  The existence of hot Jupiters has challenged theories of planetary formation
since the first extrasolar planets were detected. Giant planets are generally
believed to form far from their host stars, where volatile materials like water
exist in their solid phase, making it easier for giant planet cores to
accumulate. Several mechanisms have been proposed to explain how giant planets
can migrate inward from their birth sites to short-period orbits. One such
mechanism, called Kozai-Lidov migration, requires the presence of distant
companions in orbits inclined by more than $\sim40$ degrees with respect to the
plane of the hot Jupiter's orbit. The high occurrence rate of wide companions
in hot Jupiter systems lends support to this theory for migration. However, the
exact orbital inclinations of these detected planetary and stellar companions
is not known, so it is not clear whether the mutual inclination of these
companions is large enough for the Kozai-Lidov process to operate. This paper
shows that in systems orbiting cool stars with convective outer layers, the
orbits of most wide planetary companions to hot Jupiters must be well aligned
with the orbits of the hot Jupiters and the spins of the host stars. For a
variety of possible distributions for the inclination of the companion, the
width of the distribution must be less than $\sim20$ degrees to recreate the
observations with good fidelity. As a result, the companion orbits are likely
well-aligned with those of the hot Jupiters, and the Kozai-Lidov mechanism does
not enforce migration in these systems.
