Quintessence restrictions on negative power and condensate potentials

We study the cosmological evolution of scalar fields that arise from a phase transition at some energy scale ${\Lambda }_c.$ We focus on negative power potentials given by $V=c{\Lambda }_c^{4+n}{\phi }^{-n}$ and restrict the cosmologically viable values of ${\Lambda }_c$ and n. We make a complete analysis of V and impose SN1a conditions on the different cosmological parameters. The cosmological observations ruled out models where the scalar field has reached its attractor solution. For models where this is not the case, the analytic approximated solutions are not good enough to determine whether a specific model is phenomenologically viable or not and the full differential equations must be solved numerically. The results are not fine-tuned since a change of $45\%$ in the initial conditions does not spoil the final results. We also determine the values of $N_c$ and $N_f$ that give a condensation scale ${\Lambda }_c$ consistent with gauge coupling unification, leaving only four models that satisfy unification and SN1a constraints.