[Cai, Yi-Fu] Arizona State Univ, Dept Phys, Tempe, AZ 85287 USA
; [Cai, Yi-Fu] Chinese Acad Sci, Inst High Energy Phys, Beijing 100049, Peoples R China
; [Saridakis, Emmanuel N.] Chongqing Univ Posts & Telecommun, Coll Math & Phys, Chongqing 400065, Peoples R China
We investigate cyclic and singularity-free evolutions in a universe governed by Lagrange-multiplier modified gravity, in both scalar-field cosmology and the f(R) one. In the scalar case, cyclicity can be induced by a suitably reconstructed simple potential, and the matter content of the Universe can be successfully incorporated. In the case of f(R)-gravity, cyclicity can be induced by a suitably reconstructed second function f(2)(R) of a very simple form; however, the matter evolution cannot be analytically handled. Furthermore, we study the evolution of cosmological perturbations for the two scenarios. For the scalar case the system possesses no wavelike modes due to a dust-like sound speed, while for the f(R) case there exists an oscillation mode of perturbations which indicates a dynamical degree of freedom. Both scenarios allow for stable parameter spaces of cosmological perturbations through the bouncing point.