CosmoTransitions: Computing cosmological phase transition temperatures and bubble profiles with multiple fields☆
Introduction
Phase transitions driven by scalar fields have probably played an important role in the very early evolution of the Universe. In most inflationary models, the dynamics are driven by the evolution of a scalar inflaton field, while at later times electroweak symmetry breaking is thought to be driven by a transition in the Higgs field vacuum expectation value. Electroweak scale physics is currently being probed by the LHC, so the phenomenology of the electroweak phase transition is of particular interest. A strongly first-order electroweak phase transition would have been a source of entropy production in the early Universe (thereby changing the evolution of its scale with respect to temperature) and produced a stochastic background of gravitational radiation [1], perhaps observable by future space-based gravitational radiation observatories [2]. In addition, a strongly first-order electroweak phase transition may have satisfied the Sakharov conditions [3] and been responsible for the current baryon asymmetry of the universe (for recent studies, see e.g. Refs. [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]), or may have affected the relic density of (for example) dark matter particles [19], [20], [21].
In the standard model, the electroweak phase transition is not strongly first-order unless the Higgs mass is below ∼70 GeV[22], [23], [24], which is excluded by the current LEP bound of 114.4 GeV[25]. However, electroweak baryogenesis can be saved in extensions to the standard model, many of which include extra dynamic scalar fields (such as two-Higgs-doublet models [26], [27], [28], [29], [30], [31], [32]). The amount of produced baryon asymmetry depends crucially upon the dynamics of the phase transition, and particularly upon the bubble-wall profiles that separate the high- and low-temperature phases. These profiles are fairly easy to calculate using effective field theory if there is only one scalar field, but multiple fields greatly increase the computational complexity.
In this paper, I present an easy-to-use numerical package (CosmoTransitions) to analyze phase transitions in finite temperature field theory with multiple scalar fields. The program consists of three basic parts (see Fig. 1): modules for finding the tunneling solution (bubble wall profile) between different vacua, a module for finding critical temperatures and phase transitions, and an abstract class to define specific field-theoretic models and their effective potentials. In Section 2, I describe the algorithms for finding bubble wall profiles for both single and multiple fields. Section 3 describes the algorithm for finding phase transitions, while Section 4 describes how one can implement a specific model in a simple program. Finally, I present numerical results in Section 5 and conclude in Section 6.
To download the latest version of CosmoTransitions, visit http://chasm.ucsc.edu/cosmotransitions.
Section snippets
Calculating bubble profiles
First-order cosmological phase transitions proceed by the nucleation of bubbles of true vacuum out of metastable false vacuum states. The bubbles have both surface tension and internal pressure, so that large bubbles tend to expand and small bubbles tend to collapse. Critical bubbles—bubbles that are just large enough to avoid collapse—will drive the phase transition.
Given a Lagrangian where is a vector of scalar fields, a critical bubble can be found by extremizing
Exploring phase structures
In order to determine the characteristics of a phase transition, we must first find where, and at what temperatures, the various phases exist. In theories with spontaneously broken symmetries there is at least one zero-temperature phase and there is generally one high-temperature symmetry-restoring phase. If these phases coexistence at some temperature, then there is likely a first-order phase transition between them. However, even in relatively simple models there can be intermediate phases
Structure of a simple program
There are essentially three parts to a simple program using my code: the tunneling algorithms and phase tracing algorithms described above, and the implementation of a specific model. Much of this last task happens in the generic_potential class, which must be subclassed to study any particular theory.
From the point of view of the finite temperature effective potential, the theory is completely determined by the tree-level potential and field-dependent mass spectrum. The generic_potential class
Deformation
To test the path deformation and tunneling routines, I use a simple potential given by This has one local minimum at , and a global minimum near . For , the phases are nearly degenerate and any tunneling between them will be thin-walled.
I run the pathDeformation.fullTunneling class for both thin () and thick-walled () potentials, with results shown in Fig. 6. Each line represents 15 individual deformation steps with adaptive
Conclusion
I presented the publicly available CosmoTransitions package to analyze cosmological phase transitions. This included algorithms to find the temperature-dependent phase minima, their critical temperatures, and the actual nucleation temperatures and tunneling profiles of the transitions. I introduced a novel method of path deformation to find the profiles, which I then demonstrated in simple test cases to accuracies of order ∼ 0.1%. The deformation algorithm has been successfully tested in 2 and
Acknowledgments
I am grateful to Stefano Profumo for many helpful comments over the course of this project. I receive support from the National Science Foundation.
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This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).