Cosmography with the Einstein Telescope
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IOP PUBLISHING CLASSICAL AND QUANTUM GRAVITY
Class. Quantum Grav. 27 (2010) 215006 (9pp) doi:10.1088/0264-9381/27/21/215006
Cosmography with the Einstein Telescope
B S Sathyaprakash
1
, B F Schutz
2
and C Van Den Broeck
3
1
School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, UK
2
Max Planck Institute for Gravitational Physics, The Albert Einstein Institute,
Am M
¨
uhlenberg 1, Golm, D-14476, Germany
3
Nikhef – National Institute for Subatomic Physics, Science Park 105, 1098 XG Amsterdam,
The Netherlands
E-mail: B.Sathyaprakash@astro.cf.ac.uk, B.F.Schutz@aei.mpg.de and vdbroeck@nikhef.nl
Received 13 May 2010, in final form 3 August 2010
Published 27 September 2010
Online at stacks.iop.org/CQG/27/215006
Abstract
The Einstein Telescope, a third-generation gravitational-wave detector under a
design study, could detect millions of binary neutron star inspirals each year. A
small fraction of these events might be observed as gamma-ray bursts, helping
to measure both the luminosity distance D
L
to and redshift z of the source.
By fitting these measured values of D
L
and z to a cosmological model, it
would be possible to infer the dark energy equation of state to within 1.5%
without the need to correct for errors in D
L
caused by weak lensing. This
compares favourably with 0.3–10% accuracy that can be achieved with the
Laser Interferometer Space Antenna (where weak lensing will need to be dealt
with) as well as with dedicated dark energy missions that have been proposed,
where 3.5–11% uncertainty is expected.
PACS numbers: 04.30.Db, 04.25.Nx, 04.80.Nn, 95.55.Ym
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Standard candles are used to measure the geometry and dynamics of the Universe. A standard
candle is a source whose intrinsic luminosity L can be inferred from the observed properties
such as its spectral content, time variability of the flux of the radiation it emits, etc. Since the
observations also measure the apparent luminosity F, one can deduce the luminosity distance
D
L
to a standard candle from 4πD
2
L
= L/F. If the redshift z to the source is known, then
from a population of such sources it will be possible to measure the cosmological parameters
since the luminosity distance is related to the redshift, in a flat Universe, via
D
L
(z) =
c(1+z)
H
0
z
0
(1+z
)
−3/2
dz
[
M
+
(1+z
)
3w
]
1/2
, (1)
0264-9381/10/215006+09$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1
Class. Quantum Grav. 27 (2010) 215006 B S Sathyaprakash et al
where H
0
is the Hubble constant,
M
and
are the dark matter and dark energy densities,
respectively, at the present time and w is the dark energy equation-of-state parameter.
There is no unique standard candle in astronomy that works for all distances. A distance
scale is built by using several candles, each of which works over a limited range of distance.
For instance, the method of parallax can determine distances of a few kpc, Cepheid variables
up to 10 Mpc, the Tully–Fisher relation up to tens of Mpc, the D
n
–σ relation up to hundreds
of Mpc and type Ia supernovae up to only a few redshifts [1]. This way of building a distance
scale has been referred to as the cosmic distance ladder. For cosmography, a proper calibration
of distances to high redshift galaxies is based on the mutual agreement between the different
rungs of this ladder. It is critical that each of the rungs is calibrated with an error as small as
possible.
2. Self-calibrating standard sirens of gravity
Cosmologists have long sought standard candles that can work over large distance scales
without being dependent on the lower rungs of the cosmic distance ladder. In 1986, it was
discovered [2] that gravitational astronomy can provide such a candle, or, more appropriately,
a standard siren, in the form of an inspiraling compact binary consisting of neutron stars
(NSs) and black holes (BHs). This method needs no calibration at all, relying purely on the
modelling of the two-body problem in general relativity. Gravitational-wave observations,
therefore, provide not only a powerful distance measuring tool but also a useful check on other
distance ladders.
In simple terms, in the inspiral phase of a compact binary, the amplitude of gravitational
waves depends on the ratio of a certain combination of the component masses called the chirp
mass and the luminosity distance. Both of these can be measured for a chirping signal—a signal
whose frequency increases by a measurable amount during its observation. For such sources
one can directly infer the luminosity distance. In reality, the response of an interferometer to
such a signal, in an approximation that keeps only the dominant signal harmonic at twice the
orbital frequency, is given by
h(t) =
4A(θ, ϕ, ψ, ι)νM[πMF(t)]
2
3
cos (t)
D
L
. (2)
Here, 0 A 1 is a numerical factor that depends on the location of the binary on the
sky and its orientation relative to the detector; M and ν are, respectively, the binary’s total
mass and symmetric mass ratio; (t) and F(t) are the signal’s phase and frequency; D
L
is
the luminosity distance; (θ, ϕ) gives the source’s location on the sky; ι is the orientation of
the system relative to the line of sight; and ψ is the wave’s polarization angle. In the case
of non-spinning binaries in quasi-circular orbits, an inspiral signal is characterized by nine
parameters in all (M,ν,t
0
,
0
,θ,ϕ,ψ,ι,D
L
). Here, t
0
and
0
are the fiducial parameters
defining the epoch when the signal’s frequency reaches a certain value and its phase at that
epoch. In this approximation, the signal amplitude depends only on the chirp mass
M defined
by
M = ν
3/5
M and not separately on the two mass parameters.
The signal’s phase has been computed to high order in post-Newtonian theory [3] and
depends only on the two mass parameters; here we go to 3.5 PN. One can therefore employ
matched filtering to extract the signal and to measure the two mass parameters (M, ν) as well as
the two fiducial parameters (t
0
,
0
). Note that for a source at a redshift z the signal’s frequency
will be redshifted to F → F/(1+z) but the mappings M → (1+z)M and D
L
→ (1+z)D
L
leave the signal invariant. Thus, a source of intrinsic total mass M
i
will appear to be a binary
2
Class. Quantum Grav. 27 (2010) 215006 B S Sathyaprakash et al
of total mass M = (1+z)M
i
. One must optically identify the host galaxy and measure its
redshift to deduce its intrinsic mass M
i
.
3. Multi-messenger cosmology with binary neutron s tars and GRBs
In general, the response of a single interferometer will not be sufficient to disentangle the
luminosity distance from the angular parameters. An optical identification of the source can
determine the direction to the source (and its redshift), leaving three unknown parameters
(ψ,ι,D
L
). Additionally, if the signal is associated with a gamma-ray burst (GRB), then the
source’s orbital plane will be perpendicular to the line of sight, implying that ι 0 and ψ 0.
Thus, for inspirals detected in coincidence with GRBs, a single detector is good enough to
measure the luminosity distance. Multi-messengers like GRBs can, therefore, make precision
cosmography possible, without the need to build a cosmic distance ladder.
For signals not associated with a GRB, D
L
can be measured if the source lasts long enough
to cause a modulation in the signal’s frequency due to the motion of the detector relative to the
source, as would be the case for the Laser Interferometer Space Antenna (LISA). If the signal
lasts for only a short time, as would be the case for ground-based detectors, one would need a
network of three detectors to measure D
L
.
Over the next two decades GW detectors could provide a new tool for cosmology. The
Laser Inteferometer Gravitational-Wave Observatory (LIGO) in the USA and Virgo in Europe
have reached design goals for their initial operation. Recent science runs of LIGO and Virgo
have begun to impact our understanding of astrophysical sources and phenomena. Both are
now getting ready to upgrade to advanced sensitivities by 2014 and are expected to detect ∼40
binary neutron star (BNS) mergers each year [4, 5]. Redshift could be measured to a (small)
number of events associated with GRBs and might allow the measurement of the expansion
rate of the Universe in the ∼500 Mpc range, where optical data is scarce [6, 7]. Observation by
the LISA of extreme mass ratio inspirals could measure the Hubble constant pretty accurately
[8]. The LISA will also observe binary super-massive BH mergers with signal-to-noise ratios
(SNRs) of several thousands at z ∼ 1, enabling the measurement of the dark energy equation
of state to within several percent [9, 10].
In the rest of this paper we will discuss how well it might be possible to constrain
cosmological parameters with the Einstein Telescope (ET)—a third-generation ground-based
interferometer that is currently under a design study [11, 12]. The ET is envisaged to be
ten times more sensitive in amplitude than the advanced ground-based detectors, covering a
frequency range of 1–10
4
Hz. Achieving the sensitivity of the ET will pose challenges in
mitigating gravity gradient, thermal and quantum noises but careful analysis shows that the
technology might be within reach in the next decade [13]. One possible topology for the ET
could be an equilateral triangle, 10 km on a side, allowing the operation of three V-shaped
interferometers at a single site [14, 15]. A network of such detectors might be available over
the next 15–20 years, but we will explore how accurately it might be possible to measure the
cosmological parameters with a single such ET.
3.1. Distance reach of the ET for neutron star binaries and event rates
For a fixed SNR ρ
0
, the distance up to which an inspiral signal could be detected in the ET is
given by
D
L
(ρ
0
) =
A
(θ,ϕ,ι,ψ)M
5/6
ρ
0
F
lso
F
s
f
−7/3
S
h
(f )
df
1/2
, (3)
3
Class. Quantum Grav. 27 (2010) 215006 B S Sathyaprakash et al
10
0
10
1
10
2
10
3
10
4
Total mass (in M
O
.
)
1
2
4
10
20
40
100
Luminosity Distance (Gpc)
Sky-ave. dist. Vs. Obs. M, ν=0.25
Sky-ave. dist. Vs phys. M, ν=0.25
Sky-ave. dist. Vs Obs. M, ν=0.10
Sky-ave. dist. Vs phys. M, ν=0.10
0.20
0.37
0.66
1.37
2.40
4.26
9.35
Redshift z
Figure 1. Range of the ET for inspiral signals from binaries as a function of the intrinsic (red
solid line) and the observed (blue dashed line) total mass. We assume that a source is visible if it
produces an SNR of at least 8 in the ET.
where S
h
(f ) is the one-sided noise power spectral density (PSD), which we take to be the
‘ET-B’ PSD as in figure 1 of [15]; we also assume the 10 km triple Michelson setup as
explained in that reference. F
s
is a frequency below which the accumulated SNR is negligible
because the PSD rises far faster than the signal spectrum and F
lso
is the signal frequency
corresponding to the last stable orbit of the binary, taken to be F
lso
= 1/(6
3/2
πM). The
quantity A
is an appropriate combination of the function A in equation (2) and the antenna
pattern functions of the three independent Michelson interferometers. In computing the
distance reach of the ET one can take an average of A
over sky position and set ι = ψ = 0,
again assuming relatively strong beaming of GRBs; the corresponding root-mean-square (rms)
value is A
rms
=
2/5A
opt
, where A
opt
is the value for an optimally oriented and positioned
system [16]. However, we note that GRBs may have beaming angles up to ∼40
◦
, i.e. ι ∼ 20
◦
[17]. If one averages over all angles (θ,φ,ψ,ι) but with the constraint ι<20
◦
, then the
rms value becomes A
rms
= αA
opt
with α 0.614, which is barely different from the case
ι = ψ = 0. The distance up to which the ET might detect signals from an inspiraling BNS
with an SNR of 8 is shown in figure 1 as a function of the observed total mass (blue dashed
lines). We assume a cosmological model in which H
0
= 70 km s
−1
Mpc
−1
,
M
= 0.27,
= 0.73 and w =−1, which allows us to convert distances to redshifts by inverting
equation (1), and which we have used to convert from the observed masses in figure 1 to the
intrinsic masses (red solid lines).
By extrapolating the rate of BNS inspirals expected in advanced detectors to the ET,
whose distance reach for a BNS is z 2orD
L
16 Gpc, one might expect 4 × 10
7
events
per year. Of course, this naive extrapolation does not give the correct rate as it does not include
the cosmological evolution of compact binaries. For our purposes, however, even if the rate
is an order of magnitude lower it does not matter. As an aside, legitimate concern has been
raised that this high event rate may lead to a confusion background [18]. As it turns out, the
PSD-weighted signals are actually quite short and will not tend to have significant overlap
with each other [19].
As noted earlier, in order for BNS inspirals to be useful for cosmography, it is essential
that their location on the sky and redshift are determined separately. If, as suspected, BNSs are
4
