This paper proposes an ansatz for an expansion history of the Universe unifying the two accelerated eras: inflation (early-time acceleration) and dark energy (late-time acceleration). The ansatz is made at the level of the slow-roll factor, which effectively amounts to an ansatz at the level of the Hubble rate throughout the history of the Universe. While the approach is potentially interesting, at the end of the day, I do not think it is very well motivated because what makes more sense is to start from a well-motivated potential and work out things from there. Anyway, while reading this paper I had a few comments/suggestions for the authors, which I list below:
- On page 1, it is stated that the strong energy condition (SEC) yields another bound on the coefficients. I do not understand why one should invoke the SEC in this context. It is known that the SEC needs to be violated to get acceleration (whether inflation or dark energy)! In fact, the SEC states that $w\geq-1/3$. Maybe the authors actually meant the dominant energy condition?
- I do not understand whether the resulting form of $H(N)$ can be reconciled with the usual matter and radiation epochs between the two accelerating regimes. To correctly obtain these epochs is crucial for the model to make any sense.
- Where do the priors on $N$, $\xi$ and $\Gamma$ come from? Wouldn't it make more sense to choose flat priors on these quantities?
- The "cosmological seesaw" mechanism is potentially interesting, although at the end of the day I think it is just moving the cosmological constant problem one step further. To get a small cosmological constant, one wants a large inflationary energy scale, but one should not forget that the latter is bounded by the tensor-to-scalar-ratio. In addition, it is unclear where $\Lambda_0$ comes from as a fundamental parameter. Finally, as mentioned above, it makes more sense to start from the potential and work out things from there. This is clear from Eq. (18), where the authors reverse engineer the potential needed to make the model work. This is of course a subjective statement, but I find it hard to see this potential emerge in a sensible UV theory.
- There is a presumably incomplete sentence in Section 1: "This assumption is based on  where"?
[These comments were shared with us by a member of the cosmology community. They do not necessarily reflect the opinion of the Cosmo Comments team.]