Heavy dark matter particle annihilation in dwarf spheroidal galaxies: radio signals at the SKA telescope

TL;DR

This paper explores the potential of the upcoming SKA radio telescope to detect signals from heavy dark matter particle annihilation in dwarf spheroidal galaxies, providing a complementary approach to collider searches.

Contribution

It analyzes the detectability of trans-TeV dark matter via radio signals at SKA, incorporating particle physics, diffusion, and energy loss effects, with benchmark scenarios based on the MSSM.

Findings

Radio flux from Draco could be detectable in 100 hours at SKA for dark matter masses up to 8 TeV.

Detectability depends on astrophysical parameters, with specific regions identified.

Complementarity between collider and radio detection methods is emphasized.

Abstract

A weakly interacting dark matter candidate is difficult to detect at high-energy colliders like the LHC, if its mass is close to, or higher than a TeV. On the other hand, pair-annihilation of such particles may give rise to $e^+ e^-$ pairs in dwarf spheroidal galaxies (dSph), which in turn can lead to radio synchrotron signals that are detectable at the upcoming Square Kilometre Array (SKA) telescope within a moderate observation time. We investigate the circumstances under which this complementarity between collider and radio signals of dark matter can be useful in probing physics beyond the standard model of elementary particles. Both particle physics issues and the roles of diffusion and electromagnetic energy loss of the $e^\pm$ are taken into account. First, the criteria for detectability of trans-TeV dark matter are analysed independently of the particle physics model(s) involved. We thereafter use some benchmarks based on a popular scenario, namely, the minimal supersymmetric standard model. It is thus shown that the radio flux from a dSph like Draco should be observable in about 100 hours at the SKA, for dark matter masses upto 4-8 TeV. In addition, the regions in the space spanned by astrophysical parameters, for which such signals should be detectable at the SKA, are marked out.

Tables

Table 1. Table 1 : Lightest neutralino mass ( m χ 1 0 subscript 𝑚 subscript superscript 𝜒 0 1 m_{\chi^{0}_{1}} ) and its pair annihilation rate ( ⟨ σ ​ v ⟩ delimited-⟨⟩ 𝜎 𝑣 \langle\sigma v\rangle ) inside a dSph along with branching fractions ( B f subscript 𝐵 𝑓 B_{f} in percentage) in different annihilation channels for the selected MSSM benchmark points from Kar  et al. ( 2019a ) . Lightest neutralino is the DM candidate ( m χ 1 0 = m χ subscript 𝑚 subscript superscript 𝜒 0 1 subscript 𝑚 𝜒 m_{\chi^{0}_{1}}=m_{\chi} ).

Model	annihilation channel ($B_f$)	$m_{{\chi }_1^0}$	$⟨\sigma v⟩$
		($GeV$)	(${10}^{-26}{\text{cm}}^3{\text{s}}^{-1}$)
A1a	$b\overline{b}\left(85\%\right)$, ${\tau }^{+}{\tau }^{-}\left(14\%\right)$	1000.6	0.27
B2a	$b\overline{b}\left(76\%\right)$, ${\tau }^{+}{\tau }^{-}\left(15\%\right)$, $W^{+}{W}^{-}\left(3\%\right)$, $t\overline{t}\left(3\%\right)$, $ZZ\left(2.8\%\right)$	3368.0	1.19
E	$b\overline{b}\left(79.1\%\right)$, ${\tau }^{+}{\tau }^{-}\left(18.3\%\right)$, $t\overline{t}\left(2.5\%\right)$	8498.0	9.12

Equations

$$Q_e(E,r)=⟨\sigma v⟩\left\{\sum _f\frac{d{N}_f^e(E)}{dE}{B}_f\right\}{N}_{pairs}(r)$$

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

$$D(E){\nabla }^2\left(\frac{dn}{dE}\right)+\frac{\partial }{\partial E}\left(b(E)\frac{dn}{dE}\right)+Q_e(E,r)=0$$

$$b(E)$$

$$\frac{dn}{dE}(r,E)=\frac{1}{b(E)}{\int }_E^{m_{\chi }}d{E}^{\prime }G(r,\Delta v)Q_e(E^{\prime },r)$$

$$G(r,\Delta v)$$

$$\Delta v={\int }_E^{E^{\prime }}d\tilde{E}\frac{D(\tilde{E})}{b(\tilde{E})}$$

$$J_{Synch}(\nu ,r)=2{\int }_{m_e}^{m_{\chi }}dE\frac{dn}{dE}(r,E)P_{Synch}(\nu ,E,B)$$

$$S_{\nu }(\nu )=\frac{1}{L^2}\int dr{r}^2{J}_{Synch}(\nu ,r)$$

$${\frac{dn}{dE}(r,E)|}_{\text{mbox}NSD}=\frac{1}{b(E)}{\int }_E^{m_{\chi }}d{E}^{\prime }{Q}_e(E^{\prime },r)$$

$${\frac{dn}{dE}(r,E)|}_{\text{mbox}Nb}=\frac{1}{D(E)}f(r,r_h)Q_e(r,E)$$

$$f(r,r_h)={\int }_{r^{\prime }=r}^{r_h}d{r}^{\prime }(\frac{1}{r^{\prime 2}})\left\{{\int }_{\tilde{r}=0}^{r^{\prime }}d\tilde{r}{\tilde{r}}^2{\left(\frac{{\rho }_{\chi }(\tilde{r})}{{\rho }_{\chi }(r)}\right)}^2\right\}$$

$$⟨\sigma v⟩=\sum _f⟨\sigma v{⟩}_f{B}_f$$

Full text

Dark matter, Dwarf galaxies, SKA, Supersymmetry

††preprint: HRI-RECAPP-2019-004

I Introduction

Some yet unseen weakly interacting massive particles (WIMP) are often thought of as constituents of dark matter (DM) in our universe. Many scenarios beyond the standard model (SM) of particle physics including them are regularly proposed, with various phenomenological implications. They are constrained by the data on relic density Ade et al. (2014) of the universe as well as various direct search experiments Aprile et al. (2017); Akerib et al. (2017). It is expected that a WIMP DM candidate should also be detected at the Large Hadron Collider (LHC) in the form of missing transverse energy (MET) (see for example Carena et al. (2019); Ellis (2018); Abe et al. (2018)).

Detectability at colliders and also in direct search experiments, however, depends on the mass as well as the interaction cross-section of the DM particle. In particular, detection becomes rather difficult if the WIMP mass approaches a TeV Shchutska (2016). For a weakly interacting DM, production at the LHC has to depend mostly on Drell-Yan processes where the rate gets suppressed by the $s$, the square of the subprocess centre-of-mass energy, and also by the parton distribution function at high $x$. An exception is the minimal supersymmetric standard model (MSSM) Martin (1997); there the production of colored superparticles (squarks/gluions) via strong interaction may be followed by decays in cascade leading to DM pair-production. However, here, too, the event rates go down drastically when one looks at the current constraints on colored superparticle masses as well as the various other limits on the MSSM spectrum. On the whole, while a near-TeV DM particle still admits of some hope at the LHC in the MSSM Shchutska (2016); ATL (2014); CMS (2013); ATL (2013), the reach is considerably lower for most other scenarios where the ‘dark sector’ is at most weakly interacting Suematsu (2008); Morgante (2017). It is therefore a challenge to think of additional indirect evidence, if a trans-TeV DM particle has to be explored.

The annihilation of DM particles in our galaxy as well as in extra-galactic objects leads to gamma-ray signals Daylan et al. (2016); Calore et al. (2015); Kar et al. (2018, 2017); Karwin et al. (2017); Beck and Colafrancesco (2017); Achterberg et al. (2015) as well as positrons, antiprotons etc Hektor et al. (2014); Bai et al. (2018); Cholis et al. (2019); Cui et al. (2018). Constraints have been imposed on DM annihilation rates in various ways out of the (non)-observation of such signals Ackermann et al. (2017); Albert et al. (2017); Ackermann et al. (2015); Cuoco et al. (2018). An alternative avenue to explore is that opened by radio synchrotron emission from galaxies, arising out of electron-positron pairs generated from DM annihilation Regis et al. (2017); Natarajan et al. (2015, 2013); Spekkens et al. (2013). In this paper, we focus on the potential of the upcoming Square Kilometre Array (SKA) radio telescope Braun et al. (2017) in this regard.

While the prospect of radio fluxes unveiling DM annihilation has been explored in earlier works Regis et al. (2017); Natarajan et al. (2015, 2013); Spekkens et al. (2013); Colafrancesco et al. (2015); Cirelli and Taoso (2016); Regis et al. (2014), it was pointed out in reference Kar et al. (2019a) that SKA opens up a rather striking possibility. The annihilation of trans-TeV DM pairs in dwarf spheroidal galaxies (dSph) lead to electron-positron pairs which, upon acceleration by the galactic magnetic field, produces such radio synchrotron emission. dSph’s are suitable for studying DM, since star formation rates there are low Strigari et al. (2008); Geha et al. (2009), thus minimising the possibility of signal originating from astrophysical processes. Their generic faintness prompts one to concentrate on such galaxies which are satellites of the Milky Way. The SKA can ensure sufficient sensitivity required to detect the faint signal from the sources, and at the same time, will have high enough resolution to remove the foregrounds. It was shown in Kar et al. (2019a) that about 100 hours of observation at the SKA can take us above the detectability threshold for radio signals from the annihilation of DM particles in the 5-10 $TeV$ range. Of course, the compatibility of such massive WIMP with the observed relic density requires a dark sector spectrum with enough scope for co-annihilation in early universe, as was demonstrated in Kar et al. (2019a) in the context of the MSSM. What one learns from such an exercise is that the probe of at least some DM scenarios should thus be possible on a time scale comparable with the running period of the LHC, and that the reach of the LHC Shchutska (2016); ATL (2014); CMS (2013); ATL (2013) for WIMP detection may be exceeded considerably through such a probe. It was also found that cases where SKA could observe radio fluxes from a dSph were consistent with limits from $\gamma$-ray observations as well as antiparticles in cosmic rays.

In order to ascertain which scenarios are more accessible in such radio probes, one needs to understand in detail the mechanisms whereby high-mass DM particles can produce higher radio fluxes, and also the effects of astrophysical processes that inevitably affect radio emission. This is the task undertaken in the present work.

The spectrum as well as the dynamics of the particle physics scenario, along with the DM profile in a dSph, is responsible for DM annihilation as well as the subsequent cascades leading to electron-positron pairs.111In principle, electron-positron pairs may also be directly produced in a complete model-independent scenario. We have referred to cascade since our benchmark correspond to $b\bar{b},t\bar{t},W^{+}W^{-}$ and $\tau^{+}\tau^{-}$ as dominant annihilation channels, inspired by the MSSM where direct $e^{+}e^{-}$ is not found to dominate for high DM masses. The electron(positron) energy distribution at the source function level is also the determined by the above factors Colafrancesco et al. (2006); McDaniel et al. (2017); Linden et al. (2010). However, they subsequently pass through the interstellar medium (ISM) of the galaxy, facing several additional effects. These include diffusion as well as electromagnetic energy loss in various forms, including Inverse Compton effect, Synchrotron loss, Coulomb effect and Bremsstrahlung effect Colafrancesco et al. (2006); Beck and Colafrancesco (2016); Colafrancesco et al. (2007); McDaniel et al. (2017). Besides, the galactic magnetic field is operative all along. The way these affect the final $e^{+}e^{-}$ energy distribution is highly inter-connected and nonlinear. For example, while the magnetic field causes electrons to lose more energy in synchrotron radiation, it puts a check on the reduction of the flux through diffusion by confining them longer within the periphery of the dSph. This check applies to electrons at lower energies at the cost of those at higher energies. Also, electromagnetic energy loss enhances the population of low-energy electrons and positrons, the enhancement being more when they have higher kinematic limits, enabled by higher mass of the DM particle. Such low-energy $e^{+}e^{-}$ pairs enhance the flux in the frequency range appropriate for a radio telescope.

In the following sections, we analyse the various ingredients in the radio flux generation process, as outlined above. We first do this for fixed DM particle masses, and for values of their annihilation cross-sections fixed by hand. The relative strengths of the effects of the particle theoretical scenario as well as diffusion and radiative process are thus assessed. This also serves to evolve an understanding of the dependence on the diffusion coefficient, parameters involved in electromagnetic energy loss, and, of course, the strength of the galactic magnetic field. If the nature of the DM particle(s) is known in independent channels, then the observation of the observed radio flux may be turned around to improve our understanding of these astrophysical parameters, using the results presented here.

Finally, we use some theoretical benchmark points to demonstrate the usefulness of the SKA in probing trans-TeV DM. A few sample MSSM spectra are used, largely because they offer the scope of co-annihilation that is so essential for maintaining the right relic density Kar et al. (2019a). Using the minimum annihilation cross-section required for any DM mass for detection at SKA (in 100 hours), and the maximum value of this cross section that is compatible with limits from $\gamma$-ray and cosmic-ray data, we show that several benchmark points with DM mass in the 1-8 $TeV$ range, which are yet to be ruled out by any observation can be investigated in 100 hours of SKA observation.

We have organised the paper as follows. In section II we have discussed in somewhat brief manner the calculations of synchrotron flux originating from DM annihilation inside a dSph. In section III we have analysed the effects of various astrophysical parameters in different steps of the production of radio flux. Section IV describes the features of heavy DM. In section V we have shown the detectability curves or threshold limits for observing radio flux in SKA, in both model independent as well as model dependent way. We have also shown the final radio fluxes for some theoretical benchmarks. Finally in section VI we conclude.

II Essential Processes: Recapitulation

We start with a brief resume of sequence of processes that leads to radio flux from a dSph, as discussed, for example, in Natarajan et al. (2015, 2013); Colafrancesco et al. (2006); Beck and Colafrancesco (2016); Colafrancesco et al. (2007); McDaniel et al. (2017). Dark matter pair annihilation inside of a dSph can produce SM particle pairs such as $b\bar{b},\tau^{+}\tau^{-},W^{+}W^{-},t\bar{t}$ etc. These particles then cascade and give rise to large amount of $e^{\pm}$ flux. The resulting energy distribution of which can be obtained from the source function McDaniel et al. (2017); Linden et al. (2010):

[TABLE]

where $\langle\sigma v\rangle$ and $N_{pairs}(r)\left(=\frac{\rho^{2}_{\chi}(r)}{2m_{\chi}^{2}}\right)$ are respectively the velocity averaged annihilation rate and the number density of DM pairs inside the dSph. $m_{\chi}$ is the DM mass and $\rho_{\chi}(r)$ is the DM density profile in the dSph as a function of radial distance $r$ from the centre of the dSph. For our analysis we have taken Draco222In this paper we have used the dSph Draco to illustrate our points, primarily because the relevant parameters such as the $J$-factor are somewhat better constrained for this object Geringer-Sameth et al. (2015). However, similar conclusions apply to other dSph’s such as Seg1, Carina, Fornax, Sculptor etc. Geringer-Sameth et al. (2015); Choquette (2018); Kar et al. (2019b) dSph assuming an Navarro-Frenk-White (NFW) profile Navarro et al. (1996), with

[TABLE]

where $\rho_{s}=1.4$ $GeV.cm^{-3}$ and $r_{s}=1.0$ $kpc$ Colafrancesco et al. (2007); McDaniel et al. (2017). $\frac{dN^{e}_{f}(E)}{dE}B_{f}$ estimates the number of $e^{\pm}$ produced with energy $E$ per annihilation in any of the aforementioned SM channel ($f$) having branching fraction $B_{f}$.

Figure 1 shows the electron energy distribution for four different annihilation channels ($b\bar{b},\tau^{+}\tau^{-},W^{+}W^{-},t\bar{t}$) and for two DM masses, namely, $m_{\chi}=$ 300 $GeV$ (upper left panel) 5 $TeV$ (upper right). For all cases $\langle\sigma v\rangle$ has been assigned a fixed value ($10^{-26}\mbox{cm}^{3}\mbox{s}^{-1}$), and the branching fraction corresponding to each channel has been set in turn at $100\%$. In addition, a comparison between cases with the two masses has also been shown in the lower panel for the $b\bar{b}$ and $\tau^{+}\tau^{-}$ channel. The prediction for $t\bar{t}$ and $W^{+}W^{-}$ falls in between those curves. All these energy distributions are obtained using micrOMEGAs Bélanger et al. ; Belanger et al. (2011). Further discussions on these curves will be taken up in section IV.

The electrons produced in DM pair annihilation, diffuse through the interstellar medium (ISM) in the galaxy and lose energy through various electromagnetic processes. Assuming a steady state and homogeneous diffusion, the equilibrium distribution $\frac{dn}{dE}$ can be obtained by solving the equation Natarajan et al. (2013); Colafrancesco et al. (2006); Beck and Colafrancesco (2016); Colafrancesco et al. (2007); McDaniel et al. (2017); Storm et al. (2017)

[TABLE]

where $D(E)$ is the diffusion parameter. The exact form of $D(E)$ is not known for dSphs, hence for simplicity, we assume it to have a Kolmogorov spectrum $D(E)=D_{0}\left(\frac{E}{GeV}\right)^{0.3}$, where $D_{0}$ is the diffusion coefficient Colafrancesco et al. (2007); McDaniel et al. (2017); Spekkens et al. (2013). Very little is known about the value of $D_{0}$ for dSphs due to their very low luminosity. A choice like $D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$ is by and large reasonable for a dSph such as Draco Colafrancesco et al. (2007), although higher values of $D_{0}$, too, are sometimes used as benchmark Regis et al. (2014). We have mostly used $D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$, and demonstrated side by side the effect of other values of this largely unknown parameter. As explained in Natarajan et al. (2015, 2013); Jeltema and Profumo (2008), assuming a proper scaling, $D_{0}$ for a dSph can also in principle have a value one order lower than that in the Milky Way Donato et al. (2004). However, the present study is not restricted to this value, mainly $D_{0}\lesssim 3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$. The detectability of the radio flux for higher $D_{0}$, too, has been studied by us, as will be seen when we come to Figure 13.

The energy loss term $b(E)$ takes into account all the electromagnetic energy loss processes such as Inverse Compton (IC) effect, Synchrotron (Synch) radiation, Coulomb loss (Coul), bremsstrahlung (Brem) etc. and their combined effect can be expressed as Colafrancesco et al. (2006); Beck and Colafrancesco (2016); McDaniel et al. (2017)

[TABLE]

where $m_{e}$ is the electron mass and $n_{e}$ is the average thermal electron density in the dSph (value of $n_{e}\approx 10^{-6}$). Values of the energy loss coefficients are taken to be $b^{0}_{IC}\simeq 0.25$, $b^{0}_{Synch}\simeq 0.0254$, $b^{0}_{Coul}\simeq 6.13$ $b^{0}_{Brem}\simeq 1.51$, all in units of $10^{-16}$ $GeVs^{-1}$ Colafrancesco et al. (2006); Beck and Colafrancesco (2016); McDaniel et al. (2017). In a dSph like Draco, the first two terms (i.e. IC and Synch) in Equation 4 dominates over the last two terms (i.e. Coul and Brem) for $E>1$ $GeV$ Natarajan et al. (2013). The energy loss term $b(E)$ depends on the galactic magnetic field ($B$) through the synchrotron loss term which goes as $B^{2}$. It is extremely difficult to get insights (say, through polarization experiments) for the magnetic field properties of dSph. The lack of any strong observational evidence suggests that the magnetic fields could be, in principle, extremely low. On the other hand, there could be various effects which can significantly contribute to the magnetic field strengths in dSph. In fact, numerous theoretical arguments have been proposed for values of $B$ at the $\mu G$ level. It has been argued, for example, that the observed fall of $B$ from the centre of Milky Way to its peripheral region can be linearly extrapolated to nearby dSph’s, leading to $B\gtrsim 1$ $\mu G$. The possibility of the own magnetic field of a dSph has also been suggested. For detailed discussions we refer the reader to reference Regis et al. (2015). For the dSph considered in this work, we have mostly used $B=1$ $\mu G$ Colafrancesco et al. (2007); McDaniel et al. (2017); Spekkens et al. (2013), but more conservative values like $B\sim 10^{-1}-10^{-3}$ $\mu G$ have also been considered. The manner in which diffusion and electromagnetic processes affect our observables will be discussed in detail in sections III and V. The overall potential of the SKA in probing regions in the $D_{0}-B$ space in a correlated fashion will be reported in section V.

Equation 3 has a solution of the form Natarajan et al. (2013); Colafrancesco et al. (2006, 2007); McDaniel et al. (2017)

[TABLE]

with the boundary condition $\frac{dn}{dE}(r_{h})=0$, where $r_{h}$ is the radius of the diffusion zone. For Draco, $r_{h}$ is taken to be 2.5 $kpc$ Colafrancesco et al. (2007); McDaniel et al. (2017). $G(r,\Delta v)$ is the Green’s function of the equation and has a form

[TABLE]

with $r_{n}=(-1)^{n}r+2nr_{h}$. $\sqrt{\Delta v}$ is called the diffusion length scale, expressed as

[TABLE]

$\sqrt{\Delta v}$ determines the distance traveled by an electron as it loses energy from $E^{\prime}$ to $E$. For small galaxies like dSphs, the mean value of this length scale is expected to be larger than $r_{h}$ even for non-conservative choices of $D_{0}$ and $B$. We shall discuss this in detail in section III. After interacting with the magnetic field $B$ present inside the galaxy, the produced electron/positron distribution $\frac{dn}{dE}$ will emit synchrotron radiation (with frequency $\nu$) at a rate governed by the synchrotron emission power $P_{Synch}(\nu,E,B)$ Colafrancesco et al. (2006); Beck and Colafrancesco (2016); Colafrancesco et al. (2007); McDaniel et al. (2017); Storm et al. (2017).

In Figure 2, we have shown the dependence of the synchrotron emission power on the electron energy $E$ for two different magnetic fields ($B=$ 1 and 0.1 $\mu G$) and for two frequencies ($\nu=$ $10$ and $10^{4}$ $MHz$). For each frequency shown, it is clear that a stronger magnetic field always intensifies the power spectrum.

The synchrotron emissivity $J_{Synch}$ as a function of frequency $\nu$ and radius $r$ is obtained by folding $\frac{dn}{dE}(r,E)$ with $P_{Synch}(\nu,E,B)$ as

[TABLE]

Finally, the approximate radio synchrotron flux can be obtained by integrating $J_{Synch}$ over the diffusion size ($r_{h}$) of the dSph Colafrancesco et al. (2006); Beck and Colafrancesco (2016); Colafrancesco et al. (2007); McDaniel et al. (2017); Storm et al. (2017)

[TABLE]

where $L$ is the luminosity distance of the dSph. For Draco, $L\sim 80$ $kpc$ McDaniel et al. (2017); Geringer-Sameth et al. (2015)).

III Effect of various astrophysical parameters

Let us now take a closer look at the astrophysical effects encapsulated in equations 3 and 4. Such effects are driven by the diffusion coefficient $D_{0}$ and the electromagnetic energy loss coefficient $b(E)$, which determine the steady state $e^{\pm}$ distribution $\frac{dn}{dE}(r,E)$ for a given $\frac{dN^{e}}{dE}$. As has been already mentioned, $b(E)$ is implicitly dependent on the galactic magnetic field $B$.

We use again the two DM masses ($m_{\chi}=300$ $GeV$ and 5 $TeV$) for studying the effects of $D_{0}$ and $b(E)$. To have some idea on these effects separately, as well as their contribution in an entangled fashion, we have considered the implication of equation 3 for three different scenarios:

• •

NSD: considering only the effect of energy loss term $b(E)$ and neglecting the spatial diffusion $D(E)$ (i.e. solution of equation 3 by setting $D(E)=0$).

• •

Nb: considering the effect of diffusion parameter $D_{0}$ and neglecting the energy loss term $b(E)$ (i.e. solution of equation 3 by setting $b(E)=0$).

• •

SD+b: considering the effects of both the diffusion parameter $D_{0}$ and the energy loss term $b(E)$ (i.e. the complete solution as explained in equation 5).

For the NSD scenario, solution of equation 3 has a simpler form Colafrancesco et al. (2006):

[TABLE]

Note that, $b(E)$ increases with $E$ (from equation 4). Its effect on the number density in different energy regions is more prominent for this NSD scenario where we neglect the effect of distribution. The number density decreases in the high energy region due to the combined effect of the $\frac{1}{b(E)}$ suppression as well as the lower limit of the integration in equation 10. This can also be seen from Figure 3 where we have plotted the ratio of $\frac{dn}{dE}$ and the source function $Q_{e}$ (which determines the initial electron flux due to DM annihilation) against $E$. The term $\frac{dn}{dE}\frac{1}{Q_{e}}$ (in unit of $s^{-1}$) essentially determines how the shape of the initial distribution $Q_{e}$ gets modified due to the effect of various astrophysical processes. As discussed in section II, the energy loss term $b(E)$ depends on the square of the magnetic field $B$ through synchrotron loss. Thus $b(E)$ increases with increase of $B$, which in turn will reduces the $\frac{dn}{dE}$ at all $E$. This phenomenon can also be observed in Figure 3 where the red and magenta lines indicate the distributions for $B=1$ and 10 $\mu G$333$B=10$ $\mu G$ has been used in this Figure for the sake of comparison with 1 $\mu G$, just to see the effect of ‘large $B$’. In our prediction on observable radio flux, more conservative (and perhaps realistic) values of $B$ have been used., respectively. For comparison, we have also shown the cases for $B=0.1$ $\mu G$ (green lines). Note that, the later case coincides with the case for $B=1$ $\mu G$. This is due to the fact that the magnetic field dependence (through synchrotron loss) in the energy loss term $b(E)$ gets suppressed by the Inverse Compton ($b_{IC}$) term for lower $B$ ($B<1$ $\mu G$), which can be clearly seen from equation 4. Also, for heavier DM masses the energetic electrons are produced in greater abundance (as seen from Figure 1) which can lead to a larger $\frac{dn}{dE}\frac{1}{Q_{e}}$. The effects of two different DM masses, $m_{\chi}=300$ $GeV$ and 5 $TeV$, have been shown in this figure by dashed and solid lines respectively.

Now, for the Nb scenario, the solution becomes

[TABLE]

with the same boundary condition as what we assumed for equation 5. Here

[TABLE]

Since the effect of $b(E)$ is absent in this case, the distribution $\frac{dn}{dE}$ will not depend on the magnetic field $B$. Figure 4 shows the comparison between the NSD case (cyan lines) and the Nb case (black lines) at two different radii ($r=0.1$ $kpc$; left panel and $r=2$ $kpc$; right panel) for $B=1$ $\mu G$ and $D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$. It is clear from equation 11 that the ratio $\frac{dn}{dE}\frac{1}{Q_{e}}$ for the Nb scenario will not depend on the initial spectrum $Q_{e}$ and its energy profile will follow the energy dependence of $\frac{1}{D(E)}$. This presence of $D(E)$ in the denominator leads to $\frac{dn}{dE}\frac{1}{Q_{e}}\sim E^{-0.3}$ in the limit $b(E)\to 0$. This can also be verified from the relatively flat curves for Nb cases in this figure.

We should mention here that neglecting diffusion is not a bad assumption where the length scale ($\sqrt{\Delta v}$) over which the $e^{\pm}$ losses energy is much shorter than the typical size of the system Colafrancesco et al. (2006); McDaniel et al. (2017). However, for smaller systems like dSphs the effect of diffusion cannot be neglected. This can be justified from Figure 4 where we have plotted the SD+b case (i.e. taking diffusion into account along with energy loss effect; shown by red curves) along with the two previously mentioned scenarios. It is clear that the NSD scenario is strikingly different from the one which takes diffusion into account. Addition of diffusion in the system essentially suppresses the $e^{\pm}$ distribution, especially in the low energy energy region. As a result, the plots for the SD+b case closely match the Nb one at lower energy, but diverges following the NSD scenario at higher energies where the effect of $b(E)$ dominates. This phenomenon can also be explained in terms of the diffusion length scale ($\sqrt{\Delta v}$) which is a result of combined effects of diffusion and energy loss (equation 7). It is indicative of the length covered by an electron as it loses energy from the source energy $E^{\prime}$ to the interaction energy $E$ which is typically larger than the size of the dSph. In order to determine the minimum energy that an electron can possess before escaping the diffusion radius, we have plotted the interaction energy $E$ vs. $\sqrt{\Delta v}$ in Figure 5 for different sets of $D_{0}$ and $B$. The left and right panels correspond to the source energy $E^{\prime}=1$ $TeV$ and 100 $GeV$ respectively. The diffusion radius (for Draco) $r_{h}=2.5$ $kpc$ is shown by the black solid lines for both cases. The minimum energy is thus calculated from the point where the model crosses this line. A higher $D_{0}$ will increase this minimum value of interaction energy and as a result, it reduces the $e^{\pm}$ number density by making them leave the diffusion zone earlier. Note that, the cases with $B=0.1$ $\mu G$ practically coincides with that for $B=1$ $\mu G$. This is due to the fact that $b(E)$ remains unchanged for $B\leq 1$ $\mu G$, as already been mentioned. On the other hand, we have explicitly checked that, a larger value of $B$ (say $B=10$ $\mu G$; not shown in the plot) will help the electron in losing energy more quickly before escaping the diffusion zone, causing a suppression in the number density of high energy electrons and an enhancement of the low energy electrons.

Similar conclusions can also be drawn from Figure 6. To illustrate the effects of $D_{0}$ and $B$ in the SD+b scenario, we have plotted $\frac{dn}{dE}\frac{1}{Q_{e}}$ against $E$ for the DM masses 5 $TeV$ (upper panels) and 300 $GeV$ (lower panels) at two different radii 0.1 $kpc$ (left panels) and 2 $kpc$ (right panels). For all cases, the dominant annihilation channel has been assumed to be $b\bar{b}$ with $\langle\sigma v\rangle$ = $10^{-26}\mbox{cm}^{3}\mbox{s}^{-1}$. Cases with different $D_{0}$ and $B$ are indicated by different color coding as mentioned in the insets of corresponding figures. Similarly, Figure 7 shows the variation of the same with respect to radius $r$ for those two DM masses ($m_{\chi}=5$ $TeV$ $-$ left panel and $m_{\chi}=300$ $GeV$ $-$ right panel). For each DM mass we have assumed two different combinations of energies, one being low ($E=0.1$ $GeV$ for both DM masses) and the other, high ($E=1$ $TeV$ for $m_{\chi}=5$ $TeV$ and $E=100$ $GeV$ for $m_{\chi}=300$ $GeV$). In both panels, the black line indicates the diffusion radius (2.5 $kpc$) for Draco. One can see that, large $D_{0}$ (blue curves) implies a lower density distribution compared to the corresponding cases with smaller diffusion coefficient. The difference is more prominent for low energy.

The effects of diffusion coefficient and magnetic field on the final radio flux $S_{\nu}$ can be seen from Figure 8. The radio flux corresponding to higher diffusion coefficient ($D_{0}=10^{30}\mbox{cm}^{2}\mbox{s}^{-1}$) will be suppressed compared to the lower diffusion case ($D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$) due to the escape of a large number of $e^{\pm}$ from the stellar object, as discussed above. This suppression is slightly larger in the lower frequency region, since the number density $\frac{dn}{dE}$ decreases more in the low energy regime (see Figure 6). For, a constant $D_{0}$, say, $3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$, a relatively lower value of $B$ (0.1 $\mu G$) reduces the radio flux by about an order of magnitude in all frequency ranges. This is solely due to the fact that, although $\frac{dn}{dE}$ has similar values for different values of $B$ (1 and 0.1 $\mu G$), the synchrotron power spectrum $P_{Synch}$ decreases with decrease in $B$ (as evident from Figure 2). We further emphasise that, though this analysis assumes an NFW profile for Draco, the choice of other profiles such as Burkert Colafrancesco et al. (2007); Burkert (1996) or Diemand et al. (2005) Diemand et al. (2005) (hereafter D05) Colafrancesco et al. (2007) keep our predictions similar, as can be seen in Figure 9.

IV Heavy DM particles

The radio flux obtained in terms of DM annihilation from a dSph crucially depends on the source function $Q_{e}(E,r)$ (equation 1). Let us try to explain why one can get higher radio flux (obtained via $dn/dE$ through integration of $Q_{e}(E,r)$) for higher DM masses in some cases Kar et al. (2019a). For this to happen, $Q_{e}$ corresponding to energetic $e^{\pm}$ is intuitively expected to go up for higher $m_{\chi}$. One may consider contributions of several components in the expression of $Q_{e}(E,r)$:

• •

$\langle\sigma v\rangle$: For a trans-TeV thermal DM $\chi$, the factors affecting $\langle\sigma v\rangle$ are its mass ($m_{\chi}$) and the effective couplings to SM particle pairs. While we are concerned here with the annihilation cross sections for $\chi$ within a dSph, one also needs consistency with the observed relic density Ade et al. (2014). In general, the expression for the relic density ($\Omega h^{2}$) indicates inverse proportionality to $\langle\sigma v\rangle$, and $\frac{1}{m_{\chi}^{2}}$ occurs as the flux factor in the denominator of the latter Gondolo and Gelmini (1991). Thus a factor of $m_{\chi}^{2}$ occurs in the numerator. Therefore, $m_{\chi}\gtrsim 1$ $TeV$ may make the relic density unacceptably large. One way to alleviate this is to ensure the possibility of resonant annihilation Gondolo and Gelmini (1991), as is possible in the MSSM through the participation of a pseudoscalar of appropriate mass Gilmore (2007). This also serves to enhance the observed radio flux, as we shall see later. In addition, the annihilation of $\chi$ in the early universe may involve channels other than those in a dSph, through co-annihilation with particles spaced closely with it Gilmore (2007). Once either of the above mechanisms is effective for the relevant particle spectrum, a trans-TeV DM particle remains consistent with all data including the relic density. This ensured, the pair-annihilation of a trans-TeV DM particle in dSph’s enable the production of relatively energetic $e^{\pm}$ pairs which in turn reinforce the radio signals in the desired frequency range.

• •

$N_{pairs}(r)\left(=\frac{\rho^{2}_{\chi}(r)}{2m_{\chi}^{2}}\right)$: The numerator is supplied by observation. For higher $m_{\chi}$, the denominator suppresses the number density of DM in the dSph. Thus this term tends to bring down the flux for higher DM masses.

• •

$\frac{dN^{e}_{f}(E)}{dE}B_{f}$: This has to have a compensatory effect for higher mass if the suppression caused by the previously mentioned term has to be overcome. Of course, the branching fractions $B_{f}$ in different channels have a role. ${dN^{e}/dE}$ is often (though not always) higher for higher DM masses. This feature is universal at higher energies. This is basically responsible for a profusion of higher energy electrons produced via cascades, after annihilation in any channel has been taken place. More discussion will follow on this point later. It should be noted that ${dN^{e}/dE}$ represents the probability that an electron (positron) produced from the annihilation of one pair of DM. This gets multiplied by the available number density of DM. Thus, for comparable value of $B_{f}$ in two benchmark scenarios, higher ${dN^{e}/dE}$ along with higher $\langle\sigma v\rangle$ in the dSph (when that is indeed the case), can cause enhancement of $Q_{e}(E,r)$ for higher $m_{\chi}$.

Finally, the available electron energy distribution $\frac{dn}{dE}$ (which, convoluted with the synchrotron power spectrum, will yield the radio flux (i.e. equations 8 and 9)), is obtained by solving equation 3 which has a generic solution (equation 5). Therefore, higher values of the source function ($Q_{e}(E,r)$) can produce higher radio flux at higher $m_{\chi}$.

As mentioned above, the annihilation of heavier DM particles will produce more energetic $e^{\pm}$, as can be seen by comparing the upper-left and upper-right panels of Figure 1 where we have shown $dN^{e}/dE$ in different channels ($b\bar{b}$, $\tau^{+}\tau^{-}$, $W^{+}W^{-}$, $t\bar{t}$). The lower panel shows the comparison of this $e^{\pm}$ spectrum for two DM masses (300 $GeV$ and 5 $TeV$) arising from $b\bar{b}$ and $\tau^{+}\tau^{-}$ annihilation channels. Note that, the values of $dN^{e}/dE$ for $m_{\chi}=300$ $GeV$ and 5 $TeV$ are differently ordered for the $\tau^{+}\tau^{-}$ and $b\bar{b}$ annihilation channels. The spectrum for $b\bar{b}$ channel is the steepest one, while $\tau^{+}\tau^{-}$ is the flattest. In fact, $dN^{e}/dE$ for the $b\bar{b}$ channel is governed mostly by charged pion ($\pi^{\pm}$) decay at various stages of cascade. For the $\tau^{+}\tau^{-}$ channel, on the other hand, there is a relative dominance of ‘prompt’ electrons. There remains a difference in the degree of degradation in the two cases. Such degradation is reflected more in the low-energy part of the spectrum, and leads to different energy distributions, which in turn is dependent on $m_{\chi}$ via the energy of the decaying $b$ or $\tau$. The reader is referred to see references Colafrancesco and Mele (2001); Rudaz and Stecker (1988); Kamionkowski and Turner (1991) for more detailed explanations.

The presence of energetic $e^{\pm}$ in greater abundance, which is a consequence of higher $m_{\chi}$, enhances the resulting radio signal obtained through equation 5 and 8. As a result of this, the final radio flux $S_{\nu}$ (equation 9) gets positive contribution for higher DM masses compared to that for relatively lower masses for most of the annihilation channels (especially $b\bar{b}$, $W^{+}W^{-}$, and $t\bar{t}$). This can easily be seen if one compares the quantity $S_{\nu}\times 2m_{\chi}^{2}$ (i.e. removing the effect of the multiplicative factor $\frac{1}{m_{\chi}^{2}}$ present in the expression of $N_{pairs}$) for two DM masses, 300 $GeV$ and 5 $TeV$, in the $b\bar{b}$ annihilation channel (left panel of Figure 10). If one chooses same annihilation rate (e.g. $\langle\sigma v\rangle$ = $10^{-26}\mbox{cm}^{3}\mbox{s}^{-1}$), the higher DM mass will give higher $S_{\nu}\times 2m_{\chi}^{2}$ (mainly in the high frequency region) for this channel. We have explicitly shown this for different choices of astrophysical parameters as indicated by different colors in the figure.

In scenarios where $\langle\sigma v\rangle$ corresponding to a particular $m_{\chi}$ is calculated from the dynamics of the model, it can happen that for some particular benchmark the annihilation rate ($\langle\sigma v\rangle$) is higher for larger $m_{\chi}$ (as mentioned earlier and will be discussed further in the next section). In those cases larger $\langle\sigma v\rangle$ can at least partially compensate the effect of $\frac{1}{m_{\chi}^{2}}$ suppression (coming from $N_{pairs}$) for higher $m_{\chi}$. Thus one can get higher radio fluxes ($S_{\nu}$) for larger DM masses compared to the smaller one.

For the $\tau^{+}\tau^{-}$ channel the situation is somewhat different. As we have already seen from the lower panel of Figure 1, there is a large degradation of the $e^{\pm}$ flux ($dN^{e}/dE$) in this channel in the low energy region for higher $m_{\chi}$. This degradation in the source spectrum for higher $m_{\chi}$ will continue to present in the equilibrium distribution $\frac{dn}{dE}$. After folding this density distribution with power spectrum (see equation 8), the final frequency distribution will be suppressed for higher $m_{\chi}$, mainly in the low frequency range. In the higher frequency range, flux is still high for higher $m_{\chi}$ similar to that in other annihilation channel (e.g. $b\bar{b}$), because higher $m_{\chi}$ corresponds to higher electron distribution in the high energy range. All these phenomena are evident from the red curves (solid and dashed) of right panel in Figure 10. Note that, these curves are for the choice of astrophysical parameters $D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$, $B=1$ $\mu G$. If one decreases the magnetic field $B$ to 0.1 $\mu G$ or increases the diffusion coefficient $D_{0}$ to $10^{30}\mbox{cm}^{2}\mbox{s}^{-1}$, the corresponding effect can be seen from the green and blue curves respectively.

To summarise, one can expect high radio flux for trans-TeV dark matter annihilation from a dSph, based on the following considerations:

• •

The high-mass DM candidate has to be consistent with relic density, for which $\langle\sigma v\rangle$ at freeze-out is the relevant quantity.

• •

Such sizable $\langle\sigma v\rangle$ can still be inadequate in maintaining the observed relic density. Co-annihilation channels may need to be available in these cases, though such co-annihilation does not occur in a dSph. This in turn may necessitate a somewhat compressed trans-TeV spectrum.

• •

The high mass DM candidate should have appropriate annihilation channels which retain a higher population of $e^{\pm}$. This not only off sets the suppression due to large $m_{\chi}$ but also enhances through the energy loss term ($b(E)$) the $e^{\pm}$ density at energies low enough to contribute the radio observable at the SKA Braun et al. (2017).

• •

Higher magnetic fields will be more effective in producing synchrotron radiation by compensating the suppression caused by a large galactic diffusion coefficient. As we have checked through explicit calculation, this happens even after accounting for electromagnetic energy loss of the $e^{\pm}$ through synchrotron effects.

V Detectability Curves and Final Radio Flux

The upcoming Square Kilometre Array (SKA) radio telescope will play an important role in detecting DM induced radio signal from dwarf spheroidal galaxies. Due to its large effective area and better baseline coverage, the SKA has a significantly higher surface brightness sensitivity compared to existing radio telescopes. Since the focus of this work is on the diffuse synchrotron signal from dSphs, we estimate the instrument sensitivity corresponding to the surface brightness and compare with the predicted signal. To calculate the sensitivity for a given source dSph, we need to know the baseline distribution of the telescope. The sensitivity can also be affected by the properties of the sky around the source, e.g., whether there exists any other bright source in the field of view.

Since the detailed design of the SKA is yet to be finalised, it is difficult to estimate the noise in the direction of a given source. For our purpose, however, it is sufficient to estimate the approximate values of the sensitivity using the presently accepted baseline design. In order to do so, we make use of the documents provided in the SKA website Braun et al. (2017). The calculations presented in the document allow us to compute the expected sensitivity near zenith in a direction well away from the Galactic plane for all the frequencies relevant to the SKA, i.e., for both SKA-MID and SKA-LOW. These include contribution from the antenna receiver, the spill-over and the sky background (consisting of the CMB, the galactic contribution and also the atmospheric contribution). Note that we have assumed that all other observational systematics, such as the presence of other point sources in the field, the effect of the primary beam, pointing direction-dependence of the sky temperature etc are already corrected for and hence are not included in the calculation of the thermal noise.

From the above calculation, we find that typical values of the SKA surface brightness sensitivity in the frequency range 50 MHz – 50 GHz for 100 hours of observation time is $10^{-6}$ – $10^{-7}$ Jy with a bandwidth of 300 MHz Braun et al. (2017); Kar et al. (2019a); Colafrancesco et al. (2015). This may allow one to observe very low intensity radio signal coming from ultra-faint dSph’s. While comparing the predicted signal with the telescope sensitivity, we have assumed that the SKA field of view is larger than the galaxy sizes considered here and hence all the flux from the galaxy will contribute to the detected signal. This assumption need not be true for the SKA precursors like the Murchison Widefield Array (MWA) where the effect of the primary beam needs to be accounted for while computing the expected signal Kar et al. (2019b).

Till date, people do not have clear understanding about either the DM particle physics model which governs the production of initial stage $e^{\pm}$ spectrum, or the astrophysical parameters like galactic diffusion coefficient ($D_{0}$), magnetic field ($B$) etc. which are responsible for the creation of radio synchrotron flux. Thus in this analysis, we have constrained the DM parameter space, responsible for detecting the radio signal at SKA, for particular choices of astrophysical parameters which are on the conservative side for a typical dSph like Draco Colafrancesco et al. (2007). On the other hand, assuming some simplified DM model scenarios with trans-TeV DM masses, we have estimated the limits on the $B-D_{0}$ plane, required for the radio signal from Draco to be observed at SKA.

Figure 11 shows the minimum $\langle\sigma v\rangle$ required for any $m_{\chi}$ in four different annihilation channels ($b\bar{b}$, $\tau^{+}\tau^{-}$, $W^{+}W^{-}$, $t\bar{t}$) for detection of DM annihilation induced radio signal from Draco with 100 hours of observation at SKA assuming a typical band width 300 $MHz$. The radio signal has been taken to be detectable when the observed flux in any frequency bin rises three times above the noise so as to ensure that the detection is statistically significant and is not affected by spurious noise features. We vary the DM mass over a wide range of 10 $GeV$ to 50 $TeV$, assuming 100% branching fraction ($B_{f}$) in one annihilation channel at a time. The predictions here are for a conservative choice of the diffusion coefficient ($D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$) Colafrancesco et al. (2007). The bands are due to the variation of the galactic magnetic field $B$ from 1 $\mu G$ (lower part of the band) to a more conservative value 0.1 $\mu G$ (upper part of the band). As expected, the minimum $\langle\sigma v\rangle$ required will be larger for lower magnetic fields as lower $B$ reduces the the radio synchrotron frequency distribution ($S_{\nu}$). These limits are the detection thresholds for SKA to observe radio signal from Draco. For $m_{\chi}\sim 1$ $TeV$ this limit could be as low as $\langle\sigma v\rangle\sim 3\times 10^{-29}\mbox{cm}^{3}\mbox{s}^{-1}$. For lower values of the diffusion coefficient such as $D_{0}=3\times 10^{26}\mbox{cm}^{2}\mbox{s}^{-1}$ Natarajan et al. (2015, 2013); Jeltema and Profumo (2008), the detectability threshold band comes down even further. Moreover, we have not considered any halo substructure contributions which are expected to enhance the radio flux Beck and Colafrancesco (2016) or lower the threshold limits even more. Along with these lower limits, we have also shown the model-independent upper limits (in 95% C.L.) on $\langle\sigma v\rangle$ in various channels from cosmic-ray (CR) antiproton observation (dashed curves) Cuoco et al. (2018) and from 6 years of Fermi LAT (FL) $\gamma$-ray data (dotted curves) Ackermann et al. (2015). Note that the upper bounds from cosmic-ray antiproton observations are the strongest ones. For each annihilation channel, the area bounded by the upper and lower limits represents the region in the $\langle\sigma v\rangle-m_{\chi}$ plane which can be probed or constrained by SKA with 100 hours of observation. It is clear from the figure that even with conservative choices of astrophysical parameters for $m_{\chi}\approx 50$ $TeV$, there are significantly large regions of the parameter space which can be probed in SKA. In such extreme cases, it is of course necessary to have a dark sector that allows high co-annihilation rates, so that the observed relic density bound is not excluded.

In the context of model-dependent analysis, three benchmark points, named as Model A1a, B2a and E, from our earlier work Kar et al. (2019a) have been considered here. These benchmarks correspond to the minimal supersymmetric standard model (MSSM) scenario where the lightest neutralino ($\chi^{0}_{1}$) is the DM candidate ($\chi$). Table 1 contains the possible annihilation channels with branching fractions, DM masses ($m_{\chi^{0}_{1}}$) and annihilation rates ($\langle\sigma v\rangle$) calculated in those benchmark points. All these quantities have been calculated using publicly available package micrOMEGAs Bélanger et al. ; Belanger et al. (2002). Masses of neutralino and all other supersymmetric particles in these three cases are in the trans-TeV range. All of these benchmarks produce relic densities within the expected observational limits Ade et al. (2014); Harz et al. (2016); Klasen et al. (2016) and satisfy constraints coming from direct DM searches Aprile et al. (2017); Akerib et al. (2017), collider study Aaboud et al. (2018), lightest neutral Higgs mass measurements Allanach et al. (2004) and other experiments Aaij et al. (2017); Sandilya (2017). These benchmarks have been discussed in further detailed in reference Kar et al. (2019a). Figure 12 shows the detectability of these benchmarks for Draco dSph in SKA after 100 hours of observations. The predicted flux in each case falls within the area which is still allowed by the data and is at least 3 times above the observation threshold of SKA. The upper limits here are taken from cosmic-ray antiproton observation at 95% C.L Cuoco et al. (2018) and the lower limits have been calculated for $D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$ and $B=1$ $\mu G$. Along with these we have indicated the total $\langle\sigma v\rangle$ (listed in third column of table 1) by different points for each benchmark. It should be noted that each such benchmark allows more than one annihilation channel. Consistently, the cross-sections are obtained by taking a sum over different channels appropriately weighted by the branching ratios:

[TABLE]

It is clear that all these high mass cases, which are allowed by cosmic-ray antiproton data, can easily be probed in SKA with 100 hours of observations. This is due to the following reason:

• •

$\langle\sigma v\rangle$ is more effective in offsetting $\frac{1}{m_{\chi}^{2}}$ suppression (present in the expression for number density of DM pair; equation 1) partially, though not fully.

• •

A greater abundance of high-energy $e^{\pm}$ is created by high $m_{\chi}$. This, via electromagnetic energy loss driven by the term proportional to $b(E)$ in equation 3, generates a bigger flux of radio synchrotron emission, as evinced in the expression for $J_{Synch}(\nu,r)$ (equation 8).

• •

The cases where one predicts more intense radio flux for higher $m_{\chi}$ have DM annihilation mostly in the $b\bar{b}$ channels, as against the $\tau^{+}\tau^{-}$ channel. The corresponding cascade branching ratios as well as the three-body decay matrix elements and their energy integration limits are responsible for bigger radio flux.444Such effects can in principle be also expected if the $t\bar{t},W^{+}W^{-}$ branching ratios dominates.

Figure 11 shows how the detectability threshold at the SKA varies with the galactic magnetic field $B$, for a fixed value of $D_{0}$. However, the values of these two parameters for the dSphs are quite uncertain. In view of this, it is also important to find out which regions in the astrophysical parameter space are within the scope of the SKA with 100 hours of observation, when both $B$ and $D_{0}$ vary over substantial ranges. The detectability of the signal in the $B-D_{0}$ space is shown in Figure 13 for some illustrative DM scenarios. These correspond to DM masses of 1 (red curve) and 5 (blue curve) $TeV$. Cases where the dominant annihilation channel is $b\bar{b}$ are shown in the left panel, while the curves on the right panel capture the corresponding situations with $\tau^{+}\tau^{-}$ as the main channel. The solid line of each color corresponds to the maximum value of $\langle\sigma v\rangle$ for the chosen $m_{\chi}$, consistent with the cosmic-ray upper limit. All points above and on the left of the curve correspond to the combinations of $B$ and $D_{0}$ that make the radio signals detectable over 100 hours of observations with the SKA. Points for higher $B$ in this region correspond to models that are detectable even with lower values of $\langle\sigma v\rangle$. Similarly, the detectable models having progressively lower $\langle\sigma v\rangle$ are arrived at, as one moves to lower $D_{0}$ for a fixed value of $B$. It is evident from both the $b\bar{b}$ and $\tau^{+}\tau^{-}$ channels (chosen for illustrations in Figure 13) that consistent values of $\langle\sigma v\rangle$ can lead to detectability with 100 hours at SKA, for $B\gtrsim 5\times 10^{-4}$ $\mu$G for $D_{0}\sim 10^{27}\mbox{cm}^{2}\mbox{s}^{-1}$. On the other hand, for larger values of $D_{0}\sim 10^{30}\mbox{cm}^{2}\mbox{s}^{-1}$, we need $B\gtrsim 10^{-2}~{}\mu$G for the signal to be detectable.

Finally, we show in Figure 14 the final radio fluxes in two of the MSSM benchmarks listed in table 1. For illustration we have taken the benchmark points A1a ($m_{\chi}\sim 1$ $TeV$) and E ($m_{\chi}\sim 8.5$ $TeV$). The annihilation is $b\bar{b}$-dominated for both cases. The yellow band here represents the SKA sensitivity Braun et al. (2017), with a bandwidth of 300 $MHz$. The band is due to the variation of the observation time from 10 hours (upper part of the band) to 100 hours (lower part of the band) Braun et al. (2017); Kar et al. (2019a). The choice of the diffusion coefficient and the magnetic field is on the conservative side ($D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$ and $B=1$ $\mu G$). We have already shown the detectability of these benchmarks in Figure 12. From this figure we can see that, even for 10 hours of observation, these high-$m_{\chi}$ benchmarks can easily be observed in SKA over most of its frequency range.

One important feature of Figure 14 is that, in a wide range of frequency (300 $MHz$ – 50 $GHz$), suitable for SKA, $S_{\nu}$ for very high mass case (Model E) is higher than that for low mass case (Model A1a). Though the DM mass in Model E is larger, annihilation rate in this case is higher ($\sim 9\times 10^{-26}\mbox{cm}^{3}\mbox{s}^{-1}$) than in Model A1a ($\sim 3\times 10^{-27}\mbox{cm}^{3}\mbox{s}^{-1}$). In general, $\langle\sigma v\rangle$ should have a $\frac{1}{m_{\chi}^{2}}$ suppression, because of the energies of the colliding DM particles Gondolo and Gelmini (1991). In spite of that, Model E (higher $m_{\chi}$) has greater $\langle\sigma v\rangle$555higher value of $\langle\sigma v\rangle$ for higher $m_{\chi}$ is also needed to keep the relic density under the observed limit, even when there is scope of co-annihilation in the early universe. than Model A1a (lower $m_{\chi}$) mainly due to the closer proximity to a s-channel resonance mediated by CP-odd pseudoscalar in the annihilation process like $\chi^{0}_{1}\chi^{0}_{1}\rightarrow b\bar{b}$. For detailed information reader is referred to see reference Kar et al. (2019a)

As discussed earlier, The higher value of $\langle\sigma v\rangle$ partially offsets the effect of larger DM mass and consequently we get $\frac{\langle\sigma v\rangle}{m_{\chi}^{2}}$ (which appears in the source function; equation 1) for Model A1a and Model E as 2.7 and 1.26 (in units of $10^{-33}\mbox{GeV}^{-2}\mbox{cm}^{3}\mbox{s}^{-1}$), respectively, though the latter model has larger $m_{\chi}$ ($\sim 8.5$ $TeV$) compared to the former ($\sim 1$ $TeV$). Thus a 72-fold suppression due to $m_{\chi}^{2}$ results in a suppression just by a factor of $\approx 2$ at the level of $\frac{\langle\sigma v\rangle}{m_{\chi}^{2}}$. Also, note that the $b\bar{b}$ annihilation channel dominates for both models. These observations, together with the discussion in section IV and the contents of Figure 10, explains a higher mass DM particle generating higher radio flux for scenarios where the $b\bar{b}$ annihilation channel dominates over $\tau^{+}\tau^{-}$.

VI Conclusion

The aim of this paper has been to show that radio signal arising from a high mass (trans-TeV) WIMP DM can be detectable as radio synchrotron flux from a dSph, to be recorded by the upcoming SKA Telescope. We have analysed not only the particle physics aspects of DM annihilation and subsequent cascades leading to $e^{\pm}$ pairs, but have also included the relevant astrophysical processes the electrons/positrons pass through before emitting radio waves, upon acceleration by the galactic magnetic field. We have set out to identify the mechanism whereby a trans-TeV DM candidate can thus be visible in radio search. We found that in the SKA frequency range, enhancement of the radio flux for this case is possible mainly due to the following reasons:

• •

Larger cross section or annihilation rate (required to maintain relic density under the observed limit for a trans-TeV DM) facilitated by the dynamics of the particle physics model. This helps in compensating the $\frac{1}{m_{\chi}^{2}}$ suppression due to large DM mass.

• •

Presence of energetic $e^{\pm}$ in the DM annihilation spectrum in greater abundance. This partially reduces the $\frac{1}{m_{\chi}^{2}}$ suppression effect and on the other hand enhances, through the energy loss term, the electron/positron density at low energies which helps to produce large radio flux.

• •

Dominance of the annihilation channel $b\bar{b}$ which yields a comparatively larger abundance of $e^{\pm}$ in all of the energy range of the spectrum produced by DM annihilation.

Simultaneously, effect of various astrophysical parameters (e.g. $D_{0}$, $b(E)$, $B$) on the radio synchrotron flux produced from the annihilation of a trans-TeV DM particle has been studied in detail.

Using SKA sensitivity we have drawn the limits in the $\langle\sigma v\rangle-m_{\chi}$ plane to observe radio flux from Draco with 100 hours of observation. We found that, these limits are much more stronger than the previously obtained bounds on $\langle\sigma v\rangle$ from Fermi-LAT $\gamma$-ray Ackermann et al. (2015) or AMS-02 cosmic-ray antiproton observation Cuoco et al. (2018). Even for a conservative choice of astrophysical parameters ($D_{0}=3\times 10^{28}\mbox{cm}^{2}\mbox{s}^{-1}$ and $B=1$ $\mu G$), we found that these limits can go as low as $\langle\sigma v\rangle\sim 3\times 10^{-29}\mbox{cm}^{3}\mbox{s}^{-1}$ for a $m_{\chi}\sim 1$ $TeV$. This indicates towards a large region of WIMP parameter space which can be probed through the upcoming SKA. Along with these we have also shown the limits in the $B-D_{0}$ plane. We found that, for a DM mass in the trans-TeV range, magnetic field as low as $B\sim 10^{-3}$ $\mu G$ and diffusion coefficient as high as $D_{0}\sim 10^{30}\mbox{cm}^{2}\mbox{s}^{-1}$ are well enough to produce radio flux above SKA sensitivity in 100 hours of observation time.

Taking the minimal supersymmetric standard model (MSSM) as an illustration, we have shown that benchmark points with lightest neutralino masses ($m_{\chi^{0}_{1}}$) $\sim$ 1 - 8 $TeV$, which satisfy all the constraints from observed relic density, direct DM searches and collider searches, can lead to detectable signals at the SKA with 100 hours observation, even with conservative choice of $B$ and $D_{0}$. We have illustrated how the effects mentioned earlier can lead to a larger radio flux for a high mass DM benchmark point compared to a low mass case, thus establishing the credibility of the search for heavy DM through radio observation in SKA666After submitting this paper, we came to know of reference Cembranos et al. (2019), where a similar study has been carried out for some different scenarios. While the broad conclusions have some similarity, our conclusions are a little more optimistic, since we have predicted detectability with 100 hours of observation, with realistic benchmarks. One of the reasons that have led to the more positive predictions is the inclusion of the $b\bar{b}$ annihilation channel in our study..

**Acknowledgements

**

The work of AK and BM was supported by the funding available from the Department of Atomic Energy, Government of India, for the Regional Centre for Accelerator-based Particle Physics (RECAPP), Harish-Chandra Research Institute. TRC and SM acknowledge the hospitality of RECAPP during the formulation of the project.