Cosmology with gravitationally lensed repeating Fast Radio Bursts

High-precision cosmological probes have revealed a small but significant tension between the parameters measured with different techniques, one of which is based on time delays in gravitational lenses. We discuss a new way of using time delays for cosmology, taking advantage of the extreme precision expected for lensed Fast Radio Bursts, short flashes of radio emission originating at cosmological distances. With coherent methods the achievable precision is sufficient to even measure how time delays change over the months and years, which can also be interpreted as differential redshifts between the images. It turns out that uncertainties arising from the unknown mass distribution of gravitational lenses can be eliminated by combining time delays with their time derivatives. Other effects, most importantly relative proper motion, can be measured accurately and separated from the cosmological effects. With a mock sample of simulated lenses we show that strong constraints on cosmological parameters are potentially possible. Finally the lensed images can be used as galactic interferometer to resolve structures and motions of the burst sources with insanely high resolution and help revealing their currently unknown physical nature.


Introduction
The gravitational lens effect (Refsdal 1964a) can deflect light so strongly that it produces multiple images of a single source. Refsdal (1964b) argued that time delays (light travel time differences between images) can be used to determine distances and thus the Hubble constant (H 0 ), at a time when it was not even clear that the effect may ever be observed. For very distant systems, Refsdal (1966) showed that time delays can also be used to test cosmological theories.
Practical applications of this brilliantly simple idea turned out to be quite difficult. Even after the discovery of the first gravitationally lensed active galactic nucleus (AGN) by Walsh et al. (1979), it took many years before the time delay between the two images was agreed on with sufficient accuracy. Main reason for this difficulty are the typically slow intrinsic variations of AGN, compared to the originally proposed lensed supernovae.
An even more fundamental difficulty is the influence of the a-priori unknown mass distribution of the lens on the derived results. Parameterised mass models can often be fitted to the observed image configuration, flux ratios and relative image distortions. In the best case, extended sources with rich substructures provide a wealth of constraints for realistic multi-parameter mass distributions. However, there are fundamental degeneracies between the lensing mass distribution and the unknown intrinsic source structure.
Best known is the mass-sheet degeneracy (Falco et al. 1985;Gorenstein et al. 1988), according to which scaling a given mass distribution while adding a homogeneous mass sheet can leave e-mail: wucknitz@mpifr-bonn.mpg.de the image configuration unchanged, provided that the source structure and position is scaled accordingly. This transformation also scales the time delay and thus the derived Hubble constant. More realistic, but very similar in effect, are changes of the radial mass profile of the lens. A more general degeneracy was described by Schneider & Sluse (2014).
Additional measurements of the velocity dispersion of lensing galaxies can partly break these degeneracies, but at the price of introducing additional complicated astrophysics into the problem. Nevertheless, very competitive results have been achieved so far. Wong et al. (2019) describe results from a joint analysis of six gravitational lenses. Their result for the Hubble constant is precise to 2.4 %, but disagrees with Cosmic Microwave Background (CMB) results far beyond the formal uncertainties (Planck Collaboration et al. 2018). Millon et al. (2019) discuss systematic uncertainties in the lensing analysis. Kochanek (2020) presents a more pessimistic view and argues that accuracies below 10 % are hardly possible, regardless of the formal precision.
Even though the tension between determinations of the Hubble constant with different methods is well below 10 % now, it is still highly significant. These differences may result from limited understanding of the involved systematics, or for instance from a behaviour of dark energy that is different from simple models. Either way, there is a need for additional clean methods with preferably less, but at least with different systematics. We will argue that gravitational lensing applied in a new way is a very promising option. The unique properties of Fast Radio Bursts (FRBs) are essential for this approach.
Our approach is different from related ideas presented in the literature. Li et al. (2018) and Liu et al. (2019) propose to use the high accuracy of time delays measured from gravitationally lensed FRBs together with classical mass modelling. They argue that the absence of a bright AGN core in FRB host galaxies facilitates the use of optical substructures as lens modelling constraints.
The essential new observable, time delay changes over time, is discussed by Piattella & Giani (2017) for rather special lenses. Zitrin & Eichler (2018) explicitly investigate that timederivatives of time delays (as expected from cosmic expansion) may be measurable with lensed FRBs. They also discuss that relative proper motion of the source has a strong effect that is difficult to distinguish from the cosmological expansion.
We argue that, with coherent time-delay measurements, accuracies much better than the burst duration can be achieved, and that at this level we can not only disentangle cosmological effects from proper motion, but that we can even eliminate the unknown mass distributions of the lensing galaxies as main source of potential systematic errors.
The structure of this manuscript is as follows: In Section 2 we describe the sources (FRBs) that can be used to determine time delays at a level of precision that even allows us to measure how they evolve with time. Section 3 describes how these new observables can be used to avoid effects of the unknown mass distribution almost entirely. It turns out that relative transverse motion has a stronger effect than cosmology. We will discuss how this effect can be removed, with the 'by-product' of measuring the proper motion with unprecedented accuracy.
In Section 4 we discuss to which level we may be able to determine combinations of cosmological parameters from realistic measurements. We find that with an ensemble of lens systems we can potentially obtain competitive constraints on a number of parameters.
At the level of precision that is required and achievable, many additional small effects have to be considered. These possible caveats are the subject of Section 5. Similar to the proper motion, these are also interesting in themselves.
The use of gravitationally lensed images as arms of a galaxysize interferometer is discussed in Section 6. We can potentially resolve structures of a few kilometres at cosmological distances. At the same time the required time-delay precision is only achievable for sources that actually have structures on these scales, which is only true for Fast Radio Bursts.
A summarising discussion follows in Sec. 7.

Fast Radio Bursts (FRBs)
A new type of radio sources was discovered by Lorimer et al. (2007). These are bright radio bursts with durations less than a few milliseconds. Similar to pulsars their signals are dispersed; they arrive later at lower frequencies. In FRBs this dispersion is higher than expected from the integrated electron column density within the Milky Way, which is evidence for an extragalactic origin at cosmological distances. The first of these objects were detected only once, which makes an accurate localisation difficult. The discovery of the first repeating FRB by Spitler et al. (2016) changed the game completely. Observing the roughly known location with the Karl G. Jansky Very Large Array (VLA) allowed Chatterjee et al. (2017) to localise it to sub-arcsecond accuracy. This was sufficient to identify a host galaxy and thus determine the redshift z = 0.1927 of host and FRB (Tendulkar et al. 2017). This was the final proof that at least this FRB originates at a cosmological distance. Marcote et al. (2017) determined the position to milli-arcsec (mas) accuracy with Very Long Baseline Interferometry (VLBI).
The first direct localisation at discovery of another FRB source was achieved with the Australian Square Kilometre Array Pathfinder (ASKAP) by Bannister et al. (2019), which led to the identification of a host galaxy at z = 0.3214. The same instrument is now finding new FRBs regularly. An even higher yield is produced by the Canadian HI Intensity Mapping Experiment (CHIME, CHIME/FRB Collaboration et al. 2018), which does now find several new FRBs per day. CHIME also discovered the second repeating FRB (CHIME/FRB Collaboration et al. 2019), so far without an accurate localisation. The first VLBIlocalisation of another repeater found by CHIME was presented by Marcote et al. (2020).
Given that these instruments, even though considered widefield, can only observe a small fraction of the sky, the number of FRBs detectable by a true all-sky survey with sensitivities comparable to current facilities must at least be hundreds, if not thousands per day. At the same time, radio astronomical technology is making rapid progress, which makes a full-sky FRB monitoring feasible in the foreseeable future.
Currently the redshift distribution can only be estimated, which makes it difficult to predict the fraction that is gravitationally lensed. The Cosmic Lens All Sky Survey (CLASS, Browne et al. 2003) found that one out of 700 from their AGN source population is strongly lensed with multiple images on arcsecond scales. Marlow et al. (2000) studied the redshift distribution of the CLASS parent source population and found a wide range with a mean of z = 1.2 and an rms scatter of 0.95. The known source redshifts from Browne et al. (2003) are 0.96, 1.013, 1.28, 1.34, 1.39, 2.62, 3.214, 3.62. Even though the FRB population may be systematically closer to us, which reduces the chance of gravitational lensing, the future discovery of lensed FRBs is a realistic possibility. Some fraction of them will be repeating, which allows us to measure how time delays evolve over time and apply the approach presented in this manuscript.
We note that we have to distinguish between the concepts of bursting FRB sources, individual bursts from FRB sources, and lensed 'echoes' (or images) of individual bursts.
With lensed bursts of millisecond duration, it is obvious that time delays can be measured at least with this accuracy. Typical time delays are of the order days to months, and for order-ofmagnitude estimates we assume 10 6 s, approximately 12 days. See Table 1 for the numbers that we use for our estimates. The fractional accuracy is then 10 −9 , a huge leap forward compared to the few-percent level that is (at the utmost) achievable with lensed AGN. Millon et al. (2020) present a table of known and new time delays of quasars lensed by single galaxies. Uncertainties are mostly above one day. A particularly good time delay for B0218+357 is known from gamma-ray observations (Cheung et al. 2014) and radio monitoring (Biggs & Browne 2018). The results are consistent with each other and have uncertainties of 0.16 and 0.2 days, respectively. Significant improvement beyond this is not expected for lensed AGN.
As Li et al. (2018) argue, the uncertainty of the time delay measurements for lensed FRBs is entirely negligible in the total error budget. Because the lensed host galaxies of FRBs will generally not harbour a strong AGN, their structure in optical images may make lens modelling more accurate as well, so that a level of 1 % might be reachable with ten lensed FRBs (Li et al. 2018).
We worry that, even if this approach can certainly reduce some uncertainties, the fundamental mass-model degeneracies persist and may lead to significant systematic errors even in a joint analysis of many lenses. For this reason we propose an alternative route to cosmology with lensed FRBs. O. Wucknitz et al.: Cosmology with gravitationally lensed repeating Fast Radio Bursts We need an even higher accuracy of the time delays to follow this route. Luckily this is made possible by lensed FRBs. If we can observe not only the intensity as function of time, but the full electromagnetic wave field from a source, and if we have two coherent copies of the wave field from the same burst, then we can determine the time delay (group delay) between them with an uncertainty given by the reciprocal bandwidth. With good signalto-noise ratio (SNR), the accuracy can be improved even more. Modern radio receivers have bandwidths of hundreds of MHz up to a few GHz, so that uncertainties smaller than a nanosecond are certainly achievable. In principle one may even use the (absolute) phase delay, with uncertainties given by the reciprocal observing frequency. 1 In reality, practical problems like clock drifts and ionospheric and atmospheric delays have to be considered. We will find out below that, as result of time-varying time delays, different images will have different redshifts. For short bursts this effect is so small that it generally does not have to be taken into account in the correlation analysis, but can easily be included when needed. What has to be included, on the other hand, are differences of dispersion in the interstellar medium of the lensing galaxy that produce frequency-dependent delay differences. These follow a λ 2 law and can be corrected with coherent de-dispersion.
Achieving nanosecond accuracy over time ranges of weeks and months is not totally unrealistic, but we base our arguments on a much more conservative assumed uncertainty of one microsecond (Tab. 1), which corresponds to a fractional uncertainty of the typical time delay of the order 10 −12 . We note that this coherent way of measuring time delays does not rely on the short bursts, or even on the presence of intrinsic intensity variations, because it uses the wave field itself. It is, however, essential that the images are mutually coherent, which is not the case for any sources other than FRBs, as explained in Section 6. Of course finding the time delay between the waves is made much easier by having a good estimate from the intensity correlation.

Homogeneously expanding Universe
Our general idea is very simple: As result of the expansion of the Universe, time delays should also roughly increase at this rate, 1 When using this absolute phase delay, a frequency-independent phase shift depending on the type of image has to be taken into account. In principle this could even be used to determine the types of image. d∆t/dt ∼ H 0 ∆t, which over about three years (Tab. 1) corresponds to a change of 0.2 ms or a fractional change of 2 × 10 −10 , which is easily measurable with an accuracy of better than one percent. In the following we will derive how this time-delay increase is related to cosmological parameters and observables.
Let us, for the moment, assume that the entire Universe, including the gravitational lens, expands uniformly with a geometry according to the following metric: 2 Here t denotes cosmic time (also proper time of comoving observers), R(t) is the scale factor and dL is the comoving length interval. Comoving spatial coordinates are x j and the spatial metric γ i j does not depend on time.
Under this condition light rays follow spatial geodesics with a time-dependence defined by the speed of light. Light rays connecting comoving source and observer at different times will follow the same spatial path and are shifted by a constant interval in the conformal time coordinate η, which provides the important relation between scale factor and redshift, In case of a gravitational lens, the time delay will not vary over time if measured in terms η. Because the measured time delay ∆t is a scaled version of ∆η, its logarithmic time-derivative is a direct measurement of the current Hubble constant: 1 ∆t In this unrealistic situation, the time delay increases exactly with the expanding Universe. We could thus measure the Hubble constant directly, without mass model uncertainties and even without knowing image positions or redshifts. In reality the gravitational lens itself does not expand with the Hubble flow and we have to derive the effect in detail as shown in the following.

Lensing theory
With the image position θ, the true source position θ s and the apparent deflection angle α, the lens equation reads where all angles are small two-dimensional vectors in the tangential plane relative to some reference axis. Time delays can be derived using the methods described by Cooke & Kantowski (1975). For this we do not need to assume a certain cosmological model, as long as we parameterise it by angular size distances between observer and lens D d , between observer and source D s , and between lens and source D ds . Here we use the subscripts d and s for deflector and source, respectively, and suppressed 0 for the current epoch (observer). We assume that the global geometry of the Universe still follows the homogeneous expansion according to Eqs. (1)-(4). Later we will include the effect of radial motion into these parameters.
An angular size distance D ab is defined as ratio between a transverse offset at b and the corresponding angle measured at a, where both sides are considered at the cosmic time at which the signal passes. Because the spatial geodesics do not change in comoving coordinates, and because the length is measured at b, this distance scales with R b over time, as long as a and b are fixed in comoving coordinates. We will later relax the assumption that light rays follow geodesics of the average global spacetime, and allow for systematic over-or underdensities near the light path.
The time delay measured by an observer consists of a geometric part and the Shapiro (or potential) part, 3 Later we will absorb the θ 2 s /2 term into the constant. The Shapiro delay is described via the lensing potential ψ, which is related to the delay τ d as measured in the lens frame via The lens equation (7) results from Fermat's principle with α = ∇ θ ψ.

Lensing-induced redshifts
The time delay will slowly change with time in a way that is more complicated than the naive Eq. (6). Details are derived in the following. The time-derivative of the time delay can also be interpreted as additional lensing-induced redshift z(θ), in the sense that the time lag between subsequent bursts in image θ scales with 1 + z(θ). This scaling only has a meaning as relative redshift between images, z(θ 1 ) − z(θ 2 ), similar to the time delay itself, which only has a unique definition as delay between images.
The lag between bursts (in contrast to the delay between lensed images of the same burst) cannot be measured coherently and therefore has a much lower accuracy of the order a millisecond. Because we measure the redshift as difference of timederivatives of time delays, which are measured coherently, the achievable accuracy is actually given by the ratio of the coherent timing uncertainty and the burst separation. If we use Latin letters as indices for images and Arabic numbers for the bursts (dropping the subscript 0 in t 0 ), the relation can be written as which is correct to first order in z and thus sufficient for all practical purposes. For real measurements we can use as many bursts as available and fit them jointly.
With the values in Table 1 the measurement accuracy in z is of the order 10 −14 even under our conservative assumptions. This is better than for the relative change of the time delay, because the relevant baseline is the time between bursts, which can be many years, and which increases without limit if the FRB is repeating persistently. It is this extreme precision that makes gravitationally lensed FRBs such a promising tool. Because we want to measure a tiny effect (time-delay increase due to the expanding Universe) with extreme precision, we have to consider many other effects that may influence the results. In the analysis below we find no show-stoppers, but some of the additional effects (most importantly the proper motion) are even interesting in themselves.

Distance parameters in a Robertson-Walker Universe
For most of the final analysis we assume that the general geometry (with the exception of the lens) is expanding homogeneously according to Eqs. (1) and (4). This means that the angular size distance D ab can be written as a comoving (and thus constant) angular size distance multiplied with the scale factor R b . This is a common assumption in lensing theory, but at the level of accuracy that is required to understand the current tension in cosmological parameters, this simplification may not be sufficient anymore.
World models that are homogeneous on large scales, but have small-scale deviations near the light paths, have been introduced by Kantowski (1969), Dyer & Roeder (1972) and Dyer & Roeder (1973). The basic idea is that matter is clumped, but clumps near the line of sight are explicitly described as gravitational lenses, so that the remaining density around the light rays may be reduced relative to the global mean, which reduces the focusing and increases the angular size distances. These inhomogeneities can also be described explicitly as perturbations at certain redshifts along the line of sight. Even if we can describe the effect well (e.g. if the inhomogeneity parameter in the Dyer-Roeder distance is known), they break the symmetry of the homogeneously expanding geometry.
In the following derivations, we will generally start with the general case (described by angular size distances and their derivatives) before we consider the homogeneous case, in which the time-derivatives of distance parameters and the prefactor in Eq. (8) can be derived as follows using variants of Eq. (6): In the scenario in which the geometry of the entire Universe, including the lens, expands homogeneously, the potential and thus image positions do not change, so that Eq. (14) describes the only variation with time in Eq. (8). This confirms our general result from Eq. (6). We note that the relation always holds, even in the inhomogeneous case.

Non-expanding lens
We write the Shapiro delay in Eq. (10) in terms of a physical vector x = D d θ and assume that the function τ d (x) is invariant as result of the invariant mass distribution in physical space. By writing the equation twice, once for a reference epoch (superscript ref) and once in general, but for the same argument of τ d , we can derive how the lensing potential ψ of a constant physical mass distribution varies with time: The resulting deflection angle scales like The last form is for homogeneous expansion and means that it is invariant for fixed physical vector x, exactly what we expect for an invariant physical mass distribution. This relation automatically holds for an isothermal lens, in which the deflection angle does not vary in the radial direction. Therefore such a mass distribution produces time delays growing with H 0 , but this behaviour can only be utilised if we know that the lens is isothermal. In that situation we do not need the relative redshifts at all, but can determine the Hubble constant directly from the time delays, for instance by using the formalism presented by Wucknitz (2002). For the time-derivative of Eq. (8), we can neglect the resulting shift of image positions, because they do not affect the light travel time as result of Fermat's principle. For the time-derivative of the potential at fixed θ, we have to consider both scalings in Eq. (16) and remember that the gradient is the deflection angle: The last equation is for the homogeneous Universe.

Transverse motion
It has been known for years that transverse motion of a gravitational lens relative to the optical axis has an effect on the observed redshifts of lensed images. This effect can be interpreted in different ways, for instance as Doppler effect (Birkinshaw & Gull 1983), gravitomagnetic effect of the moving lens (e.g. Pyne & Birkinshaw 1993;Kopeikin & Schäfer 1999), as energy transfer in the scattering of photons by the lens (Wucknitz & Sperhake 2004), or directly as time delays changing with the changing geometry (Zitrin & Eichler 2018).
With transverse velocities V s , V d and V 0 of source, lens and observer, we can define the proper motionṡ and write the result of Wucknitz & Sperhake (2004) for the motion-induced redshift as As a consistency check for the difference between images, we can take the contribution to the time-derivative of Eq. (8) due to the proper motion of the source and confirm that it is consistent with Eq. (25) in the case of a fixed lens and observer. When taking the derivative, we can again neglect the shift of the images because of Fermat's principle. For a transverse speed of 300 km s −1 , the induced redshift for image separations of 1 arcsec amounts to 5 × 10 −9 , corresponding to radial Doppler speeds of 1.5 m s −1 . Already Birkinshaw & Gull (1983) suggested to measure the effect on the CMB caused by moving clusters of galaxies, but concluded that it is difficult. Molnar & Birkinshaw (2003) estimated that detecting the effect with optical spectroscopy is challenging even for massive clusters of galaxies. Radio frequencies can be measured with much higher accuracy, but redshifts can only be determined for sufficiently narrow spectral features.
With repeating FRBs we can eventually measure relative redshifts with a precision of the order 10 −14 , not by measuring the radio frequency, but the time lag between bursts, via coherent measurements of time delays as explained above. This accuracy corresponds to transverse speeds of only about one metre per second or proper motions of the order 10 −13 arcsec per year! We will see later how well the proper motion can be disentangled from the various other effects on the redshifts.
We note that in the equation for the scaling of the lensing potential of a non-expanding lens in Eq. (16), we implicitly assumed that the centre of the scaling law is the coordinate origin. An offset in this centre is equivalent to an additional proper motion that corresponds roughly to the offset over a Hubble time.

Combining delays and redshifts
Let us now try to combine the equations for the time delays, Eq. (8), with the ones for its derivatives (or redshifts), including the contribution from the non-expanding lens, Eq. (21), and the transverse motion of the source from Eq. (25). For simplicity we assume that lens and observer are at rest, which is still fully general, because only relative motion matters. For each image we have one equation for the time delay and one for the redshift, If we consider only differences between images, we can eliminate the additive constants. A lens system with n images provides n − 1 equations each of type (26) and (27). The unknowns are n − 1 potential differences, two components of the source position, two of the proper motion, and some number of cosmological parameters. Even in a quadruply lensed system there are not enough constraints to determine all unknowns, not even if the cosmology is known. Luckily, as we will see, nature is so kind that it allows us to eliminate all unwanted parameters and determine the more interesting ones.
Because the potential is the major uncertainty in classical applications of gravitational lensing for cosmology, we substitute the time delay for both occurrences of the potential in Eq. (27), which is trivial for the term in square brackets. We are then left with equations for the redshifts with only cosmological and geometrical terms: We eliminated the unknown lensing potential and find that the position and proper motion of the source only appear in the combinationθ s . Because we have to determine or eliminate the proper motion anyway, the unknown source position does not add further free parameters. When studying the proper motion itself, the deviation inθ s is still small, for galaxy-scale lenses typically 1 km s −1 . Cosmological parameters can be determined or constrained via the proxies Z t and Z θ .
In the homogeneous case, we can now apply equations (12) and (13) and find For an isothermal lens, the potential can be eliminated even without redshifts, and Eq. (26) reduces to as described by Wucknitz (2002). With this, the two terms with H d in Eqs. (29) and (32) cancel and we are left with the pure dependence on H 0 and no other cosmological parameters. Unfortunately this only helps if we know about the isothermality. After eliminating the lensing potential, we are left with effectively n − 1 equations from the redshifts less the additive constant. Unknowns are the proper motion and cosmological parameters. For a double-imaged system we cannot constrain cosmological parameters, but we can still estimate one component of the proper motion with slightly reduced accuracy. The terms with Z t and Z θ in Eq. (28) are of the same order of magnitude, about 2 × 10 −12 in Tab. 1. Because the proper-motion-induced redshift is much larger, typically 5 × 10 −9 , we can neglect the cosmological terms for a measurement of the proper motion component along the image separation with an accuracy of the order 0.1 km s −1 , but we have to know the distance parameters to convert it to physical units. For a quad system, we can determine the full proper motion vector in the same way, and in addition obtain constraints on the cosmological parameters. We can treat the additive constant in Eq. (28) as a free parameter, invert the set of equations and express our two 'cosmological parameters' Z t and Z θ and the proper motion as known linear functions f 1 , f 2 , f 3 of the constant: Here the parameters are also written for the homogeneous case in an alternative way that separates the effect of H 0 and the other cosmological parameters. For this we introduced the reduced distance parameter d eff = H 0 D eff /c, which is independent of H 0 . If the other parameters are known, H 0 andθ s can be determined uniquely. Otherwise the information from several lensed FRBs can be combined to determine all parameters (see Sec. 4). Even in the inhomogeneous case, we have good constraints on cosmology, but slightly more obscure. Eq. (35) remains unchanged, but the right hand sides of Eqs. (36) and (37) have to be replaced by their more general versions using Eqs. (30) and (31).

Radial motion
Radial motion of source, lens, and observer influences the result for two reasons. First there is the direct effect on the timederivative of distances. In Euclidean geometry we would havė D ab = v b − v a . This simple relation does not hold for cosmological distances, but the order of magnitude will be similar. The characteristicḊ/D ≈ v/D is of the order 10 −21 s −1 for 1 Gpc and 300 km s −1 . Compared to the Hubble constant, this is below one percent, so that the direct effect of radial motion shall not be of concern here, even though it should be investigated further.
The second effect of relevance is relativistic aberration, which shifts the apparent positions of sources towards the apex of motion. When approaching a source, it appears compressed, which can be interpreted as an increased angular size distance. To first order we find the corrected distancẽ which depends on the motion of the observing side only. 4 For non-relativistic velocities this is an insignificant scaling. The effect on the derived proper motion is larger than its formal precision, but still small. Radial motion also modifies the observed redshifts, which are needed to calculate distances given a cosmological model. This is a small effect that we can neglect.
4 Of course physical effects should only depend on relative motion. A situation with a moving source can be Lorentz-transformed to that of a moving observer. The relativistic length contraction does not matter here, because it is of second order. More important is the effect of redefined simultaneity. In one situation we define the distance at the moment of emission and in the other at the moment of observation. The relative motion between these two moments accounts for the difference in the two descriptions. We conclude that uniform radial motion will generally not require significant corrections. A realistic acceleration, however, can have a relevant effect on time-derivatives of the distances and therefore on the observed redshifts. For the additional contribution to the time-derivative of the time delays in Eq. (8), we need the effect on the effective distance, to first order in the velocities Here the accelerations a 0 and a d are measured in their respective comoving frames, and the latter had to be scaled for redshift to obtain the derivative of v d with respect to observed time.
Formally we also have to consider that velocities relative to the Hubble flow decrease due to the expansion, 5 but this effect is smaller than the Hubble expansion itself by a factor of v/c and therefor negligible.
For the observer we assume that the diurnal motion and orbit around the Solar system barycentre have been corrected. Another relevant effect is the Galactocentric acceleration. We will argue in Section 5.1 that these can be corrected for with sufficient accuracy.
Radial acceleration of the lensing galaxy is potentially more critical, because it cannot be measured independent of the lensing redshifts and thus acts as additional source of (probably highly non-Gaussian) statistical errors. For the Milky Way the acceleration within the local group is dominated by the gravitation of the Large Magellanic Cloud. With a total mass of 1.4 × 10 11 M (Erkal et al. 2019) and a distance of 50 kpc, the gravitational acceleration divided by c is 2.6 × 10 −20 s −1 , which corresponds to about one percent of the Hubble constant. If this is typical for lensing galaxies, the effect on the achievable accuracy is limited, but a good sample of lens systems should be used to identify possible outliers.
Finally we see that radial acceleration of the source has no effect on the derived cosmological parameters. This is essential, because values similar to the Galactocentric acceleration or even much larger have to be expected, which could otherwise ruin the method. Very high velocities of the source can still be relevant due to the direct effect on time derivatives of distance parameters. High velocities are expected in tight orbits around massive objects, which could hopefully be identified by their time dependence. In the unlikely case that FRBs are emitted from relativistically moving systems, e.g. in AGN jets, their motion would swamp the effects that we are interested in. This curse for cosmology would be a blessing for FRB physics, and time delays and relative lensing redshifts could be used to study the FRB motion with the highest precision.

Cosmology with ensembles of lensed FRBs
We learned in the analysis above that one gravitationally lensed repeating FRB provides one constraint on the combination of proxy cosmological parameters Z t and Z θ .
If all cosmological parameters with the exception of the Hubble constant are known, Z θ can be calculated and Z t is simply proportional to H 0 with a known scale factor, so that we have a direct measurement of the Hubble constant. Using our reference  numbers we expect an accuracy of 0.5 %. In reality we have to take into account the accuracy of the image positions and how accurately the system of equations can be solved for H 0 , which will depend on the image configuration of the lens. For this first investigation of the potential as cosmological probes, we are not trying to define a truely representative sample of lensed FRBs, which is impossible given that we do not even know the source redshift distribution. Instead we use a rather arbitrary sample of lenses with isothermal elliptical potentials plus external shear, all with the same Einstein radius E = 1 arcsec. 6 Adding an external convergence or using a range of power-law indices would generally make the results better by providing more diversity, which reduces degeneracies. Having external shear and ellipticity, or at least some lenses with shear and some with ellipticity, also turns out to be important, which has to be investigated further.
The potential for our simple family of models is given by in which (γ x , γ y ) represents the external shear and the ellipticity of the potential. For small ellipticities this is a good approximation of an elliptical mass distribution. Parameters were chosen A&A proofs: manuscript no. lensed_frbs_2020_04_24 from Gaussian distributions and the source positions from a uniform distribution within the central caustic to produce quadruple images. For the simulations we assume a spatially flat homogeneous cosmological model with H 0 = 70 km s −1 Mpc −1 , matter density Ω M = 0.7, dark-energy density Ω w = 0.3 and equationof-state parameter w = −1, equivalent to a cosmological constant. This cosmology is used to calculate from the lens models in angular coordinates the time delays and their time derivatives. 7 The image configurations are illustrated in Fig. 1 and parameters are listed in Table A.1. With the exact values as mock measurements we use Markov Chain Monte Carlo (MCMC) simulations to explore the parameter space that is consistent with these measurements. Practically we use the MultiNest software (Feroz et al. 2009) and its Python interface (Buchner et al. 2014;Buchner 2016). The only explicit free parameters for the MCMC simulation are the cosmological parameters. The image positions are fitted implicitly within the loop by analytically minimising the deviations of the image positions and the redshifts. This χ 2 is then used for the likelihood of the outer MCMC loop. Tests show that this accelerated approach introduces only negligible errors. The time delays themselves can be used directly, because of their extreme precision. Their uncertainty matters only in so far as the time delays are used to determine the relative redshifts.
As measurement uncertainties we assume 10 −14 for the relative redshifts and 0.5 mas for the image positions (0.35 mas for each component). It happens that these errors influence the resulting uncertainty to a very similar level, so that both have to be reduced for a significant improvement.
First we investigate how well individual parameters can be determined from individual lenses, assuming that all others are known. Table 2 shows the resulting uncertainties for the Hubble constant and the matter density, assuming the other is known and a spatially flat Universe with w = −1. We find that the uncertainties vary strongly from lens to lens, which means that more realistic samples should be investigated in the future. Generally we find that more asymmetric lenses (visible as larger caustics) are better, but details have to be worked out.
Because a measurement of the Hubble constant alone will not be able to solve the current parameter tension, we also simulate fits of sets of parameters to the full ensemble of lenses. Fig. 2 and Tab. 3 show results for the Hubble constant and matter density for a flat Universe with w = −1. Fig. 3 and Tab. 4 add w as free parameter. Some of the correlations between these parameters are significant, but none are so extreme to preclude the determination of all. This may change if we relax the condition of a spatially flat Universe.
The potentially achievable accuracy is competitive with combinations of other methods, particularly for the matter density and the equation of state of dark energy. It remains to be seen how realistic our assumptions are.

Caveats
There are a number of effects that can perturb the measured time delays and (more importantly) their time-derivatives. Most relevant is the proper motion, which can be determined and implicitly corrected for quad systems as shown above. The accuracy of this correction depends on the astrometric accuracy. Radial mo- tion is also discussed above, where we noted that the motion of the Earth relative to the background metric must be taken into account.

Motion of the observer
As explained above, we have to correct the arrival times of bursts for a number of effects. First there is the diurnal motion of the observer (which is known extremely well) and the Earth's orbit around the Solar system barycentre. Relative to the known planets and the Sun, this motion is certainly known to better than 300 m, which would correspond to the claimed timing accuracy of one microsecond. Unfortunately we need to know the motion relative to a local inertial frame. Should there be additional large unknown masses in the Solar system, for instance an unknown distant planet, its direct influence on the motion of the inner planets may be small, but it does introduce a relative motion between the true barycentre and the one derived from the known planets.
With timing observations of a binary pulsar, relative radial acceleration between the pulsar and us can be measured very accurately. Verbiest et al. (2008) did this for PSR J0437-4715 with an accuracy corresponding to a/c of 1.2 × 10 −19 s −1 , which is about 5 % of the Hubble constant. Within the measurement accuracy the result is consistent with the Galactocentric acceleration and kinematic effects, with a remaining uncertainty of any unmodelled contributions of 6 × 10 −19 s −1 . This constraint was improved by Deller et al. (2008) with a new measurement of the parallax, with a limit of 1.6 × 10 −19 s −1 at the 2-σ level. The acceleration measurement itself was also improved by Reardon et al. (2016) by an order of magnitude and is now at an uncertainty of 1.2 × 10 −20 s −1 . It is still consistent with the Galactic and kinematic contributions.
For our acceleration relative to an inertial frame, we have to consider the total Galactocentric acceleration of the observer. This amounts to a/c = 8 × 10 −19 s −1 or about one third of the Hubble constant. It is thus a very relevant effect that must be corrected for. MacMillan et al. (2019) show an overview of results from Galactic kinematics compared with VLBI measurements via the resulting apparent proper motion of extragalactic sources. These measurements include the acceleration of the Milky Way towards the average Hubble flow. Their accuracy (of the order 6 × 10 −20 s −1 ) is currently sufficient to correct lensing redshift measurements for most of the effect, but further improvements are required.
With a sufficiently large sample of lensed FRBs, our local acceleration can be measured (and corrected) from the observed lensing redshifts, because its effect is position dependent. This can potentially even be used to improve the Solar system ephemeris, find unknown masses in the outer Solar system, and to study the acceleration of the Milky Way relative to the background. This approach has the disadvantage that it can absorb potential large-scale anisotropies, which may then go unnoticed.

Evolution of lens structure
Our entire argument is based on the assumption that the mass distribution of the lens stays constant with time. According to Table 1 we are measuring relative changes of time delays of the order 2 × 10 −10 , corresponding to substantial changes over time scales of 10 Gyr, which means that even small violations of the assumption will matter. Typical rotational time scales are much smaller, but they are not relevant as long as the rotating mass distribution is symmetric. Major encounters or mergers will certainly introduce large perturbations, but we hope that such systems can be excluded based on their morphology.
Other changes on smaller scales are still of concern and have to be investigated in the future using structure formation simulations. These can not only provide the mass distribution at a given moment, but also its changes over time. Even if they do not reproduce realistic mass distributions of galaxies exactly, they will allow us to estimate typical effects on observed lensing-induced differential redshifts.

Masses near the line of sight
Our approach of combining time delays and their time derivatives (redshifts) allows us to eliminate the unknown mass distribution of the lens and break the mass-sheet degeneracy, as long as all unknown masses are at the same known redshift. Additional over-or underdensities along the line of sight will not automatically be corrected for. Because even their total static effect is usually small (e.g. Millon et al. 2019), we expect that the remaining error in our method will not introduce large systematics, but this has to be investigated in detail.
More worrying are potential effects on the time derivatives due to moving or evolving masses near the line of sight. Their influence has to be studied, probably again based on simulations of the large-scale structure formation.

Astrometry of lensed images
Being able to measure and correct for the relative proper motions relies on accurate measurements of the relative positions of the lensed images, with uncertainties on sub-mas levels. For bright persistent sources, for instance lensed AGN, this is not problematic. With VLBI at L-band near 1.4 GHz we can achieve resolutions of a few mas, even better at higher frequencies. The achievable positional uncertainty scales with the resolution divided by the SNR, so that relative positions better than 0.1 mas, at least for pointlike sources, are routinely achieved.
For FRBs the situation is more difficult. First they are short, so that the SNR will generally be lower than for AGN sources. With careful gating (SNR-based weighted averaging) it is still realistic to achieve an SNR of 100, which would be sufficient. 8 Another difficulty is the very poor image fidelity due to the sparse uv coverage for individual bursts/images. Even with proper model-fitting to the observed visibilities, this causes highly non-Gaussian errors, which can only be beaten down by combining several bursts.
Finally we can only see one image at a time, so that the positions cannot be measured directly relative to each other, but have to be relative to a nearby calibrator source using phasereferencing. This always leaves residual errors due to the atmosphere and ionosphere. In principle they can be reduced by observing a number of reference sources around the target, but if they are not within the same primary beam of the telescopes, we have to alternate scans of target and references, which means we may miss individual bursts. Alternatively one can point at the target and detect bursts in realtime to move to calibrators directly after a burst. This, however, is not a standard VLBI observing mode.
For the first repeating FRB, Marcote et al. (2017) achieve an accuracy of 2-4 mas per axis by combining four bursts. The second VLBI-localisation of another FRB by Marcote et al. (2020) had a similar accuracy even for individual bursts. There is certainly room for significant improvement, but the astrometry will nevertheless be challenging.
In addition to VLBI we can also use timing of the lensed burst images around the year to measure or improve the astrometry, similar to the technique to determine positions from pulsar timing. If we have a frequently repeating FRB with good coverage along the Earth's orbit around the Sun, effects of the orbit can be disentangled from the long-term differential redshifts. The orbit itself is then a lever arm of 1 AU. With a timing accuracy of one microsecond, 9 this corresponds to a positional accuracy of about 0.4 mas per measurement, unaffected by the atmosphere or ionosphere. The timing measurement requires much less effort than a full global VLBI experiment, so that it can be repeated for very many, maybe thousands of bursts, with the corresponding increase in precision.

Microlensing
We have to distinguish between two types of microlensing. The first is due to individual masses along the line of sight, as analysed by Chang & Refsdal (1979). Even if a large fraction of dark matter exists in the form of compact masses, the chance of lensing is small for any given line of sight, and our proposed method will generally not be affected. This type of microlensing in FRBs has already been proposed to study the abundance of compact objects by Muñoz et al. (2016), who argue that masses in the range 10-100 M can be detected or constrained using incoherent methods. Using spectral features in the emission caused by (coherent) interference between the lensed images, Eichler (2017) describe how the range probed can be extended to much lower masses. The typical order of magnitude of the time delay between lensed images of a point-mass, assuming a modest amplification ratio, is given by the light-crossing time of the Schwarzschild radius, which is 10 µs for one Solar mass and scales proportional to the mass.
Potentially more problematic for our application is microlensing caused by the dense ensemble of masses in the lensing galaxy. Because the density must be high for macrolensing to occur, macro-images will always be affected by some degree of microlensing. Microlensing by the star field can produce a high number of subimages, but generally only a few of them contribute significantly to the total flux (Saha & Williams 2011). The probability distribution of this number also depends on the type of image (minimum, maximum or saddle-point of the arrivaltime function). For known (non-FRB) lens systems, the microimages cannot be resolved, not even with VLBI observations of radio lenses. The only observed signature is the total amplification, and its variation seen in light curves. Astrometric effects are slowly getting in reach now for the GAIA satellite.
Time delays in this type of microlensing have not been investigated so far. Our preliminary simulations indicate that typical time delays between the brighter micro-images are of the same order of magnitude as the corresponding single-lens time delays, or a factor of a few larger. For a Solar mass we thus have to expect delays in the few-times 10 µs range. This is well above the assumed accuracy of one microsecond (Tab. 1) and thus potentially detrimental for our proposed cosmological application of (macro-) time delays. Luckily the situation is not hopeless. Relevant for us is not so much the accuracy of the absolute time delays, but the accuracy of their rate of change. For a rough estimate of the typical time scale, we can divide the Einstein radius of typical masses (a few times 10 11 km, angular scale a few µarcsec) by the typical transverse velocity, which results in many decades. Because we were assuming to measure changes over ∼ 3 years, the effect is reduced by at least an order of magnitude, which brings it into an acceptable range.
We can measure the wave field, which enables us to distinguish the microlensed images via their time delays. Generally there will be a few relevant subimages, so that we can monitor their relative separations over the years and from this estimate the possible error on the derived macro time delays. The micro time delays can be measured with nanosecond precision, which provides invaluable and entirely new information about the mass distribution and motion of compact objects in the lensing galaxies. In contrast to optical microlensing studies, in which most constraints are degenerate with the source size, we can directly measure micro time delays (which are a measure for the masses) and their changes over time (which are a measure of their kinematics).
Both, the potential for dark-matter studies and the possible problematic influence on the cosmological application, have to be studied in detail in the future.

Interstellar scattering
The interstellar plasma has a refractive index for radio waves that is proportional to the electron density and to λ 2 . Density fluctuations cause deflections that are similar in nature to microlensing and can produce a high number of subimages that interfere with each, which we observe as interstellar scintillation. The effect is independent for the lensed macro-images, so that it spreads out the correlation between the measured wave fields and reduces the accuracy of a macro-delay determination. Sophisticated methods have been developed to study the effect in scintillating pulsars, but these rely on a good time coverage in order to model the scattering field. For FRBs we do not have this luxury, but have to try and reduce the effect as much as possible by observing at high frequencies. Scattering sizes roughly scale with λ 2.2 , corresponding time delays with λ 4.4 .
The range of scattering delays depends critically on the line of sight, with low Galactic latitudes generally showing much stronger effects. Using the NE2001 model for the distribution of free electrons in the Milky Way (Cordes & Lazio 2002), we find that for 1.4 GHz the scattering delay is always below 1µs for Galactic latitudes above ±20 • , and mostly even above ±10 • , so that the redshift accuracy will stay within our assumed limits for most of the sky. Because of the strong frequency dependence, we can always reduce the effect by going to higher frequencies. This will generally be necessary if we want to study the delays of micro-images in detail.

Ionospheric and atmospheric delays
These are relevant for VLBI-astrometry of the lensed images, but can be neglected for the redshifts derived from measured time delay changes. The troposphere has an effect of less than 10 nanoseconds at zenith, and most of that is predictable. The effect of the ionosphere is frequency dependent and very variable. The column density of electrons is almost always below 100 TECU (one TECU corresponds to 10 16 m −2 ), which at 1 GHz causes delays of about 130 nanoseconds. The delay scales inversely proportional to the observing frequency squared and can thus generally be neglected.

Finding lensed FRBs
The main caveat is that no gravitationally lensed FRB repeaters are known so far. The ASKAP and CHIME telescopes are already now finding FRBs at a high rate, which with a lensing fraction of about one in 1000 would mean about one lensed FRB per year. However, because they cover only a small fraction of the sky at a given moment, they will almost always miss the lensed copies of bursts. One might try to identify lensed candidates by their expected microlensing signature and then try to find repeats and lensed copies with targeted observations, but this approach will hardly find many suitable systems.
Needed for the search of gravitationally lensed FRBs is a telescope that continuously monitors a significant fraction of the sky without interruptions, so that no bursts or lensed images would be missed. A good option is to concentrate on the circumpolar region, for example north of declination 50 • , which always has an elevation greater than 10 • at latitudes above 50 • . This patch corresponds to more than one tenths of the entire sky, or about 20 times the field of view of CHIME. At the same sensitivity as CHIME we expect to find 40 FRBs per day (Fonseca et al. 2020), or a significant number of lensed ones per year.
The most efficient instrument for such a wide-field monitoring is a regular array of antennas or small dishes with a beamformer based on a two-dimensional Fast Fourier Transform as proposed by Otobe et al. (1994) and Tegmark & Zaldarriaga (2009). A one-dimensional version of this approach is already being used by CHIME-FRB (Ng et al. 2017;Masui et al. 2019), and an application for the two-dimensional case is straight forward. Appropriate antenna arrays were developed as design studies for the Square Kilometre Array (SKA). The EMBRACE test array (Torchinsky et al. 2015(Torchinsky et al. , 2016 is an 8 × 8 m array consisting of 4600 dual-polarisation (only one fully equipped) elements for the frequency range 900-1500 MHz. The original analogue beamformer cannot directly be used for the FFT-based beamformer design, but we are considering options to combine the existing array or a variant of it with a CHIME-like beamformer as a first step in the direction of a sensitive wide-field FRB search instrument.

Gravitational lens as interferometer
Mutual coherence between gravitationally lensed images has already been discussed by Schneider & Schmid-Burgk (1985) and others. For our discussion we can think of the gravitational lens as a huge intergalactic interferometer, the arms of which are given by individual lensed images. Let us assume that these have an apparent separation of θ, which corresponds to a baseline in the lens plane of L = D d θ. The angular resolution at a wavelength of λ d (measured in the lens plane) is then λ d /L. The corresponding linear resolution in the source plane we call ∆x, which would be seen by the observer as the angular resolution ∆θ, with Here we assumed that the 'arms of the interferometer' are fixed. In reality, however, the images will slightly shift with position in the source plane, which adds geometric and potential parts to the phase difference between parts of the source. Luckily these effects cancel because of Fermat's principle, just as discussed in Section 3.5. A gravitational lens really is equivalent to a fixed Young's double slit.
Using our order-of-magnitude estimates from Table 1, neglecting the lens redshift and the differences between distance parameters, and using a wavelength of λ 0 = 20 cm, we get a linear resolution of 40 km and an angular resolution of 3 × 10 −16 arcsec, or about 0.3 femto-arcsec. With the actual values from our lens sample, we find (for 2 arcsec image separation) linear and angular resolutions of 4-11 km and 15-53 atto-arcsec, respectively! The images are mutually coherent only as long as this lensinterferometer does not resolve out all their structures, in other words only as long as the sources have structures on scales below about ten kilometres. We emphasise that most of this paper relies on this assumption, because otherwise we could not achieve the required precision in the measured redshifts. Cho et al. (2020) find narrow structures in FRB 181112 with rise time of down to 15 µs, or 10 µs when correcting for redshift. This corresponds to a maximum source size of 3 km, which is small enough to maintain coherence.
As long as there is at least some level of mutual coherence, we can now use the lens as interferometer with (for a quad) six independent baselines. At the very least we can compare amplitudes of correlations ("visibilities") between baselines and in this way measure the sizes of source components. It remains to be discussed how well the effects of microlensing and interstellar scattering can be corrected for in the analysis. These will be different for the different macro-images, so that they can be treated as responses of the virtual interferometer stations corresponding to the images. The resolution corresponding to the subimages caused by these effects is much poorer, so that they will not introduce differential effects across the FRB source. In other words we should be able to measure phase differences between the visibilities of subcomponents of the source, or even between subsequent bursts. Depending on the achievable SNR we may be able to measure relative positions with accuracies of a kilometre or even better. The extreme precision of such a measurement at cosmological distances cannot be admired enough.
We cannot expect to measure consistent phases between bursts that are separated more than the interstellar scintillation timescale. With the assumed reduced precision of a microsecond, we can measure positional offsets between bursts with an accuracy of the order 50 femto-arcsec or typically 10 4 km, which is still impressive for sources at Gpc distance. Uniform motion will be absorbed in the relative proper motion, but any deviation from that, e.g. due to orbital motion, will be easily measurable with exquisite precision.
Dai & Lu (2017) discuss this and other aspects of motion of the source, lens, and observer in the context of lensed FRBs, but they only consider incoherent measurements with millisecond accuracy.

Discussion
With gravitationally lensed FRBs, time delays can potentially be measured very accurately, not only to milliseconds, as expected from their short nature, but even to microseconds or even nanoseconds, because we can measure the wave fields of the images and correlate them coherently. For repeating FRBs this offers the opportunity to measure how time delays change over the months and years. These changes are described as differential redshifts between the images. We derived how the differential redshifts are related to parameters of the lens to the cosmological model. We found that in quadruply imaged systems we have sufficient information to combine the time delays themselves with their time-derivatives (or the redshifts) to eliminate uncertainties from the a-priori unknown mass distribution of the lens entirely.
In classical gravitational lensing, where mass models are fitted to the observable image configuration (and maybe other parameters like the velocity dispersion of the lens), fundamental parameter degeneracies remain that cannot be removed without further assumptions. Because the true mass distributions are unknown, it is difficult to estimate the biases introduced by the assumptions. The mass-model uncertainties are generally seen as the most serious source of systematic errors in lensing.
Removing this main uncertainty with lensed FRBs does not come for free: The differential redshifts are extremely small, and a number of additional effects have to be taken into account in the analysis. Most important is the relative transverse motion of the source, lens and observer. This contribution from proper motion is about three orders of magnitude larger than the cosmological effect, but for quad systems it can be determined and removed, as long as the relative image positions are accurately known. Such measurements are possible with VLBI. Kochanek et al. (1996) argue that proper motion measurements at cosmological distances would be very valuable to study our motion relative to a rest frame defined by external galaxies independent of the CMB dipole, or to study local kinematics. They estimate that with VLBI, normal peculiar velocities of galaxies are "too small for direct detection given current life expectancies". VLBI precision has improved since then, and life expectancies generally increased, but the statement is still true. The proper motion from Table 1 corresponds to less than one µarcsec over a decade, which is beyond detectability with VLBI. Kochanek et al. (1996) propose to measure quad lenses with VLBI to utilise the lensing magnification to boost the motion and to provide nearby reference sources, but even then a measurement is challenging. The relative redshift induced by proper motion, on the other hand, enables very accurate measurements, and any conclusions will only be limited by the sample variance. Large-scale systematic motion, for instance cosmic rotation on sub-horizon scales, can then be studied with high accuracy, as well as other unexpected effects. This is not the first time that time-derivatives of time delays are discussed in the literature. Zitrin & Eichler (2018) investigate a number of aspects of observing cosmic expansion and other secular changes (including transverse motion) in realtime. They calculate rates of change of time delays for special mass models, but do not combine time delays and their derivatives to determine cosmological parameters, nor do they describe how quad systems can be used to separate transverse motion and cosmology. Piattella & Giani (2017) calculate expected drifts of redshifts, image positions and time delay for point-mass lenses, but do not discuss which quantities can be measured with lensed FRBs.
At this moment it is not possible to estimate achievable accuracies for cosmological parameters reliably, because many important parameters are unknown. Typical source redshifts are most important, but we also found that subtle details of the mass distribution of lensing galaxies can lead to very different accuracies of our proposed method. As a first attempt we used a simulated sample of lenses, not claimed to be representative, as basis for a first assessment of the achievable results. With six lenses we find that accuracies of a few percent may be realistic for the Hubble constant, the matter density and the equation-of-state parameter w of the dark energy. We repeat that these estimates are based on strong assumptions on the lenses, and currently on the assumed spatial flatness of the Universe.
Given that the classical approach of lensing time delays is producing competitive results despite the known limitations, it should also be applied to lensed FRBs. Li et al. (2018) and Liu et al. (2019) argue that lensed FRBs provide better constraints on the lens mass distribution than lensed AGN because of the absence of a bright dominant component in the host. Combining both methods is probably the best option. At the very least we can use classical modelling for lensed FRBs and use the differential redshifts as additional constraints. Even if this does not improve the cosmological result, using the time delays themselves in addition to the redshifts provides one additional constraint on the mass distribution. It is only one, because we add three equations for the relative time delays, but need to add the source position as free parameter.
Further investigations of a number of aspects are needed, most importantly the effects of evolution of the mass distribution in the main lens and along the line of sight. These can be estimated based on structure formation simulations. Besides the unknown redshift distribution, we may use lens models of existing well-studied non-FRB lenses and determine how well those lenses would be suited for cosmology, if their sources were FRBs.
Microlensing is another important effect. According to our estimates it does not invalidate our approach, but provides entirely new information on the masses in the lenses and may in this way shed some light on the dark matter problem. Time delays between micro-images cannot be measured with other sources, and their properties still have to be investigated theoretically.
The physical nature of FRB sources is currently not understood at all, but we know that they must either be small or at least have subcomponents of at most a few kilometres in size. This means there is no chance to resolve their structures with any astronomical instrument, not even with space VLBI. Gravitational lenses can be used as a natural telescopes, not just in the classical sense that they magnify the sources, but be providing interferometric baselines on the scale of a galaxy. The electromagnetic fields of lensed images can be correlated with each other, and interferometric visibilities can be produced that will resolve structures of a few kilometres at cosmological distances. This approach cannot be used for standard AGN sources, because they would be completely resolved out. Having small structures is not only necessary in order to measure highly precise coherent delays, but the correlations will resolve structures and help studying the sources in turn.
Besides additional work on the theoretical understanding, it is now of the highest importance to build instruments that can find gravitationally lensed repeating FRBs in sufficient numbers. New arrays like ASKAP and CHIME completely changed the game regarding the number of FRBs, but they are not optimised for the discovery of lensed ones, because they will generally miss most of the lensed images. Continuous coverage of a large field on the sky appears to be the best way of finding lensed FRBs. The required technology exists, partly even the hardware, and FFT telescopes can be built as efficient lensed-FRB machines.
The potential for cosmology and other aspects of astrophysics is certainly there and worth the effort.