Stable cosmological solutions and superpotential method in two-field models

1. Stable cosmological solutions and superpotential method in two-field models Nikolay Bulatov Moscow State University Based on: “Null Energy Condition violation and classical stability in the Bianchi I metric” I. Ya. Aref’eva, N. V. Bulatov, L. V. Joukovskaya, S. Yu. Vernov. Phys. Rev. D, Vol. 80,2009, P.083532-083558 arXiv:0903.5264 “Stable exact solutions in cosmological models with two scalar fields” I. Ya. Aref’eva, N. V. Bulatov, S. Yu. Vernov. arXiv:0911.5105

2. Introduction • The Dark Energy state parameter -1.11<w<-0.86 • A time-dependent state parameter fits observational data better • The parameter of state value can be less than -1 (w<-1), therefore the Null Energy Condition can be violated • Due to this fact theories with the Null Energy Condition violation are of interest • These theories are unstable due to the presence of ghosts • The Null Energy Condition violation can be realized within an effective theory while the fundamental theory must be stable and admit quantization • The Null Energy Condition violating models can admit classically stable solutions • As the Friedmann metric is isotropic, the stability of these solutions was studied only under isotropic fluctuations

3. Introduction • Fluctuations can be anisotropic, there are strong limits on the anisotropy of the Universe • As an anisotropic metric it is possible to consider the Bianchi I metric • Stability of isotropic solutions in the Bianchi models was considered in several works, it was proven that for space-time of all Bianchi types except the IX type with a positive cosmological constant and matter satisfying the Dominant and Strong Energy Conditions initially anisotropic solutions tend to isotropic one • In our works we investigated classical stability of isotropic solutions in the Bianchi I metric in the presence of phantom scalar fields.

4. Cosmological model with N scalar fieldsand the CDM in the Bianchi I metric where ispotential, nonzero constant defines whether field is ordinary or phantom scalar field, is a cosmological constant. where

5. Introducing the new variables The shear The EOM The equations for evolution of the new variables

6. Few facts about stability • The solution is attractive or stable if for any solution that starts close enough to • The stable point is called the Lyapunov stable point if all solutions that start near it remain close to it. The stable fixed point is called asymptotically stable if all solutions that start near it converge to it.The solution that tends to a fixed point is attractive if and only if this fixed point is asymptotically stable. • The Lyapunov theorem states that to prove the stability of the fixed point of the nonlinear system it is enough to prove the stability of the fixed point of the corresponding linearized system.

7. Stability of isotropic solutions in two-field cosmological models The EOM

8. Stability of isotropic solutions in two-field cosmological models The fixed point The expansion of variables The linearized system

9. Stability of isotropic solutions in two-field cosmological models The stability conditions

10. Stability of isotropic solutions in two-field cosmological modelswith the CDM The stability conditions

11. The Superpotential method Let’s suppose that Let’s express In this case the choice of functions F and G is ambiguous We can restore the superpotential and potential solving the following equations As the choice of F and G is ambiguous then the superpotential and potential will be obtained ambiguously too, and we’ll have the set of models with different potentials for which our functions are the solutions of the EOM

12. The Superpotential method Using the stability conditions for the potential we can obtain the stability conditions in terms of the superpotential The stability conditions The Superpotential method allows us having solutions to restore the models for which these functions are the solutions and these solutions are stable

13. Quintom model with the polynomial sixth degree potential(Aref’eva, Koshelev, Vernov, 2005) The coordinates of the fixed point The Superpotential The stability conditions In the case with there is exact solution The stability conditions

14. Conclusion • We investigated the stability of isotropic cosmological solutions in the Bianchi I metric for the two field models with the Null Energy Condition violation. • We obtained that the Null Energy Condition is not necessary for the stability of isotropic cosmological solution under anisotropic fluctuations. • We obtained the conditions of stability of isotropic solution that are valid both in the Friedmann and Bianchi I metrics for models with two fields solutions for which tend to a fixed point. • We considered the Superpotential method in the case of two fields and obtained the stability conditions in terms of the superpotential. • We applied obtained results to the String Field Theory inspired model with exact solutions.