CERN Accelerating science

The spectrum of gamma-rays emerging per annihilation for DM which annihilates solely to a $W^+ W^-$ final state. Results are shown for four masses: 1 (solid lines), 10 (dashed lines), 50 TeV (dashed-dotted lines), and 100 TeV (dotted lines), and in each case we show the spectra as provided by \texttt{PPPC4DMID}~\cite{Cirelli:2010xx} (blue lines) and \texttt{HDMSpectra}~\cite{Bauer:2020jay} (red lines). We use the differences between these approaches to estimate the theoretical uncertainties in our model-independent approach.

Cumulative (left, as defined in Eq.~(\ref{eq:dmflux})) and differential (right) $J$-factor profiles, expressed in GeV$^2$cm$^{-5}$sr and GeV$^2$cm$^{-5}$, respectively, versus the angular distance from the GC, $\theta$. Results are shown for three profiles, NFW (red line), cNFW (blue line), and Einasto (black). The red and blue-shaded regions correspond to the 1$\sigma$ uncertainty band for the NFW and cNFW profile parametrizations, respectively. We will use the difference between these profiles to estimate the impact of systematic uncertainties in the Milky Way DM profile on the IACT indirect detection sensitivity.

Cumulative (left, as defined in Eq.~(\ref{eq:dmflux})) and differential (right) $J$-factor profiles, expressed in GeV$^2$cm$^{-5}$sr and GeV$^2$cm$^{-5}$, respectively, versus the angular distance from the GC, $\theta$. Results are shown for three profiles, NFW (red line), cNFW (blue line), and Einasto (black). The red and blue-shaded regions correspond to the 1$\sigma$ uncertainty band for the NFW and cNFW profile parametrizations, respectively. We will use the difference between these profiles to estimate the impact of systematic uncertainties in the Milky Way DM profile on the IACT indirect detection sensitivity.

{\it Left panel:} DM and background gamma-ray fluxes expected in the region from $0.4-0.5^{\circ}$ of the GC (our ROI 2). For DM, we show the spectra for the self-annihilation of WIMPs of masses $m_{\DM}$ = 3 TeV and 10 TeV, respectively, in the $W^+W^-$ annihilation channel and with a velocity-weighted annihilation cross section $\langle \sigma v \rangle = 10^{-27}$ cm$^3$s$^{-1}$. The hadronic (proton + helium) (solid black line) and electron (orange line) cosmic-ray fluxes are shown. The former is the largest background, and indeed in the plot we show the spectra at 1\% of the expected value, and it remains the dominant contribution. The diffuse fluxes from the H.E.S.S. Pevatron~\cite{Abramowski:2016mir} (green line), the base of the Fermi Bubbles~\cite{Moulin:2021mug} (blue line) together with expectation from the MSP-bulge population~\cite{Macias:2021boz} (gray lines) assuming a power-law electron spectrum with an energy cut-off of 1 and 10 TeV, respectively, are shown. Finally, we show a model for the Galactic Diffuse Emission (pink line), which is the sum of the $\pi^0$, bremsstrahlung, and ICS components. {\it Right panel:} Energy-differential count rates as a function of energy for the same signal and backgrounds in the same region. As opposed to the left hand plot, here the results have been convolved with the relevant instrumental effects. Residual background is shown as the sum of the rates for CR hadrons and electrons. Note that the Pevatron emission is restricted to the inner 0.5$^\circ$ of the GC~\cite{Abramowski:2016mir}.

{\it Left panel:} DM and background gamma-ray fluxes expected in the region from $0.4-0.5^{\circ}$ of the GC (our ROI 2). For DM, we show the spectra for the self-annihilation of WIMPs of masses $m_{\DM}$ = 3 TeV and 10 TeV, respectively, in the $W^+W^-$ annihilation channel and with a velocity-weighted annihilation cross section $\langle \sigma v \rangle = 10^{-27}$ cm$^3$s$^{-1}$. The hadronic (proton + helium) (solid black line) and electron (orange line) cosmic-ray fluxes are shown. The former is the largest background, and indeed in the plot we show the spectra at 1\% of the expected value, and it remains the dominant contribution. The diffuse fluxes from the H.E.S.S. Pevatron~\cite{Abramowski:2016mir} (green line), the base of the Fermi Bubbles~\cite{Moulin:2021mug} (blue line) together with expectation from the MSP-bulge population~\cite{Macias:2021boz} (gray lines) assuming a power-law electron spectrum with an energy cut-off of 1 and 10 TeV, respectively, are shown. Finally, we show a model for the Galactic Diffuse Emission (pink line), which is the sum of the $\pi^0$, bremsstrahlung, and ICS components. {\it Right panel:} Energy-differential count rates as a function of energy for the same signal and backgrounds in the same region. As opposed to the left hand plot, here the results have been convolved with the relevant instrumental effects. Residual background is shown as the sum of the rates for CR hadrons and electrons. Note that the Pevatron emission is restricted to the inner 0.5$^\circ$ of the GC~\cite{Abramowski:2016mir}.

{\it Left panel:} Mean expected 95\% upper C.L. limits on $\langle \sigma v \rangle$ as a function of the DM mass $m_{\DM}$ for six different two-body final states. All spectra are computed with \texttt{HDMSpectra}, and we show results for the assumption of the DM distributed in the inner galaxy as the Einasto profile. The dashed gray horizontal line represents the expected cross section for a conventional thermal relic. {\it Right panel:} Equivalent results for neutrino final states, and the equivalent sensitivity obtained by ANTARES for the $\nu_{\mu} \bar{\nu}_{\mu}$ channel~\cite{Albert:2016emp} (although the limits in that work are 90\% C.L.). We emphasize that although the prompt annihilation is to $\nu \bar{\nu}$, H.E.S.S. can constrain this channel as the electroweak corrections will generate photons at the considered masses. To facilitate the comparison, we adopted the NFW parameters used in Ref.~\cite{Albert:2016emp}, which we label as the aNFW profile. The limits that we obtain for the $\nu_{\tau} \bar{\nu}_{\tau}$ channel are also shown for the assumption of DM distributed according to the Einasto profile, as adopted for the left panel.

{\it Left panel:} Mean expected 95\% upper C.L. limits on $\langle \sigma v \rangle$ as a function of the DM mass $m_{\DM}$ for six different two-body final states. All spectra are computed with \texttt{HDMSpectra}, and we show results for the assumption of the DM distributed in the inner galaxy as the Einasto profile. The dashed gray horizontal line represents the expected cross section for a conventional thermal relic. {\it Right panel:} Equivalent results for neutrino final states, and the equivalent sensitivity obtained by ANTARES for the $\nu_{\mu} \bar{\nu}_{\mu}$ channel~\cite{Albert:2016emp} (although the limits in that work are 90\% C.L.). We emphasize that although the prompt annihilation is to $\nu \bar{\nu}$, H.E.S.S. can constrain this channel as the electroweak corrections will generate photons at the considered masses. To facilitate the comparison, we adopted the NFW parameters used in Ref.~\cite{Albert:2016emp}, which we label as the aNFW profile. The limits that we obtain for the $\nu_{\tau} \bar{\nu}_{\tau}$ channel are also shown for the assumption of DM distributed according to the Einasto profile, as adopted for the left panel.

The impact on our limits of the systematic uncertainty on the DM signal prediction, focusing on a representative final state, the $W^+ W^-$ channel (shown in Fig.~\ref{fig:LimitsChannels}). In the top panel we show the limit for the three different DM profiles we consider, the NFW (solid lines), cNFW (dashed lines), and Einasto (dotted lines). The variation between profiles can impact the limits by almost an order of magnitude. In each case, we also show the limit obtained when using the spectrum as computed by \texttt{PPPC4DMID}~\cite{Cirelli:2010xx} (blue) and \texttt{HDMSpectra}~\cite{Bauer:2020jay} (red), cf. Fig.~\ref{fig:GammaYield}. The impact of the spectrum is most pronounced at lower masses, and in the lower panel we show the percentage difference between the spectra for the NFW profile.

The impact of our systematic nuisance parameters on the DM sensitivity. For two different profiles, NFW and cNFW, we show the impact of profiling over the uncertainty on the $J$-factor as described in Eq.~\eqref{eq:likelihood_Jfac}, where $\sigma_J$ is determined from the width of the bands in Fig.~\ref{fig:Jfacvstheta}. We further demonstrate the impact of our treatment of an additional systematic uncertainty encoded by $\beta_{ij}$ in Eq.~\eqref{eq:likelihood_beta}, labelled as $\sigma_{\beta}$, although we only show this for the NFW profile---the impact for the cNFW profile is comparable.

{\it Top panel:} 95\% C. L. sensitivity on $\langle \sigma v \rangle$ as a function of the DM mass $m_{\DM}$ for the $W^+W^-$ channel and the NFW profile parametrization. The horizontal grey long-dashed line is set to the value of the natural scale expected for the thermal-relic WIMPs. The dashed and dotted lines show the limits when the indeces of the power laws describing the spectra of cosmic rays are changed by $\pm$ 0.2. {\it Bottom panel:} percentage difference of the limits obtained for the two uncertainty values shown in the top panel and the limits with no uncertainty.

Impact of the GDE contribution to the overall background on the 95\% C. L. sensitivity on $\langle \sigma v \rangle$ as a function of the DM mass. The DM distribution is assumed here to follow the Einasto profile and the DM particles self-annihilate into the $W^+W^-$ channel.

The impact of background systematics on the ability to reconstruct an injected DM signal. For two different DM masses -- 1 (left panel) and 10 TeV (right) -- and assuming annihilation to $W^+ W^-$ we inject a specific cross section, $\langle \sigma v \rangle_{\rm inj}$, and consider our ability to reconstruct this, labeled as $\langle \sigma v \rangle_{\rm reco}$. The orange line and band show the mean and 1$\sigma$ reconstruction in the absence of background systematics. If we add to this various background uncertainties, such as a variation in the cosmic-ray energy spectrum ($\Delta$CR), or similar variations to the MSP ($\Delta$MSP), and Fermi bubble ($\Delta$FB) contributions, they expand the uncertainty as shown in the red, gray, and blue bands, respectively. (The uncertainties from each contribution are added in that order sequentially.) See the text for further details.

Power-constrained mean expected 95\% limits on the line cross section for three canonical DM models: the Wino (left), Higgsino (middle), and Quintuplet (right). In the top panels we show the sensitivity in each case (assuming an Einasto profile), which can then be compared to various theoretical predictions for the rates. While these models can be considered for arbitrary masses, their is a unique $m_\DM$ signaled out as the mass where the correct relic abundance is obtained from a thermal cosmology, and these are labeled by the thermal vertical bands. In the lower panels we show results for the cNFW halo, as well as a breakdown of the contribution to the limit from the line only in each case, and the other contributions such as the endpoint and continuum. The sharp features in the quintuplet expected limit are physical, and explored in Fig.~\ref{fig:GammaYieldWinoHiggsinoQuintuplet} and in the text.

The DM annihilation spectrum for the Wino (left), Higgsino (middle), and Quintuplet (right), with and without the continuum contribution added in, after convolution with the H.E.S.S. energy resolution. Spectra with and without continuum are shown as solid and dashed lines, respectively. (For the Wino and Quintuplet, the spectrum without the continuum includes the line and endpoint contributions, whereas the Higgsino includes only the line.) The Wino and Higgsino spectra evolve smoothly as a function of mass, whereas the Quintuplet does not. Locations where the Quintuplet spectrum evolves sharply give rise to the sharp variations in the limit as a function of mass seen in Fig.~\ref{fig:WinoHiggsinoQuintuplet}, and are discussed further in the text.

(Left) Percentage differences between the Asimov and Monte Carlo simulation computations of the expected mean upper limits (solid line) and the 1$\sigma$ containment band (dashed line) on $\langle \sigma v \rangle$ as a function of the DM mass $m_{\DM}$. The limits are computed at 95\% C. L. on $\langle \sigma v \rangle$ for the $W^+W^-$ channel derived for the H.E.S.S.-like mock dataset of GC observations and using the computation of the gamma-ray yield from \texttt{HDMSpectra}. (Right) Similar to Fig.~\ref{fig:LimitsWWComparisonSpec}, but here showing results for both the $W^+ W^-$ and $\mu^+ \mu^-$ two-body final states.

(Left) Percentage differences between the Asimov and Monte Carlo simulation computations of the expected mean upper limits (solid line) and the 1$\sigma$ containment band (dashed line) on $\langle \sigma v \rangle$ as a function of the DM mass $m_{\DM}$. The limits are computed at 95\% C. L. on $\langle \sigma v \rangle$ for the $W^+W^-$ channel derived for the H.E.S.S.-like mock dataset of GC observations and using the computation of the gamma-ray yield from \texttt{HDMSpectra}. (Right) Similar to Fig.~\ref{fig:LimitsWWComparisonSpec}, but here showing results for both the $W^+ W^-$ and $\mu^+ \mu^-$ two-body final states.

Similar to Fig.~\ref{fig:LimitsWWComparisonSpec}, but here showing results for 500\,h and 1,000\,h of flat time exposure across the considered ON region. The sensitivity is shown for the $W^+ W^-$ and$\tau^+\tau^-$ two-body final states.