Spherical collapse in f(R) gravity and BKL conjecture

In this talk, I will discuss spherical scalar collapse in f(R) gravity in double-null coordinate. Before the collapse, the scalar degree of freedom f' is coupled to a physical scalar field, and general relativity is restored. During the collapse, the major energy of the physical scalar field moves to the center. As a result, f' loses the coupling and becomes light, and gravity transits from general relativity to f(R) gravity. Due to strong gravity from the singularity and the low mass of f', f' will cross the minimum of the potential and approach zero. f' tries to suppress the evolution of the physical scalar field, which is a dark energy effect.
As the singularity is approached, metric terms are dominant over other terms.  Mesh refinement and asymptotic analysis in the vicinity of the singularity are implemented. Kasner solution for spherical scalar collapse in f(R) theory is obtained, and is confirmed by numerical results. These results support the Belinskii-Khalatnikov-Lifschitz conjecture well.