Gamma ray emission from a baryonic dark halo

A recent re-analysis of EGRET data by Dixon et al. has led to the discovery of a statistically significant diffuse $\gamma$-ray emission from the galactic halo. We show that this emission can naturally be accounted for within a previously-proposed model for baryonic dark matter, according to which dark clusters of brown dwarfs and cold self-gravitating $H_2$ clouds populate the outer galactic halo and can show up in microlensing observations. Basically, cosmic-ray protons in the galactic halo scatter on the clouds clumped into dark clusters, giving rise to the observed $\gamma$-ray flux. We derive maps for the corresponding intensity distribution, which turn out to be in remarkably good agreement with those obtained by Dixon et al. We also address future prospects to test our predictions.


Introduction
Observations of the diffuse γ-ray emission during the last twenty years have been successfully interpreted in terms of a two-folded structure ⋆ a highly anisotropic component strongly concentrated along the galactic disk, ⋆ an apparently isotropic component. While the former is evidently galactic in nature -being actually accounted for by cosmic ray (CR) interactions in the interstellar medium (ISM) [3] -the origin of the latter still remains an open problem in high-energy astrophysics (see e.g. [1,4,5]). We will restrict our attention to the latter component throughout the present paper. ‡ We would like to dedicate this work to the memory of Dennis W. Sciama § To whom correspondence should be addressed.
A comprehensive account of these matters as well as of their theoretical explanations can be found in [2].
A question naturally arises. Where does the γ-ray emission in question come from? No doubt, its characteristic isotropy calls for an extragalactic origin -an option which is further supported by the fact that it fits remarkably well with the extragalactic hard X-ray background [7].
The next question to address is whether the considered γ-ray background arises from a truly diffuse process or rather from the contribution of very many unresolved point sources. Either option has received considerable attention. Among the theories of diffuse origin are a baryon-symmetric Universe [8], primordial black hole evaporation [9,10], early collapse of supermassive black holes [11], a new population of Gemingalike pulsars [12] and WIMP (Weakly Interacting Massive Particle) annihilation (see e.g. [13]). Models based on discrete source contribution include a variety of possibilities. What is clear since a long time is that normal galaxies fail to account for the observed isotropic background -at least as long as their disk emission is considered [14]- [17] since the corresponding intensity falls shorter by a factor ∼ 10 with respect to the detected flux. A more realistic option is provided by active galaxies [18,19]. Indeed, blazars seem to yield a successful explanation of the isotropic γ-ray emission [20]- [24]. Finally, a somewhat hybrid model has recently been proposed, in which the isotropic γ-ray background is produced in clusters of galaxies through the interaction of CRs with the hot intracluster gas [25]. However, this model has been severely criticized [26,27]. In fact, it gives rise to a γ-ray spectral index in disagreement with the observed one and relies upon a value for the CR density in the intracluster space which is too high to be plausible. More generally, it has been shown that the contribution to the isotropic γ-ray emission from clusters of galaxies is negligible [27].
Recently, Dixon et al. [1] have re-analyzed the EGRET data concerning the diffuse γ-ray flux with a wavelet-based technique, using the expected (galactic plus isotropic) emission as a null hypothesis. Although the wavelet approach does not allow for a good estimate of the errors, they find a statistically significant diffuse emission from an extended halo surrounding the Milky Way. This emission traces a somewhat flattened halo and its intensity at high-galactic latitude is [1] Φ γ (E γ > 1GeV) ≃ 10 −7 − 10 −6 γ cm −2 s −1 sr −1 .
Clearly, the comparison of eqs. (2) and (3) entails that the newly discovered halo γ-ray flux is a relevant fraction of the standard isotropic diffuse emission (at least for E γ > 1 GeV). Our aim is to show that the observed halo γ-ray emission naturally arises within a previously-proposed model for baryonic dark matter, according to which dark clusters of brown dwarfs and cold self-gravitating H 2 clouds populate the outer galactic halo and can show up in microlensing observations [28]- [32]. Basically, CR protons in the galactic halo scatter on the clouds clumped into dark clusters, giving rise to the newly discovered γ-ray flux.
Although we already pointed out that a signature of the model is a diffuse γ-ray emission from the galactic halo [28,29], a more thorough study is required to compare the predicted intensity distribution with the observed one. A short account of these results has been presented elsewhere [33]. In the present paper, we provide a more exhaustive analysis. In addition, we estimate the γ-ray emission from the nearby M31 galaxy.
The paper is organized as follows. In Section 2 we recall the main features of our model for baryonic dark matter in the galactic halo. In Section 3 we address the CR confinement in the galactic halo and we estimate the CR energy density. In Section 4 we compute the halo γ-ray flux -produced by the clouds clumped into dark clusters through proton-proton scattering -as detected on Earth. Section 5 is devoted to the study of the γ-ray flux due to Inverse Compton (IC) scattering of electrons off background photons. In Section 6 we present γ-ray intensity maps, pertaining to both proton-proton scattering and IC scattering, and discuss their interplay. Finally, in Section 7 we address future prospects to test our predictions.

Dark clusters in the galactic halo
Ever since the discovery that standard big-bang nucleosynthesis correctly accounts for the light element abundances, a lesson has become clear: most of the baryons in the Universe happen to be in nonluminous form, thereby making a strong case for baryonic dark matter.
In order to see how this comes about, we recall that the fraction of critical density contributed by luminous matter is estimated to be Ω L ∼ 0.005 [34] †. Yet, agreement between the predicted and observed abundances of nucleosynthetic yields is achieved only provided the similar contribution from baryons -in whatever form -lies in the range 0.01 < ∼ Ω B < ∼ 0.05 [35]. Actually, this conclusion has recently been sharpened by deuterium measurements in Quasi Stellar Object (QSO) absorption spectra, which probe regions of space much farther away than previously explored and give Ω B ≃ 0.05 [36]. So, about 90% of the baryonic matter in the Universe is expected to be dark.
Needless to say, one is naturally led to wonder about the distribution and form of baryonic dark matter.
Several possibilities have been contemplated over the last few years. Although no logically compelling reason in favour of any particular option has emerged so far, it looks intriguing that a naturalness argument strongly suggests that the galactic dark halos should be predominantly baryonic.
Basically, the idea is as follows. As is well known, both optical and HI observations have shown that all galactic rotation curves exhibit a universal qualitative behaviour: after a steep rise corresponding to the bulge, they stay approximately constant out to the last measured point. This feature -namely the lack of a keplerian fall-off -provides a stark evidence in favour of a spheroidal dark halo surrounding the luminous part of any galaxy. This is however not the end of the story. For, rotation curves trace the luminous -hence baryonic -matter within the optical disk, but are dominated by the halo dark matter at larger galactocentric distances. Yet, both contributions invariably turn out to match smoothly and exactly, thereby signalling a striking visibleinvisible conspiracy (also called disk-halo conspiracy). Before proceeding further, a point should be stressed. With only a rather limited sample of available rotation curves, that conspiracy was initially understood as a fine-tuning whereby the disk and the halo of spiral galaxies manage to produce a flat rotation curve [37]. Further studies have shown that such a flatness is only approximate: brighter galaxies tend to have slightly falling rotation curves, whereas fainter ones possess slightly rising rotation curves [38]. Still, what really matters for the visible-invisible conspiracy (as stated above) is the lack of any jump in the rotation curve within the disk-halo transition region, besides the approximate flatness.
A priori, only a mysterious fine-tuning could justify the conspiracy in question if the halo dark matter were different in nature from luminous matter, that is to say if it were nonbaryonic. So, baryonic dark matter looks like a natural constituent of galactic halos. Incidentally, this situation is very reminiscent of the case of grand unified theories in particle physics, where supersymmetry has been invoked as a successful way out of a similar, mysterious fine-tuning needed to stabilize the gauge hierarchy against radiative corrections [39]. Thus, we are led to the conclusion that -much in the same way as fundamental interactions ought to be supersymmetric -galactic halos ought to be predominantly baryonic! Remarkably enough, a specific model of baryonic dark halos emerges naturally from the present-day understanding of globular clusters. Indeed, a few years ago we have realized [28,29] that the Fall-Rees theory for the formation of globular clusters [40]- [42] automatically predicts -without any further physical assumption -that dark clusters made of brown dwarfs † and cold H 2 clouds should lurk in the galactic halo at † Although we concentrate our attention on brown dwarfs, it should be mentioned that red dwarfs as galactocentric distances larger than 10−20 kpc. Accordingly, the inner halo is populated by globular clusters, whereas the outer halo chiefly consists of dark clusters. ‡ Below, we summarize the main features of our model.
Although the mechanism of galaxy formation is not yet fully understood, the theory for the origin of globular clusters seems to be fairly well established -thanks to the pioneering work of Fall and Rees [40] -and can be summarized as follows. After its initial collapse, the proto-galaxy is expected to be shock heated up to its virial temperature ∼ 10 6 K. Because of thermal instability, density enhancements rapidly grow as the gas cools. Actually, overdense regions cool more rapidly than average, and so proto-globularcluster (PGC) clouds form in pressure equilibrium with the hot diffuse gas. When the PGC cloud temperature drops to ∼ 10 4 K, hydrogen recombination occurs: at this stage, the PGC cloud mass and size are ∼ 10 5 (R/kpc) 1/2 M ⊙ and ∼ 10(R/kpc) 1/2 pc, respectively (R being the galactocentric distance). Below ∼ 10 4 K, an efficient cooling can be brought about only by photon emission from roto-vibrational transitions in H 2 . Whether this mechanism is actually operative or not crucially depends on the intensity of the environmental ultraviolet (UV) radiation field, as we are now going to discuss.
In fact, in the central region of the proto-galaxy an AGN (Active Galactic Nucleus) along with a first population of massive stars are expected to form, which act as strong sources of UV radiation that dissociates the H 2 molecules. It is not difficult to estimate that the H 2 destruction should occur for galactocentric distances smaller than 10 − 20 kpc. As a consequence, cooling is heavily suppressed in the inner halo, and so here the PGC clouds remain for a long time in quasi-hydrostatic equilibrium at temperature ∼ 10 4 K, resulting in the imprinting of a characteristic mass ∼ 10 6 M ⊙ . Eventually, the UV flux decreases, thereby allowing for the formation and survival of H 2 . Accordingly, the PGC clouds can further cool, collapse and fragment, ultimately producing ordinary stars clumped into globular clusters.
What is most relevant for the present considerations is that in the outer halo -namely for galactocentric distances larger than 10 − 20 kpc -no substantial H 2 destruction should take place, owing to the distance suppression of the UV flux. Therefore, here the PGC clouds monotonically cool, collapse and fragment. When their number density exceeds ∼ 10 8 cm −3 , virtually all hydrogen gets converted to molecular form by three-body reactions (H + H + H → H 2 + H and H + H + H 2 → H 2 + H 2 ), which makes in turn the cooling efficiency increase dramatically [54]. As a result, no imprinting of a characteristic mass on the PGC clouds shows up, and the fragment well can be accomodated within the considered setting. ‡ Similar ideas have been proposed by Ashman and Carr [43], Ashman [44], Fabian and Nulsen [45,46], and Kerins [47,48]. Moreover, a scenario almost identical to the one investigated here has been put forward by Gerhard and Silk [49]. Somewhat different baryonic pictures have been worked out by Pfenniger, Combes and Martinet [50], Sciama [51], and Gibson and Schild [52] (see also [53]).
Jeans mass can drop to values considerably smaller than ∼ 1M ⊙ . The fragmentation process stops when the PGC clouds become optically thick to their own line emissionthis happens for a fragment Jeans mass as low as ∼ 10 −2 M ⊙ [54]. In this manner, dark clusters containing brown dwarfs in the mass range 10 −2 − 10 −1 M ⊙ should form in the outer halo. Typical values of the dark cluster radius are ∼ 10 pc.
In spite of the fact that the dark clusters resemble in many respects globular clusters, an important difference exists. Since practically no nuclear reactions occur in the brown dwarfs, strong stellar winds are presently lacking. Therefore the leftover gas -which is ordinarily expected to exceed 60% of the original amount -is not expelled from the dark clusters but remains confined inside them. Thus, also cold gas clouds are clumped into the dark clusters. Although these clouds are primarily made of H 2 , they should be surrounded by an atomic layer and a photo-ionized "skin". Typical values of the cloud radius are ∼ 10 −5 pc.
Besides accounting for the halo dark matter in a natural fashion -without demanding any new physical assumption -this model elegantly explains the visible-invisible conspiracy. For, whether ordinary matter is luminous or dark ultimately depends on the intensity of the environmental UV radiation field during the proto-galactic epoch -no fine-tuning is indeed involved! Moreover, the UV field in question is expected to be stronger for brighter galaxies. Accordingly, brighter galaxies should have the dark clusters lying farther away from the galactic centre than fainter galaxies, thereby making the contribution of dark matter to the rotation curve of brighter galaxies less significant than for fainter ones: this circumstance precisely agrees with the abovementioned observed pattern of rotation curves [38].
Observationally, the present model makes a crucial prediction: very high-energy cosmic ray proton scattering on the clouds should give rise to a detectable diffuse gamma-ray flux from the halo of our galaxy. This topic will be dealt with in great detail in the next Sections.
Further support in favour of the baryonic scenario in question comes from the understanding of the Extreme Scattering Events: dramatic flux changes over several weeks during monitoring of compact radio quasars [55]. It is generally agreed that ESEs are not intrinsic variations, but rather apparent flux changes caused by refraction when a (partially) ionized cloud crosses the line of sight. Recently, Walker and Wardle [56] pointed out that the first consistent explanation of ESEs requires the refracting clouds to have precisely the same properties of the cold H 2 clouds predicted by the present model (it is their photo-ionized "skin" that causes the radio wave refraction).
Last but not least is the issue of MACHOs (Massive Astrophysical Compact Halo Objects), detected since 1993 in microlensing experiments towards the Magellanic Clouds. Regretfully, their origin remains controversial. Although the events detected towards the SMC (Small Magellanic Cloud) seem to be a self-lensing phenomenon [57,58], a similar interpretation of all the events discovered towards the LMC (Large Magellanic Cloud) looks unlikely [59]. Yet -even if most of the MACHOs are dark matter candidates lying in the galactic halo -their physical nature is unclear, since their average mass strongly depends on the still uncertain galactic model, ranging from ∼ 0.1 M ⊙ for a maximal disk up to ∼ 0.5 M ⊙ for a standard isothermal sphere.
Superficially, white dwarfs look as the best explanation, but the resulting excessive metallicity of the halo makes this option untenable, unless their contribution to halo dark matter is not substantial (see [60,61]). So, some variations on the theme of brown dwarfs have been explored.
An option is that the galactic halo resembles more closely a minimal halo (maximal disk) rather than an isothermal sphere, in which case MACHOs can still be brown dwarfs. † In this connection, two points should be stressed. First, a large fraction (up to 50% in mass) can be binary systems -much like ordinary stars -thereby counting as twice more massive objects [63]. Second, within our model brown dwarfs can actually be beige dwarfs -with mass substantially larger than ≃ 0.1 M ⊙ -as suggested by Hansen [64], since a slow accretion mechanism from cloud gas is likely to occur [65].
An alternative possibility has been pointed out by Kerins and Evans [66]. Since in the present model the initial mass function obviously changes with the galactocentric distance, ‡ it can well happen that brown dwarfs dominate the halo mass density without however dominating the optical depth for microlensing. What are then MACHOs? Quite recently, faint blue objects discovered by the Hubble Space Telescope have been understood as old halo white dwarfs lying closer than ∼ 2 kpc from the Sun [68]- [70]: they look as a good candidate for MACHOs within this context.
Finally, we remark that recently ISO observations [71] of the nearby NGC891 galaxy have detected a huge amount of molecular hydrogen, which might account for almost all dark matter, at least within its optical radius. Other observations suggest that similar clouds are also present farther away [72]. In addition, Sciama [51] has argued that a known excess in the far-infrared emissivity of our galaxy (over that expected from a standard warm interstellar dust model) would be naturally accounted for by a population of cold H 2 clouds building up a thick galactic disk.

Cosmic ray confinement in the galactic halo
Neither theory nor observation allow at present to make sharp statements about the propagation of CRs in the galactic halo §. Therefore, the only possibility to get † Notice that also the H 2 clouds can give rise to microlensing events [62]. ‡ Evidence for a spatially varying initial mass function in the galactic disk has been reported [67]. § We stress that -contrary to the practice used in the CR community -by halo we mean the (almost) spherical galactic component which extends beyond ∼ 10 kpc. some insight into this issue rests upon the extrapolation from the knowledge of CR propagation in the disk. Actually, this strategy looks sensible, since the leading effect is CR scattering on inhomogeneities of the magnetic field over scales from 10 2 pc down to less than 10 −6 pc [73] and -according to our model -inhomogeneities of this kind are expected to be present in the halo as well, because of the existence of molecular clouds -with a photo-ionized "skin" -clumped into dark clusters. Indeed, typical values of the dark cluster radius are ∼ 10 pc, whereas typical values of the cloud radius are ∼ 10 −5 pc [32].
As is well known, CRs up to energies of ∼ 10 6 GeV are confined in the galactic disk for ∼ 10 7 yr [73]. It can be shown that in the diffusion model for the propagation of CRs, the escape time τ esc is given by [73] where D(E) is the diffusion coefficient, while h d and R h are the half-thickness of the disk and the radius of the confinement region, respectively. We remind that -for CR propagation in the disk -the diffusion coefficient is D(E) ≃ D 0 (E/7 GeV ) 0.3 cm 2 s −1 in the ultra-relativistic regime, whereas it reads D(E) ≃ D 0 ≃ 3 × 10 28 cm 2 s −1 in the non-relativistic regime [73]. CRs escaping from the disk will further diffuse in the galactic halo, where they can be retained for a long time, owing to the scattering on the above-mentioned small inhomogeneities of the halo magnetic field †.
Indirect evidence that CRs are in fact trapped in a low-density halo has recently been reported. For example, Simpson & Connell [75] argue that, based on measurements of isotopic abundances of the cosmic ratio 26 Al/ 27 Al, the CR lifetimes are perhaps a factor of four larger than previously thought, thereby implying that CRs traverse an average density smaller than that of the galactic disk.
A straightforward extension of the diffusion model implies that the CR escape time τ H esc from the halo (of size R H ≡ R h ∼ 100 kpc, much larger than the disk half-thickness) is given by where D H (E) is the diffusion coefficient in the galactic halo.
As a matter of fact, radio observations in clusters of galaxies yield for the corresponding diffusion constant D 0 a value similar to that found in the galactic disk [76] ‡. So, it looks plausible that a similar value for D 0 also holds on intermediate length scales, namely within the galactic halo. In the lack of any further information on the energy-dependence of D H (E), we assume the same dependence as that established for the disk. Hence, from eq. (5) we find that for energies E < ∼ 10 3 GeV the escape time of CRs from the halo is greater than the age of the Galaxy t 0 ≃ 10 10 yr (notice that below the ultra-relativistic regime τ H esc gets even longer). As a consequence -since the CR flux scales like E −2.7 (see next Section) -protons with E < ∼ 10 3 GeV turn out to give the leading contribution to the CR flux.
We are now in position to evaluate the CR energy density in the galactic halo, getting where L G ≃ 10 41 erg s −1 is the galactic CR luminosity (see, e.g., [78]). Notice, for comparison, that ρ H CR turns out to be about one-tenth of the disk value [79]. In fact, this value is consistent with the EGRET upper bound on the CR density in the halo near the SMC [80].
We remark that we have taken specific realistic values for the various parameters entering the above equations in order to make a quantitative estimate. However, somewhat different values can be used. For instance, R H may range up to ∼ 200 kpc [34], whereas D 0 might be slightly larger than the above value, e.g. ≃ 10 29 cm 2 s −1 consistently with our assumptions. Moreover, L G can be as large as 3 × 10 41 erg s −1 [81]. It is easy to see that these variations do not substantially affect our previous conclusions.

Proton-proton scattering in the galactic halo
We proceed to estimate the halo γ-ray flux produced by the clouds clumped into dark clusters through the interaction with high-energy CR protons. CR protons scatter on cloud protons giving rise (in particular) to neutral pions, which subsequently decay into photons. A highly nontrivial question concerns the opacity effects in the clouds. Quite recently, Kalberla et al. [82] have addressed precisely this issue, showing that opticaldepth effects for both protons and photons are negligible within our model. Finally, we expect an irrelevant high-energy (≥ 100 MeV) γ-ray photon absorption outside the clouds, since the mean free path is orders of magnitudes larger than the halo size.
As far as the energy-dependence of the halo CRs is concerned, we adopt the same ‡ Moreover, we note that average magnetic field values in galactic halos are expected to be close to those of galaxy clusters, i.e. between 0.1 µG and 1 µG [77].
power-law as in the galactic disk (see below) [79] The constant A is fixed by the requirement that the integrated energy flux agrees with the above value of ρ H CR . Explicitly where for definiteness we take the integration range to be 1 GeV ≤ E ≤ 10 3 GeV. A nontrivial point concerns the choice of α. As an orientation, the observed spectrum of primary CRs on Earth would yield α ≃ 2.7. However, this conclusion cannot be extrapolated to an arbitrary region in the halo (and in the disk), since α crucially depends on the diffusion processes undergone by CRs. For instance, the best fit to EGRET data in the disk towards the galactic centre yields α ≃ 2.45 [83], thereby showing that α gets increased by diffusion. In the lack of any direct information, we conservatively take α ≃ 2.7 even in the halo, but in Table 1 we report some results for different values of α for comparison . At any rate, the flux does not vary substantially. Let us next turn our attention to the evaluation of the γ-ray flux produced in halo clouds through the reactions pp → π 0 → γγ. Accordingly, the source function q γ (> E γ , ρ, l, b) -yielding the photon number density at distance ρ from Earth with energy greater than E γ -is [79] where the lower integration limit E p (E γ ) is the minimal proton energy necessary to produce a photon with energy > E γ , σ n p→π (E π ) is the cross-section for the reaction pp → nπ 0 (n is the π 0 multiplicity), ρ H 2 (ρ, l, b) is the halo gas density profile and n γ (E p ) is the photon multiplicity.
Unfortunately, it would be exceedingly difficult to keep track of the clumpiness of the actual gas distribution in the halo, and so we assume that its density is smooth and goes like the dark matter density -anyhow, the very low angular resolution of γ-ray detectors would not permit to distinguish between the two situations (evidently this strategy would be meaningless if optical-depth effects were not negligible). Accordingly, the halo gas density profile reads for x 2 + y 2 + z 2 /q 2 > R min , (R min ≃ 10 kpc is the minimal galactocentric distance of the dark clusters in the galactic halo). We recall that f denotes the fraction of halo dark matter in the form of gas, ρ 0 (q) is the local dark matter density,ã = 5.6 kpc is the core radius and q measures the halo flattening. For the standard spherical halo model ρ 0 (q = 1) ≃ 0.3 GeV cm −3 , whereas it turns out that e.g. ρ 0 (q = 0.5) ≃ 0.6 GeV cm −3 .
In order to proceed further, it is convenient to re-express q γ (> E γ , ρ, l, b) in terms of the inelastic pion production cross-section σ in (p lab ). Since eq. (9) becomes where ρ H 2 (ρ, l, b) is given by eq. (10) with x = −ρ cos b cos l + R 0 , y = −ρ cos b sin l and z = ρ sin b. For the inclusive cross-section of the reaction pp → π 0 → γγ we adopt the Dermer [84] parameterization where p is the proton laboratory momentum in GeV/c, the factor 2 comes from the fact that each pion decays into two photons, whereas 1.45 accounts for the CR composition [84], which includes also heavy nuclei. The quantity is expressed in terms of the Mandelstam variable s, while m π and m p are the pion and the proton mass, respectively. Because dV = ρ 2 dρdΩ, it follows that the observed γ-ray flux per unit solid angle is So, we find where I 1 (l, b) and I 2 (> E γ ) are defined as and m p is the proton mass. According to the discussion in Sections 2 and 3, typical values of ρ 1 (l, b) and ρ 2 (l, b) in eqs. (15) and (17) Table 1.

Inverse-Compton scattering
Another mechanism whereby γ-ray photons are produced is IC scattering of high-energy CR electrons off galactic background photons. Here we estimate the resulting flux, while the interplay between proton-proton scattering and IC scattering will be discussed in the next Section.
The electron injection spectrum which best fits the locally observed electron spectrum is given by the following power-law valid for E e > ∼ 10 GeV (see e.g. [85]) with a ≃ 2.4 and K 0 ≡ K(0) ≃ 6.3 × 10 −3 e − cm −2 s −1 sr −1 Gev a−1 (the value of K 0 is obtained by normalizing eq. (19) with the observed local CR electron spectrum at 10 GeV). Since a is somewhat model-dependent (in particular it depends on the diffusion processes), its actual value is not well determined, and indeed it could be as low as a ≃ 2 [4] or even a ≃ 1.8 [5]. However, what is relevant is the electron spectrum where the γ-ray production occurs and -due to diffusion processes -the value of a is expected to increase with the distance from the galactic plane where the electrons are mostly produced.
In order to estimate the galactic radiation field, we adopt the model of Mazzei, Xu & De Zotti [86] for the photometric evolution of disk galaxies. This model reproduces well the present broad-band spectrum of the Galaxy over about four decades in frequency, from UV to far-IR. Accordingly, the two main contributions to the galactic radiation field come from stars at wavelength λ ∼ 1µm and diffuse dust at λ ∼ 100µm. The total stellar luminosity of the Galaxy is L ⋆ ∼ 3.5 × 10 10 L ⊙ and the amount of starlight absorbed and re-emitted by dust is L d ∼ 1.2 × 10 10 L ⊙ (see e.g., [86,87]). As regards to the photon energy distribution, we can roughly approximate the emission spectrum (see Fig. 4 in [86]) with the sum of two Planck functions with temperature T ⋆ ∼ 2900 K and T d ∼ 29 K, respectively.
According to the previous assumptions, the source function q ph (E γ ) for γ-ray production through IC scattering is given by [73] Here < ǫ ph (T ⋆,d ) >≃ 8kT ⋆,d /3 is the average energy of background photons emitted by stars or dust and σ T is the Thompson cross-section. In deriving eq. (20), use is made of the fact that the γ-ray energy is related to the electron and background photon energies according to so that very high-energy electrons are needed in order to produce γ-rays. For example, a γ-ray with E γ ≃ 1 GeV produced by this mechanism requires E e ≃ 170 GeV for a target photon emitted by dust, while E e ≃ 17 GeV is demanded for starlight. The intensity of diffuse galactic γ-rays of energy > E γ produced in this way and coming to Earth along the line-of-sight (l, b) turns out to be where we have introduced the function f e (ρ, l, b) ≡ K(ρ, l, b)/K 0 as the ratio of the electron CR intensity relative to the local intensity, while < n ph (ρ, l, b) > is the average density of background photons. Let us next focus our attention on the functions f e (ρ, l, b) and < n ph (ρ, l, b) >. The electron component of CRs is galactic in origin, mainly produced by supernovae and pulsars located inside the disk. Electrons diffuse through the Galaxy and their distribution is energy-dependent and not uniform, namely, the characteristic diffusion length scale gets smaller for higher electron energy. This feature cannot be described in the framework of the widely used Leaky Box Model, and in order to obtain the electron density at an arbitrary point in the Galaxy one has to resort to the transport equation (see e.g. [4,73]). Unfortunately, several fairly unknown parameters enter this equation, like the electron diffusion coefficient, the rate at which electrons lose energy, the density of sources and the electron spectrum.
An alternative approach relies upon the experimental evidence of the thick disk †, in which high-energy electrons may be retained for a long time before escaping into the galactic halo. Indeed, the observed characteristics of the radio emission spectra of our and other galaxies lead to a relative density distribution of electrons f e (R 0 , z) ≡ n e (z)/n e (0) extending up to 5 − 12 kpc perpendicularly to the galactic plane, as shown in Figure 5.29 of [73]. These numerical results can be approximated by f e (R 0 , z) = exp[−(z/z e ) 3/2 ], with the parameter z e depending on the electron energy. From eq. (21) and the ensuing discussion, it turns out that z e ≃ 2.5 kpc for E e ≃ 170 GeV while z e ≃ 3.5 kpc for E e ≃ 17 GeV. As far as the radial dependence of the electron distribution is concerned, we assume that f e (R, 0) follows the same R-dependance of the CRs, which can be obtained by using a best fit procedure to the data in Figure 11 of [88]. This yields However, following Bloemen [88] -who suggested a stronger radial gradient for the electron component of the CRs -we also tested the effect of using a steeper radial electron distribution on the IC γ-ray flux. We anticipate that the corresponding results show that the IC γ-ray flux does not change significantly for galactic longitudes |l| ≤ 90 0 (irrespectively of the latitude values) while it increases up to a factor of two at l = 180 0 for |b| ≤ 30 0 .
The last quantity to be specified in eq. (22) is the average background photon density < n ph (ρ, l, b) > or, equivalently, the background photon flux Φ ph (ρ, l, b) emitted by stars and dust Note that the photon flux dΦ ph (ρ, l, b) at a point P (ρ, l, b) from the solid angle dΩ subtended by an infinitesimal area da ′ centered in P ′ (R ′ , φ ′ , z ′ = 0) on the galactic plane is given by where α is the angle between the normal to the area da ′ and the direction PP ′ . We can trace the surface brightness I ⋆,d (R ′ ) to the stellar/dust distribution. Assuming that visible matter makes up an exponential disk, we set where h * ,d ≃ 3.5 kpc is the scale length for the visible matter and the constant A * ,d is fixed by the total disk luminosity as In this way, we get A ⋆ = 4.71 × 10 20 γ cm −2 s −1 and A d = 1.64 × 10 22 γ cm −2 s −1 , with R d ≃ 15 kpc. By integrating eq. (25) on the galactic disk, we find Finally, by using eqs. (24), (26) and (28) -and recalling eq. (20) -eq. (22) can be rewritten in the form where we have set and Numerical values of Φ IC γ (> E γ , l, b) at high-galactic latitude are exhibited in Table  2 and plotted in Figure 3. Table 2. The galactic diffuse γ-ray intensity due to IC scattering of high-energy electrons on background photons from stars and dust is given (in units of 10 −7 γ cm −2 s −1 sr −1 ) for a = 2.0, 2.4 and 2.8. The results for a = 2, 2.8 are reported for illustrative purposes. We adopt the following values: T * = 2900 K, L * = 3.5 × 10 10 L ⊙ and T d = 29 K, L d = 1.5 × 10 10 L ⊙ .

Discussion
Our main result are maps for the intensity distribution of the γ-ray emission from baryonic dark matter (DM) in the galactic halo and from IC processes in the galactic disk. In order to make the discussion definite, we take the fraction of halo dark matter in the form of molecular clouds f ≃ 0.5. As far as the IC emission is concerned, the standard electron spectral index a = 2.4 is used. We stress that the shape of the IC maps does not depend on the value of a.
In Figures 1 we exhibit the contour plots in the first quadrant of the sky (0 0 ≤ l ≤ 180 0 , 0 0 ≤ b ≤ 90 0 ) for the halo γ-ray flux Φ DM γ (E γ > 1 GeV). Corresponding contour plots for E γ > 0.1 GeV are identical, up to an overall constant factor equal to 8.74, as follows from eq. (16). Figure 1a refers to a spherical halo, whereas Figure 1b pertains to a q = 0.5 flattened halo. Regardless of the adopted value for q, Φ DM γ (E γ > 1 GeV) lies in the range ≃ 6 − 8 × 10 −7 γ cm −2 s −1 sr −1 at high-galactic latitude. However, the shape of the contour lines strongly depends on the flatness parameter. Indeed, for q > ∼ 0.9 there are two contour lines (for each flux value) approximately symmetric with respect to l = 90 0 (see Figure 1a). On the other hand, for q < ∼ 0.9 there is a single contour line (for each value of the flux) which varies much less with the longitude (see Figure 1b).
As we can see from Table 1 and Figures 1, the predicted value for the halo γ-ray flux at high-galactic latitude is close to that found by Dixon et al. [1] (see also Table  3). This conclusion holds almost irrespectively of the flatness parameter.
Moreover, the comparison of the overall shape of the contour lines in our Figures  1a and 1b with the corresponding ones in Figure 3 of Ref. [1] entails that models with flatness parameter q < ∼ 0.8 are in better agreement with the data, thereby implying that  most likely the halo dark matter is not spherically distributed. This result has been also recently confirmed in the analysis by [82].
In Figure 2 we present contour plots for the γ-ray flux due to the IC scattering, for E γ > 1 GeV. The corresponding contour plots for E γ > 0.1 GeV are identical, up to an overall constant factor equal to 5 (this follows from eq. (31)). The contour lines decrease with increasing longitude.
Rough values of the measured residual γ-ray flux at E γ ≥ 1 GeV (after subtraction of both the isotropic background and the standard galactic diffuse component) is given for different galactic latitude and longitude values (interpolated from Fig. 3a in [1]). Nevertheless, given the large uncertainties both in the data and in the model parameters (such as for instance the electron scale height and the electron spectral index a), one might also explain the observations with a nonstandard IC mechanism [5]. Our calculation, however, seems to point out that the IC contour lines in Figure  2 decrease much more rapidly than the observed ones for the halo γ-ray emission (see Figure 3 in [1]). More precise measurements with a next generation of satellites are certainly required in order to settle the issue.

Gamma rays from the halo of M31
As M31 resembles our galaxy, the discovery of Dixon et al. [1] naturally leads to the expectation that the halo of M31 should give rise to a γ-ray emission as well. Below, we will try to address this issue in a quantitative manner, assuming that the halo of M31 is structurally similar to that of our galaxy and that our model for baryonic dark matter is correct.
We suppose that the various parameters entering the calculations in Sections 3 and 4 take similar values for M31 and for the Galaxy, apart from the M31 central dark matter density ρ(0) ≃ 2.5 × 10 −24 g cm −3 and the M31 core radiusã ≃ 5 kpc. Accordingly, the evaluation of the corresponding flux Φ M 31 γ halo proceeds as before, with only minor modifications. Specifically, we can use again eq. (16) -with I 2 still given by eq. (18) -but now I 1 is to be replaced by L 1 (see below), in order to account for the different geometry. Notice that f in eq. (16) presently denotes the fraction of halo dark matter of M31 in the form of H 2 clouds.
Consider a generic point P in the halo of M31, and let R and r denote its distance from the centre O of M31 and from Earth, respectively. Since the distance of O from Earth is D ≃ 650 kpc, we have R(r) = (r 2 + D 2 − 2rDcosθ) 1/2 , where θ is the angular separation between P and O as seen from Earth. For simplicity, we suppose that the M31 halo is described by an isothermal sphere with radius R H and density profile ρ(R) = ρ(0) 1 + (R/ã) 2 .
Note that the ensuing amount of dark matter in M31 turns out to be about twice as large as that of the Galaxy. According to the discussion in Section 2, the dark clusters should populate only the outer halo of M31. So, we compute Φ M 31 γ halo from regions of the M31 halo with R min < R < R H , with R min ≃ 10 kpc and R H ≃ 100 kpc, for definiteness. As it is easy to see, the values of θ corresponding to R min and R H are θ min ≃ 1 0 and θ H ≃ 9 0 , respectively.
Observe that regions of M31 halo with angular separation less than θ min from O do not contribute in eqs. (33) and (34), and so Φ M 31 γ halo should be regarded as a lower bound on the total γ-ray flux from M31 halo.
This value has to be compared both with the γ-ray flux from M31 disk and with the γ-ray emission from the halo of the Galaxy. The former quantity has been estimated to be ≃ 0.2 × 10 −7 γ cm −2 s −1 for E γ > 0.1 GeV [89,90] within a field of view of 1.5 0 × 6 0 , whereas the latter quantity, integrated over the entire field of view of M31 halo, is ≃ 4.3 × 10 −7 γ cm −2 s −1 for E γ > 0.1 GeV, according to our results in Section 4 and 6. † As far as observation is concerned, no γ-ray flux from M31 has been detected by EGRET. Accordingly, the EGRET team has derived the upper bound [91] Φ M 31 γ (E γ > 0.1 GeV) < ∼ 0.8 × 10 −7 γ cm −2 s −1 .
Unfortunately, a direct comparison between eqs. (35) and (36) is hindered by the fact that eq. (36) is derived under the assumption of a point-like source.
Clearly, a good angular resolution of about one degree or less is necessary in order to discriminate between the halo and disk emission from M31. So, the next generation of γ-ray satellites like AGILE and GLAST can test our predictions.