# Gravitational lensing of Continuous Gravitational Waves

Marek Biesiada<EMAIL_ADDRESS>National Centre for Nuclear Research
Pasteura 7, 02-093 Warsaw, Poland
Department of Astronomy, Beijing Normal University,
Beijing 100875, People’s Republic of China
Sreekanth Harikumar<EMAIL_ADDRESS>National Centre for
Nuclear Research
Pasteura 7, 02-093 Warsaw, Poland

###### Abstract

Continuous gravitational waves are analogous to monochromatic light and
therefore could be used to detect wave effects like interference or
diffraction. This would be possible with strongly lensed gravitational waves.
This article reviews and summaries the theory of gravitational lensing in the
context of gravitational waves in two different regimes: geometric optics and
wave optics, for two widely used lens models such as point mass lens and
Singular Isothermal Sphere (SIS). Observable effects due to wave nature of
gravitational waves are discussed. As a consequence of interference GWs
produce beat patterns which might be observable with the next generation
detectors like ground based Einstein Telescope, Cosmic Explorer or space-borne
LISA, DECIGO. This will provide us a opportunity to estimate the properties of
lensing system and other cosmological parameters with alternative techniques.
Diffractive microlensing could become a valuable method to search for
intermediate mass black holes formed in the centers of globular clusters. We
also point to an interesting idea of detecting the Poisson-Arago spot proposed
in the literature.

Gravitational Waves; Gravitational Lensing ; Poisson- Arago Spot;
Interference; microlensing

##  1 Introduction

With the first detection of gravitational waves in 2015 Abbott et al. (2016)
from coalescing compact binary system we have entered the long expected era of
GW astronomy. The new window on the Universe has been opened. First GW
detections brought the confirmation of existence of binary black hole (BBH)
systems in Nature. A half of century ago, primary candidates for chirping
signals were binary neutron stars (BNS) due to sober expectations based on
Hulse-Taylor like BNS systems discovered so far. Indeed in 2017 first such
coalescence has been registered Abbott et al. (2017) and accompanying
electromagnetic signals spanning from gamma-rays, through the optical to the
radio waves has been registered allowing the identification of host galaxy and
making a plethora of various tests possible. Further observing runs of LIGO-
Virgo-KAGRA network enriched considerably the statistics of registered events.
In the future there will be qualitative boost when the 3rd generation of
ground-based detectors like the Einstein Telescope(ET) Punturo & et al (2010)
or Cosmic Explorer(CE) Dwyer et al. (2015) as well as space-borne detectors
(LISA, DECIGO, TianQin) become operative. First, the sensitivity of ground-
based detectors will be increased by an order of magnitude over the existing
ones, allowing to probe three orders of magnitude larger volume of the
Universe. Second, the satellite detectors will probe much lower frequencies of
GW (inaccessible from the ground due to seismic noise) enabling to observe
adiabatic inspiralling signals from binary systems much earlier than the
coalescence phase probed by ground based detectors. This means that besides
the already registered chirp signals, we would gain access to almost
monochromatic continuous GW signals.

In this paper we discuss some new opportunities, which will be available when
continuous GW signals are registered. In the case of light, historically there
has been a dispute about its nature: corpuscular vs. wave. Wave nature of
light has been established with the interference and diffraction patterns,
which are best visible with monochromatic light. By analogy, interference and
diffraction patterns should be revealed in GWs as well. But what sort of
experiment could be done in this context? We point to gravitationally lensed
GW signals as a promising physical setting to reveal wave effects of GWs.

Bending of light by the Sun was the first classic test of General Relativity.
The essence of light bending by massive bodies lies in the fact that paths of
photons (light rays) are null geodesics in spacetime, which is curved by the
presence of mass. Since the Eddington’s expedition in 1919, this phenomenon
now evolved to a mature discipline of extragalactic astronomy called
gravitational lensing. Currently it is routinely used to study the structure
of galaxies (lenses) and cosmology. Gravitational waves, in the geometric
optics approximation, follow the null geodesics as well. Hence one can expect
to see gravitationally lensed GW signals. Such signals coming from unresolved
images would interfere producing characteristic patterns. In the next section
we will review strong lensing of GWs. Section 3 contains discussion of some
observable effects particularly pronounced when the source emits continuous
GWs. We summarize in Section 4.

##  2 Gravitational Lensing of Gravitational Waves

As one of the successful predictions of General Relativity, strong
gravitational lensing by galaxies has become one of the most important tools
in studying cosmology Cao et al. (2015), galaxy structure and evolution Cao et
al. (2016). With the dawn of gravitational wave astronomy, the robust
prediction suggested that a considerable number of GW signals from
inspiralling neutron stars would be gravitationally lensed, focusing on the
third generation of ground-based GW detectors Ding et al. (2015) and second
generation of space-based GW detectors Piórkowska-Kurpas et al. (2021),
respectively.

Strong gravitational lensing occurs whenever the source, the lens (a galaxy, a
black hole, a star) and the observer are almost aligned. In the particular
case of perfect alignment the source will not be obscured by the lens but
instead visible as a glowing ring – called the Einstein ring. In the case of
axisymmetric lenses, the angular radius of this ring i.e. the Einstein radius
is:

${\theta_{E}}=\sqrt{\frac{{4G{M_{l}}}}{{{c^{2}}}}\frac{{{D_{ls}}}}{{{D_{l}}{D_{s}}}}},$
(1)

where $M_{l}$ is the mass of the lens, ${D_{l}}$ is the angular diameter
distance to the lens at redshift ${z_{l}}$, ${{D_{s}}}$ is the angular
diameter distance to the source at redshift ${z_{s}}$, ${D_{ls}}$ is the
angular diameter distance between the lens and the source. Misalignment in
this optical system by the angle $\beta$ (angle between directions to the
source and to the lens) leads to the appearance of multiple images. Usually
two (spherically symmetric mass distribution) or four images are formed at
angular separations of $\theta_{i}$ from the center of the lens. The typical
separation between different images is set by the Einstein radius and their
locations is described by the lens equation:

$\beta=\theta-\alpha(\theta)$ (2)

where: $\alpha(\theta)$ is the deflection angle determined by the projected
mass distribution of the lens (for detailed derivations see Schneider et al.
(1992)). The total time delay introduced by gravitational lensing at the
angular position $\theta$ from the lens is:

$\Delta
t=\frac{1+z_{l}}{c}\frac{D_{l}D_{s}}{D_{ls}}\left[\frac{(\theta-\beta)^{2}}{2}-\phi(\theta)+\phi_{m}(\beta)\right]$
(3)

where $\phi(\theta)$ is the lens potential determining the deflection angle
$\alpha(\theta)=\nabla_{\theta}\phi(\theta)$. Term $\phi_{m}(\beta)$
corresponds to the arrival time in a non-lensed case and in practice it is a
constant adjusted to ensure the extreme value of time delay functional. Total
time delay, more precisely the time delay functional measures the delay
between lensed signal arrival from the image at position $\theta$ with respect
to (unmeasurable) arrival time from the source if the lens was absent.
Fermat’s principle implies that images correspond to stationary points of time
delay functional, and one can see that condition $\nabla_{\theta}\Delta t=0$
is equivalent to the lens equation (2). The measurable quantity is the time
delay difference $\Delta t_{ij}$ between images at $\theta_{i}$ and
$\theta_{j}$. Another measurable quantity is the ratio of image
magnifications. The (signed) magnification is the inverse Jacobian of the lens
equation:
$\mu=\left(det\left(\frac{\partial\beta}{\partial\theta}\right)\right)^{-1}$.
The sign corresponds to the parity of the image, so physically relevant are
$|\mu|$ and since usually the intrinsic luminosity is unknown, only
magnification ratios are relevant. The exception could be the lensed standard
candles. In the optical, image locations are another sort of observables, but
for the purpose of our discussion they are irrelevant.

For the purpose of this work, we refer to two simplest, commonly used,
analytic lens models: point mass model and singular isothermal sphere (SIS)
model. Their choice is dictated by two realistic scenarios: lensing by an
isolated mass like a massive (with $M_{l}<10^{7}M_{\odot}$) black hole (BH) or
an intervening galaxy ($M_{l}\sim 10^{9}-10^{11}M_{\odot}$).

In the case of axisymmetric point lenses, the typical separation between
different images is set by the Einstein radius

${\theta_{E}}=\sqrt{\frac{{4G{M_{l}}}}{{{c^{2}}}}\frac{{{D_{ls}}}}{{{D_{l}}{D_{s}}}}},$
(4)

where $M_{l}$ is the mass of the lens, ${D_{l}}$ is the angular diameter
distance to the lens at redshift ${z_{l}}$, ${{D_{s}}}$ is the angular
diameter distance to the source at redshift ${z_{s}}$, ${D_{ls}}$ is the
angular diameter distance between the lens and the source. The dimensionless
lens equation (Schneider et al., 1992) and its corresponding solutions are

$y=x-\frac{1}{x}\rightarrow\begin{array}[]{*{20}{c}}{\left\\{{\begin{array}[]{*{20}{c}}{{x_{+}}=\frac{{y+\sqrt{{y^{2}}+4}}}{2}}\\\
{{x_{-}}=\frac{{y-\sqrt{{y^{2}}+4}}}{2}}\end{array}}\right.}\end{array}$ (5)

where $y=\beta/{\theta_{E}}$, $\beta$ denoting the angle between directions to
the lens and to the source, $x=\theta/{\theta_{E}}$, where $\theta$ is the
angular position of the image actually seen by the observer.

In the case of point mass lenses, two images could be produced at angular
positions ${\theta_{+}}={x_{+}}{\theta_{E}}$ and
${\theta_{-}}={x_{-}}{\theta_{E}}$, respectively. The inverse of the
determinant of the Jacobi matrix $\frac{\partial\beta}{\partial\theta}$
defines their magnifications,

${\mu_{\pm}}=\frac{1}{1-{\left(\frac{1}{{{x_{\pm}}}}\right)^{4}}}.$ (6)

Time delay between images produced by a point mass lens is Schneider et al.
(1992):

$\Delta
t=\frac{{4G{M_{l}}(1+{z_{l}})}}{{{c^{3}}}}\left(\frac{{y\sqrt{{y^{2}}+4}}}{2}+\ln\frac{{\sqrt{{y^{2}}+4}+y}}{{\sqrt{{y^{2}}+4}-y}}\right).{\rm{}}$
(7)

Point mass lenses (also called Schwarzschild lenses) are representative to
lensing by stars (microlensing) and BHs with stellar or intermediate mass BHs.

It is well established that early-type galaxies act as lenses in the majority
of strongly gravitational lens systems detected. Even though their formation
and evolution are still not fully understood in details, a singular isothermal
sphere model (SIS) can reasonably characterize the mass distribution of
massive elliptical galaxies within the effective radius (Treu & Koopmans,
2004; Treu et al., 2006; Cao et al., 2016; Liu et al., 2020). In the case of
axisymmetric SIS model (Narayan & Bartelmann, 1996; Bernardeau, 1999), its
three-dimensional density profile could be described as
$\rho(r)={{\sigma_{v}^{2}}}/{{2\pi G{r^{2}}}}$, where $r$ is the distance from
the sphere center and ${\sigma_{v}}$ is the velocity dispersion of the lens.
In this case, the dimensionless lens equation (Schneider et al., 1992) could
be written as

$y=x-\frac{x}{{|x|}},$ (8)

where $x=\theta/{\theta_{E}}$, $y=\beta/{\theta_{E}}$ as above. There are two
cases for the solution of the lens equation: when $y<1$, the solutions are

$x_{\pm}=y\pm 1.$ (9)

The corresponding angular positions of the images are
${\theta_{\pm}}=\beta\pm{\theta_{E}}$, where the Einstein radius
${\theta_{E}}$ could be written as

${\theta_{E}}=\sqrt{\frac{{4GM(\theta_{E})}}{{{c^{2}}}}\frac{{{D_{ls}}}}{{{D_{l}}{D_{s}}}}}=4\pi\frac{\sigma_{v}^{2}}{c^{2}}\frac{D_{ls}}{D_{s}},$
(10)

where $M(\theta_{E})$ is the mass within the Einstein radius. In this case,
the magnifications of the images are

$\mu_{\pm}=1\pm\frac{1}{y}.$ (11)

On the other hand, when $y>1$, the lens equation only has one solution:
$x=y+1$. Its corresponding magnification is $\mu={{|x|}}/({{|x|-1}})$.

For SIS model, the time delay between different images reads:

$\Delta
t=\frac{32\pi^{2}}{c}\left(\frac{{{\sigma_{v}}}}{c}\right)^{4}\frac{{{D_{l}}{D_{ls}}}}{{{D_{s}}}}(1+{z_{l}})y.$
(12)

###  2.1 Wave optics regime

The above summarized results have been obtained using the geometric optics
approximation, which is excellent is most astrophysically relevant cases of
strong gravitational lensing of electromagnetic waves. Indeed this
approximation is valid as long as typical length scale of the system is much
larger than the wavelength and timescale of typical variations of the system
are much larger than the period of the wave. In the case of GWs it becomes
different: frequency range probed by ground-based detectors comprise
$10\;Hz<f<10\;kHz$ (to be lowered down to $1\;Hz$ in the future third
generation) and in the space-borne detectors it will be $0.1\;mHz<f<100\;mHz$
in LISA and $1\;mHz<f<100\;Hz$ in DECIGO. This corresponds to the GW
wavelenghts of $10^{4}\;m<\lambda<10^{7}\;m$ in ground-based and
$10^{6}\;m<\lambda<10^{12}\;m$ in space-borne detectors. The first papers
acknowledging the importance of wave regime for GW lensing were Nakamura
(1998); Takahashi & Nakamura (2003).

In the wave optics regime, we consider the gravitational wave from a distant
source as propagating in the background spacetime $g_{\mu\nu}^{(L)}$ of the
lens object characterized by the gravitational potential $U(\textbf{r})<<1$.
The total metric, including the perturbation (due to GW signal) is given by
$g_{\mu\nu}=g_{\mu\nu}^{(L)}+h_{\mu\nu}$, where $h_{\mu\nu}=\psi e_{\mu\nu}$.
Since the polarization tensor $e_{\mu\nu}$ is parallel transported along the
null geodesic, we may neglect its change (being of order $\sim
U(\textbf{r})<<1$) and study the propagation equation of the scalar wave:
$\partial_{\mu}\left(\sqrt{-g^{(L)}}g^{(L)\mu\nu}\partial_{\nu}\psi\right)=0$.
For the monochromatic scalar wave
$\psi(\textbf{r},t)=\tilde{\psi(\textbf{r})}e^{-2\pi ift}$, in the frequency
domain and remembering the weak potential assumption, this equation can be
cast to the form of Helmholtz equation:

$\left(\nabla^{2}+4\pi^{2}f^{2}\right)\tilde{\psi}=16\pi^{2}f^{2}U\tilde{\psi}$
(13)

The solution to the Helmholtz equation can be given in terms of Kirchoff
integral Schneider et al. (1992), where it is convenient to introduce the
dimensionless amplification factor:

$F(f)=\frac{\tilde{\psi}^{L}}{\tilde{\psi}}$ (14)

which is the ratio of wave amplitudes with and without lensing.Then the
Kirchoff integral allows us to calculate the amplification factor at the
observer Schneider et al. (1992) as:

$F(f,\beta)=\frac{1+z_{l}}{c}\frac{D_{s}}{D_{l}D_{ls}}\frac{f}{i}\int
d^{2}\theta\exp\left[2\pi if\Delta t(\theta,\beta)\right]$ (15)

where $\Delta t$ is lensing time delay given by equation (3) and cosmological
nature of distances in the optical system has been taken into account.
Calculation of the integral (15) can be simplified by switching from angles to
dimensionless variables $x$ and $y$ (see above after Eq.(5)) and introducing
new parameter $w=\frac{8\pi G}{c^{3}}M_{l}(\theta_{E})(1+z_{l})f$:

$F(w,y)=\frac{w}{2\pi i}\int d^{2}y\left[\exp[iwT(x,y)\right]$ (16)

where: $T(x,y)=\frac{(x-y)^{2}}{2}-\frac{\phi(x)}{\theta_{E}^{2}}$. In general
amplification factor (16) should be evaluated numerically, but in two
particular cases it can be integrated analytically. In the point mass lens
model the dimensionless lensing potential is $\phi(\textbf{x})=lnx$ and the
analytical solution to the diffraction integral is given by

$F(f)=\exp\Big{[}\frac{\pi
w}{4}+i\frac{w}{2}\Big{(}ln\Big{(}\frac{w}{2}\Big{)}-2\phi_{m}(y)\Big{)}\Big{]}\Gamma\left(1-\frac{i}{2}w\right)\;{}_{1}F_{1}\left(\frac{i}{2}w,1;wy^{2}\right)$
(17)

where $\phi_{m}(y)=0.5(x_{m}-y)^{2}/2-lnx_{m}$ with
$x_{m}=(y+\sqrt{y^{2}+4})/2$ and ${}_{1}F_{1}$ is the confluent hypergeometric
function. In the case of SIS model, the Kirchoff integral could be cast into
analytically manageable form Takahashi & Nakamura (2003):

$F(f)=-iw\;e^{iwy^{2}/2}\int_{0}^{\infty}dx\;xJ_{0}(wxy)\exp\left[iw\left(\frac{1}{2}x^{2}-x+\phi_{m}(y)\right)\right]$
(18)

where $\phi_{m}(y)=y+1/2$ and the $J_{0}$ is the Bessel function of order
zero. Geometric optics corresponds formally to $f\to\infty$, hence $w>>1$. In
such case Kirchoff integral can be calculated using stationary phase
approximation and dominant contributions will come from image positions. Hence
the amplification of the wave intensity is:

$|F(f)|^{2}=|\mu_{+}|^{2}+|\mu_{-}|^{2}+2|\mu_{+}\mu_{-}|\sin{(2\pi f\Delta
t_{d})}$ (19)

where first two terms correspond to magnifications in geometric optics, the
last one represents the interference between images.

##  3 Effects of gravitational wave lensing of continuous waves

Since the lensed images would not be resolved, they are expected to come from
the same location on the sky separated in time by the lensing time delay
$\Delta t_{ij}$. In the case of compact binary systems coalescences this has
been expected to be revealed as just two signals with similar temporal
patterns and amplitudes affected by magnification. This picture is basically
true for merging systems staying within the detectors’ sensitivity band for a
fraction of a second. However, if such a system could be monitored earlier
i.e. in adiabatic inspiral phase (almost monochromatic signal), then the
brighter image (arriving earlier) could imprint its interference patterns on
the strain of the fainter image. In this section we will discuss some of these
effects stemming from the wave nature of GWs, which could be observed. We
review here some fresh ideas that already appeared in the literature (and to
which one of us contributed).

###  3.1 Beat Patterns

In an interesting paper Hou et al. (2020) it has been discussed that if the
lensed images from the coalescing binary system could be detected
simultaneously by the detector, then beat patterns would be revealed prior to
the merger signal. The merger signal would arrive first from the brighter
(more magnified) image, then after lensing time delay the merger signal from
the fainter one would be registered. If they stayed for a couple of seconds in
the detector sensitivity band, then one would see beat patterns. First of all,
this condition is very restrictive for the strong lensing system
configuration: time delay should be of order of minutes at most. This means
that probability of detecting such an event from the ground is much lower than
probability of detecting a strong lensed GW in general. However, the payoff of
such detection should be considerable, because as the authors of Hou et al.
(2020) argue, one would be able to measure the redshifted lens mass purely
from the GW signal, and measure time delay distance without reconstructing the
Fermat potential. This is interesting enough, so that it is worth to recall
the basic points here.

Suppose that we have a signal in time domain $h(t)$, which is a superposition
of two differently magnified signals from unresolved two images:

$h=\mu_{+}[A^{+}\cos(\omega_{1}t+\phi_{1})+A^{\times}\sin(\omega_{1}t+\phi_{1})]+\mu_{-}[A^{+}\cos(\omega_{2}t+\phi_{2})+A^{\times}\sin(\omega_{2}t+\phi_{2})]$
(20)

where $\mu_{+}$ and $\mu_{-}$ are their respective magnification, $A^{+}$,
$A^{\times}$ are the amplitudes and $\omega_{1}$, $\omega_{2}$ , $\phi_{1}$,
$\phi_{2}$ are angular frequencies and initial phases respectively. It can be
rewritten as:

$\displaystyle
h=\mu_{sum}\left[A^{+}\cos(\omega_{f}t+\phi_{f})\cos(\omega_{b}t+\phi_{b})+A^{\times}\sin(\omega_{f}t+\phi_{f})\cos(\omega_{b}t+\phi_{b})\right]+$
$\displaystyle+\mu_{diff}\left[-A^{+}\sin(\omega_{f}t+\phi_{f})\sin(\omega_{b}t+\phi_{b})+A^{\times}\cos(\omega_{f}t+\phi_{f})\sin(\omega_{b}t+\phi_{b})\right]$

with $\mu_{sum}=\mu_{+}+\mu_{-},$ $\mu_{diff}=\mu_{+}-\mu_{-},$ the
frequencies $\omega_{b}=\frac{\omega_{1}-\omega_{2}}{2}$ and
$\omega_{f}=\frac{\omega_{1}+\omega_{2}}{2}$ represent beat and average
frequency, respectively and phases $\phi_{b},\;\phi_{f}$ are defined in a
similar manner. In the early inspiral phase (i.e. well before coalescence)
when the frequency evolves according to $\frac{d\omega}{dt}=\frac{192}{5}{\cal
M}^{5/3}\left(\frac{\omega}{2}\right)^{11/3}$ Jaranowski & Krolak (2009); Husa
(2009) and $\omega_{b}<<\omega_{f}\approx\omega_{1}\approx\omega_{2}$ one has:

$\omega_{b}=\frac{96}{5}\Big{(}\frac{\omega_{f}}{2}\Big{)}^{11/3}\mathcal{M}\Delta
t$ (22)

where $\mathcal{M}$ is the chirp mass of the binary system and $\Delta t$ is
the lensing time delay (3) given by formula (7) in case of point mass lens.
Fig. 1 illustrates the beat pattern originating in the lensed coalescing
system. One can see the evolution of beat pattern toward the coalescence.
During the last stages of inspiral $\omega_{f}$ increases and the beat pattern
disappears. In the case of lensed continuous GW signals beat pattern would be
rigid. Now, if one use beat pattern defined by equation (3.1) as a template in
matched filtering method, then the ratio $\frac{\mu_{sum}}{\mu_{diff}}$ can be
extracted from the data and easily converted to the magnification ratio
$r=\left|\frac{\mu_{+}}{\mu_{-}}\right|$. The lensing time delay $\Delta t$ is
directly measurable and in the case of registering two coalescence signals the
accuracy of such measurement would be unprecedented. In the case of almost
monochromatic signals from the inspiral phase long before the merger
(accessible in low frequency bands of DECIGO and LISA) time delay $\Delta t$
could still be extracted from the matched filtering using beat pattern
template. The knowledge of the magnification ratio $r$ will allow to make
interesting inferences. In the case of a point mass lens, after expressing
time delay (7) in terms of $r$, one can see that redshifted mass of the lens
can be calculated as:

$M_{l}(1+z_{l})=\frac{c^{3}}{4G}\Delta t\left[\frac{r-1}{\sqrt{r}}+\ln
r\right]^{-1}$ (23)

On the other, hand in the case of SIS lens the so called time delay distance
$D_{\Delta t}$ can be obtained:

$D_{\Delta t}\equiv\frac{{{D_{l}}{D_{ls}}}}{{{D_{s}}}}=\frac{c^{5}\Delta
t}{32\pi^{2}\sigma_{v}^{4}(1+z_{l})}\frac{r-1}{r+1}$ (24)

provided that lens is identified and its redshift is known. This is also an
important result, demonstrating the breadth of information which can be
extracted from GW signal alone. Needless to say that time delay distance
measurements have been used to determine the Hubble constant in a way
independent of the cosmic ladder or CMB techniques. It has been shown in Liao
et al. (2017) that lensed GW signals from coalescing BNS systems would allow a
percent accuracy in $H_{0}$ measurements. Moreover, time delay distance can be
used to test the cosmological model beyond the $\Lambda$CDM.

Figure 1: A schematic diagram showing the beat pattern for a binary inspiral
signal. Masses of the components are $m_{1}=30M_{\odot}$ and
$m_{2}=20M_{\odot}$, source is located at $z_{s}=2$, lens at $z_{l}=1$ is
assumed to be a point mass of $M_{l}=10^{6}\;M_{\odot}.$ Time delay has been
chosen as $\Delta t=2.1\;s$ thus fixing the $\beta$ parameter respectively.

###  3.2 Diffractive microlensing

Besides the strong lensing discussed in previous section, which in case of
electromagnetic window usually means multiple images of a quasar (or galaxy)
lensed by a galaxy, one can consider microlensing. It is again a strong
lensing case (usually starlight lensed by a star) where the images cannot be
resolved, but the relative motion of the elements of the optical system make
temporal changes of total magnification of the source detectable. It was an
ingenious idea of Paczyński Paczynski (1986) that two individually
unobservable effects (image separation and relative proper motion) can be
combined to give observable effect. One can expect, that the similar
phenomenon could be detected in GW domain. In the paper Liao et al. (2019)
such setting has been proposed and discussed. Besides Liao et al. (2019)
fringes due to GW lensing have also been studied in Lai et al. (2018); Jung &
Shin (2019); Christian et al. (2018). One can think of gravitational lensing
of GWs (in or close to wave regime) as an analog of one-slit or two-slit
experiment with light. When the light is monochromatic (a laser beam) a
characteristic fringe pattern is visible on the screen. In astrophysical
scales the “screen” has astronomical scales and is inaccessible directly. This
corresponds to fixed $y$ parameter. However, in the microlensing case $y$
changes in time (due to relative motions) and one can explore the “screen” by
watching the manifestation of fringes. The peculiar motions of the lensing
system mentioned above include the motion of the observer $\textbf{v}_{obs}$,
the source $\textbf{v}_{s}$ and the lens $\textbf{v}_{l}$\- all measured in
the CMB frame combined to obtain the effective velocity $\textbf{v}_{eff}$ as
given by Kayser et al. (1986)

$\textbf{v}_{eff}=\textbf{v}_{s}-\frac{1+z_{s}}{1+z_{l}}\frac{D_{s}}{D_{l}}\textbf{v}_{l}+\frac{1+z_{s}}{1+z_{l}}\frac{D_{ls}}{D_{l}}\textbf{v}_{obs}$
(25)

Since in lensing we are referring to angles on the sky, the effective velocity
should be converted to angular proper motion.

For the comparison between the size of “barrier” (lens) and the wavelength of
GW one uses of the parameter $w=\frac{8\pi
G}{c^{3}}M_{l}(\theta_{E})(1+z_{l})f$ introduced in section 2. Recall that if
$w>>1$ the amplification factor in (15) simplifies to (19). The last term in
(19) is the interference term. If the amplitude of the diffraction fringes are
to be detectable the parameter $w$ should be large enough (but not too big).
It has been proposed in Liao et al. (2019) that $w>1$ is the reasonable
condition for the fringe pattern to be detectable. To see a complete
diffraction or interference pattern the observation period $T_{obs}$ should be
more than the timescale over which the fringe pattern are affected due to
peculiar motion of the system, we denote the fringe timescale as $t_{f}$. This
timescale of transit of fringes can be denoted by $t_{f}=\frac{t_{E}}{w}$
where $t_{E}$ is the Einstein radius crossing time. For galactic sources and
lens it is given by

$t_{E}\approx
34.7\text{days}\sqrt{4\frac{D_{l}}{D_{s}}\Big{(}1-\frac{D_{l}}{D_{s}}\Big{)}}\Big{(}\frac{D_{s}}{8\text{kpc}}\Big{)}^{1/2}\times\Big{(}\frac{M}{M_{\odot}}\Big{)}^{1/2}\Big{(}\frac{v_{eff}}{200\text{km}s^{-1}}\Big{)}^{-1}$
(26)

Therefore the criterion for detecting a fringe pattern can be summarized in
three conditions Liao et al. (2019):

  1. 1.

w $>$1, to have enough amplification

  2. 2.

y$<$3 to detect fringes before they are damped

  3. 3.

$\Delta t>t_{f}$ to see a fringe pattern

Let us observe, that the second condition means that the probability of
microlensing is almost an order of magnitude larger than in the optical. It is
because the cross-section for lensing is proportional to the Einstein ring
area $\pi y^{2}$ and in the optical $y=1$ is the threshold for microlensing.
Estimates of Narayan & Ostriker (1990); Arnett et al. (1989) suggest that
galactic bulge contains about $10^{9}$ NSs and globular cluster contain
$10^{3}$ NSs which could be the target population for the current and next
generation gravitational detectors. In order to find the probability of
observing fringe pattern from such a population of monochromatic sources one
need to calculate optical depth which gives the probability that a given
source falls into the Einstein radius of any lens located along the line of
sight $\tau=\int_{0}^{D_{s}}n(D_{l})\pi R_{E}^{2}dD_{l}.$ If we consider a
source population in the Milky Way bulge lensed by a constant density of
stellar mass lenses in the galactic disk moving with a velocity
$v_{rot}=200\text{kms}^{-1}$, the lensing probability is given by Paczynski
(1986, 1991).

$\tau=\frac{v_{rot}^{2}}{2c^{2}}=f_{l}\times 10^{-6}$ (27)

where $f_{l}$ is the fraction of lens larger than one solar mass, which is
required to fulfill the above mentioned conditions for diffractive
microlensing. Currently, the issue of intermediate mass BHs and their
existence has been debated. They can be formed in the very cores of globular
clusters. Therefore, it would be interesting to focus on globular clusters as
hosts of gravitational lenses for lensing continuous GWs. One such candidate
is already known. M22 is a rich globular cluster with projected location
pointing the Galactic bulge – a perfect site of sources (spinning neutron
stars or binary systems). A typical velocity dispersion in a globular cluster
is $\sigma_{v}\sim 10$ km$s^{-1}$, hence the typical transverse velocity of
the lens can be very well approximated by that of a cluster as a whole $\sim
200\;\text{kms}^{-1}$. Again, Paczynski (1994) has shown that the optical
depth for microlensing in such case is:

$\tau=f_{l}\frac{\sigma^{2}}{c^{2}}\frac{2\pi}{\varphi}\frac{D_{ls}}{D_{s}}\approx
2.4\;\;10^{-4}\;f_{l}\left(\frac{\sigma_{v}}{10\;kms^{-1}}\right)^{2}\left(\frac{1^{\prime}}{\varphi}\right)\frac{D_{ls}}{D_{s}}$
(28)

where $\varphi$ denotes the angular distance from the center of the cluster.
Knowledge of the distance and transversal velocity of lenses belonging to the
cluster may result with accurate estimates of their masses, breaking the
degeneracies inherent to the microlensing technique. The microlensing of the
bulge star at $D_{s}=8.2\;kpc$ by a low-mass object in the globular cluster
M22 located at $D_{l}=2.6\;kpc$ has been reported in e.g. Pietrukowicz et al.
(2011). Let us remark that the inclusion of $\varphi$ term has been motivated
by the fact that in the optical only the outskirts of the cluster can produce
microlensing of the bulge stars. Regions close to the center would obscure the
light of bulge stars. In the GW domain this is completely different - GWs can
probe central parts of the cluster easily. This will significantly enhance the
optical depth. Moreover, the intermediate mass black holes (IMBHs) $M\sim
10^{2}-10^{4}\;M_{\odot}$ are expected to reside in the centers of globular
clusters. Hence, diffractive microlensing could open a new chapter of
searching for IMBHs. One of the candidates of continuous GW signals is a
(slightly deformed) spinning neutron star. Their detection is fairly
difficult, since the signal is strongly modulated by the Earth’s rotation and
orbital motion. Moreover, this modulation is different for every sky position.
Diffraction and interference fringes caused by intervening mass acting as a
lens and moving with respect to the source also produce modulation. Luckily,
the timescales of these effects are significantly different from timescales
involved with the motion of the detector. In the setting discussed by us here,
the directed search with the know position on the sky (in case of globular
cluster lensing) can be applied. The GW signal coming from a rotating NS is so
weak that, in order to detect it in the detector’s noise, one has to analyze
months-long segments of data. Implementation of the F-statistic Jaranowski &
Krolak (2009) consists of coherent searches over two-day periods, and is
followed by a search for coincidences among the candidates from the two-day
segments. The timescales for fringe modulation are larger, and one can expect
that respective amplitude modulation could be detectable with current
techniques. This approach should be extended to other sources of continuous
GWs like binary systems observable in low-frequency bands to be probed by
DECIGO and LISA.

###  3.3 Poisson-Arago spot

One of the observable signature of wave nature of light besides interference
is diffraction. The French physicist Fresnel presented his work on diffraction
in a competition sponsored by the French Academy of Sciences in 1818. It was a
period when Newtons idea of corpuscular theory of light was prevalent, one of
the judges in the competition Poisson in an attempt to disprove the wave
nature of light has pointed out that according to Fresnel’s theory a parallel
beam of light falling on a spherical obstacle would produce a bright spot at
the centre as if the obstacle is not present. This was considered to be an
absurd result, later another French physicist Arago successfully conducted an
experiment and the Poisson spot was seen. This phenomenon later came to be
known as the Poisson’s spot or the Spot of Arago established the wave nature
of light in 19th century. Therefore, the similar phenomenon should be expected
for GWs and strong lensing optical configuration appears to be a perfect
candidate. The reason for the appearence of the spot is that wavefronts
originating at the boundary of circular obstacle interfere constructively. The
same is expected from the Einstein ring of lensed GW. To some extent one can
see analogous behavior in wave GW regime, when the amplification calculated
from the Kirchoff integral (16) has maximal value, when $\beta=0$ Oguri (2019)

$\left|F(w,\beta=0)\right|^{2}=\frac{\pi w}{1-e^{-\pi w}}.$ (29)

An interesting and rigorous study of the Poisson-Arago spot produced when GWs
are passing in the background of a Schwarzschild BH has been presented in
Hongsheng & Xilong (2021). This strong field scenario can be understood as an
on-axis scattering problem using the Regge-Wheeler equationRegge & Wheeler
(1957); Nambu et al. (2019) which describes the axial perturbations of the
Schwarzschild metric as linear approximationFiziev (2006) and the solutions
are given by confluent Heun equation. The amplitude of the gravitational wave
passing in the background of this opaque disc is given by

$\Phi_{l}=\frac{1}{r}\sum_{l=2}^{\infty}(2l+1)e^{-i\omega
t}P_{l}(\cos\theta)R_{ls}(r)$ (30)

where $P_{l}(\cos\theta)$ denotes the Legendre polynomials and $R_{ls}$ is the
$l-th$ partial wave of the GW with spin $s=2$ and angular momentum
$\hbar\sqrt{l(l+1)}$ measured with respect to the scattering center. For a
Schwarzschild BH the gravitational waves with impact parameters $3\sqrt{3}M$
falls into the black hole and will not reach infinity i.e) BH can shield
gravitational waves.$R_{ls}/r$ in (30) is an oscillatory function in the far
zone at which the summation in $l$ in eqn(30) becomes divergent. This issue
can be solved in Hongsheng & Xilong (2021) by using the Fresnel half wave zone
according to which effective contribution to diffraction comes from the $1/4-$
zone and the residues cancel each other. In the case of small scattering angle
and at large distance $R_{ls}/r$ becomes

$\Phi_{l}=e^{-i\omega t}e^{\omega r(1-\frac{\theta^{2}}{4})+\pi
M\omega}\Gamma(1-2i\omega M)J_{0}\Big{(}2\sqrt{\frac{M}{r}\omega
r\theta}\Big{)}$ (31)

The fringe pattern is determined by the function
$J_{0}\Big{(}2\sqrt{\frac{M}{r}\omega r\theta}\Big{)}$. The time delay between
successive bright or dark spots depends on the mass of the BH, wavelength of
the gravitational wave and the relative position of the system. The most
promising obstacle we expect to see Arago spot is the SMBH located at the
centre of our Milky Way galaxy which acts like an opaque disc. For
gravitational wave of frequency 100 Hz the diffraction time delay between
bright spot and the first dark fringe is 3.86 days and the second bright
fringe is 6.2 days. These effects will be observable with the next generation
space based detectors like LISA.

##  4 Conclusions

Gravitational lensing of GW signals has been discussed in many papers since
the first detection Ng et al. (2018). In particular, Broadhurst et al. (2019)
put forward an idea that unexpectedly high masses of BBHs detected by LIGO are
consequences of the signals being lensed. These claims have been subsequently
refuted Collaboration et al. (2021), but the interest in lensed GW remained.
Some teams undertook efforts to monitor ongoing LIGO-Vrigo-KAGRA detections
for the possibility of some signals being lensed by galaxy clusters Smith et
al. (2017) with hope to repeat the Refsdal supernova story Kelly et al.
(2016).

In this paper we have discussed new opportunities of detecting wave effects in
lensed GW signals like the beat patterns (interference) and diffractive
microlensing. The former would allow independent measurement of the lens mass
and the time delay distance purely from the GW signal. The latter would open a
new opportunity to search for IMBHs inside globular clusters and more
generally, study the central parts of globular clusters – a technique
completely inaccessible in the electromagnetic window. The future 3rd
generation of detectors like the Einstein Telescope or Cosmic Explorer is
expected to detect lensed GW signals routinely at the rate of 50-100 events
per year Biesiada et al. (2014); Yang et al. (2019). The prospects are also
bright for the planned space-borne detectors like DECIGO, which will probe an
order of magnitude larger volume of the Universe that the ET. In Piórkowska-
Kurpas et al. (2021) it has been estimated that DECIGO would routinely
register about $50$ lensed GW signals per year. Remarkably, some of these
sources will later enter the band of ground-based detectors. Inspiralling
sources detectable by the DECIGO would exhibit slowly evolving frequency hence
being almost monochromatic and allowing to detect patterns discussed in this
paper. The expected rate of detecting beat patterns by DECIGO has been
assessed in Hou et al. (2021).

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$\sim 100$ MeV) are produced in collisions taking place in high-density
regions of the jet, the peak (sub-MeV to MeV part) of the spectrum is mainly
determined by collisions that dissipate most of the energy. Consequently, the
region where most of the energy is dissipated is representative for the
complete burst in that energy range; on the other hand, it is not
representative for the whole GRB’s neutrino emission, as it would overestimate
the neutrino fluence and underestimate the peak energy.

We conclude that energetic GRBs are promising candidates for multi-messenger
searchers. Baryonic loadings as high as necessary to power UHECRs probably do
not modify the spectral energy distribution as much as to rule out the UHECR
paradigm – in IC-dominated scenarios, these cannot even be uniquely
identified. However, interesting insights can be obtained in SYN-dominated
scenarios from correlated flux enhancements between the the optical-UV to soft
X-rays, and the GeV to TeV gamma-ray bands. Finally, the discovery of a
neutrino from an energetic GRB, such as GRB 221009A, would be the smoking gun
signature for our hypothesis that a class of energetic GRBs could be the
sources of UHECRs.

Note. – During completion of this work GRB 221009A was detected. As this is
likely to become the most prominent example for this class of energetic GRBs,
we discuss the implications of our model for this burst in a companion letter
(Rudolph et al., 2022b).

Acknowledgements

We would like to thank Iftach Sadeh and Marc Klinger for reading the
manuscript and useful comments. M.P. acknowledges support from the MERAC
Fondation through the project THRILL and from the Hellenic Foundation for
Research and Innovation (H.F.R.I.) under the “2nd call for H.F.R.I. Research
Projects to support Faculty members and Researchers” through the project
UNTRAPHOB (Project ID 3013). A.R. was supported by the International
Helmholtz-Weizmann Research School for Multimessenger Astronomy, largely
funded through the Initiative and Networking Fund of the Helmholtz Association
and received funding from the Carlsberg Foundation (CF18-0183).

Software: astropy(Robitaille et al., 2013; Price-Whelan et al., 2018), gammapy
(Deil et al., 2018; Nigro et al., 2019) and Python v3.9.

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##  Appendix A Time evolution in the representative collision

(a)

(b)

Figure 16: Temporal evolution of comoving photon spectra until
2$t^{\prime}_{\mathrm{dyn}}$ for our representative collision, assuming
$t^{\prime}_{\mathrm{inj}}=t^{\prime}_{\mathrm{dyn}}$. We show the result for
the (left) IC-dominated leptonic case and (right) SYN-dominated case with
$f_{p/e}=30$. Black lines indicate the emitted spectra.

For each collision we evolve the system until
$t^{\prime}=t^{\prime}_{\mathrm{inj}}+t^{\prime}_{dyn}$ and calculate the
emitted spectrum integrating over the escaping the photon spectrum. As this
approach differs from the one of past publications (e.g. Bosnjak et al., 2009;
Rudolph et al., 2022a), we illustrate its effect in Figure 16 for the
representative collision of SPE54, showing the IC-dominated leptonic scenario
on the left and the SYN-dominated lepto-hadronic scenario on the right.

Solid lines represent the comoving photon spectrum at different simulation
times $t^{\prime}$, where early (late) times correspond to light (dark) blue
colors. The emitted spectra are shown in black. We first discuss the leptonic
IC-dominated case (left plot). For $t^{\prime}<0.75t^{\prime}_{dyn}$ the
synchrotron peak around $E^{\prime}\simeq 10^{-2}m_{e}c^{2}$ and the inverse
Compton peak at higher energies are clearly visible. At later times,
synchrotron and inverse Compton radiation of secondary leptons enhances the
photon fields at low and intermediate energies. The emitted spectrum encodes
all this information. In principle, the same behaviour is observed for the
photon fields of the lepto-hadronic case (right plot). Here we further notice
how internal $\gamma\gamma$-annihilation suppresses VHE $\pi^{0}$-decay with
increasing simulation time.

The neutrino spectra (that are composed of $\nu_{\mu}$, $\bar{\nu}_{\mu}$,
$\nu_{e}$, and $\bar{\nu}_{e}$) also show an evolution with time. For example,
the peak of the spectrum evolves to lower energies with time. This may be
understood as follows: early on, the spectrum is dominated by $\nu_{\mu}$ from
pion decays. Later, the decay products of muons dominate. However, due to the
muons being subject to synchrotron cooling, those secondaries have lower peak
energies.

Overall our approach equally captures features that are produced early on or
late in the simulation. This may not be true e.g. for a steady-state solution.

##  Appendix B Impact of modeling choices

(a)

(b)

(c)

(d)

Figure 17: Examining the impact of different modeling choices in a leptonic
modeling setup for the single-pulse scenario at the time-integrated spectra
$E_{\mathrm{obs}}\mathcal{F}_{E_{\mathrm{obs}}}$. We show comparisons of (top
left) different injection schemes, (bottom left) different choices for the
fraction of accelerated electrons and (bottom right) different initial Lorentz
factors of the outflow. In all figures, we explore different
$f_{\mathrm{B/e}}$. In the (top right) panel we further show the decomposed
spectrum for $\delta t^{\prime}_{\mathrm{inj}}\rightarrow 0$ and
$f_{\mathrm{B/e}}=10^{-3}$. For the photon indices in the lower panels we
indicate the synchrotron slow- and fast-cooling predictions as well as a
photon index of $-2$ (that marks local maxima/minima of
$E_{\mathrm{obs}}\mathcal{F}_{E_{\mathrm{obs}}}$) as dotted lines.

To complement the main results, we systematically explore the impact of the
following parameter assumptions and modeling choices: (a) The injection
timescale of accelerated particles, (b) the scaling of the number fraction of
accelerated electrons throughout the fireball evolution, (c) the maximum and
minimum (initial) Lorentz factor of the outflow and (d) the magnetic field
strength (set through $f_{\mathrm{B/e}}$). For $f_{\mathrm{B/e}}$ the maximum
(minimum) value considered corresponds to the SYN-dominated (IC-dominated)
scenario presented in the main text. For simplicity we limit ourselves to
leptonic models and show the simulated full-burst spectra examining these
aspects in Figure 17. We will discuss the results in context to the parameter
and model choices generally applied in GRB radiation models and necessary to
reproduce spectra with properties similar to observed ones.

### (a) Injection timescale of accelerated particles

In Bosnjak et al. (2009); Daigne et al. (2011); Bošnjak & Daigne (2014), the
injection timescale of primary electrons (for each single-collision radiation
calculation) was considered to be much smaller than the dynamical timescale.
To scrutinise the implications of different choices of the injection
timescale, we show the simulated spectra for
$t^{\prime}_{\mathrm{inj}}=t^{\prime}_{\mathrm{dyn}}$ and
$t^{\prime}_{\mathrm{inj}}=0.01t^{\prime}_{\mathrm{dyn}}$ (labelled as $\delta
t^{\prime}_{\mathrm{inj}}\rightarrow 0$) in the top left plot Figure 17.

We find that the results for the two injection timescale agree well for strong
magnetic fields (high $f_{\mathrm{B/e}}$). For low $f_{\mathrm{B/e}}$, the
results for $\delta t^{\prime}_{\mathrm{inj}}\rightarrow 0$ differ in three
main aspects: (1) The fluence of at the sub-MeV peak does not change with
respect to the one for high $f_{\mathrm{B/e}}$, (2) the (V)HE component and
low-energy fluences are weaker than for
$t^{\prime}_{\mathrm{inj}}=t^{\prime}_{\mathrm{dyn}}$. and (3) the low-energy
photon index depends systematically on $f_{\mathrm{B/e}}$ (where high
$f_{\mathrm{B/e}}$ yield low photon indices and vice versa, see also Rudolph
et al. (2022a)). On the other hand, the low-energy photon index is almost
independent of $f_{\mathrm{B/e}}$ for
$t^{\prime}_{\mathrm{inj}}=t^{\prime}_{\mathrm{dyn}}$. As these effects appear
for low $f_{\mathrm{B/e}}$, we attribute them to inverse Compton scatterings
that are more efficient in case weak magnetic fields. We relate this to the
three mains points identified before: (1) The higher fluence at the sub-MeV
peak may be explained through the dominating energy loss mechanism of primary
electrons at $\gamma^{\prime}_{\mathrm{e,min}}$: for $\delta
t^{\prime}_{\mathrm{inj}}\rightarrow 0$ energetic primaries are present only
at early $t^{\prime}$. However, at this point the photon spectrum is still
building up and cannot serve as target for inverse Compton scatterings.
Consequently, the efficiency of inverse Compton scatterings is low(er) and
primary electrons at $\gamma^{\prime}_{\mathrm{e,min}}$ predominantly cool
through synchrotron emission also in low $f_{\mathrm{B/e}}$ scenarios. This,
in turn, also means that inverse Compton scatterings are generally less
efficient, which yields lower (V)HE photons and also less contributions from
the secondary cascade. This is further illustrated in the top right plot of
Figure 17, where we show the decomposed spectrum for $\delta
t^{\prime}_{\mathrm{inj}}\rightarrow 0$ and $f_{\mathrm{B/e}}=10^{-3}$:
Despite the fact that the parameter choices correspond to the IC-dominated
scenario, primary synchrotron emission still dominates the observed spectrum.
This implies that the low- and high-energy fluences which are produced by
cascade emission are lower than for
$t^{\prime}_{\mathrm{inj}}=t^{\prime}_{\mathrm{dyn}}$ (point (2) mentioned
before). The lower contribution of the secondary cascade also explains why
harder low-energy photon indices can be achieved: as detailed above, inverse
Compton scatterings in Klein-Nishina regime modify the low-energy slope of the
primary synchrotron such that values up to $-1$ can be achieved. This was also
observed in Daigne et al. (2011), but could not be reproduced by our results
in the main text due to the strong contribution of synchrotron emission of
secondary lepton pairs. Now choosing the same setup as Daigne et al. (2011) we
can however confirm their findings.

We conclude that SYN-dominated models (i.e. with high $f_{\mathrm{B/e}}$) are
robust against variations of the injection timescale of energetic particles,
whereas low-$f_{\mathrm{B/e}}$ scenarios are sensitive to the choice of
$t^{\prime}_{\mathrm{inj}}$. Meanwhile, the findings of past publications of
Daigne et al. (2011) can only be reproduced for the same settings of the
injection timescale.

### (b) Evolution of the number fraction of accelerated electrons
$\zeta_{\mathrm{e}}$

GRB synchrotron models within the internal shock scenario commonly set the
number fraction of accelerated electrons proportional to the dissipated energy
per unit mass
($\zeta_{\mathrm{e}}\propto\varepsilon^{\prime}_{\mathrm{diss}}$). The
intention is to keep the minimum Lorentz factor of electrons
$\gamma^{\prime}_{\mathrm{e,min}}$ constant throughout the fireball evolution
to reproduce the correct spectral evolution. Additionally, it results in
narrow(er) spectra. In the lower left panel of Figure 17, we show the results
for $\zeta_{e}\propto\varepsilon^{\prime}_{\mathrm{diss}}$ and
$\zeta_{\mathrm{e}}=\mathrm{cst.}$ (with our benchmark choice of
$t^{\prime}_{\mathrm{inj}}=t^{\prime}_{\mathrm{dyn}}$ and setting the constant
$\zeta_{e}$ such that the same peak energy is reproduced). Independent of
$f_{\mathrm{B/e}}$ the simulations with $\zeta_{\mathrm{e}}=\mathrm{cst.}$ are
different in two main points. First, the synchrotron peak is broadened and
reduced in energy fluence. Second, the low-energy and high-energy fluences are
enhanced. The broad synchrotron peak is reflected in the photon indexes, which
are relatively flat around the peak. With values that are overall below
$-1.5$, it is questionable whether $\zeta_{\mathrm{e}}=\mathrm{cst.}$ can
produce narrow spectra compatible with observations.

We conclude that
$\zeta_{\mathrm{e}}\propto\varepsilon^{\prime}_{\mathrm{diss}}$ (which yields
a constant minimum Lorentz factor of electrons
$\gamma^{\prime}_{\mathrm{e,min}}$ throughout the fireball evolution) is
necessary to reproduce spectra compatible with observed ones within a multi-
collision model. We point out that this is a somewhat fine-tuned choice and
should be confirmed by simulations of particle acceleration in (mildly)
relativistic shocks for a broad range of shock strengths and comoving energy
densities.

### (c) Initial jet Lorentz factor

Next, we examine the impact of the jet Lorentz factor. We choose three
different initial Lorentz factor distributions, characterised by their minimal
and maximal $\Gamma$:

$(\Gamma_{\mathrm{min}}^{\mathrm{inital}},\Gamma_{\mathrm{max}}^{\mathrm{inital}})\in\\{(100,500),(200,700),(400,900)\\}\,.$

For the sake of simplicity, the lower right panel of Figure 17 only shows the
results for $f_{\mathrm{B/e}}=1$ and $f_{\mathrm{B/e}}=10^{-3}$ (i.e. the SYN-
and IC-dominated scenario introduced in the main text). For all cases we
adjusted $\zeta_{\mathrm{e,0}}$ such that the peak energy remains the same. We
find that for the SYN-dominated scenario ($f_{\mathrm{B/e}}=1$), the results
are robust against variations of the initial Lorentz factor distribution.
However, the IC-dominated scenario of $f_{\mathrm{B/e}}=10^{-3}$ shows strong
dependence on the Lorentz factor distribution: the lower the Lorentz factor
is, the stronger the distortions of the spectrum are. A natural explanation
for this behaviour is the impact of lepton pairs produced in
$\gamma\gamma$-annihilation, which are produced in dense environments and
distort the spectrum by synchrotron radiation at low energies and inverse
Compton radiation at high energies. For low Lorentz factors, the comoving
densities are higher (due to smaller collision radii and a smaller effect of
boosting). Consequently, the $\gamma\gamma$-annihilation efficiency is
increased and a large number of pairs reshape the spectrum.

We point out that the initial jet Lorentz factors are only one way of altering
the comoving (energy) densities. Other parameters would be the overall
duration of the burst (short durations yield small collision radii and hence
large densities) or the isotropic equivalent energy (large
$E_{\mathrm{\gamma,iso}}$ induce large densities).

### (d) Magnetic field strength

We proceed by reviewing the impact of the magnetic field strength (set through
$f_{\mathrm{B/e}}$), which we varied for all scenarios presented above. The
most extreme choices presented as blue and yellow lines in Figure 17
correspond to the SYN- and IC-dominated scenario presented in the main text.
Varying $f_{\mathrm{B/e}}$ systematically between these two values we observe
a smooth transition between the two scenarios: the features of the IC-
dominated scenario such as an increase in low- and high-energy fluence become
more pronounced as $f_{\mathrm{B/e}}$ is decreased. As outlined above, for
$\delta t^{\prime}_{\mathrm{inject}}\rightarrow 0$ also the low-energy photon
index depends systematically on $f_{\mathrm{B/e}}$, where low (high)
$f_{\mathrm{B/e}}$ yield hard (soft) photon indices. This however cannot be
confirmed in scenarios with a longer injection timescale.

##  Appendix C Pion and muon cooling and impact on neutrino spectra

In this section we study the cooling of intermediate pions and muons and its
impact on neutrino spectra. We account for synchrotron and adiabatic cooling
and decay. For a particle of Lorentz factor $\gamma^{\prime}$ and species $i$
(with corresponding mass $m_{i}$, charge number $Z_{i}$ and rest-frame
lifetime $\tau_{0,i}$ ) the corresponding cooling/decay times in the plasma
rest frame are

$\displaystyle t^{\prime}_{\mathrm{sy,i}}(\gamma^{\prime})$ $\displaystyle=$
$\displaystyle\frac{6\pi}{Z_{i}^{4}\sigma_{t}}\frac{m_{i}c^{2}}{c}\left(\frac{m_{i}}{m_{e}}\right)^{2}\frac{1}{B^{\prime
2}\gamma^{\prime}}$ (C1) $\displaystyle t^{\prime}_{\mathrm{ad,i}}$
$\displaystyle=$ $\displaystyle t^{\prime}_{\mathrm{ad}}$ (C2) $\displaystyle
t^{\prime}_{\mathrm{dec,i}}(\gamma^{\prime})$ $\displaystyle=$
$\displaystyle\tau_{0,i}\gamma^{\prime}\,,$ (C3)

with corresponding cooling rates
$t^{\prime-1}_{\mathrm{sy,i}}(\gamma^{\prime})$,
$t^{\prime-1}_{\mathrm{ad,i}}$ and
$t^{\prime-1}_{\mathrm{dec,i}}(\gamma^{\prime})$.

###  C.1 Pion and muon cooling in the representative collision and
identification of cooling regimes

(a)

(b)

Figure 18: Pion and muon loss rates $t^{\prime-1}_{\mathrm{cool}}$ for the
representative collision, for (top) the SYN-dominated scenario and (bottom)
the IC-dominated scenario; Complemented by the corresponding evolution of
(comoving) maximal proton energies obtained for the radiative calculations
until $t^{\prime}_{\mathrm{dyn}}$.

We study the cooling of pions and muons first at the example of representative
collision. While adiabatic cooling and decay rates are independent of the
magnetic field, synchrotron cooling scales with $B^{\prime 2}$. Similar to the
results in the main text we again study the SYN- and IC-dominated scenarios.

The cooling timescales for pions and muons are shown in the upper panel of
Figure 18, calculated from Eq. C1 \- Eq. C3. For the SYN-dominated scenario,
adiabatic cooling is sub-dominant at all energies. The longer decay time of
muons shifts their critical energy $E_{\mathrm{c,sy/dec}}$ to lower energies
with respect to pions. For the IC-dominated scenario, synchrotron cooling is
less efficient. For pions however either synchrotron cooling or decay
domiantes while for muons adiabatic cooling dominates at intermediate
energies.

The figure further displays the maximum proton energies as a function of
$t^{\prime}/t^{\prime}_{\mathrm{dyn}}$ (recall that the injection time was set
to $t^{\prime}_{\mathrm{inj}}=t^{\prime}_{\mathrm{dyn}}$. For the SYN-
dominated scenario the proton energies are initially limited by adiabatic
cooling and by photo-pion production at later $t^{\prime}$. This implies that
the maximum proton energies consecutively move to lower values with evolving
$t^{\prime}$ as photo-pion production gets more efficient with the building up
of the photon fields. For the IC-dominated case, the maximum proton energy is
dominated by adiabatic cooling until almost $t^{\prime}_{\mathrm{dyn}}$ and
remains almost constant. We compare the maximum proton energies to the pion
and muon cooling rates: For the SYN-dominated scenario, secondaries are
produced at high enough energies to be able to cool prior to their decay. In
contrast to this, the pions produced in the IC-dominated scenario decay
without prior cooling. The secondary muons cool adiabatically before decaying.

We label the energies at which the timescales of two different processes are
equal as critical energies. These mark the intersections of timescale/ cooling
rates in the respective plots. Three processes imply three critical energies:
$E_{\mathrm{c,ad/dec}}$ (where
$t^{\prime}_{\mathrm{ad}}=t^{\prime}_{\mathrm{dec}}$), $E_{\mathrm{c,sy/dec}}$
(where $t^{\prime}_{\mathrm{sy}}=t^{\prime}_{\mathrm{dec}}$) and
$E_{\mathrm{c,ad/sy}}$ (where
$t^{\prime}_{\mathrm{ad}}=t^{\prime}_{\mathrm{sy}}$).

As the decay rate decreases with particle energy, it can be assumed to be the
dominating loss process at the lowest energies. Depending on the relative
importance of the three processes, two scenarios are possible and can be
identified in Figure 18:

  1. (1)

$E_{\mathrm{c,ad/dec}}<E_{\mathrm{c,ad/sy}}<E_{\mathrm{c,sy/dec}}$ : At the
highest energies synchrotron cooling dominates, at intermediate ones adiabatic
cooling and at the lowest decay. It is realised for muons in the IC-dominated
(right) plot of Figure 18.

  2. (2)

$E_{\mathrm{c,ad/sy}}<E_{\mathrm{c,sy/dec}}<E_{\mathrm{c,ad/dec}}$: Adiabatic
cooling is subdominant for all energies. At high energies synchrotron losses
dominate, at low ones decay. It is nicely illustrated for muons and pions in
the SYN-dominated (left) plot of Figure 18.

###  C.2 Cooling regimes throughout the fireball dynamical evolution

(a)

(b)

Figure 19: Critical energies $E^{\prime}_{\mathrm{c}}$ (see Eq. C4) for pions
and muons. The number indicates whether above the critical energy (1)
adiabatic or (2) synchrotron cooling dominates. For reference we further
indicate the maximal proton energies obtained by balancing acceleration with
adiabatic and synchrotron losses (in contrast to Figure 18 where also photo-
pion and photo-pair production were taken into account). The dashed line
indicates the radius of the representative collision.

From the representative collision we move to the complete fireball dynamical
evolution. We define the critical energy as

$E^{\prime}_{\mathrm{c}}=\mathrm{min}\\{E^{\prime}_{\mathrm{c,sy/dec}},E^{\prime}_{\mathrm{c,ad/dec}}\\}\,.$
(C4)

It may be re-phrased as energy below which particles do not cool prior to
their decay. The evolution of critical energy as a function of collision
radius for our educative example is shown in Figure 19. The color-coding
further indicates if above the critical energy, adiabatic or synchrotron
cooling dominate: Dark red/green means that adiabatic cooling dominates, for
light red/green synchrotron cooling is dominant above the critical energy.
Thus, light (dark) colors correspond to scenario (1) ((2)) introduced before.
For comparison we further indicate the maximal proton energy calculated by
balancing synchrotron and adiabatic losses with acceleration. We point out
that this is rather an upper limit on $E^{\prime}_{\mathrm{prot,max}}$, since
photo-pair or photo-pion production may further reduce the maximal energy
attainable.

For the SYN-dominated scenario, synchrotron cooling dominates for both pions
and muons throughout the full fireball evolution. Further, the critical
energies lie several orders of magnitude below $E_{\mathrm{p,max}}^{\prime}$.
In this scenario we thus expect strong impact of pion and muon cooling. For
the IC-dominated case, pion critical energies are dominated partially by
adiabatic, partially by synchrotron cooling. Muon critical energies are
dominated by adiabatic cooling throughout the complete fireball evolution. In
this scenario, the maximum proton energies lie below the pion critical
energies for all collisions. Pions will thus not cool before decaying. Muon
critical energies are below $E_{\mathrm{p,max}}^{\prime}$ for some part of the
fireball evolution, the difference in energy is however one order of magnitude
at most.

We conclude that synchrotron cooling of secondary pions and muons will reduce
their energies prior to their decay for high magnetic fields (high
$f_{\mathrm{B/e}}$). As this introduces a spectral break, it re-shapes the
secondaries such as lepton pairs and neutrinos. For low magnetic fields (low
$f_{\mathrm{B/e}}$) on the other hand, the maximal proton energies are too low
to enable pion cooling before their decay. Muons may cool adiabatically before
their decay. Overall, the cooling effects of pions and muons will much less
impact lepton and neutrino spectra.

###  C.3 Impact on neutrino spectra

Figure 20: Evolution of comoving photon and neutrino spectra until
$2t^{\prime}_{\mathrm{dyn}}$ for the representative collision and the SYN-
dominated scenario.

Finally, we review the impact of muon and pion cooling on neutrino spectra. We
found that for the SYN-dominated scenario the neutrino spectra peak at much
lower energies compared to the photons produced in $\pi^{0}$ decays. For the
IC-dominated scenario, neutrino and $\pi^{0}$-decay spectra extended to
approximately the same energies. In the last section we deduced that for the
SYN-dominated scenario, intermediate pions and muons are subject to
synchrotron cooling prior to their decay. To validate that this explains the
difference between neutrino and photon spectra, we perform simulations
with/without pion and muon cooling for the representative collision in the
SYN-dominated scenario.

We show the evolution of comoving photon and neutrino spectra in the left plot
of Figure 20. Throughout the complete evolution, the neutrino spectra peak at
lower energies than the HE photons from $\pi^{0}$ decays. The right plot shows
the evolution of neutrino spectra (1) with full treatment of pions/muon, (2)
without pion and muon cooling and (3) assuming an infinitesimally small decay
time of muons and pions (which implicitly means that pion and muon cooling are
not accounted for).

For the first snapshot at $t^{\prime}=0.2t^{\prime}_{\mathrm{dyn}}$, the
scenarios with/ without pion and muon cooling are little different and do not
extend to the highest energies. If however, pions and muons are assumed to
decay instantaneously (dotted lines in the right plot), energies comparable to
the photons from $\pi^{0}$-decays are reached. This means that at early times
$t^{\prime}$ the maximal energies of neutrinos are limited by the longer decay
times of high-energy pions and muons (recall the dependency of the decay time
on particle Lorentz factor, Eq. C3). As the system evolves, the decay time
even of high-energy particles gets smaller than $t^{\prime}$ and the dash-
dotted and dotted lines in Figure 20 extend to similarly high energies. We
point out that the effect of the decay timescale is not as relevant if pion
and muon cooling are accounted for: By cooling down, particles move to shorter
decay times.

Overall, Figure 20 confirms that the lower peak energies of neutrino spectra
for a SYN-dominated scenario can be attributed to cooling effects of
intermediate pions and muons.