Testing the variation of the fine structure constant with strongly lensed gravitational waves
aa r X i v : . [ g r- q c ] S e p Testing the variation of the fine structure constant withstrongly lensed gravitational waves
Xin Li, a) Li Tang, b) Hai-Nan Lin, c) and Li-Li Wang d) Department of Physics, Chongqing University, Chongqing 401331,China
The possible variation of the electromagnetic fine structure constant α e on cos-mological scales arouses great interests in recent years. The strongly lensed grav-itational waves and the electromagnetic counterparts could be used to test thisvariation. Under the assumption that the speed of photon could be modified, whilethe speed of GW is the same as GR predicated, and they both propagate in a flatFriedman-Robertson-Walker universe, we investigate the difference of time delaysof the images and derive the upper bound of the variation of α e . For a typical lens-ing system in the standard cosmological models, we obtain B cos θ ≤ . × − ,where B is the dipolar amplitude and θ is the angle between observation and thepreferred direction. Our result is consistent with the most up-to-date observationson α e . In addition, the observations of strongly lensed gravitational waves andthe electromagnetic counterparts could be used to test which types of alternativetheories of gravity can account for the variation of α e .Keywords: fine structure constant, gravitational wave, gravitational lensing I. INTRODUCTION
Gravitational waves (GW), as one of the predictions of general relativity, has been de-tected recently by the advanced LIGO detector . Up to now, the LIGO and Virgo Col-laborations have directly observed five GW events produced by the mergers of compactbinary systems . The first four events were produced by the merge of binary black holesystems. The last one, GW170817, was produced by the merge of binary neutron starsystem, and the corresponding electromagnetic (EM) counterparts have been detected bymany instruments . The observations of GW can be used to test cosmology and generalrelativity. One important cosmological quantity, i.e. the luminosity distance of the source,can be derived directly from the GW signal. Location of the source can be found from EMcounterparts and the redshift of the source can be found from the association of the sourcewith its host galaxy.These information obtained from the observations of GW can be used to constrain thecosmological parameters, such as the equation-of-state of dark energy, the Hubble constant,etc . Also, the GW signal has been used to constrain the mass of graviton , and therelative arriving time between the GW signals of GW170817 and its EM counterparts hasbeen used to constrain the Lorentz invariance violation . However, the intrinsic timedelay in the emission time of GW signal and its EM counterpart can not be measureddirectly. To test the Lorentz invariance violation more precisely, an approach using thestrongly lensed GW is proposed to cancel the intrinsic time delay . This approachrequires GW and its EM counterpart occur behind a strong gravitational lensing and twoimages are observed.Up to now, such phenomena has not yet been observed by the astronomical instru-ments. The LIGO and Virgo collaborations established a program ? about the iden-tification and follow-up of EM counterparts, which activates the campaign to find EM a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] d) Electronic mail: [email protected] counterparts . The on-going third generation detectors with higher sensitivity suchas the Einstein Telescope will discover more GW events . The plausibility of suchphenomena being observed is discussed in Ref. . It is expected that GW and its EMcounterpart could be observed behind a strong gravitational lensing in the future.Testing the constancy of the fundamental physical constants is very important . Ana-lyzing from observations of quasar absorption spectra, Webb et al. found that the finestructure constant α e varies at cosmological scales. However, debates still remain . TheGW signals provide us a new window to study the variation of α e . It is very interesting totest the variation of α e by the observations of GW signals and its EM counterparts. Manymodels have been proposed to explain the variation of α e . These models can be dividedinto two types. The first is that the EM field is coupled to other field, such as quintessencefield . Other is that our universe is anisotropic, for example, our universe is a Finslerspacetime instead of Riemann spacetime . If α e does vary at cosmological scales, thenthe observations of the strongly lensed GW signals and their EM counterparts could beused to test which type of model is valid. This is due to the fact that the method proposedby Refs. mainly considers the difference between the time delay of two images of theGW signals and their EM counterparts. The first type of models requires that the speeds ofphoton and GW are different. Then the observations of the strongly lensed GW signals andtheir electromagnetic counterparts will find the difference from the method . The othertype of model requires that both photon and GW propagate in the anisotropic universe withthe same anisotropic speed. Then such observations will not find their difference. In thispaper, we will discuss these points and show that the observations of the strongly lensedGW signals and their EM counterparts could test whether Webb’s result is valid or invalid.The arrangement of the paper is as follows: In Section 2, we introduce the basic informa-tion of GW and EM counterpart, and the method to calculate the difference of time delaysbetween the GW and EM in detail. Then in Section 3 we use such method to constrainthe variation of α e and compare it with the observational data. Finally, conclusions andremarks are given in Section 4. II. METHODOLOGY
Webb et al. showed that the variation of fine structure constant α e have a dipolarstructure in high redshift region ( z > . also confirmed that thedipolar variation of α e is still a good fit to the most up-to-date data. According to theirresults, the variation of α e can be expressed as △ α e α e = B cos θ, (1)where B represents the dipole amplitude and is assumed to be a constant, and θ is the anglebetween the dipolar direction and the observed direction.One direct reason of the variation of α e is the variation of speed of light. Therefore, fromthe (1), the speed of light c is anisotropic with dipolar structure in the cosmological scale, c γ = c / (1 + B cos θ ) , (2)where c is the speed of light at present epoch.The method proposed by Refs. considers GW and its EM counterparts propagatingthrough a strong gravitational lensing and at least two images of the GW event and twoimages of the EM counterparts are observed. The observations of this phenomena can detecttwo arriving time of GW events and two arriving time of EM events. The time delay of thetwo GW events does not depend on the initial emission time of the GW signals. The timedelay of the two EM events also does not depend on the initial emission time of the EMsignals. Thus, this method does not depend on the intrinsic separation time of GW and itsEM counterparts.In general relativity, the two time delay should be the same. If the difference of the twotime delay is observed, then it is a signal of new physics. Two reasons will deduce thedifference of the two time delay. One reason is that the speeds of graviton and photon aredifferent. The other is that the geodesics of GW and photon are different. One type ofmodel, such as Refs. , could explain Webb’s results by assuming the photon propagatein a Finslerian universe. In such model, both the graviton and photon are massless and theirgeodesic are the same. Therefore, if Webb’s results are confirmed in future astronomicalobservations and the observations show no difference between the two time delay, thenit implies our universe may be Finslerian. Webb’s results could be explained by anothertypes of model which assume the speed of photon is modified. In alternative theories ofgravity, both the speed of photon and graviton could be modified. Several approachescould lead to the modifications. For example, the speed of photon could be modified ifLagrange of electromagnetic field possesses the non-minimal coupling form . And thespeed of gravity could be modified if graviton couple to background gravitational field, suchas massive gravity . In Refs. , the graviton or scalar curvature in Lagrange does notcouple to background gravitational field, thus, the speed of graviton is unchanged in thesemodels. Due to that the speeds of light and graviton are different, the difference betweenthe two time delay should occurs. In the rest of our paper, we will mainly discuss how to testWebb’s results with a model-independent method, which is only based on the assumptionthat the speed of light is different with the graviton.The spatial geometry of FRW spacetime would not greatly affect the difference betweenthe two time delay. And the recent data of Planck satellite prefers a spatially flat universe .For simplicity, we suppose the spacetime is depicted by the flat FRW metric ds = c dt − a ( t ) (cid:2) dr + r dθ + r sin θdφ (cid:3) , (3)where a ( t ) is the scale factor. Hence the travel distance of the GW from the emitted moment t e to observed moment t is r GW = Z t t e c a ( t ) dt = c H ˜ r GW ( z ) , (4)where H is the Hubble constant, and˜ r GW ( z ) = Z z dz ′ E ( z ′ ) (5)is the reduced comoving distance travelled by the graviton, where E ( z ) = p Ω m (1 + z ) + (1 − Ω m ).As for EM counterpart, the photon travels with the speed c γ given in Eq.(2) and the cor-responding distance is given as r γ = Z t t e c γ a ( t ) dt = c H ˜ r γ , (6)where ˜ r γ = [1 − f ( B, θ )] ˜ r GW and f ( B, θ ) = B cos θ − B cos θ . Here, we have use the factthat the magnitude B of α e variation is very small and expand f ( B, θ ) to the second orderof B .In a strong gravitational lensing system, the time delay between two collinear imageswhich are observed on the opposite side of the lens is given as∆ t = 1 + z l c D l D s D ls ( θ A − θ B ) , (7)where θ A = θ E + β and θ B = θ E − β are the radial distances of two images, respectively,and β denotes the misalignment angle. Here, D ls denotes the angular diameter distancebetween the lens and source, and D l ( D s ) denotes the angular diameter distance betweenthe lens (source) and observer. In the singular isothermal sphere lens model, the Einsteinring radius takes the form θ E = 4 π D ls D s σ c , (8)where σ is the one dimensional velocity dispersion. Combining eqs.(4)(7)(8), the time delaysof two GW signals is given as∆ t GW = 32 π H (cid:18) σc (cid:19) β ˜ r ( z l )˜ r ( z l , z s ) θ E ˜ r ( z s ) . (9)The electromagnetic field with a non-minimal coupling, such as refs. implies thephoton is massive. From the geodesic equation in general relativity, one can find thatthe deflection of the photon with a small rest mass would be altered with a factor 1 +( m γ c / E γ ), where m γ and E γ are the mass and energy of the photon. Thus, the Einsteinradius is modified as θ E,γ = θ E [1 + ( m γ c / E γ )] and the time delay between the two imagesof EM is given as ∆ t γ = 32 π H (cid:18) σc (cid:19) β ˜ r γ ( z l )˜ r γ ( z l , z s ) θ E ˜ r γ ( z s ) " m γ c E γ . (10)In flat FRW spacetime, the spacetime is Minkowski spacetime in local. Thus, the dispersionrelation of massive photon is the same with other massive particles in Minkowski spacetime.Combining the eq.(2) and the dispersion relation, the relation between the mass of thephoton and its speed is derived as m γ c E γ = 12 (cid:18) − B cos θ ) (cid:19) . (11)By making use of the formulae (7,10) and eq.(11), to second order in B , we obtain thedifference of the two time delay∆ t GW − ∆ t γ = ∆ t GW B cos θ. (12) III. RESULTS
The observational accuracy δT of the difference between two time delays, i.e., ∆ t GW − ∆ t γ ,could give a constraint on the dipole variation of α e . From the eq.(12), we find that B cos θ ≤ (cid:18) δT ∆ t GW (cid:19) / . (13)In the strong gravitational lensing systems compiled in Ref. , the redshift ranges are z l ∈ [0 . , . z s ∈ [0 . , . σ ∈ [103 , β/θ E should not be to large in order to ensure the formation of multiple images. Pi´orkowskaet al. demonstrated that the maximal value of misalignment parameter β/θ E is 0 .
5. Forthe timing accuracy δT of time delay, observation has demonstrated that the GW signalcan be detected at precision < − ms . Moreover, the timing precision of promising EMcounterparts, such as SGRB and FRB, could be in the order of 10 − − ms . Thus,the accuracy of the EM time delay determine the ability of testing Webb’s result. However,since the strongly lensed gravitational waves and their electromagnetic counterparts havenot been detected, δT = 1ms could be set as a mediate timing precision of promising EMcounterparts to obtain the detecting precision of testing the α e variation.Considering the ΛCDM cosmology parameters given by the Planck data , i.e., H =68 kms − Mpc − , Ω M = 0 .
3, and using the typical parameters of strong lensing system( z l = 1, z s = 2, σ = 250 km / s, and β/θ E = 0 . δT = 1ms, we obtain the bound of α e variation as B cos θ ≤ . × − . (14)Webb et al. showed that the magnitude of α e variation is (0 . +0 . − . ) × − . Pinho et al. showed that the magnitude of α e variation is (0 . ± . × − . Thus, in the detectingprecision, the constraint on α e variation eq.(14) is consistent with the previous researcheson the α e variation. This implies that the observations of the difference between the GWtime delay and EM time delay are capable of testing whether the α e variation is valid ornot.The upper limit of the variation of α e measured in the Milky Way is | ∆ α e /α e | < . × − . Thus, if Webb’ result is correct, there should be a physical mechanism that α e varies with redshift. In fact, our previous research has shown one such possible physicalmechanism , where the speed of light depends on the redshift with the form c γ = c / (1 + B ( z ) cos θ ) , (15)where B ( z ) = b Z z z ′ p Ω m (1 + z ′ ) + (1 − Ω m ) dz ′ = b D ( z ) . (16)here D ( z ) = R z z ′ E ( z ′ ) dz ′ . To the first order of b , the difference of time delays measured bythe GW and EM windows becomes∆ t GW − ∆ t γ = ∆ t GW b cos θF ( z l , z s ) , (17)where F = x ( z s )˜ r g,ls − x ( z l )˜ r g,ls + x ( z l )˜ r g,l − x ( z s )˜ r g,s − D ( z s ) (18)here x ( z ) = R z D ( z ′ ) E ( z ′ ) dz ′ . It is different with eq.(12), where the formula represents that thedifference of time delays is proportion to B . As eq.(16) is shown that, if the variation of α e is independent on the redshift, viz, D ( z ) = 1 in eq.(16), the term which is proportionalto B would vanish in eq.(17)( F =0). Then, the formula (17) reduces to the formula (12)under the consideration of the second order of B . With the same variable setting as abovethat z s = 2 and z l = 1, the upper bound of dipolar variation b cos θ ≤ . × − , whichreaches a very high accuracy to test the variation of the fine structure constant.It should be noted that Webb’s result about the dipolar variation of α e mainly appearsin high redshift region ( z > . z >
1. Theon-going third generation detectors like the Einstein Telescope with higher sensitivity arecapable of testing Webb’s result. IV. CONCLUSIONS AND REMARKS
The associated detection of GWs and their EM counterparts provides a way to testfundamental physics. In this paper, we have used the method proposed by Refs. totest the possible variation of α e . The method considers the difference between the timedelay of two images of the GW event and its EM counterparts. The difference betweenthe speeds of photon and graviton, and the difference between the geodesic of photon andgraviton, can account for the difference between the two time delay. In an anisotropicuniverse, the geodesic and speeds of photon and graviton are modified in the same way.Therefore, if Webb’s results are confirmed by the future data and the observations show nodifference between the two time delay, then it implies our universe may be anisotropic, suchas Finslerian universe.In this paper, we consider that the speed of photon is modified, such as electromagneticfield coupling to a quintessence field . In these models the speed of graviton remains thesame as the prediction of general relativity, and both the graviton and photon propagate inthe same flat FRW universe. It is shown that the dipolar variation of α e has a upper limit,namely, B cos θ ≤ . × − , which implies that Webb’s result can be tested in currentlyaccuracy with this method. Additionally, considering the variation of α e could be a functionof the redshift as Li & Lin , we obtained a bound of α e variation, b cos θ ≤ . × − ,which is a higher detecting precision to test the result of Webb.One should notice, due to the present sensitivity of LIGO detector, one can not find GWsignals with EM counterparts locating at redshift z >
1. Since, Webb et al. found thedipolar variation of α e only appear in high redshift ( z > .
6) region. Thus, it is expectedthat the on-going third generation detectors such as the Einstein Telescope could test thevalidity of Webb’s result by observing the difference between the time delay of two imagesof the GW event and its EM counterparts. In this paper, we only give a limit of B cos θ , theconstraint on B and the preferred direction are not given. If many events of the stronglylensed GWs and their EM counterparts are observed in the future, then it is possible to usethe data to constrain the dipolar amplitude B and the preferred direction of the universe. ACKNOWLEDGMENTS
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