Density Induced Vacuum Instability

We thank the referee for the insightful comments and interesting questions on our manuscript. Below we address in detail each of the points raised and, where appropriate, list the corresponding changes we have made.

1) Classical vacuum transitions in the early universe

In our paper we consider a simple potential with two non-degenerate minima, the higher of which gets destabilised by baryonic-density effects thus triggering a phase transition. We completely agree with the referee that such a phase transition could a priori happen as well in the early universe, in particular due to finite temperature effects. This would depend on the temperature necessary to destabilize the metastable minimum, e.g. $T \sim 100 \, \text{MeV}$ at the quark-hadron transition would roughly correspond to the densities found in neutron stars $n \sim (100 \, \text{MeV})^3$, or the higher $T \sim 100 \, \text{GeV}$ if we consider a barrier that depends on the Higgs VEV. However, it could well be that such high temperatures were never reached in the early universe (if reheating temperatures were lower). In this respect, not much is known about the hot history of the universe for temperature above BBN ($T \sim \text{MeV}$), where baryonic densities were very low $\rho_{\tiny{\text{B}}} \sim 10^{-10} \, \text{MeV}^3$ (see e.g. Hook 2017). For this reason, we chose to be agnostic about the fate of the metastable vacuum in the early universe. Besides, we would like to point out as well that one of our motivations for considering such type of potentials is the relaxion, were the two minima we have focused on are embedded in a landscape containing many non-degenerate minima. It has been shown in Graham 2015 and Choi 2016 that, due to the overall flatness of the relaxion potential, even if reheating temperatures were high enough as for the barriers between minima to disappear, the universe would have cooled so fast that the relaxion would not have rolled many minima down the potential. This means that the relaxion would have eventually landed in some minimum that is susceptible to being destabilised by stellar compact objects.

We have added a couple of sentences elaborating on this point at the end of section 2.

2) Small bounce action for vacuum decay

The referee is right in that to assess if the decay probability is large enough for the in-density decay to take place, the bounce action $\textit{and}$ the time $\mathcal{T}$ and volume $\mathcal{V}$ are relevant. We did have in mind rough estimates for what the latter are, which we present in the following. Still, the decay rate is exponentially sensitive to the value of $(f/\Lambda_B)^4 = 1/\lambda$, so the message we tried to convey is that a strong coupling $\sqrt{\lambda} = O(1)$ is generically required, as we now show: The tunneling rate per volume is given by \begin{equation} \Gamma/\mathcal{V}=Ae^{-S_{\tiny{\text{B}}}}, \end{equation} where $S_{\text{B}} \sim 27 \sqrt{3} \pi^2 (1-\zeta)^2 (1-\delta^2)^{-3} \lambda^{-1}$ for a deep minimum and $A \sim f^4$, with $\zeta = \zeta(n)$ encoding the effect of a non-zero density, $\zeta(0) = 0$ and $\zeta(n>0) \neq 0$. The decay probability becomes order one, i.e. $\Gamma \mathcal{T} = O(1)$ for \begin{equation} \frac{\lambda (1-\delta^2)^3}{(1-\zeta)^2} \sim \frac{27 \sqrt{3} \pi^2}{\log(f^4 \mathcal{T} \mathcal{V})} \, . \end{equation} The volume of all the neutron stars in the observable universe can be roughly estimated as $V_{\text{NS}} \sim R_{\text{NS}}^3 N_{\text{NS/G}} N_{\text{G}} \sim 10^{29} \, \text{m}^3 $, where $R_{\text{NS}} \sim 10 \, \text{km}$ is the typical radius of a neutron star, $N_{\text{NS/G}}\sim 10^9$ the number of neutron stars per galaxy (like ours) and $N_{\text{G}} = 10^{11}$ the number of galaxies. As the relevant time scale one can just take the age of the universe, since star formation happened relatively early, $\tau\sim 1/H_0 \sim 10^{17} \, \text{s}$. One then finds for $f < M_{\tiny\text{Pl}}$, \begin{equation} \frac{\lambda (1-\delta^2)^3}{(1-\zeta)^2} \gtrsim 1.0 \,. \end{equation} Note that such a value barely changes if we consider instead the volume of the whole universe, $V_{U} \sim 1/H_0^3 \sim 10^{78} \, \text{m}^3$. Therefore, the the change in $\zeta$ is the important parameter for the decay of the metastable minimum taking place in density and not in vacuum, as we explained in the text. In the best case scenario where the metastable minimum is not very deep, e.g. $\delta \approx 0.8$, considering a mild reduction of the potential barrier in density, $\zeta \approx 0.5 < \delta^2$, is enough to catalyse the decay of a minimum that is essentially stable in vacuum. From the inequality above this means that $\sqrt{\lambda} \gtrsim 2.9$, which is certainly not a weak coupling. Deeper metastable minima require even larger coupling.

We have rewritten the sentence below Eq. (45) and added some further details to make this point more clear.

3) Vacuum decay catalysed by short but violent processes

All we wanted to say with the statement that ``...seeded nucleation of bubbles of the true ground state would generically not take place during the formation of the star...'' is that the time scales of vacuum decay ($T_B \equiv 1/\Gamma$) and star formation ($T_S$) are unlikely to coincide. This is in fact pointed out in the sentence immediately after, where we also state that the lifetime of the star (i.e.~the time since its formation, $\mathcal{T} \sim 1/H_0$) could be much larger. In addition, let us note that the violent process of a star explosion or the formation of a black hole would be a classical effect not directly related to vacuum decay, which is the focus of this section.

4) System of many stars

We completely agree with the referee. We already took this fact into account since to derive the decay rate we considered the total volume in stars, as explained in detail in point 2 above.

5) Vacuum decay catalysed by black holes

We thank the referee for bringing these recent references to our attention. We have now included the corresponding citations.

6) Pop III stars

We thank the referee for pointing out the possibility that Pop III stars could have acted as nucleation seeds for phase transitions consistent with cosmological data. We have added a sentence and provided further details on the properties of this hypothetical population of stars.

7) Gravitational waves

We agree with the referee that it would be very interesting to study the potential gravitational wave signals associated with the phase transitions we have studied. Along with their detailed cosmological dynamics, this is something we are currently investigating. We are stating this now in section 5.