Integrability of R² gravity cosmological models with radiation

We consider R² R 2 gravity cosmological models with radiation. We find the general solution to the trace equation R=0 □ R = 0 in the spatially flat Friedmann–Lemaitre–Robertson–Walker (FLRW) metric. We analyze possible evolution of the Hubble parameter depending on the sign of the radiation energy density and find conditions for the existence of a bounce solution. A scalar field Lagrangian with the induced gravity term and the fourth-order monomial potential can play a role of radiation. In this case, we also obtain the general solution to the field equation. Therefore, the resulting R² R 2 gravity model with a scalar field is integrable in the spatially flat FLRW metric. Using a conformal metric transformation, we obtain a two-field chiral cosmological model that is also integrable in the spatially flat FLRW metric.