Shortcomings of Shapiro delay-based tests of the on cosmological scales

Olivier Minazzoli,1 Nathan K. Johnson-McDaniel,2 and Mairi Sakellariadou3 1Centre Scientifique de Monaco, 8 Quai Antoine 1er, 98000, Monaco Artemis, Universit´eCˆoted’Azur, CNRS, Observatoire Cˆoted’Azur, BP4229, 06304, Nice Cedex 4, France 2Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom 3Theoretical Particle Physics and Cosmology Group, Physics Department, King’s College London, University of London, Strand, London WC2R 2LS, United Kingdom The “Shapiro delay” experienced by an astronomical messenger traveling through a gravitational field has been used to place constraints on possible deviations from the equivalence principle. The standard Shapiro delay used to obtain these constraints is not itself an observable in general relativ- ity, but is rather obtained by comparing with a fiducial Euclidean distance. There is not a mapping between the constraints obtained in this manner and alternative theories that exhibit equivalence principle violations. However, even assuming that the comparison with the fiducial Euclidean dis- tance is carried out in a way that is useful for some class of alternative theories, we show that the standard calculation of these constraints cannot be applied on cosmological scales, as is often done. Specifically, we find that the Shapiro delay computed in the standard way (taking the Newtonian potential to vanish at infinity) diverges as one includes many remote sources. We use an infinite homogeneous lattice model to illustrate this divergence, and also show how the divergence can be cured by using Fermi coordinates associated with an observer. With this correction, one finds that the Shapiro delay is no longer monotonic with the number of sources. Thus, one cannot compute a conservative lower bound on the Shapiro delay using a subset of the sources of the gravitational field without further assumptions and/or observational input. As an illustration, we compute the Shapiro delay by applying the Fermi coordinate expression to two catalogs of galaxy clusters, illustrating the dependence of the result on the completeness of the catalogue and the mass estimates.

I. INTRODUCTION As a consequence, there is a large collection of papers dealing with this theme [3–37], starting with Longo [38], In the standard model of fundamental physics, differ- Krauss and Tremaine [39], LoSecco [40], and Pakvasa et ent types of electromagnetic and gravitational waves all al. [41], although three of these papers actually compared travel on null geodesics in the geometric optics limit (see, objects following null and non-null geodesics (electro- e.g., [1], who demonstrates this for gravitational waves).1 magnetic waves and or antineutrinos); LoSecco 2 In other words, they may be different either with re- compares neutrinos and antineutrinos. All of these pa- spect to their respective energies (e.g., gamma versus ra- pers consider possible differences in the Shapiro delay [43] 3 dio electromagnetic waves), or to the fundamental field experienced by messengers with different properties. they are associated with (e.g., gravitational versus elec- The most recent surge of publications on this theme tromagnetic waves), or other properties (e.g., polariza- has followed the quasi-coincident detections of gravita- tion), and still have trajectories that follow from the same tional waves (GW170817) and a short-duration gamma- null-geodesic equation. Tests of this statement can also ray burst (GRB 170817A), from which many constraints be extended to uncharged massive objects with the same on the universality of the null-geodesic equation have

arXiv:1907.12453v2 [gr-qc] 4 Nov 2019 mass, e.g., matter versus antimatter, which will follow been derived in the literature, starting from the one of the same timelike geodesics. This property takes its root the LIGO, Virgo, Fermi GBM, and INTEGRAL collab- from the Einstein equivalence principle that led to gen- orations [46]. eral relativity minimally coupled to the standard model Although there exist various small differences between of particle physics. Therefore, being able to test the uni- these analyses, the vast majority of them—starting with versality of the geodesic equation is a test of the equiv- Longo [38]—assume that the metric perturbation is null alence principle, and any deviation from it would be an at infinity and then simply apply the usual parameterized indicator of physics beyond the current standard model of physics.

1 It has been shown that the propagation of gravitational waves 2 However, at the time those papers were written, it was not known does not follow the laws of geometric optics for lenses with masses whether neutrinos had a mass, though [42] showed that the neu- 5 −1 less than approximately 10 M (f/Hz) , where f is the gravi- trinos’ mass has a negligible effect on these constraints. tational wave frequency [2]. However, this is not relevant for the 3 Note that the “Shapiro time delay” is also known as the “gravi- cases considered here. tational time delay” in the lensing community [44, 45]. 2 post-Newtonian Shapiro delay equation: dinates. If one changes the coordinates, then one can ob- tain both positive and negative values of the time delay, Z rO 1 + γ −4 as illustrated in Gao and Wald [48]. Thus, applications of δT = − 3 U(r(l))dl + O(c ), (1) c rE this expression to equivalence principle constraints tac- itly assume that the coordinates used to obtain it are where rE and rO denote emission and observation posi- somehow preferred in the context of the equivalence- tions, respectively, U(r) is the Newtonian gravitational principle violating theories being tested, as discussed fur- potential, and the integral is computed along the tra- ther in Sec.II. jectory. γ is the parameter appearing in the space-space −2 Another issue with Eq. (2) is that on cosmological component of the c parametrized post-Newtonian met- scales—and even if the universe was flat and static— ric. In the parametrized post-Newtonian framework, γ there actually are many distant sources (all the way to will be the same for different messengers, since it is a infinity, at least in the static universe case). Therefore, property of the metric. Nevertheless, for the tests of the the assumption that the gravitational potential is null at equivalence principle we are considering, the standard infinity is at best an approximation. In what follows, we approach is to assign different values of γ to different will show that it is actually an inappropriate assump- messengers, though this is not derived from any specific tion on cosmological scales, because in some situations it alternative theory or fundamental principle—see the dis- can lead to an unphysical divergence of the Shapiro delay cussion in Sec.IIB. with the number of the gravitational sources contribut- Equation (1) is gauge dependent, so employing it to ing to the delay.5 To demonstrate this divergence, we will constrain deviations from the equivalence principle in- use an infinite homogeneous lattice toy model, because volves some assumptions, which are usually tacit and are it allows one to quantify the divergence arithmetically— discussed further below. Additionally, it implicitly as- and because at the same time, it also has been shown to sumes that the trajectory is short enough that one can be a good model of the cosmological metric in [49]. treat the region of containing it as Minkowski What we find is that assuming a null potential at in- plus a linear perturbation, to a good approximation. finity corresponds to imposing an unsuitable choice of We will see that this assumption is well justified for gauge (related to an ambiguous choice of the coordinate sources like GW170817, but not for sources with red- > time), and that the unphysical infinite quantities disap- shifts z ∼ 1, like, for instance, the gamma-ray bursts at pear as soon as one use an appropriate gauge (with a z = 1.5, z = 2.2, z = 2.6, or even z = 11.97 considered coordinate time related for instance to the in [14], [23], [19], and [25], respectively. While Nusser [10] of an observer). Nevertheless, we also find that a cor- gives a formulation of the constraint that is applicable to rected Shapiro delay equation is no longer monotonic more distant sources, the previously cited papers use the with the number of considered sources, which implies standard formulation in Eq. (1). that one cannot simply use a subset of the sources in If one assumes Keplerian potentials (which is a good order to estimate a conservative minimum of the Shapiro approximation for all sources that are sufficiently far from delay, which in turns imply that one cannot get a con- the line of sight), the previous equation reduces to [47] servative estimate of the test of the universality of the    null-geodesic equation, unless the trajectory lies in a re- X GMP rP + RPE + REO −4 gion of spacetime where one has measurements of the δT = (1+γ) 3 ln +O(c ), c rP + RPE − REO P gravitational field. (2) Overall, the goal of the present study is to show that, where rP := k~xP k and RXY := k~xY − ~xX k (k · k de- even with the usual tacit assumptions made in construct- notes the Euclidean distance).4 In this situation, one ing the test, it is not possible to obtain a conservative can check that the Shapiro delay is monotonic with re- bound on violations of the equivalence principle in cos- spect to the number of sources. As a consequence, one is mological situations with the standard method. allowed to consider only a subset of the sources in order As a consequence, we think that most (if not all) of to be able to estimate a conservative minimum value of the constraints on violations of the equivalence principle the Shapiro delay, which is necessary in order to give a from propagation over cosmological distances given in conservative limit on the violation of the equivalence of the literature so far should be taken with a great deal of the null-geodesic equation. caution. However, as mentioned above, Eq. (1) [and thus Eq. (2)] is gauge dependent, in that it compares the prop- In Sec.II, we discuss the basics of the Shapiro de- agation time in the curved spacetime to that of the back- lay constraints on the equivalence principle considered in ground Minkowski spacetime in a particular set of coor- the literature, including the tacit assumptions underly- ing them. In Sec.III, we present a preliminary discussion

4 Note that this equation can be given in various equivalent forms by re-arranging the terms in the parentheses in terms of other 5 The divergence already arises at the level of the metric pertur- geometrical quantities. See for instance Eq. (1) in [38]. bation. 3 on the relevance of considering a perturbed Minkowski Nevertheless, again assuming asymptotic flatness, one spacetime on cosmological scales. In Sec.IV, we use a can also obtain the same Shapiro delay expression (in the homogeneous lattice universe in order to arithmetically GR case), up to higher-order PN corrections, by consid- show that the gauge conditions that are usually implic- ering the (gauge-invariant) affine distance daff (discussed itly used in the literature lead to a divergence of the in, e.g., Sec. 2.4 of [52]). We compute the affine distance Newtonian potential, and we give a cure to this nonphys- for a general post-Newtonian metric in AppendixA, ical divergence. In Sec.V, using a fiducial gauge that is showing that it gives the distance one would na¨ıvely com- tied to an observer’s proper time, we derive an analytical pute using the background Minkowski metric in the co- expression of the Shapiro delay in an infinite inhomoge- ordinates in which the post-Newtonian metric is given neous flat Keplerian universe in order to show that the for an observer at rest with respect to that background Shapiro delay is no longer monotonic with the number metric. One then obtains the Shapiro delay (plus higher- of considered sources in general, contrary to the conven- order PN effects) by subtracting daff/c (where c is the tional wisdom. In Sec.VI, we apply our Shapiro delay ) from the total one-way propagation de- expression to two catalogs of galaxy clusters in order to lay. This assumes that the metric is known, which al- further show the behavior of the Shapiro delay with re- lows one to compute the affine distance. Therefore, at alistic distributions of matter. We give some concluding least in asymptotically flat and stationary cases, there is remarks in Sec.VII. Additionally, we compute the affine a gauge-invariant definition of the Shapiro delay. How- distance in a post-Newtonian metric in AppendixA and ever, this definition is observer-dependent, because the give a convergence proof for some lattice sums we con- affine distance depends on the observer 4-velocity. One sider in the AppendixB. notably gets the standard definition for a specific class of observers which are at rest with respect to the fiducial background metric. Also, note that while the affine dis- II. BASICS OF SHAPIRO DELAY tance is gauge invariant, it cannot be obtained directly CONSTRAINTS ON THE EQUIVALENCE from standard astronomical observations. PRINCIPLE However, the affine distance is also the distance one would obtain by starting from the angular distance and A. Definition of the Shapiro delay correcting for the magnification due to - ing [see, e.g., Eq. (42) in [52] for this definition of the magnification]. In any space-time geometry, and for Shapiro delay-based constraints on the equivalence any theory of in which the reciprocity relation principle only consider a portion of the full propaga- holds and the intensity is conserved, the angular dis- tion time between two spacetime points—the full prop- tance d and the luminosity distance d are related agation time is also known as the one-way propagation ang lum by d (z) = (1 + z2)d (z), where z is the redshift [53– time.6 This one-way propagation time can be obtained lum ang 56]. The luminosity distance is an observable for compact unambiguously from an observable in some situations, binary coalescence observations with gravitational waves and therefore is itself observable in those situations. In- as well as for electromagnetic observations of a source deed, if one considers a stationary spacetime, with the with known luminosity (a “standard candle”). observer and source at rest, then the one-way propaga- tion time is just half of the two-way propagation time, In practice, for negligible redshift, this means that if where the two-way propagation time would be the proper one can measure both the luminosity distance dlum, or time measured by an observer between sending a signal the angular distance dang, and the one-way propagation to a reflector and receiving the reflection. time T , the Shapiro delay (plus higher-order PN effects) can simply be defined by δT = T − d /c, provided However, even in a case where the one-way propagation lum that the magnification from gravitational lensing is neg- time is observable, one still has to define a way of split- ligible.7 This could be the case in the Solar System, for ting what one refers to as the Shapiro delay from the total instance, if one was in a situation where one can ap- propagation time. Such a splitting is, in general, gauge proximate the gravitational field in the Solar System by dependent, as discussed for instance in [48]. In asymp- the stationary gravitational field of the Sun, since the totically flat cases, one obtains the usual expression in leading contribution to the magnification for small im- Eq. (1) in the standard parameterized post-Newtonian pact parameter gravitational lensing is quadratic in the (PPN) gauge (see Sec. 2.4 in [51]) used to obtain the mass of the lens [see, e.g., Eqs. (3) and (9) in [57]], while PPN expression. This expression is the basis of the orig- the Shapiro delay is linear in the mass. One could mea- inal tests in Longo [38] and Krauss and Tremaine [39]. sure the luminosity distance by observing a spacecraft with a known intrinsic transmitter power. However, as

6 In contrast, Solar System observations of the Shapiro delay (dis- cussed in, e.g., [50]) are based on differential measurements of the delay as the path changes, and not on the comparison of the 7 Note that this definition gives the Shapiro delay in terms of the observed delay with a fiducial value. observers proper time. 4 mentioned above, this is not how current Shapiro delay III. PRELIMINARIES constraints are obtained in the Solar System. Given the strong simplifying assumptions that were In this section, in order to match with the notation in necessary to properly define the Shapiro delay, one can the literature, we set G = c = 1. expect the analysis to be much more involved in situa- In what follows, we show that one can treat the cosmo- tions in which those simplifying assumptions no longer logical spacetime as Minkowski plus a linear perturbation hold. In particular, if one wants to consider cosmo- for sufficiently nearby sources (including GW170817), logical situations, then it is no longer a good approx- and also give some simple arguments for why the stan- imation to consider a stationary spacetime: The de- dard Shapiro delay calculations are not trustworthy on partures from stationarity are sufficiently large that cosmological scales. they cannot be neglected in these calculations, even for Since the Shapiro delay is usually considered in the the relatively short propagation times appropriate for post-Newtonian framework, where one is perturbing GW170817/GRB 170817A—they correspond to about around a Minkowski background, we first want to write 10% of the na¨ıve Shapiro delay, as shown in the next the FLRW background as Minkowski plus a linear per- section. While it may be possible to relate the two-way turbation, which one can do for a sufficiently small patch propagation delay to the one-way propagation delay in of spacetime around any observer. In order for the ap- the cases of interest, this relation would be rather com- proximation of a linear perturbation to be good, the plicated, so we will not pursue this avenue here. patch has to have dimensions much less than the Hub- −1 ble radius LH := H0 ' 4.4 Gpc, here taking the cur- rent value of the Hubble parameter H(t) to be H0 = −1 −1 B. Bounds on equivalence principle violations 68 km s Mpc ; cf. [60]. Specifically, we are interested in a patch containing an observer on the Earth and the Another thing that is usually assumed in the literature source of the radiation under investigation and expand the metric in powers of Hk~xk, where ~x is the displace- is that the propagation time of waves of different nature 9 would simply be related to Eq. (1) but with a different ment vector from the Earth. This combination is the parameter γ—which by definition would be a violation of natural dimensionless expansion parameter for this sit- the equivalence principle. This should be seen as a con- uation. In fact, we write the FLRW metric in Fermi venient way to infer quantitatively how close the Shapiro normal coordinates, where we have [e.g., Eq. (4) of [61]] time delays must be for waves of different nature, rather ds2 = −{1 − [H˙ (t) + H2(t)]k~xk2}dt2 than being a parametrization that takes its root from a FLRW fundamental theory. + [1 − H2(t)k~xk2/2]kd~xk2 + O(H4k~xk4). Indeed, as far as we are aware, the method of defining (3) the Shapiro delay to use in equivalence principle tests discussed in the previous subsection is not derived from, (See [62] for related discussion.) Here overdots denote or even inspired by, any alternative theories.8 The same derivatives with respect to cosmic time, so the Friedmann seems to be true of the expression used by Nusser [10] in equations for a spatially flat cosmology give H2 = (8πρ+ the cosmological case, where the Shapiro delay is defined Λ)/3 and H˙ +H2 = (−4πρ+Λ)/3, where ρ is the average with respect to the background Friedmann-Lemaˆıtre- energy density and Λ is the cosmological constant. Robertson-Walker (FLRW) cosmology. However, since We see that the neglected term is of order 10−8 for the the standard Shapiro delay expression is used in many distance of ∼ 40 Mpc appropriate for GW170817 (see, studies, for the purposes of this study we will assume e.g., [63]) and is thus completely negligible compared to that it gives a useful measure of equivalence principle the first-order perturbation, which is of the order of 104 violations in at least some alternative theories (e.g., ones larger. However, this term is of order unity for distances > in which there is a preferred coordinate system). around LH ' 4.4 Gpc or greater, i.e., z ∼ 1, so one cannot treat the metric as Minkowski plus a first-order Again, the goal of the present study is to show that, perturbation in those cases. We shall not consider such even with the strong assumptions listed in this section, cases further here. it is not possible to obtain a conservative bound on We now want to consider the case in which we have violations of the equivalence principle in cosmological a perturbation to FLRW from, e.g., a galaxy or galaxy situations—at least not with the standard method. cluster. As a simple model, we will use the McVittie spacetime [64], given in modern notation in, e.g., Eq. (20) of [65], which describes a spherically symmetric mass em- bedded in an expanding universe. We will only consider 8 There is some motivation for this sort of expression in Coley and Tremaine [58], considering propagation governed by different connections, though it seems tailored to produce the standard Shapiro delay expression. There are also suggestions that this 9 Here we use “the Earth” as a shorthand for a geodesic of the expression can be used to test emulators [59]. FLRW metric close to the actual position of the Earth. 5 this spacetime well away from the singularity at the hori- der) by zon, so that pathology is not a concern. For the case that gives a spatially flat FLRW metric in the limit where the embedded mass is zero, the McVittie metric in isotropic " # coordinates takes the form Z r1 2m H2(t(r)) + 2H˙ (t(r)) + r2 − 4mH(t(r)) dr r0 r 4 1 − µ(τ, ρ)2 2 2 2 4 2    2 dsMcV = − dτ + a (τ) [1 + µ(τ, ρ)] kd~yk , r1 (1 − 3Ωm)H0 3 3 1 + µ(τ, ρ) = 2m ln + (r1 − r0) (4a) r0 12 3  m 3ΩmH0 4 4 µ(τ, ρ) := , (4b) + (r − r ) − 4mH0(r1 − r0), 2ρa(τ) 8 1 0 (6) where τ and ~y are cosmic time and the comoving spa- tial coordinates, respectively (using the same notation as in [61]), ρ := k~yk, a(τ) is the scale factor, and m is the 2 ˙ 2 ˙ ˙ mass parameter of the embedded mass, which is equal to where we have used H (t)+2H(t) = H0 +2H0+2(H0H0+ ¨ 2 2 2 2 2 the Schwarzschild mass when a(τ) = 1. We now want H0)t + O(H0 t ) = [1 − 3(1 − 2H0t)Ωm]H0 + O(H0 t ), to express this metric in coordinates similar to the Fermi noted that t(r) = −r, and neglected the contribution normal coordinates used for the FLRW metric in Eq. (3). from the time dependence to the spatiotemporal terms, If we apply the coordinate transformation given in Eq. (3) since these already give a small contribution. The contri- of [61], we obtain butions of the three terms in the sum (taking the terms in brackets as a single term here) to the na¨ıve Shapiro  2m  delay are ∼ 160 days, ∼ 19 years (from individual con- ds2 = − 1 − [H˙ (t) + H2(t)]k~xk2 − dt2 McV k~xk tributions of ∼ 15 and ∼ 2 years), and ∼ −8 hours, respectively, corresponding to the contribution from the 8mH(t) − ~x · d~xdt Keplerian potential, the cosmological curvature, and the k~xk spatiotemporal terms. We thus see that the spatiotempo- (5)  H2(t)k~xk2 2m  ral terms are indeed negligible, although note that they + 1 − + kd~xk2 2 k~xk contribute negatively to the effect. The small factor of 1 − 3Ω ' 0.07 is part of why the time dependence (sec-  m2  m 4 4 2 ond term in brackets) is a relatively large correction to + O H k~xk , 2 , mH k~xk . k~xk the time-independent contribution (first term in brack- ets). Except for the mixed spatiotemporal terms, this is the same metric one would obtain if one na¨ıvely super- Additionally, the contribution from the Keplerian po- posed the Fermi normal coordinate linearized FLRW tential is much smaller than that from the cosmological metric [Eq. (3)] and the standard Newtonian order post- curvature. Thus, it might seem that one can compute Newtonian metric of a point mass usually used to com- this na¨ıve Shapiro delay using only the cosmological cur- pute the Shapiro delay. Note, however, that the metric vature to a good approximation, since it should take into in Eq. (5) no longer is in Fermi coordinates, compared account the contributions from the many distant galaxies to the metric in Eq. (3), because of the perturbing mass that yield the diverging contribution to the Shapiro de- terms. lay (see Sec.IVB) in the na¨ıve calculation using Eq. (2). In fact, the spatiotemporal terms are much smaller This may in fact be the case, but the Shapiro delay ob- than the perturbations in the diagonal terms for the ex- tained in this manner would not necessarily give a con- ample case we consider. Specifically, we consider the servative bound, since there will be negative contribu- same setup used in the equivalence principle constraint tions to the metric perturbation, and thus the Shapiro from the GW170817/GRB 170817A signals in [46], using delay, from nearby underdensities. Thus, a more careful the Milky Way’s Keplerian potential contribution with a calculation is necessary to assess the size of these con- 11 mass m = 2.5 × 10 M and minimum and maximum tributions to the Shapiro delay. This could in principle distances of r0 = 100 kpc and r1 = 26 Mpc (the 90% be substantial, given the maximum size of the Newto- credible level lower bound on the distance obtained solely nian potential of ∼ 10−4 mentioned in [67], which is an from gravitational waves from the analyses performed at order of magnitude larger than the maximum of the cos- the time of that paper [66]). We consider a radial path, mological curvature term in the Fermi coordinate FLRW for simplicity, and also use the cosmological parameters metric [Eq. (3)] at the 26 Mpc distance. It is possible that −1 −1 H0 = 68 km s Mpc ,Ωm = 0.31, and ΩΛ = 1 − Ωm a statistical argument like in Nusser [10] could be used (cf. the TT,TE,EE+lowE+lensing+BAO parameters in to give constraints on deviations from the cosmological Table 2 of [60]). The na¨ıve Shapiro delay including the curvature, even in the absence of direct measurements of spatiotemporal terms is given (suppressing the remain- the gravitational field along the line of sight. 6

IV. THE INFINITE HOMOGENEOUS LATTICE one is solving the equations in a single cell, so the appro- priate boundary conditions are those at the cell bound- In order to show the unrealistic divergence of the met- ary, not at infinity. The solution can be decomposed such ric perturbation with the number of sources, it is con- that venient to use a model where the perturbation can be Λ 2 computed arithmetically. An infinite homogeneous lat- ΦΛ = r , (12) 6 tice model allows such computations. Additionally, it has been shown to produce the results of the usual standard with r2 := x2 + y2 + z2, and model of cosmology with “small” back-reaction effects GM that are due to the discreteness of the perturbations con- Φ = + Φ (t) + δΦ (t)xl, (13) sidered [49]. In such a model, the universe is filled with M rc2 t l an infinite lattice of cells. All cells possess the same mass with the tidal potential [49, 68] at the same relative location, and the metric is given in terms of a post-Newtonian expansion in each cell.10 The GM X 1 evolution of the universe then follows from junction con- Φt := + Φ0(t), (14) c2 k~x − L(t)β~k ditions between cells, known as the Israel junction condi- ~ 3 β∈Z∗ tions. Therefore, such a model allows us to compute the Shapiro delay from an infinite set of masses uniformly dis- 3 3 ~ where Z∗ := Z \{0} and Φ0(t) and δΦl(t) are gauge- tributed, in an otherwise realistically evolving universe. dependent terms. In what follows, we set the mass at the origin of the lattice equal to 0, in order to consider an observer located at the origin, which simplifies the A. Metric within each cell expressions. We thus have

Φ = Φ (t) + δΦ (t)xl, (15) Given a cell with a sufficiently small size L(t)[49, 68], M t l the metric can be expanded around Minkowski spacetime and reads: B. An unsuitable choice of coordinate time 2 2 2 i j −3 ds = (−1 + h00) c dt + (δij + hij) dx dx + O(c ), (7) When considering a problem with a finite number of −2 bodies, the gauge is often restricted to the case that sat- where h00 and hij are the c perturbations of the metric (Latin letters denote spatial indices), such that [49] isfies

lim ΦM = Φ0(t) = 0, (16) h00 ≡ 2Φ = 2 (ΦM + ΦΛ) , (8) r→∞   ΦΛ lim ∂lΦM = δΦl(t) = 0, (17) h ≡ 2Ψδ = 2 Φ − δ . (9) r→∞ ij ij M 2 ij such that the coordinate time corresponds to the proper This metric is expressed in standard post-Newtonian co- time of an ideal observer situated at infinity. Then, if ordinates, so it takes this simple diagonal form to leading one considers, for instance, Solar System observables, one order [68]. At leading order in the post-Newtonian ap- simply converts this coordinate time to the proper time proximation, the Einstein equation with a cosmological of an actual observer. However, when there are (non- constant then reduces to negligible) sources located all the way up to infinity, the metric perturbation cannot be taken to be null at infinity, 4πG X (3)  ~ 4ΦM = − Mδ ~x − L(t)β , (10) so the previous gauge restrictions simply do not make c2 sense when there are sources located all the way up to β~∈ 3 Z infinity. The infinite lattice model allows us to illustrate and this from a simple arithmetic point of view: Φt is not finite when Φ0 = 0, which is a consequence to the fact that the Epstein zeta function 4ΦΛ = Λ, (11) X 1 where 4 is the flat-space Laplacian and M is the mass in (18) k~kkn each cell. Let us stress that in the lattice model [49, 68], ~ 3 k∈Z∗

is only finite for n > 3. (While this Epstein zeta function can be analytically continued to all of C with a simple 10 Recall that the post-Newtonian expansion assumes weak fields pole at n = 3—see, e.g., [69]—we will not consider this and slow motions, as well as a Minkowski background at the here, as we are only concerned with cases for which the scale of the phenomenon considered—here, within a cell. sums we are considering converge.) Indeed, using the 7

homogeneous property of the lattice model, and defining with ξ~ := ~x/L(t), one has X ξi − βi J i(ξ~) := . (24) GM X 1 kξ~ − β~k3 Φt(t) − Φ0(t) = . (19) β~∈ 3 L(t)c2 kξ~ − β~k Z∗ ~ 3 β∈Z∗ One can verify that J i(ξ~) is also finite. Additionally, One can verify that this series is divergent for all locations J i(~0) is obviously equal to zero thanks to the symmetry ~x. As a consequence, the metric perturbation at any of the sum (if the sum is performed in a way that re- given location would not be finite if one naively imposes spects this symmetry—see AppendixB). The last touch Φ0(t) = 0 by hand. As far as we are aware, Φ0(t) = 0 is in order to select Fermi coordinates is to demand that always assumed in the literature. ∂iΦM (~xO) = 0, where ~xO is the position of the observer, thus fixing δΦi. For an observer at the origin, we thus have δΦi = 0. C. Fixing the coordinate time issue

Interestingly enough, this divergence can be renormal- V. THE INFINITE INHOMOGENEOUS FLAT ized if one chooses instead to restrict the gauge freedom KEPLERIAN UNIVERSE to Fermi coordinates11 (i.e., by demanding that the tidal potential Φt and its gradient cancel out at the location Now that we have seen arithmetically with the lattice associated with an observer). It would mean that one model why one needs to carefully chose a non-ambiguous uses a coordinate system that follows the geodesic mo- coordinate time in order to have “meaningful” potentials tion of an observer. Note that this is the usual procedure in the metric, we can derive the Shapiro delay more prop- in the framework of reference frame theory in order to erly. In most of the literature, the universe is taken to be define a proper reference frame (see for instance [70]). In flat and perturbed by either some Keplerian potentials, this situation, the tidal potential would read or by some other types of potentials that are meant to depict dark matter in halos. We are not interested in GM ~ a detailed study here because our goal is to show that Φt(t) = 2 I(ξ), (20) L(t)c a conservative estimate of the Shapiro delay cannot be obtained with these kind of calculations anyway. Hence, with we will limit our study and assume Keplerian potentials. ! Also, we now consider a parametrized post-Newtonian X 1 1 I(ξ~) := − . (21) metric in order to be able to quantify the difference of kξ~ − β~k kβ~k ~ 3 the Shapiro delays for different messengers the way it is β∈Z∗ done in the literature—see Sec.IIB. Let us note that a One can verify that I(ξ~) is indeed finite (with the appro- parametrized post-Newtonian metric is compatible with priate prescription for the lattice summation, since it is the lattice model, see [71]. The metric we consider there- a conditionally convergent series—see AppendixB) for fore reads ~ 3 all ξ∈ / Z∗, while the two members of the sum lead to ds2 = (−1 + 2Φ)c2dt2 + (1 + 2γΦ)δ dxidxj + O(c−3), divergent series when taken separately. Let us also note ij ~ (25) that, by construction, I(0) = 0. where With the previous gauge restriction, the potential re- duces to G X MP l Φ = 2 + Φ0(t) + δΦl(t)x . (26) " # c k~x − ~xP k ~ P GM 1 I(ξ) l ΦM = 2 + + δΦl(t)x , (22) c r L(t) We choose our coordinate system such that they are Fermi coordinates associated to the observer: Φ(~x = where I(ξ~) is defined in Eq. (21). ~xO = ~0) = 0 and ∂iΦ(~x = ~xO = ~0) = 0. Therefore, The gradient of the potential on the other hand reads the potential reduces to12

GM i   ~ X GMP 1 1 ~x · ~xP ∂iΦM = − 2 2 J (ξ) + δΦi, (23) L (t)c Φ = 2 − − 3 . (27) c k~x − ~xP k k~xP k k~xP k P

11 There is of course an infinite set of coordinate systems that would allow one to make the calculations without being confronted with 12 See also Eq. (28) in [70], or the leading order of Eq. (4.31) in infinities. [72]. 8

Note that, while the observer is at rest in the Fermi co- tion to the sum vanishes if the source mass density is ordinate system, the source of gravitational and electro- isotropic. We have checked that one indeed has Φ  1 magnetic waves is not. However, we assume that the in the application to galaxy clusters that we consider. source is at a cosmological distance and emits the mes- Assuming that a given wave travels on the null sengers being considered over a short time period, such geodesics of the metric (25), one has cdt = that any motion of the source has a negligible effect on [1 + (1 + γ)Φ] dl + O(c−3). Using the fact that a null the path the messengers take. Of course, this expression geodesic is a straight line at leading order when one is is only valid when Φ  1, as was assumed in the deriva- far from the “lensing regime”—defined in [73, 74] as the tion. In particular, the final term can be large for large regime for which multiple images can appear—this equa- k~xk. This term is associated with the forces acting on tion can be analytically solved, such that the propagation the observer in the Newtonian picture, and its contribu- time between an emission point ~xE and the observer is

"    2 # REO X GMP rP + RPE + REO REO 1 REO −4 T (~xE, ~xO) = + (1 + γ) 3 ln − − cos θP + O(c ), (28) c c rP + RPE − REO rP 2 rP P

where rP := k~xP k, RXY := k~xY − ~xX k, and with observer of the two types of waves X and Y . Then, ac- cos θP := (~xO −~xE)·xˆP /k~xO −~xEk. Note that, since one cording to Eq. (28), and with the assumptions about the uses Fermi coordinates associated with the observer, the nature of the equivalence principle violation mentioned propagation time is expressed in terms of the observer in Sec.IIB, the fractional difference between the two proper time—as it should be. propagation times would read

A. Constraints on the equivalence principle

Now, let us assume that one is able to measure the difference of propagation time between an event and the

"    2 # X GMP rP + RPE + REO REO 1 REO −4 ∆TXY = (γY − γX ) 3 ln − − cos θP + O(c ). (29) c rP + RPE − REO rP 2 rP P

Because the terms inside the brackets do not all have bution of the gravitational sources. the same sign, one cannot simply use a subset of the The first catalog we will consider is based on a friends- sources in order to get a conservative constraint on the of-friends finding algorithm [75]. We will refer to it as difference γY −γX , unlike what is usually assumed in the Tempel2016. The second catalog, on the other hand, cal- literature. As a consequence, unless one has an absolute ibrates the group finder from a halo occupation model knowledge of the location and the mass of all the sources, [76]. We will refer to it as Tully2015. The former cata- or the propagation occurs only in a region in which one log is built on the 2MRS, CF2, and 2M++ survey data has measurements of the gravitational field, one cannot comprising nearly 80, 000 galaxies within the local vol- give a conservative constraint on the difference γY − γX ume of 430 Mpc radius; while the latter is built from a from this type of calculation. sample of the 2MASS Redshift Survey almost complete to Ks = 11.75 over 91% of the sky, which has about 43, 000 entries, giving a maximum distance of 240 Mpc. VI. ILLUSTRATION WITH GW-GRB 170817 In Tully2015, the clusters’ mass is either obtained from AND TWO CATALOGS OF CLUSTERS adjusted intrinsic luminosity and mass to light prescrip- tion, or obtained from the virial theorem. In Tempel2016, As an illustration, we apply the analytical equation the masses are solely obtained from the virial theorem. (28) to the case of the (quasi-)coincident detection of The code that derives the following results is freely the GW170817 and the short dura- accessible [77]. tion gamma ray burst GRB 170817A. We use two recent In TableI we give the results for the Shapiro delay catalogs of galaxy clusters in order to model the distri- for GW170817/GRB 170817A using the same parameters 9

TABLE I: The Shapiro delays (with γ = 1) for GW170817/GRB 170817A using the position and sky location given in the text and computed using the different catalogs considered and either Eq. (28) or Eq. (2), i.e., either the new expression or the usual, incorrect expression (which is equivalent to taking Φ0 = 0 and δΦl = 0). We also give the number of sources used in each calculation.

catalog # of sources Shapiro delay with new Eq. (28) [yr] Shapiro delay with old Eq. (2) [yr] Tempel2016 5, 166 + 77 + 22, 985 Tully2015 (luminosities) 25, 472 + 65 + 190, 502 Tully2015 (virial) 1, 119 − 20 + 37, 282 as in [46] (a distance of 26 Mpc and the sky location instance, of the 12,106 sources in Tempel2016, only 5,166 from [78]). For comparison, the value obtained with just have their mass estimated via the virial theorem. the Milky Way’s Keplerian potential outside of 100 kpc We therefore conclude that this provides additional ev- (as in [46]) is 79 days. We find that the values obtained idence that the Shapiro delay in Eq. (28) is more ap- with the standard, incorrect expression [Eq. (2)] are more propriate than the one usually found in the literature than two or three orders of magnitude larger than the [Eq. (2)], in the sense that it gives consistent results with values obtained with the new expression [Eq. (28)], and the estimation based on a FLRW universe. even have opposite sign in the Tully2015 virial mass case.

A. Comparison with cosmology

It is interesting to compare the estimate of the Shapiro delay when using either the catalogs or the FLRW metric in Fermi coordinates. Indeed, since the FLRW metric as- sumes a homogeneous mean density, one can expect that an average of the Shapiro delay over the whole sky in the catalogs gives a value with the same order of magnitude with respect to an estimate from the matter contribu- FIG. 1: Matter contribution to the Shapiro delay in an tion alone in an FLRW universe. Note, however, that FLRW universe, as well as an average over the whole this comparison cannot be rigorous, as one compares cal- sky at a given distance of the Shapiro delay from each culations in gauges that use different prescriptions. As- catalog, plotted versus the distance to the source. suming the given in Sec.III, and the line element [Eq. (3)], the total Shapiro delay reads

(Ω − 2Ω )H2 T (~x , ~x ) = Λ m 0 (r3 − r3 ). (30) E O 12 E O B. Discussion on the estimates

The contribution from matter alone therefore reads It is important to keep in mind that we give those re- 2 sults as an illustration only, and there are not meant to ΩmH0 3 3 Tmatter(~xE, ~xO) = − (r − r ). (31) be taken either as rigorous or as conservative estimates 6 E O of the Shapiro time delay. They are not rigorous be- The contribution from matter alone in an FLRW universe cause the space-time model that has been used does not would therefore be about −152 years for a source located depict our universe accurately—in particular, it neglects at 26 Mpc. cosmology (see Secs.II andIII). In Fig.1, we plot the evolution of the Shapiro delay But even if one is in a regime and in a theoretical with the distance of the source of gravitational waves in framework that are such that the assumptions that led an FLRW universe, as well as an average over the whole to Eq. (29) hold, it would not be possible to give con- sky of the Shapiro delay from each catalog. We find that servative constraints on the violation of the equivalence the estimate with the FLRW metric lies in between the principle with this method from an incomplete set of the estimate with the catalog that determines masses from gravitational sources anyway (see the previous section). the luminosity, and the estimates with the catalogs that Indeed, adding new sources may result in a decrease of determine the masses from the virial theorem. This is the Shapiro delay, and not necessarily to an increase— consistent, given that catalogs that infer masses from the unlike what is usually found in the literature. This can virial theorem cannot estimate all the masses due to a be seen with the two catalogs considered here in Fig.2, lack of (good enough) data, and therefore they necessarily where we plotted the behavior of the total Shapiro de- tend to have an underestimated density of sources. For lay (when γ = 1) as one includes more and more re- 10

(a) From Tempel2016. completeness.

(a) From Tempel2016.

(b) From Tully2015 with masses inferred from luminosities.

(b) From Tully2015 with masses inferred from luminosities.

(c) From Tully2015 with masses inferred from the virial theorem.

(c) From Tully2015 with masses inferred from the virial theorem.

FIG. 2: Plots of the time delay as one includes more and more remote objects from a catalog. Note that the size of the portion of the universe represented in each catalog is different, with maximum radii of ∼ 400, 240, FIG. 3: Sky maps of the Shapiro delay for sources all and 220 Mpc for cases (a), (b), and (c), respectively. over the sky at a given distance (26 Mpc) estimated using the different catalogs we consider. mote objects from the catalog. Indeed, one can see that the behavior of the Shapiro delay is not monotonic with the number of sources. This behavior arises because the VII. CONCLUSION terms in the bracket of Eq. (28) can either be positive or negative depending on the geometrical configuration. The standard constraints on the equivalence principle Also, one can see in Fig.3 that the Shapiro delay can using the arrival times of astronomical messengers with be close, or even equal, to zero for several locations of the different properties emitted in close succession from the source of the gravitational and electromagnetic waves. same source suffer from a variety of issues. The most Although the results from the two catalogs are differ- basic of these is that the Shapiro delay considered is not ent in magnitude, one can see in Figs.2 and3 that their an observable in , and it is not clear behaviors are roughly consistent. However, from the dif- how to relate the standard gauge-dependent way it is ferent mass estimate used in Tully2015, one can see that calculated to equivalence principle violating alternative there are also significant variations that are caused by theories. However, even assuming that the standard cal- different mass estimate, in addition to different catalog culation is a useful way of constraining some class of 11

alternative theories, there are further problems encoun- the VizieR service was published in [81]. N. K. J.-M. tered when one applies these calculations on cosmological acknowledges support from STFC Consolidator Grant scales, as is commonly done. In this paper, we describe No. ST/L000636/1. Also, this work has received fund- these issues. ing from the European Union’s Horizon 2020 research We first use a toy model in order to show arithmetically and innovation programme under the Marie Sk lodowska- the appearance of a divergence in the metric perturba- Curie Grant Agreement No. 690904. M.S. is supported tion used to compute the Shapiro delay when one is using in part by STFC grant ST/P000258/1. This document the set of gauge restrictions [Eqs. (16-17)] that is com- is LIGO-P1900149-v3. monly implicitly used in the literature. We show that Appendix A: Computing the affine distance in the the divergence appears because of an unsuitable choice post-Newtonian metric of coordinate time in order to describe the metric, and is therefore simply cured by using an appropriate (non- The affine distance (see, e.g., Sec. 2.4 of [52]) is de- ambiguous) coordinate time—for instance, a coordinate fined for a given null geodesic and observer by the affine time that corresponds to the proper time of any given parameter λ of the past-oriented geodesic in the spe- observer. cific parameterization where λ = 0 corresponds to the However, the resulting expression for the Shapiro delay µ ν spacetime event of the observation and gµν x˙ (0)u0 = 1, after fixing the coordinate time issue is no longer mono- where Greek letters denote spacetime indices, xµ(λ) is tonic with the number of the gravitational sources. This ν the geodesic, u0 is the observer’s 4-velocity at the obser- thus prevents one from estimating a conservative min- vation, and overdots denote derivatives with respect to imum amplitude of the Shapiro delay from a subset of the affine parameter. the sources of the gravitational field. As a consequence, To calculate this distance for the post-Newtonian met- analyses based on Eq. (1) are not conservative on cosmo- ric, we start from the post-Newtonian line element given 13 logical scales. in Eqs. (7.104) of [82], which we take through O(c−3) While it might be possible to constrain the gravita- and omit the remainders: tional field along the line of sight using cosmological ob- 2 µ ν servations (e.g., of galaxy velocities, as in [79]), and thus ds = gµν dx dx avoid the need to use Eq. (1), we do not consider this = −[1 − 2U(~x)]dt2 − 83/2U (~x)dtdxk (A1) here. We feel that any further developments of this test k should be informed by a fundamental theory, to avoid the + [1 + 2U(~x)]kd~xk2, gauge dependence of the current formulation of the test, and hope that this paper encourages such work. where we have set G = c = 1 (except for order counting in powers of c−1), and introduce  = O(c−2) as our order counting parameter. Here U is the Newtonian potential ACKNOWLEDGMENTS and Uk is the vector potential, both defined explicitly in Box 7.5 of [82]. We use Roman letters to denote spatial The authors thank Shantanu Desai, Daniel Holz, and indices and raise and lower indices with the Minkowski Robert Wald for useful comments while preparing the fi- metric. Additionally, we have taken the time dependence nal version of the manuscript and Francesco Pannarale of all these potentials to be negligible, since we want to for a careful reading of the manuscript. O.M. thanks consider a situation that is stationary to a good approx- Aur´elienHees for his insightful feedback on a prelim- imation. We take the observation to occur at the origin, inary version of the manuscript. This research has with λ = 0, and the source to be at a spatial location of ~ made use of the VizieR catalogue access tool, CDS, dsource, with λ = λsource. Strasbourg, France [80]. The original description of Null geodesics in this metric have to satisfy

2 3/2 k ˙ 2 −[1 − 2U(~x(λ))]t˙ (λ) − 8 Uk(~x(λ))t˙(λ)x ˙ (λ) + [1 + 2U(~x(λ))]k~x(λ)k = 0, (A2a) 2 ˙ 2 l 2[1 + 2U(~x(λ))]¨xk(λ) − 2∂kU(~x(λ))[t˙ (λ) + k~x(λ)k ] + 4∂lU(~x(λ))x ˙ (λ)x ˙ k(λ) 3/2 l 3/2 +16 ∂[kUl](~x(λ))t˙(λ)x ˙ (λ) − 8 Uk(~x(λ))t¨(λ) = 0 (A2b) (the null geodesic condition and the spatial part of the geodesic equation), where we have written the geodesic as

13 However, from Eq. (28), one can verify that the issue, of course, outside the Solar System. In other words, what makes the two does not appear for Solar System calculations because rP  cases (cosmological versus Solar System) different is the distance REO for all the significant gravitational sources that are located traveled by light. 12 xµ(λ) = (t(λ), ~x(λ)). We write the geodesic of the metric (A1) as

µ  3/2 3/2  x (λ) = t0(λ) + t1(λ) +  t1.5(λ), ~x0(λ) + ~x1(λ) +  ~x1.5(λ) . (A3)

3/2 ˙ We then expand Eqs. (A2) to  , starting with the This means that we want to scale λ by k~x0k (recall- ˙ ˙ null geodesic condition, which gives (recalling that the ing that ~x0 is constant). Thus, with the scaled λ, ~x0 = geodesic is past-oriented) ~ ~ ~ dsource/kdsourcek and we have daff = λsource = kdsourcek. The affine distance is thus the distance one would na¨ıvely ˙ t˙0(λ) = −k~x0(λ)k, (A4a) compute using the background metric. ˙ ˆ ˙ t˙1(λ) = −2U(~x0(λ))k~x0(λ)k − ~n0(λ) · ~x1(λ), (A4b) ˙ k ˆ ˙ t1.5(λ) = −4x ˙ 0 (λ)Uk(~x0(λ)) − ~n0(λ) · ~x1.5(λ), (A4c) Appendix B: Convergence of lattice sums

ˆ ˙ ˙ where ~n0 := ~x0/k~x0k and we have used the lower-order Here we show that the lattice sums in Eqs. (21) equations freely in simplifying the higher-order equa- and (24) converge when summed on expanding cubes tions. Similarly, Eq. (A2b) gives or spheres (centered at the origin). We first note that these series are not absolutely convergent, since the mag- ¨ ~ −3 ~x0(λ) = 0, (A5a) nitudes of their terms do not fall off faster than kβk as kβ~k → ∞. Thus, their convergence (and value) depends ~x¨ (λ) = 2k~x˙ (λ)k2∇~ U(~x (λ)) − 2[~x˙ (λ) · ∇~ U(~x (λ))]~x˙ (λ), 1 0 0 0 0 0 on the order in which they are summed (see, e.g., [83] for (A5b) a discussion of this for the Madelung constant that gives k l ˙ x¨1.5(λ) = 8∂[kUl](~x0(λ))x ˙ 0(λ)k~x0(λ)k. (A5c) the binding energy of an ion of NaCl). An obvious way to sum is using expanding cubes, since this method retains ˙ From these expressions, we see that ~x0 is constant, and many of the symmetries of the underlying lattice. How- ˙ ¨ ˙ ¨ ~x0 · ~x1 = ~x0 · ~x1.5 = 0. We take ~x0(0) = ~0 and ever, this is not the only method of computing the lattice ~ ˙ ˙ sum for which it converges; unlike for the Madelung con- ~x0(λsource) = dsource. Thus, we have ~x0 · ~xA = const., for A ∈ {1, 1.5}, where the constant has to be zero, be- stant (which is more subtle, due to the alternating nature cause the perturbation to the path should not change of the function being summed), a sum using expanding spheres would also converge, though a sum using regions its endpoints, so ~xA(0) = ~xA(λsource) = 0. Thus, ˙ ˙ ˙ ˙ with less symmetry, such as expanding rectangular boxes, ~x0 · ~x1 = ~x0 · ~x1.5 = 0. µ µ would generically not converge. Thus, for u0 = (∂/∂t) (i.e., an observer at rest with To demonstrate convergence of the sum in Eq. (21), respect to the background Minkowski metric) we apply Taylor’s theorem with Lagrange remainder to

g x˙ µ(0)uν = k~x˙ (0)k + O(2), (A6) 1 1 µν 0 0 f(α) := − , (B1) kαξ~ − β~k kβ~k where we have included the remainder term explicitly, to emphasize the order to which this holds. yielding

1 1 ξ~ · β~ 3(ξ~ · β~)2 − kξ~k2kβ~k2 (¯αkξ~k2 − ξ~ · β~)[3kξ~k2kα¯ξ~ − β~k2 − 5(ξ~ · β~ − α¯kξ~k2)2] − = f(1) = + + (B2) kξ~ − β~k kβ~k kβ~k3 kβ~k5 2kα¯ξ~ − β~k7

for someα ¯ ∈ (0, 1) (depending on ξ~ and β~). The first the indices. The first of these symmetries implies that two terms vanish when summed over a cube (or sphere) the βkβl (k 6= l) terms vanish upon summation, giving centered at the origin. The first term vanishes because 2 2 2 2 2 2 ~ 2 ~ 2 ~ 5 a summand of [3(ξ1 β1 + ξ2 β2 + ξ3 β3 ) − kξk kβk ]/kβk . the set of points in the cube (sphere) centered at the This vanishes when summed over all cyclic permutations origin is symmetric under β~ → −β~. The second term of indices. vanishes because the set of points in the cube (or sphere) We now want to bound the remainder term. Since we centered at the origin is symmetric under β → −β 1,2,3 1,2,3 are only interested in its behavior for large kβ~k, we can (i.e., switching the sign of the individual components of assume that kβ~k ≥ 2kξ~k. We then apply the Cauchy- β~) and is also symmetric under cyclic permutations of Schwarz inequality and triangle inequality (a number 13

14 ~ 3 of times) to the numerator to bound its magnitude by der. The sum of this bound over β ∈ Z∗ converges, (kξ~k2 +kξ~kkβ~k)[3kξ~k2(kξ~k+kβ~k)2 +5(kξ~kkβ~k+kξ~k2)2] ≤ thus demonstrating that the sum on expanding cubes or 27kξ~k3kβ~k3. We use the assumption that kξ~k ≤ kβ~k/2 spheres of Eq. (21) indeed converges. to obtain the final bound. We also apply the reverse triangle inequality to give a lower bound on the magni- We now apply the same technique to demonstrate that tude of the denominator of kβ~k7/64, and thus an upper the sum in Eq. (24) converges, i.e., we note that Taylor’s bound of 1728kξ~k3/kβ~k4 on the magnitude of the remain- theorem with Lagrange remainder gives

ξi − βi βi kβ~k2ξi − 3(ξ~ · β~)βi 2kα¯ξ~ − β~k2(ξ~ · β~ − α¯kξ~k2)ξi − [kξ~k2kα¯ξ~ − β~k2 − 5(ξ~ · β~ − α¯kξ~k2)2](¯αξi − βi) = − + + kξ~ − β~k3 kβ~k3 kβ~k5 2kα¯ξ~ − β~k7 (B3) for someα ¯ ∈ (0, 1) (depending on ξ~ and β~). As before, we that |ζi| ≤ kζ~k for any vector ζ~, giving an upper bound find that the first two terms vanish when summed over a for the numerator of 2(kξ~k + kβ~k)2(kξ~kkβ~k + kξ~k2)kξ~k + cube or sphere centered at the origin. This is obvious for [kξ~k2(kξ~k + kβ~k)2 + 5(kξ~kkβ~k + kξ~k2)2](kξ~k + kβ~k) ≤ the first term. For the second term, we first note that the 27kξ~k2kβ~k3, for kξ~k ≤ kβ~k/2, as before. The lower bound sums of the β β (k 6= l) terms (over a cube or sphere cen- k l on the denominator is the same as before, giving an ~ 2 i 2 i ~ 5 tered at the origin) vanish, giving [kβk − 3(β ) ]ξ /kβk overall upper bound on the magnitude of the remainder (no sum), which vanishes when summed over all cyclic of 1728kξ~k2/kβ~k4, which converges when summed over permutations of indices. We can bound the remainder ~ 3 using the same techniques as before, along with noting β ∈ Z∗, thus demonstrating that the sum on expanding cubes or spheres of Eq. (24) indeed converges.

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