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Constraints on Cosmological Dark Matter Annihilation from the Fermi-LAT Isotropic Diffuse Gamma-Ray Measurement

The first published Fermi large area telescope (Fermi-LAT) measurement of the isotropic diffuse gamma-ray emission is in good agreement with a single power law, and is not showing any signature of a dominant contribution from dark matter sources in the energy range from 20 to 100 GeV. We use the absolute size and spectral shape of this measured flux to derive cross section limits on three types of generic dark matter candidates: annihilating into quarks, charged leptons and monochromatic photons. Predicted gamma-ray fluxes from annihilating dark matter are strongly affected by the underlying distribution of dark matter, and by using different available results of matter structure formation we assess these uncertainties. We also quantify how the dark matter constraints depend on the assumed conventional backgrounds and on the Universe's transparency to high-energy gamma-rays. In reasonable background and dark matter structure scenarios (but not in all scenarios we consider) it is possible to exclude models proposed to explain the excess of electrons and positrons measured by the Fermi-LAT and PAMELA experiments. Derived limits also start to probe cross sections expected from thermally produced relics (e.g. in minimal supersymmetry models) annihilating predominantly into quarks. For the monochromatic gamma-ray signature, the current measurement constrains only dark matter scenarios with very strong signals.


Abstract
The first published Fermi large area telescope (Fermi-LAT) measurement of the isotropic diffuse gamma-ray emission is in good agreement with a single power law, and is not showing any signature of a dominant contribution from dark matter sources in the energy range from 20 to 100 GeV. We use the absolute size and spectral shape of this measured flux to derive cross section limits on three types of generic dark matter candidates: annihilating into quarks, charged leptons and monochromatic photons. Predicted gamma-ray fluxes from annihilating dark matter are strongly affected by the underlying distribution of dark matter, and by using different available results of matter structure formation we assess these uncertainties. We also quantify how the dark matter constraints depend on the assumed conventional backgrounds and on the Universe's transparency to high-energy gamma-rays. In reasonable background and dark matter structure scenarios (but not in all scenarios we consider) it is possible to exclude models proposed to explain the excess of electrons and positrons measured by the Fermi-LAT and PAMELA experiments. Derived limits also start to probe cross sections expected from thermally produced relics (e.g. in minimal supersymmetry models) annihilating predominantly into quarks. For the monochromatic gamma-ray signature, the current measurement constrains only dark matter scenarios with very strong signals.

Introduction
From the early 1980's, when astrophysicists became first largely convinced that most of the mass in the Universe is dark [1], to more recent years when the ΛCDM paradigm became well established, a substantial experimental effort has been dedicated to dark matter (DM) identification. One strong candidate for the constituents of DM are Weakly Interactive Massive Particles (WIMPs) which, by having masses and couplings at the electroweak scale, naturally produce a relic abundance in agreement with current observations; for a review see, e.g., [2,3,4]. Potentially, one of the methods to indirectly detect DM is its annihilation or decay signal in gamma-rays. In that respect, the data collected by the Fermi-LAT instrument [5] have been eagerly awaited. It explores the gamma-ray sky in the 20 MeV to 300 GeV range and has an improved sensitivity by more than an order of magnitude, compared to its predecessor EGRET [6]. After its first year of mission, it has already provided a wealth of information and advanced our knowledge in many areas of astrophysics and has set new limits on the rate of DM induced gamma-rays [7,8,9,10,11,12,13]. Some of the promising DM gamma-ray targets are the Galactic center [14,15], because of its proximity and expected high density of DM; dwarf satellite galaxies [16,17], because of their low or absent astrophysical backgrounds; the extended Milky Way halo [18,19]; and individual DM substructures in our Galaxy [20,21].
In this work we use the full sky gamma-ray survey to investigate a possible isotropic DM signal in originating from annihilations summed over halos at all redshifts [22,23,24]. Most cosmological halos are individually unresolved and will contribute to an approximately isotropic gamma-ray background radiation (IGRB). Attempts to fit a cosmological DM signal to the IGRB previously measured by EGRET [25,26] have been presented in [27], where they find that a neutralino, as the lightest supersymmetric particle, with a mass M χ ≈ 500 GeV gives the best fit to the data. However, due to the large uncertainties both in the data and in the background model they do not claim a detection (see also [28,29] for other attempts to fit a DM signal). In the new IGRB measurement presented by the Fermi-LAT in [30], there are no spectral features that favor any of these DM candidates.
Using the Fermi-LAT measured isotropic diffuse gamma-ray background emission [30], we present new limits on a signal from cosmological DM. Besides such a signal, the IGRB could also receive a contribution from unresolved Galactic DM subhalos [31,32]. We will focus mainly on limiting the extragalactic DM signal in this work, but comment carefully on the possible size of Galactic contributions. A different approach to extract a DM signal from a full sky analysis, which we will not follow, is to analyze the power spectrum of the gamma-ray signal, which may contain identifiable signatures on different angular scales [33,34,35,36,37].
There are several important uncertainties inherently present when trying to constrain DM properties from the type of analysis presented in [30]. The largest comes from the theoretical modeling of the expected DM annihilation luminosity. We use recently presented results from the 'Millennium II' simulation of cosmic structure formation [38,39], as well as the approach in the Fermi-LAT pre-launch study [40], to calculate the DM contribution to the IGRB signal. Another uncertainty stems from the contribution of more conventional, astrophysical sources to the extragalactic gamma-ray signal, which is currently hard to quantify. A large contribution is believed to originate from unresolved point sources, with the most important potentially being unresolved blazars [41,42,43,44,45]. Other sources, such as ordinary star forming galaxies [46,47] and in particular starburst galaxies [48], as well as structure shocks in clusters of galaxies [49,50,51,52,53], might also contribute (see, e.g., [54] for a short review). The Fermi-LAT is expected to improve our knowledge of these sources and increase our understanding of the shape and normalization of their contribution to the IGRB in the near future (for early results, see [55]). We address these background uncertainties by presenting both very conservative and more theoretically-motivated limits on the DM contribution to the IGRB signal.
The paper is organized as follows. In section 2 we describe the calculation of the isotropic gamma-ray flux from cosmological distant DM annihilations, and comment on the potential contribution from Galactic DM. In section 3 we motivate and describe the particle physics DM models we constrain. Section 4 contains a description of our procedure for obtaining the limits, and in section 5 we present and discuss our results. Section 6 contains our summary.

Extragalactic
There are several ingredients necessary to calculate the flux of gamma-rays from cosmological DM annihilation. In addition to the gamma-ray yield per annihilation, assumptions need to be made on the distribution and evolution of DM halos in the Universe. Also, for high-energy gamma-rays, the effects of intergalactic absorption become important and has to be taken into account. The flux from DM induced extragalactic photons can be expressed as, [23], where c is the speed of light, H 0 the Hubble constant equal to 100×h km s −1 /Mpc, τ (z, E 0 ) the optical depth, σv the sample averaged DM annihilation cross section times relative velocity (hereinafter referred to as cross section), dN γ /dE the gamma-ray spectrum at emission, m DM the DM mass, andρ 0 its average density today, while h(z) = Ω M (1 + z) 3 + Ω Λ parameterizes the energy content of the Universe. The quantity ∆ 2 (z), as defined in [23], describes the enhancement of the annihilation signal arising due to the clustering of DM into halos and subhalos (relative to a uniform DM distribution in the Universe). For the Ω M , Ω Λ , and h we will consistently adopt the values used in [23] and [38]; which will be the two references we follow in order to derive ∆ 2 (z).
To quantify ∆ 2 (z) it is necessary to know the DM distribution on all length scales. Currently it is best predicted from numerical N-body simulations, which calculate the evolution of the matter distribution from an ab initio almost homogeneous distribution of DM in the early Universe. However, for N-body simulations computing resources limit the mass-resolution and the ability to properly model the effects of baryons. This prevents solid predictions for the DM structure on all scales, from cosmological down to the smallest scales of relevance. As the largest contribution to the DM induced extragalactic gamma-ray flux might come from small halos formed in an earlier, denser Universe, it is of importance to at least extrapolate down to mass scales expected for the smallest DM halos. Also, within the larger halos there should exist smaller bound structures that have survived tidal stripping, and that could contribute significantly [31,32]. The minimal DM halo size is limited by the so called free streaming and/or acoustic oscillations, and is typically in the range 10 −9 − 10 −4 M for WIMP DM (see, e.g., [56,57,58,59,60] and references therein). The needed extrapolations are therefore many orders of magnitude, since even simulations concentrated on Milky Way-size halos do not currently reach below subhalo masses of about 10 5 M at z = 0 [32]. We will take two main approaches to calculate the cosmological DM signal, and consider different extrapolations to different smallest halo masses.
In the first approach, we will use the results from one of the most recent N-body simulations, 'Millennium II' (MS-II) [39], as it was used by Zavala et al. in [38] to determine the cosmological DM annihilation signal. In their work, two basic structural properties of DM halos, the maximal rotational velocity V max and the radius where the maximum occurs, r max , are used to characterize their structure. By approximating the internal structure of halos by spherically symmetric Navarro, Frenk and White (NFW) density profiles [61], V max and r max enable calculation of the luminosity expected from halos down to a fairly low mass limit, ∼ 7 10 8 h −1 M (i.e., the resolution of resolved halos in the simulation). If instead an Einasto profile shape [62,63] was assumed, that would only introduce an increase of about 50% in the signal, as argued in [38].
To take into account the presence of host halos smaller than the ones resolved in the simulation, Zavala et al. fit a single power law (for each redshift) to the differential luminosity contribution per halo mass interval versus halo mass. The obtained luminosity function is then extrapolated down to a damping scale mass limit of 10 −6 h −1 M . 1 To also include the effects from substructures inside main halos, a similar universal power law was also independently established for the differential luminosity contribution as a function of subhalo mass. This power law for substructures was obtained by using the most massive halos in the MS-II that still have a significant number of resolved subhalos. Again this power law was used to extrapolate the luminosity contribution from subhalos down to 10 −6 h −1 M . Using reasonable uncertainty bounds, both on normalization and slope in the power law, for the undertaken extrapolation, Zavala et al. find that the enhancement factor due to substructures can be between roughly 2 and 2000 for a Milky-Way-sized halo. This is in broad agreement with the value of 232 presented in [19] for the high resolution Aq-A halo of the Aquarius project (which is a part of the Millennium simulation suite).
When placing our DM limits, we consider three cases for ∆ 2 from the derivation in [38]. The most conservative case takes into account the contribution to the DM signal only from halos and subhalos resolved in MS-II simulation (MSII-Res). As the other extreme, we consider the most optimistic case, where subhalos and halos down to ∼ 10 −6 M was taken into account as to maximize the contribution of these smallest halos (MSII-Sub2). We also consider a moderate case (MSII-Sub1), which we will adopt as our reference model, in which the extrapolation of the contribution from structures and substructures down to ∼ 10 −6 M has been done in a very conservative way.
As an alternative approach, we calculate ∆ 2 using the same 'semi-analytical' procedure as in [23]. Here the contribution from halos of all masses is integrated by using theoretically motivated analytical functional forms on the relevant properties of the DM structure, which in turn are tuned to fit results from numerical N-body simulations. The ingredients needed are the redshift dependence on the halo mass function, the DM halo density profile, and the spread in halo shapes for each halo mass. It is convenient to parametrize a halo by its virial mass, M , and concentration parameter, c. The concentration parameter is treated as a stochastic variable with a log-normal distribution for a fixed mass. Simplified, we can write the quantity ∆ 2 (z) as: where dn/dM is the halo mass function, with its functional form calculated as in the ellipsoidal collapse model [65], and P (c) is a log-normal distribution with variance σ(log 10 c) = 0.2 [66,23]. We model the dependence of concentration parameter on halo mass and redshift according to the Bullock et al. toy model [66]. We assume a NFW DM profile and fix the concentration parameter to stay constant for halo masses below 10 5 M , as it was done in [23], to minimize the risk of overestimating the DM signal from extrapolation of BullSub

MSII-Sub1
MSII-Res Figure 1: Comparison of the different models used to calculate the enhancement of DM annihilation signal due to structure formation; ∆ 2 (z) based on the Millennium II simulation (MSII-models) [38] and the semi-analytic model (BulSub) [23]. All the enhancement factors ∆ 2 (z) are multiplied by the factor (1 + z) 3 /h(z) in order to reflect the relevant part of the integrand in equation (1) [23,40], we set 10% of a halo mass in substructures and assume that the subhalo mass function has a power-law behavior in mass M −β , with a slope β = 1.9. This is in broad agreement with findings of new simulations [31,32] for Milky Way-size halos. The concentration parameter of subhalos is not constant, but depends on the subhalo mass [23] and on the distance from the center of the halos [31,32]. We here associate a concentration parameter four times higher in substructures, compared to a main halo of the same mass [23]. This is the same type of structure description also used in the Fermi pre-launch paper [40], and used in several recent works [64,67]. We dub this scenario the semi-analytical NFW Bullock et al. substructure model (BulSub).
The result of the semi-analytical (BulSub) approach lies between the extreme values found in MS-II simulation, and turns out to be quite close to the MSII-Sub1 case. We show a comparison of these four models via the quantity (1 + z) 3   parameter and halo mass function. To summarize, we note that all above scenarios could basically be related by an overall shift in their predicted signal amplitudes. We will keep all of them, however, in our exclusion plots as it is illustrative and allows easy comparisons with previous works.
For the optical depth τ (z, E 0 ), as a function of redshift z and observed energy E 0 , we use the result of Gilmore et al. [68]. 2 In figure 2, we compare this result to the older, commonly assumed, absorption model of Stecker et al. [69] by plotting the relevant part of the integrand in equation (1): . The newer absorption model in [68] gives significantly lower optical depth. For z ≥ 1 the difference to the older model [69] is large for gamma-ray energies E 0 20 GeV, and for higher energies the difference is even larger and their deviation start at much lower redshifts. We show that the choice of absorption model plays a role for the DM limits when the limits are set by the gamma-ray spectrum in the high energy end of the Fermi-LAT measurement. We comment further on this in sections 3 and 5.

Galactic
In addition to an extragalactic DM signal, there could be a significant contribution from pair annihilations along the line of sight through the DM halo in which the Milky Way is embedded. Current N-body simulations show highly galactocentric smooth DM density profiles, extending far beyond the visible Galaxy, and with the main halo hosting a large amount of substructures in form of subhalos (which themselves contain subhalos) [19,31].
The Galactic main halo's DM density profile would by itself, from an observer on Earth, give rise to a very anisotropic DM annihilation signal. 3 The DM annihilation signal from the Galactic substructures, however, has a completely different morphology and could potentially produce a fully isotropic signal. This is because the flux is proportional to the number density distribution of subhalos, and this distribution is much less centrally concentrated than the main halo DM distribution [32]. How isotropic the emission from unresolved Galactic subhalos is may vary somewhat between different high resolution N-body simulations (see, e.g., [71]). With a substructure signal spatially distributed as presented in [19] for the Aquarius A-1 simulation, we find that the substructure component itself gives a diffuse emission that varies by less than 25 % in different sky directions. Since variation in the substructure distribution between different DM halo realizations is not excluded, we conclude that the signal from unresolved Galactic substructures could be practically isotropic [19].
To illustrate the range of predictions for the isotropy of the DM-induced gamma-ray spectra, we show in figure 3 uncertainty bands for the three mentioned DM components (i.e. Galactic main halo, Galactic substructure, and the extragalactic component), for DM annihilation into bb quarks.
The vertically hatched band illustrates the span in the expected (isotropic) extragalactic signal, defined as the region between our MSII-Sub1 and MSII-Sub2 cases.
The horizontally hatched band is the (∼ isotropic) flux that can be expected from Galactic DM substructure. The band's width indicates the range in signal strength, and comes from uncertainties in how the extrapolation of the flux contribution from DM subhalos down to masses of 10 −6 M is performed. We model the gamma-ray signal from Galactic substructures, inside the galactocentric distance r (in kpc), as: This functional form is a parametrization of the result presented for the Aquarius simulation in [19]. L 200 main is the total DM-induced luminosity inside r 200 from the smooth halo (normalized through the Einasto profile in equation (4)), and B gives the relative signal enhancement inside r 200 due to substructures. The upper boarder of the vertically hatched band is obtained when a single power law relation between the substructure flux and the minimal DM subhalo mass are related as suggested in [19], which give B ∼ 230. 4 10 3 Figure 3: The vertically hatched band illustrates the span in the expected isotropic extragalactic (EG) gamma-ray signal, defined by being the region enclosed by our MSII-Sub1 and MSII-Sub2 cases. The horizontally hatched band is the flux that can be expected from Galactic substructure. The filled grey band is the signal range that could be expected from the main DM Galactic halo, at a latitude of 10 • , which would by itself produce an anisotropic signal. The data points show the measurement of the IGRB by the Fermi-LAT [30]. The gamma-ray spectra are from DM particles with mass of 400 GeV, a total annihilation cross section σv = 3 × 10 −26 cm 3 s −1 into bb quarks, and a minimal subhalo mass cut-off at 10 −6 M . See the text for more details.
The lower boarder is when the substructure signal strength instead is implemented consistently with the average substructure enhancement used in the MSII-Sub1 calculation of the extragalactic signal. Then the luminosity from all substructures inside r 200 for a Milky-Way-sized halos is merely B ∼ 2 times the luminosity of the main DM halo. This lower signal limit is also similar in amplitude to the finding in [71], where the Aquarius simulation is used, but a subhalo concentration extrapolation with a double power law approach is applied to soften the DM halo concentration for small subhalo masses. We thus find that the diffuse DM signal from Galactic substructure could be insignificant, but that, with the uncertainty bands in figure 3, Galactic substructures could also potentially enhance the DM signal by at least an order of magnitude relative to the extragalactic MSII-Sub1 signal. This range covers the result that [71] finds by self-consistently extrapolating results from two specific high resolution simulated halos. All these scenarios would obviously only increase the DM signal and would, if taken into account, only lead to stronger DM constraints than we derive from the extragalactic signal in this work.
The grey band is the (anisotropic) signal from the main DM Galactic halo, at a latitude of 10 • , where the upper and lower limits correspond to when an Einasto and a cored isothermal [72] DM density profile is adopted, respectively. These profiles are given by: where the local DM density ρ(r = r ≈ 8.2 kpc) = ρ = 0.4 GeV/cm 3 . For points in the vertically and horizontally hatched bands below the grey band, the anisotropic DM signal from the smooth main halo is expected to become stronger than the isotropic signal (at a latitude of 10 • ). 5 In such scenarios, in principle a different analysis approach should be chosen as a DM component should be included in the modeling of the Galactic (anisotropic) contribution for the IGRB measurement (which was not the case in the analysis in [30]). Such a combined spectral and morphological DM analysis is beyond the scope of this work, and instead we stay focused on the absolute size and spectral shape of the measured IGRB flux to set the limits on DM signals (see [76], for additional discussion in this direction). One could still be tempted to use the fact that, with the utilized Galactic diffuse model, the derived IGRB spectrum is observed to be isotropic within ∼ 10% outside of the Galactic plane (|b| ≥ 10 • ) [30]. This means that if an anisotropic DM annihilation signal from the Galactic main DM halo, in the region just above the Galactic plane, is sufficiently strong then this anisotropic signal could be more constrained by the found level of anisotropy than the absolute flux in the studied region. However, substantial uncertainties in the modeling of the Galactic diffuse contribution are present, and in the end a simultaneous all sky fit, with both a DM component and a model for the Galactic cosmic rays, gas, and interstellar radiation field, would have to be performed before fully reliable quantitative statements can be made. Additionally, as discussed above, the relation between the extragalactic and Galactic DM signal is not unique, and this uncertainty makes it important to also establish independent limits on isotropic extragalactic DM signals from the IGRB measurement.
We end this discussion on the Galactic DM signal by concluding that, to have a isotropic signal, one of the following scenarios should be valid: there exists a strong extragalactic DM signal, e.g. due to substructure enhancements like in the MSII-Sub2 case, or the Galactic substructures themselves give a very strong isotropic flux contribution, or that the DM signal from the inner Milky Way region is relatively low, for example by having a several kpc core in the smooth DM density profile. In the remainder of this work we will derive limits on the extragalactic signal, assuming it dominates the IGRB.

Particle Physics Models
A variety of different extensions of the Standard model of particle physics could in principle produce strong enough fluxes of gamma-rays to be observed by the Fermi-LAT [40]. We consider three generic types of DM models with distinctively different gamma-ray signatures, exemplified in figure 4: [Model 1: bb] Many DM candidates (e.g., within supersymmetry) have their dominant annihilation channel into quarks and gauge bosons. In these cases, gamma-rays are produced through the hadronization and decay of π 0 . For definiteness we assume here a 100% branching ratio into bb, but also annihilations instead into other quarks as well as into W/Z gauge bosons and higgs would all give fairly similar spectra [77]. We use the DarkSUSY package [78] to obtain the gamma-ray yield dN γ /dE, when calculating the DM signal. The gammas from inverse Compton (IC) scattering are not included in this case, as they are significantly less constraining.
[Model 2: µ + µ − ] We consider here DM models having leptons as the dominant annihilation channel. Although this is not the classical WIMP scenario, these types of models have been suggested to fit the PAMELA positron [79] and/or Fermi-LAT electron+positron [80] data, without violating anti-proton constraints [81,82]. The mechanism most often discussed is to either require direct annihilation into leptonic channels ad hoc, or include a light enough scalar/pseudoscalar which subsequently decays exclusively into leptons due to kinematic constraints (like for example in [83,84]). These types of ('leptonic') models produce continuous gamma-ray spectra by both final state radiation (FSR) [85] and inverse Compton scattering [86] on background radiation. The annihilation cross section needed is typically higher, by two to three orders of magnitude, than the one naively expected at freeze-out for thermally produced DM. A possibility to obtain such an enhanced signal today could e.g. be by the Sommerfeld enhancement effect [87,88,89,84]. We consider the case of 100% annihilation into mono-energetic µ + and µ − , which is a model that under favorable circumstances has been shown to give a very good fit to cosmic charged lepton data [90].
The yield of gamma-rays, dN γ /dE, produced by inverse Compton scattering of DM induced electron and positrons, from the decaying µ + µ − pairs, off of cosmic background radiation was calculated in [64,67]. We follow the same approach, using our ∆ 2 (z) DM structure scenarios. We convolve the differential Klein-Nishina cross section with the number density of cosmic microwave background radiation and fold it with the spectral distribution of electrons and positrons given by the DarkSUSY package [78], that utilize tabulated PYTHIA 6.154 results [91]. The flux of final state radiation photons from the prompt µ + µ − γ process is calculated by using the approximate formula given in [85,92].
Leptonic models favored by the PAMELA and Fermi measurements have been ruled out based on their contribution to radio and gamma-ray signals from the Galactic center region once either an NFW or Einasto DM density profile is chosen [93,94,95,10,11,96]. That might seem in contradiction with our approach here, where we, nevertheless, assume a NFW profile for the DM halo distributions. We note, however, that if a cored Burkert profile [97] is taken for all halos, the resulting strength of the extragalactic DM signal is only about 20 percent lower compared to the value obtained with a NFW profile [23]. On the other hand, the radio and gamma-ray constraints are derived from observation of the inner O(1) and O(100) parsec, respectively, from the Galactic center, and a significantly cored DM density profile would avoid these limits [94,95]. These inner sub-kiloparsec regions are not yet well resolved by current DM N-body simulations, and since also the interaction between DM and baryonic components could play an important role there, there is some freedom in tuning the inner DM density in the Milky Way.
[Model 3: γγ] In addition to the continuous gamma-spectra, electromagnetically neutral DM can also annihilate through loop suppressed processes and produce monochro-matic gamma-lines, either through the annihilation into two photons (γγ), a photon and Z-boson (γZ) or a photon and a Higgs-boson (γh). While branching ratios to these channels are typically less than 10 −2 , there are DM candidates, like the inert doublet model [98], non-thermal WIMPs [99] and Dirac DM models with a Z' portal only to heavy quark states in the Standard Model [100], that can have this as their main annihilation channel. For brevity, we consider only the γγ channel in this work. In contrast to the continuous photon-spectra for [Model 1] and [Model 2] that have very broad 'peaks' at energies of a around a few percent and at around (m DM /1TeV) 10 −3 of the DM particle mass, respectively, the gamma-line sharply peaks at an energy equal to the DM particle mass in the local rest-frame. Therefore, new limits on this signal are truly enabled by Fermi-LAT measurements, which for the first time provides convincing constraints for gamma-lines in the 10 to 100 GeV energy range. See also the dedicated Fermi-LAT search for Galactic DM lines [9].
The detection of a multi-GeV gamma line would be an unmistakable DM signal, as no conventional astrophysical source would produce spectral lines at these energies, and give a unique determination of the DM particle mass. However, the observed width of the gamma-line produced in cosmological DM annihilation depends strongly on the optical depth, as illustrated in figure 4 for the Gilmore et al. [68] and Stecker et al. [69] cases. We will illustrate this effect further by calculating DM cross section limits with both choices, in Section 5.
The measurement of the IGRB in [30] is not corrected for the energy dispersion of the instrument. In principle, our signal therefore needs to be convolved with the energy dispersion before comparing it to the data. It can be anticipated that the effect is very small, since the bin width of the measurement is much larger than the energy resolution. In figure 4 we convolve the signal from a 180 GeV DM particle with an energy dispersion that follows closely the one used in [9], which is also valid for the IGRB measurement. The effect on the exclusion limits is, as expected, negligible.

Upper Limit Analysis
The Fermi IGRB measurement is obtained as an isotropic component in a full sky fit (for latitudes ≥ 10 • ) including multiple diffuse Galactic foregrounds and point sources.
To reduce the number of misinterpreted charged particles in the IGRB analysis a more stringent event selection (compared to the current public Fermi Pass6V3 diffuse selection class [5,101]) was imposed. It was estimated that this selection enabled to further reduce the cosmic-ray contamination by 25 -95% in the energy range from 0.2 to 102.4 GeV, making the charged cosmic-ray particles subdominant to the IGRB flux. The remaining charged cosmic-ray background was estimated by a Monte Carlo simulation (the result is shown in figure 3 of [30]) and subtracted from the isotropic signal to obtain the IGRB. The final IGRB measurement is dominated by systematic uncertainties related to the above-mentioned subtractions as well as by the uncertainties in the effective area of the instrument. For this analysis these systematic errors were added in quadrature to the statistical errors to give the total one standard deviation error. The measured fluxes M i and bin errors Σ i are taken from the first line of Table I in [30]. There are also other systematic uncertainties attached to the Fermi-LAT measurement of the IGRB not included in the above errors; the most notable is the uncertainties in the diffuse Galactic foreground modeling. It is difficult to accurately incorporate such systematic uncertainties to the limits reported here, especially as the Galactic foreground model has highly nonlinear dependencies on its parameters. This point merits a dedicated analysis which is beyond the scope of this work. The measured IGRB energy spectrum is in a good agreement 6 with a single power law with a photon spectral index of -2.41. A single dominant source with a significantly different spectral shape can therefore be excluded in this energy range. In the light of this we derive upper limits on DM annihilation rates σv for our set of generic DM scenarios. Somewhat different limits can be derived depending on what is assumed for the background to the DM signal. To bracket this we derive limits in two ways: 1.
[Conservative] A conservative upper limit on σv (for a given cosmological DM distribution and for a given particle DM candidate) is placed by restricting DM signals to not exceed the measured intensity in any individual bin by more than some given significance. Of course, this means that to explain all the Fermi IGRB data bins a fully adjustable additional background is needed to explain the gamma-ray flux also in all the energy-bins to which the DM is not contributing enough. The limit for this conservative approach is hence taken to be the lowest σv for which the integrated DM flux, φ γ i , in any bin equals the measured flux, M i , plus n times the bin error, Σ. In other 'words', we take the upper limit to be sup{ σv : φ γ i ≤ M i + n × Σ i }. Our upper limits, corresponding to 90, 95 and 99.999% confidence level, are when n = 1.28, 1.64 and 4.3. All our upper limits are derived under the approximation that the probability distributions for the flux in each energy bin are independent and take Gaussian functional forms. The values of n simply correspond to what is needed to retrieve stated confidence levels for one-sided (upper) limit on the σv parameter (after marginalizing over all other fitting parameters; as in the case of our stringent limits that include additional background parameters).

[Stringent]
A more realistic scenario is that there is a significant astrophysical background present to which a small DM signal might be superimposed. The exact shape and normalization of such a background is to large extent still unknown. For the purpose of this analysis we use the sum of two power laws as the background. One of the power-law spectral components is motivated by theoretical calculations of the contribution of starforming galaxies to the extragalactic background [47], with a photon spectral index of -2.7. The other one is motivated by the observed spectra of blazars and has a harder photon 6 A single power law with free normalization and spectral index gives a reduced χ 2 = 1.29/(9−2) = 0.18. From a statistical point of view, a reduced χ 2 value much less than 1 is not expected and could indicate that the estimated uncertainties used are too large. Probably it is an effect of too conservative estimates of the included systematical errors (that is dominating the errors) and that the data points are treated as uncorrelated. We want to stay conservative at this stage, and keep in mind that the extra uncertainties from the Galactic diffuse foreground modeling is not folded in, and therefore hold on to these potentially too conservative error estimates. spectral index of -2.4 [45,55]. 7 How much a sum of two such power laws can be distorted by a DM signal, before deviating by more than some given amount, defines our stringent procedure to set upper limits on σv . In practice, a χ 2 test is performed by comparing the data set {M i }, with errors {Σ i }, to the best fit background (with the normalizations as free parameters) including a DM induced spectrum for a given σv ≥ 0. Our upper limits are then obtained when increasing σv forces χ 2 to deviate from its best fit value by more than ∆χ 2 = n 2 . Again the 90, 95 and 99.999% upper confidence regions are when n = 1.28, 1.64 and 4.3, respectively. The limits are derived with the absorption model by Gilmore et al. [68], as presented in section 2.1. If instead the Stecker et al. absorption model [69] is used, the cross section limits get somewhat weaker for some DM masses. The difference between the absorption models becomes only important when the limits are set by the high energy end of the DM spectra (i.e. energies 50 GeV). For DM annihilating into heavy quarks or directly into gammas, this happens for high DM masses (≥ 600 GeV), whereas for the 'leptonic' models the change in absorption model affects the lower DM mass limits since they are mostly limited by the high energy FSR part of the DM-induced gamma-ray spectra. We quantify the relative impact, from the two absorption descriptions, on our limits by showing two branches of the dash-dotted cross section lines. To not clutter the plots, the effect is only shown for MSII-Res case, but the relative effect is similar for all our ∆ 2 (z) scenarios. Figure 5 shows the exclusion regions for DM models annihilating to bb final states together with a parameter scan result for DM candidates in MSSM7 8 space where the branching ratio into final states of bb is > 80%. We also show 'bb like' states, defined as when more than 80% are hadronic fragmentation-induced gamma-rays, as is the case for, e.g., gauge bosons and quark as the prompt final states. All MSSM7 models are consistent with accelerator constraints and have neutralino thermal relic abundance corresponding to the inferred cosmological DM density by WMAP [102]. It is not always direct to compare different works on DM annihilation cross section limits; different physics assumptions, different analysis methods and different data sets are often used. We will anyway make a comparison to a few other DM constraints, as to put our cosmological DM results into context. With the MSII-Sub2 case our cross section limits are among the strongest indirect detection limits presented to date, but this setup is admittedly a WIMP structure scenario that might be overly optimistic. The structure and substructure description applied in our BulSub scenario as well as the strict analysis procedure is similar to what was used in the Fermi analysis of Galaxy clusters [13] and (with the exception of no additional inclusion of substructure) the Fermi analysis of dwarf galaxies [8], see also [7]). It is therefore worthwhile to compare those analyses with our BulSub scenario with the strict upper limit calculation procedure. Our bb cross section limits are, in this perspective, comparable to the ones presented in the Fermi analysis of dwarf galaxies [8] and somewhat stronger than the constraints from galaxy clusters in [13]. For hadronic annihilation channels, cosmic-rays, especially antiproton data, can provide comparable limits [82]. Such limits are, however, associated with additional uncertainties due the uncertainties related to charged particle propagation in the Galaxy. In the preparation of this paper, Fermi-LAT data was used in [10,11] to set cross section limits on Galactic DM induced gamma-rays. In these two papers, their data analysis method is more similar to our conservative analysis approach, and the presented limits are comparable to our conservative MSII-sub1 limits when their Galactic DM halos are described by a smooth Einasto or NFW DM density profile. As mentioned, most hadronic channels are very similar in their gamma-ray production. To within a factor of two (if final states are not very close to, or below, production thresholds) our cross section limits are also valid for prompt annihilation into the heavy gauge bosons, the other standard model quarks, gluons, as well as into the leptonic τ + τ − channel. Figure 6 shows the exclusion region for the leptonic DM model, together with the best fit region for this model to the PAMELA and Fermi-LAT positron and electron data. The sharp upper endings of the gray best fit regions come from the constrain to not overshoot HESS data [103]. Both the best fit regions and the exclusion regions for all our discussed DM scenarios are calculated in a self-consistent way, modulo minor corrections. Below a DM mass of about 500 GeV, the limits on these models are determined by the FSR signal at the high-energy end of the DM spectra, see figure 4, and therefore depend more substantially on the choice of the absorption model. We note here that this conclusion holds even if one considers the constraints that the low energy COMPTEL [104] and EGRET [25,26] data would pose on the first (IC) peak in the spectra. The difference between the Stecker et al. [69] and the Gilmore et al. [68] absorption model results in a  figure 5. Also shown is the IDM parameter space from [98].

Results and Discussions
difference in the FSR signal calculated in the two cases by a factor 2, and affects our limits correspondingly. As discussed, prior to this work many leptonic DM models adjusted to fit the PAMELA and Fermi data were already in tension with a wide range of experimental studies of gamma and radio signals [93,94,95,10,11,96], as well as neutrinos [105], from the inner Galaxy, Big Bang Nucleosynthesis [106,107] , and the non-observation of distortions of the cosmic microwave background [108,109,110,111,112]. In this context, we want to point out that our limits have a weak dependence on the DM density in the very inner part of the Milky Way (likewise the limits based on Big Bang Nucleosynthesis and non-distortions of the cosmic microwave background). Also our moderate MSII-Sub1 (at least in the stringent analysis case) and the BulSub DM scenarios exclude models that are most favored by the PAMELA/Fermi measurement. The stringent BulSub limit is somewhat stronger than the most optimistic limits set by the Fermi-LAT observation of dwarfs spheroidal galaxies [8] and galaxy clusters [13]. Similar to the cluster analysis, and in contrast to the constraints placed by the non-detection of dwarf spheroidal galaxies, our limits do not rely on the modeling of cosmic ray electron and positron transport and diffusion in DM halos. In the two recent papers [10,11], mentioned previously in this section, they find cross section limits on the µ + µ − channel from Fermi-LAT data that are fairly similar to our MSII-sub1 stringent limit. Although their limits do not require strong signal enhancements in the inner Galaxy, they still also have uncertainties related to the diffusion modeling [10,11].
For the case of a e + e − annihilation channel the constraints would become stronger by a factor of a few compared to µ + µ − , unless it is the FSR that is the constrained process, then the limits are only stronger by a factor ∼ ln 4m 2 DM m −2 e / ln 4m 2 DM m −2 µ 2. For multi lepton final states, such as µ + µ − µ + µ − , the limits get instead weaker. Figure 7 shows the cross section limits for DM models annihilating to γγ states, together with the gamma-line strength expected in the inert doublet model [98]. The cross section limit on a DM particle, with mass of m DM , annihilating instead into a photon and a heavy X particle, with mass m X , is given by The annihilation limit σv m DM γγ,limit is directly read from figure 7 at the corresponding WIMP mass m DM . For this expression to be exactly valid, the X particle is assumed to be stable, as otherwise that produce an intrinsic broadening of monochromatic lines.
Our monochromatic line constraints allows us only to constrain scenarios that predict very strong gamma-lines, such as those presented in [100]. Whereas models with relative strong gamma lines as, e.g., in the inert doublet model [98] or in [113] are still below the sensitivity of the Fermi-LAT isotropic measurement. Even if our most optimistic MSII-Sub2 stringent analysis still present much stronger annihilation cross section constraints than the dedicated line search of the Fermi-LAT collaboration [9], the more solid upper limit from the MSII-Sub1DM structure scenario are significantly weaker. Comparing to the earlier works [22,23] on cosmological DM induced gamma-lines, we find a missing factor (1 GeV/E 0 ) in their presented gamma-line signals. This is accordingly reflected in the substantially weaker flux limits we obtain than anticipated from these papers.
We note that the used a binned Fermi-LAT spectrum introduces a wiggly behavior in the upper limits for the case of DM annihilation into monochromatic gamma-rays. This is because the narrow signal is sometimes split between two energy bins. The sharp rise in the upper limits above 100 GeV is due to the fact that the Fermi-LAT measured spectrum ends there, and that DM induced gamma-lines at high redshifts do not produce very strong signal at lower observed energies because of strong absorption at these energies.

Summary and Outlock
By using the isotropic gamma-ray background measurement of the Fermi-LAT [30], we have set upper limits on the cross section (at 90, 95 and 99.999% confidence level) for cosmological DM annihilating into different final states that induce high-energy gammarays. Such cross section limits have associated uncertainties, with the largest uncertainty stemming from the modeling of the evolution of DM structure and substructure. The difficulty of estimating the isotropic background to the cosmological DM annihilation signal further increases the uncertainty in limits. Given these uncertainties, our most conservative and most optimistic limits on cross sections span three orders of magnitude. While the most conservative constraints barely reach exclusion of theoretically discussed DM cross sections, more optimistic descriptions of the DM halos and subhalos would allow to exclude all (or an interesting fraction of) such models. This demonstrates the potential the isotropic signal has to observe a DM signal, but also the importance to better understand the DM structure formation in order to reduce uncertainties and enable firm constraints on electroweak properties of DM particles. Better knowledge of the conventional extra galactic astrophysical contributions and of the galactic gamma-ray foreground model would also further improve the understanding and interpretation of the IGRB. If the main part of the Fermi-LAT measured IGRB originates from unresolved extragalactic objects, then, by continuing to resolve astrophysical contributions, the IGRB flux could be reduced. This effect would then be to lower the measured IGRB flux during the Fermi mission, which, together with improved statistics and extended energy range, would allow to probe a larger fraction of the DM parameter space.
The DM signal from our own Galaxy (both the more isotropic signal from Galactic DM substructures, and the anisotropic signal expected from the smooth Galactic main halo) could also make an important contribution. It is therefore expected that a simultaneous analysis of morphology and gamma-ray energy spectra, including conventional and DM Galactic as well as extragalactic signals, could be beneficial in the search of DM signals.
This paper also investigated the importance of the transparency of the Universe to high energy gamma-rays on the perceived DM signals. We find that, in the cases when the limits are set by the high energy part of the DM annihilation spectra (for energies 50 GeV), there is notable difference between existing models of the optical depth.