⚛️ phenomenology

Gravitational waves in the complex-singlet-extended Manohar-Wise model

This paper investigates cosmological phase transitions and stochastic gravitational-wave signals in a complex-singlet-extended Manohar-Wise model, demonstrating that both one-step and two-step thermal histories can produce observable signals from domain-wall annihilation and phase transitions within the sensitivity ranges of future space-based and pulsar-timing array detectors.

Original authors: Benhong Mei, Yabo Dong, Jun Fan, Jingya Zhu

Published 2026-10-01

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Benhong Mei, Yabo Dong, Jun Fan, Jingya Zhu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is not a static stage but a dynamic history written in the fabric of space and time. For decades, scientists have listened to this history through the ripples of gravitational waves, the faint echoes of colliding black holes and neutron stars that reach us from the violent moments of cosmic evolution. Yet, there is a silence in the early universe that these detectors have not yet broken. The first moments after the Big Bang were likely filled with dramatic shifts in the fundamental forces, where the laws of physics themselves changed as the cosmos cooled. These shifts, known as phase transitions, are similar to water freezing into ice, but occurring on a scale so vast and energetic that they could have generated a background hum of gravitational waves, distinct from the sharp chirps of merging stars. Detecting this hum would allow us to peer back into an era that light cannot reach, testing theories about the hidden particles and symmetries that shaped our reality.

A team of researchers from Henan University has recently explored how such a signal might appear if the universe followed a specific, complex path involving new types of particles. They focused on a theoretical framework that adds a mysterious, invisible particle to the known family of forces, one that carries a unique "color" charge and interacts with the Higgs field, the mechanism that gives particles their mass. In their model, this new particle sector undergoes a spontaneous breaking of a hidden symmetry, a process that creates invisible walls of energy stretching across the early universe. These walls are unstable; they eventually collide and vanish, releasing a burst of gravitational energy. The researchers simulated the entire thermal history of this scenario, calculating how the universe would have cooled and how these events would have rippled through space-time.

The study reveals two distinct ways this cosmic drama could have unfolded. In the first scenario, the universe undergoes a single, violent shift where the forces of nature change all at once. This event creates a strong, sharp signal of gravitational waves that peaks at a frequency of roughly one-thousandth to one-hundredth of a cycle per second. This frequency falls squarely within the range that future space-based observatories, such as the Laser Interferometer Space Antenna, are designed to detect. In the second scenario, the change happens in two stages. First, the invisible particle sector shifts while the rest of the universe remains unchanged, followed later by the main shift in the forces of nature. This two-step history also produces a high-frequency signal from the first stage, but it is the second stage of the story that offers a unique twist.

The invisible walls created by the symmetry breaking in these models do not vanish immediately. As they eventually collapse, they generate a second, much lower-frequency hum. The researchers found that the strength of this low-frequency signal depends heavily on the tension, or tightness, of these walls. In the single-step scenario, the walls are relatively loose, and the resulting signal is too faint for current or planned pulsar timing arrays to hear. However, in the two-step scenario, the walls are much tighter and more energetic. When these walls annihilate, they produce a signal strong enough to be detected by the Square Kilometre Array, a massive radio telescope project currently under development. This signal would appear at a frequency of about two billionths of a cycle per second, a deep, slow pulse in the nanohertz band.

The work suggests that the universe might leave behind a dual signature: a high-pitched chirp from the violent phase transition and a low, rhythmic throb from the collapse of these invisible walls. If future detectors observe both signals simultaneously, it would provide compelling evidence for this specific extension of the standard model of particle physics. The researchers emphasize that while their calculations are robust within the approximations they used, the actual detection of such a signal requires further validation of the complex dynamics involved. Nevertheless, the study highlights a powerful synergy between different types of gravitational wave detectors. By listening across a vast range of frequencies, from the millihertz band of space-based lasers to the nanohertz band of pulsar timing, astronomers may soon be able to reconstruct the hidden thermal history of the early universe and confirm the existence of these elusive, color-charged particles.

Technical Summary: Gravitational Waves in the Complex-Singlet-Extended Manohar-Wise Model

Problem Statement
The Standard Model (SM) predicts a smooth crossover for the electroweak phase transition (EWPT) given the observed Higgs mass, precluding the generation of a stochastic gravitational-wave (GW) background via bubble nucleation. Furthermore, the SM lacks mechanisms for spontaneous breaking of discrete symmetries that could lead to topological defects like domain walls. To address these limitations, this work investigates the Manohar-Wise (MW) model—an extension of the SM by a color-octet scalar doublet—further augmented by a complex scalar singlet. The study focuses on two primary phenomena: (1) the potential for a strong first-order electroweak phase transition (FOPT) capable of preserving a baryon asymmetry, and (2) the formation and subsequent annihilation of domain walls arising from the spontaneous breaking of a discrete $Z_3$ symmetry. The goal is to quantify the resulting stochastic GW signals and assess their detectability by current and future observatories.

Methodology
The authors construct a scalar sector comprising the SM Higgs doublet ( $H$ ), a complex gauge singlet ( $S$ ), and a color-octet scalar doublet ( $\Phi$ ). The scalar potential includes a cubic term ( $S^3 + S^{\dagger 3}$ ) that explicitly breaks the global $U(1)$ symmetry of the singlet sector down to a discrete $Z_3$ symmetry. A small explicit symmetry-breaking bias is introduced to ensure the eventual annihilation of the resulting domain walls, preventing them from dominating the universe's energy density.

The analysis proceeds through the following steps:

Model Implementation: The tree-level potential is defined, and vacuum stability conditions are imposed. The model is implemented in SARAH and SPheno for parameter scanning.
Thermal History Analysis: Two distinct thermal histories are investigated:
- Scenario A (One-step): A direct transition from the symmetric phase to a phase where both electroweak and $Z_3$ symmetries are broken simultaneously. This scenario is constrained by the baryon-number preservation criterion (BNPC) to ensure viable electroweak baryogenesis.
- Scenario B (Two-step): A transition first breaks the $Z_3$ symmetry (along the singlet direction) while electroweak symmetry remains unbroken, followed by a subsequent electroweak transition.
GW Calculation: The authors calculate the stochastic GW spectrum using the CosmoTransitions package to determine the nucleation temperature ( $T_n$ $T_{n}$ ), transition strength ( $\alpha$ $α$ ), and inverse duration ( $\beta/H_n$ $β / H_{n}$ ). The total GW spectrum includes contributions from:
- Sound waves and magnetohydrodynamic (MHD) turbulence generated by the FOPT.
- Domain-wall annihilation, calculated using a zero-temperature, fixed-modulus approximation.
Constraints and Scanning: The parameter space is scanned subject to theoretical constraints (boundedness from below, perturbativity, unitarity) and experimental constraints (Higgs signal strength, electroweak precision observables $S$ and $T$ ).

Key Contributions

Model Extension: The paper extends the Manohar-Wise model by a complex singlet, creating a framework where color-octet scalars and a singlet interact to modify thermal phase transition dynamics.
Dual-Source GW Prediction: The study provides a unified calculation of GW signals from both phase transitions and domain-wall annihilation within the same model framework.
Benchmark Analysis: Six benchmark points are selected to illustrate the phenomenology of both one-step and two-step thermal histories, demonstrating how the presence of color-octet scalars alters the parameter space compared to the complex-singlet-extended SM.

Results

Phase Transition Signals: For the selected benchmarks, the FOPT signals peak in the millihertz frequency range ( $10^{-3}$ – $10^{-2}$ Hz). These signals overlap with the projected sensitivity regions of space-based interferometers, specifically LISA, TianQin, and BBO.
Domain Wall Signals: The domain-wall annihilation signals are fixed to a peak frequency of approximately $2 \times 10^{-9}$ $2 \times 1 0^{- 9}$ Hz (nanohertz band) by adjusting the vacuum-energy bias.
- In Scenario A (one-step), the domain-wall tension is relatively low, resulting in GW amplitudes that remain below the projected sensitivity of pulsar timing arrays (PTAs), including the Square Kilometre Array (SKA).
- In Scenario B (two-step), the benchmarks exhibit significantly larger domain-wall tensions due to larger singlet vacuum expectation values (VEVs). Consequently, the GW signals from domain-wall annihilation in these cases reach the projected sensitivity of the SKA, while remaining below the current limits of EPTA, PPTA, and IPTA.
Parameter Space Differences: Compared to the complex-singlet-extended SM, the Manohar-Wise extension allows for larger values of the Higgs self-coupling ( $\lambda_H$ ), the singlet mass ( $m_2$ ), and the singlet-like scalar mass ( $m_\chi$ ), while accommodating smaller singlet self-couplings ( $\lambda_s$ ). This indicates that color-octet interactions significantly modify the thermal corrections and phase transition dynamics.

Significance
The paper concludes that the complex-singlet-extended Manohar-Wise model offers a testable framework for early-universe cosmology through the complementary observation of GWs in the millihertz and nanohertz bands. The detection of a two-peak stochastic background—one from a first-order phase transition and another from domain-wall annihilation—would provide strong evidence for this specific type of symmetry breaking and scalar sector extension. However, the authors note that their results are based on specific approximations (e.g., long-lived acoustic sources, fixed modulus for domain walls). For benchmarks with strong supercooling ( $\alpha \gg 1$ ), reliable predictions require further validation of transition completion, reheating history, and the lifetime of the acoustic source. The study emphasizes that while the signals are promising, a quantitative detection forecast requires a dedicated signal-to-noise analysis.

More like this