Attractivity, collisionless anisotropic inertia, and quantum-loop stability of adiabatic superhorizon solutions: the open core of Weinberg's general-case program

Weinberg (Phys. Rev. D 79, 043504 (2009)) proved that, under essentially all conditions, the nonlinear classical Einstein-matter equations admit 'adiabatic' solutions in which, far outside the horizon and in a suitable gauge, the reduced spatial metric $g_{ij}({\bf x},t)/a^2(t)$ freezes to a time-independent function ${\cal G}_{ij}({\bf x})$ while all other metric and matter perturbations vanish up to $O(a^{-2})$ corrections whose spatial dependence he gave explicitly; he then showed, by arguments linearized about the adiabatic solution, that these solutions are attractive for single-field inflation and for local thermal equilibrium with no non-zero conserved quantities. The paper itself flags three unresolved gaps, and they remain open in essentially their original generality: (1) the nonlinear basin-of-attraction problem — no proof that fluctuations of arbitrary amplitude are drawn to the adiabatic solutions even in the two treated cases, and no characterization of the matter contents for which attractivity holds (blanket attractivity for arbitrary matter is known to fail, e.g. persistent multi-field isocurvature and ultra-slow-roll second modes); (2) the construction of adiabatic solutions retaining the $O(a^{-2})$ anisotropic inertia (traceless stress) that collisionless relativistic particles generate by free streaming, which Weinberg's explicit solution had to assume away; (3) whether quantum loop corrections preserve the constancy and decay properties outside the horizon beyond the single-clock cases where soft-symmetry arguments now protect the curvature perturbation. Closing these gaps would complete the justification for using observed non-Gaussian correlations as tests of early-universe physics through epochs of unknown matter content.

Background

Inflationary cosmology seeds primordial fluctuations, and the observable targets are correlation functions of the curvature perturbation $\zeta$ (the fractional local expansion of spatial slices chosen so that the energy density is unperturbed; outside the horizon it reduces to a local difference in expansion history). A wavelength is said to be outside the horizon (superhorizon) when its comoving wavenumber satisfies $k/(aH)\ll 1$, with $a(t)$ the Robertson-Walker scale factor and $H=\dot a/a$ the Hubble rate. During the long superhorizon era the universe passes through reheating and other transitions whose microscopic constituents are unknown, so for observed correlations to test the inflationary epoch their superhorizon evolution must be independent of that unknown microphysics. For single-field (single-clock) inflation this independence is the basis of all non-Gaussianity forecasts.

The main tool is the separate-universe idea and its gradient-expansion realization, introduced by Salopek and Bond (Phys. Rev. D 42, 3936 (1990)): on scales much longer than the Hubble radius, each comoving patch evolves like an independent homogeneous universe, so spatial gradients can be counted as powers of the small parameter $\epsilon\sim 1/(aHL)$; the $\delta N$ formalism computes $\zeta$, at all orders in perturbation amplitude but leading order in $\epsilon$, as the perturbation to the local number of e-folds. Lyth, Malik and Sasaki (JCAP 05 (2005) 004) gave a general nonlinear conservation proof of $\zeta$ at this order, and Tanaka and Sasaki (Prog. Theor. Phys. 118, 455 (2007)) extended the gradient expansion through $O(\epsilon^2)$ for a single scalar field. At the purely linear level, Weinberg's 'Adiabatic modes in cosmology' (Phys. Rev. D 67, 123504 (2003)) had shown that the perturbation equations have adiabatic solutions, with a constant quantity ${\cal R}$, for arbitrary constituents of the universe — although the universal superhorizon formulas require the absence of anisotropic inertia (the traceless part of the stress perturbation).

Weinberg's two-part series 'Non-Gaussian Correlations Outside the Horizon' (Phys. Rev. D 78, 123521 (2008); Phys. Rev. D 79, 043504 (2009), the source paper of this problem) generalized this to the fully nonlinear classical theory. In a suitable gauge, the exact nonlinear field equations always have 'adiabatic' solutions in which, far outside the horizon, the reduced spatial metric $g_{ij}({\bf x},t)/a^2(t)$ tends to a time-independent but otherwise arbitrary positive matrix ${\cal G}_{ij}({\bf x})$, all perturbations of $g_{00}$, $g_{i0}$, and the matter variables vanish, and the corrections are of order $a^{-2}$ with explicit spatial dependence built from ${\cal G}_{ij}$, its Ricci tensor and curvature scalar. Since ${\cal G}_{ij}({\bf x})$ is an arbitrary function, the solution is non-Gaussian in the initial conditions, and its existence for any matter content is what makes superhorizon non-Gaussian correlations insensitive to the unknown post-inflationary constituents. The general-case paper extended the explicit multi-scalar solution of Part I to any theory whose anisotropic inertia vanishes at order $a^{-2}$, and showed — by analyzing the equations linearized about the adiabatic solution — that the adiabatic solutions are attractive (decaying perturbations) in two cases: single-field slow-roll (attractor) inflation, and a sufficiently long period of local thermal equilibrium with no non-zero conserved quantities. Three caveats were left explicit in the paper itself, and they define the problem stated here.

Problem Statement

Complete, or refute, Weinberg's program establishing that non-Gaussian correlations of superhorizon perturbations are predictable through epochs of unknown matter content. The source paper leaves three linked questions open, all at the level of the nonlinear classical (and, for the third, quantum) Einstein-matter equations:

(1) Basin of attraction. The attractivity proofs given for single-field slow-roll (attractor) inflation and for local thermal equilibrium with no non-zero conserved quantities are obtained from equations linearized about the adiabatic solution. Prove, or disprove, that these adiabatic solutions are nonlinear attractors — that superhorizon initial fluctuations of arbitrary amplitude lie in their basin of attraction — in these two cases (the first taken in the slow-roll, attractor regime, so that ultra-slow-roll regions are excluded from this two-case demand and are treated only in the general-matter characterization below); and determine, for general matter content, the necessary and sufficient conditions — at the same nonlinear, arbitrary-amplitude level — under which the adiabatic solutions are (or are not) attractors, incorporating the known obstructing mechanisms: persistent superhorizon isocurvature perturbations (fluctuations in the composition of the matter at fixed total energy density, which do not decay in multi-field systems), and non-decaying second modes such as the one appearing in ultra-slow-roll inflation (a phase with a nearly flat potential in which slow-roll attractivity is lost and a second, non-decaying solution of the superhorizon perturbation equations exists).

(2) Collisionless anisotropic inertia. Construct the adiabatic solutions of the nonlinear field equations for matter that generates anisotropic inertia $\Pi_{ij}$ of order $a^{-2}$ — as collisionless relativistic particles do by free streaming — retaining $\Pi_{ij}$ at that order in the gradient expansion (i.e. the Einstein equations coupled to the collisionless Boltzmann equation for the particle distribution function, also called the Einstein-Vlasov system, at next-to-leading order in the gradient expansion), a setting distinct from anisotropic backgrounds sourced by large-scale shear or by vector fields: establish existence, the explicit form of the $O(a^{-2})$ corrections to the reduced spatial metric and the other dynamical variables, and whether attractivity survives; or show that the adiabatic solutions fail to exist or cease to be attractive in this case, and what this implies for superhorizon non-Gaussian correlations.

(3) Quantum-loop stability. Determine whether quantum loop corrections preserve the constancy of the adiabatic solutions and the decay of non-adiabatic superhorizon corrections for general matter content — removing the 'as long as graphs with loops can be neglected' caveat beyond the single-clock cases where symmetry arguments already protect $\zeta$ — including models with isocurvature, dissipative, or collisionless components.

The common objective of the three parts is a single result: a complete statement of when the superhorizon correlations generated by inflation are conserved through all subsequent epochs regardless of their unknown constituents, so that observed non-Gaussianity (or its absence) can be used as a test of the early universe. Each part addresses a distinct assumption (attractivity scope, vanishing anisotropic inertia, tree-level approximation) of the same theorem-scale claim; partial answers that close only one part, or only a special case of one part, do not close the problem as stated.

References

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