The Effect of Gravitational Lensing on Fast Transient Event Rates

The observed number of sources as a function of flux (the ‘logN–logS’ relation) is perhaps the most fundamental quantity to population studies across astronomy. Encoded within them are the properties of the source population: luminosity function, spectra and redshift distribution (Longair & Scheuer, 1966). Constraining the population functions of fast radio bursts (FRBs) and gamma-ray bursts (GRBs) has been an area of particular interest in modern astronomy(Macquart & Ekers, 2018; Sun et al., 2015). These bursts are often produced in exotic systems or cataclysmic circumstances(Abbott et al., 2017; Woosley & Bloom, 2006; Platts et al., 2019) and therefore represent an important tracer of rare systems.

Propagation effects such as extinction, absorption or scintillation (Frontera et al., 2000; Masui et al., 2015) can however serve to obfuscate the intrinsic behaviours of a transient source, and thus must be accounted for if we are to determine the population functions from the observations. One such propagation effect is gravitational lensing.

Gravitational lensing is important to consider because it can magnify source objects, significantly amplifying their observed flux, potentially resulting in erroneously inferred luminosities. Many examples of lenses magnifying, distorting or even multiply imaging individual sources have been recorded, including for quasars (Walsh et al., 1979), GRBs (Paynter et al., 2021) and supernovae (SNe) (Kelly et al., 2015). However, the influence of gravitational lensing on the source counts of a population is typically slight; the fraction of quasars undergoing strong lensing is expected to be only a few percent of quasars beyond redshift six (Pacucci & Loeb, 2019; Yue et al., 2022) and for SNe (Porciani & Madau, 2000; Jonsson et al., 2010) the fraction of lensed bursts is constrained to be small.

The lack of observed gravitational lensing can be used to infer constraints on the population of lenses. Zumalacárregui & Seljak (2018) placed strong constraints on the fraction of dark matter in primordial black holes (PBHs) using type Ia SNe. The authors model the probability of magnification convolved with the spread in supernova magnitudes and compare with the observed spread to constrain the population of lenses at cosmological distances. They find that the observations are inconsistent with a large population of lenses and therefore restrict the fraction of dark matter in PBHs to be less than 0.3 for PBH masses greater than $0.01M_{\odot}$. Strong constraints can be placed in the case of SNe Ia observations because of the narrow distribution of intrinsic SNe Ia energies (i.e. because SNe Ia are standard candles).

These results do not mean that lensing will be unimportant for FRB and GRB source counts. The smaller angular size of GRBs and FRBs compared to SNe Ia makes them sensitive to even lower mass lenses. The uncertainty surrounding the emission mechanism of most fast transients, however, makes it difficult to separate potential propagation effects from potential emission mechanism effects. For example, it is difficult to distinguish a highly magnified event from an intrinsically luminous burst. Thus, lensing searches have been restricted to searches for multiple source images, be it in the spatial or temporal domain (Muñoz et al., 2016; Laha, 2018; Oguri, 2019; Sammons et al., 2020; Paynter et al., 2021; Leung et al., 2022; CHIME/FRB Collaboration et al., 2022; Connor & Ravi, 2022).

As we shall show, the luminosity of GRBs and FRBs are sensitive to lens masses much too small to produce resolvable multiple images: as small as $10^{-15}M_{\odot}$ and $10^{-5}M_{\odot}$ respectively, and there is little evidence to rule out the presence of a cosmological population of lenses on these low mass scales. Constraints on primordial black holes (PBHs) still allow for $100\%$ of dark matter to be comprised of PBHs in the asteroid to sub-lunar mass regime ($10^{-15}M_{\odot}\leq M_{L}\geq 10^{-10}M_{\odot}$; Carr & Kühnel, 2020). The sub-lunar to sub-stellar ($10^{-10}M_{\odot}\leq M_{L}\geq 10^{-2}M_{\odot}$) regime is also of interest as it is only constrained for our own galaxy halo, with $\sim 10^{-5}M_{\odot}$ lenses potentially existing locally in appreciable density.

To account for lensing effects on the source counts of fast transients, and estimate how this may be used to constrain the number of PBHs, we create a generic model for differential rates of fast transients with fluence, $dR/dF$, in an inhomogeneous universe and compare it to its smooth universe counterpart, considering only the total magnification caused by inhomogeneity.

This paper is structured as follows: §2.1 $\&$ §2.2 introduce the lensing theory, §2.3 $\&$ 2.4 introduce the differential event rates formalism, §2.5 discusses our numerical method and justifies our model’s assumptions, in §4 we characterise the response of our model to variation of the input parameters, §5 explores the possibility that all FRBs are highly magnified, §6 contains explicit calculations of the changes to FRB and GRB event rates in universes of varying inhomogeneity and finally, we discuss the implications of these results in §7 and explore how many FRBs would be needed to place constraints on the PBH parameter space.

Here and in following sections we illustrate $dR/dF$ using plausible values of the population functions for FRBs (James et al., 2021; Luo et al., 2020), and vary individual parameters over a broad range of typical values.

Fig. 1 plots $dR/df$ as a function of $f$ for a smooth universe, where $f$ is the observed fluence normalised to what would be observed for an $E_{\nu_{e},\text{max}}$ burst at redshift $z=1$ in a smooth universe. We normalise the rates to the expected $dR/df$ for a Euclidean universe, $R_{0}f^{-2.5}$ where $R_{0}$ is the rate density in the local universe.

Fig. 1 shows that $dR/df$ in a smooth universe has roughly two fluence domains of behaviour for event rate energy functions with $\gamma>-2.5$. We define these regions about a break fluence $f_{b,1}=10^{-1.16}$ which is the fluence corresponding to $E_{\nu_{e},\text{max}}$ at $z_{\text{max}}$. At the high fluence end ($f>f_{b,1}=10^{-1.16}$) the maximum redshift is determined by $E_{\nu_{e},\text{max}}$. Because the power law index of the energy function, $\gamma>-2.5$ is shallower than the expected Euclidean evolution $\propto f^{-2.5}$ the change in rate with fluence is dominated by the change in sources due to a restricted redshift, as opposed to having fewer bursts at higher energies. Therefore $dR/df$ in a smooth universe has a power law index at the high fluence end approaching the Euclidean expectation.

On the low fluence end ($f<f_{b,1}=10^{-1.16}$), where $z_{\text{max}}$ is instead restricted by the spatial distribution $\theta_{z}$, a change in the observed fluence does not affect $z_{\text{max}}$ and the change in the observed rate is dominated by seeing fewer bursts at higher energies. Therefore at the low fluence end $dR/df$ adopts power law index seen in the energy function of $\gamma$. If the energy function has a steeper index, $\gamma\leq-2.5$ then for a uniform spatial distribution the change in the number of bursts due to the energetics will dominate across all fluences and the Euclidean behaviour will never be recovered.

In a clumpy universe $dR/df$ depends on the convolution of the intrinsic energy function with $p(\mu)/\mu$ as per Eq. 6. For gravitational lensing this convolution kernel is generally $\propto\mu^{-4}$ in the high magnification limit. Therefore, for all energy functions with $\gamma>-4$²²2We only calculate $dR/df$ in a clumpy universe for $\gamma>-4$. Because $p(\mu)/\mu\propto\mu^{-4}$ at high $\mu$, for $\gamma\leq-4$, the inner integral of Eq. 6 would not converge. this convolution will be dominated by the intrinsic energy function, and the behaviour of $dR/df$ will be well approximated by the smooth universe behaviour described above. The exception will be the case where all events are highly magnified as we shall discuss in §5 and the edge effects we describe below.

To discern the effect of lensing we express our results as the differential event rate ($dR/df$) in a clumpy universe with a smooth matter fraction $\eta<1$, normalised by the differential event rate in a smooth universe ($\eta=1$). Fig. 2 shows the $\eta=0$ case, depicting a $1-10\%$ fractional change in $dR/df$ due to lensing that has a characteristic shape common to all values of $\gamma$.

Fig. 2 shows the low fluence regime for all values of $\gamma$ and all fluences for $\gamma\leq-2.5$ to have approximately constant fractional difference between the lensed and unlensed differential rates. $dR/df$ in these regions are dominated by the energetics and hence show similar behaviour despite any lensing, consistent with our expectation.

The fluctuation structure is comprised of an initial decrease, before a sharp increase which then tends back towards unity. To understand why this structure appears we must look to the break fluence $f_{b}$. For $f>f_{b}$, the maximum redshift becomes restricted by $E_{\text{max}}$ and $dR/df$ becomes dominated by a reduction of the volume out to which sources can be observed as discussed previously. In a smooth universe this occurs at $f_{b,1}=10^{-1.16}$, shown as a dotted line in Fig. 2. In a clumpy universe however this occurs at a lower fluence of $f_{b,\eta}=10^{-3.69}$, shown as a dashed line in Fig. 2. The lower fluence is due to a demagnification of $1/\langle\mu\rangle$ (from Eq. (2), $D_{1}^{2}(z_{\text{max}}=100)/D_{0}^{2}(z_{\text{max}}=100)=0.00295\approx 10^{-3.39}/10^{-1.16}$) associated with viewing along an empty beam in a clumpy universe compared to a smooth one. Therefore, as we increase fluence $dR/df$ becomes dominated by the reduction to $z_{\text{max}}$ in a clumpy universe before it does so for a smooth universe, resulting in the dip. Once we hit $f=10^{-1.16}$, the smooth universe also becomes dominated by reduction to $z_{\text{max}}$ resulting in an inflection point in the fractional change in accordance with $z_{\text{max}}$ decreasing faster in a smooth universe. As the fluence increases and the maximum redshift approaches the nearby universe, the mean magnification decreases and clumpy and smooth universes become indistinguishable, resulting in $dR/df$ values that converge as seen on the high fluence end of Fig. 2.

Apart from $\gamma$, our model takes in a number of input parameters as discussed in §2.3. Below we characterise the response of $dR/df$ to variation of these parameters.

Above we have describe the scenario where a burst population which could be observed in a smooth universe is altered by lensing. However, an alternative scenario which has been discussed in the FRB community is the possibility that bursts are only observed because of gravitational lensing, i.e. all observed bursts are intrinsically low energy but highly magnified. In such a situation the minimum redshift becomes important. We define $f_{\text{max}}$ as the largest fluence where an observer will see a burst with $\mu=1$, corresponding to $E_{\nu_{e},\text{max}}$ at $z_{\text{min}}$. For $f>f_{\text{max}}$ only magnified bursts are observed, and because the entire spatial domain contributes, the behaviour of $dR/df$ with fluence will be determined entirely by the inner integral over magnification in Eq. (6). As $f$ increases, the intrinsic energy required at each magnification will increase and the observed rate will decrease following $\theta_{E}$ with index $\gamma$ (fewer bursts at higher energies), assuming again that $\gamma>-4$. Additionally, the minimum observed magnification will increase with $f$, resulting in a decrease to $dR/df$. If the PDF with a factor $1/\mu$ behaves as $p(\mu)/\mu\propto\mu^{\xi-1}$ (as it does in the high magnification limit) the integration from $[\mu_{\text{min}},\infty]$ will vary with $\mu_{\text{min}}$ following a power law of index $\xi$. Therefore as $f$ increases and $\mu_{\text{min}}$ increases, $dR/df$ will also vary with index $\xi$. These behaviours will occur simultaneously, however if one is significantly steeper we expect that it will dominate the change to $dR/df$, e.g. for $\xi\ll\gamma$, $dR/df$ will be approximately $\propto f^{\xi}$, and vice versa.

Given that in the geometric optics limit for a stationary universe our chosen PDF varies as $p(\mu)\propto\mu^{-3}$ the expected behaviour for all $\gamma\geq-3$ is to have $dR/df$ vary with a power law index of $-3$. Fig. 8 shows precisely this scenario, plotting $dR/df$ for the same selection of $\theta_{E}$ functions used earlier but with $E_{\nu_{e},\text{max}}=10^{26}$ erg/Hz which is more in line with the spectral energy observed for the Galactic FRB (The CHIME/FRB Collaboration, 2020). Fig. 8 shows that for all $\gamma\geq-3$ the $dR/df$ has an index of $-3$ as expected.

If all FRBs were highly magnified in a stationary universe, the behaviour of $dR/df$ would be consistent with a $\gamma=-3$ intrinsic energy function in a smooth universe. Estimates of $\gamma$ outside the context of lensing would therefore yield $\gamma=-3$. Best estimates of $\gamma$ from observed FRBs give $\gamma\approx-2$ (James et al., 2021; Luo et al., 2020), which never has $dR/df$ behaviour consistent with a $\gamma=-3$ model and therefore allows us to refute a scenario where FRBs are only observable due to high magnifications from stationary gravitational lenses.

In appendix B we show that as a result of wave optics low mass lenses may not follow $p(\mu)\propto\mu^{-3}$. Given this potential departure from the $p(\mu)\propto\mu^{-3}$ behaviour we can only use $\gamma\neq-3$ to refute that all FRBs are highly magnified by lenses of certain mass. The range of masses which are constrained depends on the magnification required to make the bursts observable, i.e. the maximum apparent energy normalised by the maximum intrinsic energy $\mu_{\text{max}}=E_{\text{max, obs}}/E_{\text{max, int}}$. For low lens masses the maximum magnification will be insufficient to make low intrinsic energy bursts observable, allowing us to rule them out by default. For higher intrinsic energies and higher lens masses the lensing behaviour will approach the geometric expectation, $p(\mu)\propto\mu^{-3}$, which are then ruled out as FRB $\gamma\neq-3$. We plot these conditions in Fig. 9, assuming that magnifications $\mu<\mu_{\text{max}}/10^{1.5}$ have $p(\mu)\propto\mu^{-3}$ as describe in appendix B. The figure shows the excluded regions for FRBs, with the intermediate region in grey. Here lensing is of sufficient magnification to make bursts observable but close enough to the maximum magnification to have prominent fringes in the cross section that change the behaviour from $p(\mu)\propto\mu^{-3}$. In this region of the parameter space, we cannot rule out that all FRBs are highly magnified on the basis of $\gamma$ alone.

Assuming that both FRB and GRB populations are intrinsically transient, and not all highly magnified, we have shown in appendix 3 that any effect on their differential rates from lensing will be small. Therefore, estimates of their intrinsic parameters, i.e. $\gamma$, $\alpha$ and $\theta_{z}$, made without accounting for the possibility of lensing will approximate the true population parameters well even if all of the Universe’s matter were to be contained in lenses ³³3With the exception of $E_{\text{max}}$ which may be drastically affected by lensing but has little impact on the inferred value other parameters. It is therefore appropriate to use the observed population parameters as inputs to our model when calculating the expected $dR/df$ for transient populations in a clumpy universe. In this section we calculate $dR/df$ specific to each transient class for universes with varying $\eta$. We display these rates normalised to what a uniform spatial distribution at the local rate would yield, as well as the fractional differences due to lensing.

We model the effect of lensing on FRBs, long GRBs and short GRBs. We use literature values to build fiducial event rate energy functions in each case as discussed in the following sections. We stress that these models are simplified for the purpose of demonstrating the effect of lensing.

Understanding the effects of gravitational lensing on $dR/df$ is crucial if the increasing number of recorded bursts with no redshift information are to be used to constrain $\Theta_{E}$. Most of the lensing effects we have derived here are small. In the current context of observational constraints on $\Theta_{E}$, for any of the transients mentioned here, the lensing effects are negligible compared to other sources of uncertainty. Hence, lensing may in most cases be ignored when calculating the expected differential rates from a $\Theta_{E}$ model.

Provided that there is a sharp⁴⁴4We take sharp to mean steeper than the power law dependence ($\xi$) of the magnification PDF with $\mu$. In the geometric limit $\xi\approx=-3$ cutoff in the intrinsic rate at some critical energy $E_{\nu_{e},\text{max}}$, lensing will cause a fluctuation in $dR/df$ relative to what is expected in a smooth universe, effectively independent of the transients underlying $\Theta_{E}$. This fluctuation, seen in figures 10, 11 and 12, is a unique effect of lensing and could be used to constrain the value of $\eta$ which effects its scale.

A comparison between the figures in §4 shows that the scale of the fluctuation is also dependent on $\theta_{z}$, $\alpha$ and $\gamma$. Due to this degeneracy, the intrinsic population parameters must be well known if $\eta$ is to be constrained from the observed rates. Given the strong dependence of the absolute rates on these parameters however, they will require far fewer transients to be well constrained. To avoid lensing effects when constraining the intrinsic parameters of $\Theta_{E}$, they should be modelled from low fluence bursts where the effect of lensing is negligible. To define a low fluence we require $f_{b,\eta}$, the break fluence defined in §3. Without knowing $E_{\nu_{e},\text{max}}$ this becomes more difficult, however we can approximate a minimum value of $E_{\nu_{e},\text{max}}$ and $f_{b}$ by considering the magnification decomposition of the fractional change in $dR/df$ due to lensing. For the FRB case shown in Fig. 12 this magnification decomposition is plotted in Fig. 13. It shows that across all fluences there is very little contribution from magnifications above $\mu=10^{2}$. A lack of high magnification bursts means that $E_{\nu_{e},\text{max}}$ is unlikely to be lower than $1/10^{2}$ the apparent maximum. By establishing a lower limit on $E_{\nu_{e},\text{max}}$ and setting $f_{b,\eta}$ to correspond to this approximate maximum at a redshift of negligible star formation in a universe $\eta=0$ we can safely assume $f<f_{b,\eta}$ to be in the low fluence regime.

Assuming that the population parameters are well constrained by these low fluence bursts, the intrinsic model could be extrapolated to the high fluence regime for the case of a smooth universe and compared to the observed differential event rates to place a lower limit of the value of $\eta$. Averaging over the expected fluctuation for all fluences higher than $f_{b,\eta}$, and below the fluence for a maximum energy burst at $z=0.001$, we can determine the number of high fluence bursts that would be required to statistically distinguish the expected average fluctuation. We plot this number for FRBs for varying values of $\eta$ and varying intrinsic $\gamma$ and $\alpha$ values in Fig. 14. We assume that the observed bursts are distributed log-uniformly and calculate the relative error on the observed number as $1/\sqrt{N}$. We then calculate the number that would be required for the average of the absolute value of the fluctuation to be significant at the $95\%$ confidence level for a normal distribution (given the large number of bursts required we expect normality in the uncertainty).

As expected from Fig. 2 and Fig. 5 the number of bursts required to distinguish a universe with a smooth matter fraction $\eta$ from a completely homogeneous universe $\eta=1$ generally increases with decreasing $\gamma$ and $\alpha$ with smoother universes naturally requiring more bursts. For the fiducial FRB population of $\gamma=-2.0$, and $\alpha=-1.0$ a universe comprised entirely of lenses can be ruled out using 8000 high fluence FRBs. Conversely a nearly smooth universe with 5$\%$ of its matter in lenses would require some $3.5\times 10^{5}$ FRBs to distinguish from the smooth case.

Planned instruments such as CHORD or the proposed coherent all sky monitor (CASM) BURSTT (Lin et al., 2022) have predicted detection rates of $\sim 10^{4}$ per year (Connor & Ravi, 2022). Several such instruments observing over the course of ten years could reasonably achieve our desired $3.5\times 10^{5}$ high fluence FRBs, especially given the low-sensitivity – high field of view mode of operation for CASMs. This would allow formation of broad and stringent constraints over parts of the PBH space that have only been probed locally. To show which masses the constraints apply over we must consider both source extension and wave optics effects as detailed in appendix C. Doing so we calculate the PBH dark matter fraction constraints shown in Fig. 15 for $3.5\times 10^{5}$ high fluence FRBs observed at 1.4GHz. We highlight that the only observables required for each of these FRBs are the booleans $f>f_{b,\eta}$ and $\nu>\nu_{\text{min}}$⁵⁵5where this minimum frequency is used the establish the minimum probed lens mass; a precise fluence measurement is not required, neither is a localisation or redshift. If these FRBs are observed at higher frequencies, these constraints will extend down to lower masses, with an infinite frequency FRB counterpart extending all the way down to $10^{-22}M_{\odot}$. Assuming a FRB-like functional form for the GRB intrinsic $\Theta_{E}$, a similar number of GRBs could constrain PBHs down to $10^{-15}M_{\odot}$ as displayed in the figure.

Constraining $E_{\nu_{e},\text{max}}$ is an area of particular import in an inhomogeneous universe as lensing can have a large effect on the apparent maximum energy of a burst. $E_{\nu_{e},\text{max}}$ is often taken to be the greatest apparent energy amongst observed bursts, which in an inhomogeneous universe can be a large overestimation. The compact nature of fast transient sources means they are susceptible to lensing by low mass objects. Such objects may leave no observational trace of the magnification they are causing. This makes it impossible to know the magnification of an individual burst and thus impossible to confidently approximate the intrinsic $E_{\nu_{e},\text{max}}$ based on the largest apparent energy. Variation of $E_{\nu_{e},\text{max}}$ however, is not degenerate with $\eta$ and hence may be constrained if sufficient data are collected to distinguish a fluctuation in $dR/df$ due to lensing.

Gravitational lensing is one possible propagation effect to consider when modelling the differential events rate of fast transients from their intrinsic population functions. In doing so we have shown that, for a stationary universe:

1. 1.

   Except for the mass-energy range with prominent fringes shown in Fig. 9, FRBs are not all intrinsically low luminosity events highly magnified by gravitational lensing from point masses.

2. 2.

   Given current observational uncertainties, intrinsic population function parameters (other than $E_{\nu,\text{max}}$) inferred from observations without accounting for lensing will not significantly differ in a completely inhomogeneous universe ($\eta=0$)

3. 3.

   Wave optics may cause magnification PDFs to differ from the familiar $\mu^{-3}$ behaviour at high magnifications.

4. 4.

   For masses above 0.01 solar masses geometric optics will suffice for modelling the $dR/df$ of FRBs in an $\eta=0$ universe.

5. 5.

   Using low fluence ($<f_{b,\eta}$) observations of $dR/df$ to estimate $\Theta_{E}$ will be free from the effects of lensing. A further comparison with high fluence observations can be used to extract the influence of gravitational lensing. In this way the compactness of FRBs and GRBs can be exploited to constrain unexplored regions of dark matter parameter space such as low mass primordial black holes. We expect that 8000 high fluence, unlocalised FRBs would be required to rule out a completely clumpy universe, with $3.5\times 10^{5}$ required to exclude more than $6\%$ of dark matter being in PBHs in the relevant mass range.

We thank Geraint F. Lewis and Ron Ekers for productive discussions on lensing. CMT is supported by an Australian Research Council Future Fellowship under project grant FT180100321. CWJ acknowledges support from the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (project DP200102545).

The data underlying this article are available in the article and in its online supplementary material.