Abstract
In this perspective, we outline that a space borne gravitational wave detector network combining LISA and Taiji can be used to measure the Hubble constant with an uncertainty less than 0.5% in ten years, compared with the network of the ground based gravitational wave detectors which can measure the Hubble constant within a 2% uncertainty in the next five years by the standard siren method. Taiji is a Chinese space borne gravitational wave detection mission planned for launch in the early 2030 s. The pilot satellite mission Taiji1 has been launched in August 2019 to verify the feasibility of Taiji. The results of a few technologies tested on Taiji1 are presented in this paper.
Introduction
The observation of gravitational waves (GWs) enables us to explore the Universe in more details than that is currently known. By testing the theory of general relativity, it can unveil the nature of gravity. In particular, a GW can be used to determine the Hubble constant by a standard siren method^{1,2}. This method^{3} was first used by the Advanced LIGO^{4} and Virgo^{5} observatories when they discovered GW event GW170817^{6}. Despite the degeneracy problem in the groundbased GW detectors, the Hubble constant can reach a precision of 2% after a 5year observation with the network of the current surface GW detectors^{6}, although LIGO’s O3 data have shown that the chance to detect electromagnetic (EM) counterpart might be a little optimistic^{7}.
In this paper, we discuss a method to further improve the fractional uncertainty of the Hubble constant to a precision <1% by the spaceborne GW antennas. The improvement not only comes from that a spaceborne GW antenna such LISA^{8,9} and Taiji^{10,11,12,13} can avoid the degeneracy problem because of its orbital motion^{14} but also the precision of a GW source’s position and its luminosity distance can be improved by 2–3 orders of magnitude by the LISA–Taiji network^{15}, compared to the individual antenna such as LISA or Taiji. It requires a 1year overlap of LISA and Taiji missions to achieve this precision.
The joint measurement requires that the Taiji scientific collaboration establish a threestep plan to guarantee the launch of Taiji in the 2030s^{13}. The first step of this new observatory is to launch a pilot satellite, Taiji1, to prepare the necessary technology for the second step, Taiji pathfinder, also called Taiji2. Consisting of two satellites, Taiji2 will be used to demonstrate the Taiji technology around 2023–2025.
Launched in 2019, Taiji1 accomplished multiple tasks^{12,13}. It studied the manufacturing process for the Taiji payload, the onorbital working sequence of the Taiji payload, the data processing stream for the Taiji mission, and the feasibility of a few Taiji key technologies in space. In this article, we will also report the main results of the payload test for Taiji1.
The Hubble constant and the standard siren method
By observing the redshift of a specific spectral line of a remote galaxy, it has been found that the further away a galaxy is from us, the faster it appears to recede^{16}. This observation is known as Hubble’s law
Hubble’s law is interpreted as an evidence that the universe has been expanding since the Big Bang occurred ~13.8 billion years ago^{17}. In Eq. 1, H_{0} is the mean expansion rate of the Universe, called the Hubble constant, V_{H} is the receding velocity of the galaxy, and d is the distance of the receding galaxy. According to the Friedmann–Robertson–Walker cosmological model (FRW model), the dynamics of our expanding universe are governed by the density and the curvature of our universe. By measuring the Hubble constant, one can deduce the age, the size, the current state, and even the fate of our Universe^{18}. In general, there are two primary methods to measure the Hubble constant. The first way to estimate the distance is to exploit the socalled “standard candles”—Cepheid variable stars or type 1a supernovae that are at the same luminosity. The data from the first method^{19} implies that the value of H_{0} = (74.03 ± 1.42) km s^{–1} Mpc^{–1} at 68% confidence level. The other method focuses on the cosmic microwave background, and it examines how the cosmic microwave background has evolved over time. The data from the cosmic microwave background^{20} indicates that the value of H_{0} = (67.4 ± 0.5) km s^{–1} Mpc^{–1} at 68% confidence level. The mismatch between the two measurements needs to be explained since the values should agree if the models are correct.
A new and independent determination of the Hubble constant using a GW as a standard siren^{1,2} may solve this cosmic riddle. In 2017, the Advanced LIGO and Virgo detectors observed a GW signal (event GW170817) from the merger of a binary neutronstar system^{21}. The EM followup measurements of the area were then sequentially observed, which triggered the first “multimessenger” astronomical observation^{22}. With the absolute distance to the source being determined directly from the GW measurements, GW170817 was used as a “standard siren” to measure the Hubble constant^{5}. The Hubble constant was found in this case to be \(70.0_{  8.0}^{ + 12.0}\) km s^{−1} Mpc^{−1}. The uncertainty in the Hubble constant measurement (Eq. 1) largely comes from the inaccuracy of the absolute distance evaluation. The receding velocity, represented by a galaxy’s redshift, can be measured precisely by taking the spectra of the galaxy. For a groundbased GW observatory, such as the LIGO and Virgo detectors, the degeneracy between the distance D_{L} and the inclination of the GW measurement^{2} (Eqs. 2 and 3) results in a faceon or faceoff binary far away, which has a similar gravitationalwave amplitude to that of a close edgeon binary. This degeneracy contaminates the precision of the distance measurement. For simplicity, we use the geometrized unit system where c = G = 1.
where h_{+} and h_{×} are the strengths of the GW signal of two different polarizations, M_{z} is the redshifted chirp mass, f is the wave frequency, Φ is the phase of GW, \({\hat{\boldsymbol{L}}}\) is the unit vector of the source’s angular momentum, and \({\hat{\boldsymbol{n}}}\) is the unit vector pointing to the direction of the source. Taking into account more standard sirens, it leads to a N^{−1/2} convergence to the uncertainty of the Hubble constant, where N is the number of binary neutronstar mergers^{23}. It is predicted that, when more standard sirens are detected and another groundbased detector joins the network, the fractional uncertainty of the Hubble constant determined by the groundbased GW detectors will reach 2% within 5 years^{6}. This result is slightly better than that of the standard candle method.^{19}
A spaceborne GW antenna such as LISA and Taiji can avoid the degeneracy problem by virtue of its orbital motion. (For a detailed discussion of the orbital configuration for the spaceborne antenna, please refer to refs. ^{9,10,11}.) When a spaceborne GW antenna is orbiting around the sun, the position and the orientation of the source relative to the antenna are gradually changing. The motion of the detector thus modulates the measured signal and its modulation depends on the position and the orientation of the source. As a result, the distance and the inclination of a GW source are no longer degenerate^{2,8}. This reduction in ambiguity increases the space antenna’s ability to determine the luminosity distance.
For instance, let us assume a binary black hole GW source is randomly distributed in the universe, its redshift is not >1, and its total mass is <10^{6} solar masses. Then, there is a 90% likelihood that an individual space antenna can localize the GW source with an error given by δD_{L}/D_{L} <8% for the fractional distance and δΩ <4 deg^{2} for the orientation^{2,14,15}. For an individual antenna, without the EM counterparts of a GW source, the entanglement between the luminosity distance and the orientation will limit the precision to determine the distance^{2,8}. The fractional distance precision will be improved dramatically if the sky position of that GW source can be pinpointed. Considering the same type of GW sources discussed above, the distance error can be greatly reduced to 0.5% when the correlation between the distance and the orientation is combined^{2}. One way to pinpoint the GW source is to find its EM counterpart^{24,25}. Due to the poor understanding of the relation between the binary black hole merger GW event and its EM counterpart, it becomes difficult to find the EM counterpart either in advance or simultaneously^{26}.
Determining the Hubble constant with LISA–Taiji network
It was recently calculated that by the LISA–Taiji network, the localization of GW sources can be improved significantly without an EM counterpart^{15}. Taking the above example, the orientation uncertainty of such a GW source can be reduced to δΩ <0.005 deg^{2}. Consequently, the fractional distance precision will be improved to δD_{L}/D_{L} <0.5%. However, the EM counterpart is still important, as it can provide a redshift that is essential to calculate the Hubble constant. With such precise localization of a GW source, it will be relatively easy to discover its counterpart galaxy. The distributed density of a counterpart galaxy can be expressed as^{27,28}
with R as the comoving distance and R_{*} as the Hubble distance. If we only consider a small redshift such as z < 1, the exponential part of Eq. (4) will always be ~1. The projected number density dN/dΩ is ~300 galaxies/arcmin^{2} given by the Hubble Deep Field^{29}. We can normalize Eq. (4) by integrating it into projected number density, which should be <300 galaxies/arcmin^{2}. Then, we have
where R_{0} is the distance at z = 1. By assuming a cosmological model, we can convert the measured luminosity distance and its error to any other desired cosmic distance measure. By multiplying Eq. (5) with the GW error cube, the number of galaxies within the error cube can be derived. With δΩ <0.005 deg^{2} and δD_{L}/D_{L} <0.5%, the number of candidate host galaxies for a GW source with z < 1 is no >54. According to the redshiftapparent magnitude relation, the apparent magnitude of galaxies at distance z = 1 is between 24 and 25. Typically, for the spectroscopic measurement, the limiting magnitude for the device should be 3–4 magnitude greater, say 27–29, which challenges all the existing telescopes. Fortunately, future telescope such as LSST and WFIRST, assuming 2 years observation, could reach limiting magnitude of 27 and 29, respectively^{30,31}. Thus, all the candidate galaxies could be traced by a fiber spectrograph on an LSSTlike or WFIRSTlike telescope. With such a lower number, it is probable that an EM event is correlated with a GW event. In some particular cases, the spininduced precession effects may allow certain degeneracy to be broken and the analysis can achieve 1 arcmin^{−2} pointing accuracy^{32}. In such cases, the number of candidate galaxies can be reduced to a few. Thus, we can technically identify the counterpart galaxy of the GW source. Once the host galaxy of the GW source is identified, the redshift can be determined by the EM observation. With the distance of a GW source measured, the uncertainty of the Hubble constant δH_{0}/H_{0} will now be <0.5%.
Towards LISA–Taiji network
The LISA–Taiji network requires at least a 1year overlap to realize the above purpose, which means Taiji needs to advance its schedule to match LISA’s availability to collect data. As the pioneer, LISA pathfinder has been launched in 2015. The mission is a technology demonstration and has achieved great success^{33}. LISA pathfinder has paved the way for the full LISA^{34} project, which will start operating in orbit ~2032–2034. Taiji, the Chinese spaceborne GW detection mission, which has a heliocentric orbit similar to LISA, has established a threestep plan to launch in early 2030s.
Thus, it will have an overlap with LISA’s operating time^{12,13}. The first step has been accomplished by launching a pilot study satellite known as Taiji1 satellite in 2019. The second step is to launch the Taiji pathfinder (also called Taiji2) no later than 2025. Taiji2 consists of two satellites, which are planned to demonstrate most technology of Taiji and to pave the way for the full Taiji project. The final step is to launch Taiji, which is similar to the LISA constellation in 2030s. Taiji (also called Taiji3) consists of three identical satellites. The distance between the different pairs of two satellites is three million kilometers^{12,13}.
Taiji pathfinder, consists of two satellites, will be equipped with more technology related to the intersatellite laser link compared to LISA pathfinder. The high technical requirement makes the Taiji pathfinder challenging^{13}. As a completely new mission in this field, directly launching Taiji2 seems to be a quite risky task. Thus, a pilot study satellite mission Taiji1 was approved in 2018 not only to prepare the necessary technology for Taiji2 but also to verify the performance capability of Taiji mission. Taiji1 also serves as a benchmark to testify the feasibility of Taiji’s threestep plan.
Taiji1, the first step of China’s efforts
Approved on 30 August 2018 and set to fly on 31 August 2019, Taiji1, a 180 kg satellite, was a successful and quick mission. The orbit of Taiji1 was a circular Sunsynchronous dawn/dusk orbit that was inclined at an angle of 97.69 deg. The orbit provided a relatively stable sunfacing angle, which ensured that the battery could always be constantly charged and that the temperature of the satellite should not fluctuate drastically. The orbit altitude was chosen to be 600 km, a tradeoff between the launching costs and the air drag. Similar to LISA pathfinder^{33}, two major technology units were tested on Taiji1: the optical metrology system^{35} and the dragfree control system^{36}. Due to the shortterm development circle and the limited budget, the payload design was highly simplified (Fig. 1a, b). The optical metrology system consisted of an optical bench, a phasemeter and two laser sources. The dragfree control system was composed of a gravitational reference sensor (GRS) (it consisted of both a sensor head and the corresponding sensor electronics), a dragfree controller, and two types of micropropulsion systems. Figure 1 shows the distribution of the payload in Taiji1.
Taiji1 used two NdYAG lasers (first and second laser in Fig. 1a) with a wavelength of 1064.5 nm. Only one laser was working during the measurement process. The optical metrology system could switch one laser to the other under the command. The two laser beams were delivered to an optical bench by two fiber couplers (Fig. 1c). There was a frequency difference of ~1 kHz between the two delivered beams. Except for the reference interferometer, the optical bench contained two primary interferometers. The first was test mass interferometer (T.M. int.). One of the laser beams was aimed at a test mass and reflected to an optical bench (Fig. 1c). This unit measured the test mass motion. The other interferometer, called optical bench interferometer (O.B. int.), was used to monitor the optical bench noise. All of the interferometric beat notes were sensed and converted into sinusoidal voltages by the photodetectors. The phases of the sinusoidal voltages were decoded by the phasemeter^{37}.
By the data from the phasemeter, the precision of the two primary interferometers can be derived by δL = δφ·λ·(2π)^{−1/2}, where δL is the precision of the interferometer, δφ denotes the phase noise of phasemeter data, and λ is the laser wavelength.
The GRS in Taiji1, served as an accelerometer^{38}, is composed of a sensor head and the corresponding electronics. The sensor head consists of a cage and a test mass (Fig. 1c). The GRS has three axes, one nonsensitive axis and two sensitive axes (Fig. 1c). The nonsensitive axis points to the earth and the drag free is used along nonsensitive axis. The first sensitive axis is along flight direction.
By capacitive sensing, the GRS measured the disturbing acceleration of Taiji1. The data were sent to a dragfree controller. The controller then commanded the thruster to exert forces to compensate the disturbing force experienced by Taiji1. Two different types of thrusters were tested: a radio frequency ion thruster and a Hall effect thruster. Each type has four individual thrusters. They are assembled symmetrically on both sides of the satellite (Fig. 1a). These two types of thrusters, like the two lasers, backed up each other.
During the mission, all of the payloads were tested. All results fulfilled the mission requirement. Some of the measurements were shown in Fig. 2. For the T.M. int. and the O.B. int. (different lasers were used for each), the precision evaluated in the frequency band between 0.01 and 10 Hz was found to be <1 nm Hz^{−1/2}. For some frequency bins, the precision could reach 25 pm Hz^{−1/2} (Fig. 2a, b). The Taiji1 GRS noise was taken from the second sensitive axis. The dynamic range of the second sensitive axis was ±5.3 × 10^{−5} m s^{−2} and the acceleration noise of this axis measured by the readout voltage fluctuation was 10^{−10} m s^{−2} Hz^{−1/2} (Fig. 2c). The disturbance accelerations of the three axes of the Taiji1 satellite readout by Taiji1 GRS were shown in Fig. 2d. The nonsensitive axis was earth pointing, and the noise was mainly dominated by the GRS readout noise. While the first sensitive axis was along the flight direction, this noise was considered mainly caused by the air drag. The second sensitive axis was the orbit plane normal direction, and the acceleration measured by the GRS was <2 × 10^{−9} m s^{−2} Hz^{−1/2}. The noise of both thrusters measured by the GRS was found to be <1 μN Hz^{−1/2} (Fig. 2e) which was believed to be dominated by the GRS readout noise. The noise of the thrusters could also be calibrated by the data of the ion acceleration voltage, gas pressure at the supply valve, and the temperature around the thruster. By this method, the true thruster noise of the radio frequency ion thruster was derived as ~0.15 μN Hz^{−1/2} (Fig. 2e).
A dragfree control experiment was also performed along the nonsensitive axis of the GRS. We used the thrusters on one side to exert a sinusoidal force (the modulated peak in Fig. 2f), and the feedback controlled the thruster on the other side to compensate. The respective spectra densities were shown in Fig. 2f. The sinusoidal force (the modulated peak in Fig. 2f.) was well suppressed by using dragfree control, and the residue acceleration of the satellite was <10^{−8} m s^{−2} Hz^{−1/2}. The stability of the temperature control was ~±2.6 mK. A more detailed analysis of Taiji1 payload testing and the improved results would be presented in a special issue to be published soon.
The GRS noise induced by the voltage fluctuation is always proportional to its dynamic range^{39} (Eqs. 6 and 7), where S is the area of the capacitor plate, m is the mass of test mass, d is the distance of the capacitor, V_{p} is the preload voltage of the capacitor, V_{r} is the readout voltage, δV_{p} is the preload voltage noise, δV_{r} is the readout voltage noise, and ε_{0} is electrostatic constant.
It is obvious that reducing the dynamic range of GRS will in turn reduce the GRS acceleration noise (Fig. 2c).
To summarize, the first onorbit scientific run of Taiji1 showed that the spaceborne interferometers could work properly, the distance measurement noise amplitude spectra density of O.B. interferometer was at the level of 100 pm Hz^{−1/2} (10 mHz–1 Hz). In some higher frequency bin, it approached 25 pm Hz^{−1/2}. The performance of GRS also fulfilled the requirement, with the evaluated acceleration measurement noise amplitude spectra density being 10^{−10} m s^{−2} Hz^{−1/2}. The noise amplitude spectra density of the thruster exerting force was calibrated to be <0.15 μN Hz^{−1/2}. The residue acceleration of the satellite after the drag free was <1 × 10^{−8} m s^{−2} Hz^{−1/2}. The onorbit performance of Taiji1 demonstrated the feasibility of the payloads. The design, the manufacturing, the assembling, and the adjusting of payloads were effectively verified.
Outlook
A new frontier was depicted above that the data of LISA–Taiji network can be used to study the cosmology in a more precise manner. The searching for GW signals with spaceborne detectors network will help us to understand not only the nature of gravity but also the expanding history of our universe. However, a few technology challenges faced by LISA and Taiji are still needed to be tackled in near future.
The successful flight of Taiji1 has verified the feasibility of the threestep plan of Taiji. Encouraged by the achievement of Taiji1, the Taiji scientific collaboration is looking forward to flying Taiji in early 2030s. It is our optimistic expectation that LISA and Taiji will be both orbiting the sun to detect a standard siren from a massive binary black hole merger. With the LISA–Taiji network, it is highly possible that the Hubble constant can be determined with an uncertainty <0.5%.
Data availability
The data that support the findings of this study are available from the authors on reasonable request, see Author contributions for specific data sets.
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Acknowledgements
This work is supported by the “Strategic Priority Research Program of the Chinese Academy of Science” (XDA15020709, XDB23030000).
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The Taiji Scientific Collaboration. China’s first step towards probing the expanding universe and the nature of gravity using a space borne gravitational wave antenna. Commun Phys 4, 34 (2021). https://doi.org/10.1038/s4200502100529z
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DOI: https://doi.org/10.1038/s4200502100529z
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