  Using exact continuous quantum Monte Carlo techniques, we study the zero and
finite temperature properties of a system of harmonically trapped one
dimensional spin 1/2 fermions with short range interactions. Motivated by
experimental searches for modulated Fulde-Ferrel-Larkin-Ovchinikov states, we
systematically examine the impact of a spin imbalance on the density profiles.
We quantify the accuracy of the Thomas-Fermi approximation, finding that for
sufficiently large particle numbers (N > 100) it quantitatively reproduces most
features of the exact density profile. The Thomas-Fermi approximation fails to
capture small Friedel-like spin and density oscillations and overestimates the
size of the fully paired region in the outer shell of the trap. Based on our
results, we suggest a range of experimentally tunable parameters to maximize
the visibility of the double shell structure of the system and the
Fulde-Ferrel-Larkin-Ovchinikov state in the one dimensional harmonic trap.
Furthermore, we analyze the fingerprints of the attractive contact interactions
in the features of the momentum and pair momentum distributions.
