Adiabatic Duality: Duality of cosmological models with varying slow-roll parameter and sound speed

There have been thousands of cosmological models for our early universe proposed in the literature, and many of them claimed to be able to give rise to scale-invariant power spectrum as was favored by the observational data. It is thus interesting to think about whether there are some relations among them, e.g., the duality relation. In this paper, we investigate duality relations between cosmological models in framework of general relativity (GR) , not only with varying slow-roll parameter $epsilon$, but also with sound speed $c_s$, which can then be understood as adiabatic duality. Several duality relationships are formulated analytically and verified numerically. We show that models with varying $epsilon$ and constant $c_s$ can be dual in scalar spectral index, but not tensor one. On the other hand, allowing both $epsilon$ and $c_s$ to vary can make models dual in both scalar and tensor spectral indices.