An exact solution of the higher-order gravity in standard radiation-dominated era

We report that the standard evolution of radiation-dominated era (RDE) universe $a \propto t^{1/2}$ is a sufficient condition for solving sixth order gravitational field equation derived from Lagrangian containing $B R^{ab}R_{ab} + C R {R^{;c}}_{c}$ as well polynomial $f(R)$ spatially flat radiation FLRW universe. By virtue similarity between $R^{ab}R_{ab}$ and $R^2$ models up to background vanishing property ${R^{;c}}_{c}$ $ H = 1/(2t)$, analytical solution can be obtained special case general one. This proves cosmic valid even within modified theory involving higher-order terms. An application this tensor-type perturbation reduces complicated second wave. discuss possible ways discriminate gravity model on observations such wave disturbed primordial abundances.