1. Giuseppe De Risi M. Cavaglià, G.D., M. Gasperini, Phys. Lett. B 610:9-17, 2005 - hep-th/0501251 QG05, Sept. 2005

2. Plan of the talk Perturbations equation on the Kasner brane Amplification of the fluctuations Conclusions Kasner solution on the brane The braneworld scenario

3. The existence of extra dimensions in our universe was put forward in the far 20’s (Kaluza-Klein), and have became a major topic of the modern theoretical physics In the end of the past century there was a turning point of how we “implement” extra-dimension in the universe (HDD ‘98 and Randall-Sundrum ‘99) Before RS After RS The braneworld scenario

4. RS propose a solution of the form: This metric is a solution of the 5D Einstein equation only if the bulk cosmological constant and the AdS radius are related The wave equation that is obtained for the 5D transverse traceless perturbation h ij (t,x,z) of the RS metric can be expanded on a basis of 4D modes: RS model consists of a flat brane (with tension) embedded in an AdS 5 bulk: The starting point is, of course, the Einstein action The braneworld scenario

5. The massive modes contribution results in a correction to the Newtonian potential The z-dependent part of the equation decouples The braneworld scenario

6. It is possible to generalize the RS scenario to have a cosmological evolution (Binetruy, Deffayet, Langlois ’99, Shiromizu, Maeda, Sasaki ’99). The model we analyze consists of an anisotropic brane embedded in AdS We assume that the brane is rigidly located at the origin: we impose, for the metric, the ansatz d “external” dimensions x which should expand Kasner solutions on the brane n “internal” dimensions y which should contract

7. The brane and extradimensional equations can be decoupled Evolution on the brane is characterized by a Kasner inflationary regime The z-dependent part of the equation gives the AdS warp factor: Kasner solutions on the brane And the tension and the bulk cosmological constant are related by the equation: a(t)a(t) b(t)b(t)

8. To discuss metric perturbations we introduce the expansion: where the perturbation takes values only in the external space and is in the TT gauge Perturbations equation on the Kasner brane Studying the evolution of the cosmological perturbations is one of the most investigated topics in recent braneworld cosmology, because it would (possibly/hopefully) lead to direct comparison with observational data. An interesting feature in the model under discussion is that, unlike other models (see, for example Langlois, Maartens, Wands ’00, Kobayashi, Kudoh, Tanaka ’03, Easther, Langlois, Maartens, Wands ’03) the massless and massive modes are always decoupled, as in ordinary RS

9. As in the standard RS case, we expand h in eigenmodes so that the new variables satisfy Perturbations equation on the Kasner brane This equation can be obtained by the perturbed (to order o(h 2 )) action: The equation of motion for the perturbation is:

10. So the solution can be normalized in a canonical way, i.e. to the delta function Substituting this into the action we can define an action for the single massive mode: Perturbations equation on the Kasner brane Introducing the auxiliary field we obtain the Schrödinger-like equation Volcano-like potential

11. We get the action and the canonical equation for the Fourier modes u m,k This action can be put in a canonical form by describing the time evolution by the conformal time and introducing an auxiliary field u m via the pump field  m Perturbations equation on the Kasner brane

12. The massless mode The solution of the Schrödinger-like equation we obtain for m = 0, after imposing the normalization condition is From this we can get the effective value for the 4D Plank mass Amplification of the fluctuations To study the production of relic gravitons we consider a transition between the inflationary Kasner regime and a final era with a simple Minkowsky metric. Since massless and massive modes do not mix, they can be treat separately reduces to the RS result for d = 3 and n = 0

13. The auxiliary field u 0 we have defined has the correct canonical dimension, so that we can normalize its Fourier modes to an initial state of vacuum fluctuations The pump fields before and after the transition (which occurs at the conformal time  1 ) are Amplification of the fluctuations The solution of the (Bessel-like) canonical equation for m = 0 can be expressed in terms of Hankel functions. By imposing continuity at the transition epoch –  1 we get the correct solution

14. Curvature scale at the transition and we can compute the spectral distribution cutoff frequency Massive modes The solution of the SL equation we find for m ≠ 0 is Amplification of the fluctuations There is no difference with the standard result (unlike in de Sitter models studied in LMW)

15. We can then deduce the (differential) coupling parameter M m that multiplies the massive mode action and, in the light mass regime, the effective measure controlling the light modes contribution to the static two-point function Solving the canonical equation for the massive modes can be quite hard, BUT... let us consider relativistic modes In this case the amplification equation is exactly the same as in the massless case for each mode Defining we can evaluate the massive contribution to the spectrum reduces to the RS result for d = 3 and n = 0 Amplification of the fluctuations

16. The importance of the massive contribution depends on the ratio between the curvature transition scale and the AdS length massive contribution to the spectrum highly suppressed possibility that strongly enhanced amplification of the massive fluctuations Amplification of the fluctuations

17. Conclusions It is possible to have a cosmological evolution which behaves as the standard flat RS scenario Transition from an inflationary phase to a “standard” phase still produces a stochastic background of metric fluctuation If the curvature at the transition epoch is low, there is no significative difference with the standard scenario If the curvature at the transition epoch is high enough, there can be an important amplification of massive modes that contributes to the spectral amplitude Further developments can be analyzed including other fields (i.e. the dilaton) and some “bulk” influence on the brane fluctuations