*Cosmo Comments*. The following comments can also be viewed as annotations on the paper via Hypothesis.

This paper examines the possibility that a phantom crossing, i.e. a point in time where the dark energy equation of state crossed the so-called phantom divide $w=-1$ to $w < -1$, might help to address the $H_0$ tension. The case of phantom dark energy as a solution to the $H_0$ tension has been known for some time (two important works are e.g. https://arxiv.org/abs/1606.00634 and https://arxiv.org/abs/1701.08165). In fact, the current work can loosely be seen as a follow-up on the latter reference.

The authors apply a parametric reconstruction of the dark energy density, $rho_\mathrm{DE}$, by assuming the occurrence of an extremum at some scale factor $a_m$. By expanding $rho_\mathrm{DE}$ in a Taylor series around this point and keeping terms up to third order, three new parameters are introduced in addition to the LCDM parameters. As their final result, the authors claim that this model can address the $H_0$ tension, finding in particular $H_0=70.25 \pm 0.78 \mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ at 68% c.l. by combining CMB, CMB lensing, BAO, Pantheon and a prior on $H_0$ from SH0ES.

These results, if correct, are interesting since they would represent a concrete counter-example to the "no-go theorem" for late-time solutions to the $H_0$ tension, which has been nicely summarized recently in the "Hubble hunter's guide" [https://arxiv.org/abs/1908.03663]. The paper therefore brings up an important point. However, I do not agree with some aspects of the analysis and would appreciate some clarifications from the authors.

1) The phantom crossing is a statement about the dark energy equation of state only. Since the density is related non-linearly to an integral over the equation of state, it does not directly translate into properties in the dark energy density. In the standard CPL parametrization for the dark energy equation of state, no extremum in the density occurs even if $w=-1$ is crossed. However, the authors say on page 2 that an extremum in $\rho_\mathrm{DE}$ at a certain scale factor $a_m$ is "a typical feature in dark energy model with phantom crossing". Could the authors provide a concrete reference to back-up this statement? It is intriguing, but I have not seen any reference to concrete models in the paper.

2) The posterior on $a_m$ from CMB and CMB + CMB lensing data alone shows quite a prominent peak as $a \to 0$. Could the authors please clarify where this is coming from? It is not clear to me why CMB and CMB lensing data would prefer a phantom transition at extremely early times. It probably cannot solve the $A_L$ anomaly, since a phantom dark energy component behaves less like matter than a quintessence component, thus no extra lensing is induced. In conclusion, it is unclear to me why that peak is present when considering CMB and CMB+CMB lensing data alone. Naively, I would have expected a posterior which would have been rather flat. It would be nice to understand where these peaks are coming from.

3) This question relates to dataset consistency. I am not sure the authors are allowed to take the CMB+all combination, i.e. combining CMB, CMB lensing, BAO, Pantheon and a prior on $H_0$ from SH0ES (R19, it is unclear why the CCHP measurement is not mentioned in the paper). Certainly, subsets of this combination can be constructed which are in mutual tension. For example, CMB+lensing gives $H_0 > 92.8 \mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ (is this 68% c.l. or 95% c.l.?) which is clearly in tension with R19. Same for CMB+lensing vs CMB+BAO, or CMB+lensing vs CMB+Pantheon. Similarly, in Table II, I do not see the CMB+BAO+Pantheon dataset combination, raising the question whether CMB+BAO+Pantheon is compatible with R19. The sample holds true for the combination CMB+BAO+lensing+Pantheon versus R19. I would guess that both CMB+BAO+Pantheon and CMB+BAO+lensing+Pantheon are in tension with R19. However, the results for these combinations are not reported in Table II.

4) Following up on the previous point, the most interesting dataset combination to see would be CMB+BAO+Pantheon. The reason is shown in Fig. 1, of the "Hubble hunter's guide" [https://arxiv.org/abs/1908.03663]. CMB+BAO+Pantheon is the most challenging dataset to be reconciled with R19. Particularly, it is precisely the dataset that seems to point towards an early-Universe solution. Without the outcome for $H_0$ from CMB+BAO+Pantheon, I cannot have full confidence in the result that the $H_0$ tension is alleviated.

5) Another point concerns the choice of priors. It could be enlightening to use a flat prior on $\log_{10}(a_m)$ or a least informative one? This would reflect the fact that we do not know the order of magnitude of $a_m$ by putting equal weights on different decades. One suggestion might be prior boundaries between -4 and 0 so that the transition happens after matter-radiation equality? This would be similar to what Di Valentino et al. did in [https://arxiv.org/abs/1906.11255], where a log prior on $a^*$ was set between -4 and 0, as shown in Table I. Thus there is clearly some motivation to such a prior.

6) Lastly, following up on point 5, I also have some concerns about the priors chosen for the parameters $\alpha$ and $\beta$. The priors allow for rather large values of $\alpha$ and $\beta$. For these values, the Taylor expansion in Eq. (1) may no longer be consistent and higher-order terms might actually dominate the expansion. This manifests in the unphysical values especially for $\rho_\mathrm{DE}$. $\rho_\mathrm{DE}$ might change sign and go negative, for example, and when it crosses $\rho_\mathrm{DE}=0$, the equation of state in Eq. (4) will diverge. Clearly, the Taylor expansion is a reasonable phenomenology-related choice. However, I have the feeling that more physically-motivated upper prior edges on $\alpha$ and $\beta$ should be chosen.

Despite these comments, I found the paper interesting to read. It would most certainly be fascinating if phantom dark energy could solve the $H_0$ tension. However, as explained in the points above, I have some concerns about this conclusion!

*[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.]*