Possible evolution of a bouncing universe in cosmological models with   non-minimally coupled scalar fields
  We explore dynamics of cosmological models with bounce solutions evolving on a spatially flat Friedmann-Lemaitre-Robertson-Walker background. We consider cosmological models that contain the Hilbert-Einstein curvature term, the induced gravity term with a negative coupled constant, and even polynomial potentials of the scalar field. Bounce solutions with non-monotonic Hubble parameters have been obtained and analyzed. The case when the scalar field has the conformal coupling and the Higgs-like potential with an opposite sign is studied in detail. In this model the evolution of the Hubble parameter of the bounce solution essentially depends on the sign of the cosmological constant. 
Ekaterina O. Pozdeeva, Maria A. Skugoreva, Alexey V. Toporensky,   Sergey Yu. Vernov
gr-qc astro-ph.CO hep-th