Nonlinear gravitational-wave memory from cusps and kinks on cosmic strings
KKCL-PH-TH/2021-04CERN-TH-2021-016
Nonlinear gravitational-wave memory from cusps and kinks on cosmic strings
Alexander C. Jenkins ∗ and Mairi Sakellariadou
1, 2, † Theoretical Particle Physics and Cosmology Group, Physics Department,King’s College London, University of London, Strand, London WC2R 2LS, UK Theoretical Physics Department, CERN, Geneva, Switzerland (Dated: 26 February 2021)The nonlinear memory effect is a fascinating prediction of general relativity (GR), in whichoscillatory gravitational-wave (GW) signals are generically accompanied by a monotonically-increasingstrain which persists in the detector long after the signal has passed. This effect is directly accessibleto GW observatories, and presents a unique opportunity to test GR in the dynamical and nonlinearregime. In this article we calculate, for the first time, the nonlinear memory signal associated withGW bursts from cusps and kinks on cosmic string loops, which are an important target for currentand future GW observatories. We obtain analytical waveforms for the GW memory from cuspsand kinks, and use these to calculate the “memory of the memory” and other higher-order memoryeffects. These are among the first memory observables computed for a cosmological source of GWs,with previous literature having focused almost entirely on astrophysical sources. Surprisingly, wefind that the cusp GW signal diverges for sufficiently large loops, and argue that the most plausibleexplanation for this divergence is a breakdown in the weak-field treatment of GW emission from thecusp. This shows that previously-neglected strong gravity effects must play an important rôle nearcusps, although the exact mechanism by which they cure the divergence is not currently understood.We show that one possible resolution is for these cusps to collapse to form primordial black holes(PBHs); the kink memory signal does not diverge, in agreement with the fact that kinks are notpredicted to form PBHs. Finally, we investigate the prospects for detecting memory from cuspsand kinks with current and future GW observatories, considering both individual memory burstsand the contribution of many such bursts to the stochastic GW background. We find that in thescenario where the cusp memory divergence is cured by PBH formation, the memory signal is stronglysuppressed and is not likely to be detected. However, alternative resolutions of the cusp divergencemay in principle lead to much more favourable observational prospects.
CONTENTS
I. Introduction 2II. Nonlinear GW memory 3A. Late-time memory 4B. Frequency-domain memory waveforms 4III. Cosmic string burst waveforms 4IV. Memory from cusps 5A. Beaming effects 5B. Late-time memory 5C. Full waveform 6D. Time-domain waveform near the arrival time 6E. Radiated energy 7F. Ultraviolet divergence of the radiated energy 7G. Second-order memory 9H. Higher-order memory, and anotherdivergence 10I. Memory from cusp collapse 11V. Memory from kinks 14A. Beaming effects 14 ∗ [email protected] † [email protected] B. Late-time memory 14C. Full waveform 15D. Time-domain waveform near the arrival time 15E. Radiated energy 15F. Higher-order memory 16G. Caveats of our approach 17VI. Detection prospects 17A. Burst searches 17B. Stochastic background searches 18C. Consequences for the loop distributionfunction 18VII. Summary and conclusion 19Acknowledgments 19A. Understanding the origin of the higher-ordermemory divergence 191. Gaussian pulse as a toy model 192. Application to cosmic strings 203. Application to compact binaries 21B. Angular integrals for higher-order cusp memory 21C. Angular integrals for kink memory 22References 24 a r X i v : . [ g r- q c ] F e b I. INTRODUCTION
The advent of gravitational-wave (GW) astronomy hasgiven us unprecedented observational access to gravity inthe dynamical, nonlinear regime, allowing us to test thepredictions of Einstein’s General Relativity (GR) in thisregime as never before [1–3]. One such prediction is the
GW memory effect [4–6], in which the passage of a GWsignal causes a permanent offset in the distance betweentwo nearby freely-falling test masses; i.e., the system“remembers” the passage of the GW signal, rather thanjust oscillating and returning to its initial configuration.Effects like this are a generic prediction for any GWsignal whose source has unbound components that es-cape to infinity—for example, hyperbolic binary encoun-ters [7–12], core-collapse supernovae [13–17], gamma-raybursts [18–21], and particle decay [22–24]. The resultingdisplacement is then called the linear memory, as it isalready present in the GW signal obtained by solvingthe linearised Einstein equation. Gravitationally-boundsystems, such as the compact binary coalescences (CBCs)observed by LIGO/Virgo [25–28], do not source significantamounts of linear GW memory. In a now-classic paper, Christodoulou [31] used rig-orous asymptotic methods to show that there is also a nonlinear memory effect, in which essentially any
GWsignal is accompanied by a permanent strain offset (thesame effect was also discovered soon after by Blanchetand Damour [32] in the very different context of post-Minkowskian theory). Intuitively, we can understand thisby appreciating that linearised GWs carry energy andmomentum, and their effective stress-energy can there-fore source further perturbations to the spacetime metricbeyond linear order. By heuristically treating the grav-itons sourced at linear order as GW sources in their ownright, Thorne [33] showed soon after Christodoulou’s pa-per that the nonlinear memory can be interpreted as thecumulative linear memory associated with these grav-itons escaping to null infinity. By searching for nonlinearmemory with GW observatories, we can therefore directlyprobe the “ability of gravity to gravitate” (in the wordsof Ref. [34]).While originally derived in the context of classical grav-itational physics, it has since been recognised that effectsanalogous to the nonlinear GW memory are a genericfeature of field theories with massless degrees of free-dom [22, 23, 35–37], and that these memory effects areintimately related to both the asymptotic symmetry groupof spacetime and so-called “soft theorems” which governthe production of low-energy massless particles in thequantum theory [38–44]. These deep theoretical links add CBCs do generate some linear memory if the final black hole hasa non-zero recoil velocity, but this linear memory signal is likelyfar too weak to be detected [29], even for systems with maximalrecoil velocities (the so-called “super-kick” configuration) [30]. to the appeal of nonlinear GW memory as an observationalprobe of the underlying gravitational theory.With these motivations in mind, there has been a sig-nificant effort in recent years to calculate [6, 34, 45–51]and search for [52–68] memory waveforms associated withCBCs, as these are the primary target of GW observat-ories like LIGO/Virgo and pulsar timing arrays (PTAs),and are predicted to be abundant sources of nonlinearmemory.In this article, we focus instead on the nonlinear memorygenerated by another key GW source: cosmic strings [69–72]—linelike topological defects which may have formed ina cosmological phase transition in the early Universe dueto a spontaneously-broken U (1) symmetry, whose produc-tion is a generic prediction of many theories beyond theStandard Model [73]. These strings self-intersect to formclosed loops [74, 75], which oscillate at relativistic speedsand emit strong GW bursts through sharp features calledcusps and kinks [76], making them important targets forGW searches. The amplitude of these GW signals is setby the dimensionless string tension Gµ , which is related tothe symmetry breaking scale η by Gµ ∼ ( η/m Pl ) , with m Pl the Planck mass. Searches for cosmic strings withLIGO/Virgo [77–81] and pulsar timing arrays (PTAs) [82–84] have so far returned only null results, allowing usto place a conservative upper limit on the string tensionof Gµ (cid:46) − (there is some uncertainty due to themodelling of the cosmic string loop network—the moststringent constraints are at the level of Gµ (cid:46) − [81]).There are at least two reasons to expect a priori thatcosmic strings could be an important source of nonlinearmemory.
1. GW memory waveforms tend to be associated withlower frequencies than the primary GW emissionthat sources them [6] (this is because the memorygrows monotonically over a period of order the dur-ation of the primary signal, which is typically much The NANOGrav Collaboration have recently found evidence for astochastic process in the pulsar timing residuals of their 12.5-yeardataset, with a common spectrum across all pulsars [85]. Severalauthors have pointed out that this is consistent with a stochasticGW background from cosmic strings [86–90]. While this is veryexciting, we note that the NANOGrav data do not yet provideconclusive evidence for the quadrupolar cross-correlations betweenpulsars that one would expect from a GW signal, and further dataare thus needed to confidently identify the cause of the commonstochastic process. If the observed signal is indeed due to GWs,it is perfectly consistent with an astrophysical background frominspiralling supermassive black hole binaries, meaning that moreexotic interpretations must be explored with caution. We note that cosmic string loops can also source significantamounts of linear memory by emitting radiation in the under-lying matter fields. These effects are ignored by the Nambu-Goto approximation we adopt here, and can only be resolvedby field-theory simulations. In reality, we expect loops to gener-ate a combination of linear and nonlinear memory, by radiatingboth matter and GWs. This has recently been demonstrated forcollapsing circular loops, using numerical-relativity/field-theorysimulations [91]. longer than the oscillation period of the primarysignal). This means that, e.g., massive stellar bin-ary black holes which merge near the bottom end ofthe LIGO/Virgo frequency band produce memorysignals which are shifted to lower frequencies, andare thus challenging to detect with LIGO/Virgo.Cosmic string signals, on the other hand, typicallyhave very short durations and significant power atvery high frequencies, meaning that we might expecta stronger memory signal at observable frequencies.(Such “orphan” signals, where the memory emissionis detectable even though the primary signal is not,were studied in Ref. [92].)2. The angular pattern of the GW memory signal onthe sky is typically different to that of the primaryGW emission that sources it. Cosmic string cuspsand kinks emit GWs in narrow beams, meaning thatonly a very small fraction of all cusps and kinksare oriented such that their GWs can be observed.However, if the associated GW memory signal ismore broadly distributed on the sphere, this mightallow us to observe some of the many cusps andkinks whose beams are not oriented towards us.We find that both expectations are borne out by our cal-culations below: the GW memory from cusps and kinksis indeed emitted in a much broader range of directionsthan the initial beam, and the high-frequency behaviourof the primary GWs does indeed play an important rôle indetermining the strength of the memory effect. In fact, weshow that these two ingredients lead to a divergence in thememory signal from cusps on sufficiently large loops, po-tentially signifying a breakdown of the standard weak-fieldapproach for calculating the GW signal from cusps. Weattempt to clarify the root cause of this breakdown, andsuggest one possible resolution, based on the cusp-collapsescenario of Ref. [93]. Other resolutions are possible how-ever, and ultimately a fully general-relativistic treatmentwill be required to understand the true behaviour of cusps.The remainder of this article is structured as follows. InSec. II we introduce the standard expression for the nonlin-ear GW memory, Eq. (2), and develop from it some usefulformulae for calculating the late-time memory (i.e. thetotal strain offset after the GW signal has passed), Eq. (9),and the frequency-domain memory waveform, Eq. (12).In Sec. III we briefly recap the standard cusp and kinkwaveforms as derived in Ref. [76], which are the otherkey ingredient of our analysis. In Sec. IV, we calculatethe nonlinear GW memory signal associated with cusps,obtaining the simple frequency-domain waveform (21); weshow that the total GW energy radiated by this memorysignal diverges for Nambu-Goto strings, and regularisethis divergence by imposing a cutoff at the scale of thestring width δ ; we then go on to consider higher-ordermemory effects (the nonlinear GW memory sourced bythe memory GWs themselves) and show that accountingfor all such contributions leads to a divergence for largeloops, which persists even after applying the string-width regularisation; finally, we show that this divergence iscured if the cusp collapses to form a primordial black hole(PBH), as we recently proposed in Ref. [93]. In Sec. Vwe repeat the memory calculation for kinks, obtainingthe leading-order waveform (70); unlike in the cusp case,the memory signal is strongly suppressed at high frequen-cies due to interference effects, and no divergence occurs.In Sec. VI we study the observable consequences of ourmemory waveforms for GW searches, under the assump-tion that the cusp memory divergence is cured by PBHformation; we find that in this scenario the memory isstrongly suppressed, and is beyond the reach of current orplanned GW searches. Finally, we summarise our resultsin Sec. VII. We discuss the cause of the higher-ordermemory divergence in Appendix A, and give some tech-nical details of the angular distribution of the memoryGWs from cusps and kinks in Appendices B and C. Weset c = 1 throughout, but keep G and (cid:126) explicit. II. NONLINEAR GW MEMORY
We work in terms of the complex GW strain, h ( t, r ) ≡ h + − i h × = 12 ( e + ij − i e × ij ) h ij , (1)where t is the retarded time, r is a 3-vector point-ing from the source to the observer, and e Aij are thetransverse-traceless (TT) polarisation tensors. We dis-tinguish between the primary, oscillatory strain signal(sourced at linear order) and the additional strain dueto the memory effect by writing these as h (0) and h (1) respectively (reserving h ( n ) with n ≥ for the “memoryof the memory” and other higher-order memory contri-butions, which we discuss in Secs. IV G and V F). Theleading nonlinear memory correction term can be writtenin gauge-invariant form as [6, 33] h (1) ( t, r ) = 2 Gr (cid:90) t −∞ d t (cid:48) (cid:90) S d ˆ r (cid:48) ( e + ij − i e × ij )ˆ r (cid:48) i ˆ r (cid:48) j − ˆ r · ˆ r (cid:48) d E (0)gw d t (cid:48) d ˆ r (cid:48) . (2)We see immediately that since the integral is over the (non-negative) GW energy flux from the source, d E (0)gw / d t d ˆ r ,we generically obtain a nonzero memory correction whichgrows monotonically with time while the source is “on”.By projecting onto the GW polarisation tensors we ensurethat only the TT part of the angular integral contributes;this TT part vanishes if the emitted flux is exactly iso-tropic, but is generically non-vanishing for anisotropicemission.The polarisation tensors can be written as [94] e + ij = ˆ θ i ˆ θ j − ˆ φ i ˆ φ j , e × ij = ˆ θ i ˆ φ j + ˆ φ i ˆ θ j , (3)where ˆ θ and ˆ φ are the standard spherical polar unit vec-tors orthogonal to ˆ r . It is then straightforward to showthat ( e + ij − i e × ij )ˆ r (cid:48) i ˆ r (cid:48) j = [( ˆ θ − i ˆ φ ) · ˆ r (cid:48) ] . (4)We can also rewrite the energy flux (2) in terms of thelinear GW strain of the source, using the Isaacson for-mula [95, 96] d E gw d t d ˆ r = (cid:104)| r ˙ h | (cid:105) π G , (5)where the dot denotes a time derivative. The leading GWmemory term then becomes h (1) ( t, r ) = (cid:90) t −∞ d t (cid:48) r (cid:90) ˆ r (cid:48) | r ˙ h (0) ( t (cid:48) , r (cid:48) ) | , (6)where we have introduced the shorthand (cid:90) ˆ r (cid:48) [ · · · ] ≡ (cid:90) S d ˆ r (cid:48) π [( ˆ θ − i ˆ φ ) · ˆ r (cid:48) ] − ˆ r · ˆ r (cid:48) [ · · · ] , (7)for brevity. The temporal average (cid:104)· · ·(cid:105) is necessary inEq. (5) to give a gauge-invariant result, but is removeddue to the time integral in Eq. (6). Throughout we referto the oscillatory strain h (0) on the RHS that sources thememory as the “primary” GW emission. A. Late-time memory
One drawback of Eq. (6) is that it is written in termsof the time-domain primary strain, whereas the cosmicstring waveforms that we want to consider are much morenaturally expressed in the frequency domain. This prob-lem disappears when we consider the late-time memory ,i.e. the total strain offset caused by the cusp, ∆ h (1) ≡ lim t →∞ h (1) ( t ) , (8)as the time integral in Eq. (6) is then over the entire realline, and we can thus use Parseval’s theorem to write ∆ h (1) ( r ) = 2 π r (cid:90) R d f (cid:90) ˆ r (cid:48) | rf ˜ h (0) ( f, r (cid:48) ) | , (9)where ˜ h (0) is the Fourier transform of the primary strainsignal, and the extra factor of frequency comes from thetime derivative on the strain. B. Frequency-domain memory waveforms
The late-time memory (9) is useful for giving a sense ofthe total size of the memory effect, but in many cases isnot directly observable. For example, the test masses inground-based GW interferometers like LIGO and Virgoare not freely-falling in the plane of the interferometerarms; they are acted upon by feedback control systemsat low frequencies to mitigate seismic noise [97]. Theselow-frequency forces mean that the test masses cannot sus-tain a permanent displacement after the GW has passed.However, the “ramping up” of the memory signal from zero at early times to ∆ h (1) at late times can be measuredif it contains power in the sensitive frequency band of theinterferometer.We are therefore interested in calculating the fullmemory signal in the frequency domain. The simplestway of doing this is to use the result F (cid:20)(cid:90) t −∞ d t (cid:48) g ( t (cid:48) ) (cid:21) = 12 ˜ g (0) δ ( f ) − i2 π f ˜ g ( f ) (10)for a general function g ( t ) , where F [ · · · ] denotes the Four-ier transform, and ˜ g = F [ g ] . Applying this to Eq. (6), weobtain ˜ h (1) ( f ) = 12 ∆ h (1) δ ( f ) − i2 π f (cid:90) R d t r (cid:90) ˆ r (cid:48) e − π i ft | r ˙ h (0) | . (11)Note that we can neglect the term proportional to δ ( f ) , asthis only contributes at f = 0 , and we are interested herein frequencies which are accessible to GW experiments[this zero-frequency term is still captured in Eq. (9)]. Byreplacing each factor of ˙ h (0) with its Fourier transformand massaging the resulting expression, we find ˜ h (1) ( f ) = − i π rf (cid:90) R d f (cid:48) (cid:90) ˆ r (cid:48) f (cid:48) ( f (cid:48) − f ) r ˜ h (0) ( f (cid:48) , ˆ r (cid:48) )˜ h (0) ∗ ( f (cid:48) − f, ˆ r (cid:48) ) . (12)This is the simplest way of calculating the frequency-domain memory using only the primary frequency-domainsignal ˜ h (0) . III. COSMIC STRING BURST WAVEFORMS
We consider cosmic string loops in the Nambu-Goto(i.e. zero-width) approximation (although at some pointsbelow it is necessary to reintroduce a finite string width δ ).The Nambu-Goto equations of motion imply that theseloops should generically develop sharp features called“cusps”, where the tangent vector of a point on the loopinstantaneously becomes null, generating a strong burstof GWs. String intercommutations also generically lead todiscontinuities in the loop’s tangent vector called “kinks”.Nambu-Goto loops oscillate at a frequency /(cid:96) inverselyrelated to their invariant length (cid:96) , with their motionsourcing GWs at integer multiples of this base frequency.Since this frequency is typically many orders of magnitudebelow the frequency band relevant to ground-based inter-ferometers like LIGO/Virgo, we are typically interested One can show this by noting that (cid:82) t −∞ d t (cid:48) g ( t (cid:48) ) is just the con-volution of g ( t ) with the Heaviside step function Θ ( t ) . Since F [ Θ ] = (1 / δ ( f ) − i / (2 π f ) , the result follows from the convolu-tion theorem, F [ Θ ∗ g ] = F [ Θ ] F [ g ] . in very high-order harmonics of the loops. In this high-frequency regime, the GW emission of cosmic string loopsis dominated by cusps and kinks. (High-frequency GWscan also be produced through kink-kink collisions [98],but this emission mode is exactly isotropic and so doesnot contribute to the memory effect.) The asymptotichigh-frequency waveforms for cusps and kinks on a loopwith dimensionless tension Gµ and invariant length (cid:96) canbe approximated by [76] ˜ h (0)c ( f, ˆ r ) (cid:39) A c Gµ(cid:96) / r | f | / Θ ( ˆ r · ˆ r c − cos θ b ) Θ ( | f | − /(cid:96) ) , (13a) ˜ h (0)k ( f, ˆ r ) (cid:39) A k Gµ(cid:96) / r | f | / Θ ( ˆ r · ˆ r k − cos θ b ) Θ ( | f | − /(cid:96) ) , (13b)where the dimensionless pre-factors are A c ≡ Γ (1 / (cid:18) (cid:19) / ≈ . ,A k ≡ √ A c π ≈ . . (14)The fact that the frequency-domain waveforms (13a)and (13b) are real and even around f = 0 implies thatthe corresponding time-domain waveforms are real, andtherefore that the cusps and kinks are linearly polarised.The waveforms are nonzero only for frequencies above thebase mode of the loop, /(cid:96) , and only if the observer liesinside a small beam with opening angle θ b ( f ) (cid:39) / / ( | f | (cid:96) ) − / . (15)For cusps, there is a single well-defined beaming direction ˆ r c , whereas for kinks the beam is over a one-dimensional“fan” of directions on the sphere, and ˆ r k represents thedirection in this fan which is closest to the observer’s lineof sight ˆ r . IV. MEMORY FROM CUSPS
We begin by calculating the nonlinear memory fromcusps, inserting the primary waveform (13a) into Eqs. (9)and (12) to obtain the late-time memory and frequency-domain waveform, and then iterating this process to ob-tain higher-order memory corrections.
A. Beaming effects
The anisotropic beaming of the GWs from cusps is whatgives rise to a nonzero memory effect (due to its nonzeroTT projection), and is captured in the spherical integral (cid:90) ˆ r (cid:48) Θ ( ˆ r c · ˆ r (cid:48) − cos θ b ) , (16) where we are using the shorthand (7). To compute thisintegral, it is convenient to define polar coordinates ˆ r (cid:48) =( θ (cid:48) , φ (cid:48) ) such that the North pole θ (cid:48) = 0 coincides withthe centre of the beam ˆ r c . The integrand then only hassupport for θ (cid:48) ∈ [0 , θ b ] , so that Eq. (16) becomes (cid:90) ˆ r (cid:48) Θ ( ˆ r c · ˆ r (cid:48) − cos θ b )= (cid:90) θ b d θ (cid:48) sin θ (cid:48) (cid:90) π d φ (cid:48) π [( ˆ θ − i ˆ φ ) · ˆ r (cid:48) ] − ˆ r · ˆ r (cid:48) = 2 cos ι sin ι (cid:20) (1 − cos θ b ) cos θ b −
12 cos ι sin θ b (cid:21) , (17)where ι ≡ cos − ˆ r c · ˆ r is the inclination of the beam tothe observer’s line of sight. In the high-frequency regimewhere the primary cusp and kink waveforms are valid,the beam angle is very small, so we expand Eq. (17) toleading order in θ b to obtain (cid:90) ˆ r (cid:48) Θ ( ˆ r c · ˆ r (cid:48) − cos θ b ) (cid:39) θ ι ) . (18)There are a few remarks worth making about Eq. (18).First, we note that it assumes ι (cid:29) θ b ; when insteadthe inclination is much smaller than the beaming angle, ι (cid:28) θ b , the integral (16) drops to zero. The fact thatthe result (18) is purely real, despite the integrand beingcomplex, shows that the memory strain is linearly po-larised, much like the primary cusp and kink waveforms.Geometrically, the θ / factor represents the fraction ofthe sphere taken up by the beam, while the (1 + cos ι ) factor shows how the strength of the memory effect varieswith inclination. In particular, we notice that the memorystrain is nonzero when the observer lies outside of thebeam, ι > θ b . In fact, the observed memory strain van-ishes only when the beam is face-on ( ι (cid:28) θ b ) or face-off( ι = π ), as in both cases the angular pattern of the primaryGW emission is isotropic around the line of sight.It is interesting to note that this angular pattern—abroad ∼ (1 + cos ι ) distribution, except at very smallinclinations where the memory signal drops to zero—isexactly the same as that of the linear GW memory gener-ated by the ejection of an ultrarelativistic “blob” of matterfrom a massive object along a fixed axis [18]. Intuitively,this makes complete sense: the setup here is essentiallythe same, except that the “blob” is replaced by a burst ofGWs.
B. Late-time memory
Now that we have computed the angular integral (16),it is straightforward to obtain the late-time memory fromthe cusp. Inserting Eqs. (13a) and (15) into Eq. (9) andintegrating over frequency, we find ∆ h (1)c = 2 × / ( π A c Gµ ) (1 + cos ι ) (cid:96)r . (19) ± /‘ f ± /‘f / f | f | < | f − f | | f | > | f − f |−∞ + ∞ Figure 1. Schematic illustration of the different contributions to ˜ h (1)c ( f ) from the integral over f (cid:48) in Eq. (12). By introducing adimensionless dummy variable u ≡ f (cid:48) / | f | we obtain the two integrals shown in Eq. (20), one corresponding to the finite interiorregion /(cid:96) < f (cid:48) < f − /(cid:96) , and the other corresponding to the two semi-infinite exterior regions f (cid:48) < − /(cid:96) and f (cid:48) > f + 2 /(cid:96) . Inserting the numerical factors, this has a maximum valueof ∆ h (1)c ≈ . × ( Gµ ) (cid:96)/r for nearly-face-on cusps ι (cid:38) ,and smoothly tapers to zero for face-off cusps ι = π . Wenote that Eq. (19) should be taken with a pinch of salt,as it is sensitive to the low-frequency regime where theprimary waveform is less accurate; nonetheless, we expectthis to give a reasonable estimate of the magnitude of thememory effect. C. Full waveform
We now calculate the full frequency-domain cuspmemory waveform by inserting Eqs. (13a) and (15) intoEq. (12). In doing so, we must be careful to correctlyaccount for the behaviour of the two frequency arguments f (cid:48) and f (cid:48) − f ; in particular, the integrand is only nonzerowhen both of these arguments have magnitude greaterthan /(cid:96) , and the size of the beam angle θ b must alwaysbe set by whichever of the two arguments has greater mag-nitude (as this corresponds to a smaller, more restrictivebeam). These different contributions are illustrated inFig. 1 for the case where f is positive. Assuming that | f | > /(cid:96) , we find ˜ h (1)c ( f ) = − i ∆ h (1)c / π (cid:96) / f | f | / × (cid:34)(cid:90) ∞ / | f | (cid:96) d uu / (1 + u ) − (cid:90) / / | f | (cid:96) d uu / (1 − u ) (cid:35) , (20)where u is just a dimensionless dummy variable. Wecan simplify this by taking | f | (cid:29) /(cid:96) , as this is theregime where the primary waveforms are valid; in thishigh-frequency limit, both integrals can be evaluated ana-lytically. Inserting Eq. (19), the final result is ˜ h (1)c ( f ) (cid:39) − i B c ( Gµ ) (cid:96) / rf | f | / (1 + cos ι ) Θ ( | f | − /(cid:96) ) , (21)with a numerical prefactor, B c ≡ π A (3 / / (cid:20) π √ − F ( , ; ; − (cid:21) ≈ . , (22) where F is a hypergeometric function. While we as-sumed | f | > /(cid:96) in order to obtain Eq. (20), we havechecked that our final expression (21) underestimates thetrue memory signal in the region /(cid:96) < | f | < /(cid:96) , sowe can safely leave the low-frequency cutoff at /(cid:96) as inthe primary waveform. The simple expression (21) thusgives a conservative but generally accurate model of thetime-varying part of the cusp memory waveform at allfrequencies greater that the fundamental mode of the loop,while the zero-frequency offset is described by Eq. (19).Comparing Eq. (21) with the primary cusp wave-form (13a), we see that they are remarkably similar toeach other, with both being given by the same simplefrequency power law ∼ f − / and the same dependenceon the loop length (cid:96) . There are, however, some importantdifferences:1. The memory GW emission has broad support onthe sphere, while the primary waveform only hassupport inside a narrow beam.2. The memory waveform is suppressed by an addi-tional power of Gµ (this makes intuitive sense, giventhat it is a nonlinear effect sourced by the primaryGW emission).3. The numerical constant in front of the memory wave-form, B c , is an order of magnitude larger than thatin front of the primary waveform, A c .The first point is particularly crucial, as it lies at theheart of the divergent behaviour that we investigate inSecs. IV F and IV G. D. Time-domain waveform near the arrival time
While we have focused on deriving the memory signal inthe frequency domain, we can inverse-Fourier-transformEq. (21) to obtain a simple closed-form expression in thetime domain, h (1)c ( t ) − h (1)c ( t ) (cid:39) − / π B c ( Gµ ) r (1 + cos ι ) × ( t − t ) (cid:20) − Γ (2 / π | t − t | /(cid:96) ) / (cid:21) , (23) − − t/ s h + ( t ) Primary cusp signalMemory signalCusp with memory
Figure 2. The time-domain GW strain h c ( t ) from a cusp,with and without the leading-order memory contribution (23).The memory is exaggerated by a factor of ∼ /Gµ here tomake it visible. For small inclinations ι < θ b the observer lieswithin the cusp’s beam, and sees the memory superimposedon the primary cusp signal (solid red line). If the inclinationis very small, ι (cid:28) θ b , then the memory vanishes and onlythe primary cusp signal is observable (solid blue line). Inmost cases however, the observer lies outside of the beam,and only the memory is observable (red dashed line). Notethat the higher-order memory contributions (order n ≥ ) are not shown here; these would look like step functions in thetime domain, with height that either diverges rapidly with n [“large” loops, (cid:96) (cid:38) δ/ ( Gµ ) ] or converges so rapidly thatthe contribution to the total signal is negligible [“small” loops, (cid:96) (cid:46) δ/ ( Gµ ) ]. where Γ ( z ) is Euler’s Gamma function, and Γ (2 / ≈ . . (Here we have re-introduced the time of arrival ofthe primary cusp signal, t , rather than setting it to zero.)This time-domain waveform is shown in Fig. 2. Notethat this is real, which means that the signal is linearly + -polarised, just like the primary waveform h (0) .It is important to note that since Eq. (21) is only validfor high frequencies f (cid:29) /(cid:96) , Eq. (23) must only bevalid for a short duration | t − t | (cid:28) (cid:96) around the arrivaltime. However, for typical loop sizes this “short duration”is actually much longer than the relevant observationaltimescale (typically a few seconds), so the approximationin Eq. (23) may be a useful one. E. Radiated energy
Given a frequency-domain waveform ˜ h ( f ) , we can cal-culate the corresponding (one-sided, logarithmic) GW energy flux spectrum, d E gw d(ln f ) d ˆ r = π r f G (cid:16) | ˜ h ( f ) | + | ˜ h ( − f ) | (cid:17) . (24)In the context of radiation from a cosmic string loop, it ishelpful to normalise this with respect to the total energy ofthe loop, µ(cid:96) . We therefore define a dimensionless energyspectrum, (cid:15) ( f, ˆ r ) ≡ µ(cid:96) d E gw d(ln f ) d ˆ r . (25)For the primary cusp waveform (13a), this is given by (cid:15) (0)c ( f, ˆ r ) (cid:39) π A Gµ ( f (cid:96) ) / Θ ( θ b − ι ) Θ ( f − /(cid:96) ) , (26)while for the cusp memory (21), we find (cid:15) (1)c ( f, ˆ r ) (cid:39) π B ( Gµ ) ( f (cid:96) ) / (1+cos ι ) Θ ( f − /(cid:96) ) . (27)The fact that ˜ h (0)c is purely real while ˜ h (1)c is purely imagin-ary means that there is no coherent cross-energy betweenthe two contributions, so the total energy is just (cid:15) (0)c + (cid:15) (1)c .For observers in the beaming direction the energy spec-tra of the primary and memory signals scale in the exactsame way with frequency, but with a much smaller coeffi-cient for the memory signal. However, the picture changesdrastically when integrating the spectra over the sphereto compute the total emission, as the primary waveformis then suppressed by a factor of θ ∼ ( f (cid:96) ) − / . We writethese isotropically-averaged spectra as ¯ (cid:15) ( f ) ≡ (cid:90) S d ˆ r (cid:15) ( f, ˆ r ) , (28)such that ¯ (cid:15) (0)c ( f ) (cid:39) (cid:18) (cid:19) / ( π A c ) Gµ ( f (cid:96) ) − / Θ ( f − /(cid:96) ) , ¯ (cid:15) (1)c ( f ) (cid:39)
83 ( π B c ) ( Gµ ) ( f (cid:96) ) / Θ ( f − /(cid:96) ) . (29)We see that the isotropic energy spectrum due to thememory emission is blue-tilted (i.e. grows with frequency),while the primary spectrum is red-tilted. This means thatthe memory emission dominates at very high frequencies, f > (cid:96) (cid:18) A c B c Gµ (cid:19) ≈ Hz × (cid:18) (cid:96) pc (cid:19) − (cid:18) Gµ − (cid:19) − . (30) F. Ultraviolet divergence of the radiated energy
The total fraction of the loop’s energy radiated by thecusp can be found by integrating Eq. (29),
E ≡ (cid:90) ∞ d ff ¯ (cid:15) ( f ) . (31)For the primary cusp signal, this gives E (0)c (cid:39) / ( π A c ) Gµ (cid:28) . (32)For the memory signal, however, the integral (31) diverges.This is clearly unphysical, and shows a breakdown in thevalidity of Eq. (21). Note however that this breakdownis not in the low-frequency regime where we know thatEq. (21) is inaccurate; rather, the integral has an ultravi-olet divergence that goes like ∼ f / at high frequencies f → ∞ . Instead, we can understand this divergence asa breakdown of the Nambu-Goto approximation for theloop dynamics.We recall here the justification for the Nambu-Gotoaction, following Sec 6.1 of Ref. [72], so that we can clarifyhow this fails for cusps at high frequencies. Assumingthere are no non-gravitational long-range interactionsbetween widely-separated string segments, a generic cos-mic string can be described by a local worldsheet Lag-rangian of the schematic form L = − µ + α (cid:126) κ + β (cid:126) κ µ + · · · , (33)where µ is the string tension as before, κ represents space-time curvature (with indices suppressed), and α, β arenumerical constants. In most situations, the curvatureassociated with a loop is of the order κ ∼ (cid:96) − , where (cid:96) isthe length of the loop. This is many orders of magnitudelarger than the width of the loop, δ ∼ ( µ/ (cid:126) ) − / = (cid:96) Pl / (cid:112) Gµ, (34)where (cid:96) Pl is the Planck length. This implies that (cid:126) κ/µ ∼ ( δ/(cid:96) ) (cid:28) , (35)so that we can drop all but the first term in Eq. (33),leaving the Nambu-Goto action. This “zero-width” ap-proximation for the loop is equivalent to taking (cid:126) → in Eq. (33), effacing the microphysics of the underlyingfield-theory model and keeping only the classical part ofthe action. This is valid for loops larger than a criticalscale, (cid:96) (cid:38) δ/ ( Gµ ) ∼ (cid:96) Pl / ( Gµ ) / , (36)as field-theory calculations show that loops smaller thanthis lose much of their energy through matter radiation [71,99] or through topological unwinding and dispersion [91,100], such that the Nambu-Goto approximation breaksdown. Note, however, that these effects cannot resolvethe divergence identified above, as this occurs for muchlarger loops (cid:96) (cid:29) δ/ ( Gµ ) , for which matter effects shouldbe negligible.The Nambu-Goto dynamics of loops generically predictsthe formation of cusps, where the GW strain looks locallylike h ( t ) ∼ | t − t | / . As was already recognised inRef. [76], the curvature associated with this scales as κ ∼ ¨ h ∼ | t − t | − / , (37) t ˙ h c ( t ) Nambu-Goto cuspFinite-width cusp
Figure 3. The time derivative of the cusp strain signal closeto the peak, | t − t | (cid:28) (cid:96) , for an observer in the beamingdirection, ι = 0 . This diverges for the standard Nambu-Goto waveform (13a), ruining the validity of the Nambu-Gotoapproximation, and causing a divergence in the energy radiatedby the first-order memory. By introducing a frequency cutoffdue to the finite string width, f < /δ , we see that thederivative becomes finite and continuous, and the first-ordermemory divergence is regularised. which diverges at the peak, clearly ruining the validityof the Nambu-Goto approximation (35). This problemdoes not manifest itself in the primary cusp waveform,since the beaming angle θ b decreases fast enough withfrequency to ensure the total radiated energy is finite, cf.Eq. (32). However, the energy flux in the centre of thebeam is still divergent, and as we have shown here, thissources further divergences due to the nonlinear natureof gravity. Since the memory effect is not beamed, thereis nothing suppressing it at high frequencies.The simplest way to regularise this divergence is toimpose an ultraviolet cutoff at the string width scale. Bytruncating the frequency-domain cusp waveform at f ∼ /δ , the cusp is smoothed out on timescales | t − t | ∼ δ ,and the curvature reaches a finite maximum value thatscales like κ ∼ δ − / . This smoothing is shown explicitlyin Fig. 3, where we see that the time derivative of thestrain diverges in the Nambu-Goto case, but is finiteand continuous if a finite width is introduced. [In thisheuristic setup, it is not immediately clear whether ornot the smoothing afforded by a finite string width isstrong enough to prevent higher-curvature terms in theLagrangian (33) from becoming important near the cusp;we return to this point later.]We therefore consider only GWs with frequency f < /δ .This has no impact on the primary waveform, since /δ ≈ Hz × (cid:18) Gµ − (cid:19) / , (38)which is beyond the reach of any current or planned GWexperiments. However, the cutoff does impact the GWmemory. For example, setting an upper limit of f = 1 /δ in the integral (31), we find a finite value for the energyradiated by the cusp memory, E (1)c (cid:39) π B c ) ( Gµ ) ( (cid:96)/δ ) / ∼ π B c ) ( Gµ ) / ( (cid:96)/(cid:96) Pl ) / . (39)Numerically, this gives E (1)c ≈ × − × (cid:18) Gµ − (cid:19) / (cid:18) (cid:96) pc (cid:19) / , (40)which shows that the energy radiated due to the first-order memory effect is smaller than that from the primaryemission for observationally-allowed values of the stringtension, Gµ (cid:46) − . However, the factor of ( (cid:96)/δ ) / (cid:29) in Eq. (39) is concerning. Based on Eq. (39), cusps onGUT-scale strings ( Gµ = 10 − ) would radiate far more en-ergy through the memory effect than through the primaryGWs; for (cid:96) (cid:38) − pc they would radiate more than theentire energy of the loop. Such a situation would be un-physical, and would indicate the breakdown of the validityof the primary cusp waveform. One might argue that thisis not an issue, since GUT-scale Nambu-Goto stringsare already ruled out by observations; however, we showbelow that accounting for higher-order memory effectsexacerbates the problem, leading to similar unphysicalresults even for much lower string tensions.We should note that the cutoff imposed here is ratherad-hoc, and is done in a way that is agnostic to theunderlying microphysics of the string. In reality, thedivergence will be resolved in a way which may dependon the microphysics, and the GW memory observablesmay be sensitive to this; indeed, this is implied by thefact that Eq. (32) depends directly on the string width δ . G. Second-order memory
The memory waveform (21) describes the spacetimecurvature generated by the energy-momentum of theprimary GWs from the cusp. However, these memoryGWs themselves carry energy-momentum, and will in turnact as a source of their own GW memory (see Ref. [50]for a discussion of this effect in the context of binaryblack hole coalescences). We refer to this “memory of thememory” here as the second-order memory effect .The calculation of this second-order memory contri-bution is straightforward for cusps, simply substitutingEq. (21) into the RHS of Eq. (12) and following all of the
Figure 4. A cartoon illustration of the angular distribution ofthe energy radiated by a cusp. The primary emission (blue) isconcentrated in a narrow beam of width θ b ∼ ( f(cid:96) ) − / , whilethe first-order memory emission (violet) is proportional to (1 + cos ι ) , and the second-order memory emission (red) isproportional to sin ι . same steps as before. The key difference is in the angularintegral, (cid:90) ˆ r (cid:48) (1 + cos ι (cid:48) ) = 16 sin ι, (41)which, due to the broader emission of the first-ordermemory signal, is not suppressed by a factor of θ ; cf.Eq. (18). The resulting expressions for the late-timememory and frequency-domain waveform are ∆ h (2)c = 2( π B c ) r ( Gµ ) (cid:96) / sin ι (cid:90) ∞ /(cid:96) d ff / , ˜ h (2)c ( f ) = − i π B rf ( Gµ ) (cid:96) / sin ι (cid:20) (cid:90) ∞ /(cid:96) d f (cid:48) f (cid:48) / ( f (cid:48) + | f | ) / + (cid:90) | f | / /(cid:96) d f (cid:48) f (cid:48) / ( | f | − f (cid:48) ) / (cid:21) , (42)both of which contain integrals which diverge due tohigh-frequency contributions from the first-order memory.Introducing the f < /δ cutoff from Sec. IV F once again,we find to leading order in δ , ∆ h (2)c (cid:39) π (cid:96)r B ( Gµ ) ( (cid:96)/δ ) / sin ι, ˜ h (2)c ( f ) (cid:39) − i ∆ h (2)c π f Θ (1 /δ − | f | ) Θ ( | f | − /(cid:96) ) . (43)0This is interesting in that it departs from the the ∼ f − / scaling of both the primary waveform (13a) and the first-order memory waveform (21); the second-order memoryinstead has a slower decay with frequency, due to thelack of beaming in the first-order memory. In fact, forintermediate frequencies /(cid:96) < | f | < /δ the second-order memory is identical to the Fourier transform ofa Heaviside step function of height ∆ h (2)c . This meansthat on timescales much shorter than the loop oscillationperiod (cid:96)/ and much longer than the light-crossing timeof the loop width δ , the second-order memory signal lookslike a step function in the time domain; this is becausethe signal is dominated by high frequencies f (cid:46) /δ ,and therefore “switches on” in a very short time interval | t − t | (cid:46) δ . The waveform (43) can be used to calculate a second-order correction to the energy radiated by the cusp, (cid:15) (2)c = π r f Gµ(cid:96) (cid:16) | ˜ h (0)c + ˜ h (1)c + ˜ h (2)c | − | ˜ h (0)c + ˜ h (1)c | (cid:17) = π r f Gµ(cid:96) (cid:16) | ˜ h (2)c | + 2˜ h (1)c ˜ h (2) ∗ c (cid:17) = (cid:20) π B ( Gµ ) ( (cid:96)/δ ) / f (cid:96) sin ι + π B ( Gµ ) ( (cid:96)/δ ) / ( f (cid:96) ) / sin ι (1 + cos ι ) (cid:21) × Θ (1 /δ − | f | ) Θ ( | f | − /(cid:96) ) . (44)We see that, unlike for the first-order correction, thereare now nonzero cross-terms due to ˜ h (1)c and ˜ h (2)c beingexactly in phase with each other, resulting in two newcontributions to the energy. Integrating over frequencyand emission direction, the total energy is given by E (2)c = 1615 ( π B c ) ( Gµ ) ( (cid:96)/δ ) / + 4( π B c ) ( Gµ ) ( (cid:96)/δ ) . (45)Comparing with the corresponding first-order result (39)we see a clear pattern emerging, where the second term inEq. (45) is multiplied by a factor of ∼ π B c ( Gµ ) ( (cid:96)/δ ) / compared to the first-order energy, and multiplying by thesame factor again gives the first term in Eq. (45). Thisfactor is greater than unity so long as the loop length (cid:96) islarger than (cid:96) ∗ ≡ δ ( π B c ) / ( Gµ ) ≈
90 m × (cid:18) Gµ − (cid:19) − / , (46) This picture is closely related to the zero-frequency limit (ZFL)method for calculating radiation from high-energy gravitationalscattering [8, 12, 14, 101], which leverages the fact that the scatter-ing is effectively instantaneous compared to the GW frequenciesconsidered. which applies to all macroscopically-large loops.One would naively expect successive memory correc-tions for a generic GW source to become less and lessimportant at higher order; the fact that they become more important for such a large class of cosmologically-relevant cosmic string loops is surprising, and suggeststhat something unphysical is happening. Indeed, if weplug numerical values into Eq. (45), we find that loopslarger than about m × ( Gµ/ − ) − / (i.e. notmuch larger than the solar system) have E (2)c greater thanunity, meaning that they radiate away more than the totalenergy of the loop. This represents a significant worseningof the issue we identified at the end of Sec. IV F. Theproblem only gets worse when we go beyond second-ordercorrections, as we demonstrate below. H. Higher-order memory, and another divergence
Iterating the procedure described above to calculatethe third-order GW memory, it is straightforward to showthat it obeys the same step-function-like relation that wefound at second order, ˜ h (3)c ( f ) = − i ∆ h (3)c π f Θ (1 /δ − | f | ) Θ ( | f | − /(cid:96) ) , (47)with the late-time memory given by ∆ h (3)c = 2 π r (cid:90) R d f (cid:90) ˆ r (cid:48) r f (cid:16) | ˜ h (2)c | + 2˜ h (1)c ˜ h (2) ∗ c (cid:17) , (48)where we have made sure to include both of the second-order energy contributions as source terms for the third-order memory. Upon integration, this becomes ∆ h (3)c = − δ r ( (cid:96)/(cid:96) ∗ ) / sin ι (cid:18) −
13 cos ι (cid:19) − δr ( (cid:96)/(cid:96) ∗ ) sin ι (cid:18) ι (cid:19) , (49)where (cid:96) ∗ is the Gµ -dependent lengthscale defined inEq. (46).The third-order memory clearly scales differently forloops with (cid:96) (cid:29) (cid:96) ∗ (which we call “large loops”) comparedto those with (cid:96) (cid:28) (cid:96) ∗ (which we call “small loops”). Whengoing to fourth order and beyond, we obtain an increasingnumber of cross-terms in the energy at each order (startingwith | ˜ h ( n )c | , then h ( n − ˜ h ( n ) ∗ c , h ( n − ˜ h ( n ) ∗ c , and so on,down to h (1)c ˜ h ( n ) ∗ c ), resulting in a proliferation of termsin the resulting memory expressions, each with a differentpower of (cid:96)/(cid:96) ∗ . It is therefore much simpler to treat smalland large loops separately, and focus on the leading powerof (cid:96)/(cid:96) ∗ in each case. This results in a very economicalformula for iterating the memory calculation, valid for all n ≥ , ∆ h ( n )c = rδ (cid:90) ˆ r (cid:48) | ∆ h ( n − | , for (cid:96) (cid:29) (cid:96) ∗ π rδ (cid:90) ˆ r (cid:48) ∆ h ( n − | ˜ h (1)c (1 /δ ) | , for (cid:96) (cid:28) (cid:96) ∗ (50)1where we have evaluated the frequency integral in eachcase, leaving just the angular integral. The frequency-domain waveform is then given by the same quasi-step-function form as before, ˜ h ( n )c = − i ∆ h ( n )c π f Θ (1 /δ − | f | ) Θ ( | f | − /(cid:96) ) , (51)and the corresponding energy spectra are given by (cid:15) ( n )c = r f π Gµ(cid:96) | ∆ h ( n )c | , for (cid:96) (cid:29) (cid:96) ∗ r f Gµ(cid:96) ∆ h ( n )c | ˜ h (1)c | , for (cid:96) (cid:28) (cid:96) ∗ (52)Solving Eq. (50) iteratively with Eq. (43) as an input,we find for all n ≥ , ∆ h ( n )c = δr ( (cid:96)/(cid:96) ∗ ) n / L n ( ι ) , for (cid:96) (cid:29) (cid:96) ∗ δr ( (cid:96)/(cid:96) ∗ ) n/ S n ( ι ) , for (cid:96) (cid:28) (cid:96) ∗ (53)where L n ( ι ) and S n ( ι ) are polynomials in cos ι whichdescribe the angular pattern of the memory for large andsmall loops, respectively—these are described in detail inAppendix B. For small loops, all memory effects aresubdominant compared to the primary emission, andEq. (53) gives a convergent geometric series. For largeloops, on the other hand, the memory emission becomesstronger at each order, and Eq. (53) gives a lacunary serieswhich diverges extremely quickly. One might hope thatthe polynomials L n ( ι ) decrease in magnitude fast enoughto counteract the divergence, but we find empirically inAppendix B that | L n ( ι ) | ≈ / n − sin ι, (54)so the series diverges as long as (cid:96) (cid:38) (5 / (cid:96) ∗ ≈ km × ( Gµ/ − ) − / . We discuss the cause of this divergencein Appendix A, and argue that it is caused by a super-Planckian GW energy flux from the cusp. I. Memory from cusp collapse
The results of the previous section show that, even withan ultraviolet cutoff in place at the scale of the stringwidth, the standard cusp waveform (13a) leads to a di-vergence for all “large” loops with length (cid:96) (cid:38) δ/ ( Gµ ) .The fact that the divergence appears in observable, gauge-independent quantities (the memory strain and, therefore,the radiated energy) means that something unphysicalmust be going on. Since the only inputs to our calcula-tion are the cusp waveform (13a) and the GW memoryformula (2), at least one of these two ingredients mustbreak down for cusps on loops of this size.In order to track down the cause of the divergence, letus list the assumptions that go into Eqs. (2) and (13a): 1. The GW frequency is assumed to be much greaterthan the fundamental mode of the loop, f (cid:29) /(cid:96) .This is because the waveform (13a) is derived usingthe universal behaviour of the loop on scales (cid:28) (cid:96) near the cusp.2. The loop’s dynamics are assumed to follow theNambu-Goto action (33) on lengthscales larger thanthe loop width δ . (We have imposed a cutoff thateffaces scales below this, in order to regularise thefirst divergence we encountered in Sec. IV F.)3. The loop is assumed to evolve according to theflat-space equations of motion derived from (33);i.e. gravitational backreaction is assumed to benegligible.4. The GWs generated by the loop are assumed tobe well-described by linear perturbations on a flatbackground.As mentioned earlier, the first assumption cannot bethe source of the problem, as the divergence is associatedwith very high frequencies near the string width scale.The second assumption is robust so long as ( i ) thehigher-order curvature terms in the worldsheet Lag-rangian (33) are negligible, and ( ii ) the strings are createdthrough the breaking of a local gauge symmetry, so thatthe underlying field theory does not give rise to long-rangeinteractions (i.e. we are not considering global strings,which would instead be described by the Kalb-Ramondaction [72]). As mentioned in Sec. IV F, while introducinga finite string width prevents the curvature from diverging,it does not necessarily guarantee that the higher-ordercurvature terms are negligible. In principle the memorydivergence identified here could be cured by departuresfrom the Nambu-Goto action near the cusp. However,these departures would have to take place on scales muchgreater than the string width, which seems very difficultto achieve.By elimination, it seems that the problem is mostlylikely due to assumptions 3 and 4: i.e., that the flat-spacedescription of the cusp’s dynamics and GW generation isinconsistent. Indeed, we can trace the divergence back tothe fact that the integrated GW energy flux diverges inthe centre of the cusp’s beam, (cid:82) d(ln f ) (cid:15) (0)c ( f, ˆ r c ) → ∞ ,which already suggests that a flat-space description isinsufficient. This divergent flux is hidden somewhat bythe narrowness of the beam, θ b ∼ ( f (cid:96) ) − / , which ensuresthat E (0)c is finite, but we have shown that the GW memoryinherits and amplifies the divergence.One possible resolution is that the gravitational backre-action of the loop on itself could smooth out the cusp onscales much larger than δ . However, there is large body ofliterature on cosmic string backreaction [102–112] whichindicates that the backreaction timescale is ∼ (cid:96)/ ( Gµ ) ,much longer than the loop’s oscillation period. While itis true that these studies all assume linearised gravity,and are thus likely to underestimate the magnitude of2 − − − − ‘/ m10 − − − − − − − − E c ( ‘ ) primary emission1st-order memory ... n →∞ n th-order memory − − − − ‘/ m10 − − − − − − − − E c ( ‘ ) primary emission1st-order memory ... ‘ = ‘ ∗ Figure 5. The fractional energy radiated by a cusp at different orders in the memory expansion as a function of loop length,with Gµ = 10 − . The top panel shows the standard cusp case, clearly illustrating the divergence at (cid:96) (cid:38) (cid:96) ∗ ≈
90 m . The bottompanel shows the cusp collapse case, for which the radiated energy at each order drops to a small, (cid:96) -independent value for (cid:96) (cid:38) (cid:96) ∗ ,curing the divergence. f (cid:28) /δ . In Ref. [93]we proposed exactly such a mechanism. We argued thatwhen cusps form on sufficiently large cosmic string loops,they source such extreme spacetime curvature that a smallportion of the loop could collapse to form a black hole ata time ∼ Gµ(cid:96) before the peak of the cusp emission. Re-markably, the loops for which this “cusp collapse” processis predicted to occur are those with length (cid:96) (cid:38) δ/ ( Gµ ) —exactly the same loops for which the higher-order memorydivergence occurs. We suggest that this is no coincidence,and that one plausible solution to the divergence we haveidentified here is given by the results of Ref. [93].It is difficult to calculate the precise GW signal associ-ated with cusp collapse, but there are two main qualitativedifferences it introduces compared to the standard cuspwaveform: ( i ) the Fourier transform of the primary strainsignal, ˜ h (0)c , is reduced by a factor ≈ / , as it only re-ceives contributions from the half of the signal at timesbefore the peak; ( ii ) more importantly, there is a loss ofpower at frequencies f (cid:38) / ( Gµ(cid:96) ) , due to the truncationimmediately before the peak. Both of these effects in-fluence the corresponding GW memory signal. We canaccount for ( i ) by multiplying the strain at each orderin the memory expansion by the appropriate power of / , and can approximate the effect of ( ii ) by introducinga sharp cutoff at frequency f = 1 / ( Gµ(cid:96) ) , which is equi-valent to replacing δ → Gµ(cid:96) in all previous expressions.The critical lengthscale (46) which previously marked theonset of the divergence then becomes (cid:96) ∗ → (cid:96) ( π B c ) / ( Gµ ) (cid:29) (cid:96), (55)which means that we are always in the “small loop” regime, (cid:96) (cid:28) (cid:96) ∗ ; higher-order memory corrections are suppressedby powers of Gµ , and the divergence is avoided completely.Note that the cusp collapse process is only predicted totake place for loops with (cid:96) (cid:38) δ/ ( Gµ ) , and that the GWmemory from smaller loops is still described by the resultsgiven above, with (cid:96) ∗ given by Eq. (46).More explicitly, if the divergence is indeed cured byinvoking cusp collapse, then the GW observables from the cusp are as follows: the primary waveform is ˜ h (0)c (cid:39) A c Gµ(cid:96) / r | f | / Θ ( ˆ r · ˆ r c − cos θ b ) Θ ( | f | − /(cid:96) ) × (cid:40) (1 / Θ (1 /Gµ(cid:96) − | f | ) , for (cid:96) (cid:29) (cid:96) ∗ Θ (1 /δ − | f | ) , for (cid:96) (cid:28) (cid:96) ∗ (56)with (cid:96) ∗ given by (46); the first-order memory waveform is ˜ h (1)c (cid:39) − i B c ( Gµ ) (cid:96) / rf | f | / (1 + cos ι ) Θ ( | f | − /(cid:96) ) × (cid:40) (1 / Θ (1 /Gµ(cid:96) − | f | ) , for (cid:96) (cid:29) (cid:96) ∗ Θ (1 /δ − | f | ) , for (cid:96) (cid:28) (cid:96) ∗ (57)with the corresponding late-time memory given by ∆ h (1)c ≈ × / ( π A c Gµ ) (1 + cos ι ) (cid:96)r × (cid:40) / , for (cid:96) (cid:29) (cid:96) ∗ , for (cid:96) (cid:28) (cid:96) ∗ (58)and the n th-order memory waveforms for n ≥ are allstep-function-like, ˜ h ( n )c ≈ − i ∆ h ( n )c π f Θ ( | f | − /(cid:96) ) × (cid:40) Θ (1 /Gµ(cid:96) − | f | ) , for (cid:96) (cid:29) (cid:96) ∗ Θ (1 /δ − | f | ) , for (cid:96) (cid:28) (cid:96) ∗ (59)with height given by ∆ h ( n )c ≈ (cid:96)r (cid:18) π B c (cid:19) n − ( Gµ ) n +13 L n ( ι ) , for (cid:96) (cid:29) (cid:96) ∗ δr ( (cid:96)/(cid:96) ∗ ) n/ S n ( ι ) , for (cid:96) (cid:28) (cid:96) ∗ (60)It is interesting to note that loops with length just belowthe cusp-collapse threshold, (cid:96) (cid:46) (cid:96) ∗ , emit more energythrough GW memory than those above the threshold. Wecan see this already in the primary emission, where thetotal energy emission is E (0)c ≈ / ( π A c ) Gµ × (cid:40) / , for (cid:96) (cid:29) (cid:96) ∗ , for (cid:96) (cid:28) (cid:96) ∗ (61)with only cusps above the collapse threshold being subjectto the / reduction in GW power due to the truncationof the signal. However, the difference becomes moresignificant in the memory emission, E (1)c ≈
12 ( π B c ) ( Gµ ) / , for (cid:96) (cid:29) (cid:96) ∗ π B c ) / ( Gµ ) ( (cid:96)/(cid:96) ∗ ) , for (cid:96) (cid:28) (cid:96) ∗ (62)where there is an extra power of ( Gµ ) / for cusps abovethe collapse threshold, compared to those just below it.This pattern continues for higher-order memory, wherethe total radiated energy for all n ≥ is given by4 E ( n )c ≈ (cid:18) π B c (cid:19) n ( Gµ ) n +2 / (cid:90) S d ˆ r π | L n ( ι ) | , for (cid:96) (cid:29) (cid:96) ∗
12 ( π B c ) / ( Gµ ) ( (cid:96)/(cid:96) ∗ ) (2 n − / (cid:90) S d ˆ r π ι ) S n ( ι ) , for (cid:96) (cid:28) (cid:96) ∗ (63)We note briefly that Ref. [93] also discussed a furtherGW signature associated with cusp collapse: the high-frequency quasi-normal ringing of the newly-formed blackhole after the collapse. We neglect this effect here how-ever, as it is hard to say anything concrete about thephase evolution and angular pattern of this additionalGW emission, and this prevents us from calculating theassociated memory signal. It would be interesting to re-visit this contribution to the memory if and when moredetailed phase-coherent cusp collapse waveform modelsbecome available. V. MEMORY FROM KINKS
Unlike cusps (which are transient), kinks are persistentfeatures of cosmic string loops: they propagate aroundthe loop at the speed of light, continuously emitting GWsin a beam which traces out a one-dimensional “fan” ofdirections, like a lighthouse. This fact is usually unim-portant when computing the primary GW emission fromkinks, since the beam only overlaps with the observer’sline of sight for a small fraction of the loop oscillationtime, meaning that kinks are effectively transient sourcesfor a given observer. However, in order to calculate aGW memory signal, we need to know the primary GWflux in all directions over the entire history of the source,which for kinks means specifying the beaming directionas a function of time.We consider here the simplest case, where the beamof the kink traces out a great circle on the sphere at aconstant rate. We choose our polar coordinates such thatthis circle lies in the equatorial plane, θ k = π / . Theazimuthal direction of the kink’s beam is then given by φ k = 4 π σt/(cid:96) with σ = ± , where the two different signscorrespond to left- and right-moving kinks respectively. From Eq. (13b) we then have ˜ h (0)k ( f, ˆ r ) (cid:39) A k Gµ(cid:96) / r | f | / e − i σφf(cid:96)/ Θ ( θ b − | ι | ) Θ ( | f | − /(cid:96) ) , (64)where the inclination ι ≡ π / − θ describes the anglebetween the observer’s line of sight and the closest point The definition of whether a given kink is left- or right-moving issomewhat arbitrary here. For concreteness, we call kinks with σ = +1 “left-moving”; these move anti-clockwise around the loopwhen viewed from the North pole θ = 0 . Conversely, kinks with σ = − are called “right-moving”; these move clockwise whenviewed from θ = 0 . on the equatorial plane, and takes values ι ∈ [ − π / , π / (with positive/negative values corresponding to the ob-server being above/below the plane). Note that we havepicked up a phase factor e − i σφf(cid:96)/ to account for the timeat which the kink passes closest to the line of sight. Aswe show below, this direction-dependent phase ultimatelyleads to a strong suppression of the kink memory signalcompared to the cusp case. A. Beaming effects
As in the cusp case, the first step in calculating theGW memory signal is to compute the angular integralthat captures the beaming effects of the kink, (cid:90) ˆ r (cid:48) e − i σφ (cid:48) f(cid:96)/ Θ ( θ b − | ι | ) . (65)Here the small circle of radius θ b around the North polethat we considered for cusps has been replaced with anarrow band of half-width θ b around the equator, and wehave included the φ -dependent phase factor from Eq. (64).It is straightforward to integrate out the zenith angle θ (cid:48) if we keep only the leading-order term in θ b ; this gives (cid:90) ˆ r (cid:48) e − i σφ (cid:48) f(cid:96)/ Θ ( θ b − | ι | ) (cid:39) θ b K f(cid:96)/ ( − σι ) , (66)where we have defined a family of azimuthal integrals, K n ( ι ) = (cid:90) π d φ π e i nφ (sin ι cos φ − i sin φ ) − cos ι cos φ . (67)Notice that the argument n is always an integer, as theGW frequency is always an integer multiple of the loop’sfundamental mode /(cid:96) (although we often ignore thiswhen taking the continuum limit at high frequencies).Computing the integral Eq. (67) for general n thereforecorresponds to finding the Fourier spectrum of a complic-ated nonlinear function of φ ; we perform this calculationexplicitly in Appendix C. B. Late-time memory
Using Eq. (9), along with the expression for the angularintegral K ( ι ) from Eq. (C5), we find that the late-timememory from the kink is given by ∆ h (1)k = 3 / (cid:96)r ( π A k Gµ ) | ι | + cos 2 ι − ι . (68)5Inserting numerical values, the strongest effect is ∆ h (1)k ≈− . × ( Gµ ) (cid:96)/r when the observer lies in the planeof the kink ι = 0 , an order of magnitude smaller thanthe maximum cusp memory. The late-time kink memorydecreases smoothly to zero as one approaches the poles ι → ± π / . C. Full waveform
The calculation here is very similar to the cusp case inSec. IV C. Inserting Eq. (64) into Eq. (12), and takingcare to enforce the frequency cutoffs and to account forthe two competing beam angles θ b ( f (cid:48) ) and θ b ( f (cid:48) − f ) , weobtain ˜ h (1)k ( f ) (cid:39) − i2 / π ( A k Gµ ) (cid:96) / / rf | f | / K f(cid:96)/ ( − σι ) × (cid:34)(cid:90) ∞ / | f | (cid:96) d uu / (1 + u ) − (cid:90) / / | f | (cid:96) d uu / (1 − u ) (cid:35) (69)for | f | > /(cid:96) . (This is analogous to Eq. (20) from thecusp case.) Taking the limit | f | (cid:29) /(cid:96) , we can evaluatethe integrals analytically to find ˜ h (1)k ( f ) (cid:39) − i B k ( Gµ ) (cid:96) / rf | f | / K f(cid:96)/ ( − σι ) Θ ( | f | − /(cid:96) ) , (70)where the numerical constant is B k ≡ π A √ (cid:18) (cid:19) / (cid:20) π √ − F ( , ; ; − (cid:21) ≈ . , (71)and where we have set the frequency cutoff at double thefundamental mode f = 2 /(cid:96) due to the different behaviourof the angular integral K n ( ι ) for n = 1 compared to n ≥ .This waveform (70) shares many features with boththe cusp memory waveform (21) and the primary kinkwaveform (64). The most important difference from bothof those waveforms is the dependence on inclination, whichhere is a function of frequency. Using the results derivedin Appendix C, we can rewrite Eq. (70) as ˜ h (1)k ( f ) (cid:39) − i B k ( Gµ ) (cid:96) / rf | f | / | ι | cos ι × (cid:18) cos ι | ι | (cid:19) f(cid:96)/ Θ ( f − /(cid:96) ) , for σι < (cid:18) cos ι | ι | (cid:19) − f(cid:96)/ Θ ( − f − /(cid:96) ) , for σι > (72)meaning that the memory signal from left-moving kinkscontains only negative frequencies above the equatorialplane and only positive frequencies below the plane, andvice versa for right-moving kinks. For high frequencies | n | (cid:29) the angular integral K n ( ι ) has a maximum value of (cid:39) / (e | n | ) at inclination ι (cid:39) /n .This means that the kink memory signal is only observablevery close to the plane of the kink (but not in the plane,where it vanishes), and is suppressed by an extra power offrequency compared to the primary signal, ˜ h (1)k ∼ f − / . D. Time-domain waveform near the arrival time
As with the cusp case, we can reverse-Fourier-transformEq. (72) to find the time-domain memory strain aroundtime of arrival of the primary kink signal, | t − t | (cid:28) (cid:96) .Unlike the cusp case, we obtain a signal which is notlinearly polarised, but contains both + and × polarisationcontent, h (1)k , + ( t ) − h (1)k , + ( t ) (cid:39) π B k ( Gµ ) / r ( t − t ) × sin | ι | cos ι E / (cid:18) | ι | cos ι (cid:19) ,h (1)k , × ( t ) − h (1)k , × ( t ) (cid:39) π B k ( Gµ ) / (cid:96)r ( t − t ) × sin σι cos ι E − / (cid:18) | ι | cos ι (cid:19) , (73)where we have used the generalised exponential integralfunction, E n ( z ) ≡ (cid:82) ∞ d x e − zx x − n . The × -polarised com-ponent is suppressed by an additional factor of ( t − t ) /(cid:96) (cid:28) , meaning that the signal is still approximately + -polarised.An important difference with respect to the cusp case isthat since kinks are long-lived rather than transient, theirmemory signal is not concentrated around a particulararrival time, but can in principle be observed at all times.One can easily obtain expressions analogous to Eq. (73) atany point in the kink’s periodic motion by substituting inthe appropriate time when evaluating the reverse Fouriertransform; in general this leads to a mixing between the + - and × -polarisation modes. E. Radiated energy
As in the cusp case, we are interested in the total energyradiated by the kink, and how the memory emission addsto this. Using Eq. (25), we find that the dimensionlessenergy spectra for the primary emission and the first-ordermemory are (cid:15) (0)k ( f, ˆ r ) (cid:39) π A Gµ ( f (cid:96) ) − / Θ ( θ b − | ι | ) Θ ( f − /(cid:96) ) ,(cid:15) (1)k ( f, ˆ r ) (cid:39) π B ( Gµ ) ( f (cid:96) ) / sin ι cos f(cid:96) − ι (1 + sin | ι | ) f(cid:96) Θ ( f − /(cid:96) ) . (74)6 Figure 6. A cartoon illustration of the angular distribution ofthe energy radiated by a kink. The primary emission (blue)is concentrated in a narrow fan of half-width θ b ∼ ( f(cid:96) ) − / around the plane of the kink, while the memory emission (red)is concentrated in two lobes either side of this plane, whichare exponentially suppressed as one approaches either of thedirections normal to the plane. Integrating over the sphere, the primary emission is sup-pressed by a factor of the beaming angle θ b ∼ ( f (cid:96) ) − / ,giving ¯ (cid:15) (0)k ( f ) (cid:39) / / ( π A k ) Gµ ( f (cid:96) ) − / Θ ( f − /(cid:96) ) . (75)The spherical integral for the memory contribution canbe evaluated explicitly to give (cid:90) S d ˆ r sin ι cos f(cid:96) − ι (1 + sin | ι | ) f(cid:96) = 8 π f (cid:96) ( f (cid:96) − f (cid:96) + 2) , (76)so that at high frequencies, the isotropic energy spectrumis approximately ¯ (cid:15) (1)k ( f ) (cid:39) π B k ) ( Gµ ) ( f (cid:96) ) / Θ ( f − /(cid:96) ) . (77)We see that the angular pattern of the kink memorystrongly suppresses the isotropic spectrum at high fre-quencies, meaning that unlike in the cusp case, there isno frequency range where the memory contribution dom-inates, and the total radiated energy converges withoutimposing an ultraviolet cutoff, E (0)k (cid:39) / ( π A k ) Gµ, E (1)k (cid:39) π B k ) / × Gµ ) . (78) F. Higher-order memory
As with the cusp case, we can iterate the memorycalculation with our 1st-order memory waveform (70) asan input to calculate the 2nd-order memory effect (the“memory of the memory”). This involves calculating theangular integral (cid:90) ˆ r (cid:48) e − i σφ (cid:48) f(cid:96)/ K f (cid:48) (cid:96)/ ( − σι (cid:48) ) K ( f (cid:48) − f ) (cid:96)/ ( − σι (cid:48) ) , (79)which in the high-frequency regime | f (cid:48) | , | f (cid:48) − f | (cid:29) /(cid:96) iswell-approximated by (cid:39) K f(cid:96)/ ( − σι )( f (cid:48) (cid:96) − f (cid:96)/ (cid:20) Θ ( f (cid:48) − /(cid:96) ) Θ ( f (cid:48) − f − /(cid:96) ) − Θ ( − f (cid:48) − /(cid:96) ) Θ ( − f (cid:48) + f − /(cid:96) ) (cid:21) . (80)With this result to hand, the remaining steps are very sim-ilar to the calculations for the 1st-order memory describedabove, yielding ˜ h (2)k ( f ) (cid:39) − i 512 π B ( Gµ ) rf | f | / (cid:96) / K f(cid:96)/ ( − σι ) Θ ( | f | − /(cid:96) ) × (cid:90) ∞ / | f | (cid:96) d uu / ( u + 1) / ( u + 1 / ≈ − i 6664 × ( Gµ ) rf | f | / (cid:96) / K f(cid:96)/ ( − σι ) Θ ( | f | − /(cid:96) ) . (81)The fact that this has the exact same dependence on theinclination ι as the 1st-order memory makes it straightfor-ward to iterate the process to higher orders. Doing this,we find that for all n ≥ , the kink GW memory is givenschematically by ˜ h ( n )k ( f ) ∼ − i( Gµ ) n f (cid:96) r ( | f | (cid:96) ) n/ K f(cid:96)/ ( − σι ) Θ ( | f | − /(cid:96) ) , (82)multiplied by some numerical constant (for which theredoes not seem to be a simple expression for all n ).The situation here is drastically different from the cuspcase. For cusps we saw that each successive order in thememory was suppressed by larger powers of Gµ (cid:28) , butalso enhanced by larger powers of (cid:96)/δ (cid:29) , and that incertain situations the latter would dominate, causing adivergence. For kinks, we see instead that each order inthe memory is not only suppressed by powers of Gµ , butis further suppressed by a factor of ( | f | (cid:96) ) − / (cid:28) eachtime. The signal falls off quickly enough with frequencythat higher-order memory contributions are not sensitiveto the string-width scale, and no factors of (cid:96)/δ appear.This means that there is no situation where the kinkmemory diverges, and that the higher-order contributionsare negligible in any observational scenario. This lack ofdivergence is in agreement with the cusp-collapse mechan-ism we have invoked as a possible resolution for the cuspdivergence, as kinks are not predicted to form PBHs [93].7 G. Caveats of our approach
We have only considered the simplest case where kink’sbeam traverses a fixed plane at a constant rate, in orderto make detailed analytical calculations feasible. Thissituation is highly idealised, and is not representative ofthe loops one would find in a cosmological loop network,which would typically contain structure on scales smallerthan the loop length (cid:96) , causing the path of the beam tovary on those scales. We expect that such structure isonly likely to make a significant qualitative difference toour results if it is on a scale corresponding to the GWfrequency of interest. We note that small-scale structureon loops is expected to be damped over time throughgravitational backreaction [103], so our simple treatmenthere is not likely to be too unreasonable. In any case,we do not expect such considerations to change the mainconclusion of this Section: that memory from kinks ishighly suppressed compared to the cusp case.We have also neglected the fact that kinks always appearin pairs on loops, with one left-mover for every right-mover.A realistic loop is likely to have several pairs of kinks,with each of these sourcing a GW memory signal likethe one calculated here. Since the kinks travel aroundthe loop at the same average rate, it is possible thatthe superposition of their memory signals could give riseto interesting coherent effects. However, regardless ofwhether the kink memory signals are coherent or not,the GW energy flux will still be of the same order ofmagnitude, so our main conclusions are unaffected.
VI. DETECTION PROSPECTS
Having calculated the nonlinear GW memory wave-forms associated with cusps and kinks, it is natural toask whether these signals are detectable with current orfuture GW observatories. Clearly the divergent beha-viour diagnosed for cusps in Sec. IV G could, in principle,have important observational implications, depending onhow the divergence is regulated. For the purposes of thisSection we assume the divergence is resolved along thelines of the cusp-collapse scenario described in Sec. IV I.We show below that, under this assumption, the cuspand kink memory signals are suppressed so strongly thatthey are well beyond the reach of GW observatories, evenfuturistic third-generation interferometers like EinsteinTelescope (ET) [122] and Cosmic Explorer (CE) [123].
A. Burst searches
We start by calculating the expected detection horizonsfor individual bursts of GW memory from cusps andkinks—i.e. the maximum distance at which a burst can bedetected, on average. We assume a matched-filter search,such that the optimal root-mean-square SNR (averaging − − − f / Hz10 − − − − − − − − Ω g w ( f ) primary GWsleading-order memoryhigher-order memoryLIGO/Virgo O3LISAPPTA Figure 7. Contributions to the SGWB energy spectrum Ω gw ( f ) from cosmic strings at different orders in the memory expan-sion, assuming that the memory divergence is resolved throughthe cusp-collapse scenario described in Sec. IV I. We seethat the memory effect is negligible compared to the primaryemission. Here we assume “model 2” [83, 113] of the stringnetwork, and a string tension of Gµ = 8 . × − (this isthe largest string tension allowed by current constraints inthe cusp-collapse scenario [93]; smaller tensions suppress thememory effect even further). The dashed/dotted curves showthe contributions from the matter/radiation era, respectively,with the solid curves showing the combined spectra. A distinctchange in all three of the radiation-era spectra is visible ataround f ≈
10 Hz ; at frequencies above this, the spectra areincreasingly dominated by small loops which do not undergocusp-collapse, and this is why the memory effect becomes moreprominent (though still undetectable). The magenta curveshows the power-law-integrated (PI) sensitivity curve [114]from the LIGO/Virgo O3 isotropic stochastic search [80], whichis publicly available at Ref. [115]. The green curve shows theParkes Pulsar Timing Array (PPTA) PI curve [116, 117], cal-culating using the code from Ref. [114], which is publiclyavailable at Ref. [118]. The cyan curve shows the projectedLISA [119] power-law-integrated sensitivity curve, as describedin Refs. [120, 121]. over sky location and polarisation angle) for a frequency-domain waveform ˜ h ( f ) which is isotropically averagedover the source inclination is given by [94] ρ rms = (cid:34)(cid:88) I
45 sin α I (cid:90) ∞ d f | ˜ h ( f ) | P I ( f ) (cid:35) / . (83)The sum here is over different GW detectors, with P I ( f ) representing the noise power spectral density (PSD) ofdetector I , and α I the opening angle between the twointerferometer arms (this angle enters through the de-tector’s response function; LIGO, Virgo, and CE have anopening angle of π / , while each of ET’s three interfero-8meters has an opening angle of π / ). It is convenient torewrite this in terms of the fractional energy spectrum,using Eq. (25) to give ρ rms = (cid:34)(cid:88) I Gµ(cid:96) π r sin α I (cid:90) ∞ d f ¯ (cid:15) ( f s ) f P I ( f ) (cid:35) / , (84)where r ( z ) is now the comoving distance to the source,and f s = (1 + z ) f is the source-frame frequency, with z the redshift. We assume that any cosmic string sig-nal with ρ rms ≥ can be confidently detected; in real-ity, this threshold depends on the distribution of non-Gaussian noise transients (“glitches”) in the network, andhow closely these are able to mimic the waveforms ofinterest, but we ignore these details here.Even assuming an optimistic third-generation GW de-tector network consisting of Einstein Telescope plus twoCosmic Explorers, we find that given current constraintson the string tension the detection horizon for a cuspmemory signal is at most ≈ . This corresponds to anegligibly small detection rate, as very few cosmic stringsare expected within a volume of this size. For kink memorythe result is even more pessimistic, with a detection ho-rizon of ≈ .
008 AU (a few times larger than the Earth-Moon distance). Larger values of the string tension Gµ would boost the detectability of these memory bursts, butwould be in conflict with existing observational results. B. Stochastic background searches
The combined GW emission from many loops through-out cosmic history gives rise to a stochastic GW back-ground (SGWB) [70–72, 76, 78, 81, 128–134]. The intens-ity of this SGWB as a function of frequency is usuallyexpressed as a fraction of the cosmological critical density ρ crit = 3 H / (8 π G ) in terms of the density parameter, Ω gw ( f ) ≡ ρ crit d ρ gw d(ln f ) , (85)which for cosmic string loops can be written as Ω gw ( f ) = 16 π Gµ H (cid:88) i N i (cid:90) d t (cid:90) d (cid:96) a n ( (cid:96), t )¯ (cid:15) i ( f /a, (cid:96) ) , (86)where t is cosmic time, a ( t ) is the FLRW scale factor, and n ( (cid:96), t ) d (cid:96) is the comoving number density of loops with For ET we use the “ET-D” noise PSD developed in Ref. [124],while for CE we use the “CE-2” noise PSD developed in Ref. [125]. Stochastic GW background constraints from PTAs [82, 83] implythat Gµ (cid:46) . × − in the cusp-collapse scenario [93] for “model2” [113] of the loop network. The other network model consideredmost frequently in the literature (“model 3” [126, 127]) gives aneven more stringent constraint of Gµ (cid:46) . × − [81]. (SeeRefs. [78, 81, 128] for an overview of these models.) length between (cid:96) and (cid:96) + d (cid:96) . The sum is over differentGW signals (cusps, kinks, and kink-kink collisions), with N i the mean number of signals per loop per oscillationperiod, and ¯ (cid:15) i the isotropic energy spectrum of each signal,as defined in Eq. (28).In Fig. 7 we show the SGWB spectrum from a particu-lar model of the loop network, including the contributionsfrom first-order and higher-order GW memory from cuspsand kinks. We see that while the primary SGWB reachesa plateau at high frequencies, the memory contributionsto the SGWB grow with frequency above ≈
10 Hz ; thismakes sense given the slower fall-off of the cusp memoryemission at high frequency compared to the primary cuspand kink signals. However, each order in memory is sup-pressed by an additional factor of ( Gµ ) , and this suppres-sion is strong enough to render the memory contributionunobservable at all frequencies. C. Consequences for the loop distribution function
Additional energy loss due to GW memory emissionmeans that loops decay more rapidly, modifying the dis-tribution of loop sizes. This could in principle leave dis-tinguishable imprints on the two observational probesmentioned in the previous sections. However, we show be-low that such imprints are suppressed by powers of Gµ inthe cusp-collapse scenario, rendering them unobservable.Following Ref. [135], we write the loop decay rate as d (cid:96) d t = − Γ Gµ J ( (cid:96) ) , (87)where Γ ≈ is a numerical constant, and J ( (cid:96) ) is anarbitrary function of the loop length which is equal tounity in the standard Nambu-Goto case. If J differsfrom unity then the faster/slower loop decay rate leads toa smaller/greater number of loops once the equilibrium“scaling” distribution is reached; Ref. [135] shows that themodified loop distribution function n ( (cid:96), t ) can then beobtained from the standard Nambu-Goto distribution byinserting appropriate factors of J .We account for enhanced energy loss through GWmemory radiation by writing J ( (cid:96) ) = ∞ (cid:88) n =0 E ( n ) E (0) (cid:39) E (1) E (0) , (88)where E ( n ) is the dimensionless GW energy radiatedthrough n th-order memory, as given by Eqs. (61), (62),and (63) for cusps and Eq. (78) for kinks. We find that J ( (cid:96) ) − is at most ∼ Gµ for both cusps and kinks,meaning that any modifications to the loop distributionfunction are suppressed by powers of Gµ , and are thusunobservable. (This is assuming the memory divergenceis resolved through cusp collapse; of course, it is still pos-sible that the memory emission could be much larger, andthereby lead to significant imprints on the loop distribu-tion, and indeed on the other observational probes wehave discussed.)9 VII. SUMMARY AND CONCLUSION
We have thoroughly explored the nonlinear GW memoryassociated with cusps and kinks on Nambu-Goto cosmicstrings, deriving detailed analytical waveform models forthe memory GWs, including the “memory of the memory”and other higher-order memory effects. These are amongthe first memory observables computed for a cosmologicalsource of GWs, with previous literature having focusedalmost entirely on astrophysical sources.The leading-order cusp memory waveform (21) thatwe have found is strikingly similar to the primary GWsignal from the cusp (13a), with the same characteristic ∼ f − / frequency power-law. However, one very importantdifference is that this memory signal is emitted in alldirections, unlike the primary signal which is confined toa narrow beam of width θ b ∼ f − / . As a result, we findthat the total GW energy radiated by the cusp memory diverges for a Nambu-Goto (i.e., zero-width) loop. Thisdivergence can be regularised by introducing a cutoff atthe scale of the string width δ ∼ (cid:96) Pl / √ Gµ , but this thenintroduces powers of (cid:96)/δ (cid:29) to the higher-order memoryterms, causing the sum of all memory contributions todiverge for loops of length (cid:96) (cid:38) δ/ ( Gµ ) .In Sec. IV I we have argued that the most plausibleexplanation for this unphysical memory divergence isthe assumption that the spacetime containing the cos-mic string loop is well-described by a flat backgroundwith linear perturbations. We have suggested that somestrong-gravity mechanism must kick in on loops of length (cid:96) (cid:38) δ/ ( Gµ ) to suppress the high-frequency GW emissionfrom cusps, thereby curing the divergence. Our earlierwork Ref. [93] provides exactly such a mechanism: thecollapse of a small cosmic string segment near the cuspto form a PBH. Remarkably, this cusp-collapse processis predicted to occur for all loops of length (cid:96) (cid:38) δ/ ( Gµ ) —exactly the same loops for which we have diagnosed thememory divergence. We have shown explicitly that ifcusp collapse does indeed occur for these loops, the cor-responding GW memory is strongly suppressed, and thedivergence is cured.We have also calculated the memory emission associatedwith kinks, and have shown that this is suppressed due tointerference between GWs emitted by the kink at differentpoints in its history. There is thus no situation in whichthe kink memory signal diverges; this accords with thecusp-collapse description, as kinks are not predicted toform PBHs in that scenario.In Sec. VI we have investigated the detection prospectsfor these cusp and kink memory signals, calculating theexpected detection rate of individual memory signals formatched-filter searches with third-generation GW obser-vatories like Einstein Telescope and Cosmic Explorer, aswell as the contribution of memory GWs to the SGWBspectrum, and possible imprints in the cosmic stringloop distribution function. We find that by requiring thestring tension to agree with existing observational bounds( Gµ (cid:46) − ) and invoking cusp collapse to prevent the memory from diverging, the resulting memory signal isvery strongly suppressed, and is not likely to be detectedby any current or upcoming GW observatories. Of course,if cusp collapse does not occur, then it is possible thatlarge loops could source much stronger memory signals;however, one would then need an alternative means ofresolving the cusp divergence.Our work demonstrates the importance of consideringthe nonlinear memory associated with a broader class ofGW sources than just compact binaries; we have shownthat the memory effect is interesting not just from an ob-servational point of view, but also as a tool for sharpeningour theoretical understanding and modelling of said GWsources. For cosmic strings in particular, as an unexpec-ted by-product of our analysis we have shown that thestandard Nambu-Goto description of cusps is unphysical,and that strong gravity effects (possibly including PBHformation) could play an important rôle in a more com-plete understanding of their dynamics. This motivatesfurther work to better understand cusps in full GR, andthereby compute reliable waveform predictions for GWobservatories. It also motivates a broader examinationof the nonlinear memory effect in GW astronomy andcosmology, to see what other surprises may be in store. ACKNOWLEDGMENTS
We thank Josu Aurrekoetxea and Eric Thrane for theirvaluable feedback on this work. A. C. J. is supportedby King’s College London through a Graduate TeachingScholarship. M. S. is supported in part by the Science andTechnology Facility Council (STFC), United Kingdom,under research grant No. ST/P000258/1. This is LIGOdocument number P2100040.
Appendix A: Understanding the origin of thehigher-order memory divergence
Here we derive a condition (A5) on a generic GW signal h ( t ) which, we argue heuristically, is necessary for theassociated nonlinear memory to diverge. We find thiscondition by considering a simple toy model in whichwe can tune the “sharpness” of the signal to find wherethe divergence sets in. We then show that this conditionpredicts the divergence for cusps on “large” cosmic stringloops, while also providing an explanation for why thememory from compact binaries never diverges.
1. Gaussian pulse as a toy model
Our goal here is to derive a condition on how “sharp” aputative GW signal must be to give rise to a divergentnonlinear memory expansion. To investigate this, we usea toy model in which the primary GW is a Gaussian pulse0which reaches our detector at retarded time t = 0 , h (0) ( t ) = Ar exp (cid:18) − t σ (cid:19) . (A1)This has amplitude A and width σ , both of which havedimensions of length. We ignore the polarisation andangular pattern of the pulse, though these can play animportant role (e.g., for the case of kinks, as discussedabove), and focus on the pulse’s behaviour as a functionof time. The Fourier transform of Eq. (A1) is ˜ h (0) ( f ) = Ar √ π σ exp (cid:18) − (2 π f σ ) (cid:19) , (A2)so we see that the pulse has a characteristic frequency of ∼ /σ . The pulse is smooth and infinitely differentiablefor all finite σ ; however, by making σ small, we canmake the pulse arbitrarily “sharp” (equivalently, we canmake the characteristic frequency arbitrarily large). Thissharpness can be quantified by the maximum value of thetime derivative of the strain, max t | r ˙ h (0) ( t ) | = e − / A/σ ∼ [ amplitude ] × [ characteristic frequency ] , (A3)i.e. if the dimensionless ratio A/σ is much greater thanunity the pulse is “sharp”, and if
A/σ is much less thanunity the pulse is “soft”.The pulse’s first-order memory can be written as r ˙ h (1) = C | r ˙ h (0) | = C (cid:18) Atσ (cid:19) exp (cid:18) − t σ (cid:19) , (A4)where C is some constant arising from the angular integral,which we assume to be O (1) for all orders in the memoryexpansion. (This assumption can easily be violated: e.g.,if the integrand has vanishing TT component then C = 0 .)We immediately see that the memory signal is larger thanthe primary GW signal near the arrival time ( | t | ∼ σ ) ifand only if A/σ (cid:29) , or equivalently, max t | r ˙ h ( t ) | (cid:29) . (A5)Using the Isaacson formula (5), we see that this is equi-valent to max t (cid:12)(cid:12)(cid:12)(cid:12) d E gw d t (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) G = m Pl t Pl , (A6)i.e. if the GW energy flux is super-Planckian.Moving on to the second-order memory, there are twocontributions: the self-energy of the first-order memory,and its cross-energy with the primary signal, r ˙ h (2) = Cr ( | ˙ h (1) + ˙ h (0) | − | ˙ h (0) | )= C | r ˙ h (1) | + 2 Cr ˙ h (1) ˙ h (0) = C (cid:18) Atσ (cid:19) exp (cid:18) − t σ (cid:19) − C (cid:18) Atσ (cid:19) exp (cid:18) − t σ (cid:19) . (A7) The general pattern is straightforward from here. Treatingthe “sharp” regime A/σ (cid:29) and the “soft” regime A/σ (cid:28) separately (analogous to the separation between the“large loop” and “small loop” regimes for cusps), we findfor all n ≥ , r ˙ h ( n ) (cid:39) C (cid:20) CAtσ exp (cid:18) − t σ (cid:19)(cid:21) n , A/σ (cid:29) C (cid:20) − CAtσ exp (cid:18) − t σ (cid:19)(cid:21) n +1 , A/σ (cid:28) (A8)which shows that the memory expansion diverges ifEq. (A5) holds.Of course, there are many ways in which this argumentcould fail, some of which we have already mentioned (inparticular, if C (cid:28) ). However, the general takeaway isthat if the time derivative of a GW strain signal is large,then a memory divergence might occur. For max t | r ˙ h | (cid:28) on the other hand, it seems very likely that the memoryexpansion must always converge. If this is indeed the case,then we could interpret Eq. (A5) as a necessary, but notsufficient, condition for the memory divergence.
2. Application to cosmic strings
How does this fit into our results for cosmic stringcusps? Neglecting numerical constants, the time-domaincusp waveform looks like h (0)c ( t ) ∼ − Gµr (cid:96) / | t | / + constant , (A9)so the time derivative diverges at t = 0 due to the absolutevalue function. If we introduce a cutoff at the string widthscale δ , we find that max t | r ˙ h (0)c | ∼ Gµ ( (cid:96)/δ ) / . (A10)Naively applying the condition (A5), we would thus expectthe cusp memory signal to diverge for loops of length (cid:96) (cid:38) δ/ ( Gµ ) / . (A11)However, this does not happen, for the simple reasonthat there is a small factor C (cid:28) arising from the an-gular integral, and as mentioned above this causes thecondition (A5) to fail.We can circumvent this issue by considering the first-order memory signal from the cusp as the source of thedivergence: this should give C = O (1) as it has a broademission pattern, and is not concentrated into a narrowbeam. Indeed, using Eq. (23) we find that the maximumtime derivative for the first-order cusp memory signal is max t | r ˙ h (1)c | ∼ ( Gµ ) ( (cid:96)/δ ) / . (A12)Applying the condition (A5) again, we find that the cuspmemory signal should diverge if (cid:96) (cid:38) δ/ ( Gµ ) , (A13)1 − . − . . . . ι . . . . . . . ( / ) n − | L n ( ι ) | / n = 2 ...n = 6 − . − . . . . ι − − ( − ) n S n ( ι ) n = 2 ...n = 12 Figure 8. Functions describing the strength of the n th-order cusp memory signal as a function of inclination, with L n ( ι ) representing “large” loops (cid:96) (cid:29) (cid:96) ∗ , and S n ( ι ) representing “small” loops (cid:96) (cid:28) (cid:96) ∗ . which is exactly what we find from the careful analysisin Sec. IV H. This supports the idea that Eq. (A5) givesa necessary condition for the memory divergence, whichis generally applicable if the factor C arising from theangular integral is not too small.
3. Application to compact binaries
Here we use a very simple heuristic analysis to show that | r ˙ h ( t ) | is at most O (1) for compact binary coalescences, sothat the memory divergence condition (A5) never holds.We neglect numerical factors throughout.The GW signal from a CBC can be written as h ( t ) = A ( t )e i φ ( t ) , (A14)where the leading-order Newtonian contributions to theamplitude and phase are [94] A ∼ ( G M ) / rτ / , φ ∼ (cid:16) τG M (cid:17) / , (A15)with τ the time until coalescence and M ≡ η / M thechirp mass, where η ≤ / is the dimensionless mass ratio.The time derivative of the strain (A14) is therefore | r ˙ h ( t ) | = (cid:113) | r ˙ A| + | r ˙ φ A| ∼ (cid:115)(cid:18) G M τ (cid:19) / + (cid:18) G M τ (cid:19) / . (A16)Formally, this diverges in the Newtonian analysis as τ → . However, introducing a cutoff at the ISCO radiustruncates the signal at a finite value of τ , τ min ∼ η − / G M , (A17) so that max t | r ˙ h ( t ) | ∼ (cid:112) η + η . (A18)Since η is at most O (1) , we see that the condition (A5) isnever met. This makes complete sense, given that we knowthat the memory from CBC signals cannot diverge. As aby-product, Eq. (A18) leads us to conjecture that equal-mass binaries ( η = 1 / ) should give rise to stronger higher-order memory effects than extreme mass-ratio inspirals( η (cid:28) ). Appendix B: Angular integrals for higher-order cuspmemory
Here we derive the angular patterns of the higher-ordercusp memory in the “large-loop” ( (cid:96) (cid:29) (cid:96) ∗ ) and “small-loop”( (cid:96) (cid:28) (cid:96) ∗ ) limits, which are described by the functions L n ( ι ) and S n ( ι ) respectively, with ι the inclination between thecusp beaming direction and the observer’s line of sight.These are defined by inserting the n th-order memoryformula (53) into the iterative relation (50), which gives L n +1 ( ι ) = (cid:90) ˆ r (cid:48) | L n ( ι (cid:48) ) | ,S n +1 ( ι ) = (cid:90) ˆ r (cid:48) ι (cid:48) ) S n ( ι (cid:48) ) , (B1)for all n ≥ , with L ( ι ) = S ( ι ) = 2 sin ι. (B2)(The integral symbol (cid:82) ˆ r (cid:48) is defined in Eq. (7).)2We find that L n and S n can be written as polynomialsin cos ι (of order n − and n , respectively). The iterativeprocess (B1) can therefore be carried out by evaluating asimpler family of integrals, C n ( ι ) ≡ (cid:90) ˆ r (cid:48) cos n ι (cid:48) . (B3)To do so, we choose our ˆ r (cid:48) = ( θ (cid:48) , φ (cid:48) ) coordinates such that θ (cid:48) is zero along the line of sight, so that cos θ (cid:48) ≡ ˆ r · ˆ r (cid:48) ,with the cusp beaming direction defined relative to theline of sight by ˆ r c · ˆ r = cos ι . We then have cos ι (cid:48) ≡ ˆ r c · ˆ r (cid:48) = cos θ (cid:48) cos ι − cos φ (cid:48) sin θ (cid:48) sin ι, (B4)and Eq. (7) becomes (cid:90) ˆ r (cid:48) = (cid:90) S d ˆ r (cid:48) π (1 + cos θ (cid:48) )e − φ (cid:48) , (B5)so that we can use a binomial expansion of cos n ι (cid:48) =(cos θ (cid:48) cos ι − cos φ (cid:48) sin θ (cid:48) sin ι ) n to write C n ( ι ) = n (cid:88) k =0 (cid:18) nk (cid:19) ( − sin ι ) k cos n − k ι × (cid:90) +1 − d x x ) x n − k (1 − x ) k/ × (cid:90) π d φ (cid:48) π e − φ (cid:48) cos k φ (cid:48) , (B6)where we have set x = cos θ (cid:48) . Using the Beta functionidentity B( a, b ) ≡ (cid:90) d x x a − (1 − x ) b − = Γ ( a ) Γ ( b ) Γ ( a + b ) , (B7)we then obtain, for all n ≥ , C n ( ι ) = n (cid:88) k =0 √ π ( n − k )(2 n )! cos k ι sin n − k ) ι n +1 k !( n − k + 1)! Γ ( n + ) ,C n +1 ( ι ) = n (cid:88) k =0 √ π ( n − k )(2 n + 1)! cos k +1 ι sin n − k ) ι n +2 k !( n − k + 1)! Γ ( n + ) . (B8)Calculating each iteration of Eq. (B1) is then reduced towriting down the appropriate linear combination of C n ( ι ) for different n .The resulting expressions for S n ( ι ) and L n ( ι ) for thefirst few n ≥ are shown in Tables II and I respectively.We find empirically that the large-loop angular functionscan be approximated by | L n ( ι ) | ≈ / n − sin ι (B9)with an accuracy of ∼ , as is illustrated in the leftpanel of Fig 8. The small-loop angular functions S n ( ι ) donot seem to follow such a simple pattern. Appendix C: Angular integrals for kink memory
Here we compute the integral K n ( ι ) defined in Eq. (67)for all angular modes n ∈ Z . Note that while the integ-rand is complex, the integral itself is always real, as theimaginary part of the integrand is an odd function of φ .Note also that we can focus on non-negative n ≥ byexploiting the symmetry property K − n ( ι ) = K n ( − ι ) . (C1)For some special values of ι we can obtain the fullspectrum immediately from Eq. (67), K n (0) = − δ n, −
12 ( δ n, + δ n, − ) ,K n ( ± π /
2) = δ n, ± . (C2)For general values ι ∈ [ − π / , + π / , we proceed by ex-panding the denominator of Eq. (67) using the geometricseries and the binomial theorem, − cos ι cos φ = ∞ (cid:88) k =0 cos k ι cos k φ = ∞ (cid:88) k =0 k (cid:88) m =0 (cid:32) km (cid:33) cos k ι k e − i( k − m ) φ . (C3)This converges everywhere on ι ∈ [ − π / , + π / except for ι = 0 , in which case we use Eq. (C2) instead. We canthen integrate term-by-term to extract the contributionfrom each order in the series. The result, valid for all n ≥ , is K n ( ι ) = ∞ (cid:88) k =max(0 , − n ) (cid:32) k + n − k (cid:33) (cid:16) cos ι (cid:17) k + n − (cid:18) ι (cid:19) − ∞ (cid:88) k =0 (cid:32) k + nk (cid:33) (cid:16) cos ι (cid:17) k + n +2 + ∞ (cid:88) k =0 (cid:32) k + n + 2 k (cid:33) (cid:16) cos ι (cid:17) k + n +2 (cid:18) − sin ι (cid:19) . (C4)The first sum must be carried out separately for the threecases n = 0 , n = 1 , and n > , yielding the followingexpressions: K ( ι ) = 4 sin | ι | + cos 2 ι −
32 cos ι ,K ( ι ) = sin | ι | −
12 sin | ι | (cid:20) cos ι + (sin ι −
1) 1 − sin | ι | + sin ι (1 + 2 sin ι + sin | ι | )cos ι (cid:21) ,K n ( ι ) = ι cos ι (cid:18) cos ι ι (cid:19) n , ι > , ι ≤ , for n > . (C5)3 Table I. The first few large-loop angular functions, as defined by Eqs. (B1) and (B2). Note that L n ( ι ) is a polynomial in cos ι oforder n − , such that the number of terms grows exponentially with n . Despite this apparent complexity, we find that theseformulae can be approximated by the simple expression (B9). n L n ( ι )2 2 sin ι − ι
15 (5 − cos 2 ι )4 sin ι ι −
262 cos 4 ι + 7( − ι )]5 sin ι (cid:20) ι − ι + 627703754313 cos 6 ι − ι +1106354755 cos 10 ι + 1001( − − ι + 245 cos 14 ι ) (cid:21) ι (cid:20) ι − ι + 22676810284676710047197528844734128748260 cos 6 ι − ι + 114707816611411474747133898122922244740 cos 10 ι − ι + 322110616165743776447552901776475040 cos 14 ι − ι + 490882191972236519729403447127344 cos 18 ι +1463( − ι + 246246861424248720285801892 cos 22 ι +1495( − ι + 44031911264492955804 cos 26 ι +6525( − − ι + 271199768840 cos 30 ι ))) (cid:21) Table II. The first few small-loop angular functions, as defined by Eqs. (B1) and (B2). Note that S n ( ι ) is a polynomial in cos ι of order n . n S n ( ι )2 2 sin ι − ι ι )4 2 sin ι
25 (25 + 24 cos ι + 3 cos 2 ι )5 − ι
175 (154 + 201 cos ι + 42 cos 2 ι + 3 cos 3 ι )6 sin ι ι + 7932 cos 2 ι + 960 cos 3 ι + 45 cos 4 ι )7 − sin ι ι + 172824 cos 2 ι + 5(29702 + 5619 cos 3 ι + 450 cos 4 ι + 15 cos 5 ι )]8 sin ι − ι + 157771 cos 2 ι + 32552 cos 3 ι + 3490 cos 4 ι + 200 cos 5 ι + 5 cos 6 ι )9 − sin ι ι + 6432382 cos 2 ι + 5( − ι + 46508 cos 4 ι + 3565 cos 5 ι + 154 cos 6 ι + 3 cos 7 ι )]10 sin ι (cid:20) − ι + 2472686192 cos 2 ι + 5(214599584 cos 3 ι + 36187180 cos 4 ι +7( − ι + 28400 cos 6 ι + 960 cos 7 ι + 15 cos 8 ι )) (cid:21) − sin ι (cid:20) − ι − ι + 116827655852 cos 3 ι + 27832319320 cos 4 ι +3261899900 cos 5 ι + 49( − ι + 214995 cos 7 ι + 5850 cos 8 ι + 75 cos 9 ι ) (cid:21)
12 sin ι (cid:20) − ι − ι + 5( − ι +238091184664 cos 4 ι + 37145938400 cos 5 ι + 3229010267 cos 6 ι +179962104 cos 7 ι + 6563970 cos 8 ι + 147000 cos 9 ι + 1575 cos 10 ι ) (cid:21) − . − . . . . ι − . − . − . − . . . . . . K n ( ι ) n = 0 n = 1 n = 2 ...n = 20 Figure 9. The angular integral K n ( ι ) for non-negative n , asgiven by Eq. (C5). The corresponding curves for negative n are obtained by reflecting around ι = 0 . These clearly agree with Eq. (C2) for ι = ± π / . Notethat for all n , the integral K n ( ι ) is not differentiable at ι = 0 ; this is related to the fact that the series in Eq. (C3)diverges at ι = 0 . However, despite this formal divergence,we see that Eq. (C5) agrees with Eq. (C2) in the limit ι → , whether this limit is taken from above or frombelow. We therefore use Eq. (C5) over the full domain ι ∈ [ − π / , + π / . 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