Comparison of cosmic string and superstring models to NANOGrav 12.5-year results
CComparison of cosmic string and superstring models toNANOGrav 12.5-year results
Jose J. Blanco-Pillado
Department of Physics, University of the Basque Country, UPV/EHU, Bilbao, Spain andIKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain
Ken D. Olum
Institute of Cosmology, Department of Physics and Astronomy,Tufts University, Medford, MA 02155, USA
Jeremy M. Wachter
Skidmore College Physics Department,815 North Broadway Saratoga Springs,New York 12866, USA
Abstract
We compare the spectrum of the stochastic gravitational wave background produced in severalmodels of cosmic strings with the common-spectrum process recently reported by NANOGrav. Wediscuss theoretical uncertainties in computing such a background, and show that despite such un-certainties, cosmic strings remain a good explanation for the potential signal, but the consequencesfor cosmic string parameters depend on the model. Superstrings could also explain the signal, butonly in a restricted parameter space where their network behavior is effectively identical to that ofordinary cosmic strings. a r X i v : . [ a s t r o - ph . C O ] F e b . INTRODUCTION The NANOGrav collaboration has recently reported some evidence of a stochastic signalin their 12 . GWSMBHB ∼ f / . All this makes them themost likely candidate to explain a potential signal at these frequencies.There are, however, other potential sources of gravitational waves at these frequencieswhich are associated with cosmological processes in the primordial universe. One of the mostnatural and promising sources is the stochastic background of gravitational waves createdby a network of cosmic strings. Cosmic strings are effectively one-dimensional topologicaldefects that may have been produced by a phase transition in the early universe [4, 5]. Wewill be interested here in the simple case of Abelian-Higgs strings, or superstrings, with nocouplings to any massless particle other than the graviton. Such a string network is describedby a single quantity that parameterizes the characteristic energy scale of the universe at thetime of string formation. This energy scale specifies the energy per unit length of the stringas well as its tension, µ . Since we are interested in gravitational effects, we will be mostinterested in the combination Gµ , where G is Newton’s constant. We will work in unitswhere c = 1, so that Gµ is dimensionless.The equality between the energy per unit length of the strings and their tension impliesthat the dynamics of these strings are relativistic. Putting all of these facts together, onecan immediately see why cosmic strings are good candidates for gravitational waves: theyare cosmologically large relics that store very high energy densities associated with the earlyuniverse, and they move relativistically under their own tension. This explains why anaccurate computation of the SGWB from strings has been pursued for a long time in thecosmic string community [6–26].Because string models are described by a single parameter related to the universe’s energyat their time of formation, an observation of the SGWB from strings would indicate theexistence of new physics at the string scale. However, the apparent simplicity of the single-parameter model is deceiving when it comes to detecting strings. The dynamics of thecosmic string network are complicated, making it difficult to obtain detailed descriptions ofthe necessary ingredients to compute the SGWB. One has to resort to large scale simulationsto be able to establish basic facts needed in this calculation, like the number density of cosmicstring loops throughout the history of the universe, or the typical power spectrum of such2oops. These are questions that one would have to answer in any model that produces anstochastic background: how many emitters are there, and how do they emit? Knowing this,we can estimate the combined effect of all sources.Comparisons of some cosmic string models’ predictions with the NANOGrav data haverecently been made in [27–30]. We will focus here on how theoretical uncertainties in thetypical power spectrum of a cosmic string loop impact the amplitude and slope of the SGWBsignal in the NANOGrav window, and therefore how the most-likely µ (and associatedconfidence intervals) changes due to this uncertainty. We do not suggest that a confirmedcosmic string detection would resolve this theoretical uncertainty, as that requires a betterunderstanding of cosmic string networks and evolution. II. THE SGWB FROM COSMIC STRINGS
The basic idea behind the string SGWB computation is simple: for any given obser-vational frequency, collect the contributions from all the different strings throughout thehistory of the universe that emit waves with the appropriate frequency such that they areobserved at the observational frequency today. It is customary to present this informationby calculating the critical density fraction of energy in gravitational waves per logarithmicfrequency today, Ω GW (ln f ) = 8 πG H f ρ GW ( t , f ) , (1)where H is the Hubble parameter today, and ρ GW denotes the energy density in gravitationalwaves per unit frequency.The calculation of the energy density has been described in detail in [22], and a summarycan be found in Appendix A. Each loop radiates in discrete multiples n of its fundamental os-cillation frequency 2 /l , where l is the invariant loop length, given by the loop energy dividedby µ . We write the power from loop i in harmonic n as P ( i ) n Gµ , so P ( i ) n is dimensionless.We write the total radiation power Γ ( i ) Gµ , where Γ ( i ) = (cid:80) ∞ n =1 P ( i ) n . For our purposes here,we will neglect differences in Γ ( i ) between loops and just write Γ.The three main ingredients we need to compute the string SGWB are: • A cosmological model. • The number density of non-self-intersecting loops as a function of length at any mo-ment in time. • The average power spectrum of gravitational waves from non-self-intersecting loops inthe network, P n .We consider a standard cosmological history, and take the loop number density describedin [22] based on the simulations reported in [31]. This leaves the average power spectrumof non-self-intersecting loops, P n . This is probably the quantity in the calculation with thehighest uncertainty at this moment, since it depends not only on the gravitational radiationspectrum of non-self-intersecting loops at formation, but also on their evolution. This isa challenging problem, since one needs to follow the change in shape of a representative The string network contains both loops and long, horizon-spanning strings, but the contribution of longstrings to the SGWB is subdominant for all µ . We consider only the SGWB due to loops. ∼
50, which we will take as the Γ for all SGWB we study. We use P BOSn (after the authors’ initials) to indicate the average power spectrum computed by this work,and will use it as one of the models we study in the following section. It is quite smooth,and has a long tail describing the emission of a substantial amount of power at the high-frequency modes of the string. This can be traced to the presence of cusps in the final stagesof the evolution of these smoothed loops. The SGWB spectra arising from this model werediscussed in Ref. [22].Cusps are moments of the loop’s oscillation when a point on the loop formally reachesthe speed of light [33]. Cusp formation leads to the loop emitting a significant amountof radiation, which is beamed in the direction of motion of the cusp [8]. Accumulatingradiation from many such events forms a stochastic background whose power spectrum hasa long tail, of the form P cusp n ∝ n − / [8]. Because cusps are thought to be generic featuresof loops, a common model of the power spectrum is one where low modes, which describethe shape of the loop, are less important than high modes. If we focus on these high-modecontributions to gravitational waves, then we can use a model where the spectrum is simplygiven by P cusp n . We will choose a constant of proportionality so that (cid:80) ∞ n =1 P cusp n = Γ. Thisis the second model of P n we consider when discussing a possible string SGWB.Another characteristic feature on realistic loops are kinks: points along the string wherethere is a discontinuity in its tangent vector. These occur every time two segments of stringintersect one another and exchange partners. Kinks move at the speed of light along thestring, emitting a fan of radiation whose spectrum at high mode number emission goes as P kink n ∝ n − / [35]. As with cusps, we can consider a model with only kink radiation. Thisis our third model.A fourth and final model takes the reverse approach: instead of focusing on the high-harmonic tail, we consider a spectrum consisting only of the fundamental mode, P mono n = (cid:26) Γ if n = 10 otherwise . (2)Like the pure-cusp and pure-kink spectra, this is not a realistic assumption, but it serves asa limiting case for strings which radiate primarily in low harmonics.The real average spectrum should be calculated from a realistic distribution of non-self-intersecting loops obtained from a scaling simulation and evolved under their own gravity.This can be done using linearized gravity, since the force that affects each loop’s shapedepends on Gµ , which in our case is always very small. This idea was first developed in [36],and has recently been advanced both analytically [37–40] and numerically [41]. The resultsfrom these papers indicate that cusps and kinks are smoothed over time. Some of the effects Cusp bursts can be sources of transient events in gravitational wave detectors. See the discussion in [12,34, 35].
4f backreaction are captured by the smoothing procedure of [32], but there are cases wherethis approach is not so accurate. The specific results of the long-term effect of backreactionon loops produced by a scaling string network are therefore still unclear. Thus we will showthe gravitational wave amplitudes and spectral slopes to be expected for all four models andcompare them with the NANOGrav observations.
III. COMPARISON WITH NANOGRAV 12.5-YEAR DATA
The NANOGrav collaboration presents their data using the characteristic strain of theform h c ( f ) = A (cid:18) ff yr (cid:19) α = A (cid:18) ff yr (cid:19) (3 − γ ) / , (3)where f yr = 1 / year, A is the strain amplitude, and γ is the spectral index. The energy densityin gravitational waves can be obtained from this characteristic strain using the relationΩ GW ( f ) = 2 π H f h c ( f ) . (4)NANOGrav also reports likelihoods in the parameter space of ( γ, A ), which we will use toconstruct confidence regions to use for our analysis of the effect of different P n .For a given Gµ and P n , we can compute the energy density in gravitational waves withEq. (1). From this, we approximate the spectral index and amplitude using the two lowestfrequencies seen in NANOGrav, f = 1 / (12 . f = 2 f . This process provides a goodfit to compare to the 5-frequency contours because the two lowest frequencies in NANOGravare much better determined in comparison to the third through fifth lowest frequencies. Ourmethod is to calculate γ = 5 − ln(Ω GW ( f ) / Ω GW ( f ))ln(2) , (5a) A = (cid:115) H Ω( f ) f − γ yr π f − γ (5b)for each Ω GW .Figure 1 shows the curves one obtains in the ( γ, A ) plane for Gµ ∈ [10 − , − ] for ourfour models of P n . All models have been normalized so that the total power is given byΓ = 50. We report the approximate ranges of log Gµ which predict values within the 1 σ ,1 . σ , and 2 σ confidence range in Table I.The important general result of Fig. 1 is that the ( γ, A ) parameters predicted by the dif-ferent spectra are quite similar over the range we investigate. This means that the theoreticaluncertainty in the average gravitational wave power spectrum from loops will not greatlyaffect the conclusions obtained from identifying the NANOGrav result with the SGWB fromcosmic strings. In other words, assuming the actual spectrum of realistic loops is somewhereclose to the models we study here, we can infer that the constraints on Gµ are quite similarto the ones obtained from this figure. Of course, future data and analysis will likely reduceuncertainties, shrinking the range of the significance contours, and allowing us to pin downthe most likely value of Gµ . Our ability to do that will depend on reducing our uncertaintyin the loop power spectrum. This is the job of simulation, and a detection consistent with5 G µ =10 -9 G µ =10 -10 G µ =10 -11 Log ( A ) Spectral index γ monokinkcuspBOS FIG. 1. The amplitude vs. spectral index for various cosmic string tensions, Gµ , for four models ofthe average power spectrum, P n . The tic marks show steps of 0 . Gµ , from − −
11, withlarge tics every 0 .
5. The short, medium, and long dashes show the 1 σ , 1 . σ , and 2 σ contours (i.e.,they enclose 68%, 86%, and 95% of the likelihood, respectively) made from the NANOGrav 12 . P n model σ range . σ range σ range BOS ( − . , − .
40) ( − . , − .
52) ( − . , − . − . , − .
39) ( − . , − .
50) ( − . , − . − . , − .
52) ( − . , − .
64) ( − . , − . − . , − .
67) ( − . , − .
80) ( − . , − . Gµ falling within the 1 σ , 1 . σ , and 2 σ confidence intervalsof NANOGrav for the four P n models. any of the above curves should not be considered evidence for that P n being the true powerspectrum of loops in nature. IV. RELATIONSHIP TO PREVIOUS WORKA. Upper bounds
The authors of Ref. [23], including two of us, derived bounds on the possible values of Gµ from non-observation of a SGWB. We concentrated on the BOS model. Using results fromthe Parkes PTA [3, 45], we gave a limit of Gµ < . × − , and using the the NANOGrav9-year results [46], we gave Gµ < . × − . However, referring to Fig. 1, we see that the6est fit Gµ is about 5 . × − , 40% larger than the limit based on NANOGrav and about4 times the limit based on Parkes.There are two reasons for this discrepancy. First, all pulsar timing arrays include modelsof individual pulsar noise. If not treated correctly, this modeling can absorb the effectsof the SGWB, leading to incorrect upper bounds. This is discussed in detail in Ref. [47].Reference [1] compares the NANOGrav red noise process detection with their previouslygiven upper limits.Second, pulsar timing is dependent on the solar system ephemeris, which tells us howto remove the earth’s motion through the solar system from the observed data. We donot know this ephemeris to the accuracy necessary, and thus ephemeris uncertainty is anadditional source of error in gravitational wave measurements. In particular, if one allows theobservations to influence the choice of ephemeris, one may thereby absorb some gravitationalwave power and infer incorrect limits. See Ref. [48] for more detailed discussion. B. Other cosmic string SGWB results
Other recent papers [27–30] have interpreted the NANOGrav 12.5-year data as a cosmicstring signal. We discuss the similarities and differences between their approaches and ourshere, and comment generally on agreements between those approaches.The majority of the sources mentioned [27, 28, 30] employ the velocity-dependent one-scale (VOS) model for generating the cosmic string SGWB. This model has an additionalparameter: the loop size at formation as a fraction of horizon size, α , which Ref. [27] sets to0 . Reference [29] follows the sameapproach as this paper. All of the aforementioned use a cusp power spectrum in creatingtheir SGWB, and so we can only make meaningful comparisons between their results andour cusp results.The VOS model and the one we use here are in near-exact agreement when VOS takes α = 0 . α = 0 . Reference [29] considersmetastable cosmic strings, characterized by a parameter κ ; we would expect their resultsto match ours in the limit κ → ∞ , i.e., when the decay rate of cosmic strings due tomonopole–antimonopole pair production goes to zero and the strings decay only via GWs.There is one additional concern in comparing different results in the γ -log A plane. Sup-pose two different approaches generate identical SGWB, so they predict the same Ω GW atsome common reference frequency f ref , but they use different approaches to determine γ .When they extrapolate the amplitude from f ref to f yr to report A (see Eq. (5b)), the resulting A will be different. The difference in the reported logarithmic amplitude is∆(log( A )) = log( A /A ) = 12 ( γ − γ ) log (cid:18) f ref f yr (cid:19) . (6)Taking this effect into account, Ref. [27] draws very similar conclusions to ours as to thebounds on Gµ , as does Ref. [28] for the α = 0 . When not exploring the effect of varying α , taking α = 0 . Note that Refs. [28, 30] conclude that values of α < . α = 0 .
1, and so we cannot make a direct comparison. Reference [29] does notdisplay a comparison to their results with a stable string SGWB, but their bounds on Gµ as κ increases seem to be converging towards results consistent with ours (e.g., the point with Gµ = 10 − and largest κ is on the edge of the 1 σ contour). V. COSMIC SUPERSTRINGS
Until now, we have been discussing the gravitational spectrum produced by a network ofcosmic strings that exchange partners whenever they intersect. This is the expected inter-action of strings that appear as topological defects in field theory (e.g., in the Abelian-Higgsmodel [50, 51]). There are, however, other scenarios where a network of cosmologicallyinteresting string-like objects is produced. In particular, many cosmological models of su-perstring theory suggest the production of fundamental strings, which are then stretchedto cosmological size by an expanding universe [52–55]. Once stretched, these fundamentalstrings have similar dynamics to their classical counterparts, except for the crucial aspectthat their intercommution is different. This is due to the fact that their interactions arequantum mechanical in origin, and also because the strings in these models may move in aspace with additional dimensions. Both these effects may significantly reduce their chance tointercommute. That is, the strings sometimes pass through one another, rather than split-ting and rejoining to form sharply-angled kinks. This issue has been studied in Ref. [56],where the conclusion was that the probability p of reconnection could be as low as 10 − .A decrease in the intercommutation probability should have an effect on the macroscopicproperties of the network. There has been some debate in the literature about how this lowerprobability would modify the overall density of the strings [53, 57, 58]. This is importantto the calculation of the SGWB, since the density of loops has a direct impact on the sizeof Ω GW . Large scale simulations would be necessary to establish the precise modificationsthat this reduced probability will bring to the final scaling distribution of loops presentedearlier, but they have not yet been done. Here, we will assume that the effect of reducing p is to increase the loop number density by factor 1 /p , without changing the properties ofthe loops, so that Ω GW ∝ p . (7)Lowering the intercommutation probability increases the amplitude of gravitational waveswithout changing the slope, and so we may estimate the range of p which is compatible withthe current NANOGrav data. The upward displacement of the curves in the ( γ, A ) planequickly moves them away from the 1 σ region, as shown in Fig. 2, in agreement with theresult of Ref. [27].However, the current likelihood data will never completely exclude a superstring networkat the 2 σ level. The 1 /p enhancement means that for small p we are interested in a smaller Gµ . This puts us in the low- f region in the cosmic string background spectrum [22], whereΩ GW rises with frequency as f / , giving γ = 7 /
2. For any small p , there will be some Gµ giving the A that lies in the 2 σ region at the left of Fig. 2. While we only show superstringsusing the BOS model of P n , the 3 / P n , and so this effectis generally true.Despite this, the NANOGrav data as currently given is most consistent with p ≈
1. Asa consequence, superstrings are likely to explain the potential signal only if their networkproperties are very similar to those of cosmic strings.8 G µ =10 -11 G µ =10 -12 Log ( A ) Spectral index γ p=1p=10 -1 p=10 -2 p=10 -3 FIG. 2. The amplitude vs. spectral index for various superstring tensions, Gµ , for the BOS modelof the average power spectrum. The intercommutation probabilities go from 1 to 10 − in powers often, with lower p moving the curve out of NANOGrav’s significance region (c.f. Fig. 1), representedby the dashed grey lines. The other models of P n return similar results. The vertical gray lineshows the spectral index to be expected from SMBHB. A counterargument to this claim is the idea that strings are wiggly, and so each stringcrossing has multiple potential intersection events, increasing the chance that strings inter-commute and thus depressing the 1 /p enhancement to the energy density. A specific exampleof such an argument can be made using the results of Ref. [58], which found the energy den-sity to have very little enhancement down to p ≈ .
1, after which it follows Ω GW ∝ p − . .This would relax the bounds on p somewhat, allowing superstrings down to p ∼ − to fallat the edge of the 1 . σ region, roughly where p = 10 − lies in Fig. 2.Our conclusions about superstring viability change slightly if improved statistics movesthe confidence interval contours towards the left, towards the predicted SMBHB signal’svertical line at γ = 13 /
3. There, an enhancement to the amplitude due to p ∼ . Gµ < ∼ − . .It may therefore be necessary to distinguish a superstring SGWB from a supermassive blackhole binary SGWB. Because we expect the number of cosmic strings or superstrings thatcontribute to the SGWB to be large in the frequency band seen by NANOGrav, this couldbe accomplished by studying anisotropies in the reported signal, which we would not expectif strings are the source. VI. CONCLUSIONS
Regardless of the model chosen to represent the average power spectrum of a cosmicstring loop, the potential signal reported by NANOGrav could be a cosmic string stochasticgravitational-wave background. Thus, as long as these models are close to the true averagepower spectrum, a confirmation of a cosmic-string signal would predict the existence of anetwork of strings with a tension in the range of Gµ ≈ [10 − . , − . ]. Such Gµ values are9 -13 -12 -11 -10 -4 -3 -2 -1 frequency (Hz)3 ⋅ -10 ⋅ -10 ⋅ -10 ⋅ -10 Ω g w h LISA bandmonokinkcuspBOS
FIG. 3. The energy density vs. frequency of the SGWB for the four P n models we consider, asseen in the LISA band. All curves are for Gµ = 10 − , but the range of tensions which fit theNANOGrav data produce similar results. The decline in the lines, and its variation, are directconsequences of changing degrees of freedom in the universe’s past, and so LISA could measuredeviations from a standard cosmological model for such curves [65]. low enough that we would not expect such strings to be visible in the cosmic microwave back-ground [59] or to produce gravitational wave bursts that can be seen in interferometers [60] or pulsar timing arrays [62].Superstrings are less favorable as an explanation for the signal. They would either haveto have very similar network properties to cosmic strings, due to p ≈
1, or would have to berescued by changes to the confidence interval contours.If the signal is indeed from cosmic strings, then we can expect to see other parts of theSGWB in future gravitational wave telescopes. The values of Gµ we consider are too low forLIGO/VIRGO to observe the SGWB [61] , but LISA, the Einstein Telescope, or the BBOare sensitive in the correct frequency and amplitude range. In LISA, for example, we couldmeasure the section of the SGWB which contains information about cosmological history,particularly the effect of changing degrees of freedom [22, 49, 64, 65], as shown in Fig. 3.Such a measurement could be used to quantify deviations from the standard model and thusprobe new physics. ACKNOWLEDGMENTS
We would like to thank Xavier Siemens for helpful conversations. This work is sup-ported in part by the Spanish Ministry MCIU/AEI/FEDER grant (PGC2018-094626-B-C21), the Basque Government grant (IT-979-16) and the Basque Foundation for Science See also the results presented for model A in [61]. Here we only consider model A in Ref. [61]. See Ref. [63] for a critical discussion of the viability of othermodels presented in this reference.
Appendix A: Computing the gravitational wave energy density
Fundamental to Eq. (1) is ρ GW , the energy density in gravitational waves per unit fre-quency. It can be written ρ GW ( t, f ) = Gµ ∞ (cid:88) n =1 C n P n , (A1)where P n describes the average gravitational wave spectrum of the cosmic string loops inthe network and C n = (cid:90) t dt (1 + z ) nf n ( l, t ) = 2 nf (cid:90) ∞ dzH ( z )(1 + z ) n (cid:18) n (1 + z ) f , t ( z ) (cid:19) , (A2)where n ( l, t ) is the loop number density, H ( z ) is the Hubble parameter and t ( z ) the age ofthe universe at redshift z . We consider a standard cosmological history, so H ( z ) and t ( z ) aregiven by the usual expressions in terms of the components of the universe, Ω r , Ω m , and Ω Λ ,as well as the number of degrees of freedom at each moment in time (see [22] for a detailedexplanation of these functions).Finding the form of n ( l, t ), is equivalent to finding the distribution of non-self-intersectingloops at all times in the history of the universe. This sounds like a challenging problem sinceit will be impossible to simulate the evolution of the network for such a wide range of timescales. Luckily for us the evolution of a cosmic string network has a scaling solution , wherethe energy density of the string remains a small fraction of the background energy density ofthe universe. This is an important property of the model since it makes cosmological stringnetworks compatible with observations. There is a more important aspect of this scalingsolution for our calculation: in a scaling solution, the form of the loop distribution satisfies n ( l, t ) = t − n ( x ) , (A3)where x = l/t is the ratio of the loop size to the horizon size at some particular time,and n ( x ) is the number of loops per unit x in a volume t . The scaling solution simplifiesthe problem, reducing it to finding n ( x ). Finding this scaling solutions from numericalsimulations presents a big challenge, since one has to run for extremely long periods of timebefore reaching a true scaling solution for the loop’s distribution. Here, we use the resultsof the Nambu-Goto simulations presented in [31] and analyzed in [20], which allow us towrite the distribution for loops as n r ( l, t ) = 0 . t / ( l + Γ Gµt ) / (A4)for loops existing in the radiation era. Some of these loops will survive until the matter era,when they will contribute to the number of loops as n rm ( l, t ) = 0 . (cid:0) √ Ω r (cid:1) / ( l + Γ Gµt ) / (1 + z ) . (A5) See, for example, [66] for a discussion of the existence of transient solutions early in a simulation. n m ( l, t ) = 0 . − . l/t ) . t ( l + Γ Gµt ) , (A6)for l < . t , but these make no significant contribution to the gravitational wave spectrumtoday.We note that these expressions depend on the parameter Γ, which describes the aver-age total power of gravitational radiation emitted by the population of non-self-intersectingloops. The emission of energy into gravitational waves reduces the length of the loop ac-cording to l = l − Γ Gµ ( t − t ) , (A7)which is why the previous expressions depend on Γ.The final ingredient, P n , is discussed in the main text. [1] Zaven Arzoumanian et al. (NANOGrav), “The NANOGrav 12.5 yr Data Set: Search for anIsotropic Stochastic Gravitational-wave Background,” Astrophys. J. Lett. , L34 (2020),arXiv:2009.04496 [astro-ph.HE].[2] R. W. Hellings and G. S. Downs, “Upper limits on the isotropic gravitational radiation back-ground from pulsar timing analysis,” Astrophys. J. Lett. , L39–L42 (1983).[3] R. M. Shannon et al. , “Gravitational waves from binary supermassive black holes missing inpulsar observations,” Science , 1522–1525 (2015), arXiv:1509.07320 [astro-ph.CO].[4] T. W. B. Kibble, “Topology of Cosmic Domains and Strings,” J. Phys. A9 , 1387–1398 (1976).[5] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (CambridgeUniversity Press, 2000).[6] A. Vilenkin, “Gravitational radiation from cosmic strings,” Phys. Lett. , 47–50 (1981).[7] C. J. Hogan and M. J. Rees, “Gravitational interactions of cosmic strings,” Nature ,109–113 (1984).[8] Tanmay Vachaspati and Alexander Vilenkin, “Gravitational Radiation from Cosmic Strings,”Phys. Rev.
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