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PHYSICAL REVIEW RESEARCH 7, L022004 (2025)
Letter
Mechanical cosmology: Simulating scalar fluctuations in expanding
universes using synthetic mechanical lattices
Brendan Rhyno ,1,*Ivan Velkovsky ,1Peter Adshead ,2Bryce Gadway,1,3and Smitha Vishveshwara1,†
1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
2Illinois Center for Advanced Studies of the Universe & Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
3Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
(Received 7 February 2024; revised 23 September 2024; accepted 27 February 2025; published 3 April 2025)
Inspired by recent advances in observational astrophysics and continued explorations in the field of analog
gravity, we discuss the prospect of simulating models of cosmology within the context of synthetic mechanical
lattice experiments. We focus on the physics of expanding universe scenarios described by the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric. Specifically, quantizing scalar fluctuations in a background FLRW
spacetime leads to a quadratic bosonic Hamiltonian with temporally varying pair production terms. Here we
present a mapping that provides a one-to-one correspondence between these classes of cosmology models and
feedback-coupled mechanical oscillators. As proof of principle, we then perform experiments on a synthetic
mechanical lattice composed of such oscillators. We simulate two different FLRW expansion scenarios with
universes dominated by vacuum energy and matter and discuss our experimental results.
DOI: 10.1103/PhysRevResearch.7.L022004
As current day probes of the cosmos, such as the James
Webb Space Telescope [1], begin to reveal extraordinar-
ily deep glimpses of the infant universe, they lead to new
questions on our present understanding of its beginnings.
Inflation [2–6] is now well established as the leading mech-
anism for setting the initial state of our universe. An early
phase of accelerated expansion, inflation was proposed to
solve the horizon and flatness problems of the hot Big Bang
cosmology. However, it was soon realized that quantum me-
chanical fluctuations of the metric and fields during this epoch
would generate nearly scale-invariant density and gravita-
tional wave spectra [7–10]. Hand in hand with theoretical
models, computational techniques, and observational astron-
omy, a surge in the development of analog gravitational
systems serves to test, corroborate, and enhance this under-
standing [11–13]. Here, we show that synthetic mechanical
lattices composed of measurement-based feedback-coupled
mechanical oscillators are excellently poised to simulate
key features of inflationary cosmology and, more generally,
Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology.
The essence of the inflationary paradigm that lends itself to
these experimental simulations is as follows. Quantum fluctu-
ations about a Bose-condensed inflaton field lead to density
perturbations in the post inflationary universe that clump and
*Contact author: brhyno2@illinois.edu
†Contact author: smivish@illinois.edu
Published by the American Physical Society under the terms of the
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distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
collapse under the influence of local gravity to give rise to
the anisotropies in the cosmic microwave background, and the
large scale structures in the universe today. These quantum
mechanical fluctuations arise as the zero-point motion of the
fields which are then stretched to superhorizon scales due
to accelerated expansion. This quantum mechanical origin of
structure, reflected in dynamical boson pair production, has
become a crucial prediction of inflationary cosmology.
Historically, classical and quantum fluids have provided
fertile ground for analog gravitational models ranging from
black holes to the expanding universe [12,14–17]. For in-
stance, in Bose-Einstein condensates, in the hydrodynamic
approximation, phase fluctuations obey the massless Klein-
Gordon equation subject to an effective metric [18–20].
The versatile setting of ultracold atomic gases provides
close cosmological parallels [21–32] even including proto-
cols involving physical expansions [33–44]. A plethora of
alternatives exist for cosmological analogs, including in the
realms of earth sciences [45], conducting wires [46], as well
as metamaterials [47–49] where customizable Hamiltonian
dynamics can be engineered [50,51].
Mechanical systems have evolved to an extraordinary level
of sophistication in the gravitational context, finding appli-
cations in gravitational wave detectors and searches for dark
matter and dark energy candidates [52–58]. Here, we propose
that they even provide a superb arena for analog gravity. Our
main observation is that in FLRW theories, the equations of
motion of the bosonic degrees of freedom are ultimately those
of coupled harmonic oscillators. The unique feature of the
oscillator network considered here is its access to highly tun-
able feedback forces [59–61]. When precise feedback forces
can be engineered, we show there exists a natural mapping
in which classical oscillator systems are well positioned to
2643-1564/2025/7(2)/L022004(7) L022004-1 Published by the American Physical Society
BRENDAN RHYNO et al. PHYSICAL REVIEW RESEARCH 7, L022004 (2025)
perform analog simulations that directly target key properties
of FLRW cosmology. Here, we provide a proof-of-principle
experimental demonstration in which we perform a compre-
hensive study of FLRW models in a synthetic mechanical
lattice, simulating expansion scenarios driven by dark energy
or matter, and obtaining the analogs of astrophysically rel-
evant quantities, such as the power spectrum and the pair
production.
In what follows we outline standard features of FLRW
cosmology and then introduce the mechanical oscillator ex-
periment pinpointing the parallels between the two. Finally,
we present and discuss our experimental analog simulation
results.
Conceptual aspects of FLRW cosmology. We consider the
simplest effective model that captures salient features of the
quantum mechanical production of fluctuations in FLRW
cosmology; namely, a free real massless scalar field ϕin a
background spacetime (¯h=c=1):
S=d4x√−g1
2gμν∂μϕ∂νϕ,(1)
where gμν denotes the components of the metric tensor
with determinant gand inverse gμν . Here, the field ϕ
represents the fluctuations about the background inflaton
field. (Alternatively, ϕcould represent fluctuations in the
transverse-traceless part of the spatial metric—gravitational
waves.) Appropriate for large length scales in which the uni-
verse is spatially homogeneous and isotropic, we focus on the
(spatially flat) FLRW metric:
ds2=a2(τ)(dτ2−dx2),(2)
where ais the scale factor, and τis conformal time. As is
convention, primes represent derivatives with respect to con-
formal time, while overdots represent derivatives with respect
to cosmic time. The two are related by dt =adτ.
The evolution of the scale factor follows from the Fried-
mann equations [62]
3M2
PlH2=ρ, M2
Pl ˙
H=−1
2ρ(1 +w),(3)
where H=˙a/ais the Hubble parameter, MPl is the Planck
mass, and ρand w=p/ρ are the energy density and equa-
tion of state of the matter fields driving the expansion of
the universe. For vacuum energy-dominated expansions, rele-
vant for early universe inflationary cosmology as well as the
current state of our own universe, the Hubble parameter is
constant. In this case, the scale factor grows exponentially,
a∼eHt. For either a matter-dominated (w=0) or radiation-
dominated (w=1/3) expansion, the scale factor exhibits
power-law scaling with time: a∼t2/(3(1+w)).
For our purposes, the quantum mechanical production of
fluctuations in the scalar field in Eq. (1) about these expanding
backgrounds is most easily analyzed by transforming from
the Lagrangian to the Hamiltonian and canonically quantiz-
ing the theory. Particle production can then be seen from
the evolution of the creation and annihilation operators de-
scribing the instantaneous occupation of the initial vacuum
state. To quantize the theory, one can canonically normal-
ize the kinetic term in Eq. (1), by rescaling the physical
field to y(τ,x)≡a(τ)ϕ(τ,x)[63,64]. One then moves to a
Hamiltonian description of the dynamics by introducing the
canonical momentum, and performing a Legendre transfor-
mation. Canonical quantization amounts to promoting the
comoving field and its canonical momentum to quantum field
operators obeying canonical commutation relations. By virtue
of translational invariance, one Fourier transforms and intro-
duces bosonic creation (annihilation) operators, ˆ
b†
k(ˆ
bk), in
the standard way to reach the final form of the Hamiltonian
operator [63]:
ˆ
H(τ)=d3k
2k(ˆ
b†
kˆ
bk+ˆ
b−kˆ
b†
−k)
+ia(τ)
a(τ)(ˆ
b†
kˆ
b†
−k−ˆ
b−kˆ
bk),(4)
where k=|k|and a/a0 for the expansion scenarios con-
sidered here. For a time-varying scale factor (a= 0), the
instantaneous spectrum of the Hamiltonian is time dependent
and becomes unbounded from below for modes with k<a/a
[65–68].
The Heisenberg equations of motion for the bosonic
modes that follow from this Hamiltonian can be solved in
terms of a Bogoliubov transformation [69]ˆ
bk(τ)=uk(τ)ˆ
bk+
v∗
k(τ)ˆ
b†
−k, where the Bogoliubov coefficients uk(τ) and vk(τ)
evolve in conformal time according to the Bogoliubov equa-
tions:
iuk(τ)
vk(τ)=ki
a(τ)
a(τ)
ia(τ)
a(τ)−kuk(τ)
vk(τ).(5)
The 2 ×2 matrix generating the dynamics is non-Hermitian
and has imaginary instantaneous eigenvalues whenever
k<a/aand real eigenvalues otherwise. Matrices with this
structure and unitary transformations of them appear fre-
quently in the study of non-Hermitian quantum systems
[70–75]. Note that these Bogoliubov equations are simply
parametric oscillators in disguise [76]. If one defines the func-
tion Yk(τ)≡[uk(τ)+vk(τ)]/√2k, it is straightforward to see
that 0 =Y
k+(k2−a/a)Yk. In the cosmological context,
this function appears as the coefficients in the Fourier modes
of the comoving field operator, ˆyk(τ)=Yk(τ)ˆ
bk+Y∗
k(τ)ˆ
b†
−k,
and is thus intimately related to the power spectrum of the
field.
The Bogoliubov equations and their solutions offer a con-
venient formalism to analyze important physical features
associated with each expansion scenario. For instance, under
unitary time evolution generated by the Hamiltonian, Eq. (4),
the vacuum state |0evolves into a two-mode squeezed state
for each (k,−k) pair [68,77]:
|ψ(τ)∝exp d3k
2tanh(rk(τ))eiθk(τ)ˆ
b†
kˆ
b†
−k|0,(6)
with squeeze parameter rk(τ)≡arsinh(|vk(τ)|) and squeeze
angle θk(τ)≡arg(uk(τ)) −arg(vk(τ)).
It is also straightforward to calculate vacuum expectation
values of Heisenberg operators in terms of the Bogoliubov
coefficients. At conformal time τ, the vacuum expectation
value for the number of excitations in a given kmode, the
number of particle pairs produced with opposite momenta,
and the power spectrum of the comoving field are determined,
L022004-2
MECHANICAL COSMOLOGY: SIMULATING SCALAR … PHYSICAL REVIEW RESEARCH 7, L022004 (2025)
FIG. 1. Analog cosmology using coupled mechanical oscillators.
(a) A cartoon depiction of the scale factor that dictates the expansion
of a model Universe. (b) A synthetic mechanical lattice consisting
of modular mechanical oscillators. Each oscillator consists of two
springs holding a mass that is equipped with an accelerometer and
a dipole magnet embedded in a pair of anti-Helmholtz coils. Input
from the functional form of the scale factor, along with real-time
accelerometer measurements, allows one to engineer feedback forces
using the anti-Helmholtz coils which effectively makes the system
behave as an analog simulator of the equations of motion in the
cosmological theory. Analog versions of the correlation functions
of interest in cosmology can then be extracted from the mechanical
oscillator experimental data.
respectively, by
0|ˆ
b†
k(τ)ˆ
bk(τ)|0=|vk(τ)|2,(7a)
0|ˆ
b†
k(τ)ˆ
b†
−k(τ)|0=u∗
k(τ)vk(τ),(7b)
0|ˆyk(τ)ˆy†
k(τ)|0= 1
2k|uk(τ)+vk(τ)|2,(7c)
where, for simplicity of presentation, we have employed box
regularization in these expressions to suppress the Dirac delta
distribution. Dynamic scale factors are required to generate
nonzero vk(τ)inEq.(5). Hence from Eq. (7) we see that in-
flationary expansion drives both particle production and scalar
field fluctuations beyond the zero-point value.
Experimental setup. We have seen that the dynamics of the
cosmology model are encoded in the dynamics of parametric
oscillators. We now demonstrate that synthetic mechanical
lattices [51,59–61] are ideally suited to solve such systems
of equations. The experimental system consists of a pair of
mechanical oscillators. Each oscillator, a mass held between
two springs, is equipped with an accelerometer that enables
real-time measurements. The oscillator mass also features a
dipole magnet that sits at the center of an anti-Helmholtz coil
pair. By driving coil currents that depend on the real-time
measurements, we enact measurement-based feedback forces
[68] as depicted in Fig. 1.
Without additional forces, the equations of motion for the
position and momentum of the nth oscillator at time tin
the experiment are ˙xn(t)=pn(t)/mand ˙pn(t)=−mω2xn(t),
where mand ωare the mass and angular frequency of each
oscillator with overdots representing derivatives with respect
to time. The oscillators are synthetically coupled to one an-
other using measurement-based feedback forces. Monitoring
the real-time acceleration of each oscillator and numerically
differentiating this data to obtain its jerk enables us to engineer
custom forces that take into account the current state of the
system [59–61]. For this reason, it is convenient to instead
work with the acceleration and jerk of each oscillator, which
we will denote by Xn(t) and Pn(t). With the inclusion of
measurement-based feedback forces, their equations of mo-
tion are given by
˙
Xn(t)=Pn(t),(8a)
˙
Pn(t)=−ω2Xn(t)+Fn(t,{Xn,Pn}),(8b)
where Fnis the feedback “force” on the nth oscillator [68].
Formally changing variables to the complex function An(t)≡
ω
2Xn(t)+i1
√2ωPn(t), one finds that the synthetic mechani-
cal lattice, after being prepared in some initial configuration,
evolves in time according to
i˙
An(t)=ωAn(t)−1
√2ωFn(t,{A∗
n,An}).(9)
It is from these equations that we find a natural mapping
between the cosmological and experimental systems.
To perform an analog simulation of the cosmology model,
here we establish a mapping between the Bogoliubov equa-
tions, Eq. (5), and the equations which govern the synthetic
mechanical lattice, Eq. (9). Since each observable discussed
previously in the cosmological context depends only on
various combinations of the Bogoliubov coefficients, this
mapping allows us to directly construct analogs of physi-
cal quantities in cosmology using the experimental output
data (Fig. 1). Because each oscillator encodes a single com-
plex variable, we only require two oscillators, A1and A2,
to simulate the dynamics of the Bogoliubov coefficients uk
and vkfor a given momentum k. We interpret the physi-
cal time in the experiment tas the conformal time τin
the Bogoliubov equations and identify experimental feedback
“forces” F1/√2ω=−kA1−i(a/a)A2+c.c and F2/√2ω=
kA2−i(a/a)A1+c.c. A separation of timescales between
the characteristic period of oscillations and the dynamics we
simulate means “counter-rotating” terms can be neglected in
the rotating-wave approximation [68,78,79], and the Bogoli-
ubov coefficients ukand vkcan be extracted from A1and
A2, respectively, by amplitude demodulating the signals with
carrier frequency ω.
Analog simulation of FLRW cosmology. To perform an
analog simulation of the cosmology model, we set a wave
vector magnitude of interest, choose a functional form for
the scale factor, and allow the system to evolve according to
Eq. (9) with the appropriate feedback forces. From the output
acceleration data of the experiments, we perform amplitude
demodulation of the signal and construct the Bogoliubov coef-
ficients from which various physically meaningful quantities
from the cosmologic perspective can be extracted.
Here we perform analog simulations of FLRW cosmol-
ogy using scale factors that correspond to dark energy-driven
expansions as well as matter-driven expansions. Setting the
initial conformal time to zero for convenience gives the
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BRENDAN RHYNO et al. PHYSICAL REVIEW RESEARCH 7, L022004 (2025)
FIG. 2. Mechanical cosmology for expanding FLRW space-
times. Columns (a) and (b) correspond to dark energy- and
matter-driven expansions, respectively. In all images, theoretical
curves are shown in black and experimental results in red. We choose
wave vector magnitude k=0.2 and set the scale factor according
to Eq. (10), with ˜τ=76.9 in the dark energy-driven expansion
and ˜τ=5 in the matter-driven expansion. Row (i) shows the en-
ergy scales that enter into the Bogoliubov equations and appear as
couplings in the measurement-based feedback forces; in particular,
the off-diagonal a/aenergy scale is shown in bold and the wave
vector as a dashed line. The remaining rows show properties of
the time evolved initial vacuum state with theory curves, obtained
by numerically solving Eq. (5), as black dashed lines and experi-
mental results, obtained by analog simulations realizing Eq. (9), as
red solid lines. Row (ii) shows the squeeze parameter, expressed as
arsinh(|vk|), and row (iii) shows the squeeze angle, arg(uk)−arg(vk)
(mod 2π), divided by 2π. The experimental data has been amplitude
demodulated with carrier frequency 13.06 Hz, and a 0.1 sec moving
average window has been applied.
following conformal-time Hubble parameter in each expan-
sion scenario:
dark energy: a(τ)
a(τ)=1
˜τ−τ,0τ<˜τ; (10a)
matter: a(τ)
a(τ)=2
˜τ+τ,0τ<∞,(10b)
where ˜τis an expansion-specific parameter related to initial
conditions [68]. For our proof-of-principle demonstration, we
choose values of ˜τappropriate to observe nontrivial dynamics
on the order of ∼1 minute of analog simulation. The resulting
energy scales in the Bogoliubov equations (i.e., couplings in
the measurement-based feedback forces) are shown in row
(i)ofFig.2. In the matter-driven expansion, the instanta-
neous eigenvalues of the matrix in Eq. (5) start out imaginary
(k<a/a), offering a mechanism to increase the squeeze
FIG. 3. Mechanical cosmology for various expanding FLRW
spacetimes continued from Fig. 2. Once again, columns (a) and
(b) correspond to dark energy- and matter-driven expansions, re-
spectively, with theory curves shown as black dashed lines and
experimental results shown as red solid lines. The rows show the
equal-time vacuum expectation values expressed in Eq. (7): (i)
shows the number of excitations in a kmode with magnitude k, (ii)
shows the imaginary part of opposite-momentum pairs produced, and
(iii) shows the comoving field power spectrum.
parameter and the production of particles and fluctuations, but
become real as time progresses. The dark energy-driven ex-
pansion exhibits the opposite behavior as a/aonly increases
with conformal time.
The experimental results of our analog simulations of
FLRW cosmology under each expansion scenario are dis-
played in Figs. 2and 3, showing measurements for the
two-mode squeezed vacuum state and vacuum expectation
values, respectively. To compare with theoretical predictions,
numerical solutions to the Bogoliubov equations are also pro-
vided in each case. As both expansion scenarios unfold, we
observe the analogs of two-mode squeezing, particle produc-
tion, and enhanced scalar field fluctuations. In most cases,
the experimental data qualitatively agree with the theoretical
results. Namely, the frequency of oscillations and relative
heights of local minima and maxima in the experimental data
are fairly consistent with theory curves across various quanti-
ties. One of the more sensitive measures is the squeeze angle,
which depends on the phase difference between the Bogoli-
ubov coefficients. In particular, the experimental measurement
of the squeeze angle extracted from the analog simulation
of the dark energy-driven expansion is somewhat noisy in
relation to the theoretical curve, as this relative phase be-
comes ill defined when the population of the second oscillator
(|A2|2=|vk|2) gets close to zero.
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MECHANICAL COSMOLOGY: SIMULATING SCALAR … PHYSICAL REVIEW RESEARCH 7, L022004 (2025)
Another source of disagreement comes from the nor-
malization of the Bogoliubov coefficients, which in theory
is 1 =|uk(τ)|2−|vk(τ)|2. One can use this in the FLRW
models to re-express physical quantities in various ways.
This normalization is however not guaranteed in the pres-
ence of experimental imperfections. In practice we found
greater noise on the oscillator simulating uk(τ). For better
agreement then, we chose to extract the squeeze parame-
ter in Fig. 2using the data from the oscillator simulating
vk(τ).
Conclusion and outlook. In this Letter, we have demon-
strated how synthetic mechanical lattices offer precise par-
allels for simulating inflationary and, more generally, FLRW
cosmology. Through mapping the bosonic dynamics of scalar
field fluctuations onto the motion of mechanical oscillators,
we were able to simulate and measure key physical infla-
tionary quantities, such as particle production and the power
spectrum. This proof-of-principle study potentially opens up
an entire toolbox for analog gravity in the realm of the
early universe. Here, we have but exploited the physics of
two coupled oscillators in a much more powerful system
which currently contains eighteen oscillators in which one
can achieve arbitrary connectivity and nonlinearity. The scope
in this experimental system is thus vast for simulating more
complex actions and spacetimes, including interaction effects
such as those considered in cosmological collider physics [80]
coupling the inflaton to other heavy degrees of freedom, as
well as dissipation effects and more. Given the customiz-
able nature of the experiment, with prospects for entering
nonlinear regimes and coupling multiple degrees of freedom,
there is also the potential to study dynamics beyond that of
the early universe performing analog simulations into fur-
ther astrophysical domains and other branches of physics.
Synthetic mechanical lattices thus embody a highly tunable
playground for simulating different cosmological scenarios
to complement theoretical and observational astrophysics, as
probes continue to reveal more insights and mysteries about
our primordial universe.
Acknowledgments. We thank Naceur Gaaloul and Zain
Mehdi for insightful discussions. We gratefully acknowledge
support by the National Aeronautics and Space Administra-
tion, Jet Propulsion Laboratory Research Support Agreement
No. 1699891 (B.R. and S.V.), by the United States Department
of Energy, DE-SC0015655 (P.A.), and the AFOSR MURI
program under Agreement No. FA9550-22-1-0339 (I.V. and
B.G.).
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