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Implications of Graviton–Graviton Interaction to Dark Matter

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Implications of Graviton–Graviton Interaction to Dark Matter View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Physics Letters B 676 (2009) 21–24 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Implications of graviton–graviton interaction to dark matter A. Deur 1 University of Virginia, Charlottesville, VA 22904, USA article info abstract Article history: Our present understanding of the universe requires the existence of dark matter and dark energy. Received 26 February 2009 We describe here a natural mechanism that could make exotic dark matter and possibly dark energy Received in revised form 8 April 2009 unnecessary. Graviton–graviton interactions increase the gravitational binding of matter. This increase, Accepted 22 April 2009 for large massive systems such as galaxies, may be large enough to make exotic dark matter superfluous. Available online 25 April 2009 Within a weak field approximation we compute the effect on the rotation curves of galaxies and find the Editor: A. Ringwald correct magnitude and distribution without need for arbitrary parameters or additional exotic particles. PACS: The Tully–Fisher relation also emerges naturally from this framework. The computations are further 95.35.+d applied to galaxy clusters. 95.36.+x © 2009 Elsevier B.V. Open access under CC BY license. 95.30.Cq Cosmological observations appear to require ingredients beyond (dark matter) or gravity law modifications such as the empirical standard fundamental physics, such as exotic dark matter [1] and MOND model [4]. dark energy [2]. In this Letter, we discuss whether the observa- Galaxies are weak gravity field systems with stars moving at tions suggesting the existence of dark matter and dark energy non-relativistic speeds. For weak fields, the Einstein–Hilbert action could stem from the fact that the carriers of gravity, the gravitons, can be rigorously expanded in a power series of the coupling k 2 interact with each others. In this Letter, we will call the effects (k ∝ G) by developing the metric gμν around the flat metric ημν . of such interactions “non-Abelian”. The discussion parallels similar This is known (see e.g. Refs. [5,6]) but we recall it for convenience: kψ phenomena in particle physics and so we will use this terminol- gμν is parametrized, e.g. gμν = (e )μν , and expended around ogy, rather than the one of General Relativity, although we believe ημν .Itleadsto: it can be similarly discussed in the context of General Relativity. √ 1 4 μν We will connect the two points of view wherever it is useful. d x −ggμν R Although massless, the gravitons interact with each other be- 16π G cause of the mass–energy equivalence. The gravitational cou- = d4x ∂ψ∂ψ + kψ∂ψ∂ψ + k2ψ 2∂ψ∂ψ +··· pling G is very small so one expects G2 corrections to the Newto- nian potential due to graviton–graviton interactions to be small in μν + kψμν T . (1) general. However, gravity always attracts (gravitons are spin even) μν μν and systems of large mass M can produce intense fields, balancing Here, g = det gμν , R is the Ricci tensor, ψ is the gravity field, μν the smallness of G.Indeed,G2 corrections have been long ob- and T is the source (stress–energy) tensor. Since our interest served for the sun gravity field since they induce the precession is ψ self-interactions, we will not include the source term in the μν of the perihelion of Mercury. Such effects are calculable for rel- action. (We note that it does not mean that T is negligible: we 00 μν atively weak fields, using either the Einstein field equations (the will use later the fact that T is large. It means that T is not non-linearity of the equations is related to the non-Abelian na- a relevant degrees of freedom in our specific case. This will be ture of gravity [3]), or Feynman graphs in which the one-graviton further justified later.) A shorthand notation is used for the terms exchange graphs produce the Newtonian (Abelian) potential and ψn∂ψ∂ψ which are linear combinations of terms having this form higher order graphs give some of the G2 corrections. Gravity self- for which the Lorentz indexes are placed differently. For example, coupling must be included in these calculations to explain the the explicit form of the shorthand ∂ψ∂ψ is given by the Fierz– measured precession [3]. The non-Abelian effects increase gravity’s Pauli Lagrangian [7] for linearized gravity field. strength which, if large enough, would mimic either extra mass The Lagrangian L is a sum of ψn∂ψ∂ψ. These terms can be + transformed into 1 ψn 1∂2ψ by integrating by part in the ac- n+1 tion d4x L. We consider first the ∂ψ∂ψ term. The Euler–Lagrange E-mail address: [email protected]. equation of motion obtained by varying the Fierz–Pauli Lagrangian 1 1 2 μν =− 2 μν − μν 00 Present address: Thomas Jefferson National Accelerator Facility, Newport News, leads to ∂ ψ k (T 2 η Tr(T )).SincetheT compo- μν VA 23606, USA. nent dominates T within the stationary weak field approxima- 0370-2693 © 2009 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2009.04.060 22 A. Deur / Physics Letters B 676 (2009) 21–24 Fig. 1. Graviton interactions corresponding to the terms of the Lagrangian, Eq. (2). tion, so too ∂2ψ 00 dominates ∂2ψμν and one can keep only the ψ 00 terms in ψ∂2ψ,i.e.in∂ψ∂ψ. Finally, after applying the har- μ = 1 κ monic gauge condition ∂ ψμν 2 ∂νψκ ,weobtainforthefirst L → 1 00 λ term in ∂ψ∂ψ 4 ∂λψ ∂ ψ00. Higher order terms proceed 1 n+1 2 2 similarly since they are all of the form n+1 ψ ∂ ψ. The factor in front of each ψn∂ψ∂ψ (n = 0) may depend however on how gμν is expanded around ημν . For this reason, and because the higher order terms are complicated to derive, we use a different Fig. 2. Two-point Green function, after subtracting the 1/r contribution (dotted line). approach to determine the rest of the Lagrangian: we build it from r is in lattice units. the appropriate Feynman graphs (see Fig. 1) using, with hindsight of previous discussion, only the ψ 00 ≡ φ component of the field. Each term in the Lagrangian corresponds to a Feynman graph: the To quantify gravity’s non-Abelian effects on galaxies, we have terms quadratic, cubic and quartic in φ correspond respectively to used numerical lattice techniques: A Monte Carlo Metropolis algo- the free propagator, the three legs contact interaction and the four rithm was employed to estimate the two-point correlation function legs contact interactions. The forms φ∂φ∂φ and φ2∂φ∂φ (rather (Green function) that gives the potential. To test our Monte Carlo, than φ3 or φ2∂φ for example for the three legs graph) are imposed we computed the case for which the high-order terms of L are by the dimension of G. (Note that in Eq. (1), the origin of the two set to zero, and recovered the expected Newtonian potential or, derivatives in the generic form φn∂φ∂φ is from the two derivatives when a fictitious mass mφ is assigned to the field φ, the expected in the Ricci tensor and the absence of derivative in g .) − μν Yukawa potential V (r) ∝ (e mφr )/r.Wealsoinsuredtheindepen- Since we are considering the total field from all particles, then dence of our results from the lattice size and the physical system (neglecting here non-linear effects and binding energies) k2 = size. In our calculations, the usual circular boundary conditions m16π G where m is the nucleon mass and m = M with M the cannot be used. A pathological example is the one of a linear po- total mass of the system. This may be modeled with a space that is tential, for which a simulation with such conditions would return discretized with a lattice spacing d. We are interested in the attrac- an irrelevant constant rather than the potential. Instead of circu- tion between two cubes of d3 volume filled with the gravity fields lar boundary conditions, we set the boundary nodes of the lattice generated by N sources of similar masses and homogeneous dis- to be random with an average zero value. These nodes were never tribution. Since we are unable to treat N sources we consider only updated. In addition, although we update the fields on the nodes a global field. Under the field superposition principle, the magni- close to the boundary nodes, we did not use them in the calcu- tude of the total field in each cube is proportional to N.Asgravity lation of the Green function (in the results presented, we ignored always attracts, we used a global coupling N mG = MG.3 1 the 4 nodes closest to the lattice boundary. We varied this number Under these simplifications and hypotheses we obtain from and found compatible results). Eq. (1)4: The Green function is shown in Fig. 2. These results are for a √ 4 lattice of size L = 28d,withd the lattice spacing, 16π GM = 4.9 × 4 4 16π GM d d x L = d x ∂μφ∂μφ + φ∂μφ∂μφ −3 10 3! 10 (d converts the coupling to lattice units), M = 10 M and μ=1 d = 1kpc.Forr 5, the Green function is roughly linear. 16π GM 2 We now apply our calculations to the case of galaxies. For ho- + φ ∂μφ∂μφ +··· . (2) 4! mogeneous distributions with spherical symmetry, the net field distortion from the force carrier self-interaction cancels out.5 Sim- ilarly, a cylindrical symmetry reduces the effect.
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