Dependence of accessible dark matter annihilation cross sections on the density profiles of dwarf spheroidal galaxies with the Cherenkov Telescope Array

Dwarf spheroidal galaxies are excellent targets in $\gamma$-ray searches for dark matter. We consider dark matter searches in dwarf spheroidal galaxies (dSphs) with the Cherenkov Telescope Array (CTA). The aim of this work is to reveal a quantitative and precise dependence of the accessible dark matter annihilation cross sections on the dark matter density profiles of dSphs and on the distance to them. In most data analyses, researchers have assumed pointlike signals from dSphs because it is difficult to resolve the expected emission profiles with current $\gamma$-ray observatories. In future however, CTA will be able to resolve the peak emission profiles in dSphs. We take several variations of the dark matter density profile of Draco dSph as examples and analyze the simulated observations of with CTA. We derive the accessible region of the dark matter annihilation cross-section with each dark matter density profile. The accessible region of the annihilation cross section can differ by a factor of 10 among plausible profiles. We also examine the dependence on the distance to the target dSphs by assuming the same profiles of dSphs at different distances. Closer targets are better due to the higher J-factor, while their spatial extension significantly degrades our reach to the annihilation cross section compared to the value expected from a simple distance scaling of the J-factor. Spatial extension of the source affects the probable parameter region in energy-dependent ways. In some $\gamma$-ray energy ranges, this behavior becomes moderately dependent on the properties of the observation facility.

Figure 1

Examples of DM density profiles in our analyses. Left: NFW profile of model 5 in Ref. [106]. Center: Burkert profile of model 3 in Ref. [106]. Right: power-law of $(\mathrm{index}0)+\mathrm{cutoff}$ profile in Ref. [107]. The horizontal axis represents the distance measured from the center of the dSph. Numbers in the legends correspond to those in Table 1.

Figure 2

$\gamma$-ray spectrum for DM of mass $m_{\mathrm{DM}}=100\mathrm{GeV}$, 1 TeV, 10 TeV, 100 TeV, and 1 PeV annihilating into $\overline{b}b$ (left), $W^{+}{W}^{-}$ (center), and ${\tau }^{+}{\tau }^{-}$ (right). Spectrum of $m_{\mathrm{DM}}=100\mathrm{GeV}$ annihilating into $W^{+}{W}^{-}$ is shown for comparison and not used in our analyses.

Figure 3

Upper limits on the DM annihilation cross section assuming a 500 hr observation. In each panel, we show our results with dash-dotted lines using several dash lengths. For comparison, we put three lines: (a) the CTA sensitivity curve for a 500 hr observation of the Galactic halo with CTA [51] assuming a Burkert profile (long-dashed line), (b) the same but assuming an Einasto profile (short-dashed line), and (c) an observational result of the current upper limit by Fermi-LAT using 25 dSphs with kinematically derived J-factors [88] (solid line). We also show a reference for the cross section $⟨\sigma v⟩$ corresponding to the relic abundance in a horizontal band in the bottom part. Detailed calculations for the relic abundance is given in, e.g., Ref. [36]. Panels in the left column show sensitivities assuming profiles in Table 1. The distance for each profile is also shown in the same table. The severest case of $J_{\mathrm{tot}}=10^{19.15}$ (number 14 in Table 1, left panel in Fig. 1) is shown with a short dashed-dotted line, and the weakest one of $J_{\mathrm{tot}}=10^{18.58}$ (number 16 in Table 1, right panel in Fig. 1) is shown with a long dash-dotted line. Middle dash-dotted lines correspond to the case of the center panel in Fig. 1 of $J_{\mathrm{tot}}=10^{18.70}$ (number 9). We also show the upper limits, assuming a point source of $J_{\mathrm{tot}}=10^{19.15}$ with the shortest dash-dotted lines. The center (right) column shows the achievable sensitivities assuming the same profiles of the 16 sources in Table 1 at 40 kpc (160 kpc) from the Earth. In each column, the top, middle, and bottom panels correspond to the upper limits on annihilation cross sections of DM into $\overline{b}b$, $W^{+}{W}^{-}$, and ${\tau }^{+}{\tau }^{-}$, respectively.

Figure 4

The ratio between the upper limits on the annihilation cross section, assuming the same profiles of dSphs at different distances, considering profile number 14 (left) and number 16 (right). The expected value of ratios ${⟨\sigma v⟩}_{d_j}/{⟨\sigma v⟩}_{d_i}$ from the difference of J-factors are shown as dotted lines. The ratio of $(1/2{)}^2$ are expected for $(d_j,d_i)=(40\mathrm{kpc},80\mathrm{kpc})$ and (80 kpc, 160 kpc), while $(1/4{)}^2$ is expected for $(d_j,d_i)=(40\mathrm{kpc},160\mathrm{kpc})$. Deviations from the scaling from the J-factor are higher for the case of profile number 16.

Figure 5

Upper limits on the cross section for annihilation into $\overline{b}b$ pairs shown as a function of $J_{\mathrm{tot}}$. ${⟨\sigma v⟩}_{\overline{b}b}$ is evaluated at $m_{\mathrm{DM}}=630\mathrm{GeV}$. The dotted line corresponds to the relation ${⟨\sigma v⟩}_{\mathrm{UL}}\propto J^{-1}$, which is expected for point sources.

Figure 6

Ratios of upper limits on the annihilation cross section obtained with profile number 16 and number 14 in Table 1. The horizontal axis is the WIMP mass $m_{\mathrm{DM}}$, and the vertical axis is the ratio $r={⟨\sigma v⟩}_{\mathrm{No}.16}/{⟨\sigma v⟩}_{\mathrm{No}.14}$. Solid, dot-dashed, and dashed lines correspond to the case considering the DM annihilating into $\overline{b}b$, $W^{+}{W}^{-}$, and ${\tau }^{+}{\tau }^{-}$, respectively.