aa r X i v : . [ h e p - t h ] M a y Non-singular Brane cosmology with aKalb-Ramond field
G. De Risi , Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 2EG,UK Istituto Nazionale di Fisica Nucleare, 00186 Roma, Italy
Abstract
We present a model in which a 3-brane is embedded in a warped 5-dimensionalbackground with a dilaton and a Kalb-Ramond 2-form. We show that it is possible tofind static solutions of the form of charged dS/AdS-like black hole which could have anegative mass parameter. The motion of the 3-brane in this bulk generates an effective4-dimensional bouncing cosmology induced by the negative dark radiation term. Thismodel avoids the instabilities that arises for previous non-singular braneworld cosmolo-gies in a Reissner-Nordstrøm-AdS bulk.
Braneworld models [1, 2] have generated, during the past decade, enormous attention, dueto the dramatic change they inspired in our understanding of extra dimensions. Accordingto this framework, our universe is a “brane” embedded in a higher-dimensional space, onwhich the Standard Model fields are confined, while gravity is localised near the braneby the warped geometry of the extra dimension. It is possible to construct models inwhich the brane evolution mimics a Friedmann-Robertson-Walker (FRW) cosmology, withmodifications at small scales due to the gravitational effect of the bulk spacetime on thebrane [3–5]. In particular, provided the bulk is taken to be a Reissner-Nordstrøm-AdS black1ole, such modifications can lead to bouncing 4D cosmological models [6]. Unfortunatelythe brane, during its evolution in the bulk, always crosses the Cauchy horizon of the AdSblack hole, which is unstable [7, 8].In this paper we present a different model [9], in which this problem is avoided. Weconsider a brane embedded in a supergravity background in which both the dilaton andthe Kalb-Ramond 2-form are turned on (but without a dilaton potential). By dualizing the2-form, we obtain Einstein-Maxwell like equations of motion, but with a different sign forthe kinetic term of the Maxwell-like field. The static solution is therefore different, and theterm that dominates at high curvature is like “stiff matter” with positive energy density.Even though this implies that the energy contribution at high curvatures is positive, so thatit can not drive a bounce, this opens an interesting possibility of having negative energycontributions at intermediate curvatures, by letting the mass of the black hole be negative.The parameter space allows this while avoiding a naked singularity. In this case, we showthat it is possible that the brane bounces before crossing the black hole horizon, so that theeffective 4-dimensional cosmological evolution will not suffer from any instability [8].
We will consider the low-energy string effective action: S = M Z d x √− g (cid:18) R − e σ φ − H ABC H ABC e σ φ − g AB ∂ A φ∂ B φ (cid:19) − T Z d ξ √− γe lφ (1)(with H ABC = ∂ [ A B BC ] ), which describes a 3-brane embedded in a 5-dimensional bulk withdilaton and Kalb-Ramond 2-form. We will take into account the presence of the brane,which is assumed to be neutral with respect to the antisymmetric field, by implementingthe Israel junction condition in the next section [10]. The equation of motions derived from(1) can be greatly simplified if we take, for the Kalb-Ramond field, the ansatz H CAB = ǫ CABDE ∇ D A E e − σ φ . (2)by which we get the folowing equations of motion: G AB = (cid:20) − e σ φ L −
12 ( ∂ c φ ) + 12 e − σ φ F (cid:21) g AB + ∂ A φ∂ B φ − e − σ φ F AC F CB , (3) ∇ A ∇ A φ − σ e σ φ L − σ e − σ φ F = 0 , (4) ∇ B (cid:16) e − σ φ F AB (cid:17) = 0 , (5)2ith F MN = ( ∇ M A N − ∇ N A M ) being a “pseudo” Maxwell field strength dual to H ABC .A static solution with a maximally symmetric 3-space and purely electric field can be castas: ds = − (cid:18) − Q R − µR + k − L6 R (cid:19) dt + dR (cid:16) − Q R − µR + k − L R (cid:17) + R (cid:18) dr − kr + r d Ω (cid:19) A ( R ) = ± QR . (6)In this solution it is possible to set µ <
0, which means that the mass of the central bodyis negative, without having a naked singularity. In fact, assuming that the bulk cosmologicalconstant is negative, L <
0, we have an horizon located at R = (cid:18) µ L (cid:19) / s C / (cid:18) Q √− L( − µ ) / (cid:19) , (7)where C / is the Chebyshev cubic root [9]. In the next section we will describe how thisbulk affects the cosmology of an embedded moving brane. The movement of an embedded brane in the 5D bulk induces a cosmological evolution onthe brane via the Israel junction condition [10]. In the case under investigation, if we assumefor simplicity a pure tension, spatially flat brane, the modified Friedmann equation is H = L µa + Q a . (8)The behaviour of H as a function of a is depicted in Fig. 1 for different values of the ratiobetween the 4D de Sitter curvature radius and the characteristic length of the Kalb-Ramondblack hole obtained by the charge to mass ratio R KR = Q − µ ) / .The black plot have two values of a for which H = 0. It is not difficult to understand thatthe largest branch of this plot describes a bouncing universe that approaches asymptoticallyto de Sitter The bounce occurs at the following value of the scale factor: a b = √ (cid:18) − µ L (cid:19) / s cos (cid:20)
13 arccos (cid:18) − Q √ L − µ ) / (cid:19)(cid:21) . (9)Now we have to show that the bounce occurs outside the horizon. Fig. 2 shows a b ( l ), andthe corresponding value of R . We can see that there is a region, when the brane tension isclose to the minimum allowed value, in which the bounce radius is greater than the horizon3igure 1: The Hubble parameter H as a function of the scale factor a . The three coloursrepresent decreasing values of R KR ℓ : 15 .
81 (red, dotted), 1 .
41 (blue, dashed), 0 .
70 (black,solid).position, so that the entire evolution of the brane lies in the physically viable region outsidethe horizon. This feature is quite general, and the reason is easy to understand, since wecan see from (9) that a b → ∞ as L → has to satisfy the following inequalities: L < − L2 C / (cid:18) Q √ − L ( − µ ) / (cid:19) − Q √ − L ( − µ ) / C / (cid:18) Q √ − L ( − µ ) / (cid:19) + Q √ − L ( − µ ) / for Q √− L( − µ ) / < , (10) L < − L4 C − / (2 Q √− L( − µ ) / ) for Q √− L( − µ ) / > , (11)So there is always an allowed value of the brane tension for which the brane evolution liesentirely outside the horizon, and therefore free of instabilities. In this paper we presented a braneworld model in which the cosmological evolution of thebrane is non-singular, and the brane universe bounces smoothly from a phase of contractionto a subsequent expanding phase. The cosmological evolution on the brane is induced by itsmovement through a static bulk AdS black hole supported by a non-trivial Kalb-Ramondantisymmetric 2-form. The bouncing is driven by the negative dark radiation that appears4igure 2: Plot of a b as a function of the tension l for different values of R KR ℓ : 0 . .
02 (red, lighter). Dashed curves of the same colour represent the position of thehorizons for the same value of R KR ℓ .as a peculiar feature of the bulk solution, which is a negative mass black hole. Furtherinvestigations would be needed to clarify the main issues that may arise in developing thismodel, namely the overall stability of the 5D negative mass black hole solution and theinclusion of radiation on the brane, which could spoil the singular-free behaviour. Otherinteresting development would be the study of the model in presence of a DBI couplingbetween the brane and the Kalb-Ramond field, which arises naturally in the context of StringTheory, and the analysis of perturbations, in order to compare the model to observations. Acknowledgements
I would like to thank the I.N.F.N. for providing fundings to attend the “43 rd Rencontres deMoriond” conference, where this talk was given.
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