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

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## Abstract

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 \(\gtrsim 400\) GW events are observed. As for the preferred direction, \(\gtrsim 800\) 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 (\(\Lambda \)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, 13, 14, 15], the anisotropy of the distance-redshift relation of type-Ia supernovae [16, 17, 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, 20, 21, 22, 23, 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, 27, 28, 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, 31, 32]. The host galaxy NGC4993 at redshift \(z\sim 0.01\) 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 \(70.0_{-\,8.0}^{+12.0}~\mathrm{km~s}^{-1}~\mathrm{Mpc}^{-1}\) [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, 36, 37, 38, 39, 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 \(\Lambda \)CDM model. Therefore, throughout this paper we assume a fiducial flat \(\Lambda \)CDM model with Planck parameters \(\Omega _M=0.308\) and \(H_0=67.8~\mathrm{km~s}^{-1}~\mathrm{Mpc}^{-1}\) [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

*z*, the chirp mass in the observer frame is given by \(\mathcal {M}_{c,\mathrm{obs}}=(1+z)\mathcal {M}_{c,\mathrm{com}}\) [42]. In the following, \(\mathcal {M}_c\) 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]

*N*independent detectors form a network and detect the same GW source simultaneously, the combined SNR is given by

*d*is the dipole amplitude and \(\theta \) is the angle between GW source and the preferred direction of the universe. The preferred direction can be parameterized as \((\alpha _0,\delta _0)\) in the equatorial coordinate system. The three parameters \((d,\alpha _0,\delta _0)\) can be obtained by fitting the GW data to Eq. (16) using the least-\(\chi ^2\) method.

## 3 Simulation from advanced LIGO and Virgo

- 1.
Sample black hole mass from uniform distribution, \(m_1,m_2\in U[3,100]M_\odot \).

- 2.
If \(0.5<m_1/m_2<2.0\), sample redshift

*z*from the probability distribution function (21); else go back to step 1. - 3.
Sample the sky position (\(\alpha ,\delta \)) from the uniform distribution on 2-dimensional sphere. This can be done by sampling \(\alpha \in U[0,2\pi ]\), \(x\in U[-\,1,1]\), and setting \(\delta =\arcsin x\).

- 4.
Sample \((\phi _r+\Omega _rt)_H\) from \(U(0,2\pi )\), and calculate \((\phi _r+\Omega _rt)_L\) and \((\phi _r+\Omega _rt)_V\) from Eqs. (17) and (18).

- 5.
Calculate the SNR of each detector \(\rho _H\), \(\rho _L\) and \(\rho _V\) using Eq. (9), and the combined SNR \(\rho \) using Eq. (11).

- 6.
If \(\rho >8\), calculate the fiducial luminosity distance \(d_L\) and its uncertainty \(\sigma _{d_L}\) from Eqs. (8) and (13), respectively; else go back to step 1.

- 7.
Convert \(d_L\) and \(\sigma _{d_L}\) to \(\mu \) and \(\sigma _\mu \) using Eqs. (14) and (15). Calculate the anisotropic distance modulus \(\mu \) using Eq. (16).

- 8.
Sample the simulated distance modulus from Gaussian distribution \(\mu _\mathrm{sim}\sim \mathcal {G}(\mu ,\sigma _\mu )\). Go back to step 1, until the desired number of GW events are obtained.

## 4 Constrain on anisotropy

*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, \(\gtrsim 400\) GW events are needed.

To see the constraint ability of GWs on the preferred direction, in Fig. 2 we plot the average \(1\sigma \) confidence region in the \((\alpha _0,\delta _0)\) plane for different *N*. For comparison the \(1\sigma \) confidence region of Union2.1 is also plotted. This figure shows that \(\sim 400\) 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 4 shows the distribution of preferred directions in 1000 simulations when \(N=800\). The red and blue error ellipses are the \(1\sigma \) and \(2\sigma \) 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 \(1\sigma \) and \(2\sigma \) 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 Fig. 5, we plot the average dipole amplitude and \(1\sigma \) 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 \(1\sigma \) 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 \(\sim 800\) GW events, the former is smaller than the latter at about \(2\sigma \) 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 \(\bar{d}=0.50\times 10^{-3}\) and standard deviation \(\sigma _d=0.22\times 10^{-3}\). 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 \(1\times 10^{-3}\), this anisotropic signal can be tested by \(\sim 800\) 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 \(\gtrsim 400\) events if advanced LIGO and Virgo reach to the designed sensitivity. To tightly constrain the preferred direction, however, \(\gtrsim 800\) 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.

## Notes

### 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 106112016CDJXY300002.

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