Coherent Gravitational Waveforms and Memory from Cosmic String Loops

We construct, for the first time, the time-domain gravitational wave strain waveform from the collapse of a strongly gravitating Abelian Higgs cosmic string loop in full general relativity. We show that the strain exhibits a large memory effect during merger, ending with a burst and the characteristic ringdown as a black hole is formed. Furthermore, we investigate the waveform and energy emitted as a function of string width, loop radius and string tension $G\mu$. We find that the mass normalized gravitational wave energy displays a strong dependence on the inverse of the string tension $E_{\mathrm{GW}}/M_0\propto 1/G\mu$, with $E_{\mathrm{GW}}/M_0 \sim {\cal O}(1)\%$ at the percent level, for the regime where $G\mu\gtrsim10^{-3}$. Conversely, we show that the efficiency is only weakly dependent on the initial string width and initial loop radii. Using these results, we argue that gravitational wave production is dominated by kinematical instead of geometrical considerations.


I. INTRODUCTION
The detection of Gravitational Waves (GW) from black hole (BH) binaries [1] by the LIGO/Virgo collaboration marked the start of a new era of observations. Beyond astrophysical objects such as BH and neutron stars, this paved the way for the use of GW to search directly for signatures of new physics. One of the key targets of this search is the existence of a network of cosmic strings [2][3][4][5].
Cosmologically, cosmic string networks naturally arise after a phase transition in the early universe, possibly during GUT symmetry breaking [6][7][8][9][10][11]. These networks are known to be a source of gravitational waves, and there is a large literature concentrating on the stochastic background of weak field emission of GW through cusps, travelling kinks and kink-kink interactions of the strings . This signal is the total integrated power of incoherent GW from all such individual emissions, i.e. the sum of all individual emissions which themselves are too weak to be directly detected. Furthermore, these networks may manifest themselves through other channels, such as their imprints via lensing on the Cosmic Microwave Background [36,37].
Complementarily, one can also search for localized coherent events of these strings. Coherent events are those that are individually energetic enough to be detected directly. Such events can occur, for example, when the strings self-interact through the formation of sharp cusps, through the collisions of travelling kinks that are formed during the intercommutation (i.e. collisions) of cosmic strings, or when cosmic string loops collapse. Such a search requires the construction of GW waveform templates -parameterized coherent time/frequency Pl . The dotted signal was calculated using the semi-analytical approach while the solid line is from the integration of the NR signal. The strain exhibits a large memory due to the aspherical loss of matter ejecta during merger, ending with a characteristic ringdown after the black hole is formed. A summary movie of the simulation can be found on https://youtu.be/-dhYA2788LA. domain signals which can then be searched via matchfiltering in the detector signal stream or identified within a burst search. We emphasise that searches for stochastic and coherent signals are complementary -the nondetection/detection of one does not imply the nondetection/detection of the other.
In the literature, collapsing cosmic string loops have been considered as seeds in the formation of primordial black holes [38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Recently, we presented the first investigation of the collapse of circular loops with full general relativity [52]. By solving the full non-linear system of Abelian-Higgs field equations coupled to 3+1 Einstein gravity, we showed that the main two outcomes were dispersion and black hole formation. If the loop is not massive enough or thin enough, it will unwind and disperse all the energy into scalar, gauge and gravitational radiation. However, a black hole can also be a result, resulting in a large emission of gravitational waves.
In this paper, we compute this corresponding coherent GW strain in the time-domain -see fig. 1. In other words, we compute the GW strain waveform from individual GW events from the collapse to black holes of cosmic string loops, which is manifestly a strong gravity event.
We show that the coherent GW strain signals from the collapse of cosmic string loops are dominated by two major components. The first component is that of a large gravitational wave memory [53,54] effect during the merger, generated by a large aspherical "jet-like" ejection of matter radiation. The second component is that of the final ringdown phase post-BH formation, with the initial collapse stage being a subdominant contribution to the total signal. We also find that the efficiency of GW production is around O(1)% of the total cosmic string mass. This efficiency is dependent on the cosmic string tension Gµ, with lower tension producing more GW -up to 2.2% for Gµ = 2 × 10 −3 , which is the lower bound of the parameter space studied in this work. In comparison, the efficiency for head-on BH mergers and inspiral merger is 0.06% and ∼ 5% respectively. We will comment on this somewhat counter-intuitive result in section V.
Coherent GW events are categorized by its energy ("loudness") and its characteristic frequency. The distance d from which one of these events could be observed by current and future GW detectors is given by where E GW is the energy emitted in GWs and h is the strain sensitivity of the detector. Roughly speaking, interferometers are optimized to detect GW induced strain of h ∼ 10 −21 around a finite frequency domain -for the LIGO/Virgo interferometers this is f ∼ 10 − 1000 Hz. In the case of GW events when a black hole is formed, the quasinormal mode (QNM) frequency of the characteristic ringdown phase is determined by its mass. Combined, this means that LIGO/Virgo is sensitive to E GW ∼ M events at around 100 Gpc. Thus to produce coherent GW observable by LIGO/Virgo one must produce sufficiently energetic ("loud") events at its detector frequency 1 . This means that LIGO/Virgo will be sensitive to cosmic string loop events 2 of around 100M at a distance of around 1 Gpc [52].
To check the dependency of the waveforms and energy as a function of the initial conditions and parameter of the cosmic string loops, we compute the waveforms for the three main parameters of the system. The first parameter is the string tension Gµ which specifies the underlying theory. The next two parameters, the initial radius R 0 and the width of the string δ, define the initial string geometry. We find evidence that the the mass normalized waveforms depend strongly on the string tension Gµ, and weakly on the string width δ and initial string radii R 0 , for the regime Gµ > 10 −3 . Hence, it follows that the GW production efficiency of collapsing cosmic string loops is only weakly dependent on initial string loop radii R and the width of the string δ -at least for the parameter space studied in this work. Combined with the fact that the power is dependent on string tension Gµ -and this sets the loop velocity at BH formation -we argue that the generation of GW is driven by collapse kinematics instead of the geometry of the system. The paper is organized as follows. In section II, we describe the Abelian Higgs cosmic string model and recap some previous results. In section III, we describe the parametric dependences of GW power from both string geometry and string model for cosmic string collapse events. In section IV, we show how the waveform is not degenerate to other BH known processes, and we derive the full coherent time-domain GW strain waveform from a combination of semi-analytic and numerical results. We discuss the prospects and strategies for a direct detection search and conclude in section V.

II. ABELIAN HIGGS STRING LOOPS
The action of the Abelian Higgs model minimally coupled to gravity 3 is and V (φ) is the sombrero potential of the complex scalar field φ FIG. 2. Gravitational wave signals as a function of string tension Gµ : The signal is normalised with the initial mass of the system and shifted such that the maximum of rΨ4 coincides at time t = 0, for three cases from table (A 3) for Gµ = {3 × 10 −3 , 6 × 10 −3 , 1 × 10 −2 } and corresponding mass M0 = {18.85MPl, 13.19MPl, 10.05MPl}. The relationship between Ψ4 and detector strain h is given in Eqn. (16). The thickness of the line is an estimate of the numerical error. Unphysical parts of the signal are de-emphasised using ticked lines with different transparencies. We find that smaller Gµ have larger amplitudes and hence produce more gravitational wave radiation (with 2.2% for Gµ = 2 × 10 −3 with R = 1600M −1 Pl ). The rest of the initial mass goes into the black hole and matter radiation. A table summary of all the runs is shown in (A 3).
where η is the symmetry breaking scale.
For simplicity, we set the charge e and the dimensionless coupling constant λ to obey the critical coupling limit in which the Higgs and vector masses are identical and the string tension µ is related to the symmetry breaking scale as The coupling constant λ and the string tension Gµ set the width of string as In [52], we constructed the initial conditions to a circular cosmic string loop. The mass of such a configuration of radius R 0 is given by which is independent of the coupling constant λ. Also in [52], we showed that the hoop conjecture argument accurately predicts that an initially static loop with radius R 0 and tension Gµ will form a black hole as long as the condition is satisfied.

III. PARAMETRIC DEPENDENCE OF GW SIGNALS
In this section we study how the gravitational wave signal changes when we vary the parameters of the model: the string tension Gµ, the initial loop radius R 0 and the string width δ.
We first focus on the string tension Gµ. We performed a series of simulations with the string parameters shown in table (A 3) with fixed λ = 2. Since varying Gµ substantially changes the mass of the string (see Eqn. (8)), for each choice, we choose its initial R 0 to ensure that a black hole can be formed (i.e. obey the condition Eqn. (9)).
The energy radiated in GWs can be estimated from the rΨ 4 Weyl scalar by equation Eqn. (C1). The efficiency of GW production normalized over total string mass, E GW /M 0 is shown in Fig. 3. Interestingly, we find that this scales as where A is a numerical factor found to be approximately A ≈ 10 −2 . Intriguingly, this means that smaller tension leads to greater efficiency, with the caveat that we have only explored a small regime of the total possible parameter space. This scaling clearly cannot be unbounded as Gµ → 0, and must turnover at some point. We will discuss this further in section V. We can also explore the dependence of GW emissions as a function of string width δ and initial radius R 0 . In [55], using purely geometrical arguments, Hawking computed the efficiency of GW emitted from an infinitesimally thin cosmic string loop, and showed that it has an upper bound of 29%. This is obtained by assuming that the initial horizon of the black hole is a thin disk, and then computing the difference of the disk's total area with the area of the final Schwarzschild black hole. Hence, it is plausible that if the initial horizon of the black hole is less disk-like and more spheroidal, the efficiency will become smaller since the initial horizon area will then be greater (and the difference with the area of the Schwarzschild black hole is smaller). To test for this idea, we can define a dimensionless "thickness" parameter, such that a cosmic string is "thin" if δ/R 0 is small and "thick" if δ/R 0 is close to unity. In the infinitesimally thin limit, δ/R 0 → 0. Our argument above suggests that the GW efficiency should increase as δ/R 0 decrease, with the Hawking limit being δ/R 0 = 0. However, as we will show in below, this is not borne out by our numerical simulations, at least in the limited range of parameters we are able to explore. We test this argument by performing simulations with varying string width δ and radius R 0 , while keeping other parameters fixed as follows.

FIG. 3. Efficiency in GW production vs string tension:
We find that the efficiency EGW/M0 ∝ A 16π 2 Gµ −1 obey a simple power law with A = 10 −2 (solid line). The simulation parameters and results are tabulated in Tab. (A 3) while the star-dotted point on the right is the result from our previous paper [52]. Note that the last data point to the left may signal the turnover of the inverse power law 1/Gµ. Fig. 4, we see that the signals only depend weakly on string width. Initial radius R 0 dependence : We performed three simulations with varying R 0 = {160, 240, 320}M −1 Pl at fixed Gµ = 1 × 10 −2 and λ = 2. Since the mass scales with R 0 and the ringdown frequency of a black hole is inversely proportional to its mass, we normalise the signal with their initial mass. From the results shown in Fig. 5, we find that the normalised signal at most scales weakly with R 0 .
The above results suggest that the GW emission efficiency is only weakly dependent on initial string dimensionless thickness δ/R 0 .
On the other hand, the numerically obtained scaling Eqn. (10) can be suggestively rewritten as where γ is the Lorentz factor of the string infall velocity and t BH is black hole formation time, i.e.
We can derive Eqn. (13) as follows. In [52], we have shown that the dynamics of a radius R 0 cosmic string loop during the infall is well described by the Nambu-Goto approximation [56], for which the position and velocity at some given time are given by The black hole forms approximately when r BH = 2GM 0 = 4πR 0 Gµ, which using Eqn. (14) happens at time t BH = R 0 cos −1 (4πGµ), so that the velocity at black hole formation is which using γ = (1 − v 2 ) 1/2 leads to Eqn. (13). For Gµ = 1 × 10 −2 − 2 × 10 −3 , this corresponds to v(t BH ) ≈ 0.9920 − 0.9997, so it is an ultra-relativistic event. Note that the velocity equation Eqn. (15) does not depend on λ and R 0 . Physically, the smaller the string tension, for a fixed loop mass M 0 the larger the radius of the loop has to be, the longer it takes for the loop to reach the Schwarzschild radius and hence the faster the loop will be moving when the black hole is formed. Hence we conjecture that the GW emission process is dominated by the kinetic energy of the system, with the string geometry playing only a minor role 5 .

IV. GRAVITATIONAL STRAIN WAVEFORMS
Our goal in this section is to construct the strain waveform. The gravitational wave strain h as seen by a detector is related to the Weyl scalar Ψ 4 by the following equation of motion Thus we would need to integrate Eqn. (16) to obtain h. The details of this integration are described in appendix (A). Furthermore, as we have described in our previous work [52], numerically the early time infall signal is contaminated by the presence of unphysical artefacts from the numerical construction of its initial conditions 6 . To circumvent this, we note that during this early time period, the infall tracks the trajectory of a Nambu-Goto string until a distance of O(δ) [52]. We use this fact to construct a semi-analytic model of the GW emission during infall as follows. The modified trajectory is given by where the Heaviside functions ensure consistency with the initial data of our numerical simulations where the loop is static for t < t 0 (see fig. 8 and fig. 9). In Cartesian coordinates (x, y, z) such that r = x 2 + y 2 + z 2 , the stress tensor in the corresponding basis is where the velocity is Pl . Both signals are normalized to mass, but the black hole formed from the head-on collision is 16× closer to the detector. This shows that the ringdown signal of cosmic string loop is not degenerate with distance to spin-free BH merger ringdown.
The gravitational wave signal of such system is then given in the weak field limit by the standard formula [57] (20) where t ret = t − r/c is the retarded time and is valid for arbitrary velocities, and Λ ij,kl is the projector to the traceless-transverse gauge. The result and details of this calculation for various methods as well as a convergence test can be found in appendix (A 2) and Fig. 7. We plot the resulting gravitational wave strain for Gµ = 4 × 10 −3 with R 0 = 600M −1 Pl in Fig. 1. As one can see, r∆h + = rh + (∞)−rh + (−∞) > 0. This is known as the gravitational wave memory effect [53,54,[58][59][60], which is a large permanent shift in the strain waveform. The nature of this memory arises from the fact that post-merger, there is a loss of matter emitted axially in an ultra-relativistic jet (Fig. 16) -and hence is highly aspherical -while its "incoming" velocity is zero (i.e. the loop is initially static). This generates a large linear memory shift [61] akin to that of a core-collapse supernova [62].
We can estimate the magnitude of this memory using the linear memory formula [54,59] where M A an v A are the rest mass and asymptotic velocity respectively of ejecta particle A and θ A is the angle between v i A and the direction to the detector. The ∆ expresses the difference between the initial "incoming" and "outgoing" values. The initial velocity of the loop is v i A = 0. From numerical simulations, it can be seen that the outgoing ejecta is highly beamed like jets in the direction axial to the loop (see Fig. 16). In general, to use this formula, one must calculate the flux of ejecta as a function of angle. Since our goal is not to make a precise prediction of its value (we directly obtain this from numerical simulations), but to simply demonstrate that our numerical result is indeed gravitational wave memory, we approximate its magnitude as follows. We assume that all the ejecta is travelling at a constant velocity axially (i.e. perpendicular to the plane of the loop) at v i A = (0, 0, ±v z ) where v z ∼ 1 (the exact value does not affect the final answer significantly).
We express the right hand side of Eqn. 21 onto a spherical basis at radius r by first rotating each instance of the metric r∆h ij → r∆h i j (θ, φ) where (θ, φ) are the coordinates on the sphere. We then project the metric onto their traceless and transverse components to obtain where it can be shown that and E total ≈ M 0 −M BH = 1.32M Pl (see Tab. (A 3)) is the total integrated relativistic flux energy for both matter and GW we directly measured from our simulations. To compare this to our numerical result in Fig. 1, we project Eqn. (23) onto the l = 2, m = 0 mode as which about a factor of 5 smaller when compared to the numerical value we obtained, but at the right order of magnitude. We emphasise that Eqn. (24) is just an estimate of the memory assuming the interactions stay within the linear regime, and hence it is not surprising that the true memory is larger.

V. DISCUSSION AND PROSPECTS FOR DETECTION
In this work, we showed that GW production of cosmic string loops that collapse and form black holes scales as but depends weakly on its initial string width and loop radius. We argue that this strongly suggests that the GW production in such a collapse is dominated by kinematic processes, and not geometric ones. Clearly, since Gµ is theoretically not bounded from below, Eqn. (25) cannot scale without bound to smaller values as it violates the Hawking bound E GW /M 0 → 0.29 at Gµ ≈ 2 × 10 −5 . This suggests that there must exist some new scale where this turnover from the inverse power law to some other relationship. This turnover may already be hinted in fig. 3, where the Gµ = 2 × 10 −3 point is diverging from expression Eqn. (25), and will be a focus of our future investigations. Furthermore, our cosmic string loops are Planckian in their masses. To generate loops of solar masses require that the loops have large radii -for example for Gµ ≈ 10 −10 require a loop of around 100 a.u. 7 .
Observations of the CMB [37] and the LIGO/Virgo search for stochastic GW [4,5] constraints the current cosmic string tension to Gµ 10 −14 − 10 −7 -this value is dependent on the details of the cosmic strings network evolution which is uncertain (and model dependent) [63][64][65][66][67][68][69]. This regime is obviously beyond the validity of our scaling argument. While we have only explored a small regime of the possible parameter space and the amplitude of the GW signal may differ for other parameters, we do not expect the form of the GW strain signal shown in Fig. 1 to differ substantially at lower Gµ. We also emphasise that strongly gravitating strings such as fundamental strings with Gµ ∼ 10 −2 can also be produced in many popular brane inflation models [70][71][72][73]. Modulo such theoretical concerns about the probability distribution of such events which can only be estimated from large network simulations, we take the agnostic view that their existence can be put into observational test.
On the other hand, we believe that the large gravitational wave memory of these events is a robust result regardless of the string parameters, since it is sourced by the large aspherical emission of post-collapse debris which we expect to occur regardless. While GW memory are historically removed from both the detector data streams and theoretical predictions, there is now increasing interest in their search [74,75] and is currently a goal of the LIGO/Virgo collaboration [76].
Both such short signals with little GW production during the infall phase suggests that this it is best looked for in the transient short-during burst channel [77][78][79][80][81]. This channel makes only minimal assumptions on the expected signal waveform, at the cost of reduced sensitivity to weaker signals. One may wonder whether the string loop burst waveform is degenerate with other processes such as very massive binary black hole inspiral or head-on mergers -and hence can be picked up by already existing match-filtered searches. The former case is trivial since the lack of an oscillatory pre-merger signal and the fact that the black hole formed the collapse has no spin, are sufficient features to distinguish from a binary black hole inspiral system, and thus it is not degenerate.
For a more symmetric scenario such as a head-on BH-BH merger, in Fig. (6) we show that it is not degenerate. While the ringdown signal from the black hole formed from a loop is degenerate with a black hole with the same mass formed from a head-on merger 16× closer, the pre-merger and the merger itself differ considerably. Therefore, it will be distinguishable as long as one has access to the full waveform.
To detect such weaker signals, one would need to make use of the full match-filtering search, which requires the construction of a parameterised GW waveform template. In this work, we argue that the primary parameter for the construction of such waveform templates is the string tension Gµ, with secondary parameters being the initial string width and radii. We undertook the first steps in the construction of the GW strain waveform template (Fig. 1). In an upcoming publication, we will complete the construction of these templates, and use them to search for cosmic string loop collapse signatures in the LIGO/Virgo data stream. The GW strain can be obtained directly from integrating the numerically obtained Weyl Scalar Ψ 4 , with the boundary conditions that the emission in gravitational wave power stops at large times and P GW ∝ḣ We hence have the freedom to shift h where h num is the gravitational wave strain calculated using a numerical integration technique from Ψ 4 . However, we found in the simulations that the quasi-normal modes become unreliable after a certain time due to numerical resolution (see Fig. 11 for t > ∼ 4500), which causes substantial errors in the integration. To deal with this, this we substitute the signal with analytical QNMs [82] for the corresponding l = 2 mode. We performed convergence checks in resolution, courant-factor, box-radius and extraction radius, to ensure that all our numerical integrations are converged.

Weak-field gravity extension
To construct the infall signal, we will calculate ithe strain of a collapsing circular and planar cosmic string loop with energy momentum tensor given by where we define r = x 2 + y 2 and the behaviour of the pre-merger collapse in the weak-field limit is well described by where δ(t − t 0 ) is the Dirac delta and we set the starting time t 0 = 0 to be consistent with the simulations. Note that we have use the Heaviside Theta functions to impose the initial of the cosmic loop such that it is infinitely static from t < t 0 , consistent with the initial conditions of our numerical simulations. This is important as the Nambu-Goto loop is oscillating, and hence will contribute GW in the regime t < t 0 , in contradiction to our numerical simulations (see Figs. 8 and 9). We run both methods with three resolutions, which we refer as low, mid and high. The difference between them becomes smaller as the resolution is increased, indicating that our integration has converged. Both methods recover the same signal.
The effective GW generated for sources that are relativistic is given by [57] where n is the direction of the observer n = (sin θ sin φ, sin θ cos φ, cos θ) , and Λ ij,kl (n) is the projector to the TT gauge, where We define the fourier transform as To check the calculation we also calculate the same expression in the time-domain, we indeed find that both formulations converge to the same result (see Fig. 7). The loop is initially at rest with radius R0, then starts to collapse at t0 and forms a black hole at tBH. The dashed grey is the solution of an oscillating loop following the Nambu-Goto action. As shown in Fig. 9, the first signal an observer at x obs receives depends on the past history of the loop (grey shaded area). For the Nambu-Goto case, one would get gravitational radiation coming from the expansion phase of the loop (after it has shrunk to a point in the previous cycle). We cut this spurious signal off by imposing a Heaviside function in Eqn. (14).

Fitting to the NR signal
We match the strain from our numerical relativity simulations rh num with the weak gravity calculation rh weak of the previous section as follows The free shift b parameter is found by finding the best fit value over a region where both signals are valid (shaded region in Fig. 10) . We define this region of validity as, that when GM/R(t f ) ≈ 0.25, such that t f = R 0 cos −1 (4GM/R 0 ). In addition, we define the starting point as the time when most of the initial data artefacts have passed the detector (we can read this value from the rΨ 4 plot). The best fit is shown in Fig. 10. visible in Fig. 2, where we increased the radius of the ring for smaller Gµ to guarantee black hole formation. The initial peak, which is the artificial, becomes more and more separated with the signal for larger R 0 .
To calculate the total emitted GW energy we use the usual equation where S r is a sphere of radius r.

Numerics and Convergence Tests
In Fig. (14), we show that the volume-averaged Hamiltonian constraint violation where V is the simulation box coordinate volume with the interior of the apparent horizon excised, is under control throughout the simulation. We use the gradient conditions on φ and χ to tag cells for regridding. The precise criteria is chosen depending on the symmetry breaking scale η and the total mass of the system. We use the symmetry of the system to only simulate one quarter of the system, which reduces the computational cost of the problem.
We cut off our signal after some time t when the black hole has formed (and hence the QNM signal is completely determined analytically), and fit QNM modes for the l = 2 m = 0 mode [84] in Fig. 11). We test the precision of the simulation by comparing the radiated energies with the initial mass. We find that these number for the simulations in table (A 3) are consistent within the 1-5 % range.
We tested the convergence of our simulations with a cosmic string loop of Gµ = 2 × 10 −3 and R 0 = 1600M −1 Pl by using a box of size L = 3072M −1 Pl in which we improved by a factor of 1.2 between the medium and highest resolution and 1.25 between the lowest and medium resolution. The convergence of rΨ 4 is shown in Fig. 15, for different coarse grid resolutions: low (∆x = 32M −1 Pl ), medium (∆x = 38.4M −1 Pl ) and high (∆x = 48M −1 Pl ), in addition to 6 refinement levels. In colour we plot the energy density. Initially, the loop starts to collapse from rest (upper left); The energy density of the loop increases as its radius becomes shorter and accelerates to ultra-relativistic speeds, when Lorentz contraction effects emerge in the direction of the collapse (upper right). When the radius of the loop is of the same order as the width of the string, the collision happens, where high curvature effects appear (lower left). If the system is massive and thin enough, part of the initial mass of the system collapses to a black hole and high-relativistic jets are emitted axially as a result of the ultra-relativistic collision (lower right). This aspherical ejection of matter is responsible for a constant shift in the gravitational waveform known as gravitational wave memory. The full movie can be found on https://youtu.be/0sSH54gXu4U.