Breakdown of the fixed-background approximation by pump depletion and self-fragmentation: the observation window of a parametric analogue system

Analogue models of gravity describe linear excitations on a classical background solved first and held fixed. We ask when this fixed-background approximation breaks down for a dynamical pump field: a self-interacting scalar oscillating about its vacuum and coupled quadratically to a signal scalar, with the coupling g as the only free parameter. The approximation fails in two ways, by the depletion of the pump through the pairs it creates and by the self-fragmentation of the pump through its own nonlinearity, and we give both times in closed form. The vacuum-seed depletion clock μt_dep=½ln(32R), with R the number of pump quanta per resonant mode and μ the growth rate, carries a shot-to-shot jitter μΔt_dep=√[ψ′(N_det)]/2 fixed by the number of detected modes alone and independent of the seed temperature; coupled to the damping of the signal, the depletion becomes a rate, the damping the output port, and the energy does not return to the pump. The self-fragmentation sets a second edge, which crosses the depletion edge at the pump strength Q_c=0.0396 and meets the observation requirement at the edge of the operating range. The result is an observation window 1≲G<min[½ln(32R),G_frag] on the amplification exponent, set by the backreaction and not by thermal noise until the thermal detuning exceeds the resonance band. Four methods determine g, two of them independent, with an error budget and three consistency tests; a conversion table moves the window between dynamical, replenished and fixed pumps and places representative experiments in it. Preprint of the manuscript submitted to The European Physical Journal Plus on 2 November 2026; not peer reviewed. Version 2 (3 November 2026): the Note added at the end of Sec. 8 cites K. Chinni and N. Quesada, Phys. Rev. A 110, 013712 (2024), in which the closed-form time of the pump depletion of the zero-dimensional model appeared, withdraws the corresponding statements in Secs. 1.1 and 3.1, and states the relation to W. Xing and T. C. Ralph, arXiv:2610.39387. Version 3 (3 November 2026): the Note added now also cites earlier work in which the same closure and the leading logarithm of the depletion time appeared for a field pump, the stimulated decay of axion condensates into photon pairs (R. F. Sawyer, arXiv:1809.01183 and arXiv:1908.04298; P. Carenza, A. Mirizzi and G. Sigl, Phys. Rev. D 101, 103016 (2020)), estimates of the duration of classicality of degenerate scalar fields (Phys. Rev. D 97, 043531 (2018); Phys. Rev. D 100, 125010 (2019)), and work on the breakdown of mean-field theory near a dynamical instability (X. Han and B. Wu, Phys. Rev. A 93, 023621 (2016); A. Vardi and J. R. Anglin, Phys. Rev. Lett. 86, 568 (2001)); it restricts the statement of what the paper adds accordingly. References [58]–[64] are added; the text is otherwise that of the submitted manuscript. Online Resource 1 is unchanged. Code and data: https://doi.org/10.5281/zenodo.23847961.