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From $U(1) \times U(1)$ Symmetry Breaking to Majoron Cosmology: Insights from NANOGrav 15-year Data

Tathagata Ghosh, Kousik Loho, Sudip Manna

Abstract

We study the cosmology of a modified majoron model motivated by the need to protect a global $U(1)$ symmetry from gravity-induced hard explicit breaking (by $d \leq 4$ operators) at the Planck scale. The model extends the Standard Model by introducing a gauged $U(1)_{B-L}$ and an approximate global $U(1)$ symmetry, each spontaneously broken by a corresponding complex scalar singlet. This setup gives rise to a network of effectively global and local cosmic strings, whose stochastic gravitational wave signals can jointly account for the spectrum observed by the NANOGrav collaboration, particularly for majoron masses $m_χ < 10^{-23}$ eV. Although the fit is not as strong as that from supermassive black hole mergers, the model still provides an alternative explanation rooted in high-energy physics. The model also generates light neutrino masses via the seesaw mechanism and avoids cosmological constraints from $ΔN_{\text{eff}}$, CMB anisotropies, and isocurvature fluctuations. Although the majoron can contribute to dark matter through thermal, coherent oscillation, and string-induced production mechanisms, its relic abundance remains subdominant in the NANOGrav-compatible region. In contrast, the measured dark matter relic density is achievable at higher $m_χ$, though at the cost of tension with cosmological bounds. If the NANOGrav fits are viewed as constraints, given their comparatively lower Bayes factors, they yield bounds that are significantly stronger than those imposed by the CMB and other cosmological data.

From $U(1) \times U(1)$ Symmetry Breaking to Majoron Cosmology: Insights from NANOGrav 15-year Data

Abstract

We study the cosmology of a modified majoron model motivated by the need to protect a global symmetry from gravity-induced hard explicit breaking (by operators) at the Planck scale. The model extends the Standard Model by introducing a gauged and an approximate global symmetry, each spontaneously broken by a corresponding complex scalar singlet. This setup gives rise to a network of effectively global and local cosmic strings, whose stochastic gravitational wave signals can jointly account for the spectrum observed by the NANOGrav collaboration, particularly for majoron masses eV. Although the fit is not as strong as that from supermassive black hole mergers, the model still provides an alternative explanation rooted in high-energy physics. The model also generates light neutrino masses via the seesaw mechanism and avoids cosmological constraints from , CMB anisotropies, and isocurvature fluctuations. Although the majoron can contribute to dark matter through thermal, coherent oscillation, and string-induced production mechanisms, its relic abundance remains subdominant in the NANOGrav-compatible region. In contrast, the measured dark matter relic density is achievable at higher , though at the cost of tension with cosmological bounds. If the NANOGrav fits are viewed as constraints, given their comparatively lower Bayes factors, they yield bounds that are significantly stronger than those imposed by the CMB and other cosmological data.

Paper Structure

This paper contains 23 sections, 41 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model
  3. The simplest majoron model
  4. The modified majoron model
  5. Cosmic Strings and Gravitational Waves
  6. Gravitational Waves from global cosmic strings
  7. Gravitational Waves from local cosmic strings
  8. Gravitational Waves from cosmic strings in the modified majoron model
  9. Results
  10. The simplest majoron model:
  11. The modified majoron model:
  12. Constraints
  13. Relativistic degrees of freedom ($\Delta N_{\text{eff}}$)
  14. The simplest majoron model:
  15. The modified majoron model:
  16. ...and 8 more sections

Figures (9)

  • Figure 1: Posterior distributions of $\log_{10} (\frac{\eta}{\text{GeV}} )$ from comparing the simplified majoron model with the NANOGrav 15-year data when GCS is considered as the only source of stochastic GW background. Here, $\eta$ is the VEV of the complex singlet $\phi$ breaking the global $U(1)_L$ symmetry. Among the four red dashed lines the inner two depict the $1\sigma$ credible interval of $\log_{10}(\frac{\eta}{\textrm{GeV}})$ while the outer two lines are for $2\sigma$.
  • Figure 2: Posterior distributions of GCS and SMBHB parameters from comparing the simplified majoron model with the NANOGrav 15-year data in the GCS+SMBHB scenario. The darker (lighter) shade of blue signifies the $1\sigma$ ($2\sigma$) credible contours. Among the four dashed lines of both $\gamma$ as well as $\log_{10}$A the outer (inner) two describe the $2\sigma$ ($1\sigma$) credible intervals. However, for the $\log_{10}(\frac{\eta}{\textrm{GeV}})$ the left (right) one among the two dashed lines signify the upper limits of the $1\sigma$ ($2\sigma$) credible intervals.
  • Figure 3: The red bands show the GW spectrum for the GCS-only (left) and GCS+SMBHB (right) cases, obtained by varying model parameters within their $2\sigma$ Credible Intervals in the simplified majoron model. Blue violins represent the NANOGrav 15-year data.
  • Figure 4: Posterior distributions of $\log_{10} (\frac{\eta^{\prime}}{\text{GeV}} )$ and $\log_{10} (\frac{\eta}{\text{GeV}} )$ from comparing the modified majoron model with the NANOGrav 15-year data when $m_{\chi} < 10^{-23}$ eV, and Type-$\phi$ and Type-$\phi^{\prime}$ CSs are considered as the only source of GW. For the model, we show the distributions for the Type-C GW spectrum only. Types A, B, and D GW spectra provide similar posterior distributions. Here, $\eta^{\prime} \, (\eta)$ is the VEV of the complex singlet $\phi^{\prime} \, (\phi)$ breaking the local $U(1)_{B-L}$ (accidental global $U(1)$) symmetry. The red dashed lines for $\log_{10}(\frac{\eta^\prime}{\textrm{GeV}})$ represent the different $1-2\sigma$ variable intervals in a similar manner as in Fig. \ref{['fig:GCS_posterior']}. For $\log_{10}(\frac{\eta}{\textrm{GeV}})$ the $1\sigma$ and $2\sigma$ upper limits are superimposed on each other and are depicted using red dashed lines. Also, the contour shading follows the same convention as Fig. \ref{['fig:GCS+SMBHB_posterior']}.
  • Figure 5: Posterior distributions of Type-$\phi$ and Type-$\phi^{\prime}$ CSs, and SMBHB parameters from comparing the modified majoron model with the NANOGrav 15-year data for $m_{\chi} < 10^{-23}$ eV in the CS+SMBHB scenario. We show the distributions for the Type-C GW spectrum only. Types A, B, and D GW spectra provide similar posterior distributions. The contour shading and dashed lines follow the same convention as in Fig. \ref{['fig:GCS+SMBHB_posterior']}.
  • ...and 4 more figures