Asymmetric dark matter: residual annihilations and self-interactions

1 interesting model with rich phenomenology
2 very clear presentation

mentioned below

The authors study an interesting two-component asymmetric self-interacting dark matter scenario coupled to a light U(1) vector boson.
A particular focus of the current study is its astrophysical and cosmological implications to infer the viable regions of parameter space.
Dark matter self-interactions received a lot of interest lately, in particular models which allow for a velocity dependence
of the scattering cross section. The simplest realisations however are in strong tension with CMB
limits (vector mediators) or a combination of BBN and direct detection limits (scalar mediators).

The current paper is an interesting attempt to avoid these constraints, in particular the
constraints from late time energy injection due to dark matter annihilations by assuming an asymmetry in the dark sector which suppresses the annihilation rate.
The authors provide a very comprehensive analysis and the paper is very well written. It should be considered for publication in
SciPost after a couple of minor comments have been addressed.

It is not completely clear what the role of the dark electron is. If the dark electron mass is close to the dark proton mass it should
contribute sizably to the dark matter abundance. Equation (7) seems to indicate that the DM abundance is dominated by the dark protons alone.
Does this mean that the dark electrons are much lighter? If so does the electron abundance need to be reduced via annihilations (in which case
the dark photon would need to be even lighter) or is it OK (e.g. with $N_\text{eff}$) if a larger symmetric component remains?
If the dark electron is very light does it induce sizable dissipation? It would be good to discuss these points a bit more.

The kinetic mixing parameter is strongly constrained by direct detection and does not allow for thermalisation of the two sectors in large regions of the parameter space.
Without further assumptions the temperatures of the two sectors are therefore independent. The authors should comment on the implications of this.

1- Interesting model, novel results;
2- Very clear presentation;
3- Many connections to other works and open research problems.

1- Incomplete discussion of the effects of dark electrons;
2- Room for improvements in the calculation and presentation of Sommerfeld-enhanced CMB constraints;
3- Missing discussion of non-standard temperature ratios.

In their work the authors study a specific model of DM, which has three essential ingredients: a dark photon mediator from a new broken gauge group, a particle-antiparticle asymmetry in the dark sector and two oppositely charged DM species, named dark protons and dark electrons. The study provides an interesting new perspective on atomic dark matter by considering the case where the mediator is sufficiently massive that recombination in the dark sector is either impossible or inefficient, so that (repulsive) self-interactions can be very important at late times. At the same time the mediator is sufficiently light that these self-interactions are enhanced by non-perturbative effects.

Similar models without an asymmetry turn out to be strongly constrained by indirect detection and CMB constraints and the purpose of the present study is to investigate how strongly the symmetric component must be suppressed in order to evade these constraints. It turns out that a rather strong suppression is necessary, implying that even models with a strong particle-antiparticle asymmetry may be testable with these observations.

The manuscript is very well written and the presentation is clear. The results are robust and of interest to the community. In short, I can recommend publication once the following issues have been addressed.

1- It is well known that for the dark photon model direct detection constraints forbid thermal equilibrium between the dark sector and the Standard Model sector in the interesting regions of parameter space. The authors should comment, at least qualitatively, on the effect of a temperature ratio different from unity. Such a discussion would be a valuable addition to section 7.
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2- Throughout the work the authors assume that the dark electrons play no role apart from ensuring charge neutrality. It seems to me that there are a number of important caveats to this assumption. First, I would expect the dark electrons to have stronger self-interactions than dark protons (at least in regions of parameter space where the mediator is effectively massless). This could lead to observable consequences unless the dark electrons are at least an order of magnitude lighter than the dark protons, such that their contribution to the total dark matter abundance is negligible. If dark electrons are assumed to be at least an order of magnitude lighter than dark protons, this implies that also dark photons need to be this light, to ensure efficient annihilation of the symmetric component of dark electrons. This means that the inaccessible region at the top left of the plot should be much larger than currently indicated. Another potential concern is that the dark proton freeze-out will lead to an increase of the dark sector temperature relative to the temperature of the Standard Model sector. Freeze-out of dark electrons may then be less efficient, leading to a larger abundance of the symmetric component. Again, this issue presumably disappears for sufficiently light dark electrons, at the expense of requiring sufficiently light dark photons. The corresponding discussion needs to be extended.
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3- I wonder where the statement $v_\text{CMB} \sim 10^{-8}$ stems from. Presumably, this requires a calculation of the kinetic decoupling temperature, which should depend on additional details of the model, such as the mass of the dark electrons. I would suggest that the authors clarify the origin of this value and to what degree their results depend on the kinetic decoupling temperature (presumably not very much).
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4- In the parameter region where the resonances from Sommerfeld enhancement become very densely spaced, it becomes difficult to indicate excluded parameter regions, leading to CMB constraints that look quite ugly. I am wondering whether there may be a more meaningful way of drawing constraints, such as indicating in some way the density of resonances (i.e. how likely it is that a randomly chosen nearby point will be excluded because it is sufficiently close to a resonance). I realize that this may require substantial additional work, so maybe it is sufficient to add a comment on how the reader should interpret the CMB constraints in these regions.
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5- In the calculation of these CMB constraints, have the authors included the modification of the Sommerfeld enhancement factor due to unitarity restoration (see arXiv:1603.01383)? Or is this effect not relevant here?
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6- The authors state that they find a second excluded region from ANTARES with $M_{p_D} \geq 3 \times 10^3 \, \mathrm{TeV}$. This is presumably a typo -- should this read GeV?
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7- The authors encourage indirect detection experiments to perform an interpretation of their data in the context of this model. I personally think it would be even more important to encourage them to make data available in a format that allows for an independent interpretation. Maybe the authors would like to consider modifying this statement?