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Searching for Intelligent Life in Gravitational Wave Signals Part I:
Present Capabilities and Future Horizons
Luke Sellers1,2★, Alexey Bobrick1,3,4, Gianni Martire1,
Michael Andrews5, Manfred Paulini5
1Advanced Propulsion Laboratory at Applied Physics, 477 Madison Ave, New York, 10022, United States
2University of California Los Angeles, Department of Physics, Los Angeles, CA, 90095, USA
3Technion - Israel Institute of Technology, Physics Department, Haifa 32000, Israel
4Lund University, Department of Astronomy and Theoretical Physics, Box 43, SE 221-00 Lund, Sweden
5Carnegie Mellon University, Department of Physics, Pittsburgh, Pennsylvania 15213, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We show that the Laser Interferometer Gravitational Wave Observatory (LIGO) is a powerful
instrument in the Search for Extra-Terrestrial Intelligence (SETI). LIGO’s ability to detect
gravitational waves (GWs) from accelerating astrophysical sources, such as binary black
holes, also provides the potential to detect extra-terrestrial mega-technology, such as Rapid
And/or Massive Accelerating spacecraft (RAMAcraft). We show that LIGO is sensitive to
RAMAcraft of 1Jupiter mass accelerating to a fraction of the speed of light (e.g. 10%) up to
about 100 kpc. Existing SETI searches probe on the order of thousands to tens of thousands
of stars for human-scale technology (e.g. radiowaves), whereas LIGO can probe all 1011 stars
in the Milky Way for RAMAcraft. Moreover, thanks to the 𝑓−1scaling of the GW signal
produced by these sources, our sensitivity to these objects will increase as low-frequency,
space-based detectors are developed and improved. In particular, we find that DECIGO and
the Big Bang Observer (BBO) will be about 100 times more sensitive than LIGO, increasing
the search volume by 106. In this paper, we calculate the waveforms for linearly accelerating
RAMAcraft in a form suitable for LIGO, Virgo, or KAGRA searches and provide the range
for a variety of possible masses and accelerations. We expect that the current and upcoming
GW detectors will soon become an excellent complement to the existing SETI efforts.
Key words: extraterrestrial intelligence: astrobiology: gravitational waves: relativistic pro-
cesses: instrumentation: detectors: instrumentation: interferometers
1 INTRODUCTION
Gravitational wave (GW) detectors such as LIGO and Virgo (Abbott
et al. 2020) have contributed three entirely new signals to the astro-
nomical catalog, namely GWs from binary black hole (BBH) merg-
ers (Abbott et al. 2016b), neutron star-black hole (NSBH) mergers
(Abbott et al 2021), and binary neutron star (BNS) mergers (Ab-
bott et al 2017a). Each of these signals consists of a mutual orbit of
stellar-mass (1 – 100M), compact objects that LIGO and Virgo
have detected as far out as several thousand Mpc. The accumula-
tion of these detections has led to an array of significant findings,
including stringent tests of general relativity (Abbott 2019), a novel
method for determining the cosmological constant (Soares-Santos
et al. 2019), and evidence of various neutron star processes, such
as their connection to gamma-ray bursts (Abbott et al 2017a), their
equation of state (Abbott 2018), and the production of heavy r-
★Corresponding author: luke@appliedphysics.org
process elements, such as gold and uranium (Abbott et al. 2017b).
Furthermore, the detection of new signals in the future will shed
light on topics including black hole (BH) formation, the quantum
behavior of BHs, structure formation in the early universe, dark
matter, and dark energy (Castelvecchi 2018;Barausse 2020;Carson
2021;Arvanitaki et al. 2015;Leor et al 2019;Weiner et al. 2021).
The detection of these new signals will be dramatically en-
hanced by the sensitivity improvements provided by future GW
detectors. These improvements will come in two types, namely the
improvement of overall detector sensitivities and the broadening of
detector sensitivity bands. In particular, the planned Einstein Tele-
scope (Maggiore et al. 2020) and Cosmic Explorer (Reitze et al.
2019) will improve the overall sensitivity, leading to the discovery
of thousands of BBH mergers from across the observable universe
each day. At the same time, the upcoming detectors LISA (Robson
et al. 2019), TianQin (Luo et al. 2016), DECIGO (Kawamura et al.
2006), the Big Bang Observer (BBO) (Corbin & Cornish 2006),
and improvements to Pulsar Timing Arrays (PTAs) (Moore 2015;
©2022 The Authors
arXiv:2212.02065v1 [astro-ph.IM] 5 Dec 2022
2L.Sellers et al.
Hazboun et al. 2019), such as the Square Kilometer Array (SKA)
(Dewdney et al. 2009), will collectively widen the sensitivity band
of the detector network. This will lead to the detection of entirely
new sources, such as supermassive BH mergers, extreme mass-ratio
inspirals, close white dwarf and subdwarf binaries, amongst others.
Collectively, these improvements will vastly extend the number of
detected signals and lead to the discovery of multiple new types of
astrophysical sources.
Since any system involving the bulk acceleration of mass pro-
duces GWs, new signal candidates include not only astrophysical
and cosmological events, but also technological signals (technosig-
natures), such as those generated by rapid and/or massive acceler-
ating spacecraft (RAMAcraft). We can already perform targeted
searches for any such object simply by calculating its GW signal
and plugging the result into existing detection pipelines. In this pa-
per, we take a first look at the capabilities of LIGO and future GW
detectors, such as LISA, to detect these objects. Specifically, we
compute the range at which a linearly moving RAMAcraft with
constant acceleration will be detectable by either LIGO or LISA.
In the case of LIGO, we set novel constraints on the occurrence of
such trajectories while LIGO was online.
Though our focus is on artificial objects, it may be the case that
the signals we examine are mimicked by some naturally occurring
phenomena. In Section 4, we discuss potential candidates, including
gravitational wave bursts from highly eccentric BBHs.
In the artificial case, the detected GW signal may be the result
of an advanced transportation mechanism utilized by an intelli-
gent species. Thus far, the search for such objects has primarily
been restricted to the examination of electromagnetic (EM) signals
through the efforts of the Search for Extra-Terrestrial Intelligence
(SETI) (Wright 2022;Gajjar et al. 2021;Traas et al. 2021) and like-
minded research groups (Hoang & Loeb 2020;Trilling et al. 2018).
For example, it has been suggested that thermal emission due to the
surface heating of a 100-meter-scale interstellar spacecraft could be
detected by the James Webb Space Telescope (JWST) from up to
about 100 AU (Hoang & Loeb 2020). Similarly, multi-color tele-
scopes can also detect 100-meter-scale interstellar objects (ISOs)
within our solar system, which was famously demonstrated in 2017
with the detection of ‘Oumuamua’ (Trilling et al. 2018, although
the latter is not necessarily of an artificial origin).
Adding a GW detection pipeline for RAMAcraft can provide
several important benefits in the search for extraterrestrial life. First,
not all systems emit significant EM waves. Second, GW detectors
have a vast field of view, while EM images comprise a small fraction
of the sky. Third, due to the weakness of gravitational interactions,
all GW signals of all frequencies travel through space virtually
unimpeded. By contrast, substantial EM radiation absorption oc-
curs, for example, in the interstellar medium for low-frequency ra-
dio, optical and ultraviolet wavelengths and ultraviolet, microwave
and X-ray bands in the Earth’s atmosphere. Consequently, detecting
EM signals for certain frequencies may be challenging or require
space-based telescopes. The fourth advantage is that the GW quan-
tity that is observed, namely the strain, falls off like 𝑅−1where 𝑅
is the distance to the source, whereas the observed EM quantity,
namely the flux, falls off like 𝑅−2. This difference in scaling can
be significant over large distances, enabling the upcoming detec-
tors to probe the entire observable universe for BBH mergers, for
example. Finally, GW detectors monitor natural and SETI sources
simultaneously. By contrast, EM searches typically require point-
ing telescopes at one target at a time, thereby competing with the
observation time needed for natural sources and other SETI targets.
Despite these difficulties, SETI and other like-minded groups have
extended EM searches to the order of tens of thousands of stars.
For example, one of SETI’s most successful programs, the Break-
through Listen Initiative (Gajjar et al. 2021), set constraints on opti-
cal signatures from advanced civilizations for 1327 stars through a
recent search (Price C. D 2020). Furthermore, a SETI press release
announced in 2016 a search for 20,000 stars. Although these feats
are remarkable, extending these searches by orders of magnitude
will be challenging due to the aforementioned difficulties involving
EM searches.
For these reasons, we propose searching for RAMAcraft and
other technosignatures by studying GW signals. Similar to the tar-
gets of astrophysical GW searches, the prime targets for GW SETI
searches at the present moment must be objects from so-called
mega-technology, which includes objects of planetary-scale mass
and/or with changes in velocity comparable to the speed of light
(e.g. 10%). While these parameter scales go beyond anything fa-
miliar to us on Earth, systems of interest such as physical warp
drives (Jackson & Benford 2020;Bobrick & Martire 2021), Dyson
spheres (Dyson 1969;Zackrisson et al. 2018), and others necessarily
involve such masses. In a similar spirit, the study by Gourgoulhon
et al. (2019); Abramowicz et al. (2019) showed that an orbiting body
of Jupiter-mass-scale intended to extract energy from Sagittarius A
(Sgr A) would be detectable by LISA. As we show in Section 3, our
detectable parameter space for RAMAcraft is comparable to these
results. Furthermore, as we show in Section 4, what may bridge
the gap between these large mass scales and something more fa-
miliar for the case of RAMAcraft is that the GW signal spectrum
scales like 𝑓−1. Therefore, detectors that are more sensitive to lower
frequencies will be more sensitive to these objects.
This paper is structured as follows: In Section 2, we calculate
the time-domain signal produced by a RAMAcraft operating as a
newtonian rocket, then calculate the frequency spectrum, focussing
on the case of constant acceleration. In Section 3, we compute the
distance from Earth at which such an object would be detectable
by both LIGO and LISA for a given mass and change in velocity.
In Section 4, we discuss how detection feasibility can be improved
by future detector improvements and how this approach can be
generalized for future studies. Finally, we offer concluding thoughts
in Section 5.
2 GRAVITATIONAL WAVE SIGNAL FROM LINEAR
ACCELERATION
In this section, we calculate the time-domain GW signal emitted
by an object with a general linear trajectory. We then apply con-
servation of momentum ad-hoc and find that our result agrees with
the exact calculation using a Newtonian rocket, ignoring the effects
of exhaust. We show how to account for exhaust and move on to
focus on the frequency spectrum of objects undergoing constant
acceleration.
2.1 Time-Domain Signal
In general relativity, if the gravitational field is weak, then the metric
𝑔𝜇𝜈 that describes the curvature of spacetime can be decomposed
into the metric that describes flat space 𝜂𝜇 𝜈 and that which describes
a perturbation of flat space ℎ𝜇 𝜈 (Caroll 2004)
𝑔𝜇𝜈 =𝜂𝜇 𝜈 +ℎ𝜇𝜈 (2.1)
where the notion that the field is weak is communicated by the
condition that ℎ𝜇𝜈 1in natural units. The metric perturbation
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 3
constituting a GW in the Trace-Reversed Lorenz gauge for far away,
slow-moving matter is then given by the quadrupole formula
ℎ𝑖 𝑗 =2𝐺
𝑅𝑐4
𝑑2𝐼𝑖 𝑗
𝑑𝑡2(2.2)
where 𝐺is Newton’s constant, 𝑐is the speed of light, 𝑅is the
distance from the source to the detector, and 𝐼𝑖 𝑗 is the quadrupole
of the mass distribution
𝐼𝑖 𝑗 (𝑡)=∫𝑥𝑖𝑥𝑗𝑇00 (𝑡, 𝑥 , 𝑦, 𝑧)𝑑3𝑥(2.3)
where 𝑇00 is the energy density of the system and 𝑥𝑖is a Cartesian
coordinate relative to some origin. We will see later that the choice
of origin is inconsequential.
It can then be shown that, in the Transverse-Traceless (TT)
gauge, the metric perturbation ℎ𝜇𝜈 satisfies the wave equation
ℎ𝑇 𝑇
𝜇𝜈 =0(2.4)
where =−𝜕2
𝑡+𝜕𝑖𝜕𝑖is the flat space wave operator. In this gauge,
ℎ𝑇 𝑇
𝜇𝜈 has two degrees of freedom, or polarizations
ℎ𝑇 𝑇
𝜇𝜈 ∼©«
0 0 0 0
0ℎ+ℎ×0
0ℎ×−ℎ+0
0 0 0 0ª®®®¬
(2.5)
where ℎ𝑇 𝑇
𝜇𝜈 takes the form (2.5) for a wave propagating in the ˆ𝑧-
direction. Then using the geodesic deviation equation, it can be
shown that the effect of the wave passing through a group of test
masses is to cause them to expand and contract in a ‘plus’ or ‘cross’
pattern, depending on the polarization ℎ+or ℎ×. The signal that is
then perceived by the detector is
ℎdet(𝑡)=𝐹+ℎ++𝐹×ℎ×(2.6)
where 𝐹+and 𝐹×are the detector response functions that depend
on the orientation between the source and the detector (Klimenko
et al. 2005).
2.1.1 Signal Calculation
For simplicity, we define the trajectory 𝑧(𝑡)of the object to be
in the purely 𝑧-direction. Then, modeling the object as a point mass,
the energy density for an at most mildly relativistic object is given
by
𝑇00(𝑡, 𝑥 , 𝑦, 𝑧)=𝑀(𝑡) · 𝛿(𝑥)𝛿(𝑦)𝛿(𝑧−𝑧(𝑡))(2.7)
where 𝑀is the instantaneous mass of the object. Note that the
choice of origin in (2.7) corresponds to taking the origin in (2.3) as
the initial position of the object. The effect of shifting this origin to
some inertial frame, such as the center of mass frame, is to change
the quadrupole by terms that are at most quadratic in 𝑡, thereby
contributing a non-detectable overall constant to the strain (2.2) (for
details, see Section 10.5 of Weinberg 1972).
We then find that all of the components of 𝐼𝑖 𝑗 are zero except
𝐼𝑧𝑧 =𝑀(𝑡) · 𝑧2(𝑡)(2.8)
and that the strain for a general linear trajectory is given by
ℎ𝑧𝑧 =2𝐺
𝑅𝑐4n2𝑀¤𝑧2+4¤
𝑀 𝑧 ¤𝑧+2𝑀 𝑧 ¥𝑧+¥
𝑀𝑧2o(2.9)
where a dot denotes differentiation with respect to the detector frame
time 𝑡.
At this point, the astute reader will likely protest that such an
object ignores the cause of the acceleration and therefore does not
constitute a closed system. However, we can just apply the requisite
conservation laws ad-hoc by modeling our system as a discrete
rocket and invoking Newtonian arguments.
Suppose that our object achieves its acceleration by ejecting
mass at successive points in time. Then by conservation of mass,
momentum, and Newton’s Third Law at each ejection, it is neces-
sarily the case that
(i) ¤
𝑀𝑅=−¤
𝑀𝐸→¥
𝑀𝑅=−¥
𝑀𝐸
(ii) 𝑀𝑅¥𝑧𝑅=−𝑀𝐸¥𝑧𝐸
(iii) ¤
𝑀𝑅¤𝑧𝑅=−¤
𝑀𝐸¤𝑧𝐸
where quantities with subscript 𝑅and 𝐸pertain to the rocket and
exhaust, respectively. Note that the third condition follows from the
product rule combined with the conservation of momentum and
Newton’s Third Law. We are then left with
ℎ𝑧𝑧 =4𝐺 𝑀 (𝑡)
𝑅𝑐4𝑣(𝑡)2(2.10)
where 𝑣=¤𝑧. For an object with initial velocity 𝑣0that undergoes a
constant acceleration 𝐴at 𝑡=0for a period 𝑇, we have
𝑣(𝑡)=
𝑣0𝑡 < 0
𝐴𝑡 +𝑣00< 𝑡 < 𝑇
𝐴𝑇 +𝑣0𝑇 < 𝑡
(2.11)
and the strain is given by
ℎ𝑧𝑧 =
4𝐺𝑀
𝑅𝑐4𝑣2
0𝑡 < 0
4𝐺𝑀
𝑅𝑐4(𝐴𝑡 +𝑣0)20< 𝑡 < 𝑇
4𝐺𝑀
𝑅𝑐4(𝐴𝑇 +𝑣0)2𝑇 < 𝑡
(2.12)
Note that the constant velocity contributions do not emit power since
the power from GWs is given by 𝑃𝐺𝑊 ∝𝑑3𝐼𝑖 𝑗 /𝑑𝑡3. Even so, the
difference between the two constant velocity contributions should
be observable through the GW memory effect (Favata 2010).
For this study, we consider objects far enough away and ac-
celerating over sufficiently short distances such that 𝑅is effectively
constant. This condition is met when
𝑅1
2𝐴𝑇2(2.13)
As we can see from the strain (2.10), objects that accelerate for
longer durations are more easily detected, so accounting for the
object’s motion is well-motivated for future studies.
We now might like to view the metric perturbation in the TT
gauge
ℎ𝑇 𝑇
𝑖 𝑗 ∼1
2ℎ𝑧𝑧 ©«
000
0−1 0
001ª®¬(2.14)
choosing the transverse direction of 𝑛𝑖=ˆ𝑥. The form of (2.14)
means that waves that are emitted in the plane perpendicular to the
direction of motion of the object will be purely ‘+’ polarized, in
agreement with Sutton (2013). For a general direction 𝑛𝑖, we find
that
ℎ𝑇 𝑇
𝑖 𝑗 =1
2ℎ𝑧𝑧 ·(2.15)
MNRAS 000,1–18 (2022)
4L.Sellers et al.
·©«
(𝑛2
𝑥+1)𝑛2
𝑧+𝑛2
𝑥−1𝑛𝑥𝑛𝑦𝑛2
𝑧+𝑛𝑥𝑛𝑦𝑛𝑥𝑛3
𝑧−𝑛𝑥𝑛𝑧
𝑛𝑥𝑛𝑦𝑛2
𝑧+𝑛𝑥𝑛𝑦(𝑛2
𝑦+1)𝑛2
𝑧+𝑛2
𝑦−1𝑛𝑦𝑛3
𝑧−𝑛𝑦𝑛𝑧
𝑛𝑥𝑛3
𝑧−𝑛𝑥𝑛𝑧𝑛𝑦𝑛3
𝑧−𝑛𝑦𝑛𝑧(1−𝑛2
𝑧)2ª®®¬
Regardless of the normal direction, we see that the polarization
components ℎ+and ℎ×will be on the order of
ℎ+,×.1
2ℎ𝑧𝑧 (2.16)
with saturation occurring for ‘+’ polarisations of waves emitted in
the plane perpendicular to the motion of the object.
Since we are striving to attain an order of magnitude estimate
of detection ranges for these signals, we will use the form (2.14) for
the strain and set 𝐹+=1. Then the strain measured by the detector
is
ℎ(𝑡)=1
2ℎ𝑧𝑧 (𝑡)(2.17)
For future studies in which the orientation of the source with respect
to the detector needs to be examined, using (2.6) and (2.15) will be
necessary.
Here we comment on how these results change when the
propulsion mechanism is taken into account. We do this by cal-
culating the strain produced by a Newtonian rocket in Appendix A.
The cancellations that were justified using Newtonian arguments
then occur explicitly, producing (2.10) and an additional term de-
rived from the exhaust:
ℎ(𝐸)
𝑧𝑧 =4𝐺
𝑅𝑐4·∫𝑡
0−¤
𝑀(𝑡0)𝑢2(𝑡0)𝑑𝑡 0(2.18)
where ¤
𝑀 < 0and 𝑢is the constant velocity with which each ejected
mass travels, which may change between ejections. In simple terms,
the strain contribution from the exhaust is given by its total kinetic
energy, just like the rocket contribution.
The GW signal (2.17) can be seen in Figure 1for the case of
positive acceleration. We also take the mass 𝑀to be a constant and
neglect the exhaust contribution, which we do for the remainder of
the paper as well. Assuming that the mass is constant is reason-
able if the exhaust velocity is sufficiently larger than the change in
velocity of the rocket. The sample is then collected between times
𝑡1and 𝑡2. The constant velocity portions then contribute between
[𝑡1,0]and [𝑇 , 𝑡2]. Interestingly, the spectrum generated by positive
acceleration is identical to that of negative acceleration if we swap
𝑣0⇐⇒ 𝑣𝑓and |𝑡1| ⇐⇒ |𝑡2−𝑇|. In this manner, we can search
for two signals (accelerating and decelerating) at once by conduct-
ing our search in the frequency domain. The two cases can still be
differentiated by their spectrograms.
2.2 Frequency-Domain Signal
For calculating the spectrum of discrete data, we use the Fast Fourier
Transform (FFT) convention used in Allen et al. (2012),
ℎ(𝑓𝑘)= Δ𝑡
𝑁
𝑛=1
ℎ(𝑡𝑛)𝑒−2𝜋𝑖 𝑘𝑡𝑛/𝜏(2.19)
where 𝜏=𝑡1−𝑡2is the length of the time-domain data sample.
This convention matches the continuous Fourier Transform (CFT)
of the signal (2.17) with ℎ(𝑡 < 𝑡1)and ℎ(𝑡 > 𝑡2)set to zero. Then
the discrete spectrum is given by
ˆ
ℎ(𝑓)=∫∞
−∞
ℎ(𝑡)𝑒−2𝜋𝑖 𝑓 𝑡 𝑑𝑡
=∫𝑡2
𝑡1
ℎ(𝑡)𝑒−2𝜋𝑖 𝑓 𝑡 𝑑𝑡
(2.20)
100 50 0 50 100 150 200
t [s]
0
1
2
3
4
5
6
7
8
h
1e 22
Strain from Constant Acceleration
A
> 0
Figure 1. Plot of the GW signal (2.17) generated by a 10−3Mobject ac-
celerating constantly from 10−2𝑐→0.5𝑐in 100 s from 1021 m (≈30 kpc)
away. The data sample length exceeds the acceleration period, so the con-
stant velocity contributions from (2.12) are visible. Note that for a more
realistic object, the acceleration will not begin or cease instantly, so the
beginning and end of acceleration will be rounded off. However, the form
(2.12) suffices for an estimate of detection ranges.
There are then four cases that are important to consider when cal-
culating (2.20), namely
(i) The entire acceleration period is captured by the data sample:
𝑡1<0and 𝑇 < 𝑡2
(ii) The initial velocity and part of the acceleration period are
captured by the data sample: 𝑡1<0and 0< 𝑡2< 𝑇
(iii) The final velocity and part of the acceleration period are
captured by the data sample: 0< 𝑡1< 𝑇 and 𝑇 < 𝑡2
(iv) The data sample is contained entirely within the acceleration
period: 0< 𝑡1< 𝑇 and 𝑡1< 𝑡2< 𝑇
We then calculate (2.20) for Case (i), after which the results are
simple to generalize to the other three cases. For this case, the
spectrum (2.20) becomes
ℎ(𝑓)=2𝐺 𝑀
𝑅𝑐4(𝐼1+𝐼2+𝐼3)(2.21)
where
𝐼1=∫0
𝑡1
𝑣2
0𝑒−2𝜋𝑖 𝑓 𝑡 𝑑𝑡
𝐼2=∫𝑇
0(𝐴𝑡 +𝑣0)2𝑒−2𝜋𝑖 𝑓 𝑡 𝑑𝑡
𝐼3=∫𝑡2
𝑇(𝐴𝑇 +𝑣0)2𝑒−2𝜋𝑖 𝑓 𝑡 𝑑𝑡
(2.22)
The spectrum is then given by
ℎ(𝑓)=2𝐺 𝑀
𝑅𝑐4𝑖 𝐴 2
4𝜋3𝑓3n1−𝑒−2𝜋𝑖 𝑓 𝑇 o+
1
2𝜋2𝑓2n𝐴2𝑇𝑒−2𝜋𝑖 𝑓 𝑇 −𝐴𝑣01−𝑒−2𝜋 𝑖 𝑓 𝑇 o
+𝑖
2𝜋 𝑓 n𝐴2𝑇2𝑒−2𝜋 𝑖 𝑓 𝑡2+2𝐴𝑇𝑣 0𝑒−2𝜋𝑖 𝑓 𝑡2
+𝑣2
0𝑒−2𝜋𝑖 𝑓 𝑡2−𝑒−2𝜋𝑖 𝑓 𝑡1o (2.23)
where the term proportional to 𝑣2
0is zero for discrete data where
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 5
𝑓=𝑘𝜏−1with 𝑘∈Z. The spectrum (2.23) can be generalized to
the other three cases by taking
(i) 𝑇→𝑇
(ii) 𝑇→𝑡2
(iii) 𝑇→𝑇−𝑡1
(iv) 𝑇→𝑡2−𝑡1=𝜏
for each case. For future reference, we refer to these𝑇modifications
as the ‘perceived acceleration period’ of the data sample.
It is also instructive to view the spectrum (2.23)in terms of the
change in velocity Δ𝑣=𝐴𝑇. With this substitution, the spectrum
becomes
ℎ(𝑓)=2𝐺 𝑀
𝑅𝑐4 𝑖
4𝜋3𝑓3Δ𝑣
𝑇2n1−𝑒−2𝜋𝑖 𝑓 𝑇 o+
1
2𝜋2𝑓2Δ𝑣2
𝑇𝑒−2𝜋𝑖 𝑓 𝑇 +
Δ𝑣·𝑣0
𝑇1−𝑒−2𝜋𝑖 𝑓 𝑇 +
𝑖
2𝜋 𝑓 nΔ𝑣2𝑒−2𝜋 𝑖 𝑓 𝑡2+Δ𝑣𝑣0𝑒−2𝜋𝑖 𝑓 𝑡2
+𝑣2
0𝑒−2𝜋𝑖 𝑓 𝑡1𝑒−2𝜋𝑖 𝑓 (𝑡2−𝑡1)−1o
(2.24)
The form (2.24) is useful because the 𝑓−1term is independent of
𝑇. As we show next, the spectrum is dominated by this term, so the
dependency of the spectrum on 𝑇is largely mitigated using (2.24)
(see Appendix B).
2.2.1 Frequency Scaling Behavior
The form of the spectrum (2.23) is encouraging because it
suggests that objects with constant acceleration generatepar ticularly
strong signals for lower frequencies. This points to the use of low-
frequency, space-based detectors as a particularly promising avenue
for the detection of such objects. Here we parse out exactly how the
spectrum scales with frequency.
For simplicity, we consider Case (i) when discussing the spec-
trum scaling behavior. Furthermore, we assume that roughly the
same time duration of both initial and final velocity contributions
are present in the data sample. There are some slight variations
when these assumptions are relaxed, but for the purposes of this
study, this consideration is comprehensive.
We ultimately find that the spectrum scales like 𝑓−1for fre-
quencies 𝑓 > 𝜏−1and 𝑓0for frequencies 𝑓 < 𝜏 −1, which are irrele-
vant for discrete data (see Figure 2). To illustrate why this is the case,
we start by showing that the dynamic contribution 𝐼2scales like 𝑓−1
for frequencies 𝑓 > 𝑇 −1and 𝑓0for 𝑓 < 𝑇 −1, even for𝑇 > 1s. This
result is somewhat unintuitive because the dynamic contribution 𝐼2
contains 𝑓−2and 𝑓−3terms. However, for frequencies 𝑓 > 𝑇−1,
the following inequality is necessarily satisfied:
𝑇𝑛−1
𝑓≥1
𝑓𝑛(2.25)
Then, since each 𝑓−2and 𝑓−3term has a corresponding 𝑓−1term
with a relative factor of 𝑇and 𝑇2, respectively, the 𝑓−1behavior
will dominate for 𝑓 > 𝑇−1. For lower frequencies, the 𝐼2contribu-
tion plateaus because the phase in (2.20) becomes much larger than
the acceleration period 𝑇. Therefore, the integrands become rela-
tively constant as the frequency decreases further, and the spectrum
plateaus.
In the case where only the dynamic portion of the signal is
captured 𝜏 < 𝑇 (Case (iv)), the effect is to change the ‘perceived
acceleration period’ from 𝑇→𝜏, and the previous argument holds
for frequencies 𝑓 > 𝜏−1.
In the case where 𝜏 > 𝑇 and both constant velocity contribu-
tions are captured with roughly equal length, the contributions from
both 𝐼1and 𝐼3scale like 𝑓−1until the integrands begin to saturate
at 𝑓 < 𝜏−1. Then, since the dynamic contribution 𝐼2plateaus at
𝑓 < 𝑇 −1and 𝜏 > 𝑇 by assumption, the 𝑓−1contributions from 𝐼1
and 𝐼3will dominate for 𝑇−1< 𝑓 < 𝜏−1. So to summarize, the
spectrum scales like 𝑓−1for 𝑓 > 𝜏−1, which is the entire frequency
domain for discrete data.
In fact, we can show that any sufficiently smooth linear tra-
jectory will scale like 𝑓−1for 𝑓&𝜏−1if the trajectory does not
undergo rapid oscillations during the acceleration period. If we re-
main agnostic about the form of the velocity during the acceleration
duration and Taylor-expand on the interval [0, 𝑇 ],
𝑣(𝑡)=𝑣0+𝑣1𝑡
𝑇+𝑣2𝑡
𝑇2
+𝑣3𝑡
𝑇3
+... (2.26)
We can again compute the dynamic contribution to the spectrum
trajectory:
𝐼2=∫𝑇
0𝑣2(𝑡)𝑒−2𝜋𝑖 𝑓 𝑡 𝑑𝑡 (2.27)
Since the higher power contributions 𝑓−𝑛come only from 𝐼2, we
find that each of these terms has the form
ℎ(𝑓−𝑛) ∝ 𝑣𝑖𝑣𝑗𝑒−2𝜋𝑖 𝑓 𝑇
𝜋𝑛𝑓𝑛𝑇𝑛−1(2.28)
where 𝑣𝑖is the 𝑖th coefficient in the expansion (2.26). The spectrum
then clearly scales like 𝑓−1for 𝑓 > 𝑇 −1by (2.25).
Then for lower frequencies, we still have the constant veloc-
ity contributions that scale like 𝑓−1. We knew these contributions
would dominate for 𝑓 < 𝑇 −1because the dynamic contribution
plateaus for these frequencies. This need not be the case for a gen-
eral trajectory, particularly if 𝑣(𝑡)oscillates rapidly on the interval
[0, 𝑇 ]. However, for any sufficiently smooth trajectory, the previ-
ous arguments will hold, and the spectrum will scale like 𝑓−1for
𝑓&𝜏−11.
The significance of this scaling behavior is two-fold. First, since
the spectrum scales like 𝑓−1, the signal strength is clearly stronger
for lower frequencies, so detectors that are more sensitive to lower
frequencies will more easily detect these objects (for details, see
Section 4). Second, for an object that accelerates for a period 𝑇, the
𝑓−1behavior can be seen for frequencies 𝑓 < 𝑇 −1by extending
the data sample duration 𝜏 > 𝑇 , thereby capturing the gravitational
memory of the initial and final velocity. This effect can be seen in
Figure 3. As we discuss in more detail in Section 4, this is significant
because it implies that low-frequency detectors will not only be
sensitive to objects that accelerate for periods greater than or equal
to the characteristic time-scale of the detector, but for shorter times
as well. Granted, this effect is contingent on the assumption (2.13)
not being violated as 𝜏is increased, but as we show in Section 4, this
is not a particularly stringent requirement for objects at parsec-scale
distances from Earth.
1We use &as opposed to >to allow for small variations due to oscillations
in 𝑣(𝑡)
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6L.Sellers et al.
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10 13
10 11
10 9
10 7
10 5
|h(f)| [Hz 1]
Frequency Scaling of the Signal Spectrum
|h(f)|
m = 1
f = T 1
f = 1
Figure 2. Plot of the GW spectrum (2.23) for an acceleration period of 𝑇=
1000 s and a data sample length of 𝜏=105s. We can see that the spectrum
continues to scale like 𝑓−1past 𝑓−1=𝑇−1until 𝑓=𝜏−1, after which the
spectrum levels out.
10 310 210 1100101102
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v
T = 0.1 s
T = 1 s
T = 10 s
T = 100 s
T = 1000 s
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10 18
|h(f)| [Hz 1]
Spectrum h(f) for = 1000 s for a given
v
T = 0.1 s
T = 1 s
T = 10 s
T = 100 s
T = 1000 s
Figure 3. Plot of spectrum for a fixed change in velocity (2.24) for varying
acceleration periods 𝑇for two cases, namely where the data sample length
is made comparable to the acceleration period 𝜏=3𝑇and where 𝜏is fixed
at the maximum value of 𝑇plotted (𝑇=1000 s). We choose to plot these
values for a given Δ𝑣because the 𝑓−1behavior is independent of𝑇in this
case (see (2.24)). We see that for 𝜏∼𝑇, the spectrum plateaus at 𝑓∼𝑇−1,
whereas the 𝑓−1behavior is extended past these plateau points when 𝜏is
fixed to a larger value, up to some oscillations.
3 DETECTING CONSTANT ACCELERATION
We now move on to computing the distance at which LIGO and
LISA can detect RAMAcraft with constant acceleration.
3.1 Detection Statistics
Before we determine detection ranges for these signals, we first
review the detection statistics from the literature that we will use
to quantify the strength of GW signals. Specifically, we discuss the
search methods commonly used at detector collaborations, namely
transient burst and matched-filtering (MF) searches. But first, we
review detector sensitivity curves.
3.1.1 Detector Sensitivity Curves
When a GW passes through an object, it causes it to expand and
contract. Detectors then register a signal (or strain) ℎby tracking
these movement patterns. The most well-established of these detec-
tors are massive interferometers such as LIGO, Virgo, and KAGRA.
Using kilometer-scale lasers, these interferometers can detect dis-
placements down to one ten-thousandth of the width of a proton
(CaltechDataSheet 2022).
In principle, these detectors can detect strains ℎof any size.
However, as with any detector, this prospect is limited by the noise
profile. The noise profile of these detectors has been carefully scru-
tinized for many years for the purposes of noise mitigation (Abbott
et al. 2020). Such noise mitigation efforts have proven to be a de-
manding engineering challenge; the detectors are so sensitive that
distant airplanes, passing trucks, and even tumbleweeds can cause
interference (Berger 2018).
Since the noise profile sets a detection threshold, it also deter-
mines the sensitivity of the detector. For this reason, the terms ‘noise
profile’ and ‘sensitivity curve’ are interchangeable. The sensitivity
curve of LIGO (Abbott et al. 2020) and the projected sensitivity
curve of LISA (Robson et al. 2019) can be seen in Figure 4. Specifi-
cally, the curves in Figure 4are amplitude spectral densities (ASDs)
averaged over many data samples, giving them a colored Gaussian
noise profile. For real-time detection, stochastic glitches also con-
tribute to the noise 𝑛(𝑡), so the colored Gaussian noise description
is an approximation.
The ASD is the square root of the power spectral density (PSD),
which in the continuous case, is given by
𝛿(𝑓−𝑓0)𝑆𝑛(𝑓)=h𝑛(𝑓)˜𝑛(𝑓0)i (3.1)
𝑆𝑛(𝑓)is essentially an ensemble average of |𝑛(𝑓)|2where 𝑛(𝑓)
is a CFT of a pure noise input. However, in the discrete case, a
different convention is typically used depending on the convention
used to take the FFT of time-domain samples. In our case, we use
the convention from Allen et al. (2012), which matches the output
of the CFT of a time-domain signal with compact support within
the sample duration 𝜏. Then if 𝑛(𝑓𝑘)is an FFT generated using this
convention, we find that the curves in Figure 4and 𝑛(𝑓𝑘)are related
by
𝑆𝑛(𝑓𝑘)=h|𝑛(𝑓𝑘)|2i · (2Δ𝑓)(3.2)
where Δ𝑓is the frequency grid spacing (see Figure 5). The reason
for the 2Δ𝑓factor is two-fold. The factor of 2 accounts for the fact
that the sensitivity curves 𝑆𝑛in Figure 4 are one-sided PSDs. The
factor of Δ𝑓=𝜏−1accounts for the fact that the noise FFTs scale
like 𝑛(𝑓𝑘) ∝ 𝜏1/2.
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 7
100101102103104
f (Hz)
10 23
10 22
10 21
10 20
S
n
(f)1/2
LIGO Noise ASD
10 510 410 310 210 1100
f (Hz)
10 20
10 19
10 18
10 17
10 16
10 15
10 14
S
n
(f)1/2
LISA Noise ASD
Figure 4. (Top) Plot of LIGO sensitivity curve generated using the
aLIGOZeroDetHighPower method from the PyCBC library (Nitz et al.
2022). (Bottom) LISA sensitivity curve taken from the GitHub provided
in Robson et al. (2019).
3.1.2 Transient Burst Searches
One straightforward way to deduce whether or not a signal is de-
tectable is to determine whether the signal rises above the detector
sensitivity curve within some frequency band. This is the guiding
idea behind burst searches, which search for excess power within
some frequency band. This category of searches does not impose
any assumptions on the shape of the signal, so these search methods
apply to all possible GW shapes.
The well-established burst detection pipelines used, for exam-
ple, in the burst searches (The LIGO Scientific Collaboration et al.
2019a,b;Siemens et al. 2006) are the Coherent Wave Burst (cWB)
(Klimenko et al. 2008) and the Omicron-LIB (oLIB) (Robinet et al.
2020) pipelines. The statistic used in both of these pipelines is the
excess power statistic. For a continuous signal with spectrum ℎ(𝑓),
the excess power within some frequency band is given by
𝜌2
b=4∫𝑓2
𝑓1
|ℎ(𝑓)|2
𝑆𝑛(𝑓)𝑑𝑓 (3.3)
where 𝑆𝑛(𝑓)is again the power spectral density (PSD) of the de-
tector and 𝑓1and 𝑓2determine the choice of frequency band. For
discrete data ℎ(𝑓𝑘), the same statistic can be implemented as in
Anderson et al. (2001)
𝜌2
b=4
𝑘2
𝑘1
|ℎ(𝑓𝑘)|2
𝑆𝑛(𝑓𝑘)Δ𝑓(3.4)
where Δ𝑓=𝜏−1is the frequency resolution and 𝜏is the data
sample duration. In both cases, 𝑆𝑛should be normalized such that
the excess power output for a purely noise input is 𝜌2
b≈1. The
alternative would be to compute the excess power output for a pure
noise input 𝑛(𝑓)
𝜎2
b=4∫𝑓2
𝑓1
|𝑛(𝑓)|2
𝑆𝑛(𝑓)𝑑𝑓 (3.5)
and to find the signal-to-noise ratio (SNR) by taking 𝜌2
b→𝜌2
b/𝜎2
b=
(𝑆/𝑁)2
b. Since the noise will contain stochastic glitches, this is not
possible in general, but this can be done to a reasonable degree by
normalizing the output from noise data generated via the sensitivity
curves in Figure 4, as we do later in our analysis.
3.1.3 Matched Filtering
The burst SNR will only reach a detection threshold of 𝜌2
b/𝜎2
b>1
for signals whose spectrum exceeds the noise in some frequency
band. Ideally, we’d like to use a detection method that is valid for
signals that are everywhere below the noise level. Such a method is
provided by matched-filtering (MF).
The optimal detection statistic for processing signals embed-
ded in Gaussian noise is the MF SNR. Although interferometer
noise contains glitches that are non-Gaussian, glitch mitigation in
combination with MF (Allen et al. 2012) has allowed GW pipelines
to detect signals below the noise threshold (Abbott et al. 2016a).
For a strain comprised of both a signal and a noise part ℎ(𝑡)=
𝑠(𝑡) + 𝑛(𝑡), the MF statistic is given by Allen et al. (2012),
𝜌2
MF =4R∫∞
−∞
𝑠(𝑓)˜
ℎ(𝑓)
𝑆𝑛(𝑓)𝑑𝑓 (3.6)
where ˜
ℎ(𝑓)is the complex conjugate of the strain spectrum for the
signal being searched for. In the discrete case, we have
𝜌2
MF =4R
𝑘2
𝑘1
𝑠(𝑓𝑘)˜
ℎ(𝑓𝑘)
𝑆𝑛(𝑓𝑘)Δ𝑓
(3.7)
In the ideal case where the strain is comprised entirely of the signal,
the output (3.6) is the same as (3.3). However, the MF output for a
pure noise signal
𝜎2
MF =4R∫∞
−∞
𝑛(𝑓)˜
ℎ(𝑓)
𝑆𝑛(𝑓)𝑑𝑓 (3.8)
is typically much smaller than the excess power output of pure noise
(3.5) because the noise and the signal spectrum should be relatively
uncorrelated. Consequently, the discrepancy in statistic output 𝜌2
for signals and noise is typically much larger in MF than in burst
searches, making MF the more optimal method.
3.1.4 Modified Statistic and 𝜌2vs 𝜏Scaling
By convention, we choose to implement the detection statistic
such that the burst statistic 𝜎2
bfor a pure noise input is approximately
1. This is achieved by (see Figure 5)
𝜌2
b=2
𝑁
𝑁
𝑘=1
|ℎ(𝑓𝑘)|2Δ𝑓
𝑆𝑛(𝑓𝑘)(3.9)
MNRAS 000,1–18 (2022)
8L.Sellers et al.
10 210 1100
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10 36
[Hz 1]
LISA Sensitivity Curve vs |n(f
k
)|22
f
|n(f
k
)|22
f
S
n
(
f
)
Figure 5. Plot of LISA sensitivity curve 𝑆𝑛(𝑓𝑘)vs |𝑛(𝑓𝑘)|2·2Δ𝑓, where
𝑛(𝑓)is an FFT of the time-domain noise data simulated from 𝑆𝑛(𝑓)
using the colored_noise method from pycbc.noise.reproduceable. The ratio
is very close to 1 throughout the frequency interval, yielding 𝜎2≈1using
the convention (3.10).
for the signal output and
𝜎2
b=2
𝑁
𝑁
𝑘=1
|𝑛(𝑓𝑘)|2Δ𝑓
𝑆𝑛(𝑓𝑘)(3.10)
for the noise output, where 𝑁is the number of discrete points within
the sensitivity band of interest. Similarly, for MF searches we use
𝜌2
MF =2
𝑁
𝑁
𝑘=1
𝑠(𝑓𝑘) · ˜
ℎ(𝑓𝑘)Δ𝑓
𝑆𝑛(𝑓𝑘)(3.11)
for the signal output and
𝜎2
MF =2
𝑁
𝑁
𝑘=1
𝑛(𝑓𝑘) · ˜
ℎ(𝑓𝑘)Δ𝑓
𝑆𝑛(𝑓𝑘)(3.12)
for the noise output. We make such a choice because the statistics
𝜌2and 𝜎2differ from the conventions (3.3 –3.8) by the same factor
(2𝑁)−1, and their ratio is what yields the SNR.
With the discrete statistics (3.9) and (3.10) defined, we con-
clude here by explicitly clarifying the scaling of each of these statis-
tics with respect to the data sample length 𝜏.
The scaling for 𝜌b/𝜎bis fairly straightforward. As we men-
tioned earlier, we find that the noise spectrum scales like 𝑛(𝑓𝑘) ∝
𝜏1/2. From the factor of Δ𝑓in (3.10), we find 𝜎2
b∝𝜏0. Then,
assuming our signal corresponds to Case 1, the spectrum strength
does not depend on the data sample length ℎ(𝑓𝑘) ∝ 𝜏0, so 𝜌2
b∝𝜏−1
and the burst SNR scales like
𝜌b
𝜎b
=𝑆
𝑁b∝𝜏−1/2(3.13)
The scaling behavior for MF searches is slightly more complicated
because there is an extra effect that cannot be read from the defi-
nitions (3.11 -3.12). Using just these definitions and carrying out
a similar analysis to (3.13), we would find that 𝜎MF ∝𝜏−1/4and
𝜌MF ∝𝜏−1/2. However, because the noise correlation 𝜎MF is com-
puted as a discrete sum, the output depends on how many points are
used. Intuitively, if one samples a sine wave, the odds that the sum of
two sampled points is very small is quite low, but the odds increase
as many points are taken over many wavelengths. Empirically, we
find that this effect contributes an extra scaling of 𝜎MF ∝𝜏−1/4(see
Appendix D). Therefore, we find that the MF SNR scales like
𝜌MF
𝜎MF
=𝑆
𝑁MF ∝𝜏0(3.14)
Both of these results (3.13) and (3.14) are for an object that satisfies
Case (i) from the previous section. For an object where the data
sample ends before the acceleration ceases, the spectrum (2.24) is
modified to the perceived change in velocity: Δ𝑣=𝐴𝑇 →Δ𝑣0=
𝐴𝑡2. Then, as 𝜏increases, the perceived change in velocity increases
proportionally to 𝜏, and since the 𝑓−1part of the spectrum scales
like ℎ(𝑓−1) ∝ Δ𝑣02for small 𝑣0, we find
𝜌b
𝜎b
=𝑆
𝑁𝑏∝𝜏3/2(3.15)
and
𝜌MF
𝜎MF
=𝑆
𝑁MF ∝𝜏1(3.16)
for 𝑡2< 𝑇 and small 𝑣0. These results will be important to consider
in the next section where we present the ranges for both LISA and
LIGO using both the same and different data sample lengths.
3.2 Detection Analysis
We are now prepared to compute the ranges at which LIGO and
LISA can detect linearly moving RAMAcraft undergoing con-
stant acceleration. Specifically, we will compute detection ranges
for both burst and MF searches to account for the two cases in which
these signals are either passively (burst) or actively (MF) searched.
For simplicity, we compute both statistics for the ideal case where
the strain ℎis devoid of noise 𝑠(𝑡)=ℎ(𝑡). In this case, the two raw
outputs (3.9) and (3.11) are equivalent. We then differentiate the
burst SNR from the MF SNR by applying the appropriate normal-
izations (3.10) and (3.12), respectively. Then choosing a detection
threshold of (𝑆/𝑁)det =8, we determine the detectability of linearly
accelerating trajectories with small initial velocity 𝑣0based on their
mass 𝑀, change in velocity Δ𝑣, and distance from the detector 𝑅.
We use the change in velocity Δ𝑣as opposed to the acceleration 𝐴
because the spectrum is effectively independent of the acceleration
duration 𝑇for a given Δ𝑣, thereby eliminating a variable from our
analysis. We choose signals with small initial velocity 𝑣0both be-
cause the results that we find apply to objects with large changes in
velocity, and also because setting 𝑣0→0allows us to express the
range in terms of one variable 𝑀Δ𝑣2(see (3.17)).
Furthermore, since the burst detection pipelines are indifferent
to signal shape, linearly accelerating signals that fall within the
detectable regime for LIGO burst searches should have already
been detected (if the signal occurred while LIGO was online). The
lack of burst detections thereby constrains the parameter space for
linearly accelerating trajectories.
3.2.1 Methodology
For this analysis, we use discrete data and data analysis packages
from both the NumPy and PyCBC (Nitz et al. 2022) libraries. The
key steps involved for a given parameter set {𝑀, Δ𝑣 , 𝑇, 𝑅 }, detec-
tor (LIGO or LISA), and detection method (burst or MF) are the
following:
(i) Choosing some test parameter set {𝑀 , Δ𝑣, 𝑇 , 𝑅0}, generate a
time-domain signal ℎ(𝑡𝑛)according to (2.17).
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 9
(ii) Generate a frequency-domain signal ℎ(𝑓𝑘)using the con-
vention (2.19).
(iii) Import data for either the LIGO or LISA noise sensitivity
curve 𝑆𝑛(𝑓𝑘).
(iv) Simulate noise data in the time-domain 𝑛(𝑡𝑛)using the sen-
sitivity curve.
(v) Generate a frequency-domain noise signal 𝑛(𝑓𝑘)using the
convention (2.19).
(vi) Depending on the search method, compute 𝜎2for the noise
input 𝑛(𝑓𝑘)using either (3.10) or (3.12).
(vii) Depending on the search method, compute the detection
statistic 𝜌2by either (3.9) or (3.11).
(viii) ‘Divide out’ the test parameter set by calculating a ‘reduced
SNR’ (𝑆/𝑁)red for either a burst or MF search. This allows us to
factor out the parameter dependence of the SNR and solve for 𝑅for
a given 𝑀and Δ𝑣.
(ix) Choose a detection threshold (𝑆/𝑁)det .
(x) With the calculated ‘reduced SNR’ and the parameter depen-
dencies factored out, solve for 𝑅for a given 𝑀Δ𝑣2.
Here we go over these steps in more detail where necessary. To
generate the data ℎ(𝑓𝑘), we use the numpy.fft library, which uses the
convention (2.19) out of the box. To import data for the noise sensi-
tivity curves 𝑆𝑛(𝑓𝑘), we use the aLIGOZeroDetHighPower method
from the pycbc.psd library for the case of LIGO. For LISA, we use
the sensitivity curve data provided in the GitHub from Robson et al.
(2019) and interpolate the data on the frequency grid of ℎ(𝑓𝑘). To
simulate noise 𝑛(𝑡𝑛)using the sensitivity curves 𝑆𝑛(𝑓𝑘), we use the
colored_noise method from pycbc.noise.reproduceable.
The sensitivity curve constitutes an ensemble average of many
noise samples, which is why we need to generate a simulated noise
spectrum 𝑛(𝑓𝑘)when we already have a sensitivity curve 𝑆𝑛(𝑓𝑘).
Simulating a noise sample 𝑛(𝑡𝑛)and taking its spectrum 𝑛(𝑓𝑘)
better represents what one noise data sample will look like, and
therefore better represents the output 𝜎2if a pure noise data sample
is collected.
Here we explain how to compute the range 𝑅after computing
statistics for a test parameter set. By taking 𝑣0→0, our spectrum
for a given change in velocity Δ𝑣(2.24) changes like
ℎ(𝑓)=2𝐺 𝑀Δ𝑣2
𝑅𝑐4𝑖 𝑒−2𝜋 𝑖 𝑓 𝑡2
2𝜋 𝑓 +𝑒−2𝜋𝑖 𝑓 𝑇
2𝜋2𝑓2𝑇(3.17)
+𝑖
4𝜋3𝑓3𝑇21−𝑒−2𝜋𝑖 𝑓 𝑇
Then if we extract the parameter dependence of each spectrum
by
ℎ𝑟 𝑒𝑑 (𝑓)=2𝐺 𝑀 Δ𝑣2
𝑅𝑐4−1
ℎ(𝑓)(3.18)
we can rewrite the SNR quantities as
𝜌2
b
𝜎2
b
=2𝐺 𝑀Δ𝑣2
𝑅𝑐422
𝑁Í𝑁
𝑘=1|ℎ𝑟𝑒 𝑑 (𝑓𝑘)|2Δ𝑓
𝑆𝑛(𝑓𝑘)
2
𝑁Í𝑁
𝑘=1|𝑛(𝑓𝑘)|2Δ𝑓
𝑆𝑛(𝑓𝑘)
=2𝐺 𝑀Δ𝑣2
𝑅𝑐42𝑆
𝑁2
red,b
(3.19)
for burst searches and
𝜌2
MF
𝜎2
MF
=2𝐺 𝑀Δ𝑣2
𝑅𝑐42
𝑁Í𝑁
𝑘=1|ℎ𝑟𝑒 𝑑 (𝑓𝑘)|2Δ𝑓
𝑆𝑛(𝑓𝑘)
2
𝑁|R nÍ𝑁
𝑘=1
𝑛(𝑓𝑘)· ˜
ℎ𝑟𝑒 𝑑 (𝑓𝑘)Δ𝑓
𝑆𝑛(𝑓𝑘)o|
=2𝐺 𝑀Δ𝑣2
𝑅𝑐4𝑆
𝑁2
red,MF
(3.20)
for MF searches. Then after computing the statistics 𝜌2and 𝜎2, we
can calculate (𝑆/𝑁)red by dividing out the test parameter set and
solving for the range 𝑅for a given 𝑀Δ𝑣2from the equation
𝑆
𝑁2
det
=𝜌2
b
𝜎2
b
=2𝐺 𝑀Δ𝑣2
𝑅𝑐42𝑆
𝑁2
red,b
(3.21)
for burst searches and
𝑆
𝑁2
det
=𝜌2
MF
𝜎2
MF
=2𝐺 𝑀Δ𝑣2
𝑅𝑐4𝑆
𝑁2
red,MF
(3.22)
for MF searches. In particular, we see that 𝑅∝ (𝑆/𝑁)bor 𝑅∝
(𝑆/𝑁)2
MF, depending on the search method.
3.2.2 Results
Following the prescription just outlined, here we look at the
ranges that an object producing the GW signal (2.17) would be
detectable by LIGO or LISA. In particular, we take the limit 𝑣0→0,
which corresponds to objects whose change in velocity is much
larger than their initial velocity (or vice versa if 𝐴 < 0). In this case,
we can plot the range 𝑅as a linear function of one parameter 𝑀Δ𝑣2,
which gives the total change in kinetic energy of the RAMAcraft
if its mass is approximately constant.
Before presenting the results, we explain here our choice of
data sample parameters 𝜏,𝑡1, and 𝑡2. For simplicity, we limit our
considerations for now to Case (i), where the entire acceleration
duration is contained within the data sample. Specifically, we choose
𝑡1=−𝜏/3and 𝑡2=2𝜏/3for a given 𝜏to stay within the constant
spectrum regime in Figure C1 (see Appendix C).
Our choice of data sample length 𝜏then depends on the detec-
tion method. For burst searches, the SNR scales like 𝜏−1/2, so we
choose a 𝜏not much smaller than that which makes the sensitivity
band accessible at 𝑓min =𝜏−1. For LISA, we choose 𝜏=600 s,
and for LIGO, we choose 𝜏=0.1s. For MF searches, the SNR is
independent of 𝜏, up to some oscillations to within a factor of a
few (see Appendix D). We then choose 𝜏=1,200 s for LISA and
𝜏=300 s for LIGO.
The frequency band that we compute the statistics over also
depends on the detection method. For burst searches, the SNR is
essentially just an average of the ratio ℎ(𝑓)/𝑛(𝑓), so including
points only very close to the maximum value will result in the largest
output. This is the idea behind band-pass burst searches (Moreschi
2019). For our purposes, this can be achieved by sampling a small
number of points about the sensitivity band minimum. It is however
necessary to sample enough points such that the noise statistic 𝜎2is
close to 1. This is unlikely to be the case when sampling just one or
two points because the quantity |𝑛(𝑓)|2Δ𝑓oscillates about 𝑆𝑛(𝑓).
Empirically, we find that sampling 5 points about 1 mHz for the case
of LISA and 20 points about 100Hz for the case of LIGO (and the
aforementioned frequency spacings 𝜏−1) achieves this goal.
For MF searches, one generally wants a much wider frequency
range to achieve a small noise correlation (3.12). Empirically, we
find not much difference in output when using a wide-band of
MNRAS 000,1–18 (2022)
10 L.Sellers et al.
[10−3,1]Hz for LISA and [1, 200] Hz for LIGO versus a moderate
band of [10−3,10−2]Hz for LISA and [10, 200] Hz for LIGO, so
we choose the latter to conserve computational resources 2.
To conclude our review of parameter choices, the choice of
acceleration period 𝑇from the test parameter set {𝑀 , Δ𝑣, 𝑇 , 𝑅0}
should not affect the results much due to the approximate indepen-
dence of the spectrum on 𝑇for a given Δ𝑣(see Figure D2). Still,
there are some slight variations due to the oscillations occurring for
𝑓 < 𝑇 −1(see Figure 3). However, we find that the contributions
of these oscillations is to cause variations in the SNR to within a
factor of a few as well. We therefore just quote our choices in 𝑇
in generating the signals, namely 𝑇=10 s and 10ms for the LISA
and LIGO burst search, respectively, and 𝑇=100 s and 10s for the
LISA and LIGO MF search, respectively. The various acceleration
periods 𝑇were chosen such that 𝑇≤𝜏for each search, ensuring
that each test parameter set satisfied Case (i).
The signal ranges, using both burst and MF search meth-
ods, can be seen in Figure 6for LIGO and LISA. We look
at signals with parameters 𝑀and Δ𝑣ranging from 𝑀Δ𝑣2=
[10−9𝑀𝑐2,0.25 𝑀𝑐2]. The right-hand limit was chosen so
that the parameter space ranges up to a solar mass object achieving
a velocity that satisfies the mildly relativistic assumption of (2.7).
The left-hand bound was chosen such that the range of the maximal
curve (LIGO MF search) was approximately 10−1pc, around and
below which the Newtonian contribution to the strain may begin
to dominate. We leave determining exactly where this transition
occurs for a future study. Since the x-axis is the change in kinetic
energy of the object if the mass 𝑀is approximately constant, it is
instructive to quantify these bounds in terms of rest-mass conver-
sions of astrophysical objects. The left-hand bound 10−9M𝑐2then
corresponds to the rest-mass of an object one-hundredth the mass of
Mercury being converted into kinetic energy, while the right-hand
bound corresponds to the rest-mass conversion of a star a quarter
the mass of the Sun.
For a given curve, the area underneath gives the parameter
space for which detection is possible. For LIGO burst searches in
particular, since the SNR is indifferent to the shape of the signal, the
area under the burst search curve represents parameter sets {𝑀, Δ𝑣}
that would have been detected by existing burst search pipelines. In
other words, these parameter sets have likely not occurred while
LIGO was online.
To summarize the results from Figure 6, the largest ranges
are given by the LIGO MF search. On the higher end of this plot,
we see that RAMAcraft with masses 10 ·10−3M(10 Jupiters)
undergoing a change of velocity of 0.1𝑐are detectable up to 105−
106pc, or right around the distance to the Andromeda galaxy. On
the lower end of the range, we see that Mercury-scale masses ∼
10−7Mundergoing the same change in velocity are detectable up
to 1-10 pc, or at and beyond the distance to Proxima Centauri.
Similarly, using the results from the LIGO burst search, we
can rule out the occurrence of certain RAMAcraft (while LIGO
was online) by the lack of burst detections. On the upper end of
the range, we can rule out a tenth of a solar mass RAMAcraft
achieving Δ𝑣=0.1𝑐out to 105−106pc, while on the lower end,
we can rule out ten Mercury-mass objects achieving Δ𝑣=0.1𝑐at
1 pc. Since the range is proportional to the mass, we can interpolate
between these two regimes by taking 𝑀→𝑎 𝑀 and 𝑅→𝑎 𝑅 for
2Though the LIGO band extends into the kHz regime, since the signal
scales like 𝑓−1, we find not much benefit to integrating much further past
200 Hz.
𝑎 > 1. Similarly, we can interpolate between different Δ𝑣by taking
Δ𝑣→𝑎Δ𝑣and 𝑅→𝑎2𝑅.
An important distinction remains across different values of
𝑇in that the approximation of constant 𝑅(2.13) is more easily
satisfied for objects with smaller acceleration periods 𝑇. Since our
results assume that 𝑅is constant, it is important to discern which
parameters are valid for the different regions of the plots. We can
do this by plotting the curve
𝑅=5Δ𝑣𝑇 (3.23)
for varying values of 𝑇and masses 𝑀. Points above these curves
satisfy the approximation (2.13) to at least a factor of 10. Quoting
the numbers here, the results from Figure 6for the LIGO MF search
RAMAcraft are accurate for a ten Jupiter mass objects for the
entire plot domain for acceleration periods 𝑇≤1011 s (over 3000
years), while the results are accurate in the region of interest (105-
106pc) for 𝑇≤1013 s (over 300,000 years). The results for mercury-
scale mass RAMAcraft are accurate for 𝑇≤5×108s (or about
15 years). For burst searches, the results for 10−1Mobjects are
accurate for the entire plot domain for 𝑇≤1010 s (over 300 years)
and for 𝑇≤1012 s (over 30,000 years) for the region of interest
(105- 106pc). The results for ten mercury-scale mass 10−6M
objects are valid for 𝑇≤5×107s (about a year and a half). To
summarize, the results for the LIGO MF search are quite robust
to the assumption (2.13), though loosening this requirement may
be warranted for the burst search results. We leave this for future
studies
3.2.3 Burst vs MF Search
In Figure 6, we can see explicitly that the MF search method
is more sensitive than the burst search, as expected. Empirically, we
find that the ratio of ranges using an MF vs burst search is about 36
for the case of LIGO and about 6 for the case of LISA. Because the
range 𝑅is proportional to 𝜌2
MF, this discrepancy of a factor of 6 can
be explained by a √6variation in 𝜌MF, which is on par with what
we see in Figure D2.
3.2.4 Comparing LIGO and LISA
From Figure 6, we can see that LIGO and LISA perform com-
parably for both burst and MF searches, but that LIGO edges LISA
in both cases, namely by a factor of a few for burst searches and
around ten for MF searches.
This may be counter-intuitive because, for example, the signal
spectrum scales like 𝑓−1, and LISA is more sensitive for lower
frequencies. We can however see why this is the case from a simple
analytical argument. We aim to compute a ratio of ranges for LIGO
and LISA
𝑟det =𝑅LIGO
𝑅LISA (3.24)
Then for both detectors, the SNR squared accumulated at one point
is proportional to
Δ 𝜌2
b
𝜎2
b!∝|ℎ(𝑓)|2
|𝑛(𝑓)|2(3.25)
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 11
10 910 810 710 610 510 410 310 210 1
M( v)2 [M c2]
10 2
10 1
100
101
102
103
104
105
106
107
108
R [pc]
Range for a given M( v)2 (LIGO)
MF
burst
10 910 810 710 610 510 410 310 210 1
M( v)2 [M (c)2]
10 3
10 2
10 1
100
101
102
103
104
105
106
107
R [pc]
Range for a given M( v)2 (LISA)
MF
burst
Figure 6. Range 𝑅for the signal with parameters 𝑀Δ𝑣2=
[10−9M(𝑐/s)2,0.25 M𝑐2]to be detected by LIGO (Top) or LISA (Bot-
tom) using either a burst or MF search.The region under the curves yields the
parameter sets {𝑀 , Δ𝑣 , 𝑅 }for which the signal would have been detected
(LIGO burst) or can be detected (MF). The dotted horizontal lines give, in
increasing order, the distance to the nearest star, the distance to the edge of
our Galaxy, and the distance to the Andromeda galaxy.
in the case of burst searches (see (3.9-3.10)) and
Δ 𝜌2
MF
𝜎2
MF !∝|ℎ(𝑓)|2
𝑛(𝑓)˜
ℎ(𝑓)∼ℎ(𝑓)
𝑛(𝑓)(3.26)
in the case of MF searches (see (3.11-3.12)), where the last relation
holds up to a tripling of the complex phase of the strain spectrum
𝑧·𝑧
˜𝑧=𝑟𝑒𝑖 𝜙 ·𝑟 𝑒𝑖 𝜙
𝑟𝑒−𝑖 𝜙 =𝑟 𝑒3𝑖 𝜙 (3.27)
Then if we extract the 𝑅dependence of each spectrum by 𝑅·ℎ(𝑓)=
ℎ0(𝑓), we find that each statistic scales like
Δ 𝜌2
b
𝜎2
b!∝1
𝑅2|ℎ0(𝑓)|2
|𝑛(𝑓)|2
Δ 𝜌2
MF
𝜎2
MF !∝|ℎ(𝑓)|2
𝑛(𝑓)˜
ℎ(𝑓)∼ℎ(𝑓)
𝑛(𝑓)=1
𝑅
ℎ0(𝑓)
𝑛(𝑓)
(3.28)
and we again find that the range 𝑅scales like 𝑅∝ (𝜌b/𝜎b)or
𝑅∝ (𝜌2
MF/𝜎2
MF), as we saw in the methodology section. Then
moving from LIGO to LISA, our overall sensitivity drops from
about 5×10−24 Hz−1/2to 10−20 Hz−1/2, and the sensitivity band
minimum shifts from around 100 Hz to 5×10−3Hz. Since the
signal spectrum scales like 𝑓−1, the signal spectrum will be larger in
LISA’s sensitivity band than LIGO’s. Since these effects will apply
for each SNR contribution (3.25) and (3.26), they should apply to
the total SNR as well, yielding the following proportionality for 𝑟det
𝑟det ∝5×10−3Hz
100 Hz 10−20 Hz−1/2
5×10−24 Hz−1/2!(3.29)
for both search methods. The proportionality (3.29) is the same for
both search methods due to the fact that 𝑅∝𝜌b/𝜎band 𝑅∝𝜌2
b/𝜎2
b,
respectively. However, there are two more effects that we need to
take into account. The first is for burst searches, namely that since
the burst SNR scales like 𝜏−1/2, LIGO will receive a gain in range
by virtue of the fact that it can access its sensitivity band with a
smaller 𝜏. We then get that the ratio (3.24) for burst searches is
𝑟(𝑏)
det ≈5×10−3Hz
100 Hz 10−20 Hz−1/2
5×10−24 Hz−1/2!𝜏LISA
𝜏LIGO 1/2
(3.30)
Then from our results, for the burst search we chose 𝜏LISA =200 s
and 𝜏LIGO =0.1s. Plugging these values in then predicts 𝑟(𝑏)
det ≈
4.47. Using the data from Figure 6, we find that 𝑟(𝑏)
det ≈2.57, so our
results are in good agreement with this prediction.
For MF searches, recall from (3.14) that the noise correlation
𝜎MF receives an additional scaling of 𝜏−1/4, which we conjecture is
due to the changing number of points in the correlation computation.
Then for a given frequency spacing 𝜏−1, since the LIGO sensitivity
band sits at around 100 Hz vs 5mHz for the case of LISA, the LIGO
noise correlation computation contains 100 / 0.005 = 20,000 times
as many points as LISA. One may try to overcome this effect by
increasing the sample length for LISA, but this will not increase
the SNR by (3.14). Then since the range scales like 𝑅∝𝜌2
MF, this
effect enters the ratio expression like
𝑟(MF)
det ≈5×10−3Hz
100 Hz 1/2 10−20 Hz−1/2
5×10−24 Hz−1/2!
≈14.14
(3.31)
which again matches very closely what we get using the data from
Figure 6𝑟(MF)
det ≈15.
4 DISCUSSION
In this section, we discuss how sensitivity improvements for low
frequencies in particular will benefit the search for RAMAcraft,
as well as ways in which our analysis can be generalized to other
acceleration mechanisms
4.1 Sensitivity Improvements: Comparing Other Detectors
The results (3.30) and (3.31) are not to say that low-frequency
detectors aren’t more useful for detecting RAMAcraft. For the
case of LISA, the deficit in overall sensitivity as compared to LIGO
is too large to take advantage of the low-frequency sensitivity. There
are, however, low-frequency detectors for which this is not the case,
such as DECIGO and the Big Bang Observer (BBO). For the case of
DECIGO, the sensitivity band minimum sits at around 0.5 Hz with
a sensitivity of about 10−24 Hz−1/2(Moore et al. 2014). Assuming
that (3.30) and (3.31) continue to hold, plugging these numbers
in and using 𝜏DEC =20 s yields both 𝑟(𝑏)
det and 𝑟(MF)
det ≈0.01.
DECIGO searches will therefore be 100 times more sensitive than
LIGO, increasing the search volume by a factor of 106. Similarly,
MNRAS 000,1–18 (2022)
12 L.Sellers et al.
10 310 210 1100
100
101
102
103
b
b
(
b
, 0
b
, 0
)
1
Gain in SNR vs Shift Parameter (burst)
(
S
/
N
)
b
/(
S
/
N
)
b
, 0
m = 1
10 310 210 1100
100
MF
MF
(
MF
, 0
MF
, 0
)
1
Gain in SNR vs Shift Parameter (MF)
(
S
/
N
)
MF
/(
S
/
N
)
MF
, 0
m 0.26
Figure 7. Sensitivity gain achieved by shifting over the LISA sensitivity
curve by a factor 𝛿 < 1. The quantitiy 𝜌/𝜎· (𝜌0/𝜎0)−1is the ratio of the
SNR computed when shifting the sensitivity curve by the factor 𝛿to the
SNR computed when the curve is not shifted at all. The scaling behavior
𝛿−1for 𝜌bmatches nearly perfectly, while the scaling behavior 𝛿−0.25 for
𝜌MF has some oscillations about the behavior that we expect, likely due to
inconsistencies associated with computing 𝜎MF .
the sensitivity band of the BBO sits at around 0.1 Hz, also with
a sensitivity of about 10−24 Hz−1/2(Moore et al. 2014). Plugging
these values in with 𝜏BBO =40 s yields 𝑟(𝑏)
det and 𝑟(MF)
det ≈0.005,
making the range of the BBO even better than DECIGO by about a
factor of 2.
Another promising detection method to consider is the use of
Pulsar Time Arrays (PTAs), since their sensitivity bands fall within
the nHz regime (Moore 2015;Hazboun et al. 2019). Although the
sensitivity curves presented in Hazboun et al. (2019) look reason-
ably sensitive, these plots are of the characteristic strain, which
carries a factor of 𝑓1/2compared to 𝑆𝑛(𝑓)1/2. So even though the
sensitivity band minimum occurs around 5 nHz, the overall sen-
sitivity of about 10−10 Hz−1/2is too low to provide a benefit as
compared to LIGO. Plugging these values into (3.30) and (3.31)
and using 𝜏PTA =109s (around 30 years), we find that 𝑟(𝑏)
det and
𝑟(MF)
det ≈108. However, because the sensitivity band is on the order
of nHz, PTAs can be less sensitive than LIGO by a factor of about
105and still provide a benefit 𝑟det <1. Therefore, sensitivity im-
provements to PTAs, such as SKA, could significantly improve the
feasibility of detecting RAMAcraft.
4.2 Sensitivity Improvements: Longer Acceleration Periods
The preceding analysis is predicated on the assumption that the
entire acceleration period is contained within the data sample, or
in other words, that the ‘perceived change in velocity’ does not
change with 𝜏. However, this very well may not be the case. More to
the point, detectors that can record data for longer periods of time
without interruption may be much more sensitive to RAMAcraft
with longer acceleration periods, as is evident from the scaling laws
(3.15 -3.16). There is good reason to believe that LISA (and other
space-based detectors) will be capable of longer uninterrupted data
samples than LIGO due to the fact that LISA will be isolated in
space, while LIGO experiences the perturbing influence of ground-
based noise. A reasonable prediction would be that LISA will be
capable of uninterrupted samples of up to 1 week, whereas LIGO’s
capacity will be approximately 1 day (Professor Neil Cornish, pri-
vate communication).
The impact that this effect would have on 𝑟det is clear from
(3.15 -3.16). For burst searches, the scaling (3.15) will overpower
the third factor in (3.30) and yield
𝑟(𝑏)
det (𝑇 > 𝑡2) ≈ 5×10−3Hz
100 Hz 10−20 Hz−1/2
5×10−24 Hz−1/2!
×𝜏LISA
𝜏LIGO −1(4.1)
which gives ≈0.014 for 𝜏LISA =1week and 𝜏LIGO =1day. For an
MF search, we would get
𝑟(MF)
det (𝑇 > 𝑡2) ≈ 5×10−3Hz
100Hz 1
2 10−20Hz−1/2
5×10−24Hz−1/2!
×𝜏LISA
𝜏LIGO −1(4.2)
which would give ≈2.02 for objects with acceleration periods
greater than 1 week. Therefore, LISA will be superior to LIGO at
detecting bursts but still worse for MF detections.
The most extreme examples of uninterrupted data collection
come from PTAs, which can record for up to years without interrup-
tion. Plugging in the PTA parameters and 𝜏PTA =10 years into (4.1)
and (4.2), we find that 𝑟(b)
det (𝑇 > 𝑡2) ≈ 0.055 and 𝑟(MF)
det (𝑇 > 𝑡2) ≈
17,328. So, PTAs would be more sensitive to bursts from the get-go
and furthermore could be about 109times less sensitive than LIGO
and still provide a benefit for MF searches. Therefore, sensitivity
improvements to PTAs may be of particular value for detecting
RAMAcraft with long acceleration periods in the future.
4.3 Sensitivity Improvements: Overall Sensitivity and
Shifting the Sensitivity Band
So far, we have examined sensitivity improvements that come from
moving between different detectors. Here, we assess the benefit of
sensitivity improvements for a given detector, namely improvements
to the overall sensitivity or a shift of the sensitivity band.
We can quantify improvements to the overall sensitivity of a
given detector by simply scaling its sensitivity curve
𝑆𝑛(𝑓) → 𝛼𝑆𝑛(𝑓)(4.3)
for 𝛼 < 1. Given (3.9 -3.12), this would result in increases of the
burst and MF SNRs by a factor of 𝛼−1and 𝛼−1/2, respectively, both
of which would increase the range by a factor of 𝛼−1.
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 13
Similarly, we can quantify a sensitivity improvement to lower
frequencies by shifting the sensitivity curve like
𝑆𝑛(𝑓) → 𝑆𝑛(𝛿 𝑓 )(4.4)
for 𝛿 < 1. The effect that this would have on the SNR is also
straightforward to work out. For a given signal ℎ(𝑓), the effect of
shifting the sensitivity band of 𝑆𝑛by a factor 𝛿is to accumulate the
majority of the SNR for frequencies where ℎ(𝑓)is larger by a factor
of 𝛿−1. This will result in 𝜌b/𝜎bincreasing by a factor of 𝛿−1. For
MF searches, 𝜌MF/𝜎MF will increase by 𝛿−1/2due to the increase
in ℎ(𝑓). However, 𝜌MF/𝜎MF will also decrease by a factor 𝛿1/4
due to a decrease in points used to calculate the noise correlation
𝜎MF. Thus, the total increase in 𝜌MF/𝜎MF should be by a factor of
𝛿−1/4. Therefore, the range will increase by a factor of 𝛿−1for burst
searches and a factor of 𝛿−1/2for MF searches. This effect can be
seen in Figure 7.
4.4 Sensitivity Improvements: Elongating Data Samples Past
the Acceleration Period
As we mentioned in Section 2, assuming that the approximation
(2.13) is valid for 𝜏 > 𝑇 , we can extend the 𝑓−1scaling of the
spectrum past 𝑇−1by increasing the data sample length 𝜏. This
may be useful to take advantage of the low-frequency sensitivity of
some hypothetical detector. Assuming that the acceleration duration
is entirely contained in the data sample, elongating 𝜏will lengthen
the portions of constant velocity contributions. Since the integration
bounds enter these contributions only in their phase argument, this
does not result in any scaling behavior. However, as we increase 𝜏,
we do gain access to lower frequencies in our spectrum 𝑓=𝑘 𝜏−1. It
is however important to do so by encompassing both the initial and
final constant velocity contributions, otherwise the spectrum will
diminish as in Appendix C.
Since our spectrum scales like 𝑓−1, increasing data sample
times may boost the range for these objects for some combination
of data sample length and detector sensitivity band. The extent to
which we can extend 𝜏over the constant velocity contributions will
depend on the duration said contributions remain in the detector’s
field of view. More to the point, since this paper assumes that the
object is approximately a constant distance from the detector, we
can only extend 𝜏for a duration less than or equal to the point
where this approximation is no longer valid. In order to determine
this duration, we can define the ‘time in view’ 𝑇𝑅for the object at
a given distance from the detector:
𝑇𝑅=𝑅
10h𝑣i(4.5)
where we claim that the constant distance approximation is valid
for objects that travel 10% of their initial distance from the detector.
Then h𝑣iis the average velocity of the signal over the data sample
and is given by,
h𝑣i=1
𝜏∫𝑡2
𝑡1
𝑣(𝑡)𝑑𝑡
=𝑣0+𝑡2−𝑇/2
𝜏Δ𝑣
(4.6)
where again Δ𝑣=𝐴𝑇. The result clearly depends on which veloci-
ties are captured by the data sample, though we can gauge the extent
to which we can extend our data sample for objects that undergo
a very large change in velocity compared to their initial velocity
Δ𝑣𝑣0. In this case, assuming that a reasonable portion of the
final velocity is present in the data sample, the average velocity is
h𝑣i ≈ Δ𝑣.𝑇𝑅is then given by,
𝑇𝑅≈𝑅
10Δ𝑣(4.7)
At the lower end of the ranges in Figure 6, a RAMAcraft located
1 pc away undergoing a change in velocity of 0.1𝑐has a time in
view of about 3 years. At the upper end of the range, a RAMAcraft
located 105pc away achieving the same change in velocity has a
time in view of about 326,188 years. So towards the upper end of
the ranges in Figure 6, we can extend 𝜏practically indefinitely to
try to take advantage of some hypothetical low-frequency detector.
Towards the lower end of the range, we can still extend 𝜏quite
considerably up to a few years, which would likely cover the capa-
bilities of ground-based and space-based interferometers, as well as
a substantial portion of the detection lengths of PTAs.
Of course, it is not prudent to extend the data sample to fre-
quencies 𝜏−1to the left of the detector sensitivity band (unless a
detector is built whose left-hand sensitivity band boundary scales
weaker than 𝑓−1). The benefit just described is therefore contingent
on the improvement of detector sensitivities to lower frequencies.
Finally, we note that this is indeed a physical effect and not an
artifact of data sample parameters. The boost in range by extending
the 𝑓−1behavior of the spectrum into some low-frequency sensi-
tivity band is afforded by the fact that the RAMAcraft and the
gravitational memory contributed by its change in velocity is in few
of the detector for the requisite amount of time.
4.5 Including More General Objects
In our analysis, we have assumed that our RAMAcraft move in
a straight line by some propulsive mechanism, and our frequency
analysis furthermore assumes that their accelerations are constant.
Objects that would circumvent these specifications include:
(i) Objects with a non-linear trajectory
(ii) Objects with a non-constant acceleration
(iii) Hypothetical objects that accelerate by some non-propulsive
mechanism
To include these generalizations in a search network, we need only
compute their GW signal shapes and include them in the catalog of
MF templates, which we leave for future studies.
In terms of the first two generalizations, we may at the very
least consider some regime in which the acceleration and trajectory
are approximately straight and constant, respectively. For example,
for a mass falling straight into a gravitational well with an initial
separation from a much larger mass 𝑀of 𝑅, the acceleration of the
smaller mass is approximately constant for a duration:
𝑇2∼𝑅3
10𝐺 𝑀 (4.8)
However, since extreme velocities comparable to 𝑐can only be
reached near the event horizon of BHs, these approximations will
generally be applicable to short fractions of the curved trajectory.
Therefore, the waveforms of the entire burst signal will differ from
the case we considered.
Examples of generalization (iii) proposed in the literature are
Warp Drive spacetimes, e.g. Alcubierre (1994). However, none of
the warp drive spacetimes proposed thus far emit gravitational waves
by construction, see, e.g. Alcubierre & Lobo (2017); Bobrick &
Martire (2021). If a warp drive spacetime that does have a GW
signal were to be published, it would be quite interesting to include
the signal in a search network.
MNRAS 000,1–18 (2022)
14 L.Sellers et al.
4.6 Near-Earth Trajectories
The RAMACrafts considered in this study have been of astrophys-
ical scale. We might, however, be interested in finding the detectable
parameter space for near-Earth trajectories (NETs) from a distance
within our solar system 𝑅≤1016 km, or even trajectories close
to our atmosphere 𝑅≤106km. However, at these distances, the
Newtonian portion of the strain, not the far-field quadrupole, will
almost certainly dominate the signal. Therefore, to assess the detec-
tion possibilities of objects that come closer to Earth, one should
examine the Newtonian contribution of these types of signals. We
leave this for a future study.
4.7 Future Horizons
At the present moment, it may be safely stated that there havebeen no
coincident bursts detected by LIGO above (𝑆/𝑁)det =8. Therefore,
there have been no accelerating Jupiter-scale RAMAcraft in our
Galaxy in the last few years. Assuming an average trip of 100 years
(ten times the typical interstellar distance at 0.1𝑐) and accounting for
the fact that there will be two bursts per trip, we can deduce that there
are no more than a few tens of Jupiter-mass RAMAcraft in our
whole Galaxy.Such strong constraints are enabled exclusively by the
LIGO sensitivity. With the installment of low-frequency detectors
like DECIGO and the BBO, we expect that these constraints can be
made much more stringent.
As new systems of interest are postulated and the GW catalog
grows, a very large number of filters will be required to search
for all of these signals. Using conventional MF pipelines, this will
quickly lead to intractable computational demands. However, this
demand might be circumvented by training a neural network on a
large filter set (Andrews et al. 2022). The neural network could then
sift through large data sets without needing to implement each filter
for every search.
Optimism aside, a candidate detection of a RAMAcraft sig-
nal should be treated with skepticism. Burst-like signals may also
be caused by astrophysical sources that temporarily exceed the
sensitivity curve, such as highly-eccentric BBHs (Romero-Shaw
et al. 2020). Therefore, MF detection will be more conclusive than
burst searches. Even then, scrutiny should be applied to distinguish
whether other natural sources can produce similar waveforms.
On the other hand, if GW detectors become more sensitive
such that burst detections of signals emitted from these advanced
technologies become a possibility, researchers would have the op-
portunity to reverse-engineer those modes of transportation. This
is because the shape of the GW signal is entirely dependent on
the trajectory of the object. Thus, as a burst signal is detected, one
can attempt to reason the qualities of the transportation mechanism
present based on the shape of the GW signal.
5 CONCLUSION
We have computed the ranges at which a linearly moving RA-
MAcraft with constant acceleration would be detected by LIGO
or LISA using either a burst or a MF search with a detection thresh-
old of (𝑆/𝑁)det =8. We find on the upper end that RAMAcraft
with the mass of ten Jupiters undergoing a change in velocity of
Δ𝑣=0.1𝑐are detectable from 105−106pc, or right around the
distance to the Andromeda galaxy. On the lower end, we find that
Mercury-scale masses undergoing Δ𝑣=0.1𝑐are detectable from
1–10 pc, or at and beyond the distance to Proxima Centauri.
Because burst searches are indifferent to the signal shape, we
can use our results for the LIGO burst search to constrain the oc-
currence of certain RAMAcraft trajectories (while LIGO was op-
erating). At the present moment, we can rule out tenth of a solar
mass RAMAcraft achieving Δ𝑣=0.1𝑐up to 105−106pc and
ten-mercury-mass objects achieving Δ𝑣=0.1𝑐out up to 1pc.
Furthermore, our analysis shows that the sensitivity to these
objects can be increased through the development of low-frequency
detectors thanks to the 𝑓−1scaling of the signal spectrum. In par-
ticular, we find that DECIGO and the Big Bang Observer (BBO)
will increase the search volume for these objects by a factor of 106,
while both LISA and PTAs may provide benefits for detecting RA-
MAcraft undergoing long acceleration periods. More generally,
the shifting of any detector frequency band by a factor 𝛿 < 1will
increase the detection range by 𝛿−1for burst searches and 𝛿−1/2
for MF searches. The prospect of improving our sensitivity to these
objects is therefore promising, and is furthermore coupled to low-
frequency detector improvements that are of a particular interest to
the fundamental physics community as well.
GW detection is a sophisticated science, though it is still in its
infancy. As the methodology is further developed, the sensitivity
of detectors may become such that the detection of these objects
is a regular occurrence. In this spirit, it would be interesting to
complete a fully-fledged search for these objects. Our next papers
will explore relaxing the assumption of constant 𝑅in order to better
gauge the sensitivity of objects closer to Earth, a simulated MF
search to verify and improve the robustness of the templates (??),
and an investigation of real detector data for these objects. We invite
the scientific community to join us.
ACKNOWLEDGEMENTS
We thank Jerry Tessendorf, Lavinia Heisenberg, Shaun Fell, Alex
Nitz from PyCBC, Thibault Damour, Brandon Melcher, Tito Dal
Canton, Stephen Fairhurst, Edward Rietman, Haydn Vestal, Justin
Feng, Christopher Helmerich, Jared Fuchs, and Toolchest for their
extremely valuable comments and discussions at various stages of
this work.
APPENDIX A: GW SIGNAL FROM A NEWTONIAN
ROCKET
Here we calculate the GW signal produced by a Newtonian rocket
and compare the result to the simplifications made in Section 2.
For a Newtonian rocket, the rocket contribution will still satisfy
(2.9). The contribution from the exhaust, however, is slightly more
complicated to calculate. To begin, we note that the energy density
contributed by the exhaust of a discrete rocket in the detector frame
of reference is simply a sum of the contributions of each individual
ejected mass 𝑑𝑚
𝑇00
𝐸=
𝑑𝑚
𝑑𝑚 ·𝛿(𝑥)𝛿(𝑦)𝛿(𝑧−𝑧𝑑𝑚 (𝑡)) (A1)
where 𝑧𝑑𝑚 (𝑡)is the trajectory of each mass 𝑑𝑚. For a rocket that
has been expelling mass for a time 𝑡, each mass ejected at time
𝑡0≤𝑡with constant ejection velocity (which may change between
masses) 𝑢(𝑡0)has the trajectory
𝑧𝑑𝑚 (𝑡)=˜𝑧(𝑡0) − 𝑢(𝑡0)·(𝑡−𝑡0)(A2)
where ˜𝑧(𝑡0)is the position of the rocket at the time of ejection. Given
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 15
a continuous rate of change ¤
𝑀, we can generalize the discrete case
(A.1) to a rocket that continuously expels mass
𝑇00
𝐸=∫𝑡
0−¤
𝑀𝑑𝑡 0𝛿(𝑥)𝛿(𝑦)𝛿𝑧−˜𝑧(𝑡0) − 𝑢(𝑡0)(𝑡−𝑡0)(A3)
Then, to calculate the strain produced by the exhaust, we again start
with the quadrupole
𝐼(𝐸)
𝑖 𝑗 =∫𝑥𝑖𝑥𝑗𝑑3𝑥∫𝑡
0−¤
𝑀𝑑𝑡 0𝛿(𝑥)𝛿(𝑦)𝛿𝑧−˜𝑧(𝑡0) − 𝑢(𝑡0)(𝑡−𝑡0)
(A4)
Then, for finite 𝑡, we can swap the integration order by Fubini’s
Theorem and find that
𝐼(𝐸)
𝑧𝑧 =∫𝑡
0−¤
𝑀𝑑𝑡 0˜𝑧(𝑡0) − 𝑢(𝑡0)(𝑡−𝑡0)2(A5)
Then, applying the quadrupole formula, we have
ℎ(𝐸)
𝑧𝑧 =2𝐺
𝑅𝑐4𝑑2
𝑑𝑡2∫𝑡
0−¤
𝑀𝑑𝑡 0˜𝑧(𝑡0) − 𝑢(𝑡0)(𝑡−𝑡0)2(A6)
To compute both derivatives, we use the following identity
𝑑
𝑑𝑡 (∫𝑏(𝑡)
𝑎(𝑡)
𝑔(𝑡0, 𝑡)𝑑𝑡0)=∫𝑏(𝑡)
𝑎(𝑡)
𝜕𝑔
𝜕𝑡 𝑑𝑡0+𝑏0(𝑡)𝑔(𝑏 , 𝑡) − 𝑎0𝑔(𝑎, 𝑡 )
(A7)
Then, we find that the strain contribution from the exhaust is given
by
ℎ(𝐸)
𝑧𝑧 =2𝐺
𝑅𝑐4−2∫𝑡
0¤
𝑀(𝑡0)𝑢2(𝑡0)𝑑𝑡 0+2¤
𝑀(𝑡)𝑧(𝑡)𝑢(𝑡)(A8)
−¥
𝑀(𝑡)𝑧2(𝑡) − 2¤
𝑀 𝑧 ¤𝑧o
Look again at the contribution of the rocket
ℎ(𝑅)
𝑧𝑧 =2𝐺
𝑅𝑐4n2𝑀·¤𝑧2+𝑧¥𝑧+4¤
𝑀 𝑧 ¤𝑧+¥
𝑀𝑧2o(A9)
where the last two terms in (A8) cancel manifestly and the second
term 2¤
𝑀𝑧𝑢 is canceled by Newton’s Third Law:
2𝑧·𝑑𝑝 𝐸
𝑑𝑡 =2¤
𝑀𝑢 𝑧 =−2𝑧·¤
𝑀¤𝑧+𝑀¥𝑧=−2𝑧·𝑑 𝑝𝑅
𝑑𝑡 (A10)
We see that all of the terms cancel except the first terms for both
contributions, which are additive since ¤
𝑀 < 0. So, the two non-
canceling contributions are due to the kinetic energy of the rocket
and exhaust.
Finally, one might protest that the first term in (A8) is not the
change in kinetic energy of the exhaust and is missing a term from
the product rule:
𝑑
𝑑𝑡 n𝑀𝐸𝑢2o=−¤
𝑀𝑢2(𝑡) + 2𝑀𝐸𝑢¤𝑢(A11)
where ¤
𝑀=−¤
𝑀𝐸. However, the cumulative kinetic energy of the
exhaust is not given by 𝑀𝐸(𝑡)𝑢2(𝑡), since 𝑢may change between
different ejections. By modeling the rocket as the continuum limit
of a discrete rocket, we are defining the rate of change of the kinetic
energy as
𝑑𝐾𝐸
𝑑𝑡 =−¤
𝑀𝑢2(𝑡)(A12)
Then
𝐾𝐸(𝑡)=∫𝑡
0−¤
𝑀𝑑𝑡 0𝑢2(𝑡0)
=𝑀𝐸(𝑡)𝑢2(𝑡) − 2∫𝑡
0𝑀𝐸𝑢¤𝑢𝑑 𝑡0
≠𝑀𝐸(𝑡)𝑢2(𝑡)
(A13)
APPENDIX B: T SCALING OF THE STRAIN SPECTRUM
FOR A GIVEN ACCELERATION AND CHANGE IN
VELOCITY
Here we discuss the scaling of the signal spectrum when it is written
for both a given acceleration 𝐴(2.23) and a given change in velocity
Δ𝑣(2.24). This discussion will explain the relative performance of
MF and burst searches. We parameterize the 𝑇scaling by the power
law,
ℎ(𝑓) ∼ 𝑇𝑚(B1)
and plot 𝑚versus varying initial velocities 𝑣0. We compute the
curves by generating signal spectrums for a set of acceleration pe-
riods 𝑇and compute the ratio of each spectrum with the spectrum
computed using the smallest 𝑇value. The scaling of this ratio yields
the value of 𝑚, which we then compute for varying values of 𝑣0.
We compute each spectrum in the top curve with one acceleration
𝐴and express the values 𝑣0in units of 𝐴𝑇 𝑓= Δ𝑣, where 𝑇𝑓is the
largest 𝑇value used to compute signal spectrums. We use this same
set of 𝑣0for the bottom curve as well.
For a given acceleration 𝐴, the spectrum should scale like 𝑇2
for Δ𝑣𝑣0and 𝑇for 𝑣0Δ𝑣. This is indeed what we see in
Figure B1. The case for a given Δ𝑣is less clear. If the 𝑓−2and 𝑓−3
corrections to the 𝑓−1behavior are substantial, then we should find
some 𝑇dependence. In Figure B1, we see that the spectrum (2.24)
decreases slightly with increasing 𝑇, possibly due to the decreasing
𝑓−2and 𝑓−3corrections. However, the dependence is minuscule.
APPENDIX C: EFFECT OF VARYING OBSERVATION
START AND STOP TIMES FOR A FIXED SAMPLE
LENGTH
We might also like to see how the strength of our spectrum
changes as we capture different portions of the signal. Here we
reference the four cases pertaining to the overlap of the data sample
and the acceleration period. Figure C1 for Case (i) shows what
happens as we slide the data sample along the signal for a fixed 𝜏. In
particular, we can see that there is not much change in the spectrum
strength by sliding the data sample interval along the signal length
for a fixed 𝜏until around 75% of the data sample is comprised of a
constant velocity contribution. This is expected, since ℎ(𝑓) → 0as
𝑡1→𝜏. The results are more or less the same for Cases (ii) and (iii),
while the spectrum for Case (iv) will never vanish. Furthermore,
the spectrum will grow for each of these cases as their perceived
acceleration period increases. In Section 3, to stay within the regime
of relatively constant ℎ(𝑓), we choosesignals cor responding to Case
(i) that capture reasonable amounts of both 𝑣0and 𝑣𝑓.
APPENDIX D: SNR VS SAMPLE LENGTH SCALINGS
There are two pertinent results needed to explain the scaling behav-
iors (3.13 –3.16). The first is the scaling of the noise correlation
MNRAS 000,1–18 (2022)
16 L.Sellers et al.
10 310 210 1100101102103
v
0 [
v
]
1.0
1.2
1.4
1.6
1.8
2.0
m
h
(
f
) T scaling for a given
A
vs
v
0
10 310 210 1100101102103
v
0 [
v
]
1.0
0.8
0.6
0.4
0.2
0.0
m
1e 6
h
(
f
) T scaling for a given
v
vs
v
0
Figure B1. Scaling behavior of the signal spectrum ℎ(𝑓) ∼ 𝑇𝑚for a given
acceleration 𝐴(Top) and change in velocity Δ𝑣(Bottom). For a given 𝐴,
we find that the scaling starts at 𝑚=2and decreases to 𝑚=1, as expected.
For a given Δ𝑣, we find that the scaling is relatively minuscule, as expected.
The scaling does, however, increase for larger 𝑣0.
10 210 1100101102
f [Hz]
10 25
10 24
10 23
10 22
10 21
10 20
10 19
h(f)
Varying t1 and t2 for fixed
t1 = -0.000000
t1 = -0.100000
t1 = -0.200000
t1 = -0.500000
t1 = -0.750000
t1 = -0.800000
t1 = -0.900000
Figure C1. Plot of the spectrum (2.23) for a fixed data sample length 𝜏. We
can see that the spectrum length is fairly stable until around 75% of the data
sample is comprised of the initial velocity contribution, after which it starts
to dive toward 0.
101102103104
Number of Points in Frequency Band
10 1
100
MF
MF
, 0
MF
vs Number of Sampled Points (LISA)
m 0.48
104105106
Number of Points in Frequency Band
10 1
100
MF
MF
, 0
MF
vs Number of Sampled Points (LIGO)
m 0.47
Figure D1. Scaling behavior of the noise correlation 𝜎MF vs the number of
points used to compute 𝜎MF . We vary the number of points by varying 𝜏
for a fixed time-spacing 𝑑𝑡. We expect a contribution of 𝜏−1/4, though we
find an extra scaling factor of about 𝜏−1/4. We conjecture that this is due to
an increase in the number of points used to compute 𝜎MF .
𝜎MF with respect to the number of points used in the correlation
sum. To see this effect, we plot the statistic 𝜎MF for a fixed mass,
acceleration, acceleration period, and distance from the detector
for varying 𝜏∈𝑇· [1,103]. This choice of 𝜏prevents the effect
of a changing ’perceived change in velocity’ (3.15 –3.16) from
interfering.
The results can be seen in Figure D1, where we plot 𝜎MF in
units of 𝜎MF,0, which is calculated with the initial sample length
𝜏=𝑇. We expect a scaling of 𝜏−1/4contributed by the ˜
ℎ(𝑓𝑘)Δ𝑓
term in the correlation 𝜎2
MF, but there is an additional scaling of
𝜏−1/4, combining to a total scaling of 𝜎MF ∝𝜏−1/2. We conjecture
that this is a result of sampling more points by virtue of the decreased
frequency spacing Δ𝑓=𝜏−1. So, 𝜎MF decreases with increasing 𝜏,
so the SNR increases. This conjecture also agrees with the results
in Sections 3and 4, as we state there as well. The result of this
behavior is that the MF SNR is approximately independent of 𝜏, as
can be seen in Figure D2 up to oscillations within a factor of a few.
These oscillations are likely due to inconsistencies in computing the
noise-signal correlation.
The second effect that explains the results (3.15 –3.16) is that
for the case 𝑡2< 𝑇, increasing 𝜏will result in an increase in the
‘perceived change in velocity’, and thereby increase the spectrum
strength. In Figure D3, we show that the resulting scaling for the
MF SNR is 𝜌MF/𝜎MF ∝𝜏1, as predicted.
MNRAS 000,1–18 (2022)
Searching for Intelligent Life in Gravitational Wave Signals Part I 17
100101102103
[T]
100
MF
MF
(
MF
, 0
MF
, 0
)
1
MF SNR vs for
t
2>
T
(LISA)
100101102103
[T]
100
101
MF
MF
(
MF
, 0
MF
, 0
)
1
MF SNR vs for
t
2>
T
(LIGO)
Figure D2. Scaling behavior of the MF SNR 𝜌MF/𝜎MF vs 𝜏for the case
where the acceleration period is contained entirely in the data sample. We
find that the SNR is independent of 𝜏, as predicted in (3.14). However, we
also find that this prediction oscillates to within a factor of a few.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable request
to the corresponding authors.
10 210 1100
[T]
100
101
102
MF
MF
(
MF
, 0
MF
, 0
)
1
MF SNR vs for
t
2<
T
(LISA)
m 1.07
10 210 1100
[T]
100
101
102
MF
MF
(
MF
, 0
MF
, 0
)
1
MF SNR vs for
t
2<
T
(LIGO)
m 0.95
Figure D3. Scaling behavior of the MF SNR 𝜌MF/𝜎MF vs 𝜏for the case
where the acceleration period is longer than the data sample. We find that
the SNR scales close to ∝𝜏1, as predicted in (3.16).
MNRAS 000,1–18 (2022)
18 L.Sellers et al.
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