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The first full-sky image of gamma-ray emission was obtained in 1972 by the OSO-3 satellite. It consisted of 621 events detected above 50 MeV [1]. Since then, telescopes with lower sensitivities and better angular and energy resolutions have significantly improved our understanding of the gamma-ray Universe. The all-sky maps produced by the Fermi Large Area Telescope 1 (Fermi LAT from now on) after more than 6 years of data taking contain more than 5 million events above 1 GeV. These maps exhibit a rich morphology: along the Galactic plane, the diffuse Galactic foreground is the most evident feature and, overall, it accounts for ∼ 80% of the detected gamma rays. This diffuse radiation is produced by the interaction of cosmic rays (CRs) with the Galactic interstellar radiation field and with the nuclei of the Galactic interstellar medium. Also, the Third Fermi LAT catalog (3FGL) reported the detection of 3033 sources throughout the sky [2]. Extended gamma-ray emitters are discussed in Ref. [3]. Other structures, e.g. the Fermi bubbles [4], represent more complex phenomena, whose emission is not fully understood yet.

In order to reproduce the data from the Fermi LAT it is necessary to include one additional contribution, i.e. a diffuse and nearly isotropic emission called the Diffuse Gamma-Ray Background (DGRB) [1, 5, 6, 7, 8, 9]. The DGRB is thought to be predominantly of extragalactic origin: gamma-ray sources with a flux smaller than the sensitivity of Fermi LAT are not detected individually, producing instead a cumulative diffuse glow that contributes to the DGRB. Unresolved blazars [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], misaligned Active Galactic Nuclei (MAGNs) [26, 27, 28, 29], star-forming galaxies (SFGs) [30, 31, 32, 33, 34, 35] and millisecond pulsars (MSPs) [36, 37, 38] are guaranteed components to the DGRB. More uncertain source classes, e.g. galaxy clusters [39] or Type Ia supernovae [40, 41], may also play a role, together blue with diffuse phenomena, e.g. the radiation produced by the interaction of Ultra-High-Energy Cosmic Rays (UHECRs) with the Extragalactic Background Light (EBL) [42, 43]. 2. It is possible to estimate how much different source classes contribute to the DGRB, but its exact composition remains one of the main unanswered questions of gamma-ray astrophysics. Finding a definitive answer would constrain the faint end of the luminosity function of the DGRB contributors. Indeed, the study of the DGRB may represent the only source of information about those objects that are too faint to be detected individually.

The DGRB may also shed some light on exotic Physics as, e.g., on the nature of Dark Matter (DM): a huge experimental effort is currently devoted to the so-called indirect detection of DM, i.e. the search for particles (e.g. gamma rays, neutrinos, positrons or anti-protons) produced by the annihilations or decays of DM. Such a signal would constitute the first evidence that DM can interact non-gravitationally and it would represent an enormous step forward in our understanding of its nature [44, 45, 46, 47]. Targets like the center of the Milky Way (MW), local satellite galaxies or nearby galaxy clusters are considered optimal, thanks to the intensity of the expected DM signal and/or to the absence of significant competing backgrounds. Yet, no signal has been robustly associated with DM. Then, if DM annihilates or decays producing gamma rays and such a signal has not been detected up to now, it is most probably unresolved and it contributes to the DGRB. Looking for the features of a DM component in the DGRB, one can, then, hope to finally unravel the long-standing mystery of the nature of DM [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 45, 60, 61, 62, 63, 64, 65].

The most recent measurement of the intensity of the DGRB has been performed by the Fermi LAT, in the range between 100 MeV and 820 GeV [9] (see also Refs. [1, 5, 6, 7, 8] for the previous measurements). Valuable information on the nature of the DGRB can be extracted from its intensity energy spectrum. Its steepness could indicate whether a particular class of sources dominates the emission. Moreover, the transition between two energy regimes dominated by different classes could, in principle, give rise to breaks and features in the energy spectrum. Yet, since the intensity of the DGRB is only sensitive to the sum of its contributions, ultimately there is only a limited amount of information that can be extracted from it.

Fortunately, due to its excellent sensitivity and exemplary angular resolution, the Fermi LAT marked the beginning of an era in which the intensity energy spectrum is no longer the only observational data available for the study of the DGRB. In 2012, the first measurement of anisotropies in the DGRB was reported [66], and the detection of a non-null auto-correlation angular power spectrum (APS) provided complementary constraints on the composition of the DGRB [67, 68, 63]. Moreover, in Ref. [69], the authors used the photon-count probability distribution measured after 11 months of Fermi LAT data to constrain the contribution of unresolved blazars to the DGRB. More recently, the cross-correlation of the DGRB anisotropies with observables tracing the Large Scale Structure (LSS) of the Universe has also been considered. In Ref. [70] the authors measured the 2-point correlation of the DGRB with 5 different galaxy catalogs and they reported a signal with 4 of them. Ref. [71], instead, cross-correlated the DGRB with the cosmic shear observed by the Canada France Hawaii Telescope Lensing Survey (CFHTLenS) and found no significant detection. These works, together with Refs. [72, 73, 74, 75], have proved that the study of the cross-correlation of the DGRB with the LSS is a very powerful strategy that may provide access to components of the DGRB that are only subdominant in the intensity energy spectrum or in the auto-correlation APS. In particular, the technique has the potential to deliver the first detection of DM-induced gamma-ray emission.

In the near future, the data on the DGRB is expected to further increase, due to the extended run of the Fermi LAT and to the fore-coming Cherenkov Telescope Array (CTA) [76, 77]. Other frequencies and other messengers will also be crucial to improve our modeling of the DGRB and to extract complementary information about its nature. The scenario is rapidly evolving and the study of the DGRB is quickly becoming a standard tool for the characterization of unresolved astrophysical sources and of a potential DM-induced gamma-ray signal.

Therefore, we believe that this is the right moment to summarize where we stand in our understanding of the DGRB. In this article we will survey the data available on the DGRB at present and we will discuss how these observations have been used to constrain the nature of the emission. We will also enumerate the classes of sources or emission mechanisms that have been proposed as contributors to the DGRB. By sketching a snapshot of the state-of-art on the DGRB circa 2015, we intend to provide the community with a reference point from which to build on.

We end by noting that the DGRB is intrinsically an analysis- and time-dependent quantity. Indeed, its intensity depends on the sensitivity of the telescope employed to detect it and on its instrumental capability to resolve sources. Even with the same detector, an increase in statistics or, in general, any improvement in the detection sensitivity will result in a different DGRB. In the following sections, every time we mention the DGRB we will make sure to specify which measurement of the DGRB we refer to.

The paper is organized as follows: in Sec. 2 we focus on the intensity of the DGRB. Sec. 2.1 reviews the recent measurement of the DGRB energy spectrum by the Fermi LAT, while Secs. 2.2 and 2.3 are devoted to the description of the sources that have been proposed as contributors to the emission: astrophysical objects are studied in Sec. 2.2, while Sec. 2.3 discusses the case of the DM-induced emission. Sec. 3 reviews the Fermi LAT measurement of the auto-correlation APS of anisotropies and its impact on our understanding of the DGRB. The topic of Sec. 4 is the measurement and the interpretation of the photon count probability distribution, while in Sec. 5 we investigate the cross-correlation with probes of LSS, namely galaxy catalogs in Sec. 5.1, cosmic shear in Sec. 5.2 and other observables in Sec. 5.3. Finally, we present our conclusions in Sec. 6.

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2 Different names have been used in the literature to denote the DGRB, e.g. Extragalactic Gamma-Ray Background or Isotropic Gamma-Ray Background. We believe the denomination used in this review is more precise since, as we will see in the following sections, the DGRB may be not entirely extragalactic and since it exhibits a certain amount of anisotropy. We also note that, in Ref. [9], the Fermi LAT Collaboration uses the name Extragalactic Gamma-Ray Background to characterize the cumulative emission of all sources (both resolved and unresolved), while Isotropic Gamma-Ray Background refers to the unresolved component only, i.e. what we call DGRB (see Sec. 2). Back

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