Early Universe synthesis of asymmetric dark matter nuggets

We compute the mass function of bound states of asymmetric dark matter—nuggets—synthesized in the early Universe. We apply our results for the nugget density and binding energy computed from a nuclear model to obtain analytic estimates of the typical nugget size exiting synthesis. We numerically solve the Boltzmann equation for synthesis including two-to-two fusion reactions, estimating the impact of bottlenecks on the mass function exiting synthesis. These results provide the basis for studying the late Universe cosmology of nuggets in a future companion paper.

Figure 1

Approximate conditions for synthesis to begin with efficient formation of ${}^{2}X$. The top line corresponds to the condition for the formation rate to ever be larger than the Hubble rate in the ${\mathrm{BE}}_2\gg m_{\varphi }$ limit [Eq. (6)]. The bottom line shows the condition for synthesis to begin when the temperature is an order of magnitude larger than $T_{\mathrm{eq}}$, ensuring that our analysis assuming synthesis before the onset of nonlinear structure formation remains valid. The dashed lines correspond to ${\mathrm{BE}}_2=m_{\varphi }$; at least roughly speaking, ${\mathrm{BE}}_2>m_{\varphi }$ [Eq. (5)] in order for the dissociation rate to ever dip below the formation rate.

Figure 2

Contours of typical nugget number exiting big bang darkleosynthesis, ${\overline{k}}_{\mathrm{fo}}$ (dashed red line) and typical nugget mass ${\overline{M}}_{\mathrm{fo}}$ (solid purple) for ${\alpha }_{\varphi }=0.03$ (left) and ${\alpha }_{\varphi }=0.3$ (right). The temperature of the dark sector is assumed to be roughly $T_X\approx T_{\gamma }$. The blue shaded region corresponds to ${\mathrm{BE}}_2<m_{\varphi }$, where ${}^{2}X$ synthesis will not be efficient [Eq. (5)]. The upper $m_X$ cutoff corresponds to the requirement that the ${}^{2}X$ fusion rate be smaller than Hubble as in Eq. (6), and the lower $m_X$ cutoff corresponds to $T_{\text{synth}}\lesssim 10T_{\mathrm{eq}}$. (See Fig. 1.) The various kinks in the contours are results of the change in $g_{*}$ as the synthesis temperature passes through the QCD phase transition and neutrino decoupling.

Figure 3

Nugget distribution function at different interaction time $\gamma$ (right) and specific number fraction as a function of $\gamma$. Here, the different line style depicts different assumptions for the branching ratio, with the solid/dashed/dotted lines corresponding to the energy-scaling/uniform/coagulation branching ratio assumption as described in the main text.

Figure 4

Left: Nugget mass functions $k^2{y}_k(\gamma )$ for different $\gamma$ and kernel assumptions. The different colors, red/purple/blue, show the mass functions for the fusion/energy scaling/uniform assumptions for the kernel respectively. The solid/dashed/dotted lines show the curves for $\gamma =10$, 20, 100 respectively. All lines with the same color are seen to converge to a common function $s^{2}f(s)$ according to Eq. (34). The blue (uniform) and purple (energy scaling) curves converge to the same function approximately. Right: The extracted differential nugget density exiting synthesis for ${\overline{M}}_{\mathrm{fo}}={10}^9\mathrm{GeV}$ [the distribution is proportional to $sf(s)$]. A different ${\overline{M}}_{\mathrm{fo}}$ simply shifts the $x$-axis logarithmically while maintaining the shape of the differential distribution.

Figure 5

Contours of the nugget number, $k_{\mathrm{fo}}$ (dashed red line), and the typical mass of the nuggets, $M_{\mathrm{fo}}$ (solid purple) for the large nugget populations exiting darkleosynthesis when a significant bottleneck is present. The coupling is fixed at ${\alpha }_{\varphi }=0.03$ (left) and ${\alpha }_{\varphi }=0.3$ (right). Similar to Fig. 2, the blue shaded region corresponds to inefficient ${}^{2}X$ fusion due to small binding energy, and the upper (lower) cutoff for $m_X$ corresponds to an ${}^{2}X$ fusion rate always smaller than Hubble and $T_{\text{synth}}\lesssim 10T_{\mathrm{eq}}$ respectively.

Figure 6

Left: Nugget mass function at different interaction time $\gamma$, with two bottlenecks at $k=3$, 4. The solid/dashed/dotted curves correspond to $p$ equal to ${10}^{-4}/{10}^{-5}/{10}^{-6}$ respectively. Right: Nugget mass function for fixed $p={10}^{-5}$, with the solid/dashed/dotted curves corresponding to no additional bottleneck; an additional bottleneck at $k=9$; and an additional bottleneck at $k=9$, 10. The solid and dashed lines are indistinguishable at $\gamma =2$, 10, while for the dotted line only a very small population extends beyond the $k=8$ bin.