Superheavy quasi-stable strings and walls bounded by strings in the light of NANOGrav 15 year data

Composite topological structures such as superheavy"quasi-stable strings"(QSS) and"walls bounded by strings"(WBS) arise in realistic extensions of the Standard Model of high energy physics. We show that the gravitational radiation emitted in the early universe by these two unstable structures with a dimensionless string tension $G\mu\approx 10^{-6}$ is consistent with the NANOGrav evidence of low frequency gravitational background as well as the recent LIGO-VIRGO constraints, provided the superheavy strings and monopoles experience a certain amount of inflation. For the case of walls bounded by strings, the domain walls arise from the spontaneous breaking of a remnant discrete gauge symmetry around the electroweak scale. The quasi-stable strings, on the other hand, arise from a two step breaking of a local gauge symmetry. The monopoles appear from the first breaking and get connected to strings that arise from the second breaking. Both composite structures decay by emitting gravitational waves over a wide frequency range. The Bayes factors for QSS and WBS relative to the inspiraling supermassive black hole binaries are estimated to be about 60 and 30 respectively, which are comparable with that of metastable strings and cosmic superstrings.

In this paper, we would like to point out that well-known composite topological structures, known as "Quasi-Stable Strings (QSS)" [26] and "Walls Bounded by Strings (WBS)" [27], which emit [28] background gravitational radiation, can be consistent with the NANOGrav evidence.In both cases, the strings are superheavy with a dimensionless string tension Gµ ∼ 10 −6 .In the case of QSS, the structure consists of superheavy monopoles at the ends of strings, and it disappears after the gravitational wave emission by the strings.Note that the monopoles experience a period of inflation before getting attached to the strings and reentering the horizon.In contrast to metastable strings [29], the monopoleantimonopole pair creation on the string is not effective in the QSS scenario.In WBS, the domain walls arise from the spontaneous breaking of a discrete gauge symmetry at the electroweak scale.
Since the strings radiate gravitational waves over a wide frequency range, if they are superheavy they may not be compatible with the third run advanced LIGO-VIRGO [30] results.Indeed, the superheavy strings that form boundaries of the electroweak scale domain walls or connect the superheavy monopoles should experience a certain amount of inflation in order to avoid this conflict.A rough estimate shows that the strings should experience partial inflation and reenter the horizon at a cosmic time of order 10 −10 sec or later.
We do not intend here to provide details on how these topological structures may come about, but it is perhaps worth mentioning specific realistic particle physics models where they could be implemented.For an example of how the QSS structures arise, consider the following breaking of flipped SU (5) [31]: We have listed the SU (5) × U (1) X representations responsible for the breakings by their appropriate vacuum expectation values.The first breaking yields superheavy monopoles carrying color, electroweak and U (1) Z magnetic charges.The subsequent breaking of U (1) Z ×U (1) X to U (1) Y at a somewhat lower scale reveals that the monopoles are topologically unstable and get connected to strings, which we call QSS.These monopoles do not carry unconfined fluxes after the electroweak breaking.
For another example, consider the following breaking of SO (10) [31]: The first breaking yields the standard SU (5) monopoles arXiv:2306.17788v3[hep-ph] 27 Nov 2023 (which presumably will be adequately diluted by inflation) as well as superheavy monopoles carrying U (1) χ and U (1) Y magnetic charges.The subsequent breaking of U (1) χ at a somewhat lower scale reveals that the latter monopoles are topologically unstable and get connected to strings (QSS).Note that after the electroweak symmetry breaking these monopoles do not carry any Coulomb magnetic flux.
To see how the WBS system appears, consider the following breaking of the flipped SU (5) model [31]: The first breaking produces topologically stable superheavy strings but no monopoles.A subsequent breaking of the gauge Z 2 symmetry yields the desired composite structures, namely "walls bounded by strings."

II. GRAVITATIONAL WAVE BACKGROUND AND NANOGRAV 15 YEAR DATA
The strings start forming loops through intercommuting after the horizon reentry of the string network at a time t F .This network behaves like a network of stable strings before the horizon reentry of the monopoles at time t M for the QSS, and before the time R c = µ/σ (σ is the wall tension) for the string-wall network.After t fin = t M , there will be contribution from the loops formed earlier and the monopole-antimonopole pairs connected by string segments (M S M ) present within the particle horizon.On the other hand, WBS structures larger than R c will collapse and, therefore, only loops of size less than R c formed before the time t fin = R c contribute to the gravitational waves after R c .In both cases, the network disappears after a time scale ∼ t fin /ΓGµ (Γ ∼ 10 2 ) [32][33][34] and the bound from the cosmic microwave background anisotropy [35,36] is alleviated.
We follow the prescription in Ref. [26] to compute the gravitational wave spectra from the QSS, and take into account that the monopoles do not carry Coulomb magnetic flux.The gravitational wave background from the WBS network is computed following Refs.[37,38].The gravitational wave background from the string loops is the sum of the contributions from all the normal modes: The contribution from each mode in the case of WBS is given by [37,38] and a loop formed at time t i has length l at a subsequent time t given by Here, ρ c is the critical energy density at the present time t 0 , a(t) the scale factor of the universe, F ≃ 0.1, Γ ≃ 50, α ≃ 0.1 in the pure string limit [33,34], and the loop formation efficiency C eff = 5.7 in the radiation dominated universe [39][40][41][42][43][44][45].For QSS, the contribution from each normal mode of the loops formed before t fin = t M can also be given by Eq. ( 5) with the terms involving 1 2πRc removed.The contribution to the gravitational wave background from the M S M structures of QSS is computed using the burst method as described in Ref. [26]: where H 0 is the present Hubble parameter and the lower limit z * of the integral over the redshift z which separates the infrequent bursts from the stochastic background is computed from The length of an M S M structure evolves as with Γ ∼ 8 ln γ(z M ) and γ ∼ µ m M l (m M is the monopole mass).The wave form for the gravitational wave burst is given by [28,46,47] where g = 2 √ 2 sin 2 √ 2/π and r is the proper distance.The number density of the M S M structures is The burst rate can be written as where ∆(f, z) is the fraction of the gravitational wave bursts that are observed (see Ref. [26] for more details).Figs. 1 and 2 respectively show the gravitational wave background from the QSS and the WBS structures in the case of superheavy strings with Gµ = 10 −6 and t M = 3 × 10 2 sec, and Gµ = 10 −6 with v dw = 10 2 GeV, the vacuum expectation value associated with the walls.f (Hz) FIG. 1. Gravitational wave background from quasi-stable strings with Gµ = 10 −6 and the monopole horizon reentry time tM = 3 × 10 2 sec.The evidence for gravitational waves in NANOGrav is compatible with this scenario.The red violin plots show the posterior of HD correlated free spectra of PTA data.The strings experience some e-foldings and reenter the horizon at tF ∼ 10 −10 sec to satisfy the advanced LIGO-VIRGO third run (LV-3) bound [30].The gray region depicts the bound from Big Bang Nucleosynthesis (BBN) [48].We also show the power-law integrated sensitivity curves [49,50] for planned experiments, namely, HLVK [51], CE [52], ET [53], DECIGO [54], BBO [55,56], LISA [57,58], and SKA [59,60].
The string network experiences some e-foldings during inflation and the loop formation starts around t F ∼ 10 −10 sec.This alleviates the bound from the advanced LIGO-VIRGO third run (LV3) [30].It is worth mentioning that non-standard cosmology such as matter domination (MD) in the pre-BBN era or both inflation and MD can also alleviate the LV3 bound around the dec-aHertz region [61][62][63].As an example, consider models with partial inflation of the strings starting right after their formation and followed by a matter dominated period of field oscillations which ends by reheating.In this case, using Ref. [64] we find that the horizon reentry time is given by where ξ str is the correlation length at string formation, N str the e-foldings experienced by the strings, t r the reheat time, and t e the time at which inflation terminates.The gravitational wave background predicted in other frequencies can be tested in various proposed experiments including HLVK [51], CE [52], ET [53], DECIGO [54], BBO [55,56], LISA [57,58], and SKA [59,60].The strings experience some e-foldings of inflation, and reenter the horizon at tF ∼ 10 −10 sec to satisfy the bound from the third run advanced LIGO-VIRGO (LV-3) data [30].We perform a Bayesian analysis of these two models using the wrapper PTArcade [65] and the NANOGrav 15 year data.We employ the Ceffyl package [66] and take into account the quadrupolar Hellings-Downs (HD) cor-  relation [67] between the pulsars to obtain the posterior distributions of the model parameters for the stochastic gravitational wave background.To estimate the Bayes factor of the models with respect to the supermassive black hole binaries (SMBHB), we use the Enterprise code [68,69] without the HD correction.The Bayes factors are estimated to be around 60 and 30 for QSS and WBS, respectively, which are comparable with the Bayes factors for the metastable strings (∼ 20) and the superstrings (∼ 50) [2].The posterior distributions of Gµ and t M for QSS are shown in the triangular plot in Fig. 3.The plots in Fig. 4 depict the posteriors of the model parameters Gµ and v dw of WBS.Table I presents the 68% and 95% confidence level intervals of the model parameters for QSS and WBS.We find that the stochastic gravitational wave background for QSS and WBS is compatible with the NANOGrav 15 year data for Gµ ≈ 10 −6 , and t M ≈ 10 2 sec and v dw ≈ 10 2 GeV respectively.
For completeness, note that the Gµ values of interest in this paper correspond to symmetry breaking scales ∼ 10 15 -10 16 GeV.In the example of WBS given in Eq. ( 3), the proton decay ( p → e + π 0 ) is mediated by superheavy leptoquark gauge bosons associated with the symmetry breaking responsible for string formation.The lifetime is estimated to be larger than the current lower bound provided by Super-Kamiokande [70] for Gµ ≳ 10 −6 .Proton decay can be observed within the next few years at Hyper-Kamiokande [71] for Gµ ≲ 3 × 10 −6 .

III. SUMMARY
Inspired largely by the apparent evidence of a low frequency gravitational wave background by the NANOGrav and other collaborations, our main aim here is to point how realistic extensions of the Standard Model of high energy physics can now be tested by this and hopefully near future discoveries.We have focused on two distinct composite topological structures, namely, quasistable cosmic strings and walls bounded by strings.The local strings in both cases are superheavy with Gµ ≈ 10 −6 and the two composite structures are unstable and decay through the emission of gravitational waves.A certain amount of inflation of the strings is necessary in order that the gravitational emission is also compatible with the LIGO-VIRGO constraint on Gµ in the 10 − 100 Hz frequency range.The gravitational wave background from the quasi-stable strings with log 10 (Gµ) = −5.84 0.97 −0.75 and log 10 (t M /sec) = 2.28 1.08 −1.5 , and walls bounded by strings with log 10 (Gµ) = −5.97 0.66 −0.68 and log 10 (v dw /sec) = 1.96 0.56 −0.55 can explain the recent evidence in the NANOGrav 15 year data.The Bayes factors for QSS and WBS provide strong evidence for these scenarios in comparison with the simplest model for SMB-HBs, and they are comparable with the other competing models of cosmic strings such as metastable strings and cosmic superstrings.

FIG. 2 .
FIG.2.Gravitational wave background from domain walls bounded by cosmic strings with Gµ = 10 −6 and different choices for the vacuum expectation value v dw associated with the domain walls.The red violin plots show the posterior of HD correlated free spectra of PTA data.The NANOGrav evidence is compatible with this scenario for v dw ∼ 10 2 GeV.The strings experience some e-foldings of inflation, and reenter the horizon at tF ∼ 10 −10 sec to satisfy the bound from the third run advanced LIGO-VIRGO (LV-3) data[30].

FIG. 3 .
FIG. 3. Corner plot of the posterior distribution of Gµ and tM for QSS with the 1σ (dark) and 2σ (light) credible regions.The diagonal plots are the marginalized 1D distributions with the vertical lines indicating the 68% and 95% confidence level intervals.
FIG.4.Corner plot of the posterior distribution of Gµ and v dw for WBS with the 1σ (dark) and 2σ (light) credible regions.The diagonal plots are the marginalized 1D distributions with the vertical lines indicating the 68% and 95% confidence level intervals.