No Slow-Roll Inflation A La Generalized Chaplygin Gas In General Relativity

The Generalized Chaplygin Gas (GCG) model is characterized by the equation of\nstate $P = -A \\rho^{-\\alpha}$, where $A>0$ and $\\alpha < 1$. The model has been\nextensively studied due to its interesting properties and applicability in\nseveral contexts, from late-time acceleration to primordial inflation.\nNonetheless we show that the inflationary slow-roll regime cannot be satisfied\nby most of the parameter space of the GCG model when General Relativity (GR) is\nconsidered. In particular, although the model has been applied to inflation\nwith $0 < \\alpha < 1$, we show that for $-1 < \\alpha \\le 1$ there is no\nexpansion of the Universe but an accelerated contraction. For $\\alpha \\le\n-5/3$, the second slow-roll parameter $\\eta_H$ is larger than unity, so there\nis no sustained period of inflation. Only for $\\alpha$ very close to -1 the\nmodel produces enough $e$-folds, thus greatly reducing its parameter space.\nMoreover, we show that the model is ruled out by the Planck 2018 results.\nFinally, we extend our analysis to the Generalized Chaplygin-Jacobi Gas (GCJG)\nmodel. We find that the introduction of a new parameter does not change the\nprevious results. We thus conclude that the violation of the slow-roll\nconditions is a generic feature of the GCG and GCJG models during inflation\nwhen GR is considered and that the models are ruled out by the Planck 2018\nresults.\n