During inflation, cosmological perturbations are expected to undergo decoherence through interactions with unobserved degrees of freedom, and the quantum-to-classical transition is normally tracked through the purity or linear entropy of the reduced state of the observable mode. Bhattacharyya, Brahma, Haque, Lund and Paul (JHEP 05 (2024) 058) proposed a complementary diagnostic: in the curved-space Caldeira-Leggett two-field model in de Sitter space, the complexity of purification (COP) of the light field develops a bump, i.e. an abrupt change in its growth rate, whose timescale correlates with the saturation of the linear entropy, and they conjectured this growth-rate change to be a generic complexity signal of decoherence. Whether this signature survives in non-Gaussian, interacting inflationary models, exactly the regime where entropy-based methods become impractical, and how the signature maps onto realistic inflationary parameters and survives post-inflationary evolution (mode re-entry and reheating) remain open.

Background

Inflation, a phase of accelerated expansion of the very early universe, is the standard explanation for the origin of cosmic structure: quantum fluctuations of the inflaton and of the metric are stretched to cosmological scales, and after decoherence they are treated as classical stochastic perturbations. Decoherence here means the loss of quantum coherence of the observable (system) degrees of freedom due to interaction with unobserved environmental degrees of freedom, which is why the early universe is studied as an open quantum system within an open effective field theory framework. Candidate environments include additional heavy fields, UV modes beyond the cutoff, and gravitational nonlinearities.

A standard toy model is the Caldeira-Leggett model of a system oscillator bilinearly coupled to a bath of oscillators. Its curved-space version, used in the source paper, contains two minimally coupled scalar fields in a de Sitter background (the approximately constant-Hubble-rate HH spacetime describing inflation to leading order): a light field φ\varphi (mass mm, taken to vanish, modeling the adiabatic perturbation, i.e. the fluctuation along the inflaton's slow-roll trajectory) and a heavy field χ\chi (mass M>3H/2M>3H/2, modeling an entropic, transverse direction), with bilinear interaction λ2φχ\lambda^{2}\varphi\chi of coupling strength λ\lambda. Because the interaction is quadratic, each Fourier mode k\mathbf{k} of φ\varphi couples only to mode −k-\mathbf{k} of χ\chi, and the Gaussian state of each pair is exactly solvable through its covariance matrix. Starting from the Bunch-Davies vacuum (the de Sitter-invariant free-field vacuum), super-horizon modes (wavenumber k<aHk<aH, where aa is the scale factor) get squeezed, i.e. their quantum state becomes a two-mode squeezed state with exponentially growing squeezing parameter; this squeezing is the origin of the classical stochastic appearance of the perturbations.

The usual diagnostics of decoherence are the purity γ=Tr ρ2\gamma=\mathrm{Tr}\,\rho^{2} of the reduced density matrix of the system mode, the linear entropy Sl=1−γS_{l}=1-\gamma, and the entanglement negativity (a computable measure of entanglement based on the eigenvalues of the partially transposed density matrix). A separate phenomenon, recoherence, was found by Colas, Grain and Vennin in a related Gaussian two-field inflationary model: if the interaction is dynamically quenched by the expansion, the adiabatic mode can become pure again after a transient period of mixedness. This raises the question, emphasized in the source paper, of whether states decohered during inflation remain mixed after modes re-enter the horizon in the post-inflationary universe.

Circuit complexity quantifies the difficulty of preparing a target quantum state from a reference state using a set of elementary gates; in the Nielsen approach it is computed as the length of the shortest geodesic with respect to a chosen cost function on the space of states or gates. For mixed states, one can use the complexity of purification (COP): purify the mixed state by tensoring with an ancilla, minimize the complexity of the pure purified state over the free parameters of the purification, and take the minimum as the complexity of the mixed state. For Gaussian states this is tractable: the purification is a Gaussian ansatz with a free parameter (an imaginary part Im(β)\mathrm{Im}(\beta) of a rotation angle), which the source paper treats numerically. Applying complexity to cosmology, Bhattacharyya, Das, Haque and Underwood showed that the complexity of de Sitter perturbations grows without bound as the squeezing increases, so complexity saturation cannot be used as a decoherence signal in an accelerating background, unlike in flat space. In the flat-space Caldeira-Leggett model, Bhattacharyya, Hanif, Haque and Paul showed that the saturation timescale of COP coincides with the timescale at which the state becomes maximally mixed. The source paper combined these threads: in the curved-space Caldeira-Leggett model in de Sitter, although COP grows without bound, its rate of growth changes abruptly at a timescale abumpa_{\mathrm{bump}} (a bump in the COP curve), and this timescale was found numerically to track the saturation of the linear entropy in both coupling regimes λ2<3MH/2\lambda^{2}<3MH/2 and λ2>3MH/2\lambda^{2}>3MH/2, leading to the conjecture that a growth-rate change in complexity is a signal of decoherence discernible from the system's complexity alone. A later refinement by the same authors, the complexity of formation (COF, the COP of the interacting theory minus that of the free theory), subtracts the universal squeezing-driven growth to expose interaction-driven features.

The gap is that all of this rests on Gaussian, exactly solvable models with a single environment field and quadratic coupling, whereas realistic inflationary dynamics, and general relativity itself, are inherently nonlinear: cubic and higher-order couplings mix modes, and multiple environment fields may act simultaneously. For the non-Gaussian (cubic and higher-order) interactions the reduced state is not characterized by a covariance matrix, both COP and the linear entropy are hard to compute, and it is unknown whether the complexity-based signature survives at all. The source paper also reports only numerical trends for how abumpa_{\mathrm{bump}} depends on the coupling, without an analytic mechanism, and notes that the slopes of linear complexity growth may be related to Lyapunov exponents (the exponential rates quantifying sensitivity to initial conditions in chaotic dynamics) without developing the connection.

Problem Statement

The problem is to establish whether the complexity growth-rate-change signature of decoherence found in the Gaussian curved-space Caldeira-Leggett model is a robust and physically grounded diagnostic of the quantum-to-classical transition of inflationary perturbations. It comprises two linked aspects of this single objective:

  1. Robustness beyond Gaussian models. Define or adapt a complexity-of-purification-type measure applicable to the mixed state of the observable mode in non-Gaussian, interacting inflationary open systems (cubic and higher-order mode-mixing couplings between the adiabatic sector and the environment field(s), where the covariance-matrix description fails), and/or in models with multiple environment fields (which may remain Gaussian), and determine whether a change in the growth rate of the complexity of the observable mode still occurs at the decoherence timescale, as diagnosed independently (for instance by linear entropy, purity, or entanglement negativity wherever those remain computable).

  2. Physical grounding. Determine for which physically realistic inflationary parameter ranges the signature is meaningful, and what becomes of it under post-inflationary evolution, i.e. horizon re-entry and reheating, in particular whether states decohered during inflation remain mixed or can recohere.

A complete answer addresses both aspects, since each supports the same claim from a different side (robustness and physical relevance), and thereby validates or refutes the growth-rate signature as a decoherence diagnostic for the early universe. Because no canonical complexity definition exists for mixed, non-Gaussian states, the verdict attaches to the generalized measure and the models actually studied: a positive result establishes the signature for that measure and those models, while its failure in one adapted measure or model is a concrete limitation of the diagnostic in that scope rather than a refutation of every possible complexity-based signature. The source paper additionally acknowledges further ancillary directions that are recorded here as secondary and are not required for closure: an analytic understanding of the bump timescale abump(λ,M,H,k)a_{\mathrm{bump}}(\lambda,M,H,k) and its scaling, which the paper reports only through numerical trends (its Fig. 5 shows abumpa_{\mathrm{bump}} versus λ\lambda); the conjectured interpretation of the slopes of the linear complexity growth in terms of Lyapunov exponents, which the authors announce as a forthcoming study; whether the de Sitter COP behavior and its decoherence interpretation admit a precise holographic dual in the sense of de Sitter holographic complexity conjectures; and whether the numerical minimization of the purification over Im(β)\mathrm{Im}(\beta) can be replaced by an efficient or analytic strategy, including extensions to multi-mode purifications.

References

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