Unified Brane Gravity: Cosmological Dark Matter from Scale Dependent Newton Constant
aa r X i v : . [ h e p - t h ] J u l Unified Brane Gravity:Cosmological Dark Matter from Scale Dependent Newton Constant
Ilya Gurwich ∗ and Aharon Davidson † Physics Department, Ben-Gurion University, Beer-Sheva 84105, Israel
We analyze, within the framework of unified brane gravity, the weak-field perturbations causedby the presence of matter on a 3-brane. Although deviating from the Randall-Sundrum approach,the masslessness of the graviton is still preserved. In particular, the four-dimensional Newton forcelaw is recovered, but serendipitously, the corresponding Newton constant is shown to be necessarilylower than the one which governs FRW cosmology. This has the potential to puzzle out cosmologicaldark matter. A subsequent conjecture concerning galactic dark matter follows.
PACS numbers: 04.50.Kd, 11.25.Db, 95.35.+d
1. Introduction
The discovery of dark matter [1] continues to be achallenging problem for astrophysics and cosmology. Al-though many ideas from particle physics [2] have beenput forth, none of them so far have been able to providea convincing explanation for its mysterious nature and itspreponderance in the constitution of the Universe. Be-sides, the search for a suitable particle candidate so farhas proved elusive. A different approach proposes thatdark matter is not matter at all in the conventional sensebut rather an artifact originating from a deviation fromgeneral relativity. Various forms of modified theories ofgravity have been proposed to explain the phenomenon[3]. However, all such theories are beset with their ownproblems. In this context, it is important to note the re-markable coincidence in the amounts of galactic and cos-mological dark matter. This is natural for particle darkmatter but has to be explained by any full theory of arti-fact dark matter in modified theories of gravity. Recently,the idea that brane world theories could provide an expla-nation for the dark matter has been suggested [4]. Braneworld theories have recently made great breakthroughsin the area of reproducing some results of general rela-tivity on the cosmic scale, as well as in deriving the lowenergy Newtonian limit [5, 6, 7, 8]. The possibility thatbranes can naturally produce a solution to an unsolvedproblem in gravity, such as dark matter, will generate agreat boost in the theory aside from being a significantachievement and a good verification of the branes andextra-dimensions ideas.In this paper we analyze the weak-field perturbationsaround a flat background generated by matter on thebrane. We do so in the framework of unified brane grav-ity [9], following Dirac’s prescription of careful variationin the region of the brane [10] (we present the basic prin-ciples in the next section). After some remarks on thegeneral scenario, we focus on the radial case, thus study-ing the field created far from a source. Our main resultsare as follows:1. We recover a Newtonian 1 /r potential. 2. The conventional (cosmological) Newton constantis suppressed by a constant factor greater than1. This difference between cosmological and radialNewton constants gives rise to natural cosmologicaldark matter. The amount of dark matter, charac-terized by the ratio between the two constants isan arbitrary parameter at this point.We later discuss the possibility of a transition scale be-tween the two Newton constants. Such a transition willresult in an effective deviation from the Newtonian po-tential. An observer who is unaware of this transitioncould interpret it as a continuous distribution of darkmatter. To calculate the exact transition, one wouldneed to analyze weak-field perturbations around a cos-mological brane, which is very complicated. We do cal-culate roughly the scale of the above transition withoutthe exact solution. Remarkably the scale ∼ ly (lightyears) is only 1 order of magnitude above the experimen-tal scale. It remains to be seen whether this predictionis correct and whether this transition will lead to flatrotation curves.
2. The Basics of Unified Brane Gravity
Dirac has shown[10] that when performing variationof action on a surface, around which one or more of thefields are discontinuous, it is crucial to perform the vari-ation in a coordinate system where this surface remainsstatic, to preserve the linearity of the variation. Vio-lating this principle results in nonlinear variation, andif this problem is untreated, it would lead to incorrectequations of motion. Dirac demonstrated this in his pa-per, where he performed both the naive variation and thecorrect variation on a bubble model of an electron, andshowed that the naive variation results in a missing termin the equations of motion. In [11], Karasik and David-son demonstrated how the naive variation would lead toa wrong Snell law, whereas the Dirac prescription leadsto the correct equation.Unified brane gravity[9] is based on the same actionprinciple as the standard brane models (Randall-Sundrum, Dvali-Gabadadze-Porrati, and Collins-Holdom) but following carefully Dirac’s prescription forcorrect variation of the brane. We work in a coordinatesystem, where the variation of the bulk metric on thesurface of the brane has only 5 degrees of freedom, δg ab = g AB,C δy C y A,a y B,b + 2 g AB y A,a δy B,b , (1)where g AB is the bulk metric and y A are the bulk coor-dinates. This way, the brane remains undeformed duringthe variation. The other degrees of freedom are not lostbut are simply expressed by the constraints that definethe brane. The equations of motion remain covariant andare independent of the reference frame. Using this princi-ple of variation, the unified brane gravity field equationsfor a Z symmetric AdS bulk with an AdS scale b − takethe form14 πG ( K µν − g µν K ) =3 b πG g µν + 18 πG (cid:18) R µν − g µν R (cid:19) + T µν + λ µν . (2)In addition to the familiar terms (namely, the Israel junc-tion term, the brane surface tension, the Einstein tensorassociated with the scalar curvature R , and the phys-ical energy-momentum tensor T µν = δ L matter /δg µν ofthe brane), unified brane gravity introduces λ µν . Thelatter consists of Lagrange multipliers associated withthe fundamental induced metric constraint g µν ( x ) = g AB ( y ( x )) y A,µ y B,ν . In the above field equations, λ µν servesas a geometric (embedding originated) contribution tothe total energy-momentum tensor of the brane. If thevariation would be performed naively, it would yield λ µν = 0 . (3)This is reminiscent of Dirac’s missing term. A vanishing λ µν in Eq.(2) results in the familiar Collins-Holdom equa-tions (Dvali-Gabadadze-Porrati with AdS bulk), wherethese in turn contain the Randall-Sundrum and Dvali-Gabadadze-Porrati equations as special cases. FollowingDirac’s variation principle, we find that λ µν is nonzerobut is conserved, and its contraction with the extrinsiccurvature vanishes λ µν ; ν = 0 , λ µν K µν = 0 . (4)This is reduced to the Regge-Tietelboim theory in thestatic bulk limit. Since the Regge-Tietelboim theory wasderived in a static bulk and is the simplest of all branemodels, it is very reassuring to have it as a limit (thestandard brane models are unsuccessful in that).
3. General Perturbations and theGraviton
We begin with the simplest scenario of a four-dimensional flat brane of positive tension embedded infive-dimensional AdS bulk ds = dy + e − b | y | η µν dx µ dx ν . (5) b − = p − / Λ denotes the AdS scale, η µν is the four-dimensional Minkowski metric, and the brane is conve-niently located at y = 0. Before turning to the maindiscussion concerning perturbations of this brane, it isimperative to understand the full potential of the unper-turbed brane. In the conventional Randall-Sundrum andCollins-Holdom scenarios, in order to ensure its flatness,the brane has to be of positive (or negative) tension: σ = 3 b πG . (6)Unified brane gravity, although it requires the same, al-lows for one more degree of freedom.To see the point, first recall the unified brane gravityfield equations (2,4). For a flat brane embedded in a five-dimensional AdS background, which is the special caseof interest, K µν = − bη µν . In turn, Eq.(4) simply im-plies that the corresponding λ µν is traceless. A tracelessand conserved source serves as an effective (positive ornegative) radiation term.The flatness of the unperturbed brane can be achievedthe conventional way, if the energy-momentum and theembedding terms both vanish, that is, T µν = λ µν = 0.But now there exists the milder option T µν + λ µν = 0.Following the above, if (and only if) the real matteron the brane exclusively consists of radiation, one canchoose an appropriate λ µν to cancel it out. To be morespecific, let our unperturbed flat brane host a constantradiation density ρ , and choose the embedding countert-erm to be λ µν = − T ,radµν = − diag (cid:0) ρ, ρ, ρ, ρ (cid:1) . Thisreflects the peculiarity that a flat brane can in fact behot , which is unique to unified brane gravity. The per-turbations around such a brane are expected to be quitedifferent from those around a Collins-Holdom brane, thusgiving rise to new physics. To study the perturbationsinduced by an arbitrary source δT µν ≡ τ µν , we find ituseful to invoke Gaussian normal coordinates, such that δg AB = h µν δ µA δ νB are the only allowed nonzero compo-nents (and reserve the option of supplementing this gaugelater by the traceless nontransverse gauge). It is impor-tant to keep in mind that, although τ µν is arbitrary, theperturbations of the metric are accompanied by built-in perturbations of all of the brane components that are per-turbed not by the source itself but rather by the shift inthe brane space-time structure. For example, the radia-tion energy-momentum term must still satisfy the conser-vation and traceless conditions, but it must be satisfiedin the new metric. To this extent, the radiation term iscorrected via a perturbation T radµν = T ,radµν + δT radµν thatsatisfies ∂ ν δT radµν = η νλ (cid:0) Γ σλµ T ,radσν + Γ σλν T ,radµσ (cid:1) + h νλ ∂ λ T ,radµν , (7) η µν δT radµν = h µν T ,radµν , (8)to preserve conservation and tracelessness, respectively.Here Γ λµν = (cid:0) − ∂ λ h µν + ∂ µ h λν + ∂ ν h λµ (cid:1) is the affine con-nection. The above perturbation does not represent anaddition of radiation (which can be present indepen-dently via τ µν ) but rather a geometric effect. By thesame token, the embedding term is also perturbed via λ µν = λ µν + δλ µν and satisfies ∂ ν δλ µν = η νλ (cid:0) Γ σλµ λ σν + Γ σλν λ µσ (cid:1) + h νλ ∂ λ λ µν , (9) η µν δλ µν = b − δK µν λ µν , (10)However, since for a general perturbation δK µν is notproportional to h µν , the term s µν ≡ λ µν + T radµν = δλ µν + δT radµν (11)is not necessarily zero. One can furthermore verify that s µν is conserved and not necessarily traceless: s ≡ η µν s µν = 12 b λ µν (cid:18) ∂∂ | y | + 2 b (cid:19) h µν . (12)The nonlocalized part of the perturbation equations isthe same as the familiar Randall-Sundrum case, sincethe bulk still follows the normal five-dimensional Einsteinequations ∂ ∂ | y | − b + e b | y | ! h µν = 0 , (13)where ≡ η µν ∂ µ ∂ ν is the four-dimensional (unper-turbed) d’Alembertian. The localized part of the equa-tion is δ ( y ) (cid:20) πG (cid:18) ∂∂ | y | + 2 b (cid:19) + 18 πG (cid:21) h µν = δ ( y ) ( τ µν + s µν ) . (14)The propagation of modes into the bulk remains the sameas in all of the familiar cases. Thus, we will be focusingon only the perturbations on the brane. Performing sep-aration of variables, h µν = A ( y )¯ h µν ( x µ ) [12], where wehave normalized without loss of generality A (0) = 1 anddefine α = 1 + A ′ (0)2 b . Next let us separate the perturba-tion ¯ h µν = h ( m ) µν + h ( u ) µν to the standard term h ( m ) µν , whichfollows the usual brane equation and thus admits the fa-miliar solutions and the new term h ( u ) µν , which is a direct result of the additional effective source s µν . For h ( m ) µν , wecan write (cid:18) αb πG RS + 18 πG (cid:19) h ( m ) µν = τ µν , (15)where G RS = bG is the Randall-Sundrum gravitationalconstant on the brane, whereas for the new term (cid:18) αb πG RS + 18 πG (cid:19) h ( u ) µν = s µν . (16)Unfortunately, we cannot find a general Green functionto Eq.(16), because there is no closed-form expressionof s µν in terms of h ( u ) µν . To that end, the only generalprescription to solve Eq.(16) is perturbatively in ρ (seeAppendix A). Despite not being able to find a generalsolution, we can get a clue on its properties by takingthe trace of Eq.(14) and reorganizing the various terms δ ( y ) (cid:18) πG η µν − b λ µν (cid:19) (cid:18) ∂∂ | y | + 2 b (cid:19) h µν + 18 πG η µν h µν = δ ( y ) η µν τ µν . (17)Keeping in mind that λ µν ∼ − ρ and G N ∼ bG , bylooking at the first term in the equation, one may expectthat the effective Newton constant may take the followingform: 1 G N = 1 G CH + β ρb , (18)where 1 G CH = 1 G RS + 1 G (19)is the effective Newton constant in the Collins-Holdomscenario and β is a dimensionless constant. In the nextsection, we show that this prediction is indeed true and,interestingly, β is geometry dependent.Although we did not obtain a propagator for the gravi-ton, the form of the equation looks all too similar tothe usual brane equations and along with Eq.(17) sug-gests that, despite deviating from the standard Randall-Sundrum scenario, the graviton propagator remains thesame. In the following section, we show that the New-tonian potential is recovered for large r and thus provethat the graviton’s zero-mode is massless.
4. Static Radial Source
In all studies of gravitational perturbations, the point-like radial source is of special interest. Since an exactradial solution is missing in all brane theories, the bestidea we have for a radial potential comes from pertur-bative treatment. We solve the equations far from thesource, in the region where τ µν = 0.For the radial case, we show that an exact (non per-turbative in ρ ) weak-field solution can be obtained. Thisis mainly due to the fact that we are able to express s µν explicitly. We choose to work in a traceless Gaussianframe. For a radially symmetric perturbation, we canchoose, in addition to the Gaussian traceless gauge, theradial gauge [13]. It follows that s θθ = s ϕϕ = 0. Solvingthe conservation equation for s µν along with Eq.(12) andgauging following the above, we have s tt ( r ) = s rr ( r ) = − s ( r ) + 12 r Z drrs ( r ) . (20)From Eq.(16), we see that since s tt = s rr it followsthat h ( u ) tt = h ( u ) rr ≡ h ( u ) . This is the familiar form ofradial fluctuations. Finally, h ( m ) µν constitutes the famil-iar Collins-Holdom solution. The exact solution is quitecomplicated, but to first order in 1 /r , the solution simplyyields h ( m ) tt = h ( m ) rr ∼ = 2 G CH Mr , (21)where M = R d xτ tt is the mass of the source. Substi-tuting Eq.(12,20,21) along with = 1 r ddr (cid:18) r ddr (cid:19) (22)into Eq.(16), we can write the equation for h ( u ) κ rh ( u ) ′′′ + 4 κ h ( u ) ′′ + (cid:18) κ r + (cid:18) k − αρ (cid:19) r (cid:19) h ( u ) ′ +2 kh ( u ) = − G CH M αρ r , (23)where κ ≡ πG and k ≡ αb πG RS . The solution ofphysical relevance is the nonhomogeneous one, namely, h ( u ) = − G CH M b πG RS ρ r . (24)The full perturbation ¯ h µν = h ( m ) µν + h ( u ) µν is therefore¯ h tt = ¯ h rr = 11 + 4 πG RS ρ b G CH Mr . (25)It is important to note that it is only due to the solu-tion being independent of α that we can proceed withoutintegrating over all the mass modes. The Newtonian po-tential is thus recovered, giving us further reassurancethat the graviton is indeed massless, since a mass termin the propagator would have generated an exponentialdecay. The associated Newton constant is G rN = G CH πG RS ρ b , (26) where the r index stands for radial.Now that the mathematics has been understood, wereturn to physics. Alone, Eq.(26) has nothing new tooffer. However, gravitational measurements in our Uni-verse, although they began with the Solar System, whichis physically a radially symmetric system, are now quitebased in the field of cosmology as well. We recall (see Ap-pendix B) the cosmological result for expansion arounda flat background gives an FRW solution with an associ-ated Newton constant1 G cN = 1 G RS + 1 G + 4 πρ b , (27)where the c index stands for cosmological and ρ here hasthe exact same role of background radiation. Equation(26) can also be written as1 G rN = 1 G RS + 1 G + 4 πρ b (cid:18) G RS G (cid:19) . (28)Now, if we further assume that the role of radiation inour case is also played by the background radiation fromcosmology, we can compare the two results. First of all,since we do have bounds on b from both particle andgravitational localization, we can clearly state that theterm ρb is negligible in both equations. This means that G cN = G CH , whereas1 G rN = 1 G cN + 4 πρ b G RS G . (29)The last term in the radial gravitational constant wouldhave been negligible if not for the factor G RS G . We haveno experimental or theoretical bounds on the latter ratio.In fact, the proposed self-accelerated Dvali-Gabadadze-Porrati solution for the cosmological constant requiresthis quantity to be very large. If it is large enough, thenthe above term can be significant in the calculation ofthe Newton constant. Thus, in principle, we have a realdifference between the cosmological and the radial grav-itational constants, the radial constant being necessarilylower. However, historically, the Newton constant wasmeasured in radial systems (the Solar System). Thus anobserver that is unfamiliar with this physics would in-terpret this effective growth of the gravitational constantas missing cosmological mass (since, in general relativity,mass is inseparable from the gravitational constant), thusbringing him to the phenomenon of cosmological darkmatter, without facing dark matter in the Solar System .Although we have not shown it here (this is a conjec-ture subject to future research), when solving the pertur-bation equations around a cosmological background, oneexpects the two branches of the solution, one being the G rN and the other G cN , to be connected, creating somesort of transition between them. Such a transition, toan observer that is unaware of this effect, will seem asa gradual increase of mass, that may result in flat rota-tion curves. Although the exact solution to fluctuationsaround a cosmological brane is highly complex, we cangive a rough estimate to the typical scale of such a tran-sition. We assume the scale to be roughly in the regionwhere the cosmological and radial curvatures are of thesame order of magnitude, so that the cosmological andradial solutions ”mix” . The radial curvature is of theorder r s r , r s being the Schwarzschild radius and the cos-mological curvature is of the order of H , H being theHubble constant. The scale of the predicted transition istherefore r dm ∼ (cid:0) r s t Hubble (cid:1) / ∝ M / , (30)where t Hubble is the age of the Universe. When this scaleis calculated for the Sun, the result is 100 ly, which is waybeyond the scale of the Solar System. At these distances,other stars contribute, and thus the effect is unmeasur-able today. For a galactic mass, on the other hand, theresult is of the order of 10 ly, which is only 1 order ofmagnitude higher than the real galactic scale. One needsto remember that it is only a rough estimate and alsothat galaxies are not radial systems and are composed ofmany stars, each giving an effect on the scale of about100 ly, so that the combined effect may be closer thanthe above result, to give the exact scale of flat rotationcurves.
5. Summary and Conclusions
We have studied the behavior of weak-field perturba-tions around a flat brane, in the framework of unifiedbrane gravity. It was shown that, even for the most gen-eral perturbation, the novel embedding term is ”harm-less” and the graviton propagator is intact, leaving thegraviton massless. We verify this result, in particular, fora spherically symmetric source, where the conventionalNewtonian potential 1 /r is recovered. However, upona closer examination, we see that, although the func-tional form of the potential is standard, the gravitationalconstant differs from the one found in cosmology. Fur-thermore, the radial gravitational constant is necessarilylower than the cosmological one. For an observer, famil-iar only with Einstein’s general relativity, this would beimmediately interpreted as cosmological dark matter .This can also be the source of galactic dark matter. Theflat rotation curves may simply represent the transitionbetween the radial and cosmological gravitational con-stants. The scale of the suggested flat rotation curves ispredicted in this case to be of the order of (cid:0) r s t age (cid:1) / .When evaluated for a galactic mass, this is indeed closeto the galactic scale. Despite this transition being thenatural outcome of the two different gravitational con-stants, there is no reason why such a transition would generate flat (rather than some general form) rotationcurves, and the flatness of the rotation curves is wishfulthinking at this point.Although the radial dark matter solution is completelyspeculative in this paper, the cosmological dark matter isfully postulated. The only thing that is arbitrary is theamount of dark matter. This is due to the arbitrariness of G RS G . In fact, in order to account for the right amountof dark matter, we would need an extremely large G ,implying a very low five-dimensional plank mass M ≈ ( ρl ) / . Appendix A: Perturbative Method
We can expand the solution to Eq.(16) via h ( u ) µν = ∞ X i =1 h ( i ) µν (31)and (cid:18) αb πG RS + 18 πG (cid:19) h ( i ) µν = s ( i ) µν , (32)where, for i > s ( i ) ≡ s ( i ) µν η µν = αλ µν h ( i − µν (33)and s (1) ≡ s (1) µν η µν = αλ µν h ( m ) µν . (34) Appendix B: Cosmological GravitationalConstant
In [9] we have proven that, when expanding the cosmo-logical equations around a flat background with positivetension and radiation density of Eq.(76), ρ ( a ) = r − Λ ωa , (35)where ω is a constant. The resulting FRW equation wasgiven by Eq.(81): e ρ = (cid:18) πG + r − Λ (cid:18) πG + ρ b (cid:19)(cid:19) ǫ , (36)where ǫ = 3 ˙ a + ka and, therefore,1 G cN = 1 G + 1 G RS + r − Λ πω Λ a . (37)We would like to express the last term in Eq.(37) in termsof ρ and b and, therefore,1 G cN = 1 G CH + 4 πρ b . (38)The authors thank Professors Philip Mannheim, Ed-uardo Guendelman and especially our colleague ShimonRubin for enlightening discussions and constructive com-ments. A special thanks to Wali Kameshwar for con-structive comments that have significantly improved thispaper. ∗ Email: [email protected] † Email: [email protected][1] E. Komatsu et al. (WMAP Collaboration), Astrophys.J. Suppl. , 330 (2009). R.B. Tully and J.R. Fisher,Astron. Astrophys. , 661 (1977). M. Davis et al. As-trophys. J. , 371 (1985). F. Zwicky, Helv. Phys. Acta. , 110 (1933). S.M. Carroll, Nature Phys. , 653 (2006).[2] G. Jungman, M. Kamionkowski and K. Griest, Phys.Rept. , 195 (1996). R. Holman, G. Lazarides and Q.Shafi, Phys. Rev. D27 , 995 (1983). C.D. Froggatt andH.B. Nielsen Phys. Rev. Lett. , 231301 (2005).[3] M. Milgrom, Astrophys. J. , 635(1989). J.D. Bekenstein, Phys. Rev. D70 , 083509 (2004).[4] J.A.R. Cembranos, A. Dobado and A.L. Maroto, Phys.Rev. Lett. , 241301 (2003). M.K. Mak and T. Harko,Phys. Rev. D70 , 024010 (2004).[5] W. Israel, Nuovo Cimento
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