Primordial non-Gaussianity as a probe of seesaw and leptogenesis

We present the possibility that the seesaw mechanism and nonthermal leptogenesis can be investigated via primordial non-Gaussianities in the context of a majoron curvaton model. Originating as a massless Nambu-Goldstone boson from the spontaneous breaking of the global baryon (B) minus lepton (L) number symmetry at a scale vB−L, majoron becomes massive when it couples to a new confining sector through anomaly. Acting as a curvaton, majoron produces the observed red-tilted curvature power spectrum without relying on any inflaton contribution, and its decay in the post-inflationary era gives rise to a nonthermal population of right-handed neutrinos that participate in leptogenesis. A distinctive feature of the mechanism is the generation of observable non-Gaussianity, in the parameter space where the red-tilted power spectrum and sufficient baryon asymmetry are produced. We find that the non-Gaussianity parameter fNL ≳ O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(0.1) is produced for high-scale seesaw (vB−L at O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(1014−17) GeV) and leptogenesis (M1 ≳ O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(106) GeV) where the latter represents the lightest right-handed neutrino mass. While the current bounds on local non-Gaussianity excludes some part of parameter space, the rest can be fully probed by future experiments like CMB-S4, LSST, and 21 cm tomography.

Two clear evidence of physics beyond the Standard Model (SM) are the presence of nonzero neutrino mass, as demonstrated by the oscillation experiments [1][2][3] (see the current global fit [4,5]), and the observation of cosmic baryon asymmetry, which is quite precisely inferred from the abundances of light elements during Big Bang Nucleosynthesis (BBN) [6] and the measurement of the Cosmic Microwave Background (CMB) by Planck 2018 [7].
These two observations can be elegantly explained from a minimal extension of the SM with two or more Majorana right-handed neutrinos (RHNs).Tiny neutrino masses are generated through the type-I seesaw mechanism [8][9][10][11][12], and baryon asymmetry can be explained from baryogenesis via leptogenesis [13].In both cases, the Majorona nature of the RHN mass plays a crucial role.This mass could arise from an explicit or a spontaneous violation of the gauged/global lepton number (or equivalently, baryon minus lepton number) symmetry.
Due to the intimate connection between the neutrino mass scale and the CP violation for leptogenesis, the lightest RHN mass is required to exceed the Davidson-Ibarra bound, M 1 ≳ 10 9 GeV, if one assumes that the RHN mass spectrum is hierarchical, and lepton flavor effects [14][15][16] are completely absent [17,18].By considering mildly hierarchical RHN mass with lepton flavor effects, one could reduce the bound down to ∼ 10 6 GeV with certain tuning, such that the tree-level and one-loop contributions to neutrino mass are equally important [19].If we were to consider quasi-degenerate mass spectrum such that CP violation can be resonantly enhanced [20][21][22], the Davidson-Ibarra bound is completely evaded.In that case, the lower bound on RHN mass scale is set by the temperature when the electroweak sphaleron processes decouple at T = 132 GeV [23].
Only the lepton asymmetry generated above T > 132 GeV can induce a nonzero baryon asymmetry, and hence, the RHN mass scale cannot be much lower than this temperature such that a sufficient amount of lepton asymmetry can be generated from the RHN decay.For such a low scale, one should also consider another source of lepton asymmetry generation from the RHN oscillations [24].
Combining both the contributions from decays and oscillations, it is shown in refs.[25,26] that the RHN mass scale can be as low as 50 MeV.Low-scale RHNs can be searched in colliders, or in high-intensity experiments (see refs.[27,28] for the recent status).
Unfortunately, any direct detection of the theoretically motivated scenario of high-scale seesaw and leptogenesis lies beyond the energy reach of the current or foreseeable future experiments.The two indirect probes that immediately come to mind are lepton number violation through neutrinoless double beta decay [29] and CP violation in neutrino oscillation [30].From the theoretical side, one can rely on the structure of couplings that are consistent with SO (10) Grand Unified Theories [31][32][33][34][35][36][37] or from the requirement of Higgs vacuum (meta)stability in the early Universe [38,39].
In this work, we propose a novel approach to explore the scale of seesaw and leptogenesis by investigating their imprint on primordial non-Gaussianities.In our setup, the RHN masses are generated from the spontaneous breaking of a global U (1) B−L symmetry at a high scale v B−L around the GUT scale.A massless Nambu-Goldstone boson, dubbed as the "majoron", is generated in the process [59,60].We assume that the U (1) B−L is anomalous under a new force that is confined at a scale Λ below v B−L .As a result of this anomaly, the majoron experiences a periodic potential and becomes massive.We show that the majoron can act as a curvaton [61][62][63][64], which produces the entirety of the observed red-tilted scalar density perturbation without any contribution from the inflaton, following closely the scenario of an axionlike curvaton in ref. [65]. 1 In this setup, inflation [77][78][79][80][81] is driven by the inflaton and majoron remains a subdominant field during inflation, with its quantum fluctuations converted to classical perturbations at horizon exit with a flat spectrum.The perturbations in the majoron field are converted into curvature perturbations at late times in the post-inflationary era, when the inflaton decay products have redshifted away and majoron dominates the energy density of the Universe.This process generates observable local non-Gaussianity within the reach of future CMB and LSS experiments [82] and 21 cm tomography [83].Intriguingly, we find that the decay of the majoron at the end of the curvaton dynamics can produce a nonthermal population of RHNs which participate in successful leptogenesis.The scale of leptogenesis, primarily determined by the lightest RHN mass, and the seesaw scale, identified as the B − L breaking scale, leave nontrivial imprint on the non-Gaussianity parameter f NL .Although such observable f NL may arise from other scenarios, nonobservation of f NL predicted in our scenario would certainly falsify the proposed majoron-as-curvaton mechanism.
The paper is organized as follows.In section II we provide a concise overview of the majoron model, briefly examining its key aspects.In section III we discuss how the majoron can act as a curvaton and generate the curvature power spectrum and spectral index, and how this leads to observable non-Gaussianity.Section IV investigates the scenario of nonthermal leptogenesis after the majoron decays.We present our key results in section V and conclude in section VI.

II. MAJORON MODEL FOR NEUTRINO MASS GENESIS
We will consider the simplest majoron model [59,60] with a global baryon minus lepton number U (1) B−L symmetry under which three SM singlet fermions N i carry charge −1 and a complex scalar field σ carries charge 2. The relevant new interactions for us is where ℓ α and H are respectively the SM lepton and Higgs doublet (ϵ is the SU (2) antisymmetry tensor).Without loss of generality, we work in the basis where dimensionless couplings ξ i and y α are real while λ remains a complex 3×3 matrix.After σ acquires a vacuum expectation value v B−L along the radial direction, we have where M i = ξ i v B−L .We will assume that the radial field has mass m ρ > T max , where T max is the maximum temperature from reheating after inflation and ignore the contribution from ρ.At this point, the majoron χ is a Nambu-Goldstone (NG) boson which remains massless.It only couples to N i and an elegant way to deal with this nonlinear parametrization is to carry out field-dependent redefinition of fermionic fields, f → f e iq B−L f χ/(2v B−L ) , where q B−L f is the B −L charge of the fermion f .From the kinetic terms, we obtain where Carrying out integration by parts in the action and discarding the surface term, we have where the nonconservation of B − L current comes from the Majorana mass term and U (1) B−L has no anomaly with respect to the SM gauge interactions.Now, we will give mass m χ to χ by assuming that U (1) B−L is anomalous under a new gauge interaction which becomes strong at Λ < v B−L .χ will couple to the new gauge field as follows where g X is the gauge coupling of the new gauge interaction, G is its dual field strength of G (with both Lorentz and gauge indices suppressed) while n is the anomaly coefficient.In the explicit majoron model in this work, the decay of χ to the new sector through eq. ( 6) is loop-suppressed or forbidden in the absence of states with masses lighter than m χ /2.On the other hand, if m χ > 2M i for some i, the decay width of χ → N i N i is given by2 For definiteness, we will assume that Relaxing this assumption will give only order of one changes to the parameter space of our analysis.Hence we will consider only the decay χ → N 1 N 1 with decay width given by eq. ( 7).
To complete the story, after the electroweak symmetry breaking, light neutrino mass is generated through the type-I seesaw mechanism [8][9][10][11][12] with light neutrino mass matrix given by with v = 174 GeV the Higgs vacuum expectation value.

III. MAJORON AS A CURVATON
In this section we discuss how majoron can play the role of curvaton to generate the power spectrum, spectral index and non-Gaussianity.
Here, we model the majoron mass, following ref.[65], where p 0 ̸ = 0 and Θ is the Heaviside step function Θ(x) = 1 for x > 0 and 0 otherwise.If the new gauge sector is identified as QCD, p 0 = 4 [84].χ is now a pseudo-NG boson with zero temperature mass We assume that the de Sitter temperature during inflation This implies that during inflation, the mass of the majoron is m χ0 .The equation of motion of the majoron can be written as where the first term can be neglected assuming slow roll during inflation, and the Hubble rate during inflation, H inf , can be considered constant.Solving this equation, we can determine the majoron field value at the end of inflation χ end .
As soon as inflation ends, the inflaton instantaneously decays into radiation and reheats the Universe.The maximum radiation temperature is obtained by setting the inflaton energy density equal that of the radiation 3H 2 inf M 2 Pl = π 2 30 g ⋆ T 4 , and solving for the temperature where the reduced Planck mass is M Pl = 2.43 × 10 18 GeV and g ⋆ = 106.75assuming the SM relativistic degrees of freedom.We make a further assumption so that the potential of χ diminishes after inflation and its field value is approximately frozen at χ end .In the post-inflationary period, the evolution of the Hubble rate can be tracked from the Friedmann equation where the energy densities of radiation and majoron are given by ρr + 4Hρ r = 0, and The majoron field evolves according to where the first term can no longer be neglected, and H evolves with time.In the equations above, we have not included the decay terms of χ since we will be using the sudden decay approximation when H = Γ χ .This has negligible effect on the calculation of f NL , as shown in ref. [85].
As the Universe cools down sufficiently, majoron begins to oscillate about the potential minimum when H ≃ m χ .Assuming that this happens at T osc when the Universe is still radiation-dominated by setting 3m 2 χ M 2 Pl = π 2 30 g ⋆ T 4 osc , we have As long as v B−L < 90 π 2 g⋆ M Pl ≈ 0.3M Pl , T osc > Λ, and the majoron potential at the onset of oscillation would depend on p ̸ = 0.This dependence continues until At the onset of oscillation, the number density of χ can be approximated as From this point onwards, the number of χ per comoving volume is conserved where s = 2π 2 45 g ⋆ T 3 is the cosmic entropy density.In fact, Y χ is not so sensitive to p and for p → ∞, the value saturates to For our calculation, we will fix p = 4.
Assuming sudden decay approximation, χ decays at T dec when H ≃ Γ χ and from energy conservation, we have where T is the temperature after all χ particles have decayed.
The decay of the majoron injects entropy and dilutes the number density of other species.The dilution factor from entropy injection is where R is the energy density of χ in comparison to the radiation density at T dec .One can solve for T dec and hence d for a given Γ χ and Y χ .

B. Power spectrum, spectral index and non-Gaussianity
Majoron remains a sub-dominant source of energy density during inflation and in the early reheating period, at least until it starts to oscillate.The energy density of the oscillating majoron eventually supersedes the radiation energy density.The super-horizon field fluctuations of the majoron created during inflation can be converted into curvature perturbations as the majoron dominates the energy density of the Universe.The generation of the curvature perturbation continues until the majoron finally decays at H ≃ Γ χ .If the majoron is to act as a curvaton, it is crucial for it to be sufficiently long-lived so that it can dominate the energy density of the Universe before its decay.
The curvature perturbation induced by the majoron can be computed using the δN formalism [86][87][88][89][90][91].The quantity δN is defined as the perturbation in the number of e-folds between a spatially flat slice and an uniform density slice and the curvature perturbation depends linearly on δN .Expanding in terms of the field fluctuations, the curvature perturbation can be expressed as Here χ in is the initial majoron field value at time t in when the CMB modes exit the horizon, and N is the total number of e-folds between t in and the time when the majoron decays.Defining the Fourier transform of the curvature perturbation as the two-and three-point correlation functions are given by [92] ⟨ζ where prime denotes the delta function (2π) 3 δ (3) ( i k i ) stripped off the correlation functions.
Plugging eqs.( 25) and ( 26) into eqs.( 27) and ( 28), one can derive the following expressions for the power spectrum and the non-Gaussianity parameter [68,72,93], up to leading order in δχ in .Here, power spectrum of the majoron field fluctuations on spatially-flat hypersurfaces at horizon exit is assumed to be Gaussian, and is given by P δχ in = H 2 inf /(2π) 2 [94].Here we have assumed that the Hubble rate during inflation remains nearly constant.
Typically it takes 50−60 e-folds of inflation to resolve the horizon problem [95].For definiteness, we will take N = −50 at the time when the CMB modes leave the horizon.Then, identifying N = 0 as the e-folding number at the end of inflation, we denote N f as the 'final' e-folding number that marks the end of the curvaton dynamics when the majoron decays.
To calculate the power spectrum and the non-Gaussianity parameter, we need to know the first and second order derivatives of the 'total' e-folding number N with respect to the initial field value χ in .Since the number of e-folds before inflation is independent of the majoron field value χ in , we can instead take the derivatives of the final e-folding number N f at the time of majoron decay.It is done by carefully tracking the evolution of the majoron field value, Hubble rate, and the energy density of the radiation bath and of the majoron field with the coupled system of equations (13) ( 16), ( 17) and ( 18), taking the non-trivial temperature dependence of the majoron potential at different stages of evolution into account.
For convenience, we define the dimensionless field θ ≡ χ/v B−L .During inflation, the dynamics of the majoron can be described by eq. ( 13), dropping the first term assuming slow roll.In terms of the e-folding number N = log a(t), it can be written as with the initial condition θ(N Here we have taken the majoron mass to be m χ0 assuming T inf < Λ. We denote the end of inflation with N = 0. To analyze the post-inflationary dynamics, we express the Hubble rate H in terms of the e-folding number N , and introduce the dimensionless variable x ≡ m χ0 t,.Then, eqs.( 16), ( 17) and ( 18) can be condensed into the following two coupled equations, Here prime denotes derivative w.r.t.x, and ρr,ini = ρ r,ini /(3m 2 χ0 M 2 Pl ), where ρ r,ini = 3H 2 inf M 2 Pl is the radiation energy density after instantaneous reheating at the end of inflation.
For any particular initial condition π/2 ≤ θ in ≤ π at N = −50, the solution of eq. ( 30) yields θ end , the majoron field value at the end of inflation.This, together with its first derivative calculated from eq. ( 30), are fed into eqs.( 31) and (32) as initial conditions at N = 0 at x = m χ0 /(2H inf ).In order to solve eqs.( 31) and ( 32), the post-inflationary period can be divided into two regimes, (i) T > Λ where p ̸ = 0, and (ii) T < Λ where p = 0.The first regime occurs from the end of inflation at x = m χ0 /(2H inf ) to x = x p when T = Λ, where The Universe is radiation dominated during this regime, and the relation between x and temperature T can be determined from where we have used t = 1/(2H) in writing the last equality.The second regime then starts and continues until the majoron decays at H = Γ χ , corresponding to x = m χ0 /Γ χ , at which point we evaluate the 'final' e-folding number N f .Since the majoron field value oscillates until it decays, solving the system of equations ( 31) and ( 32) is numerically very challenging.We use an approximate method, elaborated and justified in appendix A, to circumvent this problem.
Scanning over 0 ≤ θ in ≤ π and determining N f for each θ in , we can approximate dN f /dθ in and d 2 N f /dθ 2 in to calculate the power spectrum and f NL from eq. ( 29), The spectral index of the power spectrum is defined as H inf /m χ0 dx, and assuming slow-roll during inflation, it can be expressed as where the first (second) term comes from the curvaton (inflaton) dynamics.
There are five free parameters in our analysis, the Hubble rate during inflation H inf , the B − L breaking scale v B−L , the majoron mass m χ0 , the decay width Γ χ , and the intial field value χ in , or its dimensionless version θ in ≡ χ in /v B−L .As long as Γ χ is sufficiently small, so that the majoron's lifetime is sufficiently large and it can dominate the energy density of the Universe, the results do not depend on Γ χ .We assume that the majoron as a curvaton is responsible for generating the entirety of the spectral index n s and the scalar power spectrum P ζ observed at  [83], nearing the "gravitational floor", f local NL ≃ 0.01 [92,103,104], which is the non-Gaussianity generated by purely gravitational interactions among inflaton fluctuations in single-field inflation.

IV. THE DOMINANCE OF NONTHERMAL LEPTOGENESIS
RHNs can be produced during two stages of the cosmological evolution in our setup.One from the thermal bath during the reheating period, and the other from the decay of the majoron at the end of its dyanamics.
After inflation, the reheating temperature ( 14) is in general higher than the mass of N i (in particularly N 1 ).Unavoidably, N i will be produced through the Yukawa interactions which depend on λ αi in eq. ( 1).As a result, thermal leptogenesis will take place and taking into account the dilution from entropy injection in eq. ( 24), the B − L asymmetry generated from N 1 can written as where ϵ 1 parametrizes the CP violation from N 1 decay, Y eq N 1 (T ≫ M 1 ) = 45 2π 4 g⋆ and η 1 ≤ 1 is the efficiency factor taking into account the washout processes.
On the other hand, nonthermal leptogenesis will occur much later at T ≪ M 1 from N 1 particles which are produced from the decays of χ.The temperature of the bath after majoron decay can be determined by combining eq. ( 7) with eq. ( 23), We have verified that for the parameter space interesting for the curvaton dynamics yields T < M 1 , for which only nonthermal production of the RHN is relevant.
Another constraint on T is that it should be greater than the temperature where the Big Bang Nucleosynthesis (BBN) commences T ≳ T BBN ∼ MeV.However, since we are interested in leptogenesis from N 1 decays, we require T > 132 GeV [23] when the electroweak sphalerons are still active, so that the generated B − L asymmetry can be converted into baryon asymmetry.
In general, since the decay width of N 1 is much larger than that of χ, Γ N 1 ≫ Γ χ , N 1 particles decay almost instantaneously after being produced.With T ≪ M 1 , the washout process is completely suppressed and the B − L asymmetry produced by these new population of N 1 is where the factor of 2 arises since each χ decays to two N 1 .From eq. ( 21), we verify that nonthermal contribution above will dominate over thermal contribution (37) as long as η 1 ≲ 0.08.This is generally realized due to either strong washout (large λ α1 ) or inefficiency in N 1 production (small λ α1 ) when the efficiency is suppressed η 1 ≪ 0.1.To highlight the role played by χ in realizing nonthermal leptogenesis, we will assume the thermal contribution is subdominant in the rest of the work.
Finally, at T ∼ 132 GeV when the electroweak sphaleron freezes out after the electroweak phase transition at 160 GeV [23], the baryon asymmetry is given by where we have taken into account the finite top mass.This value is to be matched with the observed value Y obs B ≃ 8.7 × 10 −11 [7].Assuming hierarchical masses of N i , the CP asymmetry parameter from the decays of N 1 is bounded from above by the Davidson-Ibarra bound [17] where the atmospheric mass squared splitting is ∆m 2 atm ≃ 2.5 × 10 −3 eV 2 and m h + m l is the sum of the masses of the lightest and the heaviest light neutrinos.We will consider the most optimistic scenario by saturating the bound in eq. ( 41) by setting m h + m l = 0.05 eV which gives

V. IMPRINTS OF SEESAW AND LEPTOGENESIS ON NON-GAUSSIANITY
The massive majoron acting as a curvaton to give rise to the primordial curvature power spectrum also generates observable non-Gaussianity.On the other hand, at the end of the curvaton dynamics, the decay of the majoron yields a nonthermal population of RHNs, which participate in leptogenesis to generate the observed baryon asymmetry of the Universe.Hence, this setup provides a way to explain three observables, the magnitude and spectral index of the scalar power spectrum, and the baryon asymmetry observed at CMB.A testable prediction of this mechanism is the non-Gaussianity parameter f NL that can be probed in upcoming CMB experiments and 21 GeV (middle), and 10 14 GeV (right).In each case, we show four examples of different H inf . 4For comparison, we show the region excluded by Planck 2018 data [96] for non-observation of non-Gaussianity (hatched region), and sensitivity of future experiments CMB-S4+LSST [82] and 21 cm tomography [83], and the "gravitational floor" [92,103,104] representing the f NL generated by purely gravitational interactions of the inflaton (gray regions). 4The corresponding mχ0 and θin are shown in appendix C. Keeping everything else fixed, f NL does not depend on M 1 unless it is sufficiently large.This is consistent with our expectation that smaller M 1 results in a smaller decay width, therefore, longer lifetime of the majoron, facilitating that it can dominate the energy density of the Universe before it decays.In this case the ratio of the energy density of the majoron to that of the radiation bath, R, is so large that one can effectively take the limit R → ∞.On the other hand, if M 1 is sufficiently large, energy density of the majoron dominates for a short period.In this case, the ratio of energy densities, R is not very large and one gets a non-negligible contribution to f NL that increases with larger M 1 .This effect is more prominent when v B−L ≪ M Pl .
Next, let us discuss the baryon asymmetry generated in this model.In fig. 3 GeV could never produce the observed Y B .Similarly, for a given ϵ 1 , one can determine a lower bound on M 1 for successful leptogenesis.The shaded blue region represents the Davidson-Ibarra bound assuming a hierarchical mass spectrum of N i and negligible lepton flavor effects, given by eq. ( 42), which is satisfied only for smaller ϵ 1 and larger M 1 .Beyond this assumption, we notice that there is a large parameter space for successful leptogenesis, which by itself is a nontrivial result.
Figs. 2 and 3 show that the same parameter space that generates the entirety of the curvature power spectrum and the spectral index observed at CMB scales yields testable non-Gaussianity, and results in successful leptogenesis.Furthermore, these results depend crucially on the seesaw scale v B−L .Hence, the massive majoron model provides us an interesting way to investigate the high scale of seesaw and leptogenesis through primordial non-Gaussianities observable at future CMB, LSS experiments and 21 cm tomography.While nonobservation of local non-Gaussianities will rule out our proposed scenario, a positive signature will only be a suggestive evidence since there are other mechanisms where the non-Gaussianity can be produced.

VI. DISCUSSION AND CONCLUSION
We have presented an interesting way to investigate the high scale of seesaw and nonthermal leptogenesis through their imprints on the primordial non-Gaussianity.In our setup, the right-handed neutrino Majorana masses are generated from the spontaneous breaking of a global U Here we have dropped the temperature dependent factor as typically x quad > x p .The exponential factor in the second term ensures that N ′ (x) remains same for x = x quad in both eqs.( 32) and (A1).
This approximation can be verified by evolving the system of equations exactly up to the point x max , beyond which numerical solution becomes difficult because of the oscillatory nature of the solution θ(x), and then comparing the result to the approximate solution obtained by using the strategy above for x quad < x < x max .In fig.4, we show the normalized curvaton energy density ρθ (x) = 1 2 θ ′ (x) 2 + 1 − cos θ(x) (A2) for both cases.The energy density of curvaton obtained from exact numerical solution shown tiny oscillations with a decreasing average value, which is captured remarkably well by the approximate result.
(H inf , v B−L ), m χ0 , θ in , M 1 , reproduces the observed power spectrum and spectral index at the CMB scales entirely from the majoron dynamics, and predicts observable non-Gaussianity.

I. Introduction 2 II. Majoron model for neutrino mass genesis 4 III. Majoron as a curvaton 5 A. Cosmological evolution 5 B
. Power spectrum, spectral index and non-Gaussianity 8 IV.The dominance of nonthermal leptogenesis V. Imprints of seesaw and leptogenesis on non-Gaussianity VI. Discussion and Conclusion I. INTRODUCTION

A
. Cosmological evolutionAs we discussed in the previous section, from the spontaneous breaking of global U (1) B−L symmetry at a scale v B−L , one obtains a massless majoron χ.If U (1) B−L is anomalous under a new gauge interaction which becomes strong at a scale Λ < v B−L , χ will acquire a periodic potential[65,84]

FIG. 1 :
FIG. 1: An example of determining the initial dimensionless field value θ in ≃ 2.21 and majoron mass m χ ≃ 1.18 × 10 12 GeV, for which the spectral index and the scalar power spectrum matches with Planck 2018 values n s = 0.9649 and P CMB ζ

FIG. 2 :
FIG. 2: f NL as a function of the lightest RHN mass M 1 for three choices of the B − L breaking scale v B−L = 10 17 GeV (left), 10 16 GeV (middle) and 10 14 GeV (right).In each case we show results for four choices of H inf , shown in the inset, with an appropriate choice of m χ0 that is required to generate both the scalar power spectrum, P ζ , and the spectral index, n s , observed at CMB scales, entirely from the curvaton dynamics.For comparison we show the upper bound on local non-Gaussianity from Planck 2018 [96] (hatched region), sensitivities of future experiments CMB-S4+LSST and 21 cm tomography, and the "gravitational floor".

FIG. 3 :
FIG. 3: Baryon asymmetry from nonthermal leptogenesis.Purple lines show the generated asymmetry for different CP asymmetry parameters.The gray shaded region shows the parameter space where generated asymmetry is smaller than the observed asymmetry.Blue shaded region corresponds to successful leptogenesis where the RHN mass is above the Davidson-Ibarra bound.
, we show the baryon asymmetry Y B as a function of the RHN mass M 1 for the same set of benchmark values of v B−L .Baryon asymmetry depends weakly on H inf , only through the field value at the end of inflation, χ end in eq.(21).The resulting Y B for different H inf considered in fig.(2) for each v B−L are very close.Hence, in fig.3, we only show results for one representative H inf , but for three choices of the CP asymmetry parameter ϵ 1 .The gray region represents baryon asymmetry below the observed value Y obs B .Here the case ϵ 1 = 1 represents the theoretical maximum of the baryon asymmetry that can be produced from nonthermal leptogenesis in this model.This sets a lower bound on M 1 in fig. 3.For example, for the benchmark scenario in the left panel, M 1 ≲ 3 × 10 5

1 )
FIG. 4: Comparison of the exact numerical solution vs. the quadratic approximation of the equation of motion.

FIG. 6 :FIG. 7 :
FIG. 6: m χ0 as a function of the lightest RHN mass M 1 for three choices of the B − L breaking scale v B−L = 10 17 GeV (left), 10 16 GeV (middle) and 10 14 GeV (right).In each case we show results for four choices of H inf , shown in the inset.
[96] namely n s = 0.9649 and P CMB ζ = 2.4 × 10 −9[97].3Forns,thisimplies that cos θ in < 0, hence π/2 ≤ θ in ≤ π.In fig.1, we show how n s , P ζ and f NL varies with θ in for given H inf and v B−L .We see that n s and P ζ generated entirely from the majoron acting as a curvaton matches the Planck 2018 observed values when θ in ≃ 2.2 and m χ0 ≃ 0.29H inf .This results in a prediction for the non-Gaussianity parameter f NL ≃ 0.56, which is consistent with the current estimate on local-type non-Gaussianity from Planck 2018 is f local NL = −0.9±5.1[96], and can be probed at future 21 cm tomography sensitive up to f local NL ≃ 0.03