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Matter and quasi-matter bounce scenarios are studied for an F(R) gravitymodel with holonomy corrections and a Lagrange multiplier, with a scale factor$a(t) = \left(a_0t^2 + 1 \right)^n$, where the Hubble parameter squared has alinear and a quadratic dependence on the effective energy density. Provided $n< 1/2$, it is shown that the primordial curvature perturbations are generateddeeply into the contracting era, at large negative time, which makes thelow-curvature limit a good approximation for calculating the perturbation powerspectrum. Moreover, it is shown that, for $n$ within this range, the obtainedcosmological quantities are fully compatible with the Planck constraints, andthat the "low curvature limit" comes as a viable approximation to calculate thepower spectra of both scalar and tensor perturbations. Using reconstructiontechniques for F(R) gravity with the Lagrange multiplier, the precise form ofthe effective F(R) gravity is found, from which one determines the powerspectra of scalar and tensor perturbations in such bouncing scenario.Correspondingly, the spectral index for curvature perturbations and the tensorto scalar ratio are obtained, and these values are successfully confronted withthe latest Planck observations. Further, it is shown that both the weak and thenull energy conditions are satisfied, thanks to the holonomy correctionsperformed in the theory--which are then proven to be necessary for achievingthis goal. In fact, when approaching the bouncing era, the holonomy correctionsbecome significant and play a crucial role in order to restore the energyconditions. Summing up, a cosmological bouncing scenario with the scale factorabove and fulfilling the energy conditions can be adequately described by theF(R) model with a Lagrange multiplier and holonomy corrections, which prove tobe very important.

Viable non-singular cosmic bounce in holonomy improved F(R) gravity endowed with a Lagrange multiplier