The Cosmological CPT Theorem

Please see the attached file for our response to the referee report, or the text from the LaTeX for our response to the referee report below.

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First of all, we would like to thank the referee for their comments about the paper; they helped us identify some points where our discussion was lacking clarity and also noticed some further research directions that we agree are interesting. In the following, we quote the different parts of the report and answer them in turn.

\begin{itemize}
\item \emph{In Eq. 3.53, both the operator $O(\eta)$ and the state $\Psi(\eta)$ depend on time $\eta$, which I find a bit confusing. Did the authors work with the Schrödinger picture or the Heisenberg picture here?}

As we mention below this equation all our analysis was done in the interaction picture, and so both states and operators have time dependence. However, the author is correct that in (3.53) we are not yet in this picture, and so we have changed our notation to the Heisenberg picture where the states do not have any time dependence.
\item \emph{Thus, I wonder if the discussions in the rest of the paper can be generalised to FLRW spacetime as well.}

We agree that this is an interesting question to ask and two of the authors addressed this in a \href{https://arxiv.org/pdf/2510.21701}{recent paper}. Actually, in the case of the RR symmetry, we already in this paper make arguments that are valid for a range of flat FLRW spacetimes (as long as they start in a period of initial inflation with a Bunch-Davies vacuum). The main obstacle to generalising the other symmetries, is that it is not clear that we should impose some kind of CRT condition in generic spacetimes. In the case of de Sitter, the CRT symmetry is inherited from the global extension of the spacetime, which we don't have in other cases. Also, a typical cosmology will not have a scale-invariance symmetry $\eta \to \lambda\eta$, $y \to \lambda y$.

\item \emph{I wonder whether these phase formulas could be tested using real cosmological observables. Can authors give some explicit examples in terms of observables instead of wavefunction coefficients?}

Again, this question is indeed an important one to address, and this was precisely the angle taken in a \href{https://arxiv.org/pdf/2501.06383}{follow-up paper} by one of the authors, where it was shown that this symmetry argument imposes constraints on the parity-odd part of the correlators. We felt that these results were best separated firstly to appropriately assign credit for the work to the author and secondly as we viewed this paper as a look at the formal aspects, with the second paper there for anyone just interested in the phenomenology. We do, however, address that these results could be used in this way in the conclusions.

In general, a constraint on the \emph{phase} of a wavefunction coefficient is not usually easy to observe, because it requires looking at correlators involving time derivatives $\pi$ (e.g.~at least one of $\langle \phi \pi \rangle$ or $\langle \pi \pi \rangle$), which are subleading compared to correlators involving the field $\phi$ itself (see e.g.~\href{http://arxiv.org/abs/2110.01635}{this paper by Șengör and Skordis} for the case of the two-point correlator). It is only at special values of $\Delta$, such as the one addressed by the \href{https://arxiv.org/pdf/2501.06383}{recent paper}, that there is a no-go result on certain correlators becoming large. For generic values of $\Delta$, it is more feasible to view our result as a way to constrain the unobservable part of the wavefunction, based on the observable part.

The Discussion has be edited to clarify this point. The new text begins with the phrase: "In general, a constraint on the phase".
\end{itemize}
In addition to the points raised by the referee we also made the following changes that we felt further improved the presentation of our results.
\begin{itemize}
\item In the introduction to Section 6, we have noted with appropriate references that Charlotte Sleight and collaborators have worked with similar, yet distinct, analytic continuations when relating EAdS correlators to dS correlators.
\item In Section 6, we have corrected some confusing notation where the argument of the wavefunction was the vector components of a single position or momentum vector, so that now it is more explicit that we are rotating all of the vectors upon which the function depends. We have left the notation in Section 7 unchanged for compactness, with this notation defined below (7.19).
\item In Section 6, we correct footnote 47 as we noted that our results do seem to hold for the case of principal series (heavy) fields, if one imposes self-adjoint boundary conditions.
\item In Section 7, we correct footnote 71 as we noted that not all $c=26$ two-dimensional CFTs can be duals to pure gravity in $\text{dS}_3$
.
\item Other minor corrections including added references and correcting typos.
\end{itemize}
These changes will be appearing in v3 on the arXiv within the next few days.

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