The origin of the cosmic gamma-ray background in the MeV range

There has been much debate about the origin of the diffuse $\gamma$--ray background in the MeV range. At lower energies, AGNs and Seyfert galaxies can explain the background, but not above $\simeq$0.3 MeV. Beyond $\sim$10 MeV blazars appear to account for the flux observed. That leaves an unexplained gap for which different candidates have been proposed, including annihilations of WIMPS. One candidate are Type Ia supernovae (SNe Ia). Early studies concluded that they were able to account for the $\gamma$--ray background in the gap, while later work attributed a significantly lower contribution to them. All those estimates were based on SN Ia explosion models which did not reflect the full 3D hydrodynamics of SNe Ia explosions. In addition, new measurements obtained since 2010 have provided new, direct estimates of high-z SNe Ia rates beyond $z\sim$2. We take into account these new advances to see the predicted contribution to the gamma--ray background. We use here a wide variety of explosion models and a plethora of new measurements of SNe Ia rates. SNe Ia still fall short of the observed background. Only for a fit, which would imply $\sim$150\% systematic error in detecting SNe Ia events, do the theoretical predictions approach the observed fluxes. This fit is, however, at odds at the highest redshifts with recent SN Ia rates estimates. Other astrophysical sources such as FSRQs do match the observed flux levels in the MeV regime, while SNe Ia make up to 30--50\% of the observed flux.


Introduction
The cosmic gamma-ray background is a diffuse background whose origin is diverse and remains unknown at various energy ranges. On the low energy side, from X-ray energies up to around 0.3 MeV, it has been confirmed that AGNs and Seyfert galaxies provide most of the emission (Madau et al. 1994;Ueda et al. 2003). At E > ∼ 0.3 MeV, the spectrum of AGNs and Seyfert galaxies sharply cuts off. From 50 MeV to the GeV range, blazars seem to be responsible for the observed fluxes (Zdziarski 1996;Sreekumar et al. 1998).
However, the latest results from Fermi, in the GeV range, which show a higher gamma-ray background at GeV energies than previous results from EGRET, have put into question the former attribution to blazars as the main source (see discussions in Lacki, Horiuchi & Beacom 2014). However, recent examinations of this topic find that at > 100 MeV, blazars account for the 50% of the background, while the rest 50% is contributed by star-forming and radio galaxies (Ajello et al. 2015;Di Mauro & Donato 2015).
The measurements in the MeV range have been provided by various space missions.
The first exploration of the region between 1 and 5 MeV was made by the APOLLO 15/16 missions (Trombka et al. 1977). The reanalysis of the Apollo data, the measurements from HEAO-A4 , the Solar Maximum Mission (Watanabe et al. 1999a), and COMPTEL (Kappadath et al. 1996;Weidenspointner 2000) provided the basic empirical results on the diffuse gamma-ray background, in the range from 100 keV to 10 MeV. The slope of the emission spectrum is a fast decrease with increasing energy, from a few hundred keV to 10 MeV, where it meets a flatter slope around 10 MeV and beyond, revealing the need of an intense extragalactic source in the MeV window. In this range of energies, the discussion on the origin of the background has been reinitiated at the beginning of the XXIst century and continued up to the present time (Lacki, Horiuchi & Beacom 2014).
Much of the current discussion has to do with the possibility that Type Ia supernovae are able to fill the gap of the observed cosmic gamma-ray background in the 0.3-3 MeV range. Such possibility was first suggested by Clayton and Silk (1969). The level of the flux was studied in the 90's by The et al. (1993) and Watanabe et al. (1999b) (hereafter W99b). At those times, there were no empirical rates of SNe Ia available, R Ia (z), measured at high redshifts. Therefore, the authors had to speculate on the basis of the star formation rates in galaxies and the production of SNe Ia with different delay times between the birth of the progenitor binary systems and the explosions. Depending on the delay time since the birth of the system, the SNe Ia rate could be either too low or just right, to account for the measured level of the gamma-ray background (W99b was 1.41 ± 0.35 10 −3 M ⊙ −1 h 2 65 . In that work, they found that the delay time between star formation and supernova explosion could shift the estimate up or down, but that most star formation histories would average out the effect, since the distribution of SNe Ia delay times appeared to be broad. The result of this study was that SNe Ia yield a background emission, in the MeV range, that can explain the extragalactic emission measured by COMPTEL and SMM. Just shortly afterwards, the topic was addressed by other authors (Iwabuchi & Kumagai 2001;Ahn, Komatsu & Höflich 2005 -hereafter AKH05-;Strigari et al. 2005), and it has recently been touched upon again by Horiuchi & Beacom (2010). They have obtained a variety of results concerning the SN Ia contribution, but they all agree that such -6contribution is too low to account for the MeV background. This negative result can be traced to the first high-z SNe Ia rates obtained by Dahlen et al. (2004) (hereafter D04).
Such rates peak at a smaller z than current rates (Okumura et al. 2014;Rodney et al. 2014;Graur et al. 2014).
Both W99b and RCV01 (see also previous similar findings by The et al. 1993 andby Zdziarski 1996) had found that there was good room for SNe Ia to account for the unexplained MeV background. They did so by integrating the full spectrum of SN Ia models over the time when they give significant emission. Horiuchi & Beacom (2014) integrate only the gamma-ray line emission from various models. This fact may contribute to the finding of a very low contribution from SNe Ia. AKH05 take a SNIa rate (from D04) that is one order of magnitude smaller than the one adopted by W99b. This may explain the discrepancy in the results. One interesting feature of the SNe Ia contribution, which is shared by the calculations by W99b, RCV01, and AKH05, is that the predicted SNe Ia gamma-ray background contribution has a shape (power law in energy) that runs parallel to the observed data, unlike the predictions for other sources. Currently, the disagreement is in the level of SNe Ia gamma-ray contribution to the gamma-ray background fluxes.
Nowadays, we have more extended knowledge of the SNe Ia rates, for a wide variety of redshifts (some of the most recent contributions were mentioned above). We also know well the cosmological parameters required for the calculation (H 0 , Ω M , Ω Λ ). Moreover, for the first time, a SN Ia (SN 2014J) has been detected in the MeV range by INTEGRAL (Churazov et al. 2014;Diehl et al. 2014) confirming that SNe Ia emit continuum and gamma-ray lines. The latter authors can account for the gamma-ray data with a white dwarf explosion having a small amount of 56 Ni at the outskirts and around 0.6 M ⊙ in the innermost core. This model is quite similar to those used in the previous calculations of the gamma-ray background (and also to those that will be used in the present work). Thus, we now have a firm idea of the gamma-ray flux emitted by SNe Ia.
All the precedent has led us to recalculate the gamma-ray background from SNe Ia, now based on a much better knowledge of the ingredients needed to do it. The present paper is organized in the following way. In Section 2, we discuss the different codes used to compute the gamma-ray fluxes from given SNe Ia models. The models used for the present study are described in Section 3. In Section 4, we give an update of the SNe Ia rate at various z and we show the interpolations used in the background calculations. In Section 5, we explain the gamma-ray background formulation and then we present the results.
Section 6 gives a discussion of the results and the final Section states our conclusions.

Gamma-ray escape from models
In order to derive the emergent spectrum predicted by supernova models, we use the Monte Carlo code developed in Burrows &  and The, Burrows & Bussard (1990), modified to include bremsstrahlung X-ray production and the iron fluorescence line at ∼6.4 keV (Clayton & The 1991;The, Bridgman & Clayton 1994). This code follows the gamma-ray emission of radioactive nuclei such as 44 Ti, 56 Ni, 56 Co, 57 Co. The code was used in the predictions by W99b.
In the predictions in RCV2001, a modified version of the Ambani & Sutherland (1988) code was used. The same code had been used by Lehoucq, Cassé & Cesarsky (1989) to study the radioactive output from SN1987A. Then, it was modified (Ruiz-Lapuente et al. 1993) to include positronium formation in the 56 Co decay, giving rise then to two-photon (parapositronium) and three-photon decay (orthopositronium), the energy distribution being that derived by Bussard, Ramaty & Drachman (1979). The Monte Carlo routine was then used for the study of gamma-rays from SNe Ia (Ruiz-Lapuente et al. 1993). A comparison of the predicted gamma-ray emission, for model W7 (Nomoto, Thielemann & Yokoi 1984), with that from W99b shows the gamma-ray fluxes from the code used in RCV2001 to be a 10% higher in the MeV regime than those used in W99b. Such differences among the outputs of different gamma-ray codes have been studied by Milne et al. (2004). As seen in that work, variations of 10% can arise from the different physics and methods used to predict the gamma-ray emission. With the use of the same code as in W99b, Burrows & The (1990), and The, Burrows & Bussard (1990), we are here on the conservative side concerning the predicted gamma-ray emission (see Milne et al. 2004).

Input models
A wider variety of predictions in the MeV domain can be obtained when using different input models for the SNe Ia. This has been recently shown by The & Burrows (2014). Looking at these results, one sees that there are 20% variations, in the MeV range, depending on the input model used. In this work, we pick five different models in our attempt to delineate the upper limit constraint on the gamma-ray emmision from normal SNe Ia. These models are W7 (Nomoto, Thielemann & Yokoi 1984), the fully mixed W7 model (W7fm) used in W99b, the W7dt model of Yamaoka et al. (1992) (this one predicting a larger gamma-ray escape than most of the other models explored by The & Burrows (2014), the spherically-averaged 3D model N100 of a delayed detonation Seitenzahl et al. 2013), and the (also averaged) 3D model of the violent merging of two WDs of Pakmor et al. (2012).
The first three models have already been extensively discussed in the literature. Their gamma-ray emissions are compared in Figure 1. The 3D model N100 of Röpke et al. (2012) corresponds to an initially isothermal WD, made of C and O in equal parts, and with a central density of 2.9 × 10 9 g cm −3 . It was ignited in 100 sparks (hence its name), placed randomly in a Gaussian distribution within a radius of 150 km from the WD's center. After an initial deflagration phase, a detonation was triggered at every location of the flame where the criterium for deflagration-to-detonation transition of Ciaraldi-Schoolmann et al.
The model from Pakmor et al. (2012), also a 3D model, simulates the violent merger of a 1.1 M ⊙ and a 0.9 M ⊙ WD. As the material of the tidally disrupted secondary WD hits the primary WD, a hot spot forms which leads to the ignition of C burning. The conditions reached there are sufficient to trigger a detonation that burns the merged object and leads to the explosion.
The spectral evolution of the gamma-ray emission from the two 3D models above is shown in Fig. 1 of Summa et al. (2013), together with the spread due to different viewing angles for the maximum-light epochs in gamma rays of both models. Here, however, we use the spherically-averaged models, with the same treatment of gamma-ray escape as for the three 1D models (W7, W7fm, and W7dt) considered. Their comparison with W7fm is shown in Figure 2. Ruiter et al. (2013) have shown that sub-Chandrasekhar pure detonation models can reproduce the observed peak-magnitude distribution of the SN Ia (Li et al. 2011). The brightness of the explosion is then mainly determined by the 56 Ni mass synthesized in the detonation of the primary WD. Sim et al. (2010) have derived a relationship between the mass of that WD and the expected peak bolometric brightness. Ruiter et al. (2013) use the population synthesis data from Ruiter et al. (2011) to derive the theoretical peak-magnitude distribution.
Here we estimate the contribution of this channel to the gamma-ray background, based on the fact that the distributions of WD and 56 Ni masses peak at 1.1 M ⊙ and 0.6 M ⊙ , respectively. In Figure 3 we compare the gamma-ray spectrum of a representative model of -10such mergings with that of model W7fm.
What has been shown in this Section is that the background arising from different explosion models, from very different origins but aiming at explaining normal SNe Ia, are minimal.
The convolution of the gamma-ray output from several models, here exemplified by model W7fm, with the supernova rates is explained in Section 5.  curves help to discriminate SNe Ia from SNe Ib/II.

SNe Ia rates
The final count of SNe Ia detected leads to the number of supernovae per comoving volume unit, r V (z). The number of SNe Ia expected in a redshift bin (z 1 < z < z 2 ) is given by (Okumura et al. 2014): where V (z)dz is the comoving volume in a redshift slice of thickness dz, Θ is the solid angle The D04 data are shown as red circles in Figure 4. The fit to these data appears as the red long-dashed line in the Figure. As it can be seen, we have fitted the D04 data in a similar way as in AKH05. We have considered, as in that work, that the first errorbar is reduced by taking the mean value of the upper and lower error bars of the first z. We see that there are still wide divergences in the measured SN Ia rates, in spite of the considerable increase of the data base in recent years. They reflect the differences in the procedures followed to derive the rates from the observations.

Background contributions
Based on the gamma-ray spectra in Figures 1-3  to 600 days later were cumulated. Therefore, the number of active SNe Ia per unit of comoving volume, at any time, is that of those produced during the preceding time interval: R ′ Ia (z) = const. × R Ia (z), the latter being the comoving SN Ia rate (SN yr −1 Mpc −3 ). Thus, the contribution to the gamma-ray background of the shell at comoving radius r and with thickness dr would be: where d M being the proper motion distance (in Mpc). The flux received from that shell (in photons cm −2 s −1 keV −1 ) will be: d L being the luminosity distance (in cm). The factor (z + 1), multiplying E, accounts for the redshift of the photons. We thus have: Due to time dilation, there should be a factor (1 + z) −1 multiplying the comoving SN rate, but it is cancelled by the (1 + z) factor accounting for compression of the energy bins.
Then, since d L = (1 + z)d M : d(d M ) depends on the cosmological parameters H 0 , Ω M , and Ω Λ , and We adopt H 0 = 70 km s −1 Mpc −1 , Ω M = 0.3, and Ω Λ = 0.7 in calculating our cosmic gamma-ray background. Indeed, the adopted values for Ω M and Ω Λ are the values favored by the PLANCK collaboration (Ade et al. 2013). The H 0 = 70 km s −1 Mpc −1 value is a mean value coming from the discussions of the various approaches to determine H 0 .
Thus, the universe is flat and: We take z lim = 2.5, as our various test calculations show that the contribution from SNe Ia at higher redshifts to the background is negligible. That can readily be understood because SNe Ia rates drop significantly at redshifts z larger than ∼ 2. Furthermore, in order to compare calculated fluxes with observed values, we need to divide the F γ (E) above by 4π, to convert to the units of observed fluxes (photons cm −2 s −1 keV −1 sterad −1 ).
In Figure  In order to make a consistency check, we calculate the gamma-ray background from rates resulting from the fit to the D04 data (red dashed line in Figure 4) and its 1 σ and 2 σ predictions, using the W7fm model. Though the model is not the same as that used in AKH2005, we expect our predictions should not significantly differ from theirs. Indeed, our results of cosmic gamma-ray background shown in Figure 6 as black solid histogram coincide with those of AKH05. In Figure 6, we also plot the 2 σ upper limit of cosmic gamma-ray background using the D04 SNe Ia rate as the red dashed line confirming AKH05 conclusion that SNe Ia can not account for the observed gamma-ray background. However, the conclusion is influenced mostly due to the D04 SNe Ia rate used in the calculations, as we shall see below.
Having calculated the gamma-ray background using the fit of the SNe Ia rate to the D04 data, now we take the fit of Okamura et al. (see Figure 8).
One could think that the SN Ia rates going to higher redshifts could be underestimated One can see that, after z = 2, the increase in the contribution is very minor. Here and in the next three Figures, the black squares correspond to the COMPTEL data, analyzed by Kappadath et al. (1996), while the continuous line gives the results from the Solar Maximum Mission (Watanabe et al. 1999a), the dotted lines being the 1 σ upper and lower limits. if one does not take into account the distribution of SN Ia light curve stretches, because that would disfavour the detection of the small-stretch, lower luminosity supernovae.
However, all the collaborations mentioned above have taken due account of this effect in their modeling of the observed rates.

Discussion
SNe Uncertainties on this contribution come from the SNe Ia Galactic rates (Milne et al. 2001).
Such non-fully attribution to SNe Ia of the 511 keV line has led to the suggestion that most of the contribution comes from annihilation of a dark matter candidate (Fayet 2004;Boehm et al. 2004). In the GeV regime, the lack of a full contribution from astrophysical sources such as blazars, led to the proposal that weakly interacting massive particles (WIMPs) annhilations in the Galactic halo could be the main contributor to the GeV background fluxes. Those WIMPS would have masses in the range from 0.1 GeV to 10 GeV, for a variety of dark-matter halo models (Pullen, Chary, & Kamionkowski 2007). However, recent work (Ajello et al. 2015;Di Mauro & Donato 2015) constrain very much the dark matter contributions in this energy range. These authors find that blazars, together with star-forming and radio galaxies, can account for the gamma-ray background there.
In a similar way, the early estimates of SN Ia rates at high z (D04), predicted a small contribution from Type Ia SNe to the gamma-ray background (AKH04). In that calculation, the SN Ia contribution is so small that even the 2 σ upper bound derived from the D04 SNe Ia rates cannot account for the measured gamma-ray background (AKH04,

Conclusions
We have seen, from the calculations in this paper, that at face values the SNe Ia rates at various redshifts are too low to account for the observed gamma-ray background in the MeV range.
It is found that, above z = 2-2.5, the gamma-ray emission from SNe Ia makes no significant contribution the the cosmic gamma-ray background. So, future determinations of the SNe Ia rates above that range should not make any noticeable difference. What makes a difference in the results is the overall SNe Ia rate from z = 0 to z = 2. We have shown here that the present central values of the SNe Ia rates from Okumura et al. (2014) and Rodney et al. (2014) predict a contribution that is about a factor of five below the observed background. However, the formal 1 σ upper bound of the fit by Okumura et al. (2014), which in fact lies between the 1 σ and 2 σ upper bounds to the highest rates, could produce the observed background. One might consider such upper bound, at and above z = 1.5, as being too generous, since there are indications ) of a decrease in the SNe Ia rates at z > 1.5, and also of lower rates at all z. Large contributions at z beyond 2 are, as we have seen, too distant to be significant, even in the most optimistic cases, where the rates continue to grow, in a steep power law above such z (see Figure 4).
A relevant aspect of the uncertainties in determining properly the rates of SNe Ia at high z (> 1.4) is that the detection efficiency decreases rapidly with redshift, as the observed bands shift farther into the rest-frame UV spectrum, where the SN Ia emission is scarce (see the spectra shown in Riess et al. 2007). This makes the uncertainties become very large at z ∼ 1.6 ( Barbary et al. 2012).
The currrent measurements of rates at high z concentrate on normal (cosmological) SNe Ia. They make up to about 60% of all SN Ia explosions (see Ruiz-Lapuente 2014, for a review). The average predictions for the gamma-ray background, which at face values give gamma-ray backgrounds that are too low, might slightly change if the bulk of the non-cosmological SNe Ia were included. It has been estimated that up to 30% of SNe Ia could be of the Type Iax , though a lower estimate is found in White et al. (2014). Those SNe Ia produce smaller amounts of 56 Ni, thus making an even smaller contribution to the background (although the smaller total mass should make gamma-ray escape at larger fraction). Furthermore, there is also a 15% of SN1991bg-like events, much less luminous than the cosmological SNe Ia.
In the opposite perspective of the above, a factor that could make the mean gamma-ray prediction larger would be to adopt models brighter in gamma-rays than W7 or W7fm (both models giving about the same average contribution over 600 days after explosion); some SNe Ia models can give twice as much gamma-ray escape. One should acknowledge, however, that those models might fare worse in the comparison with observed spectra and light curves in the optical range.
In summary, following many measurements and calculations of the SNe Ia rates at various z, we have a more deterministic estimate of the SNe Ia gamma-ray background in the MeV range than it was possible in the last decade of the XXth. century and beginning of the XXIst. The last ten years have produced many measurements and calculations of the SNe Ia rates at various z. Only accounting for the very large error bars of the rates one can reach the level of the observed background. The conclusions are independent from the particular model taken to represent normal SNe Ia. Attributing the gamma-ray background to SNe Ia appears tempting, among other things because it has the right spectral shape in the MeV range. However, if the current uncertainties around the central values for the SN Ia rates were to shrink, then one would need to look for another source of the gamma-ray background in the MeV range (dark matter, MeV blazars). In that case, SNe Ia could make up to 50% of the gamma-ray background emission (if we take the mean values of the present observed rates) and another source make the rest of the contribution.