Big bang initial conditions and self-interacting hidden dark matter

A variety of supergravity and string models involve hidden sectors where the hidden sectors may couple feebly with the visible sectors via a variety of portals. While the coupling of the hidden sector to the visible sector is feeble, its coupling to the inflaton is largely unknown. It could couple feebly or with the same strength as the visible sector, which would result in either a cold or a hot hidden sector at the end of reheating. These two possibilities could lead to significantly different outcomes for observables. We investigate the thermal evolution of the two sectors in a cosmologically consistent hidden sector dark matter model where the hidden sector and the visible sector are thermally coupled. Within this framework, we analyze several phenomena to illustrate their dependence on the initial conditions. These include the allowed parameter space of models, dark matter relic density, proton-dark matter cross section, effective massless neutrino species at big bang nucleosynthesis time, self-interacting dark matter cross section, where self-interaction occurs via exchange of dark photon, and Sommerfeld enhancement. Finally, fits to the velocity dependence of dark matter cross sections from galaxy scales to the scale of galaxy clusters is given. The analysis indicates significant effects of the initial conditions on the observables listed above. The analysis is carried out within the framework where dark matter is constituted of dark fermions, and the mediation between the visible and the hidden sector occurs via the exchange of dark photons. The techniques discussed here may have applications for a wider class of hidden sector models using different mediations between the visible and the hidden sectors to explore the impact of big bang initial conditions on observable physics.

Figure 1

Top left panel: Exhibition of the dependence of the dark freeze-out temperature when ${\xi }_0=0.01$ (blue) vs ${\xi }_0=1$ (red) for model (f) in Table 1. Top right panel: Zoom in of the top left panel in the region of the freeze-out. Bottom left panel: Yields of dark fermion (dark matter) and dark photon for model of the top panels for ${\xi }_0=0.01$ (blue) and ${\xi }_0=1$ (red). Bottom right panel: Zoom in of the bottom left panel to exhibit the shift of the dark fermion yield for the cases ${\xi }_0=0.01$ (blue) and ${\xi }_0=1$ (red).

Figure 2

Left panel: Exhibition of the dependence of the relic density $\mathrm{\Omega }{h}^2$ on ${\xi }_0$ in the range ${\xi }_0=(0,1)$ for the model points of Table 1. Right panel: Exhibition of the dependence of $\mathrm{\Delta }{N}_{\mathrm{eff}}$ at BBN time on ${\xi }_0$ in the range ${\xi }_0=(0,1)$ for the model points of Table 1.

Figure 3

Left panel: A scatter plot of $\delta$ vs $m_{{\gamma }^{\prime }}$ displaying the models allowed under the constraint $0.012\le \mathrm{\Omega }{h}^2\le 0.12$ for ${\xi }_0=0.01$ (blue) and ${\xi }_0=1$ (red). The solid red ellipse shows that a significant region of the parameter space in the $m_{{\gamma }^{\prime }}-\delta$ plane becomes accessible when ${\xi }_0=1$, which would otherwise be excluded when $\xi =0.01$. This is meant as an illustration that ${\xi }_0$ plays a significant role in determining the allowed parameter space of models. Right panel: Plot of the spin-independent proton-DM cross section for six model points where the vertical lines show the shift in the cross section as one moves from ${\xi }_0=0.01$ to ${\xi }_0=1$. The experiment constraints are from CRESST-III [92] and XENONIT [93].

Figure 4

A diagram exhibiting a contribution to $D\overline{D}\to D\overline{D}$ scattering beyond the Born approximation.

Figure 5

Left panel: Plot of $S$ wave Sommerfeld enhancement for an attractive potential, where $S_{E0}$ is for the Yukawa potential by solving numerically (red), $S_{C0}$ is for the Coulomb potential (black), and $S_{H0}$ is for the Hulthen potential as a function of $g_X$ for the case when $m_D=250\mathrm{GeV}$, $m_{{\gamma }^{\prime }}=45\mathrm{MeV}$, and $v=10\mathrm{km}/\mathrm{s}$. Right panel: Plot of $S$ wave Sommerfeld suppression for the case of a repulsive potential where we use the same symbols $S_{E0},S_{H0},S_{C0}$ for suppression as for enhancement to avoid a proliferation of notation.

Figure 6

Plot of $S$ wave Sommerfeld enhancement for an attractive Yukawa potential for the case ${\xi }_0=0.01$ (blue) and for ${\xi }_0=1$ (red), where for the left panel, the $x$ axis is $v$, and for the right panel, the $x$ axis is $m_{{\gamma }^{\prime }}/m_D$. Here, we allow $g_X$ to vary but keep the relic density $\sim 0.12$ for ${\xi }_0=0.01$ and ${\xi }_0=1$.

Figure 7

Plot of $S$ wave Sommerfeld suppression for a repulsive Yukawa potential for the case ${\xi }_0=0.01$ (blue) and for ${\xi }_0=1$ (red), where for the left panel, the $x$ axis is $v$, and for the right panel, the $x$ axis is $m_{{\gamma }^{\prime }}/m_D$. Here, we allow $g_X$ to vary to keep the relic density $\sim 0.12$ for ${\xi }_0=0.01$ and for ${\xi }_0=1$.

Figure 8

A fit to the galaxy data taken from [33], which studies the dependence of $\sigma v/m_D$ on ${\xi }_0$ in the range (0.01–1) for the six models of Table 1. Here, solid lines are for ${\xi }_0=1$ and the dashed line for ${\xi }_0=0.01$ exhibit the dependence of $\sigma v/m_D$ on ${\xi }_0$. The fits (in red) are done using the full analysis by numerically integrating the Schrodinger equation including identical particle effects as well as Sommerfeld enhancement. For comparison, we also exhibit the tree-level QFT cross section shown by black curves that does not consider the effect of identical scattering. In the analysis, we allow $g_X$ to vary but keep the relic density fixed at $\sim 0.12$ as ${\xi }_0$ varies. It is to be noted that “galaxy data” is itself a computed quantity based on observation as evident from [33].

Figure 9

Left: Exhibition of peaks in fits to the galaxy data (which is the same as in Fig. 8) with specific parameters. Right panel: Here, it is shown how the peaks in the left panel at $⟨v⟩\sim {10}^2\mathrm{km}/\mathrm{s}$ and at $⟨v⟩\sim {10}^3\mathrm{km}/\mathrm{s}$ arise from successive additional of higher waves. Thus, the peak at $⟨v⟩\sim {10}^2\mathrm{km}/\mathrm{s}$ arises mainly from $S$ and $P$ contributions, while the one at $⟨v⟩\sim {10}^3\mathrm{km}/\mathrm{s}$ arises from contributions from up to $l=5$.

Figure 10

A comparison of $\sigma v/m_D$ vs $v$ for different cross sections, which include the total cross section ${\sigma }_{\text{total}}$, the transverse cross section ${\sigma }_T$, the viscosity cross section ${\sigma }_V$ as defined by Eq. (6.3) and the tree-level QFT cross section with and without identical particle effects for model (f) of Table 1. The analysis is done for ${\xi }_0=1$.

Figure 11

Evolution of $\xi (T)$ with different initial condition using Eq. (2.15) of this paper (solid) and using the approximation of entropy conservation (dashed). Left panel: Here, $\delta =4\times {10}^{-8}$, and analysis is given for three widely different values of ${\xi }_0$, i.e., ${\xi }_0=0.001$, ${\xi }_0=0.01$, ${\xi }_0=0.1$. Right panel: Here, ${\xi }_0=0.001$, and an analysis for several different values for $\delta$ in the range $\delta =0$ to $\delta ={10}^{-8}$ is exhibited. The rest of parameters are chosen so that $m_D=2\mathrm{GeV}$, $m_{{\gamma }^{\prime }}=2\mathrm{MeV}$, $g_X=0.015$.

Figure 12

Evolution of $\xi (T)$ with different initial conditions using Eq. (2.15) of this paper (solid) and using the approximation of entropy conservation (dashed). Left panel: Here, $\epsilon =0.9\delta$, and the plot is for three different values of ${\xi }_0$ as shown. Right panel: Here, ${\xi }_0=0.001$, and the plots are for several different values of $\epsilon$. For both the left and the right panels, the rest of parameters are chosen to be $m_D=2\mathrm{GeV}$, $m_{{\gamma }^{\prime }}=2\mathrm{MeV}$, $g_X=0.015$, $\delta =4\times {10}^{-8}$.

Figure 13

Left panel: Yields of dark fermion and dark photon when ${\xi }_0=0.01$ with and without conversation of entropy (COE). Right panel: Evolution of $\xi (T)$ when ${\xi }_0=1$ with and without COE. The model point we used here is the same as Fig. 1, which is model (f) of Table 1.