Measuring cosmic distances with standard sirens 

The gravitational waves accompanying the merger of two massive compact objects encode the
distance to the merger without the usual appeal to a hierarchy of length scales.
Daniel E. Holz; Scott A. Hughes; Bernard F. Schutz

Physics Today 71 (12), 34–40 (2018);

https://doi.org/10.1063/PT.3.4090

CrossMark

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22 November 2023 00:18:52

 Measuring cosmic distances
WITH STANDARD SIRENS

22 November 2023 00:18:52

Upgrading the
Laser Interferometer
Gravitational-Wave
Observatory. (Courtesy
of Caltech/LIGO.)

34 PHYSICS TODAY | DECEMBER 2018

Daniel Holz is a professor of physics and of astronomy
and astrophysics at the University of Chicago. Scott
Hughes is a professor of physics at MIT in Cambridge,
Massachusetts. Bernard Schutz is a professor of
physics and astronomy at Cardiff University in the UK.

Daniel E. Holz,
Scott A. Hughes, and
Bernard F. Schutz

The gravitational waves

accompanying the merger of
two massive compact objects
encode the distance to the
merger without the usual appeal
to a hierarchy of length scales.

22 November 2023 00:18:52

D ecades of experimental effort paid
off spectacularly on 14 September
2015, when the two detectors of
the Laser Interferometer Gravita-
tional-Wave Observatory (LIGO)
spotted the gravitational waves generated by a
pair of coalescing black holes.1 To get a sense of
the effort leading to that breakthrough, consider
that the gravitational waves caused the mirrors at
the ends of each interferometer’s 4 km arms to os-
cillate with an amplitude of about 10−18 m, roughly
a factor of a thousand smaller than the classical
proton radius. The detection was also a triumph
for theory. The frequency and amplitude evolu-
tion of the measured waves precisely matched
general relativity’s predictions for the signal pro-
duced by a binary black hole merger, even though
the system’s gravity was orders of magnitude
stronger than that of any system that had been
precisely probed before that detection. As figure 1
shows, gravitational-wave astronomy began not
with a bang but with a chirp.
DECEMBER 2018 | PHYSICS TODAY 35
MEASURING COSMIC DISTANCES

Labeled GW150914, that first reported event a

was soon joined by other detections of binary
black hole mergers. Each of those events ap- 1.0
peared to be totally dark to traditional astro-

STRAIN (10 −21)

0.5
nomical instruments—the matter and electro-
magnetic fields near the merging black holes 0.0
were not sufficient to generate any signal
other than gravitational. As had long been −0.5
promised, gravitational waves have opened a
window onto an otherwise invisible sector of −1.0
the universe.
Although celebrated by the physics and as-
tronomy communities and feted by the broader 0.25 0.30 0.35 0.40 0.45
public, gravitational-wave astronomy did not TIME (s)
initially overlap significantly with more tradi-
512

NORMALIZED AMPLITUDE
tional astronomy. That changed on 17 August b
8
2017, when a gravitational-wave signal, fol-
FREQUENCY (Hz)

lowed by a burst of gamma rays, triggered one 256

of the most intense observing campaigns in 6
the history of astronomy. LIGO, joined now 128
by the Virgo detector in Pisa, Italy, recorded 4
a minute-long chirp (see figure 2) encoding 64
the final several thousand orbits in the coales- 2
cence of two neutron stars.2 The stars’ colli-
32
sion, at about 1⁄3 the speed of light, was an 0
astronomical cataclysm. Just 1.7 s after the 0.25 0.30 0.35 0.40 0.45
end of the gravitational-wave signal, the or- TIME (s)
biting Fermi Gamma-Ray Space Telescope and FIGURE 1. THE CHIRP OF GW150914, the first gravitational-wave event to be

22 November 2023 00:18:52

INTEGRAL observatory recorded a short detected. (a) On 14 September 2015, the two detectors of the Laser Interferometer
gamma-ray burst.3 Gravitational-Wave Observatory observed the effects of gravitational waves: an
oscillating change in the separation of test mirrors in their arms. When divided by the
The LIGO–Virgo alert provided the sky po-
length of a detector arm, the separation change is called strain. Shown here are the strain
sition and, importantly, the distance to the event.
results (orange) taken at the Hanford interferometer (the black curve is a theoretical
Just 11 hours later, optical astronomers identified
fit); the data from the Livingston interferometer are remarkably similar. Visual inspection
a violent event in the galaxy NGC 4993, a kilo-
reveals that the frequency and amplitude of the gravitational-wave chirp increase
nova explosion that shone 1000 times brighter with time. (b) A spectrogram obtained from the Hanford data shows the time–
than a typical nova. More than 70 teams made frequency relation explicitly. (Panels adapted from ref. 1.)
follow-up electromagnetic observations. The
effort represents the first time a source has
been detected through both its gravitational and electromag- tant object is redshifted as it propagates from its source to an
netic radiation. A significant portion of the world’s professional observer.
astronomers are coauthors with the gravitational-wave teams Consider light with a wavelength λ emitted from a source
on the summary paper describing those observations.4 Ob- that is a distance D away. Observers on Earth will measure the
servations in the x-ray and radio bands continue as we write. light to have a wavelength of (1 + z)λ, where z is the light’s red-
From the event, astronomers are learning much about gamma- shift. To leading order in z, the source distance and redshift are
ray bursts, neutron stars, and their associated physics and proportional:
astronomy. cz = H0D , (1)
Because a gravitational wave encodes the distance to its
source, GW170817 provided the astrophysical community with where the Hubble constant H0 is today’s value of the Hubble
another advance: the first measurement of the local cosmic ex- parameter H. It has dimensions of inverse time; the recipro-
pansion rate—the Hubble constant—via gravitational waves.5 cal 1/H0, known as the Hubble time, provides a rough estimate
That milestone opened up a completely new way to measure of the age of the universe. Astronomers conventionally ex-
the dynamics of the universe: the standard-siren technique. press H0 in units of km s−1 Mpc−1, because the megaparsec
(1 Mpc = 3.26 million light-years) is convenient for intergalactic
A ladder to the stars distances. As mentioned above, equation 1 is a leading-order
The Hubble constant has been the single most important pa- expression. For far distant objects, it needs to be corrected with
rameter describing cosmology since Edwin Hubble discovered higher-order terms that depend on the nature of the matter and
the expansion of the cosmos in 1929. On the largest scales, the energy that fill the universe.6
universe expands homogeneously and isotropically, so every In principle, just one object of known distance and cosmo-
part of it recedes from every other part. General relativity shows logical redshift suffices to determine the Hubble constant. The
that due to the cosmic expansion, radiation emitted from a dis- redshift of many objects can be determined from spectral mea-
36 PHYSICS TODAY | DECEMBER 2018
 NORMALIZED AMPLITUDE
surements, but determining astronomi- 0 6
2 4
cal distances is much more challenging.
For nearby objects, distance can be de-
termined using parallax—that is, measur- 400 a

FREQUENCY (Hz)
ing the apparent angular shift in the posi- 300
tion of an astronomical object as Earth
200
orbits the Sun. The technique does not
work well for larger distances, as the an-
gular shift due to Earth’s orbital motion 100
becomes too small to measure.
For objects beyond our galaxy, an im- 50
portant tool for measuring distances is −10 −8 −6 −4 −2 0 2 4 6
the standard candle: an astronomical TIME FROM MERGER (s)
source whose intrinsic luminosity is as-
sumed to be known. Suppose a star has b GW170817 GW170817
luminosity L and observers on Earth DECam observation DECam observation
measure it to have a flux F. From the in- (0.5–1.5 days post merger) (14 days post merger)
verse square law and assuming the star
radiates isotropically, you obtain the lu-
minosity distance
L
D= . (2)
4πF
Nature does not provide observers
with stars whose luminosities are pre-
cisely known. However, it does provide
stars and other objects whose luminosi-

22 November 2023 00:18:52

ties can be inferred accurately. Celebrated
examples are the Cepheid variables, giant
stars whose luminosities vary periodi- FIGURE 2. THE CHIRP OF GW170817, the gravitational-wave event accompanying the
cally. By studying a group of such stars coalescence of two neutron stars. (a) The increasing frequency of the gravitational wave
in the Small Magellanic Cloud—a dwarf with time is clearly evident in the spectrograms generated from Laser Interferometer
3
galaxy near the Milky Way—Henrietta Gravitational-Wave Observatory data. Note the marked difference in time scale between the
Leavitt discovered in 1912 that each chirps for the black hole merger (figure 1) and neutron star merger. (Courtesy of Tito Dal Canton.)
(b) The Dark Energy Camera (DECam) was one of several instruments to see an optical
star’s oscillation period correlates with
counterpart to GW170817 shortly after the neutron star merger. By two weeks after the
its intrinsic luminosity. Some Cepheids
merger, the optical counterpart had vanished. (Adapted from ref. 13, M. Soares-Santos et al.)
are close enough that their distances can
be determined using parallax, and thus
their luminosity can be calibrated. With the luminosity–period pernovae7 yields H0 = 73.24 ± 1.74 km s−1 Mpc−1. An alternative
relationship empirically established, Cepheid variable stars can method, based on the Planck satellite’s measurements8 of
serve as standard candles for measuring distances beyond the fluctuations in the cosmic microwave background gives
limits of parallax. H0 = 67.74 ± 0.46 km s−1 Mpc−1. The two values are uncomfort-
By putting together multiple methods for measuring dis- ably far apart if their uncertainties are to be believed. Given the
tances, astronomers construct what is called the cosmic distance many rungs on the distance ladder that must be empirically
ladder (see the article by Mario Livio and Adam Riess, PHYSICS calibrated, it would not be surprising for one or both of the H0
TODAY, October 2013, page 41). On each rung of the ladder, ob- determinations to be affected by undiscovered systematic er-
jects thought to be of standard luminosity are identified and rors. Or the universe might be more complicated than the com-
calibrated in terms of measurements contributing to the previ- munity now thinks. Perhaps it is more inhomogeneous; per-
ous rung. haps it is less isotropic; perhaps an important contribution
Various sophisticated methods now exist for measuring H0, to its mass–energy budget has been overlooked; or perhaps
but many depend in one way or another on the distance ladder. general relativity does not describe the universe well on the
One method relies on an important standard candle that can largest scales.
be seen very far away: the type Ia supernova explosion. Super- The discrepancy among measurements of H0 , one of the fun-
nova observations not only helped determine H0, they also im- damental quantities of cosmology, may be the result of system-
plied nonlinear contributions to equation 1 that showed the ex- atics, or it may hint at new physics. Both possibilities motivate
pansion of the universe is accelerating. That result led to the new measurements to resolve or confirm the tension.
awarding of the 2011 Nobel Prize in Physics to Saul Perlmutter,
Adam Riess, and Brian Schmidt (see PHYSICS TODAY, December Binary inspiral, a standard siren
2011, page 14). The box on page 38 highlights some important features of grav-
The most recent measurement of the expansion using su- itational waves. Its equation B3 shows that the amplitude of a
DECEMBER 2018 | PHYSICS TODAY 37
MEASURING COSMIC DISTANCES

KEY PROPERTIES OF + POLARIZATION

GRAVITATIONAL WAVES
Gravitational waves are described by a

AMPLITUDE
tensor field hμν that characterizes the dy-
namics of gravity in general relativity. The TIME
indices μ and ν range over the four coor-
dinates of space and time. Many of the
properties of hμν are analogous to those
of the vector potential that characterizes
electromagnetic radiation.
For sources moving at much less than
the speed of light, electromagnetic radi- × POLARIZATION
ation is described by the vector potential tromagnetic radiation’s electric and mag- electromagnetic basis polarizations
A that arises from a source’s time-varying netic fields, gravitational waves are or- point along two orthogonal axes in the
electric dipole moment p: thogonal to their direction of propaga- plane perpendicular to the direction of
μ 1 dpj tion. As a result, the components of hμν propagation, and the electric force that a
Aj = 0 . (B1) with time indices—h00 , h0ν and hμ0—do
4π D dt passing wave exerts on charges can be
not radiate and they can be ignored decomposed into components along
Here, D is the distance from the source, μ0
when discussing gravitational waves. The those basis directions. Gravitational
is the permeability of free space, and the
tensor Ijk is the source’s mass quadrupole waves also exert forces normal to the
index j labels one of the three spatial di-
moment propagation direction, but they act
mensions. The dipole moment, in turn, is
the integral of the charge density ρc over
the volume of the source, weighted by
source [ 1
3 ]
Ijk = ∫ ρm r ′j r ′k − (r ′)2 δjk dV ′, (B4)
tidally, stretching along one axis as they
squeeze along the perpendicular axis. If
the position r: where ρm is the source’s mass density and the wave propagates along the z-axis,

22 November 2023 00:18:52

δjk = 1 when the indices match and van- one polarization stretches and squeezes
p= ∫ ρcr ′ dV ′ . (B2) ishes otherwise. Notice that gravitational along the x- and y-directions. That polar-
source
radiation involves two time derivatives of ization is conventionally labeled h+ . The
In 1918 Albert Einstein showed that
the relevant moment rather than the sin- other polarization, h× , stretches and
the analogous result for gravitational
gle derivative appropriate for electro- squeezes along axes rotated by 45° from
waves is the quadrupole formula
magnetic radiation. Notice also that the the x- and y-axes. The figure above shows
2G 1 d 2Ijk different constants that connect source how a circular ring of freely floating par-
hjk = , (B3)
c 4 D dt 2 and radiation reflect the different funda- ticles would be distorted over time by a
where G is Newton’s constant and c is the mental forces involved. gravitational wave propagating toward
speed of light. The indices j and k in equa- Both electromagnetic and gravita- the observer with one or the other of
tion B3 are purely spatial. Much like elec- tional waves have two polarizations. The those polarization states.

gravitational wave falls off inversely with the distance to the increased energy loss due to gravitational waves, which leads
source. If it were somehow possible to learn how the source’s to a further decrease in the orbital separation, and so on. The
mass quadrupole moment (defined in the box) varies with time, binary thus chirps; the gravitational waves go from low fre-
then a measurement of the gravitational-wave amplitude would quency to high at an increasing rate, all the while increasing in
reveal that distance. amplitude. From Kepler’s law and a well-known formula that
As was first shown by one of us (Schutz), for gravitational relates the power emitted in gravitational waves to the binary’s
waves generated by binary stars it is indeed possible to take changing quadrupole moment, it is possible to show that

( )
the measured data and derive how the quadrupole moment 5/3
dΩ 96 GM
varies.9 In other words, binary inspiral allows for a determina- = Ω 11/3 , (3)
dt 5 c3
tion of the distance to the source without any reference to the
cosmic distance ladder. The only empirical calibration needed plus correction terms that don’t alter the substance of our
is of the gravitational-wave detector, to make sure it reports the story. In equation 3, we have introduced the chirp mass
amplitude of the wave correctly. Beyond that, the only assump- M = (m1m2)3/5(m1 + m2)−1/5.
tion is that general relativity is valid. The rate of change of the frequency depends only on one
Consider a binary in a circular orbit. Its members will circle parameter: the chirp mass. Once you know M, you know how
one another with a frequency Ω that depends on the binary quickly the frequency is changing at any point in the evolution
members’ separation and their masses m1 and m2. The gravita- of the binary system. All binaries with the same M will have,
tional waves it emits take energy from the orbit and cause the to leading order, the same chirping sweep from low frequency
binary’s components to spiral toward one another. As the sep- to high, although the corrections to equation 3 do introduce a
aration decreases, the orbital frequency increases, which causes dependence on the individual masses of the stars. The chirp
38 PHYSICS TODAY | DECEMBER 2018
 the source distance; the only fundamental as-
sumption is that general relativity is valid. Be-
0.04 cause gravitational-wave detection is more like
hearing a signal than seeing an image, several
of us in the field independently came up with
the name standard siren. The term first ap-
0.03
PROBABILITY

peared in print in a paper by two of us (Holz

and Hughes), and it seems to have stuck.11
The “standard” in standard siren arises be-
0.02 cause the waves’ frequency sweep and ampli-
tude both depend on the same mass M. That
is no coincidence. It follows from the fact that
0.01 the intrinsic gravitational luminosity—a com-
bination of amplitude and frequency—depends
only on the number of orbits remaining until
coalescence.12 That number, in turn, depends
0.00
only on Ω and dΩ/dt. Determining those two
50 60 70 80 90 100 110 120 130 140
quantities from the gravitational waveform
H0 (km s−1 Mpc−1)
thus yields the binary’s instantaneous luminos-
FIGURE 3. A STANDARD-SIREN DETERMINATION of the Hubble constant H0. The ity. Measuring the amplitude directly then gives
blue curve shows the probability distribution for the value of H0 , as determined from the luminosity distance D.
measurements of the event GW170817. Vertical dashed lines mark the 68% credible
interval; vertical dotted lines mark the 95% interval. Also shown are measured values
and error intervals for supernova observations (brown) and for the Planck satellite’s
A redshift degeneracy
The chirp mass enters the equations for the
determination (green) based on observations of the cosmic microwave background.
gravitational waveform and frequency evolu-
The standard-siren value is consistent with the supernova and CMB measurements;
tion in a combination with units of time: the
standard-siren measurements have the potential to resolve the tension between
chirp time GM/c3. That time scale experiences
them. (Adapted from ref. 5.)

22 November 2023 00:18:52

the usual cosmological redshift, so there is a
fundamental mass–redshift degeneracy: A bi-
time, GM/c3, characterizes how rapidly Ω changes due to nary with masses m1 and m2 at redshift zero produces gravita-
gravitational-wave emission. tional waves that fit exactly the same waveform template as the
Let’s turn now to the waves’ amplitude and consider a cir- waves from a binary with masses m1/(1+z) and m2/(1+z) at red-
cular binary oriented such that the normal to its orbital plane shift z.
makes an angle ι to our line of sight. With that convention, ι = 0° The degeneracy can be broken in several ways. Perhaps the
means the binary is viewed head on, and ι = 90° corresponds simplest is to assume values for H0 and other cosmological pa-
to an edge-on view. As described in the box, gravitational waves rameters. Then, from equation 1, a measurement of distance, as
come in two polarization states. With the convention that the determined from the amplitude of a gravitational wave, yields
normal to the orbital plane is in the xz-plane, their amplitudes an estimate of the redshift. The true masses of the binary can
are given by then be inferred. That approach has been applied for most of

( )
2c GM 5/3 the sources that LIGO and Virgo have measured to date, in-
h+ = Ω 2/3 (1 + cos 2 ι) cos2 Φ (t) , cluding GW150914.
D c3
If, however, the gravitational-wave event has an electro-
( )
4c GM 5/3 (4)
h× = Ω 2/3 cos ι sin2 Φ (t) , magnetic counterpart, a measurement of the spectrum of the
D c3 counterpart or of an associated host galaxy determines the red-
where Φ(t) is the accumulated orbital phase found by integrat- shift to the source without the need of additional assumptions.
ing the orbital frequency Ω over the duration t of the measure- In that case, telescopes and gravitational-wave detectors do the
ment, and the factor of 2 multiplying Φ(t) is due to the waves’ tasks to which they are best suited, measuring redshift and dis-
quadrupolar nature.10 The amplitudes depend on the masses tance, respectively. Given those quantities, equation 1 yields the
m1 and m2 only through the chirp mass. Hubble constant H0.
Once observers have measured gravitational waves from a
binary, they can accurately match the waves’ phase evolution Gravitational-wave event GW170817
to that of a model template, as in figure 1a. Doing so determines The field of standard-siren cosmology was conceived9 in 1986,
M, typically with high precision. If it is possible to measure and after more than 30 years of gestation, it finally arrived with
more than one polarization, then the ratio of their amplitudes triplets: the detection of GW170817 by the LIGO and Virgo ob-
determines the inclination angle ι. Once M and ι are known, the servatories, followed 1.7 seconds later by the discovery of an
distance to the source is determined, according to equation 4, associated gamma-ray burst, followed 11 hours later by the dis-
by measuring the waves’ amplitude. covery of an optical counterpart.13 GW170817 comprised two
Binary inspiral thus acts as the gravitational-wave ana- compact objects with masses in the range of 1.36–2.26 solar
logue of a standard candle, but it does not require the cosmic masses and 0.86–1.36 solar masses, consistent with a binary
distance ladder. No empirical calibrations are needed to obtain neutron star system. The coalescence of the stars occurred at a
DECEMBER 2018 | PHYSICS TODAY 39
MEASURING COSMIC DISTANCES

luminosity distance of about 40 Mpc. With an observed signal- operating within the next decade. Additional detectors im-
to-noise ratio of 32, GW170817 is by far the closest and loudest prove the signal-to-noise ratio and make it possible to better
gravitational-wave source detected to date. determine source parameters. The community has already
The optical counterpart to GW170817 (see figure 2b) was benefited from detector synergy with the measurement of
found within 10 arcseconds of the center of the galaxy NGC 4993, GW170817: The weak signal measured by Virgo indicated that
an angle that corresponds to a separation of about 2 kpc. The the event was near a null in the Virgo antenna sensitivity
redshift of the galaxy is 0.009. A Bayesian analysis5 that com- pattern. That information significantly aided in locating the
bines the galactic redshift with the LIGO–Virgo measurement event direction and greatly facilitated the search for the optical
of distance to GW170817 leads to an estimate of the value of counterpart.
the Hubble constant: H0 = 70 +12 −1 −1
−8 km s Mpc . Figure 3 shows the Planning for the next generation of gravitational-wave de-
full Bayesian probability distribution for H0. tectors is under way.17 Ground-based detectors, such as the pro-
In principle, the combined LIGO and Virgo distance esti- posed Cosmic Explorer and the Einstein Telescope, will operate
mate is limited by the detectors’ amplitude calibration. The cur- in largely the same 10–10 000 Hz frequency band as LIGO and
rent precision of a few percent will improve as the detectors Virgo but will be able to measure binary inspiral to much larger
approach their design sensitivity. However, as equation 4 distances—essentially to all neutron star mergers in the uni-
shows, the measurement of D and hence of H0 depends on de- verse. (See PHYSICS TODAY, October 2018, page 25.) The space-
termining the inclination ι. Uncertainty in ι increases uncer- based Laser Interferometer Space Antenna (LISA), a European
tainty in D and thus increases uncertainty in H0. Space Agency mission that includes NASA participation, is set
The determination of ι comes primarily by measuring to launch by 2034. LISA will operate in the millihertz frequency
gravitational-wave polarizations. The two LIGO detectors are band, have high sensitivity, be self-calibrating, and measure
closely aligned in orientation, so they do a poor job constrain- standard-siren events involving the coalescence of black holes
ing polarization. Virgo is at a different orientation, so it can with masses from about 104 up to about 107 solar masses for
provide additional constraints on polarization. Unfortunately, distances corresponding to redshifts as large as 20. Especially
Virgo had little sensitivity to the sky position of GW170817. on the low end of its mass range, LISA may be able to determine
Polarization was thus not well constrained even by the com- the source position accurately enough to pin down the galaxy
bined LIGO and Virgo measurements. The result was a large cluster or even the galaxy hosting the event.
uncertainty in ι and a roughly 15% error in the measured dis- Just over one year ago, GW170817 not only allowed a mea-

22 November 2023 00:18:52

tance to GW170817. If additional information can constrain surement of the distance to its source but also provided a proof
the inclination—for example, from studies of the associated of principle that the standard-siren technique could measure
gamma-ray burst, kilonova, or afterglow—then the distance es- the Hubble constant. Future measurements will transform stan-
timate and derived value of the Hubble constant will corre- dard sirens into an important tool for studying the expansion
spondingly improve. history of the universe.

A loud, bright future

The event GW170817 is the first in what we expect to be a rich
catalog of standard sirens. LIGO and Virgo are currently being
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40 PHYSICS TODAY | DECEMBER 2018