  The heat transport in rotating Rayleigh-B\'enard convection is considered in
the limit of rapid rotation (small Ekman number $E$) and strong thermal forcing
(large Rayleigh number $Ra$). The analysis proceeds from a set of
asymptotically reduced equations appropriate for rotationally constrained
dynamics; the conjectured range of validity for these equations is $Ra \lesssim
E^{-8/5}$. A rigorous bound on heat transport of $Nu \le 20.56Ra^3E^4$ is
derived in the limit of infinite Prandtl number using the background method. We
demonstrate that the exponent in this bound cannot be improved on using a
piece-wise monotonic background temperature profile like the one used here.
This is true for finite Prandtl numbers as well, i.e. $Nu \lesssim Ra^3$ is the
best upper bound for this particular setup of the background method. The
feature that obstructs the availability of a better bound in this case is the
appearance of small-scale thermal plumes emanating from (or entering) the
thermal boundary layer.
