Quintessence and tachyon dark energy models with a constant equation of state parameter

In this work we determine the correspondence between quintessence and tachyon dark energy models with a constant dark energy equation of state parameter, $w_e$. Although the evolution of both the Hubble parameter and the scalar field potential with redshift is the same, we show that the evolution of quintessence/tachyon scalar fields with redshift is, in general, very different. We explicity demonstrate that if $w_e\mathrm{\ne }-1$ the potentials need to be very fine-tuned for the relative perturbation on the equation of state parameter, $\mathrm{\Delta }{w}_e\mathrm{/}(1+w_e)\ll 1$, to be very small around the present time. We also discuss possible implications of our results for the reconstruction of the evolution of $w_e$ with redshift using varying couplings.