[Paper Review] The Schrodinger picture of standard cosmology

Nandinii Barbosa-Cendejas, M. Reyes|arXiv (Cornell University)|Dec 31, 2009

Cosmology and Gravitation Theories4 references3 citations

TL;DR

This paper proposes a novel Schrödinger picture for standard cosmology by mapping the dynamics of a scalar field in a flat FRW universe with a cosmological constant to a time-independent one-dimensional quantum mechanical Schrödinger equation. By reinterpreting the Hubble parameter and scalar field evolution through quantum bound state wavefunctions, the authors show that all exactly solvable quantum mechanical potentials—such as the harmonic oscillator—yield exact cosmological solutions, including those resembling standard inflationary models, with the cosmological constant directly linked to the quantum ground state energy.

ABSTRACT

We consider a time independent Schrodinger type equation derived from the equations of motion that drives a single scalar field in a standard cosmology model for inflation in a flat space-time with a Friedman-Robertson-Walker (FRW) metric with a cosmological constant. We find that all the 1-dimensional bound state solutions of quantum mechanics lead to at least one exact solution for the dynamical equations of standard cosmology, and that these solutions resemble the most recurrent inflationary solutions found in the literature. The analogies derived from this approach may be used to realize a deeper understanding of the dynamics of the model.

Motivation & Objective

• To establish a formal analogy between the dynamics of scalar field inflation in FRW cosmology and stationary quantum mechanical systems.
• To address the lack of a systematic method to derive exact solutions for scalar field potentials in inflationary models.
• To explore whether known exactly solvable quantum mechanical problems can generate physically meaningful cosmological solutions.
• To demonstrate that the cosmological constant emerges naturally from quantum energy eigenvalues in this framework.

Proposed method

• Define the Hubble parameter H(t) via a transformation of the scalar field wavefunction ψ(t), such that H(t) = (1/3) d/dt [ln ψ(t)].
• Transform the Riccati equation for H(t) into a linear Schrödinger-type equation: [−d²/dt² + 3V(t)]ψ(t) = −3Λψ(t).
• Map the scalar field potential V(ϕ) to the quantum potential U(x), and the cosmological constant Λ to the quantum ground state energy E₀ via the relation 3V(ϕ) + 3Λ ↔ 2U(x) − 2E₀.
• Use known quantum mechanical bound state solutions (e.g., harmonic oscillator) as ψ(t) to generate exact time-dependent H(t) and ϕ(t) via the derived transformations.
• Derive the scalar field potential V(ϕ) from the quantum potential and energy eigenvalue, showing that V(ϕ) = λϕ² for the harmonic oscillator case.
• Analyze the behavior of solutions under different energy regimes, distinguishing between bound states (leading to recollapsing universes) and unbound states (leading to ever-expanding universes).

Experimental results

Research questions

• RQ1 Can the dynamics of inflationary cosmology be reformulated as a time-independent Schrödinger equation?
• RQ2 Do all exactly solvable one-dimensional quantum mechanical problems yield exact cosmological solutions for the scalar field and Hubble parameter?
• RQ3 How is the cosmological constant Λ related to quantum mechanical energy eigenvalues in this framework?
• RQ4 What is the physical interpretation of the wavefunction ψ(t) in the context of cosmological energy density and volume evolution?
• RQ5 Can the slow-roll approximation in inflation be understood as a WKB approximation in the corresponding quantum mechanical problem?

Key findings

• The Schrödinger-type equation derived from the cosmological Riccati equation admits exact solutions when the Hubble parameter is constructed from quantum bound state wavefunctions.
• For the harmonic oscillator potential in quantum mechanics, the corresponding cosmological scalar field potential is V(ϕ) = λϕ² with λ = ω/2, and the cosmological constant is Λ = −2λ/3.
• The ground state solution of the quantum harmonic oscillator yields a time-varying Hubble parameter H(t) = −(ω/3)t, leading to a scale factor that vanishes at t → ∞, indicating a recollapsing universe.
• The slow-roll approximation in inflation corresponds to the WKB approximation in the quantum mechanical problem, establishing a deep dynamical analogy.
• Unbounded quantum states (with E < E₀ or E₀ < E < E₁) yield scale factors a(t) that diverge or oscillate, but only the lowest-energy unbound state (E < E₀) leads to a physical, ever-expanding universe with a(t) > 0 for all t.
• The cosmological constant is determined by the quantum ground state energy, with its sign fixed by the corresponding quantum problem, providing a direct link between vacuum energy and cosmological expansion.