Draft 1

Primary thermalisation mechanism of Early Universe observed from Faraday-wave scattering on liquid-liquid interfaces

Vitor S. Barroso
[E-mail<EMAIL_ADDRESS>School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
Nottingham Centre of Gravity, University of Nottingham, Nottingham, NG7 2RD, UK

August Geelmuyden
[E-mail<EMAIL_ADDRESS>School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
Nottingham Centre of Gravity, University of Nottingham, Nottingham, NG7 2RD, UK

Zack Fifer
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
Nottingham Centre of Gravity, University of Nottingham, Nottingham, NG7 2RD, UK

Sebastian Erne
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria

Anastasios Avgoustidis

School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

Nottingham Centre of Gravity, University of Nottingham, Nottingham, NG7 2RD, UK

Richard J. A. Hill

School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom

Silke Weinfurtner
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
Nottingham Centre of Gravity, University of Nottingham, Nottingham, NG7 2RD, UK

For the past two hundred years, parametric instabilities have been studied in various physical systems, such as fluids, mechanical devices and even inflationary cosmology. It was not until a few decades ago that this subharmonic unstable response arose as a central mechanism for the thermalisation of the Early Universe, in a theory known as preheating. Here we study a parametrically driven two-fluid interface to simulate the key aspects of inflationary preheating dynamics through the onset of nonlinear Faraday waves. We present a detailed analysis of the effective field theory description for interfacial waves through the factorization properties of higher-order correlations. Despite the intricacies of a damped and highly interacting hydrodynamical system, we show that the scattering of large amplitude Faraday waves is connected to a broadening of primary resonance bands and the subsequent appearance of secondary instabilities as predicted in preheating dynamics.

Parametric instabilities can be responsible for dramatic events, from the collapse of bridges <cit.> and rolling of ships at sea <cit.> to the thermalisation of our universe following cosmic inflation, 13.8 billion years ago <cit.>. In a leading theory for the thermalisation of the Early Universe,
known as preheating, broad parametric resonance efficiently transfers the energy of the inflaton field to other fields and particles, thus producing the hot plasma required for the Big Bang theory to proceed. However, direct observations of the non-linear dynamics of preheating in the early universe are not feasible. Here, we conduct a controlled experiment to simulate the key aspects of inflationary preheating in a parametrically driven interface between two fluids <cit.>. We study the scattering of large amplitude Faraday waves and observe a broadening of primary resonance bands and the subsequent appearance of secondary instabilities and their estimated growth rates <cit.>, as predicted in preheating. Adapting the statistical machinery from field theories, namely two-point functions and the factorisation properties of higher order correlators <cit.>, we show that the interfacial evolution is accurately captured by leading terms in an effective perturbative description. Our results demonstrate the robustness of preheating dynamics in a strongly interacting and damped system.

The study of instabilities on fluid surfaces caused by an external vibration dates back to $1831$, when Faraday first detailed the phenomenon in a cylindrical glass filled with water <cit.>. He noted that the unstable waves arising on the fluid surface oscillate with half of the external driving frequency. This sub-harmonic response characterises parametric resonance, which, since then, has been extensively examined and identified in a wide range of physical systems <cit.>.  In general, when subjected to a periodic forcing with a frequency of $\omega_\mathrm{d}$, the spectral response of these systems displays the so-called unstable resonance bands at specific frequencies <cit.>. For a common form of the coupling that we consider here, the unstable bands occur at integer multiples of $\omega_0\equiv\omega_\mathrm{d}/2$. Within such resonance bands, parametric amplification happens with a common exponential rate but is dominated by the primary instability at $\omega_0$. As amplitudes grow larger, non-linear effects come into play and limit the amplification.

The interface between two fluids undergoing a vertical oscillatory acceleration is no different to Faraday's original system and displays a similar unstable behaviour <cit.>. While two coupled sets of Navier-Stokes equations govern the fluids' motion, interfacial waves evolve according to an effective set of dynamical equations. Fifer et al. <cit.> showed that this emergent description is able to emulate the propagation of scalar fields in various cosmological scenarios. This is one of many systems, where small excitations in a fluid or superfluid, such as sound or interface waves, experience an effective spacetime geometry <cit.> provided by the fluid flow.
From an experimental standpoint, gravity simulators have been very successful in mimicking a variety of black hole (e.g. Hawking radiation <cit.>, superradiance <cit.> and ringdown <cit.>) and cosmological (e.g. Hubble friction <cit.> and excitations in  Friedmann-Roberston-Walker-type spacetimes <cit.>) processes in the lab. As such, gravity simulators open the possibility of probing fundamental processes of otherwise inaccessible physical systems. These simulators rely on the experimental study of a physically realisable “analogue system" that has, to some approximation, the same mathematical description as the fundamental phenomena of interest.

Experimental setup and interfacial dynamics.
(a): Schematic depiction of the driving setup. The fluid cell (blue rectangles) sits on a platform (in orange) supported by springs (in red) and guided by vertical rods with pneumatic air-bearings. The up-down red arrow illustrates the harmonic motion of the system. (b): A typical profile of the measured vertical acceleration $F(t)$ over time. The driver starts at $t=0$ and stops at $t=26.36$s. Damping in the spring-mass driving platform creates a transient amplitude change in $F(t)$, which takes about $4$ seconds to reach a constant value. (c): Inset of (a) portraying the geometry of the fluid cell with the experimentally reconstructed interface at the instance ($t=23.48$s) of maximal amplitude.
(d): Three rendered temporal snapshots of the experimentally observed interfacial waves. (e): Cross-section of the fluid cell in (c) at fixed azimuthal angle $\theta_0$ portraying the interfacial height $z=\xi(t,r,\theta_0)$ in comparison with the horizontal walls at $\pm h_0 = \pm 17.5mm$. When at rest, the surface is located at $z=0$. The dashed, vertical lines indicate the radii where the steep curvature of the menisci between the interface and the vertical walls at $r_1=20\mathrm{mm}$ and $r_2=40\mathrm{mm}$ cause the detection method to fail. (f): Instantaneous amplitudes $b_{m,\omega_0}$ in logarithmic scale of the azimuthal mode $m=4$, at the primary resonance frequency $\omega_0$, for $1500$ repetitions (light blue) at fixed radius $r_0=24\mathrm{mm}$. Out of the ensemble of repetitions, two qualitatively distinct runs are highlighted in bold with colored regions showing different parts of the dynamics: (dark green) detection method noise floor, (turquoise) log-linear unstable growth, (orange) transition from amplification-dominated to damping-dominated dynamics and (red) log-linear unstable decay. The difference in the two realisations is due to the randomness of the initial state.

In this context, the powerful machinery of Effective Field Theories (EFT) becomes extremely useful, allowing one to classify, and systematically study, non-linear correction terms in the effective descriptions of both the fundamental system and its experimental analogue. Generally, the EFT descriptions of the two systems are not identical, in that higher-order corrections differ. However, one often observes that the expected phenomenology still arises in the experimental analogue, thus demonstrating the universality and robustness of the physical phenomena under study. Moreover, this experimental approach allows one to tune the system's configurations and run the experiment repetitively, in effect scanning the parameter space of the EFT. The bridge between the experiment and the fundamental system is the statistical machinery of EFT, particularly, correlation functions. The latter can be readily computed in the effective description, and also accurately reconstructed from direct observations of the experimental analogue system, as demonstrated in pioneering work on ultra-cold atoms systems <cit.>.

Here, we investigate driven parametric instabilities on a liquid-liquid interface and observe their subsequent non-linear breakdown, which exhibits key features of preheating, the leading theory for the thermalisation of the early Universe. We employ methods introduced in <cit.> to develop and validate the EFT of interfacial mode-mode interactions in our system. Hence, the statistical machinery can provide a tool to extend the programme of gravity simulators from free to interacting fields, allowing us to explore intrinsically non-linear scenarios <cit.>, such as cosmic preheating <cit.>, experimentally.

We perform a series of automated and synchronised parametric resonance experiments on the interface of a biphasic solution of potassium carbonate, ethanol and water enclosed in an annular cylindrical cell (see Supplementary Information for details). A driving platform (Fig. <ref> (a)) oscillates the fluid cell vertically with measured acceleration $F(t)$ (Fig. <ref> (b)), whose frequency and amplitude are set to $\omega_{\mathrm{d}}/(2\pi)=  6.07 ~ \mathrm{Hz}$ and $F_0=0.352 g=3.45 ~ \mathrm{m}~\mathrm{s}^{-2}$, respectively. During each repetition, we observe the unstable evolution of the interfacial elevation $\xi$ reconstructed from the experiment (Fig. <ref> (c), see Supplementary Information). As depicted in Fig. <ref> (d), the resonant modes appear as azimuthal waves on the interface, i.e., fixed $m$ values in the decomposition $\xi(t,r,\theta)=\sum_m\xi_m(t,r)\exp(im\theta)$. Their radial profile and the cell's radial cross-section are shown in Fig. <ref> (e).

We observe that, throughout the system's evolution, the azimuthal mode $m=4$ is the dominant parametrically amplified wave, as seen by the four crests on the interface in Fig. <ref> (d).

The essential features of this mode's evolution around the primary resonance band at $\omega_0$ are captured by the time-dependent complex envelope $b_{m=4,\omega_0}$, defined from the frequency decomposition of the azimuthal mode, $\xi_m\propto \int d\omega [b_{m,\omega}e^{-i \omega t} + b_{-m,\omega}^*e^{i \omega t}]$. These instantaneous amplitudes are displayed in logarithmic scale in Fig. <ref> (f) for all experimental repetitions (in light blue), and two qualitatively distinct realisations are highlighted to stress distinguishable stages of the evolution.

Initially, we see the detection scheme's noise level (in dark green) overwhelming the signal until the instabilities grow out of it between $10$ and $17$ seconds. A clear log-linear trend (turquoise region) appears in all repetitions with consistent exponential growth rates throughout. Similarly, all runs are seen to decay exponentially (in red) after the driver is turned off at $26.3$ seconds. However, while some repetitions transition quickly from the amplifying region to the final damping stage (orange region), others saturate in amplitude before that. With the latter case, see upper curve in Fig. <ref> (f), we observe a bound to the growth of the unstable interfacial waves, which is inconsistent with a single-mode linear evolution and indicates that the energy continuously provided by the driver is being scattered into other modes in the system <cit.>. In the early universe, the driver effectively shuts off due to backreaction from produced particles, which drains energy out of the inflaton condensate and dampens the oscillations that had driven parametric resonance <cit.>.
In what follows, we establish an effective non-linear field theory for this interacting interfacial dynamics, and probe it experimentally by applying statistical measures to the ensemble of experimental repetitions.

From the hydrodynamical equations governing the bulk motion of each fluid layer, one can derive the linear dynamics <cit.> of the interfacial height $\xi$ in terms of its spatial eigenmodes $\xi_{mk}(t)$. The latter are coefficients of an expansion of $\xi(t,r,\theta)$ in orthogonal eigenfunctions $f_{mk}(r,\theta)$ of the 2D Laplacian in polar coordinates <cit.>, with eigenvalues $-k^2$. For a fixed azimuthal number $m$, the radial confinement ($r_1\le r\le r_2$) in Fig. <ref> (e) discretises the infinite spectrum of positive wavenumbers $k$, resulting in a reduced density of states available at the interface; see Supplementary Information for a discussion of the boundary conditions. In order to obtain the non-linear dynamics of one of these modes $\xi_{mk}$, we invoke the variational formulation described by Miles <cit.>, and derive the relevant interaction terms that contribute to this single-mode evolution. By fixing the azimuthal number $m$, the approximate driven nonlinear equation of motion of a single mode $k$ reads

\begin{equation}
\ddot{\xi}_{mk}+\left(2\gamma_k + \tilde{\gamma}_k[\xi]\right) \dot{\xi}_{mk}+\left(\omega_k^2(t)+\tilde{\delta}_k[\xi]\right)\xi_{mk}=\tilde{\eta}[\xi],\label{eq:stochasticDynamics}
\end{equation}

\begin{equation}
\omega_k^2(t) = \frac{(\rho_1-\rho_2)(g-F(t))+\sigma k^2}{\rho_1+\rho_2}k\tanh(kh_0), \label{eq:dispersion}
\end{equation}

and the amplitude-dependent nonlinear terms $\tilde{\gamma}_k$, $\tilde{\delta}_k$ and $\tilde{\eta}$ are derived in Supplementary Information. In Eq. (<ref>) the linear damping $\gamma_k$, which encompasses any viscous contributions, is a phenomenological addition to the predicted dynamics, as discussed in Kumar & Tuckerman <cit.>.

At first, we benchmark our proposed model of Eq. (<ref>) against the experimental results by simulating the single-mode evolution with the inclusion of self-interaction terms only. Due to its dominant unstable growth, the azimuthal mode $m=4$ reaches amplitudes at least one order of magnitude larger than the remaining modes and, thus, receives negligible contributions from them (see Supplementary Information). Thus, the approximate nonlinear terms are given by

\begin{align}
\tilde{\gamma}[\xi] &\approx -\gamma_k A_{k}\xi_{mk}^2k\tanh(kh_0), \\
\tilde{\delta}[\xi]&\approx \frac{1}{2} A_{k}(\dot{\xi}_{mk}^2-\omega_k^2(t)\xi_{mk}^2) k\tanh(kh_0),
\end{align}

with the numerical coefficient $A_{k}$ defined in Eq. (<ref>) of Supplementary Information for $p\equiv p'\equiv k'\equiv k$. By disregarding interactions with different modes, the source term $\tilde{\eta}[\xi]$ in Eq. (<ref>) only exhibits a stochastic noise term $\eta_k(t)$. This quantity accounts for the dynamics at microscopic scales, where the interface jitters due to the influence of random molecular (Brownian) motion <cit.> and environmental noise sources. Consequently, as each experimental repetition starts, those interfacial fluctuations set a stochastic initial state that later on evolves in a practically deterministic way. Linear evolution preserves the initial distribution for the parametrically amplified interfacial fluctuations. This picture is consistent with the experimental ensemble of amplitudes shown in Fig. <ref> (f) by the light blue lines, where we can see all repetitions growing uniformly until amplitude plateaus appear.

In Fig. <ref> (a), we display the ensemble distributions of the instantaneous amplitudes $b_{4,\omega_0}$ at the four stages of experimental repetitions illustrated in Fig. <ref> (f). As expected from white detection noise in the first region (at $7$ seconds), we observe normally distributed amplitudes, which then evolve to more intricate, non-Gaussian distributions at later times.
The average squared amplitude, $\langle|b_{4,\omega_0}|^2\rangle$, is shown in Fig. <ref> (b), for both the experimental (dark-green solid line) and simulated (red solid line) ensembles. This quantity confirms that our effective model accurately predicts the average amplitude of the dominant nonlinear unstable mode throughout the entire evolution.
To characterise the deviation from a featureless Gaussian and identify the onset of non-linearities, we employ a set of convenient measures of non-Gaussianity <cit.>, corresponding to equal-time correlators in field theories. These are defined in terms of higher-order statistical moments $\langle\cdot\rangle$ and cumulants $\langle\cdot\rangle_\mathrm{c}$ of the amplitudes $b_{m,\omega}$, see Supplementary Information, as follows

\begin{equation}
M^{(2n)}_{m,\omega}(t,r_0) \equiv \frac{
\langle \left( b_{m,\omega}^* b_{m,\omega}\right)^n \rangle_\mathrm{c}
\langle \left( b_{m,\omega}^* b_{m,\omega}\right)^n \rangle}~.
\label{eq:Gamma_correlation_def_appendix}
\end{equation}

For a normally distributed classical ensemble, the numerator vanishes at all orders of $2n$ greater than $2$, and, hence, the quantity vanishes entirely <cit.>. The denominator of the above equation is commonly referred to as the full correlation function, while the numerator is its connected part. The latter vanishes for non-interacting fields for $n>1$, and it is the fundamental quantity for computing particle scattering and decay processes <cit.>. In Fig. <ref> (c) and (d), solid dark-green lines display the non-Gaussianity measures $M^{(2n)}_{m,\omega}$ for the mode $m=4$, at the primary resonance frequency $\omega_0$, at even orders $n=2$ and $n=3$. As expected, the experimental ensemble results in nearly vanishing non-Gaussianity measures in the white noise-dominated region (before $8$ seconds). We recover this property in our model by introducing a Gaussian noise floor in our simulated ensemble, whose results are shown by the red curves in Fig. <ref>.

Ensemble properties of primary instability.
(a): Ensemble distributions of (the real part of) the amplitudes $b_{m,\omega_0}(t)$ of the $m=4$ mode at four different times (vertical dotted lines): the noise-floor (dark-green), the log-linear unstable growth (turqoise), the non-linear period (orange) and the decay (red). (b): The averaged squared amplitudes $\langle |b_{4,\omega_0}|^2\rangle$ for experiment (dark green) and simulation (red). (c) and (d) depict the statistical measures $M^{(2n)}_{m,\omega}$ for $n=2,3$ computed on both experimental data (dark green) and simulations (red), with bootstrapped bands of one standard deviation. Overall, we observe agreement between the self-interacting model and the experimental results.

As the signal of the repetitions leaves the background noise, all curves ramp up and reach a stable non-zero value, indicating an initial non-Gaussian distribution of interfacial waves. The constant value between $10$ and $21$ seconds shows the expected linear evolution of the ensemble at sufficiently small amplitudes. At around $22$ seconds, both measures experience abrupt changes, demonstrating a deviation from the linear distribution-preserving evolution of the ensemble. We see that, before the time indicated by the yellow dashed line in Fig. <ref>, the simulated and experimental curves match for $M_{m,\omega}^{(4)}$ and $M_{m,\omega}^{(6)}$ within the $1\sigma$-confidence intervals, shaded regions in Fig. <ref>. For both non-Gaussianity measures, we observe a quantitative discrepancy between simulation and experiment in the non-linear region.
Nevertheless, our model shows that the non-linearity is dominated by the self-interaction of the dominant parametrically amplified mode. The observed nonlinear dynamics signals the transition to the nonlinear Faraday resonance which may lead to stationary pattern formation on the interface in certain configurations in the long time limit <cit.>. Here, we are interested in the onset of these nonlinearities prior to the saturation of the amplitudes. In this regime, a limited number of scattering channels are determined by the dominant parametrically amplified mode, leading to secondary instabilities with higher wavenumber, in correspondence with preheating dynamics in the early universe.

Thermalisation by secondary instabilities.
(a): Radially Fourier transformed instantaneous amplitudes, $\tilde{b}_{4,\omega}(t,k_r)$,  at various $\omega$ and $k_r$, of experimentally reconstructed (dark-green lines) and simulated (red lines) data.
The primary instabilities at $(\omega_0,k_0)$ are displayed as solid lines, and the secondary instabilities at $(3\omega_0,k_1)$ are depicted by dashed lines.
Solid light blue lines indicate the log-linear fit of the growth region of the simulated $|\tilde{b}_{4,\omega_0}(t,k_0)|$ and $|\tilde{b}_{4,3\omega_0}(t,k_1)|$, with growth rates $\lambda_0=0.52~\mathrm{s}^{-1}$ and $\lambda_1=1.48~\mathrm{s}^{-1}\approx 2.85\lambda_0$, respectively, verifying the $3:1$ ratio predicted by preheating mechanism (see main text for details).
(b) and (c) show the radial spectra of the experimental amplitudes $\tilde{b}_{4,3\omega_0}(t,k_r)$ at two different times, matching the colors of the dotted vertical lines in (a). This verifies the thermalisation mechanism of preheating, i.e., the broadening of primary instabilities at low $k_0$ and the nonlinear amplification of the secondaries at higher $k_1$.

So far we have only considered a self-interacting truncated model. This reduces the complete effective Lagrangian (Eq. (<ref>) in Supplementary Information) to include only non-linear terms proportional to $(\xi_{m,k}\dot{\xi}_{m,k})^2$, with a resulting equation of motion that does not allow the amplification of any other modes. Going beyond self-interactions, the dominant primary Faraday instability $\xi_{4,k_0}$ can source the dynamics of a secondary mode $\xi_{l,k_1}$, which is captured by including interaction terms proportional to $\xi_{l,k_1}\xi_{4,k_0}\dot{\xi}_{4,k_0}^2$ and $\xi_{4,k_0}^2\dot{\xi}_{4,k_0}\dot{\xi}_{l,k_1}$ in our model. In these terms, we note that modes with azimuthal number $l=4$ exhibit relatively low damping and large non-linear coefficients. In this case, the only remaining degree of freedom is the wave-number $k_1$. Therefore, accompanying the primary instability (solid dark line in Fig. <ref>(a)) at $m=4$ with $k_0/(2\pi)\approx 0.35~\mathrm{cm}^{-1}$, we observe the growth of the secondary mode $l=4$ with $k_1/(2\pi)\approx 1.77~\mathrm{cm}^{-1}$ at the $3\omega_0$-resonance band (dashed dark line). These quantities are displayed in Fig. <ref> (a) as the radial Fourier transform $\tilde{b}_{4,\omega}(t,k_r)$ of the instantaneous amplitudes $b_{4,\omega}(t,r)$ with radial wave-number $k_r$.

Berges & Serreau <cit.> employ approximate analytical and numerical techniques to identify a signature of preheating, arising from a $\phi^4$-type interaction, as the scattering of the primary to the secondary instabilities, where the latter appears with an integer multiple of the frequency and slope of the former. As outlined above, our system provides an analogue simulator for mode-mode scattering exhibiting quartic interactions, and as such grants a platform to investigate preheating experimentally. Our theoretical model (red lines in Fig. <ref> (a)) accurately captures the non-linear features of the observed mode-mode interaction between primaries (solid lines) and secondaries (dashed lines), as depicted in Fig. <ref> (a). We extract the slopes (plotted as light blue lines) for both primary and secondary instabilities, first for a single simulated run, obtaining the ratio $2.85$, and then for the entire simulated ensemble (see Extended Data Fig. <ref>), for which we obtain $3.06$.

The behaviour of the experimental secondary instability (dark dashed curve) in Fig. <ref> (a) is due to an overlap between both low $k_0$ and high $k_1$ modes at frequency $3\omega_0$, resulting in a radial spectrum that can not be separated. Thus, when examining the slope of the experimental amplitude $\tilde{b}_{4,3\omega_0}(t,k_1)$, we observe a contribution from the primary growth to the pure secondary instability, which does follow the simulated model (red dashed curve). The relative contribution of the modes depends on their random initial state and hence varies between experimental repetitions. The outcome of this mode superposition is a damped secondary growth (see Extended Data Fig. <ref>) preventing us from reliably comparing the extracted slopes to our model. Qualitatively, in Fig. <ref> (b) and (c), we observe in our experiment the broadening of the primary resonance at $k_0$ and scattering into the secondary instability with higher $k_1$, as predicted in preheating models in Bose-Einstein condensates <cit.>. Our findings support the preheating scenario and approximate techniques put forward by Berges & Serreau <cit.> and theoretical analogue preheating proposals in ultra-cold atoms <cit.>. Additionally, the results presented here motivate the development of experiments and data analysis tools to reliably extract scattering amplitudes from analogue simulations.

By revisiting the century-old phenomenon of interfacial Faraday instabilities, we have outlined how carefully prepared, repeatable experiments can be used to simulate the key aspects of preheating. Our findings show that the mechanism of preheating prevails in our system despite the extra complications, such as dissipation and additional scattering channels. Our results are in support of universality and robustness of theoretical models tackling the thermalisation of the Early Universe and its distinct stages. As we have shown, analogue preheating simulators open a new avenue of investigation, with the potential to explore regimes beyond those we can calculate.

Acknowledgments SW is acknowledging Joerg Schmiedmayer's support and numerous eye-opening discussions on extracting effective field theories from hydrodynamical systems. SW and SE acknowledge discussions with Torsten Zache, whose insights on preheating simulations and encouraging feedback played an essential role in this project. The authors are grateful to Bill Unruh for his invaluable input in understanding the effect of temperature variations on the experimental setup. We thank Ed Copeland and David Kaiser for their detailed feedback on this work. The authors are grateful to Terry Wright, Pete Smith, Sionnach Devlin, Andrew Stuart and Tommy Napier for technical support. In particular, we are indebted to Terry Wright, whose expertise enabled the construction of the high-performance vertical acceleration platform. VB and ZF acknowledge several fruitful experimental discussions with Naresh Sampara and George Hunter-Brown.

SW acknowledges support provided by the Leverhulme Research Leadership Award (RL-2019- 020), the Royal Society University Research Fellowship (UF120112) and the Royal Society Enhancements Grant (RGF/EA/180286 and RGF/EA/181015), and partial support by the Science and Technology Facilities Council (Theory Consolidated Grant ST/P000703/1), the Science and Technology Facilities Council on Quantum Simulators for Fundamental Physics (ST/T006900/1) as part of the UKRI Quantum Technologies for Fundamental Physics programme. RJAH acknowledges support provided by the Leverhulme Trust (RPG-2018-363). SE acknowledges support through the EPSRC Project Grant (EP/P00637X/1) and an ESQ (Erwin Schrödinger Center for Quantum Science and Technology) fellowship funded through the European Union’s Horizon 2020 research and innovation program under Marie Skłodowska-Curie Grant Agreement No 801110. This project reflects only the author’s view, the EU Agency is not responsible for any use that may be made of the information it contains. ESQ has received funding from the Austrian Federal Ministry of Education, Science and Research (BMBWF).

Author contributions
VB, AG, ZF and SE performed the experiment and the
data analysis. VB and AG did the theoretical calculations. SE, AA, RJAH and SW provided scientific guidance in experimental and theoretical aspects of this work.
AA, RJAH and SW proposed the analogy and designed the experiment. All authors contributed to interpreting the data and writing the manuscript.

Supplementary Information is available for this paper.

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