In this review we have summarized the current knowledge on the Diffuse Gamma-Ray Background (DGRB). The DGRB is what remains in the gamma-ray sky after the subtraction of the diffuse Galactic foreground and of the resolved sources. It is interpreted as the cumulative emission of the objects that are not bright enough to be detected individually.
Since its first detection in 1972 by the OSO-3 satellite , this emission has been deeply investigated in an attempt to understand its composition. The Fermi LAT satellite, in operation since 2008, has greatly improved our understanding of the DGRB. This emission is measured as an isotropic template in a multi-component fit to the Fermi LAT data. The fit also includes a model for the resolved sources and for the diffuse Galactic foreground. The most recent measurement of the DGRB energy spectrum is reported in Ref.  and it covers almost 4 orders of magnitude in energy, from 100 MeV to 820 GeV. Our imperfect knowledge of the Galactic foregrounds represents the main source of uncertainty in the analysis and it induces a systematic error on the DGRB intensity of ∼ 15−30%, depending on the energy range considered.
Before the Fermi LAT, the energy spectrum was the only source of information available on the DGRB. However, the scenario drastically changed in 2012 when, for the first time, the Fermi LAT measured also the angular power spectrum (APS) of anisotropies in the DGRB . The emission was found to exhibit a Poissonian APS in the multipole range between ℓ = 155 and 504, with a significance as large as 7.2σ between 1.99 and 5.0 GeV. This signal is independent of energy between 1 and 50 GeV.
Recently, cross-correlations of the DGRB with different data sets have also revealed to be a powerful tool to unveil the composition of the DGRB. Ref.  reported a significant cross-correlation of the DGRB with 4 out of the 5 galaxy catalogs considered, namely the optically-selected quasares of the Sloan Digital Sky Survey (SDSS) Data Release 6 in Ref. , the IR galaxies of the 2 Micron All-Sky Survey (2MASS) Extended Source Catalog from Ref. , the main galaxy sample of SDSS Data Release 8 from Ref.  and the radio sources from the NRAO VLA Sky Survey (NVSS) . These cross-correlation signals, obtained after the analysis of 60 months of Fermi LAT data, are localized at small angles (below few degrees) with significances that range between 3σ (in the case of SDSS Data Release 8 main galaxies) and more than 10σ (for the NVSS catalog), for gamma-ray energies above 1 GeV. The cross-correlation with NVSS, however, is most likely contaminated by a 1-halo term not related to the Large Scale Structure (LSS) of the Universe.
In Ref. , the authors measured, for the first time, the cross-correlation of the DGRB with the cosmic shear induced by the gravitational lensing detected in the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) . Their results are compatible with a null cross-correlation signal in the angular range between 1 and 100 arcmin. Ref.  proves, though, that measuring the cross-correlation with the cosmic shear signal is not an unrealistic goal. Indeed, more promising results are expected from the cross-correlation between the DGRB and the data expected from the Dark Energy Survey (DES) or from Euclid . Finally, Ref.  reported a 3.2σ detection of the cross-correlation between the DGRB and the lensing potential of the CMB measured by the Planck Collaboration . The signal is localized in the multipole region between ℓ = 40 and 160, for gamma-ray energies between 700 MeV and 300 GeV.
The increased amount of observational data on the DGRB has allowed significant progress in the modeling of its composition. The DGRB is interpreted as the cumulative emission of unresolved sources, e.g. blazars, misaligned Active Galactic Nuclei (MAGNs), star-forming galaxies (SFGs) and MilliSecond Pulsars (MSPs). It, therefore, represents a reservoir of invaluable information on these astrophysical sources as it may be the only way to study the emission of objects that are too faint to be detected individually. In particular, the DGRB can potentially determine the faint end of the luminosity function of the aforementioned populations, i.e. a goal that would be quite difficult, if not impossible, to achieve otherwise.
Before the Fermi LAT, the predictions for the contribution of unresolved blazars to the DGRB were affected by large uncertainties [94, 10, 11, 12, 13, 15, 30, 16]. Nowadays, the wealth of new information on the DGRB, combined with the population studies of resolved blazars performed by the Fermi LAT Collaboration, have established that this source class cannot account for more than (20 ± 4)% of the DGRB in the energy range between 0.1 and 100 GeV . The subclass of blazars responsible for the bulk of the blazar contribution varies depending on the energies considered. Indeed, Refs. [23, 24, 25] showed that unresolved high-synchrotron-peaked BL Lacs can explain the whole DGRB at energies above ∼ 100 GeV. At lower energies, astrophysical populations other than blazars are required. Yet, the modeling of these other gamma-ray emitters, such as namely MAGNs, SFGs and MSPs, is not as robust as that of blazars. This is caused by difficulty in performing reliable population studies with the limited sample of resolved sources currently available in the gamma-ray range. Generally, it is useful to assume a correlation between luminosities at different wavelengths (i.e. gamma-ray with radio frequencies [28, 29] in the case of MAGNs, or gamma rays and infra-red light  for SFGs).
The overall picture indicates that the 4 classes of astrophysical sources mentioned above are enough to explain the totality of the DGRB energy spectrum measured in Ref.  (see Fig. 9). As for the APS of DGRB anisotropies, Refs. [67, 68, 63] prove that unresolved blazars alone (more specifically, high-synchrotron-peak BL Lacs) can account for the whole APS reported in Ref. . These two important results can be reconciled by the fact that the blazar component is produced by a relatively small number of bright but unresolved objects and, thus, it gives rise to significant anisotropies. On the contrary, the other source classes produce fairly isotropic cumulative emission, as their members are more numerous and fainter.
The picture gets more complicated, though, when considering also the measured cross-correlation with LSS tracers: the sum of unresolved blazars, MAGNs and SFGs provides a good fit to the cross-correlation APS detected between the DGRB and the CMB lensing potential . It is also compatible with the lack of significant cross-correlation with the CFHTLenS cosmic shear . However, Ref.  finds that the model that best fits the two-point correlation functions measured in Ref.  with 5 galaxy catalogs can only account for ∼ 30% of the DGRB intensity reported in Ref. . It is still unclear if such a limitation of the astrophysical interpretation of the DGRB can be alleviated by a more sophisticated modeling of its components. Also, it will be interesting to verify whether a similar scenario will be confirmed by new surveys or by more complete data releases of the catalogs currently employed. Alternatively, one will be forced to supplement the model with another component, which does not correlate with the LSS, such as gamma-ray emission associated with the DM halo of the MW or with its DM substructures.
Quantifying the contribution of known astrophysical populations automatically constrains the intensity of other potential contributors to the DGRB. We briefly presented, among others, the case of clusters of galaxies, Type Ia supernovae and of Ultra-High-Energy Cosmic Rays interacting with background radiation. However, the most studied scenario is that of a potential gamma-ray emission from Dark Matter (DM) annihilation or decay. Since no DM signal has been undoubtedly detected in the gamma-ray sky so far, it is expected that most of this hypothetical gamma-ray emission will contribute to the DGRB. Searching for DM in the DGRB has the advantage that the DM-induced component is sourced by the emission coming from all DM halos and subhalos around us. It will, thus, depend on the ensemble-averaged properties of the DM halo population, which can be inferred from N-body cosmological simulations [452, 343, 267, 453, 454] or predicted by the theory of structure formation [200, 201]. his is intrinsically different from the observation of a specific target, e.g. the Galactic Center or a dwarf Spheroidal satellite galaxy. Indeed, each of these targets could be very peculiar and deviate considerably from the ensemble average, potentially hamper the interpretation of any data. Another benefit of using the DGRB to search for DM is that a potential DM signal contributing to the DGRB would be sensitive to the process of assembly of DM halos and their subsequent evolution. This kind of information would be difficult (if not impossible) to extract by observing individual targets in the sky. In this regard, the DGRB may be the only cosmological non-gravitational probe of DM. In addition, it represents a fundamental source of complementary information in the study of any claimed DM signal.
The intensity of this DM-induced cosmological emission depends on the properties of the DM particle, e.g. its mass, annihilation cross section or decay lifetime. Also, it rests on the abundance and properties of DM structures. Our understanding of DM (sub)halos heavily relies on the results of N-body cosmological simulations. Yet, as of today, even the simulations with the highest resolution are far from resolving the whole DM halo hierarchy down to the predicted Mmin. The properties of low-mass DM structures, thus, need to be inferred by extrapolating the characteristics of their more massive counterparts, that are well resolved in the simulations. Heuristic power-law extrapolations, which predict very bright low-mass DM halos, have been commonly used in the literature [274, 56, 275, 276, 339, 217]. However, recent high-resolution simulations of the smallest DM halos [277, 279] suggest that those extrapolations are not well motivated, favoring, instead, models that yield a more moderate DM-induced gamma-ray emission from low-mass structures [267, 280, 281]. Overall, this results in a DM-induced cosmological signal which is substantially weaker than the one obtained when assuming power-law extrapolations. Also, and perhaps even more importantly, the improved knowledge provided by Refs. [267, 280, 281] on the structural properties of the smallest DM halos has considerably reduced the theoretical uncertainty associated with the DM contribution to the DGRB down to a factor of ∼ 20 .
Not unexpectedly, the measured DGRB energy spectrum can be used to constrain the intensity of the DM-induced emission and, thus, to derive upper (lower) limits on the annihilation cross section (decay lifetime). Figs. 11 and 12 summarize some of these results. For annihilating DM, the so-called “sensitivity-reach” limits derived in Ref.  from their fiducial Halo Model exclude thermal annihilation cross sections of 3 × 10−26 cm2 s−1 for masses below ∼ 100 GeV in the case of annihilations into b quarks. When compared to other indirect searches for DM, the upper limits inferred from the DGRB represent the strongest constraints on ⟨ σ v ⟩ currently available, for DM masses below ∼ 1 TeV. A more conservative statistical analysis would increase the upper limits by a factor of ∼ 10. 52 In the case of decaying DM, the lower limits on τ obtained in Ref.  from the DGRB measurement in Ref.  exclude decay lifetimes smaller than ∼ 3 × 1027 s for mχ ≲ 2 TeV and decays into µ+ µ−. At larger masses, the strongest lower limit comes from Ref. , which excludes lifetimes as large as 1028 s. When compared to other searches of decaying DM, these limits represent the most constraining information available on τ, up to DM masses of 20 TeV.
A similar strategy can be used when comparing the APS measured in Ref.  to the predicted anisotropies in the DM-induced emission. Nevertheless, the latter turns out to be quite isotropic [56, 62, 339] and, thus, the corresponding upper limits, although competitive with other indirect DM searches, are less stringent than those inferred from the DGRB [339, 420]. The cross-correlation with galaxy catalogs and with cosmic shear are also very promising strategies to constrain (or even to detect) a potential DM signal in the DGRB [73, 74, 72, 75]. This is possible mainly thanks to the fact that both cross-correlations are particularly sensitive to the way the matter is distributed in the local Universe and, in particular, to the most massive DM halos. We remind that unresolved blazars, the main contributors to the auto-correlation APS, do not populate neither the local volume nor the largest DM halo masses. A model of the DGRB including both astrophysical sources and annihilating DM is used in Ref.  to describe measurement of the 2-point correlation function reported in Ref. . The analysis excludes DM candidates with annihilation cross sections larger than the thermal value for masses below 40 GeV, for annihilations into b quarks and a moderate value of the subhalo boost. When assuming the same value of the subhalo boost, the obtained upper limit is currently the strongest one among all those derived from DGRB data, including measurements of the DGRB intensity and of the auto-correlation APS.
The energy spectrum, auto-correlation and cross-correlation APS of the DGRB are sensitive to different characteristics of the sources contributing to the emission. Considering at the same time all three observables provide a very powerful handle to reconstruct the composition of the DGRB. Needless to say, such an ambitious goal requires ample data sets, including information from wavelengths other than the gamma-ray energy range. Fortunately, a great wealth of new observational information is expected in the near future: the Fermi LAT will continue gathering data until, at least, 2016. The Cherenkov Telescope Array (CTA), expected to be in operation by 2020, will improve by a factor ∼ 10 the sensitivity of current Cherenkov telescopes in the energy range between a few dozens of GeV and a few dozens of TeV. Its complementarity with respect to the Fermi LAT will allow an improved precision in the determination of the DGRB energy spectrum in the sub-TeV energy range, as well as the extension of the measurement beyond the TeV. In addition, CTA will perform the first survey of a significant portion of the sky at these very-high energies [455, 456]. Combined with the Fermi LAT data gathered since Ref. , it will be possible to extend the data on the auto-correlation APS to higher energies and to decrease the size of the bins in energy. On the other hand, the cross-correlation of the DGRB with LSS tracers will hugely benefit for the imminent release of the data gathered by the DES  during its first year of operation. The near future will also see the advent of the next generation of galaxy catalogs, e.g. the extended Baryon Oscillation Spectroscopy Survey (eBOSS) 53 and the Dark Energy Spectroscopic Instrument (DESI, formerly BigBoSS) . On a longer scale, Euclid will offer weak lensing measurements with unprecedented precision after 2020 .
The increased observational data available on the DGRB will also be accompanied by significant progress in the modeling of its contributors. Improving our understanding of the emission of blazars, SFGs and MAGNs will alleviate the degeneracies currently affecting our interpretation and will reduce the uncertainty associated with each component. A fully multi-wavelength approach is required to achieve such a goal: in the X-ray band the Nuclear Spectroscopic Telescope Array (NuSTAR)  has been recently launched and ASTRO-H [, ] will follow within this year (2015). Infra-red data from Herschel and the Wide-field Infrared Survey Explorer (WISE) 54 are already public and the James Webb Space Telescope (JWST) 55 is expected to be launched in 2018. Regular observation in radio has started with Low-Frequency Array (LOFAR) 56, a pathfinder for the Square Kilometer Array (SKA) 57 planned for 2020.
We end the review by noting that the IceCube Collaboration has recently reported the detection of the first extraterrestrial neutrinos [461, 462, 463]. The 37 observed neutrino events represent an excess over the atmospheric background extending to the PeV scale, with a significance of more than 5σ . Even if the origin of these neutrinos is not clearly established yet (see Refs. [464, 465] and references therein), a possibility is that they originate from a diffuse neutrino flux similar to the DGRB. 58 If this interpretation was confirmed, many of the techniques discussed in this review could also be adopted to investigate this diffuse neutrino flux. Indeed, many of the sources contributing to the DGRB are expected to emit neutrinos as well [466, 467, 468, 469]. This implies that any constraint on their neutrino emission would indirectly constrain also their contribution to the DGRB (and viceversa). The development of a fully multi-messenger approach is certainly a very tantalizing possibility that has been already considered, e.g., in Refs. [470, 471, 472, 207, 473, 161].
In conclusion, the DGRB is a fundamental component of the gamma-ray sky, whose exact composition still remains unveiled. The recent rapid growth of available data on the DGRB (mostly thanks to the outstanding performance of the Fermi LAT) has triggered an increased attention from the scientific community. Astrophysicists and astroparticle physicists aim at reconstructing the composition of the DGRB to infer novel information on the sources contributing to the emission, especially in their low-luminosity regime. New data sets are already available (or will be soon) which can provide a significant progress to the common goal of dissecting the true nature of the DGRB. By summarizing where we stand on our current understanding of this emission, with this review we hope to have offered a useful reference to those who will analyze and interpret the data to come, as well as to help finding new avenues and opportunities for further research.
MF gratefully acknowledges support of the Leverhulme Trust. MASC is Wenner-Gren Fellow and acknowledges the support of the Wenner-Gren foundation to develop his research. We also acknowledge the project MultiDark CSD2009-00064. We thank M. Ajello, S. Ando, K. Bechtol, M. Di Mauro, A. M. Green, G. Zaharijas and J. Zavala for useful comments on the manuscript.
52 Note also that both the fiducial sensitivity-reach limits and the more conservative ones in Ref.  are affected by an uncertainty of a factor of ∼ 3. Back.
53 https://www.sdss3.org/future/eboss.php Back.
54 http://wise.ssl.berkeley.edu/ Back.
55 http://www.jwst.nasa.gov/ Back.
56 http://www.lofar.org/ Back.
57 http://www.skatelescope.org/ Back.
58 To stress the similarity with the DGRB, we propose to call this neutrino emission, the Diffuse Neutrino Background (DNB). Back.