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--<l prut·icle dark matter approach, iu which the <lark matter c;urnsists of unknown non-baryonic particles, e.g. neutralinos as predicted by super-symmetric extensions of the standard model of particle physics (sec 1 for a review). Furthermore, the dark matter triggers the formation of large-scale structures by gravitational collapse and l'Xplains the distri­ bution of baryonic matter from galaxy duster scales up to cosmological 5<'ales by the non-linear growth of i11itial perturbations. Simulatio11s suggest some universal <lark matter density profile 2 around distributions of ordinary baryonic matter • An important characteristic of dark mat­ ter, required by the necessity of clustering matter on small scales, is that it should be cold (or non-relativistic) at the epoch of galaxy formation. Together with the obsen·ational evidence of rlark em~rgy (prnsnmably a cosmological constant. A) measnrcrl from the Hubble rliagram of supernovas, the particle dark matter hypothesis has yielded the successful concordance model of cosmology called A-CDl\I, whi--:h reproduces extremely well I.he observed cosmic microwave background spectrum 3 . However, despite these successes at cosmological scales, the particle dark matler approach has sowe dHEcult.ies 4 aL explaining u1 a natural way the Oat rotat.iou curves of galaxies. one of Lhe inost persuasi"e evidence for t.he existence of dark matter, and the Tully-Fisher empirical relar.ion between the observed luminosity ancl he a..."'Ymptotic rotation velocity of spiral gala."<:ies. In orde1· to de.al with these dif!knltics rul linked witl.i the propextics of <lark matter !It glllac-tk scales an alternative paradigm has emerged in the name of ~he modified l\ewtoni..a.n dynamics 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 <lark matter au<l <lark cucrgy. This relativistic mu<lcl will be shuwu tu heudit frum both the successes of A-CDl\I at cosmological scales, and MOND at galactic scales. 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 <lata. The specific MO:\D regime corresponds to the weak gravity 5 6 7 limiL much weaker than the scale ao- In this regime (where formally g -> 0) we have • • 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<la1ne11tal µhysics. 111 a11y case the agreerne11t uf (1)-(2) with a large number of observaLions calls for a clear physical explanation. Taking the divergence of both sides of (1), and using the usual Poisson equation for the .:\ewtonian field gx, we obtain a local formulation of MOND in the form of the modified Poisson equation 10 V · (p. g) = -4rrG Pb , (3) where Pb is the density of baryonic matter. This equation can be derived from a Lagrangian, and that Lagrangian has been the starting point for constructing relativistic extensions of MOND. Such extensions postulate the existence of extra (supposedly fundamental) fields associated with gravity besides the spin-2 field of general relativity. Promoting the Newtonian potential U to a scalar field q>, scalar-tensor theories for MOND have been constructed 11 but shown to be non viable; essentially because light signals do not feel the presence of the scalar field, since the physical (Jordan-frame) metric is conformally related to the Einstein-frame metric, and the 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<ls as long as the mass density of polarizc<l 111asses appears as the <livergem;c of a \·ector fied, namely takes the dipolar form 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.