[Paper Review] Eternal Inflation is "Expensive"

Laura Mersini–Houghton, Liam Parker|arXiv (Cornell University)|May 2, 2007

Cosmology and Gravitation Theories3 citations

TL;DR

This paper argues that eternal inflation is highly 'expensive' in energy due to adiabatic regularization suppressing quantum fluctuations, requiring potential energies near or above the transplanckian scale (V ≥ 0.1Mₚ⁴) to sustain eternal inflation. This suppression narrows the Gaussian distribution of field fluctuations, making large enough quantum jumps to trigger new inflationary regions extremely unlikely at subplanckian energy scales, and further constrained by stringent requirements on the equation of state w ≪ 10⁻⁸.

ABSTRACT

The discovery of the string theory landscape has recently brought attention to the eternal nature of inflation. In contrast to the common belief that eternal inflation may be a generic feature of most inflationary models, in this note we argue that the suppressed amplitude of perturbations due to adiabatic regularization, together with a fine-tuning constraint on the equation of state of the rare inflating pockets with large fluctuations, render eternal inflation expensive in energy and may make it unlikely to occur. The energy scales of the eternally inflating pockets have to be very close to the transplanckian regime in order to compensate for the suppression of regularized perturbations.

Motivation & Objective

• To re-evaluate the likelihood of eternal inflation under the influence of adiabatic regularization, which suppresses the amplitude of inflaton fluctuations.
• To assess whether the standard assumption of generic eternal inflation holds when regularization effects are included.
• To determine the energy scale and field dynamics required for eternal inflation to occur under these corrected fluctuation amplitudes.
• To examine whether the stringent equation of state constraint (w ≪ 10⁻⁸) on inflating pockets further undermines the feasibility of eternal inflation.

Proposed method

• Apply adiabatic regularization to the inflaton fluctuation spectrum, replacing the conventional H/2π estimate with a modified expression involving Hankel functions and mass-dependent suppression.
• Use the regularized fluctuation amplitude δϕ_q ≈ 0.1m (where m² = V''(ϕ)) as the standard deviation of the Gaussian field distribution, instead of H/2π.
• Re-derive the criterion for eternal inflation using the regularized amplitude, leading to the condition m²H² / |ḋϕ|² ≥ 37 instead of the conventional H² / |ḋϕ| ≥ 3.8.
• Incorporate the flatness condition ΔV / Δϕ⁴ ≪ 10⁻⁷, which constrains the curvature of the inflaton potential and implies fine-tuning of m² relative to H.
• Derive a bound on the equation of state w using energy conservation and the requirement that ΔV / Δϕ⁴ ≪ 10⁻⁷, leading to w ≪ 10⁻⁸.
• Analyze a quartic potential V(ϕ) = ¼λϕ⁴ as a concrete example to quantify the required energy scale and field values for eternal inflation under regularization.

Experimental results

Research questions

• RQ1 How does adiabatic regularization alter the amplitude of quantum fluctuations in inflationary models?
• RQ2 What energy scale is required for eternal inflation to occur when fluctuations are suppressed by regularization?
• RQ3 Can the rare large fluctuations needed to trigger eternal inflation still satisfy the equation of state constraint w ≪ 10⁻⁸?
• RQ4 Is eternal inflation still a generic feature of inflationary models when corrected for regularization effects?
• RQ5 How does the requirement for large field values (ϕ > 750Mₚ) in the quartic model affect the feasibility of eternal inflation?

Key findings

• Adiabatic regularization suppresses the amplitude of inflaton fluctuations to δϕ_q ≈ 0.1m, drastically reducing the standard deviation of the Gaussian field distribution compared to the conventional H/2π estimate.
• The criterion for eternal inflation now requires m²H² / |ḋϕ|² ≥ 37, which translates to a potential energy V(ϕ) ≥ 0.1Mₚ⁴—four orders of magnitude higher than in the conventional treatment.
• For a quartic potential V(ϕ) = ¼λϕ⁴, the condition V(ϕ) ≥ 0.1Mₚ⁴ implies ϕ ≥ 0.75Mₚλ⁻¹/⁴, and with λ < 10⁻¹², this requires ϕ > 750Mₚ, far exceeding typical subplanckian scales.
• The requirement for large fluctuations to trigger eternal inflation leads to a stringent constraint on the equation of state: w ≪ 10⁻⁸, meaning the energy density must behave nearly like a cosmological constant.
• Even if large fluctuations occur, the energy density and pressure terms involving ȧ and ḡϕ may not be small enough to satisfy the w ≪ 10⁻⁸ condition, further undermining the feasibility of eternal inflation.
• The combination of suppressed fluctuations and extreme equation of state constraints makes eternal inflation energetically prohibitive and unlikely to occur generically at subplanckian energy scales.