Abstract
Annihilation of Dark Matter usually produces together with gamma rays comparable amounts of electrons and positrons. The e + e − gyrating in the galactic magnetic field then produce secondary synchrotron radiation which thus provides an indirect mean to constrain the DM signal itself. To this purpose, we calculate the radio emission from the galactic halo as well as from its expected substructures and we then compare it with the measured diffuse radio background. We employ a multi-frequency approach using data in the relevant frequency range 100 MHz-100 GHz, as well as the WMAP Haze data at 23 GHz. The derived constraints are of the order σAv = 10 −24 cm 3 s −1
Abstract
for a DM mass mχ = 100 GeV sensibly depending however on the astrophysical uncertainties, in particular on the assumption on the galactic magnetic field model. The signal from single bright clumps is instead largely attenuated by diffusion effects and offers only poor detection perspectives. 
I. INTRODUCTION
Cosmology and Astrophysics provide nowadays a compelling evidence of the existence of Dark Matter (DM) [1, 2] . Nevertheless, its nature still remains elusive, and Dark Matter constituents have escaped a direct detection in laboratory so far. Promising candidates are DM particles produced in thermal equilibrium in the early universe, the so-called Weakly Interacting Massive Particles (WIMPs). Theoretically, models of WIMPs naturally arise, for example, in SUSY as the Lightest Supersymmetric Particle or as the Lightest Kaluza-Klein Particle in the framework of extra-dimensions. These candidates are self-conjugate and can thus annihilate in couples to produce as final states: neutrinos, photons, electrons, light nuclei (as wells as their antiparticles), etc., which can in principle be detected.
I. INTRODUCTION
Among the indirect DM detection channels, gammaray emission represents one of the most promising opportunity due to the very low attenuation in the interstellar medium, and to its high detection efficiency. See for example Ref.s [2, 3, 4] for a review of this extensively studied issue. The expected neutrino detection rates are generally low although forthcoming km 3 detectors offer some promising prospect [5, 6] . Finally, positrons and protons strongly interact with gas, radiation and magnetic field in the galaxy and thus the expected signal sensibly depends on the assumed propagation model [7, 8, 9] . However, during the process of thermalization in the galactic * danielcole@example.net † danielcole@example.net ‡ danielcole@example.net medium the high energy e + and e − release secondary low energy radiation, in particular in the radio and Xray band, that, hence, can represent a chance to look for DM annihilation. Furthermore, while the astrophysical uncertainties affecting this signal are similar to the case of direct e + , e − detection, the sensitivities are quite different, and, in particular, the radio band allows for the discrimination of tiny signals even in a background many order of magnitudes more intense.
I. INTRODUCTION
Indirect detection of DM annihilation through secondary photons has received recently an increasing attention, exploring the expected signature in X-rays [10, 11, 12] , at radio wavelengths [13, 14, 15, 16] , or both [17, 18, 19] . In the following we will focus our analysis on the radio signal expected from the Milky Way (MW) halo and its substructures. It is worth noticing that the halo signal has been recently discussed in Ref.s [20, 21, 22] in connection to the WMAP Haze, which has been interpreted as a signal from DM annihilation. In this concern we will take in the following a more conservative approach, by assuming that the current radio observations are entirely astrophysical in origin, and thus deriving constraints on the possible DM signal. The main point will be the use of further radio observations besides the WMAP ones, in the wide frequency range 100 MHz-100 GHz, and a comparison of the achievable bounds. Furthermore, the model dependence of these constraints on the assumed astrophysical inputs will be analyzed. We will also discuss the detection perspectives of the signal coming from the brightest DM substructures in the forthcoming radio surveys.
I. INTRODUCTION
The paper is organized as follows: in section II we will discuss the astrophysical inputs required to derive the DM signal such as the structure of the magnetic field, the DM spatial distribution and the radio data employed to derive the constraints. In section III we describe in detail the processes producing the DM radio signal either when it is originated from the halo or from the substructures. In section IV we present and discuss our constraints, while in section V we analyze the detection sensitivity to the signal coming from the single DM clump. In section VI we give our conclusions and remarks.
II. ASTROPHYSICAL INPUTS

A. Dark matter distribution
Our knowledge of the DM spatial distribution on galactic and subgalactic scales has greatly improved thanks to recent high resolution zoomed N-body simulations [23, 24, 25, 26] . These simulations indicate that for the radial profile of the galactic halo the usual NavarroFrank-White (NFW) distribution [27] 
A. Dark matter distribution
still works as a good approximation over all the resolved scales. We will thus use this profile in the following. Note, anyway, that this choice is quite conservative with respect to other proposed profiles like the Moore profile [28] , which exhibits an internal cusp ∝ r −1.5 that would give in principle a divergent DM annihilation signal from the center of the halo. Observationally the situation is more uncertain: Baryons generally dominate the gravitational potential in the inner kpc's and fitting the data thus requires to model both the baryon and DM component at the same time. The NFW profile is in fair agreement with the observed Milky Way rotation curve [29] , although, depending on the employed model, it is possible to find an agreement for many different DM profiles (see also [2] and references therein). We emphasize, however, that the various profiles differ mainly in the halo center (for r < ∼ 1 kpc) where the uncertainties, both in numerical simulations and from astrophysical observations are maximal. Thus, our analysis which explicitly excludes the galactic center, does not crucially depend on the choice of the profile.
A. Dark matter distribution
A problem related to the profile of Eq.1 is that the mass enclosed within the radius r is logarithmically divergent. A regularization procedure is thus required to define the halo mass. Following the usual conventions we define the mass of the halo as the mass contained within the virial radius r vir , defined as the radius within which the mean density of the halo is δ vir = 200 times the mean critical cosmological density ρ cr which, for a standard cosmological model (Ω m ≃ 0.3, Ω Λ ≃ 0.7 [1] ) is equal to ρ cr ≃ 5 × 10 −6 GeV c −2 cm −3 . The parameters describing the halo are then determined imposing the DM density to be equal to ρ S = 0.365 GeV c −2 cm −3 near the Solar System, at a galactocentric distance of R S = 8.5 kpc.
A. Dark matter distribution
Simulations, however, predict a DM distribution sum of a smooth halo component, and of an additional clumpy one with total masses roughly of the same order of magnitude. Hereafter we will assume for the mass of the Milky Way
A. Dark matter distribution
12 M ⊙ , where M h and M cl denote the total mass contained in the host galactic halo and in the substructures (subhaloes) distribution, respectively. The relative normalization is fixed by imposing that subhaloes in the range 10 7 M ⊙ , 10 10 M ⊙ have a total mass amounting to 10% of M MW [23] . Current numerical simulations can resolve clumps with a minimum mass scale of ∼ 10 6 M ⊙ . However, for WIMP particles, clumps down to a mass of 10 −6 M ⊙ are expected [30, 31] . We will thus consider a clump mass range between 10 −6 M ⊙ and 10 10 M ⊙ . Finally, to fully characterize the subhalo population we will assume a mass distribution ∝ m −2 cl and that they are spatially distributed following the NFW profile of the main halo. The mass spectrum number density of subhaloes, in galactocentric coordinates r, is thus given by
A. Dark matter distribution
where A is a dimensional normalization constant. The above expression assumes some approximations: for example, a more realistic clump distribution should take into account tidal disruption of clumps near the galactic center. Numerical simulations suggest also that the radial distribution could be somewhat anti-biased with respect to the host halo profile. However, with our conservative assumptions the DM annihilation signal is dominated by the host halo emission within up to 20
A. Dark matter distribution
• − 30
A. Dark matter distribution
• from the galactic center so that the details of the subdominant signal from the clumps have just a slight influence on the final results. Recent results also show that mass distribution seems to converge to m −1.9 cl rather than m −2 cl [25] . This would also produce only a minor change in the following results.
A. Dark matter distribution
Following the previous assumptions the total mass in DM clumps of mass between m 1 and m 2 results to be
A. Dark matter distribution
where c h ≡ r vir /r h denotes the host halo concentration; while their number is
A. Dark matter distribution
Imposing the normalization condition M (10 7 M ⊙ , 10 10 M ⊙ ) = 10%M MW , we finally get for the mass due to the entire clumps distribution:
A. Dark matter distribution
while for the number of these clumps we obtain
A. Dark matter distribution
Finally by using the previous constraints one can fix the values of free parameters r h , ρ h and A, hence obtaining r h = 14. ⊙ . A further piece of information is required to derive the annihilation signal from the clumps, namely how the DM is distributed inside the clumps themselves. We will assume that each clump follows a NFW profile as the main halo with r cl and ρ cl replacing the corresponding quantities of Eq.1. However, for a full characterization of a clump, further information on its concentration c cl is required. Unluckily, numerical simulations are not completely helpful in this case, since we require information about the structure of clumps with masses down to 10 −6 M ⊙ , far below the current numerical resolution. Analytical models are thus required. In the current cosmological scenario [1] structures formed hierarchically, via gravitational collapse, with smaller ones forming first. Thus, naively, since the smallest clumps formed when the universe was denser, a reasonable expectation is c cl ∝ (1 + z f ), where z f is the clump formation redshift. Following the model of ref. • from the galactic center (see section III), so that the constraints on the DM signal do not crucially depend on the unresolved clumps signal, coming basically only from the smooth halo component. However, given the large uncertainties in the models, larger contributions from the unresolved population of clumps are in principle possible considering a different parametrization of the concentration (see for example the various models considered in [25, 33] ). We will not investigate further this possibility here. An enhancement of the clumps signal is also possible considering different choices of the clump profile other than the NFW: Differently from the case of the halo, in fact, the clump signal depends sensibly from the chosen profile and a Moore profile or an Isothermal profile can in principle enhance the signal of several orders of magnitude. Also in this case we choose to quote conservative constraints and we will not consider these possibilities further. 
B. Galactic Magnetic Field
The MW magnetic field is still quite uncertain especially near the galactic center. The overall structure is generally believed to follow the spiral pattern of the galaxy itself with a normalization of about ∼ 1µG near the solar system. Eventually, a toroidal or a dipole component is considered in some model.
B. Galactic Magnetic Field
We will consider in the following the Tinyakov and Tkachev model (TT) [34] which is a fair representative of the available descriptions of MW magnetic field. Within this model , the field shows the typical spiral pattern, an exponential decrease along the z axis and a 1/R behavior in the galactic plane. The field intensity in the inner kiloparsecs is constant at about 7µG. We will use the slightly modified parametrization of this model as described in [36] . Higher normalizations are in principle possible considering more complex structures as for example a dipole or a toroidal component [35] . Indeed some recent analyses [37, 38] including new available data seems to favor the presence of these further structures. We will thus consider as possible also an "high normalization model" that we simply parameterize as a constant 10 µG field. This choice is also motivated by a comparison with the results of [20, 21] where the same magnetic field is used.
B. Galactic Magnetic Field
Further, beside the regular component, the galactic magnetic field presents a turbulent random component. The r.m.s. intensity of this component is generally expected of the same order of magnitude of the regular one, but both its spatial distribution and its spectrum are poorly known, thus here we neglect its effects. Naively, this random component is expected to affect the synchrotron maps that we will show in the following producing a blurring of the otherwise regular pattern. Also, the random component contributes to increase the overall normalization of the field. Thus without this component the synchrotron signal is slightly underestimated so that we can regard this choice as conservative.
C. Radio Data
In the following we will derive constraints on the DM emission comparing the expected diffuse emission from the smooth halo and the unresolved population of clumps with all sky observation in the radio band. In the frequency range 100 MHz-100 GHz where the DM synchrotron signal is expected, various astrophysical processes contribute to the observed diffuse emission. Competing synchrotron emission is given by Cosmic Ray electrons accelerated in supernovae shocks dominating the radio sky up to ∼ 10 GHz. At higher frequencies the Cosmic Microwave Background (CMB) and its anisotropies represent the main signal. However, thanks to the very sensitive multi-frequency survey by the WMAP satellite, this signal (which represents thus a background for DM searches) can be modeled in a detailed way and can thus be removed from the observed radio galactic emission [39] . Other processes contributing in the 10-100 GHz range are given by thermal bremsstrahlung (freefree emission) of electrons on the galactic ionized gas, and emission by small grains of vibrating or spinning dust.
C. Radio Data
In the following our approach will be to compare the DM signal with the observed radio emission where only the CMB is modeled and removed. For this purpose we use the code described in [40] where most of the radio survey observations in the range 10 MHz-100 GHz are collected and a scheme to derive interpolated, CMB cleaned sky maps at any frequency in this range is described.
C. Radio Data
A more aggressive approach would be of course to try to model and subtract also the remaining emissions (synchrotron, free-free, dust) in order to compare the expected DM signal with the residual radio map. This is indeed the approach followed in [41, 42] where residual maps at the 5 WMAP frequencies are derived using spatial templates for the various expected astrophysical components. The residual maps then exhibit the feature called the WMAP Haze, which has been indeed interpreted as radio emission related to DM annihilation [20, 43] . However, the fit procedure used for the Haze extraction is crucial, and using more degrees of freedom to model the foregrounds as performed by the WMAP team [44] fails in finding the feature. We will anyway show in the following for comparison the constraints derived using the Haze residual map at the WMAP frequency of 23 GHz [45] . A map of the Haze and of the 1 GHz emission is shown in Fig.1 . We will see however that within our conservative approach comparable or better constraints can be obtained thanks to the use of multi-frequency information. For a given DM mass, in fact, 23 GHz is generally not the best frequency to use and better constraints are instead obtained using observations at lower frequencies even without further foreground modeling.
C. Radio Data
Definitely, a detailed foreground modeling at all radio frequencies would clearly give much stronger constraints on the DM signal and/or eventually confirm the DM nature of the WMAP haze. To this purpose consistent progress will be achieved in the next years with the new high quality data coming from the PLANCK mission and from low frequency arrays like LOFAR and SKA.
III. DM SYNCHROTRON SIGNAL A. Particle Physics
In a standard scenario where WIMPs experience a non exotic thermal history, a typical mass range for these particles is 50 GeV < ∼ m χ < ∼ 1 TeV, while a simple estimate for their (thermally averaged) annihilation cross section yields
III. DM SYNCHROTRON SIGNAL A. Particle Physics
for Ω cdm h 2 ≈ 0.1 as resulting from the latest WMAP measurements [1] . However, this naive relation can fail badly if, for example, coannihilations play a role in the WIMP thermalization process [46] , and a much wider range of cross sections should be considered viable. In this work we consider values of m χ from about 10 GeV to about 1 TeV, and σ A v in the range (10
III. DM SYNCHROTRON SIGNAL A. Particle Physics
The e + e − annihilation spectrum for a given supersymmetric WIMP candidate can be calculated for example with the DarkSUSY package [47] . However, the final spectrum has only a weak dependence on the exact annihilation process with the channels χχ → ZZ, W + W − , qq giving basically degenerate spectra. For leptonic channels like the τ + τ − decaying mode the spectrum differs significantly, although this channel has generally a quite low branching ratio. For simplicity we will assume hereafter full decay into qq channel, hence e − (e + ) will be emitted by decaying muons (anti-muons) produced in pions decays. In this framework, the resulting e + , e − spectrum can be written as a convolution, namely
III. DM SYNCHROTRON SIGNAL A. Particle Physics
1
III. DM SYNCHROTRON SIGNAL A. Particle Physics
In particular, Eq. (8) is the electron (positron) spectrum produced in the muon (anti-
III. DM SYNCHROTRON SIGNAL A. Particle Physics
In this approximation the final electron (positron) spectrum can be cast in a simple polynomial form of the ratio E e /m χ c 2 :
III. DM SYNCHROTRON SIGNAL A. Particle Physics
where Table I .
III. DM SYNCHROTRON SIGNAL A. Particle Physics
The main advantage of using the above analytical approximation instead of a more accurate numerical input is that, as will be clear in the next section, most of the observables for the radio emission will be expressed in an analytical form as well. This, in turn, is of help for a better understanding of the physical results. Nevertheless, the difference with the complete numerical calculation turns to be small, arising only for quite low electron energies, and thus for very low radio frequencies. At low energies, in fact, the analytical form has an asymptotic behavior ∝ E −1.5 e while the numerical spectrum has a turn down. From a comparison with the numerical output from DarkSUSY for a 100 GeV WIMP with 100% branching ratio into bb the analytical form is a fair approximation until E e ≈ 1 GeV, which for a magnetic field B ∼ 10 µG translates into a minimum valid frequency ν = 10-100 MHz, thus below the frequency window we are going to explore (see Eq.20 below).
B. Electrons equilibrium distribution
Dark matter annihilation injects electrons in the galaxy at the constant rate
B. Electrons equilibrium distribution
On the other hand, the injected electrons loose energy interacting with the interstellar medium and diffuse away from the production site. In the limit in which convection and reacceleration phenomena can be neglected, the evolution of the e + e − fluid is described by the following diffusion-loss equation [49, 50, 
B. Electrons equilibrium distribution
where dn e /dE e stands for the number density of e + , e − per unit energy, K(E e , r) is the diffusion constant, and b(E e , r) represents the energy loss rate. The diffusion length of electrons is generally of the order of a kpc (see section V) thus for the diffuse signal generated all over the galaxy, and thus over many kpc's, spatial diffusion can be neglected. This is not the case for the signal coming from a single clump for which the emitting region is much smaller than a kpc. We will further analyze this point in section V. By neglecting diffusion, the steady state solution can be expressed as
B. Electrons equilibrium distribution
where τ = E e /b(E e , r) is the cooling time, resulting from the sum of several energy loss processes that affect electrons. In the following we will consider synchrotron emission and Inverse Compton Scattering (ICS) off the background photons (CMB and starlight) only, which are the faster processes and thus the ones really driving the electrons equilibrium. Other processes, like synchrotron self absorption, ICS off the synchrotron photons, e + e − annihilation, Coulomb scattering over the galactic gas and bremsstrahlung are generally slower. They can become relevant for extremely intense magnetic field, possibly present in the inner parsecs of the galaxy [14] , and thus will be neglected in this analysis.
B. Electrons equilibrium distribution
For synchrotron emission the energy loss is given by (for ex. see [52] ) b syn (E e ) = 4/3 cσ T γ 2 β 2 U B with U B = B 2 /2µ 0 the magnetic energy density so that the time scale of the energy loss is: with τ 0 syn = 3.95 × 10 17 s . Similarly, for Inverse Compton emission the energy loss is given by b ICS (E e ) = 4/3 cσ T γ 2 β 2 U rad . The relevant radiation background for ICS is given by an extragalactic uniform contribution consisting of the CMB with U CMB = 8π 5 (kT ) 4 /15(hc) 3 ≈ 0.26 eV/cm 3 , the optical/infrared extragalactic background and the analogous spatially varying galactic contribution, the Interstellar Radiation Field (ISRF). For the latter we use as template the Galprop distribution model [53] which reduces to the extragalactic one at high galactocentric distances. In this model, the ISRF intensity near the solar position is about 5 eV/cm 3 , and reaches values as large as 50 eV/cm 3 in the inner kpc's. With this model the ICS is always the the dominant energy loss process, also near the galactic center (see Fig.2 ). We thus have b ICS we have
B. Electrons equilibrium distribution
µ( r) = B( r) µG
B. Electrons equilibrium distribution
with the function µ( r) enclosing the whole spatial dependence. By substituting the above expressions into Eq. (14) we get the following equilibrium distribution for electrons
C. Synchrotron spectrum
The synchrotron spectrum of an electron gyrating in a magnetic field has its prominent peak at the resonance frequency with ν 0 = 3.7 × 10 6 Hz. This implies that, in practice, a δ-approximation around the peaks works extremely well. Using this frequency peak approximation, the synchrotron emissivity can be defined as
C. Synchrotron spectrum
This quantity is then integrated along the line of sight for the various cases to get the final synchrotron flux across the sky.
Single clump signal
According to the description of previous section II A, let us consider a clump of mass m cl , whose center of mass is placed at R cl and with a sufficiently small size. In this case it is possible to neglect the spatial variation of µ( r) inside the clump itself, and thus the flux I ν can be calculated as: with d cl the distance between the observer and the clump. This can be rewritten as Fig. 3 shows some examples of signal, produced by three clumps of our simulation. An important feature to notice is that the synchrotron signal sensibly depends on the magnetic field both in the normalization and in the covered frequency range. In particular, the signal frequency cutoff, remnant of the energy spectrum cutoff near m χ , depends on B following Eq.20. Fig. 4 shows instead the positions and radio intensities for a realization of the clumps distribution with masses m cl > 10 7 M ⊙ . It can be seen that all the clumps with a non negligible signal lie near the galactic plane where most of the galactic magnetic field is concentrated. Few clumps are visible at high latitude just because of projection effects, being located very near and slightly up or below the solar position with respect to the galactic plane.
Diffuse signals
The diffuse halo signal is similarly given by the integral along the line of sight of Eq.(21)
Diffuse signals
where (l, b) are coordinates on the sphere and s the line of sight coordinate. To calculate the total contribution from the substructures, instead, we have to sum over all haloes
Diffuse signals
with I res ν given by Eq. (23) and r = r(s, l, b) . Interestingly, the sum of the two diffuse contributions can be rewritten as
Diffuse signals
where
Diffuse signals
GeV cm
Diffuse signals
Thus, from the point of view of DM annihilation the unresolved clumps signal behaves like a further smooth NFW component with the same scale radius of the halo profile, but with a different effective density ρ CL = 0.604 GeVc −2 cm −3 , and with an emissivity simply proportional to the density profile instead of its square.
Diffuse signals
We see that the halo component dominates in the central region of the galaxy, where
Diffuse signals
which corresponds to a disk of radius 27.3 degrees (see fig. 5 ).
IV. DM ANNIHILATION CONSTRAINTS
The pattern and intensity of the DM radio map resulting from the sum of the contributions from the smooth halo and unresolved clumps is shown in Fig. 6 for m χ = 100 GeV and σ A v = 3 × 10 −26 cm 3 s −1 . Similar maps are obtained at different frequencies and different m χ and σ A v to obtain DM exclusion plots. For our analysis we use a small mask covering a 15
IV. DM ANNIHILATION CONSTRAINTS
• ×15
IV. DM ANNIHILATION CONSTRAINTS
• region around the galactic center where energy loss processes other than synchrotron and ICS start possibly to be relevant. We include the galactic plane although this region has basically no influence for the constraints on the DM signal.
IV. DM ANNIHILATION CONSTRAINTS
In Fig.7 we show the radio constraints on the DM annihilation signal in the m χ -σ A v plane for various frequencies and various choices of the foreground. Several comments are in order. First, we can see that, as expected, the use of the haze at 23 GHz gives about one order of magnitude better constraints with respect to the synchrotron foregrounds at the same frequency. However, using also the information at other frequencies almost the same constraints can be achieved. This information in particular is complementary giving better constraints at lower DM masses. This is easily understood since a smaller DM mass increases the annihilation signal (∝ m −2 χ ) at smaller energies, and thus smaller synchrotron frequencies. In particular, the constraints improve of about one order of magnitude at m χ ∼ 100 GeV from 23 GHz to 1 GHz while only a modest improvement is achieved considering further lower frequencies as 0.1 GHz. This saturation of the constraints is due to the frequency dependence of the DM signal, that below 1 GHz becomes flatter than the astrophysical backgrounds so that the fraction of contribution from DM is maximal at about 1 GHz. Further, the constraints show a threshold behavior given basically by Eq.20 which settles a maximum emitted radio frequency for a given DM mass m χ . This threshold behavior is, for example, clearly seen at 23 GHz in the right panel of Fig.7 where only for masses above ∼ 40 GeV the cross section is constrained. • and ±60
IV. DM ANNIHILATION CONSTRAINTS
• , reflects basically the structure of the magnetic field as seen in projection from the Solar System, where the observer is located (compare also with Fig.2 ). The galactic center region and the first few magnetic arms are visible as regions of high magnetic field intensity and hence prominent synchrotron emission.
IV. DM ANNIHILATION CONSTRAINTS
Notice that although the astrophysical background which we compare with at 1 GHz is an interpolation, the derived constraints are still valid given the smooth behavior and the broad frequency extent of the DM signal, which does not exhibit narrow peaks at particular frequencies. However, effective measurements have been performed for example at 408 MHz and 1.4 GHz (see [40] ). Quoting our constraints at these exact frequencies would change the results only slightly.
IV. DM ANNIHILATION CONSTRAINTS
The DM signal has thus a broad frequency extent and also below 1 GHz is still relevant. This is a potential problem for the DM interpretation of the WMAP Haze given that, in the Haze extraction procedure, the observed radio emission at 408 MHz is used as template of the synchrotron background. In fact, naively, a DM signal at 23 GHz should be relevant at 408 MHz as well, unless either the DM mass or the magnetic field is so high to shift the DM contribution to higher frequencies and making it negligible at 408 MHz.
IV. DM ANNIHILATION CONSTRAINTS
The second relevant point to notice is that the constraint depends quite sensibly on the magnetic field assumptions. The constraints we obtain with the TT model are generally almost two orders of magnitude weaker with respect to the results reported in [21] . They are instead more in agreement with [22] where Galprop has been employed to calculate the DM synchrotron signal. For a closer comparison with [21] we choose, as they do, a constant magnetic field of 10 µG although still keeping the Galprop ISRF model. Even in this case our derived constraints are a factor of 5 weaker (despite the inclusion of the contribution from substructures). The remaining factor of 5 can be finally recovered using a constant ISRF with U rad = 5 eV/cm 3 as assumed in [21] . In this case, in fact, the smaller values of U rad reduces the ICS losses enhancing in turn the synchrotron signal. It should be said however that, while the magnetic field normalization is still quite uncertain, the ISRF is instead more constrained and a large variation with respect to the Galprop model seems unlikely. The constraints shown in Fig.7 extend down to ∼ 10 GeV, which is somewhat the mass limit for a conservative analysis. It is clear that for low masses the constraints come more and more from lower frequencies: For example for a WIMP of 30 GeV the data at 100 MHz are 2 orders of magnitude more constraining than the data at 10 GHz. However, extremely low frequencies are not experimentally accessible. For a WIMP of 1 GeV, from Eq.20 with a magnetic field of O(µG) only frequencies < ∼ 10 MHz would be useful to place constraints on the DM signal. Although observations at this frequency exist [40] , in general the survey sky coverage is quite incomplete and the data quality is non-optimal. Observations in this very low frequency range should substantially improve with the next generation radio arrays LOFAR and SKA. WIMP masses below 1 GeV still would produce observable synchrotron radiation at the galactic center where the magnetic field is likely much higher than µG scale (possibly O(mG) ). This kind of analysis would be however quite model dependent and would face further background uncertainties.
V. SINGLE CLUMPS DETECTABILITY
To have a reliable estimate of the sensitivity to a single clump detection diffusion effects cannot be neglected. Although the integrated synchrotron clump signal is given by Eq.(23), the clumps appear extended rather than pointlike with a dimension typically of several degrees. As a reasonable approximation we can assume that the signal is spread over an area of radius equal to the diffusion length of the electrons l D = K(E e )τ loss (E e ), where K is the diffusion coefficient and τ loss is the energy loss time given by Eq. (17) . We use for K the Galprop model [49] 
V. SINGLE CLUMPS DETECTABILITY
with a reference energy E e0 = 3 GeV, a Kolmogorov spectrum δ = 1/3 and K 0 = 10 28 cm 2 /s. Taking as reference the parameters of a very bright clump like the #3 in table II, we get (1pc = 3 × 10 18 cm)
V. SINGLE CLUMPS DETECTABILITY
for E e ≈ 10 GeV and for a radiation density U rad ≈ 5 eV/cm 3 . The energy losses are basically dominated by ICS thus the result is almost independent of the magnetic field value. Moreover the dependence on the electron energy and the radiation density itself is very weak. Of The chance of clump radio detection seems thus quite poor even with the next generation experiments. On the other side, the fact that the signal is anyway extended and not pointlike makes the clump signal not really complementary to the diffuse component sharing the same systematics with a much fainter signal. It is likely thus that the a role for DM investigations in the radio will be played basically by the diffuse signal.
V. SINGLE CLUMPS DETECTABILITY
1 At radio frequencies the Rayleigh-Jeans law Fν = 2ν 2 /c 2 k B T is employed to translate fluxes into brightness temperatures
VI. SUMMARY AND CONCLUSIONS
Using conservative assumptions for the DM distribution in our galaxy we derive the expected secondary radiation due to synchrotron emission from high energy electrons produced in DM annihilation. The signal from single bright clumps offers only poor sensitivities because of diffusion effects which spread the electrons over large areas diluting the radio signal. The diffuse signal from the halo and the unresolved clumps is instead relevant and can be compared to the radio astrophysical background to derive constraints on the DM mass and annihilation cross section.
VI. SUMMARY AND CONCLUSIONS
Constraints in the radio band, in particular, are complementary to similar (less stringent but less model dependent) constraints in the X-ray/gamma band [55, 56] and from neutrinos [57] . Radio data, in particular, are more sensitive in the GeV-TeV region while neutrinos provide more stringent bounds for very high DM masses ( > ∼ 10 TeV). Gammas, instead, are more constraining for m χ < ∼ 1 GeV. The combination of the various observations provides thus interesting constraints over a wide range of masses pushing the allowed window significantly near the thermal relic possibility.
VI. SUMMARY AND CONCLUSIONS
More into details, we obtain conservative constraints at the level of σ A v ∼ 10 −23 cm 3 s −1 for a DM mass m χ = 100 GeV from the WMAP Haze at 23 GHz. However, depending on the astrophysical uncertainties, in particular on the assumption on the galactic magnetic field model, constraints as strong as σ A v ∼ 10 −25 cm 3 s −1 can be achieved. Complementary to other works which employ the WMAP Haze at 23 GHz, we also use the information in a wide frequency band in the range 100 MHz-100 GHz. Adding this information the constraints become of the order of σ A v ∼ 10 −24 cm 3 s −1 for a DM mass m χ = 100 GeV. The multi-frequency approach thus gives comparable constraints with respect to the WMAP Haze only, or generally better for m χ < ∼ 100 GeV where the best sensitivity is achieved at ∼ GHz frequencies.
VI. SUMMARY AND CONCLUSIONS
The derived constraints are quite conservative because no attempt to model the astrophysical background is made differently from the case of the WMAP Haze. Indeed, the Haze residual map itself should be interpreted with some caution, given that the significance of the feature is at the moment still debated and complementary analyses from different groups (as the WMAP one) miss in finding a clear evidence of the feature. In this respect, the multifrequency approach will be definitely necessary to clarify the nature of controversial DM signals as in the case of the WMAP Haze. Progresses are expected with the forthcoming data at high frequencies from Planck and at low frequencies from LOFAR and, in a more distant future, from SKA. These surveys will help in disentangling the various astrophysical contributions thus assessing the real significance of the Haze feature. Further, the low frequency data in particular, will help to improve our knowledge of the galactic magnetic field. Progresses in these fields will provide a major improvement for the interpretation of the DM-radio connection.

{'paper_id': '38898162', 'title': 'Radio constraints on dark matter annihilation in the galactic halo and its substructures', 's2_url': 'https://api.semanticscholar.org/CorpusID:38898162', 'source': 'S2ORC'}