Cosmological boost factor for dark matter annihilation at redshifts of $z=10-100$ using the power spectrum approach

We compute the cosmological boost factor at high redshifts of $z=10–100$ by integrating the nonlinear matter power spectrum measured from high-resolution cosmological $N$-body simulations. An accurate boost factor is required to estimate the energy injection from dark matter annihilation, which may affect the cosmological reionization process. We combined various box-size simulations (side lengths of 1 kpc–10 Mpc) to cover a wide range of scales, i.e., $k=1–{10}^7{\mathrm{Mpc}}^{-1}$. The boost factor is consistent with the linear theory prediction at $z\gtrsim 50$ but strongly enhanced at $z\lesssim 40$ as a result of nonlinear matter clustering. Although dark matter free-streaming damping was imposed at $k_{\mathrm{fs}}={10}^6{\mathrm{Mpc}}^{-1}$ in the initial power spectrum, the damping disappears at later times of $z\lesssim 40$ as a result of the power transfer from large to small scales. Because the simulations do not explore very small-scale clustering at $k>{10}^7{\mathrm{Mpc}}^{-1}$, our result is a lower bound on the boost factor at $z\lesssim 40$. A simple fitting function of the boost factor is also presented.

Figure 1

Dimensionless linear matter power spectrum at $z=10–100$. Solid curves include free-streaming damping at $k_{\mathrm{fs}}={10}^6{\mathrm{Mpc}}^{-1}$, whereas dashed curves exclude free-streaming damping. The suppression at $k\gtrsim {10}^3{\mathrm{Mpc}}^{-1}$ is caused by the baryon gas pressure (i.e., the Jeans effect) after the decoupling epoch [44].

Figure 2

Dimensionless matter power spectrum, ${\mathrm{\Delta }}^2(k;z)$, at $z=10–60$. The symbols denote the simulation results with various box sizes: $L=10\mathrm{Mpc}$ (purple), 1 Mpc (blue), 100 kpc (orange), 10 kpc (green), and 1 kpc (red) from left to right. The gray symbols are the same as the red symbols but do not include the free-streaming damping. The filled circles indicate the HR results with the number of particles $N_p=5120^3$, whereas the crosses indicate the LR results with $N_p=2560^3$. Solid curves are the linear theory prediction with the free-streaming damping, and dashed curves are the same without the free-streaming damping. Dotted red lines indicate the shot noise for the red circles. Vertical dot-dashed red lines indicate the Nyquist wave number for the red circles. The discontinuity between the larger and smaller boxes, which is especially prominent at $z=17–40$, results from the lack of density fluctuations larger than the smaller box size.

Figure 3

Boost factor calculated from the simulation results of ${\mathrm{\Delta }}^2$ for the maximum wave number $1.6\times {10}^7{\mathrm{Mpc}}^{-1}$ denoted by the red circles. Orange curve represents our fit to the simulation results given in Eq. (6), whereas dashed black curve represents the linear theory prediction. Dotted green and blue curves represent the previous halo-model results [3] for the minimum halo masses ${10}^{-6}{M}_{\odot }$ and ${10}^{-3}{M}_{\odot }$, respectively.

Figure 4

Dimensionless matter power spectrum ${\mathrm{\Delta }}^2(k;z)$ for various box sizes $L$ with different number of particles $N_p$ but the same spatial resolution: $L=100\mathrm{kpc}$ with $N_p=5120^3$ (red circles), $L=50\mathrm{kpc}$ with $N_p=2560^3$ (green triangles), and $L=25\mathrm{kpc}$ with $N_p=1280^3$ (blue diamonds). Solid curves show the linear theory prediction.

Figure 5

Nonlinear power spectrum measured from the hydrodynamic simulations including the baryonic processes: TNG100-1 (blue diamonds) and TNG50-1 (purple circles) at $z=20$ (left) and 10 (right). The gray symbols are the same as the colored ones but are measured from the corresponding DMO runs. Red curve indicates the linear theory, and dashed blue and purple lines indicate the shot noise for TNG100-1 and TNG50-1, respectively. The shot noise is included in the plotted points (i.e., it is not subtracted). The bottom portions of the panels plot the ratio of ${\mathrm{\Delta }}^2$ to the corresponding value for the DMO runs.