Constraining Light Dark Matter with Diffuse X-Ray and Gamma-Ray Observations

We present constraints on decaying and annihilating dark matter (DM) in the 4 keV to 10 GeV mass range, using published results from the satellites HEAO-1, INTEGRAL, COMPTEL, EGRET, and the Fermi Gamma-ray Space Telescope. We derive analytic expressions for the gamma-ray spectra from various DM decay modes, and find lifetime constraints in the range 10^24-10^28 sec, depending on the DM mass and decay mode. We map these constraints onto the parameter space for a variety of models, including a hidden photino that is part of a kinetically mixed hidden sector, a gravitino with R-parity violating decays, a sterile neutrino, DM with a dipole moment, and a dark pion. The indirect constraints on sterile-neutrino and hidden-photino DM are found to be more powerful than other experimental or astrophysical probes in some parts of parameter space. While our focus is on decaying DM, we also present constraints on DM annihilation to electron-positron pairs. We find that if the annihilation is p-wave suppressed, the galactic diffuse constraints are, depending on the DM mass and velocity at recombination, more powerful than the constraints from the Cosmic Microwave Background.


I. INTRODUCTION
A wide variety of precision astrophysical and cosmological observations have corroborated the existence of dark matter (DM), without providing any conclusive indications of its nature or its non-gravitational couplings to the Standard Model (SM). For the past 30 years, a broad experimental program has attempted to uncover the DM properties. However, the vast majority of the existing experiments search either for Weakly Interacting Massive Particles (WIMPs) or for axions, overlooking other theoretically viable and motivated possibilities.
One interesting possibility is light dark matter (LDM) in the keV to 10 GeV mass range. In this paper, we focus on such DM and study constraints from existing indirect searches.
While DM decays are less constrained by early Universe cosmology, stringent constraints can be placed on decaying DM from observations of the galactic and extra-galactic diffuse X-ray or gamma-ray background. The lifetime of Weak-scale DM is constrained from observations with the Fermi Large Area Telescope (Fermi LAT) to τ 10 26 sec [28][29][30][31][32], many orders of magnitude larger than the age of the Universe. For DM below O(100 MeV), the usual gamma ray constraints from the Fermi LAT do not apply, although the instruments on several other satellites (listed in Table I below) are sensitive to photons with energies well below a GeV. The available data cover a photon energy range from 10's of GeV down to a few keV, providing the possibility of exploring a much broader range of DM candidates than WIMPs. Indeed, some of these data have already been utilized to constrain LDM, see, e.g., [33][34][35][36][37][38][39][40][41][42][43][44]. Sterile neutrinos with a mass ∼ O(1 − 10 keV) are a particularly popular candidate and their constraints have been explored in, e.g., [36,[38][39][40][45][46][47][48][49][50][51][52][53][54][55][56][57]. Below a few keV, thermal DM candidates become too warm to adequately explain the formation of structure in the Universe, so that such candidates necessarily have a mass above the lower energy bound accessible by these satellite experiments.
The goal of this paper is to derive constraints on light DM candidates in the keV to 10 GeV mass range, using the diffuse photon spectra data listed in Table I  NFW profile is taken from [58,59], the Moore profile from [60], and the cored isothermal profile can be found in [61]. The profiles "Ein, α" are Einasto profiles [62] with slope parameter α.
extend several results in the literature. Taking a largely model-independent approach, we discuss a wide range of DM decay topologies. We consider photons that are produced directly in the decay or from final state radiation (FSR) of charged particles that are produced in two-or three-body decays. We map our results onto several known LDM models, and show limits on the corresponding model parameter space. For example, we consider constraints on a kinetically mixed supersymmetric hidden sector (with the hidden photino decaying toGγ orGe + e − , withG the gravitino) and a sterile neutrino (with the sterile neutrino decaying to a neutrino and a photon). While the constraints we derive are robust, they are based on published data. Consequently, they can easily be improved by optimizing the search regions and taking better account of the signal and background fitting.
While our focus is on decaying DM, we also consider annihilating DM. A thermal relic with a p-wave (or velocity suppressed) annihilation cross section is less constrained from CMB data than s-wave annihilation, since DM is cold at the CMB epoch. For this case, we find that the limits from the diffuse background can be more constraining than the CMB.
The outline of the paper is as follows. In Sec. II, we review both the expected signals resulting from DM decays and annihilations as well as the relevant gamma-ray and X-ray observatories (HEAO-1, INTEGRAL, COMPTEL, EGRET, Fermi). We further discuss our methods for placing the limits on such DM. In Sec. III, we discuss models of decaying light DM such as decaying gravitinos, sterile neutrinos, and hidden photinos. For each model we map the lifetime constraints onto constraints of the model parameter space. In Sec. IV, we take a model-independent approach and constrain the lifetime for various decay topologies.
Sec. V is devoted to constraints on the annihilation cross-section of light DM to electronpositron pairs. We conclude in Sec. VI.

II. CONSTRAINING LIGHT DARK MATTER WITH DIFFUSE PHOTONS
In this section, we discuss the data and the statistical methodwe use to place constraints on decaying and annihilating LDM. We begin with a brief review of the expected signal rate.

A. Flux from Dark Matter Decays and Annihilations
Given a DM annihilation or decay spectrum, dN γ /dE γ , and a galactic DM density profile, ρ(r), the galactic contribution to the differential photon flux per unit energy is given by, Here r 8.5 kpc is the Sun's distance from the Galactic center, ρ = 0.3 GeV/cm 3 is the local DM density, α = 1 (2) for DM decays (annihilations), Γ D is the decay rate, σv is the thermally averaged annihilation rate, and is a dimensionless quantity that describes the density of decays or annihilations along the line-of-sight (l.o.s.) and over the solid angle Ω. We will present results assuming ρ(s) follows the NFW DM density profile [58,59], but in Table I we also list values of J D,A for other DM density profiles for each experimental survey region. Our results can thus be easily rescaled.
Note that the choice of ρ(s) becomes less important for survey regions farther from the galactic plane and also less important for decaying compared to annihilating DM.
In addition to the contribution to the photon flux from DM decays in the Milky Way halo, there is a contribution arising from the smooth distribution of DM throughout the whole Universe (see, e.g., [37,[45][46][47]). A photon produced at redshift z that is detected with energy E was emitted with energy E(z) = E(1 + z). Such a photon was emitted at a comoving distance, χ(z), with where κ = Ω Λ /Ω m ∼ 3 and a flat Universe, Ω m + Ω Λ = 1, is assumed. The extragalactic photon spectrum arising from DM decays at redshift z is given by dN/dE(z), so that the measured flux is Because the photon flux from DM decays scales linearly with the DM density, this contribution is not very model dependent. For dN γ /dE(z) = δ(E(z) − m DM /2), this reduces to the case that is usually considered, namely DM decaying to a redshifted monochromatic gamma-ray line, This effect implies that the spectral shape of a photon "line" from DM decays is smeared to receive contributions from a continuous range of energies.
In principle, similar extragalactic contributions exist for the annihilating DM case. However, the smooth part of extragalactic DM annihilation is subdominant compared to the galactic contribution and may be safely ignored. On the other hand, extragalactic annihilations resulting from DM substructure at low redshift may contribute a significant amount to the photon flux since it scales as the square of the DM density [68]. Since this contribution is not well known [69], we conservatively omit it from our analysis below. When stable charged particles (like electrons) appear as decay or annihilation products, photons will be emitted through final state radiation (FSR). We use the Altarelli-Parisi splitting function, to estimate the photon spectrum, where Q is the square of the momentum imparted to the photon, α EM 1/137, and dN/dE f is the differential rate of decay or annihilation to the final state particle f . For multiple charged-particles in the final state, we sum over the contributions.

B. Data
We place constraints on LDM using the data summarized in Table I and shown in Fig. 1. We emphasize that none of the datasets have been optimized for LDM searches. It is therefore likely that significantly stronger constraints may be achieved with dedicated analyses.
As mentioned above, we assume an NFW profile in all cases, but the results can easily be rescaled for other profiles using the information in Table I. For the inner-galaxy data from INTEGRAL or COMPTEL, the bounds from decaying DM can be adjusted by up to O(30%); using high-latitude data, the difference is typically less than O(10%). In contrast, the expected photon flux from DM annihilations near the galactic center can change by up to an order of magnitude for different choices of the density profile.
For our analysis we use the following datasets: • HEAO-1. We use data from observations of 3-50 keV photons made with the A2 High-Energy Detector on HEAO-1 [63]. Other datasets from the experiment are significantly weaker than those from the INTEGRAL experiment discussed below. To avoid point source contamination, the observations come from regions of the sky 20 • above the galactic plane. As is clear from different choices of density profile. The excellent energy resolution allows us to remove the well-resolved 511 keV line in our analysis.
• COMPTEL. We use the COMPTEL data from [65]. These observations are obtained by averaging over the sky at latitudes | | ≤ 60 • and |b| < 20 • . Compared to the INTEGRAL region of interest, the model predictions are about half as sensitive to the density profile at these galactic latitudes. We find an O(20%) uncertainty for DM decay bounds due to the DM density profile.
• EGRET. We use the data shown in panel E of Fig. 2 in [66], which lies in the 20 MeV are sensitive only at the few-percent level to the DM density profile.
• Fermi. We use data from the upper panel of Fig. 12 of [67], with 0 < < 360 • and 8 • < |b| < 90 • , between 200 MeV-10 GeV. We choose these latitudes to enhance the signal to background ratio while minimizing the uncertainty in the DM profile. The resulting decay bounds are only O(5%) sensitive to varying the DM density profile.

C. Statistical Methodology
Our goal is to obtain robust, conservative bounds using the above data sets. We do this by requiring that the predicted count from the DM signal in each bin does not exceed the observed central value plus twice the error bar. In all cases we use the statistical uncertainties, except for EGRET and Fermi where we take the dominant systematic uncertainties.
These bounds could be significantly strengthened with dedicated searches in the future and by including fits to different astrophysical background components, e.g., from astrophysical ICS. In Appendix A, we show the improvement that could be obtained with a goodnessof-fit test that assumes knowledge of the various backgrounds. The expected improvement varies between a factor of a few to an order of magnitude, but involves larger systematic uncertainties as the backgrounds are not precisely known. For this reason, the results we present use only this simple test described above.

III. MODELS OF DECAYING LIGHT DARK MATTER
In this section, we outline several simple scenarios that can accommodate LDM, and we place constraints on the model parameter space. The models below should be viewed as benchmarks that are not, however, complete. In particular, we do not discuss the production mechanism that results in the observed relic abundance. In the next section, we will derive "model-independent" constraints, where the results are presented as generic constraints on the lifetime versus mass for a given decay topology.

A. Hidden Photino
Consider a supersymmetric hidden sector, with an additional U (1) d gauge group [4,7,[70][71][72][73]. We assume that the SM and hidden sector can interact with each other through gauge Constraints on hidden photino decay to left: gravitino and photon and right: gravitino and hidden photon (with the latter taken to have mass m γ d = 0.9m γ d and going to final state In the left plot, the solid (dotted) lines are with √ F = 10 4 (10 2 ) TeV.
The constraints are derived from the diffuse gamma-and X-ray data taken from HEAO-1 (orange), INTEGRAL (green), COMPTEL (blue), EGRET (red), and Fermi (yellow). In the "Short-Lived" region the DM lifetime is shorter than the age of Universe. Above the solid red line, the hidden photino is stable.
kinetic mixing [74,75], where W d (W Y ) are the supersymmetrized field strength of the hidden gauge group (hypercharge). The value of may naturally be of order 10 −3 − 10 −4 when generated by integrating out heavy fields charged under both sectors. Conversely, if Eq. (7) results from higher dimensional operators, can be significantly smaller, as we will assume below in order to obtain MeV-GeV masses.
An interesting possibility is to have the hidden gaugino play the role of DM. To realize this, supersymmetry must be broken and communicated both to the visible and hidden sector. If the communication occurs through gauge mediation, the breaking in the hidden sector may be significantly smaller than in the visible sector as supersymmetry breaking is transmitted to the hidden sector through D-term mixing [71]. As a consequence, the hidden  photon mass is given by, where Here, a long lifetime requires a slightly larger SUSY breaking scale. Note that the two possibilities lead to distinct indirect detection signals. In the first case one expects a spectral line, while in the second the spectrum is dominated by the FSR photons from the kinematically accessible charged particles that arise from the decay of the hidden photon.
The constraints for both cases are shown in Fig. 2 Above the solid red line, the hidden photino is stable. The photon spectrum for a variety of different decay channels may be derived from [78]. In both panels, the "Short-Lived" region indicates that the DM lifetime is shorter than the age of Universe.
These additional constraints are shown in Fig. 3 together with the limits derived here (and shown in the left panel of Fig. 2), for the case where the hidden photon decays directly to a photon and a gravitino,γ d → γG. We note that some of these additional constraints are model dependent and may be evaded.
Due to its mixing with the active neutrinos, it may decay either via a 2-or 3-body channel.
The leading diagrams that contribute to these decay channels are shown in Fig. 4. In its simplest form, the theory at low energy is described by two parameters: • m s -the sterile neutrino mass • sin θ α -the mixing angle between ν s and active neutrinos of flavor α; in what follows, we will only consider ν s − ν e mixing.
The mixing above can be induced, for example, in supersymmetric theories with a superpotential, W = XLLE c . The two-body decay rate for a Majorana neutrino is given by [89] τ νs→νγ 9α EM sin 2 θ while the three-body decay rate is [90] τ νs→ναe + e − c α sin 2 θ 96π 3 G 2 F m 5 Here the neutrino flavor α = e, c α = 1+4 sin 2 θ W +8 sin 4 θ W 4 0.59 [90], and we are only considering decays to e + e − pairs. The resulting gamma-ray fluxes from both channels contribute at roughly similar levels once the splitting function is introduced.
The relic abundance of sterile neutrinos is model dependent and varies according to the specific production mechanism and dynamics in the early Universe. An irreducible and Eqs. (11) and (12). The constraints from the diffuse gamma-and X-ray data are HEAO-1 (orange), INTEGRAL (green), COMPTEL (blue), and EGRET (red). Within the solid black region, the neutrino energy density must be greater than the observed DM density. Above (below) the black solid line, the neutrino lifetime is shorter (longer) than the age of the Universe. Within the green boundaries, the sterile neutrino is ruled out by Ly-α forest data [48,49]. Two cases for the sterileneutrino energy density are assumed. In the left plot, the density is assumed to precisely equal the DM energy density everywhere below the dark and light gray regions. In the right plot, the density is determined by the (irreducible) DW mechanism.
UV-insensitive contribution to the abundance of sterile neutrinos arises from the so-called Dodelson-Widrow (DW) mechanism [91] in which the neutrinos are produced via oscillations.
Thus, in the absence of new dynamics at low temperature, one finds [48] Ω s 0.25 Additional contributions may arise from, e.g., non-thermal production [8] or due to an extended Higgs sector [92,93].
In order to place model-independent bounds on the parameter space of sterile neutrinos, we consider two different possibilities for the size of the sterile-neutrino relic abundance.
First, we consider an unspecified UV mechanism that contributes to the DM density in those regions where the DM is under-abundant, setting Ω νs = Ω DM . Next, we assume the relic abundance is determined solely by the DW mechanism and, depending on the mixing angle and mass, Ω νs can be greater than or less than Ω DM . We show our bounds for both these cases in the left and right panel of Fig. 5, respectively, in the m νs −sin 2 2θ plane. In addition, we show existing bounds from the observation of the Lyman-α forest [49] and the overclosure region, in which the neutrino density produced by the DW mechanism exceeds the observed DM density. We also show the region where the sterile-neutrino lifetime is shorter than the age of the Universe, and hence it cannot act as DM. Several additional constraints exist on sterile neutrinos, for example, from the power spectrum of large scale structure [94] and of the CMB [94], from BBN [95], and from Supernova-1987A [96]. However, these constraints lie in the region where either the lifetime is too short or where the DM density is too high.

C. Gravitino Dark Matter
Another interesting possibility is gravitino DM [97][98][99][100][101][102][103][104]. The gravitino may be unstable on cosmological timescales and here we consider gravitino decays induced by R-parity violating (RPV) interactions [99][100][101]. Since we are interested in light DM, we will focus on the RPV operator that allows the gravitino to decay to leptons, W = λ ijk i j e c k . A small coefficient λ in the RPV vertex can ensure that the gravitino lifetime is longer than the age of the Universe.
Gravitinos are typically produced in three processes [97]: (i) gaugino scattering, dominantly at the re-heat temperature, (ii) freeze-out and decay of the lightest ordinary supersymmetric particle (LOSP, such as a neutralino), and (iii) freeze-in production from decays of visible sector particles, dominated at temperatures of order the superpartner masses.
Once gravitinos are produced with the observed relic abundance, their decay rate is controlled by the strength of the RPV vertex, as well as by the mass of the observable superpartners. The RPV operator considered here allows decays in one of two ways, as shown in the diagrams of Fig. 6. First, through an off-shell slepton, one hasG → ν j This process is suppressed both by three-body phase space and by the slepton propagator, which gives an additional factor proportional to (m 3/2 /m) 4 , wherem is the slepton mass. One finds [101] τ a more exact expression can been found in [100].
A second, two-body, decay mode isG → γν, which usually dominates the decay width [101] and gives stronger bounds. It is induced by a mixing between the photino and the neutrino, |Uγ ν |, which occurs if the RPV terms induce a VEV for the sneutrino [99,101] or via a loop with a charged lepton and slepton. This gives a gravitino lifetime [98,99], In the left panel of Fig. 7 we show the constraints on the photino-neutrino mixing angle as a function of the gravitino mass. In deriving the bound we require that the gravitino has the observed DM relic abundance. We do not show limits from BBN as those depend strongly on the dominant production mechanism and hence on the re-heat temperature and the spectrum of the superpartners [104].
with Λ eff = Λ/λ, the effective cutoff scale of the theory. The outgoing photon has an energy E γ = (m 2 1 − m 2 2 ) /2m 1 . In the right panel of Fig. 7, we show the limits on Λ eff versus the χ 1 mass, m 1 . Since the effective operator that controls the decay is dimension 5 and not higherdimensional, the limits are exceptionally strong, constraining the effective cutoff scale to be very high (or conversely, the corresponding coupling to be small, λ 1). An approximate symmetry in the UV may be required to protect these decays.

E. Dark (Pseudo-) Scalars
As a final model for light DM, we consider two-body decays of diphotons or charged particles.
If DM is a pseudoscalar decaying to two photons, its lifetime is [105] τ π d →γγ α 2 EM m 3 (17) Here f π d is the decay constant in the hidden sector, which we assume is Abelian. This decay produces a spectral line at an energy m π d /2. We show the constraint in the left panel of Fig. 8, from which it is clear that the scale of f π d needs to be very high.
If DM is a scalar that decays to charged particles that produce photons through FSR, e.g., φ → e + e − , the lifetime is The spectrum is bounded by the energies 0 < E γ < m φ /2. The constraints on the coupling g are shown in the right panel of Fig. 8. As is apparent, tiny couplings are required for such DM to agree with observations.

IV. MODEL-INDEPENDENT BOUNDS AND SPECTRA
In the previous section, we presented limits on specific model parameters. In this section, we fill in some of the details of the analysis there, and show bounds in terms of the lifetime only, making the constraints "model-independent." Despite the wide variety of possible decays that produce a photon signal, there are very few distinct event topologies of interest: • Two-or three-body decays, with or without FSR.
• Two-body cascade decays, where one or both of the decay products themselves subsequently decay to photons or charged particles.
In the limit of small outgoing particle masses, the differential decay width at low energies for each of these topologies may be written as a function of the total width, the photon energy, E i , and the mass of the outgoing particle, m i . We will use the small parameters to expand our results.
When relevant in the model-independent bounds below, we only consider photons and electrons as SM final states. Typically these bounds will weaken moderately as new decay channels to additional charged or unstable heavier particles open up. One exception, however, is for the case where the decay products include π 0 's which consequently decay to photons. In such a case, a significant improvement in the limits is expected due to the sharp spectral feature.

A. Two-Body Decays Involving a Photon
We first consider two-body decays of DM directly to a photon and a neutral particle, or to two photons. Models that give line-like features include a hidden photino decaying to a gravitino and a photon via kinetic mixing, as discussed in Sec. III A. There are, of course, a profusion of other model-building possibilities that produce a monochromatic photon. These decays can produce one or two monochromatic photons with differential width, Here ν 2 ≡ m 2 /m DM refers to the mass of the outgoing decay partner, in the case of a single photon. The constraints on the lifetime for the decay to two photons are shown in Fig. 9.

B. Two-Body Decays with FSR
Two-body decays to charged particles produce photons through FSR. The differential width to photons is approximately given by integrating a δ-function with the Altarelli-Parisi splitting function, as shown in Eq. (6), to give where the spectrum is bounded by the energies 0 < E γ < m φ /2. We use the exact calculation of the three-body final state for the spectra and the exclusion regions in Fig. 10. In this figure, we show the dimensionless galactic photon spectrum Bounds on the DM decay lifetime for this process, with regions as in Fig. 2. as well as the redshifted extragalactic spectrum dN γ,eg /dx (dashed lines). The extragalactic spectrum is calculated by performing the integral in Eq. (4) normalized such that the total number of photons for 0 < x < 1 is equivalent for galactic and extragalactic photons.
As described above, this decay naturally arises if the DM is a light scalar. Furthermore, the decay to two leptons is a popular toy model that parameterizes possible DM decay and annihilation. The bounds for this case are shown on the right of Fig 10. As expected, they are a few orders of magnitude weaker than the bounds from the monochromatic decay shown in the previous subsection.

C. Two-Body Cascade Decays
We next consider the case of DM decay to a pair of neutral particles, one of which subsequently decays to e + e − : φ 1 → φ 2 φ 3 → φ 2 + − . An example for a decay of this type was presented in Sec. III A for the hidden photino model, where the hidden photino de- cays to a gravitino and hidden photon, which then subsequently decays to charged leptons: We derive the photon spectrum from these cascade decays from [78].
The spectrum for FSR resulting from a single boosted lepton is where x = 2m 1 E γ / (m 2 1 + m 2 3 − m 2 2 ). This spectrum, under the assumption of m 3 = 0.9m 1 and m 2 = 0.01m 1 , is shown on the left of Fig. 11 where the galactic (solid lines) and redshifted extragalactic (dashed lines) contributions are shown. As can be seen, Eq. (24) does not have a precise cutoff at E γ = m 1 /2. However, as noted in [78], the number of unphysical photons produced with E γ > m 1 /2 is second order in the expansion parameters and the effect of this error on the bounds is negligible.
The constraints on the lifetime of the decaying particle are shown on the right of those with monochromatic photons.

D. Three-Body Decays with FSR
Next we examine three-body DM decays, where the DM decays to a pair of charged particles plus a neutral particle. Our formula was specifically derived for the case of Weak decays of a sterile neutrino, ν s → νe + e − (as we discussed in Sec. III B), though only minor changes result for a more generic decay φ 1 → φ 2 e + e − .
The differential width of a fermionic DM decaying to e + e − ν via weak processes and including FSR is, Here we neglect both the neutrino and the electron masses and "..." stands for higher-order terms in ν e . For the case of a decay process mediated by a heavy neutral scalar particle, the above remains the same with the omission of the last term.
The spectrum for the above is plotted on the left of Fig. 12 where, as before, the galactic (solid lines) and redshifted extragalactic (dashed lines) contributions are shown. The con- straints on the lifetime are shown on the right of Fig. 12. We find the bounds to be similar in magnitude to the two-body + FSR case, however sensitivity to the endpoint feature in the spectrum is apparent and results in the wiggles displayed in the figure.

E. Three-Body Decays Involving Photons
Three body decays such as φ 1 → φ 2 γγ are also possible. We remain agnostic about the UV completion and do not embed this interaction in any of the theories above. Nonetheless, we include it here for completeness.
To obtain bounds, we assume that this decay is induced by the higher-dimensional oper- We see here that the width is exponentially sensitive to the energy in the limit ν 2 → 0, which means that the photons from this decay are preferentially grouped near the DM mass.
Consequently, for a given m χ , the constraint arises from a single bin in a given experiment.
We display the spectrum and constraint on the lifetime in Fig. 13, with the assumption m 2 = 0. In this limit, the differential spectrum is the same regardless of m 1 . As expected, these bounds compare favorably to the monochromatic photon lines.

V. ANNIHILATING LIGHT DARK MATTER
Here we consider bounds on annihilating DM, specializing to the case of annihilation to e + e − (see also [106]). The differential photon spectrum for this case is where we have defined δ = (1 − λ γ ) (1 − λ γ − ν 2 e ). The bounds are shown in Fig. 14. From Table I, we see that these results are sensitive (within factors of a few) to the DM density profile (we use the NFW profile for all results), especially for experiments that observe regions near the center of the galaxy such as INTEGRAL and COMPTEL. For DM masses below ∼ 100 MeV the bounds are stronger than the thermal annihilation cross-section around 3 × 10 −26 cm 3 /s. These bounds can be compared with those from CMB observations, which are very strong for s-wave processes. Indeed, for DM masses below ∼ 7 GeV, the annihilation cross-section must be smaller than the thermal annihilation cross-section of 3 × 10 −26 cm 3 /s. At first sight, it appears that the diffuse photon bounds are not competitive with the CMB bounds.
However, p-wave annihilation rates may be larger in the galaxy today relative to the CMB epoch if the velocity of the DM at recombination is smaller than the galactic velocity which we take to be, v 0 = 220 km/sec.
The velocity of the DM at recombination depends on the kinetic decoupling temperature. As long as the DM remains kinetically coupled to the plasma, its velocity is v DM ∼ 3T γ /m DM . Once the DM kinetically decouples, however, it cools much more quickly: its temperature at redshift z is T DM = T kd z z kd 2 , for a kinetic decoupling temperature T kd at redshift z kd . As a result, the DM velocity is where we define x i ≡ T i /m DM . The above is easily smaller than the observed galactic velocity, even for very light DM.
We show in Fig. 14 the CMB constraint from s-wave processes, as well as the constraint from p-wave processes for x kd = 10 −4 and 10 −6 , taking T γ = 0.235 eV at the CMB epoch (corresponding to z CMB = 1000). In order to compare the galactic and CMB constraints for both s-and p-wave annihilation, we show contours of σv ∝ (v DM /v 0 ) 2(n−1) , where n = 1(2) for s(p)-wave. We can see that the CMB constraints are always stronger than the diffuse photon constraints for s-wave annihilation. However, the diffuse constraints are stronger than the CMB constraints for p-wave annihilation, especially for larger kinetic-decoupling temperatures where the DM is colder. In addition to model-independent constraints, we also placed limits on specific benchmark models of light DM: hidden-photino DM, sterile-neutrino DM, gravitino DM, dipole DM and hidden (pseudo-) scalar DM. We found that the constraints from decaying DM are often stronger than other existing experimental, astrophysical, or cosmological constraints.

VI. CONCLUSIONS AND FUTURE IMPROVEMENTS
We conclude that X-ray and gamma-ray observatories provide a powerful and independent probe of light DM. Appendix A: Constraints with fits to astrophysical backgrounds In this paper, we derived robust, conservative constraints by only taking into account the DM signal, as described in Sec. II C. Stronger constraints can be obtained by fitting the DM signal simultaneously with the different astrophysical background components. This could improve the constraints especially if the DM signal spectrum has a sharp feature like a line or an edge (as appears in an FSR spectrum). However, for softer spectra, while the constraints may be formally stronger, they also suffer from larger systematic uncertainties, since the background components are not known precisely. Furthermore, the isotropic extragalactic flux, which contributes an O(1) amount to the diffuse galactic signal at high galactic latitudes, can smear out any spectral shapes [107].
To illustrate the improvements possible with using a simultaneous fit of signal and backgrounds, we use the background components as derived by the different collaborations in [63][64][65][66][67] and perform a naïve χ 2 goodness-of-fit test (GOF) in Fig. 15. For the GOF, we take as many distinct background components as have been identified by each collaboration, and, nent with index n s = 2.4, and again we minimize over the normalization of this background.
The EGRET and Fermi data are dominated by the systematic error on the effective area, so we take the total shapes as given by the collaborations and allow the normalizations on the entire background shape to float simultaneously. We show the comparison in Fig. 15, and we find that the GOF improves the constraints, but only by at most an order of magnitude. [