Kinematics of radion field: a possible source of dark matter
Abstract
The discrepancy between observed virial and baryonic mass in galaxy clusters have lead to the missing mass problem. To resolve this, a new, nonbaryonic matter field, known as dark matter, has been invoked. However, till date no possible constituents of the dark matter components are known. This has led to various models, by modifying gravity at large distances to explain the missing mass problem. The modification to gravity appears very naturally when effective field theory on a lowerdimensional manifold, embedded in a higherdimensional spacetime is considered. It has been shown that in a scenario with two lowerdimensional manifolds separated by a finite distance is capable to address the missing mass problem, which in turn determines the kinematics of the brane separation. Consequences for galactic rotation curves are also described.
Keywords
Dark Matter Sterile Neutrino Rotation Curve Galaxy Cluster Radion Field1 Introduction
Recent astrophysical observations strongly suggest the existence of nonbaryonic dark matter at the galactic as well as extragalactic scales (if the dark matter is baryonic in nature, the third peak in the Cosmic Microwave Background power spectrum would have been lower compared to the observed height of the spectrum [1]). These observations can be divided into two branches – (a) behavior of galactic rotation curves and (b) mass discrepancy in clusters of galaxies [2].
The first one, i.e., rotation curves of spiral galaxies, shows clear evidence of problems associated with Newtonian and general relativity prescriptions [2, 3, 4]. In these galaxies neutral hydrogen clouds are observed much beyond the extent of luminous baryonic matter. In a Newtonian description, the equilibrium of these clouds moving in a circular orbit of radius r is obtained through equality of centrifugal and gravitational force. For cloud velocity v(r), the centrifugal force is given by \(v^{2}/r\) and the gravitational force by \(GM(r)/r^{2}\), where M(r) stands for total gravitational mass within radius r. Equating these two will lead to the mass profile of the galaxy: \(M(r)=rv^{2}/G\). This immediately posed serious problem, for at large distances from the center of the galaxy, the velocity remains nearly constant \(v\sim 200~\text {km/s}\), which suggests that mass inside radius r should increase monotonically with r, even though at large distance very little luminous matter can be detected [2, 3, 4].
The mass discrepancy of galaxy clusters also provides direct hint for existence of dark matter. The mass of galaxy clusters, which are the largest virialized structures in the universe, can be determined in two possible ways – (i) from the knowledge about motion of the member galaxies one can estimate the virial mass \(M_\mathrm{V}\), second, (ii) estimating mass of individual galaxies and then summing over them in order to obtain total baryonic mass M. Almost without any exception \(M_\mathrm{V}\) turns out to be much large compared to M, typically one has \(M_\mathrm{V}/M \sim 2030\) [2, 3, 4]. Recently, new methods have been developed to determine the mass of galaxy clusters; these are (i) dynamical analysis of hot Xray emitting gas [5] and (ii) gravitational lensing of background galaxies [6] – these methods also lead to similar results. Thus dynamical mass of galaxy clusters are always found to be in excess compared to their visible or baryonic mass. This missing mass issue can be explained through postulating that every galaxy and galaxy cluster is embedded in a halo made up of dark matter. Thus the difference \(M_\mathrm{V}M\) is originating from the mass of the dark matter halo the galaxy cluster is embedded in.
The physical properties and possible candidates for dark matter can be summarized as follows: dark matter is assumed to be nonrelativistic (hence cold and pressureless), interacting only through gravity. Among many others, the most popular choice being weakly interacting massive particles. Among different models, the one with sterile neutrinos (with masses of several keV) has attracted much attention [7, 8]. Despite some success it comes with its own limitations. In the sterile neutrino scenario the Xray produced from their decay can enhance production of molecular hydrogen and thereby speeding up cooling of gas and early star formation [9]. Even after a decade long experimental and observational efforts no nongravitational signature for the dark matter has ever been found. Thus a priori the possibility of breaking down of gravitational theories at galactic scale cannot be excluded [10, 11, 12, 13, 14, 15, 16, 17].

Obtaining effective gravitational field equations on a lowerdimensional hypersurface, starting from the full bulk spacetime, which involves additional contributions from the bulk Weyl tensor. The bulk Weyl tensor in spherically symmetric systems leads to a component behaving as mass and is known as “dark mass” (we should emphasize that this notion extends beyond Einstein’s gravity and holds for any arbitrary dimensional reduction [29, 30, 38]). It has been shown in [39] that the introduction of the “dark mass” term is capable to yield an effect similar to the dark matter. Some related aspects were also explored in [40, 41, 42, 43], keeping the conclusions unchanged.

In the second approach, the bulk spacetime is always taken to be antide Sitter such that bulk Weyl tensor vanishes. Unlike the previous case, which required \(S^{1}/Z_{2}\) orbifold symmetry, arbitrary embedding has been considered in [44] following [45]. This again introduces additional corrections to the gravitational field equations. These additional correction terms in turn lead to the observed virial mass for galaxy clusters.
Further the same setup is also shown to explain the observed rotation curves of galaxies as well. Hence both problems associated with dark matter, namely the missing mass problem for galaxy clusters and the rotation curves for galaxies, can be explained by the two brane system introduced in this work via the kinematics of the radion field.
The paper is organized as follows – in Sect. 2, after providing a brief review of the setup we have derived effective gravitational field equations on the visible brane which will involve additional correction terms originating from the radion field to modify the gravitational field equations. In Sect. 3 we have explored the connection between the radion field, dark matter, and the mass profile of galaxy clusters using relativistic Boltzmann equations along with Sect. 4 describing possible applications. Then in Sect. 5 we have discussed the effect of our model on the rotation curve of galaxies while Sect. 6 deals with a few applications of our result in various contexts. Finally, we conclude with a discussion of our results.
Throughout our analysis, we have set the fundamental constant c to unity. All the Greek indices \(\mu ,\nu ,\alpha ,\ldots \) run over the brane coordinates. We will also use the standard signature \((++\cdots )\) for the spacetime metric.
2 Effective gravitational field equations on the brane
The above effective field equations for gravity have been obtained following [46], where no stabilization mechanism for the radion field was proposed. In this work as well we would like to emphasize that we are working with the radion field in the absence of any stabilization mechanism. However, as already emphasized in [46], in order to provide a possible resolution to the gauge hierarchy problem one requires stabilization of the radion field. Even though we will not explicitly invoke a stabilization mechanism, we will outline how stabilization can be achieved and argue that it will not drastically alter the results.
In such a situation with a stabilized radion field, the field \(\Phi (x)\) appearing in the above equations can be thought of as fluctuations of the radion field around its stabilized value [47]. In particular stabilization of the radion field can be achieved by first introducing a bulk scalar field following [48] and then solving for it. Substitution of the solution in the action and subsequent integration over the extra spatial dimension lead to a potential for \(\Phi \). The same will appear in the above equations through the projection of the bulk energymomentum tensor, which would involve the bulk scalar field and shall lead to an additional potential on the right hand side of the above equations, whose minima would be the stabilized value for \(\Phi =\Phi _{c}\). Choosing \(\Phi =\Phi _{c}+\Phi (x)\), where \(\Phi (x)\) represents small fluctuations around the stabilized value, one ends up with similar equations as above with bulk terms having contributions similar to \(T^\mathrm{(vis)}\) and \(T^\mathrm{(hid)}\), respectively. Thus the final results, to leading order, will remain unaffected by the introduction of a stabilization mechanism. Even though the fact that the virial mass of galaxy clusters scale with r will hold, the subleading correction terms in the case of galactic motion will change due to the presence of a stabilization mechanism due to the appearance of extra bulk inherited terms in the above equations. It would be an interesting exercise to work out the above steps explicitly and obtain the relevant corrections due to the stabilization mechanism, which we will pursue elsewhere.
As illustrated above for the two brane system the nonlocal terms get mapped to the radion field, the separation between the two branes. Hence ultimately one arrives at a system of closed field equations for a two brane system. The field equations as presented in Eq. (8) are closed since the radion field \(\Phi \) satisfies its own field equation Eq. (10). Hence the problematic nonlocal terms in a single brane approach get converted to the radion field in a two brane approach and make the system of gravitational field equations at low energy closed.
3 Virial theorem in galaxy clusters, kinematics of the radion field and dark matter
It is well known that the galaxy clusters are the largest virialized systems in the universe [2]. We will further assume them to be isolated, spherically symmetric systems such that the spacetime metric near them can be presented by the ansatz in Eq. (11). Galaxies within the galaxy cluster are treated as identical, point particles satisfying general relativistic collisionless Boltzmann equation.
The Boltzmann equation requires setting up appropriate phase space for a multiparticle system along with the corresponding distribution function f(x, p), where x is the position of the particles in the spacetime manifold with its fourmomentum \(p\in T_{x}\), where \(T_{x}\) is the tangent space at x. Further the distribution function is assumed to be continuous, nonnegative and describing a state of the system. The distribution function is defined on the phase space, yielding the number \(\mathrm{d}N\) of the particles of the system, within a volume \(\mathrm{d}V\) located at x and have fourmomentum p within a three surface element \(d\overrightarrow{p}\) in momentum space. All the observables can be constructed out of various moments of the distribution function. Further details can be found in [39].
Before concluding the section, let us briefly mention the connection of the above formalism with the gauge hierarchy problem. The separation between the two branes is denoted by d, which varies with the radion field \(\Phi \), logarithmically [see Eq. (9)]. The radion field except for a constant contribution varies weakly with radial distance and hence leads to very small corrections to the distance d between the branes. Thus the graviton mass scale for the visible brane will be suppressed by a similar exponential factor as in the original scenario of Randall and Sundrum [21, 52], leading to a possible resolution of the gauge hierarchy problem. Thus, as advertised earlier, the existence of an extra spatial dimension leads to a radion field, producing a possible explanation for the dark matter in galaxy clusters along with solving the gauge hierarchy problem.
4 Application: cluster mass profiles
5 Effect on galaxy rotation curves
6 Application to other scenarios
In this work we have used a two brane model with the brane separation being represented by the radion field \(\Phi \). We have also assumed that our universe corresponds to the visible brane. In such a setup the effective gravitational field equations on the brane, written in a spherically symmetric context, depends on the radion field and its derivatives. The use of a collisionless Boltzmann equation leads to the result that the virial mass of the galaxy clusters scales linearly with radial distance. Thus without any dark matter we can reproduce the virial mass of galaxy clusters by invoking extra dimensions.

In present day particle physics an important and long standing problem is the gauge hierarchy problem, which originates due to the large energy separation between the weak scale and the Planck scale. In our model the branes are separated by a distance d, such that the energy scale on our universe gets suppressed by \(M_\mathrm{vis}\sim M_\mathrm{Pl}e^{2kd}\), with k being related to brane tension. Thus a proper choice of k (such that \(kd\sim 10\) ) leads to \(M_\mathrm{vis}\sim M_\mathrm{weak}\) and hence solves the hierarchy problem. Along with the missing mass problem, i.e., producing a linear virial mass our model has the potential of resolving the gauge hierarchy problem as well. This is a major advantage over modified gravity models, where the modifications in gravitational field equations are due to modifying the action for gravity. These models, though able to explain the missing mass problem, usually ds not address the gauge hierarchy problem.
 The next hurdle comes from local gravity tests. This should place some constraints on the behavior of the radion field. The analysis using a spherically symmetric metric ansatz has been performed in [58] assuming dark matter to be a perfect fluid which is a perturbation over the Schwarzschild solution. We can repeat the same analysis with our radion field mass function, which is a perturbation over the vacuum Schwarzschild solution. We then can compute the correction to the perihelion precession of mercury due to dark matter which leads to the following constraint on the bulk curvature radius [58, 59, 60]:where \(\Delta \delta \phi =0.004\pm 0.0006\) arc second per century corresponds to an excess in the perihelion precession of Mercury [61]. a is the semimajor axis, e stands for eccentricity, and \(T_{\mathrm{M}}\) and \(T_{\mathrm{E}}\) are the periods of revolution of Mercury and Earth, respectively.$$\begin{aligned} \frac{2\ell (3\beta )(\beta 2)}{\kappa ^{2}M_{\odot }}a(1e^{2})\le \frac{10^{5}}{36^{2}\pi }\frac{T_{\mathrm{M}}}{T_{\mathrm{E}}}\Delta \delta \phi \end{aligned}$$(49)

Let us now briefly comment on the relation between the existence of a fifth force and dark matter. In all these models the generic feature corresponds to the existence of a scalar field which couples to dark matter and in turn couples weakly (or strongly) to standard model particles [62, 63, 64]. In our model this feature comes quite naturally, since the radion field \(\Phi \), which plays the role of dark matter, can also be thought of as a scalar field, coupled to standard model particles through the matter energymomentum tensor with coupling parameter \(\sim \kappa ^{2}/\ell (3+2\omega )^{1}\). Thus effectively we require a fifth force to accommodate modifications of gravity at small scales. There exist stringent constraints on the fifth force from various experimental and observational results (see for example [65, 66, 67]). We can apply these constraints on the fifth force for scalar tensor theories of gravity and that leads to the following bound on the composite object: \((\kappa ^{2}/12\pi G \ell )(1+\Phi )<2.5\times 10^{5}\). Hence for compatibility of the radion field presented in this work with fifth force constraints, the bulk curvature \(\ell \), the bulk gravitational constant \(\kappa ^{2}\), and the radion field must satisfy the above mentioned inequality.

Finally we address some cosmological implications of our work. In cosmology one averages over all the matter contributions at the scale of galaxy clusters and assumes all the matter components to be perfect fluids. The same applies to our model as well, in which the effect of a radion field \(\Phi \) at the galactic scale is to generate an effective dark matter density profile, with a given mass function. Since the mass function obeys the observed dark matter profile, therefore on average in the cosmological scale it reproduces the standard dark matter content and hence the standard cosmological models.
7 Discussion
Brane world models can address some of the long standing puzzles in theoretical physics, namely: (a) the hierarchy problem and (b) the cosmological constant problem. To solve the hierarchy problem we need two branes, with warped fivedimensional geometry such that the energy scale on the visible brane gets suppressed exponentially leading to TeV scale physics. For the cosmological constant the brane tension plays a crucial role. Two brane models naturally inherit an additional field, the separation between the branes (known as the radion field). The radion field is also very important in both macroscopic and microscopic physics, for it can have possible signatures in inflationary scenarios [24, 25, 26], black hole physics [68, 69], collider searches [70], etc. Along with the gauge hierarchy and the cosmological constant problem, another very important problem in physics is the missing mass problem. This appears since the baryonic and the virial mass of a galaxy cluster do not coincide. In this work using a two brane setup we have shown that, along with the gauge hierarchy and cosmological constant problem, this model is also capable of addressing the missing mass problem through the kinematics of the brane separation, i.e., the radion field. Due to the presence of this additional field, the gravitational field equations on the brane get modified and yield additional correction terms on top of Einstein’s field equations. By considering the relativistic Boltzmann equation we have derived the virial mass of galaxy clusters, which depends on an effective additional mass constructed out of the radion field. Moreover, these correction terms modify the structure of gravity and hence the motion under its influence at large distance, thereby producing a linear increase in the virial mass of the galaxy clusters. This in turn leads to the appropriate velocity law for galaxies within a galaxy cluster, solving the missing mass problem.
Footnotes
 1.
In addition to the introduction of extra dimensions we could also modify the gravity theory without invoking ghosts, which uniquely fixes the gravitational Lagrangian to be Lanczos–Lovelock Lagrangian. These Lagrangians have special thermodynamic properties and also modify the behavior of fourdimensional gravity [31, 32, 33, 34, 35, 36, 37]. However, in this work we shall confine ourselves exclusively within the framework of Einstein gravity and shall try to explain the missing mass problem from kinematics of the radion field.
Notes
Acknowledgements
S.C. thanks IACS, India, for warm hospitality; a part of this work was completed there during a visit. He also thanks CSIR, Government of India, for providing a SPM fellowship.
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