  The long-sought pair density wave (PDW) is an exotic phase of matter in which
charge density wave (CDW) order is intertwined with the amplitude or phase of
coexisting, superconducting order \cite{Berg2009,Berg2009b}. Originally
predicted to exist in copper-oxides, circumstantial evidence for PDW order now
exists in a variety of materials. Recently, scanning tunneling microscopy (STM)
studies have reported evidence for a three-component charge density wave (CDW)
at the surface of the heavy-fermion superconductor, UTe$_2$, persisting below
its superconducting transition temperature. Here, we use hard x-ray diffraction
measurements on crystals of UTe$_2$ at $T = 1.9$ K and $12$ K to search for a
bulk signature of this CDW. Using STM measurements as a constraint, we
calculate the expected locations of CDW superlattice peaks, and sweep a large
volume of reciprocal space in search of a signature. We failed to find any
evidence for a CDW near any of the expected superlattice positions in many
Brillouin zones. We estimate an upper bound on the CDW lattice distortion of
$u_{max} \lesssim 4 \times 10^{-3} \mathrm{\AA}$. Our results suggest that the
CDW observed in STM is either purely electronic, somehow lacking a signature in
the structural lattice, or is restricted to the material surface.
