Synchrotron Emission from Dark Matter Annihilation: Predictions for Constraints from Non-detections of Galaxy Clusters with New Radio Surveys

The annihilation of dark matter results in the production of a wide variety of Standard Model particles, including electrons (and positrons). We use DarkSUSY Gondolo et al. (2004) to calculate the electron injection spectrum per dark matter annihilation event. The electron injection spectrum, ${\tfrac{{dN}}{{dE}}}_{\mathrm{inj}}$, depends on the mass of the dark matter particle and on the annihilation channel. Generally, the injection spectra cut off sharply at the dark matter particle mass at high energies. The injection spectrum from annihilation to muons is very hard, while the spectra from annihilation to tau particles is softer, and the spectra from annihilation to quarks is softer still (see, e.g., Figure 4 in Gaskins 2016).

Diffusion and energy losses in astrophysical environments modify the injection spectrum from dark matter annihilation. The electron equilibrium spectrum is obtained from the following diffusion equation that takes these mechanisms into account:

$$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}\displaystyle \frac{{{dn}}_{e}}{{dE}} & = & {\rm{\nabla }}\left[D(E,r){\rm{\nabla }}\displaystyle \frac{{{dn}}_{e}}{{dE}}\right]\\ & & +\displaystyle \frac{\partial }{\partial E}\left[{b}_{\mathrm{loss}}(E,r,z)\displaystyle \frac{{{dn}}_{e}}{{dE}}\right]+Q(E,r).\end{eqnarray} \tag{ 1 }$$

Here, dn_e/dE is the electron equilibrium spectrum, $Q(E,r)$ is the source term, $D(E,r)$ is the spatial diffusion coefficient, and ${b}_{\mathrm{loss}}(E,r)$ is the energy loss term described below:

$$\begin{eqnarray}{b}_{\mathrm{loss}}(E,r,z) & = & {b}_{\mathrm{syn}}+{b}_{\mathrm{IC}}+{b}_{\mathrm{brem}}+{b}_{\mathrm{coul}}\\ & \approx & 0.0254{\left(\displaystyle \frac{E}{1\mathrm{GeV}}\right)}^{2}{\left(\displaystyle \frac{B(r)}{1\mu {\rm{G}}}\right)}^{2}\\ & & +0.25{\left(\displaystyle \frac{E}{1\mathrm{GeV}}\right)}^{2}{(1+z)}^{4}\\ & & +1.51n(0.36+\mathrm{log}(\gamma /n))\\ & & +6.13n(1+\mathrm{log}(\gamma /n)/75.0).\end{eqnarray} \tag{ 2 }$$

The energy loss term b_loss(E) has units of 1 × 10⁻¹⁶ GeV s⁻¹, where $E=\gamma {m}_{e}{c}^{2}$ is the energy of a single electron, and n is the average thermal electron density, $\approx 1\times {10}^{-3}$ cm⁻³ for clusters. For GeV electrons and positrons, synchrotron and IC losses dominate. When $B\gt {B}_{\mathrm{CMB}}$, synchrotron losses dominate over IC losses, where ${B}_{\mathrm{CMB}}\simeq 3.25{(1+z)}^{2}\,\mu {\rm{G}}$.

The source term is related to the electron injection spectrum from dark matter self-annihilation:

$$\begin{eqnarray}&&Q(E,r)=\displaystyle \frac{\langle \sigma v\rangle {\rho }_{\chi }^{2}(r)}{2{m}_{\chi }^{2}}{\displaystyle \frac{{dN}}{{dE}}}_{\mathrm{inj}},\end{eqnarray} \tag{ 3 }$$

where ${m}_{\chi }$ is the dark matter mass (in units of energy), $\langle \sigma v\rangle$ is the annihilation cross section, and ${\rho }_{\chi }(r)$ is the spatial distribution of dark matter.

In cluster environments, energy losses from synchrotron radiation and IC scattering dominate, and the diffusion timescale is long compared to these energy loss timescales (e.g., Colafrancesco et al. 2006). We therefore neglect the dependence on spatial diffusion and the time dependence in Equation (1). The expression for the electron equilibrium spectrum then reduces to:

$$\begin{eqnarray}&&\displaystyle \frac{{{dn}}_{e}}{{dE}}(E,r,z)=\displaystyle \frac{\langle \sigma v\rangle {\rho }_{\chi }{(r)}^{2}}{2{m}_{\chi }^{2}{b}_{\mathrm{loss}}(E,r,z)}{\int }_{E}^{{m}_{\chi }}{dE}^{\prime} {\displaystyle \frac{{dN}}{{dE}^{\prime} }}_{\mathrm{inj}}.\end{eqnarray} \tag{ 4 }$$

The synchrotron power per frequency radiated by a single electron in the presence of a magnetic field is as follows:

$$\begin{eqnarray}&&{P}_{\nu }(\nu ,E,r,z)={\int }_{0}^{\pi }d\theta \displaystyle \frac{\sin \theta }{2}2\pi \sqrt{3}{r}_{0}{\nu }_{0}{m}_{e}c\sin \theta F\left(\displaystyle \frac{x}{\sin \theta }\right),\end{eqnarray} \tag{ 5 }$$

where ${r}_{0}={e}^{2}/({m}_{e}{c}^{2})$ is the classical electron radius, θ is the pitch angle, and ${\nu }_{0}={eB}/(2\pi {m}_{e}c)$ is the non-relativistic gyrofrequency (where the electron mass has units of energy). The quantities x and F are defined as follows:

$$\begin{eqnarray}&&x\equiv \displaystyle \frac{2\nu (1+z)}{3{\nu }_{0}{\gamma }^{2}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&F(s)\equiv s{\int }_{s}^{\infty }d\xi {K}_{5/3}(\xi )\approx 1.25{s}^{1/3}\exp (-s){[648+{s}^{2}]}^{1/12},\end{eqnarray} \tag{ 7 }$$

where ${K}_{5/3}(\xi )$ is the modified Bessel function of order 5/3.

Given the equilibrium energy spectrum of electrons, the synchrotron emissivity is:

$$\begin{eqnarray}&&{j}_{\nu }(\nu ,r,z)=2{\int }_{{m}_{e}{c}^{2}}^{{m}_{\chi }}{dE}\displaystyle \frac{{{dn}}_{e}}{{dE}}(E,r,z){P}_{\nu }(\nu ,E,r,z).\end{eqnarray} \tag{ 8 }$$

The factor of 2 is to account for both electrons and positrons. The surface brightness is the line of sight (LOS) integral of the emissivity, and the flux density is the surface brightness integrated over the solid angle on the sky of the emission region. For small regions (where the ratio of its size to its distance is much less than 1), this equation for flux density can be approximated as follows:

$$\begin{eqnarray}&&{S}_{\nu }(\nu ,z)\approx \displaystyle \frac{1}{{D}_{A}^{2}}{\int }_{0}^{R}{{drr}}^{2}{j}_{\nu }(\nu ,r,z),\end{eqnarray} \tag{ 9 }$$

where ${S}_{\nu }$ is the integrated flux density (measured typically in Janskys), D_A is the angular diameter distance, and R is the maximum radius of the region of interest.

We assume a Navarro–Frenk–White (NFW) model for the dark matter halo density profile (Navarro et al. 1996, 1997):

$$\begin{eqnarray}&&{\rho }_{\mathrm{NFW}}(r)=\displaystyle \frac{{\rho }_{s}}{\tfrac{r}{{r}_{s}}{\left(1+\tfrac{r}{{r}_{s}}\right)}^{2}},\end{eqnarray} \tag{ 10 }$$

where ${\rho }_{s}$ is the central density and r_s is the characteristic scale radius. The central density ${\rho }_{s}$ is determined from the scale radius r_s, and the virial mass and radius of the object of interest (Hu & Kravtsov 2003):

$$\begin{eqnarray}&&{\rho }_{s}=\displaystyle \frac{{M}_{\mathrm{vir}}}{4\pi {{\boldsymbol{r}}}_{\mathrm{vir}}^{3}f({r}_{s}/{r}_{\mathrm{vir}})},\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}&&f{(x)={x}^{3}[\mathrm{ln}(1+{x}^{-1})-(1+x)}^{-1}].\end{eqnarray} \tag{ 12 }$$

We use the scaling relationship derived from X-ray observations of clusters (Buote et al. 2007) to determine r_s from the virial mass and radius:

$$\begin{eqnarray}&&{c}_{\mathrm{vir}}=9{\left(\displaystyle \frac{{M}_{\mathrm{vir}}}{{10}^{14}{h}^{-1}}\right)}^{-0.172},\end{eqnarray} \tag{ 13 }$$

where the concentration ${c}_{\mathrm{vir}}={r}_{\mathrm{vir}}/{r}_{s}$. We derive the virial mass and radius of the Coma Cluster from the M₅₀₀ and r₅₀₀, as determined by X-ray observations (Chen et al. 2007), and corrected for our adopted cosmology. The concentration-mass relation is in general redshift-dependent. However, we choose to keep the density profile and total mass fixed, and simply calculate the resulting synchrotron signal from this dark matter halo profile at different redshifts. This is because we want to focus on just how the signal varies as redshift, without introducing any evolution of the dark matter halo that would substantially complicate the redshift dependence. Of course, for actual observations of clusters, the full, redshift-dependent calculation for the dark matter density parameters should be used.

The NFW profile is a conservative choice in that it is a smooth profile with no substructure. There certainly exist dark matter subhalos down to the mass scale of at least dwarf galaxies (${10}^{7}\,{M}_{\odot }$) in clusters. However, there are only small boosts (∼10%) to the synchrotron signal when a profile with this level of substructure is used (Storm et al. 2013). Integrating down to subhalo masses of ${10}^{-6}\,{M}_{\odot }$ gave boost factors of as much as a factor of 2 in Storm et al. (2013), but the precise value depends on the profile shape of the substructure and the region of interest considered.

We only consider here smooth, azimuthally symmetric dark matter distributions. Mass distributions in clusters can be considerably more complicated than this, especially in merging clusters with more than one subcluster peak; the prototypical example of such a cluster is the Bullet Cluster (Clowe et al. 2006). The most accurate estimate of the synchrotron signal from such clusters would require density profiles specific to those objects. A single NFW profile is most appropriate for clusters that are more relaxed. Coma is in fact a merging cluster; however, a NFW profile remains a good fit to its dark matter density inferred from weak lensing (e.g., Lokas & Mamon 2003).

The detection of synchrotron radio halos in clusters proves the existence of magnetic fields distributed throughout the cluster volume. Inferred magnetic field strengths from Faraday Rotation measures are typically in the few to tens of μG range (e.g., Carilli & Taylor 2002). An extensive study of the magnetic field in the Coma Cluster by Bonafede et al. (2010) found that the radial profile of the field was best fit by one that scales with a beta-model fit to the gas density in the cluster. Faraday rotation studies and simulations of other clusters also support this scaling (Murgia et al. 2004; Vacca et al. 2012).

We therefore assume a magnetic field profile that scales with a beta-model fit to the thermal gas density in a cluster:

$$\begin{eqnarray}&&B(r)={B}_{0}{\left(1+{\left(\displaystyle \frac{r}{{r}_{c}}\right)}^{2}\right)}^{-3/2\beta \eta }.\end{eqnarray} \tag{ 14 }$$

From Faraday Rotation measures, cool-core, relaxed clusters appear to host magnetic fields that are more centrally peaked than merging, unrelaxed clusters (Carilli & Taylor 2002; Taylor et al. 2006). Given this data, we consider two sets of parameters for the magnetic field: a "cool-core"-strength field and a "non-cool-core"-strength field. We set B₀ = 25 μG and ${r}_{c}=46\,\mathrm{kpc}$ for our "cool-core" model. This choice of B₀ is the inferred central magnetic field value from Faraday rotation measures of the prototypical cool-core cluster Perseus (Taylor et al. 2006). The value for r_c is the best-fit core radius for a beta-model fit to the thermal gas density of Perseus derived from X-ray observations and corrected for our adopted cosmology (Chen et al. 2007). For our "non-cool-core" model, we set B₀ = 4.7 μG and ${r}_{c}=291\,\mathrm{kpc}$. This value for B₀ is inferred from Faraday rotation measures of the Coma Cluster (Bonafede et al. 2010). The value for r_c is similarly the best-fit value for a beta-model fit to the thermal gas density of the Coma Cluster used in Bonafede et al. (2010), derived from X-ray observations in Briel et al. (1992). For both models, we choose $\eta =0.5$ and $\beta =0.75$, following Bonafede et al. (2010). Varying β and η from 0.5 to 1 changes our results by at most 40% at low dark matter masses, annihilation to quarks, and for the "cool-core" magnetic field profile. For the "non-cool-core" model, the effect of β and η is smaller, down to $\lt 1 \%$ for annihilation to muons and ${m}_{\chi }=1000\,\mathrm{GeV}$.

ASKAP is an interferometer located in the future headquarters of the Australian part of the SKA. It consists of 30 12 m telescopes in a configuration that balances the need for high resolution and enhanced sensitivity to diffuse emission. The primary science goal of ASKAP is to produce deep surveys of the southern sky in the GHz frequency range (Johnston et al. 2008). One such survey, EMU, is planned to survey 75% of the sky, up to a declination of $+30^\circ$, at 1.4 GHz to a depth of $10\,\mu \mathrm{Jy}$ and resolution of 10 arcsec (Norris et al. 2011). EMU is expected to begin survey operations in 2017.⁴

The APERTIF system is an upgrade to the existing WSRT in the Netherlands that will provide WSRT with an instantaneous field of view of about 8^◦, roughly a factor of 30 larger than the current FOV. This will allow APERTIF to perform fast, deep surveys (Verheijen et al. 2008). Two of the planned surveys include a Shallow Northern Sky Survey with a depth of ∼13 μJy/beam, and a Medium Deep Survey covering ∼350 deg² to ∼8 μJy/beam with a resolution of 15–35 arcsec.⁵ These surveys are expected to start in 2017 and are highly complementary to the ASKAP EMU survey.

LOFAR is an international radio telescope array designed to operate in the 10–240 MHz range (van Haarlem et al. 2013). The bulk of the antennas are located throughout the Netherlands, with several international telescopes located throughout Europe. Several surveys are planned for LOFAR. The Multifrequency Snapshot Sky Survey (MSSS) is the first northern-sky survey from LOFAR and has already released preliminary results (Heald et al. 2015). The MSSS is designed to be a commissioning survey, reaching a depth of 10 mJy at a resolution of 2 arcmin over a wide range in frequency, from 30 to 160 MHz.

The planned Tier 1 survey will survey the entire northern sky in the 15–180 MHz band, and is supposed to achieve a depth of $\sim 100\,\mu \mathrm{Jy}$ at a ∼5 arcsec resolution at 120 MHz (van Haarlem et al. 2013). The Tier 1 survey will be highly complementary to the APERTIF surveys in particular, allowing for robust measurements of the spectral indices of the radio sources detected by both surveys over more than a decade in frequency.

LOFAR will also conduct deeper surveys over smaller areas. For example, the Tier 2 surveys will cover 550 deg² to a depth of $15\,\mu \mathrm{Jy}$ at 150 MHz in a set of selected extragalactic, cluster, and supercluster fields. Finally, a Tier 3 ultra-deep survey covering 83 deg² to a depth of 7 μJy at 150 MHz will be conducted and may be relevant to dark matter detection depending on if suitable target clusters fall within the survey region (van Haarlem et al. 2013).

In order to constrain the dark matter annihilation cross section, we need to calculate the signal from dark matter and estimate the sensitivity of our chosen radio surveys to diffuse emission from a galaxy cluster. We choose to consider a dark matter halo mass of 10¹⁵ M_⊙, which is roughly the mass of the Coma Cluster. For this putative halo mass, we consider how the constraints on annihilation change over a redshift range of $0\lt z\lt 0.5$ and two choices for the magnetic field profile ("cool-core" and "non-cool-core"; see Section 2.3).

The synchrotron signal scales strongly with the observational frequency. Interestingly, the LOFAR surveys will access very low frequencies. In Figure 1, we show how the flux density as calculated in Equation (9) varies as a function of the observing frequency for a 10¹⁵ M_⊙ dark matter halo at two different redshifts, and for two different magnetic field models. Note that the drop in the magnitude of the signal for larger redshifts is due to the increasing distance.

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As is clear in Figure 1, the choice of observing frequency can have a dramatic effect on the estimated signal and therefore the constraints on dark matter annihilation. At lower frequencies, the differences between magnetic field models are minimized, especially for nearby objects. This is due to a complicated set of effects. First, the magnetic field profiles we use, a strong but sharply peaked "cool-core" model and a weaker but flatter "non-cool-core" model, average out to similar values at ∼300 kpc, around our chosen ROI. Second, the dependence of the signal on the magnetic field scales with the strength of the field, the index α of the radio emission spectrum, and implicitly on the redshift via the effective magnetic field strength of the CMB. For a power-law radio emission spectrum, the dependence is as follows:

$$\begin{eqnarray}&&{S}_{\nu }\sim {\nu }^{-\alpha }\displaystyle \frac{{B}^{1+\alpha }}{{B}^{2}+{B}_{\mathrm{CMB},z=0}^{2}{(1+z)}^{4}}.\end{eqnarray} \tag{ 15 }$$

Third, assuming a power-law spectrum, the radio spectral index α is related to the ${e}^{\pm }$ spectral index p: $\alpha =(p-1)/2$ (where ${dN}/{dE}\propto {E}^{-p}$). The shape of the ${e}^{\pm }$ spectrum depends on the dark matter mass and the annihilation channel. Generally, leptons and higher dark matter masses yield harder spectra than quarks and lower dark matter masses; p typically varies from 2 to 3 (however, the actual spectra are more accurately described by power laws with cutoffs). Therefore, the dependence of the signal on the magnetic field is weaker for annihilation to muons than to quarks (see Figure 3 and the discussion in Section 4.3). Additionally, different observing frequencies essentially probe different parts of the underlying ${e}^{\pm }$ spectrum. At lower observing frequencies, the radio emission spectrum is harder than at higher observing frequencies, as seen in Figure 1. The combination of these effects accounts for the similarities in the signal for the different magnetic field profiles at low redshift and low observing frequencies. At high redshift, the effective CMB field starts to become relevant.

The signal, of course, also depends on the region of interest. We choose a region size that balances optimizing the signal-to-noise within the region, while also integrating over a physically meaningful area. In Figure 2, we show how the emissivity, as calculated in Equation (8), varies as a function of radius. There is a turnover in the radial profile around the ∼100–300 kpc range. We therefore choose a circular region with a fixed physical radius of 300 kpc as the region in which we determine our signal and calculate the minimum detectable flux as discussed in the next section. This radius of 300 kpc roughly corresponds to the core radius of the Coma Cluster. Somewhat interestingly, after 300 kpc, the emissivities calculated with a "cool-core" magnetic field profile fall off more quickly than the "non-cool-core." If we were to choose a larger region size, the differences between these different magnetic field models would be smaller. However, this would also not appreciably boost the signal, while also increasing the noise, as we discuss in the next section.

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A conservative estimate for the values of the upper limits from the non-detection of a cluster in a radio survey can be derived by estimating the minimum detectable flux density integrated within some fixed region. The primary parameters we need in order to estimate upper limits on radio emission from future radio surveys are the size of the region of interest, the sensitivity of the radio survey, and the angular resolution of the survey.

We use the quoted expected sensitivities for each survey in our calculations: for the ASKAP and APERTIF surveys, we assume a sensitivity of $10\,\mu \mathrm{Jy}$ per beam at 1.4 GHz, and for the LOFAR Tier 1 survey, we assume a sensitivity of $70\,\mu \mathrm{Jy}$ per beam at 120 MHz (Norris et al. 2011; van Haarlem et al. 2013; Shimwell et al. 2017). While these surveys will have ≲10 arcsec resolution, we assume a Gaussian beam with a FWHM size of 25 arcsec × 25 arcsec. This larger beam size increases the sensitivity to extended emission, but is also small enough so that a circular area with a 300 kpc radius at a redshift of 0.5 is still sufficiently sampled by the beam (300 kpc corresponds to 49.2 arcsec at z = 0.5).

These sensitivities are close to the confusion limit for this resolution (see, e.g., Condon et al. 2012 for a recent analysis of confusion noise in radio images, and references therein). However, when imaging diffuse emission, generally bright point sources are first removed using the full resolution data set (5 arcsec for LOFAR and ≲10 arcsec for ASKAP and APERTIF) and the remaining extended emission is imaged, which helps to mitigate issues of confusion noise from overlapping point sources in the restoring beam. In practice, the noise level of any image depends on the details of the imaging process and the quality of the data. We expect that the noise levels for different resolutions should be relatively similar for high-quality, well-imaged maps. For example, pointed observations of diffuse emission with LOFAR already reach noise levels within a factor of 3 of the assumed sensitivity above for a similar resolution (van Weeren et al. 2016a). Additionally, for a few clusters observed in the GMRT Radio Halo Survey (Venturi et al. 2007, 2008), the noise levels are within a factor of 2–3 of the confusion limit, after point-source subtraction and convolution with a larger beam (see also Vernstrom et al. 2015). The development of new calibration and imaging techniques, such as facet calibration developed for LOFAR (van Weeren et al. 2016b), will also help to improve the quality of images from upcoming surveys.

With these parameters, we estimate an upper limit on the diffuse radio emission detectable by a particular radio survey. We calculate the upper limit as the survey sensitivity per beam times the ratio of the region area over the beam area:

$$\begin{eqnarray}&&\mathrm{UL}(r,z)={\sigma }_{\mathrm{rms}}\left(\displaystyle \frac{{A}_{\mathrm{ROI}}(r,z)}{{A}_{\mathrm{beam}}}\right)\end{eqnarray} \tag{ 16 }$$

The area of the region of interest is:

$$\begin{eqnarray}&&{A}_{\mathrm{ROI}}{(r,z)=\pi {\theta }_{\mathrm{ROI}}(r,z)}^{2}\end{eqnarray} \tag{ 17 }$$

Here, r is the radius of the region of interest, fixed to 300 kpc. The angular radius ${\theta }_{\mathrm{ROI}}$ is equal to this physical radius divided by the angular diameter distance at a given redshift.

The area of the beam is:

$$\begin{eqnarray}&&{A}_{\mathrm{beam}}=\displaystyle \frac{\pi }{4\mathrm{ln}(2)}{\theta }_{\mathrm{beam}}^{2},\end{eqnarray} \tag{ 18 }$$

where ${\theta }_{\mathrm{beam}}$ is the usual quoted FWHM value for a Gaussian beam, assumed to be 25 arcsec in our calculations.

This methodology for estimating upper limits is similar to that in Cassano et al. (2009, 2012), where they estimate the minimum detectable flux density from a radio halo to determine the number of halos that are potentially detectable by upcoming radio surveys.

In the multipanel Figure 3, we show the upper limits on the predicted annihilation cross section as a function of dark matter mass for different magnetic field profiles, redshifts, and annihilation channels. Since the synchrotron signal from dark matter annihilation depends on the resulting electron and positron spectra, the strongest constraints result from annihilation to muons. This channel produces, from muon decay, a much harder spectrum of electrons and positrons compared to channels where electron–positron production stems from hadronization and charged-pion decay. Although it is not shown in the plots, the constraint curves at 120 MHz do turn over at about ∼2 GeV. (The turnover in the constraint curves at 1400 MHz is seen in the figures clearly, at around 10 GeV.)

At energies $\lesssim 30\,\mathrm{GeV}$, the predicted upper limits on the annihilation cross section are $\gtrsim 1$ order of magnitude below the thermal relic cross section.⁶

In each plot of the annihilation cross section limits, we also compare our predicted limits to those from the combined non-detection of gamma-rays from a sample of 15 dwarf galaxies with Fermi-LAT, which are the best single-channel cross section limits to date (Ackermann et al. 2015). In the case of annihilation to muons, our predicted limits are up to an order of magnitude more sensitive than the limits from dwarf galaxies even for the shallow LOFAR Tier 1 and ASKAP EMU surveys. For other annihilation channels, the limits from radio are generally less sensitive than the gamma-ray dwarf limits for the shallow, wide-area surveys, because annihilation to tau leptons and to bottom quarks yields many gamma-rays, but relatively fewer electrons and positrons. In Figure 4, we show that the predicted limits from LOFAR non-detections will rule out some of the the best-fit models to the Galactic Center Excess (Calore et al. 2015; Abazajian & Keeley 2016; Daylan et al. 2016). Similarly, deep pointed observations of appropriate clusters with LOFAR, ASKAP, or APERTIF can yield the strongest indirect constraints on a large range of dark matter annihilation models.

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The shape of the predicted constraint curves vary depending on the observing frequency for a given dark matter particle mass, since a different part of the underlying electron/positron spectrum is being probed. At low dark matter masses, the non-detection of diffuse emission from clusters yield the most constraining limits at lower observing frequencies, in the range of the LOFAR surveys, while at higher dark matter masses, the higher observing frequencies of the EMU and APERTIF surveys yield better limits. Notably, at 120 MHz, the effect of the magnetic field profile on the limits is smaller than at 1.4 GHz and virtually negligible at high dark matter masses for all annihilation channels. This makes LOFAR a better instrument to constrain dark matter from clusters, since the magnetic field profiles and strengths in clusters are somewhat uncertain.

Finally, the effect of redshift on the cross section limits is observed to be less severe than the variation in the signal from Figure 1 would imply. This is because while the synchrotron flux decreases as a function of distance and therefore redshift, the minimum detectable flux for a given survey also decreases with redshift for a fixed physical region of interest; the smaller angular size at higher redshift leads to a lower background in the cluster region. The limits from a cluster at z = 0.5 are at most an order of magnitude larger than those at z = 0.023 (the redshift of the Coma Cluster), and are only a factor of ∼3 larger in most cases. This implies that limits from radio emission are less sensitive to the distance of the object, which allows for a larger sample to be analyzed, in comparison to, for example, indirect detection studies in the gamma-ray band.