Sound Shell Model in Gravitational Waves

• The Sound Shell Model is a framework that explains gravitational wave production via sound waves generated during first-order phase transitions.
• It characterizes how expanding bubble shells, variable sound velocities, and hydrodynamic behavior jointly shape a double-broken power law gravitational wave spectrum.
• The model offers actionable insights into early Universe physics, aiding in the interpretation of detector data from observatories like LISA and TianQin.

The sound shell model captures the effects of sound waves generated during a first-order phase transition, particularly in the context of gravitational wave production. The model is vital in predicting the gravitational wave spectrum based on the acoustic phenomena associated with expanding and colliding bubbles within a plasma. It has become increasingly relevant for understanding the early Universe's physics and evaluating the gravitational wave signals detectable by observatories like LISA.

1. Key Features and Principles

The sound shell model (SSM) centers on the idea that gravitational waves (GWs) during a first-order phase transition primarily originate from sound waves. These waves arise as bubbles of a stable phase undergo rapid nucleation, expansion, and collision within a metastable background phase. The key attributes include:

• Shell Structure: Each expanding bubble forms a shell of sound, defined by a moving wall producing either compression or rarefaction waves.
• Acoustic Dominance: GWs are sourced mainly by sound waves, not from direct bubble collisions.
• Velocity Correlator: The fluid velocity field is modeled as a Gaussian random field, allowing statistical treatment to derive GW spectra.

2. Modeling Sound Velocities

The sound shell model incorporates realistic sound velocities, differing between symmetric and broken phases. These velocities can diverge from the common assumption of $c_s = 1/\sqrt{3}$, impacting the GW spectrum significantly:

• Detonation vs. Deflagration: The different sound velocities influence how detonation (wall faster than sound) and deflagration (wall slower than sound) spectra are shaped. Detonation primarily involves the broken phase sound velocity ($c_-$), while deflagration depends on both $c_+$ and $c_-$.
• Spectrum Characteristics: Variations in sound velocities modify the peak frequency and amplitude by changing fluid responses and velocity profiles.

3. Gravitational Wave Spectrum

The GW spectrum generated according to the sound shell model exhibits a distinctive structure influenced by:

• Length Scales: The GW spectrum displays breaks associated with two key scales: the mean bubble separation ($R_*$) and the shell width ($\Delta R$).
• Spectral Scaling: Includes power laws in the form of $k^5$, $k^1$, transition regimes, and high-frequency $k^{-3}$ fall-off.
• Double-Broken Power Law: The spectrum can be fitted with two breaks corresponding to $R_*$ and $\Delta R$, linked to wall velocity and transition dynamics.

4. Impact of Nucleation and Transition Parameters

The time evolution of bubble nucleation plays a crucial role in shaping the GW spectrum:

• Nucleation Rate ($\beta$): The spread and timing of nucleation affect GW amplitude and peak frequency.
• Exponential vs. Simultaneous: Exponential nucleation typically results in broader spectra and higher amplitudes, as opposed to idealized simultaneous bubble nucleation.

5. Experimental Implications

Accurate modeling of sound velocities is critical for predicting and interpreting GW signals from first-order phase transitions:

• Detector Sensitivity: Realistic sound velocities substantially alter the GW spectrum's peak features, affecting potential detection by LISA, TianQin, and other GW observatories.
• Early Universe Insights: Sound shell models facilitate exploring new physics by offering a method to infer phase transition parameters via GW observations.

6. Theoretical Developments and Comparisons

Research has expanded on the sound shell model, exploring different aspects such as:

• Forced vs. Free Propagation: Distinctions between early-stage forced sound shell propagation and later free propagation refine GW predictions.
• Inverse Phase Transitions: Novel investigations into inverse transitions, where fluid is drawn toward bubbles, offering parallels to traditional 'direct' transitions.

7. Conclusions and Future Directions

The sound shell model remains a foundational tool in the paper of gravitational waves from cosmic phase transitions. It effectively connects macroscopic observational data to microscopic phase transition characteristics, helping to reveal dynamics of the early universe. Future research may explore greater model refinements and accommodate complex hydrodynamic conditions or varying cosmological backgrounds, thus extending our understanding of both theoretical models and potential observational frameworks.