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Journal ArticleDOI

The Speed of Galileon Gravity

TL;DR: In this article, the authors analyzed the speed of gravitational waves in coupled Galileon models with an equation of state and a ghost-free Minkowski limit, and they showed that the acceleration of the expansion of the universe can not be explained by the mass of the gravitons.
Abstract: We analyse the speed of gravitational waves in coupled Galileon models with an equation of state $\omega_\phi=-1$ now and a ghost-free Minkowski limit. We find that the gravitational waves propagate much faster than the speed of light unless these models are small perturbations of cubic Galileons and the Galileon energy density is sub-dominant to a dominant cosmological constant. In this case, the binary pulsar bounds on the speed of gravitational waves can be satisfied and the equation of state can be close to -1 when the coupling to matter and the coefficient of the cubic term of the Galileon Lagrangian are related. This severely restricts the allowed cosmological behaviour of Galileon models and we are forced to conclude that Galileons with a stable Minkowski limit cannot account for the observed acceleration of the expansion of the universe on their own. Moreover any sub-dominant Galileon component of our universe must be dominated by the cubic term. For such models with gravitons propagating faster than the speed of light, the gravitons become potentially unstable and could decay into photon pairs. They could also emit photons by Cerenkov radiation. We show that the decay rate of such speedy gravitons into photons and the Cerenkov radiation are in fact negligible. Moreover the time delay between the gravitational signal and light emitted by explosive astrophysical events could serve as a confirmation that a modification of gravity acts on the largest scales of the Universe.

Summary (2 min read)

1. Introduction

  • Gravitational waves have now been predicted for nearly a century and despite decades of experimental efforts, their existence is only confirmed by indirect evidence coming from the time drift of the period of binary pulsars.
  • – 2 – Galileons have been widely studied both on purely theoretical grounds, with results showing that this kind of models arise also in the context of massive gravity [20] and braneworld models [21].
  • The authors show that these processes are negligible for allowed differences between the speed of gravitons and photons.

2.1 The Models

  • They are potential candidates to explain the late time acceleration of the expansion of the Universe.
  • Such Galileons are scalar field theories which have equations of motion that are at most second order in the derivatives.
  • Moreover they are interesting dark energy candidates where an explicit cosmological constant is not compulsory.
  • These terms play an important role cosmologically.

3.1 Screening Effects

  • When the speed of gravitons exceeds the speed of light by more than one percent, the change in the period of binaries cannot accommodate observations [13].
  • One possible way out which could reconcile both a large speed of gravitons on cosmological scales and a constrained one in the pulsar environment is the presence of screening in the – 6 – form of the Vainshtein mechanism.
  • In this situation the authors assume that the time variation is coming from the background cosmological evolution and – 7 – the radial dependence is sourced by an over-density of matter.
  • The authors choose that hij is only non-zero for hθθ and hθφ.

3.2 Cubic Galileons

  • If the authors assume that x0 is not very small, screening does not modify the speed of gravitons and the speed of gravitational waves emitted by compact objects like binary pulsars can only be small when the influence of the quartic Galileon terms is negligible.
  • The authors will see that one can preserve a positive c2 and still impose that c4 is small together with an equation of state close to -1 when the Galileon scalar field does not lead to all the dark energy of the Universe.
  • This implies that c̄2 1 in order to guarantee that the cubic term dominates.

4.1 Graviton Decay

  • The authors have seen that the speed of gravitational waves emitted by binary pulsars can deviate from unity by a one percent for almost cubic Galileon models even if they are a subdominant – 10 – component of the late universe.
  • The gravitons go faster than the speed of light and become unstable: they can decay into massless particles.
  • The authors have also summed over the initial graviton polarisations.

4.2 Cerenkov Radiation

  • The gravitons can also emit two photons by the Cerenkov effect thereby losing energy and increasing the difficulty of detecting them.
  • (4.29) The energy k is the one of one tagged photon while the other one has an energy k1.
  • The initial graviton has momentum k′ and the outgoing one k2.

5. Time Delay

  • The gravitons with a speed larger than the speed of light produced by astrophysical sources would arrive in their detector well in advance of the light signal.
  • The difference of emission times between neutrinos and gravitational waves is estimated to 2This was recently discussed for a difference choice of Horndeski scalar-tensor theory in [39].
  • (5.1) We have seen that current bounds from binary pulsars only constrain ∆cT at the 10 −2 level implying a time delay, for sources one kpc away, of order 30 years.the authors.the authors.
  • For the supernova SN19871A, gravitational waves could have reached the earth as early as 1700 years in advance.
  • For Galileons, this would lead to an extraordinarily fine tuned model, which would behave like a cubic model, with the coefficient of the quartic term suppressed by at least fourteen orders of magnitude.

6. Conclusion

  • The authors have analysed the behaviour of gravitational waves for Galileon models that include quartic terms and have a stable Minkowski limit, and shown that only subdominant Galileon models where a significant part of the dark energy is due to a cosmological constant can comply with the stringent binary pulsar bounds.
  • When this is the case, the propagating gravitons do not suffer from particle physics instabilities such as decay into two photons or Cerenkov radiation.
  • As a result, the speed of gravitons remains superluminal but the difference between the speed of propagation of gravitons and photons cannot be more than one percent.
  • In spite of this the time delay between the arrival of gravitational waves and light can be extremely large, more than a thousand years for supernovae of the SN1987A type.
  • More reasonable time delays can be expected for closer objects when tighter bounds on the parameters of the models apply.

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Preprint typeset in JHEP style - HYPER VERSION
The Speed of Galileon Gravity
Philippe Brax
Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA,CNRS,
F-91191Gif sur Yvette, France
E-mail: philippe.brax@cea.fr
Clare Burrage
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD,
United Kingdom
E-mail: Clare.Burrage@nottingham.ac.uk
Anne-Christine Davis
DAMTP, Centre for Mathematical Sciences, University of Cambridge, CB3 0WA, UK
E-mail: A.C.Davis@damtp.cam.ac.uk
Abstract: We analyse the speed of gravitational waves in coupled Galileon models with
an equation of state ω
φ
= 1 now and a ghost-free Minkowski limit. We find that the
gravitational waves propagate much faster than the speed of light unless these models are
small perturbations of cubic Galileons and the Galileon energy density is sub-dominant to
a dominant cosmological constant. In this case, the binary pulsar bounds on the speed
of gravitational waves can be satisfied and the equation of state can be close to -1 when
the coupling to matter and the coefficient of the cubic term of the Galileon Lagrangian
are related. This severely restricts the allowed cosmological behaviour of Galileon models
and we are forced to conclude that Galileons with a stable Minkowski limit cannot account
for the observed acceleration of the expansion of the universe on their own. Moreover any
sub-dominant Galileon component of our universe must be dominated by the cubic term.
For such models with gravitons propagating faster than the speed of light, the gravitons
become potentially unstable and could decay into photon pairs. They could also emit
photons by Cerenkov radiation. We show that the decay rate of such speedy gravitons into
photons and the Cerenkov radiation are in fact negligible. Moreover the time delay between
the gravitational signal and light emitted by explosive astrophysical events could serve as
a confirmation that a modification of gravity acts on the largest scales of the Universe.
arXiv:1510.03701v1 [gr-qc] 9 Oct 2015

Contents
1. Introduction 1
2. Galileons 3
2.1 The Models 3
2.2 Cosmological Galileons 4
2.3 The Speed of Gravitons 5
3. The Speed of Gravitons and Screening 6
3.1 Screening Effects 6
3.2 Cubic Galileons 9
4. Graviton Instability 10
4.1 Graviton Decay 10
4.2 Cerenkov Radiation 13
5. Time Delay 14
6. Conclusion 15
1. Introduction
Gravitational waves have now been predicted for nearly a century and despite decades of
experimental efforts, their existence is only confirmed by indirect evidence coming from
the time drift of the period of binary pulsars. New experiments such as the advanced
Laser Interferometry Gravitational-Wave Observatory (a-LIGO) [1], the advanced VIRGO
interferometer [2], the Kamioka Wave Detector (KAGRA) [3], the space based mission
DECIGO [4] or eLISA [5] will be able to test directly the existence of gravitational waves
to improved levels. Gravity waves are also important probes for theories going beyond
Einstein’s General Relativity (GR) [6]. These theories are motivated by the discovery of
the recent acceleration of the expansion of the Universe [7] whose origin is still unknown.
Models such as the quartic Galileons [8] where a coupling between a scalar field and gravity
is present predict a background dependent speed of gravitational waves.
In this work we focus on Galileon models [8]. These are a subset of the Horndeski
action [9,10] describing the most general scalar tensor model with second order equations
of motion. The Galileon terms on flat space are protected by a symmetry, the so called
Galileon symmetry, which is softly broken on a curved spacetime background [11]. In these
models the cosmic acceleration is due to the presence of higher order terms in the derivatives
1

compared to quintessence models where a non-linear potential, typically containing a term
equivalent to a cosmological constant, provides the required amount of vacuum energy.
In vacuum the scalar mediates a fifth force of at least gravitational strength. Locally
close to massive sources the scalar field is strongly influenced by matter and within the
Vainshtein radius GR is restored. On cosmological time scales, the scalar field evolves.
This cosmological time drift is screened from matter fields whilst the average density of the
universe is sufficiently high but has consequences for the dynamics of gravity locally [12].
In particular the speed of gravitational waves in a massive environment is not protected
from the evolution of the background cosmology by the Vainshtein mechanism [8], meaning
that it can differ from the speed of light in a significant manner [13]. We will review this
calculation in Section 3.
If we impose that the equation of state of the scalar field should be close to -1 now
and the existence of a stable Minkowski limit of the theory in the absence of matter, both
necessary conditions for a viable cosmology dominated by Galileons at late times and a
meaningful embedding of the model in higher dimensions
1
[14], we find that the speed
of gravitational waves would be much greater than one. This would increase the rate of
emission of gravitational waves from binary pulsars. As a result, the speed of gravity in such
a Galileon model is not compatible with the bound that positive deviations of the speed of
gravity from the speed of light cannot be more than one percent [13,15]. We then conclude
that these Galileon models cannot lead to the acceleration of the Universe on their own
and a certain amount of dark energy must be coming from a pure cosmological constant.
This forces the quartic Galileon terms to be subdominant to the cubic terms in order that
the binary pulsar bound can be satisfied. When this is the case, the time delay between
gravity and light or even neutrinos can be as large as a few thousand years for events
like the SN1987A supernova explosion. This would essentially decouple any observation of
supernovae gravitational waves from the corresponding photon or neutrino signal coming
from such explosive astrophysical events. On the other hand, a time difference as low as
the uncertainty on the difference in emission time signal between neutrinos and gravity,
e.g. up to 10
3
s for supernovae [16], would allow one to bound deviations of the quartic
Galileon model from its cubic counterpart at the 10
14
level.
One possible caveat to these results would be if the superluminal gravitational waves
do not reach our detectors because they either decay into two photons or lose all their
energy through Cerenkov radiation [17]. We will show that superluminal gravitational
waves with a speed as large as one percent higher than the speed of light are not excluded
by particle physics processes. A related possibility is at the origin of the stringent bounds
on subluminal gravitational waves which could be Cerenkov radiated by high energy cosmic
rays. As these high energy rays are observed the speed of gravitons cannot be significantly
smaller than that of the particle sourcing the cosmic ray [18, 19]. We analyse the decay
and the Cerenkov effect for superluminal gravitational waves and we find that their effects
are negligible.
1
We require this embedding in higher dimensional brane models with positive tension branes as a pre-
requisite first step towards a possible extension to fundamental theories such as string theory.
2

Galileons have been widely studied both on purely theoretical grounds, with results
showing that this kind of models arise also in the context of massive gravity [20] and
braneworld models [21]. Constraints on the allowed cosmology of Galileon theories can
be obtained from a wide variety of observations, unveiling a very rich phenomenology
[12,22–36]. Here we consider for the first time the constraints that current and near future
observations of gravitational waves can place on these theories.
In section 2, we recall details about Galileon models and show that quartic models
with an equation of state close to -1 lead to very fast gravitons. In section 3, we consider
the influence of the Vainshtein mechanism on the propagation of gravity and we check
that the screening mechanism does not protect the speed of gravity from large deviations
compared to the speed of light. We also introduce models of subdominant Galileons whose
gravitational waves have a speed which satisfies the binary pulsar bounds. In section 4
we consider the decay rate of gravitons into two photons, and the Cerenkov radiation.
We show that these processes are negligible for allowed differences between the speed of
gravitons and photons. Finally In Section 5 we discuss the time delay in the arrival time
of gravitons and photons from explosive astrophysical sources. We conclude in section 6.
2. Galileons
2.1 The Models
In this paper, we are interested in models of modified gravity with a Galilean symmetry.
They are potential candidates to explain the late time acceleration of the expansion of
the Universe. They also lead to a modification of gravity on large scales. Such Galileons
are scalar field theories which have equations of motion that are at most second order in
the derivatives. Moreover they are interesting dark energy candidates where an explicit
cosmological constant is not compulsory. Their Lagrangian reads in the Jordan frame
defined by the metric g
µν
L =
1 + 2
c
0
φ
m
Pl
R
16πG
N
c
2
2
(φ)
2
c
3
Λ
3
φ(φ)
2
c
4
Λ
6
L
4
c
5
Λ
9
L
5
. (2.1)
The common scale
Λ
3
= H
2
0
m
Pl
(2.2)
is chosen to be of cosmological interest as we focus on cosmological Galileon models which
can lead to dark energy in the late time Universe. We also require that c
2
> 0 to avoid
the presence of ghosts in a Minkowski background. This theory could be rewritten in the
Einstein frame where the conformal coupling of the scalar field to matter would be given
by
A(φ) = 1 +
c
0
φ
m
Pl
(2.3)
3

where c
0
is a constant. The complete Galileon Lagrangian depends on operators with
higher order terms in the derivatives which are given by
L
4
=(φ)
2
2(φ)
2
2D
µ
D
ν
φD
ν
D
µ
φ R
(φ)
2
2
L
5
=(φ)
2
(φ)
3
3(φ)D
µ
D
ν
φD
ν
D
µ
φ + 2D
µ
D
ν
φD
ν
D
ρ
φD
ρ
D
µ
φ (2.4)
6D
µ
φD
µ
D
ν
φD
ρ
φG
νρ
] .
These terms play an important role cosmologically. In the following and in the study of
the cosmological evolution, we focus on the coupling of the Galileon to Cold Dark Matter
(CDM) as the coupling to baryons is more severely constrained by the time variation
of Newton’s constant in the solar system, at the one percent level, and does not play a
significant role for the background cosmology [38].
This model is a subset of terms in the Horndeski action describing the most general
scalar tensor theory with second order equations of motion
L = K(φ, X) G
3
(X, φ)φ + G
4
(X, φ)R + G
4,X
(φ)
2
(D
µ
D
ν
φ)
2
+
G
5
(X, φ)G
µν
D
µ
D
ν
φ
1
6
G
5,X
h
(φ)
3
3φ(D
µ
D
ν
φ)
2
+ 2(D
µ
D
α
φ)(D
α
D
β
φ)(D
β
D
µ
φ)
i
with the particular functions
K = c
2
X, G
3
(X) = 2
c
3
Λ
3
X, G
4
(X, φ) =
A
2
(φ)
16πG
N
+ 2
c
4
Λ
6
X
2
, G
5
(X) = 6
c
5
Λ
9
X
2
(2.5)
where X =
(φ)
2
2
is the kinetic energy of the field. In the following we shall focus on
quartic Galileons with c
5
= 0 as this leads to both interesting cosmology and a non-trivial
speed for gravitational waves.
2.2 Cosmological Galileons
We focus on the behaviour of Galileon models on cosmological scales in a Friedmann-
Robertson-Walker background
ds
2
= a
2
(
2
+ dx
2
) (2.6)
where η is conformal time and we have set the speed of light c = 1. The equations of
motion of the Galileons can be simplified using the variable x = φ
0
/m
Pl
where a prime
denotes
0
= d/d ln a = d/d ln(1 + z), a is the scale factor and z the redshift. We define
the scaled field ¯y =
φ
m
Pl
x
0
, the rescaled variables ¯x = x/x
0
and
¯
H = H/H
0
where H is the
Hubble rate, and the rescaled couplings [36] ¯c
i
= c
i
x
i
0
, i = 2 . . . 5, ¯c
0
= c
0
x
0
, ¯c
G
= c
G
x
2
0
where x
0
is the value of x now. Notice that x
0
is not determined by the dynamics and is
a free parameter of the model. The cosmological evolution of the Galileon satisfies [37]
¯x
0
= ¯x +
αλ σγ
σβ αω
¯y
0
= ¯x
¯
H
0
=
λ
σ
+
ω
σ
σγ αλ
σβ αω
4

Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors consider a model with explicit breaking of the Galileon symmetry and investigate its radiative stability, showing that the quantum corrections, to one-loop, do not detune the classical Lagrangian generating suppressed counterterms.
Abstract: With recent constraints on the propagation speed of gravitational waves, the class of scalar-tensor theories has significantly been reduced. We consider one of the surviving models still relevant for cosmology and investigate its radiative stability. The model contains operators with explicit breaking of the Galileon symmetry and we study whether they harm the re-organization of the effective field theory. Within the regime of validity we establish a non-renormalization theorem and show explicitly that the quantum corrections, to one-loop, do not detune the classical Lagrangian generating suppressed counterterms. This is striking since the non-renormalization theorem is established in the presence of a genuine Galileon symmetry breaking term.

9 citations


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TL;DR: In this paper, the authors studied the effect of gravity acceleration on the propagation of GWs and proposed new tests of DE and General Relativity with GWs, and investigated the formation of black-holes (BHs) in the early universe, which has strong implications on their contribution to the dark matter and their signatures.
Abstract: Gravitational wave (GW) astronomy opens new opportunities to explore the universe and its fundamental laws. This thesis focuses on probing the pillars of the standard cosmological model with GWs, specially its most puzzling components: dark energy (DE) and dark matter (DM). We propose and apply new tests of DE and General Relativity (GR) with the propagation of GWs. We also investigate the formation of black-holes (BHs) in the early universe, which has strong implications on their contribution to the DM and on their GW signatures. Just as electromagnetic radiation can scan materials, GWs can probe the medium in which they propagate. DE models beyond Einstein’s gravity generically modify the propagation of GWs. We identify the speed of GWs as a key test of gravity and find the conditions for an anomalous speed to arise. We emphasis that a non-luminal speed can appear in cosmological models aiming at DE such as Galileons, but also in environments with a spatial profile induced by screening or scalar hair. After the multi-messenger event GW170817, we determine the consequences of the tight constraint on the speed of GWs for different classes of gravity theories and DE models, setting the dead ends and the road ahead. Standard sirens like GW170817 constrain as well the GW luminosity distance. We derive this observable in general theories of gravity and discuss its detectability with the future space-based detector LISA. Particularly distinguishable oscillatory patters are produced by GW oscillations, a phenomenon that we study in detail. Other probes of GW oscillations are modified wave-forms, induced anomalous speeds and polarization dependent signals. Primordial BHs (PBHs) could be a unique relic to unveil the physics of the early universe. We study the production of PBHs in single field model of inflation with a quasi-inflection point, showing the growth of perturbations beyond slow-roll (SR) at sub- and super-horizon scales. We propose a particle physics motivated model, critical Higgs inflation, achieving a copious PBH production with several GW signatures. However, when curvature fluctuations are enhanced, quantum diffusion dominates the classical inflationary dynamics. We develop a formalism based on stochastic inflation beyond SR to account for this effect. We encounter that the classical prediction is importantly modified, with relevant non-Gaussian contributions. To quantify better the quantum correction, we devise a method to compute directly the tail of the curvature perturbation distributions. As a first step, we apply it to SR inflation. We conclude that the abundance of PBHs is many orders of magnitude larger than the Gaussian prediction, discussing its implications for inflationary model building as well as for the GW observables. Altogether, GW astronomy stands as a powerful channel to advance forward in the quest for understanding the dark universe. We discuss the future prospects of this line of research, highlighting the theoretical challenges and observational opportunities that next generation GW detectors will provide.

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Cites background from "The Speed of Galileon Gravity"

  • ...All the cosmologically viable models have an impact of GW propagation [359], as shown in Fig....

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  • ...On the other hand covariant Galileons [90] and the covariantization of other generalizations [63, 356–358] will generically predict cg ”= c [359]....

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Journal ArticleDOI
TL;DR: In this article, the authors study the connection between self-acceleration and the presence of ghosts for a quite generic class of theories that modify gravity in the infrared, defined as those that at distances shorter than cosmological, reduce to a certain generalization of the Dvali-Gabadadze-Porrati (DGP) effective theory.
Abstract: In the Dvali-Gabadadze-Porrati (DGP) model, the "self-accelerating" solution is plagued by a ghost instability, which makes the solution untenable. This fact, as well as all interesting departures from general relativity (GR), are fully captured by a four-dimensional effective Lagrangian, valid at distances smaller than the present Hubble scale. The 4D effective theory involves a relativistic scalar pi, universally coupled to matter and with peculiar derivative self-interactions. In this paper, we study the connection between self-acceleration and the presence of ghosts for a quite generic class of theories that modify gravity in the infrared. These theories are defined as those that at distances shorter than cosmological, reduce to a certain generalization of the DGP 4D effective theory. We argue that for infrared modifications of GR locally due to a universally coupled scalar, our generalization is the only one that allows for a robust implementation of the Vainshtein effect-the decoupling of the scalar from matter in gravitationally bound systems-necessary to recover agreement with solar-system tests. Our generalization involves an internal Galilean invariance, under which pi's gradient shifts by a constant. This symmetry constrains the structure of the pi Lagrangian so much so that in 4D there exist only five terms that can yield sizable nonlinearities without introducing ghosts. We show that for such theories in fact there are "self-accelerating" de Sitter solutions with no ghostlike instabilities. In the presence of compact sources, these solutions can support spherically symmetric, Vainshtein-like nonlinear perturbations that are also stable against small fluctuations. We investigate a possible infrared completion of these theories at scales of order of the Hubble horizon, and larger. There are however some features of our theories that may constitute a problem at the theoretical or phenomenological level: the presence of superluminal excitations; the extreme subluminality of other excitations, which makes the quasistatic approximation for certain solar-system observables unreliable due to Cherenkov emission; the very low strong-interaction scale for pi pi scatterings.

2,086 citations

Journal ArticleDOI
TL;DR: In this paper, a pulsar with a pulsation period that varies systematically between 0.058967 and 0.59045 sec over a cycle of 0.3230 d was detected.
Abstract: We have detected a pulsar with a pulsation period that varies systematically between 0.058967 and 0.059045 sec over a cycle of 0.3230 d. Approximately 200 independent observations over 5-minute intervals have yielded a well-sampled velocity curve which implies a binary orbit with projected semimajor axis sin i = 1.0 solar radius, eccentricity e = 0.615, and mass function f(m) = 0.13 solar mass. No eclipses are observed. We infer that the unseen companion is a compact object with mass comparable to that of the pulsar. In addition to the obvious potential for determining the masses of the pulsar and its companion, this discovery makes feasible a number of studies involving the physics of compact objects, the astrophysics of close binary systems, and special- and general-relativistic effects.

1,553 citations

Journal ArticleDOI
TL;DR: The Advanced LIGO gravitational wave detectors (ALGWR) as mentioned in this paper are the next generation instruments which will replace the existing initial LIGA detectors and are currently being constructed and installed.
Abstract: The Advanced LIGO gravitational wave detectors are next generation instruments which will replace the existing initial LIGO detectors. They are currently being constructed and installed. Advanced LIGO strain sensitivity is designed to be about a factor 10 better than initial LIGO over a broad band and usable to 10 Hz, in contrast to 40 Hz for initial LIGO. This is expected to allow for detections and significant astrophysics in most categories of gravitational waves. To achieve this sensitivity, all hardware subsystems are being replaced with improvements. Designs and expected performance are presented for the seismic isolation, suspensions, optics and laser subsystems. Possible enhancements to Advanced LIGO, either to resolve problems that may arise and/or to allow for improved performance, are now being researched. Some of these enhancements are discussed along with some potential technology being considered for detectors beyond Advanced LIGO.

1,217 citations