Constraining the Cosmic Strings Gravitational Wave Spectra in No Scale Inflation with Viable Gravitino Dark matter and Non Thermal Leptogenesis

We revisit the hybrid inflation model gauged by $U(1)_{B-L}$ extension of the Minimal Supersymmetric Standard Model (MSSM) in a no-scale background. Considering a single predictive framework, we study inflation, leptogenesis, gavitino cosmology, and the stochastic background of gravitational waves produced by metastable cosmic strings. The spontaneous breaking of $U(1)_{B-L}$ at the end of inflation produces a network of metastable cosmic strings while, the interaction between $U(1)_{B-L}$ Higgs field and the neutrinos generate heavy Majorana masses for the right-handed neutrinos. The heavy Majorana masses explain the tiny neutrino masses via the seesaw mechanism, a realistic scenario for reheating and non-thermal leptogenesis. We show that a successful non-thermal leptogenesis and a stable gravitino as a dark matter candidate can be achieved for a wide range of reheating temperatures and $U(1)_{B-L}$ symmetry breaking scales. The possibility of realizing metastable cosmic strings in a grand unified theory (GUT) setup is briefly discussed. We find that a successful reheating with non-thermal leptogenesis and gravitino dark matter restrict the allowed values of string tension to a narrow range $10^{-9} \lesssim G\mu_{CS} \lesssim 8 \times 10^{-6}$, predicting a stochastic gravitational-wave background that lies within the 1-$\sigma$ bounds of the recent NANOGrav 12.5-yr data, as well as within the sensitivity bounds of future GW experiments.


Introduction
Spontaneous symmetry breaking in grand unified theories (GUTs) can produce variety of topological or non-topological defects [1,2,3,4,5]. These defects generically arise from the breaking of a group, G, to its subgroup, H, such that a manifold of equivalent vacua, M, G/H, exists. Monopoles form when the manifold M contains non-contractible two-dimensional spheres [6], cosmic strings when it contains non-contractible loops and domain walls when M is disconnected [7]. The monopoles can be avoided by inflation which naturally incorporates the GUT scale in supersymmetric hybrid inflation [8]. It has been shown for a large class of GUT models that in spontaneous symmetry breaking schemes curing the monopole problem, the formation of cosmic strings cannot be avoided [9].
Cosmic strings are interesting messenger from the early universe due to their characteristic signatures in the stochastic gravitational wave background (SGWB). The evidence for a stochastic process at nanohertz frequencies as reported by recent NANOGrav 12. 5 year data has been interpreted as SGWB in a large number of recent papers [10,11,12,13,14,15,16,17,18,19,20]. Relic gravitational waves (GWs) provide a fascinating window to explore the very early universe cosmology [21]. Cosmic string produce powerful bursts of gravitational radiation that could be detected by interferometric gravitational wave detectors such as LIGO, Virgo, and LISA [22,23]. In addition, the stochastic gravitational wave background (SGWB) can be detected or constrained by various observations including Big Bang Nucleosynthesis (BBN), pulsar timing experiments and interferometric gravitational wave detectors [24].
Among the various proposed extensions of the minimal supersymmetric standard model (MSSM), the U (1) B−L is the simplest [25,26,27]. Here B and L denote the baryon and lepton numbers, respectively and B − L is the difference between baryon and lepton numbers. As a local symmetry, the B − L group resides in the grand unified (GUT) gauge group SO (10). The spontaneous breaking of U (1) B−L at the end of inflation requires an extended scalar sector, which automatically yields hybrid inflation explaining the inhomo-geneities of the CMB. In the B − L breaking phase transition, most of the vacuum energy density is rapidly transferred to non-relativistic B − L Higgs bosons, a sizable fraction also into cosmic strings. The decay of heavy Higgs boson and heavy neutrino leads to an elegant explanation of the small neutrino masses via the seesaw mechanism, explaining the baryon asymmetry via thermal and nonthermal leptogenesis [28,29,30,31]. The temperature evolution during reheating is controlled by the interplay between the B − L Higgs and the neutrino sector, while the dark matter originates from thermally produced gravitinos. The embedding of U (1) B−L into a simply-connected group such as SO (10) or Pati-salam symmetry (SU (4) C × SU (2) L × SU (2) R ), produces metastable cosmic strings due to the spontaneous pair creation of a monopole and an anti-monopole. Once the string is cut, the monopoles at the two ends are quickly pulled together due to string tension, forcing them to annihilate. If the string network is sufficiently long-lived, it can generate a stochastic gravitational wave background (SGWB) in the range of ongoing and future gravitational wave (GW) experiments [32,33].
Hybrid inflation, in particular, is one of the most promising models of inflation, and can be naturally realized within the context of supergravity theories. This scenario is based on the inclusion of two scalar fields [34], with the first one realizing the slow-roll inflation and the second one, dubbed the "waterfall" field, triggering the end of inflationary epoch. While in the standard hybrid inflationary scenario [8,35,36], the GUT gauge symmetry is broken at the end of inflation, in shifted [37] and smooth variants [38,39], the gauge symmetry breaking occurs during inflation and thus, the disastrous magnetic monopoles and other dangerous topological defects are inflated away.
In this paper we study standard hybrid inflation in the context of supergravity where a no-scale Kähler potential is assumed . We consider the framework of MSSM gauge symmetry augmented by a U (1) B−L factor and investigate the implementation of hybrid inflation and its interplay with the issues of non-thermal leptogenesis, gravitino dark matter and stochastic gravitational wave background (SGWB) generated by metastable cosmic string network. For µ hybrid inflation see Ref [40]. We consider the value of monopole-string-tension ratio from √ λ 7.4 (metastable cosmic strings) to √ λ 9.0 (quasi-stable cosmic strings). The parametric space consistent with successful reheating with non-thermal leptogenesis and gravitino dark matter restrict the allowed values of string tension to the range 10 −9 Gµ CS 8×10 −6 and predicts a stochastic gravitational wave background (SGWB) that lies within the 1-and 2-σ bounds of recent NANOGrav 12.5 years data, as well as the sensitivity bounds of future gravitational wave (GW) experiments.
The layout of the paper is as follows. In Sec. 2 we describe the basic features of the model including the superfields, their charge assignments, and the superpotential con-strained by a U (1) R symmetry. The inflationary setup is described in Sec. 3. The numerical analysis is presented in Sec. 4 including the prospects of observing primordial gravity waves, non-thermal leptogenesis, gravitino cosmology and stochastic gravitational wave background (SGWB) generated by metastable cosmic string network. Our conclusion is summarized in Sec 5.

Model Description
In this section, we present basic features regarding the gauge symmetry and the spectrum of the effective model in which the inflationary scenario will be implemented. The gauge group U (1) B−L is embedded in a grand unified (GUT) gauge group SO (10) and is based on the gauge symmetry In addition to the MSSM matter and Higgs superfields, the model supplements six superfields namely; a gauge singlet S whose scalar component acts as inflaton, three righthanded neutrinos N c i and a pair of Higgs singlets H and H, which are responsible for breaking the gauge group U (1) B−L . The charge assignment of these superfields under the gauge symmetry SU (3) C × SU (2) L × U (1) Y × U (1) B−L as well as the global symmetries U (1) R , U (1) B and U (1) L are listed in Table 1.
The U (1) B−L symmetry is spontaneously broken when the H, H singlet Higgs super fields acquire vacuum expectation values (VEVs), providing Majorana masses to the right handed neutrinos. The superpotential of the model, invariant under the symmetries listed in Table 1, is given as (2. 2) The first line in the above superpotential contains the usual MSSM µ-term and Yukawa couplings supplemented by an additional Yukawa coupling among L i and N c i . These Yukawa couplings generate Dirac masses for up and down quarks, charged leptons and neutrinos. The family indices for Yukawa couplings are generally suppressed for simplicity. The first term in the second line is relevent for standard supersymmetric hybrid inflation with M being a GUT scale mass parameter and κ, a dimensionless coupling constant. The non-renormalizable term in the second line generates Majorana masses for right handed neutrinos N c i and induces the decay of inflaton to N c i . By virtue of the extra global symmetries, the model is protected from dangerous proton decay operators and R-parity violating terms.

Inflation Potential
We will compute the effective scalar potential contributions from the F -and D-sector, radiative corrections as well as the soft supersymmetry breaking terms. The superpotential terms relevant for inflation are We consider a no-scale structure Kähler potential which, after including contributions from the relevant fields in the model, takes the following form where T , T * are Kähler complex moduli fields, T = (u + i v), hence T + T * = 2u and ξ is a dimensionless parameter. Here we choose u = 1/2. For later convenience, we define so that Eq 3.2 can be written as The fields carrying SU quantum numbers are given in Table 1 and denoted collectively here with φ i . The D-term potential is given as, where D p a is defined for SU (N ) groups as (3.6) The D-flat potential can be achieve by parametrization of the fields H and H. We can rewrite the complex fields in terms of real scalar fields as where the phases θ, θ and ϑ can be stabilized at ϑ = π 4 and θ = θ = 0, (3.8) along the D-flat direction (|H| = |H| = Y 2 ). The F -term SUGRA scalar potential is given by with z i being the bosonic components of the superfields z i ∈ {S, H,H, · · · }, and we have defined Using the F -flatness condition, D z i W = 0, the minima of potential lies at Y = 2M and S = 0. Along the inflationary trajectory, Y = 0, the gauge group U (1) B−L is unbroken. After the end of inflation, the spontaneous breaking of the gauge group U (1) B−L yields cosmic strings. Defining dimensionless variable x ≡ |S|/M , we obtain the following form of potential along the inflationary trajectory, The action of our model for non-canonically normalized field x is given by Introducing a canonically normalized field χ that satisfies Since γM 2 m 2 p , integrating Eq. (3.14) in this limit, we obtain the canonically normalized field χ as a function of x. The canonically normalized potential as a function of χ can be written as, The effective scalar potential including the well-known radiative corrections and soft SUSY breaking terms, can be expressed as (3.18) and Here, Q is the renormalization scale, a and M S are the coefficients of soft SUSY breaking linear and mass terms for S, respectively, and m 3/2 is the gravitino mass. For simplicity, we set M S = m 3/2 and assume a suitable initial condition for arg S to be stabilized at zero and take a to be constant during inflation (for details see Ref [41]).

Analysis
In this section, we analyze the implications of the model and discuss its predictions regarding the various cosmological observables. We pay particular attention to inflationary predictions and stochastic gravitational waves (GW) spectrum consistent with leptogenesis and gravitino cosmology.

Inflationary Predictions
The inflationary slow-roll parameters can be expressed in terms of χ as where a prime denotes a derivative with respect to χ. The tensor-to-scalar ratio r, the scalar spectral index n s , and the running of the spectral index dns d ln k are given by The number of e-folds is given by [42], where l denotes the comoving scale after crossing the horizon, χ l is the field value at l, χ e is the field value at the end of inflation, (i.e., when = 1), and T r is the reheating temperature which will be discussed in the following section. The amplitude of curvature perturbation ∆ R is given by The results of our numerical calculations are presented in Fig. 1 where the variation of parameters is shown in the κ − M plane. In our analysis, the scalar spectral index is fixed at the central value of Planck's bounds n s = 0.9655. To keep the SUGRA expansion, parameterized by γ, under control we impose S 0 ≤ m p . We restrict M ≤ 2 × 10 16 GeV and T r ≤ 10 10 GeV to avoid the gravitino problem. We further restrict our numerical results by imposing the following conditions  Fig. 1 shows the variation of tensor to scalar ratio r, whereas, the right panel shows the variation of the field value S 0 . The color bar depicts the range of string tension Gµ CS obtained in our model. It should be noted that the parameter γ which controls the SUGRA corrections, makes this model more predictive than the standard hybrid model of inflation. Using leading order slow-roll approximation, we obtain the following analytical expressions for n s in the small κ limit, It can readily be check that for γ = 0.162292, we obtain n s ∼ 0.9655 which is in excellent agreement with the numerical results displayed in Fig. 1. The above equation therefore gives a valid approximation of our numerical results. For the scalar spectral index n s fixed at Planck's central value (0.9655), we obtain the following ranges of parameters Using the Plank's normalization constraint on ∆ R , we obtain the following explicit dependence of r on κ and M r 2κ

Reheating with non-thermal leptogenesis
At the end of inflation epoch, the vacuum energy is transfered to the energies of coherent oscillations of the inflaton S and the scalar field θ = (δH +δH)/ √ 2 whose decays give rise to the radiation in the universe. The inflaton decay to right handed neutrino is induced by the superpotential term where β ij is a coupling constant and Λ represents a high cut-off scale (in a string model this could be identified with the compactification scale). Heavy Majorana masses for the right-handed neutrinos are provided by the following term Also, Dirac neutrino masses of the order of the electroweak scale are obtained from the tree-level superpotential term The small neutrino masses supported by neutrino oscillation experiments, are obtained by integrating out the heavy right-handed neutrinos and read as The neutrino mass matrix m ν D αβ can be diagonalized by a unitary matrix U αi as m ν D αβ = U αi U βi m ν D , where m ν D is a diagonal mass matrix m ν D = diag(m ν 1 , m ν 2 , m ν 3 ) and M i represent the eigenvalue of mass matrix M ν c ij . The lepton asymmetry is generated (inducing also baryon asymmetry [28,29]) through right-handed neutrino decays. The lepton number density to the entropy density in the limit where cp is the CP asymmetry factor and is generated from the out of equilibrium decay of lightest right-handed neutrino and is given by [43], 3 Im y ν y ν † 1i and T r is reheating temperature which can be as estimated as where g is 228.75 for MSSM. The Γ is the decay width for the inflaton decay into righthanded neutrinos and is given by [43] with the inflaton mass given by Assuming a normal hierarchical pattern of light neutrino masses, the CP asymmetry factor, cp , becomes where m ν 3 is the mass of the heaviest light neutrino, v u = H u is the VEV of the up-type electroweak Higgs and δ eff is the CP-violating phase. The experimental value of lepton asymmetry is estimated as [44], In the numerical estimates discussed below we take m ν 3 = 0.05 eV, |δ eff | = 1 and v u = 174 GeV, while assuming large tan β. The non-thermal production of lepton asymmetry, n L /s, is given by the following expression

(4.22)
A successful baryogenesis is usually generated through the sphaleron processe where an initial lepton asymmetry, n L /s, is partially converted into a baryon asymmetry [45,46]. However, the right handed neutrinos produced in inflaton decays are highly boosted, which affects the estimate of the final baryon asymmetry as given in [32]. Eq. (4.22) is used in our numerical analysis to calculate inflationary predictions which are consistent with leptogenesis and baryogenesis. The color bar on the right displays the range of string tension parameter Gµ CS , while the shaded region corresponds to the stable gravitino, as discussed in the following section.

Gravitino Dark Matter
An important constraint on the reheat temperature T r arises, when gravitino cosmology is taken into account, that depends on the SUSY breaking mechanism and the gravitino mass m 3/2 . As noted in [25,47], one may consider the case of α) a stable LSP gravitino; β) unstable long-lived gravitino with mass m 3/2 < 25 TeV; γ) unstable short-lived gravitino with mass m 3/2 > 25 TeV.
We first consider the case of stable gravitino, in which case it is the lightest SUSY particle (LSP) and assuming it is thermally produced, its relic density is estimated to be [48] Ω 3/2 h 2 = 0.08 T r 10 10 GeV where mg is the gluino mass parameter and h is the present Hubble parameter in units of 100 km sec −1 Mpc −1 and Ω 3/2 = ρ 3/2 /ρ c . 5 6 . A stable LSP gravitino requires mg > m 3/2 while current LHC bounds on the gluino mass are around 2.2 TeV [52]. It is found from Eq. 4.24 that the overclosure limit Ω 3/2 < 1 puts a severe upper bound on the reheating temperature T r , depending on the gravitino mass m 3/2 . Here, we have omitted the contribution from the decays of squarks and sleptons into gravitinos. Using the lower bound of relic abundance Ω 2 h = 0.144 [53] in the above equation, we display a gray shaded region in Fig. 3 that satisfies the condition (mg > m 3/2 ) and hence, in this region, gravitino is the lightest supersymmetric particle (LSP) and acts as a viable dark matter 5 ρ 3/2 and ρ c are the present energy density of the gravitino and the critical energy density of the present universe, respectively. 6 Eq. (4.24) contains only the dominant QCD contributions for the gravitino production rate. In principle there are extra contributions descending from the electroweak sector as mentioned in [49], [50] and recently revised in [51]. If we consider these type of contributions in our analysis, we estimate that (depending on gaugino universality condition) our results will deviate ∼ (10 − 15)%. candidate. For gravity mediated SUSY breaking, the constraints on gravitino mass and reheat temperature from BBN are [54] T r 10 7 GeV for m 3/2 = (100 − 5000) GeV , T r ∼ (10 7 − 2.5 × 10 9 ) GeV for m 3/2 ≥ 5000 GeV. (4.25) The shaded region in above Fig. 3 describes the gravitino dark matter and the value of gravitino mass varies in the range 1.5 GeV m 3/2 4.2 × 10 5 GeV with reheat temperature T r 10 7 GeV. However, the gravitino mass in the range 100 GeV m 3/2 5000 GeV requires a reheat temperature T r < 10 7 GeV, which is not achieved in our model and therefore, this small range of m 3/2 is ruled out by BBN.
An unstable gravitino, could be either long-lived or a short-lived. The lifetime of a long-lived gravitino with mass m 3/2 < 25 TeV is aboutτ 1 sec. A long-lived gravitino leads to the cosmological gravitino problem [55] that originates due to the fast decay of gravitino which may affect the light nuclei abundances and thereby ruin the success of BBN theory. To avoid this problem, one has to take into account the BBN bounds (Eq. 4.25) on the reheating temperature. Nevertheless for all range of 5000 ≤ m 3/2 ≤ 25000 GeV, a long-lived gravitino scenario is viable and consistent with the BBN bounds (4.25).
For short-lived gravitino, the BBN bounds on the reheating temperature are not ef-fective and gravitino decays into the LSP neutralinoχ 0 1 , for which the abundance is given by where Y 3/2 is the gravitino Yield and is defined as, Since the LSP neutralino density produced by gravitino decay should not exceed the observed DM relic density, choosing the upper bound of relic abundance Ωχ0 1 h 2 = 0.126 and using equations (4.27) and (4.26), we find a relation between the reheating temperature T r and mχ0 1 , given by For gravity mediation scenario mχ0 1 ≥ 18 GeV [56], which is easily satisfied in the current model. Therefore, the short-lived gravitino scenario is also a viable possibility in this model. The region above the shaded area in Fig. 3 corresponds to short-lived gravitino. Finally, we obtain the following ranges of string tension Gµ CS and gravitino mass for stable and unstable gravitino consistent with BBN bounds, 10 −9 Gµ CS 8 × 10 −6 , (−3.2 × 10 9 am 3/2 5 × 10 8 ) GeV. (4.29) In next sub section, we analyze stochastic gravitational waves (GW) spectrum, consistent with leptogenesis and gravitino cosmology.

Gravitational Waves From Cosmic Strings
The superposition of GW sources, such as inflation, cosmic strings and phase transition, would generate a stochastic GW background (SGWB). The tensor perturbations upon horizon re-entry give rise to the inflationary SGWB [15,16,57,58] which imprint a distinctive signature in the CMB B-mode polarization. The amplitude and scale dependence of the inflationary SGWB is parameterized via the tensor-to-scalar ratio r and the tensor spectral index n T , which satisfy the inflationary consistency relation r = −8n T [63], within single-field and hybrid slow-roll models. Since r ≥ 0, this requires n T ≤ 0 (red spectrum) [64]. With current constraints on tensor-to-scalar ratio r, the amplitude of the inflationary SGWB on PTA and interferometer scales is far too small to be detectable by these probes and would instead require a strong blue tilted (n T > 0) primordial tensor power spectrum [15]. For a detailed study on SGWBs from first-order phase transition associated with the spontaneous U (1) B−L gauge symmetry breaking, see Refs [59,60,61,62].
In this section, we study SGWB spectra produce by the decay of cosmic string network [10,11,12,13]. The breaking of U (1) B−L gauge symmetry generates stable cosmic string network that can put severe bounds on model parameters. These bounds can be relaxed if the cosmic strings are metastable. The embedding of U (1) B−L group in SO(10) GUT gauge group leads to production of metastable cosmic string network which can decay via the Schwinger production of monopole-antimonopole pairs, generating a stochastic gravitational wave background (SGWB), in the range of ongoing and future gravitational wave (GW) experiments.
The MSSM matter superfields reside in the 16 (spinorial) representation, whereas the MSSM Higgs doublet reside in 10 representations of SO (10). The SO(10) symmetry breaking to MSSM gauge group is achieved by non-zero VEV of 45 multiplet; with x being a real constant. As a special case of x = 0, the model, after spontaneous breaking of SO (10), can be effectively realized as B − L extended MSSM, U (1) χ=B−L . The Higgs superfield pair (H,H) belong to (16 +16) representation of SO (10) and is responsible for breaking G B−L to MSSM. The first step gauge symmetry breaking produces magnetic monopoles which are inflated away during inflation, whereas the second step breaking produces metastable cosmic string network.
If cosmic strings form after inflation, they exhibit a scaling behavior where the stochastic GW spectrum is relatively flat as a function of the frequency, and the amplitude is proportional to the string tension µ CS . For our case, µ CS can be written in term of M as [65], where Υ = κ 2 2g 2 with g = 0.7 for MSSM. The CMB bound on cosmic string tension, reported by Planck 2018 [66,67] is where Gµ CS denotes the dimensionless string tension with the gravitational constant G = 6.7×10 −39 GeV −2 . The observation of GWs from cosmic strings is crucially dependent on two scales; the energy scale of inflation Λ inf , and the scale at which cosmic string generate the GW spectrum Λ CS ≡ √ µ CS . The amplitude of the tensor mode cosmic microwave background (CMB) anisotropy fixes the energy scale of inflation as Λ inf ∼ V 1/4 ∼ 3.3 × 10 16 r 1/4 [68]. Using Planck 2-σ bounds on tensor-to-scalar ratio r we obtain the upper limit on scale of inflation, Λ inf < 1.6×10 16 GeV [69]. In our model, strings form after inflation, namely Λ inf > Λ CS , for which a stochastic gravitational wave background (SGWB) is generated from undiluted strings. The SGWB arising from meta-stable cosmic strings network are expressed relative to critical density as [70] Ω where ρ gw denotes the GW energy density, ρ c is the critical energy density of the universe, and H 0 = 100 h kms −1 Mpc −1 is the Hubble parameter. The parameter P n 50 ζ(4/3)n 4/3 is the power spectrum of GWs emitted by the n th harmonic of a cosmic string loop and C n (f ) indicates the number of loops emitting GWs that are observed at a given frequency f which is a function of number density of cosmic string loops N ( , t), with = 2n/((1+z)f ). For the number density of cosmic string loops, N ( , t), we use the approximate expressions of Blanco-Pillado-Olum-Shlaer (BOS) model given in [70,71] N r ( , t) = 0.18 (4.37) For our region of interest, the dominant contribution is obtained from the loops generated during the radiation-dominated era. For t(z) and H(z), we use the expressions for ΛCDM cosmology assuming a standard thermal history of universe, while ignoring the changes in the number of effective degrees of freedom with z The integration range in the above equation corresponds to the lifetime of the cosmic string network, from its formation at z max Tr 2.7K until its decay at z min given by [72,73,74], where Γ 50, m M is the monopole mass, µ is the string tension, and we fix the reheat temperature at T r = 10 8 GeV. The dimensionless parameter λ is the hierarchy between the GUT and U (1) B−L breaking scales. Fig. 4 shows gravitational wave spectra from metastable cosmic strings for the predicted range of cosmic string tension, 10 −12 Gµ CS 10 −6 . The curves in left panel are drawn for the GUT and the B − L breaking scales ratio, √ λ = 8. The parametric space consistent with successful reheating with non-thermal leptogenesis and gravitino dark matter restricts the value of Gµ CS in the range 10 −9 Gµ CS 10 −6 . This is shown in the right panel of Fig. 4 where the curves are drawn for √ λ = 7, 8, 9. It can be seen that the GW spectrum for the entire range of Gµ CS passes through most GW detector sensitivities. LIGO O1 [22] has excluded cosmic strings formation at Gµ CS 10 −6 in the high frequency regime 10-100 Hz. The low frequency band, 1-10 nHz, can be probed by NANOGrav [75], EPTA [76] and other GW experiments at nano Hz frequencies. Planned pulsar timing arrays SKA [77], spacebased laser interferometers LISA [23], Taiji [78], TianQin [79], BBO [80], DECIGO [81], ground-based interferometers, such as Einstein Telescope [82] (ET), Cosmic Explorer [83] (CE), and atomic interferometer AEDGE [84], will probe GW generated by metastable cosmic string in a wide regime of frequencies.

Explaining the NANOGrav results
We now discuss the SGWB signal predicted by metastable cosmic strings for recent NANOGrav 12.5 yr results [75], which constrain the amplitude and slope of a stochastic process. The amplitude of the SGWB is obtained in terms of dimensionless characteristic strain h c = A(f /f yr ) α at the reference frequency f yr = 32 nHz as [12] Ω GW (f ) = 2π 2 f 2 yr A 2 The dark (light) yellow region represent the 1-σ (2-σ) bounds reported by NANOGrav 12.5 yr data [75].
where A is the strain amplitude. At low GW frequency, Ω GW behaves as ∼ f 3/2 , whereas at high GW frequencies, Ω GW ∼ 1. NANOGrav uses a power law fit with 5−γ = 2+2α = n gw and constrain the parameters A and γ. This allows us to directly translate the 1-and 2-σ NANOGrav bounds given in [75] into the Ω yr gw -n gw plane, as displayed by the yellow shaded regions in Fig. 5. Following [11], we extract the amplitude Ω yr gw and slope n gw using Eq. (4.41) by comparing the amplitude at the pivot scale f * and taking the logarithmic derivative of Ω gw (f ) at the desired frequency scale f * 7 , (4.43) Fig. 5 shows the comparison of the predictions from metastable cosmic strings (mesh of solid and dotted curves) with the constraints on the amplitude and tilt from [75] (yellow shaded region). We vary Gµ CS from 10 −11 to 10 −6 , however the CMB constraint Gµ CS ≤ 1.3 × 10 −7 only applies to cosmic strings with a life-time exceeding CMB decoupling, corresponding to √ λ 8.6. For each value of Gµ CS , we consider the GUT and the B − L breaking scales ratio in the range √ λ = 7.4 − 9.0, where smaller values lead to a small spectrum at nHz frequencies that can be detected by future experiments, while all values √ λ 8.8 quickly converge towards the result for stable cosmic strings and can be observed by NANOGrav and PPTA experiments [75]. The parametric space in the above model, consistent with successful reheating with non-thermal leptogenesis and gravitino dark matter restrict the allowed values of string tension to the range 10 −9 Gµ CS 8 × 10 −6 that lies within the 1-and 2-σ bounds of NANOGrav, as well as the sensitivity bounds of future gravitational wave (GW) experiments.

Summary
To summarize, we have investigated various cosmological implications of a generic model based on the U (1) B−L extension of the MSSM gauge symmetry in a no-scale Kähler potential setup, highlighting the issues of inflation, leptogenesis and baryogenesis, gravitino as well as the stochastic gravitational wave background (SGWB) from metastable cosmic sting network. The embedding of U (1) B−L into a simply-connected group SO(10), produces metastable cosmic string due to the spontaneous pair creation of a monopole and an anti-monopole, which can generate a stochastic gravitational wave background (SGWB) in the range of ongoing and future gravitational wave (GW) experiments. The interaction between U (1) B−L Higgs and the neutrino superfields generate heavy Majorana masses for the right-handed neutrinos. The heavy Majorana masses explain the tiny neutrino masses via the seesaw mechanism, a realistic scenario for reheating and non-thermal leptogenesis. A wide range of reheat temperature (10 7 T r 10 10 ) GeV and U (1) B−L symmetry breaking scale (1.3 × 10 13 M 2.0 × 10 16 ) GeV is achieved here with successful non-thermal leptogenesis and stable gravitino as a possible dark matter candidate. The metastable cosmic string network admits string tension values in the range 10 −12 Gµ CS 8 × 10 −6 . A successful reheating with non-thermal leptogenesis and gravitino dark matter restrict the allowed values of string tension to the range 10 −9 Gµ CS 8 × 10 −6 , predicting a stochastic gravitational-wave background that lies within the 1-σ bounds of the recent NANOGrav 12.5-yr data, as well as within the sensitivity bounds of future GW experiments.