Bounds on GUP parameters from GW150914 and GW190521
aa r X i v : . [ g r- q c ] J a n Bounds on GUP parameters from GW150914 and GW190521
Ashmita Das ∗ Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
Saurya Das † Theoretical Physics Group and Quantum Alberta,Department of Physics and Astronomy, University of Lethbridge,4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada
Noor R. Mansour ‡ and Elias C. Vagenas § Theoretical Physics Group, Department of Physics,Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
We compute bounds on the GUP parameters for two versions of GUP using gravitational wave datafrom the events GW150914 and GW190521. The speed of the graviton and photon are calculatedin a curved spacetime modified by GUP, assuming that these particles have a small mass. Theobservational bound on the difference in their speeds translates to bounds on the GUP parameters.These bounds are some of the best obtained so far in the context of quantum gravity phenomenology.
I. INTRODUCTION
A completely consistent quantization of the gravitational field has remained as one of the main problems of TheoreticalPhysics for decades. In this search, there are many promising candidates such as String Theory, Loop QuantumGravity, Causal Dynamical Triangulations, Doubly Special Relativity (DSR). It is expected that a theory of quantumgravity (QG) will lead us to the unification of the fundamental forces of nature. However, there has not been anyexperimental or observational support for any theory of QG so far. Efforts to obtain signatures of QG constitute QGPhenomenology [1, 2].QG theories on the other hand, predict the existence of a minimum measurable length scale O ( ℓ P l ) [3–5], whichconstrains the measuring device from probing an arbitrarily small length scale. This in turn implies the modifi-cation of the Heisenberg Uncertainty Principle (HUP) to the Generalized Uncertainty Principle (GUP), which hasplayed an important role in the development of QG phenomenology [6–9]. Its implications have been explored inthe context of Hawking radiation from a black hole (BH) spacetime [10–12], BH thermodynamics [13–15], Friedman-Robertson-Walker (FRW) cosmology, FRW thermodynamics [16, 17], condensed matter systems [18], Neutrino oscil-lation probabilities [19], and other quantum mechanical systems [20] etc. In a separate context, studying solutionsof the GUP-modified Schr¨odinger, Klein-Gordon, and Dirac equations in a one, two, and three dimensional box, theauthors of Refs. [21–24] have shown that a measured length must be quantized in units of ℓ P l .In this paper, we focus on two specific GUPs. The first, proposed by Kempf et al. in Ref. [5], consists of thefollowing commutator and the corresponding resulting GUP (cid:2) x , p (cid:3) = i ~ (cid:0) + β p (cid:1) (1a)∆ x ∆ p ≥ ~ (cid:20) β (∆ p ) (cid:21) (1b)where ( x , p ) are the 3-position and 3-momentum operators and ∆ x and ∆ p are their uncertainties, respectively. Theparameter β is given as β = β ℓ Pl ~ = β M Pl c > x, ∆ p while β is the dimensionless GUPparameter. In the above, M P l is the Planck mass, such that M P l c ∼ TeV. As is well known, Eqs.(1a) and (1b)imply a minimum position uncertainty: ∆ x = ~ √ β = √ β ℓ P l .Next, we consider the generalization proposed in Refs. [21, 25–28][ x i , p j ] = i ~ (cid:20) δ ij − α (cid:18) p δ ij + p i p j p (cid:19) + α (cid:18) p δ ij + 3 p i p j (cid:19)(cid:21) . . (2) ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] In the above, i, j = 1 , , p = P j =1 p j p j , and α = α /M P l c = α ℓ P l / ~ . In this case, α is a dimensionlessparameter which will henceforth be referred to as the LQGUP parameter since in this version there are both linearand quadratic terms in momentum. Eq. (2) is consistent with DSR theories. In addition, it can also be proposedfrom a purely phenomenological point of view, and has also been shown to follow as an effective theory from verygeneral grounds [29]. This gives rise to the following modified position-momentum uncertainty relation∆ x ∆ p > ~ (cid:20) − α h p i + 4 α h p i (cid:21) > ~ (cid:20) (cid:18) α p h p i + 4 α (cid:19) ∆ p + 4 α h p i − α p h p i (cid:21) . (3)The modified commutation relation, namely Eq. (2), and the modified uncertainty principle, namely Eq. (3), implya minimum measurable length and a maximum measurable momentum of the form∆ x > (∆ x ) min ≈ α ℓ P l , ∆ p (∆ p ) max ≈ M P l cα . (4)Note that for both GUPs under consideration, one assumes [ p i , p j ] = [ x i , x j ] = 0. A couple of comments are in orderhere. First, GUP gives rise to a modified energy momentum dispersion relation E ( p ) which has potential observationalconsequences, as shown by a number of authors [30–36]. Second, the dimensionless GUP parameters β and α aresometimes assumed to be O (1) . However, in this work, we will not make this assumption a priori . We keep themarbitrary, examine the consequences, and let experiments and observations decide on their values. We note however,that the above parameters give rise to the length scales α ℓ P l and √ β ℓ P l . Assuming that the length scales areno smaller than the Planck length ℓ pl ≈ − m (as the physics beyond that scale is completely unknown) anddo not exceed the electroweak scale, which is about 10 − m, one gets a set of natural bounds 1 ≤ α ≤ and1 ≤ β ≤ . The work of various authors, e.g. Ref. [20], and their exploration of low energy effects on GUPset stricter bounds on the parameters, and lend further credence to the possible existence of an intermediate scalebetween the Planck and electroweak scales.Returning to the issue of modified dispersion relations (MDR), the authors in Refs.[39–41] have shown that thisresults in the difference between the speed of gravitational waves (GW) and electromagnetic waves from the sameastrophysical event. This is compatible with data from the GW event GW150914, as reported by the Laser Interfer-ometer Gravitational-Wave Observatory (LIGO) Scientific and Virgo Collaborations [42, 43]. Ref. [40] also reportedan observationally compatible version of MDR. These potential departures from the standard dispersion relation showthat it is important to pursue the search for the QG signatures in GW data, which may shed important light on QGtheories. For example, the authors in Refs. [44, 45] have constrained the parametric space of several GUPs andobtained the upper bounds on the corresponding GUP parameters. In particular, in Ref. [44] the observational dataof GW150914 were used while in Ref. [45] the observational data of GW170814 were used. Both groups have followeda similar approach to constrain the GUP parameters by calculating the difference between the speed of GWs, i.e.,gravitons, and that of the light waves, i.e., photons.In the aforementioned papers there are two limitations. The first limitation is due to the fact that the photon speedis considered unmodified (constant). However, the speed of the photon can be modified due to QG effects when it hasa small mass. The second limitation of the aforementioned papers is that their analysis was done in flat spacetimes,although the GWs from the events GW150914 and GW170814 originated in a spacetime region of high curvature,from the merging of the two stellar mass BHs [42, 43, 46]. Although spacetime is almost flat at the site of the detectorsand, hence, their results, e.g. those of Refs. [44, 45], should be valid at least approximately. Strictly speaking, theanalysis should be carried out in a curved background spacetime, if at least to simply establish the limits of validityof the earlier results. Moreover, curvature effects may indeed be important for future events (those from regions ofeven higher curvatures) and for future detectors, with even higher accuracies.Motivated by the above, in the present manuscript, we consider a curved spacetime, reasonably approximated bythe Schwarzschild black hole metric at large distances, and obtain the modifications in the velocity of the gravitonsand photons due to the GUP effects. Our goal is to study the parametric space of the two well-known GUP versions,as described above, in the background of the specific curved spacetime.The remainder of this paper is organised as follows. In Section II, we calculate the difference in the speed ofgravitons and photons in the Schwarzschild background without GUP effects. The effect of the curved background Recently, in Ref. [37] and utilizing the GUP version, a numerical value of the dimensionless GUP parameter, namely β , was obtained,namely 82 π/
5. Furthermore, in Ref. [38] and utilizing the LQGUP version, the dimensionless GUP parameter, namely α , was shownto be proportional to powers of the dimensionless ratio ( M Pl /M ). In this work, we use the terms “velocity” and “speed” interchangeably. is expressed through the dependence of the difference in the speed of gravitons and photons on the metric element g ( r ) of the Schwarzschild black hole spacetime. In the limit of r → ∞ , our result reproduces the ones for flatspacetime, as expected. Moreover, utilizing the data from GW150914, we get the upper bound for the difference inthe speed of the gravitons and photons. This upper bound will be the reference point in the next sections in whichthe QG effects will be taken into consideration. In Section III, we obtain the difference in the speed of gravitonsand photons in a Schwarzschild black hole background including the GUP effects, which are expressed through theterms involving the dimensionless GUP parameter β . Utilizing the data from GW150914, we bound the differencein speeds and, thus, we get an upper bound for the GUP parameter β . In addition, we consider the case in whichthe photon speed is GUP-modified as is the graviton speed. In this case, in order to get an upper bound, we use thedata from GW190521 [47], since for this GW event, we have apart from the data from LIGO Scientific and VirgoCollaborations, its electromagnetic counterpart from the data of the Zwicky Transient Facility (ZTF) [48]. In SectionsIV, we follow exactly the same analysis with that in Section III, but for the case of the LQGUP version and, thus,we obtain the upper bound on the dimensionless LQGUP parameter α which is the lowest in the literature. InSection V, we conclude and present our results obtained here. Finally, we provide three appendices in which we havereconsidered all the previous three sections, namely II-IV, when the very speculative case of massive photons andgravitons is included without using the small mass approximation. II. SPEED OF GRAVITON AND PHOTON IN THE SCHWARZSCHILD BACKGROUND
As mentioned in the introduction, we model a generic curved spacetime at large distances from the source by a4-dimensional Schwarzschild metric ds = − f ( r ) c dt + 1 f ( r ) dr + r d Ω (5)where f ( r ) = (cid:18) − GMc r (cid:19) and d Ω = r dθ + r sin θ dφ . We write the squared 4-momentum of a particle of mass m in the aforesaid background as p µ p µ = g ( p ) + g ij p i p j | {z } p (6)The standard dispersion relation dictates that p µ p µ = − m c and, therefore, the above equation becomes( p ) = 1 g (cid:18) − p − m c (cid:19) . (7)As usual, one defines the energy of a particle in this background using the timelike Killing vector field ξ in theSchwarzschild background Ec = − ξ a p a = − g ab ξ a p b (8)where ξ a = (1 , , ,
0) is the timelike Killing vector. Therefore, from Eq. (8) we get E = − g cp . (9)We now specify the particle as graviton with its rest mass and energy to be m = m g and E = E g , respectively. FromEq. (7), we get the energy of the graviton to be E g = √− g ( p c + m g c ) / . (10)It may be noted that extreme cosmic phenomena cause the fluctuations in spacetime, which in turn produce gravitons.These can then propagate as a GW and hit the GW detectors such as the LIGO and Virgo detectors. The speed ofgravitons can be calculated by using the group velocity of the wave front v g = 1 √− g ∂E g ∂p . (11)Using Eq. (10), we obtain v g = c (cid:20) − m g c E g (cid:18) − GMr c (cid:19)(cid:21) / . (12)Expanding the term within the square brackets in Eq. (12) yields v g = c (cid:20) − m g c E g (cid:18) − GMr c (cid:19) − m g c E g (cid:18) − GMr c (cid:19) − O ( G ) (cid:21) . (13)Since ℓ P l = p G ~ /c and, thus, O ( G ) ∼ O ( ℓ P l ), neglecting these higher order terms, the graviton speed reads v g = c (cid:20) − m g c E g (cid:18) m g c E g (cid:19) + m g c E g GMr c (cid:18) m g c E g (cid:19)(cid:21) . (14)At this point a number of comments are in order. First, it is evident that ∆ v g = v g − c = 0 only when the gravitonsare massive, i.e., m g = 0. Second, by the same token, the speed of light remains intact as long as the photon isconsidered massless. Third, in Refs. [42, 43] the signal of the event GW 150914 is peaked at ν = 150 Hz, which leadsto the maximum energy of gravitons to be E g = hν ≈ . × − eV. In addition, we know that if the dispersionrelation for the GW is modified, the upper bound on the Compton wavelength is constrained to be λ g > m and,thus, one obtains an upper bound for the mass of gravitons to be m g ≤ . × − eV /c [49] . With these values,the term m g c E g in Eq. (14) becomes m g c E g = 5 . × − m/sec [44]. Therefore, one can neglect terms of higher orderin m g c E g and the speed of graviton from Eq. (14) is v g = c (cid:20) − m g c E g (cid:18) − GMrc (cid:19)(cid:21) . (15)Furthermore, if one considers the spacetime under consideration to be practically flat and ignores the curvature terms,then from Eq. (15), the difference between the speed of graviton and photon becomes∆ v g = c − v g = m g c E g (16)which agrees with the bound obtained in Ref. [44] in which a flat spacetime was considered. Finally, utilizing thebound on the graviton mass, one gets the bound on the difference between the graviton and photon as∆ v g ≤ . × − m/sec . (17) III. BOUND ON THE GUP PARAMETER FROM GW150914
In this section, we will study the modifications in the difference between the speed of graviton and photon, i.e., ∆ v g ,in Schwarzschild spacetime while incorporating the GUP defined by Eqs. (1a) and (1b). For this reason, the followingvariables are defined x i = x i , p = k (18a) p i = k i (1 + β k ) (18b)where x and p are the physical position and momenta, while x and k are auxiliary “canonical variables”, such that[ x i , k j ] = i ~ δ ij . Next, we expand the squared 4-momentum, using Eq. (18b) as follows p α p α = g ( p ) + p = g ( p ) + k [1 + 2 β k + O ( β )] . (19)It is easily seen that the last term is of O ( ℓ P l ) and, thus, it can be ignored compared to the linear term in β . Inaddition, the physical 4-momentum does not satisfy the standard dispersion relation, namely p α p α = − m c , whilethe non-GUP-modified 4-momentum satisfies the standard dispersion relation, namely k α k α = − m g c . Therefore,employing the above equation, one ends up with g ( p ) = − m g c − p + 2 β k k . (20) In different contexts, one can propose even smaller bounds on the graviton mass, e.g. m g ≤ − eV /c [50]. Now one can take the inverse transformation of Eq. (18b) and write k as a function of p in the form k = p (1 − β p ) . (21)Next we substitute Eq. (21) into Eq. (20) and we expand in terms of the GUP parameter. Then, due to the smallnessof the QG corrections, we ignore the higher order terms in β , and, thus, we get( p ) = ( − g (cid:20) m g c + p (1 − β p ) (cid:21) . (22)Employing Eq. (9), the energy of the graviton becomes of the form E g = √− g (cid:20) m g c + p c (1 − β p ) (cid:21) / . (23)The above equation now implies the following GUP-modified group velocity of the graviton˜ v g = 1 √− g ∂E g ∂p = (cid:20) m g c + p c (1 − β p ) (cid:21) − / (cid:20) pc (1 − β p ) − β pp c (cid:21) . (24)Recalling the graviton mass bound m g ≤ . × − eV /c , we neglect the graviton mass term in comparison withthe remaining terms in Eq. (24) and obtain ˜ v g = c (1 − β p ) . (25)At this point, we need to express the speed of the graviton, i.e., ˜ v g , in terms of E g , thus we implement the iterationmethod. First, we get the zeroth order solution for p by setting β = 0 in Eq. (23) which leads to p = (cid:20) E g ( − g ) c − m g c (cid:21) / . Then, substituting the zeroth order solution in Eq. (23), employing g = − (cid:18) − GMr c (cid:19) , and adoptingthe small m g approximation , we finally obtain p as a function of E g . So the speed of the graviton will read˜ v g = c (cid:20) − β E g c (cid:18) − GMr c (cid:19) − (cid:21) = c (cid:20) − β E g c (cid:18) GMr c + O ( ℓ P l ) (cid:19)(cid:21) = c (cid:20) − β E g c (cid:18) GMr c (cid:19)(cid:21) . (26)Therefore, for the GUP under consideration, the difference between the speed of the graviton and photon, i.e., ∆˜ v g ,is given by ∆˜ v g = c − ˜ v g = 3 β E g c (cid:18) GMr c (cid:19) . (27)In Section II, we derived a bound (see Eq. (17)) for the difference in the speed of photon and graviton, which yields∆˜ v g = 3 β E c (cid:18) GMr c (cid:19) ≤ ∆ v g = ⇒ β ≤ ∆ v g M P l c E g (cid:18) − GMr c (cid:19) . (28)At this point, we note a couple of comments. First, taking the r → ∞ limit in Eq. (27), one reproduces the flat spaceresult, i.e., ∆˜ v g = β E g M Pl c , as expected [44]. Second, for the event GW150914, we take the final black hole mass to be M = 62 M ⊙ and the luminosity distance to be r = 410 Mpc. Substituting the aforesaid values in Eq. (28) and usingthe upper bound from Eq. (17), i.e. ∆ v g = 5 . × − m/sec, we obtain β ≤ . × . (29)It should be noted that this bound is in agreement with the corresponding one obtained in Ref. [44]. In addition, it isone of lowest bounds among those obtained from observations in the sky [45, 51] and expected to improve significantlyover time with increasing accuracies of GW observations. A full analysis, with no small m g approximation, is given in Appendix B. For a more detailed derivation see Appendix B. A. ∆˜ v g with the GUP-modified speed of photon and GW190521 One now considers the case in which the velocity of the photon is also GUP-modified similar to the velocity of thegraviton. Therefore, adopting the previous analysis for the velocity of the graviton with the small mass approximation,the velocity of the photon takes the form (see Eq. (26))˜ v γ = c (cid:20) − β E γ c (cid:18) GMr c (cid:19)(cid:21) (30)where ˜ v γ is the GUP-modified velocity of the photon and E γ is the energy of the photon. Hence, the difference in thespeeds, i.e., ∆˜ v g , reads now ∆˜ v g = | ˜ v γ − ˜ v g | = 3 βc (cid:12)(cid:12)(cid:0) E g − E γ (cid:1)(cid:12)(cid:12) (cid:18) GMr c (cid:19) . (31)At this point, it should be stressed that since the speed of the photon may vary due to the curvature and QG effects,the speed of the graviton may become larger than the modified speed of photon. Therefore, from now on, we willtake the absolute value of the difference between the speed of the graviton and the speed of the photon in order todefine the quantity ∆˜ v g . Finally, employing the bound on the graviton mass in Eq. (31), we obtain a bound on theGUP parameter β β ≤ ∆ v g M P l c (cid:12)(cid:12)(cid:0) E g − E γ (cid:1)(cid:12)(cid:12) (cid:18) − GMr c (cid:19) . (32)A number of comments are now in order. First, in order to compute the bound given in Eq. (32), one needs to usedata from a GW event for which the GW signal and its EM counterpart have both been detected. Until now, the onlysuch GW event is the GW190521, a result from the merger of two black holes [47]. Second, for the event GW190521,we take the final black hole mass to be M = 142 M ⊙ and the luminosity distance to be r = 5 . E g = 2 . × − eV.Therefore for the previously mentioned upper bound on the graviton mass, namely 1 . × − eV /c , the differencebetween the speed of the photon and graviton becomes: ∆ v g = c − v g = m g c E g ≤ . × − m/sec. Third, theEM counterpart of GW190521 was detected by the ZTF [48]. In particular, the ZTF “sees” two frequency bands andspecifically for the GW190521, in the g-band the observed wavelength is λ γg = 4686 × − m while in the r-bandthe observed wavelength is λ γr = 6166 × − m. Therefore, the energies corresponding to these observed photonwavelengths are E γg = 2 .
65 eV and E γr = 2 .
01 eV. Finally, substituting the above numerical values in Eq. (32), theupper bounds on the GUP parameter using the data for the r- and g- band read, respectively, β ≤ ∆ v g M P l c (cid:12)(cid:12)(cid:0) E g − E γr (cid:1)(cid:12)(cid:12) (cid:18) − GMr c (cid:19) = 1 . × (33) β ≤ ∆ v g M P l c (cid:12)(cid:12)(cid:0) E g − E γg (cid:1)(cid:12)(cid:12) (cid:18) − GMr c (cid:19) = 8 . × (34)where we have used the upper bound, obtained above, i.e. ∆ v g = 3 . × − m/sec.In the above, the curvature term GMr c is approximately equal to 0 . × − and, therefore, it was neglected inobtaining the numerical values in Eqs. (33) and (34). IV. BOUND ON THE LQGUP PARAMETER FROM GW150914
In this section, we will study the modifications in the difference between the speed of graviton and photon, i.e., ∆ v g ,in Schwarzschild spacetime while incorporating the LQGUP version. We define the physical position and momentain terms of the canonical auxiliary variables as follows x i = x i , p = k (35a) p i = k i (1 − α k + 2 α k ) . (35b)We follow a procedure similar to Section III and allow terms up to the quadratic order of the parameter α , as well asof the LQGUP parameter α . Employing the expression for the GUP-modified squared 4-momentum and substitutingEq. (35b), we obtain g ( p ) = − m g c − p − α kk + 5 α k k . (36)We perform an inverse transformation of Eq. (35b) and write k as a function of p in the form k = p (1 + 2 α k − α k ) . (37)Solving the above quadratic equation, we get k = αp ± (cid:20) α p p + p (cid:21) / (1 + α p ) (38)which upon simplifying yields k = α p ± p . (39)The above, substituted in Eq. (36), gives( p ) = 1 g (cid:20) − m g c − p ∓ α pp − α p p (cid:21) . (40)Following the analysis in Section III, we obtain the energy of the graviton to be E g = √− g (cid:20) m g c + p c (1 ± α p ) (cid:21) / (41)which is the modified dispersion relation for the LQGUP under consideration. At this point, we follow the analysis assimilar to Section III for the iteration method (see below Eq. (25)), and the group velocity of the graviton becomes ˜ v g = 1 √− g ∂E g ∂p = c (cid:20) ± α E g c (cid:18) GMrc (cid:19)(cid:21) . (42)Let us now consider separately the two cases in Eq. (42)Case 1 : ˜ v (1) g = c (cid:20) α E g c (cid:18) GMrc (cid:19)(cid:21) (43)Case 2 : ˜ v (2) g = c (cid:20) − α E g c (cid:18) GMrc (cid:19)(cid:21) . (44)Eq. (43) gives the speed of the graviton to be greater than c which is the speed of photon in vacuum. We ignore suchsuperluminal propagation and, thus, we proceed with Case 2 associated with a subluminal graviton.For this case, the difference between the speed of the graviton and the speed of the photon is now of the form∆˜ v g = | c − ˜ v g | = 2 α E g (cid:18) GMrc (cid:19) . (45)As in Section III, the difference in the speed of graviton and photon is bounded from above by ∆ v g , and this sets abound on the LQGUP parameters as follows α ≤ ∆ v g E g (cid:18) GMrc (cid:19) − α ≤ ∆ v g M P l c E g (cid:18) − GMrc (cid:19) (46)where we have neglected terms of O ( ℓ P l ). Finally, if we use the data set from the event GW150914 [42, 43], as we didin Section III, we obtain an upper bound on the LQGUP parameter α ≤ . × . (47)At this point, it is worth mentioning that this bound agrees with the corresponding ones obtained in Ref. [44, 45]. Inaddition, it is expected to improve significantly over time with increasing accuracies of GW observations. A full analysis, with no small m g approximation, is given in Appendix C. A. ∆˜ v g with the LQGUP-modified speed of photon and GW190521 As in Subsection IIIA, we now consider the case that the velocity of the photon is also LQGUP-modified similar tothe velocity of the graviton. Therefore, adopting the previous analysis for the velocity of the graviton with small massapproximation (see Eq. (42)), we get ˜ v γ = c (cid:20) ± α E γ c (cid:18) GMrc (cid:19)(cid:21) . (48)We now explore all possible cases of ˜ v γ and ˜ v g .Case 1 : ˜ v (1) g = c (cid:20) α E g c (cid:18) GMrc (cid:19)(cid:21) ˜ v (1) γ = c (cid:20) α E γ c (cid:18) GMrc (cid:19)(cid:21) . (49)Since the above equation gives the speed of the graviton to be greater than c , i.e., it is superluminal, we drop thiscase from future considerations and proceed to the remaining Cases.Case 2 : ˜ v (2) g = c (cid:20) α E g c (cid:18) GMrc (cid:19)(cid:21) ˜ v (2) γ = c (cid:20) − α E γ c (cid:18) GMrc (cid:19)(cid:21) . (50)As in Case 1, Case 2 also introduces a superluminal and hence unphysical graviton. Thus, we drop Case 2 as well.Case 3 : ˜ v (3) g = c (cid:20) − α E g c (cid:18) GMrc (cid:19)(cid:21) ˜ v (3) γ = c (cid:20) α E γ c (cid:18) GMrc (cid:19)(cid:21) . (51)In this case, the GUP-modified speed of photon is greater than c, i.e., it is superluminal, thus we drop this case fromfuture considerations. Case 4 : ˜ v (4) g = c (cid:20) − α E g c (cid:18) GMrc (cid:19)(cid:21) ˜ v (4) γ = c (cid:20) − α E γ c (cid:18) GMrc (cid:19)(cid:21) . (52)The above equations gives us a difference between the speed of the graviton and photon of the form∆˜ v (4) g = 2 α (cid:12)(cid:12)(cid:2) E g − E γ (cid:3)(cid:12)(cid:12) (cid:18) GMrc (cid:19) (53)which leads to the bound on the LQGUP parameter of the form α ≤ ∆ v g M P l c (cid:12)(cid:12)(cid:0) E g − E γ (cid:1)(cid:12)(cid:12) (cid:18) − GMrc (cid:19) . (54)Finally, we utilize the data for the event GW190521 as given by LIGO Scientific and Virgo Collaborations [47] and byZTF [48]. Using the data for the r- and g- bands, we obtain the upper bound on the LQGUP parameter, respectively,as follows α ≤ ∆ v g M P l c (cid:12)(cid:12)(cid:0) E g − E γr (cid:1)(cid:12)(cid:12) (cid:18) − GMrc (cid:19) = 3 . × (55)and α ≤ ∆ v g M P l c (cid:12)(cid:12)(cid:0) E g − E γg (cid:1)(cid:12)(cid:12) (cid:18) − GMrc (cid:19) = 2 . × (56)where we use the upper bound ∆ v g = 3 . × − m/sec. V. CONCLUSION
The existence of a minimum measurable length has been predicted by candidate theories of QG as well as fromother considerations such as from the physics of black holes. This necessitates the modification of HUP to GUP. Animplication of GUP is the modification of the standard dispersion relation and, consequently, of the speed of particles.From theoretical studies on GWs, one can bound the mass of the graviton and, thus, the speed of the graviton. Inour analysis here, we consider two versions of GUP: one with a quadratic term in momentum (GUP) and the otherwith linear and quadratic terms in momentum (LQGUP). It should be noted that we employed LQGUP, since it canbe viewed as an “effective theory” from a Lorentz covariant theory, e.g. as described in Ref. [52].In the current work, we consider a curved spacetime background and employing GUP, we obtain upper bounds forthe GUP parameters. Using the data from GW150914, the GUP parameter is bounded as β < . × while theLQGUP parameter is bounded as α < . × . From these results, it is evident that the effects of the curvedbackground are negligible since our results are the same with the existing ones in the literature, which were derivedfor a flat background. However, one can consider the speed of the photon to be modified as well, along with the speedof the graviton. In this case, one can use the data from GW190821, since this is the only gravitational event betweentwo black holes which has electromagnetic part that was detected by ZTF. So taking into consideration the observedenergies of the photons, the GUP parameter is bounded as β < . × for the r-band observed by ZTF, and β < . × for the g-band observed by ZTF. The LQGUP parameter is bounded as α < . × for ther-band observed by ZTF, and α < . × for the g-band observed by ZTF. These bounds are among the bestcompared to the ones existing in the literature, and specifically the bounds on the dimensionless LQGUP parameter α are the tightest. Therefore, GW observations give strict bounds on the GUP parameters, especially when employingdata from different “messenger” signals which describe the same gravitational event. This underscores the need formore and further advanced gravitational wave detectors and, that of multimessenger observations. We note that inour analysis, we have assumed small but non-zero masses for the graviton and photon, consistent with observationalbounds on these masses.Finally, it should be noted that recently, the LIGO Scientific and Virgo Collaborations have released their updatedcatalogue of GW detections, i.e., GWTC-2, in which the bound on the mass of the graviton is even stricter and, forthe first time, based on observational data. The new upper bound on the graviton mass is m g ≤ . × − eV/c [53]. This bound is tighter by a factor of 10 which means that all the bounds derived in our work can be improvedby a factor of 10. VI. ACKNOWLEDGMENTS
ECV would like to thank E. Berti and M. Kasliwal for useful correspondences and fruitful comments. This work wassupported by the Natural Sciences and Engineering Research Council of Canada.
Appendix A: Speed of photon and graviton in the Schwarzschild background with massive photon
We consider the photon with a small but non-zero mass. Then, following the analysis in Section II and employingEq. (15), adapted to the massive photon, we obtain v γ = c (cid:20) − m γ c E γ (cid:18) − GMrc (cid:19)(cid:21) (A1)where m γ is the mass of the photon and we have reasonably assumed m γ c E γ ≪
1. Therefore, the difference in thespeed of the graviton and photon assumes the form∆ v g = c (cid:18) − GMrc (cid:19)(cid:18) m g E g − m γ E γ (cid:19) . (A2) Appendix B: Bound on GUP parameter with massive photon and graviton
One may now want to include QG effects due to the GUP version, as we did in Section III, when the photon andgraviton are both massive. So, we get the zeroth order solution for p by setting β = 0 in Eq. (23) which leads to0 p = ± (cid:20) E g ( − g ) c − m g c (cid:21) / and consider only the positive root for the momentum. For our convenience, we define E g = (cid:20) E g ( − g ) c − m g c (cid:21) and substitute it in Eq. (23) in order to get the momentum in the form p = p ( E g ). Finally,we substitute in the (LHS) of Eq. (23) the zeroth order expression E g √− g = ( m g c + p c ) / and get( m g c + p c ) / = (cid:20) m g c + c E g (1 − β E g ) (cid:21) / (B1)which yields p = E g (1 − β E g ) (B2) p = E / g (1 − β E g + O ( β )) . (B3)Since β ∼ ℓ P l , we neglect terms of higher order of β , so Eq. (B3) becomes p ( E g ) = E / g (1 − β E g ) and substitutingit in Eq. (24), we get ˜ v g = pc (1 − β p ) (cid:20) m g c + p c − β p p c (cid:21) / = c (1 − β p ) (cid:20) (cid:18) m g c p − β p (cid:19)(cid:21) − / . (B4)Now, we expand, the last term in the (RHS) of Eq. (B4), which is in the square brackets with respect to the parameter β and neglecting the terms of higher order of β , we obtain˜ v g = c (1 − β p ) (cid:20) − m g c p + β p + 3 m g c p − βm g c (cid:21) = c (cid:20) − m g c E g − β m g c m g c E g − β E g (cid:21) . (B5)We evaluate the second term in the (RHS) of Eq. (B5) m g c E g = m g c (cid:20) E g c ( − g ) − m g c (cid:21) = m g c (cid:20) E g c (cid:18) − GMrc (cid:19) − m g c (cid:21) = m g c (cid:20) E g c (cid:18) GMrc (cid:19) − m g c (cid:21) (B6)in which, using the data from GW190521, the terms below assume the values2 GMrc ∼ . × − , m g ∼ . × − eV /c , E g = 2 . × − eV . (B7)So, utilizing the above numerical values the second and fourth terms in the (RHS) of Eq. (B5) become m g c E g ∼ . × − and m g c E g ∼ . × − . Therefore, neglecting the terms m g c E g and m g c E g with respect to unity in Eq.(B5), the speed of the graviton reads ˜ v g = c − β m g c − β E g c . (B8)Following exactly the same analysis for the case of the massive photon, the speed of the photon will be of the form˜ v γ = c − β m γ c − β E γ c . (B9)At this point a number of comments are in order. First, we use the data for the event GW190521, and, specifically, forthe photon energy, we employ the numerical values given by ZTF, thus E γg = 2 .
65 eV and E γr = 2 .
02 eV. It is clear1that these numerical values are much higher than the energy of the graviton, i.e., E g , and, thus, the correspondingquantity E γ is larger than E g . Second, taking the above-mentioned comments into consideration, similar to the massivegraviton, we reasonably neglect the terms m γ c E γ and m γ c E γ with respect to unity.Therefore, the difference in the speed of the graviton and photon now reads∆˜ v g = | ˜ v γ − ˜ v g | = β (cid:20) c ( m γ − m g )2 + 3 c (cid:18) GMrc (cid:19) (cid:12)(cid:12) ( E g − E γ ) (cid:12)(cid:12) (cid:21) . (B10)Therefore, as in Section III, the difference in the speed of the graviton and photon is bounded from above by ∆ v g asgiven by Eq. (A2), and this sets a bound on the GUP parameter as follows β (cid:20) c ( m γ − m g )2 + c (cid:18) GMrc (cid:19) (cid:12)(cid:12) ( E g − E γ ) (cid:12)(cid:12) (cid:21) ≤ ∆ v g = ⇒ β ≤ ∆ v g M P l c (cid:20) c ( m γ − m g )2 + c (cid:18) GMrc (cid:19) (cid:12)(cid:12) ( E g − E γ ) (cid:12)(cid:12) (cid:21) − . (B11)A couple of comments are in order. First, if we had a bound for the mass of the photon in the context of GWobservations, then from Eq. (B11) the upper bound on β can be estimated. This is because all other quantities areknown for the event GW190521 by the data given for the gravitons by the LIGO Scientific and Virgo Collaborations[47] and for the photons by ZTF [48]. Second, if the masses of graviton and photon are taken to be zero, i.e.,( m γ , m g ) →
0, one obtains Eq. (31) from Eq.(B10), as expected.
Appendix C: Bound on LQGUP parameter with massive photon and graviton
One may now want to include QG effects due to LQGUP version, as we did in Section IV, when the photon andgraviton are both massive. So, we get the zeroth order solution for the momentum p of Eq. (41) without the smallmass approximation for the graviton. For our convenience, we use again the quantity E g = (cid:20) E g ( − g ) c − m g c (cid:21) andsubstitute it in Eq. (41) in order to get the momentum in the form p = p ( E g ). Then, we follow a similar analysis asin Appendix B to get p = E g (1 ± α E / g ) (C1)and the speed of the graviton now reads˜ v g = 1 √− g ∂E g ∂p = ∂∂p (cid:20) m g c + p c (1 ± α p ) (cid:21) / = pc + 2 α c pp ± αc p (cid:20) m g c + p c (1 ± α p + α p ) (cid:21) / . (C2)We now expand the denominator in the (RHS) of Eq. (C2) and keeping terms up to O ( α ), we get (cid:20) m g c + p c (1 ± α p + α p ) (cid:21) − / = ( pc ) − (cid:20) (cid:18) m g c p ± α p + α p (cid:19)(cid:21) − / (C3)= ( pc ) − (cid:20) − (cid:18) m g c p ± α p + α p (cid:19) + 38 (cid:18) m g c p ± α p + α p (cid:19) + . . . (cid:21) (C4)= ( pc ) − (cid:20) − m g c p + 3 m g c p ∓ α p ± α m g c p + α p + 3 α m g c (cid:21) . (C5)It is easily seen that there are two cases to be studied in Eq. (C2), one for each sign. Therefore, we take the first caseand write the Eq. (C2) as follows˜ v g = c (cid:20) α p + 3 α p (cid:21) (cid:20) − m g c p + 3 m g c p − α p + 3 α m g c p + α p + 3 α m g c (cid:21) (C6)= c (cid:20) − m g c p + 3 m g c p + 2 α p + 9 α m g c p + 17 α m g c α m g c p (cid:21) . (C7)2Now, we write the above equation in terms of E g and utilize the same analysis as the one adopted in the previousAppendix. We neglect the terms m g c E g and m g c E g with respect to unity and so the speed of the graviton becomes˜ v g = c (cid:20) α E / g + α m g c E / g − α m g c E / g + 2 α E g + 9 α m g c E g + 11 α m g c (cid:21) . (C8)Similarly, the speed of photon becomes˜ v γ = c (cid:20) α E / γ + α m γ c E / γ − α m γ c E / γ + 2 α E γ + 9 α m γ c E γ + 11 α m γ c (cid:21) . (C9)Therefore, the difference between the speed of graviton and photon turns out to be | ∆˜ v g | = | ˜ v γ − ˜ v g | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) αc ( E / γ − E / g ) + α c (cid:18) m γ E / γ − m g E / g (cid:19) − α c (cid:18) m γ E / γ − m g E / g (cid:19) + 2 α c ( E γ − E g ) +9 α c (cid:18) m γ E γ − m g E g (cid:19) + 11 α c (cid:18) m γ − m g (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (C10)As in Section IV, the difference in the speed of graviton and photon is bounded from above by ∆ v g as given by Eq.(A2), and this sets a bound on the LQGUP parameter as follows (cid:20) αc ( E / γ − E / g ) + α c (cid:18) m γ E / γ − m g E / g (cid:19) − α c (cid:18) m γ E / γ − m g E / g (cid:19) + 2 α c ( E γ − E g ) +9 α c (cid:18) m γ E γ − m g E g (cid:19) + 11 α c (cid:18) m γ − m g (cid:19)(cid:21) ≤ ∆ v g . (C11)It should be noted that both particles, namely graviton and photon, in this case appear to be superluminal particles.Next, we take the second possible case of Eq. (C2) which leads to the speed of the graviton to be of the form˜ v g = c (cid:20) − α p + 2 α p (cid:21) (cid:20) − m g c p + 3 m g c p + α p − α m g c p + α p + 3 α m g c (cid:21) (C12)= c (cid:20) − m g c p + 3 m g c p − α p − α m g c p + 17 α m g c α m g c p (cid:21) . (C13)Employing now p = E g (1 − α E / g ) in the above equation and proceeding as before, we obtain˜ v g = c (cid:20) − α E / g − α m g c E / g + 3 α m g c E / g + 2 α E g + 9 α m g c E g + 11 α m g c (cid:21) (C14)and ˜ v γ = c (cid:20) − α E / γ − α m γ c E / γ + 3 α m γ c E / γ + 2 α E γ + 9 α m γ c E γ + 11 α m γ c (cid:21) . (C15)In this second case, the difference between the speed of graviton and photon reads | ∆˜ v g | = | ˜ v γ − ˜ v g | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) α c (cid:18) m γ E / γ − m g E / g (cid:19) − αc ( E / γ − E / g ) + 3 α c (cid:18) m γ E / γ − m g E / g (cid:19) + 2 α c ( E γ − E g ) +9 α c (cid:18) m γ E γ − m g E g (cid:19) + 11 α c (cid:18) m γ − m g (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . 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