# GRAVITATIONAL SELF-LENSING OF FAST RADIO BURSTS IN NEUTRON STAR
MAGNETOSPHERES:
I. THE MODEL

Simone Dall’Osso Istituto Nazionale di Fisica Nucleare - Roma 1, Piazzale Aldo
Moro 2, 00185, Roma, Italy  GSC - Goethe University of Frankfurt, Max von Laue
Strasse 1, Frankfurt, Germany  Dipartimento di Fisica, Università La Sapienza,
Piazzale Aldo Moro 5, 00185 Roma, Italy  Riccardo La Placa INAF – Osservatorio
Astronomico di Roma, via Frascati 33, I-00078, Monte Porzio Catone, Italy
Dipartimento di Fisica, Università La Sapienza, Piazzale Aldo Moro 5, 00185
Roma, Italy  Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via
della Ricerca Scientifica, 00133 Roma, Italy  Luigi Stella INAF – Osservatorio
Astronomico di Roma, via Frascati 33, I-00078, Monte Porzio Catone, Italy
Pavel Bakala Research Centre for Computational Physics and Data Processing,
Silesian University in Opava, Bezručovo nám. 13, CZ-746 01, Opava, Czech
Republic  M. R. Štefánik Observatory and Planetarium, Sládkovičova 41, 920 01
Hlohovec, Slovak Republic  Andrea Possenti INAF – Osservatorio Astronomico di
Cagliari, Via della Scienza 5, 09047 Selargius, Italy

###### Abstract

Fast Radio Bursts (FRBs) are cosmological sub-second bursts of coherent radio
emission, whose source is still unknown. To date, the galactic magnetar SGR
1935+2154 is the only astrophysical object known to emit radio bursts akin to
FRBs, albeit less powerful, supporting suggestions that FRBs originate from
magnetars. Many remarkable properties of FRBs, e.g. the dichotomy between
repeaters and one-off sources, and their power-law energy distributions (with
typical index $\sim 2-3$), are not well understood yet. Moreover, the huge
radio power released by the most active repeaters is challenging even for the
magnetic energy reservoir of magnetars. Here we assume that FRBs originate
from co-rotating hot-spots anchored in neutron star magnetospheres and get
occasionally amplified by large factors via gravitational self-lensing in the
strong NS field. We evaluate the probability of amplification and show that
(i) a power-law energy distribution of events $\propto E^{-(2-3)}$ is
generally expected, (ii) all FRB sources may be regarded as repeating, their
appearance as one-off sources or repeaters being determined by the critical
dependence of the amplification probability on the emission geometry and
source orientation relative to Earth and (iii) the most active repeaters, in
particular, correspond to extremely rare and finely-tuned orientations ($\sim$
one in $10^{6}$), leading to large probabilities of amplification which make
their bursts frequently detectable. At the same time, their power release
appears enhanced, typically by factors $\gtrsim 10$, easing their energy
budget problem.

Fast radio bursts – Neutron stars: magnetars – Gravitational lensing –
Pulsars: radio emission

##  1 Introduction

Fast Radio Bursts (FRBs) are bright, sub-second transients in the radio sky,
with extreme brightness temperatures indicative of a coherent emission
mechanism. Since the earliest discoveries, their dispersion measures (DMs) and
dispersive patterns suggested an extragalactic origin (see, e.g., Cordes &
Chatterjee, 2019; Petroff et al., 2022), which was later confirmed by the
identification of host galaxies and redshift measurements in a few FRBs. Still
eluding us after 15 years of debate, the source of these events must explain
many peculiar observational properties, first among which are the huge
luminosities and energies of the brightest FRBs: many events with a precisely
determined distance reveal isotropic burst energies $E_{\rm
iso}>10^{39}-10^{41}$ erg, with characteristic millisecond durations.

A second crucial feature of FRBs is that some of them are observed as
repeating sources: since the second detection of FRB 20121102A in 2015
(Spitler et al., 2014, 2016), over fifty repeaters have been identified111See,
e.g., the CHIME repeater catalogue at https://www.chime-
frb.ca/repeater_catalog.. This rules out the possibility that all FRBs are
produced in catastrophic events, and leads to different scenarios invoking
either two distinct populations for one-off and repeating FRBs, or a single
population with a very wide range of emission rates and energy budgets.

The number of known FRBs as a function of $E_{\rm iso}$ appears to follow a
power-law distribution $\propto E_{\rm iso}^{-\alpha}$, with a cut-off above
$\sim 10^{41}$ erg and $\alpha\sim 2-3$ (e.g. James et al. 2022; Shin et al.
2023, Lu et al. 2020, Gourdji et al. 2019). Two well-studied repeaters with
very large event rate ($\gtrsim 30$ hr-1 during active phases) show bimodal
energy distributions, with the high-energy parts following similar power-
laws222The low-energy part resembles a log-normal; the full energy
distribution will be addressed in an accompanying paper (La Placa et al., in
preparation). (Li et al., 2021; Zhang et al., 2022). Observational evidence
led several authors to propose that (at least some) FRB sources are related to
a cosmic population of magnetars (e.g. Popov & Postnov, 2010; Lyutikov et al.,
2016; Beloborodov, 2017; Katz, 2017; Margalit & Metzger, 2018; Petroff et al.,
2022). This was later supported by the detection of FRB-like events from the
galactic magnetar SGR 1935+215 (CHIME/FRB Collaboration et al., 2020; Bochenek
et al., 2020; Dong & Chime/Frb Collaboration, 2022; Lu et al., 2020; Giri et
al., 2023), The coherent radio emission might be produced inside the
magnetosphere (e.g. Lyutikov et al. 2016; Lyutikov 2017; Lyutikov & Rafat
2019; Lu et al. 2022, Zhu et al. 2023). In fact, it has been shown (Lyutikov,
2021a, b; Qu et al., 2022; Lyutikov, 2023) that FRB radio waves can travel
through neutron star (NS) magnetospheres suffering little/no absorption in a
super-strong $B$-field, or if they travel (i) at a small angle ($\lesssim
0.1-0.3$ rad) relative to the local B-field and/or (ii) in a relativistically-
moving plasma ($\Gamma\gtrsim 10$). Both conditions can be met on the open
field-lines of the magnetosphere, regardless of the B-field strength. Within
the closed field-line region, where neither condition is met, a strong B-field
may prevent absorption only up to a maximal distance $R_{\rm abs}\approx
3.5\times 10^{8}$ cm $\mu^{1/2}_{33}/L^{1/4}_{42}$, due to its steep radial
decline. Here $L$ is the isotropic-equivalent luminosity333In this work we
adopt the shorthand $Q_{x}=Q/10^{x}$ for various quantities expressed in cgs
units, unless otherwise stated. of the FRB wave and $\mu$ the NS magnetic
moment (Beloborodov, 2021). Waves travelling through the closed-field-lines
may thus be absorbed at large distances from the NS, provided the spin period
is long enough ($P>0.1$ s) for the closed field-line region to extend beyond
$R_{\rm abs}$ (cf. Lyutikov, 2023). Such factors may introduce a natural
selection on the orientation and properties of observable FRB
sources444Additionally, it can lead to a frequency-dependent temporal
broadening of FRB pulses (Sobacchi et al., 2022)..

In this work, we argue that the large energies and power-law energy
distribution of bright FRBs admit a straightforward interpretation, in terms
of gravitational (self-)lensing of flares produced in the magnetospheres of
highly magnetic NSs, at a small distance from the surface. At the same time,
this scenario naturally predicts the existence of repeaters and of one-off
FRBs, as a result of geometry/orientation effects in a single population of
otherwise homogeneous sources. Our scenario is agnostic about the emission
mechanism; it only requires a magnetospheric origin.

##  2 Preliminary remarks

Most applications of gravitational lensing make use of the weak deflection
limit (WDL), owing to the characteristically large distance, $R$, between
celestial bodies (e.g. Albert et al., 2006; Jetzer et al., 2010). The lens
amplification $a$ rises sharply as the source gets closer to the caustic line,
by which we refer to the continuation of the line-of-sight behind the lens,
with respect to the observer. Here, instead, we consider lensing in the
vicinity of an NS, as approximated by a Schwarzschild lens. Since the emitting
source lies close to the lensing body and the latter is a compact object,
strong field gravity effects will come into play ($R\lesssim
200GM/c^{2}=200R_{g}$; Bakala et al., 2023); we denote the regimes of large
and small angular separations ($\theta$) from the caustic line, respectively,
as weak lensing regime (WLR) and extreme lensing regime (ELR).

In the ELR the amplification scales roughly as $a\propto\theta^{-1}$ and, for
point-like sources, it is only limited by diffraction to a maximum
amplification555This expression holds exactly in the WDL and to within one
order of magnitude in the ELR (see, e.g. Deguchi & Watson, 1986a, b; Nambu,
2013). $a_{\rm diff}\simeq 4\pi^{2}R_{g}/\lambda\approx 4\times
10^{5}(M/M_{\odot})\nu_{\rm GHz}$ for radiation at wavelength $\lambda$, or
frequency $\nu$ (in GHz), and a lens of mass $M$. With finite-size sources, of
linear scale $\ell$ and angular size $\theta_{s}=\ell/R$ relative to the
center of the lens, the amplification saturates as the angular separation is
reduced below $\theta_{s}$. In these cases, the source finite size may become
the main factor limiting the amplifications to values lower than $a_{\rm
diff}$.

The amplification provided by NS self-lensing naturally reduces the energy
requirements of bright FRBs without invoking highly relativistic motion.
Moreover, as we will show, it offers a key to explaining the dichotomy between
repeaters and one-off FRBs. Indeed, under the assumption that the seed events
occur, in each source, at a fixed magnetospheric location co-rotating with the
NS, we find that orientation/viewing angle effects naturally lead to such a
dichotomy, even for a source population made entirely of identical repeaters.
This stems from the fact that in a random sample most orientations have
extremely small probabilities of achieving large amplifications, while at the
same time there exists a minute range of finely-tuned orientations which
guarantee large amplification probabilities. In this picture, self-lensing can
also ease problems with the (time-integrated) energy budget of the most
frequent repeaters, e.g. FRB 20121122A and FRB 20201124A, which are known to
challenge even the energy reservoir of magnetars magnetospheres (Li et al.
2021, Zhang et al. 2022).

Gravitational self-lensing naturally produces a truncated power-law
probability distribution of the amplifications, $P(a)\propto a^{-(2-3)}$, with
the index determined by the location of the emitting source in the
magnetosphere. Accordingly, the same power-law will shape the energy
distribution of the amplified events, (almost) regardless of the seed energy
distribution. The latter may become apparent in the low energy tail of the
observed distribution, if it happens to be above our detection threshold.
Interestingly,

Figure 1: Left Panel: The amplification $a=\mu g^{4}$ vs. angle $\theta$ from
the caustic line, for a 1.4$M_{\odot}$ NS and (i) a point-like source at an
emission radius $r=10$ (green, dashed), (ii) a finite-size source of linear
size $\ell=30$ cm at $r=10$ (blue), (iii) the same finite-size source at
$r=50$ (orange). The dashed green and solid blue curves overlap at an angular
distance $\gtrsim\theta_{s}=\ell/r$, where the source size plays no role. The
orange curve reaches larger amplifications than the blue one because
$\theta_{s}$ is smaller at larger radii, hence the source gets closer to the
caustic line before its size becomes relevant. Right Panel: Probability
distribution $P(a)$ for each curve of the left panel. The peaks at low
amplification ($a<1$, due to gravitational redshift) indicate the large
probability for random bursts to occur far from the caustic line. Such peaks
occur, in the blue and orange curves, at $a\approx 0.65$ and 0.95,
respectively. For reference, the blue and orange curves have probabilities
$\lesssim 0.2$% and $\lesssim 0.07\%$, respectively, of having $a>5$.

the rare Giant Pulses (GPs) observed in some radio pulsars also show a similar
power-law energy distribution (e.g. McKee et al., 2019; Bilous et al., 2022;
Lin & van Kerkwijk, 2023), in contrast to typical pulses which generally
follow broad log-normal distributions (Cairns et al., 2001; Burke-Spolaor et
al., 2012; Mickaliger et al., 2018): a possible link between FRBs and GPs was
already proposed by Cordes & Wasserman (2016) and Connor et al. (2016),
motivated by their very large brightness temperatures.

This is the plan of the paper: sec. 3 introduces general features of
gravitational self-lensing that will leave an imprint on FRB observed
properties, presents the weak and extreme lensing regimes, and calculate the
overall effect of lensing when the whole NS magnetosphere can produce
localised bursts at random times. Sec. 4 extends this calculation to a
different configuration, assuming a small magnetospheric active region, stably
anchored to the NS, and compares the results. Sec. 5 describes the differences
between the observed (apparent) and actual source properties that are caused
by gravitational self-lensing. In sec. 6 we outline the main astrophysical
implications of our model, and sec. 7 shows that a subset of the cosmic
population of NS can account well for the observed number of FRBs, both
repeating and non-repeating.

##  3 Probability distribution of amplifications in active magnetospheres

In our proposed scenario, FRB emission is generated by a small hot-spot in the
NS magnetosphere, at a small distance ($R<100R_{g}$) from the star. The
emission may be beamed by a factor $f_{b}\ll 1$, in a direction assumed to be
random at each event (more details in Appendix B). If the emission occurs
close to the caustic line, and beamed in a favorable direction, it gets
amplified by a large factor $a$, which greatly enhances its detectability. If
little or no amplification takes place, though, FRB detection is hampered, and
only occurs if the intrinsic luminosity of the event is large enough. The
bolometric amplification $a$, results from the geometrical magnification given
by gravitational lensing, $\mu=\left|(\beta/\theta)d\beta/d\theta\right|$,
modified by the energy decrease due to the NS gravitational redshift,
$g^{4}=(1-2R_{g}/R)^{2}$. Here $\theta$ and $\beta$ indicate, respectively,
the angular distance of the source, and of its image in the sky, from the
caustic. For a Schwarzschild lens, the amplification of a point-like source in
the ELR scales with $\theta$, and with the radial distance from the star, $r$,
as (Bakala et al., 2023)

$a(r,\theta)=2\displaystyle\frac{\left[1-\displaystyle\left(2/r\right)^{0.85}\right]^{1.35}}{r^{1/2}\theta}\equiv
2~{}\displaystyle\frac{f(r)}{r\theta}\,,$ (1)

where $r=Rc^{2}/GM=R/R_{g}$ is dimensionless, $R$ being the distance in
physical units and $R_{g}$ the gravitational radius, and
$f(r)=r^{1/2}\left[1-\displaystyle\left(2/r\right)^{0.85}\right]^{1.35}$.

At large angular separations the WLR is reached, in which $a$ has a much
flatter scaling with $\theta$. In order to derive a simple, analytical
approximation that could incorporate all lensing regimes, we generalized eq.
(1) as

$a(r,\theta)=2~{}\displaystyle\frac{f(r)}{r\theta}~{}\left[1+\displaystyle\left(\frac{\theta}{\theta_{\rm
we}}\right)^{S}\right]^{1/S}\,,$ (2)

Here $\theta_{\rm we}$ and $S$ depend on $r$ and describe, respectively, the
angle at which the transition from extreme to weak lensing occurs, and the
sharpness of that transition. They were derived by approximating the exact
expression for $a$ as described in appendix A.

With these provisions eq. (2) gives an accurate fit to the full solution of
the Schwarzschild lens equation in any regime, while the much simpler eq. (1)
gives the correct limit in the ELR (Fig. 1). The above definitions allow us to
calculate the probability distribution for $a$, under differerent assumptions
for the geometry of the emission region.

###  3.1 Point-like sources

The simplest possible situation is that of point-like sources going off at any
time and at random positions in the NS magnetosphere, up to some maximum
radial distance $R_{\rm max}$. The probability that their emission gets
amplified by a factor $a$ will depend on the ratio between the volume within
which that particular $a$-value can be achieved, and the whole available
volume (i.e. the region with $R\leq R_{\rm max}$). For given $R$ and $\theta$
the volume element within $dR$ and $d\theta$ is $dV(\theta,R)=2\pi
R^{2}\sin\theta d\theta dR$, from which the probability distribution for $a$
in the spherical shell between $R$ and $R+dR$ is obtained

$\displaystyle P(a;r)da$ $\displaystyle=$
$\displaystyle\displaystyle\frac{dV[\theta(a)]}{4\pi
R^{2}dR}=\frac{\sin\theta\left(d\theta/da\right)}{2}~{}da$ (3)
$\displaystyle\simeq$
$\displaystyle\displaystyle\frac{f(r)\sin\left[2f(r)/a\right]}{a^{2}}~{}da\,.$

The last step in eq. (3) has the well-know form $P(a)\propto a^{-3}$: it is
valid in the ELR, where $d\theta/da$ is given by eq. (1). In practice this
means that, e.g., at $r=10$ the $a^{-3}$ scaling holds for
$\theta\leq\theta_{\rm we}\approx\pi/6$, corresponding to $a\gtrsim 0.93$. At
larger $\theta$, hence lower amplification, the marked flattening of the
scaling ($a\sim$ const.) in the WLR leads instead to a pronounced steepening
of $P(a)$ at low amplifications (Fig. 1, right panel).

Finally, by integrating eq. (3) over the volume of the active magnetosphere,
the global $P(a)$ is obtained. To this aim, we assumed that emission may occur
at different radii following a probability distribution $P(r)\propto r^{-k}$,
and set: (i) the inner edge at $r_{\rm in}\approx 10$, to avoid the inner
region likely occulted by the NS (see Appendix A), (ii) the outer edge at
$r_{\rm max}=50$. Note that the result depends only weakly on the value of the
outer edge, owing to the fast decrease of the lensing probability with growing
distance from the lens. Regardless of the value of $k$, we find that $P(a)$
maintains the $a^{-3}$ scaling at large amplifications (Fig. 2, left panel),
while the radial extension of the active magnetosphere only affects the low-
amplification end (see also next sub-section). This too is a result of the
fast decrease of the lensing probability with growing $r$, which gives a
growing fraction of weakly or non-amplified events.

###  3.2 Finite-size sources

The finite dimension ($\ell$) of actual sources will of course limit the
maximum achievable amplification. In particular, $a$ is expected to saturate
when the angular separation becomes comparable to, or smaller than, the source
angular size relative to the lens, i.e.
$\theta\lesssim\theta_{s}\sim~{}\ell/R$. We parameterize this effect as

$a_{\theta_{s}}(r,\theta)=\frac{2f(r)}{r\theta_{s}\left[1+\left(\displaystyle\frac{\theta}{\theta_{s}}\right)^{p}\right]^{1/p}}\,.$
(4)

The saturation occurs in the ELR, as long as the emitting region is relatively
small ($\ell<0.1R$). Hence, we set the smoothness parameter $p=3$ in order to
match eq. (23) in Bakala et al. (2023), which is exact for finite-size sources
in the ELR. We incorporated this saturation by adding the square parenthesis
of eq. (4) in the denominator of eq. (2) as detailed in Appendix A. The
general relation $a(\theta)$ for finite-size sources thus derived can be
roughly described as a double broken power-law (see, e.g., Fig. 1, left
panel). At small angular separations, $\theta<\theta_{s}$, a flat segment with
$a\simeq a_{\rm max}=2f(r)/(r\theta_{s})$ is followed, in the range
$\theta_{s}<\theta<\theta_{\rm we}$, by a power-law decay
$a\propto\theta^{-1/3}$. After the transition to the weak WLR, at
$\theta\geq\theta_{\rm we}$, it turns to another shallow segment where
$a\rightarrow{\rm const}$ as $\theta\rightarrow\pi$. The right panel of of
Fig. 1 illustrates the $P(a)$ distributions corresponding to each of the cases
shown in the left panel (see caption for details).

Figure 2: Left: The amplification probability distribution, $P(a)$, for point-
like sources in an active magnetosphere, in which events may occur at any
radius in the range $(10-50)R_{g}$, with probability $P(r)\propto r^{-k}$
(blue, $k=3$; orange, $k=-2$). The $P(a)$ at a fixed $r=10$ (green, dashed) is
shown for reference. The radial distribution of the events affects mostly the
low-amplification peak, which receives contributions from individual peaks at
different radii, with different relative weights (depending on $k$). The inset
also shows the other extreme of $P(a)$ at $r=50$ (red dashed). At large
amplifications the $P(a)\propto a^{-3}$ scaling is unaffected for any $k$,
besides a tiny decrease in the normalization. Right: same as left panel, but
for finite-size sources ($\ell=30$ cm). The effect at low-amplifications is
identical (for the same $k$-value), but the source finite-size may leave an
additional imprint: if $k>0$, the high-amplification end of the distribution
steepens to $P(a)\propto a^{-(3+k)}$, beyond $a_{\rm max}(r=10)$ (shown by the
green dotted curve). The steeper power-law cuts off sharply at $a_{\rm
max}(r=50)\approx 8.9\times 10^{4}$
($M/1.4M_{\odot}$)${}^{1/2}/(\ell/30~{}{\rm cm}$), the maximum possible
amplification. If $k<0$, though, the slope in the high-amplification end
remains unchanged all the way to $a_{\rm max}(r=50)$.

Moreover, we calculated the $P(a)$ resulting from allowing the hotspot, with
fixed size $\ell$ and colatitude $\xi$, to go off at any radius in the range
$r=10-50$ with probability $P(r)\propto r^{-k}$. Results are depicted in Fig.
2, right panel, for two values of $k=-2,3$.

Contrary to point-like sources, the finite size $\ell$ introduces new features
in the $P(a)$, due to the $r$-dependence of the maximum amplification, $a_{\rm
max}(r)=2f(r)/r\theta_{s}(r)$. First, the $P(a)$ spans a wider range extending
to $a_{\rm max}(50)>a_{\rm max}(10)$. Second, the scaling $\propto a^{-3}$ is
preserved only up to $a_{\rm max}(10)$. Larger amplifications can only be
achieved at larger radii, thus the corresponding section in the $P(a)$ is, in
general, sensitive to the shape of $P(r)$. However, only $k>0$ determines a
steepening to $P(a)\propto a^{-(3+k)}$ above $a_{\rm max}(10)$. The $a^{-3}$
slope is preserved, instead, if $k<0$. Eventually, a cut-off at $a_{\rm
max}(r=50)$ appears, as larger radii - hence larger amplifications - are ruled
out in our example. Note that, since the amplification is mostly sensitive to
$\theta_{s}=\ell/R$, the effect of a distribution in $\ell$ will be (almost)
degenerate with that in $r$: however a small difference will remain, due to
the residual mild radial dependence of all coefficients in eqs. (2) and (4).

##  4 Probability of amplification for an active hotspot in the magnetosphere

The configuration discussed in this section is key to our scenario. It
envisages an active region, anchored to the magnetosphere, which as a result
of the NS rotation describes an annulus of width $\ell$ centered on the spin
axis. This, e.g., may be also consistent with the occurrence, in the Crab
pulsar, of GPs always in the trailing edge of the main radio pulse, if GPs
were interpreted as gravitationally-lensed ‘regular’ pulses. Possible
implications for GPs will be discussed elsewhere. Here we describe the
probability of lensing in this configuration.

Figure 3: Active hotspot geometry. The center of the hotspot, Phs, has angular
distance $\xi$ from the rotation axis, with which the caustic line forms an
angle $i$. The observer lies at the opposite of the caustic line, relative to
the NS. As the NS rotates, the angular distance of Phs from the caustic goes
from a minimal $\theta_{\rm min}$ (maximal amplification $a_{\rm max}$) at
phase $\varphi=0$, to a maximal $\theta_{\rm max}$ (minimal amplification
$a_{\rm min}$) at $\varphi=\pi$. The rest of the rotation is symmetric
(barring the slight asymmetry due to relativistic Doppler, which is negligible
for NS spin periods longer than a millisecond; see Bakala et al. 2023).

Consider an emission region $P$ of finite size $\ell$, co-rotating with the
star at radius $r$ and colatitude $\xi$ (Fig. 3). The probability of a given
amplification will reflect the probability of the corresponding separation
angle, $\theta$, between $P$ and the caustic line throughout the rotation,

$\cos{\theta}=\cos{\xi}\cos{i}+\sin{\xi}\sin{i}\cos{\varphi}\,.$ (5)

Here i is the angle between the rotation axis and the caustic, and $\varphi$
the rotation phase measured from the point where $P$ is closest to the caustic
line.

The amplification $a$ now depends, through $\theta$, on $\varphi$, $i$ and
$\xi$, the latter two angles being fixed for a given FRB source. A key

Figure 4:  Examples of $a$-modulations vs. rotational phase, $\varphi$, for
different source geometries. Left Panel: a small inclination ($i=3$ deg), and
a co-latitude of the emitting region $\xi=3.002,3.5,5,10$ deg for the solid,
dashed, dot-dashed and short-dashed curves, respectively. Right panel: a
larger inclination ($i=30$ deg) and the same $|\xi-i|$ as in the left panel.
The maximum amplification is the same in both panels. However, the
amplification drops quickly below unity at large $i$, while at low
inclinations, it is relatively high over a wide $\varphi$-range. The lower,
solid black curves are representatative of more common configurations in a
random sample, i.e. those a large value of $|\xi-i|$.

point is that there exist some (rare) combinations of $i$ and $\xi$, for which
$\theta$ is small throughout one rotation, and the amplification
correspondingly large, producing sources with frequent and highly-amplified
bursts. Conversely, amplifications will rarely occur if $\theta$ is large most
of the time, resulting in very sporadic or altogether unobservable bursts. It
is precisely this dependence on the geometrical configuration that delineates
the difference between repeating and non-repeating sources, as discussed
extensively in sec. 6.1. Fig. 4 illustrates the variation of $a$ with the
rotational phase, for different source orientations, when $r=10$. The left
panel shows variations of $a$ with $\varphi$ for a fixed (small) inclination
of the source, $i=5$ deg, and various co-latitudes of the emitting spot. The
right panel shows the same variation for a more inclined source ($i=30$ deg).

As implied by eq. (5), $i+\xi$ sets the maximum distance from the caustic
line, thus a large $i$ (or $\xi$) will suffice to keep the minimum $a$ close
to, or below, unity. On the other hand, very large amplifications are only
achieved when $i$ and $\xi$ are extremely close, regardless of their values.
However, when both angles are small, the high-amplification peak in the phase
profile is broad, while it gets very narrow and sharp if both angles are large
(and approximately equal). In NS where both $i$ and $\xi$ are small, the
emission can reach high amplifications more frequently, and over a wider phase
interval; thus, for a given detection threshold, finite-duration events will
on average remain detectable for a longer time span. Conversely, when the peak
is narrow, large amplifications occur rarely and, even when they do, they only
last for a tiny phase interval. This is fully consistent with the bursts of
repeating sources in the CHIME catalogue, which show longer durations than the
rest of the population (Pleunis et al., 2021).

###  4.1 General properties of $P(a)$

Equation (5) allows to define the probability of a particular amplification,
$a$, as a function of $i$ and $\xi$

Figure 5: Left panel: Amplification probability, $P(a)$, for various
configurations with $\ell=30$ cm. Blue solid: $r=10$, $i=3$ deg, $\xi=3.5$
deg; Orange, short-dashed: $r=10$, $i=3$ deg, $\xi=3.002$ deg. Green, long-
dashed: $r=30~{}$, $i=3$ deg, $\xi=3.002$ deg; Red, solid, thin: $r=10$,
$i=70$ deg, $\xi=70.2$ deg. Right panel: effect on the original $P(a)$, i.e.
the solid blue curve from the left panel, of the hot-spot spanning a range of
radii at fixed co-latitude $\xi$. At low amplifications, the sharp edge of the
blue curve is replaced by a much smaller peak and a smooth decay towards the
minimum $a$, which becomes lower due to the decrease of $a$ with growing $r$,
at a fixed $\theta$. At large amplifications a steepening occurs, due to the
same dependence, followed by a sharp cut-off at the (unchanged) $a_{\rm max}$.
The three modified curves are for three probability distributions of the
emission radius, indicated in the plot legend.

$\begin{cases}dl=R\sin\xi d\varphi\\\ \sin\theta d\theta=\sin
i\sin\xi\sin\varphi d\varphi\\\
P\left(\theta\right)d\theta=\displaystyle\frac{dl\left[\varphi(\theta)\right]}{\pi
R\sin\xi}=\displaystyle\frac{d\varphi}{d\theta}\displaystyle\frac{d\theta}{\pi}\,,\end{cases}$
(6)

where $dl$ is the arc length spanned by the emitting region for a change
$d\varphi$ in rotation phase. The last of eqs. (6) leads to

$P(a)=\frac{1}{\pi}\displaystyle\frac{d\varphi}{d\theta}\displaystyle\frac{d\theta}{da}\,,$
(7)

where we used the dependence $\theta(a)$ and restricted $\varphi$ to the
interval $(0,\pi)$ where $\theta$ is uniquely determined (see Fig. 3).

From Eq. (5) we get $d\theta/d\varphi=\sin i\sin\xi\sin\varphi/\sin\theta$,
hence

$P(a)da=\displaystyle\frac{\left(d\theta/da\right)\sin\theta}{\sin
i\sin\xi\sin\varphi}\displaystyle\frac{da}{\pi}\,,$ (8)

again determined by the relation $a(\theta)$. Through a full rotation $\theta$
goes from $\left|i-\xi\right|$ to $i+\xi$ (Eq. 5); the emission annulus is
thus entirely in the ELR in all configurations with $i+\xi<\theta_{\rm
we}(r)$. In all other configurations, parts of the emission annulus will be in
the weak lensing regime, and the corresponding $d\theta/da$ must be obtained
through eq. (2).

The left panel of Fig. 5 shows our results for four selected configurations
and a source size $\ell=30$ cm. Three of these are entirely in the ELR
($i+\xi<10$ deg): their $P(a)$’s have (i) an extended power-law $\propto
a^{-2}$, as opposed to the $a^{-3}$ scaling of the previous section, due to
the reduced dimensionality of the locus of the hot-spot potential positions
relative to the caustic (a ring, as opposed to a spherical surface) and (ii)
two sharp rising edges at the minimum and maximum amplification of each
configuration. Such edges are due to the dependence of $P(a)$ on
$d\varphi/d\theta$, which steepens sharply at the extrema, where
$\sin\varphi\approx 0$. The orange, solid curve shows a more generic
configuration, with $\theta$ varying from 0.2 deg (ELR) to 140.2 deg (WLR);
correspondingly, the low-amplification end rises more gradually towards the
edge.

The right panel illustrates the result of allowing the hot-spot to span a
range of radii ($r=10-50$) with $P(r)\propto r^{-k}$ and three different
values of $k=-2,1$ and 3. In this case, extending the radial range does not,
in general, allow the system to reach larger amplifications since the maximum
amplification depends on $\theta_{\rm min}$, and the latter is set by
$|i-\xi|$ alone, as long as this exceeds the hot-spot angular size, i.e.
$|i-\xi|>\theta_{s}\approx 0.03~{}{\rm deg}~{}(\ell/10{\rm m})(r/10)^{-1}$.
Only for smaller values of $|i-\xi|$ can the changing radius affect $a_{\rm
max}$. However, since the amplification depends on $r$ at fixed $\theta$,
albeit weakly, the minimum amplification is somewhat reduced. As a result,
both sharp edges of the distribution obtained with $r$ fixed get now spread
over a range of $a$-values. In particular, the low-$a$ end of the distribution
now contains a larger probability, at the expense of the high-amplification
end: this happens because lower amplifications are increasingly more likely at
larger radii. To avoid adding more degrees of freedom, we will keep $r$ fixed
at $10$ in the following.

###  4.2 Latitudinal width of the active region

If the emitting spot is not static, but jitters randomly within an area of
latitudinal extension $\delta\xi>\theta_{s}$, then it describes an annulus
with an effective width $\delta\xi$. The resulting $P(a)$ is

$P(a;i,\xi,r)=\int_{-\delta\xi}^{+\delta\xi}P(a;i,\xi+\xi^{\prime},r)P(\xi^{\prime})d\xi^{\prime}\,,$
(9)

for which it is necessary to model the probability distribution associated to
jittering, i.e. $P(\xi^{\prime})$. To this aim, we parametrized
$P(\xi^{\prime})\propto\xi{{}^{\prime}}^{w}$ and tested three values,
$w=-1/2,0,1$, applying eq. 9 to one of the configurations of the previous
section, which has $i=3$ deg and $\xi=3.5$ deg, hence $\theta_{\rm min}=0.5$
deg (blue curve in the left panel of Fig. 5). Moreover, for each $w$-value, we
tried two possible widths, $\delta\xi=0.1$ deg or $\delta\xi=0.4$ deg, for the
annulus. Results illustrating how the original $P(a)$ (blue, long-dashed
curve) gets modified by each of the six combinations of $w$\- and
$\delta\xi$-values are reported in the two panels of Fig. 6 (see plots and
caption for details).

The $P(a)$ of the ring gets now stretched at both ends, due to jittering
taking place both outwards or inwards relative to the original latitude $\xi$.
The case of a constant probability of jittering ($w=0$) is depicted in the
left panel: we note an interesting

Figure 6: Examples of hot-spot jittering applied to the first configuration of
Fig. 5 (blue curve). The modified $P(a)$ reaches a somewhat lower minimum and,
at the same time, extends to larger amplifications, steepening towards the
large amplification end. Left panel: a constant probability of jittering
($w=0$). Curves are simply understood in terms of a transition from a thin
annulus, with $P(a)\propto a^{-2}$, to a wide annulus, $P(a)\propto a^{-3}$.
Right panel: the shape of $P(\xi^{\prime})$ leaves a specific imprint in the
modified $P(a)$, causing slope variations in the high-amplification end of the
distribution. The slope may get flatter than $a^{-2}$ at intermediate points,
or steeper than $a^{-3}$ around $a_{*}$ and close to the maximum
amplification.

transition of the $P(a)$ from $\propto~{}a^{-2}$ to the steeper $\propto
a^{-3}$ scaling, which occurs at a characteristic amplification, $a_{*}$,
determined by the width of the annulus. This transition can be understood as
follows: as long as $\delta\xi\ll\theta_{\rm min}$, the annulus is “thin” and
$P(a)$ maintains the characteristic $a^{-2}$ scaling of ring-like structures
in the ELR. As $\delta\xi>\theta_{\rm min}$, on the other hand, the annulus
becomes “wide” and increasingly akin to a two-dimensional surface. The
transition to the $a^{-3}$ scaling thus occurs at $a_{*}\approx a(\theta_{\rm
min}+\delta\xi,r)$. In the left panel of Fig. 6, the transition for the thin
annulus ($\delta\xi=0.1$ deg) is at $a_{*}\lesssim 40$, and is closely
followed by a cut-off since $\theta_{\rm min}$ only decreases from 0.5 to 0.4
deg. For the wider annulus, on the other hand, we have a lower $a_{*}\approx
25$, and the $a^{-3}$ slope extends much further before cutting off at $a_{\rm
max}(\theta=0.1{\rm deg})\approx 250$.

Additional striking features appear at both extremes of the $P(a)$ when the
probability of jittering is a function of $\xi$ (right panel in Fig. 6).
Focusing on the high amplification end, we observe an early drop steeper than
$a^{-3}$ around $a_{*}$, followed by a flattening of the profile up to $a_{\rm
max}(\theta=\xi-i-\delta\xi)$, at which the $P(a)$ eventually cuts off to
zero. A wider range $\delta\xi$ for the jittering makes this feature more
apparent and extended. These properties are likely to play a role in detailed
models of the energy distribution in individual sources (see, e.g., results on
FRB 20201124A presented by Kirsten et al., 2024). A thorough investigation
will be carried out in a future study. In the rest of this paper, however, we
restrict our attention to the case of a fixed $\xi$, to avoid introducing
additional new assumptions to our calculations.

###  4.3 A population of randomly-oriented sources: P̂(a)

We have shown that the amplification probability for an individual source in
the ring-like geometry is $P(a)\propto a^{-2}$. What happens, though, to the
probability of amplifications in a large population of sources, each with its
own random orientation ($i$) and random hot-spot co-latitude ($\xi$)? Within
such a population,

Figure 7: Collective probability distribution of amplifications, $\hat{P}(a)$,
in a population of randomly-oriented sources, each having a co-rotating hot-
spot at random co-latitude $\xi$. For $a>2$ we have $\hat{P}(a)\propto
a^{-3}$, even though in each individual source $P(a)\propto a^{-2}$. This
$\hat{P}(a)$ is almost identical to that of a single source in which the hot-
spot can occur randomly within a spherically symmetric region of the
magnetosphere. This stems from the spherical symmetry introduced, within the
population, by the random orientation of individual sources.

each $(i,\xi)$-configuration will occur with probability 1/4 $\sin i\sin\xi$.
Thus, multiplying the probability of a given $(i,\xi)$-configuration by its
$P(a)$, and integrating over all possible configurations, we obtain
$\hat{P}(a)$, the probability distribution of $a$ for the ensemble of
randomly-oriented sources,

$\hat{P}(a)=\int_{0}^{\pi/2}\int_{i}^{\pi-i}P(a;i,\xi)\sin
i\sin\xi\,d\xi\,di\,,$ (10)

where we used the symmetries in the system to limit the integration range of
each variable. The resulting $\hat{P}(a)$, depicted in Fig. 7 (solid blue
curve), scales like a power-law $\propto a^{-3}$, despite the individual
$P(a)$’s being flatter ($\propto a^{-2}$) in the adopted geometry. This
steepening, which is possible because the $P(a)$ of individual sources are
truncated power-laws, can be understood as a result of larger amplifications
being achieved only by sources with favourable combinations of $i$ and $\xi$,
and the latter becoming increasingly more rare in the population. The
$\hat{P}(a)$ calculated here is almost identical to the $P(a)$ of sec. 3.1 for
a spherically-symmetric emission volume (right panel of Fig. 1): effectively,
the spherical symmetry of random $(i,\xi)$ combinations re-introduces, at the
level of the population, the characteristic $a^{-3}$ scaling of spherically-
symmetric sources. As a result, in our model it is possible, or even expected,
that the amplification distribution of events from an individual source
differs from (be flatter than) that of the whole population.

##  5 Observed vs. intrinsic properties

Knowing the form of $P(a)$ for various emission geometries, we can now
calculate its imprint on the luminosity distribution of lensed events. In the
simplest possible scenario we consider standard candle hot-spot events with
intrinsic luminosity $\hat{L}_{0}$, hence a probability distribution
$P_{0}(L_{0})\equiv\delta(L_{0}-\hat{L}_{0})$ for their intrinsic
luminosities. The probability distribution for amplified events with lensed
luminosity $L$ will be

$\displaystyle P(L)dL$ $\displaystyle=$ $\displaystyle\int_{L_{0,{\rm
min}}}^{L_{0,{\rm
max}}}P_{0}(L_{0})dL_{0}~{}P\left[a\left(L\right)\right]\displaystyle\frac{da}{dL}dL=$
(11) $\displaystyle=$
$\displaystyle\int\delta(L_{0}-\hat{L}_{0})dL_{0}~{}P\left(\displaystyle\frac{L}{L_{0}}\right)\displaystyle\frac{dL}{L_{0}}=$
$\displaystyle=$
$\displaystyle\displaystyle\frac{P\left(L/\hat{L}_{0}\right)}{\hat{L}_{0}}dL=\displaystyle\frac{P\left[a\left(L\right)\right]}{\hat{L}_{0}}~{}dL\,,$

showing that the observed luminosity distribution will not track the intrinsic
one but, rather, the probability distribution of the amplification factor. The
resulting $P(L)$ translates to an observed “luminosity function” $F(L)$, i.e.
the rate of observed events at luminosity $L$, according to $F(L)=AP(L)$,
where the normalization constant $A$ is determined by

$\int_{S_{\rm thr}}^{S_{\rm max}}F(S)dS=A\int_{S_{\rm thr}}^{S_{\rm
max}}P(S)dS\equiv{\cal R}_{b}$ (12)

where ${\cal R}_{b}$ is the rate of observed events above the detection
threshold, $S_{\rm thr}=a_{\rm thr}\hat{S}_{0}$. From the above it obtains
that

$F(S)={\cal R}_{b}\displaystyle\frac{P(S)}{\displaystyle\int_{S_{\rm
thr}}^{S_{\rm max}}P(S)dS}=\displaystyle\frac{{\cal
R}_{b}}{\hat{S}_{0}}~{}\displaystyle\frac{P\left(S/\hat{S}_{0}\right)}{\displaystyle\int_{a_{\rm
thr}}^{a_{\rm max}}P(a)da}\,.$ (13)

In a more general case in which events have a range of intrinsic luminosities,
the probability of observing one with luminosity between $L$ and $L+dL$ is
still given by eq. (11). We do not specify the functional form of
$P_{0}(L_{0})$, but only assume that it extends from a minimum intrinsic
luminosity $L_{0,{\rm min}}$ to a maximum $L_{0,{\rm max}}$. Thus

$P(L)dL=\int_{L_{0,{\rm l}}}^{L_{0,{\rm
h}}}P_{0}(L_{0})dL_{0}P(a)\frac{da}{dL}dL\,,$ (14)

where the integral extends over all $L_{0}$-values that can achieve $L$ if
amplified. Hence, $L_{0,{\rm l}}~{}=~{}{\rm max}\left[L/a_{\rm max},L_{0,{\rm
min}}\right]$ and $L_{0,{\rm h}}={\rm min}\left[L/a_{\rm min},L_{0,{\rm
max}}\right]$. Again, the function $P(L)$ must be normalized to the observed
event rate (Eq. 12) in order to obtain the corresponding luminosity function,
$F(S)$. Further details and discussion are presented in Appendix B.

##  6 Repeating and non-repeating FRBs from a single source population

In order to highlight the role of gravitational lensing in shaping the
appearence of FRBs, we first discuss in an approximate fashion some of the key
constraints imposed by observations. In our scenario, the observed event rate
per source, ${\cal R}_{b}$, is lower than the intrinsic rate, ${\cal R}$, by
the probability of amplification above the detection threshold, i.e. a factor
$P(a>a_{\rm thr})$. If the emission is also beamed by a factor $f_{b}<1$, then
the two event rates are related as666See Appendix B.2 for further details.
${\cal R}_{b}=f_{b}{\cal R}P(a>a_{\rm thr})$. Accordingly, the number of
observed events from a source ${\cal N}_{b}={\cal R}_{b}T_{\rm obs}$, in the
observing time $T_{\rm obs}$, will be smaller than the intrinsic number of
events emitted and beamed towards us, $N_{\rm int}$ $={\cal R}f_{b}T_{\rm
obs}$.

We constrain ${\cal R}_{b}$ following Cordes & Wasserman (2016), who estimate
a birth rate $\sim 10^{4}$ day-1 for the cosmic NS population within the
Hubble volume $V_{H}=4/3\pi(c/H_{0})^{3}$, when scaled from the galactic
value. As this is $\sim 10$ times higher than the all-sky FRB rate,
$\Gamma_{b,{\rm obs}}\sim 500$ day-1 (CHIME/FRB Collaboration et al. 2020),
this population is evidently sufficient to account for $\Gamma_{b,{\rm obs}}$,
e.g. requiring that $\sim$ 5-10% of all NS within $V_{H}$ produce one
observable FRB in their active lifetime, $T_{b}$. Because magnetars are
estimated to represent $\lesssim$ 10% of all NS formed in core-collapse (e.g.
Gaensler et al. 2005; Beniamini et al. 2019) let us suppose, for the sake of
the argument, that all FRBs are due to magnetars, each leading on average to 1
observed event in a time $T_{b}\sim 10^{4}$ yr, the characteristic age for
known active magnetars. The average observed rate per source within the Hubble
volume is thus

$\langle{\cal R}_{b}\rangle\approx 10^{-4}~{}{\rm
yr}^{-1}~{}\displaystyle\frac{\left(\Gamma_{b,{\rm obs}}/10^{3}~{}{\rm
day}^{-1}\right)}{\left(T_{b}/10^{4}~{}{\rm yr}\right)}\,.$ (15)

This is much lower than the $\sim 30$ hr-1 (2.6 $\times 10^{5}$ yr-1) rate of
the most active repeaters currently known, FRB 202121102A and FRB 20201124A
(averaged over active and ‘off’ states). This huge rate mismatch represents a
severe constraint for any model aimed at explaining FRB observations. Taken at
face value, it implies that most sources may never be observed to repeat, and
that just a handful of the most active repeaters may exist in the observable
universe, in order not to exceed the all-sky rate, $\Gamma_{b,{\rm obs}}$.

Figure 8: Two different orientations of identical sources, corresponding to an
active repeater (left) and to a one-off source (right). In the left panel, the
hot-spot is located at a small angular distance from the rotation axis,
$\xi\lesssim$ a few degrees, nearly identical to the misalignement of the
line-of-sight ($i$): the hot-spot stays close to the caustic line during most,
or even all, of its rotation, hence a majority of its flares are sufficiently
amplified to be detectable. The right panel depicts a more common geometry and
orientation, in which both $i$ and $\xi$ are relatively large and comparable
to each other: the hot-spot spends most of the time far from the caustic line,
and only rarely its flares get amplified - hence detected. Even more common,
and more numerous, are orientations for which no amplification and no
detection are ever achieved.

Such a dichotomy between repeaters and non-repeaters can be accounted for in
our NS self-lensing scenario if the hot-spot is stably anchored in the
magnetosphere, i.e. as in the annular geometry of sec. 4. In that case the
probability of exceeding a given amplification threshold, $P(a>a_{\rm thr})$,
depends very strongly on the source orientation (instead, no such effect
arises in the spherical geometry of sec. 3.1). Thus, sources with a favourable
orientation, i.e. a large probability of exceeding $a_{\rm thr}$, may lead to
a much larger number of observed events than less favourably oriented sources,
even if they had the same intrinsic rate and produced the same total events,
$N_{\rm int}$. In this framework, one-off FRBs are typical randomly-oriented
sources, with very low probability of achieving amplifications above
threshold. Thus, even in the simplest assumption of identical FRB sources
(i.e. same event rate and energy), the ratio between $\langle{\cal
R}_{b}\rangle$ in eq. 15 and the $\sim$ 30 hr-1 events of active repeaters has
a clear interpretation: it represents the characteristic probability with
which we detect typical events from unfavourably oriented sources (one-off
FRBs) when they get sufficiently amplified. Therefore, we estimate $P(a>a_{\rm
thr})\sim 10^{-4}/(2.6\times 10^{5})\sim 4\times 10^{-10}$ which, as can be
read off Fig. 7, implies a characteristic threshold $a_{\rm thr}\sim 10^{3}$.
The most energetic FRB ever recorded had $E_{\rm iso}\gtrsim 10^{41}$ erg, for
which our argument implies a true energy $\sim E_{\rm iso}/a_{\rm thr}\sim
10^{38}$ erg, and $(4\times 10^{-10})^{-1}\sim(3\times 10^{9})$ identical
events that were missed as they didn’t achieve $a\geq~{}10^{3}$. The resulting
energy budget is $E_{\rm TOT}\sim 2.5\times 10^{47}$ erg for the most
energetic one-off FRB. Formally, this represents a lower limit to the actual
energy budget, as the efficiency of radio emission is expected to be $\ll 1$
(it was $\sim 10^{-5}$ in the FRB-like event from the galactic magnetar SGR
1935+215; e.g. Mereghetti et al. 2020). However, the above calculation
implicitly assumes that the source maintains its extreme burst rate for
$T_{b}=10^{4}$ yrs. Evolutionary effects may well shorten the duration of such
an intense bursting phase, in turn reducing the energy estimate. For instance,
if the burst rate were to remain constant for $\sim 300$ yr and then decrease
like $t^{-1}$, while keeping the same energy per event, the radio energy
budget would decrease by a factor 10 (more details in Appendix C).

Conversely, active repeaters are ideally-oriented sources with $P(a>a_{\rm
thr})\approx 1$, for which most bursts are detected. We know only a handful of
very active repeaters while, adopting the cosmic star formation rate of Madau
& Dickinson (2014), we estimate $\sim 10^{7}$ magnetars younger than $\sim
10^{2}$ yrs within $z\approx 0.193$, the distance and likely age of FRB
20121102A (Metzger et al., 2017; Yang & Zhang, 2017; Hilmarsson et al., 2021).
Hence, if all FRB sources are identical, the most active repeaters must be
very unlikely occurrences ($\sim 1$ in $10^{6}$ sources), although their
enhanced detectability likely boosts their incidence among observed FRBs. A
more accurate assessment would involve the modelling of a population of non-
identical sources with a range of event rates and energies: our crude
estimate, though, establishes the extreme rarity of very active repeaters.
These may exist only for finely-tuned combinations of $i$ and $\xi$, in turn
requiring777A more realistic calculation may slightly relax this constraint on
$(i,\xi)$. both angles to be small ($<$ 4 deg) and almost identical
($|\xi-i|<0.05$ deg). Note that the energy budget of the very active FRB
20121102A is estimated to be888If most events of such active repeaters are
indeed detected, and at least mildly amplified, their true energetics may be
somewhat lower. $\sim 3\times 10^{47}$ erg for $T_{b}\sim 10^{4}$ yrs (e.g. Li
et al. 2021), close to the above estimate for the most powerful one-off. In
this case too, evolutionary effects may significantly reduce the source energy
budget.

###  6.1 Formal definitions for repeaters and one-off sources

To keep our treatment simple we consider in following the case in which
hotspots are standard candles. Under this assumption we can express via eqs.
(5) and (6) the probability that bursts in a randomly oriented source overcome
a given detection threshold (i.e. achieve $a\geq a_{\rm thr}$) at least
$1/N_{\rm int}$ of the times.

Figure 9: Left Panels: Contours in the $(i,\xi)$-plane illustrating the
parameter range for repeaters (red) and for one-off FRBs (blue), assuming
$a_{\rm thr}=10$ (upper) or $a_{\rm thr}=100$ (lower), and $N_{\rm int}=50$
(solid) or $N_{\rm int}=150$ (dashed). Values were chosen for illustration
purposes. For a given source, i.e. a fixed ${\cal R}$, increasing $N_{\rm
int}$ is equivalent to assuming a longer observation time (see text). In both
plots the dot-dashed horizontal green line separates the region of sources
that can be amplified above the chosen $a_{\rm thr}$ (below the line), from
those that cannot (above the line) and, thus, cannot be detected. For
illustration purposes the $y$-axis reports ($\xi-i$): thus the locus $i=\xi$
lies along the $x$-axis while the locus $\xi=180$°$-i$ is the black diagonal
on the right-hand side (only visible in the upper plot, due to a different
scale on the $x$-axis in the lower plot). For symmetry reasons, the $(i,\xi)$
combinations lying between these two loci are sufficient to describe the whole
plane - Right Panels: probability of obtaining a repeater (red solid) or a
one-off FRB (blue dashed) vs. observing time ($T_{\rm obs}$), for $a_{\rm
thr}=10$ (upper) or 100 (lower). For each $T_{\rm obs}$, we integrated the
probability of individual ($i,\xi$)-combinations, with both angles randomly
distributed, over the area in parameter space that allows for repeaters or
one-off FRBs (left panels), respectively. The reference time is T${}_{\rm
ref}=N_{\rm int}/{\cal R}$, with $N_{\rm int}=50$.

By construction, the largest amplification along the path of the hotspot
occurs at $\varphi=0$, then decreases as $\varphi$ grows towards $\pi$. At a
given emission radius, the condition $a\geq a_{\rm thr}$ imposes
$\theta\leq\theta_{\rm max}(r,a_{\rm thr})$, where $\theta_{\rm max}$ is
obtained by solving eq. (2). In turn, $\theta_{\rm max}$ selects a symmetric
range of $\varphi$-values around $\varphi=0$, from $-\pi/N_{\rm int}$ to
$+\pi/N_{\rm int}$. We thus obtain via eq. (5)

$\arccos\left(\cos i\cos\xi+\sin i\sin\xi\cos\displaystyle\frac{\pi}{N_{\rm
int}}\right)\leq\theta_{\rm max}(r,a_{\rm thr})\,,$ (16)

which defines the region, in the $(i,\xi)$ plane, where at least one of the
$N_{\rm int}$ events overcomes the detection threshold.

Repeaters will be those sources for which, given an intrinsic burst rate
(${\cal R}$) and a particular orientation ($i,\xi$), two or more events will
be observed in time Tobs. Similarly, one-off sources will produce one observed
event under the same conditions. Thus, we can formally define

$\displaystyle\displaystyle\int_{a_{\rm thr}}^{a_{\rm max}}P(a;i,\xi)da$
$\displaystyle\equiv$ $\displaystyle{\cal I}_{a_{\rm
thr}}\geq\displaystyle\frac{2}{{\cal R}f_{b}T_{\rm obs}}\,,~{}{\rm repeaters}$
$\displaystyle 1\leq{\cal I}_{a_{\rm thr}}{\cal R}f_{b}T_{\rm obs}$
$\displaystyle<$ $\displaystyle
2\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
one-offs}$ (17) $\displaystyle{\cal I}_{a_{\rm thr}}{\cal R}f_{b}T_{\rm obs}$
$\displaystyle<$ $\displaystyle
1\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
non-FRBs}$

Figure 10: Left Panels: Probability of obtaining a repeater (solid, red) or a
one-off FRB (dashed, blue) as a function of redshift, for standard candle hot-
spots having $E_{\rm iso}=10^{37}$ erg and total events $N_{\rm int}={\cal
R}f_{b}T_{\rm obs}=10$ (upper) or $N_{\rm int}=1000$ (lower). The detection
threshold was taken from CHIME at Fthr = 5 Jy ms (Li et al., 2021),
corresponding to $a_{\rm thr}=1$ at $z=0.145$ for the assumed standard candles
- Right Panels: Same as left panels, but for 10 times weaker standard candles.
The CHIME detection threshold corresponds, in this case, to $a_{\rm thr}=10$
at $z=0.145$, and has $a_{\rm thr}=1$ at $z=0.047$.

The integral ${\cal I}_{a_{\rm thr}}$ gives the probability that an event gets
amplified above a given $a_{\rm thr}$, and is uniquely determined by $a_{\rm
thr}$ and the particular combination of $i$ and $\xi$. For a given source,
with a fixed set $(a_{\rm thr};i,\xi)$, the parameter $N_{\rm int}={\cal
R}f_{b}T_{\rm obs}$ may thus be regarded as the minimum number of events
required in time $T_{\rm obs}$, in order for a least one of them to be
detected.

###  6.2 Repeaters vs. non-repeaters vs. non-FRBs

For a source producing $N_{\rm int}$ events in time Tobs, and for a given
threshold $a_{\rm thr}$, one can use the definitions of eq. 6.1 to find
contours, in the $(i,\xi)$ plane, inside which at least one event or at least
two events are expected, respectively. The latter contour defines the area in
parameter space in which repeaters will occur, while one-off FRBs will fill
the area between the two contours. Exterior to both contours is the area of
undetected sources.

The left panels in Fig. 9 show such contours in the $i$ vs. $\xi$ plane
(repeaters in red, one-off FRBs in blue), for specific values of $a_{\rm
thr}=10$ (upper) or 100 (lower) and $N_{\rm int}=50$ (solid) or 150 (dashed),
chosen for illustration purposes. Because $N_{\rm int}={\cal R}f_{b}T_{\rm
obs}$, changing the value of $N_{\rm int}$ for a given source (i.e. a fixed
rate) is tantamount to changing the observation time. Thus, the dashed
contours in Fig. 9 may be regarded as the evolution of the solid ones, for the
same source, as $T_{\rm obs}$ is increased by a factor 3. For example, for a
source with an intrinsic rate ${\cal R}=1$ hr-1, the values $N_{\rm
int}=50(150)$ correspond to the observing times $T_{\rm obs}=50(150)$ hrs.

We see that, as the observing time is increased, both the one-off and the
repeater regions expand, but the latter grows faster, resulting in an
increased probability for repeaters over one-off FRBs. This expansion, though,
hits a wall indicated by the green, dot-dashed horizontal line. The
combinations of $i$ and $\xi$ above this line cannot achieve amplifications
$>a_{\rm thr}$: thus, further increasing the observing time will not produce
any new sources. It will, however, eventually turn all one-off events into
repeaters. Note that most of the parameter space in Fig. 9 corresponds to
undetected sources (non-FRBs) which lie above the green-dashed line and,
hence, will never become FRBs.

We also calculated the probability of observing repeaters or one-off FRBs vs.
redshift for standard candle hot-spot events. To this aim we defined a
redshift-dependent $a_{\rm thr}(z)$, which depends on the standard candle’s
characteristic energy ($E_{\rm iso}$) and on the detector sensitivity, and
then adopted the cosmological parameters measured by the 2018 Planck
collaboration (Table 2, last column in Planck Collaboration et al., 2020). For
the sake of our argument, we used the FAST fluence threshold, F${}_{\rm
thr}=0.015$ Jy ms (Li et al., 2021) and tried two different energies,
E${}_{{\rm iso},37}=1$ or 0.1, which imply $a_{\rm thr}=10$ at $z\approx 0.61$
or at $z\approx 0.23$, respectively. Moreover, we selected two widely
different values of $N_{\rm int}=10$ or 1000: these can either represent
sources observed for the same time but whose intrinsic rates differ by a
factor 100, or sources with the same ${\cal R}$ but very different observing
times. With this set up we calculated, at each redshift, the contours shown in
Fig. 9, and then integrated the probability of a generic
($i,\xi$)-combination, $1/4\sin i\sin\xi$, over the parameter space
corresponding to repeaters or one-off FRBs, respectively.

Figure 11: Contours of the lensing gain ${\cal C}=\dot{E}_{\rm
app}/\dot{E}_{\rm true}$, for standard candles at a fixed distance, as a
function of the angles $i$ and $\xi$. The four cases shown are for (a)
amplification threshold $a_{\rm thr}=5$ and $N_{\rm int}=10$, representing a
low-threshold and low-rate case, (b) $a_{\rm thr}=50$ and $N_{\rm int}=10$, a
high-threshold and low-rate case, (c) $a_{\rm thr}=5$ and $N_{\rm int}=100$, a
low-threshold and high-rate case and (d) $a_{\rm thr}=50$ and $N_{\rm
int}=100$, a high-threshold and high-rate case. Configurations below the solid
red curve have a lower power release than the apparent one because most of
their flares are amplified and detected, i.e. the probability of exceeding
$a_{\rm thr}$ is high during most of the rotation. Also shown are contours for
an apparent-to-true power ratio of 3, 10, 30 or 100. These latter finely-tuned
configurations correspond to strong repeaters: both their line of sight and
emitting hot-spot lie close to the rotation axis, with very similar offsets.
Their probability of occurrence by random distributions of $i$ and $\xi$ is
very low. For reference, we calculated the probability of sources with ${\cal
C}\geq 10$ and obtained, respectively $1.2\times 10^{-6}$, $2\times 10^{-8}$,
$1.2\times 10^{-6}$, $8.7\times 10^{-8}$ for cases (a)-(d).

Fig. 10 depicts the resulting probability for sources of either type versus
redshift, assuming a population of standard candles, for four combinations of
$a_{\rm thr}$ and $N_{\rm int}$. The probability of repeaters is maximal
($\leq 1$) at small $z$ and drops off steeply at $z\sim 0.015-0.1$ (depending
on the adopted parameters). Just beyond the same $z$-values, the probability
of one-off events has a pronounced peak, then drops off with the same steep
slope of repeaters. As a result, the one-off-to-repeater probability ratio
stays constant ($\sim$ 8) at larger $z$, which is a specific expectation of
the model999The exact value of their probability ratio, though, may be
affected by further details not addressed in our standard candle
approximation.. A lower energy of the events shifts the probability profiles
to lower $z$, maintaining their overall shape. A low number of events (i.e. a
shorter observing time or a lower burst rate) suppresses the one-off
distribution at small redshift, and shifts both distributions to lower $z$.

###  6.3 Apparent vs. true energy budget

Another crucial implication of gravitational lensing is that observed events
should have a lower intrinsic energy. In most configurations, significant
amplifications will have very low probabilities, in turn implying a large
number of non-amplified (undetected) events. The combination of these two
factors determines a substantial difference between the observed (or
“apparent”) energy budget of a source and its emitted one. In extremely rare
combinations of $i$ and $\xi$, though, significant amplifications may be
achieved with a large probability and the source will appear as a very active
repeater. In such cases the apparent energy budget of a source will exceed,
sometimes by a large factor, its true energy budget. This is a specific and
key expectation of our model. In our simple assumption of standard candle hot-
spot events, we calculated the ratio of apparent to true power release of a
given source as a function of $i$ and $\xi$, for two fixed amplification
(detection) thresholds, $a_{\rm thr}=5$ or 50, and for two different values of
$N_{\rm int}=10$ and $100$. We label this ratio, ${\cal C}=\dot{E}_{\rm
app}/\dot{E}_{\rm true}$, the lensing gain and provide details of its
calculation in Appendix B.1 (Eq. B5). Fig. 11 illustrates our results,
highlighting the region in parameter space where ${\cal C}>1$ (see caption for
details). For each of the four panels, we calculated the probability of
configurations with a gain ${\cal C}\geq 10$, assuming a random distribution
of both angles $i$ and $\xi$: the two panels on the left have nearly
coincident probabilities $\approx 10^{-6}$, the upper right panel has a
probability $\approx 8.7\times 10^{-8}$ and the lower right panel of $\approx
2\times 10^{-8}$. In all cases, these are extremely unlikely configurations,
implying a large population of sources to explain the existence of very active
repeaters. Note that in sect. 6 we estimated for FRB 20121102A and FRB
20201124A a probability $\sim 10^{-6}$ and a combination of angles which would
correspond to ${\cal C}\geq 10$, e.g. in the left panels of Fig. 11. A
detailed characterization of the active repeaters will be presented in an
accompanying paper (La Placa et al., in preparation). Finally, Tab. 1 shows a
few examples for selected values of $a_{\rm thr},i,\xi$.

##  7 Gravitational lensing of a cosmic population of sources

In Sec. 6.1, 6.2 and 6.3, we only considered standard candle hotspots in order
to highlight the role of lensing and of geometrical effects in shaping their
possible appearence. In such a population of identical hotspots, the observed
energy distribution will directly track $\hat{P}(a)$, as discussed in sec. 5.
Instead, hotspots are likely to have an intrinsic (“seed”) energy
distribution, $P(E_{0})$, in which case the observed $\hat{P}(E)$ will result
from the convolution of $\hat{P}(a)$ with that seed (eq. 14). Here we
considered two possible seeds (i) a power-law
$P(E_{0})\propto~{}E_{0}^{-5/3}$, inspired by the known energy distribution of
magnetar X-ray bursts/flares, ranging from $E_{0,\rm min}=10^{33}$ erg to
$E_{0,\rm max}=10^{37}~{}{\rm erg}$ and (ii) a log-normal, in analogy to the
low-energy distribution observed in FRB 20121102A and 20201124A and to the
typical pulses of radio pulsars, centered at $E_{0,{\rm c}}=10^{36}~{}{\rm
erg}$ and with $\sigma_{\log E_{0}}=0.5$. As in previous sections, we
calculate the resulting $\hat{P}(E)$ for hotspot of linear size $\ell=30$ cm
at $R=10~{}r_{g}$. Our results are illustrated in Fig. 12: for both seed
distributions, lensing extends significantly the high-energy tail, shaping it
into the characteristic power-law with index $\lesssim 3$ (more details are
provided in the caption).

Figure 12: Predicted energy distribution of observed events, $\hat{P}(E)$
(blue, solid), for two different seed distributions (orange, short-dashed
curves). Upper left: power-law seed, $P(E_{0})\propto E_{0}^{-5/3}$, from
$10^{33}$ erg to $10^{37}~{}{\rm erg}$, with superposed the function
$\hat{P}(a)$ (magenta, dotted) producing a high-energy tail $\propto E^{-3}$
and a gradual transition in the range $0.64\leq a<1$, due to gravitational
redshift; Upper right panel: log-normal seed, with expectation value
$E_{0,{\rm av}}=10^{36}~{}{\rm erg}$ and $\sigma_{E_{0}}=0.5$. At high
energies, lensing produces the power-law tail $\propto E^{-3}$. At low
energies, $\hat{P}(E)$ is offset relative to the seed, due to gravitational
redshift; Lower panels: The (average) rate of observed events per yr per
source ($R_{b}={\cal N}_{b}/T_{b}$) vs. redshift for each of the seed
distributions. The two curves give very similar rates, with the power-law case
declining earlier at low-$z$ but remaining somewhat larger at high redshifts.
Note that the event rate per source is $<10^{-4}$ yr-1 (the estimated average
rate per magnetar of eq. 15) at $z\gtrsim 0.05$: the all-sky rate is thus
dominated by local sources, and the observations of FRBs at high redshifts
requires the existence of a large source population (e.g. at least $10^{8}$
sources at $z\approx 0.5$, in line with the estimated number assuming
$T_{b}\approx 10^{4}$ yrs, in order to observe one event from that redshift in
$T_{\rm obs}=1$ yr).

###  7.1 Impact of cosmology

The energy distribution of the observed population of FRBs will be affected by
additional factors, primarily the sensitivity of the detector and the non-
uniform spatial distribution of sources. The former introduces a $z$-dependent
minimum energy, $E_{\rm thr}(z)$, in the observable $\hat{P}(E)$, selecting an
increasingly higher-energy tail of events at higher $z$. As a result, the
average number of observed bursts per source at redshift $z$ is

$N_{b}(z)={\cal R}f_{b}T_{b}\int_{E_{\rm thr(z)}}^{E_{\rm
max}}\hat{P}(E)dE\,,$ (18)

where we still assumed a unique ${\cal R}$ for all sources. The maximum
observed energy, $E_{\rm max}$, is $E_{0,{\rm max}}$ times the maximum
amplification ($a_{\rm max}$) and is independent of $z$. The number of
observed events per source per yr, ${\cal N}_{b}(z)/T_{b}$ for either seed
distribution is depicted in the lower panels of Fig. 12.

The spatial distribution of sources, on the other hand, determines their total
number, hence the total number of detected events, at each redshift. We assume
that FRB sources trace the star formation history of the universe, e.g. they
are associated to young NS101010This is supported, for $z\gtrsim 0.05$, by
recent studies (Zhang & Zhang, 2022) which also found hints of an additional
population in the local universe. with age $\leq~{}T_{b}$. The NS birth rate
per unit volume is related to the star formation rate (SFR) $\psi(z)$ through

$\dot{n}_{\rm NS}(z)=\psi(z)\nu_{\rm NS}\,,$ (19)

where $\nu_{\rm NS}\approx 0.0068$ M${}_{\odot}^{-1}$ is the number of NSs
formed per unit stellar mass, taken to be independent on $z$, and the SFR is
(Madau & Dickinson 2014)

$\psi(z)=0.015\displaystyle\frac{(1+z)^{2.7}}{1+\left[(1+z)/2.9\right]^{5.6}}~{}\displaystyle\frac{M_{\odot}}{\rm
yr~{}Mpc^{3}}\,.$ (20)

The rate of observable events up to redshift $z_{\rm max}$ results by
multiplying $\dot{n}_{\rm NS}(z)/(1+z)$, the NS birth rate per unit
volume111111The $(1+z)$ term accounts for cosmological time dilation., by the
number of observable events per source at that redshift, $N_{b}(z)$, and
integrating over cosmological volume.

Table 1: Ratio of apparent to true power release (lensing gain) for three $a_{\rm thr}$-values and a few selected ($i,\xi$)-combinations. The last columns show the range of amplifications spanned by each configuration. $a_{\rm thr}$ | $i$ | $\xi$ | ${\cal C}$ | $P(a>a_{\rm thr})$ | $a_{\rm max}$ | $a_{\rm min}$
---|---|---|---|---|---|---
| (deg) | (deg) | ($\dot{E}_{\rm app}/\dot{E}_{\rm true}$) | (${\cal I}_{\rm athr}$) |  |
2 | 3 | 3.01 | 20 | 1 | 2437 | 4.05
2 | 30 | 30.01 | 2.1 | 0.137 | 2437 | 0.76
2 | 3 | 4 | 7.4 | 1 | 24.4 | 3.48
2 | 30 | 31 | 0.85 | 0.133 | 24.4 | 0.76
10 | 3 | 3.01 | 16.1 | 0.263 | 2437 | 4.05
10 | 30 | 30.01 | 1.7 | 0.027 | 2437 | 0.76
10 | 3 | 4 | 3.5 | 0.208 | 24.4 | 3.48
10 | 30 | 31 | 0.4 | 0.024 | 24.4 | 0.76
100 | 3 | 3.01 | 10.0 | 0.026 | 2437 | 4.05
100 | 30 | 30.01 | 1.05 | 0.0027 | 2437 | 0.76
100 | 3 | 3.2 | 1.6 | 0.014 | 121.9 | 3.93
100 | 30 | 30.2 | 0.18 | 0.0015 | 121.9 | 0.76

The only two unknowns in this calculation are the value of ${\cal
R}T_{b}f_{b}$ in Eq. 18, i.e. the normalization of $N_{b}(z)$, and the
calibration of $E_{\rm thr}(z)$, which is set by the detector sensitivity. For
a fluence-limited detector, we translate redshifts to energy thresholds
through (see, e.g., Zhang, 2018)

$E_{\rm thr}(z)={\cal{F}}_{\nu}^{\rm thr}~{}\nu_{c}~{}\frac{4\pi
D^{2}_{L}(z)}{1+z}\,,$ (21)

where ${\cal{F}}_{\nu}^{\rm thr}$ is the detector fluence threshold, $\nu_{c}$
the central observing frequency, and $D_{L}(z)$ the luminosity distance. We
considered here ${\cal{F}}_{\nu}^{\rm thr}=5$ Jy ms, which is the CHIME
90%-completeness fluence limit (Nan et al., 2011; Li et al., 2017).

The value of ${\cal R}T_{b}f_{b}$ is set by requiring that the calculated all-
sky rate of observable events equals the actual one.

###  7.2 Results: cosmic populations with a universal rate ${\cal R}$

With the above prescriptions, we evaluated the properties of the source
population implied by our model under the simple assumption of identical
sources, each emitting bursts with the same seed energy distribution, either a
power-law or a log-normal, as described above. Again we assumed that $\sim$
10% of all NS, i.e. magnetars, are FRB sources, and obtained that:

(i) for the power-law seed distribution, the average number of events per
source is ${\cal R}T_{b}f_{b}\sim 2\times 10^{7}$ over the active lifetime
$T_{b}$. To estimate the overall energy budget of each source, we assumed
$\epsilon_{r}~{}\sim~{}10^{-5}$ for the unknown radio emission efficiency,
comparable to the measured radio-to-X-ray ratio in the brightest FRB emitted
by SGR 1935+2154. This assumes an X-ray efficiency of order unity in that
event, and implies an average energy budget $E_{\rm tot}\sim 10^{47}$ ergs per
source in FRB-related events, and ${\cal R}\sim 6$ day${}^{-1}(10^{4}~{}{\rm
yrs}/T_{b})$. The inferred value of ${\cal R}$ scales with the average energy
per burst $\propto E_{\rm av}^{-3/2}$ ($E_{\rm av}\sim 5\times 10^{34}$ ergs
with our assumptions), while $E_{\rm tot}\propto E^{-1}_{\rm av}$. Moreover,
only $\sim 10^{3}$ of the $10^{7}$ events are in the amplified tail above
$10^{37}$ ergs (1 every 10 yrs). Among them, only a few events amplified above
$a_{\rm thr}(z)$ \- as set by the detector’s sensitivity - will eventually be
detected;

(ii) for the log-normal seed, the number of events per source is reduced to
${\cal R}T_{b}f_{b}\sim 10^{6}$, as their average energy is larger ($10^{36}$
ergs), the energy budget is roughly the same, $E_{\rm tot}\sim 10^{47}$ ergs,
and the event rate is lower, ${\cal R}\sim 0.3$ day-1. Also in this case, a
minority of the events ($\approx 4500$) are in the amplified tail at $E_{\rm
iso}>4\times 10^{36}$ ergs (see Fig. 12), or one every 2 yrs.

For both seed distributions, the lower panels in Fig. 12 clarify that the
expected number of observed events per source is much lower than unity at
$z>0.01$ (300 Mpc). Thus, detecting a source beyond the local universe is
intrinsically unlikely, and only the very large number of NSs at increasing
redshifts guarantees that some events will be observed out to $z\lesssim 0.5$
(see caption).

The paucity of FRB sources in the local universe, where detection is in
principle possible without lensing, places an additional constraint on their
population, with the caveat that selection criteria for the CHIME catalog are
likely to reject a fraction of nearby sources, i.e. those with a DM $<$ 1.5
times the galactic value. In Appendix C we provide some illustrative examples
on how this issue can be addressed, and the important conclusions that can be
drawn. More detailed results and constraints based on the proposed scenario
will require a thorough population synthesis study, which is deferred to a
future work.

##  8 Summary and Conclusions

We presented a new scenario for FRBs generated in NS magnetospheres and
amplified, via gravitational self-lensing in the strong field limit, by the NS
itself. We developed a semi-analytical approximation of the complex dependence
of the amplification $a(\theta)$ for all values of $\theta$ in different
lensing regimes, in order to obtain a numerically accurate and straightforward
calculation of the amplification probabilities in different geometries (see
sec. 3). Events occurring close to the caustic line can be amplified in
intensity by several orders of magnitude, thus alleviating the energy cost of
individual bursts without invoking highly relativistic motion. Large
amplifications occur with low probability, implying that cosmological FRBs
represent the tip of the iceberg of a very wide population of undetected
events. In particular, if detected FRBs are only those events amplified above
a minimal threshold ($a_{\rm thr}$), the relation ${\cal
R}_{b}=f_{b}P(a>a_{\rm thr}){\cal R}$ holds between the rate of detections
from a given source (${\cal R}_{b}$) and its intrinsic event rate (${\cal
R}$), where $P(a>a_{\rm thr})$ represents the probability of exceeding the
minimal threshold, and $f_{b}<1$ the emission beaming factor.

To determine $P(a)$, we explored the possibility that bursts occur either at
random positions in an extended, spherically-symmetric region of the
magnetosphere, or in a hot-spot corotating with the NS, thus sweeping an
annular (ring-like) region. For individual sources, the probability of large
amplifications is a power-law $P(a)\propto a^{-n}$ with $n=3$ in the
spherically-symmetric case, and $n=2$ for a ring-like emission region. The
energy distribution of amplified events follows the same power-law, whose
slope is consistent with that observed from individual well-studied sources
(FRB 20121102A and FRB 20201124A); this lends support to the rotating hot-spot
scenario.

For the population as a whole, on the other hand, the randomly distributed
viewing angles introduce spherical symmetry, thus favouring a power-law index
$n=3$ regardless of the emission geometry, i.e. even if individual sources
follow a flatter distribution. Additional factors, e.g. a significant radial
extension and/or angular structure of the emitting region, or a physical
constraint to the hot-spot latitude, can affect these slopes, making them
steeper (or flatter) at both the low- and high-amplification ends, in
individual sources and in the population. Through a few illustrative examples
we showed that these results are robust against the additional details of the
model that we introduced, as they are primarily a consequence of the behaviour
of $a(\theta)$.

A key result of our scenario is that the dichotomy of one-off FRBs and
repeaters is found to emerge naturally in the rotating hot-spot scenario,
provided that $a_{\rm thr}>1$, as a result of the extreme sensitivity of both
the amplification probability and range to the emission geometry. For random
distributions of the source orientation and hot-spot colatitude, and with the
simple assumptions of standard candle hot-spots and identically-repeating
sources, we find that (i) for observing times $T_{\rm obs}<100/{\cal R}$, the
probability of a source appearing as a repeating FRB is lower than for one-
offs, and both probabilities are low within the underlying population, (ii)
very active repeaters correspond to the rarest cases in which most of the
emitted bursts are amplified and, thus, detected. They require both the line-
of-sight and the hot-spot to lie at nearly identical and small angles
($\lesssim$ a few degrees) from the rotation axis: thus, their probabilities
are $<10^{-5}$. Since their properties make them easier to detect than more
typical sources, very active repeaters will appear in a much larger proportion
in FRB observations: a quantitative assessment of this selection effect is,
however, beyond the scope of this paper and will be addressed in future work;
(iii) as $T_{\rm obs}$ is increased, the relative probability of repeaters
steadily increases at the expense of one-off sources, which become
increasingly more unlikely.

For identically-repeating sources emitting identical events with energy
$E_{0}$, we evaluated the probability of repeaters and one-off FRBs vs.
redshift, for different values of $E_{0}$ and of $N_{\rm int}=f_{b}{\cal
R}T_{\rm obs}$. Adopting the CHIME detection threshold (${\cal F}_{\nu}^{\rm
thr}\approx$ 5 Jy ms), we find that sources at small distance,
$D^{2}_{L}(z)/(1+z)<{\rm a~{}few}\times E_{0}/(4\pi\nu_{c}{\cal F}_{\nu}^{\rm
thr})\sim$10-100 Mpc for $E_{0}=10^{36-37}$ erg and $\nu_{c}=1$ GHz, do
require little or no amplification for detection, hence they are likely to
appear as repeaters if they emit at least a few bursts during $T_{\rm obs}$.
The repeater probability drops quickly beyond such distances, at first leading
to a rise in the probability of one-off FRBs. The latter, though, peaks in a
narrow distance range beyond a few times $E_{0}/(4\pi\nu_{c}{\cal
F}_{\nu}^{\rm thr})$, and then shows a sharp drop at $z>0.05-0.1$ (depending
on the intrinsic energy $E_{0}$, and on $N_{\rm int}$). Beyond such redshifts,
the one-off probability drops with the same slope of repeaters, remaining
higher and at a constant ratio $\sim 7$ in our simple calculation. The
resulting probability of observing repeaters or one-off FRBs will be
determined by a trade-off between its dependence on $T_{\rm obs}$ and the
dependence on redshift. Moreover, observational biases may further affect the
observed population (e.g. CHIME/FRB Collaboration et al. 2021; McGregor &
Lorimer 2024).

The apparent energy budget of individual sources in our scenario is typically
smaller than their true one. However, in the ring-like geometry, we
demonstrate that there is always a special subset of “ideally-oriented”
systems in which we happen to observe most of the emitted flares with a
significant amplification: hence, their apparent energy budget is actually
larger than their true one. This is the subset to which the most active
repeaters, e.g. FRB 20121102 and FRB 20201124A, belong. The area in the
$(i,\xi)$-plane available for such active repeaters is determined by their
intrinsic burst rate ${\cal R}$ and distance, and by our detection threshold,
where a higher (lower) rate, smaller (larger) distance and lower (higher)
threshold all favour a wider (smaller) area. For the parameter range explored
here, inspired by the characteristics of the above-mentioned repeaters, the
probability of such configurations never exceeds $10^{-5}$ and can be much
lower, particularly at larger redshifts. An immediate implication is that,
even for the huge source population of cosmological NSs, such active repeaters
will appear in very small numbers, and only out to a limited redshift.

Finally, as a first step towards understanding the cosmological population, we
studied the case of identical sources with the same rate ${\cal R}$, and in
which events follow a (universal) seed energy distribution, e.g. a power-law
or a log-normal. We assumed that young ($\leq 10^{4}$ yrs of age) magnetars
represent the cosmic population of FRB sources, which then tracks the cosmic
SFR, and found they account well for the observed all-sky rate of FRBs
requiring (i) a characteristic energy budget of $10^{47}$ erg per source, for
a radio efficiency $\sim 10^{-5}$, perfectly in line with the magnetic energy
reservoir of magnetars and (ii) an intrinsic burst rate per source which is
$100-10^{3}$ times lower than observed in the most active repeaters. We note
this is yet another peculiarity of the latter sources clearly indicating that,
in order to match the properties of the FRB population, a realistic model will
have to allow for a range of ${\cal R}$-values, reflecting a range of birth
parameters and possibly their secular evolution. We conclude that the neutron
star self-lensing scenario introduced here provides a novel and promising
avenue for interpreting many of the most relevant and yet elusive features of
FRBs.

##

This work was funded by the European Union’s Horizon2020 research and
innovation programme under the Marie Skłodowska-Curie (grant agreement
No.754496). SD also acknowledges support from the GSC at the Goethe University
of Frankfurt, where this work was completed. RLP acknowledges support from
INAF’s research grant Uncovering the optical beat of the fastest magnetised
neutron stars (FANS) and from the Italian Ministry of University and Research
(MUR), PRIN 2020 (prot. 2020BRP57Z) Gravitational and Electromagnetic-wave
Sources in the Universe with current and next-generation detectors (GEMS). AP
acknowledges the contribution of the NextGenerationEU funds within the
National Recovery and Resilience Plan (PNRR), Mission 4 - Education and
Research, Component 2 - From Research to Business (M4C2), Investment Line 3.1
- Strengthening and creation of Research Infrastructures, Project IR0000026 –
Next Generation Croce del Nord. The visualization of functions in this
manuscript was carried out with Wolfram Inc. Mathematica, gnuplot, and
GeoGebra.

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##  Appendix A Parametrization of lensing for a finite-size source

In sec. 3 we presented an approximation (eq. 2) to the amplification of a
point-like source which holds in all lensing regimes (extreme, strong and
weak), here transcribed for ease of reference:

$a(r,\theta)=2~{}\displaystyle\frac{f(r)}{r\theta}~{}\left[1+\displaystyle\left(\frac{\theta}{\theta_{\rm
we}}\right)^{S}\right]^{1/S}\,.$ (A1)

Like eq. (1), which was presented in Bakala et al. (2023), eq. (A1) is
expressed as a function of the radial distance from the lens, $r$, and of the
angular separation from the caustic, $\theta$, instead of the more common, in
WDL studies, angular coordinate on the observer’s sky. Both eq. (1) and (A1)
account for the first two images of the source, i.e. the first direct image
(FDI) and first indirect image (FII), since higher order images increasingly
become negligible.

At the close distances we are considering, $r\leq 100$, the conditions for the
WDL are never fully satisfied, and the shape parameters $\theta_{\rm we}$ and
$S$, both functions of $r$, are required to properly approximate the
large-$\theta$ range, especially for NSs, as their surface can occult some of
the emission around them. In particular, this happens to the FII of sources at
small $r$ and close to $\theta=\pi$, i.e. exactly between the lens and the
observer, which would in principle provide the main contribution to their
amplification (see, e.g., Fig. 2 in Bakala et al., 2023). However, as $\theta$
grows, the periastron of the photon trajectory, i.e. the distance of closest
approach to the center of the lens, decreases, eventually becoming smaller
than the NS radius for these FIIs. Therefore, their contribution must be
discarded in all such cases. Fig. 13 shows the total amplification from the
first two images as a function of $\theta$ for three different emission radii
(solid lines, left panel), in the case of a NS with a $6~{}r_{g}$ radius: the
sudden amplification drop at intermediate $\theta$-values is due to the
occultation of the FII by the NS surface. This effect is apparent for $r=10$
(green solid curve), while it is already negligible at $r\gtrsim 20~{}r_{g}$,
as it takes place at larger values of $\theta$ and $r$, where the contribution
of the FII quickly drops.

Figure 13: Left: The approximate expression of eq. A1 for $a(\theta)$ (dashed
curves) vs. the exact solution, for three different emission radii. The step-
like decrease of the green solid curve at intermediate $\theta$-values is due
to the occultation of the FII by the NS surface (see main text) Right:
Behaviour of the functions $\alpha(r)$ and $S(r)$ appearing in our
approximations (eq. A1 and A2). The gray shaded area represents sources for
which even the FDI can be occulted by the NS surface.

In light of these considerations, we construct our approximated formula for
$a(\theta)$ to be valid at all angular separations from the caustic for
$r\gtrsim 10$ as a smoothly broken power-law: its derivative for
$\theta\rightarrow\pi$ must go to zero to mimic the behaviour of the FDI, and
its small-$\theta$ limit must follow the total amplification in the extreme
lensing regime (eq. 1). This results in eq. (A1), in which the transition
between the extreme and weak regime in amplification takes place at the angle
$\theta_{\rm we}$, which at close radii will differ from the weak-strong
transition angle $\psi_{\rm ws}(r)=2/\sqrt{r}$, as calculated in the
WDL121212That is, where the angular coordinate of the source on the observer’s
sky corresponds to the apparent angular radius of the first Einstein ring.:
for ease of comparison with the WDL, we set $\theta_{\rm
we}(r)\equiv\alpha(r)\,\psi_{\rm ws}(r)$. The values of $\alpha(r)$ and $S(r)$
can be then found at any radius by imposing the following conditions: (i)
$a(r,\pi)~{}=~{}g^{4}=\left(1-2/r\right)^{2}$ and (ii) $a(r,\theta_{\rm
we})=a_{\rm WDL}\left(\psi_{\rm
ws}(r)\right)=3\sqrt{5}/5\left(1-2/r\right)^{2}$, where $a_{\rm WDL}$ is the
amplification in the WDL approximation (see, e.g., sec. 3.1 in Bakala et al.,
2023). We show the behaviour of the two parameters $S$ and $\alpha$ in the
right panel of Fig. 13: as expected, at larger radii the value of $\alpha$
tends to 1, indicating that the approximation tends to the WDL, while it
increases at smaller radii, highlighting the deviation from the WDL in a
stronger gravitational field. For reference, we also shaded the region of the
graph at emission radii lower than $r\simeq 8.36$: below it, even the FDI
trajectories can have a periastron below the NS surface (for a star with
radius $6~{}r_{g}$; see, e.g., La Placa et al., 2020). Since the occultation
of the FDI in the shaded region would cut out the high amplification regions
behind the lens, sources at those radii are uninteresting for the purposes of
our FRB model.

The resulting approximate amplification curves for three different radii
(dashed lines) are compared to the exact solutions (solid lines) in the left
panel of Fig. 13: the step-like decrease for the case of $r=10$ mentioned
above, is the only feature not modelled by our approximation, and does not
impact on any of the conclusions in this work, as it mildly affects a region
of relatively low amplifications. With that exception, the agreement of eq. A1
with the exact solutions is excellent at all angular separations.

In sec. 3.2 we introduced the saturation of amplifications, for sources with
finite size $\ell$, at angular distances $\theta<\theta_{s}=\ell/r$ (eq. 4).
Combining that equation with (A1), the amplification of finite-size sources in
any lensing regime is well approximated by

$a_{\ell}(r,\theta)=2~{}\displaystyle\frac{f(r)}{r\theta_{s}~{}\left[1+\left(\displaystyle\frac{\theta}{\theta_{s}}\right)^{p}\right]^{1/p}}\left[1+\displaystyle\left(\frac{\theta}{\theta_{\rm
we}}\right)^{S}\right]^{1/S}\,,$ (A2)

where we set the ‘smoothness parameter’ $p=3$ to match the behaviour of the
exact solutions at values of $\theta$ comparable to the angular size of the
source (see sec. 3.3 of Bakala et al., 2023).

##  Appendix B Observed vs. intrinsic properties

Here we elaborate further on the concepts introduced in Sec. 5, where we
illustrated the crucial implication of gravitational lensing, i.e. the
substantial difference between a source’s apparent and real properties.

###  B.1 Luminosity function and source energetics for standard candle hot-
spot events

In Sec. 5 we showed that the probability distribution $P(S)$ translates to an
observed rate distribution, $F(S)$,

$F(S)={\cal R}_{b}P(S)=\displaystyle\frac{{\cal
R}_{b}}{\hat{S}_{0}}~{}\displaystyle\frac{P\left(S/\hat{S}_{0}\right)}{\displaystyle\int_{a_{\rm
thr}}^{a_{\rm max}}P(a)da}\,,$ (B1)

where ${\cal R}_{b}$ is the rate of observed events above the detection
threshold and $P(a)$ is normalized such that $\int_{a_{\rm min}}^{a_{\rm
max}}P(a)da=1$. Here $a_{\rm max}$ is the highest possible amplification and
$a_{\rm min}$ the minimum, typically $\lesssim 1$ (although special
configurations may occur in which all events get amplified, i.e. $a_{\rm
min}>1$; see, e.g., Sec. 4).

The threshold amplification for detection, $a_{\rm thr}$, depends on the
intrinsic energy of the events, on the source distance, and on the detector’s
sensitivity. Hence, its value is not known a priori. When all events from a
source are observable, i.e. $a_{\rm thr}\leq a_{\rm min}$, then the observed
event rate ${\cal R}_{b}$ will equal the intrinsic rate, ${\cal R}$. More
generically, though, only a fraction of bursts are expected to be amplified
above threshold, hence $a_{\rm thr}~{}>~{}a_{\rm min}$ and ${\cal R}_{b}<{\cal
R}$. Accordingly, we may define an apparent source luminosity as

$\dot{E}_{\rm app}=\int_{{\rm S}_{\rm thr}}^{{\rm S}_{\rm
max}}(f_{b}S)\left(\frac{F(S)}{f_{b}}\right)dS={\cal
R}_{b}\displaystyle\frac{\displaystyle\int_{{\rm S}_{\rm thr}}^{{\rm S}_{\rm
max}}S\displaystyle\frac{P\left(S/\hat{S}_{0}\right)}{\hat{S}_{0}}dS}{\displaystyle\int_{a_{\rm
thr}}^{a_{\rm max}}P(a)da}={\cal
R}_{b}\hat{S}_{0}\displaystyle\frac{\displaystyle\int_{a_{\rm thr}}^{a_{\rm
max}}aP(a)da}{\displaystyle\int_{a_{\rm thr}}^{a_{\rm max}}P(a)da}\,,$ (B2)

where, by definition, $S=a\hat{S}_{0}$. In the integrand, we have explicitly
accounted for a possible beaming of the emission, multiplying the observed
fluence by $f_{b}<1$, and dividing by $f_{b}$ the observed rate of bursts at
fluence $S$. This shows explicitly that beaming does not affect the estimated
power of a source. Along a similar line, the true source luminosity can be
expressed as

$\dot{E}_{\rm true}=\left(f_{b}\hat{S}_{0}\right){\cal
R}=\frac{\displaystyle{\cal R}_{b}\hat{S}_{0}}{\displaystyle\int_{a_{\rm
thr}}^{a_{\rm max}}P(a)da}\,,$ (B3)

where we adjusted the true isotropic fluence, $\hat{S}_{0}$, for beaming, and
converted ${\cal R}$ into the observed burst rate ${\cal R}_{b}$ (including
beaming), according to the general relation

${\cal R}_{b}=f_{b}{\cal R}\int_{a_{\rm thr}}^{a_{\rm max}}P(a)da\,,$ (B4)

valid as long as the direction of beaming is random at each event, and the
beaming angle is wider than the narrow range of directions over which lensed
rays can reach the observer ($\delta\gamma\approx 10^{-4}$ deg).

Note that, contrary to the case of the source power, beaming does affect the
estimated event rate of the source, ${\cal R}$. Therefore, both lensing and
beaming affect the estimated event rate of a source, but only lensing has an
impact on its energy budget. The latter can be easily quantified in this
simple example, by writing the ratio between the apparent and actual power
release

${\cal C}=\displaystyle\frac{\dot{E}_{\rm app}}{\dot{E}_{\rm
true}}=\displaystyle\int_{a_{\rm thr}}^{a_{\rm max}}aP(a)da\,.$ (B5)

For standard candle hot-spots, ${\cal C}$ is simply defined by the range of
observable amplifications and by the probability distribution $P(a)$.

###  B.2 Events with an intrinsic luminosity function

Here we generalize the calculation of the apparent and true source luminosity
(power release) to the case in which the events have an intrinsic luminosity
function, $P_{0}(S_{0})$, at the source. The apparent power is obtained by
multiplying a given fluence by the rate of observed events at that fluence,
and then integrating over all fluences, from the detection threshold to the
maximum. As for the true source power, we must first convert the intrinsic
probability $P_{0}(S_{0})$ into an intrinsic luminosity function,
$F_{0}(S_{0})=BP_{0}(S_{0})$, by means of an expression similar to Eq. 12,
only with ${\cal R}$ instead of ${\cal R}_{b}$. We then proceed to integrate
$F_{0}(S_{0})$ between its minimum and maximum intrinsic values. Hence

$\begin{cases}\dot{E}_{\rm app}=\displaystyle\int_{S_{\rm thr}}^{S_{\rm
max}}SF(S)dS=A\displaystyle\int_{S_{\rm thr}}^{S_{\rm max}}SdS\int_{S_{0,{\rm
l}}}^{S_{0,{\rm h}}}\displaystyle\frac{P_{0}(S_{0})}{S_{0}}dS_{0}P(a)\\\
\dot{E}_{\rm true}=~{}~{}~{}~{}Bf_{b}\displaystyle\int_{S_{0,\rm
min}}^{S_{0,\rm max}}S_{0}P_{0}(S_{0})dS_{0}\,.\end{cases}$ (B6)

In this case, the ratio ${\cal C}$ has a more complicated expression

${\cal C}=\int_{a_{\rm thr}}^{a_{\rm
max}}P(a)da\displaystyle\frac{\displaystyle\int
P_{0}(S_{0})dS_{0}}{\displaystyle\int
S_{0}P_{0}(S_{0})dS_{0}}\displaystyle\frac{\displaystyle\int
SP(S)dS}{\displaystyle\int P(S)dS}=\displaystyle\frac{\langle
S\rangle}{\langle S_{0}\rangle}~{}\int_{a_{\rm thr}}^{a_{\rm max}}P(a)da$ (B7)

which depends on the functional form of $P_{0}(S_{0})$, as well as on $P(a)$
and the range of observable amplifications. Note that, in deriving Eq. B7, we
also used Eq. B4 to express the ratio $A/B$.

##  Appendix C General constraints on the local (unlensed) population

The number of unlensed repeaters detectable at the CHIME sensitivity (Eq. 21)
is strongly dependent on the intrinsic energy per event ($E_{0}$) and the
intrinsic burst rate per source (${\cal R}$), both of which may be decaying
with time. By comparing the most active repeaters with the few local sources
that are known to repeat (e.g. SGR 1935+215 and FRB 20200120E), and with the
estimated value of ${\cal R}$ in Sec. 7.2, the presence of such evolutionary
effects appear to be likely.

Here we sketch this evolution as a simple power-law decay of the burst rate
with a characteristic time $t_{0}$, such that

${\cal R}(t)=r_{0}\left(1+\displaystyle\frac{t}{t_{0}}\right)^{-\alpha}\,,$
(C1)

where $r_{0}$ is the initial burst rate, and $t_{0}$ determines a plateau,
during which ${\cal R}$ stays nearly constant before decaying $\propto
t^{-\alpha}$. Given a local SN rate $\sim 10^{-4}$yr-1Mpc-1 and a 10% magnetar
fraction, the number of sources detectable as unlensed repeaters in a 1-yr
long CHIME survey ($\sim$ 2 days per source) can be calculated using Eq. 21.
We estimate this number to be less than 10 provided that $E_{0}\lesssim
10^{36}$ erg, which limits the horizon for unlensed events, and that ${\cal
R}(t)<1$ day-1 within $\sim 300/E^{3/2}_{0,36}$ yrs of the source birth, where
we normalized $E_{0}$ in units of $10^{36}$ erg.

The more evolved members (age $>300/E^{3/2}_{0,36}$ yrs) of the same
population of nearby sources will produce unlensed one-off events in one year
worth of CHIME data, which should provide at most a small contribution to the
observed all-sky rate $\sim 500$ day-1.

The emerging scenario, which was very simply outlined here, provides a
consistent - though necessarily approximate - picture of the FRB source
population. With a characteristic $E_{0}\lesssim 10^{36}$ erg, and a typical
rate ${\cal R}<$ a few per year per source at an age $>10^{3}$ yrs, the
population of nearby sources here identified is perfectly in line with the
observed properties of the two local FRBs, SGR 1935+215 and FRB 20200120E.
Moreover, the youngest (age $\lesssim$ 100 yrs) and most active (${\cal
R}\gtrsim 0.25$ hr-1) members of that population are rare and only become
apparent from larger distances, where some level of amplification is required
for detection.

Finally, a more realistic expectation is that FRBs have a (broad) distribution
of intrinsic energies, as opposed to a single $E_{0}$ value.
In particular, the most active repeaters may produce intrinsically more
energetic bursts than the rest of the population, e.g. due to a stronger
interior B-field. We note, though, that their absence in the local universe
would impose such repeaters to be outliers even in terms of their intrinsic
properties: in order to match the CHIME observations, we estimate that no more
than 1% of the young magnetar population may be characterized by
E${}_{0}\gtrsim 2\times 10^{37}$ erg. A more detailed population study is well
beyond the scope of this work and is in the focus of a future publication.