Source-linked AI summary
CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields
Carroll L. Wainwright
TL;DR
Multiple scalar fields make cosmological phase-transition dynamics and bubble-wall calculations computationally difficult. CosmoTransitions provides modules for phase analysis, transition temperatures, tunneling profiles, and model definition, using path deformation for multidimensional bubbles. The package was tested on thin- and thick-walled profiles, achieving approximately 0.1% accuracy in simple test cases and supporting analyses across multiple dimensions.
Problem
Multiple scalar fields greatly increase the computational complexity of calculating bubble-wall profiles, which are important for phase-transition dynamics and electroweak baryogenesis.
Method
CosmoTransitions combines modules for tunneling solutions, transition temperatures, phase transitions, and model definitions with path deformation for multidimensional bubble profiles.
Results
The path-deformation algorithm was tested in two and three dimensions on thick- and thin-walled profiles, with simple test cases reaching accuracies of approximately 0.1%.
Takeaways & Limitations
CosmoTransitions is publicly available and designed to be extensible through a generic potential class requiring only basic model specifications.
Takeaways & Limitations
The one-dimensional overshoot/undershoot method can fail or be inaccurate for some multi-minimum potentials and may return only the first transition in a sequential tunneling process.
Abstract
from arXiv · showhide
I present a numerical package (CosmoTransitions) for analyzing finite-temperature cosmological phase transitions driven by single or multiple scalar fields. The package analyzes the different vacua of a theory to determine their critical temperatures (where the vacuum energy levels are degenerate), their super-cooling temperatures, and the bubble wall profiles which separate the phases and describe their tunneling dynamics. I introduce a new method of path deformation to find the profiles of both thin- and thick-walled bubbles. CosmoTransitions is freely available for public use.
PROGRAM SUMMARY
CosmoTransitions is a Python program for analyzing early-Universe finite-temperature phase transitions with multiple scalar fields. It determines phase structure, supercooling, and bubble-wall profiles.
- CosmoTransitions is a Python package for analyzing finite-temperature cosmological phase transitions with multiple scalar fields.It was designed and tested on Mac OS X 10.6.8 and requires Python.
- The manuscript was submitted to Computer Physics Communications on November 27, 2024.
- The program analyzes an input theory’s phase structure, supercooling at each transition, and bubble-wall profiles of nucleated bubbles.
1. Introduction
Scalar-field phase transitions are relevant to early-Universe evolution and electroweak phenomenology, but calculating bubble-wall profiles becomes substantially more difficult with multiple fields. CosmoTransitions addresses this need with modules for tunneling solutions, transition temperatures, and model specification.
- Scalar-field phase transitions likely influenced the early Universe, including inflationary dynamics and electroweak symmetry breaking.
- A strongly first-order electroweak phase transition could produce entropy and stochastic backgrounds, and may affect dark-matter relic densities.
- In the Standard Model, a strongly first-order electroweak transition requires a Higgs mass below approximately 70 GeV, excluded by the 114.4 GeV LEP bound.
- Multiple scalar fields greatly increase the computational complexity of calculating bubble-wall profiles relevant to electroweak baryogenesis.
- CosmoTransitions provides modules for bubble-wall tunneling solutions, critical temperatures and phase transitions, and field-theoretic model definitions.
- The package is publicly available for download from the CosmoTransitions website.
2. Calculating bubble profiles
CosmoTransitions computes critical bubble profiles by solving one-dimensional tunneling problems and deforming guessed paths for multiple fields. Its methods address thin-wall accuracy and limitations involving intermediate vacua or unsuitable deformation stepsizes.
- Bubble profiles: Critical bubbles are stationary tunneling configurations separating true and false vacua; their nucleation rates determine whether a phase transition occurs.At finite temperature, the nucleation rate depends exponentially on the Euclidean action divided by temperature.
- One-dimensional solution: The one-dimensional overshoot/undershoot method treats the field as a particle moving in the inverted potential −V(φ), stopping at the false vacuum at infinite radius.The method varies the initial position near the true vacuum until the required asymptotic boundary condition is met.
- One-dimensional solution: Thin-walled bubbles require special treatment because the field remains near the true vacuum for a long distance before rapidly crossing the wall.A transformed initial variable and approximate wall-radius calculation allow integration to begin near the wall, improving accuracy and reducing computation time.
- One-dimensional solution: Overshoot/undershoot can fail or lose accuracy when an intermediate minimum changes the available tunneling route or requires sequential transitions.The implementation may return only the first transition rather than a direct transition to the target vacuum.
- Multi-dimensional solution and path deformation: For multiple fields, CosmoTransitions assumes a path, solves motion along it, and deforms the path using the perpendicular force until it approaches the tunneling solution.The path is represented with spline basis functions plus a linear component connecting its endpoints.
- Multi-dimensional solution and path deformation: Deformation stepsizes must remain small enough for errors to self-correct; excessive stepsizes reverse and amplify path wiggles.The algorithm checks for such reversals to maintain an appropriate stepsize.
3. Exploring phase structures
CosmoTransitions traces phase minima across temperature to identify phase coexistence and changes such as phase disappearance or second-order transitions.
- Intermediate phases can occur between the zero-temperature and high-temperature symmetry-restoring phases, complicating phase-structure analysis.
- The local minimum is approximated from the gradient b_i and Hessian Mij through φmin = −M^−1b.
- Tracing a minimum as temperature changes enables finding all minima of a phase across temperatures.
- A singular Hessian M signals rapid minimum changes caused by phase disappearance or a second-order phase transition.
- The traceMinimum algorithm combines Nelder-Mead minimization with the temperature-evolution equation, using minimization as an error check and enabling adaptive temperature steps.
4. Structure of a simple program
A simple CosmoTransitions program combines tunneling and phase-tracing algorithms with a subclass defining the model’s potential and particle spectrum.
- A program has three parts: tunneling algorithms, phase-tracing algorithms, and implementation of a specific model.
- The generic potential class computes one-loop corrections from the tree-level potential and field-dependent mass spectrum using MS renormalization.
- The one-loop expression uses each species’ degrees of freedom and field-dependent masses, with Q as the renormalization scale and sign choices distinguishing bosons from fermions.
- Finite-temperature functions J∓(x) are implemented through direct integration and cubic interpolation, while subclasses can add structures such as counter-terms.
- A functioning model requires overriding initialization, the tree-level potential, boson and fermion mass-spectrum functions, and preferably approximate zero-temperature minima.
5. Numerical results
Numerical tests show that CosmoTransitions finds tunneling paths and bubble profiles for thin- and thick-walled transitions, and identifies multiple temperature-dependent phases and transitions in a multiscalar model.
- 5.1. Deformation: The path-deformation routine was tested on thin-walled δ = 0.02 and thick-walled δ = 0.4 potentials, with 15 adaptive deformation steps per displayed line.The thick-walled case converges more slowly because φ0 changes during each step.
- 5.1. Deformation: The deformed paths agree with manually initialized integrations within 0.1% for thin-walled and 0.3% for thick-walled bubbles.The integrated solutions are used as a numerical check of the deformation results.
- 5.2. Calculating transition temperatures: The model develops a new phase at 128.2 GeV, tunnels to it at 128.1 GeV through a first-order transition, and loses the high-temperature phase by 127.6 GeV.At high temperatures the model has a single phase at s1 = s2 = 0; the new phase appears at (s1, s2) = (43, −33).
- 5.2. Calculating transition temperatures: Two distinct phases separated by ∆φ ≈ 2 GeV occur over 112.14–112.22 GeV, although the default resolution registers the nearby discontinuity as second-order.Increasing resolution corrects analogous classification issues in higher-dimensional tests.
- 5.2. Calculating transition temperatures: The phases associated with (+v, −v) and (+v, +v) become degenerate at 75.2 GeV, but tunneling occurs only at 54.4 GeV and is very thick-walled.The metastable phase persists until just above the temperature at which it disappears.
6. Conclusion
CosmoTransitions provides publicly available algorithms for phase minima, transition temperatures, and tunneling profiles in multiple scalar-field models. Its path-deformation method handles thin- and thick-walled profiles and is designed for extensibility to higher-dimensional theories.
- 6. Conclusion: The package finds temperature-dependent phase minima, critical and nucleation temperatures, and tunneling profiles for cosmological phase transitions.It is publicly available and includes algorithms for both single- and multiple-field problems.
- 6. Conclusion: The path-deformation algorithm was tested in two and three dimensions with both thick- and thin-walled profiles, achieving accuracies of order ∼0.1% in simple test cases.The paper states that the method should work in higher dimensions as well.
- 6. Conclusion: CosmoTransitions is designed to be extensible: model builders specify the dimensions, tree-level potential, and field-dependent particle spectrum through a generic potential subclass.The stated design goal is to enable quick analysis of phase transitions in many extensions to the standard model.