#### Testing the anisotropy of the universe using the simulated gravitational wave events from advanced LIGO and Virgo

Eur. Phys. J. C
Testing the anisotropy of the universe using the simulated gravitational wave events from advanced LIGO and Virgo
Hai-Nan Lin 0
Jin Li 0
Xin Li 0
0 Department of Physics, Chongqing University , Chongqing 401331 , China
The detection of gravitational waves (GWs) provides a powerful tool to constrain the cosmological parameters. In this paper, we investigate the possibility of using GWs as standard sirens in testing the anisotropy of the universe. We consider the GW signals produced by the coalescence of binary black hole systems and simulate hundreds of GW events from the advanced laser interferometer gravitational-wave observatory and Virgo. It is found that the anisotropy of the universe can be tightly constrained if the redshift of the GW source is precisely known. The anisotropic amplitude can be constrained with an accuracy comparable to the Union2.1 complication of type-Ia supernovae if 400 GW events are observed. As for the preferred direction, 800 GW events are needed in order to achieve the accuracy of Union2.1. With 800 GW events, the probability of pseudo anisotropic signals with an amplitude comparable to Union2.1 is negligible. These results show that GWs can provide a complementary tool to supernovae in testing the anisotropy of the universe.
1 Introduction
The cosmological principle, which states that the universe
is homogeneous and isotropic on large scales, is one of the
most basic assumptions of modern cosmology. This
assumption is proven to be well consistent with various observations,
such as the statistics of galaxies [
1
], the halo power
spectrum [
2
], the observation on the growth function [
3
], the
cosmic microwave background from the Wilkinson Microwave
Anisotropy Probe (WMAP) [
4,5
] and Planck satellites [
6,7
].
Based on the cosmological principle, the standard
cosmological model, i.e. the cold dark matter plus a
cosmological constant ( CDM) model is well constructed. However,
some other observations show that the universe may deviate
from the statistical anisotropy. These include but not
limited to the large scale bulk flow [
8,9
], the CMB
temperature anisotropy [
10,11
], the spatial variation of the
electromagnetic fine-structure constant [
12–15
], the anisotropy of
the distance-redshift relation of type-Ia supernovae [
16–18
].
If the universe is indeed anisotropic, it implies that there
are new physics beyond the standard model. Whether these
anisotropic signals come from the intrinsic property of the
universe or merely the statistical fluctuation is extensively
debated [
19–24
].
The gravitational waves (GWs) provide an alternative tool
to testing the cosmology. The greatest advantages of GWs
is that the distance calibration is independent of any other
distance ladders, i.e. it is self-calibrating. Since Einstein
predicted the existence of GWs a century ago, extensive efforts
have been made to directly detect GWs but without success.
The breakthrough happens in September 2015, when the
laser interferometer gravitational-wave observatory (LIGO)
and Virgo collaborations reported a GW signal produced by
the coalescence of two black holes, which was late named
GW150914 [
25
]. Since then, four more GW events have been
observed [
26–29
]. The first four events are produced by the
merge of binary black hole systems and no electromagnetic
counterpart is expected. The last one event, GW170817, is
produced by the merge of binary neutron star system and it is
associated with a short gamma-ray burst GRB170817 [
30–
32
]. The host galaxy NGC4993 at redshift z ∼ 0.01 is
identified by the follow-up observation [33]. The simultaneous
observations of GW signal and electromagnetic counterparts
open the new era of multi-messenger astronomy. Using the
GW/GRB170817 event as standard siren, the Hubble
constant is constrained to be 70.0−+182.0.0 km s−1 Mpc−1 [
34
],
showing that GW data are very promising in constraining
the cosmological parameters. Several works have used the
simulated GW data to constrain the cosmological parameters
and showed that the constraint ability of GWs is comparable
or even better than the traditional probes if hundreds of GW
events have been observed [
35–40
].
In this paper, we investigate the possibility of using GW
data to test the anisotropy of the universe. Unfortunately,
there is only five GW events observed up to date. With such
a small amount of data points, it is impossible to do
statistical analysis. Therefore, we simulate a large number of
GW events from the advanced LIGO and Virgo detectors. It
is expected that hundreds of GW events will be detected in
the next years. We use the simulated GW data to test how
many GW events are needed in order to reach the accuracy
of type-Ia supernovae. The present astronomical
observations imply that the intrinsic anisotropy of the universe is
quite small and could be treat as a perturbation of CDM
model. Therefore, throughout this paper we assume a fiducial
flat CDM model with Planck parameters M = 0.308 and
H0 = 67.8 km s−1 Mpc−1 [
7
].
The rest of the paper is organized as follows: in Sect. 2,
we describe the method of using GW data as standard sirens
in cosmological studies. In Sect. 3, we illustrate how to
simulate the GW events from the advanced LIGO and Virgo.
In Sect. 4, we investigate the constraint ability of GW data
on the anisotropy of the universe. Finally, discussions and
conclusions are given in Sect. 5.
2 GWs as standard sirens
GW is the fluctuation of spacetime metric, as a prediction of
general relativity it has two polarization states often written
as h+(t ) and h×(t ). GW detectors based on the
interferometers such as advanced LIGO and Virgo measure the change
of difference of two optical path (which is often called the
strain) caused by the pass of GWs. The strain is the linear
combination of the two polarization states,
h(t ) = F+(t )h+(t ) + F×(t )h×(t ),
where the coefficients F+(t ) and F×(t ) are called the
beampattern functions, which depends on the location and
orientation of the detector, as well as the position of GW source. For
detectors built on the Earth, due to the diurnal motion of the
Earth the beam patterns are periodic functions of time with
a period equal to one sidereal day. The explicit expressions
of the beam patterns are given by [
41
]
F+(t ) = sin ζ [a(t ) cos 2ψ + b(t ) sin 2ψ ],
F×(t ) = sin ζ [b(t ) cos 2ψ − a(t ) sin 2ψ ],
where ζ is the angle between the two interferometer arms, ψ
is the polarization angle of GW, and
(1)
(2)
(3)
1
a(t) = 16 sin 2γ (3 − cos 2λ)(3 − cos 2δ) cos[2(α − φr − r t)]
1
− 4 cos 2γ sin λ(3 − cos 2δ) sin[2(α − φr − r t)]
1
+ 4 sin 2γ sin 2λ sin 2δ cos[α − φr − r t]
1
− 2 cos 2γ cos λ sin 2δ sin[α − φr − r t]
3
+ 4 sin 2γ cos2 λ cos2 δ,
b(t) = cos 2γ sin λ sin δ cos[2(α − φr − r t)]
1
+ 4 sin 2γ (3 − cos 2λ) sin δ sin[2(α − φr − r t)]
+ cos 2γ cos λ cos δ cos[α − φr − r t]
1
+ 2 sin 2γ sin 2λ cos δ sin[α − φr − r t],
where γ is measured counterclockwise from East to the
bisector of the interferometer arms, λ is the latitude of the
detector’s location, (α, δ) are the right ascension and
declination of the GW source in the equatorial coordinate system,
r is the rotational angular velocity of the Earth, and φr is
the initial phase characterizing the position of the Earth in its
diurnal motion at t = 0. Therefore, φr + r t represents the
local sidereal time of the detector’s location. For GW
transients such as the five events observed by advanced LIGO
and Virgo, the duration of GW signal is much smaller than
one sidereal day. In such a short time the motion of the Earth
can be neglected and the pattern functions are approximately
time-independent.
In this paper, we focus on the GW signals produced by
the coalescence of binary systems. Consider a binary system
consists of component masses m1 and m2 in the comoving
frame, define the total mass M = m1 + m2, the symmetric
mass ratio η = m1m2/M 2, and the chirp mass Mc = M η3/5.
For the GW source locating at cosmological distance with
redshift z, the chirp mass in the observer frame is given by
Mc,obs = (1 + z)Mc,com [
42
]. In the following, Mc always
refers to the chirp mass in the observer frame unless
otherwise stated. In the post-Newtonian and stationary phase
approximation, the strain h(t ) produced by the inspiral of
binary, is given in the Fourier space by [
36,43
]
H( f ) = A f −7/6 exp[i (2π f t0 − π/4 + 2ψ ( f /2) − ϕ(2,0))],
(4)
(5)
(6)
where t0 is the epoch of merger. The explicit expressions
of the phase terms ψ ( f /2) and ϕ(2,0) can be found in
Ref. [43], but these are unimportant in the following
calculation because we are only interested in the inner product
of H( f ), so the exponential term on the right-hand-side of
Eq. (6) is canceled out. The Fourier amplitude A is given by
1
A = dL
F 2 (1 + cos2 ι)2+4F 2 cos2 ι×
+ ×
59π6 π −7/6Mc5/6,
where ι is the inclination of the binary’s orbital plane, i.e. the
angle between the binary’s orbital angular momentum and
the line-of-sight, and
dL =
1 + z
H0
0
z
d z
M (1 + z)3 + 1 −
M
is the luminosity distance of the GW source to the detector.
The signal-to-noise ratio (SNR) of a detector is given by
the square root of the inner product of the strain in Fourier
space [
43
],
ρi =
H, H ,
where the inner product is defined as
a, b
= 4
flower
fupper a˜ ( f )b˜∗( f ) + a˜ ∗( f )b˜( f )
2
d f
Sh ( f )
,
where˜represents the Fourier transformation and * represents
the complex conjugation, Sh ( f ) is the one-side noise power
spectral density (PSD) characterizing the sensitivity of the
detector on the spacetime strain, flower and fupper are the
lower and upper cutoffs of the frequency. Bellow flower the
noise is uncontrollable and Sh ( f ) is often assumed to be
infinity. fupper is the highest frequency of the GW signal
during the inspiral epoch. Following Ref. [
36
], we assume
fupper = 2 fLSO, where fLSO = 1/(63/22π Mobs) is the orbit
frequency at the last stable orbit, Mobs = (1 + z)(m1 +
m2) is the total mass in observer frame. If N independent
detectors form a network and detect the same GW source
simultaneously, the combined SNR is given by
ρ =
N
i=1
ρi2
.
We require the SNR to be larger than 8 to ensure that this is
indeed the GW signal rather than the noise.
The uncertainty of luminosity distance extracted from the
GW signals can be obtained using the Fisher matrix [
36
].
Note that the distance of source dL is correlated with other
parameters, especially the inclination angle ι. In principle,
all values of ι ∈ [0◦, 180◦] are possible. It is pointed out
that the maximal effect of inclination angle on the SNR is
a factor of 2 [
39, 44
]. Here we only consider the simplified
case where the binary’s orbital plane is nearly face on, hence
the amplitude A is independent of the polarization angle ψ .
Following Ref. [39], we assume that dL is uncorrelated with
Such a treatment, although is not accurate, is reasonable
because we are only interested in the constraining ability
of GW events on the anisotropy of the universe. A more
accurate measurement of distance gives a tighter constraint
on the anisotropy. We also add an additional uncertainty
σdleLns = 0.05zdL caused by the weak lensing of galaxies
alone the line-of-sight. Therefore, the total error on dL is
given by
σdL =
2
+ (0.05zdL )2 .
The luminosity distance is often converted to the
dimensionless distance modulus by
and the uncertainty of μ is propagated from that of dL by
2dL
ρ
dL
μ = 5 log Mpc + 25,
5 σdL .
σμ = ln 10 dL
(7)
(8)
(9)
(10)
(11)
that of other parameters and then double the uncertainty of
dL calculated from the Fisher matrix as the upper limit of
instrument error on dL , i.e.,
σdinLst = 2ρdL
.
(12)
(13)
(14)
(15)
(16)
To use GWs as the standard sirens to test anisotropy of the
universe, we assume that the universe has dipole structure in
distance-redshift relation, i.e.
μ = μ CDM(1 − d cos θ ),
where d is the dipole amplitude and θ is the angle between
GW source and the preferred direction of the universe.
The preferred direction can be parameterized as (α0, δ0)
in the equatorial coordinate system. The three parameters
(d, α0, δ0) can be obtained by fitting the GW data to Eq. (16)
using the least-χ 2 method.
3 Simulation from advanced LIGO and Virgo
In this section we simulate GW events based on the advanced
LIGO and Virgo detectors. The advanced LIGO consists of
two detectors locating at Hanford, WA (119.41◦W, 46.45◦N)
and Livingston, LA (90.77◦W, 30.56◦N), respectively. Each
detector contains a laser interferometer with two orthogonal
arms of about 4 km. The Virgo detector has two arms of
3 km long and locates near Pisa, Italy (10.50◦E, 43.63◦N).
The instrument parameters are [
41
]: λH = 46.45◦, γH =
171.80◦, λL = 30.56◦, γL = 243.00◦, λV = 43.63◦, γV =
(φr +
r t )H − (φr +
r t )L = −28.64◦.
Similarly, we have
(φr +
r t )H − (φr +
r t )V = −129.91◦.
The PSD of the advanced advanced LIGO is given by [
45
],
Sh ( f ) = S0 x −4.14 − 5x −2 +
111(1 − x 2 + 0.5x 4)
1 + 0.5x 2
where x = f / f0, f0 = 215 Hz, S0 = 1.0 × 10−49 Hz−1, and
flower = 20 Hz. The PSD of the advanced Virgo is given by
[
45
]
Sh ( f ) = S0 2.67 × 10−7x −5.6 + 0.59x −4.1 exp(−α)
+ 0.68x 5.34 exp(−β) ,
where x = f / f0, f0 = 720 Hz, S0 = 1.0 × 10−47 Hz−1,
α = (ln x )2[3.2 + 1.08 ln x + 0.13(ln x )2], β = 0.73(ln x )2
and flower = 20 Hz.
We consider the merge of binary black hole systems as
the source of GW. The mass of each black hole is assumed
to be uniformly distributed in the range [
3, 100
]M . We also
require that the mass difference of two component black holes
is not to large, and restrict the mass ratio q = m1/m2 in the
range [0.5, 2.0]. Furthermore, we assume that the sources are
randomly distributed in the sky. Taking the time evolution of
the burst rate into consideration, the probability distribution
function of GW sources reads [
36
]
P(z) ∝
4π dC2 (z)R(z)
H (z)(1 + z)
,
where H (z) = H0 M (1 + z)3 + 1 − M is the Hubble
parameter, dC = 0z 1/H (z)d z is the comoving distance,
R(z) = 1 + 2z for z 1, R(z) = (15 − 3z)/4 for 1 < z < 5,
and R(z) = 0 otherwise. The advanced LIGO and Virgo is
expected to be able to detect GW signals produced by the
merge of binary black hole at reshift z ∼ 1 if it reaches to
the ultimately designed sensitivity [
43
].
We simulate a set of GW events, each of which contains
the parameters (z, α, δ, μ, σμ) that we are interested in. Since
116.50◦ and ζH = ζL = ζV = 90◦. The symbols have been
clarified in the last section, and the subscripts ‘H’, ‘L’ and
‘V’ stand for Hanford, Livingston and Virgo respectively.
Since Hanford locates at the west of Livingston by longitude
difference 28.64◦, the local sidereal time of Hanford is later
than that of Livingston by 28.64◦ (corresponding to 1.91 h),
i.e.
(17)
(18)
,
(19)
(20)
(21)
we will use the simulated data to study the anisotropy of the
universe, the sky position of GW source (α, δ) are required.
The detailed procedures of simulation are as follows:
1. Sample black hole mass from uniform distribution,
m1, m2 ∈ U [
3, 100
]M .
2. If 0.5 < m1/m2 < 2.0, sample redshift z from the
probability distribution function (21); else go back to step 1.
3. Sample the sky position (α, δ) from the uniform
distribution on 2-dimensional sphere. This can be done by
sampling α ∈ U [0, 2π ], x ∈ U [
− 1, 1
], and setting
δ = arcsin x .
4. Sample (φr + r t )H from U (0, 2π ), and calculate (φr +
r t )L and (φr + r t )V from Eqs. (17) and (18).
5. Calculate the SNR of each detector ρH , ρL and ρV using
Eq. (9), and the combined SNR ρ using Eq. (11).
6. If ρ > 8, calculate the fiducial luminosity distance dL and
its uncertainty σdL from Eqs. (8) and (13), respectively;
else go back to step 1.
7. Convert dL and σdL to μ and σμ using Eqs. (14) and
(15). Calculate the anisotropic distance modulus μ using
Eq. (16).
8. Sample the simulated distance modulus from Gaussian
distribution μsim ∼ G(μ, σμ). Go back to step 1, until
the desired number of GW events are obtained.
Some issues should be clarified. Since we only consider
the case where the binary’s orbital plane is nearly face on, our
result does not depend on the polarization angle ψ . The GW
events can happen at any time during the run of detectors. We
assume that one GW event is uncorrelated with other GW
events, and the GW events are uniformly distributed with
time. Due to the periodicity of the diurnal motion, we can
sample (φr + r t )H from U (0, 2π ), whereas (φr + r t )L
and (φr + r t )V are determined by the difference of local
sidereal times of detectors’ locations. To ensure the
significance of GW signals, we require the combined SNR of
three detectors to be larger than 8. In step 7, when
calculating the anisotropic distance modulus, we choose the
fiducial dipole amplitude to be the same to Union2.1, i.e.
d = 1.2 × 10−3 [
24
]. The fiducial direction is arbitrarily
chosen to be (α0, δ0) = (310.6◦, − 13.0◦), which is also the
same to Union2.1 but changed from the galactic coordinates
to equatorial coordinates.
4 Constrain on anisotropy
In this section we use the simulated GW data to constrain
the anisotropy of the universe. We want to see, with current
accuracy of detectors, how many GW events are needed in
order to correctly reproduce the fiducial dipole amplitude
and direction. We simulate N = 100, 200, 300, . . . , 900 data
2.4 ×10-3
2.2
2
points, respectively. For each N we repeat the simulation
1000 times and calculate the average dipole amplitude and
preferred direction.
In Fig. 1, we plot the average dipole amplitude as a
function of the number of GW events N . The central value is
the mean of dipole amplitudes in 1000 simulations, and the
error bar is the root mean square of uncertainty of dipole
amplitudes in 1000 simulations. The dipole amplitude of
Union2.1 is also plotted for comparison. From this figure,
we can see that, as the number of GW events increases, the
constrained dipole amplitude gets more close to the
fiducial dipole amplitude, and at the same time the uncertainty
is reduced. To reach the accuracy of Union2.1, 400 GW
events are needed.
To see the constraint ability of GWs on the preferred
direction, in Fig. 2 we plot the average 1σ confidence region in the
(α0, δ0) plane for different N . For comparison the 1σ
confidence region of Union2.1 is also plotted. This figure shows
that ∼ 400 GW events are not enough to tightly constrain the
preferred direction. More than 800 events are needed in order
to reach the accuracy of Union2.1. As is expected, increasing
the number of GW events can tighten the constraint.
Figure 3 shows the distribution of dipole amplitudes in
1000 simulations when N = 800. The distribution can be
fitted by Gaussian function centering at 1.28 × 10−3, with
the standard deviation σd = 0.31 × 10−3. This implies that
the fiducial dipole amplitude can be correctly reproduced
with ∼ 800 GW events.
Figure 4 shows the distribution of preferred directions in
1000 simulations when N = 800. The red and blue error
ellipses are the 1σ and 2σ confidence regions of Unin2.1,
respectively. From this figures, we can see that the simulated
directions are clustered near the fiducial direction. The
probabilities of falling into the 1σ and 2σ confidence regions of
Union2.1 are 32.4 and 75.4% respectively, implying that with
about 800 GW events the preferred direction can be correctly
recovered.
In order to test if the statistical noise can lead to pseudo
anisotropic signals, we simulate some isotropic data sets.
The simulation procedure is the same to the anisotropic case
except that the fiducial dipole amplitude is fixed to zero now.
The simulated data is then fitted by the dipole model of
Eq. (16). As was done above, we simulate different number
of GW events, repeat the simulation 1000 times, and
calculate the mean value of dipole amplitude. We expect that the
fitted dipole amplitude is consistent with zero. The results
are shown in Figs. 5 and 6.
-60
-90
1.8
1.6
1.4
1.2
e
iltud 1
pm0.8
A
0.6
0.4
0.2
00
0.16
0.14
0.12
100 200 300 400 500 600 700 800 900 1000
Number of points
Fig. 5 The same to Fig. 1 but with the isotropic GW data
P (θ < 1σ) = 32.4%
P (θ < 2σ) = 75.4%
0
30
60
In Fig. 5, we plot the average dipole amplitude and 1σ
uncertainty as the function of the number of GW events
N , together with the dipole amplitude and its uncertainty of
Union2.1 for comparison. From the figure we may see that
the dipole amplitude decreases as the number of GW events
increases, but it is not zero within 1σ uncertainty even if
the number of GW events increases to 900. This means that
the noise may lead to pseudo anisotropic signal. However,
the amplitude of pseudo anisotropic signal is always smaller
than the dipole amplitude of Union2.1 if N > 200. With
∼ 800 GW events, the former is smaller than the latter at
about 2σ confidence level.
In Fig. 6, we plot the distribution of dipole amplitudes
in 1000 simulations in N = 800 case. The distribution
is well fitted by Gaussian function, with the average value
d¯ = 0.50 × 10−3 and standard deviation σd = 0.22 × 10−3.
The probability of pseudo dipole amplitude being larger than
the dipole of Union2.1 is only 0.9 percent. Therefore, if
the universe really has an anisotropy with amplitude larger
1 × 10−3, this anisotropic signal can be tested by ∼ 800 GW
events.
5 Discussions and conclusions
In this paper, we have investigated the constraint ability of
GW events on the anisotropy of the universe using the
simulated data from advanced LIGO and Virgo. It is found that
the GW data can tightly constrain the anisotropy amplitudes
of the universe with 400 events if advanced LIGO and
Virgo reach to the designed sensitivity. To tightly constrain
the preferred direction, however, 800 events are needed.
The simulated GW events have average uncertainty 0.4 mag
on distance modulus, which is about two factors larger than
the Union2.1 compilation of type-Ia supernovae. Here we
only considered the GW signals in the inspiral epoch. The
GW frequency in the inspiral epoch is about tens Hz, which
is bellow the most sensitive frequency of the detectors. The
uncertainty can be reduced by consider the GW signals in
the merger and ringdown epochs. Here we have assumed
that the GW source can be precisely localized and the orbital
plane of inspiral is nearly face on. Otherwise the distance of
GW source may be correlated with other parameters and the
accuracy on the determination of distance gets worse.
Therefore, in practise more GW events may be needed in order to
achieve the accuracy of Union 2.1. As the improvement of
sensitivity, advanced LIGO is expected to detect hundreds
of GW events produced by the coalescence of binary black
hole systems in the next few years. Therefore, GWs provide
a promising complementary tool to supernovae in testing the
anisotropy of the universe.
In this paper, we only considered the binary black hole
systems as the sources of GWs, while the binary neutron star
and binary of neutron star – black hole systems are not
considered. This is because the binary of neutron star – black hole
systems have not been observed yet, and with the designed
sensitivity advanced LIGO and Virgo can only observe the
binary neutron star systems at very low redshift. The biggest
challenge of using GWs as the standard sirens comes from
the localization of GW source. With two detectors, the source
can only be localized on a strip of the sky. Even if with three
or more detectors, the localization accuracy is at the order
of several degrees with present sensitivity. Such an accuracy
is far from accurate enough to identify the host galaxy, thus
hampers the measurement of redshift. If the GW has
electromagnetic counterparts such as short gamma-ray bursts,
then the host galaxy can be identified and the redshift can be
determined accurately by the follow-up observations.
Unfortunately, the merge of binary black hole is expected to have
no electromagnetic counterparts. Chen [
46
] pointed out that
the redshift can be obtained statistically by analyzing over all
potential host galaxies within the localization volume. The
redshift inferred in this way, however, adds additional
uncertainty to the constraints. This disadvantage of the
measurement of redshift can be improved by the on-going third
generation detectors with higher sensitivity such as the Einstein
Telescope [
47
]. However, at present, our researches provide
a possible approach to test the standard cosmological model
by the advanced LIGO and Virgo detectors in the next years.
Acknowledgements This work has been supported by the National
Natural Science Fund of China under grant Nos. 11603005 and
11775038, and the Fundamental Research Funds for the Central
Universities project Nos. 106112017CDJXFLX0014 and 106112016
CDJXY300002.
Open Access This article is distributed under the terms of the Creative
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to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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