BRANEWORLD COSMOLOGY WITH A KALB-RAMOND FIELD

1. BRANEWORLD COSMOLOGY WITH A KALB-RAMOND FIELD Giuseppe De Risi 43rd Rencontres de Moriond La Thuile (Val d'Aosta, Italy) March 15 - 22, 2008 Phys.Rev.D77:044030,2008, arXiv:0711.3781 [hep-th]

2. Plan of the talk • Introduction to the braneworld scenario • Bouncing cosmology supported by a EM field • Bouncing cosmology from a KR field • Comments and conclusions

3. The braneworld scenario In the past decade the idea that our universe is a brane embedded in a higher dimensional manifold has become very popular. The Randall-Sundrum (RS) scenario, in particular, has been very successful in accommodating all the well-established theoretical and observational results of standard cosmology (inflation, perturbations etc.) RS model consists of a flat brane (with tension) embedded in an AdS5 bulk. Newtonian gravity is recovered at low energies because the e.o.m. of the tensor fluctuation has a volcano-like potential that bounds the zero mode. The massive modes contribution results in a correction to the Newtonian potential

4. Going to the non-linear level, it is possible (Shiromizu, Maeda, Sasaki) to show that the Einstein equation are modified Matter induced from the bulk High energy correction “Dark radiation” from the bulk The effective 4D theory is still GR with corrections that become effective at high energies Cosmology on the brane can be captured in a less general, though simpler way (Kraus): allow the brane to move trough a static bulk. In this case the Israel junction conditions becomes dynamical, and give the modified Einstein equations. High energy correction “Dark radiation” from the bulk

5. Bouncing cosmology on the brane The standard cosmological model, even in the inflationary extension, despite its success in explaining a wide range of cosmological and astrophysical observation, still suffer from a major problem: the big bang. The presence of the singularity indicates that we are extending GR beyond its validity regime We need a modification that take place at high energies Brane cosmology?! In 2003 Mukherij and Peloso proposed a braneworld model in which the cosmological evolution on the brane was non-singular. In their model, the bulk solution was a Reissner-Nordström-AdS black hole, i. e. a black hole with electric charge.

6. By letting the brane move one obtain, following the procedure sketched before, a Friedman-like equation of the form. “Dark stiff matter” The additional term sourced by the electric charge is repulsive, and it is dominant at small scales, i.e. at high energies, so it could hopefully drive a bounce in the cosmological evolution. In fact, it is possible to find exact bouncing solution for the critical case in which the brane cosmological constant is zero: closed universe flat universe open universe

7. However, this proposal was subject to several criticisms (Kanti and Tamvakis, Hovdebo and Myers) Perhaps the strongest pathology that has been pointed out about this kind of model is that the brane encounter a singularity before undertaking the bounce. During its journey trough the bulk, the brane crosses both horizons of the black hole. But it is known that the inner horizon of a Reissner-Nordström black hole is unstable. In particular, fluctuations which are normalized it the outer horizon, blow up at the inner one.

8. Brane with a Kalb-Ramond field G.D.R., Phys.Rev.D77:044030,2008, arXiv:0711.3781 [hep-th] Our aim is to study the braneworld setup in presence of bulk supergravity fields. The action is The brane is neutral with respect to U(1) charge of the KR field Variation of the action leads to the e.o.m. for the metric, the dilaton and the Kalb-Ramond field

9. The equation for the KR field is solved by the Ansatz Duality This allow us to write the equations in terms of the dualized fields (Copeland, Lahiri, Wands) Field source terms are opposite in sign! We seek for a static solution, with the usual ansatz for the metric, and assume that the dual “Maxwell” field is purely electric.

10. We found that the dilaton is constant, therefore it can be set to zero without any loss of generality. The solution for the metric and the Maxwell field are The term proportional to the U(1) charge is negative No repulsive gravity at high energies. However, the negative sign in the charge term give rise to a rather “exotic” possibility: set m < 0 without having a naked singularity. In fact, for a negative bulk cosmological constant we have one horizon located at.

11. Let us now consider the cosmological evolution of the brane. For simplicity, we will consider a pure tension spatially flat brane. The modified Friedman equation can be obtained in the usual way, following the lines depicted before The universe undergoes a bounce if the scale factor initially decreasing, reaches a minimum and start to expand again, i. e. H = 0 Here is a picture of the behavior of H as a function of a for different values of the parameter Analytically, one find that the bounce occurs at the value

12. ab R0 l Having found that the universe actually undergoes a bounce, the crucial requirement is that it occurs before the brane crosses the horizon. In fact, possible instabilities can occur only on the horizon. The figure represents the behavior of ab as a function of the brane tension l for different values of RKR. There is always a range of the tension in which the bounce occurs outside the horizon. It is possible to find an analytical solution of the allowed values for the brane tension (and thus for the induced cosmological constant)

13. Conclusions… • We study braneworld models with a bulk Kalb-Ramond field, and we find that cosmology on the brane is free from singularities • In addition, the cosmogical do not suffer from instabilities, such as other models presented in the literature BUT A crucial assumption was to assume m < 0, where m is proportional to the mass of the central body who sources the black hole • Difficult interpretation of the Newtonian limit Not sure that it is an issue… No definite answer even for the standard Reissner-Nordström black hole (Kodama-Ishibashi) • Problems related to the overall instability of the space-time

14. …and outlook Further developments include • Studying models with matter on the brane • Presence of enough “ordinary” radiation could spoil the singular-free behavior • Test the model against astrophysical bounds • Introduce a DBI coupling between the brane and the Kalb-Ramond field • The dual ansatz could be no longer valid • Studying 4D scalar and tensor perturbation • WMAP data favour a scale-invariant spectrum, difficult to have with bouncing models.