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Gamma-Ray Background from Structure Formation in the Intergalactic Medium

The universe is filled with a diffuse and isotropic extragalactic background of gamma-ray radiation, containing roughly equal energy flux per decade in photon energy between 3 MeV-100 GeV. The origin of this background is one of the unsolved puzzles in cosmology. Less than a quarter of the gamma-ray flux can be attributed to unresolved discrete sources, but the remainder appears to constitute a truly diffuse background whose origin has hitherto been mysterious. Here we show that the shock waves induced by gravity during the formation of large-scale structure in the intergalactic medium, produce a population of highly-relativistic electrons with a maximum Lorentz factor above 10^7. These electrons scatter a small fraction of the microwave background photons in the present-day universe up to gamma-ray energies, thereby providing the gamma-ray background. The predicted diffuse flux agrees with the observed background over more than four decades in photon energy, and implies a mean cosmological density of baryons which is consistent with Big-Bang nucleosynthesis.

in the remnant shocks up to an energy ∼ 100 TeV, and are confined to the collisionless fluid by magnetic fields. These shocks have a velocity of order 10 3 km s −1 , similar to the velocity of the intergalactic shocks we consider here. The inferred energy density in relativistic electrons constitutes 1-10% of the post-shock energy density in these remnants, a fraction consistent with the global ratio between the mean energy density of cosmic-ray electrons and the turbulent energy density in the interstellar medium of our galaxy.
Since the physics of shock acceleration can be scaled up to intergalactic distances, it appears natural to assume that a similar population of relativistic electrons was also produced in the intergalactic medium. The maximum Lorentz factor of the accelerated electrons, γ max , is set by equating their acceleration and cooling times. The e-folding time for the acceleration of the relativistic electrons is t acc ∼ (r L c/v 2 sh ), where v sh = (Γ + 1)[kT /(Γ − 1)m p ] 1/2 is the shock velocity relative to the unshocked gas, m p is the proton mass, and r L is the electron Larmor gyration radius. For an electron with a Lorentz factor γ e , r L = 5.5 × 10 −2 (γ 7 /B −7 ) pc, where γ 7 = (γ e /10 7 ), and B −7 is the magnetic field strength in units of 0.1µG. A magnetic field amplitude of 0.1-1µG is often detected in the halos of galaxy clusters 15−17 . In particular, a magnetic field amplitude of ∼ > 0.1µG was inferred on the multi-Mpc scale of the Coma-A1367 supercluster 18 , which provides a good example for the intergalactic structures of interest here. For such magnetic field amplitude we get, t acc ∼ 2 × 10 4 yr (γ 7 /B −7 )(kT /keV) −1 .
The acceleration time is much shorter than the lifetime of the intergalactic shocks, which is comparable to the age of the universe. The maximum Lorentz factor of the electrons, γ max , is therefore not limited by the lifetime of their accelerator, but rather by their cooling, primarily due to Compton scattering off the cosmic microwave background 19 . Synchrotron cooling is negligible for B −7 ∼ < 10. The characteristic cooling time due to inverse-Compton scattering is t cool = [m e c/(4/3)σ T γ e u cmb ] = 1.2 × 10 10 yr (γ e /200) −1 , where m e is the electron mass, σ T is the Thomson cross-section, and u cmb is the energy density of the cosmic microwave background. By equating the acceleration and cooling times, t acc = t cool , we find γ max = 3.7 × 10 7 [B −7 (kT /keV)] 1/2 . The estimates given above are valid as long as the electron Larmor radius is smaller than the coherence length of the magnetic field. Since the Larmor radius of the relativistic electrons is extremely small, r L ∼ 0.1 pc, this condition is satisfied even if the field coherence length is much shorter than the ∼ Mpc scale of the intergalactic shocks.
All electrons with γ e > 200 lose their energy over the age of the universe. The energy of these electrons is converted to a diffuse background of photons, which are produced through inverse-Compton scattering of microwave background photons. The initial energy of a microwave photon, hν 0 , is boosted by the scattering up to an average value of hν = (4/3)γ 2 e hν 0 . Substituting the mean frequency of the microwave background photons for ν 0 , we find that the accelerated intergalactic electrons produce a diffuse background of radiation extending from the UV [hν = 36(γ e /200) 2 eV] and up to extreme γ-ray energies [hν = 89γ 2 7 GeV]. For a power-law distribution of relativistic electrons, dn e /dp e = Kp −2 e , the energy density of the scattered radiation is predicted to be constant per logarithmic frequency intervals, ν(du/dν) = Kc/2 = u e /2 ln γ max .
Here u e = Kc ln γ max is the energy density in relativistic electrons produced by the intergalactic shocks. If a fraction f sh of the baryons in the universe were shocked to a (mass-weighted) mean temperature T , and a fraction ξ e of the shock thermal energy was transferred to relativistic electrons, then u e = 3 2 ξ e f sh n p kT = 5.1 × 10 −16 ξ e f sh Ω b h 2 70 /0.04 (kT /keV) erg cm −3 , where n p is the average proton density, Ω b is the cosmological baryon density parameter, and h 70 is the Hubble constant in units of 70 km s −1 Mpc −1 . By substituting ln γ max = ln(4 × 10 7 ), we then get where E = hν, and J is the number flux of photons per solid angle. Adopting the result from hydrodynamic simulations of 4 f sh (kT /keV) ∼ 1, the Big-Bang nucleosynthesis value of 20,21 Ω b h 2 70 = 0.04 +.014 −.027 (based on the deuterium abundance), and the value of ξ e ∼ 0.05 inferred from non-relativistic collisionless shocks in the interstellar medium, we obtain a background flux of ∼ 1 keV cm −2 s −1 sr −1 in the photon energy range of 3 MeV to 100 GeV. As Figure 1 illustrates, this result is in excellent agreement with the γ-ray background detected by the EGRET instrument aboard the CGRO satellite 1 .
The sky-averaged contribution of identified extragalactic sources, such as blazars (γ-ray loud active-galactic-nuclei), amounts to only ∼ < 7% of the extragalactic γ-ray flux 3 . As shown in Figure  1, the recent EGRET data is consistent with a luminosity function of faint, undetected blazars, which could account for only ∼ < 25% of the unresolved γ-ray background 2,3 . The remainder of the hard γ-ray background is expected to be highly isotropic in our model because it is integrated over a large volume throughout the universe. Its large-angle fluctuations should be comparable (to within a factor of a few) to that of the hard X-ray background, of which 25 ∼ 40% is contributed by early-type galaxies at z ∼ < 1, since these galaxies trace the same large scale structure as painted by the comic web of intergalactic shocks. This implies a root-mean-square fluctuation amplitude smaller than ∼ 5% on angular scales larger than a degree 26 . The predicted high level of isotropy can serve as a test of our model, as soon as the isotropy of the γ-ray background is measured to a higher precision than presently available. Another possible test is the spectral distortion induced in the microwave background band by the low-energy tail of the non-thermal intergalactic electrons. This distortion is, however, well below the upper limit inferred from the COBE satellite data 27 .
At photon energies ∼ < MeV the scattered background is sub-dominant relative to the cumulative flux produced by discrete sources, such as active galactic nuclei. In particular, the predicted scattered flux amounts only to ∼ 10-15% of the soft X-ray background measured by ROSAT at 1-2 keV, and is consistent with the upper limit of ∼ 20-30% on its unresolved fraction 25,28 .
Most of the γ-ray background is produced during the latest episode of shock heating of the intergalactic gas at z ∼ < 1, since the mean gas temperature is expected to decrease rapidly with increasing redshift 4 . Furthermore, the photons emitted at redshifts z ∼ > 0.5 lose a significant fraction of their energy due to the expansion of the universe. A more accurate calculation of the γ-ray background can be achieved by simulating the hydrodynamic evolution of the intergalactic medium and injecting a population of accelerated electrons into each fluid element which passes through a shock, at all redshifts. Such a simulation can yield both the background flux and its statistical fluctuations on the sky. For a given cosmological model of structure formation, the derived background flux will be proportional to ξ e , and will not depend on any other free parameters.
Although the warm ( ∼ < 10 7 K) intergalactic medium is expected theoretically to include a major fraction of the baryons in the present-day universe, it has not been observed directly as of yet. The soft X-ray emission from intergalactic structures, such as sheets and filaments or galaxy groups, is often too faint to be detectable with current telescopes 4 . The observed cosmological density of stars 21 , Ω ⋆ h 70 = 3.5 × 10 −3 , is an order of magnitude smaller than the baryon density predicted by Big-Bang nucleosynthesis. If the γ-ray background indeed results from keV shocks in the intergalactic medium, then we can derive a lower limit on the mean baryon density required to produce its observed flux. This minimum is obtained by substituting ξ e = 1/2 (equipartition) and f sh = 1 in equation (1), yielding Ω b h 2 70 ∼ > 4 × 10 −3 . The inferred lower limit is larger than the observed mass density in stars. Thus, if the reality of our model is proven, then this would not only explain the origin of the diffuse component of the γ-ray background, but also provide the first evidence for the "missing baryons" (note that even if the truly diffuse component amounts to only ∼ 20% of the γ-ray background flux, the more plausible range of ξ e ∼ < 10% yields a lower limit on the baryon density which is still larger than the mass density in stars).
In our model, the γ-ray background is produced in the dense filaments and sheets which channel gas from converging bulk flows in the intergalactic medium. The densest and hottest shocks occur at the intersections of these filaments around the locations of clusters of galaxies. Although the richest accreting clusters may be rare and contribute a small fraction of the background, they contain the brightest shocks in the sky and produce the strongest fluctuations in the diffuse γ-ray background. Direct detection of these shocks can be used to calibrate ξ e in our model. Even a statistical detection of a cross-correlation signal between background fluctuations and other sources that trace the same large scale structure, such as galaxies, X-ray gas, the Sunyaev-Zel'dovich effect, or synchrotron emission from the high energy electrons in the intergalactic medium, can be used to prove the reality of our model. We predict that as cold gas goes through the strong virialization shock of an accreting cluster, it emits non-thermal radiation with photon energies between 30 eV and TeV. The shocked gas loses a fraction ∼ (4.5/7) × ξ e ∼ 3% of its thermal energy in this process. For hν ≫ 10 keV, the cooling time of the corresponding relativistic electrons is shorter than a billion years. In this regime the total non-thermal luminosity of the cluster, L nt , is limited by the time it takes the cluster gas to cross the virialization shock, i.e. t vir ∼ (2 Mpc/2 × 10 3 km s −1 ) = 10 9 yr. For a young cluster which shocks gas of total mass M gas to a virial temperature T gas , the total (bolometric) non-thermal luminosity is (2) (The total cluster mass, including the dark matter, is typically an order of magnitude larger than M gas .) The luminosity per each logarithmic interval in photon energy, anywhere between 30 eV and TeV, is ∼ 0.1L nt . This emission component would affect estimates of the gas temperature and the thermal luminosity of young clusters which form through shocking of cold gas. However, the high-energy emission would be suppressed in the weak shocks that result from mergers of pre-existing clusters which contain pre-heated gas, since the energy distribution of the accelerated electrons is steeper for weak shocks 8 . As a cluster relaxes to hydrostatic equilibrium and ages, its non-thermal luminosity declines. The emission of hard radiation disappears almost entirely as soon as there is no strong shocking of gas. At a perfectly quiescent state of hydrostatic equilibrium, the non-thermal radiation spectrum cuts-off above a photon energy, hν ∼ 0.5 keV(τ /3 × 10 9 yr) −2 , for which the electron cooling time equals the cluster age, τ . Detection of this cut-off can be used as a method of dating the time that has passed since the last major accretion episode of cold gas by the cluster. Most relaxed clusters with ages τ ≫ 10 9 yr, are expected to show only weak non-thermal emission at hν ∼ > 10 keV. We note that the detection of nonthermal X-ray emission has been reported recently for several clusters 16,17,29 . In addition, some X-ray clusters possess radio halos 30 which may result from synchrotron emission by the residual population of relativistic electrons in the cluster magnetic field.  1) with ξ e = 0.05 and f sh kT = 1 keV. The upper shaded region shows the sum of these contributions, which provides an excellent fit to the data. The blazar contribution was calibrated 2,3 only by the total flux in photons with energies E > 10 5 keV; hence, the small deviations of some data points from the upper shaded region might be due to uncertainties in the cumulative blazar spectrum. Although the background flux predicted by Eq. (1) is independent of the magnetic field strength in the intergalactic shocks, the maximum energy of scattered photons does depend on the field strength and is given by, hν max ∼ 1.2 B −7 (kT /keV) TeV. Photons with energies ∼ > 1 TeV produce an electron-positron pair as they scatter on the infrared background 22,23 . The pair cools again by scattering microwave photons, which may in turn produce new pairs. The energy originally stored in photons with hν ∼ > TeV is spread smoothly over the 100 MeV to TeV energy range 24 , and might raise the existing flux there. However, since in our model only a small fraction of the electrons are accelerated to the energy required for the production of hν ∼ > TeV photons, we expect this effect to be small.