  Competing and intertwined orders are ubiquitous in strongly correlated
electron systems, such as the charge, spin, and superconducting orders in the
high-Tc cuprates. Recent scanning tunneling microscopy (STM) measurements
provide evidence for a charge density wave (CDW) that coexists with
superconductivity in the heavy Fermion metal UTe2. This CDW persists up to at
least 7.5 K and, as a CDW breaks the translational symmetry of the lattice, its
disappearance is necessarily accompanied by thermodynamic phase transition.
Here, we report high-precision thermodynamic measurements of the elastic moduli
of UTe2. We observe no signature of a phase transition in the elastic moduli
down to a level of 1 part in 10^7, strongly implying the absence of bulk CDW
order in UTe2. We suggest that the CDW and associated pair density wave (PDW)
observed by STM may be confined to the surface of UTe2.
