"Expansion" around the vacuum: how far can we go from Lambda?

The cosmological constant ($\Lambda$), i.e., the energy density stored in the true vacuum state of all existing fields in the Universe, is the simplest and the most natural possibility to describe the current cosmic acceleration. However, despite its observational successes, such a possibility exacerbates the well known $\Lambda$ problem, requiring a natural explanation for its small, but nonzero, value. In this paper we discuss how different our Universe may be from the $\Lambda$CDM model by studying observational aspects of a kind of "expansion" around the vacuum given by the equation of (EOS) $p_{d}=-\rho_{d} - A \rho_{d}^{\alpha}$. In different parameter regimes such a parametrization is capable of describing both quintessence-like and phantom-like dark energy, transient acceleration, and various (non)singular possibilities for the final destiny of the Universe, including singularities at finite values of the scale factor, the so-called "Big Rip", as well as sudden future singularities. By using some of the most recent cosmological observations we show that if the functional form of the dark energy EOS has additional parameters very little can be said about their values from the current observational results, which postpones, until the arrival of more precise observational data, a definitive answer to the question posed above.