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The reconstruction of a k-essence inflationary universe, considering theunification between the swampland criteria and the attractor given by thescalar spectral index $n_S(N)$ together with the slow roll parameter$\epsilon(N)$in terms of the number of $e$-folds $N$ is studied. In the contextof a coupling of the form $L(\phi)\,X$ in the k-essence model, we find theeffective potential $V$ and the coupling parameter $L$ in terms of the scalarspectral index and the slow roll parameter under a general formalism. To applythe unification in our model, we consider some examples in order to rebuild theeffective potential $V(\phi)$ and the coupling parameter $L(\phi)$ as afunction of the inflaton field $\phi$. Here, we find that the reconstructiongives rise to an exponential potential and also to natural and hyperbolicinflation, respectively. Thus, in this article we show that it is possible tounify the theoretical foundations from the swampland criteria and theobservational parameters corroborated by observations, in the reconstruction ofan inflationary universe.

Reconstructing k-essence: Unifying the attractor $n_S(N)$ and the swampland criteria