Gamma-ray and Radio Constraints of High Positron Rate Dark Matter Models Annihilating into New Light Particles
The possibility of explaining the positron and electron excess recently found by the PAMELA and ATIC collaborations in terms of dark matter (DM) annihilation has attracted considerable attention. Models surviving bounds from, e.g, antiproton production generally fall into two classes, where either DM annihilates directly with a large branching fraction into light leptons, or, as in the recent models of Arkani-Hamed et al., and of Nomura and Thaler, the annihilation gives low-mass (pseudo)scalars or vectors which then decay into or . While the constraints on the first kind of models have recently been treated by several authors, we study here specifically models of the second type which rely on an efficient Sommerfeld enhancement in order to obtain the necessary boost in the annihilation cross section. We compute the photon flux generated by QED radiative corrections to the decay of and show that this indeed gives a rather spectacular broad peak in , which for these extreme values of the cross section violates gamma-ray observations of the Galactic center for DM density profiles steeper than that of Navarro, Frenk and White. The most stringent constraint comes from the comparison of the predicted synchrotron radiation in the central part of the Galaxy with radio observations of Sgr A*. For the most commonly adopted DM profiles, the models that provide a good fit to the PAMELA and ATIC data are ruled out, unless there are physical processes that boost the local anti-matter fluxes more than one order of magnitude, while not affecting the gamma-ray or radio fluxes.
There have recently been indications of a very interesting enhancement in the amount of cosmic ray electrons and positrons detected near the Earth, both seen by PAMELA in the ratio of positrons to the sum of electrons and positrons between a few GeV and 100 GeV pamela (), and by ATIC in the sum of electrons and positrons at several hundred GeV to 1 TeV atic (). While these so far unexplained excesses might be due to standard astrophysical processes astrophysical (), positrons also constitute one of the promising channels in which to search for dark matter (DM; for reviews, see reviews ()), and these new experimental findings have therefore already triggered a large number of theoretical analyses trying to explain the data as being induced by DM annihilation or decay (see e.g. Ref. BBE08 () and references therein for supersymmetric DM, Refs. post-pamela () for alternative DM scenarios and Refs. decaying () for decaying DM scenarios). In general, these analyses seem to point at the need for DM particles with masses in the TeV range that annihilate, with a very large rate, dominantly into charged light leptons.
The bremsstrahlung process, falling like , is generally regarded in particle physics as having a “soft” spectrum. In the astrophysical context, this is, however, on the contrary a quite hard spectrum, since most of the background -ray spectra like those from acceleration near supernova remnants usually fall like or faster. Gamma-rays from DM generally feature a spectrum that is somewhere in between these two at low energies () and drops even faster close to the DM particle mass BUB () (for important exceptions see, however, BBE ().) If the DM particles annihilate directly into a pair of charged leptons, the photon distribution from the process , for , is to a good approximation of the Weizsäcker-Williams form (see, e.g., FSR ()):
where and is the annihilation rate for the lowest order process (Note that the above approximation also breaks down when there is a symmetry that suppresses the annihilation into two-body, but not into three-body final states bergstrom89 ()).
This case has recently been treated by Bell:2008vx (); Bertone:2008xr (); Cholis:2008wq (). (The last of these references also briefly treats, but leaves for a more detailed calculation, the kind of processes we will compute here.) It was found that the gamma-rays produced in DM models with these annihilation modes lead to rather severe constraints. Even more stringent bounds on this type of DM models that try to explain the PAMELA and ATIC data arise from the synchrotron radiation produced by the resulting population of electrons and positrons, in realistic models of the DM density distribution and for a wide variety of assumptions about the magnetic field in the inner Galaxy Regis:2008ij (); Bertone:2008xr ().
It remains to consider another possibility, where DM annihilates into a new type of light (sub-GeV) particles that in turn dominantly decay into light leptons (see arkani-hamed () for a general account of this idea). The advantage of this type of models is that the strongly constrained decay into hadronic modes (see, e.g., Cirelli:2008pk ()) is kinematically forbidden and that Sommerfeld enhancements in the limit of the small galactic DM velocities expected today allow for the very large annihilation cross sections that are needed to explain the PAMELA/ATIC results, but which at first seem to be at odds with the cross sections required to get the right thermal relic density for the DM. Another interesting feature of the Arkani-Hamed et al. model arkani-hamed () is that it encompasses ideas that have been proposed to explain the WMAP haze wmap-haze () and the INTEGRAL excess Finkbeiner:2007kk ().
|Arkani-Hamed et al. type||Nomura-Thaler type|
As pointed out in arkani-hamed (); Cholis:2008qq (), one may basically distinguish between scalar and vector and whether or not (in which case it dominantly decays into ). For , even decays into pions should be taken into account (which we neglect here). While MeV is roughly needed not to be in conflict with Big Bang Nuclesynthesis, one has to require MeV in order to get Sommerfeld enhancements of the order that are needed to explain the PAMELA/ATIC result with these types of DM models. Based on this discussion, we adopt the four benchmark settings A1–A4 summarized in Tab. 1.
While arkani-hamed () describes a rather general set-up, nomura () introduces a concrete realization of this idea; the proposed model has the appealing feature of containing a “standard” Peccei-Quinn axion and can be embedded in a fully realistic supersymmetric scenario. Here, DM annihilates into a scalar and a pseudoscalar , . With a mass scale of , the latter mostly decays into muons, which subsequently decay into electrons or positrons. The benchmark models for this setup N1–N5 are also given in Tab. 1.
For the first particle created in the annihilation, we analytically compute the photon multiplicity from in the rest frame of . We then make a Lorentz boost back to the DM frame, i.e. the Galactic rest frame, to get
with since the annihilation takes place essentially at rest (typical galactic velocities are ). Axions resulting from we treat in a similar way, boosting them first to the -frame and from this to the DM frame. Since may have a mass up to 50 GeV, the gamma-ray spectrum may even receive important contributions from its decay into bottom quarks or tau leptons, a possibility which we will shortly return to. (Bremsstrahlung from electrons in the muon decay will give s of lower energies and will thus not be important for our constraints.)
Summing up all these contributions, we arrive at the total photon spectrum in the DM frame that we show in Fig. 1 for the models N1–N5 in Tab. 1. We also include the corresponding spectra obtained in the Arkani-Hamed et al. set-up (models A1–A4) and, for comparison, the case of 1 TeV DM particles directly annihilating into or . Please note that, from Eq. (2), the quantity for the models listed in Tab. 1 is independent of as long as ; the direct annihilation of DM into leptons, on the other hand, does contain a logarithmic dependence on . Let us mention that while Eq. (1) provides a rather good approximation to our analytic results for photons radiated from pairs, it overestimates the photon yield from muons (especially when the mass of the decaying particle is close to like, e.g., in model AH4).
Once a DM profile is assumed, it is straightforward to estimate the corresponding gamma-ray flux from a solid angle towards the galactic center:
where is the line of sight distance. In Fig. 2, we compare the resulting flux to the gamma-ray data from the galactic center taken by the H.E.S.S. telescope hessgc (), which has an angular resolution of about , thus sr. We here show the spectrum for model N3 and, for comparison, the case where decays not only to axions but with a branching ratio of 5% to or (which is the typical case for the model presented in nomura ()). By comparison with Fig. 1, it is straightforward to arrive at the corresponding spectra for the other models in Tab. 1. We have here adopted a so-called Navarro-Frenk-White profile nfw (), with the same parameters as in Ref. Bertone:2008xr (). Note that the gamma-ray spectra in this case are consistent with the HESS data, unlike the case of the annihilation modes discussed in Bertone:2008xr (), for the same density profile. Assuming a profile , as needed to explain the WMAP ’Haze’ (see Ref. wmap-haze ()), the constraints become much more stringent. However, at the same time they become much more sensitive to the dependence of on the velocity dispersion of DM, which inevitably increases in the vicinity of the supermassive black hole at the Galactic center. As we shall see soon, however, it is possible to derive even tighter constraints without making assumptions on the small-v behaviour of .
Before that, however, let us note that another potential source of gamma rays from DM annihilations are dwarf galaxies, like the Sagittarius dwarf galaxy, observed by HESS hess-sgr (). The HESS observations put an upper bound on the integrated gamma flux above 250 GeV of cms. Assuming an NFW (isothermal) profile in the Sagittarius dwarf galaxy, this can be translated to the limit () cm s for model N3. For the other models in Tab. 1, the limits differ by a factor of a few as indicted by the spectra in Fig. 1. For other dwarf galaxies, the limits are similar: using a conservative estimate of the line of sight integral from Ref. strigari (), the limits on the gamma flux from Willman 1 as observed by Magic magic-wil1 (), e.g., translate to cm s. However, the uncertainties from dynamical constraints strigari () are large and improved future data might result in better constraints. As one typically needs a boost of order to explain the PAMELA data, we note that the limits derived here are very close to the required . This means that for some models, like AH1–AH3, the more optimistic scenarios for the halo profile of e.g. the Sagittarius dwarf are excluded.
A rather stringent constraint on the rate of injection of high energy in the Galaxy comes from the analysis of the synchrotron radiation produced by these particles as they propagate in the Galactic magnetic field. Although observations of different targets and at different wavelengths provide interesting constraints Bertone:2002ms (), the most stringent ones come from radio observations of the Galactic center, where the DM density is highest Bertone:2002ms (); radiogc (); Regis:2008ij ().
The synchrotron luminosity generated by a distribution of electrons and positrons produced by a DM distribution with profile in a magnetic field is
where , and we have adopted the monochromatic approximation for the synchrotron emission, assuming , with , for its spectrum.
By comparing the predicted synchrotron radiation with radio observations, we can set limits on the annihilation cross section for any given annihilation channel, following a procedure similar to Ref. Bertone:2008xr (). The most stringent constraint comes from the upper limit on the radio emission from a cone with half-aperture of towards Sgr A* at Davies (), which we translate in Fig. 3 to the vs. mass plane. Let us stress that the plotted in Fig. 3 is the effective annihilation cross section, including both Sommerfeld enhancements and boosts due to substructures. The only way to avoid our constraints would thus be to boost the local anti-matter fluxes by more than one order of magnitude without affecting the gamma-ray or radio fluxes. Although this theoretical possibility cannot be ruled out (e.g. Refs.boost ()), it appears to be unlikely for a realistic distribution of substructures in the Milky Way halo. Numerical simulations seem to indicate that the boost factors due to substructure is rather small Diemand:2008in (). How big the boost factors could be are still under debate and recent simulations Diemand:2008in () indicate that locally they are at most a factor of a few. A recent study Afshordi () develops a model that indicates that the local boost could be about a factor of ten. The details of the mechanism giving such large boosts are yet to be presented, however. For more discussion about boost factors, see Ref. Bertone:2008xr ().
The two sets of curves give the maximum annihilation cross section compatible with radio observations of Sgr A* for two different DM profiles: Einasto and NFW. The shaded region, corresponding to the range of annihilation cross sections that provide a good fit to the PAMELA and ATIC data, appears to be in conflict with observations, unless the DM profile is more shallow than expected in current models of structure formation. However, if the DM interpretation of the PAMELA data was corroborated by additional evidence, then our result can be interpreted as a hint of the shallowness of the DM profile.
Profiles steeper than NFW – like the needed to explain the WMAP ’Haze’ wmap-haze () – are ruled out by a rather larger margin. This confirms the dramatic importance of the multi-wavelength approach to DM studies Bertone:2008xr (); Bertone:2002ms (); radiogc (); Regis:2008ij (), especially for DM models tailored to explain anomalies in astrophysical observations.
LB and JE thank the Swedish Research Council (VR) for support. LB, TB and JE wish to thank IAP, Paris, for hospitality when this work was initiated.
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