[Paper Review] Domain walls and strings formation in the early Universe

А. А. Кириллов, Boris S. Murygin|arXiv (Cornell University)|Nov 9, 2020

Cosmology and Gravitation Theories13 references4 citations

TL;DR

This paper demonstrates that in a (2+1)-dimensional early Universe model with two scalar fields, both domain walls and cosmic strings can form dynamically from classical field evolution, even when the scalar fields are inflatons. The formation depends critically on initial field configurations: domain walls arise when fields start near a saddle point and roll down symmetrically, while strings form when fields begin at the peak and evolve radially outward, with the potential's slope and initial energy determining the outcome. The key contribution is showing that a single model with one minimum and one saddle point can produce both topological defects, depending on initial conditions.

ABSTRACT

Solitons formation through classical dynamics of two scalar fields with the potential having a saddle point and one minimum in (2+1)-space-time is discussed. We show that under certain conditions in the early Universe both domain walls and strings can be formed even if scalar fields are inflaton ones.

Motivation & Objective

• To investigate whether topological solitons like domain walls and cosmic strings can form in a (2+1)-dimensional early Universe model with two scalar fields.
• To determine the conditions under which both domain walls and strings emerge from the same potential landscape, particularly when scalar fields are inflatons.
• To analyze how initial field configurations—specifically the starting position relative to the potential's saddle point—affect the type of soliton formed.
• To assess the implications of such soliton production for primordial black hole formation and observational cosmology.

Proposed method

• The study uses a Lagrangian with two real scalar fields φ and χ, coupled to a Friedmann-Robertson-Walker metric with a constant Hubble parameter H, modeling the early Universe's expanding spacetime.
• The field dynamics are governed by second-order partial differential equations derived from the Lagrangian, including a friction term proportional to H, which acts as a damping force during evolution.
• Two distinct potentials are analyzed: a modified potential with one minimum and one saddle point (V = d(φ² + χ²) + a exp[−b(φ−φ₀)² − c(χ−χ₀)²]), and the tilted Mexican hat potential (V = λ(φ² + χ² − g²/2)² + Λ⁴(1 − φ/√(φ²+χ²))).
• Initial conditions are set such that the fields are distributed in a circular disk in field space, with radial and angular dependence via R(r) = R₀ / cosh(r₀/r) and Θ = θ, allowing for vortex-like or wall-like initial configurations.
• Boundary conditions enforce vanishing field gradients at spatial infinity, ensuring stable, localized field configurations during numerical evolution.
• Energy density is computed via ρ = ½∑[(∂ᵢφ)² + (∂ᵢχ)²] + V(φ,χ), and final configurations are analyzed to identify domain walls (planar structures) and strings (line-like ridges).

Experimental results

Research questions

• RQ1 Can domain walls and cosmic strings both form in a single scalar field model with only one minimum and one saddle point in (2+1) spacetime?
• RQ2 How do initial field configurations—specifically the starting position relative to the potential peak—affect the type of soliton formed?
• RQ3 Under what parameter regimes (e.g., potential height, slope, initial energy) do domain walls or strings dominate the final configuration?
• RQ4 Can inflaton fields in multi-field inflation models lead to the formation of topological defects like domain walls or strings, even without explicit symmetry breaking?
• RQ5 What role does the potential's tilt or asymmetry play in stabilizing or destabilizing soliton structures like strings with ridges?

Key findings

• Domain walls form when the initial field configuration starts near the saddle point and evolves symmetrically toward the minimum, resulting in a planar structure with high energy density localized at the wall.
• Strings form when the initial configuration is centered at the peak of the potential, leading to radial field evolution and a ring-like structure with a ridge, indicating a vortex-like soliton.
• The transition between domain wall and string formation depends solely on the initial value φ₁: φ₁ = -8 leads to domain walls, while φ₁ = -5 leads to strings, despite identical potentials and other parameters.
• Domain wall formation requires the local maximum height (parameter a) to be significantly larger than the potential slope (parameter d), setting a critical threshold for stability.
• For string formation, the initial field amplitude R₀ must be bounded to prevent excessive initial energy that could disrupt soliton formation.
• In the tilted Mexican hat potential, a very small tilt (Λ = 5×10⁻¹³) leads to string formation with a barely visible ridge; increasing Λ makes the ridge more prominent, confirming the role of potential asymmetry in soliton morphology.