Cosmological models with non-positive definite potentials and bounce solutions

In the General Relativity cosmological models the initial period of the Universe evolution with energies above the Planck scale should be described by quantum gravity because the classical evolution includes the initial singularity. Important question of theoretical cosmology is whether the entire Universe evolution can remain classical and has no singularity. Modified gravity models of bouncing universes with a period of contraction followed by a bounce and a resent period of expansion attract a lot of attention. At the bounce point the Hubble parameter is equal to zero, therefore, the potential of the scalar field minimally or non-minimally coupled to gravity should be negative. The dynamics of non-minimally coupled scalar field cosmological models with non-positive definite potential has been studied by I.Ya. Aref'eva, N.V. Bulatov, R.V. Gorbachev, and S.Yu. Vernov (Class. Quantum Grav. 31 (2014) 065007). Using the results of this paper, we consider cosmological models including the Hilbert-Einstein curvature term and the induced gravity term with a negative coupled constant. The considering models generalize the integrable cosmological model with bounce solutions proposed by B. Boisseau, H. Giacomini, D. Polarski, and A.A.Starobinsky (J. Cosmol. Astropart. Phys. 1507 (2015) 002). The case when the scalar field has the conformal coupling and the Higgs-like potential with an opposite sign is studied in detail. We show that in the proposed model the evolution of the Hubble parameter of the bounce solutions can be non-monotonic and essentially depends on the sign of the constant term in the potential.