[Paper Review] Applications of Elliptic and Theta Functions to Friedmann-Robertson-Lemaitre-Walker Cosmology with Cosmological Constant

Jennie D’Ambroise|ArXiv.org|Aug 18, 2009

Cosmology and Gravitation Theories13 references3 citations

TL;DR

This paper demonstrates that Jacobi and Weierstrass elliptic functions, along with theta functions, provide exact analytical solutions to the Einstein field equations in Friedmann-Robertson-Lemaítre-Walker (FRLW) cosmology with a cosmological constant. By transforming to conformal time and analyzing scale factor dynamics under specific energy density scalings, the authors derive solutions in terms of elliptic functions and their theta function representations, revealing connections to integrable systems and Bose-Einstein condensate models.

ABSTRACT

Elliptic functions are known to appear in many problems, applied and theoretical. However, a lesser known application is in the study of exact solutions to Einstein's gravitational field equations in a Friedmann-Robertson-Lemaitre-Walker (FRLW) cosmology. We will show explicitly how Jacobi and Weierstrass elliptic functions arise in this context, and will additionally show connections with theta functions.

Motivation & Objective

• To establish the role of elliptic and theta functions in solving the Einstein field equations for FRLW cosmology with a cosmological constant.
• To demonstrate how specific energy density scalings (ρ ∝ a⁻³, a⁻⁴) lead to differential equations solvable by Jacobi and Weierstrass elliptic functions.
• To derive equivalent representations of these solutions using theta functions and identify limiting cases where solutions reduce to elementary functions.
• To explore connections between FRLW cosmology and quantum systems, particularly Bose-Einstein condensates, via a generalized Ermakov-Milne-Pinney equation.
• To provide explicit analytical solutions in both cosmic and conformal time for various initial conditions and curvature parameters.

Proposed method

• Transform the cosmic time equation for the scale factor ȧ(t)² = f(ȧ(t)) into a conformal time framework via a(η) = â(t(η)), leading to a′(η)² = g(a(η)) with non-negative powers of a(η).
• Identify the resulting differential equation as a cubic in a(η), which matches the standard form solved by Weierstrass ℘-functions and Jacobi elliptic functions.
• Use the inverse relation between elliptic integrals and elliptic functions to express the scale factor a(η) in terms of Jacobi elliptic functions (e.g., dn, cn) and their parameters.
• Express the same solutions using theta functions via known identities, particularly relating ℘-functions and theta functions through modular parameters.
• Apply the generalized Ermakov-Milne-Pinney equation to link FRLW dynamics to quantum condensate dynamics, enabling a cosmological-BEC correspondence.
• Derive explicit solutions for ρ(t) = D/â(t)³ and ρ(t) = D₁/â(t)³ + D₂/â(t)⁴, with D, D₁, D₂ > 0, and analyze special cases where solutions simplify to elementary functions.

Experimental results

Research questions

• RQ1 How do elliptic and theta functions emerge as exact solutions to the Einstein field equations in FRLW cosmology with a cosmological constant?
• RQ2 What is the role of conformal time in transforming the scale factor dynamics into a solvable form involving elliptic functions?
• RQ3 For which values of energy density scaling (ρ ∝ â⁻³, â⁻⁴) do the resulting differential equations admit solutions in terms of Jacobi or Weierstrass elliptic functions?
• RQ4 How can the same physical solutions be equivalently expressed in terms of theta functions, and what are the modular parameter relations involved?
• RQ5 What is the connection between FRLW cosmology and the dynamics of Bose-Einstein condensates via the Ermakov-Milne-Pinney equation?

Key findings

• Solutions for the scale factor a(η) in FRLW cosmology with ρ(t) = D/â(t)³ are expressed in terms of Jacobi elliptic functions dn(u,k) and cn(u,k), parameterized by modulus k ∈ (0,1), with explicit dependence on the cosmological constant and energy density D.
• For specific values of D, the elliptic solutions reduce to elementary functions: when D = 3Λ/8πG, the solution becomes a(η) = √(3/Λ) * dn(√(Λ/3)η, 1/√2), which simplifies to a(η) = √(3/Λ) * sech(√(Λ/3)η) in the limit k → 1.
• The same solutions for ρ(t) = D/â(t)³ and ρ(t) = D₁/â(t)³ + D₂/â(t)⁴ are equivalently expressed using theta functions, with identities involving θ₁, θ₂, θ₃, θ₄ and modular parameter τ related to the cosmological constant and energy densities.
• The Weierstrass ℘-function provides a unified solution framework for general curvature k′ and arbitrary D₁, D₂, D > 0, with the scale factor a(η) expressed as a rational function of ℘(η) and its derivative.
• The paper establishes a direct mapping between FRLW cosmology and Bose-Einstein condensate dynamics via the Ermakov-Milne-Pinney equation, showing that the square of the wave function's second moment satisfies the same equation as the scale factor in FRLW models.
• For ρ(t) = D/â(t)³ and ρ(t) = D/â(t)⁴, the authors derive explicit solutions in both conformal and cosmic time, with the latter obtained by inverting the conformal time relation using elliptic integrals, confirming consistency with known cosmological solutions in special cases.