The bouncing universe provides a possible solution to the Big Bang singularity problem. In this paper we study the bouncing solution in the universe dominated by the Quintom matter with an equation of state (EoS) crossing the cosmological constant boundary. We will show explicitly the analytical and numerical bouncing solutions in three types of models for the Quintom matter with an phenomenological EoS, the two scalar fields and a scalar field with a modified Born-Infeld action. *

We study the cosmology of a Lee-Wick type scalar field theory. First, we consider homogeneous and isotropic background solutions and find that they are nonsingular, leading to cosmological bounces. Next, we analyze the spectrum of cosmological perturbations which result from this model. Unless either the potential of the Lee-Wick theory or the initial conditions are finely tuned, it is impossible to obtain background solutions which have a sufficiently long period of inflation after the bounce. More interestingly, however, we find that in the generic non-inflationary bouncing cosmology, perturbations created from quantum vacuum fluctuations in the contracting phase have the correct form to lead to a scale-invariant spectrum of metric inhomogeneities in the expanding phase. Since the background is non-singular, the evolution of the fluctuations is defined unambiguously through the bounce. We also analyze the evolution of fluctuations which emerge from thermal initial conditions in the contracting phase. The spectrum of gravitational waves stemming for quantum vacuum fluctuations in the contracting phase is also scale-invariant, and the tensor to scalar ratio is not suppressed.PACS numbers: 98.80.Cq

We present nonsingular, homogeneous and isotropic bouncing solutions of the conformal Galileon model. We show that such solutions necessarily begin with a radiation-dominated contracting phase. This is followed by a quintom scenario in which the background equation of state crosses the cosmological constant boundary allowing for a nonsingular bounce which in turn is followed by Galilean Genesis. We analyze the spectrum of cosmological perturbations in this background. Our results show that the fluctuations evolve smoothly and without any pathology, but the adiabatic modes form a blue tilted spectrum. In order to achieve a scale-invariant primordial power spectrum as required by current observations, we introduce a light scalar field coupling to the Galileon kinetically. We find two couplings which yield a scale-invariant spectrum, one of which requires a fine tuning of the initial conditions. This model also predicts a blue tilted spectrum of gravitational waves stemming from quantum vacuum fluctuations in the contracting phase.

A Quintom universe with an equation-of-state crossing the cosmological constant boundary can provide a bouncing solution dubbed the Quintom Bounce and thus resolve the Big Bang singularity. In this paper, we investigate the cosmological perturbations of the Quintom Bounce both analytically and numerically. We find that the fluctuations in the dominant mode in the post-bounce expanding phase couple to the growing mode of the perturbations in the pre-bounce contracting phase. *