self accelerating universe from non-linear massive gravity

Self accelerating universe from non-linear massive gravity

Chunshan LinKavli IPMU, Univ of Tokyo

Based on collaborations withA.Emir Gumrukcuoglu & Shinji Mukohyama

 arXiv:1109.3845

arXiv:1206.2723arXiv:1111.4107

Introduction

Cosmic acceleration

Introduction Can we give graviton a mass?

• Fierz and Pauli 1939

• Vainshtein 1972 non–linear interactions• Boulware–Deser (BD) ghost 1972

van Dam–Veltman–Zakharov discontinuity

Lack of Hamiltonian constrain and momentum constrain

6 degrees of freedom

Helicity ±2, ±1, 0 5 dof？ 6th dof is the BD ghost!

Introduction • C. de Rham, G. Gabadadze and A. Tolly 2011

Source of MASS !

4 Stukelberg scalars respect Poincare symmetry

Eliminate a helicity-0 mode, the so called BD ghost in the decoupling limit.

It is also BD ghost free away from decoupling limit. (Hassan & Rosen 2011)

Self accelerating FRW universe No go result for FRW solution (G. D’Amico et al 2011 Aug.) However… (A.Gumrukcuoglu, C. Lin, S. Mukohyama: 1109.3845)

It does not extend to open FRW universe

The 4 Stukelberg scalars motivated by…

Minkowski metric

Open FRW chart

Both physical metric and fiducial metric respect FRW symmetry !

reads

• 1st solution

• 2nd and 3rd solutions

Please notice that these 2 solutions do not exist when K=0.

Self accelerating FRW universe

Freedmann equation

Self accelerating FRW universe

where

The effective cosmological constant

Poincare Symmetry (arXiv:1111.4107) FRW symmetry in field space Flat/Open/Closed self-accelerating FRW universe is allowed with the same effective c.c .

Self accelerating FRW universe

Sign of the effective cosmological constant

Ghost instability

See Emir’s talk after a little while

Break FRW symmetry

Some extended theories

Anisotropic FRW universe We are going to look for a health region with large anisotropy, and then

look for an attractor in this health region

Consider the simplest Bianchi I type metric

As for the fiducial metric

Varying the stuckelburg scalars

arXiv:1206.2723

Anisotropy

EoM of Stuckelburg scalar

Where

Einstein eqs

Where

Anisotropic FRW universe

Eintein Eqs EoM

Anisotropic FRW universe

Einstein Equations

+EoM Fixed points

Local Stability

Global Stability

Isotropic Fine Tuning

Non-trivial anisotropic

solution

Local stability condition

Anisotropic FRW universe We consider an example with

For which local stability condition is satisfied. There is only one set of solution to 3 eqns of fixed point:

Remarks:Anisotropy differs from GR;Ghost free around fixed point;Coordinates redefinition

Anisotropy cosmological perturbation spectrum

1isotropic universe

Summary We introduce such two cosmological

solutions in this talk: Self-accelerating FRW universe; Anisotropic FRW universe more exciting work is in progress……

full analyse on the perturbations of the attractor solution, observational effect…

THANK YOU!