Multi-Messenger Time Delays from Lensed Gravitational Waves
MMulti-Messenger Time Delays from Lensed Gravitational Waves
Tessa Baker , ∗ and Mark Trodden † Denys Wilkinson Building, University of Oxford, Keble Road, Oxford OX1 3RH, UK. Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104, USA. (Dated: October 2, 2018)We investigate the potential of high-energy astrophysical events, from which both massless andmassive signals are detected, to probe fundamental physics. In particular, we consider how stronggravitational lensing can induce time delays in multi-messenger signals from the same source. Ob-vious messenger examples are massless photons and gravitational waves, and massive neutrinos,although more exotic applications can also be imagined, such as to massive gravitons or axions. Thedifferent propagation times of the massive and massless particles can, in principle, place bounds onthe total neutrino mass and probe cosmological parameters. Whilst measuring such an effect maypose a significant experimental challenge, we believe that the ‘massive time delay’ represents anunexplored fundamental physics phenomenon.
PACS numbers: 04.80.Nn, 98.62.Sb, 14.60.Lm, 95.30.Sf.
I. INTRODUCTION
Photons are no longer our only window onto the universe. The recent detections of GW150914 and GW151226 [1, 2]announced the arrival of the powerful new tool of gravitational wave astronomy. In addition, particle messengers –astrophysical neutrinos and cosmic rays – have been detected at Earth for some decades. However, modelling theproduction of gravitational waves (hereafter GWs) and particles during extreme astrophysical events is a challengingscientific problem, requiring advanced numerical work.In contrast, the subsequent propagation of such signals across the universe is comparatively straightforwards todescribe. This prompts us to ask if the propagation of different multi-messenger observables can be used as a new probefor cosmology and/or fundamental physics, independent of the complex details of particle or waveform generation.In particular, small relativistic corrections that accumulate with propagation distance may become measurablefor sources at high redshifts, revealing information about the difference between null and non-null geodesics of theintervening spacetime. This has the power to tell us about both the expansion rate of the universe, and also propertiesof the massive particles being used to trace the geodesics [3, 4].The simplest example that springs to mind is to compare the arrival times of photons and neutrinos from supernovae,first put forwards by Zatsepin in 1968 [5]. However, in such a system astrophysical complexities are likely to dominateeffects of interest to fundamental physics. For example, neutrinos from the famous supernovae 1987A arrived fourhours earlier than the appearance of the optical counterpart, because of the prolonged escape time of photons froma dense supernova remnant [6].Gravitational waves, on the other hand, suffer no such setbacks. Their minimal interaction with matter – and hencenegligible scattering and absorption – makes them arguably a cleaner probe, if the source itself is not the chief objectof interest. For example, one could ask the following, simplistic question: given that we know neutrinos have mass,whilst GWs are massless (in GR, at least), how much later would the neutrinos arrive at Earth – assuming the twowere emitted simultaneously?A simple calculation (presented in Appendix A) of propagation times shows that for a source in the redshift range z = 0 . −
5, the difference in arrival times between a GW and a typical neutrino would be of order one second (seealso [7]). Unfortunately, in a realistic scenario, there will be an additional contribution imprinted on this delay bythe structure of the astrophysical source, i.e. the fact that emission of particles and GWs may not commence exactlysimultaneously. This intrinsic source delay could be of order seconds or longer [8]. Without detailed knowledge andmodelling of the source, it would be impossible to know how to split the measured difference in arrival time into itsintrinsic and particle-mass contributions.As we will show in this paper, the difficulty above can be resolved if the multi-messenger signals encounter agravitational lensing event(s) en route to Earth. Gravitational waves are subject to gravitational lensing in almost ∗ [email protected] † [email protected] a r X i v : . [ a s t r o - ph . C O ] M a y exactly the same manner as photons [9]. A key part of the derivations presented here is to develop a description ofthe lensing of massive particles, a topic that seems to be curiously absent from current literature. We will find thatlensing imparts an additional delay in arrival times that is sensitive to the mass-squared of the messenger particles.This massive time delay depends on a number of quantities of interest to both cosmology and particle physics,namely the mass of the particle involved, the redshift of the source, and the expansion history of the universe. Givenhow small neutrino masses (for example) are expected to be, it is clear that the massive time delay will remain smallfor them ( < ∼
10 MeV [17, 18].2) Other compact object mergers may produce particle emission in addition to their GW signals, although thereis a higher degree of uncertainty here. A merger between a NS and a black hole (BH) has the necessary mattercomponent, although without the formation of a long-lived HMNS the neutrino luminosity may be much lower [19].A BH-BH merger could produce particle emission if there is an accretion disc close enough to be strongly affected bythe merger [20].3) We have already mentioned supernovae above; if these are significantly asymmetric, they can produce GWs inaddition to particle emissions [21–23]. For recent discussions of lensed extragalactic supernovae, see [24–26].In this work we will treat a simplified scenario, considering the lensing of massless and massive relativistic particlesby a single, isolated source – the strong lensing regime. In reality, our multi-messengers are likely to experiencemany additional small deflections along their path, analogous to weak lensing in electromagnetic astronomy. Weacknowledge from the start the existence of such complicating factors in any realistic scenario; this work is intendedto be a first step in fleshing out the key features of a hitherto unexplored phenomenon. We will discuss our omissionsin § IV, and leave a detailed comparison to projected experimental sensitivities for future investigation.The structure of this paper is as follows: in § II we derive the correction to the ‘flight time’ of a massive particle,relative to a massless one, that encounters a strong gravitational lens. In § III we explain how a strategy of differencingthe massive and massless arrival times can ameliorate the unknown intrinsic delay between their emission. We thenevaluate this differential massive time delay for some simple lens models: the singular isothermal sphere and the power-law lens. § IV discusses some additional features of the phenomenon, which would likely complicate a measurement ofthe effects described here. We conclude in § V. Several calculations tangential to our principal discussion are presentedin the appendices.
II. THE MASSIVE TIME DELAYA. Structure of the Calculation
In Fig. 1 we show the basic features of the system under consideration. An energetic event at redshift z S and conformaldistance D S releases both massless emissions (electromagnetic and/or gravitational radiation) and relativistic, massiveparticles of mass m . For argument’s sake we will sometimes refer to these massive particles as neutrinos, though ourformalism applies more generally. Note that the massless and massive fluxes will not typically commence exactlysimultaneously – we discuss how to deal with this in § III.At redshift z L and conformal distance D L , both types of emission encounter a gravitational lens with three-dimensional density profile ρ ( (cid:126)r ). Throughout this paper we will make use of the thin-lens approximation, which treats SO Lensing plane β ɑ ɑ θ θ ξ ξ χ L L D L D LS D S A ɣ ɣ FIG. 1. Diagram illustrating angles and conformal distances relevant for the derivation of § II. Two lensed paths are shown(intersecting the lensing plane at L and L ), and also the undeflected path that the rays would all follow if the lens wereabsent, OS . ξ and ξ are two-dimensional position vectors of the images in the lensing plane, and χ is the position vector ofthe source in the plane AS . α and α are the deflection angles experienced by these two rays at the lens. all the deflection as occurring instantaneously at a single plane. Quantities relating to the lens, such as its densityand gravitational potential, will be projected onto a two-dimensional plane at redshift z L .The emissions travel on multiple paths i around this lens, with each path experiencing a total angular deflection α i and finally being received by an observer O at angle θ i to the optical axis OA (defined as the axis connecting theobserver to the centre of the lens). In optical strong lensing of extended sources, these multiple paths and detectionangles correspond to multiple images of the same source, often distorted in an informative way that constrains thestructure of lens.The emissions we are interested in here originate from point sources. Hence, though we expect to detect multiple,identical point-sources of our massive and massless messengers, there will be no equivalent of the spatial imagedistortion seen in traditional optical lensing. However, in some cases it might be possible to associate the lensedpoint sources with lensed optical images of a host galaxy, although this will require significant advances in sourcelocalization of GW and neutrino detectors. That said, even if localization techniques do not improve sufficiently toallow the spatial resolution of lensed GW sources, emissions that have travelled along different lensed paths may stillbe temporally resolvable.Within the simplified model outlined above, the total conformal travel time for a massive, relativistic particle hasthe structure: η total ( θ, m ) = η undeflected ( m ) + η massless ( θ ) + η massive ( θ, m ) . (1)The first term in this expression is the travel time from the source to the observer for a particle in the absence of anylens. It is the same for all paths (hence no dependence on θ ), and provides the largest contribution to η total ; notehowever, that it will depend on the mass of the particle ( § II B). The remaining terms describe corrections to thisminimum travel time induced by the presence of the lens; in the case of a massless particle only the first correctionterm in eq.(1) exists.In principle, a massive and massless particle will be deflected by slightly different amounts at the lens – the derivationof this effect is given in Appendix B. We would therefore expect there to be a small offset in the position of theneutrino and GW sources on-sky, as shown in Fig. 2. To account for this we compare flight times along two differentbut very close paths, by writing the reception angle of the massive particle as θ = ¯ θ + δθ , where ¯ θ is the receptionangle in the massless case [27]. SLensing plane ɑ AO ɑθθ LL FIG. 2. As massive and massless particles are deflected by different amounts at the lens (¯ α and α respectively), one shouldevaluate their travel times along slightly different paths SLO and S ¯ LO . However, this turns out to produce corrections thatare second-order with respect to the differential massive time delay that is the main focus of this paper. Hence, in what follows,we can consider both massless and massive particles corresponding to the same image to have propagated along identicaltrajectories. Performing a Taylor expansion of (1), we then have: η total ( θ, m ) (cid:117) η undeflected ( m ) + η massless (¯ θ ) + ∂η massless ( θ ) ∂θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ =¯ θ δθ + η massive (¯ θ, m ) + O (cid:18) δθ , m p δθ (cid:19) , (2)where the last term – using notation to be introduced shortly in § II B – indicates that significantly suppressed termshave been neglected [28]. For this reason, it is sufficient to evaluate η massive at ¯ θ .The principle of least action (which for a massless particle, in the geometric optics limit, becomes Fermat’s principle)tells us that a particle follows a path along which the travel time is stationary; thus the third term in eq.(2) mustvanish. Hence, to first order in δθ , it is sufficient to evaluate the flight time of the both massive and massless particlesat the same reception angle ¯ θ . Any difference in arrival times then arises purely from their null/non-null nature,rather than path differences. It is interesting to note, though, that this difference in deflection angle does exist inprinciple, and would need to be accounted for in high-accuracy calculations.Now consider the conformal time delay between two emissions (either massive or massless) that travel about thelens on widely -separated paths, like those shown in Fig. 1. These are received at angles θ and θ , corresponding toseparate point source images on the sky. The conformal time interval between their arrivals is:∆ η ( θ , θ ) = η total ( θ , m ) − η total ( θ , m ) . (3)The term η undeflected in eq.(1) makes no contribution to ∆ η ( θ , θ ), as it is the same for both paths and thereforecancels out. Hence, although η total may be a significant fraction of the age of the universe, the difference betweenthe (conformal) travel time of two rays received at z = 0 is much smaller than the Hubble time, typically between afew days and a few years. Since cosmological expansion is negligible over these timescales, the conformal time delayis equivalent to the physical time delay to an extremely good approximation [29], i.e.∆ t ( θ , θ ) = a ( z = 0) ∆ η ( θ , θ ) = ∆ η ( θ , θ ) . (4)Therefore, following [30], we can safely formulate our calculation in terms of conformal distances and conformal times. B. Derivation
Our goal here is to calculate the travel time of a particle along the paths shown in Fig. 1; in the thin-lens approxi-mation each of these consists of two straight-line segments, with a total deflection angle α i incurred instantaneouslyat the lens. We use the letter i to label an unspecified number of different deflected paths; only two are shown inFig. 1 for clarity.We begin from the line element for a particle moving in a spacetime containing a single linear perturbation Φ in anotherwise homogeneous universe [31]: ds = − (cid:15)c dλ = − c a ( η ) dη (cid:18) c (cid:19) + g ij dx i dx j . Path γ LS γ OL α − θ + β θ − β α − θ − β θ + β TABLE I. Values of γ , the angle between a deflected ray and the observer-source axis OS , for the two paths shown in Fig. 1.The second column shows the value on the path segment between the source and the lens; the third column is the value on thesegment between the lens and observer. For a massive particle dλ would be an element of proper time; we have chosen slightly unusual notation here to allowus to unify the massive and massless cases (see below). Note that we have introduced a binary parameter in the firstequality: (cid:15) = 1 for a massive particle and (cid:15) = 0 for a massless one. We have also chosen to keep the spatial part ofthe metric general for now. A little rearrangement of the second equality brings this line element to the form: dη = (cid:18) c (cid:19) − dλa (cid:20) (cid:15) + g ij c dx i dλ dx j dλ (cid:21) . (5)In the massless case, the quantity λ in the expression above is simply an affine parameter (not proper time). Next, weTaylor-expand the first bracket (since Φ /c is a small quantity) and introduce the spatial three-momentum magnitude: p = m g ij dx i dλ dx j dλ . (6)Using eq.(6) in (5) then leads us to dη ≈ (cid:18) − Φ c (cid:19) (cid:20) (cid:15) + p m c (cid:21) dλa . (7)To find the (conformal) travel time of the particle from source to observer, we need to integrate dη along one of thepaths SL i O shown in Fig. 1. To do this, we first eliminate the affine parameter element dλ in favour of a conformaldistance element along the lensed path, d(cid:96) = (cid:112) δ ij dx i dx j [32]. Expanding g ij = δ ij a (1 − /c ) and performing aTaylor expansion in Φ /c , eq.(6) rearranges to become dλ ≈ m ap (cid:18) − Φ c (cid:19) d(cid:96) . Substituting this into eq.(7), we then obtain dη ≈ (cid:18) − c (cid:19) (cid:20) (cid:15) m c p (cid:21) d(cid:96)c . (8)We can now write d(cid:96) = dy/ cos γ ( y ), where y measures distance along the undeflected ray OS , and γ i ( x ) is the anglethe path element d(cid:96) makes with OS , giving d(cid:96) = dy cos γ ( y ) ≈ (cid:18) γ ( y ) (cid:19) dy , (9)where the large distances involved ensure that the small angle approximation used in the second equality is valid.Taking eqs.(8) and (9) together, and neglecting terms that are second-order in the small quantities Φ and γ , we finallyreach dη ≈ (cid:18) γ ( y ) − c (cid:19) (cid:20) m c p (cid:21) dyc . (10)Note that now that the mass of the particle is explicitly present, the parameter (cid:15) is surplus to requirement and hasbeen absorbed into m in the line above.Integrating eq.(10) along the ray OS will yield the conformal time taken for the particle to travel the lensed path. Inthe thin-lens approximation, the integration path breaks into two stages, with the value of γ ( x ) a constant along each(see Table 1). However, when integrating eq.(10) we must also remember that the spatial three-momentum p redshiftsin proportion to 1 /a ; this is true for both massive and massless particles [33]. Rearranging eq.(10), multiplying outthe first bracket and integrating, the conformal time taken to travel one of the lensed paths is given by η ( θ, m ) = (cid:90) D S (cid:20) m c a p (cid:21) dyc + (cid:90) D S (cid:18) γ (cid:19) (cid:20) m c a p (cid:21) dyc + (cid:90) D L + δD L − δ (cid:18) − c (cid:19) (cid:20) m c a p (cid:21) dyc (11) (cid:117) (cid:90) D S (cid:20) m c a p (cid:21) dyc + γ OL (cid:90) D L dyc + γ LS (cid:90) D S D L dyc − (cid:90) D L + δD L − δ c dyc + 12 (cid:18) mcp (cid:19) (cid:40) γ OL (cid:90) D L a dyc + γ LS (cid:90) D S D L a dyc − (cid:90) D L + δD L − δ c a dyc (cid:41) . (12)In the first line above we have written the three-momentum as p = p /a , where p is the value at redshift zero. Formost of the results in § III we will used a fixed value p , so we have not included it as an explicit argument of η here.The third integral has a restricted integration range, since the integrand is only non-zero in a small region of size 2 δ near the potential well Φ. In moving to the second line we have performed a Taylor expansion in the small quantity mc/p ( mc (cid:28) p since we are dealing with relativistic particles), and have broken the second integral of eq.(11) intothe two sections OL and LS indicated in Fig. 1. Since γ is a constant along these sections, it can be factored outof the integrals. Note that the integrands in the final line above pick up a factor of a from the redshifting of thethree-momentum.Eq.(12) is valid for any lensed path in the thin-lens approximation. However, for the rest of this paper we willspecialize to the two-image case illustrated in Fig. 1. For now, let us evaluate the travel time along the upper pathshown in Fig. 1, using the values of γ given in Table 1. We identify the first term of eq.(12) as the unlensed traveltime of a particle, i.e. η undeflected in eq.(1). Proceeding to evaluate the remaining integrals, we obtain η ( θ , m ) = η undeflected ( m ) + ( θ − β ) D L c + ( α − θ + β ) D S − D L ) c − D L D S c D LS ψ ( θ )+ 12 (cid:18) mcp (cid:19) (cid:26) ( θ − β ) (cid:90) a L daH ( a ) + ( α − θ + β ) (cid:90) a S a L daH ( a ) − D L D S c D LS ψ ( θ ) a L (cid:27) , (13)where we have changed the integration variables for the integrals in the second line, and in the final term havetaken the limit δ (cid:28) D S implied by the thin-lens approximation. We have also defined the two-dimensional projectedpotential as ψ ( θ ) = D LS D L D S (cid:90) D L + δD L − δ θ, y ) c dy . (14)Let us briefly focus on the lensing contribution to the travel time that exists for both massive and massless particles,i.e. the second, third and fourth terms of eq.(13). We make use of the simply-named lens equation [34] (cid:126)α scal = D LS D S (cid:126)α = (cid:126)θ − (cid:126)β , (15)where the first equality defines the scaled deflection angle. The second equality is a standard relation that can bederived by consideration of equivalent triangles in Fig. 1. Notice that it is important to consider the direction ofdeflection here. If we let ˆ (cid:126)φ denote a unit vector in the clockwise direction with respect to OA, then (referring toFig. 1) (cid:126)γ = γ ˆ (cid:126)φ , but (cid:126)γ = − γ ˆ (cid:126)φ . This results in the sign differences seen in Table 1 for the two paths.Isolating the non-mass-dependent contribution to the time delay and substituting in eq.(15) we obtain η massless ( θ , m ) = ( θ − β ) D L c + 12 (cid:20)(cid:18) D S D LS − (cid:19) ( θ − β ) (cid:21) D LS c − D L D S c D LS ψ ( θ ) (16)= ( θ − β ) D L c (cid:20) D L D LS (cid:21) − D L D S c D LS ψ ( θ ) (17)= D L D S c D LS (cid:20)
12 ( θ − β ) − ψ ( θ ) (cid:21) . (18)Eq.(18), is equivalent to the standard expression for the time delay of lensed photons, often expressed in terms of theFermat potential [35]. There are two effects that contribute to the lensed travel time: a geometric delay that arisespurely from the increased path length (first term in eq.18) and a Shapiro delay that occurs as particles pass througha gravitational potential well (second term). Note that the Shapiro delay is incurred at a single redshift, z L . We notein passing that our derivation of this expression – beginning from a line element – differs substantially from the mostwidely-used presentation, which involves deducing the geometric and Shapiro terms from Fermat’s principle.For a massive particle, the second line in eq.(13) also comes into play. Using the lens equation once more, themassive correction to the travel time along the upper path of Fig. 1 can be written as η massive ( θ , m ) = 12 (cid:18) mcp (cid:19) (cid:40)
12 ( θ − β ) (cid:20)(cid:90) a L daH ( a ) + D L D LS (cid:90) a S a L daH ( a ) (cid:21) − a L D L D S c D LS ψ ( θ ) (cid:41) . (19)This expression merits a few comments. First, note that the correction to the travel time of a massive, relativisticparticle has an overall prefactor of ( mc/p ) , as might be intuited from Special Relativistic considerations. For allthe scenarios discussed in this paper, the initial energy, E , of the massive particle is substantially greater than itsrest mass. Hence in what follows we will implicitly take ( mc/p ) = m c / ( E − m c ) ≈ ( mc /E ) . In § III, wherewe evaluate this correction numerically, we will see that this ratio is the single most important factor controlling themagnitude of the effects derived here.Second, distinct geometric (first two terms) and Shapiro (last term) contributions are still identifiable in eq.(19),even though the final form is not as elegantly compact as eq.(18). We note that the Shapiro-like correction for amassive particle in a Schwarzchild metric is derived in [36].Third, η massive depends on the cosmological expansion history in a more complicated manner than its masslesscounterpart; note that the expansion history of the universe only enters eq.(18) via the overall prefactor D L D S /D LS .This difference occurs because the redshifting three-momentum of massive particles affects their propagation speed,and hence their travel time. Whilst clearly massless particles experience energy-momentum redshifting as well, it doesnot alter their propagation speed and hence does not affect the massless time delay in such an intricate way.Finally, given the complicated dependence of eq.(19) on D L , D S and D LS , the redshift-dependence of η massive isnot easy to predict. In particular, it may not necessarily peak when the lens is halfway between the observer andsource, as typically occurs for lensing kernels. We will study this further in § III.We have found expressions for the three contributions to the travel time identified in eq.(1): η undeflected ( m ), η massless ( θ ) and η massive ( θ, m ). We now have all the tools to calculate the relative time delay between the arrivalof two massive particles that have travelled on different paths around a lens, or between a massive and masslessparticle traveling the same path. III. APPLICATION & LENS MODELSA. Differencing Strategy
For convenience we summarize here the results of the previous section, now generalized to apply to both paths inFig. 1: η total ( θ ) = η undeflected ( m ) + η massless ( θ ) + η massive ( θ, m ) + O (cid:32)(cid:20) mcp (cid:21) (cid:33) (20) η undeflected ( m ) = (cid:90) D S (cid:20) m c a p (cid:21) dyc (21) η massless ( θ ) = D L D S c D LS (cid:20)
12 ( θ ± β ) − ψ ( θ ) (cid:21) (22) η massive ( θ, m )) = 12 (cid:18) mcp (cid:19) (cid:40)
12 ( θ ± β ) (cid:34) (cid:90) a L daH ( a ) + D L D LS (cid:90) a S a L daH ( a ) (cid:35) − a L D L D S c D LS ψ ( θ ) (cid:41) The ± signs correspond to the paths below (+, i =2) and above ( − , i =1) the optical axis shown in Fig. 1. The simplelens models considered in this paper produce only two images, such that we can always orient the system as shownin the figure. We leave the treatment of more realistic system – which, famously, can only produce odd numbers ofimages [37] – to a future work.Let us explain here the most sensible way to combine these particle flight times. As we described in the introduction,a major systematic error when trying to measure the delay in arrival between massive and massless particles wouldbe the unknown relative emission time. So far we have implicitly assumed that all our messenger particles set offexactly simultaneously, but this is unlikely for any realistic source. For example, in a supernova the neutrino diffusiontimescale in a collapsing stellar core is of order a second [38]. In the case of a binary BH system, there will be a similarlight-speed propagation time for information about the merger to reach the surrounding accretion disc. Hence thereis an intrinsic component of the time delay set by the details of a high-energy astrophysics event, which is unknowablewithout detailed numerical modelling of the event and high levels of certainty for the source parameters.Fortunately, this is where our strong lensing formalism can help. Consider a futuristic experimental scenario whichdetects the following four events, all confirmed as originating from the same sky location: • t a : time in a massless signal (e.g. a GW waveform) identified as the merger event. • t b : peak flux in the accompanying massive particle signal. • t c : merger time in a massless signal, with the same structure as the previous massless signal (‘massless echo’). • t d : peak flux in a second massive particle signal, with the same flux variations as the previous massive particlesignal (‘massive echo’).The first two events here correspond to messengers arriving from the same image, which have travelled along thesame lensed path. The latter two events correspond to messengers arriving from the second image, having traverseda different lensed path. For example, messengers travelling the upper path in Fig. 1 could give rise to t a and t b , andtheir lensed echoes travelling along the lower path give rise to t c and t d . We focus on the following time intervals: t b − t a = δη intrinsic + η total ( θ , m (cid:54) = 0) − η total ( θ , m = 0) (23) t d − t c = δη intrinsic + η total ( θ , m (cid:54) = 0) − η total ( θ , m = 0) (24) T = ( t d − t c ) − ( t b − t a )= η massive ( θ , m ) − η massive ( θ , m ) (25)where δη intrinsic represents the delay in emission between massive and massless messengers that is intrinsic to thesource, as described above. The first equality of eq.(25) defines the quantity T , and the second equality uses eq.(20).The intervals t d − t c and t b − t a correspond to emissions arriving from the same image of the source; they are expectedto be small compared to t d − t b or t c − t a , which correspond to emissions of the same kind arriving from differentlensed images.The intrinsic delay between the emission of massive and massless particles is a property of the source, and is notaffected by the subsequent strong lensing. Hence it is the same for both lensed images, and therefore can be cancelledout by the differencing strategy outlined above. However, this strategy assumes an idealistic situation in which theflux variations of both the massive and massless signals are well-sampled. In a real-world scenario this may not bepossible – we will discuss this issue further in § IV.With a major source of error thus circumvented, we can now proceed to estimate the magnitude of the quantity T ,which we will term the differential massive time delay . In particular, we are interested to study the sensitivity of T to neutrino mass and the late-time cosmological acceleration (for those sources at cosmological redshifts). In the restof this section we do this using two simple lens models. Although these may not be realistic as a global descriptionof (say) galaxy clusters, our work here only requires a sufficient description of the innermost region of the lens.In Appendix C we give a few relevant formulae for describing the properties of a lens model. These belong to thestandard formalism of strong lensing and can be found in many introductory texts. B. The Singular Isothermal Sphere Lens
The singular isothermal sphere (SIS) is one of the most commonly used toy lens mass models, and is a goodapproximation for the central regions of early-type galaxies [39]. It has the spherically symmetric three-dimensionaldensity profile ρ ( r ) = σ v πGr , (26)where the particles that constitute the lens have a Maxwellian velocity distribution with one-dimensional velocitydispersion σ v . The SIS has some unusual properties: it has a constant 3D gravitational potential throughout, adensity singularity at r → r → ∞ . These pathologies do not usually cause problems when weconsider particles passing the lens at intermediate distances; however, it is the presence of the central singularity thatallows SIS lenses to evade Burke’s odd number theorem [37].The constant potential of the SIS results in further interesting features. In particular, one finds that all rays reachingthe lens undergo the same deflection towards the lens centre [34]. That is, α ( θ ) scal is a constant: α SISscal (¯ θ ) = θ SIS E = 4 π (cid:16) σ v c (cid:17) D LS D S , (27)where θ SIS E is the Einstein radius of the lens; we will drop the label ‘SIS’ for the remainder of this subsection. Usingthe formulae of Appendix C, it is fairly easy to derive the projected mean surface density and projected (i.e. 2D)gravitational potential of this model:¯ κ ( θ ) = θ E θ ψ ( θ ) = θ E θ . (28)If the source angular position ( β ) lies within the Einstein radius, the SIS lens forms two images on opposite sides ofthe lens. Using eq.(15), these are located at angular radii θ ± = θ E ± β , (29)where for convenience we have labelled the two images { θ + , θ − } instead of { θ , θ } (note that the ‘plus’ path in eq.(29)corresponds to the upper path in Fig. 1, which actually incurs the minus signs in Table 1). For the SIS lens only, thefollowing relations then hold:2 θ E = θ + + θ − β = θ + − θ − . (30)For a real strongly lensed system the image angular positions θ + and θ − can be measured. In order to progress withour theoretical calculation, we will introduce an asymmetry parameter, A , that quantifies the offset of the sourceposition from θ E , as follows: θ − = A θ E , where 0 ≤ A ≤ ⇒ θ + = θ E (2 − A ) , (32)where A = 1 would imply a perfect Einstein ring system, for which the two images merge.We can now evaluate eq.(25) for the SIS model, using the results summarized above. The terms η undeflected and η massless (¯ θ ) cancel out when we difference the time delays of massive and massless particles, as expected. The onlyterm that contributes to the differential massive time delay in the SIS case is then η massive . We obtain T = 12 (cid:18) mcp (cid:19) a L D L D S c D LS θ E ( θ + − θ − ) (33)= (cid:18) mcp (cid:19) a L D L D S c D LS θ E (1 − A ) (34)= (cid:18) mcp (cid:19) a L D L D LS c D S (cid:18) π σ v c (cid:19) (1 − A ) , (35)where in fact even the geometric contribution to the differential massive time delay has cancelled, and we are left withonly a pure Shapiro-like contribution. This vanishing of the geometric-like term is a unique feature of the SIS lens,due to its constant deflection angle; it does not occur for other lens models.The left panel of Fig. 3 shows the evaluation of T for parameter values m = 0 . p = 10 MeV, A = 0 . m = 0 . (cid:80) m ν = 0 .
23 eV from the Planck satellite [40]. An energy of 10 MeV isconsistent with the typical neutrino energies produced by accretion onto hyper-massive neutron stars after a NS-NSmerger [17, 18] and in supernovae.We note that a realistic situation would likely involve some spread in particle emission energies, and hence adispersion in arrival times. In a similar vein, neutrino oscillations will ensure that even neutrinos emitted as aninstantaneous burst are received with a spread of arrival times. The size of this dispersion could well be comparableor larger than the differential massive time delay we are pursuing. However, our quantity T is defined as a difference of multiple event timings, all of which will be dispersed in the same manner. Therefore, as long as the massive and0 z L -11 -10 -9 -8 -7 -6 T σ v =1000kms − σ v =700kms − σ v =350kms − z L ∆ η m a ss l e ss ( s ) σ v =1000 kms − σ v =700 kms − σ v =350 kms − FIG. 3.
Left:
Differential massive time delay T (eq.25) for a SIS lens. Solid curves are for a source galaxy at z S = 3 .
0, whilstdashed curves are for z S = 1 .
5. Note that z L , the redshift of the lens, cannot exceed z S . All curves are evaluated for parameters m = 0 . A = 0 . p = 10 MeV. Planck 2015 cosmological parameters are used. Right:
The massless part of the timedelay, as would be measured in standard strong lensing studies. The velocity dispersion σ v acts as a proxy for the mass of thelens. Only curves for z S = 3 are shown. massless fluxes are well-sampled – so that the peak of a dispersed signal can be located – our calculation remainsunaffected [41]. We assume a futuristic scenario where such sampling is possible; of course, this may not be achievable,see § IV. We do not consider here any effects relating to the structure of the neutrino hierarchy; see [22] for a discussion.For comparison, the right-hand panel of Fig. 3 shows the standard, massless part of the time delay. For theparameter values under consideration here, this ranges between tens of days and tens of years. Note that, in orderto maximise the small corrections of interest, we are considering higher source and lens redshifts than most opticalstrong lensing studies. This is the cause of some of our unusually large massless time delay values.We see that, irrespective of the source redshift, the differential massive time delay peaks when the lens is located atredshifts around 0.2–0.5. The shape of the curves in the left-hand panel can be understood using eq.(34) as follows:the prefactor of D L D S /D LS is shared with the massless time delay (see eq.22), and imparts the broad, flat shape seenin the right panel of Fig. 3. However, this shape is modulated by the appearance of θ E in eq.(34): for a fixed sourceredshift, the Einstein radius – being an angular scale measured by the observer – decreases as the lens is moved tohigher redshifts. This decline is responsible for the skew towards low z L in the left panel of Fig. 3.The differential massive time delay is also somewhat sensitive to cosmological parameters, as shown in Fig. 4. Fora fixed source redshift, an increase in Ω Λ boosts the conformal distances appearing in the numerator of eq.(35). Inthe ideal scenario of having multiple well-understood multi-messenger lensing systems with z L > .
5, the differentialmassive time delay could provide a new method to probe the equation of state of the dark energy sector (if assumptionsare made about the neutrino mass). This complements existing cosmological parameter constraints made using themassless time delay of photons [42, 43]. There is also the possibility of testing for novel effects such as the violationof C, P and CP symmetries in gravity [44], which we will treat in a future investigation.The velocity dispersion, σ v has a very strong influence on the magnitude of T – note that it appears to the fourthpower in eq.(35). In the SIS case σ v acts as a proxy for the mass of the lens, suggesting that lensing by galaxy clusters(which generally have larger σ v than individual galaxies) may be a more promising, albeit still challenging, candidatefor a measurable differential massive time delay.However, the selection of systems with a high velocity dispersion or mass must be balanced against the correspondinginterval between the massless and massive echoes (i.e. the interval t c − t b in eq.24). The lowest curve in the righthandpanel of Fig. 3 ( σ v = 350 kms − ) has a window of a few months between echoes, whilst for the uppermost curve ( σ v =1000 kms − ) it is of order thirty years(!) We note that similar precision timing experiments spanning decades havealready been carried out, for example, monitoring the inspiral rate of the Hulse-Taylor pulsar [45]. Unquestionably,though, this makes for an inconveniently slow experiment.One can speculate on more exotic scenarios: if neutral particles heavier than neutrinos were emitted in conjunctionwith GWs or photons, then the effects discussed here could be orders of magnitude larger. As an illustrative example,consider a situation in which particles with the mass of a nucleon are produced during an event with energy similarto that of a gamma-ray burst (GRB). Taking m = 938 MeV and p ∼ mc/p ) is boosted by1 z L T
1e 7 Ω Λ =0 . Λ =0 . Λ =1 . FIG. 4. The dependence of the differential massive time delay on the late-time expansion history, controlled via the energydensity Ω Λ . A flat cosmology is assumed in all cases. Solid lines represent a system with z S = 3 .
0, whilst dashed lines are for z S = 1 .
5. Particle properties are the same as in Fig. 3; the lens velocity dispersion used is σ v = 930 kms − and the asymmetryparameter is A = 0 . a factor of 10 and differential massive time delays of order tens of seconds become possible, see Fig. 5. Note thatcharged particles would be deflected by both Galactic and intergalactic magnetic fields, destroying the signals we areinterested in here. See [46] for a discussion of neutral cosmic ray candidates.Another hypothetical scenario would be to consider the time delays experienced by WIMPs such as axions, theo-retical particles hypothesized to solve the strong CP problem of QCD, and appearing generically in string theory [47].Most attention focuses on ultra-light axions (10 − eV - 10 − eV) as dark matter candidates, but heavier axions arepossible (not as dark matter) and could be produced in high-energy events such as supernovae and NS mergers [48].Axions with a mass of order 1 keV would experience a differential massive time delay of order seconds; however, wewill not pursue such exotic scenarios further here.We note that shortly after the present work appeared online, a letter by Fan et al. [49] was released, applyingsimilar concepts to constraining the speed of propagation of gravitational waves. C. The Power-Law Lens
The first step in complexity beyond the SIS is the power-law (PL) lens model. This has the spherically symmetricdensity profile ρ ( r ) = ρ (cid:16) r r (cid:17) n , (36)where the SIS lens is recovered for n = 2. Like the SIS, the PL lens has an infinite central cusp that is not problematicfor our current work. This can be alleviated, if desired, by the use of softened power-laws such as ρ ∝ ( r + s ) n ,where s is a constant [50].For some values of n , the PL lens can produce more than two images, e.g. for n = 1 .
5, three images are presentwhen the source lies inside the tangential critical curve of the lens plane. To facilitate discussion with the SIS case, inthis paper we will use only two of these images (specifically, the two at greatest radial distance from the lens centre).We note, though, that a third image – if resolvable – offers further differencing possibilities that could be used eitheras a check of nuisance parameters, or to provide multiple measurements of the differential massive time delay.Referring to the standard lensing definitions given in Appendix C, the PL lens has the following potential, scaled2 z L -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 T m =0 . , p =10 MeV m =3eV , p =10 MeV m =30eV , p =10 MeV m =938 MeV , p =1 TeV FIG. 5. The differential massive time delay for particles of varying mass and energy. The lowest curve corresponds to ourfiducial case of m = 0 . p = 10 MeV, representing a typical neutrino from an NS-NS merger. The uppermost curvecorresponds to the case of a high-energy, neutral particle with the mass of a nucleon. Two intermediate cases are also shown.Solid and dashed curves are the same as previous figures. All curves use σ v = 930 kms − and Planck cosmological parameters. deflection and convergence profiles [51]: α scal ( θ ) = θ E (cid:18) θθ E (cid:19) − n ψ ( θ ) = θ E − n (cid:18) θθ E (cid:19) − n (37) κ ( θ ) = 3 − n (cid:18) θθ E (cid:19) − n ¯ κ ( θ ) = (cid:18) θθ E (cid:19) − n . (38)Eqs.(30) no longer hold, but we will define analogous quantities (though note a factor of 2 difference in the seconddefinition): 2 (cid:104) θ (cid:105) = θ + + θ − ∆ θ = θ + − θ − . (39)In what follows, we will assume that the annulus enclosed by the two images is narrow compared to their offset fromthe lens centre, i.e. ∆ θ (cid:28) θ + , θ − . This is reasonable, since highly asymmetric lensing systems usually have at leastone strongly demagnified image, and are therefore less likely to be identified.The Einstein radius is now given by [51] θ E = (cid:32) θ + + θ − θ − n + + θ − n − (cid:33) n − . (40)As we did for the SIS lens, we will parameterize one of the image positions in terms of the Einstein radius as θ − = A θ E .The narrow-annulus approximation above then allows us to expand in the quantity ∆ θ/θ E (which will also be smallfor values of A close to 1. See § z L T g e o
1e 7
Geo n =1 . . . z L T Sh a p
1e 7
Shapiro n =1 . . . FIG. 6. Contributions to the relative time delay for a power-law lens, with density profile ρ ∝ r − n . Both panels have particleparameters p = 10 MeV, m = 0 . z S = 3 . A = 0 . ρ = 2 × M (cid:12) Mpc − and r = 0 . Left:
The geometric contribution, i.e. the first two terms of eq.(44). This vanishes at n = 2, the SIS case, and changes signeither side of this value. Right:
The Shapiro contribution, i.e. the third term of eq.(44). The difference between the n = 2 casehere and in Fig. 3 is due to the different values of A used. Note that in the case of n = 1 . relations: 2 (cid:104) θ (cid:105) = A − n θ E (cid:20) − n ) ∆ θ A θ E (cid:21) (41)∆ θ = 2 A θ E [ A − n − − A − n (2 − n ) (42) T ≡ T geo + T Shap (43)= 12 m c p ∆ θ θ E A − n (cid:40) ( n − A − n (cid:34) (cid:90) a L H da + D L D LS (cid:90) a S a L H da (cid:35) + a L D L D S c D LS (cid:41) . We see that the geometric contribution to T vanishes for n = 2, in agreement with § III B.Fig. 6 shows the influence of the power-law index, n , on the geometric and Shapiro-like contributions to thedifferential massive time delay. The Shapiro contribution always remains positive, whilst the geometric contributionswitches sign about its vanishing point at n = 2. For n < n > . α ( θ ) in eq.(37). Return-ing to the simple two-image picture of Fig. 1, one image will sit inside the Einstein radius and one exterior to it. For n >
2, rays from the image inside the Einstein radius will have experienced the greatest deflection at the lens. Thiscorresponds to the intuitive picture that a ray passing close to the centre of the lens, where the density is highest,will be more strongly deflected than one passing ‘further out’. We then expect signals from the θ − image (lower pathin Fig. 1) to arrive after those from the θ + image.In the case of n < θ > θ E experiences the greater deflection (see the firstof eqs.37). The n = 1 case corresponds to a uniform critical sheet, whilst the central regions of galaxies are sometimesmodelled using 1 (cid:46) n (cid:46) θ − image to arrive first; hence T geo changes sign. However, the Shapiro-like contribution to the differential massivetime delay has the potential to contradict this intuition, if it is large enough to outweigh the geometrical term.4 D. Magnifications
As well as deflecting emissions onto multiple paths, gravitational lenses are able to focus (or sometimes defocus)a bundle of rays en route to the observer. For extended electromagnetic sources, the focussing of rays results in adecreased image area and hence, due to the conservation of surface brightness, a boost in flux. Magnification occurssimilarly for point-like sources, though the situation is slightly different (as clearly there can be no change in imagearea): the brief explanation is that rays which otherwise would not intersect the observer now do so, due to theirdeflection at the lens.The magnification is defined as the ratio of the lensed to the unlensed flux. Although the source flux is clearly afunction of frequency, the spectral shape is preserved by lensing and hence µ is independent of frequency. As detailedin standard lensing texts, if the mapping of a point from the image plane to the source plane is given by the 2D vectorfunction (cid:126)β ( (cid:126)θ ), then the magnification factor is: µ = 1 | det A | , where A = ∂ (cid:126)β∂(cid:126)θ . (44)For an axisymmetric source we have [34]: det A = (1 − ¯ κ ) (1 + ¯ κ − κ ) , (45)where the dimensionless surface densities (equivalent to convergence), κ and ¯ κ , are defined in Appendix C.Since they follow the same null geodesics as photons, these expressions should apply equally well to GWs. The onlytwo requirements are that the geometric optics approximation remains valid, and that we avoid the exceptional case ofa perfect Einstein ring system, for which κ →
1, and hence the magnification above formally diverges. Takahashi [52]estimates that for a GW of characteristic frequency f , the geometric optics approximation holds for lens masses above10 M (cid:12) ( f /Hz) − ; since we are using galaxy clusters ( ∼ − M (cid:12) ) as our lenses and NS binaries ( f ∼ − θ = θ E exactly; this requires a full waveoptics treatment to remove the apparent divergence [53]. We will not make this digression here, but one can rest easythat the singularities seen in Fig. 7 below remain finite in a fully correct treatment.Taking eqs.(37), (38) and (45) together, we will calculate the magnification of the images produced by a power-lawlens. We will continue to use the notation of § III, that is, we write θ = A θ E (note that the asymmetry parameter A should not be confused with the Hessian matrix A ). Formally, for a given value of A , the location of the otherimage can be determined (in terms of θ E ) by solving eq.(40). In practice, this is awkward to do analytically except inspecial cases such as n = 2 ,
3, etc. Hence we shall make the same restrictions and approximations as used in § III C,and study systems for which the images are separated by a narrow annulus such that ∆ θ = θ + − θ − (cid:28) θ + , θ − . Underthese conditions, we quickly arrive at:det A (cid:12)(cid:12) θ + ≈ (cid:20) − A (1 − n ) (cid:26) − n ) ∆ θ A θ E (cid:27)(cid:21) (cid:20) n − A (1 − n ) (cid:26) − n ) ∆ θ A θ E (cid:27)(cid:21) det A (cid:12)(cid:12) θ − ≈ (cid:104) − A (1 − n ) (cid:105) (cid:104) n − A (1 − n ) (cid:105) , (46)where ∆ θ is given by eq.(42). Note that the above two lines then depend solely on the alignment of the lensing systemand the density profile of the lens. As usual, we can see that the SIS case, n = 2, simplifies the above expressionsconsiderably.Figure 7 shows the total magnification of the power-law lens, which sums over the individual magnification of allimages: µ Tot = (cid:88) i (cid:16)(cid:12)(cid:12)(cid:12) det A (cid:12)(cid:12) θ i (cid:12)(cid:12)(cid:12)(cid:17) − (47) ≡ (cid:16)(cid:12)(cid:12)(cid:12) det A (cid:12)(cid:12) θ + (cid:12)(cid:12)(cid:12)(cid:17) − + (cid:16)(cid:12)(cid:12)(cid:12) det A (cid:12)(cid:12) θ − (cid:12)(cid:12)(cid:12)(cid:17) − . (48)For our fiducial case of n = 2 and A = 0 .
95 the total magnification is just under 40. This is split roughly evenlybetween the two images, though the image inside the Einstein radius is slightly brighter: µ + (cid:39) . µ − (cid:39) A = 1, as we expect the number of images to changeat critical curves in the image plane (such as θ E = 1). However, the constraints of eqs.(30) imply that in the SIScase the two images are equally displaced from θ E , so we can think of A > A µ n=1.5n=2.0n=3.0 FIG. 7. The total magnification (summed over both images) for the power-law lens. A = 1 corresponds to a perfect Einsteinring system; in the geometric optics limit, the magnification of a point source then becomes infinite. This breakdown signalsthe need for a wave optics treatment. Note that the magnification factor depends solely on A and n . outermost image instead of the innermost one. The corresponding magnification plot would then just be a reflectionof Fig. 7 about the axis A = 1. We note in passing that the magnification values discussed here are comparable tothe recent detection of SN iPTF16geu, the first multiply-imaged Type Ia SN, with a total magnification µ ∼
56 [54].For massive particles, the relevant observable is the specific particle intensity, measured in units of m − s − J − sr − (though the sr − is irrelevant for an effective point source), or more conveniently the flux. This too will be boostedby a magnification factor very similar to that of the GWs and photons. Any small differences in the magnificationfactor of non-null particles compared to null ones are likely to be dominated by uncertainties in the particle luminosityof the source; hence, to a first approximation, we can assume massless and massive particles to experience the samemagnification factor.Dai et al. [55] have emphasized that for LIGO there exists a degeneracy between a lensed GW from a high-redshift,low-mass source and an unlensed GW from a low-redshift, high-mass source. They argue that in the lensed case theGW echo will be registered as a separate GW event, and hence will not be of use in breaking the degeneracy. We notethat, in the more futuristic scenario considered here, detection of the corresponding massive echo could in principlehelp to confirm candidate lensed GWs and hence break this degeneracy. IV. SYSTEMATICS & SUBTLETIES
Thus far, our calculations and discussion have been based around idealized toy models. We stress that this paperis intended to be a largely theoretical discussion of an interesting phenomenon in fundamental physics, and is not anobservational call to arms. Nevertheless, in this section we will discuss some confounding factors that would requirecareful treatment in a futuristic attempt to measure the differential massive time delay.
1. Source Redshifts
Our expression for the differential massive time delay (eq.19) depends on the redshifts of the source and lens. Thelens redshift is expected to be measurable from an electromagnetic counterpart (e.g. a massive galaxy cluster at lowredshift), but the same is not necessarily true of the source redshift. In the case of an asymmetric supernovae this is ∼ − of the NS-NS merger GWsignals we receive should be accompanied by a GRB [56].In the absence of an electromagnetic counterpart, and with current ground-based detectors, the mass and redshift ofa compact merger are famously degenerate. Specifically, the waveform constrains only the redshifted mass combination M z = M (1 + z ). As we discussed in § III D, this degeneracy is preserved even in the case of gravitationally lensedGWs.However, the authors of [56] have identified a method to break this degeneracy that may be achievable by futureGW detectors, at least in the NS-NS merger case. In brief, the method relies on studying two effects in the GWwaveform: i) corrections to the orbital phase due to tidal effects during the inspiral stage, and ii) prominent spectralfeatures during the post-merger HMNS phase. These two effects have different dependencies on the true total massof the binary and its redshift, leading to near-orthogonal contours in the (
M, z ) parameter plane (see Fig. 1 of [56]).Assuming progress in determining the NS equation of state, the authors of [56] report a mass determination within1% accuracy for all the cases they considered. It is reasonable to speculate that when/if experiments become sufficientlymature to measure the differential massive time delay, either a) the redshift of NS-NS mergers will be measurablefrom GWs alone, or b) optical counterparts will be frequently available.
2. Lens Identification
Though it may not be necessary to have an electromagnetic counterpart for the source, it is essential for the lens.Inferring the lens mass distribution from sheared optical images is now a mature field [57], and is crucial to stepbeyond the simple analytic density prescriptions used in the present work.Furthermore, the optical lens image will help with sky localization of the lensed GWs and neutrinos. For example,the Large Synoptic Survey Telescope (LSST [58]) is expected to detect cluster-scale strong lenses at roughly 0 . − . − [59], whilst a future GW detector network of LIGO+VIRGO+KAGRA [60, 61] should be able to localizeGW sources to less than 10 sq. degrees [62]. Hence the combined observations should be able to pin down a cluster-scale lens to within a handful of candidates. In the case of more than one candidate lens, estimates of their redshiftsand masses will be necessary to further ascertain which belongs to the system of interest. However, once the correctlens has been identified, the source of the lensed GWs and neutrinos can then be localized to broadly lie within thearea occupied by the optical images, at most tens of arcseconds. Triangulation using a network of neutrino detectorsis not expected to provide significant enhancements over the localization capabilities of a single neutrino detector [63],making the process of lens identification particularly crucial.The influence of multiple lensing events would also need to be modelled. In this paper we have considered a situationin which the multi-messenger signals encounter only a signal, large lens during their propagation. In reality, they arelikely to additionally encounter many smaller weak lensing events by intervening matter structures [7, 30, 64], andsome of these will vary over the spatial region spanned by our effective images. This will add some intrinsic scatterto the arrival time of the multi-messenger echoes.In addition, one should also model the Shapiro delay induced by the gravitational potential well of the Galaxy[65, 66]. Whether uncertainties in these effects eradicate the differential massive time delay signal requires a detailedstudy [4].
3. Detectors & Count Rates
Solar and atmospheric neutrino backgrounds are an obvious systematic for detection of the lensed neutrino signalsdiscussed here; for example, the dominant solar neutrino background at ∼
10 MeV is a flux of roughly 5 × cm − s − from the decay of Boron-8. Fortunately, extracting neutrino candidates of interest from underneath these backgroundsis a well-studied topic [67]. The solar neutrino background can be minimized by concentrating on neutrino eventswith energies (cid:38)
20 MeV, though this comes at the expense of a lowered count rate from our sources of interest. Wenote in passing that the SuperKamiokande collaboration were able to identify and dismiss four neutrino events thatwere candidates for association with the gravitational wave events GW150914 and GW151226 [68].Our method for cancelling out the unknown intrinsic source delay ( § III A) should work well in the case of well-sampled fluxes, e.g. if the emissions occurs as short, intense bursts. If the emissions occur over a longer timescale andis poorly sampled, there will be difficulties in recovering the matching shapes of the original signal and its echo(es).7Not only does this make successful identification of the massive echo less likely, but it will also introduce an additionalsource of error in the measurements of times t b and t d , and hence into T .Undoubtedly low neutrino flux counts are a major obstacle for the extragalactic sources we have talked about inthis paper. It may be that the differential massive time delay will only ever be measurable for Galactic sources, forwhich up to ∼ neutrino events are expected with future kiloton-scale detectors. Local-group sources should alsobe within reach: a future neutrino experiment of several hundred kilotons is expected to detect a few dozen neutrinosfrom an event in the Andromeda galaxy [69]. For these sources there is no possibility of constraining cosmologicalparameters, but arguably any bounds on the neutrino mass should be cleaner without such degeneracies. Beyondthe Local Group begins a battle between the flux scaling as 1/distance and the number of sources increasing asapproximately distance . We will not attempt a prediction of the expected outcome in the present work.Note that the typical neutrino energies considered in this paper ( ∼
10 MeV) are below the detection thresholdof some current major detectors such as IceCube and ANTARES [70, 71], though SuperKamiokande, KamLANDand the Sudbury Neutrino Observatory all have thresholds of order a few MeV [72–74]. Whilst GRBs associated tocompact object mergers may produce high-energy neutrinos ( ∼ TeV – PeV) detectable by IceCube and ANTARES,the differential massive time delay associated to these will be miniscule. Still, detection of such high-energy neutrinosmay help with on-sky source localization for the lower-energy neutrino counterparts [75].Given the magnitude of the differential massive time delay ( ∼ µ s), it will be necessary to know the distancebetween all detectors involved in the measurement to within a metre or so (simply considering the magnitude of c T ).Based on current and improving GPS sensitivities, this should not prove problematic. V. DISCUSSION
We have begun here an exploration of what might be learned from extreme but rare astrophysical events byfuture observatories detecting massless carriers, such as photons or gravitational waves, as well as massive ones,such as neutrinos or other neutral particles. In particular, we have considered the information provided by stronggravitational lensing of such signals by large massive bodies close to the line of sight to such events.We can draw an analogy between the present status of GW astronomy and the early days of CMB detection. Inthe year 2000, shortly after the first CMB acoustic peak was detected by BOOMERANG and MAXIMA [76, 77], itwould have seemed absurdly optimistic to consider measuring the eigth acoustic peak with the precision now achievedby the Planck satellite. Yet, thanks to continual improvements in detector technology and data analysis techniques,intervening experiments (WMAP [78]), and increasingly sophisticated understanding of systematics, such exquisitemeasurements are now possible. Similarly, though measuring the effects discussed in this paper is unfeasible withpresent understanding and experiments, we envision an equally rapid progression for the forthcoming decade(s) ofmulti-messenger astronomy. One of our goals here is to stimulate forward-thinking about the novel science that mightbe possible in future.To this end we have derived an expression for the differential arrival time of massive and massless particles with acommon origin. The resulting expression is sensitive to particle properties, cosmological parameters, and the massesand separations of elements in the lensing system. Though we have only evaluated the magnitude of this correction forsimplified lens models, it could be applied to real lensing systems whose mass distribution is relatively well-constrained.In the examples studied here, the differential time delay is found to have a magnitude of order 0 . µ s. We note apowerfully general paper by Fleury [79] offers explanation as to why time delay effects remain small, even when theangular deflection and magnification of messengers is substantial.Neutrino detectors are already capable of measuring intervals of order 0 . µ s, having at present a time resolutiondown to 100 ns. GW detectors lag behind somewhat – the timing resolution of the LIGO detectors is currently atthe order of 100 µ s [80, 81]. However, with the broader frequency ranges proposed for next-generation detectors likethe Einstein telescope, and improvements in detector technology, it seems reasonable to speculate that the necessaryprecision might be available to future GW experiments.In addition, we note that a key feature of the differential massive time delay is a near-coincident feature in multiplemedia and multiple detectors. Although clearly this requires a global coordination of experiments, the simultaneousnature of events should assist with the selection and rejection of candidate detections.For ease of discussion, we have generally referred to extragalactic merging neutron stars as a candidate source.However, we remind the reader that – under the right conditions – NS-BH mergers, BH-BH mergers, and asymmetricsupernovae are all potential alternative candidates. Likewise, we have generally focussed on GWs as the relevantmassless messenger; photons can also take this role (and would arguably be easier to work with), if one is certain thatprompt emission is being detected.We have not attempted here a forecast of the constraints attainable on neutrino masses or cosmological parametersfrom measurements of the differential massive time delay. To do so would require a detailed description of the8experiments involved; we suspect that instruments with the required sensitivity are not even at the blueprint stageyet. Although designs for the LISA observatory [82] are progressing rapidly, the most promising sources listed abovefall outside of its frequency range.Even if the differential massive time delay is not within foreseeable experimental reach for any source types, or infact will always be lost to source uncertainties and systematics, the lensing of massive particles seems an inherentlyinteresting and under-explored counterpart to the extensive research in optical lensing (and the much less-studiedfield of GW lensing).Several logical extensions of the present paper would be: • A detailed study of the joint redshift distributions of candidate sources and massive galaxy clusters, resultingin an estimate for the number of simultaneous GW-neutrino lensing systems in principle detectable at Earth; • Further investigation into the projected sensitivities of future GW and neutrino detectors, their timing resolu-tions and sky localization errors, particularly when operating as a network; • A more sophisticated treatment of the massive particle flux expected from the sources listed in the introduction,accounting for the energy spectrum of different species and the effects of poor sampling of the burst at detection; • The possibility that stacking signals from separate events could alleviate the issue of low count rates fromextragalactic sources. For example, one might imagine stacking signals from all systems that have a source andlens in the same redshift bin. This might allow a first measurement of the extragalactic differential massive timedelay, even if the stacking technique means that constraining fundamental parameters is not possible; • A consideration of the further differencing opportunities presented by lensing systems with more than two images( § III C); • Investigation into the potential of the differential massive time delay and related phenomena to constrain theviolation of C,P and CP symmetries in gravity.We hope to take up some of these questions in future work.
ACKNOWLEDGMENTS
TB is would like to thank the Center for Particle Cosmology at the University of Pennsylvania, where most of thiswork was carried out, for their kind hospitality. TB is supported by an award from the US-UK Fulbright Commissionand All Souls College, Oxford. The work of MT was supported in part by NASA ATP grant NNX11AI95G and byUS Department of Energy (HEP) Award DE-SC0013528. We are grateful for useful discussions with Gary Bernstein,Alessandra Buonanno, Malcolm Fairbarn, Eanna Flanagan, Bhuvnesh Jain, Joshua Klein, Eugene Lim, Julian Merten,John Miller, Chris Moore, Uros Seljak, Ulrich Sperhake, Leo Stein and Aprajita Verma.
Appendix A: Unlensed Massive Time Delay
Here we provide a basic calculation of the difference between the propagation time of massive and massless particles inthe absence of any lensing effects. See [7] for a calculation that incorporates the effects of cosmological perturbations.We begin from the familiar flat FRW line element: ds = − d t + a ( t ) (cid:2) d r + r (cid:0) d θ + sin θ d φ (cid:1)(cid:3) . (A1)For pure radial motion, the timelike component of the geodesic equation isd t d λ + a ˙ a (cid:18) d r d λ (cid:19) = 0 , (A2)where λ is an affine parameter. The normalization of the four-velocity for a massive particle gives us u µ u µ = − (cid:18) d t d λ (cid:19) + a (cid:18) d r d λ (cid:19) = − , (A3)9and combining the two equations above to eliminate d r/ d λ , we obtaind t d λ + ˙ aa (cid:34)(cid:18) d t d λ (cid:19) − (cid:35) = 0 . (A4)Integrating this leads to (where C is a constant): (cid:18) d t d λ (cid:19) − Ca . (A5)We multiply this by m and use the definition of four-momentum P µ = mu µ = m d x µ / d λ , P = E to yield E − m ≡ g ij P i P j = C m a , (A6)from which we see that the magnitude of the spatial three-momentum redshifts as 1 /a : p = (cid:113) g ij P i P j = √ C ma . (A7)We will need this result shortly. Now, to find the time taken for a neutrino to travel a cosmological distance, eq.(A3)can be rewritten as: 1 − (cid:18) d λ d t (cid:19) = 1 − m E = a (cid:18) d r d t (cid:19) (A8) ⇒ d r d t = √ E − m a E . (A9)Using our result from eq.(A7), we can write p = a p /a where a and p are defined at some fixed time, so thatd r d t = pa (cid:112) m + p = a p a (cid:113) m + a a p . (A10)For convenience we define y = a p , in terms of which the above expression then rearranges as:d r = (cid:34) a (cid:115) a m y (cid:35) − d t . (A11)Integrating this expression would give us the conformal distance travelled by a massive particle with initial spatialmomentum p in a time interval t .Now we relate this to the analogous, simpler expression for a massless particle; for linguistic convenience we willrefer to a photon, but our results apply equally to GWs. We know that the total conformal distances travelled by thephoton and the neutrino are the same. In principle, the physical distances travelled by the photon and the massiveparticle are different, since the universe continues to expand during the small time interval between their arrival atEarth. Equating the conformal distances, then: r = (cid:90) t ν t (cid:113) a m y dta ≡ (cid:90) t γ t dta , (A12)where t is the (idealized) simultaneous time of emission, t γ is the time the photon arrives at Earth, t ν is the time ofarrival of the massive particle, and all times correspond to those measured by a comoving observer.For all the examples discussed in this paper, the mass of the particle is much smaller than its initial energy. Thiscorrespondingly implies m (cid:28) y , such that: (cid:34) (cid:18) amy (cid:19) (cid:35) − ≈ − (cid:18) aa mp (cid:19) + O (cid:20) m p (cid:21) . (A13)0 z δ t ( s ) FIG. 8. The difference in arrival time between an unlensed massless particle (photon or gravitational wave) and unlensedmassive particle with m = 0 . E = 10 MeV, as a function of source redshift. Even for high-redshift sources, the differencein arrival times remains less than a second. Substituting this into eq.(A12) yields r = (cid:90) t ν t (cid:34) − (cid:18) aa mp (cid:19) (cid:35) dta ≡ (cid:90) t γ t dta . (A14)This integral can of course be evaluated exactly. However, for analytic purposes it is helpful to use the fact that t γ (cid:39) t ν − δt . Then the RHS can be written: (cid:90) t ν − δtt dta = (cid:90) t ν t dta − (cid:90) t ν t ν − δt dta (A15) ≈ (cid:90) t ν t dta − a ( t ν ) δt , (A16)where in the second line we have used the fact that the time interval δt is very small compared to the cosmologicalexpansion time. Using this in eq.(A14) and cancelling terms on either side: (cid:90) t ν t (cid:34) − (cid:18) aa mp (cid:19) (cid:35) dta = − a ( t ν ) δt (A17)Normalizing the scale factor such that a ( t ν ) = 1 today and converting the integral to be with respect to redshift, weobtain δt = 12 (cid:18) mp (cid:19) (cid:90) z (cid:18) z z (cid:19) dzH ( z ) , (A18)where z is the source redshift. To the accuracy that we are working here, we can take p ≈ E in the above expression,where E is the initial energy of the particle.The resulting difference in arrival time between a massless particle and one with m = 0 . E = 10 MeV is shown in Fig. 8. As can be seen, even for high-redshift sources the difference in arrivaltimes remains of order a second.1 d ɑ ɸɸ * r R min ɸ v ∞ v ɸ FIG. 9. Diagram indicating the quantities needed for the calculation of Appendix B.
Appendix B: Lensing of Massive Particles by a Point Mass
In this appendix we derive the modification to the well-known formula for the lensing of a massless particle by a pointmass M , i.e. α = GMc , for a massive particle. The classic derivation for the massless case can be found in manyintroductory GR texts, e.g. [83]. Thanks to the two-dimensional equivalent of Birkhoff’s theorem, the result extendsto any axially symmetric mass distribution interior to the trajectory of the lensed particle. Extended lenses thatencompass the particle trajectory (e.g. a galaxy cluster) require a more sophisticated treatment, though the essentialconclusions of this section remain the same.As in § II, we can can account for both massless and massive particle cases by writing the normalization of thefour-velocity as: u µ u µ = − (cid:15) (B1)and specifying (cid:15) = 1 for a massive particle, (cid:15) = 0 for a photon, say. Expanding the above expression in a Schwarzchildmetric and setting G = c = 1: − (cid:18) − Mr (cid:19) (cid:18) dtdλ (cid:19) + (cid:18) − Mr (cid:19) − (cid:18) drdλ (cid:19) + r (cid:18) dφdλ (cid:19) = − (cid:15) , where λ is an affine parameter and we have chosen the motion of our lensed particle to be in the θ = π/ (cid:126)ξ and (cid:126)χ to find the usual energy and angularmomentum conserved quantities: (cid:126)ξ = { , , , } (cid:126)χ = { , , , } (B2) e = − (cid:126)ξ · (cid:126)u = (cid:18) − Mr (cid:19) dtdλ (B3) (cid:96) = (cid:126)χ · (cid:126)u = r dφdλ . (B4)For a massive particle, (cid:126)u is the four-velocity. For a photon, one can choose the affine parameter such that (cid:126)u coincideswith the four-momentum of the photon. Substituting these conserved quantities into eq.(B2) and rearranging, weobtain: (cid:18) drdλ (cid:19) = e − (cid:18) − Mr (cid:19) (cid:18) (cid:96) r + (cid:15) (cid:19) . (B5)Let us replace e by something directly measurable (in principle). When the particle is at its closest approach to thelensing object we have dr/dλ = 0, leading to: e = (cid:18) − MR min (cid:19) (cid:18) (cid:96) R + (cid:15) (cid:19) , (B6)where R min is the distance of closest approach. Note that when (cid:15) = 0 we recover the fact that only the ratio e/(cid:96) ismeasurable for a photon. Substituting the above expression back into eq.(B5), and dividing the result by eq.(B4):21 r (cid:18) drdφ (cid:19) = 1 (cid:96) (cid:34)(cid:18) − MR min (cid:19) (cid:18) (cid:96) R + (cid:15) (cid:19) − (cid:18) − Mr (cid:19) (cid:18) (cid:96) r + (cid:15) (cid:19) (cid:35) . (B7)Changing variables to u = R min /r , inverting and rearranging gives dφdu = (cid:20)(cid:18) − MR min (cid:19) (cid:18) (cid:15) R (cid:96) (cid:19) − (cid:18) − M uR min (cid:19) (cid:18) u + (cid:15) R (cid:96) (cid:19)(cid:21) − (B8)= (cid:20) − u − MR (cid:18) (cid:15) R (cid:96) (1 − u ) − u (cid:19)(cid:21) − (B9)= (1 − u ) − (cid:20) − MR min (1 − u ) − (cid:18) (cid:15) R (cid:96) (1 − u ) − u (cid:19)(cid:21) − . (B10)Another change of variables, u = cos α , then yields dφ = − dα (cid:20) − MR min − cos α ) (cid:18) (cid:15) R (cid:96) (1 − cos α ) − cos α (cid:19)(cid:21) − . (B11)Using the identity 1 − cos α − cos α ≡ (1 − cos α )(1 + cos α + cos α )(1 − cos α )(1 + cos α ) (B12) ≡
11 + cos α + cos α (B13)to simplify the integrand, eq.(B11) then becomes: dφ = − dα (cid:34) − MR min (cid:32) cos α + (1 + (cid:15) R (cid:96) )1 + cos α (cid:33)(cid:35) − . (B14)For all the situations discussed in this paper, the lensed particles remain far from the Schwarzchild radius of the lens.Therefore we can perform a Taylor expansion in the small parameter 2 M/R min ≡ r S /R min (cid:28) r S is theSchwarzchild radius), obtaining dφ ≈ − dα (cid:34) MR min (cid:32) cos α + (1 + (cid:15) R (cid:96) )1 + cos α (cid:33)(cid:35) + O (cid:18) M R (cid:19) . (B15)In fact, in the case of the massive particle, an additional assumption is needed for the Taylor expansion performedabove to remain valid: that the ratio R min /(cid:96) does not grow very large. In flat space (cid:96) has the interpretation ofthe angular momentum per unit rest mass. Hence can roughly estimate (cid:96) ∼ dv , where d is the impact parameterbetween the particle and the lens and v is the initial 3-velocity of the particle. Temporarily re-instating factors of c for dimensional clarity, we require: R c (cid:96) ∼ R d c v (cid:46) . (B16)For small deflection events, the impact parameter b and distance of closest approach R min are comparable in magnitude.So our condition for the Taylor expansion to be valid then reduces to c /v (cid:46)
1. Since c/v < v ∼ c . Note this somewhat unusual situation – the results that follow here are only valid for particles that are at least moderately relativistic.Integrating from α = π/ α = 0 corresponds to moving along the particle trajectory from infinity to its closestapproach to the central mass. By the symmetry of the particle’s approach and retreat, the deflection angle α will3then be 2 φ ∗ − π (see Fig. 9) φ ∗ ≈ (cid:90) π dα (cid:34) MR (cid:32) cos α + (1 + (cid:15) R (cid:96) )1 + cos α (cid:33)(cid:35) + O (cid:18) M R (cid:19) (B17)= (cid:20) α + MR sin α + MR (cid:18) (cid:15) R (cid:96) (cid:19) tan (cid:16) α (cid:17)(cid:21) π (B18)= π MR (cid:18) (cid:15) R (cid:96) (cid:19) , (B19) ⇒ α = 4 MR (cid:18) (cid:15) R (cid:96) (cid:19) + O (cid:18) M R (cid:19) . (B20)Recall the definition of (cid:96) (eq.B4) is in terms of the affine parameter λ , which is equivalent to the proper time τ fora massive particle. Since (cid:96) is a constant, we can choose to evaluate it anywhere along the particle trajectory. Forconvenience, we choose to do this at a location that is a) sufficiently far from the lens that we can neglect the potentialwell Φ at leading order, but b) close enough not to be separated from the lens by a cosmological distance (i.e. ∼ Gpc). We approximate this intermediate regime as Minkowski space, and use it to link the particle motion in thelarge-scale FRW space to its local motion near the lens.In this Minkowski patch, the particle proper time and the time measured by an observer at rest with respect tothe lens are related by dτ = dt/γ ( v ), where v is the particle velocity (constant in the patch). Then we have (withreference to Fig. 9): (cid:96) = r dφdτ = γr dφdt = γrv φ = γrv sin φ (B21)= γrv (cid:18) dr (cid:19) = γv d , (B22)where v φ is the velocity component in the azimuthal direction. Now using this in eq.(B20): α = 4 GMR min c (cid:20) (cid:15) R d (cid:18) c v − (cid:19)(cid:21) , (B23)where we have reinstated factors of G and c . One can show fairly easily (we do not do so here for brevity) that thedifference between R min and d is a number of order r S /R min , and hence, to the order at which we are working, wecan re-write the above as: α = 4 GMdc (cid:20) (cid:18) c v − (cid:19)(cid:21) + O (cid:18) r S R (cid:19) . (B24)Note that we recover the standard result for a massless particle in the limit v → c (so we do not need the (cid:15) parameterany more). We remind the reader that we specialized to relativistic particles in eq.(B16), so this expression is notvalid in the limit v → (cid:126)α ( (cid:126)θ ) = 1 π (cid:90) d θ (cid:48) κ ( (cid:126)θ (cid:48) ) (cid:20) (cid:18) c v − (cid:19)(cid:21) (cid:32) (cid:126)θ − (cid:126)θ (cid:48) | (cid:126)θ − (cid:126)θ (cid:48) | (cid:33) , (B25)where (cid:126)θ is the angular position in the lens plane, the integral is taken over the entire lens, and κ ( (cid:126)θ ) is the dimensionlesssurface mass density, to be defined in the next appendix.From eqs.(B24) and (B25), we can see that the standard formulae for deflection of photons incur a small correctionsensitive to the velocity (equivalently, the mass and momentum) of a massive particle. One therefore might expectnull and non-null effective images of the same source to be slightly misaligned in the sky. However, as explainedin § II, the effect of this misalignment on the differential massive time delay can be neglected to the accuracy usedthroughout this paper.4
Appendix C: Lensing by a Mass Distribution
Some readers of this paper may be unfamiliar with the formalism of strong gravitational lensing; here we provide abrief summary of some of the standard definitions. Further details and excellent pedagogical introductions may befound in [34, 51, 53].The expressions here make use of the thin-lens approximation. To simplify their formulation, we will assume axialsymmetry about the optical axis (the line connecting the observer to the centre of the lens). More general, vectorialversions can be found in the references above.As discussed in § II A and derived in Appendix B, in general the deflection angle experienced by a massive particleis slightly different to that experienced by a massless particle. However, this correction only becomes relevant atorder ( mc/p ) , and so is not needed for the present work. Hence all expressions in this appendix relate to lensing ofmassless particles.We start with a lens model with three-dimensional density profile ρ ( (cid:126)r ). Under the thin-lens approximation weproject this onto a surface at z = z L [84]. The projected surface mass density is:Σ( (cid:126)ξ ) = (cid:90) dr ρ ( (cid:126)r ) , (C1)where (cid:126)r = (cid:110) (cid:126)ξ, r (cid:111) is a 3D position vector centred on the lens that can be decomposed into a component r along theoptical axis, and a 2D position vector in the lensing plane, (cid:126)ξ (see Fig. 1).One can easily show that the vectorial deflection resulting from a 2D distribution of mass elements is [34]: (cid:126)α ( (cid:126)ξ ) = 4 Gc (cid:126)ξξ (cid:34) π (cid:90) ξ dξ (cid:48) Σ( ξ (cid:48) ) ξ (cid:48) (cid:35) , (C2)where, for example, ξ is the magnitude of (cid:126)ξ . The square bracket gives the mass contained within a radius ξ in thelensing plane. Eq.(C2) is loosely comparable to the standard formula for deflection by a point mass with impactparameter b ; α = 4 GM/bc . The prefactor of (cid:126)ξ/ξ is analogous to the factor 1 /b , but also indicates that the deflectionis towards the centre of the lens.The quantity appearing in the lensing equation (15) is in fact the ‘scaled deflection angle’, ( D LS /D S ) α . To obtainthis, we take the magnitude of the equation above and replace the 2D position vectors by angular positions using ξ = D L θ : α scal ( θ ) = D LS D S α ( θ ) (C3)= D LS D L D S πGc θ (cid:34) (cid:90) θ dθ (cid:48) Σ( θ (cid:48) ) θ (cid:48) (cid:35) (C4)= 1 θ (cid:34) (cid:90) θ dθ (cid:48) κ ( θ (cid:48) ) θ (cid:48) (cid:35) , (C5)where the convergence κ ( θ ) and critical density Σ cr are defined as κ ( θ ) = Σ( θ )Σ cr Σ cr = c πG D S D L D LS . (C6)One final simplification is helpful. The second square bracket in eq.(C5) is, up to a factor of π , the dimensionlessmass of the lens contained within angular radius θ (the dimensions having been removed by Σ cr in the denominatorof κ ). Defining the dimensionless mean surface mass density by¯ κ ( θ ) = M ( < θ ) πθ , (C7)eq.(C5) then becomes α scal ( θ ) = ¯ κ ( θ ) θ . (C8)5Finally, recall that the scalar form of the lens equation is α scal ( θ ) = θ ± β . (C9)In this way the factors of θ − β (and similar) that appear in our calculations can be calculated from ρ ( r ). Notealso that the projected 2D potential ψ ( θ ) can be related to the density profile via (cid:126)α scal ( θ ) = ˜ ∇ θ ψ ( θ ), where ˜ ∇ θ is aderivative in the lensing plane. [1] B. P. Abbott et al. , Physical Review Letters , 061102 (2016).[2] B. P. Abbott et al. , Physical Review Letters , 241103 (2016).[3] L. Stodolsky, Physics Letters B , 61 (2000), astro-ph/9911167.[4] L. Stodolsky, in Neutrino Physics - Its Impact on Particle Physics, Astrophysics and Cosmology , edited by J. Bahcall,W. Haxton, K. Kubodera, and C. Poole (2001) pp. 14–20, astro-ph/0006384.[5] G. I. Zatsepin, Soviet Journal of Experimental and Theoretical Physics Letters , 205 (1968).[6] K. Hirata, T. Kajita, M. Koshiba, M. Nakahata, Y. Oyama, N. Sato, A. Suzuki, M. Takita, Y. Totsuka, T. Kifune, T. Suda,K. Takahashi, T. Tanimori, K. Miyano, M. Yamada, E. W. Beier, L. R. Feldscher, S. B. Kim, A. K. Mann, F. M. Newcomer,R. Van, W. Zhang, and B. G. Cortez, Phys. Rev. Lett. , 1490 (1987).[7] G. Fanizza, M. Gasperini, G. Marozzi, and G. Veneziano, Physics Letters B , 505 (2016), arXiv:1512.08489.[8] X. Li, Y.-M. Hu, Y.-Z. Fan, and D.-M. Wei, Astrophys. J. , 75 (2016), arXiv:1601.00180 [astro-ph.HE].[9] K. S. Thorne, “Gravitational radiation.” in Three Hundred Years of Gravitation , edited by S. W. Hawking and W. Israel(1987) pp. 330–458.[10] A. A. Ruffa, The Astrophysical Journal , L31 (1999).[11] Y. Wang, A. Stebbins, and E. L. Turner, Physical Review Letters , 2875 (1996).[12] M. Sereno, P. Jetzer, A. Sesana, and M. Volonteri, Monthly Notices of the Royal Astronomical Society , 2773 (2011).[13] M. Sereno, A. Sesana, A. Bleuler, P. Jetzer, M. Volonteri, and M. C. Begelman, Phys. Rev. Lett. , 251101 (2010).[14] A. Pi´orkowska, M. Biesiada, and Z.-H. Zhu, JCAP , 022 (2013), arXiv:1309.5731.[15] S. Nissanke, M. Kasliwal, and A. Georgieva, Astrophys. J. , 124 (2013), arXiv:1210.6362 [astro-ph.HE].[16] J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. Abbott, M. R. Abernathy, T. Accadia, F. Acernese, C. Adams, T. Adams,and et al., The Astrophysical Journal Supplement , 7 (2014), arXiv:1310.2314 [astro-ph.IM].[17] Y. Sekiguchi, K. Kiuchi, K. Kyutoku, and M. Shibata, Phys. Rev. Lett. , 051102 (2011).[18] F. Foucart, E. O’Connor, L. Roberts, L. E. Kidder, H. P. Pfeiffer, and M. A. Scheel, ArXiv e-prints (2016),arXiv:1607.07450 [astro-ph.HE].[19] O. L. Caballero, G. C. McLaughlin, and R. Surman, Phys. Rev. D , 123004 (2009), arXiv:0910.1385 [astro-ph.HE].[20] O. L. Caballero, T. Zielinski, G. C. McLaughlin, and R. Surman, Phys. Rev. D , 123015 (2016), arXiv:1510.06011[nucl-th].[21] K. N. Yakunin, P. Marronetti, A. Mezzacappa, S. W. Bruenn, C.-T. Lee, M. A. Chertkow, W. R. Hix, J. M. Blondin, E. J.Lentz, O. E. B. Messer, and S. Yoshida, Classical and Quantum Gravity , 194005 (2010), arXiv:1005.0779 [gr-qc].[22] C. D. Ott, E. P. O’Connor, S. Gossan, E. Abdikamalov, U. C. T. Gamma, and S. Drasco, Nuclear Physics B ProceedingsSupplements , 381 (2013), arXiv:1212.4250 [astro-ph.HE].[23] H. Andresen, B. Mueller, E. Mueller, and H.-T. Janka, (2016), arXiv:1607.05199 [astro-ph.HE].[24] R. M. Quimby, M. C. Werner, M. Oguri, S. More, A. More, M. Tanaka, K. Nomoto, T. J. Moriya, G. Folatelli, K. Maeda,and M. Bersten, The Astrophysical Journal Letters , L20 (2013), arXiv:1302.2785.[25] R. M. Quimby, M. Oguri, A. More, S. More, T. J. Moriya, M. C. Werner, M. Tanaka, G. Folatelli, M. C. Bersten, K. Maeda,and K. Nomoto, Science , 396 (2014), arXiv:1404.6014.[26] P. L. Kelly, S. A. Rodney, T. Treu, R. J. Foley, G. Brammer, K. B. Schmidt, A. Zitrin, A. Sonnenfeld, L.-G. Strolger,O. Graur, A. V. Filippenko, S. W. Jha, A. G. Riess, M. Bradac, B. J. Weiner, D. Scolnic, M. A. Malkan, A. von der Linden,M. Trenti, J. Hjorth, R. Gavazzi, A. Fontana, J. C. Merten, C. McCully, T. Jones, M. Postman, A. Dressler, B. Patel,S. B. Cenko, M. L. Graham, and B. E. Tucker, Science , 1123 (2015), arXiv:1411.6009.[27] The Newtonian intuition that a massive particle should experience greater deflection that a massless one – because it’sslower speed implies less momentum to ‘escape’ – turns out to be correct.[28] We will see in § II B that we are effectively doing a doing a double Taylor expansion here, in the small parameters δθ and m/p .[29] This may seem counterintuitive initially. If the source is at a cosmological redshift, surely the expansion of the universehas a significant affect on its travel time? How, then, can the conformal time delay be equivalent to the physical one? Thenuance here is that cosmological expansion affects all lensed paths in the same way, so does not contribute to their relativedifference at z = 0. That is, t total ( θ ) (cid:54) = η total ( θ ), but ∆ t ( θ , θ ) (cid:39) ∆ η ( θ , θ ) to a high degree of accuracy.[30] U. Seljak, Astrophys. J. , 509 (1994), astro-ph/9405002.[31] Note that a more correct way to describe this situation would be to use the McVittie metric instead. However, in the regimewe are considering – far outside the gravitational radius of the lens – the McVittie metric reduces to a form equivalent toeq.(II B).[32] This Euclidean notion of distance is justified since we are working in the thin-lens approximation and on a conformal grid. [33] D. Baumann, “Cosmology Lecture Notes, Cambridge Mathematical Tripos.”.[34] P. Schneider, ArXiv Astrophysics e-prints (2005), astro-ph/0509252.[35] R. Blandford and R. Narayan, The Astrophysical Journal , 568 (1986).[36] S. K. Bose and W. D. McGlinn, Phys. Rev. D , 2335 (1988).[37] W. L. Burke, Astrophys. J. , L1 (1981).[38] W. K. Rose, Advanced Stellar Astrophysics, by William K. Rose, pp. 494. ISBN 0521581885. Cambridge, UK: CambridgeUniversity Press. (1998) p. 351.[39] L. V. E. Koopmans, A. Bolton, T. Treu, O. Czoske, M. W. Auger, M. Barnab`e, S. Vegetti, R. Gavazzi, L. A. Moustakas,and S. Burles, ApJ Letters , L51 (2009), arXiv:0906.1349 [astro-ph.CO].[40] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday,R. B. Barreiro, J. G. Bartlett, and et al., Astronomy & Astrophysics , A13 (2016), arXiv:1502.01589.[41] Expressed another way, neutrino oscillations mean that a delta-function emission burst will be received with some arrivaltime distribution. The same is true for the lensed echoes of the original signal. Our calculation essentially relies on accuratelydifferencing these distributions; their width is unimportant.[42] T. Treu and P. J. Marshall, ArXiv e-prints (2016), arXiv:1605.05333.[43] V. Bonvin, F. Courbin, S. H. Suyu, P. J. Marshall, C. E. Rusu, D. Sluse, M. Tewes, K. C. Wong, T. Collett, C. D. Fassnacht,T. Treu, M. W. Auger, S. Hilbert, L. V. E. Koopmans, G. Meylan, N. Rumbaugh, A. Sonnenfeld, and C. Spiniello, ArXive-prints (2016), arXiv:1607.01790.[44] J. Klein and S. Thorsett, Physics Letters A , 79 (1990).[45] I. H. Stairs, Living Reviews in Relativity (2003), 10.1007/lrr-2003-5.[46] P. G. Tinyakov and I. I. Tkachev, Soviet Journal of Experimental and Theoretical Physics , 481 (2008), astro-ph/0612359.[47] D. J. E. Marsh, Physics Reports , 1 (2016), arXiv:1510.07633.[48] G. G. Raffelt, in Axions , Lecture Notes in Physics, Berlin Springer Verlag, Vol. 741, edited by M. Kuster, G. Raffelt, andB. Beltr´an (2008) p. 51, hep-ph/0611350.[49] X.-L. Fan, K. Liao, M. Biesiada, A. Piorkowska-Kurpas, and Z.-H. Zhu, ArXiv e-prints (2016), arXiv:1612.04095 [gr-qc].[50] R. Barkana, Astrophys. J. , 531 (1998), astro-ph/9802002.[51] P. Schneider,
Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro , edited by G. Meylan,P. Jetzer, P. North, P. Schneider, C. S. Kochanek, and J. Wambsganss (Springer-Verlag: Berlin, 2006) astro-ph/0407232.[52] R. Takahashi, ArXiv e-prints (2016), arXiv:1606.00458.[53] P. Schneider, J. Ehlers, and E. E. Falco,