Gravitational waves from walls bounded by strings in $SO(10)$ model of pseudo-Goldstone dark matter

We explore the gravitational wave spectrum generated by string-wall structures in an $SO(10)$ ($Spin(10)$) based scenario of pseudo-Goldstone boson dark matter (pGDM) particle. This dark matter candidate is a linear combination of the Standard Model (SM) singlets present in the 126 and 16 dimensional Higgs fields. The Higgs $126$-plet vacuum expectation value (VEV) $\left<126_H\right>$ leaves unbroken the $\mathbb{Z}_2$ subgroup of $\mathbb{Z}_4$, the center of $SO(10)$. Among other things, this yields topologically stable cosmic strings with a string tension $\mu \sim \left<126_H\right>^2$. The subsequent (spontaneous) breaking of $\mathbb{Z}_2$ at a significantly lower scale by the $16$-plet VEV $\left<16_H\right>$ leads to the appearance of domain walls bounded by the strings produced earlier. We display the gravitational wave spectrum for $G \mu$ values varying between $10^{-15}$ and $10^{-9}$ ($\left<126_H\right>\sim 10^{11}$ - $10^{14}$ GeV), and $\left<16_H\right>\sim 0.1$ - $10^2$ TeV range ($G$ denotes Newton's constant.) These predictions can be tested, as we show, by a variety of (proposed) experiments including LISA, ET, CE and others.

In the SO(10) model the dark matter candidate is a pseudo-Goldstone particle formed from a suitable linear combination of the SM singlets contained in the Higgs 126-plet and 16-plet fields. We briefly summarize here the salient features of this model. 1. The dark matter can decay via gauge interactions. The VEV along the SM singlet direction in 126 H should be above 10 11 GeV [14,15] in order to satisfy the lifetime bound, τ DM ≳ 10 27 sec [21], for decaying dark matter. The right-handed neutrinos (ν C L ) acquire Majorana masses from this VEV and it is worth noting that the above requirement ⟨126 H ⟩ ≳ 10 11 GeV from DM considerations coincides with the requirement of the right-handed neutrino masses preferred by the fitting of neutrino data and successful leptogenesis in SO(10) GUT [22][23][24][25][26][27][28][29][30][31][32][33][34][35].
In Sec. 2, we provide a brief outline of the model and the formation of walls bounded by strings. Sec. 3 discusses the generation of the stochastic gravitational wave background from the string-wall system and their observational prospects. Our conclusions are summarized in Sec. 4.

Pseudo Goldstone dark matter model and walls bounded by strings
The scalar sector of the minimal pGDM model contains an SM-singlet complex scalar S in addition to the SM Higgs doublet H. The scalar potential is given by [5], The Lagrangian possesses a Z 2 symmetry, the subgroup of a global U (1) symmetry S → e iα S which is softly broken by the last term in Eq. (1). As the radial component of S gets a VEV, its angular component will be a pseudo Nambu-Goldstone boson (pNGB) due the spontaneous and explicit breaking of the U (1) symmetry, and becomes a viable dark matter candidate stabilized by the CP symmetry (S → S * ) [5]. The right amount of DM relic density can be obtained for the DM mass m χ ≈ [m h /2, 10 TeV], where m h is the SM Higgs mass. There are two resonances around m χ = m h /2 and m χ = m h 2 /2, where m h 2 is the mass of the second BSM Higgs which comes from a linear combination of CP-even states. The direct detection cross-section is vanishingly small in the limit of zero momentum transfer due to its pseudo-Goldstone nature, which can alleviate the direct detection bounds [2][3][4] on WIMP-like dark matter. However, the breaking of Z 2 generates stable domain walls which contradict the standard cosmology [46].
An ultra-violate (UV) completion of the minimal model with a gauge U (1) B−L was proposed in Refs. [10,11] with complex scalars S and Φ carrying one and two units of B − L charges respectively. In this model, the spontaneous breaking of U (1) B−L gauge symmetry by the Φ VEV leaves a remnant gauge Z 2 symmetry and generates the soft breaking term in Eq. (1) from the trilinear term A unified approach to implement these ideas is based on SO(10) with S ∈ 16 H and Φ ∈ 126 H , and the trilinear term arising from the coupling (126 H (16 H ) 2 + h.c.) in the scalar potential.
This realistic SO(10) model of pGDM [14,15] includes the electroweak Higgs doublet coming from a linear combination of bi-doublets in 126 H and a complex 10 H . The Yukawa couplings of the fermion 16 F with 126 H and 10 H to produce realistic fermion masses have been extensively studied in the literature [22][23][24][25][26][27][28][29][30][31][32][33][34][35]47]. A VEV along the SM singlet direction of 126 H or 16 H breaks a diagonal generator orthogonal to the hypercharge (Y ) and reduces the rank of the gauge symmetry from five to four. An example is the symmetry breaking of SU is broken. The dark matter candidate is the pseudo-Goldstone mode coming from a linear combination of the CP-odd components of the SM singlets. The gauge boson associated with the broken generator orthogonal to the hypercharge absorbs the massless would-be Goldstone mode.
The VEV of the SU (5) singlet component in the scalar multiplet 126 H (⟨126 H ⟩ ≳ 10 11 GeV) leaves an unbroken Z 2 and therefore generates topologically stable cosmic strings [48]. Subsequently, however, the VEV ⟨16 H ⟩ in the range [10 2 , 10 5 ] GeV for the right amount of DM relic, breaks this Z 2 symmetry, which leads to the formation of domain walls bounded by strings [49][50][51]. These walls bounded by strings are distinct, of course, from those [49] arising from the breaking of C-or D-parity even intermediate  or left-right symmetric gauge symmetry achieved by the VEV of (1, 1, 1) ∈ 54 H or (1, 1, 1, 0) ∈ 210 H respectively. In these latter cases, the GUT-scale strings are produced along with topologically stable monopoles [48,53,54]. The strings become the boundary of the domain walls generated during the subsequent breaking of this C-parity at an intermediate scale [49]. In GUTs, the C-parity breaking scale is equal to or higher than the right-handed neutrino mass scale (m R ≳ 10 11 GeV) [22]. Therefore, the proposed or ongoing experiments, sensitive upto kHz frequency, cannot observe the gravitational waves. The breaking of SO (10)

Gravitational wave background
In this section we discuss the gravitational waves from the domain walls bounded by strings and their observational prospects. The cosmic strings formed at ⟨126 H ⟩ have a tension (mass per unit length) given by [57,58], where λ str and g str are the relevant quartic and gauge coupling constants, and the function The tension on the domain walls (mass per unit area) associated with the breaking of Z 2symmetry by ⟨16 H ⟩ is given by, where ξ χ is the correlation length of the pNGB field χ, ∆V χ is the potential height along the direction, m χ is the mass of the field, and λ dw denotes the quartic coupling of the associated radial mode field. Note that in the last expression of Eq. (5), considering the parameter space for the right amount of dark matter as a thermal relic, we restricted ourselves to the case in which m χ is comparable to the mass of the radial mode.
For a wall bounded by a string of radius of curvature R, the force per unit length on the string boundary ∼ µ/R dominates over the wall tension σ for R < R c = µ/σ. The maximum radius of curvature R is of the order of the cosmic time t. Therefore, the string dynamics dominates before the formation of the domain walls (t dw ) until time R c > t dw . On the other hand, if t dw > R c , the domain wall dynamics starts dominating right after their formation. We define the time t * = max[R c , t dw ] to be the maximum timescale for domination by the string dynamics [36,37,59,60]. The cosmic strings inter-commute, form loops before t * with R < t * and can produce gravitational waves [36,[61][62][63]. The domain wall dynamics become dominant for t > t * , and the string-wall networks collapse as the walls pull the strings (see Ref. [37] for a detailed analysis).
The time for the domain wall formation t dw in the radiation-dominated universe is given by where g * accounts for the effective number of massless degrees of freedom, and we take T dw ≈ ⟨16 H ⟩ / √ λ dw as the background temperature at the time of domain wall formation. We have compared R c and t dw in Fig. 1 with Gµ = 10 −12 for the relevant range of the VEV ⟨16 H ⟩ ∈ [10 2 , 10 5 ] GeV. Since R c is larger than the domain wall formation time t dw , the string-wall network starts collapsing at t * = R c and loop formation ceases. After t * ≈ R c , there will be walls bounded by strings of curvature R ≲ R c formed on or before t * . The maximum radius of curvature of the wall bounded by string dominated by the wall dynamics could be R c . The lifetime of such an object is τ ws ∼ π(ΓGσ) −1 , with Γ ∼ O(10 2 ) being a numerical factor [37,64] which is not larger than the lifetime of decay τ s ∼ 2π(ΓGσ) −1 of a string loop with radius R c . Moreover, the string-wall system oscillates relativistically after R c and can be chopped into smaller pieces making its lifetime much smaller. Therefore, the contribution to the gravitational wave background will arise dominantly from the cosmic strings in the case of t dw ≪ R c [36,37].
The gravitational wave background is represented as a relic energy density as a function of frequency f in the present time, where ρ c is the critical energy density of the present universe. The stochastic gravitational wave background from domain walls bounded by strings is estimated in Ref. [37] as a sum of the contributions from normal modes given by, where and a loop of initial length l i = αt i decays with its length (l) at any subsequent time t given by In our case, t * = R c ≫ t dw , and therefore we have ξ = 2 and Γ ≃ 50 corresponding to the pure string limit [64,65]. We have taken F ≃ 0.1, α ≃ 0.1, C eff = 5.7 for the radiationdominated universe [66][67][68][69][70][71], and q = 4/3 because of cusp domination in the gravitational wave spectrum [72][73][74][75]. We have included the sum of normal modes upto n = 10 7 . Fig. 2 shows the gravitational wave background for a typical value of Gµ = 10 −12 and ⟨16 H ⟩ = 10 3 GeV. There is a scale-invariant component (f 0 ), an f 3 power-law IR spectrum, and f −1/3 UV tail which agrees with [37,74]. The loop formation ceases at t * and contribution to the gravitational wave background at time t > t * comes from the decaying loops formed before t * . These decaying loops during t > t * contribute to the lower frequency region of the spectrum with an f 3 IR tail. The UV tail arises since we assume that the network appears at time t F = 10 −22 sec as an example, enters the scaling regime soon after t F and the earliest loops of size αt F decay at time αt F (ΓGµ) −1 . We have shown the sensitivity curves [76,77] for various ongoing and proposed experiments, namely, PPTA [78], SKA [79,80], CE [43], ET [44], LISA [38,39], DECIGO [40], BBO [41,42] and HLVK [45]. For comparison, we also display the gravitational wave background from topologically stable cosmic strings for Gµ = 10 −12 in big dashed line. Fig. 3 shows the gravitational wave background for the domain wall formation scale ⟨16 H ⟩ varying from 10 2 to 10 5 GeV, with Gµ = [10 −15 , 10 −9 ], which corresponds to the breaking scales ⟨126 H ⟩ ∼ [10 11 , 10 14 ] GeV. The gravitational wave background will be detected [76,77] in several proposed experiments namely, CE [43], ET [44], LISA [38,39], DECIGO [40], BBO [41,42] and HLVK [45], for Gµ values from 10 −15 to 10 −9 . For comparison we show the gravitational wave spectrum from topologically stable cosmic strings for Gµ = 10 −12 in big dashed line. We can see that the lower frequency regions in the gravitational wave spectra from walls bounded by strings are absent since large loops with l > 2πR c cannot be formed. The NANOGrav [81,82] and other pulsar timing array experiments [83][84][85] have recently reported evidence of stochastic gravitational wave background in the nano-Hertz frequency range. A recent study [86] has shown that the gravitational waves from superheavy strings (Gµ ≈ 10 −6 ) bounding the domain walls associated with a symmetry breaking scale in the ballpark of 10 2 GeV can explain the data. It is worth mentioning that specific breaking chains of pGDM GUT models can give rise to similar topological structures and be compatible with the PTA data.

Conclusions
Pseudo-Goldstone dark matter (PGDM) models based on realistic grand unified gauge symmetries such as SO(10) predict the existence of composite topological structures known as 'walls bounded by strings'. In the SO(10) model that we explore here, there exist a 126-plet as well as 16-plet of Higgs fields with suitable VEVs in order to obtain this dark matter particle. The 126-plet VEV at an intermediate scale yields a topologically stable Z 2 string, which then forms the boundary of the domain wall created when the 16-plet acquires a VEV in the 10 2 -10 5 GeV range. We display the gravitational wave spectrum produced by this string-wall system and show that it will be accessible in the foreseeable future at a number of proposed experiments. We also point out that the gravitational wave emission from the topological structures discussed here with a string tension Gµ close to 10 −6 is compatible with the recently released NANOGrav 15 year data.