A three-dimensional laser interferometer gravitational-wave detector

The gravitational wave (GW) has opened a new window to the universe beyond the electromagnetic spectrum. Since 2015, dozens of GW events have been caught by the ground-based GW detectors through laser interferometry. However, all the ground-based detectors are L-shaped Michelson interferometers, with very limited directional response to GW. Here we propose a three-dimensional (3-D) laser interferometer detector in the shape of a regular triangular pyramid, which has more spherically symmetric antenna pattern. Moreover, the new configuration corresponds to much stronger constraints on parameters of GW sources, and is capable of constructing null-streams to get rid of the signal-like noise events. A 3-D detector of kilometer scale of such kind would shed new light on the joint search of GW and electromagnetic emission.


Scientific RepoRtS
| (2020) 10:16285 | https://doi.org/10.1038/s41598-020-72850-6 www.nature.com/scientificreports/ pattern functions Such a 3-D laser interferometer can greatly improve the detector's response to GW strain. As an interferometer has different sensitivity to GWs from different directions, a 3-D detector owns specific pattern functions deviating a L-shaped one. The response of an interferometer to GWs is the difference between relative length changes of the two arms which can be computed from the formula h = 1 2 (n 1 · [Hn 1 ] − n 2 · [Hn 2 ]) 9 , where n 1 and n 2 denote the unit vectors of arm direction. The three arms of the 3-D interferometer are along the x, y and z axes defining the coordinate system of the detector. Another reference frame (x ′ , y ′ , z ′ ) represents the GW's coordinate, in which the z ′ axis stands for the propagation direction of GW. Then in the frame (x ′ , y ′ , z ′ ) GW has the form, The frame (x ′ , y ′ , z ′ ) can be achieved through the rotation of the frame (x, y, z), with a rotation matrix of (1)   Assuming ψ = 0 , the pattern functions for the (+) polarization are illustrated by a spherical polar plot, as is shown in Fig. 3. The response of a tradiational L-shaped detector to the (+) polarization is as shown in panel a of Fig. 3. In contrast, the total response of the 3-D detector to the (+) polarization is as shown in panel d of Fig. 3, apparently the response to the (+) polarized GW in directions perpendicular to Z axis is strengthened. Consequently, the 3-D detector can provide a more isotropic antenna pattern than the conventional L-shaped detector (as shown in the panel a of Fig. 3.), which corresponds to wider field of view to GW sources comparing to previous detectors.

Benefits
An important benefit that the 3-D detector can bring about is the construction of null-streams 10 . A troublesome problem encountered GW detection is how to distinguish the signals from the noise events. A simple but powerful method called construction of null-streams can solve such a problem, in which all the data are linearly combined in order to eliminate the GW signals. Then the output of null-streams will only be noise. Comparing with the pattern functions, one can easily find the relation which shows that the 3-D detector is redundant and null-streams can be generated by the linear combination of the output from three sub-interferometers.
Another attractive benefit of such a 3-D detector is more stringent constraints on parameters of GW sources comparing those given by previous detecors. As we know, an individual GW signal (non-spinning) depends on nine parameters: two masses M z and µ z , position angles θ and φ , orientation angles of the binary ι and ψ , time at coalescence t c , phase at coalescence c , and luminosity distance D L . Measuring the GW phase can determine for X and Y arms,  Fig. 4. Due to the extra pattern functions presented in the 3-D interferometer, the positional uncertainty can be significantly reduced comparing with the traditional L-shaped interferometers. In particular, the position of the GW source is restricted into four regions by the 3-D detector so that the position angles will become much more explicit. While a complete reconstruction of the parameters of GW sources will need a joint observation of another detector at a different location.

Sensitivity estimation
The level of total noise determines the weakest GW signals detectable. As a matter of fact, the detector will suffer several fundamental noises, including quantum noise 12 , thermal noise 13 , seismic noise 14 , and gravity gradient noise 15 et al. Here we estimate the sensitivity limit of the 3-D detector based on the idealized parameters which are likely to come true in the future.
Referring to other laser interferometer detectors, the parameters of the 3-D detector are assumed as follows. The length of each arm is set as 8 km, the mass of each mirror 200 kg, the loss angle of the coating 5 × 10 −5 , the loss angle of the substrate 5 × 10 −9 . And the whole system should be operated at temperature 290 K. Moreover, each mirror is suspended by the quadruple pendulum with the resonant frequency of 10 Hz and loss angle of 10 −9 . Besides, a Fabry-Perot cavity with a fineness of 1000 is placed in each arm, and the laser power in the cavity will be 2 MW. In this case, the corresponding noise estimation can be calculated, as shown by the black curve in Fig. 5. In such a case, the sensitivity of the 3-D detector is improved roughly by a factor of 10 in contrast with the designed sensitivity of Advanced LIGO in high frequency range 4 . The strain noise curve in Fig. 5 corresponds to the binary neutron star (BNS) range around 1000 Mpc.

Discussion
The high sensitivity of such a detector can support broader scientific goals relative to the second generation detectors. The higher signal-to-noise ratio (SNR) and higher probability of identification of GW events will be useful in answering questions, like the origin and evolution of black hole and the inner structure of NS. The detector may even detect the signals from new astrophysical sources like core collapse supernovae and isolated rotating NSs. Because of its high sensitivity in high frequency band, the detection of post-merger signals spanning from 1 to several kHz from the BNSs is expected. Undoubtedly, the detection of the peak frequencies of the post-merger stage can play a crucial role in constraining the equation of state of neutron stars [16][17][18] . While, the behavior of such a detector at low frequency is not as good as at high frequency due to the thermal noise. To suppress the low frequency noise, one may consider the xylophone configuration by adding another instrument operated with low power and cryogenic mirrors 19 .
As for the implementation, the construction of a kilometer-scale 3-D detector and keeping the stability of such a detector maybe a tough problem to handle. Here we put forward a plausible way to build the detector by taking advantage of the mountain's terrain, then two arms in each direction will be packaged by a tunnel, and the tunnel can be extended to under-ground to accommodate long arms needed by this design. However, this scheme may have some issues related with seismic noise and gravity noise which mainly dominate the low frequency range. The geometry of the 3-D detector is far different from the traditional L-shaped detector. Therefore, a 10-m prototype may be useful to test the new way of suspension and the alignment control system.
Although the triangular shaped ET can also provide the null-streams and strong constraints on the parameters of the GW sources, the 3-D detector can have a more isotropic antenna pattern excepting for few blind spots compared with ET. Due to the short duration of the merged signals (up to several minutes), it is not realistic to only count on the earth rotation to widen the field of view. Therefore, a more isotropic antenna pattern is www.nature.com/scientificreports/ important in the increase of detection rate. In addition, the opening angle of the sub-interferometers in ET is 60 • , while for the 3-D is 90 • , which resulting in the SNR ET SNR 3D = sin(60 • ) ≈ 0.87 if assuming the same arm length. So the 3-D detector can achieve more isotropic antenna pattern without losing sensitivity.

conclusion
We propose a 3-D GW detector with three Michelson interferometers setting in a regular triangular pyramid. The original motivation of the geometry is that such a detector would have a more spherically symmetric antenna pattern. In this paper, we have shown that the detector is fully redundant and able to generate null-streams by data from three sub-interferometers. And the detector can also provide much stronger constraints on the parameters of GW sources. So a network of the 3-D detectors will make the detection and parameter estimation more efficient and accurate. As a consequence, our understanding of fundamental physics will be enhanced with the combination of the electromagnetic signals and accurate GWs parameters at the same time.