# Annihilation of positrons from AGN jets as a possible source of cosmic
gamma-ray background at energies below 511 keV

B.A. Nizamov1<EMAIL_ADDRESS>M.S. Pshirkov1,2,3
1 Sternberg Astronomical Institute  Lomonosov Moscow State University
Universitetsky pr  13  Moscow  119234  Russia
2 Faculty of Physics  Lomonosov Moscow State University
Leninskie Gory 1-2  119991 Moscow  Russia
3 Lebedev Physical Institute  Pushchino Radio Astronomy Observatory

###### Abstract

The origin of the diffuse gamma-ray background in the range from hundreds keV
to several MeV is not known conclusively. From current models and observations
it is believed that, at least partially, this background is formed by blazars
and remnants of supernovae (SN) of type Ia in distant galaxies. However, these
contributions are not sufficient to reproduce the observed level of the
signal. In this work we propose another source which could contribute to this
background, namely the jets of active galactic nuclei (AGN). The composition
of jets is not known, but there are observational hints that the fraction of
positrons there is substantial. Positrons are partially evacuated to the
intergalactic medium and partially mix with the circumgalactic medium and
annihilate there comparatively quickly. Using the AGN luminosity function, we
estimated the positron production rate and the contribution of the positron
annihilation to the cosmic background below 511 keV. We also estimated the
analogous contribution from positron annihilation within SN Ia remnants in
distant galaxies. The contribution of AGNs is estimated to be a factor of 5–10
smaller than the observed background intensity, and the contribution from SNe
is yet smaller by one order of magnitude. Nevertheless, the contribution of
AGNs appeared to be larger than the contribution of blazars estimated from
Swift-BAT and Fermi-LAT observations. The main uncertainty in our model is the
fraction of positrons remaining in the circumgalactic medium which makes our
estimation an upper limit.

active galactic nuclei, cosmic gamma-ray background

00.00.2018

##  1 Introduction

Isotropic electromagnetic background in the MeV–GeV range is a manifestation
of high energy processes in the Universe. Understanding the composition of
this background not only helps to reveal the details of the physical processes
at play, but is also needed for studies of compact sources where it is
important to disentangle the contribution from the background. Over the years
of investigation, various types of objects have been proposed as sources of
the gamma-ray background, GRB from now on (see a review [[1]] and Figure 9 in
[[2]]). In [[3]] it was shown from XMM-Newton and Chandra observations below
10 keV and theoretical knowledge on AGN spectra in a wide energy range that
the X-ray emitting active galactic nuclei (AGN) can explain GRB up to $\sim
100$ keV. Also, the flat spectrum radio quasars (FSRQ) observed by Swift/BAT
can provide up to 100% of GRB in the MeV range [[4]]. However, [[5]] found
that FSRQs observed by Fermi-LAT can provide only $\sim 30$% of GRB in the
same range.

Another possible contributor at $\sim 1$ MeV are supernova (SN) explosions,
mainly SN Ia. Their contribution has been estimated for decades by different
researchers despite considerable obstacles due to poor knowledge of SN Ia rate
at large redshifts ($\gtrsim 1.5$) and differences in the models of SN
explosions itself. Among the latest works on the topic, [[6]] found that the
intensity at MeV energies from SN Ia is almost an order of magnitude smaller
than the observed background. Later, [[7]] carried out a new analysis and
confirmed that SNe Ia are insufficient to account for MeV background,
providing from one-third to one-half of the GRB in this range.

GRB could be produced by cosmic rays in distant galaxies, as proposed in
[[8]]. The authors considered normal and star-forming galaxies and found that
cosmic rays in them could produce a large fraction of GRB at GeV energies, but
still a very small fraction at MeV energies.

In the context of AGN, it is usually assumed that the gamma-ray emission is
due to inverse Compton scattering. An alternative possibility was put forward
in [[9]]: gamma rays could stem from the annihilation of positrons contained
in the AGN jet. In [[9]] it is assumed that such positrons escape into the
intergalactic medium (IGM) and annihilate there with the electrons of the hot
gas. The authors did not consider this process as a contributor to GRB, and
they only predicted the corresponding gamma-ray flux from the Virgo cluster.
In the present work, we explore an analogous possibility, namely that the
positrons contained in the AGN jet do not escape into IGM, but remain in the
gaseous halo of the host galaxy, subsequently annihilating with its electron
population. This is a continuation of our previous work [[10]] (Paper I) where
we applied the above idea to an individual galaxy M31. Under assumption that
it hosted an AGN in the past, we calculated the flux of 511 keV photons from
it, which originate from annihilation of positrons accumulated in the gaseous
halo over the lifetime of the galaxy. The resulting flux appeared to be of the
order of INTEGRAL point source sensitivity. However, the source is expected to
span approximately $1\degree$ in the sky which could hinder the detection. In
the present work, we aim to apply the same idea to the whole population of AGN
in the Universe to estimate the potential contribution of the described
process to GRB at sub-MeV energies. In what follows, we assume $\Lambda$CDM
cosmology with $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$.

##  2 Positron production in individual AGN

In calculation of the positron production in an individual AGN, we follow
Paper I. The calculation is based on the idea that the jet is composed of
protons, electrons and positrons (and maybe ions, but we do not take them into
account). Kinetic energy of the jet is assumed to be contained in protons, as
argued in [[11]]. The main argument for this is that pure electron-positron
jet should stop very quickly due to the ”Compton rocket” effect. On the other
hand, there are indications that the positrons should be present as well. For
example, this could reconcile jet energy estimations from different methods
[[12]].

To derive the positron production rate, we use statistical relations given in
[[11]], namely the authors found that the total jet power is close to the
total accretion power, on average:
$P_{\mathrm{j}}=\eta\dot{M}_{\mathrm{acc}}c^{2}$ with the coefficient
$\eta\sim 1$ representing the efficiency of energy transfer from accreting
material to the jet. The radiative power of the jet $P_{\mathrm{rad}}$ is
found to be approximately an order of magnitude smaller than the total power.
The authors argue that the total power is dominated by the kinetic energy of
protons in the jet. They assumed that, apart from radiation and magnetic
field, the jet consists of electrons and protons only (no positrons). Further,
all the electrons participate in radiation and for each electron, there is a
proton. When we allow for pairs in the jet so that there are
$n_{\mathrm{pair}}$ positrons per each proton, the kinetic power estimated
from the radiation decreases by $2n_{\mathrm{pair}}$, because the number of
protons per lepton decreases by this number, so
$P_{\mathrm{j}}=\eta\dot{M}_{\mathrm{acc}}c^{2}/2n_{\mathrm{pair}}$. On the
other hand, the jet kinetic power equals the total energy of the protons
traversing the jet cross-section per unit time:
$P_{\mathrm{j}}=\dot{N}_{\mathrm{p}}\Gamma m_{\mathrm{p}}c^{2}$ where $\Gamma$
is the jet bulk Lorentz factor. From the last two equations, noting that
$\dot{N}_{+}=n_{\mathrm{pair}}\dot{N}_{\mathrm{p}}$, we obtain

$\dot{N}_{+}=\frac{\eta\dot{M}_{\mathrm{acc}}}{2\Gamma m_{\mathrm{p}}},$ (1)

Note that the above reasoning is valid for $n_{\mathrm{pair}}$ smaller than
10–15, because if $n_{\mathrm{pair}}$ is large, the jet power is not dominated
by protons anymore. However, the value of $n_{\mathrm{pair}}$ reported in
literature is typically less than 20. Note also that the above expression
allows in principle to derive the positron production rate. Combined with the
total lepton content from SED fitting it could provide an estimate of the
relative positron content in the jet. Of course, for such a calculation to be
viable, the signal from positron annihilation should be detectable. For the
jet bulk Lorentz factor we adopt the value of 10 which is typically found in
statistical studies and modeling of blazars [[13; 14; 15]]. Note also that the
factor $\eta$ might be quite different for different objects, but we are
interested in the contribution from the whole AGN population, and
statistically this factor is close to 1 [[11]].

In Paper I, we tried to estimate $\dot{N}_{+}$ for an individual galaxy (M31)
at various cosmic times. To do this, we had to solve the continuity equation
for the supermassive black hole mass distribution function and assign an
average mass evolution to M31 because it is impossible to reliably recover
this time evolution for an individual object. In contrast, in the present work
we calculate the positron production for the whole population of AGNs in the
Universe using their luminosity function from [[16]]. In this case we
obviously know the luminosity of AGNs and can relate it to the mass accretion
rate via the radiative efficiency parameter $\epsilon$:
$L=\epsilon\dot{M}_{\mathrm{acc}}c^{2}$. We adopt $\epsilon=0.1$ [[17; 18]].
Now Eq. 1 reads

$\dot{N}_{+}(t)=\frac{\eta\int L\phi(L,t)d\log L}{2\epsilon c^{2}\Gamma
m_{\mathrm{p}}},$ (2)

where $\phi(L,t)$ is the AGN luminosity function at cosmic time $t$ and $L$ is
the bolometric luminosity of an AGN.

To annihilate efficiently, positrons must slow down from relativistic to
thermal velocities. Below 1 GeV, the main mechanism of cosmic ray positron
energy loss is Coulomb collisions [[19]]. However, the material of the jet is
subject to adiabatic cooling which affects the positron energy much stronger
than collisions. Adiabatic losses can be expressed as follows [[20]]:

$-\frac{d\gamma}{dt}=\frac{\gamma}{R}\frac{dR}{dt}$ (3)

where $\gamma$ is the particle Lorentz factor, $R$ is the source size
expansion factor (i.e. $R\equiv 1$ at the start of expansion). Hence, in
absence of additional energy supply, the particle energy decreases
proportionally to the expansion. From observations of radio lobes of 3C 84
with VSOP [[21]], the age of the lobes is estimated as $\sim 45$ yr in 2001,
and it is shown to have doubled its size in approximately 25 years. Numerical
simulations of radio lobes show that the lobe size can increase by orders of
magnitude in tens of Myr [[22; 23]]. In contrast, Coulomb collisions take
order of $10^{9}$ yr to slow down a $\sim$ GeV positron, and annihilation
times are even longer (see Paper I). Thus, we conclude that the braking time
can be neglected and we assume that annihilation starts immediately when
positrons are produced.

The annihilation signal from each AGN depends on the amount of positrons
accumulated at a given redshift. Therefore, we need to calculate this amount
from the earliest time up to the present. We do it using the Eq. 16 from Paper
I:

$\frac{dN_{+}(t)}{dt}=\dot{N}_{+}(t-t_{\mathrm{br}})-nN_{+}(t)(\langle\sigma_{\mathrm{a}}v\rangle+\langle\sigma_{\mathrm{r}}v\rangle)$
(4)

where $\dot{N}_{+}$ is now taken from Eq. 1, $n$ is the number density of non-
relativistic electrons in the halo, $\langle\sigma_{\mathrm{a}}v\rangle$ and
$\langle\sigma_{\mathrm{r}}v\rangle$ are respectively the direct annihilation
rate and the annihilation rate due to positronium formation, $t_{\mathrm{br}}$
is the positron braking time which we set to 0. We tried two values of the
temperature, $10^{6}$ and $10^{5}$ K, and one value of the density, $10^{-4}$
cm-3. Solving the last equation, we obtain the comoving positron number
density as a function of cosmic time or redshift. Note that $N_{+}(t)$ was
dimensionless in Paper I, because it referred to the total number of positrons
in the halo. Now this quantity has a dimension of cm-3 and refers to the mean
comoving number density of positrons. The resulting $N_{+}(z)$ for the two
adopted values of the halo temperature is shown in Fig. 1.

Figure 1: The mean comoving number density of positrons.

Positrons annihilate due to collisions with electrons of gaseous halo via two
processes: direct annihilation and positronium formation. Positronium decays
into two 511 keV photons or three photons with energies below 511 keV, the
branching ratio of the two processes being 1/4 and 3/4. Therefore, the
comoving production rate density of 511 keV photons is

$\varepsilon_{\mathrm{2phot}}(z)=2nN_{+}(z)\left(\langle\sigma_{\mathrm{a}}v\rangle+\frac{1}{4}\langle\sigma_{\mathrm{r}}v\rangle\right)$
(5)

where $n$ is the gaseous halo density and the factor 1/4 takes into account
that only one fourth of positronium decays into two gamma rays.111Note that
$n$ is the proper density while $N_{+}$ is the comoving density. It should be
converted to the proper density via multiplication by $(1+z)^{3}$. However, in
this case Eq. 7 changes as well: a factor $(1+z)^{-3}$ appears in the
integral. So, the result remains the same. Analogously, for the three photon
decay, the comoving production rate density is222Although three photons are
born in this process, there is no factor 3, because it is taken into account
in the decay spectrum which is normalized so that $\int\tfrac{dN}{dE}dE=3$.

$\varepsilon_{\mathrm{3phot}}(z)=\frac{3}{4}nN_{+}(z)\langle\sigma_{\mathrm{r}}v\rangle$
(6)

##  3 Integration over AGN population

It is possible to obtain the radiation intensity of the sources with a known
redshift distribution. Let us start from Eq. 4 in [[24]]:

$I_{\mathcal{E}}=\frac{c}{4\pi
H_{0}}\int\frac{L[(1+z)\mathcal{E},z]dz}{(1+z)E(z)}$ (7)

Here $I_{\mathcal{E}}$ is the specific intensity in erg/cm2/s/sr/erg,
$\mathcal{E}$ is the photon energy in the observer’s frame, $L$ is the source
luminosity density in erg/s/cm3,
$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}$. In the case of two photon
annihilation radiation, the luminosity is monochromatic and equals
$L(\mathcal{E},z)=\mathcal{E}_{0}\varepsilon_{\mathrm{2phot}}(z)\delta(\mathcal{E}-\mathcal{E}_{0})$
where $\mathcal{E}_{0}=511$ keV. Inserting this to Eq. 7, we get

$I_{\mathcal{E}}=\frac{c\mathcal{E}_{0}}{4\pi
H_{0}}\int\frac{\varepsilon(z)\delta[(1+z)\mathcal{E}-\mathcal{E}_{0},z]dz}{(1+z)E(z)}=\\\
\frac{c\mathcal{E}_{0}}{4\pi
H_{0}}\int\frac{\varepsilon(z)\delta[(1+z)\mathcal{E}-\mathcal{E}_{0},z]d(1+z)\mathcal{E}}{\mathcal{E}(1+z)E(z)}=\\\
\frac{c\mathcal{E}_{0}}{4\pi
H_{0}}\left.\frac{\varepsilon(z)}{\mathcal{E}(1+z)E(z)}\right|_{(1+z)\mathcal{E}=\mathcal{E}_{0}}=\\\
\frac{c\mathcal{E}_{0}}{4\pi
H_{0}}\frac{\varepsilon(\frac{\mathcal{E}_{0}}{\mathcal{E}}-1)}{\mathcal{E}\tfrac{\mathcal{E}_{0}}{\mathcal{E}}E\left(\tfrac{\mathcal{E}_{0}}{\mathcal{E}}-1\right)}=\\\
\frac{c}{4\pi
H_{0}}\frac{\varepsilon(\frac{\mathcal{E}_{0}}{\mathcal{E}}-1)}{E\left(\tfrac{\mathcal{E}_{0}}{\mathcal{E}}-1\right)}.$
(8)

In the case of three photon positronium decay, the luminosity has the form
$L(\mathcal{E},z)=\mathcal{E}\varepsilon_{\mathrm{3phot}}(z)\varphi(\mathcal{E})$
where $\varphi(\mathcal{E})$ is the three photon decay spectrum [[25]]. The
resulting annihilation spectrum is the sum of the two components; it is shown
in Fig. 2 in the form $\mathcal{E}I_{\mathcal{E}}$ with the observational data
indicated in the description.

Figure 2: Diffuse GRB spectrum. The observation data are HEAO-1 A4 [[26]],
Swift-BAT [[27]], SMM [[28]]. The model curves represent the contribution from
AGNs (Gilli+2007) [[3]] and FSRQs (Ajello+2009) [[4]] and (Ajello+2012) [[5]]
. Our estimations from AGNs and SNe Ia are also shown.

One can see that, although the possible annihilation component makes a
subdominant contribution, it can provide up to $\sim 20$% of the total GRB in
the 400–500 keV range. One should note however that the estimate we just
obtained is an upper limit because it does not take into account the fraction
of positrons which escape from the gaseous halo into IGM. We will return to
this point in Sec. 5.

There also could be contribution from the radiationally inefficiently
accreting AGNs, which could not be accurately accounted for in our approach.
Still, given that the characteristic kinetic power of the jets from these AGNs
is relatively modest, $<10^{42}-10^{43}\leavevmode\nobreak\
\mathrm{erg\leavevmode\nobreak\ s}$ (see, e.g. [[29]]), we would not expect
that this contribution would dominate the total signal.

##  4 Positron production in SNe Ia

As we mentioned in Sec. 1, a possible source of positrons in the Universe are
explosions of SNe Ia. We decided to make a rough estimate of their possible
positron yield, because the calculation is basically the same as we just
presented. The two quantities that we need in this case are the SN Ia rate as
a function of redshift and the average positron yield per explosion.

The SN Ia rate $R_{\mathrm{Ia}}(z)$ is usually computed as a convolution of
the star formation rate (SFR) $\rho(t)$ with the delay time distribution (DTD)
$\Phi(\Delta t)$ which represents the time gap between the progenitor star
formation and its explosion [[6]]:

$R_{\mathrm{Ia}}(z(t))=\xi\int_{t_{10}}^{t}\Phi(t-t^{\prime})\rho(t^{\prime})dt^{\prime}$
(9)

where $\xi$ is the normalization showing the number of SNe Ia per $M_{\odot}$
of stars formed, $t_{10}$ is the cosmic time corresponding to $z=10$, the time
when the first stars were born [[6]]. Following [[30]], we adopted DTD of the
form $\Phi(\Delta t)\sim\Delta t^{-1}$ and we took the SFR fit from [[31]].
Then we chose a value for $\xi$ so that the resulting SN rate corresponded to
that shown in [[6]].

The positron yield per explosion depends on the explosion model. After [[6]],
we assume that $0.58M_{\odot}$ of 56Ni is produced in an explosion. After 56Ni
decays into 56Co*, the latter decays by electron capture or positron emission
with the branching ratio 81% and 19%. Therefore, the positron production rate
can be expressed as

$\dot{n}_{+,SN}=R_{\mathrm{Ia}}(z(t))\cdot 0.19\cdot
0.58M_{\odot}\frac{N_{A}}{M_{\mathrm{Ni}}}\alpha$ (10)

The photon production rate for the two photon decay is simply

$\dot{n}_{2phot,SN}=\frac{1}{2}\dot{n}_{+,SN}$ (11)

where the factor 1/2 results from the fact that in the SN envelope (where the
temperature is far below $10^{6}$ K) positrons annihilate mainly via
positronium formation, of which only 1/4 decays into two 511 keV quanta. In
Eq. 10, the factor $\alpha$ is the fraction of positrons which eventually
produce observable gamma quanta. We will discuss it in some detail in the
following section, but for now, let us put it equal to unity to find the upper
limit of the SNe Ia contribution. $M_{\mathrm{Ni}}$ is the atomic mass of
56Ni. In analogy with the AGN case, the intensity for the three photon decay
is obtained from Eq. 7 where
$L(\mathcal{E},z)=\frac{3}{4}\mathcal{E}\dot{n}_{+,SN}(z)\varphi(\mathcal{E})$.
The resulting spectrum is shown in Fig. 2. It is evident that the SN Ia
positron contribution is still smaller than that of AGN. In fact, it is
smaller than the gamma-ray lines emerging from SNe, which is evident from the
comparison with [[6]]. And like the AGN contribution, this is again an upper
limit because of the parameter $\alpha<1$.

##  5 Discussion

In the calculation of positron production in galactic halos, we took the same
value of electron temperature and density as in Paper I, which can be regarded
as typical values for a present-time spiral galaxy. In case of cosmological
sources, their evolution should be taken into account. In [[32]], the authors
used the cosmological EAGLE zoom simulations to compare physical properties of
gaseous halos around star forming galaxies at $z=2-3$ and $z=0$. They found
that the halos in the past and present are different in the relative
proportion of cold and hot gas (i.e. gas at the temperature less than or
greater than $10^{5}$ K), in the metallicity and in the kinematics. Only the
density and temperature of the gas are important for our calculation of the
positron annihilation rate. Data given in [[32]] indicate that 10 Gyr ago,
i.e. at the redshifts $z\approx 2-3$ the halos were already dense and
contained both cold and hot components. The mass of the latter corresponds to
the electron number density of the order $10^{-3}$ cm-3. Similar results were
obtained in [[33]] using cooling flow simulations.333Note that our results
hold for relatively large halo densities. We tried $n_{\mathrm{halo}}=2\times
10^{-3}$ cm-3 and obtained slightly lower signal: although higher annihilation
rate in the dense material results in larger intrinsic luminosity, positrons
most actively annihilate in high $z$ sources the signal from which is
suppressed by the huge distance.

Direct observations of galactic halos at large redshifts are difficult and not
abundant. Observations of galaxies at $z=2-3$ in absorption lines of O, C, Si
are published in [[34; 35]]. It is not straightforward to link them to the
total halo mass. In [[35]] estimates of the total C, Si and O mass are given,
but one has to know individual abundances of these elements. Using as an
approximation the global metallicity of $10^{-3}$ from [[32]] we can derive
the total halo mass and density which appears close to $10^{-4}$ cm-3. In
[[34]], the total column densities are given only for particular ions and
cannot be translated to the total mass. Observations of hydrogen Ly$\alpha$
line are given but the corresponding column density is not provided because of
the line saturation.

X-ray observations of the hot gas in distant galaxies are also difficult.
Moreover, in the recent paper on the Galactic halo X-ray emission observed by
eROSITA [[36]] it is shown that the Milky Way’s gaseous halo most probably
consists of both a disk-like and a spherical components, and the first one
dominates the X-ray emission, while the second one contains the most of the
halo mass. If the same applies to other galaxies, this renders halo mass
determination in spiral galaxies from X-ray observations alone rather
problematic.

X-rays from local elliptical galaxies were first reported in [[37]]. In this
work, halos in a number of galaxies were modeled with a $\beta$-distribution
with the density of $10^{-1}-10^{-2}$ cm-3 in the center and few $\times
10^{-5}$ cm-3 at 100 kpc distance, which is close to what we suppose in this
work. There is evidence that these galaxies have only slightly evolved since
$z=1-1.5$. In [[38]] it is shown that the relation between X-ray and optical
luminosity of local elliptical galaxies established in [[39]] also holds for
ellipticals at $z<1.4$, once a number of factors are taken into
account.444Actually, the authors found that high redshift ellipticals are more
X-ray luminous, but it can be due to a number of galaxies residing in the
center of groups/clusters, or due to hidden AGNs, or due to the slight
evolution of X-ray properties of the galaxies. In [[40]] it is shown that the
X-ray to $B$-band optical luminosity ratio does not evolve strongly in the
range $z=0-0.7$, and the same conclusion is drawn for the range $z=0-1.2$ in
[[41]]. Summing up, existence of hot gaseous halos around the early type
galaxies is established with a great degree of certainty.

Another point of concern is the interaction between the AGN jet and the halo.
We assumed that the positron population of the jet remains in the halo and
annihilates there eventually. The jet-halo interaction can be studied in
numerical simulations. Such a simulation was performed in [[42]]: the authors
explored how much energy AGN jets can inject into the intracluster medium. In
their calculations, jet was active for 50 Myr and the calculation spanned the
time period of 336 Myr. The jet power varied from $10^{44}$ to $10^{45}$
erg/s. It is evident from their results that the material of lower power jets
($10^{44}$ erg/s) stays within 100 kpc from the central engine until the end
of calculation. For a power of $3\times 10^{44}$ erg/s, only about 10% of the
injected mass remains within this radius, and slightly less for a $10^{45}$
erg/s power jet. Another simulation [[43]] analogously showed that low-power
jets ($\lesssim 10^{43}$ erg/s) are practically trapped inside the
interstellar medium (ISM), whereas stronger jets drill through ISM relatively
easily. From AGN luminosity function of [[16]] it can be seen that the
majority of kinetic energy injection is due to more powerful ones, with
kinetic luminosities in excess of $10^{45}$ erg/s.

Taking into account the results of [[42]], it means that most positrons
produced by AGNs are probably evacuated outside the gaseous halo into IGM.
There is growing evidence for interaction between AGN jets and interstellar or
circumgalactic medium, referred to as AGN feedback, e.g. [[44; 45; 46; 47]].
In the course of this feedback, material is dragged by the jet to substantial
distances, but eventually some fraction of it falls back, participating in
larger baryon cycle. However, we could not find quantitative description of
this process. Overall, the escape of jet material outside the circumgalactic
medium makes our estimation of the GRB an upper limit.

The same is true for the estimation of the signal from SNe Ia due to the
factor $\alpha$ in Eq. 10. This factor takes into account that not all the
photons produced in the positron annihilation within SN envelope have chance
to be observed. It should not be confused with the positron survival fraction
which is used in studies of Galactic diffuse gamma ray emission, such as
[[48]] and [[49]]. In those works, survival fraction is the fraction of
positrons which escape into ISM, because if a positron annihilates within the
envelope, then the corresponding photon is either not observed at all or
attributed to a point source. In other words, such a positron cannot
contribute to the Galactic diffuse radiation. In [[48]], the survival fraction
is found to be $\sim 0.01$. In our case, the $\alpha$ parameter is larger
because supernovae reside in distant galaxies, so we only demand that the
positron annihilates in an optically thin medium, probably within the
envelope. Let us estimate the optical depth of the envelope for some typical
parameters of its mass $M=1M_{\odot}$ and velocity $v=10^{4}$ km/s. It takes
120 days for 56Ni to decay into 56Co and then to 56Fe [[50]]. With known mass,
expansion velocity and time and the attenuation factor $8.63\times 10^{-2}$
cm2/g [[50]] we obtain the optical depth of the envelope $\tau=0.13$ and
$\alpha=e^{-\tau}=0.88$. Note however that due to this exponential dependency,
$\alpha$ is sensitive to the explosion parameters. If ejecta mass is larger
and velocity smaller, it would decrease $\alpha$. Moreover, the velocity is
not uniform throughout the ejecta, but rather grows from the inner edge to the
outer. A more thorough estimation of $\alpha$ requires modelling of the
explosion which is beyond the scope of this work.

Our results could be compared to other models in this energy range, as one can
see from Fig. 2. The MeV-range background estimated from Fermi observations of
FSRQs [[5]] appears to be lower than our AGN estimate, but higher than our SN
Ia estimate. On the other hand, the hard X-ray background originating from
Compton-thin and Compton-thick AGN computed in [[3]] and the background
modeled in [[4]] on the basis of Swift observations of FSRQs are above all our
estimations. Note however that in [[4]] observations in the 15–55 keV band
were extrapolated to higher energies and only 18 FSRQs were used for the
background estimation. In [[5]] observations of both Swift-BAT and Fermi-LAT
were used to infer SED and the sample comprised of 186 FSRQs. It can be seen
from Fig. 2 that the observed background intensity in the range $\sim 300-500$
keV is substantially larger than the current models, including the one
considered in this work. However, the mechanism we proposed can contribute a
non-negligible part of this diffuse radiation.

##  6 Conclusions

In this paper we estimated the contribution to the diffuse gamma ray
background which can be made by cosmological sources of positrons via
subsequent annihilation of these particles. We considered two types of
sources. First, positrons from AGN jets which annihilate in gaseous halos
around AGN host galaxies. Second, SNe Ia in distant galaxies which mostly
annihilate in the SN ejecta. A narrow line near 511 keV due to two photon
annihilation is smeared because of cosmological expansion. Moreover, part of
the signal is due to three photon annihilation which has a continuous spectrum
below 511 keV. Our calculations show that the signal due to AGN jets is
approximately an order of magnitude below the observed intensity at 511 keV
and even smaller at lower energies. The contribution from SNe Ia is
approximately two orders of magnitude below the observed emission level. Both
estimates are upper limits. In the case of AGNs, we do not know the fraction
of positrons which remain in the halo, while in the case of SNe we do not know
the fraction of positrons which annihilate in sufficiently transparent medium.
In the latter case this proves that the contribution from SNe Ia cannot be a
substantial part of the diffuse GRB. In fact, this has been shown in previous
works, such as [[6]]. In this work the contribution from SNe Ia was estimated
to be at most 10–20%, but the calculation included nuclear gamma-ray lines.
From our calculation, it is evident, that the contribution from SN positrons
is subdominant when compared to that from SN nuclear lines.

The contribution from AGN jet positrons seems more promising. in the range
200-500 keV, it is below the estimate of [[4]], but above the estimate of
[[5]] which, we believe, is more reliable due to larger statistics and use of
two instruments, Swift-BAT and Fermi-LAT. We emphasize once again that our
estimation is an upper limit, but still the mechanism itself could contribute
a considerable fraction of diffuse GRB in the sub-MeV range.

## Acknowledgements

The work of the authors was supported by the Ministry of Science and Higher
Education of Russian Federation under the contract 075-15-2020-778 in the
framework of the Large Scientific Projects program within the national project
”Science”.

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