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The published version of this article contained a computational error in the calculation of the mass accretion rate J0 (mass current crossing the shock surface in units of M☉ yr−1) at redshift z0 of the galactic cluster to which we normalize our models of structure formation cosmic rays. The mass accretion rate J0 was calculated using Pavlidou & Fields (2006):
where Ωb = 0.04 is the baryonic matter energy density parameter, ρc, 0 is the critical density at the present epoch, cs, 1 is the adiabatic sound speed of the pre-shocked material, δs is the overdensity in which the accretor is located, and rv is the virial radius of the accretor. The Mach number of the accretion shock is 
. We chose to normalize to Coma cluster, so the correct value of the accretion rate for this cluster is J0 = 417.86 M☉ yr−1. This value is an order of magnitude lower than the value used in the published version of the paper. This does not change our main conclusion that structure formation cosmic rays can make an important contribution to the extragalactic gamma-ray background, but it does change the parameter values of the best-fit model. All of our models now give an order of magnitude higher gamma-ray flux, which now puts a tighter constraint on the ones that are allowed by the date.
The observed extragalactic gamma-ray background is, as in the published version, best matched by the structure formation cosmic-ray component where source evolution is based on the most simple model with no environmental effects taken into account, Model 1 of Pavlidou & Fields (2006), while other models overshoot the observed data. In the Results section of the published version, we have adopted as our default case the spectrum where the initial gas mass parameter was taken to be 
= 0, while the adopted spectral index was that of the strong shocks αγ = 2.1. Since our curves are now higher, even our simplest model is above the extragalactic gamma-ray background observed by the Fermi for strong shocks spectral indices αγ ≈ 2. Our most probable scenario is now for the spectral index 2.7 (Model 1). This is close to Brunetti et al. (2012), where they found best fit of the Coma spectrum to be derived using spectral index ≈2.6. Although even in this case of the softer spectra, the model does overshoot the data at some energy ranges, we note that our normalization was based on the Coma gamma-ray upper limit as reported by Fermi and thus leaves room for downward correction once the detection is made or when a specific emission model is used.
We plot Figure 1 with the same parameters as in the published version of the paper, but with now corrected value of J0 (top panels). Here we also add the same plots of our best-fit scenario with spectral index 2.7 (bottom panels). Figure 1 shows the contributions of structure formation cosmic rays, normal galaxies, and blazars shown separately (left), as well as their summarized contribution (right). We also correct Figure 2 from our published paper using our best-fit spectral index 2.7 instead of 2.1 which was used before. Figure 3 is also corrected and plotted using the same parameters as in the published version.
Figure 1. Contribution of different components to the extragalactic gamma-ray background (data points) observed by Fermi (Abdo et al. 2010). Left: all components shown separately—blazars (solid line), normal star-forming galaxies based on two limiting cases given in Fields et al. (2010; red dash dotted line represents luminosity evolution and blue dash dot dotted line represents density evolution), and structure formation cosmic-ray contribution calculated as in the published article, but with the corrected value for J0, normalized to the Coma cluster gamma-ray flux limit, with initial gas mass parameter = 0, for three different source models derived in Pavlidou & Fields (2006; long dashed, Model 1; short dashed, Model 2; dotted line, Model 3). Top panel shows structure formation cosmic-ray spectra derived using spectral index αγ = 2.1 and bottom panel αγ = 2.7. Right: the combined contribution of all components where different curves reflect different normal galaxy emission models (thick red curves, luminosity evolution; thin blue curves, density evolution) and different structure formation cosmic-ray emission models (three different line types correspond to the same models as on the top panel).
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Figure 2. This plot shows the sensitivity of our model on the adopted initial gas mass fraction parameter . For the purpose of demonstration, we plot the structure formation cosmic-ray gamma-ray emission for out best-fit model with spectral index αγ = 2.7, based on Model 1 from Pavlidou & Fields (2006), and derive adopting different initial gas mass fraction values, = 0, 1, 10. The top curve is approximately a factor of two higher than our best-fit case plotted in Figure 1. For all > 10, all curves converge and are overlapping with the = 10 curve.
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Figure 3. Contribution of structure formation cosmic rays to the extragalactic gamma-ray background (data points) observed by Fermi (Abdo et al. 2010) for different choices of the cosmic-ray spectral index ranging from αγ = 2 to αγ = 3, for Model 1 (top panel), Model 2 (middle panel), and Model 3 (bottom panel). The Coma cluster is again taken as the normalizing object with = 0.
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Our models still show that structure formation cosmic rays can make a substantial contribution to the extragalactic gamma-ray background, but the data are now more constraining to our models. Resulting contributions are now higher than those derived in some of the papers mentioned in the Discussion and Conclusion section of our published article (Miniati 2002; Colafrancesco & Blasi 1998; Kuo et al. 2005).