Scalar quantum fields in cosmologies with 2+1 spacetime dimensions

Abstract

Motivated by the possibility to use Bose-Einstein condensates as quantum simulators for spacetime curvature, we study a massless relativistic scalar quantum field in spatially curved Friedmann-Lemaître-Robertson-Walker universes with d = 2 + 1 spacetime dimensions. In particular, we investigate particle production caused by a time-dependent background geometry, by means of the spectrum of fluctuations and several two-point field correlation functions. We derive new analytical results for several expansion scenarios.

Curved and expanding spacetime geometries in Bose-Einstein condensates

“…wherein P m il−1/2 (cosh σ) are conical functions corresponding to analytically continued Legendre functions. The functions in equations (68), (69) and (70) are normalized with respect to a scalar product as discussed in detail in [43].…”

“…for R ∼ 10 −5 m, h(k) turns out to be practically independent of the spatial curvature κ for k/ √ n0 0.1. The mode functions are further constrained as a result of the canonical commutation relation (72), the orthonormality properties of the functions H km (u, ϕ) (see [43]), and the bosonic commutation relations fulfilled by the creation and annihilation operators (76), leading to a normalization condition in terms of the Wronskian,…”

“…In this sense, we extend on earlier considerations by providing an exact mapping to a more general class of cosmologies and by showing how the effect of particle production is accessible experimentally. Note that some technical details on the cosmology side are provided in our companion paper [43].…”

Quantum field simulator for dynamics in curved spacetime

“…The term 1/2 corresponds to vacuum fluctuations, the term |β k | 2 describes the populations of a mode k and the oscillating term contains the contributions from coherences. The Bogoliubov coefficients α k and β k , as well as the phase θ k = Arg(α k β k ), depend on the expansion history and are calculated by relating the quantum states before and after the expansion [32,48]. The initial distribution is taken to be of thermal form…”