Testing Weak Equivalence Principle with Strongly Lensed Cosmic Transients

Current constraints on Einstein's weak equivalence principle (WEP) utilize the observed time delay between correlated particles of astronomical sources. However, the intrinsic time delay due to particle emission time is impossible to measure and is simply assumed to be zero in previous studies. Here we propose a robust method to test WEP for relativistic messengers using strongly lensed cosmic transients, which can naturally overcome the intrinsic time delay problem. This can be achieved in two ways: by comparing the time delays between lensed images seen in different energy bands or in gravitational waves (GWs) and their electromagnetic (EM) counterparts. The power of our method mainly depends on the timing accuracy of cosmic transient and the strong lensing time delay. If the time delay of cosmic transient can be measured with accuracy about 0.1 s (e.g. gamma-ray bursts), we show that the upper limit on the differences of the parameterized post-Newtonian parameter $\gamma$ value is $\Delta \gamma<10^{-7}$ with a one-month strong lensing time delay event. This accuracy of WEP can be improved by a factor of $10^{7}$, if the lens is galaxy cluster and the strongly lensed cosmic transients have much shorter duration, such as fast radio bursts.


Introduction
The Einstein's weak equivalence principle (WEP) is one of the cornerstone of general relativity. It states that the trajectories of any freely falling, uncharged test bodies are independent of their energy, internal structure, or composition [1]. In the theory of parameterized post-Newtonian (PPN), the WEP requires that the PPN parameter γ of different particles or the same type of particle with different energies (hereafter, "different particles" represents both of cases) should be the same. WEP has been tested by time delay of different a Corresponding author: fayinwang@nju.edu.cn particles. In 1964, [2] proposed that one could use the time delays between transmission of radar pulses towards either of the inner planets and detection of the echoes. The Shapiro time delay can be formulated as t Shapiro where r e and r 0 are the emitting position of the particle and the position of observer respectively, c is the speed of light, and Ψ (r) is the gravitational potential. If the WEP is violated, the arrive times of two different particles emitted simultaneously travel in a same gravitational potential will be different. The relative Shapiro time delay is given by Up to date, the observed time delays of different type particles (e.g. photons, neutrinos, or gravitational waves), or the same type of particles with different energies have been used to test WEP, such as the different arrival times of photons and neutrinos from SN1987A [3,4], the time delay of a PeV-energy neutrino event associated with a giant flare of the blazar PKS B1424-418 [5], the photons in different energy bands of gamma-ray bursts (GRBs) [6,7], polarized photons of GRBs [7], radio signals at different frequency bands of fast radio bursts (FRBs) [8][9][10] and the Crab pulsar [11], and gravitational wave (GW) sources [12][13][14][15]. However, the observed time delay should include several terms: where ∆t int is the intrinsic time delay between two particles due to different emission times, ∆t LIV is the time delay caused by Lorentz invariance violation, ∆t spec is the potential time delay due to photons have a rest mass, and ∆t DM is the time delay contributed by the dispersion of the line-ofsight free electrons.
In the high-energy range (from keV to GeV), ∆t DM can be negligible. However, most of previous works omitted all of other potential effects based on following assumptions. First, the observed time delay is mainly attributed to the gravitational potential so the time delays caused by other effects are omitted. Second, the time delays caused by other effects and ∆t gra have the same sign. So they can give an upper limit of the violation of WEP. These two assumptions are unreasonable. For the first assumption, in some cases, the ∆t int term would dominate the observed time delay especially when the observed time delay is very small, which leads to a non-physical constraint on the WEP. In addition, ∆t spec and ∆t LIV are strongly correlated with ∆t gra in the total observed time delay. For the second assumption, in a special case that the signs of ∆t other caused by other effects and ∆t gra are opposite (e.g. ∆t other = −0.99 s and ∆t gra = 1 s), a much smaller time delay is obtained. This will lead to a much tighter, but incorrect constraint on WEP. Therefore, the intrinsic time delay ∆t int and ∆t other can severely limit the ability of this method. To avoid these effects, we propose that the strongly lensed cosmic transients can be used to test WEP. This is the first time to correct the other effects in WEP constraint. The time delay due to strong gravitational lensing between different particles is a powerful tool, which has been used to test the Lorentz-invariance violation [16] and the speed of GW [17,18].

The Method and Constraints on WEP
Gravitational lens is a prediction of the general relativity. After the first observation example of gravitational lensing, the quasar QSO 0957+561A, B [19], it has become a powerful tool in many fields of astronomy, such as probing dark matter halo, large scale structures, Hubble constant, and parameters of universe. Generally, there will be multiple images of the source when it is strongly lensed. The differences of arrival times for images are caused by the Shapiro time delay and geometric delay due to the bending of light rays [20]. In a general model of the lens, the time delay of the images relative to the case that source, lens and image are on a straight line is where θ θ θ and β β β are the position vectors of the image and source, z l is the redshift of the lens, d l and d s are the angular diameter distances of the lens and source, d ls is the angular diameter distance from the lens to source and the ψ(θ θ θ ) = Ψ (d l θ θ θ , z)dz is the projected gravitational potential [20]. Actually, t(θ θ θ ) can be divided into t geo = 1+z l c d l d s d ls  object. S 1 and S 2 are the two signals which are associated with each other in this transient event, for example a GW and its electromagnetic (EM) counterpart or particles at two different energy bands in a cosmic explosion. d l , d s , d ls and β β β are the same as those in equation (3). θ θ θ 1 1 1 and θ θ θ 2 2 2 are the positions of two images formed by gravitational lens. P 1 and P 2 are the trajectories of the light rays of the two images. We assume that the intrinsic time delay between S 1 and S 2 is ∆t int (S 1 is earlier than S 2 ) and the first and second signals of S 1 arrive at t 11 and t 12 , respectively. Here we just assume that the ∆t spec and ∆t LIV can be omitted and discuss them in the following section. If the WEP is valid, one can expect that the first and second signals of S 2 will arrive at t 21 = t 11 + (1 + z s )∆t int and t 22 = t 12 + (1 + z s )∆t int , where z s is the redshift of the source. If there is some small violation of WEP, positions of the images have small changes δ θ 11 , δ θ 12 , δ θ 21 , δ θ 22 and the arrival time of the signals become into t 11 , t 12 , t 21 , and t 22 . Similar as [21], we perform the Taylor expansion of these new arrival times and consider the first order term Because the Fermat's principle, which we assume is still valid in this case, requires that the lensed image position makes the travel time stationary, the ∂t ∂ θ | θ 1 and ∂t ∂ θ | θ 2 are equal to 0 and all the second terms of the right parts of these equations must be vanished. Then comparing t 21 − t 11 and t 22 − t 12 , the effect of the intrinsic time delay t int can be removed naturally. The difference between γ 1 and γ 2 is where α = ∆t geo /∆t gra is the ratio of time delays caused by geometric effect and Shapiro delay effect. Hereafter, we define 'time delay' as t 22 − t 12 or t 21 − t 11 , which is the time difference between different particles in the same light path. The 'strong lensing time delay' is defined as t 22 −t 21 or t 12 − t 11 , which is the time delay between two different paths due to strong lensing. The value of α depends on the choice of lens model. For the purpose of illustrating our method, we use the singular isothermal sphere (SIS) model, which has been proved to be a reliable model for lenses. In the SIS model, the distribution of stars and other mass components in galaxies are thought to be like that of particles in idea gas. The projected potential and Einstein radius of a SIS lens are ψ(ξ ) =  [20]. If the lensing is strong, β < θ E , there are two images of the source at the positions θ ± = β ± θ E . Then, from equation (3), the time delay between the two images is which is all caused by the Shapiro time delay effect (i.e. ∆ T SIS = t gra, SIS ). The difference in arrival time of two images caused by geometric time delay is ∆t geo = 0 which means that the length of the different light trajectories of the two images P 1 and P 2 are equal to each other. It should be pointed out that the SIS model is an idealized lens model. The real strong lensing by a galaxy is likely not to be a SIS case. For a certain strong lensing event, if there are other observation of the properties of the lens object, we can calculate the value of α in a suitable lens model. Generally, both of ∆t geo and ∆t gra are on the order of GM/c 3 , where M is the mass of lens [22]. Therefore, one can expect that the parameter α should be on the order of one. GRBs are promising tools to constrain the WEP for photons with different energies [6]. Due to their large luminosities, GRBs can be observed at very high redshifts [23]. Therefore, there is much more possibility for a GRB to be lensed by a galaxy or galaxy cluster in the universe. Due to the success of BATSE, Swift and Fermi satellites, the number of detected GRBs keeps increasing. There are lots of works to search the potential strong lensed GRBs in several GRB's catalogs. Although no such event was ever found [24,25]. Li & Li (2014) searched the potential lensing events in BATSE GRB data. They found four candidates. The second couple of GRBs, 2044 and 2368, has closest properties of a strong lensed GRB event. We use this GRB as an example, although they excluded the possibility in their work [25]. The flux ratios in four considered energy channels seem similar. In addition, the angular separation of them is ∆ θ = 3.88 • while the location uncertainties of them are 2.88 • and 6.06 • . The detected time delay between 2044 and 2368 is about 1.77 × 10 7 s and the time delays of the photons in energy channels of 25-60 keV and 60-110 keV are 0.085 ± 0.042 s and 1.730 ± 0.162 s for 2044 and 2368 respectively. Assuming there is a lensed GRB event with similar time delay, we constrain the WEP with our method. From equation (4), the constraint on the violation of WEP between the photons in these two energy bands is ∆ γ ≤ 2(1 + α) 1.730−0.085 1.77×10 7 = 1.86(1 + α) × 10 −7 . If we choose the SIS lens model, α = 0, it has ∆ γ ≤ 1.86 × 10 −7 . For Fermi GBM, the expected time to observe one lensed GRB is about 11 years [25]. Since Fermi GBM has served about 9 years and will be in service for another more than 10 years, it is reasonable to expect a lensed GRB event in the operating period of Fermi GBM.

Discussion
In this section, we discuss two potential biases. The first point is the time delay potentially caused by the non-zero rest mass of photons ∆t spec and the LIV ∆t LIV . Generally, these two effects are similar since both of them will cause the energy-dependent speed of photons. If the photon has non-zero rest mass, then the higher-energy photons will travel faster than lower-energy photons. On the contrary, LIV will lead to an opposite effect that higher energy photons have slower traveling speed because the so-called vacuum dispersion [26][27][28][29]. Therefore, the time delay terms ∆t spec and ∆t LIV are both caused by the potential difference of traveling speed of photons with different energies.
In figure 1, there are two traveling paths and for each path there are two images formed by two different kinds of particles. Let's assume the speeds of those two different particles are v 1 and v 2 and the lengths of the two traveling paths are L 1 and L 2 , respectively. Therefore, the time delays caused by this effect are L 1 /v 1 − L 1 /v 2 and L 2 /v 1 − L 2 /v 2 respectively. Their contribution to the difference of time delays |(t 22 −t 12 ) − (t 21 −t 11 )| which we use in equation (4) to constrain the WEP is ( . Considering a strong lensing system with strong lensing time delay is about 1 year, and the time delays of different particles t 22 − t 12 and t 21 − t 11 are about 1 second, the difference of time delays |(t 22 − t 12 ) − (t 21 − t 11 )| should be also at order of 1 second. For this kind of strong lensing system, the difference between L 1 and L 2 should be at a order of 1 light year. However the distance of a typical cosmic source should be at a order of 1 Gpc, so it has Therefore, even though the time delay caused by non-zero rest mass of photons and LIV effect is hundreds of times larger than the observed time delay between two different particles, only contributes very small part of |(t 22 − t 12 ) − (t 21 − t 11 )|. Therefore, it is reasonable to omit the terms ∆t spec and ∆t LIV , which means our method can also exclude the potential effects of non-zero rest mass of photons and LIV.
The second point is that the two traveling paths will perhaps introduce some other contributions to the observed time delay. Actually, for a typical strong lensing system, the Einstein radius is about 4 × 10 −6 ( M 10 11 M ) 0.5 ( D 1Gpc ) −0.5 [20], which is very smaller compared to the relativistic beaming angle θ ∼ Γ −1 ∼ 10 −3 where Γ is the Lorentz factor GRB's jet [30]. Therefore, the difference of the special-relativistic boost factor caused by different viewing angles can be omitted reasonably. In addition, the different paths may also lead to different effects of gravitational potential along the paths, such as the weak lensing by the large scale structure and also the effects of our Galaxy and local galaxy cluster, which may also contribute into the observed time delay. However, from equation (1), the difference between the Shapiro time delays for two paths relies on the total difference of the gravity potential along the whole paths which includes the gravitational potential of large scale structure, the local galaxy cluster and also our galaxy. Therefore, the effect is considered even though there is some small difference of gravitational potential along two different paths.

Conclusions
In this paper, we have proposed a method to constrain the violation of WEP with strongly lensed cosmic transients. Our method dose not need to make any assumption and understand physical mechanism of the transient. Moreover, because it utilizes the difference of time delays of the two lensing images, the potential effect of intrinsic time delay ∆t int , LIV time delay ∆t LIV and non-zero rest mass time delay ∆t spec can be naturally removed. By analyzing the properties of time delay of gravitational lens, we find that the parameter α, which represents the ratio of time delays between two lensing images caused by geometric and Shapiro delay effects, will be zero for the SIS lens model. Therefore, the lengths of light ray paths for two lensing images are the same in the SIS lens model, which means any test depending on the difference of the path length can not work in SIS lens model, such as testing the difference between the speeds of light and GW. Even though for a realistic lens, one should consider the parameter α when constraining the speed of GW. Otherwise an unreasonable tighter constraint will be obtained.
Due to the number of GRBs increasing, the detection of a strong lensed GRB is promising. Although there is still no such event at present. Assuming a strong lensed GRB with time delay between images is about 1.77×10 7 s and the time delays of the photons in two energy channels are 0.085 ± 0.042 s and 1.730 ± 0.162 s, one can give a constraint on WEP at about 10 −7 level. Besides GRBs, the lensed FRBs and the GW events with their EM counterparts are also potential candidates to test the WEP using our method [31][32][33]. Interestingly, lensing of FRBs has been proposed to probe dark matter [34,35].
From equation (4), with the time delay measurements of a single strong gravitational lensing event one can give a tight constrain on the ∆ γ. It can also be found that the efficiency of the constraint on WEP is proportional to the accuracy of time delay measurement and inversely proportional to the strong lensing time delay. A typical strong lensing by a galaxy will give multiple images with days to months strong lensing time delays [36]. Because the strong lensing time delay is proportional to the mass of lens, the time delay due to galaxy cluster lens is much longer [37], which will give much more strict constraint on ∆ γ. If a several-months strong lensing time delay is observed and the time delay of two images is about 0.1 s, the constraint on the violation of WEP will be up to ∆ γ < 10 −8 . For the measurement of time delay between two different particles, it depends on the accuracy of timing measurement of the transient event.
The accuracies of time measurement of FRB, GW and GRB are about 0.01 ms [38], 10 −4 ms [31] and 0.1 s [39]. Recently, the detections of GW170817 and its electromagnetic counterparts [14,40] also encourage us to make a more reliable constraint on the WEP using strong lensing of GWs and electromagnetic counterparts. Therefore, the accuracy of WEP can be improved by several orders of magnitude in the future, if the lens is galaxy cluster and the strongly lensed cosmic transients have much more precise time delay measurement, such as the GW events and FRBs.