  The primary work presented in this paper focuses on the calculation of
density-density dynamical correlations in an attractive two dimensional Fermi
gas in several physically interesting regimes, including the strongly
correlated BEC-BCS crossover regime. We use state-of-the-art dynamical BCS
theory and we address the possibility to renormalize the interaction strength,
using unbiased Quantum Monte Carlo results as an asset to validate the
predictions. We propose that a suitable interplay between dynamical BCS theory,
which is computationally very cheap and yields results directly in real time
domain, and Quantum Monte Carlo methods, which are exact but way more demanding
and limited to imaginary time domain, can be a very promising idea to study
dynamics in many body systems. We illustrate the idea and provide quantitative
results for a few values of the interaction strength in the cold gas.
