Recent direct detections of gravitational waves from merging binary black hole systems at cosmological distances of z = 0.1−0.2, performed by two Advanced LIGO detectors (Abbott et al, 2016d, Abbott et al, 2016c, Abbott et al, 2017a), first binary black-hole merger observation with the global network of three detectors of Advanced LIGO and the Advanced Virgo (Abbott et al, 2017d), as well as the first Advanced LIGO and Advanced Virgo detection of a nearby (at a distance of 40 Mpc) binary neutron-star merger (Abbott et al, 2017e), followed by a short gamma-ray burst (Abbott et al, 2017c) and broad-band electromagnetic emission observational campaign (Abbott et al, 2017f) create unprecedented opportunities for studying the Universe through a novel, never before explored channel of spacetime fluctuations. Gravitational wave astronomy is often compared to `listening to' rather than `looking at' the skies. By design which is motivated by the choice of potential sources, the ground-based gravitational wave detectors, kilometer-size laser interferometer antennas of Advanced LIGO (Aasi et al, 2015) nd Advanced Virgo (Acernese et al, 2015), are sensitive in the range of frequencies similar to the audible range of human ears - between 10 Hz and a few kHz. As in the case of an ear, a solitary laser interferometric detector is practically omnidirectional (has a poor angular resolution), and has no imaging capabilities. It registers a coherent signal emitted by a bulk movement of large, rapidly-moving masses. Once emitted, gravitational waves are weakly coupled to the surrounding matter and propagate freely without scattering. This has to be contrasted with the electromagnetic emission which originates at the microscopic level, is strongly coupled to the surroundings and often reprocessed; it carries a reliable information from the last scattering surface only. It seems therefore that gravitational wave detectors are the perfect counterpart to the electromagnetic observatories as they may provide us with information impossible to obtain by other means.
Very shortly after announcing the general theory of relativity in , Albert Einstein realized that a linearized version of his equations resembles the wave equation (Einstein 1916). The solution is interpreted as a short-wavelength, time-varying curvature deformation propagating with the speed of light on an otherwise slowly-varying, large-scale curvature background (a gravitational-wave "ripple" propagating through the four-dimensional spacetime); from the point of view of a metric tensor, it represents a small perturbation of a stationary background metric. Linear approximation corresponds to the waves propagating in the far-field limit. By exploiting the gauge freedom of the theory one may show that the solution has features similar to electromagnetic waves: it is a transverse wave which may be polarized (has two independent polarizations). Over the next 40 years, during which Einstein changed his mind to argue against their genuineness, a controversy persisted over the true nature of gravitational waves. Only in the late 50s and early 60s the works of Felix Pirani (1956), Herman Bondi (1957), Ivor Robertson and Andrzej Trautman (1960) convincingly showed that gravitational waves are indeed physical phenomena that carry and deposit energy.
A realistic wave phenomenon (and not, say, a coordinate artifact) must be
capable of transmitting energy from the source to infinity. If the
amplitude of an exemplary isotropic field at a radial distance r
from the source is h(r),
then the flux of energy over a spherical surface at r is
(r) ∝
h2(r), and the
total emitted power (the luminosity)
is 
(r) ∝
4π r2
h2(r). Since the energy has to be
conserved, the amplitude h(r) falls like 1/r,
irrespectively of the multipole character of the source (the lowest
radiating multipole in the gravity theory is the quadrupole
distribution, because for an isolated system a time-changing monopole
would correspond to the violation of the mass-energy conservation, and a
time-changing dipole would violate the momentum conservation law).
Gravitational waves are related to the changes in the spacetime distance (the proper time interval), therefore they cannot be detected by a local measurement - one has to compare the spacetime positions of remote events (Pirani 1956). The detection principle in the case of the laser interferometric detector is to measure the difference of the relative change in its perpendicular arms' lengths Lx and Ly, δLx − δLy = ∆L / L, by measuring the interference pattern in the output port located at the apex of the device. Due to the quadrupolar nature of a gravitational wave, shortening of one arm corresponds to elongation of the other. This change of lengths is reflected in varying paths that the laser light has to cross before the interference. The dimensionless gravitational-wave amplitude h = ∆L /L (the "strain") is proportional to the amount of outgoing laser light. The fact that the directly-measurable quantity is the amplitude h ∝ 1 / r, not the energy of the wave as in the electromagnetic antennæ, has a direct consequence for the reach of the observing device: one-order-of-magnitude sensitivity improvement corresponds to one-order-of-magnitude growth of distance reach r, as opposed to the factor of √10 in the electromagnetic observations (consequently, the volume of space grows like r3 in case of gravitational-wave observations, encompassing hundreds of thousands of galaxies for a distance reach of the order of hundreds of Mpc (see Abbott et al, 2016b).
Among promising sources of gravitational radiation are all asymmetric collapses and explosions (supernovæ), rotating deformed stars (gravitational-wave `pulsars' of continuous and transient nature), and tight binary systems of e.g., neutron stars and black holes. In the following we will focus on the binary systems, since their properties make them the analogues to standard candles of traditional astronomy. The fittingly descriptive term "standard siren" was first used in the work of Holz and Hughes (2005) in the context of gravitational waves from super-massive binary black holes as a target for the planned spaceborne LISA detector (Danzmann 1996). Dalal et al (2006) were the first one to point out that short GRBs produced by merging binary neutron-star systems would constitute ideal standard siren. The idea of using well-understood signals to infer the distance and constrain the cosmological parameters, e.g., the Hubble constant, was however proposed much earlier (Schutz 1986, 2002). Recently it was demonstrated in practice for the first time with the observations of a binary neutron-star merger in both gravitational waves and photons (Abbott et al, 2017e, b).
Magnitudes of the gravitational-wave strain h and the luminosity
may be estimated using
dimensional analysis and Newtonian
physics. As the waves are generated by the accelerated movement of
masses and the mass distribution should be quadrupolar, one may assume
that h is proportional to a second time derivative of the
quadrupole moment Iij = ∫ ρ(x)
xi xj d3x
for some matter distribution ρ(x). For a binary composed of
masses m1 and
m2, orbiting the center of mass at a separation
a with the orbital angular
velocity ω, h is proportional to the system's moment of inertia
μa2 and to ω2, as well as
inversely proportional to the distance,
h ∝ μa2 ω2 /
r, where μ = m1 m2 /
M is the reduced mass,
and M = m1 + m2 the total
mass. In order to recover the dimensionless h, the characteristic
constants of the problem, G and c, are used to obtain
 |
(20) |
with the use of Kepler's third law (GM = a3
ω2) in the second equation. The expression in brackets
represents the strain tensor hij in
the non-relativistic quadrupole approximation
(Einstein 1918).
Similarly, the luminosity
(the rate of energy loss
in gravitational waves, integrated over a sphere at a distance r)
should be proportional to h2 r2 and
some power of ω. From dimensional analysis one has
 |
(21) |
 |
with the proportionality factor of 32/5. Again, the expression in
brackets refers to the quadrupole approximation; the angle brackets denote
averaging over the orbital period.
Waves leave the system at the expense of its orbital energy
Eorb = − Gm1
m2 / (2a).
Using the time derivative of the Kepler's third law,
= −2a
/ (3ω),
one gets the evolution of
the orbital frequency driven by the gravitational-wave emission:
 |
(22) |
The system changes by increasing its orbital frequency; at the same time the
strain amplitude h of emitted waves also grows. This characteristic
frequency-amplitude evolution is called the chirp, by the
similarity to birds' sounds, and the characteristic function of
component masses
=
(μ3 M2)1/5 is called
the chirp mass. Orbital frequency is
related in a straightforward manner to the gravitational-wave frequency
fGW: from the geometry of the problem it is evident
that the frequency of
radiation is predominantly at twice the orbital frequency,
fGW = ω / π. By combining the equations for
GW and
h, one recovers the distance to the source r. It is a
function of the frequency and amplitude parameters, which are
directly measured by the detector:
 |
(23) |
where h21 denotes the strain in the units of
10−21 and τ = fGW /
GW denotes
the rate of change of the gravitational-wave chirp frequency. Note that
within the simple Newtonian approximation presented here (at the leading
order of the post-Newtonian expansion) the product hτ is
independent of the components' masses
(Królak and
Schutz 1987).
The simplified analysis presented above does not take into account the
full post-Newtonian waveform, polarization information, network of
detectors analysis etc., but is intended to demonstrate that binary
systems are indeed truly extraordinary "standard sirens". Their
observations provide absolute, physical distances
directly, without the need for a calibration or a `distance ladder'. Among
them, binary black hole systems occupy a special position. In the
framework of general relativity, binary black holes waveforms are
independent from astrophysical assumptions about the systems' intrinsic
parameters and their environment. At cosmological scales the distance
r has a true meaning of the luminosity distance. However,
since the vacuum (black-hole) solutions in general relativity are
scale-free, the measurements of their waveforms alone cannot determine
the source's redshift. The parameters measured by the detector are
related to the rest-frame parameters by the redshift
z: fGW =
f rfGW / (1 + z), τ
= τrf(1 + z),
=
rf(1 +
z). Independent measurements of the redshift which would
facilitate the cosmographic studies of the large-scale Universe requires
the collaboration with the electromagnetic observers i.e., the
multi-messenger
astronomy. This may be obtained by assessing the redshift of the galaxy
hosting the binary by detecting the electromagnetic counterpart of the event
(Bloom et al, 2009,
Singer et al,
2016,
Abbott et al,
2017f),
or by performing statistical study for galaxies'
redshifts correlated with the position of the signal's host galaxy
(MacLeod and
Hogan 2008).
The omnidirectional nature of a solitary detector is
mitigated by simultaneous data analysis from at least three ground-based
detectors in order to perform the triangulation of the source position,
hence the crucial need for the LIGO-Virgo collaboration, which will be
in the future enhanced by the LIGO-India detector, and the KAGRA
detector in Japan
(Aso et al, 2013).
The above principles were validated by the observations
of a binary black-hole merger with three detector network
(Abbott et al,
2017d),
resulting in improved distance and sky localization.
Compact binaries containing matter are perfect sources to produce the multi-messenger events. Neutron-star binaries are now firmly established as sources of short gamma-ray bursts (idea originally proposed in Paczynski 1986), thanks to simultaneous gravitational and electromagnetic-wave observations from the same event (Abbott et al. 2017f, c and references therein). Although the complete waveform which includes the merger requires the knowledge of the material properties of the components (the presently poorly-known dense-matter equation of state physics), the chirp waveforms are understood well enough to facilitate a firm detection and a distance measurement. Binaries involving neutron stars are the ideal "standard sirens" insofar as they naturally provide both loud gravitational-wave and bright electromagnetic emission. Short gamma-ray bursts occur frequently within the reach of ground-based detectors, at redshifts z < 0.2. (Nissanke et al, 2013) show that observing a population of the order of 10 of gravitational-wave events related to short gamma-ray bursts would allow to measure the Hubble constant with 5% precision using a network of detectors that includes advanced LIGO and Virgo (30 beamed events could constrain the Hubble constant to better than 1%). In both cases of double black-hole binaries and those involving neutron stars, rapid electromagnetic counterpart observations and precise catalogs of galaxies are needed (Abbott et al, 2016b, Singer et al, 2016).
The main source of error in the distance measurement is the limited sensitivity of the detectors (finite signal-to-noise), which translates to a limited knowledge of the source's direction and orientation (see e.g., (Nissanke et al, 2010) for a short gamma-ray burst related study). This may be improved with the measurements of gravitational-wave polarizations with a network of three or more detectors. Second limiting factor is the detectors' calibration uncertainties. Recent direct detectors of binary black-hole mergers by two Advanced LIGO detectors established the distances with rather large error bars mostly due to these factors (see e.g., Abbott et al, 2016a); the order-of-magnitude improvement in sky localization was evident for the first triple-detection with the global network of Advanced LIGO and Advanced Virgo detectors (Abbott et al, 2017d). Presence of a third detector in the network proved to be crucial for the precise sky localization of a binary neutron-star merger (Abbott et al, 2017f), allowing for rapid electromagnetic follow-up. For redshifts larger than z = 1 weak gravitational lensing will contribute to the distortion of the luminosity distance measurements at the order of 10% (Bartelmann and Schneider 2001, Dalal et al, 2006). For a detailed discussion of limiting factors in the case of a network of detectors see Schutz (2011) and references therein.
To conclude, observations of gravitational-waves from cosmological distances with current and planned detectors (e.g., spaceborne LISA, sensitive to low frequencies corresponding to chirping super-massive black hole binaries, Danzmann 1996, or a third-generation cryogenic underground Einstein Telescope, with an extended low frequency range compared to the Advanced LIGO and Advanced Virgo, Abernathy et al, 2011) promise a wealth of new astrophysical information, as demonstrated by the first multi-messenger gravitational-wave event (Abbott et al, 2017f, c). Future detectors will reach cosmological distances and redshifts of a few, being sensitive to practically all the chirping binaries in their sensitivity band in the Universe and providing precise distance measurements (see e.g., Lang and Hughes 2008). In addition to precisely measuring the Hubble constant, cosmological observations would help determine the distances to galaxies, thus contributing to building the standard 'distance ladder' (calibrating electromagnetic standard candles), establish the distribution of galaxies and voids, characterize the evolution of the dark energy and mass density of the Universe, mass distribution through the gravitational lensing, as well as the chemical evolution effects i.e., establishing the onset of star formation (Królak and Schutz 1987, Sathyaprakash and Schutz 2009). The truly multi-messenger era of astronomy is just beginning.