Perturbations of Cosmological and Black hole solutions

Transcription

1 Perturbations of Cosmological and Black hole solutions in Bi-Gravity Daisuke Yoshida, Tokyo Institute of Technology Based on collaboration with Tsutomu Kobayashi, Masaru Siino, Masahide Yamaguchi (In progress) RESCEU Summer School Green palace, 3rd August 2015

2 What is bi-gravity?

3 What is bi-gravity? One of the theory of modified gravity General Relativity non-derivative interaction Massive gravity de Rham, Gabadadze, Tolley (2011) bi-gravity Hassan, Rosen (2011) metric matter S int f μρ f νσ g μν g ρσ another metric matter External field dynamical field 1/10

4 Cosmology in bi-gravity Known result about cosmology Comelli et al (2011) Comelli et al (2014) De Felice et al (2014) Starting from the simple bi-diagonal assumption the self accelerated cosmological solutions can be obtained, but these solutions suffer from ghost or gradient instability. Does bi-gravity rejected from cosmology? No! There are many cosmological solutions in bi-gravity. Other solution may express the real universe. This is a motivation of our work! 2/10

5 Contents 0. Action and Equation of Motion 1. Construct back ground solutions 2. Investigate its stability which different from the bi-diagonal solution

6 Action and Equation of Motion

7 Action and Equation of Motion Action By ghost free requirement, S int is limited following form: de Rham, Gabadadze, Tolley (2011) with Key Point Bi-gravity has 5 free parameters Interaction term is described by 3/10

8 Action and Equation of Motion Action Equation of motion for g μν : for f μν : Key Point Bi-gravity has 5 free parameters Interaction term is described by 3/10

9 Back Ground Solutions

10 Metric Ansatz Ansatz of g μν Spherically symmetric solution Cosmological solutions Spherically symmetric Black hole solution Ansatz of f μν The bi-diagonal solution corresponds to 4/10

11 Cosmological Constant Solution The prediction of GR with c.c. coincide with observation at least back ground level. Then, we look for the case that the effect of bi-gravity behave a cosmological constant. impose remainder EoM: G ν μ + X ν μ = 8πG T ν μ m, α 3, α 4, Λ g, Λ f : parameters A 2 t, r : = f θθ /g θθ γ ν μ = g 1 f ν μ (A is constant) or bi-diagonal case 5/10

12 Summary of the Background Solution Choice of the parameter of theory Metric Equation of Motion μ X ν Any spherically symmetric solution of GR can be a solution of bi-gravity!! 6/10

13 Perturbation

14 Linear Perturbation To investigate the stability of the solution, we consider the linear perturbation remainder Equation of motion can be obtained as Einstein equation I, J = θ, φ 7/10

15 Solution of Conservation Equation Key observation: must satisfy the conservation equation, 8πG μ T ν μ μ G ν μ, if matter is conserved. Laplace equation on sphere, then solution is If is non-singular at, the solution is From remaining equation, 8/10

16 Solution of Conservation Equation Key observation: must satisfy the conservation equation, 8πG μ T ν μ μ G ν μ, if matter is conserved. I, J = θ, φ The equation of motion reduces to Perturbed Einstein equation for δg μν : 3 Constraints between perturbations δg IJ, δf IJ : Perturbed Einstein equation for δf μν : Dynamics of metric perturbations is completely same as GR!! There is no unphysical instability peculiar to bi-gravity. 8/10

17 Quadratic Perturbations The same result is true for quadratic perturbations With the solutions of linear order The dynamics of quadratic perturbations is also same as GR!! 9/10

18 Summary We showed that 1. Any spherically symmetric solution of GR can be a solution of bi-gravity. 2. Linear and quadratic perturbation of this solution is obeyed by Einstein equation. There is no instability peculiar to bi-gravity. Bi-gravity can reproduce the prediction of GR, Bi-gravity GR about spherically symmetric space time. 10/10