# Stellar prospects for FRB gravitational lensing

Liam Connor  Cahill Center for Astronomy and Astrophysics, MC 249-17
California Institute of Technology, Pasadena, CA 91125, USA Liam Connor
<EMAIL_ADDRESS>Vikram Ravi  Cahill Center for Astronomy and
Astrophysics, MC 249-17 California Institute of Technology, Pasadena, CA
91125, USA

###### Abstract

Gravitational lensing of fast radio bursts (FRBs) offers an exciting avenue
for several cosmological applications. However, it is not yet clear how many
such events future surveys will detect nor how to optimally find them. We use
the known properties of FRBs to forecast detection rates of gravitational
lensing on delay timescales from microseconds to years, corresponding to lens
masses spanning fifteen orders of magnitude. We highlight the role of the FRB
redshift distribution on our ability to observe gravitational lensing. We
consider cosmological lensing of FRBs by stars in foreground galaxies and show
that strong stellar lensing will dominate on microsecond timescales. Upcoming
surveys such as DSA-2000 and CHORD will constrain the fraction of dark matter
in compact objects (e.g. primordial black holes) and may detect millilensing
events from intermediate mass black holes (IMBHs) or small dark matter halos.
Coherent all-sky monitors will be able to detect longer-duration lensing
events from massive galaxies, in addition to short time-scale lensing.
Finally, we propose a new application of FRB gravitational lensing that will
measure directly the circumgalactic medium of intervening galaxies.

fast radio bursts, cosmology, gravitational lensing

##  1 Introduction

Gravitational lensing is the deflection of light rays by intervening matter
inhomogeneities between a source and an observer. If the deflection angle is
sufficiently large, one can observe multiple images of the source. Each image
will traverse a different path, leading to arrival time delays between the
lensed copies. Therefore, variable sources and astrophysical transients allow
one to detect gravitational lensing in the time domain. The differential
arrival time of lensed images enable valuable cosmological applications, for
example measuring the Hubble constant $H_{0}$ with a technique known as time-
delay cosmography (Refsdal, 1964; Treu & Marshall, 2016; Suyu et al., 2017).

Gravitational lensing in the time domain has primarily been observed in
distant quasars (Vanderriest et al., 1989; Treu & Marshall, 2016), the
brightness of which fluctuates on human timescales due to their compact
emitting regions. There are also several gravitationally lensed events or
candidates from explosive transients (Oguri, 2019) such as supernovae (Kelly
et al., 2015; Rodney et al., 2021), gamma-ray bursts (GRBs) (Paynter et al.,
2021), and gravitational waves events from coalescing binary black holes.

Fast radio bursts (FRBs) are bright, brief radio transients whose exact
origins remain unknown (Petroff et al., 2019; Cordes & Chatterjee, 2019). FRBs
offer a uniquely precise probe of gravitational lensing in the time domain for
two reasons: They are ubiquitous, with volumetric rates that may exceed those
of core-collapse supernovae. And they are very short in duration, allowing for
extraordinary measurements of the lensing time-delay. Typical bursts are
$\sim$ millisecond duration, but radio telescopes can preserve electric field
information about the FRB on timescales of nanoseconds, well below the
already-narrow pulse widths. This is in contrast to other time-delay lensing
events, which are fundamentally limited in the delays—and therefore lens mass-
scales—to which they are sensitive. In gravitational lensing, the image that
arrives first will be brighter than subsequent copies. A schematic diagram of
FRB lensing is shown in Figure 1.

In the past several years, a number of groups have explored the application of
FRB gravitational lensing to cosmology and fundamental physics (Zheng et al.,
2014; Li et al., 2018; Wucknitz et al., 2021; Chen et al., 2021b). Muñoz et
al. (2016) outlined how searching for FRBs lensed on timescales of
milliseconds would constrain the fraction of dark matter in massive compact
halo objects (MACHOS), particularly in the mass range 20–100 M⊙. Relatedly,
FRBs have been proposed as a probe of the mass distribution function of
primordial black holes (PBHs) between 10 and 103 M⊙ (Zhou et al., 2022). These
methods are incoherent in that phase information about the electric field
(i.e. the burst’s recorded complex voltage data) was not considered. Such
methods have an identification problem because many FRBs repeat, so one must
distinguish a genuine lensing event from a distinct burst from the same
source. However, if voltage data—rather than just total intensity—of the burst
are preserved, lensing delays can theoretically be detected down to the
instrument’s inverse radio bandwidth ($B$ $\sim$ 1 GHz or $\Delta t$ $\sim$
10-9 s) (Eichler, 2017; Pen, 2018). Barring instrumental and propagation
effects, lensed copies of the same burst ought to have identical waveforms.
The same will not be true for intrinsically different bursts, whether from a
repeating source or from a different FRB source along a similar sight line.
The unprecedented access to time delays from nanoseconds to years means FRBs
could probe lens mass scales from Jupiter-like objects up to massive galaxies,
spanning many orders of magnitude in lens masses. Indeed, techniques for
coherent time-delay gravitational lensing of compact sources such as pulsars
and FRBs have been proposed for lensing by free-floating planets in the Milky
Way (Jow et al., 2020). For longer time delays, propagation effects due to
plasma, such as scattering, have deleterious effects on the coherent lensing
signal. In those cases, being able to spatially resolve the multiple images
using very-long baseline interferoemtry (VLBI) will enable new science for
lenses that are galaxy-scale and above. The large range of time delays, image
separations, and lens masses accessible to FRBs is shown in Figure 2.

Recently, these ideas have been developed more extensively and put into
practice by the CHIME/FRB collaboration (Kader et al., 2022; Leung et al.,
2022). The authors have built coherent methods for searching voltage data of
CHIME/FRB sources for lensing events, which they refer to as “gravitational
lens interferometry”. They are able to search for lensing delays between 2.5
ns and 100 ms, corresponding to $10^{-4}$–$10^{4}$ M⊙ lenses. They have
applied these techniques to CHIME/FRB data to constrain the fraction of dark
matter in PBHs using 172 bursts for which voltage data was preserved,
accounting for decoherence from scattering (Leung et al., 2022). This amounted
to 114 distinct sightlines, meaning a positive detection would have required a
very high cosmic abundance of PBHs.

Despite the clear value of having a collection of gravitationally lensed FRBs
from cosmological distances, it remains an open question how future surveys
should optimally search for them or how many they will detect. The purpose of
this work is to compare different telescope designs and to produce a realistic
forecast for how many lensed FRBs future and current experiments might find.
This allows us to discuss several science cases at a wide range of lensing
timescales (and thus lens masses). We first offer a basic formalism for
gravitational lensing in the time domain and calculate lensing optical depths
at different mass scales. We then forecast total FRB detection rates on
DSA-2000 (Hallinan et al., 2019), CHORD (Vanderlinde et al., 2019), and a
coherent all-sky monitor (for example, BURSTT (Lin et al.,
2022)111https://www.asiaa.sinica.edu.tw/project/burstt.php), finding they will
detect tens of thousands of new FRBs. We combine these rates and their
modelled redshift distributions with lensing optical depths to estimate the
number of gravitationally lensed FRBs that will be detected on a range of
timescales. We describe the science that can be done with lens masses of
$10^{-1}$–$10^{14}$ $M_{\odot}$. Finally we describe the application of FRB
lensing to the circumgalactic medium (CGM), providing a clean measurement of
halo gas properties along different lines of sight.

Figure 1: Diagram of fast radio burst gravitational lensing.

##  2 Gravitational lensing background

If a source is at a true sky position, $\bm{\beta}$, its image will appear at
$\bm{\theta}$, deflected by the angle $\bm{\alpha}$ in the presence of a
gravitational lens. The mapping between the true and apparent source position
is given by the lens equation,

$\boldmath{\beta}=\mathbf{\theta}-\mathbf{\alpha}(\mathbf{\theta}).$ (1)

The deflection angle, $\bm{\alpha}$, is given by an integral of the projected
surface mass density, $\Sigma(\bm{\theta})$ over angular position,

$\bm{\alpha}(\bm{\theta})=\frac{1}{\pi}\int\mathrm{d}\bm{\theta^{\prime}}\frac{|\bm{\theta}-\bm{\theta^{\prime}}|}{|\bm{\theta}-\bm{\theta^{\prime}}|^{2}}\frac{\Sigma(\bm{\theta^{\prime}})}{\Sigma_{cr}}.$
(2)

Here $\Sigma_{cr}$ is the critical surface density and is given by the
geometry of the lensing system,

$\Sigma_{cr}=\frac{c^{2}}{4\pi\,G}\frac{D_{s}}{D_{l}\,D_{ls}},$ (3)

where $D_{s}$, $D_{l}$, and $D_{ls}$ are the distances to the source, the
lens, and the distance between the lens and the source, respectively.
$\Sigma_{cr}$, and therefore the solution to the lens equation, also depend on
the constituents of the Universe because $D_{s}$, $D_{l}$, and $D_{ls}$ are
all angular diameter distances.

The Lens Equation is non-linear in $\bm{\theta}$, so there can be more than
one $\bm{\theta}$ that satisfies Eq 1. Hence, gravitational lenses produce
multiple images or multiple lensed copies in time. A special case is when the
source is directly behind the lens where $\bm{\beta}=0$. In that scenario, the
deflection angle is equal to the image position. That angle is known as the
Einstein radius, $\theta_{E}=\alpha$.

The next useful quantity for our purposes is the lensing optical depth, which
is the probability that a source at redshift $z_{s}$ is lensed. It will be an
integral of the cross sections ($\sigma\approx\pi\,\theta^{2}_{E}$) of all
lenses between the observer and the source.

$\tau(z_{s})=\int_{0}^{z_{s}}\mathrm{d}z_{l}\int
n(\sigma,z_{l})\,\sigma\frac{d^{2}V}{d\Omega dz_{l}}$ (4)

Here, $n(\sigma,z_{l})$ is the number density of sources at redshift with
cross-sectional area $\sigma$ and $\frac{d^{2}V}{d\Omega dz_{l}}$ is the
comoving volume element per redshift per steradian.

We are concerned with the rate of FRB lensing events, for which we must
incorporate the source redshift distribution. Assuming an FRB detection rate
per redshift (in the absence of lensing) of $\mathcal{R}_{det}(z_{s})$, the
rate of observed lensing events, $\mathcal{R}_{L}$, will be the following:

$\mathcal{R}_{L}=\int^{\infty}_{0}dz_{s}\,\mathcal{R}_{det}(z_{s})\int_{0}^{z_{s}}\mathrm{d}z_{l}\int\,B(\gamma)\,n(\sigma,z_{l})\,\sigma\frac{d^{2}V}{d\Omega
dz_{l}}.$ (5)

The parameter $B$ accounts for a phenomenon known as magnification bias
(Turner, 1980). Since gravitational lensing can magnify the intensity of
lensed copies, faint sources that are otherwise below a survey’s detection
threshold can be made observable. This increases the number of lensed objects
and must be accounted for when computing optical depth. Assuming a power-law
luminosity function, $N(L)\propto L^{-\gamma}$, and a singular isothermal
sphere (SIS) lens,

$B(\gamma)=\frac{2^{\gamma}}{3-\gamma}.$ (6)

The equation shows magnification bias is stronger for steeper luminosity
functions, i.e. large $\gamma$. This is because steep power-laws indicate an
abundance of low-luminosity sources that can only be observed in the presence
of lensing magnification for a flux-limited survey.

Figure 2: The phase space of FRB gravitational lensing for a source at $z=1$
and lens at $z=0.5$. The dashed black curves in the top and bottom panels show
the fiducial time delay and image separation, respectively, as a function of
lens mass. The solid shaded regions give examples of the types of lenses at
each mass scale.

Evidently, the main ingredients that impact the detection of FRB lensing will
be the abundance of lenses, the mass distribution of those lenses, and the
redshift distribution of FRBs that a given survey observes. Another important
factor for transient lensing will be sky coverage and observing strategy,
which is captured by the “time-delay selection function”.

####  2.0.1 Time-delay selection function

In the image-domain, only one image is needed to search for evidence of
gravitational lensing. But for time-domain events such as FRBs, GRBs, SNe, or
time-variable AGN, one must be pointing at the same patch of sky when the
lensed signal arrives. This poses a challenge for transient gravitational
lensing. We call the probability that the lensed copy will be in the
telescope’s field-of-view (FoV) $P_{\rm FoV}(\Delta t)$. This determines the
likelihood that a lensed image will be recorded by a time-domain survey. When
calculating the true detection rate of lensed FRBs with Eq 5, we must multiply
the integrand by $P_{\rm FoV}(\Delta t)$.

For a transit instrument with east west beamwidth, $\theta_{\rm EW}$, lensed
copies will be recorded if the delay is less than a beam crossing time,
$t_{\rm trans}$. If the delay is greater than a transit time but less than one
day, it cannot be detected. For delays longer than one day, the probability of
detecting the lensed copy is the fractional sky coverage, or
$\frac{\theta_{\rm EW}}{2\pi}$. Examples of such FRB surveys are CHIME/FRB
(CHIME/FRB Collaboration et al., 2018),
DSA-110222https://www.deepsynoptic.org/team-110, and CHORD (Vanderlinde et
al., 2019). The probability that the lensed copy will be in the beam is
therefore,

$P_{\rm FoV}(\Delta t)=\left\\{\begin{matrix}1&\Delta t\leq t_{\rm trans}\\\
0&t_{\rm trans}<\Delta t<1\,\rm day\\\ \frac{\theta_{\rm EW}}{2\pi}&\Delta
t\geq 1\,\rm day\end{matrix}\right.$ (7)

For CHORD, $P_{\rm FoV}(\Delta t\geq 10^{5}\,\mathrm{s})\approx 9\times
10^{-3}$ at the center of the band, assuming the instrument is parked at the
same declination. For a steerable all-sky survey such as DSA-2000, $P_{\rm
FoV}(\Delta t)=1$ for $\Delta t$ less than a pointing time. For longer time-
delays, $P_{\rm FoV}(\Delta t)$ will be roughly the fraction of time spent on
each patch of sky, or the primary beam size divided by $3\,\pi$ ($P_{\rm
FoV}(\Delta t\geq 10^{3}\,\mathrm{s})\approx 3\times 10^{-4}$ on DSA-2000). In
practice, CHIME/FRB, CHORD, DSA-110, and DSA-2000 will have a difficult time
detecting gravitational lensing events caused by massive galaxies unless
special survey strategy is undertaken. Instead, they will be able to search
for lensing by compact objects and halos with $M_{L}\lesssim
10^{8}\,M_{\odot}$.

An ultra-widefield FRB survey, which observes a similar region of sky at all
times, will be able to search for lensing delays up to the survey duration. We
refer to such surveys as coherent all-sky monitors, or CASMs, of which the
proposed experiment BURSTT is an example (Lin et al., 2022). However, such a
design will inevitably be less sensitive than DSA-2000 or CHORD and will
therefore probe a more nearby population of FRBs. In Section 4 we compare this
trade-off. Another way to access longer lensing time-delays is to point one’s
telescope at the north or south celestial pole such that the same field is
observed at all times or observe a circumpolar region with a tracking
telescope.

###  2.1 Lensing by a point mass

We start with the simplest possible mass distribution, that of a point mass.
We calculate both the Einstein radius and typical time-delay for a
cosmological source. This will be relevant for stellar lenses, primordial
black holes / MACHOS, and intermediate-mass black holes. For a point mass, the
Einstein radius scales as the square root of mass as,

$\theta_{E}=\sqrt{\frac{4\,G\,M}{c^{2}}\,\frac{D_{ls}}{D_{l}\,D_{s}}}$ (8)

$\approx 1.6^{\prime\prime}\times
10^{-6}\left(\frac{M}{M_{\odot}}\right)^{1/2}\left(\frac{D_{l}\,D_{s}/D_{ls}}{3\,\mathrm{Gpc}}\right)^{-1/2}$
(9)

To compute a time-delay between images, we follow Oguri (2019) by defining a
fiducial time-delay, $\Delta t_{fid}$, which is the difference in arrival time
between the unlensed line-of-sight and an image deflected by $\theta_{E}$. In
practice, the true observable between images $i$ and $j$ is the difference
between their time-delays, i.e. we measure $\Delta t_{ij}=\Delta t_{i}-\Delta
t_{j}$. It is still useful to consider the typical lensing delay timescale.
This is given by,

$\Delta t_{fid}\approx
20\,\mu\mathrm{s}\times(1+z_{l})\left(\frac{M}{M_{\odot}}\right).$ (10)

The point lens will produce images separated by the following,

$\theta_{\pm}=\frac{1}{2}\left(\beta\pm\sqrt{\beta^{2}+4\theta^{2}_{E}}\right),$
(11)

where $\beta$ is angular impact parameter. The time delay will be,

$\Delta
t=\frac{4\,GM_{l}}{c^{3}}\,(1+z_{l})\,\left(\frac{y}{2}\sqrt{y^{2}+4}+\log\left(\frac{\sqrt{y^{2}+4}+y}{\sqrt{y^{2}+4}-y}\right)\right),$
(12)

where $y\equiv\beta/\theta_{E}$ (Muñoz et al., 2016).

In order to calculate the lensing optical depth, one must usually account for
the fact that the area is an annulus between $y_{min}$ and $y_{max}$ rather
than simply a cross section of $\pi\theta^{2}_{E}$. The areal scale of the
lens is then,

$\sigma(z_{l},M_{l})=\frac{4\pi\,G\,M_{l}}{c^{2}}\,\frac{D_{l}\,D_{ls}}{D_{s}}\,\left(y^{2}_{max}-y^{2}_{min}\right).$
(13)

Here, $y_{max}$ is set by the largest acceptable image pair flux ratio. The
minimum value, $y_{min}$, is set by the minimum detectable time delay between
the two images. Unlike with GRBs (Paynter et al., 2021, e.g.) or previous
incoherent treatments of FRB lensing (Muñoz et al., 2016; Zhou et al., 2022),
we assume a coherent lensed search that can find delays below the pulse width
scale, $\Delta t<<t_{FRB}$ (Eichler, 2017; Wucknitz et al., 2021). This
renders $y_{min}$ negligible so we set it to zero going forward. The source
number density is,

$n(z_{l},M_{l})=\frac{\rho_{c}\,f_{l}\,\Omega_{c}}{M_{l}}(1+z)^{3},$ (14)

where $\rho_{c}$ is the critical density of the Universe, $\Omega_{c}$ is the
cosmological density parameter of cold dark matter at $z=0$, and $f_{l}$ is
the fraction of dark matter in compact lenses (e.g. in primordial black holes
or IMBHs). Combining these terms, we find that the lensing optical depth is
independent of the mass of compact halos. It is given by,

$\tau(z_{s})=\frac{3}{2}f_{l}\Omega_{c}H^{2}_{0}\int^{z_{s}}_{0}\mathrm{d}z_{l}\,B\,\frac{(1+z_{l})^{2}}{c\,H(z_{l})}\frac{D_{l}\,D_{ls}}{D_{s}}y^{2}_{max}.$
(15)

As a useful guide, the classic Press-Gunn approximation relates the cosmic
abundance of lenses to their mean optical depth as $\tau\approx\Omega_{l}$ for
high redshift sources, where $\Omega_{l}$ is the cosmological parameter for
that type of point-mass lens (Press & Gunn, 1973). For lower redshifts, the
scaling is closer to $\tau\approx\Omega_{l}\left(\frac{z}{2}\right)^{2}$
(Narayan & Bartelmann, 1996). Notably, the relation is independent of lens
mass.

In Figure 3 we plot lensing probability curves for two point mass scenarios as
a function of redshift. The dotted curve assumes 0.1$\%$ of dark matter is
made from compact halos and the dot-dashed curve assumes this number is
$10^{-4}$, both of which are currently permitted in the relevant lens mass
range. Each uses a magnification bias of 2.

###  2.2 Strong lensing by massive galaxies

The lensing optical depth for distant objects is dominated by the dark matter
halos of massive foreground galaxies. Lensing halos are typically taken to be
singular isothermal spheres (SIS) or ellipsoids (SIE). These objects are
parameterized by their Einstein radius given by,

$\theta_{E}=4\pi\frac{\sigma_{v}^{2}}{c^{2}}\frac{D_{ls}}{D_{s}},$ (16)

where $\sigma_{v}$ is the halos’ velocity dispersion. For reference, with
$\sigma_{v}=200$ km s-1 and $D_{ls}/{D_{s}}$=0.426, the Einstein radius is
roughly 0.5”, or a typical image separation of one arcsecond. The optical
depth is then,

$\tau(z_{s})=\int_{0}^{z_{s}}\mathrm{d}z_{l}\int
d\sigma_{v}B(\gamma)\,\phi(\sigma_{v},z_{l})\frac{d^{2}V}{d\Omega
dz_{l}}\pi\theta^{2}_{E}(\sigma,z_{l},z_{s})D^{2}_{l},$ (17)

where $\frac{d^{2}V}{d\Omega dz_{l}}$ is the differential comoving volume at
$z_{l}$ and $\phi(\sigma,z_{l})$ is the number density of galaxies with
velocity dispersion $\sigma$, also known as velocity dispersion function
(VDF).

To model the lensing optical depth from massive galaxies we use a model based
on a recent empirical galaxy VDF (Yue et al., 2022). Another quick
approximation for lensing probability is given by (Oguri, 2019),

$\tau(z_{s})=\frac{5\times 10^{-4}\,z_{s}^{3}}{(1+0.041z^{1.1}_{s})^{2.7}}$
(18)

Clearly, the probability that a given FRB is lensed by an intervening galaxy
is strongly dependent on redshift, rising as $z^{3}$ for $z\lesssim 1.5$. This
is apparent in the solid black curve of Figure 3. Therefore, any FRB survey
that hopes to detect gravitational lensing would benefit from detecting
distant bursts.

Figure 3: The modeled CHIME/FRB redshift distribution (green histogram, left
vertical axis) and the probability that a source at redshift $z_{s}$ is lensed
(right vertical axis). The lensing optical depth, $\tau(z_{s})$ increases
quickly with redshift such that most CHIME/FRBs have a very low chance of
strong lensing. We have assumed a magnification bias $B=2$.

###  2.3 Coherent gravitational lensing

Radio telescopes measure directly the electric field of incoming
electromagnetic waves, sampling voltages roughly one billion times per second
for $\sim$ decimeter wavelengths. FRB search pipelines “detect” this data by
effectively squaring the voltages. They then downsample in time and frequency
to a manageable data rate, and search the lower-resolution intensity data for
dispersed pulses. Many surveys now preserve the raw voltage data with a buffer
that can be triggered and saved to disk, allowing astronomers to analyze the
radio pulse’s waveform. This is true for current surveys such as CHIME/FRB
(Michilli et al., 2021), ASKAP (Bannister et al., 2019), DSA-110, and will be
true for nearly all upcoming surveys.

Having access to the waveform itself is a major advantage for gravitational
lensing (Eichler, 2017). If two pulses arrive from a similar sky position but
at different times, their voltage timestream can be cross-correlated to test
whether they are the same pulse (i.e. lensed copies of one another) and not
just repeat bursts from the same source. Crucially, this can be done on
timescales shorter than the burst width using an autocorrelation. A “single”
FRB can be correlated with itself, and one can search for power at non-zero
time-lags, down to delays of the reciprocal bandwidth (nanoseconds). Similar
techniques have been successfully applied to the voltage timestreams of giant
pulses, effectively descattering Galactic pulsars (Main et al., 2017). Despite
the theoretical limit of 1/$\Delta B$, for this paper we follow Wucknitz et
al. (2021) and assume a practical time-lag (lensing delay) lower limit of 1
microsecond.

There are a number of practical issues related to coherent gravitational
lensing searches. These include inverting the instrument’s polyphase
filterbank, removing dispersion measure (DM) with high precision, radio
frequency interference (RFI), and the deleterious effects of interstellar
scattering. However, these are beyond the scope of our work and we point the
reader to a detailed description and application of coherent lensing searches,
or FRB gravitational lens interferometry, by the CHIME/FRB collaboration
(Kader et al., 2022; Leung et al., 2022). We are here focused on detection
rates and applications to cosmology and fundamental physics for upcoming
surveys.

##  3 FRB surveys

###  3.1 Detection rates

Despite considerable advances in constraining the source counts and all-sky
rate of FRBs, forecasting detection rates on new surveys remains challenging
(Connor, 2019). This is due to disparate RFI environments and detection
pipelines between telescopes as well as the unknown frequency dependence of
the FRB rate. However, thanks to a large collection of FRBs from CHIME at
400–800 MHz and $\mathcal{O}(100)$ events from surveys near 1.4 GHz, we are no
longer limited by small number statistics. These surveys have also implemented
careful injection tests to measure pipeline completeness, thus we are in a far
better position to estimate future survey detection rates than we have ever
been in before. We also now have cumulative rate measurements spanning more
than three orders of magnitude in flux density threshold, giving us a handle
on the abundance of FRBs at 10 mJy up to 100 Jy (Niu et al., 2021; Shannon et
al., 2018; James et al., 2019). Figure 4 shows current FRB source counts and
effective detection rates for several surveys. FAST has shown that the
milliJansky radio sky is full of FRBs (Niu et al., 2021), which is promising
for sensitive future surveys such as DSA-2000 and CHORD.

Here we extrapolate from known surveys to estimate the detection rates at
DSA-2000, CHORD, and the Omniscope. This will allow us to forecast
gravitational lensing science that they can do. We start with the standard
simplified rate equation for a transient survey, which is the product of the
survey’s FoV, $\Omega$ and areal source density above some minimum flux
density, $n(>S_{min})$,

$\mathcal{R}=\Omega\,n(>S_{min}).$ (19)

Assume a survey $k$ has a system-equivalent flux density, $SEFD_{k}$,
$n_{p,k}$ polarizations, and radio bandwidth, $B_{k}$. We can now extrapolate
from a survey $l$ with a known detection rate of survey $\mathcal{R}_{l}$,
giving,

$\mathcal{R}_{k}=\mathcal{R}_{l}\,\frac{\Omega_{k}}{\Omega_{l}}\left(\frac{SEFD_{l}}{SEFD_{k}}\sqrt{\frac{B_{k}\,n_{p,k}}{B_{l}\,n_{p,l}}}\right)^{\alpha}.$
(20)

In Table 1 we list the parameters for current and upcoming telescopes; for
existing surveys we have estimated the FRB detection rates. For parabolic
reflectors, we assume $\Omega\approx
1.13\,\left(\frac{\lambda}{D}\right)^{2}$. Both DSA-2000 and CHORD utilize
ultrawideband receivers, forcing us to make a choice about our treatment of
the frequency dependence of FRB rates. Rather than modelling an FRB spectral
index, we can break the surveys up into two sub-bands that are assumed to find
different FRBs. This is partly because FRBs are often band-limited and because
the FoV mismatch across an ultrawideband survey: For CHORD, the primary beam
at the lowest frequencies is 25 times larger than at the top, so most FRBs
arriving in the beam at the bottom of the band will not be detected at the top
of the band, even if the pulses spanned $\sim$ 1.2 GHz. We combine empirical
rates at 1.4 GHz from FAST (Niu et al., 2021) and Apertif (van Leeuwen et al.,
2022) with the CHIME/FRB detection rates at 600 MHz. We extrapolate directly
from the CHIME rate (below 1 GHz) and from the FAST/Apertif rate (above 1 GHz,
denoted by subscript “$L$”). Breaking an ultra-wideband into two sub-bands,
the rate equation becomes,

$\mathcal{R}_{k}=\mathcal{R}_{CH}\,\frac{\Omega_{k}}{\Omega_{CH}}\left(\frac{SEFD_{CH}}{SEFD_{k}}\sqrt{\frac{B_{k}}{B_{CH}}}\right)^{\alpha}+\\\
\mathcal{R}_{L}\,\frac{\Omega_{k}}{\Omega_{L}}\left(\frac{SEFD_{L}}{SEFD_{k}}\sqrt{\frac{B_{k}}{B_{L}}}\right)^{\alpha}$
(21)

Survey | FoV (deg2) | SEFD (Jy) | Frequency (MHz) | $\mathbf{N}_{ant}$ | Diameter (m) | Rate (year-1) | $\left<\mathbf{z_{s}}\right>$
---|---|---|---|---|---|---|---
CHIME | 200 | 50 | 400–800 | 1024 | N/A | 500–1000 | 0.5
Apertif | 9 | 75 | 1220–1520 | 10 | 25 | 30–90 | 0.5
FAST | 0.019 | 1.33 | 1000–1500 | 1 | 500/300† | 4–40 | 1.35
DSA-2000 | 18–3 | 2.5 / 6† | 700–2000 | 2000 | 5 | 1000-4000 | 1.2
CHORD | 72–2.9 | 9 | 300–1500 | 512 | 6 | 1100-6000 | 1.0
CASM | 5000 | 50 | 400-800 | 5000 | N/A | 12500-25000 | 0.5
Table 1: The parameters for current and upcoming (italicized) FRB surveys. The
rate column gives the FRB detections per one year on sky, and not the
effective detection rate which requires multiplying by observing duty-cycle.
The final column, $\left<\mathbf{z_{s}}\right>$, is the mean redshift. This is
estimated from the DMs of current surveys and from modelling for the three
upcoming surveys (see Section 3.2).
$\dagger$ corresponds to an effective value for the FRB search.

####  3.1.1 DSA-2000

The DSA-2000 is a proposed wide-field radio camera that will have
2000$\times$5 m steerable antennas and will observe between 700–2000 MHz
(Hallinan et al., 2019). It will have a system-equivalent flux density (SEFD)
of 2.5 Jy and a field-of-view (FoV) given by,

$\Omega=10.8\,\left(\frac{\nu}{1100\,\mathrm{MHz}}\right)^{-2}\,\,\mathrm{deg}^{2}.$
(22)

Utilizing the full FoV for pulsar and FRB search requires forming and
searching a large number of beams, which is computationally challenging. This
number is given by $N_{beam}=\left(d_{max}/D\right)^{2}$, where
$d_{max}\approx 15.3$ km is the longest baseline and $D=5$ m is dish diameter,
giving $N_{beam}\approx 10^{7}$. For a reduced number of beams there is a
trade-off between the effective FoV that can be searched, $\Omega_{eff}$, and
effective sensitivity. If baselines longer than $d_{cut}$ are discarded, the
effective SEFD will be 2.5 Jy $\sqrt{f(d<d_{cut})}^{-1}$, where $f(d<d_{cut})$
is the fraction of baselines shorter than $d_{cut}$. The effective FoV at 1100
MHz will then be,

$\Omega_{eff}=min\left\\{10.6\,\mathrm{deg}^{2},N_{beam}\times
1.13\,\left(\frac{c/1100\,\mathrm{MHz}}{\,d_{cut}}\,180/\pi\right)^{2}\right\\},$
(23)

where $N_{beam}$ is the total number of beams that can be searched. We use a
$min$ operator because we cannot usefully search beyond the instrument’s
primary beam. Assuming Euclidean source counts, detection rate is maximized
when $d_{cut}$ is such that the full primary beam is tiled by $N_{beam}$. We
find that if DSA-2000 can afford to search $10^{6}$ beams, $d_{cut}$ should be
chosen such that the synthesized beamwidth is 11” and the effective SEFD is 5
Jy. For the current antenna configuration, this corresponds to $d_{cut}\approx
4.1$ km. For the remaining forecasts, we assume that DSA-2000 will have
SEFD$=$6 Jy and $\Omega_{eff}=10.6$ deg2 at 1100 MHz. We further assume that
DSA-2000 will only search for FRBs between 700 MHz and 1600 MHz.

Following Equation 21, we find that DSA-2000 will detect 830–4600 FRBs per
year. Over the course of its nominal five year survey (and an estimated four
years on sky), this would result in $\mathcal{O}(10^{4})$ localized FRBs in
which to search for gravitational lensing.

Figure 4: All sky FRB rates and survey detection rates. The left panel shows
the cumulative all-sky event rate of FRBs as a function of fluence threshold.
The long baseline in fluence provided by FAST and ASKAP in Fly’s Eye mode
allow for source counts constraints over nearly four orders of magnitude. The
right panel shows effective detection rates and frequency coverage for four
surveys currently on sky (solid boxes), as well as three upcoming surveys for
which we have run forecasts. The FRB detection rate here includes observing
duty-cycle. For example Apertif only observed $\sim$ 25$\%$ of the time so its
effective detection rate was one quarter of its rate per time on sky.

####  3.1.2 CHORD

CHORD is funded transit radio telescope that is preparing for construction
(Vanderlinde et al., 2019). The FRB search on CHORD will use 512$\times$6 m
antennas organized in a compact grid, comprising the array’s core. CHORD
employs ultra-wideband antennas covering 300–1500 MHz and has an expected SEFD
of 9 Jy. Its FoV is given by,

$\Omega=7.7\,\left(\frac{\nu}{1100\,\mathrm{MHz}}\right)^{-2}\,\,\mathrm{deg}^{2}.$
(24)

Though it has less collecting area and sensitivity than DSA-2000, the dense
antenna configuration of its core is optimal for searching the full FoV at
nominal sensitivity. It will only need to form and search 512 beams. Two
outrigger stations separated from the core by more than $10^{3}$ km allow for
VLBI localization of FRBs at milli-arcsecond-level precision. CHORD has an
unprecedented 5:1 radio bandwidth that will require sub-band searching, due to
the FoV mismatch as a function of frequency (Vanderlinde et al., 2019). Rather
than break the full band into just two sub-bands as we have done with
DSA-2000, we choose to sum FRB detections in three geometric sub-bands:
300–515 MHz, 515–880 MHz, and 880–1500 MHz. In the top sub-band, between
880–1500 MHz, we extrapolate from the FAST detection rate because of the match
between sensitivity and frequency between the two telescopes. In the bottom
two sub-bands, we extrapolate from CHIME/FRB and use $\alpha=1.5$. We find
that CHORD will discover 1000–6000 FRBs per year, all with voltage dumps and
precise localizations.

####  3.1.3 A coherent all-sky monitor (CASM)

The logical endpoint of the large-$N$ small-$D$ paradigm for radio
interferometers is an aperture array, in which radio receivers are pointed
directly at the sky without a light-focusing dish. A dense aperture array with
a large number of feeds could survey thousands of square degrees
simultaneously at high sensitivity, detecting an enormous number of FRBs. Such
an instrument would be a coherent all-sky monitor, so we refer to this
hypothetical survey throughout the remainder of the paper with the acronym
“CASM”.

Front-end electronics and digitization would be challenging for such a large
number of antennas, but beamforming would be made feasible by FFT beamforming
(Peterson et al., 2006; Tegmark & Zaldarriaga, 2009). For example, a feed with
a $\sim$ 70 deg opening angle would have a FoV that is $\sim$ 104 times larger
than that of Parkes and 25 times greater than CHIME. Each feed would have an
effective collecting area of $\lambda^{2}$. To build up the same collecting
area as CHIME at 600 MHz, roughly 25,000 feeds would be required.

Dense aperture arrays already exist, but not with the ability to use the full
FoV coherently. For example, the Electronic Multi Beam Radio Astronomy ConcEpt
(EMBRACE) was designed and constructed in the Netherlands in order to
demonstrate phased array technology for the Square Kilometer Array (SKA) (Kant
et al., 2009). EMBRACE has over 20,000 elements and roughly 160 m2 of
collecting area. It uses hierarchical analog beamforming. The dense aperture
array design once proposed for the SKA, and for which EMBRACE was a
pathfinder, will not be built. This has led some work to mistakenly infer that
SKA-Mid will detect as many as 107 FRBs per year (Hashimoto et al., 2020; Chen
et al., 2021a), based on the dense aperture array proposal.

There is at least one proposed FRB survey that falls within the dense aperture
array design. The Bustling Universe Radio Survey Telescope in Taiwan (BURSTT)
is an example of a coherent all-sky monitor that would have a high FRB
detection rate and outriggers for localization (Lin et al., 2022). It has been
considered explicitly in the context of coherent gravitational lensing. Beyond
the high FRB detection rate afforded by an ultra-widefield telescope, CASM
experiments can go after longer time-delays from lensing because a large patch
of sky can be observed continuously. Outrigger stations will provide VLBI
localizations that will enable new applications of FRB gravitational lensing
by massive galaxies. BURSTT-2048 is expected to have an SEFD of 600 Jy and
will spend a significant portion of its time near the north celestial pole
(Lin et al., 2022).

For the purposes of this paper, we will assume a future CASM survey that has
the sensitivity of CHIME/FRB. The SEFD is taken to be 50 Jy, observing in the
band 400–800 MHz, with a FoV of 5,000 deg2. In other words, a CHIME/FRB-like
survey but with 25 $\times$ the sky coverage. These survey parameters make
forecasting FRB detection rates very simple, because it will be roughly 25
times the CHIME/FRB detection rate; one does not have to assume anything about
spectral index nor source counts. The major uncertainty, instead, is whether
such an interferometer can be built to spec. We do not attempt to model system
performance uncertainty, and instead use CASM as a place holder for an all-sky
survey with CHIME/FRB sensitivity observing at 400–800 MHz. The inferred
detection rate of CASM would be 7,500–25,000 FRBs per year, reaching
$\mathcal{O}(10^{5})$ FRBs after 5–10 years on sky.

###  3.2 Redshift distribution

Figure 5: The modeled redshift distributions of DSA-2000, CHORD, and CASM. The
latter has the highest total detection rate, but DSA-2000 and CHORD will probe
higher redshifts thanks to their high sensitivity. The shaded histograms
represent each survey’s lensing probability density function.

The number of lensed FRBs in upcoming surveys will be a strong function of the
source redshift distribution: lensing optical depth increases as the cube of
redshift for $z_{s}\lesssim 1.5$ for strong galaxy lensing. It is imperative
that a forecast for the lensing detection rate includes a realistic model for
the FRB redshift distribution. To address this, we make use of two important
relationships. One is the dependence of observed extragalactic dispersion
measure (DMex) on redshift, and the other is the relationship between distance
and burst brightness.

Macquart et al. (2020) established that FRBs with higher DMex are typically
farther away, as expected if the IGM dominates dispersion. The so-called
“Macquart relation” is given by the following,

$\left<\mathrm{DM}_{IGM}(z)\right>\approx
865\,z\,\,\,\mathrm{pc}\,\mathrm{cm}^{-3}.$ (25)

While this approximate linear relationship holds, we caution that the
$\mathrm{DM}_{ex}$ dependence on $z$ is impacted by the host galaxy DM
distribution (James et al., 2022). We solve for $z$ with,

$\mathrm{DM}_{ex}=865\,z+\frac{100}{1+z}\,\,\,\,\,\,\mathrm{pc}\,\mathrm{cm}^{-3},$
(26)

which can be rearranged into a quadratic equation,

$865\,z^{2}+(\mathrm{DM}_{ex}-865)z+\mathrm{DM}_{ex}-100=0.$ (27)

The CHIME/FRB Catalog 1 (CHIME/FRB Collaboration et al., 2021) estimates
extragalactic DM by subtracting the expected contribution of the Milky Way
along that line of sight from the observed DM. We use these to solve for $z$
in Equation 27. The resulting redshift distribution for the first CHIME/FRB
release is shown in Figure 3

The relationship between FRB brightness and distance is critical for
predicting the redshift distribution for more sensitive surveys. Fortunately
for telescopes such as DSA-2000, CHORD, and FAST, there is evidence that
dimmer FRBs come from higher redshifts, meaning more sensitive surveys will
probe a deeper redshift distribution. This was not guaranteed, as a
sufficiently flat luminosity function would result in the brightest FRBs being
farther away (Macquart & Ekers, 2018). The positive correlation between
fluence and DM was demonstrated by Shannon et al. (2018) when comparing the
Parkes DMex distribution with that of ASKAP in Fly’s eye mode. The
relationship has also been borne out by the large DMs of FAST-discovered FRBs,
which can detect pulses down to tens of milliJanskys (Niu et al., 2021). The
exact mapping between observed fluence and redshift will depend on the FRB
luminosity function, the true source distribution in $z$, and selection
effects (Connor, 2019), but for our purpose this simple model is adequate. We
take the DM and redshift distribution of the hypothetical CASM survey to be
the same as CHIME/FRB, since it will have similar parameters to CHIME but with
25 times more FoV.

In a Euclidean volume, the mean FRB distance in a given survey scales as the
square-root of a survey’s sensitivity (Li et al., 2019), so cutting in half a
telescope’s SEFD will lead to it detecting FRBs that are, one average, $\sim$
40 $\%$ farther away. For a cosmological population, the relationship is
slightly weaker. We simulate $n(z)$ for DSA-2000 and CHORD based on the
modeled redshift distribution of CHIME/FRB. We assume FRBs have a constant
comoving volume density and a single cumulative power-law luminosity function
that has $\gamma=1$.

##  4 Results

| DSA-2000 | CHORD | CASM | CHIME/FRB
---|---|---|---|---
$N_{FRB}$ (5 years) | 4,000–16,000 | 4,400–24,000 | 50,000–100,000 | 2,500–5,000
$N_{lens,*}$ | 6–32 | 8–48 | 50–100 | 0.5–5
$f_{PBH}$ (0.1–100 $M_{\odot}$)† | $\leq 1.2\times 10^{-3}$ | $\leq 1.1\times 10^{-3}$ | $\leq 1.0\times 10^{-3}$ | $\leq 0.06$
$\Omega_{IMBH}$ ($10^{3}$–$10^{5}$ $M_{\odot}$)∗ | $\leq 10^{-2}$ | $\leq 10^{-2}$ | $\leq 10^{-3}$ | $\leq 10^{-1}$
$N_{lens,IMBH}$ ($10^{3}$–$10^{5}$ $M_{\odot}$)∗ | 0.1–7.3 | 0.1–7.0 | 0.3–7.2 | 0.01–0.29
$N_{lens,gal}$ ($10^{10}$–$10^{12}$ $M_{\odot}$)∗∗ | - | - | 5–40 | -
Table 2: The results of our FRB gravitational lensing forecasts for four
surveys, assuming five years of operation with $\sim$ 80$\%$ duty-cycle.
$\dagger$ The constraints on the fraction of matter in primordial black holes
without a detection of FRB microlensing.
$*$ We take $\Omega_{IMBH}$ to be the constraints on the density of
intermediate mass black holes in the absence of a lensing detection.
$N_{lens,IMBH}$ corresponds to the expected number of FRB lensing detections
with $\Delta t$ between 1–100 seconds assuming the rate from BATSE GRBs
(Paynter et al., 2021).
$**$ The expected number of detections of FRBs lensed by massive galaxies.
DSA-2000, CHORD, and CHIME/FRB cannot access time delays longer than a
pointing duration/transit time, so they will not see strong lensing by
foreground galaxies. The CHIME/FRB stellar lensing uncertainty is large
because it is not clear what fraction of total events will have preserved
voltage data and high DM.

###  4.1 Strong lensing by stars

Considerable attention has been paid to short-duration FRB gravitational
lensing by MACHOs and PBHs (Muñoz et al., 2016; Zhou et al., 2022; Kader et
al., 2022; Leung et al., 2022). To our knowledge, nobody has considered strong
lensing of FRBs by stars in the cosmological context. Stars at $z_{l}=0.5$
will have an Einstein radius of $\theta_{E}\approx
1.6^{\prime\prime}\times\left(\frac{M}{M_{\odot}}\right)^{1/2}$ for a source
at $z_{s}=1$. This corresponds to a physical impact parameter of roughly
$10^{16}$ cm, much larger than stellar radii. The geometry gives a fiducial
lensing timescale of 20 microseconds per solar mass, which is already within
the region of lags that CHIME/FRB is capable of searching (Kader et al., 2022;
Leung et al., 2022). If the cosmological stellar density parameter is
$\Omega_{*}\approx 0.0025$, then this is equivalent to
$f_{*}=\frac{\Omega_{*}}{\Omega_{c}}\approx 10^{-2}$ (Fukugita & Peebles,
2004; Aghanim et al., 2020). From Figure 3, we see that the optical depth for
such a population of point-masses is significant, even for a relatively nearby
CHIME/FRB catalog.

However, unlike PBHs and MACHOs, stars are highly concentrated near the cores
of galaxies. Wyithe & Turner (2002) have shown that the strong spatial
clustering of stars renders the Press-Gunn approximation inadequate for
cosmological stellar microlensing. In other words, if the isolated lens
assumptions is broken and $\tau$ approaches or exceeds 1, point-mass lenses
may be over-counted (Koopmans & Wambsganss, 2001). An extreme example is if
stellar lenses in the Universe were all aligned radially; the probability of
strong microlensing would be effectively zero. Wyithe & Turner (2002) found
that roughly 1$\%$ of sources at or beyond redshift 2 will be lensed by stars.
Their work, and related papers, were in the context of GRBs, where multiple
copies from stellar microlensing cannot be detected unless the delay is longer
than the burst width. FRBs do not have this issue. Here we consider how stars
in elliptical and spiral galaxies will impact FRB gravitational lensing for
realistic FRB redshifts, correcting for the spatial distribution of stellar
mass in galaxies. In Section 4.2, we discuss how lensing by stars can be
distinguished from other point-masses such as PBHs.

For our forecast of FRB stellar lensing, we make the conservative assumption
that only stars within a relatively small threshold impact parameter of a
foreground galaxy will contribute to the optical depth. The probability of
stellar lensing is roughly 1 for sources that pass within $\sim$ 3 kpc of a
Milky Way like galaxy. Such galaxies have a volumetric density of 0.01 Mpc-3,
which means the optical depth for stellar lensing would be $\tau_{*}\approx
3\times 10^{-4}\,\left(\frac{D_{l}}{1\,\mathrm{Gpc}}\right)$ for spiral
galaxies. For elliptical galaxies the impact parameter within which
microlensing optical depth is $\sim$ 1, is larger. Assuming sightlines within
6 kpc of elliptical galaxies are microlensed, we find $\tau_{*}\approx
10^{-3}\,\left(\frac{D_{l}}{1\,\mathrm{Gpc}}\right)$. This leads to the
striking fact that roughly one per thousand CHIME/FRB sources could be lensed
by stars on timescales between 1–100 microseconds.

Similar to strong lensing by galaxies, the majority of stellar microlensing
events will be due to elliptical galaxies. Wyithe & Turner (2002) find that
for spiral galaxies, roughly 70$\%$ of high-$z$ sources microlensed by stars
will also be strongly lensed by the galaxy itself. For elliptical/S0 galaxies,
this number is closer to half. Therefore, detecting an FRB with microsecond
lensing is a good indicator that it is also lensed on days to weeks timescales
and that its host galaxy will be lensed in an optical image. If the FRB source
is a repeater, then the lensing system could be monitored in order to
constrain $H_{0}$ (Wucknitz et al., 2021).

Current and future wide-area optical/IR surveys will prove excellent resources
for identifying the host galaxies of stellar microlenses. The extant Legacy
Survey delivers images over 14,000 deg2 with a 1.2″ point-spread function
(PSF) FWHM in the $g$, $r$, and $z$ bands, detecting objects to $g=24.0$ and
$r=23.4$ (Dey et al., 2019). Space-based imaging surveys with Euclid
(Scaramella et al., 2021) will span 15,000 deg2 across several visible and
near-IR bands, with detection limits of 26.2 mag in the visible and 24.5 in
the near-IR, and a PSF FWHM of 0.225″ in the visible and 0.3″ pixels in the
near-IR that undersample the PSF. The Nancy Grace Roman Space Telescope
(Roman; Doré et al., 2018) will cover only 2,000 deg2 in the near-IR with a
0.28″ PSF FWHM, but will detect objects to 26.6 mag. Finally, the SPHEREx
mission will deliver accurate redshifts of a few $\times 10^{8}$ galaxies over
the whole sky, with a detection limit of 18.4 mag in the near-IR in each
spectral channel.

Figure 6: The fraction of the cosmic stellar mass density recovered within the
cones of the Legacy Survey (solid purple), the Euclid (solid orange) and Roman
(solid red) wide-area surveys, and SPHEREx (dashed black). For SPHEREx we
assume that objects must be detected in at least one spectral channel, and for
the remainder of the surveys we assume that objects must be detected in at
least one imaging band. See text for further details on the surveys, and the
simulated galaxy catalog used to derive the results.

Figure 6 shows the fraction of the stellar mass density of the Universe
recovered within each survey footprint, as a function of redshift. We used the
survey detection thresholds to identify detectable galaxies at each redshift
in the output catalog of a recent semi-analytic galaxy formation model based
on the Millennium simulation rescaled to the latest Planck cosmology
(Henriques et al., 2020). We used predicted galaxy magnitudes that included
the simulated effects of internal dust extinction, and required detection in
at least one band for each survey (at least one spectral channel for SPHEREx).
These results suggest that a significant fraction of the host galaxies of
stellar microlenses will be identified within the Euclid and Roman survey
areas, potentially enabling modeling of the galaxy-scale lens mass
distribution using the lensed FRB host galaxy, as well as studies of the
distant FRB host galaxy itself. The former can be used to predict the
potential observation of future images of the FRB lensed on the scale of the
galaxy (e.g., Rodney et al., 2021).

Astronomers must note the risk of mis-identifying a lens galaxy as the FRB
host. Lensing searches should be done on microsecond timescales to
differentiate the host from the lens, especially if the extragalactic DM is
larger than expected. This will affect roughly $10^{-3}$ sight lines for
typical FRB distances. Since most lenses will be elliptical galaxies but most
FRB hosts appear to be star-forming, a tight association with an elliptical
galaxy may suggest lensing. Such localization precision can currently be
achieved for CHIME/FRB VLBI, DSA-110, ASKAP, and Meerkat(Rajwade et al.,
2022). Another practical issue is that scattering in the lens galaxy may
diminish the signal’s significance, particularly in the case of spiral
galaxies, but less so with ellipticals.

###  4.2 Primordial black holes

Figure 7: The current constraints on the mass of primordial black holes along
with future constraints from FRBs. If no FRB is lensed on timescales 1 $\mu$s
to 10 ms, $5\times 10^{4}$ sources could constrain $f_{PBH}$ to less than
$8\times 10^{-4}$ between 0.1–100 $M_{\odot}$. If CHIME/FRB can coherently
search for lensing in $\sim$ 500 unscattered FRBs, a non-detection will rule
out the region $f_{PBH}/M_{PBH}$ space that is grey and double hatched. The
three horizontal lines correspond to the value of $f_{PBH}$ that would produce
on average 1 microlensing event per year from PBHs for three upcoming FRB
surveys. We created this figure by modifying existing plotting code
https://github.com/bradkav/PBHbounds.

Primordial black holes (PBHs) are a theoretical class of black holes that form
in the early Universe. They may form via direct collapse from primordial
fluctuations, or though other mechanisms (Carr & Kühnel, 2020). Unlike black
holes that are formed after stellar collapse above the
Tolman–Oppenheimer–Volkoff limit of $\sim$ 3 M⊙, PBHs can be produced with a
wide range of masses. Constraining this mass spectrum is an active area of
research and uses a variety of inputs, including evaporation timescale, small-
scale CMB fluctuations, and Ly$\alpha$ forest (Carr & Kühnel, 2020). For
simplicity, we consider here only a monochromatic mass distribution (MMD) such
that all PBHs have the same mass, $M_{\rm PBH}$.

We take $f_{\mathrm{PBH}}\equiv\frac{\Omega_{\mathrm{PBH}}}{\Omega_{c}}$ to be
the fraction of cold dark matter that resides in primordial black holes, and
estimate the region of $f_{\mathrm{PBH}}/M_{\mathrm{PBH}}$ parameter space
that current and upcoming FRB surveys will be able to search. In Figure 7 we
show constraints on the abundance of primordial black holes assuming each
survey will be able to detect lensing events with delays from microseconds to
milliseconds ($M_{L}\sim\,0.1-100\,M_{\odot}$).

We find that a non-detection from 5 years on sky with DSA-2000, CHORD, and
CASM produce a similar upper-limit of $f_{\mathrm{PBH}}\lesssim 10^{-3}$.
While the CASM considered here would find many more FRBs, the deeper redshift
distribution of DSA-2000 and CHORD result in a similar mean lensing optical
depth. We plot the value of $f_{\mathrm{PBH}}$ that would produce an average
of 1 lensed FRB per year for DSA-2000 (brown, dotted), CHORD (green, dot-
dashed), and CASM (purple, dashed).

In Zhou et al. (2022), the authors calculate the $f_{\mathrm{PBH}}$
constraints for the non-detection of gravitational lensing in 593 CHIME/FRBs.
However, their limits are weakened by the (then valid) assumption of
incoherent lensing searches, where the minimum lensing delay is set by the
CHIME FRB pulse width. They could not constrain PBH masses that are less than
roughly 10 $M_{\odot}$. The CHIME/FRB collaboration has now produced the first
observational constraints from coherent gravitational lensing (Kader et al.,
2022; Leung et al., 2022). They have used the voltage data for 172 bursts from
114 independent sightlines to constrain the fraction of dark matter in PBHs
with masses between $10^{-4}$ and 104 M⊙. They consider the impacts of
scattering from plasma local to the source and conservatively assume that high
DM events have significant dispersion in the host galaxy, putting DM $>$ 500
pc cm-3 at smaller $z_{s}$. Due to memory constraints in their data buffer,
voltage data for sources with DM $>$ 1000 pc cm-3 are not included in the
sample (Leung et al., 2022). The authors place an upper limit of 0.8 on
$f_{\mathrm{PBH}}$ for $M_{\mathrm{PBH}}\sim 10^{-3}$ M⊙.

CHIME/FRB has now detected over 3000 FRBs, so just $\sim$ 3$\%$ have gone into
constraining $f_{\mathrm{PBH}}$ (Leung et al., 2022). If CHIME/FRB can search
coherently for lensing events using the preserved voltage data of $\sim$ 500
FRBs, the PBH constraints will be given by the double-hatched grey region
shown in Figure 7. We have assumed that FRBs with DM$>$1000 pc cm-3 or
$z_{s}\gtrsim 0.8$ do not contribute to the optical depth, because CHIME/FRB
cannot store their voltages. For 50,000 FRBs with the same redshift
distribution as CHIME/FRB, the grey singly-hatched block shows the region of
$f_{\mathrm{PBH}}/M_{\mathrm{PBH}}$ parameter space that will have been ruled
out (i.e. several years of observing with CASM). Note our upper-limits are
less conservative than those of the CHIME/FRB collaboration because we have
not included instrumental and propagation effects (Kader et al., 2022; Leung
et al., 2022).

There is a further practical problem associated with searching for point-mass
lensing events. Lensing by stars will likely dominate in the microseconds lag
range, as explained in the previous section. Therefore, with only a time-delay
measurement, it will be difficult to know if a PBH has been detected rather
than a stellar lensing event in an intervening galaxy. This degeneracy can be
mollified—if not fully eliminated—by noting that PBHs ought to be less
concentrated than stellar mass, and they will be more uniformly distributed in
the cold dark matter halos of galaxies. Thus, a strong microlensing event that
occurs within $\sim$ 10 kpc of the host galaxy center will likely be due to a
star. Constraints on $f_{PBH}$ could be limited to sources with larger impact
parameters. Using the spatial offset information requires that the telescope
has $5^{\prime\prime}$ localization precision or better. At present, CHIME/FRB
cannot achieve such localizations, so if there were a positive detection
between 1–100 $\mu$s, they could not attribute it to a PBH rather than a star.

###  4.3 Intermediate mass black holes

IMBHs are black holes whose mass lies between stellar mass black holes and the
supermassive black holes that reside at the centers of galaxies (Greene et
al., 2020). They are often defined to have masses between 100 M⊙ and 105 M⊙.
There is currently no concrete observational evidence for their existence,
though there are a number of candidates in that range (e.g. ultra-luminous
X-ray sources (Kaaret et al., 2017)). Theoretically, IMBHs ought to exist in
relatively high abundance, as there must be an evolutionary bridge between
stellar and supermassive black holes. Directly observing IMBHs and
constraining their number density is major outstanding problem in astronomy.

Lensed FRBs will probe the volumetric density of IMBHs. For a lens at redshift
0.5, the lensing time delay is between 3 ms and a few seconds for lens masses
of 102–105 M⊙. We parameterize the cosmological density of IMBHs in the
standard way as,

$\Omega_{IMBH}=\frac{\rho_{IMBH}}{\rho_{c}},$ (28)

where $\rho_{IMBH}$ is the mean density of IMBHs and $\rho_{c}$ the critical
density of the Universe,

$\rho_{c}=\frac{3H^{2}_{0}}{8\pi\,G}.$ (29)

To forecast “millilensing” events on upcoming surveys, we consider two
scenarios. First, we apply a similar treatment of PBHs, where we ask how well
$\Omega_{IMBH}$ can be constrained if no FRBs are lensed on timescales of 3 ms
to a few seconds. Our second approach is to assume previous candidates in the
relevant mass range (Paynter et al., 2021; Vedantham et al., 2017) were true
gravitational lensing events and then extrapolate from their inferred optical
depths.

If no millilensing event is found, we can constrain the cosmic density of
IMBHs at $90\%$ confidence to,

$\tau\lesssim 2.3\,B^{-1}\,N_{FRB}^{-1}.$ (30)

The factor of 2.3 is from the 90$\%$ Poissonian confidence limits, having seen
zero events. $B$ is magnification bias. We can then use the Press-Gunn
relation to get,

$\Omega_{\mathrm{IMBH}}\lesssim
2.3\,B^{-1}\,N_{FRB}^{-1}\left(\frac{\left<z\right>}{2}\right)^{2}$ (31)

We use the modeled redshift distribution means of DSA-2000, CHORD, CASM, and
CHIME/FRB to compute these upper limits. They are displayed in Table 2.

Next, we will assume that gravitationally lensed fast transients have already
been observed and extrapolate rates directly from them. Paynter et al. (2021)
found evidence that GRB 950830 was lensed by a $\sim$ $5\times 10^{4}$ M⊙
black hole at $z\approx 1$. They also use the approximation (Press & Gunn,
1973) that $\Omega_{IMBH}\sim\tau(\left<z_{s}\right>=2)$ and calculate optical
depth from one lensed event in $\sim$ 2700 BATSE GRBs. For a mean source
redshift of $\sim$ 2 they claim,

$\Omega_{\mathrm{IMBH}}(M\sim 10^{4-5}\,\mathrm{M}_{\odot})\approx
4.6^{+9.8}_{-3.3}\times 10^{-4}.$ (32)

A related result comes from “Symmetric Achromatic Variability” seen in active
galaxies. Vedantham et al. (2017) proposed that the achromatic temporal
variation observed in BL Lac object J1415+1320 was a millilensing event,
plausibly caused by a dark matter subhalo or black hole located within an
intervening galaxy. The relatively large density of IMBHs implied by GRB
950830 and J1415+1320 would be promising for upcoming FRB surveys, which will
be sensitive to a wider range of lens masses and will have a larger collection
of transients than BATSE. FRB lensing will be less ambiguous than GRB events,
because of the coherent temporal methods described in Section 2.3. Again, we
use the modeled mean redshifts of four surveys in Table 1 and the
$\tau(z_{s})$ relation to estimate the optical depth from IMBHs for FRBs,
scaling from the mean redshift of GRBs. We find the DSA-2000, CHORD, CASM, and
CHIME/FRB will have effective optical depths that are 2.8, 4, 16, and 16 times
lower than BATSE GRBs, respectively. This is because FRBs in those surveys
will be more nearby than typical GRBs. We find that, contingent on GRB 950830
having been lensed, DSA-2000 will find 0.1–7.3 FRBs lensed by IMBHs after five
years of observing at 80$\%$ efficiency. Similarly, CHORD will find 0.1–7.0
events, assuming the BATSE GRB was lensed and the sample’s mean redshift was
roughly 2. CASM would detect 0.3–7.2 and CHIME/FRB could expect 0.01–0.29
IMBH-lensed bursts.

###  4.4 Massive galaxy lenses

FRBs lensed by massive galaxies will experience time-delays from days to
months. The FRB’s host galaxy should also be gravitationally lensed and will
allow for modeling of the lensing galaxy’s mass profile. Unlike time-variable
non-transient sources like quasars, the FRB is only “on” for a millisecond and
will not obscure the lensed host galaxy. Furthermore, FRBs lead to time-delay
uncertainties that are more than ten orders of magnitude smaller than previous
applications of time-delay cosmography. For these reasons, FRB lensing by
massive galaxies has been suggested as a new tool for measuring the Hubble
parameter (Li et al., 2018). Wucknitz et al. (2021) even proposed using
gravitational lenses as a galaxy-scale interferometer to resolve structure and
motion in the FRB itself. Most of these methods require repeating FRBs to
break the mass-sheet degeneracy (Falco et al., 1985). Interferometric
localization will alleviate the need for temporal coherence, as the lensed FRB
images could be spatially resolved for typical image separations.

The probability of detecting a lensed event whose delay is longer than a
survey pointing time or a transit time ($\sim$ $10^{3}$ s for DSA-2000, CHORD,
and CHIME/FRB), is very low. If a telescope is always pointing at the same
patch of sky, either because it is ultra-widefield (a CASM such as BURSTT) or
through special survey strategy, it can in principle detect lensing delays up
to the survey duration ($\sim$ years). For this reason, we consider only CASM
when forecasting detection rates of FRB lensing by massive galaxies.

We use the lensing rate formalism presented in Equation 5 to estimate the
number of FRBs lensed by massive galaxies. We take the magnification bias for
an SIS dark matter halo and source power-law luminosity function with
cumulative index $\gamma=1$. We find that after five years of observing, our
putative CASM would detect 5–40 FRBs lensed by massive galaxies. The lower
limit comes from assuming no magnification bias and 12,500 CASM FRB detections
per year. The upper bound assumes $\gamma=1.0$ and $B=4$ with 25,000 FRBs per
year

###  4.5 Probing the circumgalactic medium

Figure 8: A schematic diagram of using gravitationally lensed FRBs as probes
of the circumgalactic medium of lensing galaxies. Each image will have the
same host DM, IGM DM, and contribution from the Milky Way, so the only
difference in observed DM between multiple copies will be attributable to the
lensing galaxy’s CGM.

Most proposed methods studying the circumgalactic medium (CGM) using FRBs have
relied on the ensemble statistics of large numbers of sources, using DM and
its line-of-sight statistics to study the distribution halo gas (McQuinn,
2014; Connor & Ravi, 2021). Another approach is to model the contribution to
DM from the host galaxy, the intergalactic medium (IGM), and the Milky Way to
infer the component imparted by the halos of intervening galaxies (Prochaska
et al., 2019; Ravi, 2019). Both approaches are limited by modelling
uncertainties. But an FRB lensed by a massive intervening galaxy will probe
that galaxy’s halo gas along multiple sight-lines. This would be a clean
measure of that galaxy’s CGM, as the only difference in propagation for the
multiple paths will arise due to the galaxy lens. The method would constrain
both the radial profile of halo gas density, as well as the CGM’s
inhomogeneity. A diagram of this idea is shown in Figure 8.

Figure 9: The observed difference in DM between two copies of an FRB lensed by
a $10^{12}$ $M_{\odot}$ galaxy. $\Delta$DM is plotted as a function of image
separation, $\Delta\theta$, assuming the first image is at $\theta_{1}=1$”. We
have used four simple, spherically symmetric models for the free electron
distribution in the lensing galaxy CGM. The $y_{0}=4$ curve has two zero
points because its DM curve is non-monotonic.

A similar technique has been used for many years in quasar absorption studies,
where multiply-imaged quasars have been used to measure the structure and
differential composition of the CGM of intervening lensing galaxies (Rauch,
2002), even on spatial scales as small as 400 pc (Rudie et al., 2019).
However, such studies measure metal-bearing gas and cannot easily constrain
the total baryonic matter in the halo, due to large uncertainties when
extrapolating from the relatively rare metals to total gas content.

With strong lensing of an FRB detected with very long baseline interferometry
(VLBI), the halo of the galaxy lens could be studied directly by comparing the
DM, RM, and scattering properties of the multiple copies. Even if the lensing
galaxy is a massive elliptical, Zahedy et al. (2019) has shown that such
quiescent halos still have a rich CGM. We assume that the angular positions of
the lensed copies can be measured to $\lesssim$0.25 arcseconds with VLBI
outriggers. From this, one would have both the angular separation of the
lenses images and the lensing time-delay. VLBI localization would give the
angular impact parameter, $\theta$, allowing one to determine where in the
intervening galaxy’s halo the FRB passed through. The DM from an intervening
halo will be given by the following integral,

$\mathrm{DM}_{CGM}=2\,\int\displaylimits^{\sqrt{r^{2}_{max}-b^{2}}}_{0}\frac{n_{e}(r)}{1+z_{l}}\,dr,$
(33)

where $r_{max}$ is a halo cutoff radius, which we will take to be the virial
radius, $r_{200}$, and,

$b\approx\theta D_{l}.$ (34)

The observed quantity is the difference in DM between the two images. This is
given by the difference in sightlines from the lensing galaxy’s CGM,
$\Delta\mathrm{DM}=\mathrm{DM}_{CGM}(\theta_{1})-\mathrm{DM}_{CGM}(\theta_{2})$
given by,

$\Delta\mathrm{DM}=\frac{2}{1+z_{l}}\int\displaylimits^{r_{1}}_{r_{2}}n_{e}(r)\,dr.$
(35)

Here we have written $\Delta\,\mathrm{DM}$ as a single definite integral with
$r_{1}=\sqrt{r^{2}_{max}-\theta^{2}_{1}\,D^{2}_{l}}$ and
$r_{2}=\sqrt{r^{2}_{max}-\theta^{2}_{2}\,D^{2}_{l}}$. At present, we do not
know $n_{e}(r)$. But we will have measured $\Delta\mathrm{DM}$, the lens
redshift, $z_{l}$, and $\theta_{1}$ and $\theta_{2}$. In Figure 9 we plot
$\Delta\mathrm{DM}$ for a range of $\Delta\theta$, holding $\theta_{1}$ fixed
at 1”. We use four models from Prochaska & Zheng (2019), but this a non-
exhaustive list. The first is a classic Navarro-Frenk-White (NFW) profile
(Navarro et al., 1997), which can be generalized to an modified NFW (mNFW)
profile as,

$\rho(r)=\frac{\rho_{b}}{y^{1-\alpha}\,(y_{0}+y)^{2+\alpha}},$ (36)

with $y\equiv c\frac{r}{r_{200}}$ where $r$ is radius, $r_{200}$ is the virial
radius, and $c$ is a concentration parameter. For the standard NFW, $\alpha=0$
and $y_{0}=1$. We also consider two mNFWs with ($y_{0}=2$, $\alpha=2$) and
($y_{0}=4$, $\alpha=2$). The final model, MB04, comes from Maller & Bullock
(2004). The DMs and halo models were calculated with publicly available
code333https://github.com/FRBs/FRB.

The NFW profile is unrealistically steep at small radii, leading to large DM
differences even for small angular separations (see Figure 9). The other three
models are distinguishable for $\Delta\theta\geq 0.5^{\prime\prime}$, since
observed FRB DM uncertainties with voltage data are often less than 0.5 pc
cm-3. Combining $\Delta\,\mathrm{DM}$ with the difference in rotation measure,
$\Delta\,$RM, will give a clean measure of the CGM line-of-sight magnetic
field difference between the two impact parameters. This measurement will be
valuable because the magnetic field in galaxy halos remains largely
unconstrainted observationally (van de Voort et al., 2021).

##  5 Conclusions

Fast radio bursts offer a unique probe of the Universe’s dark matter via
gravitational lensing, thanks to their abundance, short duration, and the
coherent nature of their detection. In principle, FRBs can give us access to
$\sim$ fifteen orders of magnitude in lens mass, corresponding to lensing time
delays of microseconds to years. This is made possible by saving phase-
preserving voltage data (the electric field waveform of the radio pulse
itself). We have provided an overview of FRB gravitational lensing spanning a
wide range of time-delays and lens masses. We have forecasted detection rates
for upcoming and current surveys. These include DSA-2000, CHORD, a coherent
all-sky monitor (CASM) such as BURSTT, as well as CHIME/FRB, which is
currently finding large numbers of FRBs and searching for lensing events. The
FRB redshift distribution is critical to any lensing optical depth calculation
because lensing probability is a strong function of source distance. This
creates a trade-off between deep, sensitive telescopes such as DSA-2000 and
CHORD, and ultra-widefield but less sensitive CASM surveys.

On 1–100 microseconds timescales, FRBs will be lensed by stars in foreground
galaxies at a significant rate and could plausibly be detected by CHIME/FRB.
If lensing by stars can be distinguished from compact dark objects, FRB
surveys will limit the abundance of primordial black holes in the mass range
0.1–100 $M_{\odot}$. This mass range is under-explored and is favoured by some
theorists to be the most relevant mass scale for PBHs (Carr & Kühnel, 2020).
All surveys we have considered will also constrain the cosmological density of
intermediate mass black holes, $\Omega_{IMBH}$. If recent claims of GRB
millilensing by IMBHs (Paynter et al., 2021) are correct, DSA-2000, CHORD, and
the putative CASM survey could all detect a few lensed FRBs on timescales of
0.01–10 s over the course of their surveys. On longer delay timescales (days
to years), FRBs will be lensed by massive galaxies. Several cosmological
applications have been proposed for these systems (Li et al., 2018; Wucknitz
et al., 2021). Detecting strong lensing by galaxies is difficult because it is
not known when the lensed copy will arrive, so the radio telescope must be
pointing at the same location most of the time. Of the telescopes and survey
strategies considered here, only a CASM-like survey (e.g. a BURSTT-like
telescope but with a CHIME/FRB SEFD) could plausibly find FRB lensing by
massive galaxies, with 5–40 such events after five years of observing.

Finally, we have proposed a new method for studying the circumgalactic medium
of lensing galaxies. Multiply imaged FRBs will traverse the lensing halos’ CGM
at different impact parameters and will be differentially dispersed, Faraday
rotated, or even scattered. Unlike with previous methods, this provides a
clean model-independent measure of the total baryonic material in dark matter
halos as a function of radius. Our method requires sub-arcsecond localization
of each lensed copy of the FRB, which will only be possible for CASM-like
surveys that have VLBI outriggers.

Data availability statement This work was based on publicly available data and
can be reproduced with the Jupyter notebook found in
https://github.com/liamconnor/frb-grav-lensing.

We thank Sterl Phinney and Minghao Yue for valuable insights on strong
lensing. We also thank Matt Dobbs, Keith Vanderlinde, Zarif Kader, and Ue-Li
Pen for helpful discussions on coherent gravitational lensing.

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