[Paper Review] Non-Gaussian Correlations Outside the Horizon in Local Thermal Equilibrium

Joel Meyers|arXiv (Cornell University)|Dec 18, 2012

Cosmology and Gravitation Theories3 citations

TL;DR

This paper demonstrates that tree-level correlation functions of the reduced spatial metric $χ_{ij}$ remain constant outside the cosmological horizon during local thermal equilibrium, provided there are no non-zero conserved quantum numbers. By extending Weinberg's (2009) classical adiabatic solution with quantum field theory techniques, it proves that time dependence in correlation functions vanishes in the tree approximation, ensuring consistency between inflationary predictions and late-time observations.

ABSTRACT

Making a connection between observations of cosmological correlation functions and those calculated from theories of the early universe requires that these quantities are conserved through the periods of the universe which we do not understand. In this paper, the results of [0810.2831] are extended to show that tree-approximation correlation functions of Heisenberg picture operators for the reduced spatial metric are constant outside the horizon during local thermal equilibrium with no non-zero conserved quantum numbers.

Motivation & Objective

• To establish that cosmological correlation functions derived from inflationary theories remain unchanged after the universe enters a phase of local thermal equilibrium.
• To resolve the challenge of time evolution in correlation functions across unknown early-universe phases such as dark matter decoupling and baryogenesis.
• To show that tree-level correlation functions of the Heisenberg picture operators for the reduced spatial metric $χ_{ij}$ are time-independent far outside the horizon.
• To provide a rigorous quantum field theory foundation for the classical intuition that late-time correlation functions should match those from inflation.
• To justify using tree-level calculations in inflationary models for comparison with observational data, even without full knowledge of post-inflationary dynamics.

Proposed method

• Adopts the ADM formalism for the spacetime metric, decomposing it into spatial and temporal components with $g_{ij} = a^2 \tilde{g}_{ij}$, where $\tilde{g}_{ij}$ is the reduced spatial metric.
• Applies the results of Weinberg (2009) on adiabatic solutions to the non-linear Einstein equations, showing that $\tilde{g}_{ij}$ approaches a time-independent function $\mathcal{G}_{ij}(\mathbf{x})$ outside the horizon.
• Uses the Heisenberg picture formalism to define quantum operators for $\tilde{g}_{ij}$, ensuring unitary time evolution in the presence of interactions.
• Analyzes the generating functional $W_{\text{tree}}[J,t_1]$ in the tree approximation, showing that its time dependence vanishes asymptotically as $a(t_1) \to \infty$.
• Evaluates functional derivatives of the Lagrangian and constraints, demonstrating that the leading imaginary parts of the momentum conjugate to $\tilde{g}_{ij}$ are $O(a^{-3})$, ensuring convergence of time integrals.
• Establishes that the constraint equations and correlation functions become $t_1$-independent in the large-$a(t_1)$ limit, proving constancy of correlation functions.

Experimental results

Research questions

• RQ1 Can tree-level correlation functions of the reduced spatial metric $\tilde{g}_{ij}$ remain constant outside the horizon during local thermal equilibrium?
• RQ2 Does the absence of non-zero conserved quantum numbers ensure that correlation functions do not evolve over time in the late-time universe?
• RQ3 To what extent can inflationary predictions be reliably compared with observations if the universe passes through unknown phases after inflation?
• RQ4 How does the tree approximation in quantum field theory on curved spacetime ensure time-independence of correlation functions despite non-linear interactions?
• RQ5 What role does the adiabatic solution of the Einstein equations play in stabilizing correlation functions in the long-wavelength limit?

Key findings

• Tree-approximation correlation functions of the reduced spatial metric $\tilde{g}_{ij}$ are constant outside the horizon during local thermal equilibrium with no non-zero conserved quantum numbers.
• The time integral in the generating functional $W_{\text{tree}}[J,t_1]$ converges to a finite, $t_1$-independent limit as $a(t_1) \to \infty$, due to the $O(a^{-2})$ decay of relevant terms.
• The functional derivative of the Lagrangian with respect to $\dot{\tilde{g}}_{ij}$ has a leading imaginary part of order $a^{-3}$, ensuring that the constraint equations become time-independent in the large-$a$ limit.
• The leading-order time dependence in the correlation functions vanishes because the momentum conjugate to $\tilde{g}_{ij}$ contains only $O(a^{-3})$ imaginary contributions, which suppress time evolution.
• The constancy of correlation functions is established without assuming classicality, relying instead on explicit quantum field theory and asymptotic analysis in the tree approximation.
• This result justifies using tree-level inflationary predictions for comparison with late-time observations such as the CMB power spectrum, even when intermediate phases like baryogenesis or dark matter decoupling are not fully understood.