pyCE.instantons module

Generation and configurational-entropy analysis of instantons.

Solves for the O(d)-symmetric bounce (“bubble”) profile of a real scalar field in the asymmetric double-well potential

V(B) = (1/2) B**2 - (alpha/3) B**3 + (1/4) B**4,

with asymmetry parametrized by alpha = (3/sqrt(2)) * (1 + g). The bounce satisfies the Euclidean equation of motion

B’’ + (d/r) B’ = V’(B) = B - alpha B**2 + B**3,

with B’(0) = 0 and B -> 0 (the false vacuum) as r -> infinity, and is found by a shooting method: bisect on the core value B(0) until the profile decays into the false vacuum instead of over- or under-shooting.

From the profile, the class computes the energy densities, the Euclidean action, and the configurational entropy of the energy-density power spectrum.

class pyCE.instantons.instanton(asymmetry_factor, dimension, N)[source]

Bases: object

O(d)-symmetric bounce profile and its configurational entropy.

Instantiation does all the work: it solves for the bounce by shooting, builds the energy profiles, and computes the Euclidean action and the configurational entropy.

Parameters:

  • asymmetry_factor (float) – Asymmetry g of the potential; alpha = (3/sqrt(2)) * (1 + g). g = 0 corresponds to the degenerate (thin-wall) limit.

  • dimension (int) – Number of SPATIAL dimensions d.

  • N (int) – Maximum number of radial integration steps (grid spacing dr = 0.01).

Variables:

  • r (ndarray) – Radial grid on which the bounce was accepted.

  • DB (B,) – Bounce profile B(r) and its radial derivative B’(r).

  • B0 (float) – Core value B(0) found by the shooting method.

  • Bmax (Bmin,) – Initial bracket for the shooting method.

  • rho (PEdens, GEdens,) – Potential, gradient, and total energy densities.

  • E (PE, GE,) – Radially integrated potential, gradient, and total energies.

  • Se (float) – Euclidean action: rho integrated in d+1 dimensions (the O(d+1) bounce interpretation of the static d-dimensional profile).

  • k (denFT,) – Radial Fourier transform of rho and its k-grid.

  • mf (ndarray) – Modal fraction |denFT|**2 / max|denFT|**2.

  • Sc (float) – Configurational entropy, -Integral[ mf * ln(mf) ] over k-space with the d-dimensional radial measure.