An unbinned likelihood fit for the measurement of $\phi_s$ in the decay $B_s \to J/\psi\phi$

The interference between $B^0_s$ decays to $J/\psi\phi$ with or without $B^0_s-\overline{B}^0_s$ oscillation gives rise to a CP violating phase $\phi_s$. In the Standard model, this phase is predicted to be $-2\beta_s$ = $-0.0368\pm0.0017$. New Physics could significantly modify this prediction. The measurement of this phase is one of the key goals of the LHCb experiment. This document details one of the vital components of the analysis of the decay $B^0_s \to J/\psi\phi$, an unbinned likelihood fit method to extract the phase $\phi_s$. The fit method is decay time and decay angle dependant. Angular acceptance and proper time acceptance and proper time resolution effects are included. We also perform systematic studies to estimate effects due to uncertainties of the angular acceptance and flavor tagging. The expected sensitivity on $\phi_s$ after one year of LHCb running at the nominal luminosity of 2 fb$^{-1 }$ is (0.032 $\pm$0.0001) rad.