# Compton driven beam formation and magnetisation via plasma
microinstabilities

Bertrand Martinez<EMAIL_ADDRESS>Thomas Grismayer  Luís
O. Silva  GoLP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior
Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal

###### Abstract

Compton scattering of gamma rays propagating in a pair plasma can drive the
formation of a relativistic electron positron beam. This process is
scrutinised theoretically and numerically via particle-in-cell simulations. In
addition, we determine in which conditions the beam can prompt a beam-plasma
instability and convert its kinetic energy into magnetic energy. We argue that
such conditions can be met at the photosphere radius of bright Gamma Ray
Bursts.

##  1 Introduction

The interaction of gamma-rays with a pair plasma is a fundamental problem in
astrophysics. For instance, it is present in the context of Gamma-ray Bursts
(GRBs). The latter involve the explosion of a stellar mass object, which
energy is expelled in the form of a relativistic ejecta. As this ejecta
propagates, its interaction with the ambient medium generates a strong
emission which is detected by satellites as well as ground-based
observatories. Recent efforts demonstrated emission over a large range of
frequencies including the radio, optical, x-ray and gamma-ray bands, up to TeV
energies (Acciari et al., 2019).

For most GRBs, the temporal evolution of the observations shows that their
spectra consists of two parts, a prompt emission, which is followed by an
afterglow. The prompt emission of a GRB is defined as an initial emission
phase in the keV-MeV band, lasting a few milliseconds to several minutes. The
afterglow is a counterpart of the prompt emission extending in the optical and
radio ranges and its duration can span from a few hours to several days. While
it is acknowledged that a large part of the afterglow can be attributed to a
forward external shock driven by the ejecta propagating in the ambient medium
(Meszaros & Rees, 1997), the origin of the prompt emission is not fully
understood yet. The main reason is that there remains unanswered questions
related to the composition of the ejecta, how its energy is dissipated and how
particles are accelerated.

All models agree that there is an ejecta formed by a compact object of radius
$\sim 10^{7}\,\rm cm$. This ejecta is a relativistic plasma in expansion. Most
of its energy comes from photons, but it also contains pairs, an unknown
fraction of baryons and may even be magnetised. At this early stage of
expansion it is so dense that photons cannot escape. At the photospheric
radius $\sim 10^{12}\,\rm cm$, its density has decreased enough to enable a
fraction of gamma-rays to propagate in the ambient medium (Cavallo & Rees,
1978). One of the pending questions related to GRBs is how does the ejecta
dissipates its energy above such a large distance of $\sim 10^{12}\,\rm cm$ ?
This question has been addressed in two ways, as detailed in this recent
review on GRBs (Kumar & Zhang, 2015).

Firstly, they brought forward models that describe the overall evolution of
the ejecta. The most used is the hot fireball model which assumes the energy
of the ejecta is dissipated at the photosphere radius and in internal/external
shocks (Narayan et al., 1992; Rees & Meszaros, 1994; Piran, 2005). However, it
is worthwhile to mention that some GRB spectra are better explained with a
model assuming a magnetized ejecta, where the energy is dissipated via
current-driven instabilities (Lyutikov, 2006; Zhang & Yan, 2010). The gamma-
ray emission processes are manifold and are widely discussed in these models.
It is usually synchrotron (Meszaros & Rees, 1993; Rees & Meszaros, 1994),
Inverse Compton (Gruzinov & Mészáros, 2000) or inverse Compton up-scattering
of synchrotron photons by high-energy electrons (synchrotron-self-Compton).
In the Klein-Nishina regime, the Compton process is also held responsible for
a bulk plasma acceleration called Compton drag or radiative acceleration
(Madau & Thompson, 2000; Thompson & Madau, 2000). Our work here goes beyond
these, by assuming that collective plasma processes can be triggered as the
energetic photons propagate through the plasma. On another hand, some hadronic
processes (Bethe-Heitler and photo-pion) are discussed as possible sources of
high energy positrons, which then radiate gamma-rays via synchrotron.  Pair
production processes are also proven to play a significant role into the
development of key features of GRBs such as their high radiative efficiency
(Stern, 2003) or their flat spectrum from the infrared to the ultraviolet
(Beloborodov, 2005).

Secondly, researchers also performed numerical simulations of basic plasma
processes in the extreme conditions of GRBs using Particle-In-cell (PIC)
simulations. Employing this framework enables to account for collective plasma
dynamics, radiative processes and, more importantly, for any dynamical
feedback between both. The first example relevant in the context of GRB
(Medvedev & Loeb, 1999) is the simulation of relativistic collisionless shock
formation by the Weibel instability (Silva et al., 2003) and the subsequent
Fermi-type particle acceleration (Spitkovsky, 2008; Martins et al., 2009). The
second example is particle acceleration in magnetic reconnection regions
(Mehlhaff et al., 2020) which is known to be of interest for GRBs (Uzdensky,
2011). The third one is electron acceleration in Compton driven plasma
wakefields during the GRB interaction with its Circum Burst Medium (CBM)
(Frederiksen, 2008; Del Gaudio et al., 2020a).

In his study, Frederiksen considers the interaction of a planckian photon
distribution with a tenuous plasma (Frederiksen, 2008). The photon energy lies
between $30\,\rm keV$ and $3\,\rm MeV$ and he observes electron acceleration
in a plasma wakefield. According to the author, this wakefield is excited by
an electrostatic force acting to restore the charge separation induced by
Compton deflections. Contrary to this work, our investigation focuses on a
regime where the Compton cross section is beamed, above $10\,\rm MeV$. In
addition, we consider a range where Compton scattering prevails over the
Bethe-Heitler pair production ($\gamma e\rightarrow e^{+}e^{-}e$), for
energies below $100\,\rm MeV$. Formulas for all these processes can be found
in Lightman (1982). For this gamma-ray range $10-100\,\rm MeV$, one expects
Compton scattering to deflect electrons mainly forward such that they form a
relativistic beam (Del Gaudio et al., 2020a).

In light of these previous results, we investigate the interaction of gamma-
rays (10-100 MeV) with a background pair plasma. In section 2, with the
support of a 1D theoretical model and Particle-In-Cell (PIC) simulations, we
evidence how a gamma-ray beam (10-100 MeV) propagating in a pair plasma can
drive the formation of a relativistic and dense electron positron beam via
Compton scattering. In section 3, we specify under which conditions the pair
beam can trigger a beam-plasma instability, thus generating a small scale
magnetic field which extent is discussed. Section 4 confirms the robustness of
this process for various photon sources. Finally, we argue in section 5 that
these conditions can be met for radii $r$ such that $r\geq r_{\rm ph}$, with
$r_{\rm ph}$ the photospheric radius of a GRB.

##  2 Pair beam formation

We first discuss a simplified physical picture. Let us consider a semi-
infinite gamma-ray beam of density $n_{\omega 0}$ with a monoenergetic
distribution. We denote by $\epsilon=\hbar\omega/mc^{2}$ the photon energy
normalized by the electron rest mass. The photon beam propagates along the $x$
direction in an infinite cold pair plasma at rest, with density $n_{p0}$
associated to an angular frequency $\omega_{p}$. In the frame of this work, we
focus on the energy range $1\ll\epsilon<1/\alpha_{f}=137$, where $\alpha_{f}$
denotes the fine structure constant. In this range, Compton scattering
prevails over the two photon Breit-Wheeler process, and pair creation in the
Coulomb field of an electron or positron (Lightman, 1982).

The formation process of the pair beam relies on the beaming of the Compton
cross section for high-energy photons. Let us consider an electron at rest
experiencing Compton deflection from a gamma-ray of energy $\epsilon$, with
$d\sigma_{\rm kn}/d\Omega$ and $\sigma_{\rm kn}(\epsilon)$ the angular-
differential and total Compton cross sections (Klein & Nishina, 1928), and
$\theta$ the polar angle associated to $\Omega$, the photon angle after one
scattering. The Lorentz factor of the electron after the deflection $\gamma$
can be deduced from the energy and momentum balance as
$\gamma/\epsilon=1+1/\epsilon-1/\left[1+\epsilon(1-\cos\theta)\right]$. We
introduce $\phi$ the angle between the deflected electron momentum and the
incident photon direction and obtain
$\mathrm{cotan}(\phi)=(1+\epsilon)\tan(\theta/2)$. Averaging those quantities
over the Compton angular cross section, we obtain for $\epsilon\gg 1$,
$\langle\gamma\rangle_{\Omega}\simeq\epsilon$ and
$\langle\sin^{2}\phi\rangle_{\Omega}\simeq 4/\epsilon$ (Blumenthal & Gould,
1970). This shows that there is a simultaneous beaming of photons, electrons
and positrons centered on the direction of the incident gamma-ray. We will
discuss later the effects induced by a more realistic gamma-ray beam
distribution.

We assume the longitudinal momentum of the pair beam ($p_{x}$) can be
approximated by its average over the Compton cross section
$p_{x}\simeq\sqrt{\langle p_{x}^{2}\rangle_{\Omega}}$, and we define the
transverse momentum spread induced by Compton scattering as $\Delta
p_{\perp}=\sqrt{\langle p_{\perp}^{2}\rangle_{\Omega}}$. For $\epsilon\gg 1$:

$\frac{p_{x}}{mc}\simeq\epsilon\qquad\text{and}\qquad\frac{\Delta
p_{\perp}}{mc}\simeq\sqrt{\frac{7\epsilon}{6\ln\epsilon}}\;.$ (1)

Equation (1) shows that the deflected electron energy can be as high as the
incoming photon energy and that the transverse momentum spread of the pair
beam is typically a few percent of its longitudinal momentum $\Delta
p_{\perp}/p_{x}\simeq 5-20\%$, for $1\ll\epsilon<1/\alpha_{f}$.

The density of the photons decreases as they are scattered at a frequency
$\tau_{\omega}^{-1}=2n_{p0}c\sigma_{\rm kn}$. It thus follows that
$dn_{\omega}/dt=-n_{\omega}/\tau_{\omega}$. The solution reads
$n_{\omega}/n_{\omega 0}=\exp(-t/\tau_{\omega})$. In the high energy limit
$\epsilon\gg 1$ one has $\sigma_{\rm kn}(\epsilon)\simeq\pi
r_{e}^{2}\ln(\epsilon)/\epsilon$, with $r_{e}$ is the classical electron
radius. As a result, the photon density can be approximated as constant
$n_{\omega}/n_{\omega 0}\simeq 1$, where in fact
$\omega_{p}\tau_{\omega}\propto
r_{e}^{-3/2}n_{p0}^{-1/2}\epsilon/\ln(\epsilon)\gtrsim 10^{11}$ for any plasma
density $n_{p0}\leq 10^{18}\,\rm cm^{-3}$, a conservative upper bound for
astrophysical systems.

We now consider a more detailed model for the pair beam formation. Let us
denote $n_{p}(x,t)$ the background plasma density. The electrons (and
positrons) of the beam are the ones experiencing at least one Compton
scattering. Their density is denoted $n_{b}(x,t)$ and they have a velocity
$v_{b}$. At the initial time $t=0$, we assume the photons are located in the
$x<0$ half space and propagate toward the background plasma at rest in the
other half-space $x>0$. The evolution of the background plasma density (beam
density) is given by the continuity equation with a source term accounting for
a depletion (loading) at the Compton frequency $\nu=\sigma_{\rm
kn}cn_{\omega}$

$\displaystyle\partial_{t}n_{p}+\nabla.(n_{p}\vec{v}_{p})$
$\displaystyle=-\sigma_{\rm kn}cn_{\omega}n_{p}$
$\displaystyle\partial_{t}n_{b}+\nabla.(n_{b}\vec{v}_{b})$
$\displaystyle=\sigma_{\rm kn}cn_{\omega}n_{p}$

We introduce the Heaviside function $H$, along with the variables $\xi,\tau$
defined as $\xi=ct-x$ and $\tau=t$. The background plasma is assumed to be at
rest during the interaction ($\vec{v}_{p}=0$), and the beam to have a constant
velocity $\vec{v}_{b}=v_{x}\vec{x}=c\beta_{x}\vec{x}$. The evolution of the
background plasma and beam densities simplifies to

$\displaystyle\left[c\partial_{\xi}+\partial_{\tau}\right]n_{p}(\xi,\tau)$
$\displaystyle=-\nu n_{p}(\xi,\tau)\left[H(\xi)-H(\xi-ct)\right]$ (2)
$\displaystyle\left[(c-v_{x})\partial_{\xi}+\partial_{\tau}\right]n_{b}(\xi,\tau)$
$\displaystyle=\nu n_{p}(\xi,\tau)\left[H(\xi)-H(\xi-ct)\right]$ (3)

We first solved Eq. (2) by assuming the solution is of the form
$f(\xi)g(\tau)$. As far as Eq. (3) is concerned, its solution was derived
using the method of Laplace transform. The solutions are obtained for all
$\xi\leq c\tau$ as

$\displaystyle n_{p}(\xi,\tau)=n_{p0}e^{-\frac{\nu\xi}{c}}$
$\displaystyle\left[H(\xi)-H(\xi-c\tau)\right]$ (4) $\displaystyle
n_{b}(\xi,\tau)=\frac{n_{p0}}{1-\beta_{x}}\Big{[}$
$\displaystyle\left(1-e^{-\frac{\nu\xi}{c}}\right)H(\xi)$ (5)
$\displaystyle+\left(e^{-\frac{\nu}{v_{x}}\left(\xi-c\tau(1-\beta_{x})\right)}-1\right)H(\xi-c\tau(1-\beta_{x}))\Big{]}$

The normalized beam velocity $\beta_{x}$ can be estimated as
$\beta_{x}\simeq\sqrt{\langle\beta_{x}^{2}\rangle_{\Omega}}$. In the high
energy limit $\epsilon\gg 1$, $\beta_{x}\simeq 1-1/(2\epsilon\ln\epsilon)$.
For large times $t\gg 1/\left[\nu(1-\beta_{x})\right]$, Eq. (5) implies that
the beam cannot exceed the maximum density $n_{p0}/(1-\beta_{x})\simeq
2n_{p0}\epsilon\log\epsilon$. Since we expect collective processes for such
time scales, we underline that this limit is unlikely to be reached and should
be considered as an upper bound of the maximum achievable density. This
maximum density could however be achieved in cases where collective effects
are damped on large distances, for example either with a high magnetic field
oriented in the flow direction or a large background plasma temperature.

In order to evidence some of the properties of this solution, we recast it
with $(x,t)$ variables and in the limit $\nu t\ll 1$ and $\beta_{x}\simeq 1$
and therefore get for all $0\leq x\leq ct$

$\displaystyle n_{p}(x,t)$ $\displaystyle=n_{p0}\left(1+\nu x/c-\nu t\right)$
(6) $\displaystyle n_{b}(x,t)$ $\displaystyle=n_{p0}\nu x/c$ (7)

The result in Eq. (6) illustrates that the background plasma density evolves
linearly with the position $x=0\rightarrow ct$, increasing from $n_{p0}(1-\nu
t)$ to its maximum value $n_{p0}$. Eq. (7) evidences that the beam density is
expected to grow linearly with the propagation distance, starting from $0$ and
increasing to its maximum value $n_{p0}\nu t$ at the position of the photon
front $x=ct$.

We stress this model is valid whatever the photon and plasma densities. It
remains applicable as long as collective plasma effects do not play a
significant role. In terms of photon energies, it is strictly limited to the
range $1\ll\epsilon<1/\alpha_{f}=137$. However, we will discuss in section 4
why it can be extended to a larger range of $20\lesssim\epsilon\lesssim 200$.
In order to account for photon in the energy range $1\lesssim\epsilon\lesssim
20$, one would need to account for the much larger energy spread induced by
the Compton cross-section and also to include the two photon Breit-Wheeler
pair creation.

Figure 1:  (a1-b1-c1) illustrate the 2D electron beam density profile at three
instants (a1) $\omega_{p}t=0$ (b1) $\omega_{p}t=90$ (c1) $\omega_{p}t=130$.
(a2-b2-c2) display the 2D electron background density profile for the same
three instants. (a3-b3-c3) are 1D projections along propagation direction $x$
for the same instants. The orange curve is the background plasma density
$n_{p}$, the green thick curve is the total density $n_{p}+n_{b}$ and the
black curves are the theoretical estimates from Eqs. (4)-(5). The gamma-ray
beam propagates from left to right and its front is marked by the dashed red
line. (d1-d2) $f(p_{x})$ and $f(p_{\perp})$ distributions of electrons
scattered by Compton at the beam front for time $\omega_{p}t=130$. For the
plasma density $n_{p0}=1\,\rm cm^{-3}$, one has $c/\omega_{p}=5.3\times
10^{5}\,\rm cm$.

We have reported how a pair beam can be created when gamma-rays propagate in a
background pair plasma via Compton scattering. This process is corroborated
with theory and is now confronted to simulation results. We have run 2D
Particle-In-Cell (PIC) simulations with Osiris (Fonseca et al., 2002). It was
recently enriched with a Compton scattering module (Del Gaudio et al., 2020b),
similarly to earlier work (Haugbølle et al., 2013). It proceeds through two
key steps: first a random pairing of macro particles at every time step and in
every cells and second a Monte Carlo sampling of the angle-resolved Klein-
Nishina cross section (Klein & Nishina, 1928). For every binary collisions
treated, the cross section is evaluated in the electron rest-frame, an energy
and momentum balance is ensured and a splitting technique is employed to
handle the varying weight of macro particles. This implementation relies on
macro particle pairing and therefore stands out from other works (Levinson &
Cerutti, 2018) where the Compton cross section is evaluated from the intensity
of the ambient radiation field.

We detail the relativistic pair beam formation on an illustrative simulation
in Fig. 1. We consider a cold pair plasma ($1\,\rm eV$) with a uniform density
$n_{p0}=1\,\rm cm^{-3}$, as displayed in Figs. 1(a1-a3). In this pair plasma,
we propagate a monoenergetic ($\epsilon=100$) gamma-ray beam along the $x$
direction from left to right. It presents a uniform density $n_{\omega
0}=10^{17}\,\rm cm^{-3}$ with a front indicated by the red dashed line in
Figs. 1(a3,b3,c3). Periodic boundary conditions are assumed in the transverse
direction $y$ for all species and fields and open boundary conditions are
employed in the longitudinal $x$ direction. The domain extends on
$(160\,c/\omega_{p})^{2}$ with $2000^{2}$ cells. The cell dimensions are
$\delta x=\delta y=0.08\,c/\omega_{p}$ and the time step is $\delta t=\delta
x/2$. We initialize $80/80/160$ particles per cell for electrons, positrons
and photons.

The process of pair beam formation is illustrated in Fig. 1. Figs. 1(a1-b1-c1)
exhibit 2D profiles of the electron beam density $n_{b}$ at three instants
$\omega_{p}t=0$, $90$ and $130$.  They correspond to electrons which
experienced at least one Compton scattering. Figs. 1(a2-b2-c2) represent 2D
profiles of the electron background density $n_{p}$ for the same three
instants $\omega_{p}t=0$, $90$ and $130$. Figs. 1(a3-b3-c3) are averages of
theses 2D densities along $y$ the transverse direction.  To clarify the plot,
only the total density $n_{b}+n_{p}$ (green thick curve) and background
density $n_{p}$ (orange thin curve) are represented. Based on these figures,
we can observe the formation and propagation of an electron beam (positrons
are superimposed by symmetry). We found excellent agreement of the simulations
with the theoretical estimates (black lines) of Eqs. (4)-(5).  The simulation
times considered here ($\omega_{p}t\leq 130$) lie within the limit $\nu t\ll
1$ where we derived Eqs. (6)-(7). For such early times, the beam and
background densities have profiles increasing linearly with the propagation
distance, which explains the triangular shape of the total density profile in
Figs. 1(b3-c3).  The momenta distributions $f(p_{x})$ and $f(p_{\perp})$ of
deflected electrons are exemplified in Figs. 1(d1-d2) at time
$\omega_{p}t=130$. The distribution $f(p_{x})$ exposes a peak at
$p_{x}/mc\simeq 100=\epsilon$, exactly as our theoretical estimate in Eq. (1).
The transverse momentum profile $f(p_{\perp})$ includes both directions
$p_{y}$ and $p_{z}$. The two distributions are centered and characterized by a
$\simeq 3\,mc$ standard deviation, in agreement with Eq. (1). This is
conducted in the range of validity of the theoretical model with periodic
conditions in the transverse direction, and does not evidence collective
plasma processes with a significant impact on the beam formation.

Another set of 2D simulations was run to assert the relevance of this process
of beam formation. Our goal is to check the range of validity of the scaling
inferred for its peak density in Eq. (5). We consider two sets of simulations.
The first set is characterised by a pair plasma of density
$n_{p0}=10^{18}\,\rm cm^{-3}$, with photon densities $n_{\omega
0}=10^{25}$-$10^{27}\,\rm cm^{-3}$. The second presents a pair plasma density
of $n_{p0}=1\,\rm cm^{-3}$, with photon densities $n_{\omega
0}=10^{16}$-$10^{18}\,\rm cm^{-3}$. Connections of these prameters to
laboratory or astrophysical parameters will be discussed in section 5. Given
the symmetry of our problem, we reduce the domain size to
$(24\,c/\omega_{p})^{2}$ and follow the photon beam in a moving window with a
velocity equal to $c$. The domain has $600^{2}$ cells with dimensions $\delta
x=\delta y=0.04\,c/\omega_{p}$ and a time step $\delta t=\delta x/2$.

Figure 2:  Pair beam density $n_{b}/n_{p0}$ versus the photon density
$n_{\omega 0}$ in unit of $\sqrt{n_{p0}/r_{e}^{3}}$ for plasma densities
$n_{p0}=1\,\rm cm^{-3}$ (X) and $n_{p0}=10^{18}\,\rm cm^{-3}$ (O) at two times
in the simulations. $\omega_{p}$ is the plasma angular frequency.

Figure 2 represents the pair beam density as a function of the incident photon
density for two instants, $\omega_{p}t=46$ and $184$. The normalization by the
quantity $(n_{p0}/r_{e}^{3})^{1/2}$ comes from a factor $\nu/\omega_{p}$ in
Eq. (7). It enables to reveal a general scaling of the normalized pair beam
density $n_{b}/n_{p0}$, independent of the initial plasma density $n_{p0}$.
Despite such a wide gap of initial plasma densities considered in Fig. 2,
simulations evidence that the normalized pair beam density at a given time $t$
remains exactly the same as long as $n_{\omega 0}(r_{e}^{3}/n_{p0})^{1/2}$ (or
$\nu/\omega_{p}$) is constant. This scaling matches exactly with the
prediction of Eq. (5), illustrated as a black line in Fig. 2.

We have generalised the validity of the results obtained in Fig. 2, to account
for baryon loading in the background plasma. Firstly, we ran all the
simulations presented before with a fraction $f$ of proton density up to
$f=0.1n_{p0}$ in the pair plasma.  As the initial electron and positron plasma
densities are not equal, the beam driven by Compton now presents a non-uniform
charge and current densities. This initiates the start of the two-stream
instability. The growth rate of such instability is much lower than the
oblique instability (Bret et al., 2005b) by a factor $\gamma_{b}^{2/3}$, with
$\gamma_{b}$ the Lorentz factor of the beam. As a consequence, the exponential
growth of the oblique modes we report in the next section still takes place,
but is initiated in a slightly perturbed plasma.  Secondly, we ran one 3D
simulation, confirming our previous findings. We performed it for a plasma
density of $n_{p0}=1\,\rm cm^{-3}$ and photon density of $n_{\omega
0}=10^{18}\,\rm cm^{-3}$. The domain size is $(24\,c/\omega_{p})^{3}$ and the
photon beam is followed in a moving window with a velocity equal to $c$. The
domain has $600^{3}$ cells of dimensions $\delta x=\delta y=\delta
z=0.04\,c/\omega_{p}$ and the time step is $\delta t=\delta x/2$. We
initialize $2/2/4$ particles per cell for electrons, positrons and photons.
The transverse boundary conditions for particles and fields are all periodic.
For this 3D simulation (not shown here), the evolution of the peak beam
density is the same as for the 2D simulation with the same parameters. While
we will discuss this in detail in section 3, we add that the growth rate of
the beam plasma instability in this 3D simulation is also the same as in the
corresponding 2D simulation.

##  3 Onset of a beam-plasma instability

The normalized pair beam density $n_{b}/n_{p0}$ increases with time and is
expected to trigger an electromagnetic beam-plasma instability when
$n_{b}/n_{p0}\lesssim 1$. Two types of modes can compete and lead to an
exponential growth of magnetic fields (Bret et al., 2005a). Firstly, the modes
associated to the Current Filamentation Instability (CFI), with growth rate
$\omega_{p}^{-1}\Gamma_{\rm CFI}=(n_{b}/n_{p0}\gamma_{b})^{1/2}$. Secondly,
the modes corresponding to the OBlique filamentation Instability (OBI), with
growth rate $\omega_{p}^{-1}\Gamma_{\rm
OBI}=\sqrt{3}(n_{b}/n_{p0}\gamma_{b})^{1/3}/2^{4/3}$. We can predict that our
prevailing modes will be oblique since we consider Lorentz factors
$1\ll\gamma_{b}\lesssim 1/\alpha_{f}$ and densities $n_{b}/n_{p0}\leq 1$.
Above some critical angle in $\mathbf{k}$ space, such modes are expected to be
damped like CFI modes due to thermal effects (Bret et al., 2005a). The
physical reason is that a transverse thermal expansion of filaments competes
with the magnetic pinching force maintaining them (Silva et al., 2002). The
latter theoretical work provides a density threshold above which the
instability can still set off despite this initial spread. It reads
$\alpha_{\rm th}=n_{b}/n_{p0}\simeq\gamma_{b}(p_{\perp}/mc\gamma_{b})^{2}$.
Using the estimates from Eq. (1), we get $n_{b}/n_{p0}\simeq 7/(6\ln\epsilon)$
and

$\omega_{p}^{-1}\Gamma\simeq(7\sqrt{3}/16)^{1/3}[\epsilon\log(\epsilon)]^{-1/3}\;.$
(8)

The minimum density required to trigger the instability is reached after a
propagation time of $1/\left(\nu\ln\epsilon\right)$. Using Eq. (5), this
implies the photons should propagate through a plasma of length greater than
$r_{\rm min}\geq c/\left(\nu\ln\epsilon\right)$. This condition can be
recasted as

$r_{\rm min}\,[\mathrm{cm}]\geq 1.6\times
10^{25}\,n_{\omega}^{-1}\,[\mathrm{cm^{-3}}]\;.$ (9)

The beam density threshold $\alpha_{\rm th}=6/(7\ln\epsilon)$ is independent
from the initial plasma density, and so is the plasma length $r_{\rm min}$
given in Eq. (9). It is worthwhile to stress that provided the photon density
and the background plasma length satifies the condition in Eq. (9), the
instability will take place.  As far as the effect of the background plasma
temperature is concerned, we can rely on previous investigations (Bret et al.,
2005b). For the non relativistic background plasma temperature considered here
($1\,\rm eV$), we do not expect any mitigation effect on the onset of CFI-like
modes.

In addition, it seems meaningful to evaluate the energy conversion efficiency
from the incident photons to magnetic fields. We define
$\eta=\mathcal{E}_{B_{z}}/\mathcal{E}_{\omega 0}$, the ratio between the
magnetic energy $\mathcal{E}_{B_{z}}\simeq\int
c^{2}B_{z}^{2}/8\pi\,d\mathbf{x}$ and the incident photon energy. For a semi-
infinite beam (and 1D), the ratio $\eta$ is also the ratio of the energy
densities. For the sake of simplicity, we estimate an upper bound for the
magnetic field energy $\mathcal{E}_{B_{z}}\lesssim\int
c^{2}|B_{z}|^{2}/8\pi\,d\mathbf{x}$. Using Parseval theorem, it can be
expressed as $\mathcal{E}_{B_{z}}\lesssim\int
c^{2}|B_{z}|^{2}/8\pi\,d\mathbf{k}$. The value of the saturated magnetic field
can be approximated by equating the bounce frequency of trapped particles to
the growth rate of the instability (Yang et al., 1994). This maximum is
$eB_{z}(k)/m\omega_{p}\simeq(\gamma_{b}^{2}/p_{x})(\omega_{p}^{-1}\Gamma)^{2}\omega_{p}/kc$.
The energy conversion efficiency is then deduced by introducing the incident
photon energy density $\epsilon n_{\omega 0}$

$\eta\lesssim\frac{1}{4\pi}\left(\frac{7\sqrt{3}}{16}\right)^{4/3}\frac{n_{p0}}{n_{\omega
0}}\frac{1}{\left(\epsilon\ln^{4}(\epsilon)\right)^{1/3}}\;.$ (10)

It is important to underline this inequality only expresses an upper bound for
the conversion efficiency.  For any given plasma of density $n_{p0}$, Eq. (10)
predicts that $\eta$ can be maximized for smaller incident photon densities
$n_{\omega 0}$. However, decreasing the photon density limits the growth of
the beam density through Compton scattering and therefore increases the
propagation distance required to trigger the instability, as evidenced in Eq.
(9). As a consequence of this low conversion efficiency $\eta$, the generation
of magnetic fields can in principle take place over long distances. For a
perfectly collimated photon source emerging from the photosphere ($\simeq
10^{12}\,\rm cm$), this length can be as high as the Compton mean free path
$L_{\omega}\,[\mathrm{cm}]=c\tau_{\omega}\simeq
10^{25}n_{p0}^{-1}\,\rm[cm^{-3}]$. In fact, the typical opening angle of the
GRB ejecta is in the range $\theta\leq 20^{\circ}$, as estimated in Racusin et
al. (2009).  With this more realistic assumption, the photon density and the
beam density are expected to decrease with the propagation distance as
$1/r^{2}$ due to this transverse dilution effect. We estimate under which
condition this depletion of the beam may remain negligible compared to the
loading via Compton scattering as

$\frac{\nu}{c}(r-r_{ph})\left(\frac{r_{ph}}{r}\right)^{2}\geq
1\quad\rightarrow\quad r\leq\sigma_{\rm kn}n_{\omega 0}r_{ph}^{2}=r_{\rm max}$
(11)

The inequality (11) states that the density increase due to the Compton
loading between $r_{ph}$ and $r$ is $\nu(r-r_{ph})/c$ and remains larger than
the density decrease between $r_{\rm ph}$ and $r$ due to the transverse
dilution, which is $\left(r_{ph}/r\right)^{2}$. The inequality (11) provides a
maximum distance, denoted $r_{\rm max}$, over which the instability can be
sustained.

Given a gamma-ray density between $10^{14}\,\rm cm^{-3}$ and $10^{22}\,\rm
cm^{-3}$ that we inferred at the exit of the photosphere of GRBs (see section
5), the condition (11) is fulfilled for distances ranging from a few
photospheric radii $\sim 10^{12}\,\rm cm$ up to larger values of $10^{20}\,\rm
cm$.  We did not verified this estimate in simulations due to the
computational cost for very long propagation lengths $L\,[\mathrm{cm}]\geq
10^{8}n_{p0}^{-1/2}\,\rm[cm^{-3}]$.

##  4 Generalisation for various photon distributions

In the previous section, we have considered idealized photon distributions. We
now examine more realistic photon distribution functions.

We investigate a simulation with plasma density $n_{p0}=10^{8}\,\rm cm^{-3}$
and photon density $n_{\omega 0}=10^{22}\,\rm cm^{-3}$.  We employ the
$\delta$ notation for a Dirac delta function, and can write the momentum
distribution of the photons as $f=f_{x}=\delta(p_{x}-100\,mc)$. Figure 3
reports the energy of the $B_{z}$ field, denoted by $\mathcal{E}_{B_{z}}$ and
normalized by the incident photon energy $\mathcal{E}_{\omega 0}$ (blue solid
curve). We first focus on the monoenergetic photon distribution (blue solid
curve). The growth rate for all modes $\mathbf{k}$ is
$\omega_{p}^{-1}\Gamma=7.4\times 10^{-2}$. The $B_{z}$ field profile is
displayed during the linear phase of the instability (see the inset) and shows
the dominant mode is oblique:
$\mathbf{k}=(k_{x},k_{y})\simeq(1.8,1.4)\,\omega_{p}/c$. Its growth rate is
$\omega_{p}^{-1}\Gamma=0.12$ close to $\omega_{p}^{-1}\Gamma\simeq 0.118$, the
theoretical prediction from Eq. (8). The linear stage of the instability
starts after a propagation distance $\simeq 90\,c/\omega_{p}$, which is
consistent with the minimum inferred theoretically of $34\,c/\omega_{p}$ in
Eq. (9). The theoretical upper bound of the energy conversion efficiency given
by Eq. (10) is $\eta\lesssim 1.5\times 10^{-17}$, which is on the order of the
simulation result $\eta=5\times 10^{-18}$. We also checked that the pair beam
density rises slowly ($+30\%$) on the instability time scale for
$\omega_{p}t=90-150$. This legitimates a posteriori the use of estimates from
linear theory. However, during the saturation stage, for
$\omega_{p}t=150-500$, the front beam density increases from $n_{p0}$ up to
$\sim 4n_{p0}$, a value which is in agreement with Eq. (7). As a consequence,
the beam keeps on filamenting and the magnetic field energy keeps increasing.

Figure 3:  Magnetic field energy $\mathcal{E}_{B_{z}}$, normalized by the
incident photon energy $\mathcal{E}_{\omega 0}$. We denoted
$\eta=\mathcal{E}_{B_{z}}/\mathcal{E}_{\omega 0}$ and derived an upper bound
in Eq. (10). Simulation for a monoenergetic photon distribution (blue solid
curve), a Maxwell-Jüttner transverse profile (orange dashed curve), a front
density gradient (red dotted curve), and a synchrotron distribution (green
dash-dotted curve). The growth rate of the field is determined from the slope
of the black dashed curves. Inset: typical $B_{z}$ field profile during the
linear phase.

The aforementioned discussions are focused on a monoenergetic gamma-ray beam.
We intend to extend these results for more complex distributions. Firstly, we
consider a photon distribution $f=f_{x}f_{y}$ with
$f_{x}=\delta(p_{x}-100\,mc)$ and $f_{y}$ a Maxwell-Jüttner distribution of
temperature $T_{\gamma,\perp}$. In the case where
$T_{\gamma,\perp}/mc^{2}\ll\Delta p_{\perp}/mc$, with $\Delta
p_{\perp}/mc=\sqrt{7\epsilon/(6\ln\epsilon)}$ given by Eq (1), the results we
have discussed before are not changed since the transverse momentum spread of
the beam is only due to the Compton scattering kinematics. In the other limit
$T_{\gamma,\perp}/mc^{2}\gtrsim\sqrt{7\epsilon/(6\ln\epsilon)}$, the
transverse momentum spread of the beam is determined by $T_{\gamma,\perp}$.
Since the Compton cross section is beamed for gamma-rays, the pair beam
presents a similar transverse momentum spread as the photon beam, which is
known to lower the growth rate of the OBI (Bret et al., 2010).  The latter
behavior is confirmed in Fig. 3 where we choosed $T_{\gamma,\perp}=3mc^{2}$.
This implies a transverse momentum spread of $\Delta p_{\perp}/mc=6$ for the
photon source, while the Compton-induced spread is $\Delta
p_{\perp}/mc=\sqrt{7\epsilon/(6\ln\epsilon)}=5$. As expected, this leads to a
reduction of the growth rate $\omega_{p}^{-1}\Gamma=7.4\times
10^{-2}\rightarrow 4.5\times 10^{-2}$, see the orange dashed curve. Secondly,
we consider the photons have a monoenergetic distribution $f=f_{x}$ with
$f_{x}=\delta(p_{x}-100\,mc)$ but their density profile has an exponential
shape $\propto\exp(-x/100)$ of scale length $100\,\rm c/\omega_{p}$, followed
by a flat profile of density $n_{\omega 0}$. Since the pair beam density is
proportional to the photon density, its profile first presents a density
gradient and then the triangular shape observed in Figs. 1(b3-c3). As a
result, the growth of the OBI is delayed in time, slightly lower but
noticeable ($\omega_{p}^{-1}\Gamma\simeq 1.0\times 10^{-2}$). This time delay
is comparable to the gradient length, as can be seen on the red dotted curve
in Fig. 3. Thirdly, we model a synchrotron distribution for the incident
photon beam. We set up this energy distribution as a longitudinal momentum
distribution for the photons $f=f_{x}=F(\eta,\chi)$ (Klepikov, 1954).  The
function $F$ represents the synchrotron spectra of an electron with a quantum
parameter $\eta$, $\chi$ being the photon quantum parameter. It has a peak at
an energy of $\hbar\omega/mc^{2}=100$. Photons with an energy of
$10$-$100\,\rm MeV$ are Compton scattered and contribute to increase the pair
beam density compared to the monoenergetic case. Indeed, the Klein-Nishina
cross section is a decreasing function of photon energy. As a result, the
growth of the $B_{z}$ field energy is faster: $\omega_{p}^{-1}\Gamma=7.4\times
10^{-2}\rightarrow 1.2\times 10^{-1}$, as seen on the green dash-dotted curve
in Fig. 3.

##  5 Relevance for astrophysics and laboratory environments

We now discuss under which conditions this process can be observed in
astrophysics and in the laboratory.

Figure 4 covers a large range of photon densities $n_{\omega 0}$ and distances
$r-r_{\rm ph}$, with $r_{\rm ph}$ the photospheric radius of a GRB. It reports
whether the propagation distance is large enough to enable the instability to
be triggered, see Eq. (9). The latter limit is plotted by the continous line
in Fig. 4 and enables to distinguish two regimes of interaction. The first one
is represented by all the area on the left side of the continuous line and is
denoted I. In this case, the photons are expected to form a relativistic
electron positron beam via Compton scattering. However, the pair beam density
is too low and its transverse momentum spread is too high to enable the onset
of the instability as detailed in Sec. 3. The second interaction regime is
depicted by the area between the continuous and dash-dotted lines and is
denoted II. In this regime, the pair beam density grows high enough to allow
the instability to develop. The dash-dotted line illustrates the maximum
length of the filament, as estimated in Eq. (11). It corresponds to the radius
above which the beam density is depleted by dilution effects faster than it is
loaded via Compton scattering.

Figure 4:  Relevance of the pair beam formation and magnetisation in the frame
of the GRB-CBM interaction at the photospheric radius $r_{\rm ph}\sim
10^{12}\,\rm cm$.  On the left side of the continuous line (area I): one
expects only pair beam formation. Between the continuous and dash-dotted lines
(area II), one also awaits the beam-plasma instability. On the right side of
the dash-dotted (area III), the instability is shutdown due to the transverse
dilution of the beam.

For the photospheric radius of a GRB, we evaluated a range of relevant gamma-
ray densities. With estimates from the fireball model, this gamma-ray density
(10-100 MeV) is determined to be $n_{\omega}\in(10^{14},10^{22})\,\rm
cm^{-3}$. From this order of magnitude, we observe in Fig. 4 that such
conditions can trigger the instability and its development on large distances.
To be more precise, the instability can be expected for GRBs whenever their
gamma-ray density at the photosphere radius is above $10^{14}\,\rm cm^{-3}$,
which corresponds to a GRB with an isotropic equivalent luminosity above
$3\times 10^{50}\,\rm erg/s$. The second sizeable result is that the beam
density remains high enough to sustain the instability on large distances,
despite its transverse dilution. For example, the instability can be
maintained to distances $\geq 10^{17}\,\rm cm$ for gamma-ray densities $\geq
10^{19}\,\rm cm^{-3}$, and it corresponds to an isotropic equivalent
luminosity $\geq 10^{52}\,\rm erg/s$.

We now detail the estimates to infer what is the gamma-ray density at the
photosphere radius, following Kumar & Zhang (2015). Let us consider a compact
object with a radius $\sim 10^{7}\,\rm cm$ and an isotropic equivalent
luminosity $\mathcal{L}$ in $\mathrm{erg/s}$. For simplicity we provide the
estimate in the compact object frame, thus neglecting the red shift of its
host galaxy. The fireball emerges from the compact object and experiences an
adiabatic expansion.  At the photospheric radius $\sim 10^{12}\,\rm cm$, the
emergent thermal radiation from the photosphere has a luminosity which is
equivalent to a blackbody temperature of
$[1.3\,\mathrm{MeV}]\times\mathcal{L}_{52}^{1/4}$.  The notation
$\mathcal{L}_{52}$ is defined as $\mathcal{L}_{52}=\mathcal{L}/(10^{52}\,\rm
erg/s)$. We assume typical GRBs have an isotropic equivalent luminosity in the
range $\mathcal{L}\in(10^{47}-10^{54})\,\rm erg/s$, in line with data recently
gathered (Abbott et al., 2017). This leads to the conclusion that this
blackbody equivalent temperature for the luminosity at the photospheric radius
ranges from $70\,\mathrm{keV}$ up to $4\,\mathrm{MeV}$. However, the isotropic
equivalent luminosity is defined for photons in the standard energy band
$1\,\mathrm{keV}$ up to $10\,\mathrm{MeV}$. We then deduce the fraction of
photons in the energy range $10$-$100\,\mathrm{MeV}$, where our results are
valid, by assuming a black body energy distribution.  For instance, this ratio
of photons (10-100 MeV) becomes $\gtrsim 5\%$ if the GRB isotropic equivalent
luminosity is $\gtrsim 10^{52}\,\rm erg/s$. The final step is to deduce a
gamma-ray density from this total luminosity in gamma-rays. To this purpose,
we assumed the GRB ejecta is a spherically expanding shell and its thickness
is the GRB duration, typically ranging between $0.1$ to $100\,\rm s$.

We now discuss the role of an external magnetic field that can be expected in
an astrophysical scenario. It is awaited that an external and strong enough
magnetic field aligned with the flow will prevent the development of the
current filamentation instability (Molvig, 1975). It was however demonstrated
that if the field is not striclty aligned with the flow, then the instability
is ensured to take place with a reduced growth rate for a non-relativistic
beam temperature (Bret & Alvaro, 2011). This growth rate is a fraction of the
growth rate without any magnetic field. Its detailed value in the low and high
magnetisation limits can be found in Eqs.(8) and (10) of the article of Bret &
Alvaro (2011).

It is worth to question whether this Compton-driven beam formation and
magnetisation can take place in the laboratory. Despite the ongoing worldwide
efforts, no large size and confined pair plasmas have been formed in a
laboratory environment yet. One can still mention that pair jets of high
density $n_{p0}\sim 10^{16}\,\rm cm^{-3}$ can be generated by fast electrons
going through a mm-sized high-Z targets. The fast electrons can be generated
by an intense laser (Chen et al., 2015; Liang et al., 2015), or by a plasma
wakefield accelerator (Sarri et al., 2015; Xu et al., 2016). Recent numerical
work bring forward a new scheme to create dense pair jets $n_{p0}\sim
10^{13}\,\rm cm^{-3}$ with size of several skin-depths in all directions from
the interaction of relativistic protons ($400\,\rm GeV/c$) with a solid
beryllium/lead target (Arrowsmith et al., 2020). Regarding gamma-ray sources
(10-100 MeV), we estimated a density $n_{\omega 0}\sim 10^{17}\,\rm cm^{-3}$
for the experimental photon source recently obtained by Non Linear Inverse
Compton scattering (Poder et al., 2018; Cole et al., 2018). For such photon
density, one would need a plasma length of $10^{8}\,\rm cm$ to witness the
instability, which cannot be achieved in a laboratory environment. We also
consider the results of a recent simulation from an extremely dense gamma-ray
source obtained during the interaction of a dense electron beam with multiple
micron-sized foils (Sampath et al., 2021). Although it remains a numerical
result and its experimental realisation may lie far in the future, it is
achieving a record gamma-ray density of $n_{\omega 0}\sim 10^{23}\,\rm
cm^{-3}$ with a distribution peaked at 100 MeV. One can see in Fig. 4 that
even this high gamma-ray density might be enough to witness the onset of the
Compton-driven instability in a laboratory frame. One would need a plasma
length of $150\,\rm cm$ to observe it.  Overall, our results indicate that
exploration of the process of beam formation and magnetisation in the
laboratory is unlikely in the short term due to the combined requirements of
plasma length and gamma-ray densities.

##  6 Conclusions

To summarize, we reported a study of gamma-ray propagation in a background
pair plasma. We showed that it can lead to the formation of a relativistic
pair beam, thanks to the beaming of the Compton cross section for photons
above 10 MeV. A theoretical model to describe the formation of the beam driven
by Compton scattering has been developed, and compared with PIC simulations.
We showed the pair beam can achieve a relativistic Lorentz factor with a
density comparable to the background plasma. In addition, we quantified the
transverse momentum spread of the beam, which is induced by the Compton cross
section. We demonstrated that the pair beam, as it propagates, can convert its
kinetic energy to magnetic energy via the oblique instability, although
limited by its transverse momentum spread. The conversion efficiency of this
process is low, but it can occur over long distances. We extrapolated from PIC
simulations that it could lead to the generation of magnetic fields on
distances larger than a parsec (up to $\simeq 10^{25}\,\rm cm$) for a
perfectly collimated GRB ejecta. For a less collimated one ($\theta\leq 20$),
we estimated the instability can be sustained over long distances
$10^{12}\rightarrow 10^{17}\,\rm cm$ for GRBs with an isotropic equivalent
luminosity ranging from $3\times 10^{50}\rightarrow 10^{52}\,\rm erg/s$. We
showed that these results can be used to address the energy dissipation of
gamma-rays at the photospheric radius of GRBs. We checked the robustness of
our results for various types of photon sources, with non-uniform density
profiles and transverse and longitudinal spreads in their momentum
distribution. The simulations done in the frame of this study are performed
for the typical orders of magnitude for plasma and photon densities expected
at the photospheric radius of a GRB, using estimates from the hot fireball
model as well as data recently gathered on GRBs.

This work offers two directions to explore in the future. The first could be
to consider photon distributions with energies below 10 MeV and connect with
previous works on Compton driven plasma wakes (Frederiksen, 2008; Del Gaudio
et al., 2020a). The second is to understand the impact of the beam formation
and magnetization on global GRB models.

## Acknowledgments

The authors acknowledge fruitful discussions with Dr. Del Gaudio and Dr.
Schoeffler. This work was supported by the European Research Council
(ERC-2015-AdG Grant 695088). We also acknowledge PRACE for awarding access to
MareNostrum based in Spain.

## References

  * Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R. X., Adya, V. B. & Collaboration, LIGO VIRGO 2017 Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817a. The Astrophysical Journal 848 (2), L13.
  * Acciari et al. (2019) Acciari, V. A., Ansoldi, S., Antonelli, L. A., Engels, A. Arbet, Baack, D., Babić, A., Banerjee, B., Barres de Almeida, U., Barrio, J. A., González, J. Becerra & MAGIC Collaboration 2019 Observation of inverse compton emission from a long $\gamma$-ray burst. Nature 575 (7783), 459–463.
  * Arrowsmith et al. (2020) Arrowsmith, C. D., Shukla, N., Charitonidis, N., Boni, R., Chen, H., Davenne, T., Froula, D. H., Huffman, B. T., Kadi, Y., Reville, B., Richardson, S., Sarkar, S., Shaw, J. L., Silva, L. O., Trines, R. M. G. M., Bingham, R. & Gregori, G. 2020 Generating ultra-dense pair beams using 400 gev/c protons, arXiv: 2011.04398.
  * Beloborodov (2005) Beloborodov, A. M. 2005 Afterglow Emission from Pair-loaded Blast Waves in Gamma-Ray Bursts. The Astrophysical Journal 627 (1), 346–367.
  * Blumenthal & Gould (1970) Blumenthal, G. R. & Gould, R. J. 1970 Bremsstrahlung, synchrotron radiation, and compton scattering of high energy electrons traversing dilute gases. Review of Modern Physics 42, 237–270.
  * Bret & Alvaro (2011) Bret, A. & Alvaro, E. Perez 2011 Robustness of the filamentation instability as shock mediator in arbitrarily oriented magnetic field. Physics of Plasmas 18 (8), 080706.
  * Bret et al. (2005a) Bret, A., Firpo, M.-C. & Deutsch, C. 2005a Characterization of the initial filamentation of a relativistic electron beam passing through a plasma. Physical Review Letters 94, 115002.
  * Bret et al. (2005b) Bret, A., Firpo, M.-C. & Deutsch, C. 2005b Electromagnetic instabilities for relativistic beam-plasma interaction in whole $k$ space: Nonrelativistic beam and plasma temperature effects. Phys. Rev. E 72, 016403.
  * Bret et al. (2010) Bret, A., Gremillet, L. & Bénisti, D. 2010 Exact relativistic kinetic theory of the full unstable spectrum of an electron-beam–plasma system with Maxwell-Jüttner distribution functions. Physical Review E 81, 036402.
  * Cavallo & Rees (1978) Cavallo, G. & Rees, M. J. 1978 A qualitative study of cosmic fireballs and gamma-ray bursts. Monthly Notices of the Royal Astronomical Society 183 (3), 359–365.
  * Chen et al. (2015) Chen, H., Link, A., Sentoku, Y., Audebert, P., Fiuza, F., Hazi, A., Heeter, R. F., Hill, M., Hobbs, L., Kemp, A. J., Kemp, G. E., Kerr, S., Meyerhofer, D. D., Myatt, J., Nagel, S. R., Park, J., Tommasini, R. & Williams, G. J. 2015 The scaling of electron and positron generation in intense laser-solid interactions. Physics of Plasmas 22 (5), 056705.
  * Cole et al. (2018) Cole, J. M., Behm, K. T., Gerstmayr, E., Blackburn, T. G., Wood, J. C., Baird, C. D., Duff, M. J., Harvey, C., Ilderton, A., Joglekar, A. S., Krushelnick, K., Kuschel, S., Marklund, M., McKenna, P., Murphy, C. D., Poder, K., Ridgers, C. P., Samarin, G. M., Sarri, G., Symes, D. R., Thomas, A. G. R., Warwick, J., Zepf, M., Najmudin, Z. & Mangles, S. P. D. 2018 Experimental Evidence of Radiation Reaction in the Collision of a High-Intensity Laser Pulse with a Laser-Wakefield Accelerated Electron Beam. Phys. Rev. X 8, 011020.
  * Del Gaudio et al. (2020a) Del Gaudio, F., Fonseca, R. A., Silva, L. O. & Grismayer, T. 2020a Plasma wakes driven by photon bursts via compton scattering. Phys. Rev. Lett. 125, 265001\.
  * Del Gaudio et al. (2020b) Del Gaudio, F., Grismayer, T., Fonseca, R. A. & Silva, L. O. 2020b Compton scattering in particle-in-cell codes. Journal of Plasma Physics 86 (5), 905860516\.
  * Fonseca et al. (2002) Fonseca, R. A., Silva, L. O., Tsung, F. S., Decyk, V. K., Lu, W., Ren, C., Mori, W. B., Deng, S., Lee, S., Katsouleas, T. & Adam, J. C. 2002 Osiris: A three-dimensional, fully relativistic particle in cell code for modeling plasma based accelerators. In Computational Science — ICCS 2002 (ed. Peter M. A. Sloot, Alfons G. Hoekstra, C. J. Kenneth Tan & Jack J. Dongarra), pp. 342–351. Berlin, Heidelberg: Springer Berlin Heidelberg.
  * Frederiksen (2008) Frederiksen, J. T. 2008 Stochastically induced gamma-ray burst wakefield processes. The Astrophysical Journal 680 (1), L5–L8.
  * Gruzinov & Mészáros (2000) Gruzinov, A. & Mészáros, P. 2000 Photon acceleration in variable ultrarelativistic outflows and high-energy spectra of gamma-ray bursts. The Astrophysical Journal 539 (1), L21–L24.
  * Haugbølle et al. (2013) Haugbølle, T., Frederiksen, J. T. & Nordlund, Å. 2013 photon-plasma: A modern high-order particle-in-cell code. Physics of Plasmas 20 (6), 062904.
  * Klein & Nishina (1928) Klein, O. & Nishina, Y. 1928 The scattering of light by free electrons according to Dirac’s new relativistic dynamics. Nature 122 (3072), 398–399.
  * Klepikov (1954) Klepikov, N. P. 1954 Emission of photons or electron-positron pairs in magnetic fields. Journal of Experimental and Theoretical Physics 26, 19.
  * Kumar & Zhang (2015) Kumar, P. & Zhang, B. 2015 The physics of gamma-ray bursts and relativistic jets. Physics Reports 561, 1 – 109\.
  * Levinson & Cerutti (2018) Levinson, A. & Cerutti, B. 2018 Particle-in-cell simulations of pair discharges in a starved magnetosphere of a Kerr black hole. Astronomy & Astrophysics 616, A184.
  * Liang et al. (2015) Liang, E., Clarke, T., Henderson, A., Fu, W., Lo, W., Taylor, D., Chaguine, P., Zhou, S., Hua, Y., Cen, X., Wang, X., Kao, J., Hasson, H., Dyer, G., Serratto, K., Riley, N., Donovan, M. & Ditmire, T. 2015 High e+/e- Ratio Dense Pair Creation with 1021W.cm-2 Laser Irradiating Solid Targets. Scientific Reports 5, 13968.
  * Lightman (1982) Lightman, A. P. 1982 Relativistic thermal plasmas - Pair processes and equilibria. The Astrophysical Journal 253, 842–858.
  * Lyutikov (2006) Lyutikov, M 2006 The electromagnetic model of gamma-ray bursts. New Journal of Physics 8 (7), 119–119.
  * Madau & Thompson (2000) Madau, P. & Thompson, C. 2000 Relativistic winds from compact gamma-ray sources. i. radiative acceleration in the klein-nishina regime. The Astrophysical Journal 534 (1), 239–247.
  * Martins et al. (2009) Martins, S. F., Fonseca, R. A., Silva, L. O. & Mori, W. B. 2009 Ion Dynamics and Acceleration in Relativistic Shocks. The Astrophysical Journal 695 (2), L189–L193.
  * Medvedev & Loeb (1999) Medvedev, M. V. & Loeb, A. 1999 Generation of magnetic fields in the relativistic shock of gamma-ray burst sources. The Astrophysical Journal 526 (2), 697–706.
  * Mehlhaff et al. (2020) Mehlhaff, J M, Werner, G R, Uzdensky, D A & Begelman, M C 2020 Kinetic beaming in radiative relativistic magnetic reconnection: a mechanism for rapid gamma-ray flares in jets. Monthly Notices of the Royal Astronomical Society 498 (1), 799–820.
  * Meszaros & Rees (1993) Meszaros, P. & Rees, M. J. 1993 Relativistic Fireballs and Their Impact on External Matter: Models for Cosmological Gamma-Ray Bursts. The Astrophysical Journal 405, 278.
  * Meszaros & Rees (1997) Meszaros, P. & Rees, M. J. 1997 Optical and long-wavelength afterglow from gamma-ray bursts. The Astrophysical Journal 476 (1), 232–237.
  * Molvig (1975) Molvig, K. 1975 Filamentary instability of a relativistic electron beam. Phys. Rev. Lett. 35, 1504–1507.
  * Narayan et al. (1992) Narayan, R., Paczynski, B. & Piran, T. 1992 Gamma-Ray Bursts as the Death Throes of Massive Binary Stars. The Astrophysical Journal 395, L83.
  * Piran (2005) Piran, T. 2005 The physics of gamma-ray bursts. Review of Modern Physics 76, 1143–1210.
  * Poder et al. (2018) Poder, K., Tamburini, M., Sarri, G., Di Piazza, A., Kuschel, S., Baird, C. D., Behm, K., Bohlen, S., Cole, J. M., Corvan, D. J., Duff, M., Gerstmayr, E., Keitel, C. H., Krushelnick, K., Mangles, S. P. D., McKenna, P., Murphy, C. D., Najmudin, Z., Ridgers, C. P., Samarin, G. M., Symes, D. R., Thomas, A. G. R., Warwick, J. & Zepf, M. 2018 Experimental signatures of the quantum nature of radiation reaction in the field of an ultraintense laser. Phys. Rev. X 8, 031004.
  * Racusin et al. (2009) Racusin, J. L., Liang, E. W., Burrows, D. N., Falcone, A., Sakamoto, T., Zhang, B. B., Zhang, B., Evans, P. & Osborne, J. 2009 Jet breaks and energetics of Swift gamma-ray burst x-ray afterglows. The Astrophysical Journal 698 (1), 43–74.
  * Rees & Meszaros (1994) Rees, M. J. & Meszaros, P. 1994 Unsteady outflow models for cosmological gamma-ray bursts. The Astrophysical Journal 430, L93.
  * Sampath et al. (2021) Sampath, A., Davoine, X., Corde, S., Gremillet, L., Gilljohann, M., Sangal, M., Keitel, C. H., Ariniello, R., Cary, J., Ekerfelt, H., Emma, C., Fiuza, F., Fujii, H., Hogan, M., Joshi, C., Knetsch, A., Kononenko, O., Lee, V., Litos, M., Marsh, K., Nie, Z., O’Shea, B., Peterson, J. R., Claveria, P. S. M., Storey, D., Wu, Y., Xu, X., Zhang, C. & Tamburini, M. 2021 Extremely dense gamma-ray pulses in electron beam-multifoil collisions. Physicsl Review Letters 126, 064801.
  * Sarri et al. (2015) Sarri, G., Poder, K., Cole, J., Schumaker, W., Di Piazza, A., Reville, B., Doria, D., Dromey, B., Gizzi, L., Green, A., Grittani, G., Kar, S., Keitel, C. H., Krushelnick, K., Kushel, S., Mangles, S., Najmudin, Z., Thomas, A. G. R., Vargas, M. & Zepf, M. 2015 Generation of a neutral, high-density electron-positron plasma in the laboratory. Nature Communications 6.
  * Silva et al. (2003) Silva, L. O., Fonseca, R. A., Tonge, J. W., Dawson, J. M., Mori, W. B. & Medvedev, M. V. 2003 Interpenetrating plasma shells: Near-equipartition magnetic field generation and nonthermal particle acceleration. The Astrophysical Journal 596 (1), L121–L124.
  * Silva et al. (2002) Silva, L. O., Fonseca, R. A., Tonge, J. W., Mori, W. B. & Dawson, J. M. 2002 On the role of the purely transverse Weibel instability in fast ignitor scenarios. Physics of Plasmas 9 (6), 2458–2461.
  * Spitkovsky (2008) Spitkovsky, A. 2008 Particle acceleration in relativistic collisionless shocks: Fermi process at last? The Astrophysical Journal 682 (1), L5–L8.
  * Stern (2003) Stern, B. E. 2003 Electromagnetic catastrophe in ultrarelativistic shocks and the prompt emission of gamma-ray bursts. Monthly Notices of the Royal Astronomical Society 345 (2), 590–600.
  * Thompson & Madau (2000) Thompson, C. & Madau, P. 2000 Relativistic winds from compact gamma-ray sources. II. pair loading and radiative acceleration in gamma-ray bursts. The Astrophysical Journal 538 (1), 105–114.
  * Uzdensky (2011) Uzdensky, D. A. 2011 Magnetic reconnection in extreme astrophysical environments. Space Science Reviews 160 (1), 45–71.
  * Xu et al. (2016) Xu, T., Shen, B., Xu, J., Li, S., Yu, Y., Li, J., Lu, X., Wang, C., Wang, X., Liang, X., Leng, Y., Li, R. & Xu, Z. 2016 Ultrashort megaelectronvolt positron beam generation based on laser-accelerated electrons. Physics of Plasmas 23 (3), 033109.
  * Yang et al. (1994) Yang, T.-Y. Brian, Arons, Jonathan & Langdon, A. Bruce 1994 Evolution of the Weibel instability in relativistically hot electron–positron plasmas. Physics of Plasmas 1 (9), 3059–3077.
  * Zhang & Yan (2010) Zhang, B. & Yan, H. 2010 The internal collision induced magnetic reconnection and turbulence (ICMART) model of gamma-ray bursts. The Astrophysical Journal 726 (2), 90.