Implications of Graviton-Graviton Interaction to Dark Matter

Our present understanding of the universe requires the existence of dark matter and dark energy. We describe here a natural mechanism that could make exotic dark matter and possibly dark energy unnecessary. Graviton-graviton interactions increase the gravitational binding of matter. This increase, for large massive systems such as galaxies, may be large enough to make exotic dark matter superfluous. Within a weak field approximation we compute the effect on the rotation curves of galaxies and find the correct magnitude and distribution without need for arbitrary parameters or additional exotic particles. The Tully-Fisher relation also emerges naturally from this framework. The computations are further applied to galaxy clusters.

Cosmological observations appear to require ingredients beyond standard fundamental physics, such as exotic dark matter [1] and dark energy [2]. In this Letter, we discuss whether the observations suggesting the existence of dark matter and dark energy could stem from the fact that the carriers of gravity, the gravitons, interact with each others.
In this Letter, we will call the effects of such interactions "non-Abelian". The discussion parallels similar phenomena in particle physics and so we will use this terminology, rather than the one of General Relativity, although we believe it can be similarly discussed in the context of General Relativity. We will connect the two points of view wherever it is useful.
Although massless, the gravitons interact with each other because of the mass-energy equivalence. The gravitational coupling G is very small so one expects G 2 corrections to the Newtonian potential due to graviton-graviton interactions to be small in general. However, gravity always attracts (gravitons are spin even) and systems of large mass M can produce intense fields, balancing the smallness of G. Indeed, G 2 corrections have been long observed for the sun gravity field since they induce the precession of the perihelion of Mercury. Such effects are calculable for relatively weak fields, using either the Einstein field equations (the non-linearity of the equations is related to the non-Abelian nature of gravity [3]), or Feynman graphs in which the one-graviton exchange graphs produce the Newtonian (Abelian) potential and higher order graphs give some of the G 2 corrections. Gravity selfcoupling must be included in these calculations to explain the measured precession [3]. The non-Abelian effects increase gravity's strength which, if large enough, would mimic either extra mass (dark matter) or gravity law modifications such as the empirical MOND model [4].
Galaxies are weak gravity field systems with stars moving at non-relativistic speeds. For weak fields, the Einstein-Hilbert action can be rigorously expanded in a power series of the coupling k (k 2 ∝ G) by developing the metric g µν around the flat metric η µν . This is known (see e.g. refs. [5], [6]) but we recall it for convenience: g µν is parametrized, e.g. g µν = e kψ µν , and expended around η µν . It leads to: Here, g = det g µν , R µν is the Ricci tensor, ψ µν is the gravity field, and T µν is the source (stress-energy) tensor. Since our interest is ψ self-interactions, we will not include the source term in the action. (We note that it does not mean that T µν is negligible: we will use later the fact that T 00 is large. It means that T µν is not a relevant degrees of freedom in our specific case. This will be further justified later.) A shorthand notation is used for the terms ψ n ∂ψ∂ψ which are linear combinations of terms having this form for which the Lorentz indexes are placed differently. For example, the explicit form of the shorthand ∂ψ∂ψ is given by the Fierz-Pauli Lagrangian [7] for linearized gravity field.
The Lagrangian L is a sum of ψ n ∂ψ∂ψ. These terms can be transformed into 1 n+1 ψ n+1 ∂ 2 ψ by integrating by part in the action d 4 xL. We consider first the ∂ψ∂ψ term. The Euler-Lagrange equation of motion obtained by varying the Fierz-Pauli Lagrangian leads to ∂ 2 ψ µν = −k 2 (T µν − 1 2 η µν T r(T )). Since the T 00 component dominates T µν within the stationary weak field approximation, so too ∂ 2 ψ 00 dominates ∂ 2 ψ µν and one can keep only the ψ 00 terms in ψ∂ 2 ψ, i.e. in ∂ψ∂ψ. Finally, after applying the harmonic gauge condition ∂ µ ψ µν = 1 2 ∂ ν ψ κ κ , we obtain for the first term in L ∂ψ∂ψ → 1 4 ∂ λ ψ 00 ∂ λ ψ 00 . Higher order terms proceed similarly since they are all of the form 1 n+1 ψ n+1 ∂ 2 ψ [8]. The factor in front of each ψ n ∂ψ∂ψ (n = 0) may depend however on how g µν is expanded around η µν . For this reason, and because the higher order terms are complicated to derive, we use a different approach to determine the rest of the Lagrangian: we build it from 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 terms quadratic, cubic and quartic in φ correspond respectively to the free propagator, the three legs contact interaction and the four legs contact interactions. The forms φ∂φ∂φ and φ 2 ∂φ∂φ (rather than φ 3 or φ 2 ∂φ for example for the three legs graph) are imposed by the dimension of G.
(Note that in Eq. 1, the origin of the two derivatives in the generic form φ n ∂φ∂φ is from the two derivatives in the Ricci tensor and the absence of derivative in g µν ).
Since we are considering the total field from all particles, then (neglecting here non-linear effects and binding energies) k 2 = m 16πG where m is the nucleon mass and m = M with M the total mass of the system. This may be modeled with a space that is discretized with a lattice spacing d. We are interested in the attraction between two cubes of d 3 volume filled with the gravity fields generated by N sources of similar masses and homogeneous distribution. Since we are unable to treat N sources we consider only a global field. Under the field superposition principle, the magnitude of the total field in each cube is proportional to N. As gravity always attracts, we used a global coupling N 1 mG = MG [9]. Under these simplifications and hypotheses we obtain from Eq. 1 [10]: To quantify gravity's non-Abelian effects on galaxies, we have used numerical lattice techniques: A Monte Carlo Metropolis algorithm was employed to estimate the two-point correlation function (Green function) that gives the potential. To test our Monte Carlo, we computed the case for which the high-order terms of L are set to zero, and recovered the expected Newtonian potential or, when a fictitious mass m φ is assigned to the field φ, the expected Yukawa potential V (r) ∝ (e −m φ r )/r. We also insured the independence of our results from the lattice size and the physical system size. In our calculations, the usual circular boundary conditions cannot be used. A pathological example is the one of a linear potential, for which a simulation with such conditions would return an irrelevant constant rather than the potential. Instead of circular boundary conditions, we set the boundary nodes of the lattice to be random with an average zero value. These nodes were never updated. In addition, although we update the fields on the nodes close to the boundary nodes, we did not use them in the calculation of the Green function (in the results presented, we ignored the 4 nodes closest to the lattice boundary. We varied this number and found compatible results).
The Green function is shown in Fig. 2 We now apply our calculations to the case of galaxies. For homogeneous distributions with spherical symmetry, the net field distortion from the force carrier self-interaction cancels out [11]. Similarly, a cylindrical symmetry reduces the effect. To first order we treat a spiral galaxy as a thin disk with a cylindrical symmetry and approximate V (r) in Fig. 2 as linear: the force is of constant value b. We are interested in the force between the disk center and the circumference points. The field lines point evenly outward, so the force at any point on the circumference is reduced by 2πr: the force is then b/(2πr) and V (r) = b ln(r)/(2π).
Adding back the (unaffected) "Abelian" part a/r, we obtain: (For a homogeneous spherical distribution, the constant force becomes b/4πr 2 , leading to a Newtonian potential (a + b/4π) /r ∝ 1/r, as expected. We note that typically, a ≫ b/4π.) We can now look at rotation curves for spiral galaxies. Those, shown in Fig. 3, are obtained by calculating a and b for given galaxy masses and sizes and assuming an exponential decrease of the galaxy density with its radius: ρ(r) = M 2πr 2 0 e −r/r 0 . Galaxy luminous masses and sizes being not well known, we adjusted M and r 0 to best fit the data. They can be compared to the luminosity L of the galaxies and the values of r SL from Ref. [12] also given in Fig. 3 We did not use L in the simulation but report it since it indicates a lower bound for M (consequently, NGC7331 for which M < L pauses a problem within our simple spiral galaxy model). The curves reproduce well the data given our simple model of galaxy.
In addition to our rough approximation in of modeling galaxies, it should be emphasized that while our results should conservatively be viewed as indicating quantitatively the selfcoupling effects [13] of the gravity field, there are several caveats: 1) The particular choice of boundary conditions may generate a non-physical artifact, although we checked within the means of our lattice simulation that this was not the case; 2) There are approximations inherent to a lattice calculation, in particular the cut-off on the high energy modes due to the lattice finite spacing; 3) Approximations are used to go from the Einstein-Hilbert action to the polynomial scalar action; 4) We have used an approximate magnitude for the field self-coupling of √ GM , which neglects non-linear effects and the specific distribution of sources in the studied system. Ref. [12] for the galaxy radii, and the values given on each plots for the parameters M and r 0 . The luminosity L of the galaxies and the values of r 0 (noted scale length, SL) from Ref. [12] are also given for comparison (units are 10 9 M ⊙ for M and L and kpc for r 0 ).
The calculation applies similarly to dwarf galaxies. Results for galaxies DDO 170 and DDO 153 are shown in Figure 4. The results agree with the observation that the luminous mass together with a Newtonian potential contributes especially little to dwarf galaxy rotation curves. Dark matter was first hypothesized to reconcile the motions of galaxies inside clusters with the observed luminous masses of those clusters. Estimating the non-Abelian effects in galaxy clusters with our technique is difficult: 1) the force outside the galaxy is suppressed since the binding of the galaxy components increases (this will be discuss further at the end of the Letter), but 2) the non-Abelian effects on the remaining outside field could balance this if the remaining outside field is strong enough. Since clusters are made mostly of elliptical galaxies for which the approximate sphericity suppresses the non-Abelian effects inside them, we ignore the first effect. We assume furthermore that the intergalactic gas is distributed homogeneously enough so that non-Abelian effects cancel (i.e. the gas does not influence our computation). Finally, we restrict the calculation to the interaction of two galaxies, assuming that others do not affect them. With these three assumptions, we can apply our calculations. Taking 1 Mpc as the distance between the two galaxies and M=40×10 9 M ⊙ as the luminous mass of the two galaxies, we obtain b = −0.012 in lattice units. We express this from the dark matter standpoint by forcing gravity to obey a Newtonian form: with M ′ /M = 1 − r 2 b/a = 251. Gaseous mass in a cluster is typically 7 times larger than the total galaxy mass. Assuming that half of the cluster galaxies are spirals or flat ellipticals for which the non-Abelian effects on the remaining field are neglected, we obtain for the cluster a ratio (M ′ /M) cluster = 18.0, that is our model of cluster is composed of 94% dark mass, to be compared with the observed 80-95%.
Non-Abelian effects emerge in asymmetric mass distributions. This makes our mechanism naturally compatible with the Bullet cluster observation [15] (presented as a direct proof of dark matter existence since it is difficult to interpret in terms of modified gravity): Large non-Abelian effects should not be present in the center of the cluster collision where the intergalactic gas of the two clusters resides if the gas is homogeneous and does not show large asymmetric distributions. However, the large non-Abelian effects discussed in the preceding paragraph still accompany the galaxy systems.
In addition to reproducing the rotation curves and cluster dynamics and to explain the Tully-Fisher relation, our approach implies several consequences that can be tested: 1) Since the Non-Abelian distortions of the field are suppressed for spherically homogeneous distributions, rotation curves closer to Newtonian curves should be measured for spherical galaxies; 2) Two spiral galaxies should interact less than a similar system formed by two spherical galaxies. 3) In a two-body system, we expect a roughly linear potential for large enough effective coupling ( 10 −3 ). This may be testable in a sparse galaxy cluster; 4) The past universe being more homogeneous, and density fluctuations being less massive, the non-Abelian effects should disappear at a time when the universe was homogeneous enough; 5) Structure formations would proceed differently than presently thought since dark matter is an ingredient of the current models, and since those assume an Abelian potential. Particularly, models of mergers of galaxies using a linear potential rather than dark matter constitute another test.
Although the consequences of non-Abelian effects in gravity for galaxies are not familiar, m rather than displaying an α ef f s (r)/r 2 dependence, see e.g. [17]. We also remark that a relation akin to the Tully-Fisher one exists for the strong force in the confinement regime: the angular momenta and squared masses of hadrons are linearly correlated. These "Regge trajectories" are at the origin of the string picture of the strong force. Lattice techniques are a well developed tool to study gluon-gluon interactions at large distances. Hence, it was a ready-to use tool for our purpose. The simplest lattice QCD calculations displaying quark confinement are done in the "gluonic sector", that is without dynamical quark degrees of freedom). Similarly, our calculation excluded T µν , the sources of ψ in the Lagrangian L. We also note that the QCD Lagrangian has a similar structure as L in Eqs. 1 and 2. The close analogy between gravity and QCD is the reason we used the particle physics terminology in this Letter. This analogy has been already noticed and discussed, see e.g. [18].
Before concluding, we exploit further the QCD-gravity analogy, now on a qualitative level. The confinement of gluons inside a hadron not only changes the 1/r quark-quark potential into an r potential, but also causes two hadrons to not interact through the strong force [19] since there is no strong force carriers outside the hadrons. Similarly, the increased binding inside a galaxy would weaken its interaction with outside bodies. Such reduction of the strength of gravity is opposite to what we would conclude by explaining galaxy rotation curves with hallos of exotic dark matter or with gravity modifications, and may be relevant to the fact that the universe expansion is accelerating rather than decelerating. This is currently explained by the repulsive action of a dark energy, see e.g. [2]. However, if gravity is weakened, the difference between the assumed Abelian force and the actual strength of the force would be seen as an additional repulsive effect. Such effect would not explain a net repulsion since it would at most suppress the force outside of the mass system (as for QCD).
Thus, it would not be directly responsible for a net acceleration of the universe expansion.
Nevertheless, it may reduce the need for dark energy. To sum up, the gravity/QCD parallel propounds that dark energy may be partly a consequence of energy conservation between the increased galaxy binding energy vs. the outside effective potential energy. This would implies a quantitative relation between dark energy and dark matter, which might explain naturally the cosmic coincidence problem [2].
To conclude, the graviton-graviton interaction suggests a mechanism to explain galaxy rotation curves and cluster dynamics. Calculations done within a weak field approximation agree well with observations involving dark matter, without requiring arbitrary parameters or exotic particles. The Tully-Fisher relation arises naturally in our framework. Our approach hints that dark energy could partly be a consequence of energy conservation between the increased galaxy binding energy and the outside potential energy.