Integrability of $R^2$ gravity cosmological models with radiation

We consider cosmological $R^2$ gravity models with radiation.

We find the general solution to the trace equation $\Box R=0$ 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 $R^2$ 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.