# Dyakonov surface waves in dielectric crystals with negative anisotropy

Dmitry A. Chermoshentsev  e-mail<EMAIL_ADDRESS>Skolkovo
Institute of Science and Technology, 143025 Moscow Region, Russia  Moscow
Institute of Physics and Technology, 141701 Institutskiy pereulok 9, Moscow
Region, Russia  Russian Quantum Center, Skolkovo, Moscow 143025, Russia
Evgeny V. Anikin  Skolkovo Institute of Science and Technology, 143025 Moscow
Region, Russia  Sergey A. Dyakov  e-mail<EMAIL_ADDRESS>Skolkovo
Institute of Science and Technology, 143025 Moscow Region, Russia  Nikolay A.
Gippius  Skolkovo Institute of Science and Technology, 143025 Moscow Region,
Russia

(August 27, 2024)

###### Abstract

We report the prediction of a new type of Dyakonov surface waves that
propagate along the flat strip of the interface between two dielectrics with
negative anisotropy. It is shown that the surface waves condition is satisfied
for negatively anisotropic dielectrics due to specific boundaries of the strip
waveguide confined between two metallic plates. Such modes are studied by the
perturbation theory in the approximation of weak anisotropy. The existence of
Dyakonov surface waves in negative uniaxial crystals motivates us to
reconsider the list of materials suitable for their practical implementation.
We believe that this work opens a new unexplored research area in the field of
surface waves.

††preprint: APS/123-QED

##  I Introduction

Dyakonov surface waves (DSWs) is an important class of electromagnetic surface
waves which exist at the interface of two dissimilar materials, at least one
of which is anisotropic. In contrast to surface plasmon polaritons, DSWs have
no theoretical limit in propagation length as they can exist at the interface
of two lossless dielectrics. As shown by M. Dyakonov and N. S. Averkiev in
late 80s [1, 2], this type of DSWs can propagate in a certain range of angles
relative to the optical axis and only in a system with positively anisotropic
material(s). A positive optical anisotropy means that
$\varepsilon_{\parallel}>\varepsilon_{\perp}$, where $\varepsilon_{\parallel}$
(or $\varepsilon_{\perp}$) is the principal value of dielectric permittivity
tensor parallel to (or perpendicular to) the optical axis. Since then,
extensive research has been performed toward the theoretical studies of
Dyakonov-like surface waves at interfaces of different combinations of
isotropic, uniaxial, biaxial, and chiral materials with positive anisotropy
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. A
narrow range of propagation angles and the requirement of positive anisotropy
sufficiently decreases the number of materials suitable for practical
realization of DSWs. Nevertheless, experimental observation of such waves was
demonstrated by O. Takayama et al. for the interface formed by a dielectric
liquid, and biaxial KTP crystal [23]. Later, the same group have
experimentally demonstrated the existence of Dyakonov-like guided modes in
thin aluminum oxide nanosheets placed between anisotropic crystal and
dielectric liquid [24].

Figure 1: (a) Side view and (b) top view of the Dyakonov waveguide. Optical
axes (OA) of anisotropic dielectrics 1 and 2 are perpendicular to each other
and form the angle of $45^{\circ}$ to the waveguide boundaries.

The existence of DSWs has been also theoretically predicted for interfaces
between isotropic material and metamaterial with artificially designed
anisotropy [25, 4, 26, 27, 28, 29, 30, 31, 32]. In such structures the angular
existence domain of DSWs can be sufficiently extended. The experimental
realization of such DSWs has been reported in Ref. [33].

Very recently, a practically important case of Dyakonov-like surface waves in
finite-size resonator structures has been studied for cylindrical waveguides
[34, 35] as well as for flat interfacial strip waveguides [36, 37]. It has
been demonstrated that due to a non-zero curvature of cylindrical waveguides,
Dyakonov surface waveguide modes (DSWMs) in them inevitably have radiative
losses. At the same time, in flat interfacial strip waveguides such modes can
propagate without radiative losses like classical DSWs.

Because in the pioneering works [1, 2] it has been shown that the main
requirement for materials to support DSWs at their infinite interfaces is
being positevely anisotropic, DSWs community have been focused on positive
crystals and up to now there are no literature data on DSWs in negative
anisotropic crystals. In this work we demonstrate that the interface between
two negative anisotropic crystals can support DSWs, if it is confined between
two metallic plates.

##  II Low anisotropy approximation

Figure 2: (a) The dispersions of the DWM (red line) and the TE and TM
waveguide modes of an anisotropic waveguide (black solid and dashed lines)
shown as a difference with the dispersion of the TM waveguide mode. (b)
Theoretically calculated fields $E_{y}(0,z)$ and $E_{z}(0,z)$ (yellow and
magenta lines) shown together with COMSOL simulation results (black dashed
lines). (c) Theoretically calculated electric field intensity in DWM. Please
note that in panel (c) the limits of $y$\- and $z$-axes are different.
Calculations in (b) and (c) are made for $\lambda=1550$ nm, $k_{x}d=15.705$
and $d=1265$ nm. Dielectric permittivities for all panels are
$\varepsilon_{o}=9.75$, $\varepsilon_{e}=9.0$, $\delta\varepsilon=0.375$.

The structure under consideration consists of two spliced dielectric slabs
confined by thick metallic plates at $|y|=d/2$ (Fig.1a,b). The slabs are
infinite in $x$-direction and semi-infinite in $z$-direction; the interface
between them is at $z=0$. The principal values of the permittivity tensor of
both slabs are $\varepsilon_{e},\varepsilon_{o},\varepsilon_{o}$ with
$\varepsilon_{e}<\varepsilon_{o}$, i.e. we consider the case of a negative
anisotropy. The values of $\varepsilon_{e},\varepsilon_{o}$ are assumed to be
real and positive. The optical axes of the upper and lower slabs are rotated
by an angle of $\pm 45^{\circ}$ about the $z$-axis as shown in Fig.1b.

We represent the dielectric tensor $\hat{\varepsilon}(z)$ inside the waveguide
as $\hat{\varepsilon}(z)=\hat{\varepsilon}_{0}+\delta\hat{\varepsilon}(z)$,
where

$\begin{gathered}\hat{\varepsilon}_{0}=\operatorname{diag}(\varepsilon_{1},\varepsilon_{1},\varepsilon_{2}),\,\,\,\,\,\,\,\delta\hat{\varepsilon}(z)=\operatorname{sign}(z)\begin{pmatrix}0&\delta\varepsilon&0\\\
\delta\varepsilon&0&0\\\ 0&0&0\end{pmatrix}\end{gathered}$ (1)

with $\varepsilon_{1}=\frac{\varepsilon_{o}+\varepsilon_{e}}{2},\
\varepsilon_{2}=\varepsilon_{o}$ and
$\delta\varepsilon=\varepsilon_{2}-\varepsilon_{1}$. In addition to this model
we use two important approximations which allows us to develop a theory for
DSWMs in negative anisotropic crystals. First, we assume that materials have
low anisotropy, $\delta\varepsilon\ll\varepsilon_{1,2}$ (the most often case
for natural anisotropic crystals), that allows us to consider the Dyakonov
waveguide as a small perturbation of a homogeneous anisotropic waveguide with
a dielectric tensor $\hat{\varepsilon}_{0}$. Second, we replace metallic
plates with a perfect electric conductor (PEC) plates, which means that the
metallic cladding can be accounted for simply by using specific PEC boundary
conditions. For the sake of generality, in the end of this paper, we will show
that DSWMs exist in a more general case when these two approximations are not
valid.

The waveguide modes of unperturbed anisotropic waveguide with the PEC walls
can be analytically described for arbitrary $k_{x}$ and $k_{z}$ as shown in
Supplemental Materials. The dispersion curves of the lowest TM- and TE-modes
at $k_{z}=0$ are set by the following equations:

$\begin{gathered}(\omega^{\scriptscriptstyle\mathrm{TE}}_{k_{x},0})^{2}=\frac{c^{2}}{\varepsilon_{2}}\left[\left(\frac{\pi}{d}\right)^{2}+k_{x}^{2}\right],\,\,\,\,\,\,\,\,(\omega^{\scriptscriptstyle\mathrm{TM}}_{k_{x},0})^{2}=\frac{c^{2}k_{x}^{2}}{\varepsilon_{1}}\end{gathered}.$
(2)

One can see from Eqs. 2 that these modes intersect and the wavevector of
intersection is the following:

$k_{x}=\frac{\pi}{d}\sqrt{\frac{\varepsilon_{1}}{\varepsilon_{2}-\varepsilon_{1}}}.$
(3)

It is important to note that the intersection of the lowest TE- and TM-modes
of the unperturbed uniaxial waveguide with $\varepsilon_{1}<\varepsilon_{2}$
occurs only due to specific PEC boundary conditions. We will show that close
to the intersection, the perturbation $\delta\hat{\varepsilon}$ leads to
considerable mixing of TE- and TM-modes giving rise to a special type of the
surface mode. Hereinafter, we call this electromagnetic mode as Dyakonov
surface wave of the II type (DSW-II), while a classical Dyakonov surface wave
is referred to as Dyakonov surface wave of the I type (DSW-I). DSW-II does not
exist at the infinite interface of two negatively anisotropic plates; it also
does not exist at the strip interface of two spliced negatively anisotropic
slabs confined between air plates.

Figure 3: Domains of existence of (a) the I type and (b) the II type DSWs in
the structure shown in Fig. 1 with dielectrics with positive or negative
anisotropy. In (a) $\varepsilon_{o}=9.0$, $\varepsilon_{e}=9.75$, in (b)
$\varepsilon_{o}=9.75$, $\varepsilon_{e}=9.0$.

In the approximation of a low anisotropy, the intersection of the dispersion
curves of the lowest TM and TE modes is far from of the next order waveguide
modes (see Supplemental Materials). It enables us to expand the field of DSW-
II over the lowest TE and TM modes of unperturbed waveguide:

$\vec{E}(y,z)=\int\frac{dk_{z}}{2\pi}\left[\alpha(k_{z})\vec{E}^{\mathrm{\scriptscriptstyle
TE}}_{k_{x},k_{z}}(y)+\beta(k_{z})\vec{E}^{\mathrm{\scriptscriptstyle
TM}}_{k_{x},k_{z}}(y)\right]e^{ik_{z}z}$ (4)

where $\alpha(z)$ and $\beta(z)$ are the slowly varying envelopes and
$\vec{{E}}^{\mathrm{\scriptscriptstyle TE}}_{k_{x},k_{z}}$ and
$\vec{{E}}^{\mathrm{\scriptscriptstyle TM}}_{k_{x},k_{z}}$ are the normalized
lowest TE and TM modes, which are described as follows:

$\begin{gathered}\vec{E}^{\mathrm{\scriptscriptstyle
TE}}_{k_{x},k_{z}}(y)=\sqrt{\frac{2}{d}}\frac{1}{\varepsilon_{2}^{3/2}\varepsilon_{1}}\frac{c^{2}}{\omega^{2}}\begin{pmatrix}\varepsilon_{2}k_{x}k_{z}\cos{\frac{\pi}{d}y}\\\
i\varepsilon_{2}\frac{\pi}{d}k_{z}\sin{\frac{\pi}{d}y}\\\
-\varepsilon_{1}[k_{x}^{2}+(\frac{\pi}{d})^{2}]\cos{\frac{\pi}{d}y}\end{pmatrix}\\\
\vec{E}^{\mathrm{\scriptscriptstyle
TM}}_{k_{x},k_{z}}(y)=\frac{1}{\varepsilon_{1}\sqrt{d}}\frac{c}{\omega}\begin{pmatrix}0\\\
k_{x}\\\ 0\end{pmatrix}\end{gathered}$ (5)

Below we present the result of applying the perturbation theory for finding
the expressions of the amplitudes $\alpha(z)$ and $\beta(z)$ and the
dispersion for DSW-II (see Supplemental Materials for details on perturbation
theory). After plenty of algebraic manipulations one can obtain

$\begin{gathered}\alpha(z)=\frac{\omega^{2}\sigma\delta\varepsilon
e^{-\kappa_{2}|z|}}{\left(\gamma^{\mathrm{\scriptscriptstyle
TE}}_{k_{x},0}-\frac{\kappa_{2}^{2}}{2m_{1}}\right)}-\frac{\omega^{2}\sigma\delta\varepsilon
e^{-\kappa_{1}|z|}}{\left(\gamma^{\mathrm{\scriptscriptstyle
TE}}_{k_{x},0}-\frac{\kappa_{1}^{2}}{2m_{1}}\right)}\\\
\beta(z)=-i\left(\frac{e^{-\kappa_{2}|z|}}{\kappa_{2}}-\frac{e^{-\kappa_{1}|z|}}{\kappa_{1}}\right)\end{gathered},$
(6)

where decay constants $\kappa_{1,2}$ have positive real and are found as two
roots of the characteristic equation

$\left(\gamma^{\mathrm{\scriptscriptstyle
TE}}_{k_{x},0}-\frac{\kappa^{2}}{2m_{1}}\right)\left(\gamma^{\mathrm{\scriptscriptstyle
TM}}_{k_{x},0}-\frac{\kappa^{2}}{2m_{2}}\right)+\omega^{4}\sigma^{2}\delta\varepsilon^{2}\kappa^{2}=0,$
(7)

where

$\begin{gathered}\sigma=\frac{2\sqrt{2\varepsilon_{2}}}{\varepsilon_{1}^{3/2}\pi}\frac{k_{x}}{k_{x}^{2}+(\frac{\pi}{d})^{2}},\,\,\,\,\,\,\,\,m_{1}^{-1}=m_{2}^{-1}=\frac{2c^{2}}{\varepsilon_{1}}\end{gathered}.$
(8)

In Eqn.(7) we use the notation
$\gamma_{k_{x},k_{z}}^{\mathrm{\scriptscriptstyle
TE(TM)}}=\left(\omega_{k_{x}k_{z}}^{\mathrm{\scriptscriptstyle
TE(TM)}}\right)^{2}-\omega^{2}$ for brevity.

The dispersion equation of the surface mode can be obtained from the boundary
conditions and have the following form:

$\sqrt{m_{1}m_{2}\gamma^{\mathrm{\scriptscriptstyle
TE}}_{k_{x},0}\gamma^{\mathrm{\scriptscriptstyle
TM}}_{k_{x},0}}=2m_{1}m_{2}(\omega^{2}\sigma\delta\varepsilon)^{2}-m_{2}\gamma^{\mathrm{\scriptscriptstyle
TM}}_{k_{x},0}$ (9)

It turns out that the transcendental equation (9) has a solution (red line in
Fig. 2a) and, hence, the DSW-II exists. As expected, the DSW-II dispersion
curve is close the intersection of the lowest TE- and TM-modes. One can see
from Fig. 2a that there is an upper cut-off for the DSW-II which is due to the
fact that the solution of Eq. (9) exists when the right-hand side is non-
negative. In the cut-off point, the DSW-II transforms into the TE-mode and
becomes completely delocalized. The existence of the upper cut-off point of
the dispersion curve leads to fact that there is an upper cut-off waveguide
thickness $d$ at which the DSW-II cease to exist. After substitution of the
coefficients (8) into the right hand side of Eq. (9) we obtain that the cut-
off thickness depends on the frequency $\omega$ or vacuum wavelength $\lambda$
as follows:

$d\leq\frac{c\pi}{\omega}\sqrt{\frac{\varepsilon_{1}}{\varepsilon_{2}\delta\varepsilon}}=\frac{\lambda}{2}\sqrt{\frac{\varepsilon_{1}}{\varepsilon_{2}\delta\varepsilon}}.$
(10)

In Ref. [37], where a DSW-I in positively anisotropic materials between two
air plates are considered, the situation is opposite. Namely, there is a lower
cut-off point for the dispersion curve. Fig. 3 visually illustrates the
domains of existence of DSW-I and DSW-II in terms of the waveguide thicknesses
$d$ for different frequencies $\omega$. Since for DSW-II there is the upper
cut-off waveguide thickness, DSW-II does not exist in the case of an infinite
($d\to\infty$) interface between two anisotropic materials, in contrast to
DSW-I.

The distribution of electric field intensity of DSW-II within the waveguide’s
vertical cross-section calculated by formula (4) is shown in Fig. 2c. One can
see that the field decays away from the interface and has the maximum in the
middle of the waveguide. To verify the developed theory, we calculate the
$z$-dependence of electric field by formula (4) and compare it with the
results of full-wave electromagnetic simulations made in COMSOL (Fig. 2b). We
obtain that apart from a generally very good agreement between two field
profiles, there is the slight discrepancy between two curves near the
interface which is caused by the influence of the higher-order waveguide modes
of the unperturbed waveguide (see Supplemental Materials for details).

Figure 4: (a) Electric field intensity of DSW-II calculated in LiNbO3/Cu
structure at the vacuum wavelength $\lambda=4.04$ $\mu$m. Please note that in
panel (a) the limits of $y$\- and $z$-axes are different. (b) Dispersions of
DSW-II, TM hybrid mode and TE guided mode. (c) Figure-of-merit for DSW-II and
TM hybrid mode as a function of photon energy. Calculations for all panels are
made at the waveguide thickness $d=7$ $\mu$m. Optical axes of the upper and
lower LiNbO3 plates are oriented as shown in Fig. 1b. Please note that in
panels (b) and (c) the cut-off points for DSW-II are determined approximately
due to limitation of the computational domain size in COMSOL. Dielectric
parameters of LiNbO3 and Cu are taken from [38, 39].

As was previously mentioned, the model described above is developed for the
low anisotropy limit. Nevertheless, the analytical description of DSWs-II can
also be made for the arbitrary anisotropy in the approximation of a thin
waveguide $d\ll\lambda$. In this limit, only the lowest TM mode exists with
the electric field components $E_{x}$ and $E_{z}$ being small. As a result, a
surface-wave-like solution of Maxwell’s equations can be analytically obtained
(see Supplemental materials for details) and the corresponding propagation
constant is the following:

$k_{x}=\sqrt{k_{0}^{2}\varepsilon_{1}+\kappa^{2}}\approx
k_{0}\sqrt{\varepsilon_{1}}+\frac{\kappa^{2}}{2k_{0}\sqrt{\varepsilon_{1}}}$
(11)

where $\kappa$ is described as follows:

$\kappa=\frac{k_{0}^{4}d^{3}\delta\varepsilon^{2}}{3\sqrt{3}}\left(\sqrt{\frac{\varepsilon_{2}}{\varepsilon_{1}}}-1\right)$
(12)

##  III II-type DSW in real materials

So far we have studied DSWs-II in the dispersionless anisotropic materials
confined between two idealized PEC plates. Let us take one step towards
practical implementation of DSW-II and consider a structure consisting of
lithium-niobate-based Dyakonov waveguide placed between two copper plates. We
employ lithium niobate (LiNbO3) because it is lossless negative anisotropic
material that is widely used in various linear and non-linear optical
applications including optical waveguides. The COMSOL simulation results for
the modes of this structure are shown in Fig. 4. First, we note that DSW-II in
the LiNbO3/Cu structure exists, and like in the ideal case, the cross-
sectional field profile (Fig. 4c) has a global maximum at the interface
between upper and lower LiNbO3 waveguides. There are also local maxima of the
field distribution at some distance from the interface. Further, in contrast
to the ideal case, the initial TM mode of LiNbO3/Cu structure is a hybrid mode
resulting from the mixing between the surface plasmon polaritons at LiNbO3/Cu
interface and the TM guided mode of LiNbO3 waveguide. Due to more complex mode
structure of the LiNbO3/Cu waveguide in comparison to the ideal case, the cut-
off point of DSW-II does not lie on the TE guided mode dispersion curve
anymore (Fig. 4b). The hybrid TM mode evanescently decays into the absorptive
Cu plate, thus having a non-zero imaginary part of the propagation constant.
Eventually, this leads to the fact that in the considered example, the DSW-II
has absorption losses. The corresponding propagation length of DSW-II can be
quantitatively estimated from the figure-of-merit calculated as:

$\mathrm{FOM}=\frac{\mathrm{Re}\>k_{x}}{\mathrm{Im}\>k_{x}}$ (13)

which has a meaning of the decay length expressed in the units of the DSW-II
wavelength. As one can see from Fig. 4b, the propagation length of the DSW-II
exceeds that of the hybrid TM mode and increases when approaching the cut-off
point. In the far-infrared range or radiowaves, the skin depth in metals
become very small and much larger propagation length of DSW-II can be
expected. Another approach for increasing the propagation length of DSW-II can
be based on using photonic crystals as cladding.

##  IV Conclusions

In conclusion, we have studied a new type of Dyakonov surface waves that
propagate along the flat strip of the interface between two uniaxial
dielectrics with negative anisotropy. We have shown that the surface waves
condition is satisfied for negatively anisotropic dielectrics due to specific
boundaries of the strip interfacial waveguide confined between two metallic
plates. We have studied such modes using the perturbative approach under the
assumption of weak anisotropy. We generalized our results made for idealized
materials to the case of real materials and found out that DSWs-II should have
a finite propagation length due to absorption in metal. We suggest that the
propagation length of DSWs-II can be increased by switching to FIR or
radiowaves or by using photonic crystals as cladding. Finally, the existence
of Dyakonov surface waves in negative uniaxial crystals motivates us to
reconsider the list of materials suitable for the practical realization of
Dyakonov surface waves. Thus, we believe that this work opens a new unexplored
research area in the field of surface waves.

###### Acknowledgements.

This work was supported by the Russian Foundation for Basic Research (Grant
No. 18-29-20032).

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# Long-term radio monitoring of the fast X-ray transient EP240315a: evidence
for a relativistic jet

Roberto Ricci Dipartimento di Fisica, Universitá di Tor Vergata, Via della
Ricerca Scientifica, 1, 00133 Rome, Italy  INAF-Istituto di Radioastronomia,
Via Gobetti, 101, 40129, Bologna, Italy  Eleonora Troja  Dipartimento di
Fisica Universitá di Tor Vergata Via della Ricerca Scientifica, 1, 00133 Rome,
Italy  Yu-Han Yang Dipartimento di Fisica Universitá di Tor Vergata Via della
Ricerca Scientifica, 1, 00133 Rome, Italy  Muskan Yadav Dipartimento di Fisica
Universitá di Tor Vergata Via della Ricerca Scientifica, 1, 00133 Rome, Italy
Yuan Liu National Astronomical Observatories, Chinese Academy of Sciences,
100101 Beijing, China  Hui Sun National Astronomical Observatories, Chinese
Academy of Sciences, 100101 Beijing, China  Xuefeng Wu Purple Mountain
Observatory, Chinese Academy of Sciences, 210023 Nanjing, China  He Gao
Institute for Frontier in Astronomy and Astrophysics, Beijing Normal
University, 100875 Beijing, China  School of Physics and Astronomy, Beijing
Normal University, 100875 Beijing, China  Bing Zhang Nevada Center for
Astrophysics and Department of Physics and Astronomy, University of Nevada Las
Vegas, Las Vegas, NV 89154, USA  Weimin Yuan  National Astronomical
Observatories, Chinese Academy of Sciences, 100101 Beijing, China  School of
Astronomy and Space Sciences, University of Chinese Academy of Sciences,
Beijing, 100049, China

###### Abstract

The recent launch of Einstein Probe (EP) in early 2024 opened up a new window
onto the transient X-ray sky, allowing for real-time discovery and follow-up
of fast X-ray transients (FXRTs). Multi-wavelength observations of FXRTs and
their counterparts are key to characterize the properties of their outflows
and, ultimately, identify their progenitors. Here, we report our long-term
radio monitoring of EP240315A, a long-lasting ($\sim 1000$ s) high redshift
($z=4.9$) FXRT associated to GRB 240315C. Our campaign, carried out with the
Australian Telescope Compact Array (ATCA), followed the transient’s evolution
at two different frequencies (5.5 GHz and 9 GHz) for three months. In the
radio lightcurves we identify an unusual steep rise at 9 GHz, possibly due to
a refreshed reverse shock, and a late-time rapid decay of the radio flux,
which we interpret as a jet break due to the outflow collimation. We find that
the multi-wavelength counterpart of EP240315A is well described by a model of
relativistic jet seen close to its axis, with jet half-opening angle
$\theta_{j}\approx 3^{\circ}$ and beaming-corrected total energy $E\simeq
4\times 10^{51}$ erg, typical of GRBs. These results show that a substantial
fraction of FXRTs may be associated to standard GRBs and that sensitive X-ray
monitors, such as Einstein Probe and the proposed HiZ-GUNDAM and Theseus
missions, can successfully pinpoint their relativistic outflows up to high-
redshifts.

X-ray:transients – Radio:transients – Radio:interferometry – Gamma-ray: bursts

##  1 Introduction

The core-collapse of the most massive stars is known to produce short-lived
flashes of high-energy radiation, known as gamma-ray bursts (GRBs; Woosley &
Bloom 2006), traditionally discovered using wide field of view (FoV) gamma-ray
and hard X-ray monitors.

Since their discovery in the early 70s (Klebesadel et al., 1973), our physical
understanding of these explosions has been based on their observed properties
in the gamma-ray band (e.g. Kouveliotou et al., 1993). In particular, the
duration of the prompt gamma-ray emission is widely used to guide the
identification of the GRB progenitors and to constrain the lifetime of the GRB
central engine (e.g. Kumar & Zhang, 2015; Petropoulou et al., 2020).

The discovery of X-ray flares (Burrows et al., 2005), sudden rebrightenings of
low-energy ($<$10 keV) radiation that follow many GRBs, suggested instead that
the central engine could remain active on much longer timescales than the
observed GRB duration (Fan & Wei, 2005; Zhang et al., 2006; Liang et al.,
2006; Lazzati & Perna, 2007; Maxham & Zhang, 2009). Although multiple models
were proposed to explain X-ray flares, ranging from delayed MHD instabilities
(Giannios, 2006) to structured jets (Duque et al., 2022), those requiring a
prolonged activity of the engine better reproduce their broadband properties
(Falcone et al., 2007; Troja et al., 2015; Yi et al., 2016). This aspect was
recently demonstrated by the X-ray transient discovered by the Einstein Probe
(EP) mission on March 15, 2024, and dubbed EP240315A (Zhang et al., 2024). At
a redshift $4.9$ (Saccardi et al., 2024), EP240315A is among the most distant
high-energy transients ever known. Its temporal and spatial coincidence with
GRB240315C (DeLaunay et al., 2024), as well as the large energy output implied
by its distance scale ($E_{\gamma,\rm iso}\approx 6\times 10^{53}$ erg, Liu et
al., 2024), identify this event as a GRB.

A notable feature of EP240315A is the striking difference between its high-
energy and low-energy time profiles (Liu et al., 2024). At energies above 15
keV, observed by Swift/BAT and Konus-Wind, it displays a multi-peaked light
curve with a duration of approximately 40 s (observer’s frame), which is
fairly typical for GRBs. At lower (0.3-4 keV) energies, observed by the Wide
X-ray Telescope (WXT) aboard EP, the outburst is seen to last over 1,000 s,
which is among the longest GRB durations ever measured. Even when correcting
for time dilation effects, an intrinsic duration of $\approx$200 s remains
among the top 1.5% of GRB durations (Minaev & Pozanenko, 2020). Some of the
observed X-ray peaks occur after the prompt gamma-ray episode, and are thus
similar to X-ray flares. Some other X-ray pulses are detected even before the
gamma-ray signal, setting back the time of the explosion by at least six
minutes (Liu et al., 2024).

Observations of EP240315A naturally link X-ray flares to the prompt gamma-ray
emission as part of the same outburst from a long-lived central engine.
Although the gamma-ray phase remains dominant in terms of energetics, this
case shows that the standard gamma-ray window might greatly underestimate the
lifetime of the GRB engine, both before and after the main prompt emission
episode. This has fundamental implications for our understanding of these
extreme explosions: powering such long-lasting activity is non-trivial for
standard GRB central engines, including accreting black holes (BHs) and
magnetars (Perna et al., 2006; Dai et al., 2006; Proga & Zhang, 2006). Studies
of X-ray flares suggest that this phenomenon may affect over 30% of the GRB
population (Chincarini et al., 2007; Yi et al., 2016). The occurrence of soft
X-ray emission preceding the prompt gamma-rays by hundreds of seconds is
instead less constrained by observations. Only additional cases of joint
X-ray/gamma-ray detections would clarify whether this behavior is common or is
a peculiar trait of EP240315A.

In this paper, we explore whether EP240315A/GRB240315C is representative of
the general GRB population by studying the properties of its relativistic
outflow constrained through multi-wavelength afterglow observations. Whereas
the GRB afterglow faded fast in the X-ray and optical band, it remained
visible for weeks at radio wavelengths, allowing us to detect an achromatic
steepening of its flux. We interpret this feature as a jet-break (Rhoads,
1999; Sari et al., 1999; van Eerten et al., 2012) and use it to constrain the
geometry and total energy release of the GRB outflow.

The paper is organized as follows: in Section 2 we describe the
multiwavelength observations, focusing on the ATCA monitoring campaign; in
Section 3 we detail the afterglow modelling first in empirical terms and then
following a physical parametrization; in Section 4 we discuss the implications
of our results.

We adopt flat cosmology with parameters $H_{0}=67.4$ km s-1 Mpc-1 and
$\Omega_{m}=0.315$ (Planck Collaboration et al., 2020), which gives the
luminosity distance of $D_{L}=46.1$ Gpc.

##  2 Observations

###  2.1 Australian Telescope Compact Array

Figure 1: Smoothed double power law fit to the light curve of the 5.5 GHz
(upper panel) and 9 GHz (lower panel) radio data. Downward triangles represent
3 $\sigma$ upper limits from ATCA. The open square shows the MeerKAT data
point, shifted from 3 GHz to 5.5 GHz.

EP240315A was observed with the ATCA on seven separate runs as part of a
Target-of-Opportunity program CX564 between March 19th and June 18th 2024. The
radio interferometer was in array configuration 6A and later 6D (the maximum
extension). The long hiatus between the 6th and 7th Epoch was due to a change
in the array configuration (H168) between May 16th and June 4th, which was
unfavourable for the target detection at high resolution. All observing runs
were carried out in continuum mode with 1-MHz channel correlator mode in
C/X-band at the standard centre frequencies of 5.5 and 9 GHz, with a bandwidth
of 2 GHz each. The primary and bandpass calibrators were 1934$-$638 and
0823$-$500, and the complex gains calibrator was 0941$-$080\. The data were
reduced in Miriad (Sault et al., 1995) with standard procedures for flagging,
calibration and imaging. A robustness parameter value r=0 was used at the
Fourier inversion stage as a good trade-off between natural and uniform
weighting in order to minimize the map rms noise in source detection
experiments.

All images were Gaussian fitted and the peak flux densities values were
extracted at the transient source position, being the best estimate of flux
density in the case of point-like source observations. The rms noise in the
maps were evaluated in areas near the transient source position but far away
from bright sources. As the target was quite faint in all images the
multiplicative term (5% of the peak flux density, accounting for the residual
complex gain error) turned out to be negligible in computing the flux density
uncertainty budget. The resulting measurements are reported in Table 1, which
lists the elapsed time since the event trigger, the flux density and
corresponding 1 $\sigma$ error bar at 5.5 GHz and at 9.0 GHz.

###  2.2 Other radio facilities

In the radio band, EP240315A was also observed by e-MERLIN (Bruni et al.,
2024; Rhodes et al., 2024) and by MeerKAT (Carotenuto et al., 2024). The
e-MERLIN observations resulted in $5\,\sigma$ upper limits of 75 $\mu$Jy at
5.5 d and 105 $\mu$Jy at 12.5 d, both at a frequency of 5 GHz. These
measurements are lower than the fluxes derived with ATCA at a similar
frequency and epoch (Table 1), and point to a day-to-day variability of the
observed radio flux. Strong flux modulations are not uncommon in GRB radio
light curves and are often ascribed to the effects of interstellar
scintillation (ISS; Granot & van der Horst 2014). To estimate these effects we
use the NE2001 model of the Galactic distribution of free electrons (Cordes &
Lazio, 2002). From this we derive a scattering measure SM = $3.11\times
10^{-4}$ kpc m-20/3 and a critical frequency $\nu_{0}$ = 11.3 GHz below which
ISS might be relevant.

At lower frequencies, MeerKAT reported a weak detection of the radio
counterpart at 2.9 d with a flux density of $\approx 30~{}\mu$Jy at 3 GHz in a
map with an rms noise of 8.5 $\mu$Jy/beam. The measured flux is lower by a
factor of $\approx$2 than the simple extrapolation of the ATCA spectrum. This
is consistent with ISS effects although, as discussed below, could also be an
intrinsic feature of the afterglow spectrum due to self-absorption.

###  2.3 X-ray and optical follow-up

X-ray observations of EP240315A were performed with the Follow-up X-ray
Telescope (FXT) on board EP, starting about 42 hr after the trigger, and with
the Chandra X-ray Observatory, starting about 72 hr after the trigger (Liu et
al., 2024). A fading X-ray afterglow was detected at a flux of
1.8${}_{-0.7}^{+1.4}\times 10^{-13}$ erg s-1cm-2 (90% confidence level; c.l.).
and remained detectable for approximately 8 days after the trigger.

At optical and near-infrared (nIR) wavelengths, the counterpart was detected
by multiple facilities for about 16 days, as reported by Liu et al. (2024).
Magnitude corrections for Galactic extinction were implemented using a
reddening of $E(B-V)=0.05$ mag (Schlegel et al., 1998) and the extinction law
described by Cardelli et al. (1989) with $R_{V}\approx$3.1.

Table 1: ATCA radio flux densities in the EP240315a monitoring campaign. Col. 1: elapsed time since WXT trigger T0 in days; cols. 2-3: flux density and its error bar at 5.5 GHz; cols. 4-5: flux density at 9 GHz with its error bar. Upper limits are reported at 3 $\sigma$ level. T$-$T0 | S5.5GHz | $\pm$ | S9GHz | $\pm$
---|---|---|---|---
(d) | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy)
3.95 | 74 | 21 | 82 | 20
10.5 | 97 | 14 | 158 | 12
19.6 | 140 | 27 | 328 | 31
37.5 | 90 | 11 | 88 | 8
45.4 | 59 | 13 | 46 | 11
52.4 | 65 | 11 | 44 | 9
94.5 |  $<$ 48 | $-$ |  $<$ 39 | $-$

##  3 Afterglow Modeling

###  3.1 Basic properties

The GRB afterglow is a broadband synchrotron radiation produced by a
population of shock-accelerated electrons, whose energy distribution can be
described as a simple power-law function with index $p$. At any time, the
afterglow spectral shape consists of smoothly joint power-law segments (Sari
et al., 1998; Granot & Sari, 2002), fully characterized by four quantities:
the cooling frequency $\nu_{c}$, the characteristic synchrotron frequency
$\nu_{m}$, the self-absorption frequency $\nu_{a}$, and the peak flux $F_{\rm
pk}$.

Preliminary constraints on these parameters can be derived from the observed
afterglow evolution (e.g. Wijers & Galama, 1999) by assuming that the main
emission mechanism is the standard forward shock (FS), arising from the
interaction between the relativistic outflow and its surrounding interstellar
material (ISM) (Rees & Meszaros, 1992). When possible, we describe the
monochromatic afterglow flux as a power-law in both frequency and time,
$F_{\nu}\propto\nu^{-\beta}t^{-\alpha}$. For the radio data, a simple power-
law cannot reproduce the long-term temporal evolution, and we adopt a smoothly
broken power-law (Beuermann et al., 1999).

First, we consider the optical and X-ray data, which follow a similar temporal
decay with slope $\alpha=$1.66$\pm$0.05. This indicates that they lie on the
same segment of the synchrotron spectrum with slope
$\beta_{OX}=(2\alpha+1)/3\approx 1.5$ for
$\max(\nu_{m},\nu_{c})\lesssim\nu_{O}<\nu_{X}$ during the time frame of
observations 1 d$\lesssim t\lesssim$10 d. Within the standard GRB scenario,
the resulting electrons’ spectral index is rather steep, $p$ $\approx$
2$\beta_{OX}$ $\approx$ 3.

Once established that the optical and X-rays belong to the same segment, we
can infer the amount of intrinsic extinction from the spectral energy
distribution. According to the SED around 2 d after the burst, including $H$-,
$z$-, $I$-band and X-ray data, we derive ${E(B-V)}_{z}<0.011$ ($3\sigma$ c.l.)
for the extinction law described by Cardelli et al. (1989).

Then, we turn to the radio afterglow which is seen to rise with a slope of
$\alpha_{r}=-0.4^{+0.3}_{-0.2}$ at 5.5 GHz, consistent with the spectral
segment $\nu_{a}<\nu_{r}<\nu_{m}$. The rising trend favors a circumburst
medium with uniform density since in a medium with a wind-like density profile
the expected radio light curve is flat. At the higher frequency (9 GHz), the
observed slope of $-1.4^{+0.4}_{-0.3}$ is steeper than predictions of a simple
FS model and may include some contributions from a fast rising cooling reverse
shock (RS). To test this interpretation, we model the 9 GHz light curve with
two broken power-law models: the former tracks the FS emission and is
characterized by the same parameters of the 5.5 GHz curve, the latter tracks
the possible RS and its parameters are free to vary. This fit constraints the
rising RS slope to $\alpha_{\mathrm{RS}}$ steeper than $-$2.5, which,
considering also the lack of RS features at lower frequencies, is only
consistent with standard closure relations (e.g. Gao et al., 2013) for
$\nu_{a,\mathrm{RS}}\gtrsim 9$ GHz (observer’s frame). However, this scenario
faces some problems. For instance, the collision that occurred at such late
time should have taken place far away from the central engine, when both the
blast wave and the density of the jet ejected later would have been very low.
In this case, it would be difficult to achieve such large
$\nu_{a,\mathrm{RS}}$.

At $t\approx 40$ d the radio emission is seen to steeply decline at both
frequencies with slope $3.1^{+1.0}_{-1.2}$ and $3.7^{+0.6}_{-0.7}$,
respectively. Very few mechanisms can account for such large change in slope
($\Delta\alpha_{r}\gtrsim$3) and rapid decline. For instance, a factor of ten
drop in the density of the external medium would induce at most
$\Delta\alpha_{r}\approx$0.5 (e.g. Gat et al., 2013), whereas the passage of
the peak frequency $\nu_{m}$ would cause $\Delta\alpha_{r}\approx$2 for
$p\approx 3$ (Sari et al., 1998). A bright RS crossing the 9 GHz band could
produce a steep decay (e.g. Dichiara et al., 2022), although not as fast as
the observed value.

A late-time drop in flux can only be explained with a jet-break, that occurs
when the jet edges become visible to the observer (Rhoads, 1999; Sari et al.,
1999). We consider synchrotron radiation from the FS as the main mechanism
powering the observed radiation at 5 GHz. Within this scenario, the post-break
flux is expected to decline as $\propto t^{-1/3}$ if $\nu_{r}<\nu_{m}$ and as
$\propto t^{-p}$ if $\nu_{r}>\nu_{m}$. As we observe a value that is in
between the two predictions, the peak frequency $\nu_{m}$ is likely passing
through the observed radio frequencies between 20 d and 40 d.

Assuming a jet-break occurs at $t_{j}\gtrsim 20/(1+z)\approx 3.3$ d, we can
infer the jet opening angle (e.g. Sari et al., 1999; van Eerten et al., 2012):

$\theta_{j}\approx 3.6^{\circ}\left(\frac{t_{j}}{\rm
3.5~{}d}\right)^{3/8}\left(\frac{E_{\rm K,iso}}{\rm
10^{54}erg}\right)^{-1/8}\left(\frac{n}{\rm 0.001~{}cm^{-3}}\right)^{1/8}$ (1)

where $E_{\rm K,iso}$ is the isotropic-equivalent kinetic energy of the blast-
wave, and $n$ the density of the environment. The narrow collimation of the
jet implies that the true gamma-ray energy release is $E_{\gamma}\approx
10^{51}$ erg, in line with the population of GRBs (e.g. Wang et al., 2018).

###  3.2 Energy Injection

Between 250 and 450 ks ($\approx$13 hr rest-frame) a rebrightening of the
optical and X-ray light curves suggests that EP240315A experienced a late
episode of energy injection. The bump displays a small amplitude and a smooth
profile with a short rise time ($\Delta\,t/t\sim 0.5$) and shallower decay
($\Delta\,t/t\sim 1$), which does not necessarily require a reactivation of
the central engine. Various other mechanisms can reproduce similar small bumps
in an afterglow light curve, such as density jumps in the external medium,
complex jet structures, and refreshed shocks arising from the collision
between fast-moving and slow-moving ejecta (Rees, 1999; Granot et al., 2003).
Since the complex temporal profile and strong spectral evolution of the prompt
emission point to multiple episodes of ejections from the central engine, we
focus on the latter model. A refreshed shock would also induce a reverse shock
back into the ejecta, which could explain the steep rise of the radio light
curve at 9 GHz.

To estimate the bulk Lorentz factor of the fast moving ejecta, we use the time
of the first afterglow detection as an upper limit to the deceleration time
$t_{d}$ (Piran, 1999; Zhang, 2018):

$\Gamma_{0}>76\left(\frac{t_{d}}{\rm 0.1~{}d}\right)^{-3/8}\left(\frac{E_{\rm
K,iso}}{\rm 10^{54}erg}\right)^{1/8}\left(\frac{n}{\rm
0.1~{}cm^{-3}}\right)^{-1/8}$ (2)

Using a simple internal shock model (Moss et al., 2023), the average Lorentz
factor of the slow moving ejecta can then be estimated as $\Gamma\approx
10(t/13{\rm hr})^{3/8}(n_{-2}/E_{52})^{1/8}$ , where $n_{-2}$ is the density
of the circumburst medium in units of $10^{-2}$ cm-2, and $E_{52}$ is the
isotropic equivalent energy of the fast ejecta in units of $10^{52}$ erg. To
reproduce the relatively fast rise of the bump, a narrow spread in Lorentz
factors is also required, $\Delta\Gamma/\Gamma\approx 0.2$.

###  3.3 Broadband modeling

We model the GRB afterglow using the open-source Python package afterglowpy,
based on the semi-analytic models of van Eerten et al. (2010) and Ryan et al.
(2020). In its simplest version, the afterglow is fully described by five free
parameters: isotropic kinetic energy $E_{\rm K,iso}$, density of the ISM $n$,
distribution index of electrons $p$, the fractions of shock energy into
electrons $\epsilon_{e}$ and magnetic field $\epsilon_{B}$. The beaming of the
outflow adds an additional parameter, half-opening angle $\theta_{j}$. The
fraction of shock-accelerated electrons $\xi_{N}$ is a poorly constrained
parameter, and is generally fixed to 1 to avoid degeneracy. In our fit we
explore three different cases: $\xi_{N}=1$ for comparison with the majority of
GRB afterglow fits, $\xi_{N}=0.1$ motivated by particle-in-cell simulations of
relativistic shocks, and $\xi_{N}$ free to vary.

Table 2: Best-fit parameters from the afterglow model fitting. Case 1: $\xi_{N}=1$. Case 2: $\xi_{N}=0.1$. Case 3: $\xi_{N}$ is a free parameter. We fix $\theta_{\rm obs}=0$ for all the cases. BIC is the Bayesian Information Criterion (Schwarz, 1978) for the best-fit results: the model with lower BIC is the preferred one. Parameter | Prior | Posterior
---|---|---
|  | Case 1 | Case 2 | Case 3
log $E_{k,\mathrm{iso}}$ (erg) | (50, 60) | $53.3^{+0.5}_{-0.4}$ | $54.4^{+0.6}_{-0.4}$ | $56.7^{+0.9}_{-1.4}$
log $\theta_{j}$ (rad) | (-3, -0.5) | $-1.2^{+0.2}_{-0.2}$ | $-1.3^{+0.2}_{-0.3}$ | $-1.6^{+0.3}_{-0.3}$
log $n$ (cm-3) | (-6, 2) | $-2.3^{+1.3}_{-1.4}$ | $-2.5^{+1.3}_{-1.6}$ | $-2.4^{+1.9}_{-1.6}$
p | (2, 3.5) | $2.9^{+0.2}_{-0.06}$ | $2.9^{+0.2}_{-0.06}$ | $2.9^{+0.2}_{-0.05}$
log $\epsilon_{e}$ | (-6, -0.5) | $-1.2^{+0.3}_{-0.7}$ | $-2.4^{+0.3}_{-0.3}$ | $-4.7^{+1.4}_{-0.9}$
log $\epsilon_{B}$ | (-6, -0.5) | $-0.9^{+0.4}_{-0.8}$ | $-0.9^{+0.4}_{-0.8}$ | $-1.6^{+0.8}_{-1.4}$
log $\xi_{N}$ | (-5, 0) | 0 (fixed) | -1 (fixed) | $-2.9^{+1.2}_{-0.8}$
$\chi^{2}$/dof | $-$ | 42.81/32 | 42.12/32 | 41.28/31
BIC | $-$ | 64.6 | 63.9 | 66.7

Figure 2: Broadband modelling results. Left panel: best-fit light curves with
observing points from the radio to the X-ray band. Downward arrows represent
upper limits. Empty points are used for observations in the exclusion time
span (250-450 ks after trigger time). Right panel: corner plot of the
posterior distribution of the modelled parameters. In both panels Case 1 is
presented.

Additionally, the code allows the user to account for viewing angle effects
through the parameter $\theta_{\rm obs}$ ($\theta_{\rm obs}$=0 corresponds to
an on-axis observer), and to explore multiple angular structures of the GRB
jet (e.g. uniform, Gaussian, and power-law). In our fits, we explore the
simple standard case of a uniform jet viewed on-axis, which is already
sufficient to provide a good description of the dataset.

The code does not include the RS emission, which may contribute above 9 GHz,
and the synchrotron self-absorption regime, which may be relevant below 3 GHz.
For these reasons, we only include the observations at 5.5 GHz to model the FS
afterglow. Energy injection is included in afterglowpy but with limited
options, thus we exclude from our fit the X-ray and optical data between 250
ks and 450 ks. The results of the afterglow model fitting are summarized in
Table 2 for the three cases explored: $\xi_{N}=1$ (Case 1), $\xi_{N}=0.1$
(Case 2) and $\xi_{N}$ free to vary (Case 3). The model fit light curves from
the radio to the X-ray band and posterior parameter distributions are shown in
Fig. 2 for the Case 1 only.

The fit identifies a jet-break occurring at around 20 d after the GRB, which
translates into a jet opening angle $\theta_{j}\approx 3.6^{\circ}$ (Case 1)
and $2.9^{\circ}$ (Case 2). The evidence of a jet-break is mainly driven by
the radio data, as it occurs after the end of the X-ray and optical
observations. The two fits with fixed $\xi_{N}$ yields similar values of
$\theta_{j}$ and are equivalent in terms of reduced $\chi^{2}$ and BIC values.
However, Case 1 with $\xi_{N}$=1 yields an isotropic-equivalent kinetic energy
that is three times lower than the value measured in gamma-rays, implying a
high gamma-ray efficiency $\eta_{\gamma}\gtrsim$ 70%. This would be
challenging for most prompt emission mechanisms and, in particular, for
internal shocks in hydrodynamic jets (Daigne & Mochkovitch, 1998; Beniamini &
Piran, 2013). Instead, Case 2 ($\xi_{N}$=0.1) eases the requirement on the
gamma-ray production with a dissipative efficiency of $\eta_{\gamma}\sim$ 20%.
By using the best fit values of $E_{k,\mathrm{iso}}$ and $\theta_{j}$, we
infer a beaming-corrected kinetic energy $E_{K}$ = $(1-cos\
\theta_{j})E_{k,\mathrm{iso}}\approx 4\times 10^{50}$ erg (Case 1) and
$3\times 10^{51}$ erg (Case 2) , in keeping with cosmological GRBs (Wang et
al., 2018). By leaving $\xi_{N}$ free to vary (Case 3), the fit does not
substantially improve and the total kinetic energy tends to unphysically high
values.

All our fits tend to prefer a tenuous environment with a uniform density
between $10^{-4}$ and $10^{-2}$, which is consistent with the interstellar
medium. Using these best fit values, we infer a self-absorption frequency
$\nu_{\mathrm{a}}\approx 1$ GHz (observer’s frame), close to the range covered
by MeerKAT. Finally, it is worth noting the soft value of the electron’s
spectral index $p\approx 3$. Basic shock acceleration principles set this
value to 2.23 (Keshet & Waxman, 2005), although observations of GRB afterglows
display slightly softer spectra with an average $p\approx 2.4$ (Curran et al.,
2006). Warren et al. (2017) found that if acceleration is efficient, then
nonlinear effects linking the shock structure to the accelerated particles
lead to a harder electron’s spectrum. Our soft $p$ value is not consistent
with these findings and may point instead to non-efficient particle
acceleration.

In Figure 3 we present a comparison between the radio afterglow of EP240315A
and a representative sample of GRBs. As derived from our broadband modeling,
EP240315A displays properties fully consistent with the population of
cosmological GRBs. Whereas its prompt emission is relatively faint in the soft
X-ray band (Liu et al., 2024), its radio counterpart occupies the brightest
tail of the afterglow distribution, with a luminosity comparable to GRB 050904
(Chandra & Frail, 2012) and higher than GRB190114C (MAGIC Collaboration et
al., 2019).

Figure 3: Radio luminosity light curve of EP240315A compared with a sample of
cosmological GRBs (black) and sub-energetic GRBs (gray) (Dichiara et al.,
2022; Chandra & Frail, 2012).

##  4 Conclusions

At the time of writing, EP has publicly reported 18 FXRTs. Of these, only two
were detected by Swift/BAT, including EP240315A, whereas most of the other
ones happened outside the BAT’s field of view (FoV). Additional constraints
come from Fermi/GBM observations: only two possible gamma-ray counterparts
were reported, although in 13 cases the FXRT happened during normal operations
and had good visibility ($\theta_{\rm NaI}<60$ deg). This suggests that a
substantial fraction of the FXRT population ($\gtrsim$25%) may overlap with
GRBs.

We used the case of EP240315A to characterize the relativistic outflow
powering some of these FXRTs and compare it to the properties of cosmological
GRBs. Our study finds that EP240315A was produced by a collimated relativistic
jet with a narrow opening angle $\theta_{j}\approx 3^{\circ}$ and total energy
$E\approx 4\times 10^{51}$ erg, in keeping with the population of long GRBs.
Whereas in the case of EP240315A, its temporal and spatial coincidence with
GRB240315C unambiguously classifies it as GRB, our study shows that similar
relativistic explosions can be discovered by sensitive X-ray surveys and
successfully identified through the evolution of their multiwavelength
counterparts. Although more events are needed to better understand the
interplay of GRBs and FXRTs, this case confirms that missions such as Einstein
Probe and the proposed hiZ-GUNDAM (Yonetoku et al., 2020) and THESEUS (Amati
et al., 2021) will help us pinpoint the explosions of the most massive stars
up to the highest redshifts.

We thank Simone Dichiara for sharing the data on GRB radio afterglows. ET, RR,
YY and MY acknowledge the support of the European Research Council through the
Consolidator grant BHianca (Grant agreement ID: 101002761). The Australia
Telescope Compact Array is part of the Australia Telescope National Facility
(https://ror.org/05qajvd42) which is funded by the Australian Government for
operation as a National Facility managed by CSIRO. We acknowledge the Gomeroi
people as the Traditional Owners of the Observatory site.

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