Looking for MACHOs in the Spectra of Fast Radio Bursts

We explore a novel search strategy for dark matter in the form of massive compact halo objects (MACHOs) such as primordial black holes or dense mini-halos in the mass range from $10^{-4}$ to 0.1 solar masses. These objects can gravitationally lens the signal of fast radio bursts (FRBs), producing a characteristic interference pattern in the frequency spectrum, similar to the previously studied femtolensing signal in gamma ray burst spectra. Unlike traditional searches using microlensing, FRB lensing will probe the abundance of MACHOs at cosmological distance scales (~Gpc) rather than just their distribution in the neighborhood of the Milky Way. The method is thus particularly relevant for dark mini-halos, which may be inaccessible to microlensing due to their finite spatial extent or tidal disruption in galaxies. We find that the main complication in FRB lensing will be interstellar scintillation in the FRB's host galaxy and in the Milky Way. Scintillation is difficult to quantify because it heavily depends on turbulence in the interstellar medium, which is poorly understood. We show that, nevertheless, for realistic scintillation parameters, FRB lensing can set competitive limits on compact dark matter object, and we back our findings with explicit simulations.


Introduction
Massive compact halo objects (MACHOs) have been widely discussed in the literature as possible dark matter candidates. In particular, a lot of attention has been paid to two classes of objects: primordial black holes (PBHs) [1][2][3] and ultra-compact mini-halos (UCMHs) [4][5][6][7][8]. 1 For concreteness, we will focus our discussion on the case of PBHs, though most of our findings will apply also to other types of MACHOs.
The interesting mass range for PBHs is between 10 −16 M and 10 3 M (10 17 grams to 10 36 grams), with M being the solar mass. On the lower end of this range, the PBH abundance is strongly constrained by non-observation of the γ-ray flux that would be produced by PBHs due to their Hawking evaporation [12], and by measurements of the cosmic microwave background (CMB) anisotropies that would be affected by Hawking radiation in the early universe [13]. At large PBH masses, constraints are due to accretion onto PBHs, which would again affect the CMB anisotropies [14]. A massive effort has been put into exploring the available parameter space of PBH dark matter, in particular via microlensing searches like MACHO [15], EROS [16], OGLE [17], and most recently SUBARU-HSC [18]. Other methods to search for MACHOs in the abovementioned mass range exist, but their sensitivity is typically inferior to microlensing. Note also that several published constraints have not withstood more careful analysis and are therefore now considered invalid or controversial. Thus, big portions of MACHO parameter space remain unconstrained, while others are only loosely constrained and can still accommodate a sizable fraction of dark matter in the form of MACHOs.
It is important to note that existing microlensing searches -which are responsible for essentially all the non-controversial constraints on MACHOs in the interesting mass rangeare based on observations in the Milky Way or the Local Group of galaxies. Drawing conclusions on the MACHO abundance on cosmological scales thus requires extrapolation, which suffers from several uncertainties. First, translating observational results into constraints on the MACHO abundance requires assumptions about the dark matter density profile of the Milky Way, and for some observations like those by SUBARU-HSC, of the other galaxies in the Local Group. Second, the mass function of MACHOs and their abundance in the Local Group may differ from those on cosmological scales. Indeed, for the case of PBHs, one can expect the mass distribution to evolve in time due to accretion and mergers. Composite MA-CHOs such as UCMHs can be tidally disrupted inside big galactic halos [19][20][21][22][23][24][25][26] so that more of them can be found at larger redshifts and in less dense objects than in today's Milky Way. Finally, microlensing searches cannot constrain UCMHs or other MACHOs of finite extent if their size exceeds their Einstein radius at galactic distances [27].
In this paper, we investigate a search strategy that probes MACHOs directly at cosmological distance scales. It is based on an effect which we will call diffractive gravitational lensing. The basic idea of diffractive lensing is that a MACHO placed between a distant source and the observer and whose mass is too small to produce two resolved images of the source still causes a phase difference between the unresolved images. This leads to characteristic interference patterns in the energy spectrum of the source. Diffractive lensing was originally proposed in the context of gamma-ray burst (GRB) sources, where it is known as femtolensing [28,29]. 2 Diffractive lensing of GRBs could be expected to constrain MACHOs in the mass range from 10 −17 to 10 −14 M [30]. However, due to the non-negligible angular sizes of typical GRBs, it turns out that the interference pattern in the spectrum would be washed out [31], so that femtolensing does not currently yield bounds on the MACHO abundance [32].
We propose here to apply the diffractive lensing technique to fast radio bursts (FRBs, see e.g. reviews [33][34][35]) rather than GRBs. As the photon wavelengths for FRBs are much longer than for GRBs, the method will be sensitive to much heavier MACHOs, namely those between 10 −4 and 0.1 M . The Einstein radii of MACHOs in this mass range are much larger than those of the lenses considered in the context of GRBs, therefore the requirement on the source size is much less stringent and easily satisfied by putative FRB progenitors.
Gravitational lensing of FRBs has been considered previously in the literature: strong lensing, in particular the observation of two consecutive bursts separated in time, has been proposed in [36] and further considered in [37,38]. Such an observation would be sensitive to MACHOs of 10 M to 100 M . Diffractive lensing of FRBs has been suggested in [39,40], where it was also pointed out that the main complication in applying this technique is the distortion of FRB spectra by scintillation [41][42][43]. The latter appears due to scattering of radio waves on the turbulent interstellar medium (ISM), leading to multi-path propagation of radio signals and thus to random interference patterns in the frequency spectrum. Nevertheless, ref. [40] argued that the lensing signal can be disentangled from scintillation by considering the temporal autocorrelation of the electromagnetic field of the radio wave.
In the present paper we take further steps to explore this idea. We show that one does not need to measure the full electromagnetic field amplitude to extract the lensing signal or confirm the absence thereof. In fact, the entire analysis can be carried out in terms of the intensity (fluence) spectrum of the wave, without any need for phase information. This has practical importance as most FRB detections occur in survey mode where the full field information may not be recorded. We also demonstrate that the extraction of the lensing signal from scintillation is possible whenever these two phenomena are characterized by disparate frequency scales. Apart from the regime considered in [40] when the two lensed images are unresolved by the scintillation screen, this also includes the case when the scintillation screen does resolve the images and distorts them incoherently, provided the lensing time delay is longer than the inverse of the decorrelation bandwidth of the scintillation. Finally, we take into account scintillation in the intergalactic medium (IGM). For realistic parameter choices we find IGM scintillation to be less relevant than scintillation in the Milky Way and in the host galaxy of the source, as long as the line-of-sight does not cross any major concentrations of ionized plasma, such as galaxy clusters. Our analytic estimates are supported by numerical simulations of FRB spectra with lensing and scintillation, using a simplified model in which the ISM and IGM are described by one-dimensional scintillation screens. We propose a data analysis procedure to search for the lensing signal by looking for peaks in the Fourier transform of the FRB intensity spectrum. We find that the sensitivity is improved if the smooth component of the spectrum is first divided out in order to reconstruct an approximation to the transfer function (the ratio between the emitted and recorded intensity). This procedure also reduces our sensitivity to imperfect modeling of the FRB spectrum. The search for the peaks in the Fourier transform of the transfer function is similar to the one used for resonance searches on a smooth background in collider experiments. We validate our procedure using the simulated data.
A precise modeling of scintillation is problematic due to the poor understanding of the turbulent ISM and IGM, leading to large uncertainties in the parameters governing scintillation. In spite of these complications, we will see below that there is realistic hope of extracting the lensing signal from FRB data. Moreover, the reach of the method will significantly improve in the future with an increased number of FRB detections, in particular thanks to the CHIME and SKA telescopes, which are expected to detect a few tens of FRBs per day [44,45]. For the time being, we will explore the sensitivity as a function of the scintillation parameters and the number of FRBs.
The paper is organized as follows. In section 2 we briefly review the formalism of gravitational diffractive lensing as well as the potential and the limitations of MACHO searches using this technique. In section 3 we address the problem of scintillation. We review the relevant properties of the ISM and IGM, and we analytically estimate the impact of scintillation on diffractive lensing of FRBs. We then proceed in section 4 to the discussion of numerical results obtained by explicitly simulating the propagation of FRB signals. In this section, we also present our main results in the form of sensitivity estimates. We conclude in section 5.

Review of Diffractive Lensing
Consider a distant source such as an FRB, and a gravitational lens close to the line of sight. For our purposes, it is sufficient to consider the source and the lens to be point-like. In general, a point-like lens produces two images of the source, but if the lens has a small mass, the observer will not be able to resolve them. However, the photons corresponding to the two images travel different distances and experience different gravitational potentials, so their travel times will differ by an amount ∆t. This leads to a relative phase shift ∆φ = ω ∆t, where ω is the photon energy. If ∆φ O(1), an interference pattern will appear in the photon energy spectrum, and may allow detection of the lens even when the two images are not resolved.
More quantitatively, the time delay between the images is given by 3 where D L , D S , D LS are the comoving distances between the observer and the lens, between the observer and the source, and between the lens and the source, respectively; β is the angle under which the observer would see the source in the absence of the lens, whereas θ is the angle of the lensed image; ψ( θ) stands for the lensing potential, which depends on the density profile of the lens. The lensing potential of a point-like mass M is where the Einstein angle is a measure for the typical angle under which the images are observed relative to the lens. In eq. (2.3), Z L denotes the redshift of the lens. The positions of the images are given by the solutions to the lens equation, which for a point-like lens gives two locations Different signs of the two angles here mean that the images occur on opposite sides of the lens. Denoting y ≡ β/θ E , the magnifications of the two images are [46] µ ± = y 2 + 2 2y y 2 + 4 ± 1 2 . (2.6) Their interference leads to the observed image, which, for a point-like source and lens, is magnified by a factor µ = y 2 + 2 y y 2 + 4 + 2 y y 2 + 4 sin Ω y y 2 + 4 2 + log y + y 2 + 4 compared to the unlensed source. Here, the frequency dependence that creates the interference pattern enters through the dimensionless parameter . The diffractive lensing signal as a function of photon frequency. The vertical axis displays the ratio of the observed intensity with and without lensing, the bottom horizontal axis shows the dimensionless frequency defined in eq. (2.8), and the top horizontal axis translates Ω into a physical frequency for a specific choice of lens mass and redshift. We compare two different angular separations between the lens and the source (red vs. green lines), and we also compare the geometric optics approximation (dashed) to a full wave optics calculation (solid), concluding that, for our purposes (ν ∼ GHz) geometric optics is always a valid approximation.
The requirement ω ∆t O(1) for the observability of lensing-induced interference patterns immediately tells us the lens masses to which a given search will be sensitive. For typical values of θ, β ∼ θ E , the two terms in parentheses in eq. (2.1) are of similar order, and using eq. (2.3) for the Einstein angle, we find ∆t ∼ R s , where R s is the Schwarzschild radius of the lens. This explains why femtolensing of GRBs [28], where photons have energies of order 10 keV to 1 MeV, is most sensitive to lenses between 10 −17 M and 10 −14 M . For fast radio bursts, with typical photon energies of order 10 −6 eV (frequencies of order GHz), lens masses between 10 −4 M and 0.1 M will be relevant. This mass range extends to higher masses than one might naively expect thanks to the excellent frequency resolution of radio telescopes, which allows us to resolve the interference pattern even if ω ∆t is significantly larger than one.
Let us mention some caveats to the discussion above. First, we note that at ω ∆t ∼ 1 the geometric optics approximation with two well-defined lensed images breaks down and wave optics effects need to be taken into account [47]. The observed signal is then found by evaluating the full Fresnel integral over the lens plane, which leads to the observed magnification where ∆t is given by eq. (2.1). For a point source and a point-like lens, eq. (2.9) can be evaluated analytically [32,48], but in the general case the integral needs to be performed numerically. We compare the magnification in the geometric optics approximation to the result of a full wave optics calculation in fig. 1. As expected, the effect of wave optics is most pronounced only at small frequencies, Ω 1. The typical frequencies at which we observe FRBs are around 1 GHz, which for lens masses 10 −4 M corresponds to Ω 1. We illustrate in fig. 1 that in this regime wave optics corrections are negligible.
A second potential caveat is related to the angular size of the source. If the latter is too large, but unresolved by the observer, the integral over signals from different regions in the source will wash out any lensing-induced interference patterns. More precisely, this happens when the argument of the sine in eq. (2.7) varies by an amount 1 over the angular diameter of the source. Denoting this diameter, normalized to θ E , as σ y , interference fringes are thus only observable if Ωσ y y 2 + 4 1 . (2.10) For a 10 −4 M lens about half-way between the observer and a source at 1 Gpc, and for Ω ∼ 10, this condition implies that wash-out is avoided for sources with a physical diameter 10 13 cm. This is a conservative estimate in the sense that, for heavier lenses, the maximum allowed diameter will only increase. FRBs are variable on short time scales t var msec. Allowing for relativistic expansion of the emitting matter with a bulk Lorentz factor Γ , one can estimate the apparent transverse size of the source as a S t var Γ (see e.g. appendix A of ref. [32] for a derivation). One concludes that for realistic values of Γ < 10 5 the size of the FRB source is irrelevant for the lensing signal.
We are going to see below that this conclusion may change in the presence of strong scintillation in the FRB host galaxy. The scintillation increases an effective size of the source and thereby suppresses the lensing pattern. We defer the study of this effect to the next section.
The above discussion can be generalized to the case of non-point-like lenses along the lines of ref. [32]. In particular, it applies to UCMHs as long as their density slope is steep enough to produce multiple images of the source. 4

Scintillation
We now turn to the physics of scintillation in interstellar and intergalactic medium. Scintillation leads to distortions of the signals from distant FRBs that can look very similar to the distortions caused by diffractive gravitational lensing. To disentangle the two, we must therefore carefully model the physics of scintillation.

Scintillation Primer
When the radio waves from a distant FRB pass through the turbulent ISM and IGM environments, they will suffer refraction and diffraction, implying that signals can reach the observer along many different trajectories. Photons traveling along different paths will interfere, leading to chaotic modulation of the observed FRB frequency spectrum. The physics here is similar to the physics of atmospheric scintillation which makes the stars flicker; as FRBs emit at different wavelength, their signals are not significantly affected by the Earth's atmosphere, but mainly by the ISM.
In the subsequent discussion, we follow refs. [43,49,50]. Scintillation can be modeled by assuming that the turbulent plasma which is responsible for multi-path propagation is } D ScO confined to a thin screen. The position of the screen is chosen to be about half-way through the scintillating medium. For scintillation in the Milky Way's ISM, this typically means a scintillation screen O(kpc) away from the observer; scintillation in the IGM can be described by a scintillation screen at a distance of O(Gpc), for example half-way between the source and the observer. For simplicity, we perform the analysis in Minkowski spacetime, neglecting redshift effects.
Consider a plane wave of amplitude f in (ω) falling perpendicularly onto a scintillation screen which distorts the wavefront by adding to it a random phase ϕ(ω, x), where x is the coordinate on the screen, see fig. 2. The signal which an observer at a point O, located at a distance D ScO from the screen, receives is given by the Fresnel integral, Here, the first term accounts for the geometric phase, which arises because of the different distances traveled by photons passing the screen at different positions x. The phase perturbation ϕ(ω, x) in the second term is characterized by the diffractive scale r diff , which is the distance scale on the scintillation screen over which ϕ(ω, x) varies by O(1). More precisely, r diff is defined using the phase structure function, through the equation The averaging in eq. (3.3) is performed over an ensemble of realizations of the random phase ϕ(ω, x). Due to statistical homogeneity of the scintillation screen, this is equivalent to the spatial averaging over the screen. The diffractive scale derives its name from the fact that it sets the angle θ diff over which radiation is typically diffracted by the screen: As we will see below, r diff is frequency-dependent.
In the case of weak scintillation, where the scintillation phase ϕ(ω, x) varies more slowly with x than the geometric phase, the radiation the observer receives from a given angular position on the sky comes from the first Fresnel zone of the screen, namely a region of size This is the distance over which the geometric phase changes by O(1), so the weak scintillation regime is characterized by the condition r diff r F . In the opposite limit of strong scintillation (r diff r F ), the observed signal comes instead from a region of size r ref is called the refractive scale. The different length scales on the scintillation screen are related by r F = √ r diff r ref .
In this paper we are only interested in the energy spectra of FRBs, not in the spatial or temporal variation of their intensity. Still, it is instructive to consider the radiation intensity as a function of position in the observer plane to understand the designation "refractive scale" for r ref and "diffractive scale" for r diff [51]. As the scintillating medium moves, the picture in the observer plane will move as well leading to temporal variations in the observed signal. In the case of strong scintillation, we expect to see a diffraction pattern of intensity variations over scales of order r diff ("diffractive scintillation"), because this is by definition the scale over which ϕ(ω, x) changes by O(1). We also expect to see intensity variation over larger scales of order r ref ("refractive scintillation"). As discussed above, each point on the screen emits radiation predominantly into a cone of the angular size θ diff = r ref /D ScO . Depending on the gradient of the refractive index, the cones from neighboring points will be refracted towards each other, which leads to an increase in observed radiation intensity, or away from each other, which leads to a decrease in observed radiation intensity. These intensity variations are only observable at scales larger than the cone size, that is where v ⊥ is the transverse velocity of the scintillating medium. For the study of FRBs, refractive scintillation is not relevant as the displacement of the screen over the duration of the burst is negligible.
For weak scintillation, the observer will only register intensity variations on the scale r F . Lines of sight separated by larger scales will effectively receive radiation from disjoint regions on the screen.

Separating Lensing and Scintillation
Let us now assume that the signal from a distant FRB is gravitationally lensed by a compact object such as a PBH close to the line of sight before it is distorted by a scintillation screen (see illustration in fig. 3). For the purposes of this discussion we will restrict ourselves to lensing } D ScO Figure 3. A fast radio burst gravitationally lensed by a compact object. The superposition of the signals corresponding to the two images passes through a scintillating medium before reaching the observer.
in the geometric optics limits. We have already seen in section 2 that this approximation is sufficient for the lens masses we are interested in.
We describe the time-dependent FRB signal at the source as where f in (ω) is the amplitude of the radiation at frequency ω. Lensing in the geometric optics approximation splits this signal into two parts with different amplitudes A and B, and with a relative propagation time delay ∆t (see eq. (2.1)): A scintillation screen between the lens and the observer additionally imparts random noise on the lensed signal. The observed signal from a distant source, after passing a gravitational lens (e.g. a PBH) and then a scintillation screen (e.g. due to the Milky Way's ISM), can be written as Here, the terms δt A,B ( x) account for the variation of the lensing-induced time delays with x. This variation originates from the fact that the lens is seen under a different angle relative to the source from different points on the screen. Taking the square of this expression we obtain the intensity spectrum, and similarly for B(ω). We observe the presence of interference terms characteristic of diffractive lensing. If the amplitudes A, B were constant, these terms would lead to regular periodic modulation of the observed intensity, as discussed in section 2. Scintillation complicates the picture by making the amplitudes frequency dependent. Nevertheless, the lensing signal can still be extracted if the product of the amplitudes A * (ω)B(ω) contains a slowly varying component that survives upon averaging over frequency intervals ∆ω (∆t) −1 . Indeed, in this case the Fourier transform of the spectrum (3.11) will have a peak coresponding to the lensing time delay ∆t.
To estimate the size of the slowly varying component we consider the product of the amplitudes smeared with a Gaussian function of width ∆ω, Its characteristic value is given by the ensemble average |Q(ω, ∆ω)| 2 over random realizations of the scintillation phase. This average involves the correlator of four amplitudes, In the case of strong scintillation the integral (3.12) extends over many Fresnel regions. The contributions of different regions are essentially statistically independent. Then, by the central limiting theorem, the statistics of the amplitudes A(ω) and B(ω) are close to Gaussian and we can replace (3.14) by where we have used that the correlator A(ω )B(ω ) vanishes. Our next task is to estimate the two-point correlators entering into this expression. Let us start with where on the r.h.s. we introduced the frequency autocorrelation function for the signal coming from a single image. A similar equation holds for the correlator of the B amplitudes. An important characteristic of the scintillation-induced noise is its decorrelation bandwidth ω dec . The noise amplitude is correlated at frequencies separated by less than ω dec and uncorrelated otherwise. An estimate for ω dec is obtained by recalling that the observer sees a patch of diameter r ref on the scintillation screen. Over this distance, the geometric phase (first term in eq. (3.2)) varies by an amount ωr 2 ref /(2D ScO ). A shift in frequency by 2D ScO /r 2 ref changes the phases at different points of the patch by order unity resulting in decorrelation of the scintillation noise. Thus we arrive at the estimate Note that we assumed above that the random phase ϕ(ω, x) (second term in eq. (3.2)) changes with frequency in the same way or more slowly than the geometric phase. This is indeed the case for scintillation in ionized plasma, where ϕ is inversely proportional to frequency (see the next subsection). The decorrelation bandwidth sets the range of the autocorrelation function (3.16): one expects that C 0 is of order 1 if |ω − ω | ω dec and quickly decreases outside this range. The second term in eq. (3.15) contains the cross-correlation between the amplitudes of the two lensed images at fixed frequency A * (ω)B(ω) . Its magnitude depends on whether the two images are distorted by scintillation screen in the same way (coherently) or independently (incoherently). In the first case the correlation is order-one, whereas in the second case it is suppressed. To determine the conditions for coherent/incoherent distortion, let us consider the variation δt B ( x) − δt A ( x) of the lensing-induced time delay over the scintillation screen. This can be found by noting that a change of position on the screen changes the difference between the viewing angles of the lens and the source by δβ = (xD LS )/(D L D S ). Hence, the variation of the time delay is given by where ∆ θ is the angular distance between the A and B images. In the second equality we have used eq. (2.1). Due to scintillation, the observer receives photons from a region of size r ref on the screen. If the variation of the lensing-induced phase over this region is small, we can neglect it in the integral (3.12) for the amplitudes and obtain, In the opposite regime a fast lensing-phase variation leads to strong suppression of the correlation between the amplitudes. Thus, in general we can write, where the function U is of order unity when its argument is less than 1 and quickly vanishes outside this range. 5 We now return to the smeared amplitude of the interference term, eq. (3.13). Collecting our previous results from eqs. (3.16) and (3.20) we obtain, where we have denoted byŪ the correlator (3.20) averaged over a range of frequencies ∆ω. Extraction of the lensing signal is possible whenever either of the two terms in brackets is sizable, implying that the interference amplitude has a piece that does not vanish when averaged over frequency intervals ∆ω (∆t) −1 . If ∆ω is smaller than the decorrelation bandwidth, the autocorrelation function in the first term can be replaced by unity and the whole integral equals 1. In the opposite regime ∆ω ω dec it becomes suppressed as ω dec /∆ω. Recalling that we want the averaging interval ∆ω to be at least as large as (∆t) −1 , we conclude that the first term in eq. (3.21) is O(1) for long lensing delays ∆t (ω dec ) −1 . The magnitude of the second term in eq. (3.21) is controlled by the angular separation between the lensed images. Given that the characteristic size of ∆θ is set by the Einstein angle θ E , we can rephrase the condition for this term to be significante as ω θ E r ref < 1. In this way we arrive at four possible scenarios, which are illustrated in fig. 4: (region above the red and blue lines in fig. 4). In this case, both terms in eq. (3.21) are sizeable. The lensing phase ω∆t varies much faster with frequency than the scintillation phase Φ(ω, x), and does not change substantially  . Observability of FRB lensing by compact dark matter objects as a function of the lens mass M and of the diffractive scale corresponding to interstellar scintillation in the Milky Way. We assume an FRB at a comoving distance of 1 Gpc, and a lens at 0.5 Gpc. The angular separation between the lens and the (unlensed) source, normalized to the Einstein angle, is y = β/θ E = 0.5. We consider 1 GHz radio waves, which are perturbed by a scintillation screen at 1 kpc form the observer.
across the scintillation screen. Therefore, a clear lensing pattern can be observed in the form of rapid periodic variations of the radiation intensity with ω. The envelope of the lensing wiggles in the frequency spectrum is modulated over frequency intervals while in the middle part, we plot the autocorrelation function of the normalized signal amplitude, (3.23) Here ν = ω/(2π) is the radiation frequency and averaging is performed over a large frequency interval. In the right part of fig. 5, we show the Fourier transform of |T (ν)| 2 . As expected, the lensing signal is clearly visible, with its envelope only marginally modulated due to scintillation.
(region below the blue, but above the red line in fig. 4). The variation of the lensing phase over the scintillation screen is fast, so the two lensed images are distorted incoherently. Thus, the second term in eq. (3.21) is suppressed. Nevertheless, the first term remains of order unity as the amplitudes A(ω) and B(ω) vary slowly as functions of ω. A clear lensing signal still survives both in the transfer function and in the autocorrelation function. This is illustrated in the second row of panels in fig. 5.  Figure 5. From top to bottom, we illustrate the four qualitatively different scintillation/lensing regimes discussed in the main text. The panels on the left show in each case the transfer function |T (ν)| 2 , that is the observed radiation intensity, normalized to the radiation intensity without lensing and scintillation (see eq. (3.22)). In the middle panels we plot the autocorrelation function, eq. (3.23). The panels on the right contain the Fourier transform of |T (ν)| 2 , with the lensing peak highlighted by a vertical dotted line. In all panels, the horizontal axis is normalized in terms of the lensing time delay ∆t, and we use ν 0 = 1 GHz. Values of r diff are quoted at the midpoint of the spectrum. The plots have been produced using the simulation code described in section 4 [52], assuming an interstellar scintillation screen at 1 kpc, and a lens at D L = D S /2, y = β/θ E = 0.5.
(c) ∆t (ω dec ) −1 , ω θ E r ref 1 (region below the red, but above the blue line in fig. 4). When the time delay ∆t between the two lensed images is shorter than the inverse of the decorrelation bandwidth ω dec , the scintillation factor e iΦ(ω, x) in eq. (3.10) varies much faster than the lensing factor e iω∆t . The first term in eq. (3.21) is then small. On the other hand, we can neglect the variation of the lensing phase over the screen, ω δt( x), so the two lensed images are distorted coherently and the second term in eq. (3.13) is sizable. The lensing signal will be discernible as a modulation of the envelope of an otherwise chaotic spectrum. Indeed, the panels in the third row in fig. 5 illustrate this behavior: we observe high-frequency scintillation noise, superimposed on regular periodic oscillations due to lensing. As expected, the signal disappears from the autocorrelation function, but remains in the Fourier transform of the transfer function. We conclude that a lensing signal is in principle observable in regimes (a), (b) and (c), but strongly suppressed or completely unobservable in regime (d).
However, there are practical considerations that may render the signal unobservable even in regimes (a), (b), (c). The distance between subsequent lensing fringes in the spectrum, (∆t) −1 , should be significantly larger than the instrumental frequency resolution, but significantly smaller than the instrumental bandwidth. The first condition is violated for too large lens masses, the second one for lens masses that are too small. 6 This effectively restricts the search for compact objects using diffractive lensing of FRBs to the mass range between ∼ 10 −4 M and ∼ 0.1 M .
The above discussion applies also to the case of a scintillation screen placed between the source and the lens, see fig. 6. This setup describes scintillation in the ISM of the FRB host galaxy or in the IGM between the lens and the host. The amplitude of the observed signal is again given by eq. (3.10), with the screen-observer distance D ScO replaced by the screen-source distance D ScS . The rest of the analysis proceeds without change. Notice that strong scintillation effectively spreads the source into a patch of radius r ref on the scintillation screen. The condition ω θ E r ref < 1 for coherent distortion of the lensed images can then be reinterpreted as the restriction (2.10) on the size of the source required for diffractive lensing. One may wonder why one can discern the lensing pattern in regime (b) even if this condition is violated. The reason is that the signal from each point on the lensing screen is correlated at different frequencies within a finite decorrelation bandwidth, which is not the case for a truly incoherent source.

The Turbulent Interstellar and Intergalatic Medium
For a quantitative description of scintillation and its effect on FRB signals, we need to discuss in more detail the properties of the interstellar medium (ISM) and intergalactic medium (IGM). We focus in particular on interstellar and intergalactic plasma, neglecting the neutral gas. The reason is that the refractive index of a plasma deviates from unity by much more than the refractive index of neutral gas, so that its impact on the phase fluctuations that cause scintillation is much larger.

The Interstellar Medium
Propagation in plasma modifies the dispersion relation of electromagnetic waves with wavenumber k, Here the plasma frequency is expressed in terms of the electron number density n e , the electron mass m e and the electromagnetic fine structure constant α, As a consequence, a wave propagating in plasma acquires a frequency dependent phase shift, with the integral taken along the line of sight. The phase shift has strong frequency dependence and causes a time delay between signals in different frequency bands. In practice, the net time delay due to the average electron densityn e is removed by the dedispersion procedure that is typically applied to FRB data. Scintillation is due to a residual phase shift generated by random fluctuations in the electron density δn e = n e −n e . We assume that on the scales we are interested in the density fluctuations obey statistical homogeniety and isotropy. Hence they are characterized by an isotropic power spectrum related to the density correlation function as, 7 δn e ( X + ∆ X) δn e ( X) = d 3 κ e i κ∆ X P n (κ) , Here, X = ( x, z) is a three-dimensional coordinate vector, and κ ≡ | κ|. Observations indicate that in a wide range of wavenumbers the power spectrum obeys Kolmogorov-like scaling characteristic of a turbulent behavior [53,54], The outer scale of interstellar turbulence is estimated to be l out 10 ÷ 100 pc and the inner scale l in is smaller than 10 8 cm; the level of turbulence is [55,56]  wheren e = (0.01 ÷ 0.1) cm −3 [57]. The dominant energy source for ISM turbulence is most likely supernova remnants, but other sources like stellar winds or protostellar outflows can also play a role [55,58]. Even though Kolmogorov scaling (3.28) is widely used for the description of the turbulent ISM, its origin remains a matter of debate. The original Kolmogorov theory [59,60] predicts the spectrum of energy fluctuations in the inertial range of an incompressible fluid P E (κ) ∝ κ −11/3 . It is not entirely clear why the spectrum of density fluctuations in the compressible ISM should follow the same power-law, for a review of existing proposals see [55]. Following the common practice, we will adopt (3.28) in our estimates.
We can now express the phase structure function ξ(ω, r) from eq. (3.3) in terms of P n (κ) by plugging the relation (3.26) between ϕ(ω, x) and n e ( X), as well as the density correlation function eq. (3.27), into eq. (3.3) [42]: Here, L is the thickness of the ISM layer traversed by the wave and J 0 (z) is the Bessel function. Substitution of the Kolmogorov spectrum (3.28) yields, ξ(ω, r) = r r diff (ω) According to fig. 4, this belongs to the range of values admitting separation of the lensing effects from scintillation. Note a rather steep growth of the diffractive scale with frequency which results in a rapid suppression of scintillation for frequencies above a few GHz.

The Intergalactic Medium
While turbulence and thus scintillation in the ISM is already fraught with large uncertainties, even less is known about the properties of the IGM. In fact, much of what we know about IGM turbulence comes from observations of FRBs [61][62][63]. While to the best of our knowledge IGM scintillation has not yet been measured experimentally, there are indications that it might be accessible to SKA [64].
In the absence of direct measurements, we are forced to rely on theoretical estimates. Assuming Kolmogorov turbulence, we can estimate the magnitude of IGM density fluctuations, C 2 n , by equating the density fluctuation at the outer scale to the average IGM electron densitȳ n e 2.2 × 10 −7 cm −3 , corresponding to fully ionized hydrogen and helium [64]. 8 Considering the balance between heating and dissipation of energy in the IGM, ref. [65] estimated the outer scale to be of order l out ∼ 10 24 cm (∼ 0.3 Mpc). This leads to This is comparable to the Fresnel radius for a scintillation screen 0.25 Gpc away, r F 6.3 × 10 13 cm. We conclude that integralactic scintillation is at the borderline between the weak and strong scintillation regimes.
The observability of the lensing signal in the presence of IGM scintillation is illustrated in fig. 7 as a function of r diff and of the lens mass. We see that for realistic parameters and most of the interesting values of the lens mass the signal will be in regime (b). In other words, the two lensed images will be distorted incoherently, but the decorrelation bandwidth of the scintillation will be large enough to accommodate multiple lensing fringes.
The above estimate is based on the assumption that there are no baryon overdensitiessuch as an interjacent galaxy or galaxy cluster -along the line of sight. As r diff scales with n −6/5 e , such an overdensity can greatly increase the strength of the scintillation. While this might offer interesting opportunities for studies of the IGM using scintillation [64,66,67], it is undesirable for searches of FRB diffractive lensing. Fortunately, for sources at small redshifts Z S < 1, where most observed FRBs are expected to occur, the probability that the line of sight crosses a major baryon overdensity is low. Namely, the probability of crossing a galaxy cluster is less than 20% and about 3% for galaxies [61]. We will therefore assume the absence of overdensities and use r diff ∼ 4.5 × 10 13 cm for the IGM. In a real sample of FRBs, one could use the known locations of galaxies and galaxy clusters along the various lines of sight to reject those FRBs which are likely to be affected by baryon overdensities. Note that this restriction implies that the lens itself cannot lie inside a galaxy or a cluster. This is not a problem for dark matter, most of which is distributed in diffuse halos, filaments and voids. On the other hand, it essentially precludes searching for diffractive lensing by compact objects of astrophysical origin, such as brown dwarfs [40]. 9

Scintillation summary
When studying realistic lensed FRB signals, we should in principle consider the effect of intergalactic scintillation, of interstellar scintillation in the FRB's host galaxy, and of interstellar scintillation in the Milky Way. From the discussion of the previous subsection, we expect the effect of the IGM to be typically less important than that of the ISM. We have verified this using a sample of simulations where we combined IGM scintillation screens with ISM screens (see section 4 for the description of the simulation code [52]). However, the numerical cost of the simulations grows rapidly with the number of screens, so to reduce the computation time, we neglect IGM scintillation in the rest of our analysis.  Figure 7. Impact of intergalactic scintillation on FRB lensing by compact objects, shown here as a function of the lens mass M and the diffractive scintillation scale r diff . We assume an FRB at a comoving distance of 1 Gpc, and a lens at 0.5 Gpc. The scintillation screen is placed between the lens and the observer at 0.25 Gpc. The angular separation between the lens and the (unlensed) source, normalized to the Einstein angle, is y = β/θ E = 0.5. The frequency of the radio signal is taken to be 1 GHz.
The properties of the FRB host galaxies remain largely unknown, despite localization of several burst sources [68][69][70][71]. In the absence of any detailed information, we assume that the properties of the interstellar medium in the host galaxy are similar to that in the Milky Way. Then, barring a chance alignment of the line of sight with the galactic disk of the host, one can assume the scintillation screen of the host galaxy to have similar a effect on the lensing signal as the ISM screen of the Milky Way. 10 Therefore, for our exploratory study of the detectability of FRB lensing we choose to consider only the ISM screen of the Milky Way, which allows us to further reduce the required computing resources. It is worth emphasizing, however, that an actual future search for lensing signal in the data should include modeling of scintillation both in the host galaxy and in the IGM.

Formalism
To better understand the effect of scintillation on the spectra of astronomical radio sources in general and FRBs in particular, we have developed a simulation code to describe both lensing and scintillation. It is capable of evolving the signal from a point-like or extended radio source through an arbitrary stack of lensing and scintillation screens, and onwards to an observer. Each screen corresponds to a two-dimensional plane, oriented perpendicular to the line of sight. It associates each point x = (x 1 , x 2 ) on the plane with a characteristic phase factor ϕ j ( x) that is imparted to photons passing through that point on the screen. Here, the index j numbers the successive screens. The photon field A j ( x) on the j-th screen is obtained from the field A j−1 ( x) on the previous screen according to the Fresnel integral, (in analogy to eqs. (3.1) and (3.2)). Here, z j is the comoving line-of-sight distance of the source and the j-th screen, and the wave number k is given by the dispersion relation in a plasma, eq. (3.24). In evaluating the plasma frequency ω p , we assume a mean ISM plasma density of n e = 0.03 cm −3 . The plasma density in the IGM is several orders of magnitude smaller, but as we neglect scintillation in the IGM following the discussion in section 3.3, its precise value does not affect our analysis. Without IGM scintillation, there is no multi-path propagation through the IGM, so all photons of a given frequency are dispersed by the same amount. The screen A 0 ( x) corresponds to the source; for a point-like source, it is simply a δ-function centered at the coordinates x 0 of the source, A 0 ( x) = δ (2) ( x − x 0 ). The last (n-th) screen in the stack is placed at the location of the observer, so that the photon field A n ( x) corresponds to the observed image. We will not consider the spatial extent of the image here and just evaluate A n (0) for simplicity. As in section 3.1, the first term in the exponential factor under the integral in eq. (4.1) describes the additional geometric phase that photons experience when traveling from x to x (in the limit that | x| z j − z j−1 ). For lensing screens, the phase factor ϕ j ( x) describes the non-geometric part of the Shapiro time delay where ψ( x) = θ 2 E log | x|/(z j+1 − z j−1 ) is again the lensing potential for a point-like lens, and θ E is the Einstein angle defined in Eq. (2.3).
For scintillation screens, ϕ j ( x) = ϕ scint j ( x) describes the variation in the dispersion relation due to random fluctuations of the plasma density in the scintillating medium. As discussed in section 3.3, it is the modeling of these fluctuations that is the main source of uncertainty in our calculations. We assume Kolmogorov turbulence [55,59,60], i.e. we assume that the fluctuation mode with wave number κ on the screen has an amplitude |f ( κ)| drawn from a Gaussian distribution of width 11 Here, κ diff ≡ 2π/r diff is an arbitrarily chosen reference scale. The phase off ( κ) is also a random number, uniformly distributed between 0 and 2π. We emphasize that κ describes density fluctuations on a scintillation screen and should not be confused with the wave number k of the photon field. The scintillation phase ϕ scint j ( x) is then proportional to the Fourier transform off ( κ): The normalization of ϕ scint j ( x) is chosen such that O(1) variations in the phase occur over distance scales of order r diff , where · denotes averaging over the screen, and n is a unit vector, which we choose to be oriented in the x 1 direction. 12 We assume the incoming amplitude of the radiation at frequency ω to be of the form: This expression is an attempt to roughly fit an observed FRB spectrum from ref. [74]. While we do not claim that our extrapolation to higher frequencies than observed in ref. [74] is necessarily valid, our further analysis will rely mainly on a reconstruction of the transfer function, largely reducing the uncertainties associated with the exact shape of the FRB spectrum (see section 4.3).

Numerical Implementation
To compute the Fresnel integral in eq. (4.1) numerically, we employ several simplifications: 1. As the Fresnel integral on a lensing screen is numerically very badly behaved, we treat lensing analytically. In other words, if the j-th screen is a lensing screen, we evolve the photon field directly from screen j − 1 to screen j + 1, multiplying by the appropriate magnification, see eq. (3.9). For the lens masses considered in this paper, we always work in the geometric optics limit, even though our code (released together with this paper [52]) is also able to handle the wave optics regime, and switches to it automatically if ω(D L D S /D LS ) θ 2 E < 10. 2. We discretize scintillation screens, i.e. we evaluate the integrand in eq. (4.1) on a square grid of points, and then evaluate the integral as a simple sum. The size of the screen in both directions is chosen r ref to make sure the region | x| r ref , from which most of the signal is received, is well within the simulated region. (At distances r ref from the line of sight, the geometric phase varies much faster than the scintillation phase, so that photons emitted from this regions mostly interfere destructively.) The grid spacing dx is chosen much smaller than r diff to make sure the phase fluctuations on the screen are well resolved. Moreover, we require dx to be small enough to still resolve the fastvarying geometric phase at the edges of the screen. (The last requirement is dropped when the grid becomes so large that it would not fit into GPU memory any more.) To further optimize memory consumption, we do not simulate the whole screen at once. Instead, we first simulate the long wavelength fluctuations, discretized with a coarser grid spacing. The geometric phase factor in the Fresnel integral from eq. (4.1) is evaluated only once for each cell on this coarse grid. The cell size of the coarse grid is chosen r F to ensure that this is a good approximation. For simulating shorter wavelength fluctuations, we divide the screen into smaller patches and simulate the fluctuations on each of them separately, but at full resolution dx. The patches are always chosen much larger than r diff .
3. For most of our results, we reduce the scintillation screen to one dimension. In other words, we assume all photons to travel in the x 1 -z plane. This greatly reduces memory consumption and computational effort, while all the essential features of scintillation are preserved, as evidenced by the good agreement with the qualitative analytic estimates from section 3.2.
Our code [52] is written in Python 3, making heavy use of the CuPy library [75] to run it on CUDA-enabled GPUs and thus optimally benefit from the vectorizability and scalability of our approach. In order to vectorize the evaluation of the magnification function for gravitational lensing in the wave optics regime, we had to implement vectorized versions of certain special functions, notably the gamma function Γ(z) and the Laguerre polynomials L n (z), for the case of complex z and n. In implementing the Gamma function, we follow Lanczos' approximation [76,77], while our implementation of the Laguerre polynomials is based on the implementation in the mpmath package [78], see also [79].

Data Analysis
Let us now outline the procedure we use to extract the lensing signal from our simulated data. Our objective is to identify the periodic modulation of the frequency spectrum due to lensing on top of the stochastic background from scintillation. We would like to work in particular with the Fourier transform of the transfer function, eq. (3.22), where the lensing signal materializes as a peak at τ = ∆t.
Of course, we do not have direct access to the transfer function and its Fourier transform as we do not know the initial spectrum at the source. Moreover, observing the lensing peak is complicated by the fact that it is typically located on top of a steeply falling (or rising) background. The first step in our analysis chain is a fit of the logarithm of the "observed" radiation intensity, log I obs (ν) ≡ log |f obs (ν)| 2 with a high-order polynomial (we use d = 15). To ensure the fit is smooth, we use polynomial lasso regression with regularization parameter 0.05 [80,81]. We then subtract the fit function from log I obs (ν) to obtain an estimate log |T (ν)| 2 of the log of the transfer function |T (ν)| 2 , eq. (3.22). Besides lasso regression with a 15th order polynomial, we tried several alternative fitting functions and regularization methods (for instance ridge regression). We found them to all perform roughly comparably. The minor effects that various fitting strategies might induce on the reconstructed spectrum, can be studied by the reader independently using our public code [52].
At the second stage of our analysis, we look for peaks in |T (τ )|, defined in eq. (4.7). For a fixed peak time delay τ 0 and peak width w τ , we fit a polynomial P (τ 0 , w τ ; τ ) to the Fourier power spectrum (in linear space), excluding the points in the interval [τ 0 − w τ /2, τ 0 + w τ /2). The fitting is performed by polynomial ridge regression, with regularization parameter 10 −4 . As previously, fitting with other parameters yields comparable results and the small differences can be traced by using alternative methods in our code [52]. We then define a test statisticσ as the difference between |T (τ )| 2 at the peak and the polynomial at this time delay, normalized by the width of the peak and by the standard deviation of all samples outside the peak region: σ(τ 0 , w τ ) = N out peak j in peak |T (τ )| 2 − P (τ 0 , w τ ; τ ) N in peak j outside peak |T (τ )| 2 − P (τ 0 , w τ ; τ ) .  Figure 8. The analysis of a single, diffractively lensed FRB spectrum following the steps described in the text. The upper left plot shows the radiation power spectrum, and the lower left plot shows the Fourier transform thereof, where the lensing peak is clearly visible at the expected time delay (vertical black line). However, we also notice a strong enhancement of the power at small τ , which in some cases spoils the detectability of the lensing peak. Similarly, the middle column shows the reconstructed transfer function and its Fourier transform, and the column on the right shows the same after averaging procedure described in the text. The enhancement at small τ has disappeared and only the lensing peak remains. We have assumed a source at a comoving distance D S = 1 Gpc and a lens at D L = 0.5 Gpc, y = 0.5. An ISM scintillation screen is placed at D ScO = 1 kpc, and r diff = 10 9 cm at the spectral midpoint.
Here the sum in the numerator runs over the N time delay samples inside the peak region, while the sum in the denominator runs over all samples outside the peak region. We repeat this procedure for each time delay τ 0 and each width w τ , and we consider as a potentially interesting peak any local maximum in the two-dimensional manifold defined byσ(τ 0 , w τ ).
We quote a non-zero test statistic for the lensing signal if a peak is found located within 15% from the "real" ∆t of the lens. Otherwise, we setσ = 0. Note that the lensing peak is typically narrow, so the highest significance is always found for the minimum value of w τ , which corresponds to a peak spanning two time delay bins. The procedure is illustrated in fig. 8. In the right-hand column of that figure, we also show how the lensing peak in |T (τ )| 2 is affected if we average I obs (ν) over frequency intervals ∆ν av = 17 MHz before taking the Fourier transform. We see that high-τ fluctuations are suppressed, but also the height of the lensing peak is somewhat reduced. Averaging is therefore a useful tool only when high-τ fluctuations are worrisome. In our subsequent analysis, we will not use averaging.
To translate the test statisticσ into a statistical significance, which can then be interpreted as a quantile of the normal distribution, we have simulated 10 000 pseudo-experiments.

Relative occurrence
Pseudo-experiments sample 2σ Figure 9. Relative occurrence of statistical measuresσ in the FFT-squared of the power spectrum in the pseudo-experiments sample. The distribution can be reasonably approximated by Gaussian. We calibrate our statistical significance based on this curve.
Each of them corresponds to an unlensed FRB distorted by a scintillation screen with r diff = 10 10 cm at 1 kpc. In each pseudo-experiment, we look for a spurious peak at the ∆t that would correspond to lensing by a 10 −3 M PBH located at D L = 0.5 Gpc, y = 0.5. We plot the distribution of theσ values of the spurious peaks in fig. 9. The quantiles of this distribution correspond to the real statistical significance σ, which we will use below to set limits on the PBH abundance. One comment is in order regarding this calibration procedure. The distribution ofσ values can depend on the diffractive scale r diff of the scintillation screen. Nonetheless, as generating a large number of pseudo-experiments is computationally expensive, we use the curve in fig. 9 for all values of r diff . For r diff > 10 10 cm, applying calibration based on fig. 9 is too conservative, while for smaller r diff , it may somewhat overestimate the sensitivity. However, we are going to find that for r diff ∼ 10 9 cm, the sensitivity is weak anyway.

Results
To anticipate the sensitivity of FRB lensing to compact dark matter objects, we now investigate the significance of the lensing signal as a function of the model parameters, in particular the lens mass M , its position D L along the line of sight and angular distance β from the line of sight, as well as the diffractive scintillation scale r diff in the Milky Way. For simplicity, we will keep the distance of the source fixed at 1 Gpc (comoving), we will neglect intergalactic scintillation as well as scintillation in the FRB's host galaxy, and we will assume the Milky Way's scintillation screen to be located at a distance of 1 kpc.
As a first result, we show in fig. 10 the local statistical significance of the lensing signal as a function of M and r diff , keeping D S = 1 Gpc, D L = 0.5 Gpc, and y ≡ β/θ E = 0.5 fixed. Note that in this scan we choose the instrumental frequency band and frequency resolution as needed to well resolve the lensing peak. This assumption may be unrealistic at very small  Reconstructed transfer function lens masses, which require very large frequency bands, and at very large lens masses, which require very high frequency resolution. Therefore, at high and low lens masses fig. 10 should be thought of as proof of principle rather than as a realistic expectation. Figure 10 confirms our expectations from fig. 4: the sensitivity is good except at the smallest r diff values. In the left panel of fig. 10, we observe a loss of sensitivity at low masses. There, the lensing peak in |T (τ )| 2 falls in the low-τ region and is buried underneath the Fourier transform of the smooth component of the intensity spectrum. As expected, reconstruction of the transfer function by dividing out the smooth component significantly increases the sensitivity, as illustrated in the right-hand panel.
In order to proceed further, we focus specifically on a future SKA-type observatory 13 . This is expected to have a wide frequency band extending above a few GHz. Going to higher frequencies is beneficial for detecting diffractive lensing by dark matter as it makes scintillation less important. Of course, our analysis can be repeated for radiotelescopes operating at lower frequencies, in particular, for CHIME 14 and ASKAP. 15 As our fiducial parameters we will assume a frequency resolution no better than 1 MHz and a band width no larger than 10 GHz. This is based on the frequency range of the SKA-MID array, and on the expected resolution of the instrument [82,83]. With these assumptions, we simulate the lensing signal as a function of the lens mass, the lens distance D L from the observer and its angular distance y from the line of sight. Once again, we assume D S = 1 Gpc, and we introduce an interstellar scintillation screen at 1 kpc, with r diff = 10 10 cm. For each combination of M , D L and y, we then estimate the expected local significance of the signal, as described in section 4.3. Note that we do not run simulations for lens masses above 0.1 M because detecting a lensing signal in that case would require a frequency resolution better than 1 MHz. Similarly, we do not consider lens masses below 10 −4 M , where the spacing of lensing fringes in the spectrum becomes comparable to the maximum bandwidth.
To obtain a bound on the abundance of point-like lenses such as PBHs, we largely follow the statistical procedure described in [32]. We define the likelihood function for a single FRB source at redshift Z S , Here, L 0 ( µ) is the likelihood of the (simulated) data -assumed not to contain a lensing signal -when compared to a model that does contain a lens with parameters µ ≡ (M, Z L , y). We set L 0 ( µ) = 1−erf(σ/ √ 2), where σ is the significance of the lensing signal discussed in section 4.3 above. This immediately implies that L 0 (0), the likelihood of the data when compared to the theoretical prediction in the absence of a lens, equals unity. L FRB (M, ρ PBH , Z S ) then measures the likelihood of data which does not contain a lensing signal when compared to the prediction assuming a population of lenses of mass M and mass density ρ PBH . The factor d 3 x (1 + Z L ) 3 ρ PBH /M measures the probability of finding a lens in a small volume element d 3 x at redshift Z L . The integral in eq. (4.9) runs from the source to the observer in the longitudinal (z) direction, and out to infinity in the transverse direction. (In practice, we cut it off at y = | x ⊥ |/(D L θ E ) = 5, as there will be no sensitivity to lenses that are further away from the line of sight.) Note that x here refers to a three-dimensional physical (not co-moving) coordinate. In writing down eq. (4.9), we have assumed that the probability for a single source to be lensed by two lenses simultaneously is negligible.
To compute L FRB (M, ρ PBH , Z S ) for a given lens mass M , lens density ρ PBH , and source redshift Z S , we evaluate σ as function of the lens coordinates. We then evaluate the integral over d 3 x on the resulting grid.
Of course, observing just a single FRB is not sufficient to set a meaningful limit on compact dark matter. Obvserving a large number N of them, however, will greatly boost the sensitivity of the method. We estimate the expected 95% confidence level exclusion limit on ρ PBH for fixed M by solving the equation Note that L i FRB (0, 0, Z i S ) = 1. For simplicity, we will in the following assume that all sources are at the same redshift, and that also the properties of the interjacent scintillation screens are the same. When the probability for lensing a single FRB is small (L FRB (M, ρ PBH , Z i S ) ≈ 1), the reach for the PBH density ρ PBH is then inversely proportional to the number of sources N . This is because increasing the number of sources linearly decreases the effective optical depth.
The resulting bound on the PBH parameter space is shown in fig. 11 and compared to other constraints on primordial black holes. We see that, for a realistic number of FRB observations in SKA (a few tens per day [45]), our method can be expected to yield highly competitive constraints. Even more importantly, the expected limits are complementary to those from microlensing searches because FRBs probe the distribution of dark matter over   Figure 11. Expected limits on primordial black holes from diffractive lensing of FRBs. We assume an interstellar scintillation screen at a distance of 1 kpc, with diffractive scale r ISM diff = 10 10 cm at 1 GHz, and negligible intergalactic scintillation. We compare our projections (in blue) to microlensing constraints from Subaru HSC [18], 17 Kepler [84], MACHO [15], EROS [16], and OGLE [17], to CMB constraints due to accretion onto PBHs [14], to limits based on the dynamics of ultra-faint dwarf galaxies [85], and to bounds on the contribution of Hawking radiation from PBHs to the extra-galactic gamma background [12]. We emphasize that CMB and gamma-ray background constraints are based on specific properties of PBHs, and that the other existing constraints probe the local distribution of dark matter in the Milky Way and its immediate neighborhood. By contrast, diffractive lensing of FRBs offers a model-independent probe of MACHOs at cosmological distances.
cosmological distance scales, while microlensing is sensitive to dark matter in the Milky Way and its immediate vicinity. This makes diffractive lensing of FRBs a particularly interesting probe for compact dark matter mini-halos, which may be inaccessible to microlensing searches due to their non-negligible spatial extent [27]. Indeed, the condition r L < r E on the lens size r L , under which the lens can be treated as point-like, is more readily satisfied at cosmological distances. Besides, mini-halos may exist in regions of low baryon density, but may have suffered tidal disruption in the Milky Way. The sensitivity of diffractive lensing ends at PBH masses below 10 −4 M because at such low masses the lensing peaks in the frequency spectrum are too far apart for a significant number of them to be contained in a realistic instrumental frequency band. At high PBH mass, the limiting factor is the instrumental frequency resolution.
In general, it should be mentioned, that in vast majority of the models the mass distribution of the dark matter can be relatively broad. While we are assuming here a very narrow distribution, generalization to realistic mass distributions is desirable [13,38,[88][89][90][91].
Finally, we briefly mention other proposals for probing PBHs in the mass range between 10 −4 M and 1 M . A novel method to probe compact objects in the Milky Way is pulsar timing [27]. An alternative, indirect, approach involves primordial gravitational waves produced by adiabatic density fluctuations which also give rise to PBH formation [92,93]. The method is thus specific to PBH production via enhanced adiabatic Gaussian density fluctuations. 18 At somewhat larger PBH masses, one can look for gravitational waves from PBH mergers [95][96][97]. The expected gravitational wave signal is, however, subject to large uncertainties in the evolution of PBH populations over cosmological history [95,[98][99][100][101]. In the region of lighter masses, PBHs can lead to the destruction of neutron stars [102] and white dwarfs [103]. Exploting these mechanisms to set limits is, however, complicated by uncertainties in the dark matter concentration in globular clusters [104,105] and in the treatment of energy deposition by PBHs crossing stars [106,107].

Conclusions and Outlook
In this work we have analyzed the sensitivity of diffractive gravitational lensing of Fast Radio Burst (FRB) signals to compact dark matter objects (MACHOs). Diffractive lensing (sometimes called femtolensing or nanolensing in the literature) means the interference of the two lensed images of a source, which leads to periodic modulation of the observed frequency spectrum.
We have studied the interplay between the lensing signal and distortions caused by interstellar scintillation, and we have concluded that the lensing signal can often be extracted in spite of the distortion. We have developed a data analysis procedure to extract the lensing signal and showed its efficiency on simulated data. Our results indicate that the existing and future FRB surveys carried out by ASKAP, CHIME and SKA will be able to strongly constrain the cosmological MACHO abundance. As an illustration, we studied the sensitivity of an SKA-like survey to primordial black hole (PBH) dark matter. We concluded that it will be able to probe the parameter space of PBHs in the mass range between 10 −4 M and 0.1 M and constrain their abundance down to about 1% of the total dark matter content of the Universe.
Compared to microlensing searches, which also probe primordial black holes in this mass range, our method is unique because it tests the distribution of dark matter at cosmological distances rather than in our local cosmic neighborhood. This distinction is particularly important when generalizing our bounds to other compact dark matter candidates such as dark matter mini-halos. The local abundance of dark matter mini-halos may be reduced due to tidal disruption in the Milky Way, suppressing the probability of microlensing events. Moreover, microlensing searches may be insensitive to dark matter mini-halos due to their non-negligible spatial extent, which is larger than the Einstein radius at Galactic distances. At cosmological scales, on the other hand, the Einstein radius is significantly larger and may well exceed the mini-halo's size, so that it acts effectively as a point-like lens. In this case, our results are directly applicable. Investigating the sensitivity of FRB lensing to more extended compact objects will be an interesting direction for future work.
Apart from dark matter, the range of lens masses considered in this paper may also contain objects of astrophysical origin, such as brown dwarfs. However, it appears unlikely that they can produce a discernible lensing signal in FRB spectra. The reason is that they are expected to reside in regions of high baryonic density, namely galaxies, characterized also by high concentrations of free electrons. The scattering of radio waves produced by this ambient plasma will be much stronger than the deflection of light due to gravitational lensing, so that the effect of the latter will be completely swamped by scintillation. In principle, the situation might improve for FRBs with spectra extending to high frequencies 5 GHz, where the scintillation effects become weaker. We leave the study of this interesting possibility for future.
Before concluding, we emphasize again that the reach of diffractive lensing of radio signals depends on our understanding of interstellar and intergalactic scintillation, and thus of turbulence in the interstellar and intergalactic medium. While the scintillation parameters we have assumed in this work represent our current best understanding of these phenomena, future observations should allow us to much better quantify the impact of scintillation. The rapid progress in the study of FRBs and in radio observations in general will contribute to reducing this uncertainty.