SKA-Phase 1 sensitivity for synchrotron radio emission from multi-TeV Dark Matter candidates

TL;DR

This paper evaluates SKA's potential to detect synchrotron radio signals from multi-TeV dark matter annihilation in Draco, providing new constraints on various dark matter models and highlighting SKA's promising sensitivity.

Contribution

It presents the first detailed analysis of SKA-Phase 1 sensitivity to synchrotron emission from TeV dark matter candidates in Draco, including model-independent and branon scenarios.

Findings

SKA1-MID can exclude 10 TeV thermal DM annihilating into W+W- and b-bbar channels with 1000 hours.

Including a DM spike with an intermediate-mass black hole enhances detection prospects.

SKA can set competitive constraints on branon parameters consistent with AMS-02 data.

Abstract

In the era of radio astronomy, the high sensitivity of the Square Kilometre Array (SKA) could play a decisive role in the detection of new radio sources. In this work, we study the SKA sensitivity to the synchrotron radio emission expected by the annihilation of TeV DM candidate in the Draco dwarf spheroidal galaxy. On the one hand, we consider model-independent DM candidates: we find out that with 1000 hours of data-taking, SKA1-MID will be able to exclude up to 10 TeV thermal DM candidates that annihilate in $W^+W^-$ and $b\bar b$ channels. We also study as these constraints improve by including a density enhancement due to a DM-spike associated with an intermediate-mass black hole in Draco. On the other hand, we consider extra-dimensional brane-world DM candidates, dubbed branons. In this specific scenario, SKA allows us to set constraints on the branon parameter space ($f$,$M$), where $f$ is related to the coupling of the branon to the Standard Model particles and $M$ is the mass of the branon itself. In particular, we consider two different branon DM candidates. We find out that SKA will be able to set more stringent constraints on the branon DM candidate required in order to fit the AMS-02 data, yet the sensitivity of the instrument should be improved in order to study the branon candidate for the Galactic Centre. Nonetheless, we show that SKA represents - among other detectors - the most promising instrument for multi-wavelength detection of synchrotron radio emission by annihilating multi-TeV DM.

Equations

$$-\nabla \cdot \left[D\left(\text{r},E\right)\nabla \psi \right]-\frac{\partial }{\partial E}\left[b(\text{r},E)\psi \right]=Q_e(\text{r},E),$$

$$Q_e(\text{r},E)=\frac{1}{2}⟨\sigma v⟩{\left(\frac{{\rho }_{\text{DM}}(\text{r})}{M}\right)}^2\sum _j{\beta }_j\frac{\mathrm{d}{N}_e^j}{\mathrm{d}E},$$

$$\psi (\text{r},E)=\frac{1}{b(\text{r},E)}{\int }_E^M\mathrm{d}{E}_{s}G\left(r,E,E_s\right)Q_e(\text{r},E).$$

$$G\left(r,E,E_s\right)=\frac{1}{\sqrt{\pi {\lambda }_D^2(E,E_s)}}\sum _{n=-\infty }^{\infty }{\left(-1\right)}^n{\int }_0^{r_h}\mathrm{d}{r}^{\prime }\frac{r^{\prime }}{r_n}$$

$$\times {\left(\frac{{\rho }_{\text{DM}}(r^{\prime })}{{\rho }_{\text{DM}}(r)}\right)}^2\left[\exp (-g_n^{-})-\exp (-g_n^{+})\right],$$

$${\lambda }_D^2(E,E_s)=4{\int }_E^{E_s}\mathrm{d}\epsilon \frac{D(\epsilon )}{b(\mathbf{r},\epsilon )},$$

$$b(\text{r},E)=b_{\mathrm{brem}}(E)+b_{\mathrm{Coul}}(E)+b_{\mathrm{ICS}}(E)+b_{\mathrm{syn}}(\text{r},E).$$

$$P_{\text{syn}}(\nu ,\text{r},E)$$

$$F_i(s)$$

$${\nu }_c(\text{r},E)=\frac{3eB(\text{r})}{4\pi m_e}{\gamma }^2(E),$$

$$B(r)=B_0\exp (-r/r_c),$$

$$j(\nu ,\text{r})$$

$$I(\nu ,\theta )={\int }_{l.o.s.}\mathrm{d}l\frac{j(\nu ,l,\theta )}{4\pi },$$

$$S(\nu )={\int }_{\Omega }\mathrm{d}\Omega I(\nu ,\theta ),$$

$$S_{\text{min}}=\frac{2k_b{T}_{\text{sys}}}{{\eta }_s{A}_e({\eta }_{\text{pol}}\tau \Delta \nu {)}^{1/2}},$$

$$T_{\text{sys}}=T_{\text{sky}}+T_{\text{rcvr}},$$

$${\rho }_{\text{DM}+\text{BH}}(r)=\{\begin{array}{ll}\frac{M}{⟨\sigma v⟩(t-t_f)}& \text{if r<rcut}\\ {\rho }_{\text{DM}}(r_{sp}){\left(\frac{r}{r_{sp}}\right)}^{-{\gamma }_{sp}}& \text{if rcut≤r<rsp}\\ {\rho }_{\text{DM}}(r)& \text{if r≥rsp},\end{array}$$

$$\mathrm{d}{s}^2={\tilde{g}}_{\mu \nu }\left(x\right)W\left(y\right)\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }-g_{mn}^{\prime }\left(y\right)\mathrm{d}{y}^m\mathrm{d}{y}^n,$$

$${\mathcal{L}}_{Br}$$

Full text

I Introduction

Dark Matter (DM) is a fundamental ingredient to explain several observations in our Universe at both astrophysical and cosmological scales Clowe et al. (2004, 2006); Rubin and Ford Jr (1970); Rubin et al. (1980); Massey et al. (2010); Anderson et al. (2014), yet its nature still remains elusive Zwicky (1933); Ade et al. (2014, 2016); Bergstrom (2000); Bertone et al. (2005). Indirect DM searches conform to different detection strategies to disentangle the main features of DM, e.g. multimessenger and multi-wavelength approaches. In the first approach, DM particles may annihilate in galactic halos and produce cosmic rays that can be detected by observatories of a different kind. Nonetheless, the interaction of charged cosmic rays with the environment (i.e. galactic magnetic field) and their energy loss would generate photons in a large range of frequencies. The study of these signals sets the basis of DM indirect searches through multiwavelength astronomy; depending on the energy loss mechanism distinctive signals could be detected at different frequencies, either via synchrotron emission, Inverse Compton Scattering (ICS), bremsstrahlung or Coulombian interactions. In particular, synchrotron emission from secondary electrons and positrons produced by annihilations of Weakly Interacting Massive Particles (WIMPs) could be detected at radio frequencies with the Square Kilometre Array (SKA) Dewdney et al. (2013). The SKA-Phase $1$ (SKA1) is claimed to be one of the most promising instruments in radio astronomy. Its high sensitivity and resolution could be key to solve open questions in both Astrophysics, Cosmology and Particle Physics, including Physics beyond the Standard Model (SM) (c. f. Dewdney et al. (2013) for further details).

Radio emission from GeV DM candidates has been already investigated in numerous studies Spekkens et al. (2013); Natarajan et al. (2015); Colafrancesco et al. (2007); McDaniel et al. (2017); Colafrancesco et al. (2006). In this work, we will focus on radio emission expected by the annihilation of TeV DM particles. In particular, we study the SKA sensitivity to prospect radio signals that could be produced by synchrotron emission due to secondary fluxes of $e^{+}/e^{-}$ Feng and Zhang (2018); Cheng et al. (2017); Carquin et al. (2016); Delahaye et al. (2008); Feng et al. (2014); Ibarra et al. (2014) after the annihilation of TeV DM particles. Among other DM candidates, it has been argued that SKA1 would be able to detect radio synchrotron emission from minimal-supersymmetric (MSSM) DM candidates up to masses of tens TeV Kar et al. (2019a). Nonetheless, only a limited number of theories could naturally produce DM particles with masses heavier than tens TeV (c. f. dark atoms Belotsky et al. (2014, 2017), minimal DM models Cirelli and Strumia (2009); Garcia-Cely et al. (2015) or fermionic DM Chua and Hsieh (2013)). In this regard, brane-world theories may also naturally produce thermal DM candidates up to masses of 100 TeV Cembranos et al. (2003a).

In this work, we will focus on the Draco dwarf spheroidal (dSph) galaxy, i.e. a benchmark target for DM searches. Due to the low percentage of baryonic matter with respect to the DM abundance, also the radio background component - due to the interaction of cosmic rays of astrophysical origin with the environment - is expected to be subdominant in dSph. In fact, dSph are faint radio sources requiring detectors with high sensitivity. For instance, no detection limits of several dSphs with the Australia Telescope Compact Array (ATCA) Regis et al. (2015) and the Green Bank Telescope (GBT) Spekkens et al. (2013); Natarajan et al. (2015), have already achieved sensitivities as low as the order of mJy and the SKA would be able to reach $\mu$Jy orders of magnitude Dewdney et al. (2013).

The manuscript is organised as follows: in Section II we introduce the physics of diffusion of charged particles in the galaxy and the generalities of DM indirect detection by synchrotron emission. In Section III we briefly characterise the most relevant SKA1 technicalities. In Section IV we discuss the SKA1 sensitivity to synchrotron radio emission from TeV DM candidates. Firstly, we adopt a model-independent approach and we consider prospective boosts in the DM distribution due to a Black Hole (BH)-related DM spike. Secondly, we will consider brane-world theories and branons as prospective TeV WIMPs. In Section V we will briefly discuss the possibility to detect multiwavelength emission from TeV DM candidates with current and future generation of telescopes. Finally, Section VI is dedicated to draw the main conclusions of this study.

II Synchrotron emission

During the transport of cosmic rays in the galactic environment, the deflection of charged particles by the galactic magnetic field would result in the emission of electromagnetic radiation. Contingent upon the velocity regime of those particles, two distinct emissions could be detected. On the one hand, in the case of non-relativistic particles, emission lines are associated with the cyclotron frequency with a two-lobe distribution around the direction of acceleration. In this regard, only some environments with high magnetic fields, such as neutron stars, would render this kind of radiation detectable. On the other hand, in the case of ultra-relativistic particles, the emission is produced through synchrotron radiation in a continuous frequency range. Indeed, ultra-relativistic $e^{+}/e^{-}$ are responsible for a large number of signatures in the sky, being the synchrotron emission one of the main mechanisms of energy losses. Such an $e^{+}/e^{-}$ propagation is dominated by the diffusion equation

[TABLE]

where $\psi\left(\textbf{r},E\right)$ is the number density of $e^{+}/e^{-}$ in equilibrium per unit of energy at the point r of the galaxy, i.e., $\psi$ represents the spectrum of products after the propagation. $b(\textbf{r},E)$ represents the energy loss term and $D\left(\textbf{r},E\right)$ holds for the diffusion coefficient, which becomes $D\left(\textbf{r},E\right)\sim D\left(E\right)=D_{0}E^{\delta}$. In the following, we shall use $\delta=1/3$ according to the Kolmogorov description. It is important to highlight that the diffusion Eq. (1) portrays a simplification of the Ginzburg-Syrovatsky transport equation. The latter takes into consideration some other mechanisms such as re-acceleration of cosmic rays (negligible in the case of ultra-relativistic $e^{+}/e^{-}$), spallation of cosmic rays, radioactive decay of nuclei of the ISM as well as eventual interactions with the galactic wind Strong et al. (2007); Yuan et al. (2017). However, all these mechanisms turn out to be negligible in dSph galaxies111The mass-to-light ratio in the case of Draco is $M/L\sim 300M_{\odot}/L_{\odot}$ Lokas et al. (2005)..

Ultra-relativistic $e^{+}/e^{-}$ are expected to be produced as a consequence of the annihilation of DM particles. Thus, the source term $Q_{e}(\textbf{r},E)$ in Eq. (1) becomes:

[TABLE]

which describes how DM particles with a thermally averaged cross section $\langle\sigma v\rangle$ annihilate and inject $e^{+}/e^{-}$ to the environment with a characteristic injection spectrum $\frac{{\rm d}N_{e}^{j}}{{\rm d}E}$ that depends on the $j$ annihilation channel. Once DM particles annihilate with a probability $\beta_{j}$ into a specific SM channel $j$, these SM particles could either decay or hadronise into cosmic rays (in our case $e^{+}/e^{-}$) that are injected into the galactic medium. In this work, the injection spectra $\frac{{\rm d}N_{e}^{j}}{{\rm d}E}$ have been computed using the functions provided by the software ${\rm PPC4DMID}$ Cirelli et al. (2011), including the electroweak effect that becomes relevant for multi-TeV energy events Ciafaloni et al. (2011). Such spectra establish, together with $\langle\sigma v\rangle$, $\beta_{j}$ and the mass of DM, $M$, the Particle Physics description for the DM model under study. In the source term (2), the DM density profile $\rho_{\text{DM}}(\textbf{r})$ takes into account the spatial distribution of DM in a particular target. In the following, we shall focus on the isolated emission from the Draco dSph galaxy. For Draco dSph description, we adopt a Navarro-Frenk-White (NFW) DM density profile Colafrancesco et al. (2007)222In this work we considered a NFW profile, $\rho_{{\rm NFW}}(r)=\frac{\rho_{s}}{\frac{r}{r_{s}}\left(1+\frac{r}{r_{s}}\right)^{2}}$, with $\rho_{s}=1.40$ $\text{GeV}/\text{cm}^{3}$ and $r_{s}=1$ kpc., although more sophisticated halo descriptions, such as the Kazantzidis profile Kazantzidis et al. (2004), have also been applied to this specific galaxy Sanchez-Conde et al. (2007); Sánchez-Conde et al. (2011).

After a suitable change of variables, the diffusion equation, Eq. (1), can be rewritten as a heat-like equation Colafrancesco et al. (2006), and thus, an analytical solution can be obtained in terms of a Green’s function $G\left(r,E,E_{s}\right)$, i.e.

[TABLE]

The spherical symmetry of the problem and its boundary conditions suggest the use of the image charge method applied to the Green’s function, as explained in Baltz and Edsjo (1998); Colafrancesco et al. (2006). Such a method considers charges positioned at $r_{n}=(-1)^{n}r+2nr_{h}$, where the radius of diffusion $r_{h}$ is around $\sim 2.5$ kpc for dSphs similar to Draco Colafrancesco et al. (2007). Thus $G\left(r,E,E_{s}\right)$ becomes

[TABLE]

with $g_{n}^{\pm}(r^{\prime},E,E_{s})=\frac{(r^{\prime}\pm r_{n}r)^{2}}{\lambda_{D}^{2}(E,E_{s})}$. In the above expression, $\lambda_{D}(E,E_{s})$ represents the mean free path of $e^{+}/e^{-}$ from the coordinates of injection $(\textbf{r}_{s},t_{s})$ with kinetic energy $E_{s}$ to a coordinate (r, $t$) with kinetic energy $E$ and satisfies

[TABLE]

showing that the mean free path depends on both the diffusion $D(E)$ of cosmic rays in the medium, and the loss of energy mechanism encapsulated in $b({\rm{\bf r}},E)$. The latter quantity comprises electromagnetic interactions affecting the $e^{+}/e^{-}$ propagation via bremsstrahlung, Coulombian, ICS and synchrotron emission and is defined as Sarazin (1999); Rybicki and Lightman (2008); Delahaye et al. (2010)

[TABLE]

Depending on the astrophysical environment, some of the above mechanisms prevail over others at different energies as illustrated in Figure 1, where energy losses due to bremsstrahlung, Coulombian, ICS and synchrotron emission have been calculated333$b_{{\rm brem}}(E)=1.51\cdot 10^{-16}n_{e}E\left[\log(E/m_{e})+0.36\right],\\ b_{{\rm syn}}(\textbf{r},E)=0.0254\cdot 10^{-4}B^{2}(\textbf{r})E^{2},\\ b_{{\rm Coul}}(E)=6.13\cdot 10^{-16}n_{e}\left[1+\log(E/n_{e}m_{e})/75\right],\\ b_{{\rm ICS}}(E)=0.25\cdot 10^{-16}E^{2}\,\,\text{if}\,\,\frac{\gamma k_{b}T_{0}}{m_{e}c^{2}}<3.8\cdot 10^{-4},\\ b_{{\rm ICS}}(E)=\frac{E^{2}m_{e}c^{2}\left(k_{b}T_{0}\right)^{3}}{\gamma}\exp\left[\sum^{5}_{i=0}c_{i}\left(\ln\frac{\gamma k_{b}T_{0}}{m_{e}c^{2}}\right)^{i}\right]\,\,\text{if}\,\,3.8\cdot 10^{-4}<\frac{\gamma k_{b}T_{0}}{m_{e}c^{2}}<1.8\cdot 10^{3},\\ b_{{\rm ICS}}(E)=\frac{\sigma_{T}}{16}\frac{\left(m_{e}k_{b}T_{0}\right)^{2}}{\hbar^{3}}\left(\ln\frac{4\gamma k_{b}T_{0}}{m_{e}c^{2}}-1.9805\right)\,\,\text{if}\,\,1.8\cdot 10^{3}<\frac{\gamma k_{b}T_{0}}{m_{e}c^{2}},$

with $c_{i}$ taken from Delahaye et al. (2010). The constants $\sigma_{T},k_{b}$ and $T_{0}$ are the Thomson cross section, the Boltzmann constant and the temperature of the gas of photons respectively.. There, colour bands allow us to visualise the dependence of the energy losses both with the magnetic field (between $1$ $\mu$G and $10$ $\mu$G) and the electron density of the medium, that we assume $n_{e}=10^{-6}\text{cm}^{-3}$ Colafrancesco et al. (2007) for Draco. Figure 1 also shows that synchrotron and ICS are predominant at TeV - and higher - scales, whereas bremsstrahlung and Coulombian losses dominate the sub-GeV range of energies. This thorough analysis of predominances in Eq. (6) turns out to be useful to accelerate the numerical determination of $\psi$ in Eq. (3). Indeed, since the exponentials in Eq. (4) mix spatial coordinates with energy, the computation of integrals in Eq. (4) might become difficult. Following the usual approach in Colafrancesco et al. (2007); McDaniel et al. (2017), in order to compute the Green’s function in Eq. (4) we shall approximate $b_{{\rm syn}}(\textbf{r},E)\sim b_{{\rm syn}}(E)=2.54\cdot 10^{-6}B_{\text{avg}}^{2}E^{2}$, with an averaged magnetic field $B_{\text{avg}}=0.5\,\mu\text{G}$. This approximation is solely considered in the determination of Eq. (5), being the spatial dependence in $b_{\rm syn}(\textbf{r},E)$ fully retaken whenever Eqs. (1) and (3) are considered. Indeed, as a result of the interaction of $e^{+}/e^{-}$ with the magnetic field, the predominant emission of secondary photons at radio frequencies is expected to happen through synchrotron emission. In more detail, each $e^{+}/e^{-}$ produces a radiative emission. The emitted power of a single $e^{+}/e^{-}$ at redshift $z\approx 0$ is

[TABLE]

where

[TABLE]

being $K_{\frac{5}{3}}(\xi)$ a modified Bessel function of the second kind, $\epsilon_{0}$ the vacuum permittivity and $\alpha$ the angle formed by the perpendicular component of the magnetic field with respect to the $e^{+}/e^{-}$ momentum. Moreover, $\nu_{c}(\textbf{r},E)$ is the critical frequency

[TABLE]

around which $e^{+}/e^{-}$ emit most of their energy. Also, the magnetic field $B(\textbf{r})$ considered throughout this work is spherically symmetric yielding

[TABLE]

where $B_{0}$ is the magnetic field strength and $r_{c}=0.22$ kpc is the core radius of Draco.

The electron/positron number density $\psi_{e^{-}/e^{+}}(\textbf{r},E)$ produced by DM annihilations and responsible for emitting such synchrotron radiation, is obtained from Eq. (3). Now, we can define the total emissivity Colafrancesco et al. (2007); McDaniel et al. (2017); Natarajan et al. (2015)

[TABLE]

that takes into account the contribution of all $e^{+}/e^{-}$ at different energies for a given frequency of emission.

Thus, the specific intensity $I_{\nu,\theta}$ can be determined just by integrating $j(\nu,\textbf{r})$ along the line of sight (l.o.s.), namely

[TABLE]

where the limits of integration are $l_{\text{max}/\text{min}}=d\cos\theta\pm\sqrt{r_{h}^{2}-d^{2}\sin^{2}\theta}$. Furthermore, the flux density over the solid angle $\Omega$ is given by

[TABLE]

where $\Omega=2\pi(1-\cos\theta)$. In Figure 2 we show how the radio emission strongly depends on the magnetic field. In the case of Draco, the magnetic field lies around $\sim 1$ $\mu$G Chyży et al. (2011). The flux density $S_{\nu}$ can be computed by integrating over the selected beam size (for more details on the SKA telescope and detection strategies, see Sec. III). In fact, in radio astronomy the wavelengths (from mm to a thousand km) are usually comparable to the size of both antennas and baselines, allowing us to adopt interferometric strategies. Different antenna configurations, and hence different baselines, are instrumental to reconstruct the desired image of the sky. Radio telescopes can map the specific intensity $I_{\nu}(\theta)$ by resorting the so-called van Cittert-Zernike theorem Thompson et al. (2017). Such a result relates to the cross-correlation map of detected signals between antenna pairs in the projected baseline plane, and the specific intensity $I_{\nu}(\theta)$ in the sky coordinates through the aperture synthesis technique W. (2015a, b). Images with high resolution are also crucial in the $S_{\nu}$ determination in order to neither overestimate nor underestimate the signal and to not loose information of the source when integrating along the beam coordinates. However, high resolutions would imply a decrease of the sensitivity, crucial to detect faint radio sources as dSphs.

In the upcoming sections we shall consider the following two solid angle scales:

$\bullet\,\,\,\,\,$The solid angle subtended by Draco dSph, dubbed $\Omega_{\text{DRACO}}$, ensuring that all the emission from Draco occurs within this solid angle. In our study we took $\theta_{\text{DRACO}}=1.2$ deg.

$\bullet\,\,\,\,\,$ The largest angular scale of the radio interferometer $\Omega_{\text{SKA}}(\lambda)$ that is determined by the minimal baseline through the expression $\theta_{\text{max}}(\lambda)=58.61\lambda/D_{\text{min}}$ (in deg) in which we take $D_{\text{min}}=30$ m for SKA1. Indeed, $\Omega_{\text{SKA}}(\lambda)$ would represent more realistic scenarios than $\Omega_{\text{DRACO}}$, since the former includes the SKA limitations at different frequencies given by $\theta_{\text{max}}(\lambda)$. In this regard, an illustrative value for $\theta_{\text{max}}$ at 150 MHz lies on 3.90 deg, while for $\nu=14$ GHz $\theta_{\text{max}}$ becomes 0.042 deg. This fact indicates that the response of SKA with the frequency would affect the received flux density $S_{\nu}$ according to the integration limits for $\Omega_{\text{SKA}}(\lambda)$ in Eq. (13).

III SKA telescope

In this Section, we will provide some technicalities on the SKA telescope and detection strategies.

In particular, the SKA1-LOW consists of a low-frequency aperture array located in Australia, while the SKA1-MID is a mid-frequency array of reflector antennas, placed in South Africa. The SKA1-LOW is conformed by approximately $131,000$ log-periodic dual polarised antennas, some of them forming a compact core of $1$ km and the rest grouped in about $512$ stations of $10$s meters of diameter each distributed over $40$-km. Its range of frequencies, characterised by the lengths of the shortest and longest dipole antenna elements, lies between $50$ MHz and $350$ MHz. Once antennas collect the radio waves, a time delay is introduced with the purpose of reconstructing the signal in a particular direction of detection constituting a signal beam and then, the signals between pairs of antennas are cross-correlated through the correlator. In addition, by combining different sets of timing delays it is possible to construct independent signal beams covering a large FoV and increasing the survey speed. Once the signal is processed, a visibility map can be set and hence the information of the specific intensity of the source obtained. In the case of the SKA1-MID the frequencies of detection are divided into five bands ranging from $350$ MHz to $\sim 13.8$ GHz, though in principle the dishes are prepared to measure frequencies until $20$ GHz. In addition, SKA1-MID includes $64$ $13.5$-m diameter dishes from the MeerKAT (SKA1 precursor) and $190$ $15$-m built specifically for the SKA1. Its antennas will be distributed in a compact core with a diameter of  1 km, a further 2-dimensional array of randomly placed dishes out to $\sim 3$ km radius and three spiral arms until a radius of  80 km. After measure, signals need to be cross-correlated with each other and processed for further analysis.

Both high resolution and sensitivity are critical in the detection of radio faint sources, being SKA1 able to attain both goals. Concerning the former, the spatial configuration of the SKA1 allows us to develop interferometry strategies to reach a high angular resolution. Even though radio astronomy searches present more limitations in this matter than searches performed in other frequencies, the SKA1 long baselines ensure one of the highest angular resolutions given by the Rayleigh criterion, $\theta_{\text{res}}(\lambda)=58.61\lambda/D_{\text{max}}$ (in deg) where $\lambda$ is the wavelength of the incoming radiation and $D_{\text{max}}$ is the maximum baseline, for a specific array configuration and Point Spread Function (PSF). At $350$ MHz, in the limit between SKA1-LOW and SKA1-MID, the angular resolution $\theta_{\text{res}}$ $\sim 0.001$ deg for the SKA1-LOW, taking into account $D_{\text{max}}=50$ km and $\theta_{\text{res}}$ $\sim 2.5\cdot 10^{-4}$ deg for the SKA1-MID with a baseline of $D_{\text{max}}=200$ km. Even though larger baselines mean higher resolutions, the sensitivity could be compromised unless both the spatial configuration and the effective area $A_{e}$ of the interferometer are optimised with respect to the frequency of observation. In fact, given a bandwidth $\Delta\nu$ in an interval of time $\tau$, one can determine the sensitivity $S_{\text{min}}$, i.e. - roughly speaking - the minimal detectable flux of a signal, via the so-called radiometer equation:

[TABLE]

where $k_{b}$ is the Boltzmann constant, $\eta_{\text{pol}}$ is the number of polarisation states, and $\eta_{s}=0.9$ the efficiency of the system. In principle, as mentioned in Dewdney et al. (2013), SKA detection would be able to analyse integration times of order $1000$ hours and hence, this would be the longest integration time $\tau$ we use to determine the minimal detectable flux along a bandwidth $\Delta\nu$ of $300$ MHz.

Concerning the system noise temperature,

[TABLE]

it includes the contribution of the sky noise $T_{\text{sky}}$, i.e., all the emission from the sky which does not correspond to the target we want to measure - in the case of a specific observation, or by using the sky noise defined by a uniform model $T_{\text{sky}}=60\lambda^{2.55}$ assumed in the baseline design document Dewdney et al. (2013), and the receiver noise, also dubbed instrument noise, $T_{\text{rcvr}}$. The latter only dominates for SKA1 in frequencies smaller than $250$ MHz and it is taken as a $10\%$ of $T_{\text{sky}}$ plus a constant temperature of $40$ K. Finally, the effective area $A_{e}$ depends on the frequency and the gain of the receptor. In the case of SKA1-LOW, two different behaviours have to be considered for frequencies lower and higher than $110$ MHz, depending on the sparse-dense transition of the detector. The sparse transition is produced when antenna elements are more distant than around one wavelength and they act independently making the $A_{e}$ proportional to $\lambda^{2}$. In the dense transition (separation smaller than about $0.5$ times the wavelength) the interaction between the antenna elements force $A_{e}$ to be constant. Once again, we refer to Dewdney et al. (2013) for further details.

IV Sensitivity Analysis

In this Section, we study the sensitivity of SKA1 to the synchrotron radio emission expected by thermal TeV DM annihilation events in the Draco dSph. More specifically, we compute the specific intensity $I_{\nu}(\theta)$ and the flux density $S_{\nu}$ for the Draco dSph galaxy. In order to plot the sensitivity curves for 10, 100 and 1000 hours we tabulate the System Equivalent Flux Density SEFD=$2k_{b}T_{\text{sys}}/A_{e}$ from Braun et al. (2017) so Eq. (14) can be used to determine the minimal detectable flux, $S_{\text{min}}$.

IV.1 Model-independent thermal TeV DM

Herein we consider model-independent TeV DM candidates annihilating $100\%$ in a single SM channel, i.e., $\beta_{j}=1$ in Eq. (2), with the benchmark thermally averaged annihilation cross section $\langle\sigma v\rangle=3\cdot 10^{-26}\text{cm}^{3}/\text{s}$.

First of all, let us shed some light on key aspects of specific intensity $I_{\nu}(\theta)$. In Figure 3, we plot the specific intensity $I_{\nu}(\theta)$ against the angle $\theta$. As mentioned in Section II, the diffusion model is spherically symmetric. In both panels, we present the specific intensity for 1 TeV DM particles annihilating into the $b\overline{b}$ channel. The left panel enables a straightforward comparison of the diffused emission at different frequencies that either SKA1-LOW (150 and 350 MHz) or SKA1-MID (350, 3050 and 14000 MHz) will cover. For high frequencies the angular extension of the emission is confined in a small region while the signal at lower frequencies reaches larger angles. Considering this fact, even though the highest intensity lies inside the core of Draco, estimated as $\sim 0.067$ deg Fomalont and Geldzahler (1979), a large proportion of the signal extends further in radius. Nonetheless, the signal follows the expected peak at the centre given by the DM density profile smoothed by the diffusion. In the right panel we also show the specific intensity at $350$ MHz (which establishes the limit between SKA1-LOW and SKA1-MID) with and without spatial diffusion. Therein, it is possible to observe that neglecting diffusion results in a signal concentrated at the centre of Draco, since in this case the $e^{+}/e^{-}$ number density follows the DM density profile strictly. On the other hand, it is also worth noting that in either case (with and without diffusion) the profile of the radio signal would have a spatial dependence if we consider a spatially-dependent magnetic field. Here, this effect is assumed to be negligible due to the small radii considered in this work. The right panel of Figure 3 shows the angular resolution of SKA1 with respect to the extension of the source. Once the diffusion is considered, the radio emission extends further in distances, decreases slightly with the angle $\theta$. In fact, we have seen - in the left panel of Figure 3 - that whenever diffusion is considered, most of the signal does not come from the centre and hence the source cannot be thought of as being point-like. In this regard, we conclude that both SKA1-LOW and SKA1-MID can resolve part of the structure of the emission for Draco assuming an angular resolution $\theta_{\text{res}}$ $\sim 0.001$ deg for SKA1-LOW and $\sim 2.5\cdot 10^{-4}$ deg for SKA1-MID at $350$ MHz.

The specific intensity $I_{\nu}(\theta)$ and the high resolution angle typical of radio telescope, gives information about the spatial distribution of the radio emission detected in the target - if any. In the case of no detection of any signal, we are more interested in the sensitivity of the instrument and the flux density $S_{\nu}$: by integrating $I_{\nu}(\theta)$ over the solid angle $\Omega$ according to Eq. (13), we obtain the radio flux density $S_{\nu}$ expected by TeV DM candidates annihilating into a specific channel and we can set constraints on the DM mass and annihilation cross section.

First, we check the agreement of our results with previous studies and we plot in the upper-left panel of Figure 4 the densities $S_{\nu}$ for two sub-TeV DM masses annihilating into $b\overline{b}$ channel, using the benchmark thermally averaged cross section value $\langle\sigma v\rangle=3\cdot 10^{-26}\text{cm}^{3}/\text{s}$. Dashed lines correspond to the flux density integrated over the solid angle subtended by Draco, $\Omega_{\text{DRACO}}$, and solid lines represent the emission integrated over the SKA angle, $\Omega_{\text{SKA}}(\lambda)$, defined in Section II. In more detail, given a certain baseline $D$, the angle of the radio detector increases with the wavelength, in other words, the integration beam is smaller for higher frequencies. Due to the strategy of this analysis, the loss of the signal at high frequencies is clear in the four panels of Figure 4. At this stage, let us emphasise that the flux density shows a steepness at lower frequencies, implying that SKA1-LOW (blue region) will be more competitive features in order to detect DM candidates of lower masses. As the DM mass increases, (see e.g. the DM candidates with $M=1$ TeV and $M=10$ TeV in the upper-right panel for the $t\overline{t}$ channel) the slope is less pronounced and SKA1-MID (yellow region) will be more competitive in order to set constraints on the DM candidate, due to the slightly better sensitivity. Therein, dotted-dashed lines of different colours hold for different exposition times: in particular cyan (10h), magenta (100h) and blue (1000h). This change in the slope occurs because heavy DM mass enables $e^{+}/e^{-}$ to reach higher energies and thus the frequency of the peak, which is proportional to the kinetic energy $E$ according to Eq. (9), is shifted to higher frequencies. Finally, In the lower-left and lower-right panels in Figure 4, we show the SKA detectability to TeV-DM annihilating into $W^{+}W^{-}$ and $\tau^{+}\tau^{-}$ channels, respectively. TeV DM annihilating in the leptonic channel, i.e. $\tau^{+}\tau^{-}$ seems to be the most disfavored scenario for radio detection. In fact, according to the $S_{\text{min}}$ limit as given by Eq. (14), lower-right panel shows that masses of 10 TeV remain inaccessible even if the integration time $\tau$ is increased of one order of magnitude. Instead, DM particles annihilating in the $t\overline{t}$ and $W^{+}W^{-}$ channels, are more favored to be detected by SKA1 - e.g. radio signal from 1 TeV DM particles could be detected with 10 hours of data-taking. Nonetheless, for DM masses around 10 TeV, 1000 hours would be necessary.

In Figure 5 we show the constraints on the averaged annihilation cross section $\left\langle\sigma v\right\rangle$ vs the DM mass $M$ for model-independent DM candidates annihilating into $b\overline{b}$ (dotted-dashed line), $\tau^{+}\tau^{-}$ (dotted line) and $W^{+}W^{-}$ (full line) channels. As previously stressed, SKA1-MID (orange lines) will be slightly more competitive than SKA1-LOW to constrain DM candidates with masses heavier than 1 TeV.

IV.2 Black-hole effect in annihilating DM signals

Generally speaking, several aspects on both the astrophysical and particle physics side could generate some boost in the secondary fluxes produced by the annihilation of DM particles. In particular, sub-clumps in DM halos Beck and Colafrancesco (2016), DM-spike associated to a Black Hole (BH) Gonzalez-Morales et al. (2014) and Sommerfeld enhancement Sommerfeld (1931); Feng et al. (2010) have been widely considered in several studies devoted to DM indirect searches. In fact, DM sub-clumps in galactic satellites could enhance the flux of a factor $2-3$, in both NFW and Burkert DM density distribution profile Beck and Colafrancesco (2016). Nonetheless, under the assumption that subhalos and sub-subhalos are tidally stripped some authors estimate a smaller multiplicative factor, $\sim 1.3$ (if one assumes $M_{\text{vir}}=7\cdot 10^{7}M_{\odot}$ Beck and Colafrancesco (2016)) times the smooth halo contribution Moliné et al. (2017).

In this Section, we will show that the presence of a BH in Draco would result in a radio emission enhancement by several orders of magnitude. The possibility to have an intermediate-mass BH has been considered in previous work Colafrancesco et al. (2007); Gonzalez-Morales et al. (2014).

Authors in Wanders (2014) studied the case where a BH grows adiabatically and thus, the inner part of the DM profile $\rho(r)\sim r^{-\gamma}$ is adiabatically contracted into a final profile $\rho(r)\sim r^{-\gamma_{sp}}$, with $\gamma_{sp}>\gamma$. Consequently, the modified DM density profile $\rho_{\text{DM}+\text{BH}}(r)$ in the presence of a BH becomes

[TABLE]

where $\rho_{\text{DM}}(r)$ is the standard DM density profile (NFW for us) without a BH. As seen in Eq. (16), the BH effect turns into a spike in the DM profile. In the region $r<r_{cut}$, a plateau $\frac{M}{\langle\sigma v\rangle(t-t_{f})}$ exists in the DM density distribution due to a more efficient DM annihilation rate. In this expression, $M$ is the mass of DM particle and $(t-t_{f})$ holds the BH age, taken as $10^{10}$ yr for Draco. The transition between the aforementioned upper limit and the density in the region where the BH has no influence ($r\geq r_{sp}$) is gradually modulated by a power law with exponent $\gamma_{sp}=(9-2\gamma)/(4-\gamma)$. In the following we will consider a BH mass of $10^{3}M_{\odot}$ according to Reines et al. (2013); Moran et al. (2014); BH masses bigger than $10^{7}M_{\odot}$ are dynamically excluded for Draco Colafrancesco et al. (2007). A full discussion on how the presence of a BH in Draco would affect the multi-wavelength analysis can be found in Colafrancesco et al. (2007).

In Figure 6 left panel, we show the specific intensity for a model-independent DM candidate of $M=1$ TeV that annihilates into $b\bar{b}$ channel. The enhancement of the signal due to the presence of a BH of $10^{3}M_{\odot}$ in the centre of Draco appears clearly also with strong diffusion, although higher values of the diffusion coefficient would reduce the enhancement in the flux of several orders of magnitude.

For outer regions, the specific intensity for both cases (with and without BH) tends to the same limit. Regarding the flux density, in Figure 6 right panel, we study the emission spectra for a model-independent DM candidate with the maximum value of masses considered in this study - i.e. $M=100$ TeV - annihilating into $W^{+}W^{-}$ channel. Even considering 1000 hours, SKA1 would not be able to detect such a DM mass. However, for the case of a BH mass $10^{3}M_{\odot}$, would be detectable with only 10 hours of data-taking. We conclude that the presence of a central BH would enhance the synchrotron radio signal significantly without any qualitative differences in the emission spectra, as showed in the right panel of Figure 6.

IV.3 Branons and extra-dimensions

In this Section, we will consider an extra-dimensional model for TeV DM candidates. Roughly speaking, brane-world scenarios address the hierarchy problem in Physics by suggesting that our Universe is a $(3+1)$-dimensional brane, where the SM fields propagate. Such a brane is embedded in a higher extra-dimensional compact space - the bulk - that hosts the propagation of gravity. Thus, brane-world theories are defined through the manifold $\mathcal{M}_{D}$=$\mathcal{M}_{1+3}\times B$, being $\mathcal{M}_{1+3}$ the brane four-dimensional space-time embedded in a $D-$dimensional bulk, both of them characterised by their respective metric tensors, $\widetilde{g}_{\mu\nu}$ and $g^{\prime}_{mn}$.The global space-time of such a manifold can be represented by

[TABLE]

where $x^{\nu(\mu)}$ and $y^{m(n)}$, with $\nu,\mu=0,1,2,3$ and $m,n=4,5,6...,D-1$, denote the coordinates for the four-dimensional brane and the compact extra-dimensional space respectively. Here, $W(y)$ is the warp factor along the extra-dimensions. In the case where $W(y)\neq 1$, a curvature is produced that explicitly breaks the isometry of the brane - that is considered as gauge symmetry - and the brane fluctuation can be parametrised by a massive (M) pseudo-Goldstone field, $\pi^{\alpha}$, dubbed branon. In the range of low tension $(f\gtrsim M)$ the branon dynamic is given by the Nambu-Goto action added to the usual SM action, associated with the following Lagrangian Sundrum (1999); Bando et al. (1999); Dobado and Maroto (2001); Cembranos et al. (2001); Alcaraz et al. (2003); Cembranos et al. (2004a);

[TABLE]

As Eq. (18) shows, the coupling between branons and SM particles is highly suppressed by a factor $f^{-4}$. Also, the brane Lagrangian (18) conserves parity and terms with odd number of branons are not allowed, hence, branons are stable. Being branons weakly interacting, massive and stable, they are natural candidates for DM Cembranos et al. (2003a); Kugo and Yoshioka (2001); Cembranos et al. (2004b); Maroto (2004a, b). Moreover, branons have the possibility of annihilating into SM particles, whose probability of annihilation is expressed by the branching ratio and depends on the branon mass $M$ and the tension of the brane $f$ Cembranos and Maroto (2016).

The branons parameter space $(f,M)$ has been studied through different methods in previous works. For instance, collider detections by the ILC, LHC or CLIC have been taken into account Alcaraz et al. (2003); Cembranos et al. (2004a); Achard et al. (2004); Creminelli and Strumia (2001); Cembranos et al. (2006); Khachatryan et al. (2016). Astrophysical and cosmological bounds for brane-world theories were obtained in Cembranos et al. (2003a); Kugo and Yoshioka (2001); Cembranos et al. (2004b); Maroto (2004a, b). Other constraints are obtained in previous studies for gamma-ray detectors, e.g. EGRET, FERMI and MAGIC Cembranos et al. (2013, 2012) and by studying the positrons flux recently detected by AMS-$02$ Cembranos et al. (2019). Here, we focus on the annihilation of TeV branons producing $e^{+}/e^{-}$ as a final state. For this range of mass, $e^{+}/e^{-}$ are mainly produced via a combination of $W^{+}W^{-}$ and $ZZ$ gauge bosons and $t\overline{t}$ quark channel, due to the branon branching ratios Gammaldi et al. (2011). In the left panel of Figure 7 we show the synchrotron radio emission produced by the annihilation of thermal branons, i.e. with ${\langle\sigma v\rangle}=3\cdot 10^{-26}\text{cm}^{3}/\text{s}$. Without any boost, the SKA1 detection of thermal branons for 1000 hours of integration time becomes feasible for masses smaller than 6 TeV. Thus, only boosted radio signals could be detected for thermal branon with mass $M>6$ TeV (Right panel Figure 7, dashed black line).

Finally, we set constraints on the $\left(f,M\right)$ parameter space of extra-dimensional brane-world theories. In fact, when cosmological and astrophysical limits are imposed on brane-world models, i.e., the thermally averaged cross section is fixed, the tension $f$ becomes a function of the $M$ Cembranos et al. (2003b), so that the mass of the DM particle remains as the only free parameter of the model. In the right panel of Figure 7 we show the constraints on the $\left(f,M\right)$ parameter space that could be obtained with the SKA1 telescope. There, the dashed black line represents the curve $f(M)$ associated to the thermal relic with $\left\langle\sigma v\right\rangle_{th}=3\cdot 10^{-26}\text{ cm}^{3}/\text{s}$. The region in red would be detectable by SKA1 without any boost. This means that SKA1 could detect thermal branons with masses below 10 TeV and tensions $f<10$ TeV. Blue region in Figure 7 right panel represents the detectable parameter space considering a BH mass of $10^{3}M_{\odot}$. Concluding, we consider two branon candidates: 1) thermal branon ($f=27.5$ TeV, $M=50.6$ TeV) that well fits the gamma-ray cut-off observed by HESS at the Galactic Centre with $\langle\sigma v\rangle=1.14\cdot 10^{-26}$ ${\text{cm}^{3}/\text{s}}$ (yellow star in Figure 7, right panel) Cembranos et al. (2013); 2) a non-thermal branon ($f=4.99$ TeV, $M=38.1$ TeV) with averaged cross section of $\langle\sigma v\rangle=1.76\cdot 10^{-21}$ ${\text{cm}^{3}/\text{s}}$ fitting AMS-$02$ positron fraction excess Cembranos et al. (2019) (pink star in Figure 7, right panel). Although some works justified such high boost Bovy (2009), it is unlikely that the branon models with such parameters provide the correct relic density within the standard freeze-out mechanism.

V Multiwavelength TeV DM

As thoroughly discussed in the previous sections, although SKA1 represents one of the most promising telescopes for radio astronomy and thus for multi-wavelength DM searches, the peak of synchrotron radio emission from TeV DM does not lie on the radio frequencies. In Figure 8 we show how the higher the DM mass, the more the synchrotron emission peak is displaced to frequencies beyond radio, while the amplitude of the emission decreases with the mass. In such a figure, one can see how the peak of emission for DM masses of 1, 10, and 100 TeV lie on frequencies around $10^{12}$, $10^{14}$ and $10^{16}$ Hz respectively, quite away from radio frequencies and up to X-rays. However, in terms of sensitivity, SKA1 still remains as one of the most competitive detectors according to Figure 8. In this sense, prospective boosts due to a BH-induced DM-spike, could be important to detect synchrotron signatures beyond the radio frequencies with the current generation of experiments, such as ALMA, JWST and Chandra.

Concerning radio frequencies, we have mentioned in the introduction GBT, whose frequency band (67-115 GHz) is suitable to set constraints on TeV DM candidates with masses slightly heavier than what we can study with SKA, yet its sensitivity does not reach the detectability threshold for TeV DM candidates. Among the others, the FAST and VLA telescopes should be considered: with their present designs, both of them are suitable for detection of DM masses lighter than $10$ TeV. However, the incorporation of new technology in both detectors is expected to improve their sensitivities and allow us to eventually detect heavier DM Selina et al. (2018). Finally, at low radio frequencies, LOFAR targets a frequency band ranging from 10 to 50 MHz (Figure 8 in yellow) not achievable by other radio telescopes. Given the flux $\nu S_{\nu}$ dependence with the DM mass, we observe that LOFAR would be able to detect only tens GeV DM candidates.

VI Conclusions

In this work, we predict the prospective constraints we will be able to set on TeV DM candidates via the observation of the Draco dSph galaxy in radio frequencies with SKA.

In the framework of radio telescopes, the specific intensity provides the angular distribution of the radio emission along the size of the source, while the flux density, refers to the integrated flux over either the angular size of the source or the beam selected for the analysis, as discussed in the text. In this sense, we can set constraints on a TeV DM candidate by studying the sensitivity of SKA with respect to the flux density expected by the annihilation of TeV DM particle in the selected source.

As shown in Figures 1 and 2, the crucial role that magnetic fields play in this kind of analyses should not be dismissed. For instance, the flux density times frequency for a 100 GeV DM candidate annihilating into $b\overline{b}$ channel provides a value of $7.02\cdot 10^{-20}\,\,\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$ at $6\cdot 10^{9}\,\,\text{Hz}$ when a magnetic field of $1\,\,\mu\text{G}$ is considered while, for the same frequency, the value becomes $2.42\cdot 10^{-18}\,\,\text{erg}\,\text{cm}^{-2}\,\text{s}^{-1}$ if $B=5\,\,\mu\text{G}$. Thus an increment of just $4\,\,\mu\text{G}$ for the Draco magnetic field would enhance the radio signal around two orders of magnitude. In this regard, throughout our work, we used a conservative value of 1 $\mu$G for the magnetic field. We also show how the uncertainty related to the estimation of the diffusion coefficient affects both the intensity (Figure 2) and the angular extension (Figure 1) of the radio emission.

In Figure 4, we have compared the sensitivity of SKA with the synchrotron radio emission expected by the annihilation of model-independent thermal DM particles. We have concluded that SKA1-LOW optimizes the detection of sub-TeV DM, while the emission from heavier DM particles would be better detected by SKA1-MID. After $1000$ hours, SKA1 would make possible to detect DM candidates heavier than 10 TeV, provided DM annihilates mainly via either $t\overline{t}$ or $W^{+}W^{-}$ channels, while for DM annihilating via $\tau\overline{\tau}$ channel, only masses below 10 TeV would be detectable. In particular, in Figure 5 we set sensitivity constraints on the DM annihilation cross section and mass for model-independent DM candidates. We have also considered the case in which an intermediate-mass BH would be present in Draco, and we present the results of this study in Figure 6. In this case, the flux density would be enhanced by more than three orders of magnitude, rendering very heavy DM candidates detectable by SKA1. In particular, $100$ TeV DM would be detectable within the 10-hours of data-taking.

Finally, we consider extra-dimensional brane-world theories, i.e. branons as potential TeV WIMPs candidates. In the left panel of Figure 7 we show the flux density expected from branon DM candidate for three different masses we compare it with the sensitivity of the SKA telescope with three different observation time. We found that similar to the model-independent case, the SKA could constrain branon DM particles with masses less than 10 TeV and without any enhancement factor. On the other hand, in the right panel of Figure 7 we set constraints on the parameter space of branon candidates, i.e. the tension of the brane $f$ and the mass $M$ of the branon. In this plot, we show two branon DM candidates able to explain either the gamma rays cut-off detected by the HESS telescope (yellow star) or the AMS-02 excess in positron data (magenta star). This plot also shows the effect of a DM spike on constraining the mentioned parameter space.

Concluding, we have discussed the multi-wavelength approach as a complementary strategy to detect TeV DM candidates. Although the electromagnetic radiation - emitted by secondary products of TeV DM annihilation events via synchrotron emission - cover frequencies up to the X-ray band, the current generation of detectors have no competitive sensitivity in order to set constraints on this TeV DM candidate, being SKA the more promising detector in this sense so far. The first SKA1 operations are expected to start in 2020 and subsequently, with the addition of new technology, the progression to SKA2 will follow gradually Dewdney et al. (2013), we expect SKA detectability regions to become even more competitive in the pertinent range of frequencies. In this regard, we shall expect an improvement of a factor 10 improvement in the detection of radio signal (with respect to the one computed in this work) is expected whenever the SKA2 project will be fully operational.

Authors’ note: A preprint Kar et al. (2019b) including a complementary model- independent study was submitted to arXiv simultaneously to our communication by an independent group of authors. Unalike specific multi-TeV DM candidates are considered in each preprint.

Acknowledgements.

This work was partly supported by the projects FIS2014-52837-P (Spanish MINECO) and FIS2016-78859-P (AEI / FEDER, UE). AdlCD acknowledges financial support from projects FPA2014-53375-C2-1-P Spanish Ministry of Economy and Science, CA15117 CANTATA and CA16104 COST Actions EU Framework Programme Horizon 2020, CSIC I-LINK1019 Project, Spanish Ministry of Economy and Science, University of Cape Town Launching Grants Programme and National Research Foundation grants 99077 2016-2018 (Ref. No. CSUR150628121624), 110966 Ref. No. BS1705-09230233 and the NRF Incentive Funding for Rated Researchers, Ref. No. IFR170131220846. AdlCD would like to thank the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste) for its hospitality in the latest stages of the manuscript. VG’s contribution to this work has been supported by Juan de la Cierva-Formación FJCI-2016-29213 grant, the Spanish Agencia Estatal de Investigación through the grants FPA2015-65929-P (MINECO/FEDER, UE) and IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, INFN project QGSKY, the Agencia Estatal de Investigación (AEI) and partially by the H2020 CSA Twinning project No.692194 ORBI-T-WINNINGO. VG thanks S. Camera, D. Gaggero, P. Ullio and and the DAMASCO group for useful discussions and also acknowledges the support of the Spanish Red Consolider MultiDark FPA2017-90566-REDC. MMI would like to thank the SKA Pathfinder HI Survey Coordination Committee and the FAST-SKA Pathfinders Synergies meeting organising committee, specially to C. Carignan for its hospitality in first stages of the manuscript. MMI thanks G. Beck, S. Blyth, J. Delhaize, M. Regis, H. Villarubia Rojo and S. White for useful discussions and acknowledges financial support from the University of Cape Town Doctoral Fellowships and the National Astrophysics and Space Science Programme (NASSP).