# Refractive lensing of scintillating FRBs by sub-parsec cloudlets in the
multi-phase CGM

Dylan L. Jow,2,3,6 Xiaohan Wu,2 and Ue-Li Pen1,2,3,4,5,6
1Institute of Astronomy and Astrophysics, Academia Sinica, Astronomy-
Mathematics Building, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan
2Canadian Institute for Theoretical Astrophysics, University of Toronto, 60
St. George Street, Toronto, ON M5S 3H8, Canada
3Department of Physics, University of Toronto, 60 St. George Street, Toronto,
ON M5S 1A7, Canada
4Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo,
ON, Canada N2L 2Y5
5Canadian Institute for Advanced Research, CIFAR program in Gravitation and
Cosmology
6Dunlap Institute for Astronomy & Astrophysics, University of Toronto, AB
120-50 St. George Street, Toronto, ON M5S 3H4, Canada E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We consider the refractive lensing effects of ionized cool ($T\sim
10^{4}\,{\rm K}$) gas cloudlets in the circumgalactic medium (CGM) of
galaxies. In particular, we discuss the combined effects of lensing from these
cloudlets and scintillation from plasma screens in the Milky Way interstellar
medium (ISM). We show that, if the CGM comprises a mist of sub-parsec
cloudlets with column densities of order $10^{17}\,{\rm cm}^{-2}$ (as
predicted by McCourt et al. (2018)), then FRBs whose sightlines pass within a
virial radius of a CGM halo will generically be lensed into tens of refractive
images with a $\sim 10\,{\rm ms}$ scattering timescale. These images will be
resolved by scintillating screens in the Milky Way ISM, and therefore are
expected to suppress scintillation. From this, we argue that positive
detections of FRB scintillation can constrain the properties of these cool-gas
cloudlets, with current scintillation observation disfavouring the cloudlet
model. We propose that sheet-like geometries for the cool gas in the CGM can
reconcile quasar absorption measurements (from which we infer the presence of
the cool gas with structure on sub-parsec scales) and the unexpected lack of
lensing signals from this gas thus far observed.

###### keywords:

††pubyear: 2022††pagerange: Refractive lensing of scintillating FRBs by sub-
parsec cloudlets in the multi-phase CGM–Refractive lensing of scintillating
FRBs by sub-parsec cloudlets in the multi-phase CGM

##  1 Introduction

Despite containing a substantial fraction of the total number of baryons in
the Universe, the physics of the multiphase gas in the circumgalactic medium
(CGM) of galaxies is poorly understood. Contrary to previous expectations
drawn from simple physical models, absorption line studies of the CGM on large
scales regularly detect small amounts of cool gas ($T\sim 10^{4}\,{\rm K}$)
with a small total volume, but a large covering fraction (see e.g. Prochaska &
Hennawi, 2009; Stocke et al., 2013; Lau et al., 2016). Fluorescent emission
profiles rule out the possibility of the cool gas being distributed in, e.g.,
a thin shell surrounding the galaxy (Cantalupo et al., 2014; Hennawi et al.,
2015). A natural conclusion from these observations is that galactic halos are
filled with tiny but dense structures of cool gas. One potential physical
process by which this could occur is the shattering picture proposed by
McCourt et al. (2018), in which clouds of pressure-confined, hot gas rapidly
fragment as they cool into smaller cloudlets with a characteristic length
scale of $\sim c_{s}t_{\rm cool}$, where $c_{s}$ is the sound speed for the
cool gas and $t_{\rm cool}$ is the cooling time. As McCourt et al. (2018)
argue, this universal length scale will lead, in the case of the CGM, to the
formation of many tiny cloudlets with a characteristic radius of $r_{c}\sim
0.1\,{\rm pc}$ and characteristic column density $N_{H}\sim 10^{17}\,{\rm
cm}^{-2}$. This mist formed of sub-parsec cloudlets of cool gas suffusing the
CGM can explain the ubiquitous detection of a low volume of cool gas with a
high covering fraction.

Vedantham & Phinney (2019) note that if the CGM is indeed filled with sub-
parsec scale cloudlets with $N_{H}\sim\,10^{17}\,{\rm cm}^{-2}$, then,
assuming some fraction of the gas is ionized, there may be observable
scattering of radio sources such as fast radio bursts (FRBs). Assuming a
uniform density profile of ionized plasma within the cloudlets, Vedantham &
Phinney (2019) show that cloudlets with size $r_{c}\sim\,1\,{\rm pc}$ may
produce diffractive scattering of FRBs with a scattering timescale of $\sim
1\,{\rm ms}$, which is broadly consistent with observed FRB scattering
timescales. However, observations of FRB scintillation effectively rule out
the possibility that the ubiquitously observed scattering tails in FRBs can be
due to scattering in the CGM of intervening galaxies (Masui et al., 2015;
Ocker et al., 2022; Sammons et al., 2023). Essentially, a scattering screen
somewhere along the line of sight of the FRB that produces a $\sim{\rm ms}$
scattering timescale will be resolved by the scattering screens responsible
for scintillation located within the Milky Way. It is assumed that when the
scintillation screen resolves the scattering screen, scintillation is
suppressed. Thus, a positive detection of FRB scintillation allows for
constraints to be placed on how far away the scattering screen can be from the
FRB. In the case of FRB 110523, the first FRB to be confirmed to scintillate
due to scattering in the Milky Way ISM, the scattering screen responsible for
the millisecond scattering tail is constrained to be within $44\,{\rm kpc}$ of
the FRB source, and, therefore, cannot be associated with any galaxy’s CGM
(Masui et al., 2015). Since a substantial number of FRBs have been confimred
to scintillate, it is unlikely that the observed scattering tails come from
diffractive scattering in the CGM. Moreover, McCourt et al. (2018) predict,
using basic physical arguments and hyrodynamic simulations, the characteristic
scale of the cool gas cloudlets to be $r_{c}\sim 0.1\,{\rm pc}$. In principle,
the diffractive scattering picture employed by Vedantham & Phinney (2019) also
rules out scattering by clouds of size $0.1\,{\rm pc}$, as the scattering time
induced by clouds of this rise rises above $0.1\,{\rm s}$, which is
inconsistent with observations. However, as the diffractive scale (the length
scale below which phase variations in the scattered wavefront lead to
diffractive scattering) is many orders of magnitude smaller than the size of
the cloudlets ($r_{\rm diff}\sim 10^{11}\,{\rm cm}$, Vedantham & Phinney,
2019), diffractive scattering will be highly dependent on the internal
structure of the ionized plasma within the cloudlets. If, indeed, the ionized
plasma is not effectively a Kolmogoroz turbulent medium with outer-scale
$r_{c}$, as proposed by Vedantham & Phinney (2019), then the observable
signature of diffractive scattering may be made arbitrarily different
depending on the model of the cloudlets’ internal structure.

Thus, despite the fact that sub-parsec cloudlets of cool gas in the CGM,
should they exist, will almost certainly scatter FRBs, the absence of any
confirmed detection of scattering associated with the CGM has thus far not
been taken as strong evidence for the lack of these cloudlets. In this work,
we will discuss the refractive scattering effects of sub-parsec, cool-gas
cloudlets. The refractive scale for scattering by these cloudlets is
effectively set by the size of the cloudlets themselves, $r_{c}$, and,
therefore, refractive scattering is not sensitive to the details of the
internal structure of the cloudlets. We will show that even if refractive
scattering leads to relatively small scattering times, e.g. down to
$10\,\mu{\rm s}$, – which would be challenging to disentangle from the
observed ms scattering of FRBs from their local environment – nevertheless,
refractive scattering in the CGM will lead to a suppression of FRB
scintillation. While scintillation has been used by several authors (e.g.
Masui et al., 2015; Ocker et al., 2022; Sammons et al., 2023) to argue that
the ms scattering tails observed in FRBs must be associated with the host
environment, we will go further and argue that there can be no significant
scattering of the FRBs along the line of sight for scintillating FRBs. In
particular, whereas it is typically assumed that if the scintillation screen
in the ISM resolves the primary scattering screen then scintillation is
suppressed, we will quantitatively investigate this effect to clarify the
magnitude of the suppression, and under what precise conditions this
suppression occurs. We will use this to argue that the ubiquitous existence of
cool-gas cloudlets of size $r_{c}\sim 0.1\,{\rm pc}$ and electron column
density $N_{e}\sim 10^{17}\,{\rm cm}^{-2}$ is disfavoured by current
observation evidence. Note, however, that since the amount of published FRB
scintillation data is small (e.g. Table 1 shows all of the currently published
FRB modulation indices), this argument is based on extrapolation from the
small number of data points available. A robust conclusion can only come from
further observational studies of FRB scintillation.

The paper is organized as follows: in Section 2, we introduce the multi-plane
lensing formalism that will be necessary to discuss the two-screen scattering
model for FRBs that are both scattered by the CGM and scintillate in the ISM.
In Section 3, we will quantitatively study the suppression of scintillation
when the CGM scattering screen is resolved by the scintillation screen, and in
Section 4, we will use the phenomenon described in the previous section to
place constraints on sub-parsec, cool-gas cloudlets in the CGM. While we argue
that there cannot be strong scattering of FRBs due to the CGM, as this would
lead to the total suppression of FRB scintillation, we note that FRB
scintillation does appear to be relatively weaker than pulsar scintillation.
That is, the few published modulation indices for scintillating FRBs appear to
be in the range $0.1<m<0.9$ (see Table 1), indicating that FRBs are generally
weakly scintillated, as opposed to pulsars, which have been observed to
undergo strong scintillation. Since presumably the same structures in the ISM
are responsible for both pulsar and FRB scintillation, it remains to be
explained why FRBs appear to scintillate more weakly than pulsars. While many
effects may lead to this modest suppression of scintillation in FRBs (for
example, nanoshot structures in the intrinsic burst profile of FRBs may lead
to lower modulation indices, as has been observed for giant pulses in the Crab
pulsar (Lin et al., 2023)), it remains a possibility that scattering in the
CGM produces a small number of refractive images, leading to a modest
suppression of the modulation index. We further discuss the implications of
this in Section 5.

##  2 Multi-plane Lensing

Understanding how scintillation due to scattering in the ISM is affected by
scattering due to intervening CGM halos requires a multi-plane lensing
formalism. Lensing is fully described by the Fermat potential, or time delay
function, between generic paths through the lens screens and the un-perturbed
path connecting source and observer. The time delay for a multi-plane lens
system, with $n$ lenses, is given by (Schneider et al., 1992; Feldbrugge,
2023)

$T(\bm{\theta_{1}},...,\bm{\theta_{n+1}})=\sum_{j=1}^{n}T_{j}(\bm{\theta_{j}},\bm{\theta_{j+1}}),$
(1)

where $T_{j}$ is the time delay between the $j^{\rm th}$ and the $(j+1)^{\rm
th}$ plane, given by

$T_{j}(\bm{\theta_{j}},\bm{\theta_{j+1}})=\frac{1+z_{j}}{c}D_{j\,j+1}\Big{[}\frac{1}{2}(\bm{\theta_{j}}-\bm{\theta_{j+1}})^{2}+\beta_{j\,j+1}\psi_{j}(\bm{\theta_{j}})\Big{]}.$
(2)

The relevant distances and angles are shown in Fig. 1, and $z_{j}$ refers to
the redshift of the $j^{\rm th}$ plane. The second term in the bracket is the
time-delay induced by the $j^{\rm th}$ lens and we have defined the geometric
terms

$\displaystyle D_{ij}$ $\displaystyle\equiv\frac{d_{0i}d_{0j}}{d_{ij}},$ (3)
$\displaystyle\beta_{ij}$
$\displaystyle\equiv\frac{d_{ij}d_{0\,n+1}}{d_{0j}d_{i\,n+1}}.$ (4)

Note that the $(n+1)^{\rm th}$ plane is the source plane, and
$\bm{\theta}_{n+1}$ is the angular position of the source. For a plasma lens,
which will be the focus of this paper, the induced time delay is given by

$\psi_{j}(\bm{\theta_{j}})=-\frac{d_{j\,n+1}}{d_{0\,n+1}d_{0j}}\frac{\Sigma^{e}_{j}(\bm{\theta_{j}})e^{2}}{m_{e}\epsilon_{0}\omega^{2}(1+z_{j})^{2}},$
(5)

where $\Sigma^{e}_{j}$ is the (relative) electron surface density of the
plasma in the lens plane, $\omega$ is the observed angular frequency of the
light, and $\omega(1+z_{j})$ is the angular frequency at the lens.

Figure 1: Lensing geometry for a multi-plane lens.

In this paper, we will be considering a double lens. For two lenses, the time
delay can be written as

$T=\frac{1+z_{2}}{c}D_{23}\Big{[}\frac{1+z_{1}}{1+z_{2}}\overline{D}\Big{(}\frac{1}{2}(\bm{\theta_{1}}-\bm{\theta_{2}})^{2}+\beta_{12}\psi_{1}(\bm{\theta_{1}})\Big{)}+\frac{1}{2}(\bm{\theta_{2}}-\bm{\theta_{3}})^{2}+\psi_{2}(\bm{\theta_{2}})\Big{]},$
(6)

where we have defined $\overline{D}\equiv D_{12}/D_{23}$. It is typical to
define some characteristic angular scale, which we will call $\theta_{*}$, so
that we can write the time delay in dimensionless coordinates,
$\bm{x_{1}}\equiv\bm{\theta_{1}}/\theta_{*}$,
$\bm{x_{2}}\equiv\bm{\theta_{2}}/\theta_{*}$, and
$\bm{y}\equiv\bm{\theta_{3}}/\theta_{*}$. Also, in this work, we will be
concerned with lensing where the first lens is much closer to the observer
than the second lens, i.e. $d_{01}\ll d_{02},d_{03}$, and in particular such
that $z_{1}\approx 0$. That is, we will restrict our attention to the regime
where $D\ll 1$ and $z_{1}=0$. Noting that $\beta_{12}=(1+\overline{D})^{-1}$,
to first order in $\overline{D}$ the time delay becomes

$\displaystyle T$
$\displaystyle=\frac{1+z_{2}}{c}\theta_{*}^{2}D_{23}\hat{T},$ (7)
$\displaystyle\hat{T}$
$\displaystyle=\overline{D}.\Big{[}\frac{1}{2}(\bm{x_{1}}-\bm{x_{2}})^{2}+\theta^{-2}_{*}\psi_{1}(\bm{x_{1}})\Big{]}+\frac{1}{2}(\bm{x_{2}}-\bm{y})^{2}+\theta^{-2}_{*}\psi_{2}(\bm{x_{2}}),$
(8)

where $\hat{T}$ is a dimensionless quantity that depends only on dimensionless
parameters.

In principle, the observed flux is given by an integral over the function
$\exp[i\omega T]$; however, in this work we will be primarily concerned with
the refractive effects of plasma lenses in the CGM. That is, we restrict our
attention to the refractive / geometric optics regime, in which the observed
flux is given by a discrete number of images and their magnifications. The
image locations and their magnifications are encoded in the lens map which
determines how points in the lens plane $\bm{x_{1}}$ are mapped to the source
plane $\bm{y}$, which we will call $\bm{y}=\bm{\xi}(\bm{x_{1}})$. The lens map
is determined by Fermat’s principle, or the stationary phase principle, which
states that the geometric images are given by solutions to the equations
$\nabla_{\bm{x}_{1}}T=0$ and $\nabla_{\bm{x}_{2}}T=0$. From this it follows
that

$\displaystyle{\bm{x}_{2}}={\bm{x}_{1}}+\nabla_{\bm{x}_{1}}\psi_{1}({\bm{x}_{1}})$
(9)
$\displaystyle{\bm{y}}={\bm{\xi}}({\bm{x}}_{1})={\bm{x}_{1}}+(1+D)\nabla_{\bm{x}_{1}}\psi_{1}({\bm{x}_{1}})+\nabla_{\bm{x}_{2}}\psi_{2}\big{(}{\bm{x}_{1}}+\nabla_{\bm{x}_{1}}\psi_{1}({\bm{x}_{1}})\big{)}.$
(10)

In other words, for a given source position $\bm{y}$, we can find the ray that
connects the source to the observer by inverting the equation
$\bm{y}=\bm{\xi}(\bm{x}_{1})$ to find the point on the first lens plane the
ray passes through. The point on the second lens plane that the ray passes
through is given Eq. 9. In general, for a single source position, there may be
many solutions to the lens equations, i.e. the lens may produce multiple
images. The magnification of an image located at ${\bm{x}_{1}}^{j}$ is given
by

$\mu^{j}=|{\rm det}\nabla_{\bm{x}_{1}}{\bm{\xi}}({\bm{x}_{1}}^{j})|^{-1}$ (11)

and the observed electric field is given by

$F({\bm{y}})=\sum_{{\bm{x}_{1}}\in{\bm{\xi}}^{-1}({\bm{y}})}\frac{1}{\sqrt{\nabla_{\bm{x}_{1}}{\bm{\xi}}({\bm{x}_{1}})}}e^{i\omega
T({\bm{x}_{1}},{\bm{x}_{2}}({\bm{x}_{1}}),{\bm{y}})}.$ (12)

Note that in general $\sqrt{\nabla_{\bm{x}_{1}}{\bm{\xi}}({\bm{x}_{1}})}$ may
be complex, so that each image may have a phase factor, in addition to the
phase induced by the time delay, known as the Morse phase. The observed
intensity is $I=|F|^{2}$.

##  3 Scintillation in the presence of a resolving lens

The basic effect which we wish to describe is the suppression of scintillation
in the ISM due to the presence of an extra-galactic lens which resolves the
scintillation screen. First, we will clarify a few key terms. “Scintillation"
refers to the presence of coherent modulation of the observed intensity of a
source as a function of frequency, $I(f)$. The modulation index, $m$, which we
will define quantitatively later, characterizes the size of the intensity
fluctuations, with $m=1$ indicating strong scintillation. Scintillation of
coherent radio sources such as pulsars and FRBs is known to be due to multi-
path propagation through plasma structures in the ISM. Thus, scintillation is
a lensing effect; turbulent plasma structures in the ISM lens the source so
that the observed flux is the result of an interference pattern between many
images. The term “scattering" is also used to describe multi-path propagation
and is equivalent to “lensing" in this context. Note, however, that a source
may be strongly scattered without necessarily scintillating. For example, if
the scattering time scale (i.e. the typical difference between arrival times
of scattered/lensed images) is larger than the burst duration of the source,
there will be no overlap between the images upon arrial at the observer, which
is a requirement for the interference that leads to scintillation. Instead,
the images will appear as a scattering tail in the time domain. Thus, while
FRBs are ubiquitously observed to have exponential scattering tails, not all
FRBs are observed to scintillate.

Fig. 2 shows a basic sketch of the effect we describe in the following
sections. In the absence of an intervening lens, scintillation arises due to
multi-path propagation through a nearby scintillation screen. However, in the
presence of a background resolving lens which itself causes the source to be
multiply imaged, each of these images is scintillated independently. If the
scattering time scale associated with the background lens is larger than the
scattering time scale associated with the scintillation screen, the result is
that the intensity spectrum is modulated on a frequency-scale that is much
smaller in the case of just the scintillation screen alone. If the
characteristic frequency scale is smaller than the frequency resolution of the
observation, then the intensity spectrum will appear less strongly modulated
due to the presence of the additional lens.

Figure 2: Diagram of the two-screen lensing. (a) When only the scintillation
screen is present, the multi-path propagation of light leads to intensity
modulations as a function of frequency, $I(f)$. The black intensity curves
show the intrinsic intensity modulation, whereas the blue curve shows the
observed intensity modulation taking into account finite frequency resolution
(the observed intensity is the intrinsic intensity smoothed over the frequency
resolution). If the characteristic scale of these modulations (the
“scintillation bandwidth") is smaller than the frequency resolution, then the
observed intensity is effectively the same as the intrinsic intensity. (b) In
the presence of a background resolving lens, an addition timescale is present,
due to the multi-path propagation through the lens, as well as the
scintillation screen. This leads to intensity modulations on a smaller
frequency scale, which, if below the frequency resolution, results in the
observed intensity modulations being suppressed relative to the case where the
background lens is absent.

###  3.1 Scintillation as a Gaussian process

In Section 3.2, we introduce a simple double lens model in order to
demonstrate how an additional lens can, under certain circumstances, suppress
scintillation. However, we would first like to give a basic argument to
clearly illustrate how this effect arises and why it is generic.

For a strongly scintillated source, the observed voltage can effectively be
described by a set of complex, normally distributed numbers, $V_{i}$, where
$i$ indexes the frequency bin (Salpeter, 1967; Johnson et al., 2012). The
observed intensity is

$I_{i}=|V_{i}|^{2}={\rm Re}[V_{i}]^{2}+{\rm Im}[V_{i}]^{2},$ (13)

where the distribution of $I_{i}/\sigma^{2}$ is $\chi^{2}(2)$, since ${\rm
Re}[V_{i}],{\rm Im}[V_{i}]\sim\mathcal{N}(0,\sigma^{2})$.

Defining $\delta T\equiv(I_{i}-\overline{I})/\overline{I}$, the modulation
index is defined by

$m\equiv\langle\delta T^{2}\rangle=\frac{{\rm
Var}[I_{i}]}{\overline{I_{i}}^{2}}=\frac{{\rm
Var}[I_{i}/\sigma^{2}]}{\overline{I_{i}/\sigma}^{2}}=1,$ (14)

where the last equality follows from the fact that
$I_{i}/\sigma^{2}\sim\chi^{2}(2)$ and $\chi^{2}(k)$ distributed variables have
a variance of $2k$ and mean of $k$. Thus, for a strongly scintillated source,
the modulation index (which measures the typical size of the intensity
fluctuations) is unity.

Now let us imagine that, in addition to a scintillation screen, there is a
background lens. That is, the background lens, on its own, results in $N$
distinct images each of which is then modulated by the scintillation screen.
We write the voltage and resultant intensity as

$\displaystyle V_{i}$ $\displaystyle=\sum_{j}^{N}V^{j}_{i},$ (15)
$\displaystyle I_{i}$ $\displaystyle=|\sum_{j}^{N}V^{j}_{i}|^{2},$ (16)

where, again, each of these $V^{j}_{i}$ is a complex, normal variable. Here
$i$ indexes the frequency bin, and $j$ indexes the images from the background
lens. Since a sum of Gaussians is Gaussian itself, the intensity remains
$\chi^{2}(2)$-distributed, and the modulation index is unity.

Now, however, say the actual intensity is a smoothed version of the underlying
intensity due to finite frequency resolution. That is, the observed intensity
is

$I_{k}=\Big{\langle}\Big{|}\sum_{j}^{N}V^{j}_{i}\Big{|}^{2}\Big{\rangle}_{i\in[k+dk,k-dk]},$
(17)

where $k$ indexes the re-binned frequency after smoothing. Expanding this
gives

$\displaystyle I_{k}$
$\displaystyle=\Big{\langle}\sum_{j}^{N}|V^{j}_{i}|^{2}+\sum_{j<\ell}V^{j}_{i}(V^{\ell}_{i})^{*}\Big{\rangle}_{i\in[k+dk,k-dk]}$
(18)
$\displaystyle=\sum_{j}^{N}\big{\langle}|V^{j}_{i}|^{2}\big{\rangle}_{i\in[k+dk,k-dk]},$
(19)

where the last equality holds if we assume that $V^{j}_{i}$ and $V^{\ell}_{i}$
are uncorrelated for $j\neq\ell$ (which holds when the scintillation screen
resolves the background lens), and we assume the frequency scale of the
intensity modulations due to the background lens is finer than the smoothing
scale, $dk$. This occurs when the scattering timescale of the background lens
is large copmared to $dk^{-1}$.

Now, since each $|V^{j}_{i}|^{2}$ is a $\chi^{2}(2)$ random number, it follows
that $I_{k}\sim\Gamma(k=NK,\theta=2/K)$ where $K$ is the number of frequency
bins we smooth over. From this we obtain

$m=\frac{{\rm
Var}[I]}{(\overline{I})^{2}}=\frac{4N/K}{(2N)^{2}}=\frac{1}{NK}.$ (20)

Thus, if the background lens produces $N$ images, the modulation index is
suppressed by at least a factor of $N$. The parameter $K$ represents the ratio
between the finite frequency resolution of the observation, $\Delta f_{\rm
res.}$ and the intrinsic scale of the frequency modulation due to the
background lens (assuming the latter is smaller than the former). A lens with
scattering timescale $\tau_{\rm scatt.}$ will induce intensity modulations on
a frequency scale of $\sim\tau_{\rm scatt.}^{-1}$, from which we obtain:
$K\sim\Delta f_{\rm res.}\tau_{\rm scatt.}$. Note that here we have assumed
that these lensed images are equally bright, which will generically not be the
case. Therefore, in detail, the amount of suppression will depend on not only
the number of images but also the relative flux contained in each image.
Nevertheless, the basic effect is clear: a background resolving lens
suppresses scintillation. The more images the lens produces, or the larger the
scattering time scale, the more the modulation index is suppressed.

###  3.2 A more realistic lens model

Here we will introduce a more realistic lens model that will capture the key
effects we are trying to illustrate. For simplicity, we will consider both
lenses to be one-dimensional and oriented in the same direction so that
${\bm{\theta}_{j}}\to\theta_{j}$. For the lenses, we will take a simple
sinusoidal phase screen for the nearby lens,
$\psi_{1}(\theta_{1})=-\alpha_{1}\sin(\theta_{1}/\theta_{s})$, and for the
second lens we will take a simple rational lens of the form
$\psi_{2}(\theta_{2})=-\alpha_{2}/(1+(\theta_{2}/\theta_{l})^{2})$. The
parameters $\theta_{s}$ and $\theta_{l}$ set the angular scales of the lenses,
and the $\alpha$’s set the amplitudes. We have studied rational lenses such as
$\psi_{2}$ extensively elsewhere (see e.g. Jow et al., 2021). We choose a
sinusoidal phase screen for the nearby lens in order to model scintillation in
the ISM, which has previously been described as being due to corrugated plasma
sheets (Pen & Levin, 2014; Simard & Pen, 2018), similar to a sinusoidal sheet.
Ultimately, we choose a sinusoidal potential because it allows us to generate
many images on the sky, thereby reproducing the Gaussian statistics of strong
scintillation, while keeping the number of free parameters small. We will
refer to the nearby lens, $\psi_{1}$, as the “scintillation screen", and
$\psi_{2}$ as the “background lens".

We will choose the reference angle to be the angular scale of the background
lens, $\theta_{*}=\theta_{l}$. Thus, the time delay for this model can be
written as

$\displaystyle T$
$\displaystyle=\frac{1+z_{2}}{c}\theta^{2}_{l}D_{23}\hat{T},$ (21)
$\displaystyle\hat{T}$
$\displaystyle=\overline{D}\Big{[}\frac{1}{2}(x_{1}-x_{2})^{2}-\theta^{-2}_{l}\alpha_{1}\sin(x_{1}/\lambda)\Big{]}+\frac{1}{2}(x_{2}-y)^{2}-\frac{\theta^{-2}_{l}\alpha_{2}}{1+x_{2}^{2}},$
(22)

where $\lambda=\theta_{s}/\theta_{l}$ is the ratio of the charateristic
angular scales of the scintillation screen and the lens.

Note that for a plasma lens, the amplitudes of the lens depends on the
frequency as $\alpha\sim f^{-2}$ (see Eq. 5). For the sake of simplicity, we
will take the amplitudes to be fixed, independent of frequency, as would be
the case for gravitational lenses. This makes the numerical calculations
easier as one only needs to solve the lens equation once to the find the
geometric images, as opposed to finding a new set of images for every
frequency. For the purposes of elucidating the effect of multiple images from
a background lens on scintillation, it will suffice to assume the images are
roughly stationary with frequency. In reality, however, there will always be
some frequency above which the lensed images disappear. We include the
frequency scalings of the relevant quantities in our discussion of CGM lensing
in Section 4.

It will now be convenient to re-parameterize in terms of the convergences of
the lenses

$\displaystyle\kappa_{1}$ $\displaystyle\equiv\alpha_{1}/\theta^{2}_{s},$ (23)
$\displaystyle\kappa_{2}$ $\displaystyle\equiv\alpha_{2}/\theta^{2}_{l},$ (24)

which are defined such that $\kappa_{j}={\rm
max}|d^{2}_{\theta_{j}}\psi_{j}(\theta_{j})|/2$. Note that $\alpha_{j}$ has
units of ${\rm rad}^{2}$, so that $\kappa_{j}$ is dimensionless. This is the
convergence that may be familiar from gravitational lensing. With this re-
parametrization, one can write

$\displaystyle\hat{T}=\overline{D}\Big{[}\frac{1}{2}(x_{1}-x_{2})^{2}-\lambda^{2}\kappa_{1}\sin(x_{1}/\lambda)\Big{]}+\frac{1}{2}(x_{2}-y)^{2}-\frac{\kappa_{2}}{1+x_{2}^{2}}.$
(25)

Thus, the dimensionless time delay, $\hat{T}$, depends only on five
dimensionless parameters: $\overline{D}$, $\lambda$, $\kappa_{1}$,
$\kappa_{2}$, and $y$.

Now, we would like to determine the typical timescales associated with both
the scintillation screen and the lens. Consider the case where $\kappa_{1}=0$,
so that the scintillation screen vanishes, and there is only the lens. The
typical angular separation between images is given by
$\Delta\theta_{2}\sim\kappa_{2}\theta_{l}$. It follows that the typical
scattering time induced by the lens is

$\tau_{l}=(1+z_{2})\frac{D_{23}\theta^{2}_{l}}{2c}\kappa^{2}_{2}.$ (26)

Similarly, when $\kappa_{2}=0$, and there is only the scintillation screen,
the typical angular separation between images is given by
$\Delta\theta_{1}\sim\kappa_{1}\theta_{s}$, so that the scintillation
timescale is given by

$\tau_{s}=\frac{D_{13}\theta^{2}_{s}}{2c}\kappa^{2}_{1}.$ (27)

Taking the ratio of these two scattering times, and noting that
$D_{13}/D_{23}=\overline{D}/(1+\overline{D})$, we define

$\rho\equiv\frac{\tau_{s}}{\tau_{l}}=(1+z_{2})^{-1}\frac{\kappa_{1}^{2}}{\kappa_{2}^{2}}\frac{\lambda^{2}\overline{D}}{1+\overline{D}}.$
(28)

Using this model, we can calculate the effect a background lens has on
scintillation. Fig. 3 shows a summary plot of the lensing computed for three
different values of $\rho$: 0.01, 1, 10. The other parameters are kept fixed
at $\kappa_{1}=200$, $\kappa_{2}=10$, $\lambda=1$. A large value of
$\kappa_{1}$ is chosen so that the sinusoidal phase screen produces many
($\sim 100$) images in order to mimic scintillation. We choose a value of
unity for $\lambda$, as the typical angular scale of a sub-parsec cloud at
cosmological distances will be $\theta_{l}\sim 0.1\,{\rm pc}/1\,{\rm Gpc}\sim
10\,\mu{\rm as}$, which is the same as the typical angular scale of the
scintillated images, $\theta_{s}\sim 0.1\,{\rm au}/1\,{\rm kpc}\sim
10\,\mu{\rm as}$. Keeping these parameters fixed and varying $\rho$ causes
$\overline{D}$ to change. We choose to keep the source redshift fixed,
$z_{3}=0.1$, and the ratio between the source and lens distance fixed,
$d_{03}/d_{02}=2$. Thus, changing $\rho$ is equivalent to changing the
distance to the scintillation screen, $d_{01}$. To set the absolute timescale,
we choose the scintillation timescale to be fixed at $\tau_{s}=10\,\mu{\rm
s}$. We choose a fixed source position of $y=\beta/\theta_{l}=5.7$. Note that
we have relabled the source position $\theta_{3}\to\beta$ to reduce the number
of indices floating around and to be consistent with the lensing literature.

The top row of Fig. 3 shows the lens map between the background lens plane and
the source plane, $\theta_{2}\to\beta$. When $\rho$ is small, the lens map is
dominated by the background lens, and one obtains the usual $S$-shape lens
map, typical for single-peaked lenses of this kind (see e.g. Clegg et al.,
1998; Pen & King, 2012). As we increase $\rho$, the sinusoidal oscillations
from the scintillation screen become more prominent. The second row of Fig. 3
shows the magnifications, $\mu$, and the relative time delays, $\tau$, between
the images produced by the lens. The black dots show the images that form when
only the scintillation screen is present, and the blue dots show the images
formed by both lenses. One can see that when $\rho$ is small, there are three
distinct clusters of images, corresponding to the three images formed by the
background lens. Each of these distinct images is independently split into
many images by the scintillation screen. Indeed, for $\rho=0.01$, one can
confirm that the scattering timescale associated with the background lens is
roughly one-hundred times larger than the scintillation timescale
($\tau_{s}\sim 10\,\mu{\rm s}$ and $\tau_{l}\sim 1000\,\mu{\rm s}$). When
$\rho$ increases to unity or greater, the scintillation timescale dominates
and the images formed by the background lens can no longer be distinguished.
The effect of this on the modulation can be seen in the third row of Fig. 3.
The black curve shows the intensity modulation as a function of frequency,
$I(f)$, for the scintillation screen alone. As expected, the black curve is
essentially the same across the columns since we have fixed the scintillation
timescale to be $\tau_{s}=10\,\mu{\rm s}$, which corresponds to a
scintillation bandwidth of $\delta f=(2\pi\tau_{s})^{-1}\sim 1\,{\rm MHz}$.
The blue curve shows the intensity modulation in the presence of the
background lens. When $\rho\geq 1$, the scintillation dominates and the
frequency scale of the modulations is again set by the scintillation
bandwidth. However, when $\rho<1$, the frequency scale of the modulations is
set by the scattering timescale associated with the background lens. In the
case of $\rho=0.01$, this corresponds to a frequency scale of $0.01\,{\rm
MHz}$. The green curve shows what happens when the observation has a finite
frequency resolution which is larger than the frequency scale of the intensity
modulations. To obtain the green curve, we convolve the blue curve with a
Gaussian with a width of $0.1\,{\rm MHz}$, which is a typical value for the
frequency resolution of radio observations (see Table 1). This leads to
intensity modulations which have a similar frequency scale as the un-lensed
black curves, but with a visibly reduced modulation index. To see this more
clearly, the bottom row of Fig. 3 shows the auto-correlation function of these
different curves, $\langle\delta T(f)\delta T(f+\Delta f)\rangle$, where
$\delta T\equiv(I-\overline{I})/\overline{I}$. The zero-lag of this function
is the modulation index $m$. Essentially, the addition of a new (larger) time
scale by the background lens leads to oscillations in the auto-correlation
function, which, when averaged over due to finite frequency resolution,
results in a suppressed modulation index. In particular, the modulation index
is suppressed by about a factor of two.

Figure 3: Lensing by the double lens with a sinusoidal phase screen and
rational lens with $\kappa_{1}=200$, $\kappa_{2}=10$, $\lambda=1$,
$z_{3}=0.1$, $d_{03}/d_{02}=2$, $\tau_{s}=10\,\mu{\rm s}$, and fixed source
position $y=\beta/\theta_{l}=5.7$. Columns show the results for different
values of $\rho=0.01,1,10$, which corresponds to varying the distance to the
sinusoidal screen, $d_{01}$. The top row shows the lens map from the
background lens plane to the source plane, $\theta_{2}\to\beta$. The grey
vertical line corresponds to the source position. The second row shows the
magnification and time delay of the images formed by this lens model. The blue
dots show the images formed by the background lens and scintillation screen
together, whereas the black dots show the images formed by the scintillation
screen on its own. The third row shows the intensity modulation as a function
of frequency, $I(f)$. Again the blue curves show the full lens model, whereas
the black curves show the intensity modulation induced by the scintillation
screen on its own. The green curve in the leftmost column shows the result of
the blue curve smoothed over some finite frequency resolution, $\delta
f=0.1\,{\rm MHz}$. The bottom row shows the auto-correlation function of these
curves, $\langle\delta T(f)\delta T(f+\Delta f)\rangle$, where $\delta
T\equiv(I-\overline{I})/\overline{I}$. The zero-lag of this auto-correlation
function is the modulation index, $m$.

Whereas Fig. 3 shows the suppression of the modulation index for a particular
source position, in general the amount of suppression will vary depending on
where the source is relative to the background lens. This is because the
images formed by the lens vary in relative time delay and magnification as a
function of source position. For example, while when the source is directly
aligned with the background lens ($\beta=0$), the relative time delay may be
maximized, the images have small magnifcations. Near the caustics of the lens,
the images become bright, but two of the three images nearly overlap in
position on the sky. Fig. 4 shows the modulation index, after smoothing over a
finite frequency resolution of $0.1\,{\rm MHz}$, as a function of source
position. Again the scintillation timescale is fixed at $\tau_{s}=10\,\mu{\rm
s}$, as well as $\kappa_{1}=200$, $\kappa_{2}=10$, $\lambda=1$, $z_{3}=0.1$,
and $d_{03}/d_{02}=2$. The different colour curves correspond to different
choices of the background lens scattering timesale, $\tau_{l}$. We see that
the modulation index is more suppressed the larger the background lens
scattering timescale is. However, even when $\tau_{l}=\tau_{s}=0.01\,{\rm
ms}$, there is still some suppression at the caustics of the lens, which are
shown in the plot by the grey vertical lines. The important point to note is
that while the exact amount of suppression may depend on the source position,
the cross section is not low.

Figure 4: The observed modulation index of the double lens with a sinusoidal
phase screen and rational lens with $\kappa_{1}=200$, $\kappa_{2}=10$,
$\lambda=1$, $z_{3}=0.1$, $d_{03}/d_{02}=2$, $\tau_{s}=10\,\mu{\rm s}$, as a
function of source position $y=\beta/\theta_{l}$. The modulation index is
obtained by computing the intensity modulation as a function of frequency,
$I(f)$, for arbitrary frequency resolution and then smoothing over a finite
frequency resolution of $\delta f=0.1\,{\rm MHz}$. The different curves
correspond to different scattering timescales associated with the background
lens, $\tau_{l}$. The vertical grey lines correspond to the caustics of the
background rational lens. Note that even when the scattering timescale of the
background lens is small, the presence of the caustics manifest as a
suppression of the modulation index.

So far we have been assuming that the background lens resolves the
scintillation screen. That is, a lens that produces geometric images with a
physical separation of $\ell$ in the lens plane has an angular resolving power
of $\theta_{\rm res.}=\lambda/\ell$. For our background lens with a scattering
disk of size $\Delta\theta_{2}$ at a distance of $d_{02}$, the angular
resolving power of the lens is

$\theta_{\rm
res.}=\frac{\lambda}{d_{02}\Delta\theta_{2}}\sim\frac{\lambda}{d_{02}\theta_{l}\kappa_{2}}.$
(29)

We have thusfar assumed that the background lens resovles the scintillation
screen, which is satisfied when the resolution of the background lens is
smaller than the spacing between the geometric images of the scintillation
screen; i.e., $\theta_{\rm res.}d_{12}<\Delta\theta_{1}d_{01}$. Noting that we
also assume the scintillation screen is close to the observer such that
$d_{12}\approx d_{02}$, this latter condition reduces to
$\lambda/\Delta\theta_{2}<d_{01}\Delta\theta_{1}$. An equivalent condition is
that the scintillation screen resolves the background lens. It is only when
this condition is satisfied that we may use geometric optics (a.k.a. the
Eikonal limit) to treat the double-lens system. Essentially, if the
scintillation screen does not resolve the images formed by the background
lens, then each image is scintillated in the same way, meaning the background
lens has no effect on the observed interference pattern. Alternatively, one
may say that if the background lens does not resolve the scintillation screen,
then the coherence length of the wavefront scattered by the background lens is
smaller than the scattering structures in the scintillation screen (Ocker et
al., 2022).

##  4 Lensing by the CGM

We demonstrated in the previous section that scintillation is suppressed in
the presence of a resolving lens. The effect arises simply from the fact that
the addition of a second lensing time scale that is larger than the
scintillation timescale results in coherent modulation of the intensity on a
frequency scale smaller than the scintillation bandwidth. If this new
frequency scale is also smaller than the frequency resolution of the
observation, the result is that the intensity modulations are smaller than
they would be in the absence of the background lens. We demonstrated this
effect using a simple rational lens, $\psi_{2}(x)\propto(1+x^{2})^{-1}$;
however, in general, the background lens may be significantly more
complicated. The purpose of this paper is to investigate the effect of sub-
parsec, cool-gas cloudlets ($T\sim 10^{4}\,{\rm K}$) in the CGM on FRB
scintillation. To that end, we need to describe the refractive scattering
properties of these cool gas cloudlets. In particular, we need to determine
how many geometric images are formed by these cloudlets and what the typical
time-delays between these images are. Following Vedantham & Phinney (2019), we
will assume a model in which the individual cloudlets have a uniform size,
$r_{c}$, and column density, $N^{c}_{e}$, with a volume-filling factor
throughout the CGM obeying a simple radial power law:

$f_{V}(r)=f^{200}_{V}\big{(}\frac{r}{r_{200}}\big{)}^{-\beta},$ (30)

where $r_{200}$ is the virial radius and $f^{200}_{V}$ is the volume-filling
factor of the cloudlets at the virial radius. As in Vedantham & Phinney
(2019), we will typically assume $f^{200}_{V}=10^{-4}$ unless otherwise
stated. Our goal in this paper is to place constraints on this model based on
observations of FRB scintillation.

###  4.1 Lenisng properties of CGM cloudlets

To understand the lensing properties of a halo of cool gas cloudlets, we first
want to understand lensing by a single cloud. For a single cloudlet lens of
size $r_{c}$ and electron column density $N^{c}_{e}$, the convergence of the
lens is given by

$\displaystyle\begin{split}\kappa_{c}&=\frac{d_{02}d_{23}}{d_{03}}\frac{N^{c}_{e}e^{2}}{m_{e}r^{2}_{c}\epsilon_{0}\omega^{2}(1+z_{2})^{2}}\\\
&=0.3\,(1+z_{2})^{-2}\Big{(}\frac{d_{02}d_{23}/d_{03}}{1\,{\rm
Gpc}}\Big{)}\Big{(}\frac{N^{c}_{e}}{10^{17}\,{\rm
cm}^{-2}}\Big{)}\Big{(}\frac{r_{c}}{0.1\,{\rm
pc}}\Big{)}^{-2}\Big{(}\frac{f}{1\,{\rm GHz}}\Big{)}^{-2}.\end{split}$ (31)

In order for a lens to form multiple geometric images, the convergence needs
to be of order unity or greater. Thus, it is already possible for a typical
cool-gas cloudlet that forms via the shattering mechanism described in McCourt
et al. (2018) to lens a background radio source into multiple images. An
ensemble of many such cloudlets will produce even more images.

The typical angular separation between refractive images for a lens of angular
size $\theta_{l}$ and convergence $\kappa_{c}$ is given by
$\theta_{l}\kappa_{c}$, so that the scattering time for a single cloudlet lens
is given by

$\displaystyle\begin{split}&\tau^{c}_{l}=(1+z_{2})\frac{D_{23}\theta^{2}_{l}}{2c}\kappa^{2}_{c}\\\
&=35\,\mu{\rm s}\,(1+z_{2})^{-3}\Big{(}\frac{d_{02}d_{23}/d_{03}}{1\,{\rm
Gpc}}\Big{)}\Big{(}\frac{N^{c}_{e}}{10^{17}\,{\rm
cm}^{-2}}\Big{)}^{2}\Big{(}\frac{r_{c}}{0.1\,{\rm
pc}}\Big{)}^{-2}\Big{(}\frac{f}{1\,{\rm GHz}}\Big{)}^{-4}.\end{split}$ (32)

Typical scintillation timescales are $\sim 10\,\mu{s}$, therefore even lensing
by a single cloudlet may lead to a suppression of scintillation following the
effect we describe in Section 3. In general, any given sightline through a CGM
halo will intersect many such cloudlets. Assuming a volume filling factor of
the form Eq. 30, the average number of cloudlets that a sightline through the
CGM with impact parameter $b$ intersects is given by

$\displaystyle\begin{split}N(b)&=\frac{2\pi r^{2}_{c}}{\frac{4}{3}\pi
r^{3}_{c}}\int_{-r_{200}}^{r_{200}}f_{V}(\sqrt{\ell^{2}+b^{2}})d\ell\\\
&=3f^{200}_{V}\frac{r_{200}}{r_{c}}\Big{[}\frac{(1+\tilde{b}^{2})^{1-\frac{\beta}{2}}}{\tilde{b}^{2}}{}_{2}F_{1}(1,\frac{3-\beta}{2},\frac{3}{2},-\frac{1}{\tilde{b}^{2}})\Big{]},\end{split}$
(33)

where $\tilde{b}=b/r_{200}$. Note that within the virial radius, $N(b)\sim
3f^{200}_{V}r_{200}/r_{c}$ for $\beta\lesssim 1$.

For an ensemble of cloudlets, projected onto the lens plane, the total column
density scales with the number of intersected cloudlets, $N_{e}\sim
N(b)N^{c}_{e}$. However, since for lensing what matters is the relative
electron surface density, the typical convergence of a column of cloudlets
will scale with the rms fluctuations of the column density, i.e.
$\kappa\sim\kappa_{c}\sqrt{N}$. Thus, we have the total scattering time from
an ensemble of cloudlets is $\tau_{l}\sim N\tau^{c}_{l}$, which is

$\displaystyle\begin{split}\tau_{l}\sim 10.5\,{\rm
ms}\,&(1+z_{2})^{-3}\Big{(}\frac{f^{200}_{V}}{10^{-4}}\Big{)}\Big{(}\frac{r_{200}}{100\,{\rm
kpc}}\Big{)}\Big{(}\frac{d_{02}d_{23}/d_{03}}{1\,{\rm Gpc}}\Big{)}\\\
&\Big{(}\frac{N^{c}_{e}}{10^{17}\,{\rm
cm}^{-2}}\Big{)}^{2}\Big{(}\frac{r_{c}}{0.1\,{\rm
pc}}\Big{)}^{-3}\Big{(}\frac{f}{1\,{\rm GHz}}\Big{)}^{-4}.\end{split}$ (34)

This refractive scattering time is only sensible when the lens in question
produces multiple images (i.e. there needs to be more than one image in order
for the time delay between images to be well defined). Thus, we would like to
know how many images an ensemble of cloudlets is likely to produce. For an
ensemble of one-dimensional lenses, each with convergence $\kappa$, spaced
uniformly along a line with a spacing given by $\theta_{l}$, the number of
additional images formed scales like $\kappa$, since $\kappa$ sets the maximum
angular separation, $\beta/\theta_{l}$, from the lens the source can attain
such that the lens forms more than one image. For an ensemble of two-
dimensional lenses, such as a ensemble of cool-gas cloudlets projected onto
the plane of the sky, the number of additional images will scale like
$\kappa^{2}$. Thus, the number of images formed by a halo of CGM cloudlets,
$N_{\rm im.}$, will scale like $N_{\mathrm{im.}}-1\sim N\kappa_{c}^{2}$, where
again $N$ is the typical number of cloudlets intersected by a given sightline.
It follows that

$\displaystyle\begin{split}N_{\mathrm{im.}}-1\sim
20\,&(1+z_{2})^{-4}\Big{(}\frac{f^{200}_{V}}{10^{-4}}\Big{)}\Big{(}\frac{r_{200}}{100\,{\rm
kpc}}\Big{)}\Big{(}\frac{d_{02}d_{23}/d_{03}}{1\,{\rm Gpc}}\Big{)}^{2}\\\
&\Big{(}\frac{N^{c}_{e}}{10^{17}\,{\rm
cm}^{-2}}\Big{)}^{2}\Big{(}\frac{r_{c}}{0.1\,{\rm
pc}}\Big{)}^{-5}\Big{(}\frac{f}{1\,{\rm GHz}}\Big{)}^{-4}.\end{split}$ (35)

Thus, for typical parameters of the CGM cloudlets, an FRB with a sightline
passing within $r_{200}$ of a galaxy will be lensed into roughly twenty
refractive images, with a lensing time-scale of $10.5\,{\rm ms}$. This would
lead to intensity modulation with a typical frequency scale of $10^{-3}\,{\rm
MHz}$, which is well bellow typical frequency resolutions. Thus, if the
observed cool gas in the CGM were to exist primarily in cloudlets of this
form, we would expect FRB scintillation to be strongly suppressed. Following
our calculation in Section 3.1, the modulation index will be suppressed by
roughly $N_{\rm im}\Delta f_{\rm res.}\tau_{l}\sim 10^{4}$ for a frequency
resolution of $0.1\,{\rm MHz}$.

So far, we have computed the refractive scattering properties of the CGM
cloudlets using simple scaling arguments. We can, however, verify these
results numerically by simulating the lens potential that an FRB with impact
parameter $b$ relative to the galactic centre of some galaxy would see. For a
given impact parameter $b$, we divide the sightline from $-r_{200}$ to
$+r_{200}$ into 40 boxes of dimensions $100r_{c}\times 100r_{c}$ transverse to
the line-of-sight, and a height of $0.05r_{200}$ along the line-of-sight. For
each box, we uniformly distribute $n$ cloudlets, where $n$ is a Poisson random
variable with mean given by $f_{V}(r)V_{\rm box}/V_{c}$, where $r$ is the co-
ordinate for the centre of the box, $V_{\rm box}=100r_{c}\times 100r_{c}\times
0.05r_{200}$, and $V_{c}=\frac{4}{3}\pi r^{3}_{c}$. In doing so, we ensure
that the volume filling density of the randomly generated cloudlets obeys Eq.
30. To determine the lens potential, we then project the generated ensemble of
clouds onto the plane of the sky, assuming that each cloudlet has a Gaussian
density profile of
$N_{e}({\bm{r}})=N^{c}_{e}e^{-2({\bm{r}}-{\bm{r}}_{i})^{2}/r^{2}_{c}}$, where
${\bm{r}}_{i}$ is the co-oordinate of the centre of the cloud.

Once we have generated the projected electron density and thereby the lens
potential via Eq. 5, we can compute the number of images formed by the lens
and the associated time delay. Fig. 5 shows the result of this calculation for
a galaxy with virial radius $r_{200}=100\,{\rm kpc}$, $\beta=0.2$, and an
impact parameter of $b=r_{200}$. We choose the cloudlet sizes to be
$r_{c}=0.1\,{\rm pc}$, the observing frequency to be $f=1\,{\rm GHz}$, and the
distance parameter to be $d_{02}d_{03}/d_{23}=1\,{\rm Gpc}$. We compute the
number of images and mean time delay between images for multiple sightlines in
a range of $[b-100r_{c},b+100r_{c}]$ and report the average of these values in
Fig. 5. In order to vary the number of clouds the sightline intersects, we
vary $f_{V}^{200}$, and computing $N(b)$ using Eq. 33, we plot the scattering
time and number of images as a function of the number of clouds intersected.
In doing so, we can confirm the scalings: $\tau_{l}\sim N\kappa_{c}$ and
$N_{\rm im}-1\sim N\kappa$. The blue curves in Fig. 5 show the numerically
computed values and the dashed lines show the analytic approximations in Eqs.
34 and 35.

Figure 5: Average scattering time (top) and number of refractive images
(bottom) formed by an ensemble of cool gas cloudlets as a function of the
average number of cloudlets intersected by a given sightline (Eq.33). We take
the cloudlets to have uniform size, $r_{c}=0.1\,{\rm pc}$, with a uniform
maximum column density, $N^{c}_{e}=10^{17}\,{\rm cm}^{-2}$, and a symmetric
Gaussian density profile. We take the galaxy to have a virial radius of
$r_{200}=100\,{\rm kpc}$ and the clouds to have a power-law volume filling
factor given by Eq. 30, with $\beta=0.2$. We vary the number of intersected
cloudlets by varying the overall amplitude of the volume filling factor,
$f^{200}_{V}$. We choose a distance parameter of $d_{02}d_{23}/d_{03}=1\,{\rm
Gpc}$ for the lensing geometry, an observing frequency of $1\,{\rm GHz}$, and
an impact parameter of $b=r_{200}$.

###  4.2 Constraints on CGM cloudlets

Thus far, we have demonstrated that when a lens produces multiple refractive
images that are optically resolved by the scintillation screen, the observed
modulation index of scintillation is suppressed when the scattering timescale
of the lens is larger than $\sim\Delta f_{\rm res.}^{-1}$, where $\Delta
f_{\rm res.}$ is the frequency resolution of the observations. In particular,
the modulation index is expected to be suppressed by a factor of $\sim N_{\rm
im}\Delta f_{\rm res.}\tau_{l}$, where $N_{\rm im}$ is the number of images
produced by the lens and $\tau_{l}$ is the scattering timescale associated
with the lens. In the previous section, we showed that spherical CGM cloudlets
of cool, ionized gas with a characterstic radius of $r_{c}=0.1\,{\rm pc}$ and
electron column density of $N_{e}=10^{17}\,{\rm cm}^{-2}$ (consistent with the
shattering picture proposed by McCourt et al. (2018)) will generically result
in refractive scattering of FRBs, yielding $N_{\rm im}\sim 20$ refractive
images with a typical scattering timescale of $\tau_{l}\sim 10\,{\rm ms}$.
Given that typical frequency resolutions of FRB observations are on the order
of $\Delta f_{\rm res.}=0.1\,{\rm MHz}$, or $\Delta f_{\rm
res.}^{-1}=10\,\mu{\rm s}$, then we can infer that if the CGM is ubiquitously
populated by sub-parsec cloudlets, then any FRB with a sightline that passes
within the virial radius of a galactic halo along its line of sight will not
scintillate, or will do so very weakly.

Fig. 6 shows potential constraints that may be placed on the physical
parameters of cool-gas, CGM cloudlets based on the preceding discussion. Using
Eqs. 34 and 35, we can place constraints on the physical size and column
density of the cloudlets given an observation of scintillating FRB at a given
redshift, $z_{s}$, whose sightline passes within the virial radius of an
intervening galactic halo. That is, if the inferred scattering time is greater
than $\Delta f_{\rm res.}^{-1}\sim 10\,\mu{\rm s}$, and the number of
refractive images is $N_{\rm im}>1$, then scintillation will be suppressed for
that FRB. The top panel of Fig. 6 shows the constraints on the electron column
density of the CGM cloudlets that could be derived from a scintillating FRB as
a function of source redshift: column densities above the grey and blue curve
yield refractive images with scattering times greater than $10\,\mu{\rm s}$
and $N_{\rm im}>1$, respectively. Thus, column densities above those two
curves for a given redshift would be ruled out by a scintillating FRB at that
redshift. As we have discussed, another criterion that must be met is that the
scintillation screen in the ISM be resolved by the refractive images (or vice
versa). Since pulsar scintillation observations reveal typical scattering
length scales of $\sim 1\,{\rm AU}$ (see e.g. Brisken et al., 2010), then
using Eq. 29, we also require $d_{01}\theta_{\rm res.}>1\,{\rm AU}$, shown by
the green curve in Fig. 6. Note, however, that generically whenever $N_{\rm
im}>1$, the scintillation screen is always resolved. The bottom panel of Fig.
6 likewise shows constraints one might obtain on the characteristic radius of
the cloudltes, $r_{c}$, for fixed column density $N_{e}$. Of course, in
general, the physical mechanism that forms and sustains these cloudlets will
cause the radius and column density of the cloudlets to be related to each
other. Nevertheless, for the values of $r_{c}=0.1\,{\rm pc}$ and
$N_{e}=10^{17}\,{\rm cm}^{-2}$ predicted by McCourt et al. (2018), we note the
detection of a scintillating FRB whose sightline passes through a galactic
halo at most source redshifts would disfavour the spherical cloudlet model.

Figure 6: Conditions under which FRB scintillation is expected to be
suppressed for an FRB at a given source redshift, $z_{s}$, with a sightline
passing within the virial radius of galactic halo. We assume the galactic halo
comprises an ensemble of cool-gas cloudlets of uniform radius $r_{c}$ and
electron column density $N_{e}$, with a volume filling factor of
$f^{200}_{V}=10^{-4}$ at the virial radius. Scintillation is suppressed when
the cloudlets produce many refractive images, $N_{\rm im}>1$, which produce
frequency modulations on a scale finer than the frequency resolution of the
observation, which we take to be $\Delta f_{\rm res.}=0.1\,{\rm MHz}$. This
latter condition corresponds to a scattering timescale of
$\tau_{d}>10\,\mu{\rm s}$. Inverting Eqs. 34 and 35, we compute the values of
$N_{e}$ for which scintillation is suppressed as a function of source redshift
for fixed $r_{c}=0.1\,{\rm pc}$ (top panel), as well as the values of $r_{c}$
for which scintillation is suppressed for fixed $N_{e}=10^{17}\,{\rm
cm}^{-2}$. The shaded regions show the excluded values of these parameters
given a positive detection of scintillation. The green curves show the
condition that CGM lens resolves the scintillation screen; however, note that
whenever $N_{\rm im}>1$, this condition is always satisfied. The horizontal
dashed line shows the fiducial values of the parameters $r_{c}$ and $N_{e}$ as
predicted by the shattering picture proposed by McCourt et al. (2018). The
vertical grey line shows the inferred redshift of FRB 110523, which was the
first FRB to be confirmed to scintillate due to multipath propagation in the
Milky Way ISM, with a modulation index of $m=0.27$.

We argue that current FRB scintillation observations – while not nearly
comprehensive enough to place precise constraints on the properties of cool-
gas cloudlets in the CGM – already disfavour the ubiquitous presence of such
cloudlets. In particular, Vedantham & Phinney (2019) show that most FRB
sightlines will pass within the virial radius of at least one galactic halo
(and, in some cases, many times more). Thus, if these sub-parsec cloudlets of
cool gas were ubiquitous, then our calculations suggest that FRBs should
rarely scintillate, and should do so only very weakly. Yet, FRB scintillation
does not seem to be a rare, or weak phenomenon. For example, Schoen et al.
(2021) find that within a sample of twelve FRBs, ten have observable
scintillation. Sammons et al. (2023) find that of a sample of nine FRBs, three
show positive evidence for scintillation, while two show ambiguous evidence
for scintillation. While these are by no means unbiased samples, and therefore
one must be careful in drawing quantitative conclusions about the relative
frequency of FRB scintillation, nevertheless, it is clear that FRB
scintillation is not a rare phenomenon. Table 1 shows a list of scintillating
FRBs with published modulation indices. Two of the FRBs listed, FRBs 20220912A
(Wu et al., 2023) and 20201124A (Main et al., 2023), have modulation indices
consistent with unity, and are thus, strongly scintllated, highly disfavouring
any amount of scattering along the line of sight at the observing frequencies.

Another case of particular relevance to this discussion is FRB 20190520B,
which has been confirmed to scintillate, although the low siganl-to-noise of
the burst makes it challenging to measure a precise modulation index (Ocker et
al., 2022). Recently, the sightline of this FRB was found to pass within the
virial radius of at least two foreground galaxy clusters, and, as a result,
the FRB has a particularly high dispersion measure (DM) given its redshift
(Lee et al., 2023). The fact that this FRB both scintillates and is known to
have excess DM due to passing through multiple CGM halos presents a challenge
to the existence of sub-parsec, cool-gas cloudlets in the CGM. Therefore, we
argue that the present observational status of FRB scintillation does not
support the ubiquitous presence of such cloudlets. More detailed analyses of a
larger sample of scintillating FRBs will be needed to make more robust
conclusions.

FRB | Publication | Type | DM (pm cm-3) | Inferred redshift | f (MHz) |  $\Delta f_{\rm res.}$ (MHz) |  $\Delta f_{s}$ (MHz) | $m$
---|---|---|---|---|---|---|---|---
110523 | Masui et al. (2015) | Single burst | 623.30 | <0.5 | 700 - 900 | 0.05 | 1.24 | 0.27
121101 | Hessels et al. (2019) | Repeater | 560.57 | 0.2 | 1100 - 1700 | 0.0039 | 0.058 | 0.25
180916.J0158+65c | Marcote et al. (2020) | Repeater | 348.76 | 0.03 | 1630 - 1760 | 0.0156 | 0.059 | 0.06
20220912A | Wu et al. (2023) | Repeater | 220 | 0.077 | 1100 - 1500 | 0.1 | 0.39 | 1
20190608B | Sammons et al. (2023) | Single burst | 338.7 | 0.118 | 1200 - 1400 | 0.1 | 1.4 | 0.78
20210320C | Sammons et al. (2023) | Single burst | 384.8 | - | 700 - 950 | 0.1 | 0.91 | 0.68
20201124A | Sammons et al. (2023) | Repeater | 410.83 | 0.098 | 695 - 735 | 0.01 | 0.14 | 0.35
20201124A | Main et al. (2023) | Repeater | 410.83 | 0.098 | 1270 - 1470 | 0.195 | 1.34 | 1
Table 1: Scintillating FRBs with published modulation indices. The inferred
redshifts for 20190608B and 20201124A are from Cordes et al. (2022). The
quantity $\Delta f_{s}$ refers to the measured scintillation bandwidth. The
frequency range, $f$ over which the observations were made, as well as the
frequency resolution, $\Delta f_{\rm res.}$, are reported.

##  5 Discussion

We have argued that the existence of a mist of sub-parsec, cool-gas cloudlets
throughout the CGM of galaxies is broadly inconsistent with observations of
FRB scintillation. Nevertheless, we know from absorption studies that cool gas
with a low total volume, but large covering factor is common in the CGM of
galaxies, suggesting that the cool gas is not diffuse throughout the CGM, but
exists in many isolated clumps. How do we reconcile these two observations?
The most straightforward possibility is that the cool gas does not form
roughly spherical cloudlets, but rather highly anisotropic structures, such as
sheets or filaments. Both cool-gas sheets and filaments are equally as
plausible as spherical cloudlets as an explanation for the low total volume
but high covering factor observed in absorption measurements, as these
absorption observations cannot distinguish between different geometries of
cool-gas structures. However, as has been shown, cool-gas confined to sub-
parsec cloudlets will yield strong lensing phenomena that have not been
observed. By relaxing the confinement of the gas along one or two dimensions
(i.e. by allowing the gas to form filaments or sheets) alleviates this
tension, as, for example, a sheet of ionized plasma will only lead to mulipath
propagation of an FRB when the inclination angle of the sheet is close to
aligned with the line-of-sight of the FRB. Filaments are a less promising
solution to the problem, as filaments still lead to large transverse phase
gradients in the scattered wavefront in at least one direction regardless of
the orientation of the filament. Thus, we propose that the commonly observed
cool ($T\sim 10^{4}\,{\rm K}$) gas in the CGM forms sheet-like structures, as
opposed to cloudlets. We do not, however, present here a plausible physical
picture for how these sheets might form and persist in the CGM; we simply
point out that this geometry may resolve the present observational conundrum.

Extrapolating from the few FRBs with published modulation indices (see Table
1), many FRBs appear to scintillate with modulation indices less than unity by
a factor of two or three. This is in contrast with scintillating pulsars. In
recent decades, it has been discovered that a large fraction of pulsars
scintillate due to scattering dominated by a single thin, highly anisotropic
plasma sheet in the ISM along the pulsar’s line of sight (Stinebring et al.,
2001; Walker et al., 2004; Brisken et al., 2010). These scintillating sheets
generally induce intensity modulations in pulsars in both the frequency and
time domain with modulation indices close to unity (see e.g. Stinebring et
al., 2022, for a survey of scintillating pulsars). A detailed analysis of the
scintillation of the Vela pulsar revealed a modulation index that was within
$0.01\%$ of unity (Johnson et al., 2012). Some pulsars are known to
scintillate only very weakly ($m\ll 1$). In the case of the Crab pulsar
($m<0.02$), the modulation index is known to be highly suppressed due to a
secondary scattering screen in the Crab nebula which is resolved by the
scintillation screen in the ISM (the same effect we have been describing for
FRBs) (Main et al., 2021). However, modulation indices with only a modest
suppression, as observed in FRBs, are less frequently seen in pulsars; that
is, pulsars are seem to scintillate either strongly, or very weakly. If the
same structures in the ISM are responsible for both pulsar and FRB
scintillation, as they presumably are, then this discrepancy in the relative
strength of scintillation between the two source populations is unexpected.

If the CGM comprises a network of sheets of cool gas with sub-parsec
thickness, then this may already provide an explanation for the fact that FRBs
appear to scintillate more weakly than pulsars. We have shown that the
suppression of the modulation index scales as the number of refractive images,
$\sim N_{\rm im}$. While lensing by a single CGM halo comprising an ensemble
of spherical cloudlets with sub-parsec radii may result in many tens of
refractive images (thereby effectively washing out any scintillation), an
ensemble of sheets with sub-parsec thickness may produce only a small number
of refractive images, as only a small number of the sheets will ever
contribute to the lensing. This follows from the fact that only sheets closely
aligned with the line-of-sight of the FRB will induce phase gradients large
enough to lens the FRB at all. Moreover, while absorption observations suggest
that such sheets must have sub-parsec thickness, the transverse
inhomogeneities of the sheets may have an entirely different characteristic
scale, leading to smaller scattering times. Thus, cool-gas sheets in the CGM
may well lead to only modest suppression of FRB scintillation, consistent with
the some of the FRB modulation indices published so far. A specific case of
interest is FRB 20201124A, which was found to have a modulation index
consistent with unity in the $1270-1470\,{\rm MHz}$ band (Main et al., 2023),
and a modulation index of $m=0.35$ in the $695-735\,{\rm MHz}$ band (Sammons
et al., 2023). Since plasma lenses increase in strength as $\kappa\sim
f^{-2}$, a relative suppression of the modulation index at lower frequencies
may indicate the presence of scattering. Further analysis of this burst would
be needed to confirm this picture.

Whether or not lensing explains the observed modulation indices of FRBs is
highly speculative. There are other possibilities for why FRB modulation
indices are modestly suppressed which do not invoke any kind of lensing; for
example, nano-structure in the intrinsic emission profile can suppress
intensity correlations in scintillation (Lin et al., 2023). However, nano-
structure can at most suppress scintillation by a factor of $1/\sqrt{3}$, and
in the case of the repeater FRB 20201124A, the spectra of nearby bursts are
found to correlate at $\sim 100\%$, suggesting that there is no intrinsic fine
spectral structure (Main et al., 2023). We hope, therefore, that this provides
additional impetus for the detailed study of FRB scintillation as a potential
probe of CGM physics and lensing in general.

##  6 Conclusion

Scintillation of FRBs is caused by scattering in plasma screens in the Milky
Way ISM. When a refractive lens along the line of sight forms multiple images
that resolve the scintillation screen, the amount of scintillation observed,
as measured by the modulation index, $m$, may be suppressed. This suppression
occurs when the scattering time associated with the lensed images leads to
intensity modulations on a frequency scale that is smaller than the frequency
resolution of the observation. Essentially, the intensity modulations induced
by the lens are smoothed over due to the finite resolution, leading to a
smaller modulation index than in the absence of the lens. In this paper, we
have shown that sub-parsec, cool-gas cloudlets in the CGM of the type proposed
by McCourt et al. (2018) will generically lens FRBs in such a way as to
suppress scintillation by the Milky Way ISM. Therefore, since most sightlines
are expected to pass through at least one CGM halo, and since FRB
scintillation is neither a rare nor weak phenomenon, we argue that the current
observations disfavour the existence of these cloudlets. We suggest a sheet-
like geometry for the cool gas in the CGM as a way to accommodate both
absorption studies, which demonstrate the existence of a cool-gas phase in the
CGM with a large covering fraction, and the lack of lensing signals from this
gas.

## Data Availability

No new data were generated or analysed in support of this research.

## Acknowledgements

We receive support from Ontario Research Fund—research Excellence Program
(ORF-RE), Natural Sciences and Engineering Research Council of Canada (NSERC)
[funding reference number RGPIN-2019-067, CRD 523638-18, 555585-20], Canadian
Institute for Advanced Research (CIFAR), Canadian Foundation for Innovation
(CFI), the National Science Foundation of China (Grants No. 11929301), Thoth
Technology Inc, Alexander von Humboldt Foundation, and the Ministry of Science
and Technology(MOST) of Taiwan(110-2112-M-001-071-MY3). Computations were
performed on the SOSCIP Consortium’s [Blue Gene/Q, Cloud Data Analytics, Agile
and/or Large Memory System] computing platform(s). SOSCIP is funded by the
Federal Economic Development Agency of Southern Ontario, the Province of
Ontario, IBM Canada Ltd., Ontario Centres of Excellence, Mitacs and 15 Ontario
academic member institutions. Cette recherche a été financée par le Conseil de
recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de
référence 523638-18,555585-20 RGPIN-2019-067].

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