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### [2004.00339] Lorentzian Quintessential Inflation

Posted: April 11 2020
This paper was commented on through Cosmo Comments. The following comments can also be viewed as annotations on the paper via Hypothesis.

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:
1. 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?
2. 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.
3. Where do the priors on $N$, $\xi$ and $\Gamma$ come from? Wouldn't it make more sense to choose flat priors on these quantities?
4. 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.
5. There is a presumably incomplete sentence in Section 1: "This assumption is based on [51] 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.]

### Re: [2004.00339] Lorentzian Quintessential Inflation

Posted: June 19 2020
We thank the writer for his/her points, since he/she addressed for us final modification before the essay will be published in IntJMPD. Our comments are attached:

"I do not think it is very well motivated " - The motivation for the anzats is a function that for any logarithmic derivative has a small slow roll parameter value, for the beginning and the end of inflation (see eq. 3).

1. The writer has a fair point. This is a dominant energy condition. We parameterize the slow roll parameter and later on implement this idea with a canonical scalar field, which satisfies this energy condition.

2. As we state in the abstract: "The form corresponds to the vacuum energy both in the inflationary and in the dark energy epochs". The matter sector is procured by a reheating mechanism that was studied in 2006.04129 (accepted in EPJC). The advantage of this idea is the connection between the inflationary and in the dark energy epochs with a small number of parameters. The inflationary observables, and the small value of the dark energy density is predicted from this small amount of parameters.

3. The writer is correct. In the second version on the arxiv we addressed a uniform prior, and still the inflationary observables stay in good agreement with the Planck 2018 data.

4. Let us separate between the model perspective and the field theory perspective: The model perspective has an advantage to reach many predictions from a few fundamental parameters. For instance, the difference between vacuum energies is related to their logarithm values according to this model , and gives \xi \sim 100, which does not represent a fine tuning. We can state that the separate observables in the cosmological standard model: r, ns, As, \Omega_\Lambda, are based on \xi, \Gamma, N_0 from this model.
In the field theory perspective there are some papers that discuss the same qualitative behavior for the potential from breaking scale invariance. See for example: Gen.Rel.Grav. 47 (2015) no.2, 10 by Guendelman et al.
\Lambda_0 is an integration from the assumption of \epsilon. The asymptotic values of the energy density read: \Lambda_0 exp(+- \xi/2), which emphasizes the see-saw mechanism: one region is very large and the other is very small.
5. In the second version we modified this sentence.

Best Regards,
Benisty and Guendelman