  The dependence of the heat transfer, as measured by the nondimensional
Nusselt number $Nu$, on Ekman pumping for rapidly rotating Rayleigh-B\'enard
convection in an infinite plane layer is examined for fluids with Prandtl
number $Pr = 1$. A joint effort utilizing simulations from the Composite
Non-hydrostatic Quasi-Geostrophic model (CNH-QGM) and direct numerical
simulations (DNS) of the incompressible fluid equations has mapped a wide range
of the Rayleigh number $Ra$ - Ekman number $E$ parameter space within the
geostrophic regime of rotating convection. Corroboration of the $Nu$-$Ra$
relation at $E = 10^{-7}$ from both methods along with higher $E$ covered by
DNS and lower $E$ by the asymptotic model allows for this range of the heat
transfer results. For stress-free boundaries, the relation $Nu-1 \propto (Ra
E^{4/3} )^{\alpha} $ has the dissipation-free scaling of $\alpha = 3/2$ for all
$E \leq 10^{-7}$. This is directly related to a geostrophic turbulent interior
that throttles the heat transport supplied to the thermal boundary layers. For
no-slip boundaries, the existence of ageostrophic viscous boundary layers and
their associated Ekman pumping yields a more complex 2D surface in $Nu(E,Ra)$
parameter space. For $E<10^{-7}$ results suggest that the surface can be
expressed as $Nu-1 \propto (1+ P(E)) (Ra E^{4/3} )^{3/2}$ indicating the
dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative
prefactor $(1+ P(E))$ where $P(E) \approx 5.97 E^{1/8}$. It follows for
$E<10^{-7}$ that the geostrophic turbulent interior remains the flux bottleneck
in rapidly rotating Rayleigh-B\'enard convection. For $E\sim10^{-7}$, where DNS
and asymptotic simulations agree quantitatively, it is found that the effects
of Ekman pumping are sufficiently strong to influence the heat transport with
diminished exponent $\alpha \approx 1.2$ and $Nu-1 \propto (Ra E^{4/3}
)^{1.2}$.
