Constraints on dark photon dark matter using data
from LIGO
’
s and Virgo
’
s third observing run
R. Abbott
etal.
*
(LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration)
(Received 27 May 2021; accepted 8 March 2022; published 31 March 2022)
We present a search for dark photon dark matter that could couple to gravitational-wave interferometers
using data from Advanced LIGO and Virgo
’
s third observing run. To perform this analysis, we use two
methods, one based on cross-correlation of the strain channels in the two nearly aligned LIGO detectors,
and one that looks for excess power in the strain channels of the LIGO and Virgo detectors. The excess
power method optimizes the Fourier transform coherence time as a function of frequency, to account for the
expected signal width due to Doppler modulations. We do not find any evidence of dark photon dark matter
with a mass between
m
A
∼
10
−
14
–
10
−
11
eV
=c
2
, which corresponds to frequencies between 10
–
2000 Hz,
and therefore provide upper limits on the square of the minimum coupling of dark photons to baryons, i.e.,
U
ð
1
Þ
B
dark matter. For the cross-correlation method, the best median constraint on the squared coupling is
∼
1
.
31
×
10
−
47
at
m
A
∼
4
.
2
×
10
−
13
eV
=c
2
; for the other analysis, the best constraint is
∼
2
.
4
×
10
−
47
at
m
A
∼
5
.
7
×
10
−
13
eV
=c
2
. These limits improve upon those obtained in direct dark matter detection
experiments by a factor of
∼
100
for
m
A
∼
½
2
–
4
×
10
−
13
eV
=c
2
, and are, in absolute terms, the most
stringent constraint so far in a large mass range
m
A
∼
2
×
10
−
13
–
8
×
10
−
12
eV
=c
2
.
DOI:
10.1103/PhysRevD.105.063030
I. INTRODUCTION
Dark matter has been known to exist for decades
[1]
, yet
its physical nature has remained elusive. Depending on the
theory, dark matter could consist of particles with masses as
low as
10
−
22
eV
=c
2
[2]
, or as high as (sub-) solar-mass
primordial black holes
[3
–
6]
. Furthermore, dark matter
clouds could form around black holes that deplete over time
and emit gravitational waves
[7,8]
. Here, we focus on a
subset of the
“
ultralight
”
dark matter regime, i.e., masses of
O
ð
10
−
14
–
10
−
11
Þ
eV
=c
2
[9]
, in which a variety of dark
matter candidates may interact with gravitational-wave
interferometers. Scalar, dilaton dark matter could change
the mass of the electron and other physical constants,
causing oscillations in the Bohr radius of atoms in various
components of the interferometer
[10]
; axions
[11]
could
alter the phase velocities of circularly polarized photons in
the laser beams traveling down each arm of the detector
[12]
; dark photons could couple to baryons in the mirrors,
causing an oscillatory force on the detector
[13]
; tensor
bosons could also interact with the interferometer in an
analogous way as gravitational waves
[14]
. Here, we focus
on dark photon dark matter whose relic abundance could be
induced by the misalignment mechanism
[15
–
17]
, the
tachyonic instability of a scalar field
[18
–
21]
, or cosmic
string network decays
[22]
. Cosmic strings, in particular,
also offer a promising way to probe physics beyond the
standard model with gravitational-wave detectors at ener-
gies much larger than those attainable by particle accel-
erators
[23]
, which complements the kind of direct dark
matter search we perform here. Independently of the
formation mechanism, analyses of gravitational-wave data
could make a statement on the existence of dark photons.
A search for dark photons using data from Advanced
LIGO/Virgo
’
s first observing run
[13,24]
has already been
performed, resulting in competitive constraints on the
coupling of dark photons to baryons. Furthermore, scalar,
dilaton dark matter was searched for recently using data
from GEO600
[25]
, and upper limits were placed on the
degree to which scalar dark matter could have altered the
electron mass or fine-structure constant
[26]
.
Other experiments that have probed the ultralight dark
matter regime include the Eöt-Wash experiment, which
aims to find a violation to the equivalence principle of
general relativity resulting from a new force acting on test
masses in a dark matter field, by looking for a difference in
the horizontal accelerations of two different materials using
a continuously rotating torsion balance
[27,28]
; the
MICROSCOPE satellite
[29]
, which measures the accel-
erations of two freely-floating objects in space made of
different materials to look for a violation of the equivalence
principle and hence a new force
[30]
; the Axion Dark
Matter Experiment (ADMX), which searches for
O
ð
μ
eV
=c
2
Þ
dark matter by trying to induce an axion-to-
photon conversion in the presence of a strong magnetic
field in a resonant cavity
[31]
; and the Any Light Particle
*
Full author list given at the end of the article.
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=
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© 2022 American Physical Society
Search (ALPS), which looks for particles with masses less
than
O
ð
meV
=c
2
Þ
(that could compose dark matter) by
subjecting photons to strong magnetic fields in two
cavities, separated by an opaque barrier, to cause a
transition to an axion and then back to a photon
[32]
.
Ultralight dark matter has also been constrained by observ-
ing gravitational waves from depleting boson clouds
around black holes
[8,33
–
38]
, and by analyzing binary
mergers, e.g., GW190521, which is consistent with the
merger of complex vector boson stars
[39]
.
Compared to the analysis on data from LIGO/Virgo
’
s
first observing run
[24]
, we use two methods, one based on
cross-correlation
[13]
, and another that judiciously varies
the Fourier Transform coherence time
[40,41]
, to search
for dark photons in Advanced LIGO and Virgo data from
the third observing run (O3). Additionally, we include
the signal induced by the common motion of the mirrors
[42]
—
see Sec.
II
. Although we do not find any evidence for
a dark photon signal, we place stringent upper limits on the
degree to which dark photons could have coupled to the
baryons in the interferometer.
II. DARK MATTER INTERACTION MODEL
Ultralight dark photon dark matter is expected to cause
time-dependent oscillations in the mirrors of the LIGO/
Virgo interferometers, which would lead to a differential
strain on the detector. We formulate dark photons in an
analogous way to ordinary photons: as having a vector
potential with an associated dark electric field that causes a
quasisinusoidal force on the mirrors in the interferometers.
The Lagrangian
L
that characterizes the dark photon
coupling to a number current density
J
μ
of baryons or
baryons minus leptons is
L
¼
−
1
4
μ
0
F
μν
F
μν
þ
1
2
μ
0
m
A
c
ℏ
2
A
μ
A
μ
−
ε
eJ
μ
A
μ
;
ð
1
Þ
where
F
μν
¼
∂
μ
A
ν
−
∂
ν
A
μ
is the electromagnetic field
tensor,
ℏ
is the reduced Planck
’
s constant,
c
is the speed
of light,
μ
0
is the magnetic permeability invacuum,
m
A
is the
dark photon mass,
A
μ
is the four-vector potential of the dark
photon,
e
is the electric charge, and
ε
is the strength of the
particle/dark photon coupling normalized by the electro-
magnetic coupling constant.
1
If the analysis observation time exceeds the signal
coherence time, given by Eq.
(3) [41]
, we can write the
acceleration of the identical LIGO/Virgo mirrors in the dark
photon field as
[24]
:
⃗
a
ð
t;
⃗
x
Þ
≃
ε
e
q
M
ω
⃗
A
cos
ð
ω
t
−
⃗
k
·
⃗
x
þ
φ
Þ
;
ð
2
Þ
where
ω
,
⃗
k
, and
⃗
A
are the angular frequency, propagation
vector, and polarization vector of the dark photon field,
⃗
x
is
the position of a mirror,
φ
is a random phase, and
q
and
M
are the charge and the mass of the mirror, respectively. If
the dark photon couples to the baryon number,
q
is the
number of protons and neutrons in each mirror. If it couples
to the difference between the baryon and lepton numbers,
q
is the number of neutrons in each mirror. For a fused Silica
mirror,
q=M
¼
5
.
61
×
10
26
charges
=
kg for baryon cou-
pling and
q=M
¼
2
.
80
×
10
26
charges
=
kg for baryon-lep-
ton coupling. Practically, we cannot distinguish between
the two types of coupling, though the baryon-lepton
coupling would lead to half the acceleration relative to
that of the baryon coupling.
Because we observe for almost one year, significantly
longer than the assumed dark photon coherence time, and
the dark photons travel with nonrelativistic velocities, we
model the signal as a superposition of many plane waves,
each with a velocity drawn from a Maxwell-Boltzmann
distribution
[43]
. The superposition of dark photon plane
waves with different velocities leads to a frequency
variation of the signal
[13,41]
:
Δ
f
¼
1
2
v
0
c
2
f
0
≈
2
.
94
×
10
−
7
f
0
;
ð
3
Þ
where
v
0
≃
220
km
=
s is the velocity at which dark matter
orbits the center of our galaxy, i.e., the virial velocity
[44]
,
and the frequency
f
0
is
f
0
¼
m
A
c
2
2
π
ℏ
:
ð
4
Þ
Dark photons cause small motions of an interferometer
’
s
mirrors, and lead to an observable effect in two ways. First,
the mirrors are well-separated from each other and hence
experience slightly different dark photon dark matter
phases. Such a phase difference leads to a differential
change of the arm length, suppressed by
v
0
=c
. A simple
relation between dark photon parameters and the effective
strain
h
D
can be written as
[13]
:
ffiffiffiffiffiffiffiffiffiffi
h
h
2
D
i
q
¼
C
q
M
v
0
2
π
c
2
ffiffiffiffiffiffiffiffiffiffiffi
2
ρ
DM
ε
0
s
e
ε
f
0
≃
6
.
56
×
10
−
27
ε
10
−
23
100
Hz
f
0
;
ð
5
Þ
where
ε
0
is the permittivity of free space, and
C
¼
ffiffiffi
2
p
=
3
is
a geometrical factor obtained by averaging over all possible
dark photon propagation and polarization directions.
Equation
(5)
can be derived by integrating Eq.
(2)
twice
over time, dividing by the arm length of the interferometer,
and performing the averages over time and the dark photon
polarization and propagation directions.
1
We note that the dark photon in our scenario is a different
from the one which couples to the standard model via kinetic
mixing.
R. ABBOTT
et al.
PHYS. REV. D
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063030 (2022)
063030-2
Second, the common motion of the interferometer
mirrors, induced by the dark photon dark matter back-
ground, can lead to an observable signal because of the
finite travel time of the laser light in the interferometer
arms. The light will hit the mirrors at different times during
their common motions, and although the common motions
do not change the instantaneous arm length, they can lead
to a longer roundtrip travel time for the light, equivalent to
arm lengthening, and therefore an apparent differential
strain
[42]
. Instead of being suppressed by
v
0
=c
as shown
in Eq.
(5)
, such an effect suffers from a suppression factor
of (
f
0
L=c
), where
L
is the arm length of the interferom-
eters. Similarly to Eq.
(5)
, the common motion induces an
observable signal with an effective strain
h
C
as:
ffiffiffiffiffiffiffiffiffi
h
h
2
C
i
q
¼
ffiffiffi
3
p
2
ffiffiffiffiffiffiffiffiffiffi
h
h
2
D
i
q
2
π
f
0
L
v
0
;
≃
6
.
58
×
10
−
26
ε
10
−
23
:
ð
6
Þ
h
D
maps to
h
2
in
[42]
, and
h
C
is the result of a Taylor
expansion of
h
1
in
[42]
. The interference between the two
contributions to the strain averages to zero over time, which
indicates that the total effective strain can be written
as
h
h
2
total
i¼h
h
2
D
iþh
h
2
C
i
.
III. SEARCH METHOD
A. Cross-correlation
Cross-correlation has been widely used in gravitational-
wave searches
[45
–
47]
, but is employed differently here.
Because we are interested in ultralight dark matter, the
coherence length of a dark photon signal, given by Eq. (2)
in
[41]
, is always much larger than the separation between
earth-based detectors
[19]
. Therefore, the interferometers
should experience almost the same dark photon dark matter
field, and the signals at any two detectors are highly
correlated
[19]
.
Because the dark photon signal is quasimonochromatic,
we analyze the frequency domain by discrete Fourier
transforming the strain time series. Given a total coincident
observation time,
T
obs
, for two detectors, we divide the time
series into
N
FFT
smaller segments, with durations
T
FFT
, i.e.,
T
obs
¼
N
FFT
T
FFT
. For the
i
th time segment,
j
th frequency
bin, and interferometer
k
(1 or 2), we label the complex
discrete Fourier transform coefficients as
z
k;ij
. The one-
sided power spectral densities (PSDs) of interferometer
1(2) can be estimated by taking a (bias-corrected) running
median of the raw noise powers
P
k;ij
from 50 neighboring
frequency bins: PSD
k;ij
¼
2
P
k;ij
=T
FFT
.
The cross-correlated signal strength is
S
j
¼
1
N
FFT
X
N
FFT
i
¼
1
z
1
;ij
z
2
;ij
P
1
;ij
P
2
;ij
;
ð
7
Þ
where
“
”
is the complex conjugate, and the variance is
σ
2
j
¼
1
N
FFT
1
2
P
1
;ij
P
2
;ij
N
FFT
;
ð
8
Þ
where
h
...
i
N
FFT
is the average over
N
FFT
time segments.
Therefore, the signal-to-noise ratio (SNR) is
SNR
j
¼
S
j
σ
j
:
ð
9
Þ
In Gaussian noise without a signal, SNR
j
has zero mean
and unit variance. The presence of a signal would lead to a
nonzero offset in the mean SNR proportional to
ε
2
[see
Eqs.
(5)
–
(6)
]. We note that we will include the overlap
reduction function (ORF) in our upper limit calculation,
which accounts for the relative orientation and overlap of
two detectors and the responses of the detectors to a signal.
As indicated in
[13]
, the ORF is constant (
∼−
0
.
9
) for the
LIGO Hanford (H1) and LIGO Livingston (L1) detectors
because the dark photon coherence length always exceeds
the detector separation.
Here, we analyze only time segments satisfying standard
data quality requirements used in gravitational-wave
searches (see Sec.
IV
), and further restrict to contiguous,
coincident intervals of good data spanning the fast Fourier
transform coherence time. As in the analysis performed
using data from the first observing run (O1)
[24]
, we set
T
FFT
¼
1800
s, a pragmatic compromise between recov-
ering signal power at high frequencies with shorter-than-
optimal coherence times, and reducing noise contamination
at low frequencies for longer-than-optimal coherence times.
An important constraint at low frequencies is that requiring
longer (contiguous) coherence times necessarily reduces
total available livetime, especially given the need for
coincident H1 and L1 data. In total, we analyze 7539 pairs
of 1800-second coincident time segments from H1 and L1.
B. BSD analysis
In addition to cross-correlation, we employ an indepen-
dent method
[41]
to search for dark photon dark matter. The
method relies on band sampled data (BSD) structures,
which store the detector
’
s downsampled strain data as a
reduced analytic signal
[40]
in
10
-Hz
=
1
-month chunks. In
each 10-Hz band, we change the fast Fourier transform
coherence time
[40]
based on the expected Maxwell-
Boltzmann frequency spread of dark photons, Eq.
(3)
.
Although this frequency spread is given as a function of
v
0
,
we instead use the escape velocity from the galaxy,
v
esc
≃
540
km
=
s
[44]
, to determine the maximum allowed
T
FFT
,
T
FFT
;
max
, by requiring that the frequency spread be con-
tained in one frequency bin in
T
FFT
;
max
:
CONSTRAINTS ON DARK PHOTON DARK MATTER USING DATA
...
PHYS. REV. D
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T
FFT
;
max
≲
2
f
0
c
2
v
2
esc
≃
6
×
10
5
f
0
s
:
ð
10
Þ
Based on simulations
[41]
, we found that the sensitivity of
the search improves when taking
T
FFT
¼
1
.
5
T
FFT
;
max
,
because the power lost due to over-resolving in frequency
is less than that gained by increasing
T
FFT
.
After selecting
T
FFT
, we create time/frequency
“
peak-
maps
”
[48,49]
, which are collections of ones and zeros that
represent when the power in particular frequency bins has
exceeded a threshold in the equalized spectrum. Because
we choose
T
FFT
to confine a signal
’
s power to one
frequency bin, we project the peakmap onto the frequency
axis and look for frequency bins with large numbers of
peaks, which we call the
“
number count
”
.
We should uniformly select candidates in the frequency
domain. In Fig.
1
shows how many candidates to select in
each 10-Hz frequency band such that we would obtain, on
average, one coincident candidate every one Hz in
Gaussian noise. We also show in color how log
10
T
FFT
changes with frequency [Eq.
(10)
].
Our detection statistic is the critical ratio CR:
CR
¼
y
−
μ
σ
;
ð
11
Þ
where
y
is the number count in a particular frequency bin,
and
μ
and
σ
are the mean and standard deviations of the
number counts across all frequency bins in the band. The
CR has a normal distribution with an expectation value of 0
and unit variance in Gaussian noise, and a normalized
noncentral
χ
2
distribution with two degrees of freedom
when a signal is present.
IV. DATA
We use data from the third observing run (O3) of the
Advanced LIGO
[51]
and Virgo
[52]
gravitational-wave
detectors between 10
–
2000 Hz. O3 lasted from 2019 April
1 to 2020 March 27, with a one-month pause in data
collection in October 2019. The three detectors
’
datasets,
H1, L1, and Virgo (V1), had duty factors of
∼
76%
,
∼
77%
,
and
∼
76%
, respectively, during O3.
In the event of a detection, calibration uncertainties
would limit our ability to provide robust estimates of the
coupling of dark matter to the interferometers. Even with-
out a detection, these uncertainties affect the estimated
instruments
’
sensitivities and inferred upper limits. The
uncertainties vary over the course of a run but do not
change by large values, so we do not consider time-
dependent calibration uncertainties here
[53]
.
For the LIGO O3 data set, the analyses use the
“
C01
”
calibration, which has estimated maximum amplitude and
phase uncertainties of
∼
7%
and
∼
4
deg, respectively
[53]
.
Because of the presence of a large number of noise artifacts,
gating
[54,55]
has also been applied to LIGO data. This
procedure applies an inverse Tukey window to LIGO data
at times when the root-mean-square value of the whitened
strain channel in the 25
–
50 Hz band or 70
–
110 Hz band
exceeds a certain threshold. The improvements from gating
are significant, as seen in stochastic and continuous
gravitational-wave analyses in O3
[46]
. For the Virgo
O3 dataset, we use the
“
V0
”
calibration with estimated
maximum amplitude and phase uncertainties of 5%
and 2 deg, respectively.
V. RESULTS
A. Cross-correlation
The output of the cross-correlation analysis is a value of
the SNR in every frequency bin analyzed. At this point, we
remove frequency bins with noise artifacts, i.e., bins within
0.056 Hz of known noise lines
[56]
. To further estimate the
non-Gaussian background from artifacts, control samples are
constructed using frequency lags, i.e., examining the corre-
lations among a set of offset bins. We apply ten lags
of the frequency bin offsets, i.e., (
−
50
;
−
40
;
...
;
−
10
;
þ
10
;
...
;
þ
50
). If any frequency bin in the control sample
has a
j
Re
ð
SNR
Þj
or
j
Im
ð
SNR
Þj
larger than 4.0 within 0.1 Hz
oftheoutlier,theoutlierisvetoedascontaminatedbyspectral
leakage from a nearby non-Gaussian artifact. We choose a
band of 0.1 Hz because within that band, spectral leakage
causes non-physical correlated amplitudes and phases.
Furthermore, ten lags allows us to compare frequency bins
that are not too far from each other to construct an estimation
of the noise in the chosen frequency bin.
After removing these instrumental artifacts, we look for
frequency bins with Re
ð
SNR
Þ
<
−
5
.
8
, which corresponds
to an overall
∼
1%
false alarm probability after including
the trial factor in Gaussian noise, and is negative because
FIG. 1. Number of candidates selected as a function of
frequency in the BSD analysis, with log
10
T
FFT
colored. We
select enough candidates in each 1-Hz band such that one
coincident candidate between two detectors would occur in
Gaussian noise. The changing number of candidates as a function
of frequency ensures that we select uniformly in frequency. See
Supplemental Material
[50]
.
R. ABBOTT
et al.
PHYS. REV. D
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063030 (2022)
063030-4
H1 and L1 are rotated 90 deg with respect to each other. We
find no outliers that pass this threshold.
Finally, as a cross-check, between [5.0, 5.8] for
j
Re
ð
SNR
Þj
or
j
Im
ð
SNR
Þj
, we find four nonvetoed outliers,
which are shown in Table
I
. The number of outliers is
consistent with the Gaussian noise expectation of 4.1. We
consider the absolute value of the real and imaginary
components of the SNR because we are checking consis-
tency with the expected number of outliers in Gaussian
noise, which does not depend on the sign of the SNR. We
show the distribution of the real and imaginary parts of the
SNR in the Appendix.
B. BSD analysis
Before selecting candidates, we remove any frequencies
that fall within one frequency bin of known noise lines from
each detector
’
s data
[56]
. We subsequently require coinci-
dent candidates between two or more detectors to be within
one frequency bin of each other. At this stage, our analyses
of the Hanford-Livingston (HL), Hanford-Virgo (HV), and
the Livingston-Virgo (LV) baselines return 5801, 5628, and
5592 candidates, respectively.
In all baselines, we veto coincident candidates if one of
the candidates
’
critical ratios is less than five or one of the
candidates
’
frequencies is too close, i.e., within 5 bins, to
the edges of the 10 Hz-band analyzed. The latter veto is
necessary because the construction of the BSDs introduces
artifacts in some bands at the edges. For the HL baseline,
we remove candidates whose critical ratios differ by more
than a factor of two because the sensitivity of each
interferometer is comparable, so we do not expect a dark
photon signal to appear with vastly different critical ratios
in each detector. In the HV and LV baselines, we reject
candidates whose critical ratios in V1 are higher than those
in L1 or H1 because Virgo is less sensitive than LIGO
[57]
.
We show distributions of CR in the Hanford and Livingston
detectors across all frequencies in the Appendix, Figs.
5
and
6
, respectively, as well as the CR distribution of the
number of coincident candidates in Fig.
7
.
We are then left with eleven surviving candidates across
the three baselines, given in Table
II
, that are all due to
instrumental noise or artifacts in the peakmap. Peakmap
artifacts occur because when there are strong lines at
particular frequencies, we tend to select peaks that corre-
spond to those lines. This causes a
“
depletion
”
of peaks
nearby, and thus, a candidate could result because the level
of the noise in the projected peakmap is lower on one side
than on the other. No candidate has been found to be
coincident in all three interferometers. These surviving
candidates do not overlap with the list of known lines used
in this search
[56]
, although line artifacts or/and combs
regions are clearly visible when using a different resolution
to construct the spectra. In Fig.
2
, we show an example of
the disturbances near an outlier at 1498.76 Hz, where a
family of combs is present in both the H1 and L1 detectors.
C. Upper limits
Finding no evidence of a signal, in Fig.
3
we place
95% confidence-level upper limits on the square of the
minimum detectable dark photon/baryon coupling,
U
ð
1
Þ
B
,
using the HL baseline. The cross-correlation limits are
TABLE I. Four sub-threshold outliers returned by the cross correlation analysis of the HL baseline. We report the
(complex) signal-to-noise ratio (SNR) for each outlier and the associated background (Bkg) SNR. For the
background SNR, we include the range of the real part (Re) and imaginary part (Im) among ten lagged results. These
four events are consistent with the Gaussian noise expectation over all of the clean bands in the analysis.
Frequency (Hz)
SNR
SNR(Bkg)
483.872
0
.
53
þ
5
.
03
i
Re
∶
½
−
3
.
62
;
3
.
62
Im
∶
½
−
3
.
52
;
3
.
51
853.389
−
0
.
18
þ
5
.
02
i
Re
∶
½
−
3
.
85
;
3
.
85
Im
∶
½
−
3
.
55
;
3
.
90
1139.590
−
5
.
21
þ
0
.
67
i
Re
∶
½
−
3
.
54
;
3
.
39
Im
∶
½
−
3
.
61
;
3
.
58
1686.598
5
.
01
þ
1
.
63
i
Re
∶
½
−
3
.
50
;
3
.
70
Im
∶
½
−
3
.
65
;
3
.
89
FIG. 2. We discarded all surviving outliers because they were
due to instrumental noise or artifacts. In this figure, we can see the
comb affecting the power spectral density (PSD) of H1 and the
line in L1 responsible for the production of an outlier near
1498.8 Hz. Frequency resolution:
δ
f
¼
3
.
47
×
10
−
5
Hz.
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...
PHYS. REV. D
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shown in red for every 0.556-mHz bin, while the BSD
limits are given in black with cyan
1
σ
shading in frequency
bins in which coincident candidates were found. To
calculate these limits, we employ the Feldman-Cousins
[59]
approach, in which we assume that both CR and SNR
follow Gaussian distributions, and map the measured
detection statistics to
“
inferred
”
positive-definite statistics
based on the upper value of Table 10 of
[59]
at 95% con-
fidence. As shown in
[5]
, this approach produces consistent
limits with respect to those that would be obtained by
injecting simulated signals. With our estimates of the noise
power spectral density and
T
FFT
, we can translate the
inferred SNR and CR at each frequency to the
corresponding signal amplitude using Eq. (9) in
[13]
and
Eq. (30) in
[41]
, respectively. This amplitude is then
converted to a coupling strength using Eq.
(5)
, and adjusted
for the common mode motion effect
[42]
.
The limits from the cross-correlation analysis are more
stringent than those from the BSD method because the
former employs the phase information of the signal, while
the latter only looks at power. Furthermore, though the
choice of
T
FFT
is
“
optimal
”
in the BSD method, it is still
shorter than that used by cross correlation by as much as a
factor of six above
∼
330
Hz, and the definition of optimal
depends on whether we consider the escape or virial
velocity of dark matter as responsible for the frequency
TABLE II. Outliers returned by the BSD analysis. The frequency resolution of each outlier is
1
=T
FFT
.Wehave
determined the origin of all outliers to be from instrumental lines or peakmap artifacts. No outlier was found to be in
triple coincidence. A list of unidentified lines can be found in
[58]
.
Frequency (Hz)
Average CR
T
FFT
(s)
Baseline
Source
15.9000
5.29
44762
HL
Unknown line in L
17.8000
28.93
44762
LV
Unidentified line in L (17.8 Hz)
36.2000
8.90
22382
HV
Unidentified line in H (36.2 Hz)
599.324
12.38
1492
HV
Peakmap artifact; no significant candidate in L
599.325
12.33
1492
HV
Peakmap artifact; no significant candidate in L
1478.75
6.47
604
HL
Noisy spectra in H
1496.26
7.12
596
HL
Noisy violin resonance regions
1498.77
8.73
596
HL
Noisy violin resonance regions
1799.63
7.40
498
HV
Unidentified line in H (1799.63904 Hz)
1936.88
7.96
462
HL
Noisy violin resonance regions
1982.91
6.34
450
HL
Noisy violin resonance regions
FIG. 3. Upper limits derived using a Feldman-Cousins approach for both searches on dark photon/baryon coupling,
U
ð
1
Þ
B
. The limits
from each method are comparable, noting that the BSD-based analysis takes an optimally chosen
T
FFT
and can observe for twice as long
than the cross-correlation method can. We plot for comparison upper limits from MICROSCOPE given in
[30]
, though other weaker
limits exist
[60
–
62]
, that have been converted from the coupling constant to gravity,
α
,to
ε
2
, using the equation below Fig. 3 in
[63]
, and
from the Eöt-Wash torsion balance experiment
[28]
. To produce limits on dark photon/baryon-lepton coupling,
U
ð
1
Þ
B
−
L
, our limits
should be multiplied by four. See Supplemental Material
[50]
.
R. ABBOTT
et al.
PHYS. REV. D
105,
063030 (2022)
063030-6
variation. Additionally, cross-correlation of two data
streams can achieve better sensitivity than coincidence
analysis of the same streams (for the same livetime)
because coincidence analysis is limited by the less sensitive
of the two detectors at a given frequency.
VI. CONCLUSIONS
We have presented strong constraints on the coupling
strength of dark photon dark matter to baryons by using
data from LIGO
’
s and Virgo
’
s third observing run. In the
mass range
m
A
∼
½
2
–
4
×
10
−
13
eV
=c
2
, we improve upon
previous limits derived using data from the first observing
run of LIGO
[24]
by a factor of
∼
100
in the square of the
coupling strength of dark photons to baryons. This
improvement is due to more sensitive detectors and to
accounting for the finite light travel time
[42]
. Additionally,
our limits surpass those of existing dark matter experi-
ments, such as the Eöt-Wash torsion balance and
MICROSCOPE, by orders of magnitude in certain fre-
quency bands, and support new ways to use gravitational-
wave detectors as direct probes of the existence of ultralight
dark matter. As the sensitivities of current ground-based
gravitational-wave detectors improve, and third generation
detectors, such as Cosmic Explorer
[64]
and Einstein
Telescope
[65]
, come online, we will dig even more deeply
into the noise. Furthermore, once future-generation space-
based detectors, such as DECIGO
[66]
, LISA
[67]
, and
TianQin
[68]
, are operational, we will probe dark photon
couplings at masses as low as
m
A
∼
10
−
18
eV
=c
2
.
ACKNOWLEDGMENTS
This material is based upon work supported by NSF
’
s
LIGO Laboratory which is a major facility fully funded by
the National Science Foundation. The authors also gratefully
acknowledge the support of the Science and Technology
Facilities Council (STFC) of the United Kingdom, the Max-
Planck-Society (MPS), and the State of Niedersachsen/
Germany for support of the construction of Advanced
LIGO and construction and operation of the GEO600
detector. Additional support for Advanced LIGO was pro-
vided by the Australian Research Council. The authors
gratefully acknowledge the Italian Istituto Nazionale di
Fisica Nucleare (INFN), the French Centre National de la
Recherche Scientifique (CNRS) and the Netherlands
Organization for Scientific Research, for the construction
and operation of the Virgo detector and the creation and
support of the EGO consortium. The authors also gratefully
acknowledge research support from these agencies as well as
by the Council of Scientific and Industrial Research of India,
the Department of Science and Technology, India, the
Science & Engineering Research Board (SERB), India,
the Ministry of Human Resource Development, India,
the Spanish Agencia Estatal de Investigación, the
Vicepresid`
encia i Conselleria d
’
Innovació, Recerca i
Turisme and the Conselleria d
’
Educació i Universitat del
Govern de les Illes Balears, the Conselleria d
’
Innovació,
Universitats, Ci`
encia i Societat Digital de la Generalitat
Valenciana and the CERCA Programme Generalitat de
Catalunya, Spain, the National Science Centre of Poland
and the Foundation for Polish Science (FNP), the Swiss
National Science Foundation (SNSF), the Russian
Foundation for Basic Research, the Russian Science
Foundation, the European Commission, the European
Regional Development Funds (ERDF), the Royal Society,
the Scottish Funding Council, the Scottish Universities
Physics Alliance, the Hungarian Scientific Research Fund
(OTKA), the French Lyon Institute of Origins (LIO), the
Belgian Fonds de la Recherche Scientifique (FRS-FNRS),
Actions de Recherche Concert ́
ees (ARC) and Fonds
Wetenschappelijk Onderzoek
—
Vlaanderen (FWO),
Belgium, the Paris Île-de-France Region, the National
Research, Development and Innovation Office Hungary
(NKFIH), the National Research Foundation of Korea, the
Natural Science and Engineering Research Council Canada,
Canadian Foundation for Innovation (CFI), the Brazilian
Ministry of Science, Technology, and Innovations, the
International Center for Theoretical Physics South
American Institute for Fundamental Research (ICTP-
SAIFR), the Research Grants Council of Hong Kong, the
National Natural Science Foundation of China (NSFC), the
Leverhulme Trust, the Research Corporation, the Ministry of
Science and Technology (MOST), Taiwan, the United States
Department of Energy, and the Kavli Foundation. The
authors gratefully acknowledge the support of the NSF,
STFC, INFN and CNRS for provision of computational
resources. This work was supported by MEXT, JSPS
Leading-edge Research Infrastructure Program, JSPS
Grant-in-Aid for Specially Promoted Research 26000005,
JSPS Grant-in-Aid for Scientific Research on Innovative
Areas 2905: No. JP17H06358, No. JP17H06361 and
No. JP17H06364, JSPS Core-to-Core Program A.
Advanced Research Networks, JSPS Grant-in-Aid for
Scientific Research (S) No. 17H06133, the joint research
programofthe InstituteforCosmicRayResearch,University
of Tokyo, National Research Foundation (NRF) and
Computing Infrastructure Project of KISTI-GSDC in
Korea, Academia Sinica (AS), AS Grid Center (ASGC)
and the Ministry of Science and Technology (MoST) in
Taiwan under grants including AS-CDA-105-M06,
Advanced Technology Center (ATC) of NAOJ, and
Mechanical Engineering Center of KEK.
APPENDIX: DISTRIBUTION
OF DETECTION STATISTICS
We provide here more details on our detection statistics
for both methods. When we calculate upper limits, we
assume that these statistics follow Gaussian distributions,
CONSTRAINTS ON DARK PHOTON DARK MATTER USING DATA
...
PHYS. REV. D
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which is actually true only in clean bands. But, because we
showed the Feldman-Cousins approach to be robust toward
noise disturbances in
[41]
, we are confident that the limits
are reflective of what we would have obtained if we
performed software injections.
For the cross-correlation search, the distributions of the
real and imaginary parts of the SNR are shown in Fig.
4
after vetoing frequency bins within 0.056 Hz of the known
noise lines
[56]
and after vetoing the instrumental artifacts
as described in the main text above.
We show the distributions of the CR in Hanford (Fig.
5
)
and Livingston (Fig.
6
), over all frequency bins between
10
–
2000 Hz. We also overlay a Gaussian on the plot to
show the extent to which the distributions differs from a
Gaussian distribution. In both detectors, the number of
frequency bins whose CRs deviate from Gaussianity is
of
O
ð
10
2
Þ
, which is a small fraction of the total number of
bins analyzed.
We also include a plot to characterize the
coincident
candidates between Hanford and Livingston that are
FIG. 4. Distribution of the real (top) and imaginary (bottom)
parts of the SNR in the cross-correlation search, with those
corresponding to on-source (with zero lag) results in magenta,
background (with frequency lags) in black and the ideal Gaussian
distribution in green.
FIG. 5. Histogram of critical ratios in all frequency bins in the
Hanford detector, with a Gaussian (in red) overlayed.
FIG. 6. Histogram of critical ratios in all frequency bins in the
Livingston detector, with a Gaussian (in red) overlayed.
FIG. 7. Histogram of coincident critical ratios, after our
selection of candidates in each 10-Hz frequency band. We
performed the coincidences between the candidates returned
after analyzing Hanford and Livingston data.
R. ABBOTT
et al.
PHYS. REV. D
105,
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selected in our search. Figure
7
shows a histogram of all the
coincident candidates
’
critical ratios that we select, as well
as a black line that indicates the threshold on the critical
ratio that we impose, equal to 5. We can see that very few
candidates are coincident relative to the number of candi-
dates plotted in Figs.
5
and
6
, and that the total number of
coincident candidates that are subject to further study is
of
O
ð
1000
Þ
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