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

TL;DR

This study investigates how the dark matter density profiles of dwarf spheroidal galaxies influence the sensitivity of the Cherenkov Telescope Array in detecting dark matter annihilation signals, emphasizing the importance of spatial resolution.

Contribution

It provides a quantitative analysis of how different dark matter density profiles and distances of dwarf galaxies affect CTA's ability to constrain annihilation cross-sections.

Findings

Accessible cross-section limits vary by a factor of 10 among plausible profiles.

Closer dwarf galaxies offer better detection prospects due to higher J-factors.

Spatial extension impacts sensitivity differently across gamma-ray energy ranges.

Abstract

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 point-like 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 behaviour becomes moderately dependent on the properties of the observation facility.

Tables

Table 1. Table 1: DM density profiles for dSphs used in our analysis. We assign numbers in the first column for convenience. We adopt the center value for the parameters in each case.

No.	Reference	expression	type	log10J	log10Jtot	distance[kpc]
1	Acciari et al. Acciari:2010ab	$\left(\frac{1.7\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{0.79\mathrm{kpc}}\right)}^{-1}{\left(1+\frac{r}{0.79\mathrm{kpc}}\right)}^{-2}$	NFW	18.40	18.45	80
2	Geringer-Sameth et al.Geringer-Sameth:2014yza	$\left(\frac{0.69\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{3.7\mathrm{kpc}}\right)}^{-0.71}{\left(1+{\left(\frac{r}{3.7\mathrm{kpc}}\right)}^{2.01}\right)}^{-2.80}$	generalized NFW	19.00	19.44	76
3	LokasLokas:2001mf	$\left(\frac{16.3\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(1+\frac{r}{0.67\mathrm{kpc}}\right)}^{-3}$	generalized NFW	19.08	19.29
4		$\left(\frac{1.23\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{1.30\mathrm{kpc}}\right)}^{-1}{\left(1+\frac{r}{1.30\mathrm{kpc}}\right)}^{-2}$	NFW	18.80	18.91	72
5		$\left(\frac{0.18\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{1.99\mathrm{kpc}}\right)}^{-1.5}{\left(1+\frac{r}{1.99\mathrm{kpc}}\right)}^{-1.5}$	generalized NFW	18.88	18.90
6	Lokas et al. Lokas:2004sw	$\left(\frac{5.9\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{0.32\mathrm{kpc}}\right)}^{-1}\exp \left[-\frac{r}{0.32\mathrm{kpc}}\right]$	PL + cutoff	18.53	18.53	80
7	Mashchenko et al. Mashchenko:2005bj	$\left(\frac{4.76\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(1+\frac{r}{1.41\mathrm{kpc}}\right)}^{-1}{\left(1+{\left(\frac{r}{1.41\mathrm{kpc}}\right)}^2\right)}^{-1}$	Burkert	19.08	19.56
8		$\left(\frac{13.4\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(1+\frac{r}{0.35\mathrm{kpc}}\right)}^{-1}{\left(1+{\left(\frac{r}{0.35\mathrm{kpc}}\right)}^2\right)}^{-1}$	Burkert	18.65	18.70
9		$\left(\frac{37.8\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(1+\frac{r}{0.18\mathrm{kpc}}\right)}^{-1}{\left(1+{\left(\frac{r}{0.18\mathrm{kpc}}\right)}^2\right)}^{-1}$	Burkert	18.69	18.70
10		$\left(\frac{0.60\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{2.82\mathrm{kpc}}\right)}^{-1}{\left(1+\frac{r}{2.82\mathrm{kpc}}\right)}^{-2}$	NFW	18.95	19.15	82
11		$\left(\frac{1.70\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{1.00\mathrm{kpc}}\right)}^{-1}{\left(1+\frac{r}{1.00\mathrm{kpc}}\right)}^{-2}$	NFW	18.67	18.73
12		$\left(\frac{4.76\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{0.50\mathrm{kpc}}\right)}^{-1}{\left(1+\frac{r}{0.50\mathrm{kpc}}\right)}^{-2}$	NFW	18.70	18.72
13		$\left(\frac{13.4\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{0.25\mathrm{kpc}}\right)}^{-1}{\left(1+\frac{r}{0.25\mathrm{kpc}}\right)}^{-2}$	NFW	18.70	18.70
14		$\left(\frac{37.8\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{0.18\mathrm{kpc}}\right)}^{-1}{\left(1+\frac{r}{0.18\mathrm{kpc}}\right)}^{-2}$	NFW	19.15	19.15
15	Sanchez-Conde et al. SanchezConde:2007te	$\left(\frac{0.95\mathrm{GeV}}{{\mathrm{cm}}^3}\right){\left(\frac{r}{1.19\mathrm{kpc}}\right)}^{-1}\exp \left[-\frac{r}{1.19\mathrm{kpc}}\right]$	PL + cutoff	18.58	18.69	80
16		$\left(\frac{12.7\mathrm{GeV}}{{\mathrm{cm}}^3}\right)\exp \left[-\frac{r}{0.24\mathrm{kpc}}\right]$	PL + cutoff	18.56	18.58

Table 2. Table 2: J-factors ( J tot subscript 𝐽 tot J_{\mathrm{tot}} ) of profiles No.14 and No.16 in Table  1 assuming distance from the Earth to be 40kpc, 80kpc and 160kpc

Profile	log10$J_{\mathrm{tot}}$ (40kpc)	log10$J_{\mathrm{tot}}$ (80kpc)	log10$J_{\mathrm{tot}}$ (160kpc)
No.14	19.79	19.17	18.53
No.16	19.18	18.58	17.98

Equations

$$\varphi =\frac{1}{4\pi }\frac{⟨\sigma v⟩}{2m_{\mathrm{DM}}^2}{\int }_{E_{\mathrm{th}}}^{m_{\mathrm{DM}}}dE\frac{d{N}_{\gamma }}{dE}\cdot J$$

$$J=\int d\Omega \frac{dJ}{d\Omega }=\int d\Omega \int ds{\rho }_{\mathrm{DM}}^2.$$

$$\rho (r)={\rho }_s{\left(\frac{r}{r_s}\right)}^{-\gamma }{\left(1+{\left(\frac{r}{r_s}\right)}^{\alpha }\right)}^{-\frac{\beta -\gamma }{\alpha }},$$

$$\rho (r)={\rho }_s{\left(1+\frac{r}{r_s}\right)}^{-1}{\left(1+{\left(\frac{r}{r_s}\right)}^2\right)}^{-1},$$

$$\rho (r)={\rho }_s{\left(\frac{r}{r_s}\right)}^{-\alpha }\exp \left[-\frac{r}{r_s}\right].$$

Full text

I Introduction

Dark matter (DM) is a massive and invisible matter-component of the Universe Bergstrom:2000pn ; Bertone:2004pz ; Gaskins:2016cha . Rotation curves of galaxies Zwicky:1933gu ; Zwicky2009 ; vanAlbada:1984js ; Salucci:2002jg and bullet-cluster like encounters Barrena:2002dp ; Clowe:2003tk are examples that indicate the existence of DM. Standard cosmology also requires DM, since non-relativistic matter components different from baryons are necessary to form structures of the Universe Peebles:1982ff ; Ade:2015xua . Cosmological observations indicate that DM occupies approximately a quarter of the total energy in the Universe Komatsu:2010fb ; Ade:2015xua ; Akrami:2018vks ; Aghanim:2018eyx .

Varieties of candidates for DM are proposed. One possibility is that DM is a new particle: weakly interacting massive particles (WIMPs) (e.g.  Bringmann:2006mu ; Rott:2012gh ), strongly interacting massive particles (e.g. Mohapatra:1999gg ; Hochberg:2014dra ), axions (e.g. Preskill:1982cy ; Rosenberg:2000wb ; Visinelli:2009zm ), or sterile neutrinos (e.g. Dodelson:1993je ; Shi:1998km ; Abazajian:2001nj ; Boyarsky:2009ix ) are examples. Non-particle solutions like primordial black holes (e.g. Afshordi:2003zb ; Carr:2016drx ; Carr:2016hva ; Carr:2018rid ; Kohri:2018qtx ) are also considered. In this paper, we focus on DM categorized as WIMPs. WIMPs are one of the best-studied candidates proposed in theories beyond the standard model like supersymmetric extensions (e.g. Haber:1984rc ; Jungman:1995df ; Edsjo:1997bg ; Feng:2000gh ; Giudice:2004tc ). For WIMPs to be DM particles that explain non-relativistic, electromagnetically neutral, invisible components in Ade:2015xua ; Akrami:2018vks , their mass must be around $m_{\rm DM}\sim{\cal O}$(GeV) to ${\cal O}$(TeV), and have a velocity-averaged freeze-out annihilation cross-section $\left<\sigma v\right>\sim 3\times 10^{-26}$ cm3/s Steigman:2012nb . This value of cross-section is referred to as the canonical cross-section.

WIMPs as DM can be detectable through their feeble interaction with standard model particles. Three kinds of strategies are pursued: productions of DM with colliders (e.g. Aaboud:2017aeu ; Sirunyan:2017xgm ); measuring the scattering between DM particles and nuclei (e.g. Akerib:2016vxi ; Amole:2017dex ; Aprile:2018dbl ), called direct detection experiments; and the search for standard model particles produced after DM self-annihilation in the Universe, called indirect detection experiments. There has been no confirmed detection of particle DM neither DM yet. For WIMP models of $m_{\mathrm{DM}}\sim{\cal O}(1-10)$ GeV, $\gamma$-ray observations already constrain the DM annihilation cross-section to be smaller than the canonical value  Fermi-LAT:2016uux . Lighter DM is constrained from structure formation (e.g  Tremaine:1979we ; Abazajian:2005xn ; Horiuchi:2013noa ; Boyarsky:2008xj ). WIMPs heavier than $m_{\mathrm{DM}}\sim{\cal O}(10)$ GeV are less constrained and expected to be discovered or excluded in ongoing and future experiments.

Indirect detection experiments have advantages in DM searches at higher energy ranges of $m_{\rm DM}\gtrsim{\cal O}(1)$ TeV. Techniques for astrophysical observations to detect high-energy emissions are already developed Verzi:2017hro ; Aab:2018chp ; Aartsen:2017sml ; Abeysekara:2017hyn ; Acharya:2017ttl . A plethora of projects searching DM signals in the Universe with charged cosmic-rays (e.g.  Aguilar:2015ooa ; TheDAMPE:2017dtc ; Motz:2015cua ), neutrinos (e.g.  Aartsen:2017ulx ; Albert:2016dsy ) and $\gamma$-rays are ongoing. In general, astrophysical emissions dominate over DM signals, and elaborate strategies are required in the indirect DM search. Spectral and morphological information of emissions help to identify the sources. Considering DM searches in $\gamma$-rays with a facility of threshold energy $E_{\rm th}$, the flux from DM annihilations is

[TABLE]

where

[TABLE]

In Eq. (1), all quantities except for $J$ are determined from particle physics. The part shown as $J$ in Eq. (2) is referred to as the “(astrophysical) J-factor”. Since the J-factor is defined as the line-of-sight integral over the squared DM density $\rho_{\rm DM}^{2}$, the signal sensibly depends on the density profile and precise information about the DM distribution at the source is necessary to reliably derive the WIMP properties. The distribution of the DM is determined from stellar kinematics in optical observations (e.g. Posti2019 ).

The Galactic center is considered as one of the best targets to search DM signals in $\gamma$-rays (e.g.  Gondolo:1999ef ) because it is expected to have the highest J-factor among known targets with $J\sim{\cal O}(10^{21-22})$ GeV2 cm*-5*. Attentive strategies in the separation of DM signals from astrophysical emissions are required since the galactic center is very bright in astrophysical $\gamma$-ray emissions TheFermi-LAT:2017vmf . Also, the determination of the precise shape and a normalization of the DM density distribution at the very center of the Milky Way galaxy are remaining issues HESS:2015cda ; Pierre:2014tra ; Gammaldi:2016uhg ; Taylor:2015jaa . Dwarf spheroidal galaxies (dSphs) are satellites of the Milky Way galaxy and also good regions to focus on as first pointed out by Ref. Lake:1990du and later in Ref. Evans:2003sc . They are spatially extended objects of ${\cal O}(1)$ degrees located in high-latitude regions of the Milky Way galaxy. Several tens of dSphs are already identified with available stellar kinematics data, and the number of confirmed dSphs is continuously increasing Martin:2015xla ; Kim:2015ila ; Laevens:2015una ; Laevens:2015kla ; Luque:2015txp ; Koposov:2015jla ; Bechtol:2015cbp ; McConnachie:2012vd ; Gaia2018 ; Simons2018 . Stellar motions in dSphs indicate that they are dense and DM dominated objects  Strigari:2006rd ; Charbonnier:2011ft ; Hayashi:2016kcy with mass-to-luminosity ratios reaching $\sim 10^{3}M_{\odot}/L_{\odot}$ Mateo:1998wg ; Strigari:2007at ; Battaglia:2013wqa ; Pace:2018tin ; Irwin:1995tb . No significant $\gamma$-ray emissions have been confirmed in dSphs although possibilities that some of them contain $\gamma$-ray sources could not be excluded Geringer-Sameth:2015lua ; Fermi-LAT:2016uux .

Stacking analyses on dSphs by the ${\it Fermi}$ collaborations give the tightest upper limits on DM annihilation cross-sections Abdo:2010ex ; Ackermann:2011wa ; Ackermann:2013yva ; Ackermann:2015zua ; Charles:2016pgz ; Fermi-LAT:2016uux . For DM of $m_{\rm DM}$ $\lesssim$ ${\cal O}(100)$ GeV, the upper limits already reach to the canonical value  Abdo:2010ex ; Ackermann:2011wa ; Ackermann:2013yva ; Ackermann:2015zua ; Charles:2016pgz ; Fermi-LAT:2016uux . At higher mass ranges, ground-based Cherenkov telescopes have advantages over observations with satellite detectors. Since most of those ground-based $\gamma$-ray facilities are pointing telescopes, upper limits on DM annihilation cross-sections are obtained by observations on a few well-selected dSphs. Almost the same level of upper limits is obtained by observations with different facilities Aleksic:2011jx ; Aleksic:2013xea ; Doro:2017dqn ; Ahnen:2017pqx ; Aharonian:2007km ; Aharonian:2008dm ; Abramowski:2010aa ; Aleksic:2013xea ; Abramowski:2014tra ; Acciari:2010ab ; Aliu:2012ga ; Archambault:2017wyh ; Albert:2017vtb ; Essig:2010em ; Ahnen:2016qkx . In the very near future, the Cherenkov Telescope Array (CTA) starts its operations and is expected to improve the sensitivity to probe DM annihilation cross-sections by about one order of magnitude Acharya:2017ttl .

The designed angular resolution of CTA for $\gamma$-rays around 1TeV is ${\cal O}(0.05^{\circ})$ degree, which is finer than the typical spatial extension of dSphs hence the consideration of the DM density profile shape becomes crucial. This has been pointed out in earlier works (e.g. Charbonnier:2011ft ; Ambrogi:2018skq ). In the latest analyses with atmospheric Cherenkov telescopes, spatial extensions of DM for dSphs are taken into account Ahnen:2017pqx ; Archambault:2017wyh and tend to give upper limits milder than those assuming point sources. However, DM density distributions in the dSphs are still under discussion (see Appendix of Ref. Charbonnier:2011ft or Ref. Bonnivard:2015vua for examples). Different models for DM distributions lead to the divergence of derived upper limits.

In this paper, we examine accessible parameter regions of the DM annihilation cross-section with CTA, probing different extended DM density distributions in dSphs. We sample DM density profiles of the Draco dSph as examples. Draco is one of the well-known classical dSph galaxies. So far, its several profiles have been provided for it in the literature Acciari:2010ab ; Strigari:2006rd ; Geringer-Sameth:2014yza ; Lokas:2001mf ; Lokas:2004sw ; Mashchenko:2005bj ; SanchezConde:2007te . We consider the observation of dSphs with CTA and analyze simulated data using ${\tt ctools}$  Knodlseder:2016nnv . The sensitivity calculations for DM annihilation cross-sections are conducted with16 different profiles and compared to give a quantitative estimate of uncertainties in the searches towards dSphs. The dependence on the distances to dSphs is also investigated.

The structure of this paper is as follows. Sec. II explains our methods. In Sec. III we show a comparison of the sensitivity for annihilation cross-sections obtained with various profiles and distances. Sec. IV is devoted to discussions. We summarise in Sec.V.

II Methods

II.1 Dark matter density profiles of the source

A point source is the simplest model for a target dSph when the angular size of the target is small enough compared to the angular resolution of observational facilities. Future ground-based atmospheric Cherenkov telescopes can resolve typical dSphs, so they are to be treated as extended sources. Profiles of dSphs are sampled to investigate how their spatial extension affects the accessible region of the DM annihilation cross-section. Draco dSph is taken as the example, and we limit our analyses to spherical profiles for simplicity. Three types of DM density profiles are considered in this work:

generalized NFW profile Hernquist:1990be ; Zhao:1995cp :

[TABLE]

where ($\alpha$, $\beta$, $\gamma$)=(1, 3, 1) corresponds to the original NFW profile in Navarro:1996gj . 2. 2.

Burkert profile Burkert:1995yz :

[TABLE] 3. 3.

power law (PL) profile with an exponential cutoff:

[TABLE]

$\rho_{s}$ is the normalization of the DM density, and $r_{s}$ is the scale radius of the profile measuring the distance $r$ from the center of the target. More detailed profiles such as non-spherical cases or profiles with substructures are discussed in Ref. Bonnivard:2014kza ; Bonnivard:2015pia ; Hayashi:2016kcy ; Hutten:2016jko . Table 1 summarises our reference profiles with explicit expressions of each profile, profile type (corresponding to Eqs. (3), (4), and(5)), J-factor integrated to solid angle of $0.5$ degrees ($J_{<0.5^{\circ}}$), J-factor integrated to $4.0^{\circ}\times 4.0^{\circ}$ region ($J_{\mathrm{tot}}$) which corresponds to the size of the region of interest (RoI) in our analyses, and distance from the Earth. We also assign identification numbers in the first column in Table 1 for convenience. Note that the truncation radius for the profiles is not introduced in our analyses. The truncation radius is usually determined by the location of the outermost member star or the virial radius of the DM halo. If we take the former for the truncation radius, then it corresponds to $\theta=1.3^{\circ}$ for Draco Geringer-Sameth:2014yza . On the other hand, the virial radius is highly model-dependent. Actual radial extension of the dSphs is still under discussion. We chose our RoI to cover the outermost member star, avoiding the introduction of an additional model parameter. The J-factor integrated to 1.3∘ and $J_{\rm tot}$ defined as J-factors in our RoI differ by at most 10%. Templates of the J-factor centered on the target are generated adopting the median value of the parameters for each profile provided in references. The spatial resolution of our template is 0.01∘. In practice, we produce templates larger than the RoI, then use parts corresponding to the RoI. $J_{<0.5^{\circ}}$ values in Table 1 are shown just to make a comparison with previous works easier and are not used in our analyses.

II.2 Spectrum of the DM annihilation at the source

Three channels are considered as final states, $\bar{b}b$, $W^{+}W^{-}$ and $\tau^{+}\tau^{-}$. Those are representatives of DM annihilations into quarks, weak bosons, and leptons. The maximum mass of the DM particle in our calculation is set to $m_{\mathrm{DM}}$ = 1PeV, while the minimum to $m_{\mathrm{DM}}=$ 25GeV for lepton and quark channels, and to $m_{\mathrm{DM}}=$ 160GeV for the weak boson channel. At lower energies, contributions from residual cosmic-rays are significant. We set our minimum mass so that to avoid these contaminations. The spectra of each annihilation channel are calculated with ${\tt pythia8.2}$ Sjostrand:2014zea ; Sjostrand:2006za ; Sjostrand:2007gs . Figure 2 shows examples of spectra from $m_{\mathrm{DM}}=$100GeV to 1PeV. The spectra shown in Figure 2 include final state radiations like Bremsstrahlung of charged leptons, which are electroweak corrections different from interactions with external fields. We consider contributions from secondary $\gamma$-rays produced during propagations of charged leptons to be negligible Belikov:2009cx ; Cirelli:2009vg ; Profumo:2009uf ; Blanco:2017sbc ; Bartels:2017dpb . The treatment of the secondary $\gamma$-rays and the spectra in our calculation would be consistent with those available in Ref. Cirelli:2010xx which are computed by the old version pythia8.1 and widely used in $\gamma$-ray searches of dark matter.

The differences in the gamma-ray spectra between $W^{+}W^{-}$ (or $\bar{b}b$) and $\tau^{+}\tau^{-}$ modes come from differences of the particle multiplicity among those modes. $\gamma$-rays are produced mainly by decaying neutral pions, and partly by other decaying mesons. In the $W^{+}W^{-}$ or $\bar{b}b$ modes, emitted quark-pairs immediately fragment into a lot of mesons and baryons, which are dominant modes. The number of the multiplicity into pions would be approximately $\sim$ 30 for the center-of-momentum energy being $\sqrt{s}$ = O(1) TeV. In this case, the $\gamma-ray$ spectrum becomes broader with its mean energy being lower. On the other hand, in the $\tau^{+}\tau^{-}$ mode, the number of the multiplicity into neutral pions is much smaller (a few in $\sqrt{s}$ = O(1)TeV). In this latter case, the energy of $\gamma$-rays tends to be higher, which gives the steeper spectrum than that of the $W^{+}W^{-}$ or $\bar{b}b$ emission mode.

II.3 General procedures of our analysis

The procedure for sensitivity calculations is as follows. The software package ${\tt ctools}$ Knodlseder:2016nnv is used for the analysis. First, we simulate events assuming a 500-hour observation. The instrumental response function (IRF) ${\tt prod3b}$  Cumani:2017aca , the latest publicly available version of the CTA IRF package, is used. Assuming the northern CTA site (La Palma), we select the IRF optimized for the long-time observation at a zenith angle of 20 degrees. In the event generations, no $\gamma$-ray sources are included. Only residual charged cosmic rays as background events are simulated. After the event generation, selections and binnings are performed in energy and space. We select a 4${}^{\circ}\times 4^{\circ}$ square region centered on the target. Spatial binning is 0.03∘. In energy, events from 0.03 TeV to 180 TeV are selected and binned with 5 bins per decade on a logarithmic scale. We conduct likelihood analyses with the binned data. Median upper limits on the $\gamma$-ray flux are defined as to decrease the likelihood corresponding to a 95% confidence level. Throughout the procedure, we calculate with ${\it ctools}$ following the method in Knodlseder:2016nnv . The dependence between the $\gamma$-ray flux and annihilation cross-section is given in Eq. 1.

III Results

We conduct likelihood analyses on the simulated 500-hour observation of a dSph with DM density profiles listed in Table 1. ${\it Top}$, ${\it middle}$ and ${\it bottom}$ rows in Figure 3 show the cases of DM annihilating into $\bar{b}b$, $W^{+}W^{-}$ and $\tau^{+}\tau^{-}$, respectively. Panels in the left column show the sensitivities assuming the DM density profiles, distances, and J-factors ($J_{\rm tot}$) in Table 1. Each line is the 95% level upper limit corresponding to the profile in Figure 1. Upper limits assuming profile No.14 (NFW model 5 in Ref. Mashchenko:2005bj ) are the strongest while No.16 (PL of index 0 + cutoff model in Ref. SanchezConde:2007te ) are the weakest in our sample. Other profiles give the upper limits in the shaded regions of the Figure 3, like middle-dash dotted lines corresponding to the cases of No.9. Sensitivities with a point source of $\log_{10}(J)=19.15$, which is the same as the J-factor of profile No. 14, are also shown in a thin dash-dotted line. If we assume a point source, the upper limit always gets stronger. With the angular resolution of CTA, extended source structures are clearly resolved. Our results are consistent with the analytical discussion in Ref. Ambrogi:2018skq . Comparing between annihilation channels, wider regions of the cross-section parameter space can be covered for DM annihilating into $\tau^{+}\tau^{-}$ than for $\bar{b}b$ or $W^{+}W^{-}$ channels. This is due to the hard spectral feature which can be seen in the right panel of the Figure 2. The tendency is consistent with the latest results in Ref. Abdalla:2018mve , who assume line+broad spectra in specific WIMP models. Features in the sensitivity curves in Figure 3 at $m_{\rm DM}={\cal O}(10)-{\cal O}(100)$ TeV result from the properties of the telescope. Center (Right) columns show the sensitivities for sources at smaller (larger) distances. We adopt the same distance among the profiles here. Differences between profiles are larger (smaller) for cases assuming 40 (160) kpc due to the angular extensions. In each panel, we also show the current limit by Fermi using 25 dSphs Ackermann:2015zua with a solid line and the expected sensitivities of the Galactic halo observations using CTA with dashed lines Acharya:2017ttl ; Pierre:2014tra ; Wood:2013taa ; Silverwood:2014yza ; Roszkowski:2014iqa . We show two cases assuming different DM density profiles for the expectations of G.C. observations because the DM density profile there is under discussion. The accessible annihilation cross-section is about two orders-of-magnitude smaller for the case assuming the Einasto profile (short-dashed line) than that assuming the Burkert profile (long-dashed line) as shown in these figures.

We can expect better constraints when we adopt profiles based on the latest and detailed modelings of the dSphs. For example, we do not include the contribution from subhalos in dSphs since it is still under discussion (e.g. Hutten:2016jko ; Sanchez-Conde:2013yxa ; Moline:2016pbm ; Stref:2016uzb ; Hiroshima:2018kfv ; Charbonnier:2012gf ; Bonnivard:2015pia ; Hutten:2018aix ). Subhalos should enhance the annihilation signal, although little subhalo boost is expected in dSphs. Still, our results in this work provide conservative estimates.

In each channel of DM annihilation, the sensitivity achieves its best at $m_{\mathrm{DM}}=$ 630GeV, 1TeV and 250GeV for $\bar{b}b$, $W^{+}W^{-}$ and $\tau^{+}\tau^{-}$, respectively. These masses are universal among the profiles. By defining the rank of the profiles with the best points of the sensitivity in the DM mass range, we examine the relation between the annihilation channel final-state spectrum and the profile. There is no change in ranks of profiles between channels. No.14 in Table 1 is the strongest, No.16 is the weakest, and all other profiles lie between them in the same order.

The dependence on the distance to the source is clarified in Figure 4. Assuming the source of profile No.14 and No.16 at 80kpc, 40kpc, 160kpc, we calculate the sensitivity and take the ratio of the upper limits on the annihilation cross-section. A source distance of 80kpc is chosen to be consistent with the distance for each model within the 1-$\sigma$ error. Corresponding J-factors are shown in Table 2, which show good agreements with the scaling law of $J\propto d^{-2}$ for point sources Pace:2018tin ; Evans:2016xwx . The left (right) column corresponds to the profile No.14 (No.16). Profile No.14 (${\it left}$) almost follows the ratio expected from the scaling of J-factors in Table 2, while profile No.16 (${\it right}$) does not. For profile No.16, upper limits on $\left<\sigma v\right>$ get lower in milder ways than those expected from the scaling of the J-factor. Also, the DM mass dependence of the ratio differs between annihilation channels.

IV Discussion

IV.1 Dependences on profiles

The difference of $\left<\sigma v\right>_{\mathrm{UL}}$ between profiles (short-dashed and long-dashed lines in Figure 3, for example) is caused by two effects. Subscript “UL” denotes the upper limit here. The values of $J_{\rm tot}$ affect the sensitivity to the annihilation cross-section $\left<\sigma v\right>_{\rm UL}$ in a direct way like cases analysing point sources with different J-factors. For analyses of extended sources, upper limits on the $\gamma$-ray flux $\phi_{\rm UL}$ are also affected by the details of DM density profiles hence $\left<\sigma v\right>_{\rm UL}$ is determined by combinations of these effects. The width of the shaded regions in Figure 3 corresponds to this fact. When sources are at large distances (e.g. $d=$160kpc compared to $d=80$kpc), their density profile could not be resolved. Then the behaviour of the sensitivity curve becomes like that of a point source.

To clarify this point, we show the relation between $\left<\sigma v\right>_{\rm UL}$ and $J_{\rm tot}$ in Figure 5. $\left<\sigma v\right>_{\rm UL}$ is evaluated with $\left<\sigma v\right>_{\bar{b}b}$ at $m_{\rm DM}$=630GeV. Each marker corresponds to a profile in Table  1. The relation derived for cases of $W^{+}W^{-}$ or $\tau^{+}\tau^{-}$ is similar. The obtained $\left<\sigma v\right>_{\rm UL}$ does not follow the inverse of $J_{\rm tot}$, which is different from the case of $\phi_{\rm UL}$ independent of the DM distribution in dSphs like analyses of point sources. Therefore, a better understanding of the DM density profile is required in determining the goodness of the targets.

We also investigate the dependence of the resultant limits on the DM density profile parameters. We search the relation between the upper limits of the annihilation cross-section $\left<\sigma v\right>$ and the DM density at a certain radius (0.1∘, 0.3∘, 0.5∘ and 1.0∘), the scale radius $r_{s}$, or the index $\gamma$ defined as $\rho(r)\propto r^{-\gamma}$ at the inner part ($r<r_{s}$). We find no correlations between either of the parameters and the achievable upper limits. Hence none of the single profile parameters can be used to select the target dSphs and we should select the targets based on the whole properties of their profiles.

The dependence on the DM density profile also appears in the shape of the sensitivity curve. Figure 6 shows the ratio of the obtained upper limits assuming profile No.16 and No.14 in Table 1, $\left<\sigma v\right>_{No.16}/\left<\sigma v\right>_{No.14}$. For each annihilation channel, the ratio is about $\sim$10 and depending on the DM mass. A broad bump of the ratio at around m_{\rm DM}$$\sim 10TeV to $\sim$500TeV is seen in the case of ${\bar{b}b}$, while a dip at around $m_{\rm DM}\sim$20TeV appears in the ratio for $\tau^{+}\tau^{-}$. For $W^{+}W^{-}$, a broad bump ranges from ${\cal O}(1)$ TeV to a few hundreds of TeV and it peaks at $m_{\rm DM}\lesssim{\cal O}(100)$TeV. The DM masses at around the bumps correspond to the $\gamma$-ray emission peaks of $E_{\gamma}\sim 1-10$TeV (see Figure 2). The presence of the bumps can be interpreted as follows. The angular resolution of the CTA facility gets better as the energy increases. For example, it corresponds to about 0.1 degrees at $E_{\gamma}\sim$ 200 GeV and improves to about 0.04 degrees at $\gtrsim$1 TeV Hassan:2017paq . Therefore, the changes of $\phi_{\rm UL}$ are more significant at higher energies. On the other hand, in the very-high-energy regime at $E_{\gamma}>$10 TeV, almost no residual background events are expected. In such a case, the sensitivity is more determined by the detected number of signal events rather than the signal-to-noise ratio. It is a so-called “signal-dominant case”. In such cases, the angular resolution contributes less to the $\phi_{\rm UL}$, and $\phi_{\rm UL}$ is less affected by the spatial extension of the source. Then $\left<\sigma v\right>_{No.16}$ can get close to the expected values for those of point sources. The behaviour of the ratio $\left<\sigma v\right>_{No.16}/\left<\sigma v\right>_{No.14}$ is a manifestation of these effects since profile No.14 almost corresponds to a point source. Combining those two effects, the ratio between the profiles has the bump structures seen in Figure 6.

IV.2 Dependences on distances

If the same profiles of dSphs are located at different distances $d_{j}$ and $d_{i}(>d_{j})$, the J-factor increases by a factor of $(d_{i}/d_{j})^{2}$ for closer ones. Then $\left<\sigma v\right>_{\rm UL}$ should be simply improved by $(d_{i}/d_{j})^{2}$ when the target objects are point sources. However, improvements of $\left<\sigma v\right>_{\rm UL}$ are less than those expected from the scaling of J-factors as shown in Figure 4, due to the changes in $\phi_{\rm UL}$. Deviations from the scaling of J-factors are higher in the analyses assuming cored targets. Bumps are clearly seen in the right panels of Fig. 4. They peak at $m_{\rm DM}\lesssim$ ${\cal O}$(100)TeV for $\bar{b}b$ or $W^{+}W^{-}$, while at ${\cal O}$(10) TeV for $\tau^{+}\tau^{-}$. The features correspond to a peak at $E_{\gamma}\lesssim 10$TeV in the annihilation spectrum and a similar explanation of Sec. IV.1 holds. The sources at closer distances become more spatially extended such that $\phi_{\rm UL}$ gets worse at higher energies. As a result, the ratios $\left<\sigma v\right>_{d_{j}}/\left<\sigma v\right>_{d_{i}}$ for heavier WIMPs are more deviated from the expectations for point-sources. Contributions from the noise get lower at the very-high-energy regime, and consequently at $E_{\gamma}>$10 TeV $\phi_{\rm UL}$ are less affected by the spatial sizes of the source. Combining those two effects, the ratio between the upper limits on annihilation cross-section assuming the same profiles of dSphs at different distances have the bump structures as shown in Figure 4.

Possibilities of the uncertainties in dSph analyses due to the modelings of isotropic background events are discussed in Ref. Calore:2018sdx . In our analyses, the normalization of the background is fitted simultaneously with the dark matter signals hence the additional uncertainties due to the modelings of the background would not appear. However, the background events become Poisson-like at $E_{\gamma}\gtrsim$ a few TeV where we expect signals from DM of $m_{\rm DM}\gtrsim{\cal O}(1)$ TeV. This might induce additional uncertainties of which contributions are small compared to those in DM spatial distributions in target dSphs. We quantify this point in future works.

V Conclusion

Dependences of the accessible regions of the DM annihilation cross-section $\left<\sigma v\right>$ on the density profile of dSphs have been examined and quantified. Since the DM density profile of each dSph is still actively debated, we have taken those of Draco dSph in the literature as examples. Based on the likelihood analyses on simulated 500-hour observations with CTA assuming the 16 profiles, we have shown that the achievable upper limits on DM annihilation cross-sections are highly dependent on the details of the spatial extensions of target dSphs. We have revealed that the probable region of the annihilation cross-section can differ by a factor of $\sim$10 if we change the profile models. The dependence is different from the case of a point source whose merit is fully described with a single J-factor value. To extract information about the nature of DM from $\gamma$-ray observations with CTA, we therefore conclude that it is crucial to better constrain the density profiles of the targets.

The dependence of upper limits on the distance to the target dSphs have been also considered. J-factors get higher for closer targets if profiles are the same. However, achievable upper limits are always worse than those expected from the scaling of J-factors due to the larger spatial extensions of sources. This effect is significant at around $\gamma$-ray energies around 10TeV. At around the same energy, the effect of the spatial extension of targets is also apparent in the comparison between the annihilation channels. Improved angular resolution and the signal-dominant situation in the higher $\gamma$-ray energy regions determine the behaviour of the sensitivity curve in combination.

Acknowledgements.

We appreciate the careful reading and helpful suggestions for our earlier draft by Moritz Hütten and Vitor de Souza. Also, we thank Nicolas Produit and Subir Saker for comments. This research has made use of the CTA instrument response functions provided by the CTA Consortium and Observatory, see http://www.cta-observatory.org/science/cta-performance/(version prod3b-v1) for more details. This work is partly supported by the JSPS KAKENHI Grant Number JP17H01131 (K.K.), and MEXT KAKENHI Grant Numbers JP15H05889, JP18H04594 (K.K.), and 18K03665 (M.H.). This paper has gone through internal review by the CTA consortium.