PHENOMENOLOGY OF THE MODIFIED NEWTONIAN DYNAMICS AND THE CONCORDANCE COSMOLOGICAL SCENARIO
L. BLAl'\CHET & A. LE TIEC ylR;olCO. lnstitut d'Astrophysique de Paris - UMR 709.5 du CNRS. 1 Universite Pierre fj Marie Curie, 98 ''-' boulevard Arago, 7.5014 Paris, France
After re,;ewing the modified Newtonian dynamics (l\10:.XD) proposal, we ru:lvocate that the associated phenomenology may actually not result from a modification of Newtonian gravity, but from a mechanism of "gra,;tational polarization" of some dipolar medium pla);ng the role of dark matter We then build a relativistic model within standard general relativity to describe (at some phenomenological level) the dipolar dark matter polarizable in a gravita tional field. The model naturally involves a cosmological constant and is shown to reduce to the concordance cosmological scena1;0 (.\-COM) at early cosmological times. From the mechanism of gravitational polarization, we recover the phenomenology of MO:\ll in a typi cal galaxy at low redshift. Furthermore, we show that t.he cosmological constant A scales like
a~, where a0 is the constant MONO acceleration seal<>, in good agreement with observations.
1 Introduction
The mysteries of the nature of dark matter and dark energy arc perhaps the most important ones of contemporary cosmology. Dark matter, which accounts for the observed discrepancy between the dynamical and luminous masses of bounded astrophysical systems, is usually for mulated within the sv-eallt-- 255 5 6 7 (MO:'\D) · • . Although MO::'llD in its original fornmlation ca1mot be considered as a viable physical model, it is now generally admitted that it does capture in a very simple and powerful "phenomenological recipe·· a large number of observational facts, that any pertinent model of dark matter should explain. It. is frustrating that the two allernat.i\-es A-CD~1 and MOND. which are sucressful in com plementary domains of validity (say the cosmological scale for A-CDM and the galactic scale for MOND), seem to be fnndament.ally incompatible. In the present paper, we shall propose a different approach, together with a r.2w interpretation of tile phenomenology of MONf', which has the potential of bringing together A-CDM and MOND into a single unifying relativistic model fur 2 The modified Newtonian dynamics (MOND) 5 6 7 The original idea behind 1\102\:1) • • is Lhal there is no clark ma.LI.er, and we witness a viola.Lion of the fundamental law of gravity (or of inertia). MOXD is designed to account for the basic features of galactic. dark matter halos. It states that the "true" gravitational field gexperienced by ordinary matter, say a test particle whose acceleration would thus be a = g, is not the Xewtonian gravitational field gr-;, but is actually related to it by µ(g)g=gx. (1) 11-0 Hereµ is a function of the dimensionless ratio g/0-0 between the norm of the gravitational field 2 g = \gl, and the c.:oustant 1\10:\D acceleration scale ao ~ 1.2 x lU ·JO m/s , whose numerical value is chosen to fit the µ -g) =-+O(g-).g ·) (2) ( ao ao Ou the other hand, when g is much larger than a0 (formally g-> +oo) the usual Newtonian law is recovered, i.e. µ. -> l. Various functions µ. interpolating between the MOND regime and the :\"ewtonian limit are possible, but most of thEom appear to be rather ad hoc. Taken for granted, the :VIOND "r<'cipe" (1)-(2) b<'autifully predicts a Tully-Fisher relation and is very successful at fitting the detailed shape of rotation curves of galaxies from the observed distribution of stars and gas (see~ . H for reviews) So MOND appears to be more than a simple recipe and may well be related tu sorne uew fu11 256 ~Iaxwcll equations arc conformally invariant. This is contrary to observations: huge amounts of dark matter are indeed observed by gravitational (weak and strong) lensing. Relat.ivistic extensions of l\IOND t.hat pass the problem or light. deflection by galaxy clust.ers have been shown to require the exist.e11C'e of a time-like vector field. The prototype of such 12 1 theories is the tensor-vector-scalar (TeVeS) theory • J ,l1, whose non-relativistic limit reproduces ~IOND, and which has been extensively investigated in cosmology and at the intenncdiatc scale of galaxy clusters 15 . l\Iodified gravity theories such as TeVeS have evolved recently toward 7 Einstein-<£thcr like t heorics 16· 1 , lM. Still. recovering the level of agreement of the A-CD Y1 scenario with observations at cosmological scales remains an issue for such theories. In the present paper we shall follow a completely different route from that. of modified gravit.y and/or Einstein-rel her theories. \\<-e shall propose an alternative to these theories in the form of a specifiC' mod·ifieri matter theory based on an elementary interpretation of the l\IOND equation (3). 3 Interpretation of the phenomenology of MO:N"D The physical motivation behind our approach is the striking (and presumably deep) analogy 19 between MOND and the electrostatics of dielectric media . From electromagnetism in dielectric media we know that the :Maxwell-Gauss equation can he written in the two equivalent forms 1 V·E = - (O'free + O'po!) (4) ~o where E is the electric field, O'free and O'pol are the densities of free ond polarized (electric) charges respectively, and c:, = 1 +Xe is the relative permittivity of the dielectric medium. Such an equivalence is ouly possible because the density of polari;r,ed charges reads O'pol = - V · P, where the polarization field P is aligned with the electric field according to P = co Xe E, the proportionality coefficient Xc(E) being known as the electric susceptibility. By full analogy, one can .write the YIOI\D equation (3) in the form of the :.isual Poisson equation but sonrced by some additional distribution of "polarized gravitational masses" Ppol (to be interpreted as dark matter, or a component of dark matter), namely V- g = -47rG (Pb+ Ppo!) <===} V ·(µg) = -47rG Pb. (5) This rewriting stan Ppol = - V. II , (6) where II denotes the (gravitational analogue of the) polarization field. It is aligned with the gravitational field g (i.e. Lhe dipolar medium is polarized) according to II= --g.'\'. (7) 47rG Here the coefficient x, which dep.mds on the uorm of the gravitational field y = 191 in complete analogy with the electrostatirs of dielectric media, is related to the l\IOND function by µ=I+x. {8) Obviously x can be interpreted as a "gravitational susceptibility" coefficient, while µ itself cau rightly be called a "digravitational" coefficient. It was show11 in 19 that in the gravitational case the sign of x should be negative, in perfect agreement with what 1'IOI\D predicts; indeed, we haveµ < 1 in a straightforward interpolation between the MOXD aucl Newtonian rcgim,,s, h<'nce X < 0. This finding is in contrast to electromagnetism where Xe is positive. It can be viewed as 257 5j1Ile "anti-screening" of J:?;ravitatioual (baryon.ic) masses by polarization !Ila5S('S - the opposite effect of the usual screening of electric: (free) charges by polarization charges. Such anti-screening n .,,chanism re.suits in an enhancement. of the gravitational field a la MO.\D, and offers a very nice interpretation of the ::vtOXD phenomenology. Furthermore, it was pointed ant that the stability of the dipolar dark matter medium requires the existence of some internal force, which turned out to be simply (in a crude quasi-K'.!wtonian model 19 ) that of an harmonic: oscillator. This force could then be interpreted as the restoring force in the gravitational analogue of a plasma oscillating at its natural plasma freqneucy. Finally, it. seems from this discussion that the dielectric interpretation of MO.\TD is deeper than a mere formal analogy. However the model 19 is clearly non-viable because it. is non-relativistic, and it involves negative gravitational-type masses am! t.h<'refore a violation of the equivalence principle at a fundamental level. 4 Relativistic model for the dipolar dark fluid Here we shall Lake seriously the physical inLuiLion Lhat MO.\D has something Lo do with a 111ec:hanism of gravitational polarization. We shall build a fully rela.tivi5tic: model based 011 a matt.er act.ion in standard general rdativity. .\otc that this mPans we arc changing the point of view of the original :'.l.10ND proposal. Instead of requiring a modification of the laws of gravity in the absence of dark matter, we advocate that the phenomenology of ::\10KD results from a physically well-motivated Illt.x:hanism acting on a new type of dark matter, very exotic: compared to standard partic:le dark matter. Thus, we are proposing a modification of the dark matter sector rather than a modification of gravity as in TeVeS like theories. 4.1 Ar.tion and P.qrtatio11.~ of m.otio11 From the previous discussion, the necessity of c;1dowing dark matt.er with a new vector field to build the polarization field is clear. However this vector field will not be expected to be 20 fundamental a::. in TeVeS like theories. Extending previous work , our model will be based on a matter action (in ~ulerian fluid formalism) in general relativity of the form '. 9) This action is to be added to the Einstein-Hilbert action for gravity, and to the actions of all the other matter fields. It contains two dynamical variables: (i) a conserved current Jµ = auµ satisfying \Iµ.!µ = 0, where uµ is the normali7.ed four-velocity and a = (-.fv.Tv) 112 is I.he rest mass energy density (we poser= 1 throughout); (ii) the V('{'tor field f,µ representing th<; dipole four-vector moment carried by the fluid particles. This extra field being dynamical. the Lagrangian will also depend on its covariant derivative \7 vf.µ ; hut in our model this dependence will occur only through the c;o\-ariaut time derivative ~µ =u"\i' ,,f,µ. The Lagrangian explicitly reads (sce 21 for details) 1 . . ] L=a -l-=:+ E,µE,µ -W(II.i), (10) [ 2 112 where :=: ={ (uµ - ~1 ,)( uµ - ~'')} . The first term is a mass term in the ordinary sense (i .e. the Lagrangian of a pressureless perfect fluid), the second one is inspired by the action of spinning particles in general relativity 22 . and the third term is a kinetic term for the dipole moment. Finally, the last term represents a potential W describing some internal interaction, fuuc:iion of the polarization (scalar) field fl.:_ = (.lµv nµn") 112 , where nµ = af,µ is the polarization four-vector, and .lµv = 9µv + 1Lµ1Lv is the projector orthogonal to the four-velocity. By varying the adion (9)-(10) with respect to both the current .!µ and the dipole momrnt f,µ , we get two dynamical equations: an equation of motion for the di polar fluid, and an evolution 258 equation for the dip.·le moment(µ. From these equat.ions it c:an be shown 21 that we can impose the coustra.iut :=: = 1 as a particular way of sel2ctiug a physically interesting solution, such that the final equations depend only on the space-like project.ion ~j_ = 1-µv ~·' of the dipole moment ~µ. The final equat.ions we obtain are tiµ= -:P =-€'; W', (11) nµ = ~V'µ (w - n.LW') (12) a where we denote nµ = uµ(l +l;.lW') +1-~ ~~'and employ the notations ~i. = t;'J_/t:1 = Ili/Il.!. aud W' =dW /drI.l. The motion of the di polar Huid as given by (11) is no11-geodesic, and drive11 by the internal force F'' derived from the potential W. Observe the coupling to the Riemann c:nrvature tensor in the equation of evolution (12) of the dipole moment. By varying the >~c:tiou 1 with respect to the metric g1iv we obtain the stress-energy tensor T w. using the canonical decomposition yµv = r u"11." + P J_µv + ~ Q(µuv> +IP", we find the energy density r, pressure P, heat flux Qµ (such that uµQµ = 0) and anisotropic stresses :Eµv (u,,:Eµv = 0 and :E~ = 0) as r=W-Il.LW'+p, (13a) P = -w + n.lw' (13b) ~3 ' 1 1 Q ' = u~i_ + Il1W'v." - TI:lV'.>.u ', (13c) :Eµv = ( ~ .L"" - €'; €~) Il.l W'. (13d) Here the contributi.1n p to the energy density involves a monopolar term u and a dipolar term ·· V'.>.l1:l which dearly appears as a relativistic: generalisation of (6), and will play the nudal role when recovering MOND: (11) 4.2 Weak field expansion of the internal polenl.ial The dipolar fluid dynamics in a given background metric, and its influence on spacetime are now known: in the following we shall apply this model to large-scale cosmology and to galactic halos. For both applications we shall need to consider the model in a regime of weak gravity, which will be either first-order perturbations around a Fricdman-Lcmaitrc-Robcrtson-Walkcr (FLRW) background in cosmology, or the non-relativistic limit for galaxies. A crucial assumption we make is that the potential function W admits a minimum when the polarization II.i is zero, and can be Taylor-expanded around that minimum, with coefficients being entirely specified (modulo an overall factor G) by lhe single surface density scale '.milt from the l\IOND acceleration ao , (15) These coefficients in the expansion of W when Il.l -> 0 will be fine-t?Lned in order to recover the relevant physics at cosmological and galactic scales. Physically, this expansion corresponds to II.l « :E and is valid in the weak gravity limit g « ao. Clearly, the minimum of W is nothing but a cosmological constant A, and we find (16) The expansion is thereby determined up to third order inclusively. 259 Our assumption that the function W involves the single fundamental scale I: implies in particular that the cosmological constant A should scale with G 2 'E. 2 ~ a5. We thus introduce a dimensionless parameter a through (17) This parameter represents a conversion factor between ao and the natural acceleration scale al\. associated with the cosmological constant A. Posing x = II.l/I:, we find that (16) ca:ri be recast 2 in the form W(ll.l) = G7TG E w(x). where 2 2 2 3 w(x) = o 7T + ~x + ~x + O(x~). (18) The present model should be considered only as "effective" or phenomenological, in the sense that t.he weak-gravity expansion (16) or (18) should come from a more fundamental (presum ably qua.11tu111) umlerlyi11g theory. Therefore, the coefficieut:; i11 (18) should 11ot be give11 by 10 exceedingly large or small numbers (like 10 or 10 · 10 ), but rather be numerically of the order of one, up to a factor of, say, ten. Hence, we expect that n itself should be around one, and we sec by ( 17) that. the cosmological constant A. should naturally be numerically of the order of a6. Notice though that our modd docs not provide a way to compute the exact value of o. However, one can say that the "cosmic coincidence" (see e.g. 8 ) between the values of ao and a/\ - with the measured conversion factor being a '.::'. 0.8 - finds a natural explanation if dark matter is ma.de of a fluid of pola.riza.ble gravitational dipole moments. 5 Recovering the A.-CDM scenario at cosmological scales Consider a small perturbation of a background FLRW metric valid between, say, the end of I.he inflationary era and t.he recombination. The dipolar llnid is de.scribed by its four-velocity uµ = uµ+6uµ, where 6uµ is a perturbation of the background cornoving four-velocity uµ = (1, 0), and hy its rest mass density er = 7f + 6cr, where 60" is a perturbation of the mean cosmological valnc a. The crucial point in our analysis of the di polar fluid in cosmology is that the background value of the dipole moment field ~'.{ (which is orthogonal to the four-velocity and therefore is space-like) must vanish in order to presen-e the spatial isotropy of the FLRW background. We shall thus write ~'.{ = t5~'.{, and similarly II';_ = 15Ilj_ for the polarization. At first pertnrbat.ion order, the stress-energy tensor with explicit component.s (13) can be naturally recast, using also (16), i11 the form yµv = r:;; + r:;:;,, where the explicit expressions of the dark energy and dipolar dark matter components are Tµv _ A. ,.µv (19) de -- 87rG Y ' (20) At that order, we find that the dipolar (21) which is that of a perfect fluid with four-velocity ·uP, vanishing prcssmc and energy density (14). In the linear cosmological regime, the dipolar fluid therefore behaves as standard cold dark 260 matter (a pressureless fluid) plus standard dark energy {a cosmological constant). Adjusting the background value O' so that ndm ~ 23%, the model is thus consisten~ with the standard A-CDM scenario and the cosmological observations of the Cl\IH nur.tuations (see 21 for more details). 6 Recovering the MOND phenomenology at galactic scales .\'ext, we turn to the s~udy of the dipolar dark lluid in a typical galaxy at low redshift. We have to consider the non-relativistic limit ( c -t +oc) of the model; we consistently neglect all relativistic terms CJ(c-2 ). It is straightforward to check that in the non-relativistic limit the <" dv -- W' dt - g IT .L , {22) where g is the local gravitational field generated by both the baryonic matter in the galaJ {23) where Pb and p arc respectively the baryonic and dipolar dark matter mass densities. From (i4) the dipolar dark matter density reduces in the non-relativistic limit to p=a-V·II.L. {21) The first term is a usual monopolar contribution: the rest mass density a of the fluid, while the second term is the dipolar contribution, and can be interpreted as coming from the fluid's internal energy. Here Il.L denotes the spatial components of the polarization. In order to recover MO:'J'D, we need two things: {i) to find a mechanism for the alignment of the polarization field II.L \Yith the gravitational fidd g, so that a relation similar t.o (7) ,,ill apply: (ii) lo justify that the rest mass density u of dipole moments in {24) is small with respect to the baryonic density Pb, hence the galaxy will mostly appear as haryonic in MOND fits of rotatio11 curves. Wt• have proposed iu 21 a siugle mcchauism able to answer positively these two points. We call it the hypothesis of weak clustering of dipolar dark matter; it is an hypothesis because it has been conjectured but not proved, and should be checked using numerical simulations. The weak clustering hypothesis is motivated by a solution of the full set of equations describing the non-relativistic motion of dipolar dark mRttcr in a t.ypicRl baryonic 21 galaxy whose mass distribution Pb is spherically symmetric {see the Appendix in ). This partkular solution corresponds to an equilibrium configuration in spherical symmct ry, for which v = 0 and a= ao(r). The dipole moments rcruain at rest because the gravitational field J is balanced by the internal force :F = ft.L W'; for that solution the right-hand-side of {22) vanishes. During the cosmological evolution we expect that the dipolar medium will not cluster much because the internal force may balance part of the local gravitational field generated by an overdensity. The dipolar dark matter densit.y contrast in a typical galaxy at low redshift. should thus be small, at lea~t smaller than in the standard A-CD::'vl scenario. Hence a « Pb. and we could even envisage that a stays around its mean cosmological value, a ~ O' « Pb· Now, because of its size and typical time-scale of evolution, a galaxy is almost unaffected by the cosmological expansion of the Universe. The cosmological mass density O' of the dipolar dark matter is not 011ly homogeneous, but also almost co11sta11t iu this galaxy. The contiuuity e4uation rt·duc.:es to V·(O'v) ~ 0, and the most simple solution corresponds lo a static Iluid verifying v ~ 0. By {22) we see that the polarization field II.L is then al;gnerl with the gravitat.ional field g , namely (25) 261 On the other haud, bc'<:ausc CT « fib by the same meC'hanism, we observe that the ~ravitational field equation (23) wiLh (24) becomes V · (g - 4r.G' II.i) '.::::'. -47rU Pb, (26) which &cording to (5) is found to be rigorously equivalent to the MO'.'JD equation. Finally, by inserling the expression of the potential (16) into (25) and comparing with the defining equaLion (7), we readily find the l\IOND behaviour of the gravitational susceptibility coefficient as x(y) = -1 + .!!__ + O(g2), (27) ac in complete agreement with (2). 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