violated in the case of second definition in equation (20). Howeve, this is not the case. In fact, all bodies with diffeent inetial masses at the same adius > R 0 ae still falling with a same centipetal acceleation v 2 / = /R 0 towads the cente of galaxy while otating aound it. The geneal assumption of an -dependent cosmological inetial mass like in equation (20) becomes plausible if we ecall the cosmological adial velocity ṙ = H, namely the Hubble law. In fact, such an -dependent cosmological inetial mass is justified if we demand fo the consevation of cosmological adial momentum fo a typical body of inetial mass, m i, in the cosmological backgound > R 0 and at each given cosmological ea with constant Hubble paamete, as follows m i ṙ = Const. (22) 4 MODIFIED NEWTON S EQUATION DUE TO COSMIC EXPANSION It is well known that ou univese is expanding and accoding to Hubble law a body at distance fom an obseve expeiences a adial velocity v = H, (23) whee H is the Hubble paamete. Usually, this expansion is assumed to be consideable at lage scales vey much lage than the scale of galaxies. Howeve, let us suppose that this expansion is vey close to the galaxies so that the haloes of galaxies ae subject to the cosmic expansion. Now, we investigate the effect of this expansion on the galactic halo dynamics. To this end, we fist emind a simple poblem in elementay mechanics, namely a system with vaiable mass. In mechanics, a vaiable-mass system is a system which has mass that does not emain constant with espect to time. In such a system, Newton s second law of motion cannot diectly be applied; instead, this system can be descibed by modifying Newton s second law and adding a tem to account fo the momentum caied by mass enteing o leaving the system as dm v el dt, (24) whee v el is the elative velocity of the enteing o leaving mass dm with espect to the cente of mass of the body. Fo a moment, suppose we tun off the otation of galaxy and conside a body of constant mass m beyond the bulge in the halo and unde the influence of the gavitational foce exeted by the bulge having a mass M as GmM/ 2. Also, we assume the adial cosmic velocity (equation 23) to be esponsible fo escaping the matte fom the halo of galaxies towads the oute void space. So, the galaxy s halo becomes a typical example of a vaiable-mass system with v el = v = H. (25) It is easonable to suppose that in ode fo the v el (namely H) be constant with espect to time (Hubble paamete is almost constant), the physical powe W exeted by the cosmic expansion on the vaiable-mass system in the halo of the galaxy should be constant as well. Theefoe, we have W = (H) 2 dm = C, (26) dt whee C is a constant. This esults in dm = C(H) 2. (27) dt Obviously, we ae concened about the dynamics of the individual mass m. But, in pinciple, this mass in the halo of galaxy belongs to the same leaving mass dm. Hence, it is plausible to suppose that dm m, (28) and accoding to equation (26), we find C m. (29) This equation easonably states that the powe exeted by the cosmic expansion on the mass m to pull it out fom the halo of galaxy is popotional to the mass m itself. Now, by putting equations (25) and (27) into equation (24) and using (29), we obtain the foce exeting by the cosmic expansion on the mass m to pull it out fom the halo, as follows dm v el = mc (H) 1, (30) dt whee C is anothe constant. Actually, the mass m mediates this foce to the bulge of the galaxy (as the main gavitational souce) to which it is gavitationally bound. In fact, the cosmic expansion tends to beakdown the whole gavitational stuctue of the galaxy by pulling adially out all the individual masses in the galaxy. Hence, accoding to the thid law of action and eaction, an equal and adially inwad foce is exeted by the gavitational souce of the galaxy, namely the bulge, on the mass m to keep it in the halo stuctue. Now, let us tun on the otation of galaxy. The modified Newton s equation of motion fo the mass m inside the halo while otating with the obital velocity v at a distance fom the cente of the galaxy is witten as m v2 = GmM + mc 2 H, (31) whee the second tem in RHS emeges due to the cosmic expansion. Equation (31) leads to the following esult: v = + C H. (32) This fomula again poduces an almost flat otation cuve in the halo > R 0 povided that the second constant tem in the squae oot is consideably lage. Being the Hubble paamete in the denominato indicates that at ealy times in the univese s age when the young galaxies wee constucted, the second tem in the squae oot was so small that the obital velocity could be assumed obeying the Kepleian otation cuve v. (33) Downloaded fom

6 1734 F. Daabi Howeve, at the pesent when the univese is old enough and the Hubble paamete is small enough, the second tem in the squae oot is consideably lage and can cause fo a flat otation cuve. 5 COSMOLOGICAL SCALING We know that accoding to Newtonian dynamics, the otation cuve within the halo is theoetically expected to be Kepleian v =, (34) whee M is the total mass of the bulge and is the physical coodinate. Howeve, the obsevations indicates a flat otation cuve. To solve this poblem, we assume that the halo (without dak matte) still obeys the Kepleian dynamics (equation 34). But, on the othe hand, we assume that the halo is affected by the cosmic expansion such that the physical coodinate of a body in the halo is a instead of, a being the cosmological scale facto and is the comoving coodinate. In othe wods, we assume that in the halo the physical coodinate is eplaced by a comoving coodinate. Theefoe, while the otation cuve within the bulge is plotted with espect to the physical adius (as a linea elation) the otation cuve in the halo should be plotted with espect to the comoving coodinate accoding to a Kepleian elation v =. (35) Accoding to ou assumption, if the halo s extent would be just a consequence of the cosmic expansion, the comoving coodinate ove the halo is then limited to a vey small domain by the following condition R 0 R 0 + ɛ, (36) whee ɛ is vey small. Note that = = R 0 coesponds to the bounday of the halo (beginning of the effective cosmological domain) whee the comoving coodinate is scaled by the scale facto a 0 = 1, as a bounday value at R 0. But, the end of halo denoted by the adius R H (fom the cente of galaxy) is scaled as R H (R 0 + ɛ)a H, (37) whee a H denotes the value of scale facto at = R 0 + ɛ. In explicit wods, the appaent adial coodinate in the halo, as the physical distance fom the cente of galaxy, may be nothing but a comoving coodinate limited to a vey small domain (see equation 36) which is scaled by an -dependent expansion facto a( ) ove the halo. In fact, we assume that the scaling ate of the space in the halo (with massive stuctue) as a( ) is diffeent fom the fixed scaling ate of the empty space (with no stuctue) in the univese, namely a. Using equations (36), (37) and = a( ), we obtain R 0 R H. (38) Taking the vaiation of both sides of equation (35) leads to v = v. (39) 2 Inseting = ɛ and R 0 fom equation (36) leads to v 0, (40) which accounts fo a flat otation cuve thoughout the halo. 6 CONCLUSION In this pape, we have given fou poposals each of which is theoetically a possible solution fo the otation cuve poblem. Ou motivation was the assumption that the otation cuve poblem may be elated to the cosmic expansion which is next to the galactic stuctue, athe than the dak matte, MOND o modified gavity. In explicit wods, we have tied to answe the impotant questions: Can cosmic dynamics influence anyway the galactic dynamics? Do we expect at all that the galactic dynamics be potentially engaged with the cosmic dynamics? Is it time to conside such an engagement afte billions of yeas, o we should still wait fo the univese to coss the phantom divide and expeience a consideable acceleation? If the answes to the above questions ae positive, then we should concen about this subject seiously. In ou opinion, such an engagement is inevitable and soone o late it happens. In this study, we have tied to show the possibility that the galactic flat otation cuve may be a sign and diect consequence of such an engagement. In the fist poposal, by a genealization of equivalence pinciple in an acceleating univese, we showed that the cuent acceleation of the univese may be esponsible fo the flat otation cuve. The impotant point is that, even if this poposal is not coect fo the justification of flat otation cuve, its consequences must be consideed seiously when the univese entes the phantom phase of acceleation and the halo is inevitably affected by the consideable acceleation of this phase. In othe wods, if the flat otation cuve has its eal oigin in the dak matte, MOND, o modified gavity, one should concen about the impact of cosmic acceleation on the dynamics of halo (and even bulge) when the univese undegoes the phantom phase acceleation. The fist poposal at least pedicts a modified dynamics of halo due to the possible engagement with the cosmic dynamics, at pesent o in the futue at phantom ea. The velocity pofile obtained in this poposal is vey much like that of suggested initially by Gumille (2010) and discussed late by Lin et al. (2013) in fitting to the otation cuve data of halo fo some galaxies, except fo the diffeent oigins of the exta acceleation which is appeaed in Gumille model as Rindle acceleation, and hee as cosmic acceleation. It is appealing to investigate the possible elation between Rindle acceleation and cosmic acceleation fom the pesent point of view. In the second poposal, we discussed on the possible impact of the Mach pinciple on the dynamics of halo in an expanding univese. We showed that the dilution of inetial mass, as a consequence of the Mach pinciple in an expanding univese, may poduce a flat otation cuve ove the halo. In the thid poposal, we assumed the halo as a vaiable-mass system whose mass is subject to the cosmic expansion which is tying to pull the massive bodies out of the halo. We showed that the eaction foce of the gavitational stuctue in the galaxy against the pulling out foce of cosmic expansion can cause fo a flat otation cuve. Finally, in the fouth poposal, we assumed that the halo obeys the Kepleian dynamics with espect to the comoving coodinate, athe than the physical adius. Since the comoving coodinate in the halo is limited to a vey small domain, the Kepleian dynamics pedicts an almost constant otation velocity with a vey small vaiation ove the halo. ACKNOWLEDGEMENTS The autho would like to thank the anonymous efeee fo the enlightening comments. Downloaded fom

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