Novel signatures of dark matter in laser-interferometric gravitational-wave detectors
H. Grote1, ∗ and Y. V. Stadnik2, 3, †arXiv:1906.06193v3 [astro-ph.IM] 29 Sep 20191

Cardiff University, School of Physics and Astronomy, The Parade, CF24 3AA, United Kingdom
2Helmholtz Institute Mainz, Johannes Gutenberg University, 55128 Mainz, Germany
3
Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan(Dated: October 1, 2019)

Dark matter may induce apparent temporal variations in the physical “constants”, including theelectromagnetic fine-structure constant and fermion masses. In particular, a coherently oscillatingclassical dark-matter field may induce apparent oscillations of physical constants in time, while thepassage of macroscopic dark-matter objects (such as topological defects) may induce apparent transient variations in the physical constants. In this paper, we point out several new signatures of theaforementioned types of dark matter that can arise due to the geometric asymmetry created by thebeam-splitter in a two-arm laser interferometer. These new signatures include dark-matter-induced
time-varying size changes of a freely-suspended beam-splitter and associated time-varying shifts ofthe main reflecting surface of the beam-splitter that splits and recombines the laser beam, as wellas time-varying refractive-index changes in the freely-suspended beam-splitter and time-varying sizechanges of freely-suspended arm mirrors. We demonstrate that existing ground-based experiments
already have sufficient sensitivity to probe extensive regions of unconstrained parameter space inmodels involving oscillating scalar dark-matter fields and domain walls composed of scalar fields. In
the case of oscillating dark-matter fields, Michelson interferometers — in particular, the GEO 600
detector — are especially sensitive. The sensitivity of Fabry-Perot-Michelson interferometers, including LIGO, VIRGO and KAGRA, to oscillating dark-matter fields can be significantly increasedby making the thicknesses of the freely-suspended Fabry-Perot arm mirrors different in the two arms.
Not-too-distantly-separated laser interferometers can benefit from cross-correlation measurements
in searches for effects of spatially coherent dark-matter fields. In addition to broadband searches
for oscillating dark-matter fields, we also discuss how small-scale Michelson interferometers, suchas the Fermilab holometer, could be used to perform resonant narrowband searches for oscillating
dark-matter fields with enhanced sensitivity to dark matter. Finally, we discuss the possibility ofusing future space-based detectors, such as LISA, to search for dark matter via time-varying sizechanges of and time-varying forces exerted on freely-floating test masses.

I.

INTRODUCTION

While the existence of dark matter (DM) is well established from astrophysical and cosmological observations, the elucidation of its precise nature remains one of
the most important problems in contemporary physics.
Since extensive searches for DM particles of relatively
high masses (e.g., WIMPs) through their possible nongravitational effects have not yet produced a strong positive result, in recent years the possibility of searching
for low-mass (sub-eV) DM candidates has been receiving
increased attention. There are numerous well-motivated
DM candidates of this type, including the canonical axion, axion-like particles and dilatons, which may forma coherently oscillating classical field and/or stable solitonic field configurations known as topological defects(such as domain walls). Searches for such low-mass DM
candidates via possible particlelike signatures (such as recoils, energy depositions and ionisations) are practically
impossible, since the individual non-relativistic DM par-∗ Electronic address: groteh@cardiff.ac.uk† Electronic address: yevgenystadnik@gmail.com

ticles in this case carry very small momenta. Instead, one
can take advantage of the fact that low-mass DM particles must have large occupation numbers if they comprisethe observed DM content of the Universe (the average local cold DM density is given by ρDM ≈ 0.4 GeV/cm3 [1])and look for wavelike and other coherent signatures ofthese DM fields. In recent years, a number of novel ideas
have emerged to search for low-mass DM using precision
measurement techniques from the fields of atomic andoptical physics; see [2, 3] for recent overviews.
DM may induce apparent temporal variations in thephysical “constants”, including the electromagnetic finestructure constant α, as well as the electron and nucleon
masses me and mN , via certain non-gravitational interactions with standard-model (SM) fields [4]. In particular, a coherently oscillating classical DM field may induce apparent oscillations of physical constants in time[5, 6], while the passage of topological defects may induce
apparent transient variations in the physical constants[7, 8]. Some possible effects of such time-varying physical constants in laser interferometers and optical cavities, including time-varying changes of solid sizes andlaser frequencies, were explored in [5, 9]. Several clockcavity comparison experiments searching for DM-induced
time-varying physical constants have been conducted re-2cently [10–12]. Experiments of this type are mainly sensitive to oscillation frequencies up to ∼ Hz (equivalently
timescales down to ∼ s). However, we would also like toprecisely probe even higher oscillation frequencies (audioband frequencies and beyond), which too are interestingfrom the point of view of current astrophysical observations.
In this work, we propose new ways of searching for
DM with laser interferometers. A two-arm laser interferometer is typically used to detect small changes in the
difference of the optical path lengths in the two armsof the interferometer. Since the two arms of an interferometer are practically equal in terms of optical path
length, time-varying arm length fluctuations and changes
in the laser frequency due to a homogeneous DM field arecommon mode, and their effects are therefore strongly
suppressed in the output signal. There is, however, ageometric asymmetry created by the beam-splitter in a
two-arm laser interferometer. We point out that thebeam-splitter and arm mirrors of an interferometer, if
freely suspended, can produce differential optical-pathlength changes if one or more of the physical constantsof nature vary in time (and space). A non-zero output
signal, namely a phase difference between the two armsof the interferometer, can arise in several ways. If theDM field is homogeneous across the entire interferometer,
then the main observable effect will generally arise fromthe freely-suspended beam-splitter. A freely-suspended
beam-splitter would experience time-vaying size changes
about its centre-of-mass, thus shifting back-and-forth themain reflecting surface that splits and recombines thelaser beam (see the inset in Fig. 1). Additionally, (generally smaller) time-varying changes in the refractive index of the beam-splitter would change the optical path
length across the beam-splitter. On the other hand, if theDM field is inhomogeneous over an interferometer, thensubstantial observable effects may also arise from timevarying size changes of the freely-suspended arm mirrors(see Figs. 1 and 2). In some situations, the output signal can be significantly enhanced if the arm mirrors havedifferent physical characteristics (in particular, different
thicknesses).
Laser interferometry has been optimised over decades
to develop ultra-sensitive gravitational-wave detectors,
which have recently been employed spectacularly to directly observe gravitational waves on Earth for the firsttime [20, 21]. Additionally, smaller-scale interferometers have more recently been utilised to search for
non-gravitational-wave phenomena, such as quantumgeometry effects that may arise at the Planck scale[15, 16].
Future space-based laser-interferometricgravitational-wave detectors, such as LISA [22], are currently under development. In this paper, we explore
novel signatures of DM in ground- and space-based laserinterferometers. We estimate the sensitivities of these detectors to the physical parameters of models of DM consisting of a coherently oscillating classical field or domain
walls. Searches for coherently oscillating classical DM

ETMY

l

50

R=

Ly

%
AR

Laser

Lx
PRM

BS

ETMX

SRM
PD

FIG. 1: Simplified layout of a dual-recycled Michelson interferometer, such as GEO 600 [13, 14] and the Fermilab holometer [15, 16]. Dual recycling denotes the combination of power
recycling and signal recycling. PRM: power recycling mirror;
BS: beam-splitter of thickness l; ETMX, ETMY: end arm
mirrors (test masses); SRM: signal recycling mirror; PD: photodetector. The inset shows the beam routing through thebeam-splitter. The beam-splitting surface typically has apower reflectivity of R = 50%. The opposing face of thebeam-splitter, denoted by AR, is anti-reflective coated. For
clarity, we have omitted the single folding of the arms in
GEO 600, as well as the second co-located interferometer of
the Fermilab holometer; furthermore, the Fermilab holometer
does not have a signal recycling mirror SRM.

ETMY

Ly

ITMY
BS

Laser

ITMX

PRM

Lx
ETMX

SRM
PD
FIG. 2: Simplified layout of a dual-recycled Fabry-PerotMichelson interferometer, such as Advanced LIGO [17],VIRGO [18] and KAGRA [19]. PRM: power recycling mirror; BS: beam-splitter; ITMX, ETMX, ITMY, ETMY: arm
mirrors (test masses); SRM: signal recycling mirror; PD: photodetector. The arm mirrors (test masses) are separated by
the distances Lx and Ly , which are 4 km in the case of LIGO.

fields share similarities with searches for continuous as3well as for stochastic gravitational waves, while searches
for domain-wall objects share similarities with searches
for gravitational-wave bursts. Based on our estimates,
we emphasise that existing laser interferometers, particularly the GEO 600 Michelson interferometer [13, 14], already have sufficient sensitivity to probe extensive regions of unconstrained parameter space in these models,
including scalar DM fields oscillating at frequencies in therange ∼ 100 Hz − 10 kHz and scalar-field domain wallswith transverse sizes of up to several km. The sensitivity of Fabry-Perot-Michelson interferometers, including
LIGO [17], VIRGO [18] and KAGRA [19], to oscillating DM fields can be significantly increased by making
the thicknesses of the freely-suspended Fabry-Perot arm
mirrors different in the two arms. In this case, the sensitivity of these experiments to conventional gravitationalwave searches, which can be performed simultaneouslywith our suggested DM searches, would not necessarily
be degraded. Not-too-distantly-separated laser interferometers can benefit from cross-correlation measurements
in searches for effects of spatially coherent dark-matterfields. In addition to broadband searches for oscillating DM fields, we also discuss how small-scale Michelsoninterferometers could be used to perform resonant narrowband searches for oscillating DM fields with enhanced
sensitivity to underlying DM interactions.
The structure of our paper is as follows. In Sec. II,
we discuss how DM can induce apparent temporal variations in the physical constants and derive the effectsof such variations on freely-suspended beam-splitters, aswell as freely-suspended and freely-floating test masses.
In Sec. III, we consider the specific model of a coherently
oscillating classical DM field; we derive the effects of an
oscillating DM field on ground- and space-based laser interferometers and estimate the sensitivities of existing,modified and future experiments to the underlying DM
parameters. In Sec. IV, we consider the specific modelof topological defects in the form of domain walls; wederive the effects of domain walls on ground- and spacebased laser interferometers and estimate the sensitivities
of existing and future experiments to the underlying DM
parameters. Finally, in Sec. V, we summarise our findings and discuss DM searches with laser interferometersin the context of other measurements.
Throughout this work, unless explicitly stated otherwise, we shall adopt the natural system of units ~ = c =1, where ~ is the reduced Planck constant and c is the
speed of light in vacuum. In this paper, we express theinterferometer output in terms of the difference of the optical path lengths in different arms of an interferometer.

II. THEORY AND EFFECTS OF
DARK-MATTER-INDUCED VARYINGPHYSICAL “CONSTANTS”
A.

Non-gravitational interactions of scalar fields

A scalar (spinless, even-parity) field φ can couple to
the SM fields in a number of possible ways. Generally,
the simplest possibility involves linear-in-φ interactions:
φ Fµν F µν X φ
Llin−
mf f¯f ,
(1)
int =
Λγ
4
Λf
f

where the first term represents the coupling of the scalar
field to the electromagnetic field tensor F , while the second term represents the coupling of the scalar field to
the SM fermion fields f , with mf the “standard” mass ofthe fermion and f¯ = f † γ 0 the Dirac adjoint. The linear
couplings in (1) can be generated, e.g., via the superrenormalisable interaction of φ with the Higgs field; see[23, 24] for more details. These linear couplings, however, may be absent, e.g., as a result of an underlying Z2 symmetry (invariance under the transformation
φ → −φ). In this case, the simplest possibility would
involve quadratic-in-φ interactions:
!2
 2
φ
Fµν F µν X φquad
Lint =−
mf f¯f . (2)
Λ0γ
4
Λ0f
f

Comparing the terms in Eqs. (1) and (2) with the relevant terms in the SM Lagrangian:XFµν F µν X−
qf Jµ Aµ −mf f¯f ,(3)
LSM ⊃ −
4f

f

where qf is the electric charge carrried by the fermion
f , J µ = f¯γ µ f is the electromagnetic 4-current and Aµ
is the electromagnetic 4-potential, we see that the linear interactions in (1) effectively alter the fine-structure
constant and fermion masses according to:




α
φ
φ
α→
≈α 1+
, mf → mf 1 +,1 − φ/Λγ
Λγ
Λf(4)while the quadratic interactions in (2) effectively alter
the constants according to:" 2 #αφα→,
2 ≈ α 1 +Λ0γ
1 − φ/Λ0γ

!2 
φ
.
mf → mf 1 +
(5)
Λ0f

B.

Size changes of beam-splitter and test masses

Time-varying α and particle masses alter the geometricsizes of solid objects. The length of a solid is given by4
L ∼ N aB , where N is the number of lattice spacings andaB = 1/(me α) is the atomic Bohr radius. In the adiabatic
limit, the size of a solid body thus changes according to:
 
δL
δaB
δα δme
≈
=−
−
.
(6)
L 0
aB
α
me
Additionally, there are also small relativistic corrections
associated with electromagnetic processes and finitenuclear-mass effects [5, 25]. The former affect the α2dependence in Eq. (6) and scale roughly as ∝ (Zα) with
the nuclear charge Z, while the latter affect the me /mN dependence and are of the order ∼ me /mN ≈ 5 × 10−4 .
We thus see that the relativistic corrections are negligibly small in most common materials (including silica andsapphire) and only become non-negligible in the heaviest
stable elements [where their size is ∼ 20% of the nonrelativistic contribution in Eq. (6)].
The expression in Eq. (6) is only valid in the adiabatic limit, when the solid can optimally respond to theslow perturbations induced by the DM field. To model
more general perturbations, we can treat the responseof a solid to perturbations associated with a particularFourier-driving-frequency component f within the simple model of a strongly-underdamped driven harmonic
oscillator (damping parameter ζ  1), noting that thelength perturbations of the solid are induced by an external field [26]. In this case, the steady-state responseof the solid size changes is given by:
 
 
δL
δL
1
=
× rh
,
i2
L f
L 0
2
2
1 − (f /f0 )+ (2ζf /f0 )(7)
where (δL/L)0 is the fractional size change in the adiabatic limit [see Eq. (6)], and f0 is the frequency of afundamental vibrational mode of the solid. There arethree limiting cases in Eq. (7):• When f  f0 , the size changes of the solid are
independent of the driving frequency and coincidewith the adiabatic case in Eq. (6). The size oscillations of the solid are in phase with the oscillating
DM field.
• When f ≈ f0 , the size changes of the solid areenhanced by the large quality factor Qmech =1/(2ζ)  1 compared with the adiabatic case. The
size oscillations of the solid lag behind the oscillating DM field by a phase factor of π/2.
• When f  f0 , the size changes of the solid aresuppressed by the factor (f0 /f )2  1 comparedwith the adiabatic case. The size oscillations of thesolid lag behind the oscillating DM field by a phasefactor of π.
All of the relevant optical components in ground-based
gravitational-wave detectors are approximately cylindrically symmetric. In this case, the fundamental frequencyof the longitudinal vibrational mode is given by:
f0 =

vs
,
2Li

(8)

where vs is the sound speed in the solid component (typically of the order of half-dozen km/s in most commonlyused materials) and Li is the length of the component.
C.

Refractive-index changes in beam-splitter

Time-varying α and particle masses also affect the
propagation of a light beam through the beam-splitter
via alteration of the refractive index of the beam-splitter.
To estimate the size of these effects, we assume that thelaser (angular) frequency ω is much larger than all of thephonon-mode frequencies of the beam-splitter and adopt
a simple Lorentz model with a single electronic mode of
frequency ω0 , in the regime of normal dispersion, ω0 > ω.
In this case, far away from the electronic resonance, theexpression for the refractive index of a dielectric material
reads:√(9)n ≈ εr ,with the relative permittivity given by:ξN α1
me ω02 − ω 2


ξN α
ω2
≈1+
1+ 2 ,me ω02ω0

εr ≈ 1 +(10)

where N is the number density of atoms in the dielectric material and ξ is a numerical constant that is independent of the physical constants. Let us first consider
the non-dispersive term ξN α/(me ω02 ) in Eq. (10). Since3
N ∝ 1/a3B = (me α) and ω0 ∝ me α2 , the combinationof parameters ξN α/(me ω02 ) is independent of the physical constants. Hence the main effects of varying physicalconstants on the index of refraction arise through thedispersive term ω 2 /ω02 in (10):




δn
ω ∂n δω δω0ω ∂n δωδα δme
≈
−
≈
−2
−
.
n
n ∂ω ω
ω0n ∂ω ω
αme(11)
Most experiments use a laser of wavelength λ ≈ 1 µm
and a silica beam-splitter, for which n ≈ 1.5 andω/n · ∂n/∂ω ≈ 5 × 10−3 . If the laser is stabilised toa high-finesse reference cavity, in which the length of thesolid spacer between the mirrors is allowed to vary naturally, and the cavity length changes are adiabatic, then by
Eq. (6) we have δω/ω = −δLcav /Lcav ≈ δα/α + δme /me ,giving:δnδα≈ −5 × 10−3
.
n
α

(12)

On the other hand, if the laser is stabilised to a highfinesse reference cavity, in which the cavity length is independent of the length of the spacer between the mirrors

5(e.g., through the use of a multiple-pendulum suspension
system for the mirrors), or if the cavity length changes
are sufficiently rapid [see Eq. (7)], then δω/ω is approximately independent of changes in the physical constants.
In this case, we instead have:δnδα δme−3≈ −5 × 10
2
+
.
(13)
n
α
me
In all existing ground-based gravitational-wave detectors,
the laser is ultimately stabilised to the common-mode interferometer arm length, which is isolated against length
fluctuations (via the suspension points) via multiplependulum suspension systems for the mirrors, and so inthis case Eq. (13) applies [27].
D.

Centre-of-mass displacements of test masses

By analogy with the acceleration that a test particle ortest mass experiences in the presence of a spatial gradientin a potential, spatial gradients in α and the particle
masses give rise to accelerations on test particles and test
masses of mass Mtest [28]:δatest = −∇Mtest
.
Mtest

(14)

The overall mass of an atom with Z  1 consists of threedifferent types of contributions:
Matom ≈ AmN + Zme +aC Z 2
,
A1/3

(15)

where A is the total nucleon number of the nucleus.
The last term in (15) denotes the energy associatedwith the electrostatic repulsion between protons in a
spherical nucleus of uniform electric-charge density, withaC ≈ 0.7 MeV.
Most ground-based experiments employ beam-splittersand test masses made of silica. In this case, the relative contributions to the total test mass from the nucleon masses, electron mass and Coulomb energy are ≈ 1,
≈ 3 × 10−4 and ≈ 1.4 × 10−3 , respectively. LISA employs
Au-Pt alloy (≈ 60:40 ratio) test masses. In this case,the relative contributions to the total test mass from thenucleon masses, electron mass and Coulomb energy are≈ 1, ≈ 2 × 10−4 and ≈ 4 × 10−3 , respectively.
III.

COHERENTLY OSCILLATING CLASSICALDARK-MATTER FIELDS
A.

Dark-matter theory

Feebly interacting, low-mass (sub-eV) spinless particles are well-motivated candidates for DM. Perhaps themost renowned particle of this category is the canonical axion, which is a pseudoscalar (odd-parity) particle. Apart from the axion, low-mass scalar particles(such as the dilaton) may also exist in nature. Lowmass spinless particles can be produced non-thermallyin the early Universe via the “vacuum misalignment”
mechanism [29–31], and they can subsequently form a
coherently oscillating classical field [32]: φ ≈ φ0 cos(ωφ t),where the angular frequency of oscillation is given byωφ ≈ mφ c2 /~, with mφ being the mass of the spinless particle. Although these DM particles are typically produced with very small kinetic energies, they become virialised during the formation of galactic structures (vvir ∼ 300 km/s locally), giving these particles the2
finite coherence time: τcoh ∼ 2π/(mφ vvir
) ∼ 106 tosc ; i.e.,∆ωφ /ωφ ∼ 10−6 (see [33] for details of the expected lineshape). In other words, the oscillations of this galactic
DM field are practically monochromatic, with a quality
factor of Qφ ∼ 106 . The oscillating DM field carries thenon-zero time-averaged energy density:hρφ i ≈ ρφ ≈m2φ φ20,2(16)

and satisfies the non-relativistic equation of state hpφ i hρφ i, making it an ideal candidate for cold DM. If spinless particles comprise the entirety of the observed DM,then their reduced de Broglie wavelength cannot exceed the DM halo size of the smallest dwarf galaxies(Rdwarf ∼ 1 kpc). This places the following lower boundon their mass: mφ & 10−22 eV, which can be relaxed if
these particles make up only a sub-dominant fraction of
the observed DM. In this section, we focus on the linear
interactions of the field φ in (1). We mention that one
may also separately consider the case of quadratic interactions of the field φ in (2), see [5, 6, 34] for the various
intricacies of such types of interactions.
B. Michelson interferometers(GEO 600, Fermilab holometer)

Consider a power- and possibly signal-recycled (dualrecycled if both) laser interferometer without FabryPerot resonators in the two arms, as illustrated by thesimplified layout in Fig. 1. Archetypes of this Michelson configuration include the GEO 600 interferometer(L = 600 m without account of the single folding of thearms, l = 8 cm) and the Fermilab holometer (L = 40 m,
l = 1.3 cm). The input laser beam is fed into the power
recycling cavity consisting of the mirror ‘PRM’ and theMichelson interferometer consisting of the beam-splitter‘BS’ and mirrors ‘ETMX’ and ‘ETMY’. When operating at destructive interference at the dark port, thepower recycling cavity enhances the circulating power,
thus enhancing shot-noise-limited sensitivity. The signal recycling mirror ‘SRM’ (if present) increases the
low-frequency sensitivity of the Michelson interferometer.
The inset in Fig. 1 shows how the laser beam traverses
the beam-splitter.
The interferometer output can be expressed in terms ofthe difference of the optical path lengths in the two arms

6of the interferometer, ∆L = Lx − Ly , with Lx ≈ Ly .
DM-induced time-varying changes in the size of a freelysuspended beam-splitter of thickness l will shift the mainreflecting surface (power reflectivity of R = 50%) by theamount δl/2 in the frame of the interferometer. Assuming a nominal angle of the beam-splitter with respectarms of 45◦ , we have δLx ≈√√ to the interferometerδ[ 2nl − l/(2 2) − w/2], where for simplicity we haveomitted a geometric correction√ factor from Snell’s law ofrefraction, and δLy = −δl/(2 2) − δw/2, where w is the
thickness of the freely-suspended arm mirrors ‘ETMX’and ‘ETMY’ (w = 10 cm for the GEO 600 detector, while
w = 1.3 cm for the Fermilab holometer). Hence we have:√δ (Lx − Ly ) ≈ 2(n · δl + l · δn) .
(17)Michelson interferometers typically do not use a usualreference cavity to stabilise the laser. Instead, the laser
is stabilised to the common-mode interferometer arm
length, which is isolated against length fluctuations (via
the suspension points) via multiple-pendulum suspension
systems for the mirrors. In this case, Eq. (13) applies for
δn, while δl is governed by Eq. (7). Hence for a DM oscillation frequency fDM well below the frequency of thefundamental vibrational mode of the beam-splitter f0,BS ,we have:√δ (Lx − Ly )2nl
δα δme
≈
−
−
,
(18)
L
L
α
me
which is the case for the entire optimal frequency rangeof the GEO 600 detector. On the other hand, for fDM 
f0,BS , we have:√δ (Lx − Ly )
2nl
δα
δme
≈
−10−2
− 5 × 10−3
.
L
L
α
me
(19)
In the case of the Fermilab holometer, f0,BS ≈ 200 kHz.
We note that the interferometer arm length L = (Lx +Ly )/2 can be eliminated from Eqs. (18) and (19), but
is included here to be consistent with the conventionalstrain calibration of the interferometer output.
The sensitivity of measurements with a single interferometer (such as the GEO 600 detector) to an oscillating
DM field with finite coherence time τcoh improves with
the integration time tint as ∝ (tint )−1/2 in the temporally coherent regime tint . τcoh , and then continues toimprove as ∝ (tint τcoh )−1/4 in the temporally incoherent regime tint & τcoh . On the other hand, the sensitivity of cross-correlation measurements with a pair ofindependent and isolated interferometers (such as thetwo co-located interferometers of the Fermilab holometer) improves as ∝ (tint )−1/2 for all integration times,
provided that the DM field is spatially coherent over theentire apparatus (meaning that two identical interferometers would see the same DM signal in this case). In thecurrent Fermilab holometer setup, L = 40 m and thespatial separation between the two independent interferometers is 0.9 m. Since both of these length scales aremuch smaller than the reduced de Broglie wavelength ofan oscillating DM field for the entire optimal frequency
range of the Fermilab holometer, the DM field is therefore
spatially coherent over the entire holometer.
In Michelson interferometers of the type discussed thus
far, the DM effects on the beam-splitter give the main
contribution to the output signal. Let us briefly also discuss some possible subleading contributions to the output
signal. There are common-mode effects associated withtemporal variations of the laser frequency and interferometer arm lengths. Since the two arms of an interferometer are practically equal in length, these common-mode
effects are strongly suppressed in the output signal (andmoreover the latter common-mode effect is further suppressed in the detector’s optimal frequency range due tothe multiple-pendulum suspension systems for the mirrors). An oscillating DM field also induces common-modetime-varying size changes of the arm end mirrors ‘ETMX’and ‘ETMY’, which cancel for a homogeneous DM field
φ = φ0 cos(mφ t) if these mirrors have the same thickness.
In reality, there is a small non-zero contribution to the
output signal due to a phase difference in the oscillating
DM field between the two arm end mirrors, since in thelaboratory frame of reference an oscillating DM field contains a position-dependent term: φ = φ0 cos(mφ t−pφ ·r),
where pφ ≈ mφ hv φ i is the average momentum of a DM
particle as seen in the laboratory frame. In this case,there is an effect suppressed by ∼ L/λdB . |v φ | ∼ 10−3in the optimal frequency range of ground-based detectors.
Incidentally, since l, w  L, this also justifies treating thebeam-splitter and test masses as pointlike objects.
Using Eqs. (4), (16), (18) and (19), we estimate thecurrent sensitivities of GEO 600 [13, 14] and the Fermilab holometer using both of its co-located interferometers[15, 16] to the linear interactions of the DM field φ withthe photon and electron in (1). We present these estimates as solid lines in Fig. 3 (red = GEO 600, purple= Fermilab holometer using both of its co-located interferometers), assuming that the φ particles saturate theaverage local cold DM density of ρDM ≈ 0.4 GeV/cm3 .
We note that, because the amplitude and effects of theoscillating DM field scale as ∝ 1/mφ , the sensitivitiesof the detectors to the underlying DM interactions in(1) peak at frequencies somewhat lower than the optimal
frequencies for characteristic strains.

C.

Fabry-Perot-Michelson interferometers(LIGO, VIRGO, KAGRA)

Consider now a dual-recycled Fabry-Perot-Michelsoninterferometer of the type shown in the simplified layout in Fig. 2. Archetypes of this configuration include
the LIGO (L = 4 km, l = 6 cm), VIRGO (L = 3 km,
l = 5.5 cm) and KAGRA (L = 3 km, l = 8 cm) interferometers. In contrast to the layout of a dual-recycled
Michelson interferometer (see Fig. 1), the additional mirrors ‘ITMX’ and ‘ITMY’ form Fabry-Perot resonators in7each arm. While these Fabry-Perot cavities increase thestrain sensitivity of the interferometer to gravitational
waves, they generally reduce the effective sensitivity of
the interferometer to DM effects on the beam-splitter,
since in this case for each to-and-back passage across thebeam-splitter, the laser beam encounters N  1 to-andback passages within the Fabry-Perot cavities. In thiscase, Eq. (17) is modified accordingly:√2
δ (Lx − Ly )eff ≈(n · δl + l · δn) .
(20)
Neff
The bandwidth of the Fabry-Perot-Michelson interferometer is determined by the total propagation time ofphotons through the Fabry-Perot arm cavities, but isincreased (in the case of LIGO, VIRGO and KAGRA)
by the signal recycling mirror (which in this configuration effectively shortens the number of round-trips for
those photons that carry signal information in the measurement band). For DM oscillation frequencies below
this modified bandwidth of the Fabry-Perot-Michelsoninterferometer, we have 1/Neff ≈ 1/N . However, forDM oscillation frequencies above the modified bandwidthof the Fabry-Perot-Michelson interferometer, 1/Neff increases approximately linearly with fDM due to the finite propagation speed of photons. If the thicknesses ofthe freely-suspended Fabry-Perot arm mirrors in botharms are equal (and comparable to the beam-splitter
thickness), then the quasi-common-mode effects of an oscillating DM field on the arm mirrors, as discussed inSec. III B, are subleading in the optimal frequency rangeof the detector, provided that N . 103 . In the current LIGO setup, l = 6 cm, w = 20 cm and N ∼ 102 ,so the beam-splitter effect in Eq. (20) indeed gives themain contribution to the output signal in this case; however, the quasi-common-mode effects on the arm mirrors
in this case are not as strongly suppressed (comparedto the beam-splitter effect) as they are in the case of aMichelson interferometer (see Sec. III B).
While some Fabry-Perot-Michelson interferometers use
a small-scale reference cavity to initially lock the laser,
the laser is ultimately locked and stabilised to thecommon-mode interferometer arm length, which is isolated against length fluctuations (via the suspension
points) via multiple-pendulum suspension systems for the
mirrors. In this case, Eq. (13) applies for δn. The entire optimal frequency range of the LIGO detector lies
well below the frequency of the fundamental vibrationalmode of the beam-splitter, and so δl is governed by the
adiabatic formula (6). Thus, we have:√δ (Lx − Ly )effδα δme
2nl
≈
−
−
.
(21)
L
Neff Lαme
If the thicknesses of mirrors ‘ETMX’ and ‘ITMX’ differ by an amount ∆w with respect to the thicknesses ofmirrors ‘ETMY’ and ‘ITMY’, then there will be an additional contribution to the output signal given by:δ (Lx − Ly )∆w
δα δme
≈−
−
−
.
(22)
L
L
α
me

In the current LIGO interferometers, |∆w| ≈ 80 µm, andso in this case the contribution in (22) will be smaller
than the DM effect on the beam-splitter in Eq. (21) forthe entire optimal frequency range of the LIGO detector.
However, it is possible to significantly increase the sensitivity of a Fabry-Perot-Michelson interferometer to timevarying α and me by making the thicknesses of the freelysuspended Fabry-Perot arm mirrors sufficiently different
in the two arms.
Using Eqs. (4), (16), (21) and (22), we estimate thesensitivities of LIGO [17] in both its current configurationand in a modified configuration where the arm mirrors
in the two arms have appreciably different thicknesses(we take ∆w/w = 10% for concreteness) to the linear
interactions of the DM field φ with the photon and electron in (1). We present these estimates as blue lines
in Fig. 3 (solid line = current configuration and single
detector, dashed line = modified configuration and single detector, thin dashed line = modified configuration
and both detectors [45]), assuming that the φ particles
saturate the average local cold DM density. Note the difference in the shapes of the curves for single detectors at
higher frequencies, due to the dominant effect from thearm mirrors in the modified case, compared to the casewhere the beam-splitter effect dominates. The sensitivity of cross-correlation measurements using both LIGO
detectors is enhanced compared to measurements using a
1/4
single LIGO detector by the factor ∼ (tint /τcoh ) 1.
Laser interferometers of different types (e.g., one Michelson interferometer and one Fabry-Perot-Michelson interferometer) can also benefit significantly from crosscorrelation measurements, provided that their individual
sensitivities to the underlying DM interaction parameters
are similar.

D.

Resonant narrowband experiments

In Secs. III B amd III C, we considered broadband detection strategies. In this section, we consider the possibility of resonant narrowband searches with laser interferometers. The crucial observation is that the oscillationsof the galactic DM field are expected to be practicallymonochromatic, with a quality factor of Qφ ∼ 106 . We
begin with small-scale Michelson interferometers, havingin mind the Fermilab holometer as a possible platform.
If the DM oscillation frequency matches the fundamental frequency of the longitudinal vibrational mode of thebeam-splitter (≈ 200 kHz for the Fermilab holometer),then, according to Eq. (7), the DM-induced time-varying
size changes of the beam-splitter will be enhanced by thefactor Q = min{Qφ , Qmech }:√δ (Lx − Ly )2Qnl
δα δme
≈
−
−
.
(23)
L
L
α
me
Materials with quality factors comparable to the DM
quality factor of Qφ ∼ 106 are available. However, in8order to achieve the desired mechanical quality factor ofat least Qmech ∼ 106 (and hence an overall quality factorof Q ∼ 106 ), one would need to ensure that the clamps
which support the beam-splitter are designed in such a
way as not to degrade the overall quality factor.
For simplicity, we neglect higher-order-harmonic vibrational modes in the ensuing discussion and, furthermore, assume that the measurements are limited byBrownian thermal noise. By the equipartition theorem, the potential energy associated with the longitudinal vibrational mode of the beam-splitter is given by2
MBS ω0,BS
x2 /2 = kB T /2, where x2 is the meansquare displacement of the reflecting surface, T is thetemperature and kB is the Boltzmann constant. In thevicinity of the longitudinal vibrational mode resonance,
the thermal-noise amplitude spectral density is hence
given by:
s
QkB T
.
(24)
ASD (thermal noise) =32πMBS f0,BSWe thus see that beam-splitters of larger transverse sizes
are advantageous with regards to thermal noise (in thecurrent Fermilab holometer setup, MBS = 130 g). By
cooling the system from room temperature to liquidhelium temperature, the thermal noise can be reduced bya factor of ∼ 10. Brownian thermal noise, being broadband, scales as ∝ Q1/2 , and so the signal-to-noise ratio insuch resonant narrowband experiments scales as ∝ Q1/2 .
In order to scan over a range of different DM particle masses, one must incrementally vary the fundamental
resonance frequency. One can alter f0,BS by changing the
thickness of the beam-splitter via ablation and polishing.
For very small incrementations of f0,BS , it would generally be more efficient to incrementally change the temperature of the system. Near room temperature, silica
has a thermal expansion coefficient of ∼ 10−6 K−1 and asound speed of vs ≈ 6 km/s, while sapphire has a thermal
expansion coefficient of ∼ 10−5 K−1 and a sound speed ofvs ≈ 10 km/s. The relative sound speed change in silica
is ∼ 10−3 over a temperature interval of ∆T ∼ 10 K nearroom temperature, which would cover a frequency rangeof ∆f0,BS /f0,BS ∼ 10−3 over the same temperature interval, see Eq. (8). Beam-splitter thicknesses in the range∼ 1 − 10 cm are routinely used in existing Michelson interferometers. In dedicated experiments, beam-splitter
thicknesses in the range few mm − 30 cm may be achievable. Thus, fundamental frequencies of the beam-splitterin the range ∼ 10 kHz − 1 MHz can reasonably be covered. This frequency range overlaps with the frequency
range of proposed resonant narrowband experiments withresonant-mass detectors [46].
Just like the broadband searches discussed inSecs. III B amd III C, the sensitivity of such narrowband
searches with a single Michelson interferometer improves
with the integration time as ∝ (tint )−1/2 for tint . τcoh ,then as ∝ (tint τcoh )−1/4 for tint & τcoh , while the sensitivity of these narrowband searches with a pair of co-locatedMichelson interferometers improves as ∝ (tint )−1/2 for
all integration times. Using Eqs. (4), (16), (23) and(24), we estimate the sensitivities of a single small-scale
Michelson interferometer and a pair of co-located smallscale Michelson interferometers using the above narrowband approach and operating near room temperature
to the linear interactions of the DM field φ with thephoton and electron in (1), assuming that the measurements are limited by Brownian thermal noise and that
all of the dimensions of the beam-splitter are altered ina proportional manner. We present these estimates as
the dashed and thin dashed purple lines, respectively, inFig. 3, for Q ∼ 106 , covering a DM particle mass rangeof ∆mφ /mφ ≈ 1 and assuming that the φ particles saturate the average local cold DM density. We note that,
because the amplitude of the DM field scales as ∝ 1/mφand that the resonant condition fDM = f0,BS implies thescaling relation l ∝ 1/mφ for a fixed sound speed [seeEq. (8)], the size of the resonantly-enhanced DM effects
in our proposed narrowband experiments scales roughly
as ∝ 1/m2φ . This scaling strongly favours such narrowband searches at lower DM particle masses.
As mentioned in Sec. III C, some Fabry-PerotMichelson interferometers use reference cavities to initially lock the laser. In such cases, there can be a similar resonant enhancement when the DM oscillation frequency matches the fundamental frequency of the longitudinal vibrational mode of the reference cavity. For atypical reference cavity length of Lcav ∼ 0.5 m and mostcommonly used materials, the fundamental frequency ofthe longitudinal vibrational mode within the cavity is
f0,cav ∼ 6 kHz, see Eq. (8). Since it is generally preferable to lock and stabilise the laser to the common-mode
interferometer arm length, rather than to a small-scalereference cavity, it seems difficult to take advantage ofthis resonant enhancement factor in practice for mostFabry-Perot-Michelson interferometers. It may be possible though for the VIRGO interferometer, where thereference cavity is suspended within the main vacuum
envelope and thus is in a lower-noise environment than
the reference cavity in LIGO. For this method to work,
the read-out of the reference cavity length would have tobe sufficiently sensitive.
One may alternatively modify broadband clock-cavity
comparison experiments of the type considered in [9–12,
42] to scan over a range of DM particle masses where thereference cavity length changes are resonantly enhanced.
In this case, we would have [47]:δ (ωcavity /ωatom )δα δme≈Q+,(25)
ωcavity /ωatomαmewhere we have assumed that the relative sensitivity coefficients of the atomic transition frequency ωatom to apparent changes in α and me are much smaller than Q.
Most reference cavities have quality factors well in excessof the DM quality factor of Qφ ∼ 106 , giving an overall
quality factor of Q ∼ 106 in this case. Reference cavities
with lengths in the range ∼ 0.1 − 1 m are readily avail-9able, meaning that fundamental frequencies of the cavity
in the range ∼ 3 − 30 kHz can reasonably be covered.

E.

Space-based experiments (LISA)

Let us briefly discuss space-based laser interferometry experiments. The archetypal example in this caseis LISA, which is a three-arm interferometer in a triangular geometry (L = 2.5 × 109 m). On board each of thethree spacecrafts is a pair of quasi-freely-floating cubic
test masses of side length s = 4.6 cm. At the low DM
oscillation frequencies that lie in the optimal frequency
range of the LISA detector (∼ 10−4 − 10−1 Hz), the test
cube size changes are described by the adiabatic formula(6):δα δme
δs
≈−
−
s
α
me
p


2ρφ
1
1
≈−
cos(mφ t)
+
,
mφ
Λγ
Λe

(26)

where we have used Eqs. (4) and (16) in the second line.
Additionally, the freely-floating Au-Pt test cubes will experience a time-varying acceleration in accordance with
Eq. (14). For the oscillating DM field φ = φ0 cos(mφ t −pφ · r), we find:δatest ≈ −p
×2ρφ v φ sin(mφ t)

4 × 10−3
2 × 10−4
1
+
+
Λγ
Λe
ΛN


,

(27)

where we have again used Eqs. (4) and (16). The freelyfloating test masses will, therefore, undergo the following
time-varying centre-of-mass displacements:p2ρφ v φsin(mφ t)
δxtest ≈ +m2φ
4 × 10−3
2 × 10−4
1
×
+
+
.
(28)
Λγ
Λe
ΛN
We note that the time-varying size changes and centreof-mass displacements of the test cubes in Eqs. (26) and(28), respectively, are out of phase with respect to eachother by the factor π/2, and that the former effect scalesas ∝ 1/mφ , while the latter scales as ∝ 1/m2φ . Additionally, in contrast to the DM-induced time-varying
size changes of the test cubes, the DM-induced timevarying centre-of-mass displacements of the test cubes
are anisotropic, meaning that the resulting observablesignatures will strongly depend on the orientation of thedetector and its components with respect to v φ .
The LISA interferometer operates on the principle oftime-delay interferometry, which basically involves measuring a particular linear combination of the three arm
lengths to cancel the laser phase noises that would otherwise be imprinted in length measurements of the unequal(and naturally time-varying) arm lengths of the interferometer. Hence the common-mode effects of a homogeneous DM field φ = φ0 cos(mφ t) on test masses in different spacecrafts will cancel to leading order. In particular,
centre-of-mass displacements of identical test masses bya homogeneous DM field correspond to the translation of
the entire system. The leading non-vanishing contribution to the output signal arises due to phase differencesin the oscillating DM field, φ = φ0 cos(mφ t − pφ · r), between pairs of spacecraft, and so the DM-induced effects
on the pairs of test masses in Eqs. (26) and (28) will beeffectively suppressed by the factor ∼ L/λdB  1. It is
possible to significantly increase the sensitivity of LISA
to time-varying α and me by replacing some of the freelyfloating Au-Pt alloy test masses by test masses made ofmuch lighter elements (such as Be, Al and/or Ti) [48]. In
this case, some of the common-mode suppression would
be lifted due to the maximally different mass-energy contributions of elements from different regions of the periodic table, see Eq. (15). Using Eqs. (26) and (28), we
estimate the projected sensitivities of LISA [22] in bothits standard configuration and with our suggested modification (we suppose that some of the Au-Pt alloy test
masses are replaced by Be test masses for concreteness) tothe linear interactions of the DM field φ with the photon,
electron and nucleons in (1). We present these estimatesas green lines in Fig. 3 (solid line = standard configuration, dashed line = modified configuration), assumingthat the φ particles saturate the average local cold DM
density.
In space-based laser-interferometric detectors, thetime-varying centre-of-mass displacements of the test
masses are generally more important than the timevarying size changes of the test masses. Indeed, the ratioof the two effects, modulo different material-dependent
sensitivity coefficients, is of the order ∼ vφ /(mφ s),which is  1 in the optimal frequency range of the
LISA detector. In contrast, in ground-based Michelsoninterferometers, the (common-mode-suppressed) timevarying centre-of-mass displacements of (identical) test
masses and beam-splitter are generally less important
than time-varying size changes of the beam-splitter. In
this case, the ratio of the two effects, modulo different
system/material-dependent sensitivity coefficients andgeometric factors, is of the order ∼ vφ2 L/l, which is  1
for a typical ground-based detector [49]. Using Eq. (28),
together with the relevant sensitivity coefficients for silica test masses presented following Eq. (15), we estimate
the current sensitivities of ground-based laser interferometers to the linear interaction of the DM field φ with
nucleons in (1). We present these estimates as solid lines
in Fig. 3 (red = GEO 600, blue = LIGO, purple = Fermilab holometer using both of its co-located interferometers), assuming that the φ particles saturate the average
local cold DM density. Using the results of the recent
data analysis in Ref. [44] that searched for analogous
time-varying centre-of-mass displacements of LIGO’s test
masses in the LIGO-O1 data due to dark-photon interac-

10
tions (instead of scalar interactions), we place bounds on
the linear interactions of the DM field φ with the photon,
electron and nucleons in (1). These LIGO-O1 bounds
are denoted by the blue region in Fig. 3. We note that,in contrast to the beam-splitter effect in Eq. (21), there
is no 1/Neff suppression factor in Fabry-Perot-Michelsoninterferometry searches for time-varying centre-of-massdisplacements of the beam-splitter and test masses. We
also note that DM-induced time-varying centre-of-massdisplacements of freely-suspended interferometer components are phenomenologically more interesting for darkphoton interactions [44, 50], due to the lack of an extra
velocity suppression factor vφ  1 compared with thescalar interactions considered in the present work [seeEqs. (27) and (28), as well as [51, 52]].

F.

Local dark-matter overdensities

For DM particle masses corresponding to the optimal
frequency ranges of ground-based laser interferometers, it
is possible for the DM density near the surface of Earth
to be many orders of magnitude greater than the average local cold DM density of ρDM ≈ 0.4 GeV/cm3 inferred from galactic rotation curve measurements for our
Galaxy. Such a situation may arise, e.g., due to the capture of an overdense region of DM by the gravitational
well of Earth or the Sun. This is in stark contrast tolaboratory experiments that search for DM with particle
masses mφ ∼ 10−22 eV (see, e.g., [39, 40, 53]), where
the Heisenberg uncertainty principle prevents the gravitational collapse of such ultra-low-mass DM fields onlength scales shorter than their reduced de Broglie wavelength (which is astronomical in this case). We focus
on two specific cases of static and uniformly-distributed
overdensities of DM that are centered on the Sun andEarth, respectively.
For a spherical DM overdensity centered on the Sun,the most stringent bounds on the largest allowable DMdensity near Earth’s surface generally come from planetary ephemeris measurements. For a spherical DM overdensity of radius R ≈ 1 AU, the largest allowable DMdensity near Earth’s surface is ∼ 105 times the average
local cold DM density [54]. In this case, the DM particles are assumed to be gravitationallybound to the Sun.
D E
2
By the virial theorem, vφ ≈ GM /R ∼ 10−8 at theposition of Earth. Since the DM particles are also assumed to be localised within the sphere of radius R, we
further require that λdB . R, which sets the requirement mφ & 10−14 eV. This is serendipitous, because
this includes the entire optimal frequency ranges of current ground-based laser interferometers. Additionally,
the quality factor associated with the DM oscillations is
Qφ ∼ 108 in this case, which is ∼ 100 times larger thanin the usual “galactic picture”.
For a spherical DM overdensity centered on Earth, themost stringent bounds on the largest allowable DM den-sity near Earth’s surface generally come from a combination of lunar laser ranging and geodetic surveyancemeasurements. For a spherical DM overdensity of radiusR ≈ 60R⊕ , the largest allowable DM density near Earth’s
surface is ∼ 1011 times the average local cold DM density [55]. In this case, the DM particles are assumed to be
gravitationallybound to Earth. By the virial theorem,D E
2vφ ≈ GM⊕ /R⊕ ∼ 10−9 near Earth’s surface. Since
the DM particles are assumed to be localised within thesphere of radius R, we require that λdB . R, which sets
the requirement mφ & 10−11 eV. This is fortunate, as
this includes the entire optimal frequency range of thecurrent Fermilab holometer. In this case, the quality
factor associated with the DM oscillations is Qφ ∼ 109 ,which is ∼ 103 times larger than in the usual galactic
picture.
We note that the sensitivities of ground-based laser interferometers to the underlying DM interactions are enhanced with respect to non-DM-based experiments forthe scenarios discussed in this section, not only becauseof the increased DM density (the size of the DM effects√
scale as ∝ ρφ for linear interactions, while non-DM
effects are independent of ρφ ), but also because of theincreased coherence time τcoh ∝ Qφ (which diminishes
the role of incoherent averaging for measurements witha single interferometer when tint & τcoh and, in the caseof resonant narrowband experiments, may increase thesignal-to-noise ratio by increasing the overall quality factor Q). Another possible way for the DM density near
the surface of Earth to be greatly enhanced is via the formation and subsequent capture of DM objects that arebound by their own self-gravity and self-interactions [56].

IV.

TOPOLOGICAL DEFECTS

A.

Theory of topological defects

Topological defects are stable solitonic configurationsof DM fields that may be produced as a result of a phasetransition in the early Universe [57]. These (possibly
macroscopic) objects may come in a variety of dimensionalities: 0D (monopoles), 1D (strings) or 2D (domain
walls). As a simple illustrative example, consider a real
spinless field φ in one spatial dimension with the selfpotential V (φ) = σ(φ2 − η 2 )2 , which has two energetically equivalent vacua at φ = +η and φ = −η. In thiscase, a stable domain wall with the√ transverse profile
φ(x) = η tanh(mφ x), where mφ = ση, will form between the two vacua. The transverse size of this domain
wall is set by the reduced Compton wavelength of theunderlying field, d ∼ 1/mφ , and any physical effects produced by this wall arise only at the boundary between
the two vacua. The energy density stored inside a domain wall is given by ρinside ∼ m2φ φ2m ∼ φ2m /d2 , where
φm is the amplitude (relative to the vacuum states) of the
field φ inside the wall. A network of finite-sized domain

11
walls can account for the observed DM. We can express
the amplitude φm in terms of the energy density associated with a domain-wall network ρTDN , the typical speedof a wall vTD (locally, vTD ∼ 300 km/s), and the averagetime between encounters of a system (e.g., Earth) with awall T :φ2m ∼ ρTDN vTD T d .

(29)

If domain walls of a single type comprise the entirety ofthe observed DM, then their largest dimension(s) cannot
exceed the DM halo size of the smallest dwarf galaxies.
This places the following upper bound on their transverse
size: d  1 kpc, which can be relaxed if these objects
make up only a sub-dominant fraction of the observed
DM. In this section, we focus on the quadratic interactions of the field φ in (2). We mention that one may also
separately consider the case of Yukawa-type interactionsof DM with SM matter and the effects of the resulting
Yukawa force between a passing topological defect andthe test masses of a detector, see [58] for more details.
B.

Experiments

The passage of a domain wall through a laserinterferometric detector can result in similar signatures
to those produced by conventional gravitational waves.
One key difference between the two types of signatures isthat domain walls are expected to pass through a detector with a relative speed of vTD ∼ 10−3 , rather than at
the speed of light. Also, since the velocity distributionof domain walls is expected to be Maxwell-Boltzmannian(with a local average velocity of approximately zero), the“event” rate should be maximal for domain walls coming from the direction of the dark-matter “wind” (thatis, from the direction towards which the Solar System ismoving).
The form of the output signal due to the passage ofa domain wall through a laser interferometer will depend on several factors, including the size and geometry of the detector, the relative speed and direction of
motion of the domain wall with respect to the detector,
as well as the transverse size and cross-sectional profileof the wall. Rather than performing detailed numericalsimulations of expected output signals, which we defer
for future studies, let us consider the general features ofdomain-wall searches with laser interferometers, in orderto estimate the sensitivities of these types of detectors to
the quadratic interactions of the field φ in (2).
From Eqs. (5) and (29), we see that the magnitude ofthe domain-wall effects on the freely-suspended components of a ground-based detector (or freely-floating test
masses of a space-based detector) scale as ∝ d. Ford  L, the output signal will generally contain appreciable power at and above the characteristic frequencyof f ∼ vTD /L, which typically lies in the optimal frequency range of the detector. Additionally, there is generally no common-mode suppression in this case, in con-trast to searches for oscillating DM fields (see Sec. III).
On the other hand, for d  L, the output signal will
generally be peaked at the characteristic frequency off ∼ vTD /d, which typically lies well below the optimal
frequency range of the detector, and there will also be
common-mode suppression in this case. Hence we expect
the sensitivity of a particular detector to the quadraticinteractions of φ to be maximal for defects of transverse
size d ∼ L. We also point out that, in contrast to
searches for oscillating DM fields, there is generally no
1/Neff suppression of the output signal due to transient
α or me variations induced by a passing domain wall on
a Fabry-Perot-Michelson interferometer (compare with
Sec. III C).
For domain walls of transverse size greater than the
thicknesses of the freely-suspended (or freely-floating)components of a detector, the wall can envelop the entirety of the regions of the components that are relevant
for the interferometry measurement. In this case, from
Eqs. (5), (6) and (29), the maximum size change of a test
mass is given by:"  2 #
2
δLtest1
1∼ −ρTDN vTD T d+.
Ltest max
Λ0γ
Λ0e
(30)
For domain walls of transverse size less than the thicknesses of the freely-suspended (or freely-floating) components of a detector, the wall cannot envelop the components in their entirety. For a domain wall travelling parallel (in terms of its normal vector) to one of the armsof a ground-based detector, the size changes of the test
masses in that arm will be suppressed compared to thatin (30) by the factor ∼ d/Ltest  1. On the other hand,for a domain wall travelling perpendicular to one of thearms of a ground-based detector, the size changes of thatarm’s test masses in the region through which the laser
beam traverses will only be suppressed if the transverse
size of the domain wall is smaller than the diameter of thelaser beam. In the latter case, the duration of the passage will also be smaller compared with the case when
the domain wall travels parallel to the arm.
The freely-suspended components of a ground-based
detector (or freely-floating test masses of a space-based
detector) will also experience transient accelerations inaccordance with Eq. (14). The passage of a domain wallwith d  Ltest through a test mass occurs within atime interval of ∆t ∼ d/vTD . In this case, the freelysuspended silica components of a ground-based detector
will undergo the following maximum centre-of-mass displacements:
ρTDN T d2v#" TD
1.4 × 10−3
3 × 10−4
1
×+ 0 2 , (31)2 +2(Λ0e )(ΛN )Λ0γ|δxtest |max ∼while on the other hand the Au-Pt test cubes of LISA
will undergo the following maximum centre-of-mass dis-

12
placements:

V.

ρTDN T d2v" TD#
4 × 10−3
2 × 10−4
1
×+ 0 2 , (32)2 +2(Λ0e )(ΛN )Λ0γ|δxtest |max ∼where we have again used Eqs. (5) and (29). On theother hand, the passage of a domain wall with d  Ltest
through a test mass occurs within a time interval of∆t ∼ Ltest /vTD , but the test mass (more precisely, asmall portion ∼ d/Ltest  1 thereof) sees an appreciably
non-zero gradient of the domain wall only during a total
time interval of ∆t ∼ d/vTD . In this case, the maximum
centre-of-mass displacement of a test mass will be suppressed compared to that in (31) or (32) by the factor∼ d/Ltest  1.
Using Eqs. (5), (30), (31) and (32), and noting the suppression factors for thin domain walls discussed above,
we estimate the current sensitivities of GEO 600 [13, 14],LIGO [17] and the Fermilab holometer using both of
its co-located interferometers [15, 16], as well as the
projected sensitivity of LISA [22], to the quadratic interactions of the domain-wall field φ with the photon,
electron and nucleons in (2). We present these estimates as solid lines in Fig. 4 (red = GEO 600, blue
= LIGO, purple = Fermilab holometer using both of
its co-located interferometers, green = LISA), assuming an average time between encounters of Earth anda domain wall of T ∼ 1 year and that the domain-wall
network saturates the average local cold DM density ofρDM ≈ 0.4 GeV/cm3 . For simplicity, in arriving at theseestimates, we have neglected any possible excitation ofunderlying mechanical resonances associated with the detector that may arise due to the passage of a sufficiently
thin (and/or quickly moving) domain wall. Furthermore,in the case of the Fermilab holometer, we have also neglected the fact that when the transverse size of a passing domain wall is smaller than the spatial separation
between the two independent interferometers (0.9 m inthe current setup), if the domain wall at some momentin time simultaneously envelops one pair of test masses,
then at another moment in time the domain wall might
not simultaneously envelop another pair of test masses.
In this case, the cross-correlation signal would be diminished; however, the individual interferometers would still
respond in the usual manner described above. We note
that, in contrast to oscillating DM fields (see Sec. III), thetransient centre-of-mass displacements of the test masses
due to the passage of a domain wall generally dominate
over the transient size changes of the test masses, notonly in space-based detectors but also in ground-based
detectors, since in the case of domain walls of transverse size d  L there is generally no common-mode
suppression and hence no suppression factor of the form
∼ L/λdB  1.

DISCUSSION

In this paper, we have pointed out and explored in detail several new effects of DM on the components of laserinterferometers. The estimated sensitivities of existing,modified and future laser-interferometry experiments tooscillating DM fields and domain walls are presented in
Figs. 3 and 4, respectively. We see that existing groundbased laser interferometers already have sufficient sensitivity to probe extensive regions of unconstrained parameter space in both of these DM models.
In the case of oscillating DM fields, Michelson interferometers are especially sensitive. In particular, the existing GEO 600 detector already offers up to 2 orders ofmagnitude more sensitivity than the best current (nonDM-based) constraints from fifth-force experiments [35–38] in the scalar particle mass range few × 10−13 eV .
mφ . few × 10−11 eV, and at least 8 orders of magnitude more sensitivity than recent DM searches via
clock-cavity comparison experiments [12] in the same
mass range. For scalar particle masses in the range
several × 10−11 eV . mφ . several × 10−9 eV, dedicated resonant narrowband searches using small-scale
Michelson interferometers operating near room temperature may improve in sensitivity by up to 2 orders of magnitude compared with previous fifth-force experiments
and by at least 6 orders of magnitude compared withthe recent DM searches of Refs. [12, 41, 42]. The sensitivity of Fabry-Perot-Michelson interferometers to oscillating DM fields can be increased by making the thicknesses of the freely-suspended Fabry-Perot arm mirrors
different in the two arms of the interferometer, offering
up to 5 orders of magnitude more sensitivity than previous fifth-force experiments for a relative difference inthickness of only 10% and using a pair of LIGO interferometers operating at the design sensitivity of Advanced
LIGO. The sensitivity of LISA to oscillating DM fields
can be increased by replacing some of the freely-floating
Au-Pt alloy test masses by test masses made of muchlighter elements, offering up to a few-dozen times moresensitivity than previous fifth-force experiments.
In the case of domain walls, existing ground-based laserinterferometers are particularly sensitive to domain wallswith transverse sizes of up to several km, offering a sensitivity of up to 8 orders of magnitude beyond all otherexisting experiments and measurements [10, 59–62]. The
Fermilab holometer, with its two co-located Michelsoninterferometers, and especially a global network of laserinterferometers would benefit from their ability to disentangle correlated domain-wall-induced signatures from
uncorrelated noise sources. The space-based LISA interferometer, with its enormous “aperture size”, will besensitive to domain walls with transverse sizes of up to
several million km.
We emphasise that our newly suggested signatures of
DM in laser interferometers scale to the first power ofthe underlying DM interaction parameters (∝ 1/ΛX in
the case of linear interactions and ∝ 1/(Λ0X )2 in the case13of quadratic interactions), whereas conventional non-DM
signatures scale to the second power of the same interaction parameters (∝ 1/Λ2X in the case of linear interactions and ∝ 1/(Λ0X )4 in the case of quadratic interactions). This more favourable scaling for our proposedsearches will be especially advantageous for improving
the sensitivities of future laser interferometers to similar
DM signatures.
Finally, we briefly discuss the issue of technical naturalness, which formally requires the corrections to thescalar particle mass mφ from radiative processes involving the non-gravitational interactions in (1) and (2) tobe smaller than the “bare” mass contribution. For the
linear couplings in (1), the 1-loop corrections to m2φ areδm2φ ∼ [me Λ/(4πΛe )]2 for the scalar-electron couplingand δm2φ ∼ [Λ2 /(4πΛγ )]2 for the scalar-photon coupling,where Λ is a new-physics cut-off scale, which we assume
to be independent of the other mass-energy scales appearing in (1). Technical naturalness thereby requires
Λe & me Λ/(4πmφ ) for the scalar-electron coupling andΛγ & Λ2 /(4πmφ ) for the scalar-photon coupling. We

present these technically-natural regions as pale green
regions in Fig. 3 for Λ ∼ 10 TeV. We see that existing
and modified ground-based laser interferometers have theability to probe sizeable regions of technically-natural parameter space for the scalar-electron coupling.

[1] M. Tanabashi et al. (Particle Data Group), Review ofParticle Physics, Phys. Rev. D 98, 030001 (2018).
[2] Y. V. Stadnik and V. V. Flambaum, Searches for New
Particles Including Dark Matter with Atomic, Molecularand Optical Systems, arXiv:1806.03115.
[3] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,
A. Derevianko and C. W. Clark, Search for new physics
with atoms and molecules, Rev. Mod. Phys. 90, 025008(2018).
[4] We stress the term ‘apparent’ here, since DM fields are
external fields that perturb physical systems, in a similarmanner to how, say, an external electric field perturbs
the energy levels of an atom.
[5] Y. V. Stadnik and V. V. Flambaum, Searching for Dark
Matter and Variation of Fundamental Constants with
Laser and Maser Interferometry, Phys. Rev. Lett. 114,
161301 (2015).
[6] Y. V. Stadnik and V. V. Flambaum, Can Dark Matter
Induce Cosmological Evolution of the Fundamental Constants of Nature?, Phys. Rev. Lett. 115, 201301 (2015).
[7] A. Derevianko and M. Pospelov, Hunting for topologicaldark matter with atomic clocks, Nature Phys. 10, 933(2014).
[8] Y. V. Stadnik and V. V. Flambaum, Searching for Topological Defect Dark Matter via Nongravitational Signatures, Phys. Rev. Lett. 113, 151301 (2014).
[9] Y. V. Stadnik and V. V. Flambaum, Enhanced effectsof variation of the fundamental constants in laser interferometers and application to dark-matter detection,
Phys. Rev. A 93, 063630 (2016).
[10] P. Wcislo, P. Morzynski, M. Bober, A. Cygan, D. Lisak,
R. Ciurylo and M. Zawada, Experimental constraint ondark matter detection with optical atomic clocks, Nature
Astron. 1, 0009 (2016).
[11] C. Kennedy, E. Oelker, T. Bothwell, D. Kedar, L. Sonderhouse, E. Marti, S. Bromley, J. Robinson and J. Ye,

Constraints on Ultralight Dark Matter with an Optical
Lattice Clock, Bulletin of the American Physical Society,
H06.00005 (2018).
[12] S. Aharony, N. Akerman, R. Ozeri, G. Perez, I. Savoray and R. Shaniv, Constraining Rapidly Oscillating Scalar Dark Matter Using Dynamic Decoupling,
arXiv:1902.02788.
[13] H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel,
J. Slutsky and H. Vahlbruch, First Long-Term Application of Squeezed States of Light in a Gravitational-Wave
Observatory, Phys. Rev. Lett. 110, 181101 (2013).
[14] K. L. Dooley et al., GEO 600 and the GEO-HF upgrade program: successes and challenges, Class. Quantum
Grav. 33, 075009 (2016).
[15] A. S. Chou et al., First Measurements of High Frequency
Cross-Spectra from a Pair of Large Michelson Interferometers, Phys. Rev. Lett. 117, 111102 (2016).
[16] A. S. Chou et al., The Holometer: an instrument to probe
Planckian quantum geometry, Class. Quantum Grav. 34,065005 (2017).
[17] D. V. Martynov et al., Sensitivity of the Advanced
LIGO detectors at the beginning of gravitational wave
astronomy, Phys. Rev. D 93, 112004 (2016); Erratum
Phys. Rev. D 97, 059901 (2018).
[18] F. Acernese et al., Advanced Virgo: a second-generationinterferometric gravitational wave detector, Class. Quantum Grav. 32, 024001 (2015).
[19] Y. Aso, Y. Michimura, K. Somiya, M. Ando,O. Miyakawa, T. Sekiguchi, D. Tatsumi and H. Yamamoto (The KAGRA Collaboration), Interferometer design of the KAGRA gravitational wave detector,
Phys. Rev. D 88, 043007 (2013).
[20] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Observation of Gravitational Waves from a Binary Black Hole Merger,
Phys. Rev. Lett. 116, 061102 (2016).

Acknowledgments

We are grateful to Valery Frolov for helpful discussions on the reference cavities of laser-interferometricgravitational-wave detectors and the signal-to-noise ratio scaling of resonant narrowband experiments, as wellas helpful comments on the manuscript. We thank Harald Lück for comments on the cross-correlation analysis of co-located interferometers. We thank Kenneth
A. Strain and GariLynn Billingsley for information aboutthe LIGO test masses. We thank Peter Wolf for helpful discussions. Y.V.S. was supported by the Humboldt
Research Fellowship from the Alexander von Humboldt
Foundation.

14[21] B. P. Abbott et al. (LIGO Scientific Collaboration andVirgo Collaboration), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,
Phys. Rev. Lett. 119, 161101 (2017).
[22] K. Danzmann et al., LISA: Laser Interferometer Space
Antenna. A proposal in response to the ESA call for L3
mission concepts, (2017).
[23] F. Piazza and M. Pospelov, Sub-eV scalar dark
matter through the super-renormalizable Higgs portal,
Phys. Rev. D 82, 043533 (2010).
[24] Y. V. Stadnik and V. V. Flambaum, Improved limits on
interactions of low-mass spin-0 dark matter from atomic
clock spectroscopy, Phys. Rev. A 94, 022111 (2016).
[25] L. F. Pasteka, Y. Hao, A. Borschevsky, V. V. Flambaum and P. Schwerdtfeger, Material Size Dependence
on Fundamental Constants, Phys. Rev. Lett. 122, 160801(2019).
[26] We note that an externally-driven damped harmonic oscillator and a parametrically-driven damped harmonic
oscillator exhibit the same qualitative behaviour in thepresent context.
[27] The finite thicknesses of the power recycling mirror andend arm mirrors, which form the reference cavity for thelaser, do in fact give very small corrections to δω/ω andδn/n. These small corrections to δω/ω, coupled with thedeliberately small geometric arm length difference in actual laser interferometers, also give a negligibly small contribution to the output signal.
[28] We note that anisotropic inhomogeneities in the elemental composition of a test mass can also result in an acceleration on the test mass without there necessarily beinga spatial gradient in α or the particle masses. However,for high-quality test masses that are made of practicallyhomogeneous materials, the resulting observable effects
are generally subleading compared to those due to spatial gradients in α or the particle masses.
[29] J. Preskill, M. B. Wise and F. Wilczek, Cosmology of theinvisible axion, Phys. Lett. B 120, 127 (1983).
[30] L. F. Abbott and P. Sikivie, A cosmological bound on theinvisible axion, Phys. Lett. B 120, 133 (1983).
[31] M. Dine and W. Fischler, The not-so-harmless axion,
Phys. Lett. B 120, 137 (1983).
[32] In order for particles to form a classical field, there must
be a large number of such particles within their reducedde Broglie volume; i.e., nφ λ3dB  1. For DM particles
that saturate the observed average local cold DM density of ρDM ≈ 0.4 GeV/cm3 , this requirement is readilysatisfied by DM particles with sub-eV masses.
[33] A. Derevianko, Detecting dark-matter waves with a network of precision-measurement tools, Phys. Rev. A 97,
042506 (2018).
[34] A. Hees, O. Minazzoli, E. Savalle, Y. V. Stadnik and
P. Wolf, Violation of the equivalence principle from light
scalar dark matter, Phys. Rev. D 98, 064051 (2018).
[35] P. Touboul, G. Metris, M. Rodrigues, Y. Andre,
Q. Baghi, J. Berge, D. Boulanger, S. Bremer et al., MICROSCOPE Mission: First Results of a Space Test of
the Equivalence Principle, Phys. Rev. Lett. 119, 231101(2017).
[36] J. Berge, P. Brax, G. Metris, M. Pernot-Borras,
P. Touboul and J.-P. Uzan, MICROSCOPE Mission: First Constraints on the Violation of the Weak
Equivalence Principle by a Light Scalar Dilaton,
Phys. Rev. Lett. 120, 141101 (2018).

[37] G. L. Smith, C. D. Hoyle, J. H. Gundlach, E. G. Adelberger, B. R. Heckel and H. E. Swanson, Short-range testsof the equivalence principle, Phys. Rev. D 61, 022001(1999).
[38] N. Leefer, A. Gerhardus, D. Budker, V. V. Flambaum
and Y. V. Stadnik, Search for the Effect of Massive Bodies on Atomic Spectra and Constraints on Yukawa-Type
Interactions of Scalar Particles, Phys. Rev. Lett. 117,
271601 (2016).
[39] K. Van Tilburg, N. Leefer, L. Bougas and D. Budker,
Search for Ultralight Scalar Dark Matter with Atomic
Spectroscopy, Phys. Rev. Lett. 115, 011802 (2015).
[40] A. Hees, J. Guena, M. Abgrall, S. Bize and P. Wolf,
Searching for an Oscillating Massive Scalar Field as aDark Matter Candidate Using Atomic Hyperfine Frequency Comparisons, Phys. Rev. Lett. 117, 061301(2016).
[41] E. Savalle, B. M. Roberts, F. Frank, P.-E. Pottie,
B. T. McAllister, C. B. Dailey, A. Derevianko and
P. Wolf, Novel approaches to dark-matter detection using space-time separated clocks, arXiv:1902.07192.
[42] D. Antypas, O. Tretiak, A. Garcon, R. Ozeri,
G. Perez and D. Budker, Scalar dark matter in theradio-frequency band: atomic-spectroscopy search results,
arXiv:1905.02968.
[43] A. Branca et al., Search for an Ultralight Scalar
Dark Matter Candidate with the AURIGA Detector,
Phys. Rev. Lett. 118, 021302 (2017).
[44] H.-K. Guo, K. Riles, F.-W. Yang and Y. Zhao, Searching for Dark Photon Dark Matter in LIGO O1 Data,arXiv:1905.04316.
[45] The spatial separation between the two LIGO detectors
is 3000 km, and so an oscillating DM field is spatiallycoherent over both detectors for all DM particle masses
of interest.
[46] A. Arvanitaki, S. Dimopoulos, and K. Van Tilburg,
Sound of Dark Matter: Searching for Light Scalars withResonant-Mass Detectors, Phys. Rev. Lett. 116, 031102(2016).
[47] We note that for phenomenologically interesting interaction strengths of an oscillating DM field with a quality factor of Qφ ∼ 106 , even resonantly-enhanced DMinduced cavity length fluctuations are not sufficiently
large to change the standing-wave mode number of thelaser field inside the cavity (typical mode numbers are∼ 106 ).
[48] Obviously, ‘exchanging’ test masses on a satellite mission
such as LISA would be the scope of a new mission.
[49] We note that, for the optimal frequency ranges of groundbased laser interferometers, the non-common-mode components of DM-induced time-varying displacements ofthe parts of Earth directly in contact with the structuresconnected to the pivot points of two different test-masssystems correspond to time-varying size changes of (thesolid) Earth at frequencies well above the fundamental
vibrational frequency of Earth, and so are strongly suppressed (as are isotropic DM-induced time-varying sizechanges of Earth), meaning that the pivot points arepractically unaffected by the DM interactions under consideration. Furthermore, the optimal frequency ranges ofground-based detectors lie well above the normal-mode
frequencies of their double-pendula suspension systems
for the test masses, meaning that the test masses respondto DM-induced time-varying forces as though they were15
free (i.e., the test masses respond inertially in this case).
[50] A. Pierce, K. Riles and Y. Zhao, Searching for Dark
Photon Dark Matter with Gravitational-Wave Detectors,
Phys. Rev. Lett. 121, 061102 (2018).
[51] A. Arvanitaki, J. Huang, and K. Van Tilburg, Searchingfor dilaton dark matter with atomic clocks, Phys. Rev. D
91, 015015 (2015).
[52] S. Morisaki and T. Suyama, On the detectability of ultralight scalar field dark matter with gravitational-wave
detectors, arXiv:1811.05003.
[53] C. Abel, N. J. Ayres, G. Ban, G. Bison, K. Bodek,
V. Bondar, M. Daum, M. Fairbairn, V. V. Flambaum
et al., Search for Axionlike Dark Matter through Nuclear Spin Precession in Electric and Magnetic Fields,
Phys. Rev. X 7, 041034 (2017).
[54] E. V. Pitjeva and N. P. Pitjev, Relativistic effects anddark matter in the Solar system from observations ofplanets and spacecraft, MNRAS 432, 3431 (2013).
[55] S. L. Adler, Placing direct limits on the mass of earthbound dark matter, J. Phys. A 41, 412002 (2008).
[56] A. Banerjee, D. Budker, J. Eby, H. Kim and G. Perez,
Relaxion Stars and their detection via Atomic Physics,

arXiv:1902.08212.
[57] A. Vilenkin, Cosmic strings and domain walls,
Phys. Rep. 121, 263 (1985).
[58] E. D. Hall, R. X. Adhikari, V. V. Frolov, H. Mueller
and M. Pospelov, Laser interferometers as dark matter
detectors, Phys. Rev. D 98, 083019 (2018).
[59] K. A. Olive and M. Pospelov, Environmental dependence of masses and coupling constants, Phys. Rev. D
77, 043524 (2008).
[60] E. G. Adelberger, B. R. Heckel, S. Hoedl, C. D. Hoyle,
D. J. Kapner and A. Upadhye, Particle-Physics Implications of a Recent Test of the Gravitational Inverse-Square
Law, Phys. Rev. Lett. 98, 131104 (2007).
[61] B. M. Roberts, G. Blewitt, C. Dailey, M. Murphy,
M. Pospelov, A. Rollings, J. Sherman, W. Williams and
A. Derevianko, Search for domain wall dark matter withatomic clocks on board global positioning system satellites, Nature Commun. 8, 1195 (2017).
[62] P. Wcislo et al., New bounds on dark matter coupling froma global network of optical atomic clocks, Sci. Adv. 4,
eaau4869 (2018).

16
-5

-20

LIGO

LISA
(modified)ΛγHolometer

GeV

LIGO-O1(displacements)-15

GEO 600-25

Clock/cavity
Clock/clock-15

log10 

mϕ
eV

GEO
600

-12

-11

-14

Clock/
cavity

LIGO

Λe

GeV

Holometer

GEO 600-18

AURIGA-15

log10 

mϕ

-24
-14

eV

-10

-8

-7



LIGO
(modifed)

Fifth-force searches(non-DM)

Holometer(narrowband)
GEO
600

AURIGA
-13

-12

-11

log10 



-5

eV

-9

-20

-22

-30

mϕ

LIGO-O1(displ.)-16

LISA(modified)-10

log10 LIGO-O1(displacements)

-20

-13



log10

Λe

GeV

log10

Holometer(narrowband)

-20

-24
-14

-10

Fifth-force searches(non-DM)-25-10Fifth-force searches(non-DM)AURIGA

-5

-20

LIGO
(modifed)

AURIGA

-20

-15

-18

-22

-30

-10

LIGO-O1(displacements)

-16

log10

Λγ

log10

GeV

-10

-14

Clock/cavity

Fifth-force searches(non-DM)

-10

mϕ
eV

-9

-8

-7



-14

Fifth-force searches(non-DM)

Holometer

LIGO-O1-16

GEO 600

LISA(modified)

-25

ΛN

GeV

LIGO
-20

GEO 600

log10

ΛN

log10

GeV

Holometer
LIGO-O1

-15

-18

-20

Fifth-force searches(non-DM)

LIGO

-22

-30
-20

-15

log10 

mϕ
eV

-10



-24
-14

-13

-12

-11

log10 

-10

mϕ
eV

-9

-8

-7



FIG. 3: (Color online). From top to bottom: Physical parameter spaces for the linear interactions of an oscillating DM field φwith the electromagnetic field (photon), electron and nucleons, as functions of the DM particle mass mφ . The solid lines denotethe estimated sensitivities of current ground-based laser interferometers (red = GEO 600, blue = LIGO, purple = Fermilab
holometer). The dashed blue line denotes the projected sensitivity of a single modified LIGO interferometer, in which the
thicknesses of the Fabry-Perot mirrors in one of the interferometer arms are changed by 10%, and operating at the design
sensitivity of Advanced LIGO, while the thin dashed blue line denotes the analogous sensitivity for a pair of modified LIGO
interferometers. The dashed purple line denotes the estimated sensitivity of a single small-scale Michelson interferometer
operating in the resonant narrowband regime near room temperature, with Q ∼ 106 and covering a DM particle mass range of∆mφ /mφ ≈ 1, while the thin dashed purple line denotes the analogous sensitivity for a pair of co-located interferometers. The
solid green line denotes the projected sensitivity of the space-based LISA interferometer in its standard configuration, while thedashed green line denotes the projected sensitivity of LISA with some of its Au-Pt alloy test masses replaced by Be test masses.
All of these sensitivities assume a total integration time of tint ∼ 108 s and saturation of the average local cold DM density ofρDM ≈ 0.4 GeV/cm3 . The region in grey denotes existing non-DM-based constraints from fifth-force experiments [35–38]. The
regions in yellow denote existing DM-based constraints from experiments involving clock-clock comparisons [39, 40], clock-cavity
comparisons [11, 12, 41, 42] and the AURIGA resonant-bar detector [43]. The region in blue denotes constraints derived in thepresent work from the consideration of time-varying centre-of-mass displacements of LIGO’s test masses in the LIGO-O1 datausing the results of data analysis taken from [44]. The region in pale green represents the region of parameter space that is
technically natural for a new-physics cut-off scale of Λ ∼ 10 TeV.

170

Astrophysical bounds and short-range gravity tests
Clocks andcavitiesγ

log10 GeV

Λ′

-5

GEO 600

-10

LIGO
-15

Fermilab holometer
-20

LISA
-25

-5

0

5

10

log10 md 
0Astrophysical bounds and short-range gravity testselog10 GeV

Λ′

-5

ClocksGEO 600

-10

LIGO
-15

Fermilab holometer
-20

LISA
-25

-5

0

5

10

log10 md 
0Astrophysical bounds and short-range gravity tests

N

log10 GeV

Λ′

-5

ClocksGEO 600

-10

LIGO
-15

Fermilab holometer
-20

LISA
-25

-5

0

5

10

log10 md 
FIG. 4: (Color online). From top to bottom: Physical parameter spaces for the quadratic interactions of a domain-wall field φwith the electromagnetic field (photon), electron and nucleons, as functions of the transverse size of a domain wall d. The solidlines denote the estimated sensitivities of current ground-based laser interferometers (red = GEO 600, blue = LIGO, purple =Fermilab holometer). The solid green line denotes the projected sensitivity of the space-based LISA interferometer. All of thesesensitivities assume a domain-wall transit speed of vTD ∼ 300 km/s, an average time between encounters of Earth and a domain
wall of T ∼ 1 year, and that the domain-wall network saturates the average local cold DM density of ρDM ≈ 0.4 GeV/cm3 .
The region in grey denotes existing model-independent constraints from astrophysical observations [59] and short-range testsof gravity [59, 60]. The regions in yellow denote existing constraints from domain-wall searches using a pair of co-located clocks
referenced to a common cavity [10] and via networks of clocks [61, 62].