  Circumstantial evidence suggests that most known extra-solar planetary
systems are survivors of violent dynamical instabilities. Here we explore how
giant planet instabilities affect the formation and survival of terrestrial
planets. We simulate planetary system evolution around Sun-like stars from
initial conditions that comprise: an inner disk of planetesimals and planetary
embryos, three giant planets at Jupiter-Saturn distances, and a massive outer
planetesimal disk. We then calculate dust production rates and debris disk SEDs
assuming that each planetesimal particle represents an ensemble of smaller
bodies in collisional equilibrium. We predict a strong correlation between the
presence of terrestrial planets and debris disks, mediated by the giant
planets. Strong giant planet instabilities destroy all rocky material -
including fully-formed terrestrial planets if the instabilities occur late -
along with the icy planetesimals. Stable or weakly unstable systems allow
terrestrial planets to accrete and significant dust to be produced in their
outer regions. Stars older than ~100 Myr with bright cold dust emission (at ~70
microns) signpost the dynamically calm environments conducive to efficient
terrestrial accretion. We predict that while the typical eccentricities of
terrestrial planets are small, there should exist a novel class of terrestrial
planet system whose single planet undergoes large amplitude oscillations in
eccentricity and inclination. By scaling to the observed semimajor axis
distribution of giant exoplanets, we estimate that terrestrial exoplanets in
the same systems should be a few times more abundant at 0.5 AU than giant or
terrestrial exoplanets at 1 AU. Finally, we discuss the Solar System, which
appears to be unusual in combining a rich terrestrial planet system with a low
dust content.
