Thermal squeezeout of dark matter

We carry out a detailed study of the confinement phase transition in a dark sector with an $SU(N)$ gauge group and a single generation of a dark heavy quark. We focus on heavy enough quarks such that their abundance freezes out before the phase transition, and the phase transition is of first order. We find that during this phase transition, the quarks are trapped inside contracting pockets of the deconfined phase and are compressed enough to interact at a significant rate, giving rise to a second stage of annihilation that can dramatically change the resulting dark matter abundance. As a result, the dark matter can be heavier than the often-quoted unitarity bound of $\sim 100\mathrm{TeV}$. Our findings are almost completely independent of the details of the portal between the dark sector and the Standard Model. We comment briefly on possible signals of such a sector. Our main findings are summarized in a companion paper, while here we provide further details on different parts of the calculation.

Figure 1

Left panel: yield (number density normalized by the entropy density of the universe), $Y=\frac{n_q}{s}$, of the quarks for a quark mass of ${10}^3\mathrm{TeV}$ and confinement scale of 1 TeV. Right panel: average separation of quarks as defined in Eq. (2.5) at the onset of the phase transition for various values of $\mathrm{\Lambda }$ and $m_q/\mathrm{\Lambda }$. In each plot, we vary the cross section by a factor of 10 above or below the central value (dashed lines) in Eq. (2.3) to produce the shaded regions.

Figure 2

The degree of supercooling prior to percolation. The temperature supercools until bubbles nucleate efficiently. The nucleated bubbles quickly expand, depositing latent heat that drives $T$ back up to $T_c$ and eliminates further bubble nucleation. The temperature then stays roughly constant as latent heat deposited from bubble growth nearly cancels Hubble cooling. This plot ends at percolation, when half of the universe is in each phase. Notice that the timescale of the phase transition is much shorter than the Hubble timescale.

Figure 3

Left panel: typical radius of bubbles just before percolation (blue line) and the characteristic coalescence radius $R_1$ (orange line). For any $\mathrm{\Lambda }$ with $R_0\le R_1$, we assume that bubbles quickly coalesce and grow to radius $R_1$ at percolation. Right panel: asymptotic velocity of the pocket wall during its contraction as a function of the confinement scale when quark pressure is ignored. In the more realistic case where internal quark pressure is allowed to resist the contraction of the pocket, we expect $v_w$ to be much smaller and to not necessarily asymptote to a constant value at small radii.

Figure 4

Depiction of a quark interacting with a phase boundary. As bubbles of the confined phase grow (the wall moves to the right in the figure), their walls run into quarks that move with typical velocity $v_q$, which is much larger than the wall velocity $v_w$ (left configuration). The quark locally deforms the wall (center configuration), introducing an opposing force via surface tension. One can think of the quark as connected to the deconfined phase through a gluon string [60]. Since string breaking is shut off, i.e., the quark-gluon configuration does not have sufficient energy to pull a heavy $\overline{q}q$ pair out of the thermal background, eventually the quark comes to a halt and then rebounds back into the deconfined phase with its initial speed (right configuration).

Figure 5

Schematic illustration of different stages of the phase transition and its effect on the DM abundance. Violet indicates the confined phase and light blue the deconfined phase. Top-left panel: Once the temperature drops slightly below $T_c=\mathrm{\Lambda }$, bubbles of the confined phase begin nucleating everywhere. The nucleated bubbles start growing and push quarks (black dots) around. Top-middle panel: The bubbles grow to a point where a $\mathcal{O}(1)$ fraction of the universe has converted into the confined phase. At this point, bubbles start coalescing and quickly grow larger. Top-right panel: As the bubbles keep growing and combining, eventually we are left with isolated pockets of the deconfined phase submerged in a sea of the confined phase. Bottom-left panel: A single isolated pocket with quarks trapped inside it. Each pocket contracts as the phase transition continues. Bottom-middle panel: The particles in the pocket are compressed, and their interactions recouple. During this phase, the particles can either annihilate or bind with one another. Nonsinglet states cannot enter the confined phase, but once they form color-neutral bound states (orange dots), they can escape into the confined phase. Bottom-right panel: In the end, the pockets vanish, and only the fraction of quarks that ended up inside color-neutral baryons survive. These particles diffuse away from the original pocket’s position due to their local overdensity.

Figure 6

Top panel: evolution of the fractional number of free quarks (red), diquarks (green), and baryons (black) inside the pocket, normalized to the initial quark number. We neglect the effect of quark pressure on $v_w$ in solving the Boltzmann equations for this plot, meaning that we use Eq. (2.9) for $v_w$. As the phase transition proceeds, the pocket radius decreases. Initially, almost all the quarks are unbound. As the pocket contracts, more bound states are formed and fewer quarks are found as free particles. As their numbers increase, the various states’ annihilation rates increase as well. At some point, their production and annihilation rates are comparable, and the number of bound states inside the pocket reaches its maximum. The accumulative surviving fraction assuming zero pocket asymmetry predicted by Eq. (3.12) is denoted by the dotted purple line. The asymptotic value of this line is equal to ${\mathcal{S}}_{\text{symm}}$. We denote the asymmetric lower bound on $\mathcal{S}$ from Eq. (2.16) by the orange dashed line. Bottom panel: DM abundance evolution for this mass and confinement scale. The $T>\mathrm{\Lambda }$ region is similar to Fig. 1; the confinement takes place at $T=\mathrm{\Lambda }$ and gives rise to the abundance suppression predicted by Eq. (3.13).

Figure 7

Contours of constant survival factor $\mathcal{S}$ (green contours) in the two limiting scenarios that we consider: (i) assuming the asymmetry bound on $\mathcal{S}$ is saturated (on the left) or (ii) neglecting the quark pressure effect on the pocket wall velocity (on the right). The contours of constant DM mass in TeV are shown as well (black dashed). To obtain the right plot, we solve the full set of Boltzmann equations in Eq. (3.10) using the values for the initial pocket radius $R_i$ and the pocket wall velocity $v_w$ based on our simulation results discussed in Sec. 2 and Appendix pp1. In the light blue region on the right plot, we find that the suppression is so severe that only the accidental asymmetric abundance of quarks in each pocket survives after the phase transition. For the left plot, we simply assume $\mathcal{S}={\eta }_{\mathrm{rms}}$ for every point in the parameter space. We observe orders-of-magnitude suppression in the DM abundance due to the second stage of annihilation during the phase transition in either scenario. Note that the small difference in the ${10}^{-7}$ contours in the overlap region is a plotting artifact.

Figure 8

Produced abundance of dark baryons, the DM candidate, in the asymmetry abundance scenario. The black dashed lines are contours of constant DM mass in TeV. The relic abundance line (with the initial radius fixed to its central value, i.e., $f_R=1$) is plotted (purple line) along with its uncertainty (light purple shades) corresponding to an order-of-magnitude variation of the initial radius. The shaded red region is excluded, as it produces too much DM, while the unshaded region produces too little DM. The baryons can therefore constitute a subcomponent of the DM within the unshaded regions of parameter space. The survival factor is determined by the accidental asymmetry of the pocket, which is independent of the wall velocity as long as the asymmetry bound is saturated. Thus, the major source of uncertainty in the location of the relic abundance line is the initial pocket radius. We also find that the uncertainty from microscopic quantities is subdominant to those of the initial pocket radius. This figure clearly shows that the baryon masses accounting for the observed DM abundance can be much heavier than the unitarity bound [53].

Figure 9

Similar to Fig. 8 but now with zero quark pressure. We vary the pocket initial radius (top) or the wall velocity (bottom) within 1 order of magnitude of the central values in Eqs. (2.8) and (2.9) to characterize the uncertainty in the final relic abundance calculation stemming from these quantities. The relic abundance line using the analytic approximation of Eq. (3.20) for the survival factor is denoted by the orange curve, too. Even with the quark pressure neglected, we still find a substantial suppression in the DM abundance during the phase transition. We also find that the baryon masses accounting for the observed DM abundance can be much heavier than the unitarity bound [53].

Figure 10

Wall velocity (blue) and “global threshold” (black dashed) above which phase conversion is so fast that the universe experiences net heating. The right panel displays the same data as the left panel, but it zooms in to the very end of the phase transition when pockets have contracted significantly. The discontinuity in the middle is an artifact of our modeling. It occurs when the spectrum of bubbles, which peaks at $R_0$, discontinuously jumps to a delta function spectrum of pockets centered at $R_1$.

Figure 11

Degree of supercooling as a function of time (left), confined phase fraction as a function of time (middle), and spectrum of bubble radii at percolation, when $x=1/2$ (right).

Figure 12

Degree of supercooling as a function of time (left), confined phase fraction as a function of time (middle), and pocket wall velocity as a function of bubble radius during the second half of the phase transition (right).

Figure 13

Pocket wall velocity (left) and particle abundances (right) within the pocket using a model that crudely incorporates the effects of quark pressure on $v_w$. We choose a confinement scale of 1 TeV and dark matter mass of ${10}^3\mathrm{TeV}$. We begin the simulation by neglecting the quark pressure, allowing us to use Eq. (a26) to determine the contraction rate (red-dotted line). Eventually, the quark pressure is so large that it is able to come into mechanical equilibrium with the other forces acting on the wall, which happens at the discontinuity near $R\mathrm{\Lambda }={10}^5$. At this point, we switch over to a contraction rate given by Eq. (a30), leading to the discontinuous drop in $v_w$ in the left panel and the sudden depletion of all particle abundances in the right panel.

Figure 14

Reproducing the results of [25, 26, 61] for the $\zeta$ factor in Eq. (b1) for various annihilation or capture processes entering our full set of Boltzmann equations.