The Friedman-Lemaître-Robertson-Walker (FLRW) spacetime is often called the Standard Model of modern cosmology. In this talk we discuss small data global existence of solutions to a family of nonlinear wave equations propagating on a fixed FLRW spacetime. The initial value problem here takes the form \begin{equation} \label{semiwave}\begin{cases}-\partial_t^2\phi-n\dot a({a}^{-1}\partial_t\phi) +a^{-2} \Delta \phi = \mathcal{N}(\partial\phi) , \\ \phi(t_0,x)=\phi_0(x), \quad\partial_t\phi(t_0,x)=\phi_1(x),
\end{cases}\end{equation} where $n\gt 2$, $t_0\gt 1$, and $\mathcal{N}$ is a nonlinear function depending (at most) quadratically on derivatives of $\phi$. We consider cosmologies undergoing an accelerated expansion in the direction of positive time \[ a(t)\gt 0 , \qquad \dot a(t)\gt 0, \quad {\text{ and }} \int_{t_0}^{\infty} \frac{1}{a(s)} ds\lt \infty.\] By pairing weighted energy estimates with favorable structure on $\mathcal{N}$, we show how one can establish global wellposedness and sharp decay for solutions to a suitable class of equations of the form \eqref{semiwave}. We also discuss how this method can be extended to the case of nonaccelerated expanding FLRW spacetimes.