  The search for diffuse non-thermal inverse Compton (IC) emission from galaxy
clusters at hard X-ray energies has been undertaken with many instruments, with
most detections being either of low significance or controversial. Background
and contamination uncertainties present in the data of non-focusing
observatories result in lower sensitivity to IC emission and a greater chance
of false detection. We present 266ks NuSTAR observations of the Bullet cluster,
detected from 3-30 keV. NuSTAR's unprecedented hard X-ray focusing capability
largely eliminates confusion between diffuse IC and point sources; however, at
the highest energies the background still dominates and must be well
understood. To this end, we have developed a complete background model
constructed of physically inspired components constrained by extragalactic
survey field observations, the specific parameters of which are derived locally
from data in non-source regions of target observations. Applying the background
model to the Bullet cluster data, we find that the spectrum is well - but not
perfectly - described as an isothermal plasma with kT=14.2+/-0.2 keV. To
slightly improve the fit, a second temperature component is added, which
appears to account for lower temperature emission from the cool core, pushing
the primary component to kT~15.3 keV. We see no convincing need to invoke an IC
component to describe the spectrum of the Bullet cluster, and instead argue
that it is dominated at all energies by emission from purely thermal gas. The
conservatively derived 90% upper limit on the IC flux of 1.1e-12 erg/s/cm^2
(50-100 keV), implying a lower limit on B>0.2{\mu}G, is barely consistent with
detected fluxes previously reported. In addition to discussing the possible
origin of this discrepancy, we remark on the potential implications of this
analysis for the prospects for detecting IC in galaxy clusters in the future.
