  We show that the inherently large interatomic interactions of a Bose-Einstein
condensate (BEC) can enhance the sensitivity of a high precision cold-atom
gravimeter beyond the shot-noise limit (SNL). Through detailed numerical
simulation, we demonstrate that our scheme produces spin-squeezed states with
variances up to 14 dB below the SNL, and that absolute gravimetry measurement
sensitivities between 2 and 5 times below the SNL are achievable with BECs
between $10^4$ and $10^6$ in atom number. Our scheme is robust to phase
diffusion, imperfect atom counting, and shot-to-shot variations in atom number
and laser intensity. Our proposal is immediately achievable in current
laboratories, since it needs only a small modification to existing
state-of-the-art experiments and does not require additional guiding potentials
or optical cavities.
