Hot and heavy dark matter from a weak scale phase transition

We thank the referees for their reports. We give respond to the points raised by the referees below.

REPLY TO REPORT 1

The referee made a good point that we should have been clearer regarding the gravitational wave (GW) signal in the T_n > T_infl case. We have now included a LISA contour in the summary plot for the T_n > T_infl case, i.e. Fig. 2 left. As can be seen, this is outside the region of validity for our DM parameter space, but should allow the reader to better understand what is going on. We have also included example GW spectra for the T_n > T_infl case in the new Fig. 3 left. The captions of the figures have been suitably adjusted to provide some more information. We have also expanded our discussion at the end of Sections 4 and 5 to qualitatively describe the behaviour of the gravitational wave signal regions and discuss the complementarity of the GW signal and current/future DM free streaming constraints.


REPLY TO REPORT 2


Regarding the backreaction of the pair production on the wall, it is true we have not attempted to solve the scalar equations of motion in the presence of such an effect, in order to calculate any local deformation of the wall etc. On the other hand, as a more modest first check, we can estimate the corrections to the pressure given in Eq. (4), due to the pair creation effect (they were also pointed out in https://arxiv.org/abs/2010.02590). We have now added an explanation regarding both these points at the end of Sec. 3.

The very interesting recent study pointed out by the referee, 2203.05750, uses some distinct features of Fuzzy Dark Matter (FDM) to put constraints on such models from the Segue 1 and Segue 2 Ultra-Faint Dwarf (UFD) galaxies. The analysis exploits the wave interference effects in FDM which leads to dynamical heating of stellar orbits. This effect is not present in warm dark matter (WDM) models. Although the latter do suppress the formation of the low mass halos which would host Milky Way satellite galaxies. This effect is what allows one to place constraints on WDM (and our model) via observations of Milky Way satellites, e.g. as done in 2008.00022 (our Ref. [8]), which in part uses Pan-STARRS1 data including Segue 1 and Segue 2 (see 1912.03302). Although the UFD galaxies are (partly) the same, the heating effect allowed 2203.05750 to place a much more stringent lower bound ~10^-19 eV on FDM from the UFD, compared to the bound of ~10^-21 eV, coming from Halo suppression derived in 2008.00022.

OTHER COMMENTS

Independently, we also uncovered a factor of 1/2 correction to the production probability, Eq. (7), and have updated all plots/equations/appendices as required. In order to display a non-cold DM region in Fig. 2 with the choice lambda = 1, we have changed the fiducial choice of the warm dark matter (WDM) limit from m_wdm = 3 keV to m_wdm = 5 keV, which is supported by a number of the papers given in Refs. [3-14], although some caution is required, see in particular [7]. (Note with a choice of m_wdm = 3 keV, the non-cold DM region in Fig. 2 would sit almost precisely on the T_n = T_infl contour, when setting lambda = 1. Somewhat larger choices of lambda would still allow for the region to appear with m_wdm = 3 keV, but we prefer to keep the coupling as before.)

Finally, we have also added references to 1911.02663, which derives limits on warm dark matter from stellar streams, and 1912.02830, 1912.04238, which present an alternative mechanism in which DM interacts with bubble walls during a phase transition.

The changes to the manuscript have been indicated in red. A list of changes is given below.

- included the stellar stream limit in the abstract and introduction.
- changed the benchmark/fiducial limit used from m_wdm = 3 keV to m_wdm = 5 keV.
- corrected the production probability, Eq. (7), and dependent quantities: Eqs. (9), (30), (167)-(174), Figs. 2, 3, and 8.
- added discussion at the end of Sec. 3 regarding friction/backreaction on the wall from the pair production.
- added the LISA contour to Fig. 2 left and adjusted caption.
- added Fig. 3 left to show the gravitational wave signal in the T_n > T_infl regime.
- added discussion in Secs. 4 and 5 regarding the signal in the T_n > T_infl regime and complementarity between the gravitational wave and warm dark matter searches.