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Short distance and initial state effects in inflation: stress tensor and decoherence

Paul R. Anderson, Carmen Molina-Paris, Emil Mottola

TL;DR

The authors present a consistent low-energy EFT approach to parameterize short-distance and initial-state physics during inflation, focusing on general homogeneous and isotropic RW initial states for a free scalar field. They combine covariant renormalization with a Gaussian density-matrix (Schrödinger) description, and identify UV-allowed states as those whose high-$k$ behavior matches the fourth-order adiabatic vacuum, ensuring finite $igra T_{ab}igra_R$ and unitary evolution via $H_{ m eff}$. Key findings include that the Bunch-Davies state is the unique UV-allowed de Sitter invariant vacuum among α-states, while generic α-states and boundary-action modifications can induce observable power-spectrum changes but are constrained by backreaction through well-defined bounds; they also develop an adiabatic particle-number framework and a renormalized decoherence functional, showing decoherence is absent in massless conformally coupled and in massless minimally coupled de Sitter cases. Together, the work provides a robust, predictive framework linking early-universe short-distance physics to CMB signatures while preserving the consistency of semiclassical gravity.

Abstract

We present a consistent low energy effective field theory framework for parameterizing the effects of novel short distance physics in inflation, and their possible observational signatures in the Cosmic Microwave Background. We consider the class of general homogeneous, isotropic initial states for quantum scalar fields in Robertson-Walker (RW) spacetimes, subject to the requirement that their ultraviolet behavior be consistent with renormalizability of the covariantly conserved stress tensor which couples to gravity. In the functional Schrödinger picture such states are coherent, squeezed, mixed states characterized by a Gaussian density matrix. This Gaussian has parameters which approach those of the adiabatic vacuum at large wave number, and evolve in time according to an effective classical Hamiltonian. The one complex parameter family of $α$ squeezed states in de Sitter spacetime does not fall into this UV allowed class, except for the special value of the parameter corresponding to the Bunch-Davies state. We determine the finite contributions to the inflationary power spectrum and stress tensor expectation value of general UV allowed adiabatic states, and obtain quantitative limits on the observability and backreaction effects of some recently proposed models of short distance modifications of the initial state of inflation. For all UV allowed states, the second order adiabatic basis provides a good description of particles created in the expanding RW universe. Due to the absence of particle creation for the massless, minimally coupled scalar field in de Sitter space, there is no phase decoherence in the simplest free field inflationary models. We apply adiabatic regularization to the renormalization of the decoherence functional in cosmology to corroborate this result.

Short distance and initial state effects in inflation: stress tensor and decoherence

TL;DR

The authors present a consistent low-energy EFT approach to parameterize short-distance and initial-state physics during inflation, focusing on general homogeneous and isotropic RW initial states for a free scalar field. They combine covariant renormalization with a Gaussian density-matrix (Schrödinger) description, and identify UV-allowed states as those whose high- behavior matches the fourth-order adiabatic vacuum, ensuring finite and unitary evolution via . Key findings include that the Bunch-Davies state is the unique UV-allowed de Sitter invariant vacuum among α-states, while generic α-states and boundary-action modifications can induce observable power-spectrum changes but are constrained by backreaction through well-defined bounds; they also develop an adiabatic particle-number framework and a renormalized decoherence functional, showing decoherence is absent in massless conformally coupled and in massless minimally coupled de Sitter cases. Together, the work provides a robust, predictive framework linking early-universe short-distance physics to CMB signatures while preserving the consistency of semiclassical gravity.

Abstract

We present a consistent low energy effective field theory framework for parameterizing the effects of novel short distance physics in inflation, and their possible observational signatures in the Cosmic Microwave Background. We consider the class of general homogeneous, isotropic initial states for quantum scalar fields in Robertson-Walker (RW) spacetimes, subject to the requirement that their ultraviolet behavior be consistent with renormalizability of the covariantly conserved stress tensor which couples to gravity. In the functional Schrödinger picture such states are coherent, squeezed, mixed states characterized by a Gaussian density matrix. This Gaussian has parameters which approach those of the adiabatic vacuum at large wave number, and evolve in time according to an effective classical Hamiltonian. The one complex parameter family of squeezed states in de Sitter spacetime does not fall into this UV allowed class, except for the special value of the parameter corresponding to the Bunch-Davies state. We determine the finite contributions to the inflationary power spectrum and stress tensor expectation value of general UV allowed adiabatic states, and obtain quantitative limits on the observability and backreaction effects of some recently proposed models of short distance modifications of the initial state of inflation. For all UV allowed states, the second order adiabatic basis provides a good description of particles created in the expanding RW universe. Due to the absence of particle creation for the massless, minimally coupled scalar field in de Sitter space, there is no phase decoherence in the simplest free field inflationary models. We apply adiabatic regularization to the renormalization of the decoherence functional in cosmology to corroborate this result.

Paper Structure

This paper contains 14 sections, 220 equations, 2 figures.

Table of Contents

  1. Introduction and Overview
  2. General RW initial states
  3. Density Matrix and Hamiltonian
  4. Energy-Momentum Tensor of UV Allowed RW States
  5. Short Distance Effects in Inflation
  6. de Sitter Invariant $\alpha$ States
  7. Adiabatic Order Zero States
  8. Boundary action approach
  9. Adiabatic Particle Creation and Dephasing
  10. Decoherence
  11. Summary and Conclusions
  12. Equations for $\zeta_k$ and $\pi_k$ and their effective Hamiltonian
  13. Evaluation of Integrals
  14. Comparison of Squeezing in Different Bases

Figures (2)

  • Figure 1: The oscillatory function $F^{(p)}$ for $p=4$ defined by Eq. (\ref{['eq:Fp']}), as a function of $x=k_M(\eta-\eta_0)$.
  • Figure 2: The imaginary part of the one-loop vacuum polarization given by Eqs. (\ref{['eq:secvar']}) and (\ref{['eq:ImPi']}), which enters the decoherence functional at second order in the metric variation, $\delta g_{ab}$ (represented by the wavy lines). The shaded part of the diagram represents the complex conjugation of the unshaded part, and the resulting squared amplitude is proportional to the probability for the creation of a scalar particle/anti-particle pair, represented by the diagonal cut through the diagram.