Tailoring cosmologies in cubic shift-symmetric Horndeski gravity

We present a method for furnishing flat Friedman-Robertson-Walker spacetimes with nearly arbitrary dynamics in an important subclass of cubic Horndeski theory -- specifically shift-symmetric, cubic Horndeski theory with a vanishing conserved current. This builds on insight from previous work on the construction of static and spherically-symmetric hairy spacetimes in the same sector. The method is explicitly demonstrated by deriving exact analytical solutions describing an inflating universe and several power-law expansion scenarios, and by showing how the predicted evolution of the Hubble parameter in $\Lambda$CDM can be fit to a particular choice of Horndeski model function. We fully characterize the classes of cosmological models that cannot be generated purely by selecting a Horndeski model function.