Spacebased gravitational wave detection is based on the astrodynamical equations derived from gravitational theory to detect changes in distance between spacecraftcelestial bodies andor their state changes caused by gravitational waves. The fundamental method involves using electromagnetic waves including radio waves microwaves light waves Xrays gamma rays etc. for Doppler tracking and comparing to the stable frequency standards sources at both the transmitting and receiving ends. Examples include microwave Doppler tracking optical clock gravitational wave detection atom interferometry gravitational wave detection and laser interferometry gravitational wave detection. If the frequency sources at both ends are not sufficiently stable a generalized dualpath Michelson interferometer based on Doppler tracking combinations is needed. Currently the main spacebased gravitational wave detectors under construction or planning are laser interferometers which cover medium frequency 0.110 Hz and lowfrequency millihertz 0.1100 mHz and microhertz 0.1100 muHz gravitational wave detection bands. This article reviews the current status and prospects of these gravitational wave detection methods.
['General Relativity and Quantum Cosmology', 'Astrophysics - Instrumentation and Methods for Astrophysics', 'Physics - Instrumentation and Detectors']
Space gravitational wave detection Progress and outlook
{'No Header': "This is a quick English translation with the assistance of ChatGPT and Baidu from the Chinese article ' 倪维斗 . 空间引力波探测现状与展望 . 中国科学 : 物理学力学天文学 ,  2024, 54: 270402, Ni W-T. Space gravitational wave detection: Progress and outlook (in Chinese). Sci Sin-Phys Mech Astron, 2024, 54: 270402, doi: 10.1360/SSPMA-2024-0186' ( Open Access ). Comments and feedback are welcome.", 'Wei-Tou Ni 1,2*': "- 0 International Centre for Theoretical Physics Asia-Pacific, University of Chinese Academy of Sciences, Beijing 100190, China;\n- 2 Innovation Academy of Precision Measurement Science and Technology (APM), Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China  \n* Corresponding author (email: wei-tou.ni@wipm.ac.cn)  \n( Original extended English abstract ) Space gravitational wave (GW) detection is to detect and measure the distance change between spacecraft/celestial bodies or status change intra spacecraft/celestial body according to the astrodynamical equations of general relativity or a specific gravitational theory. The basic method is using electromagnetic waves (including radio, microwave, light, X-ray, γ-ray, etc.) to Doppler track the spacecraft/celestial body and compare them with the two stable frequency standards (sources)  at  the  emission  end  and  the  receiving  end,  e.g.  microwave  Doppler  tracking,  optical  clock Doppler tracking, atom-interferometry GW detection, laser-interferomatic GW detection. If the emission phases of the electromagnetic waves are unknown, the statistic Doppler tracking method can be used as in the Pulsar Timing Arrays. If the frequency standards at the emission and the receiving ends are not stable enough in the desired detection frequency band, then it is necessary to exploit the generalized Michelson interferometry based on two paths each consisting of multi-segments of Doppler tracking. In this case, the phase (length metrology) noise at the combination end is proportional to the product of laser source frequency noise times the pathlength difference of the two paths, and the two paths need to be carefully designed and evaluated. Each set of two paths is called a TDI (Time-Delay Interferomerty) configuration. The study of TDI configurations together with the orbit design and the noise requirement at each optical link and the final spacecraft is called TDI interferometry. The final mission products for scientists to use are TDI phase (range) sequences/spectra. These products are also useful for other gravity measurements  or  testing  specific  gravitational  theories,  e.g.  measuring  gravitomagnetic  effects.  The current projects under construction and/or study are mainly using this method of generalized Michelson laser-interferometry which  includes  AMIGO  (Astrodynamical  Middle-frequency  Gravitational Observatory), BBO (Big Bang Observer), B-DECIGO, DECIGO (Deci-Hertz Gravitational Observatory) and DO (Deci-Hertz Observatory) in middle frequency band (0.1-10 Hz), LISA (Laser-Interferometric Space Antenna) and TAIJI/TianQ in the mHz low frequency band (0.1-100 mHz), and ASTROD-GW (Astrodynamical  Space  Test  of  Relativity  using  Optical  Devices  dedicated  for  Gravitational  wave detection), Folkner's mission, LISAmax, μAries and Super-ASTROD (0.1- 100 μHz) in the  Hz lowfrequency band. In this article, we review the current status quo of these space detection methods and present an outlook.  \nKey words : gravitational waves, space gravitational wave detection, time-delay interferometry (TDI), extended Michelson interferometry, milli-Hertz gravitational waves, micro-Hertz gravitational waves, deci-Hertz  gravitational  waves,  Doppler  tracking,  optical  clock  gravitational  wave  detection,  atom interferometry gravitational wave detection  \nPACS: 04.80.Nn, 04.80.-y, 95.30.Sf, 95.55.Ym  \ndoi:  \n10.1360/SSPMA-2024-0186 (Open Access)  \n倪维斗 . 中国科学 : 物理学力学天文学 2024 年第 54 卷第 7 期 : 270402", '1 Introduction': '- 1.1 Historical Background\n- 1.2 The Difference Between the Gravitational Wave Amplitude Predicted by General Relativity and the Detectable Precision of the Technology at That Time\n- 1.3 The Origin of Space Gravitational Wave Detection\n- 1.4 LISA (Laser Interferometric Space Antenna) Mission for Gravitational Wave Detection  \n1.5 ASTROD: Using Optical Instruments for Astrodynamic Space Relativity Tests and the  \nASTROD-GW Laser Interferometric Space Gravitational Wave Detection Mission  \n- 1.6 DECIGO and BBO Laser Interferometric Space Gravitational Wave Detection Missions\n- 1.7 Taiji and TianQin Laser Interferometric Space Gravitational Wave Detection Missions 1.8 Mid-Frequency Gravitational Wave Detection Planning\n- 1.9 Joint Observations with Gravitational Wave Detector Networks\n- 1.10 Micro-Hertz Gravitational Wave Detection Planning, U.S. Astro2020 APC White Paper, and ESA Post-LISA Space Gravitational Wave Detection Preliminary Solicitation White Paper\n- 1.11 Doppler Tracking, Optical Clock Gravitational Wave Detection, Atom Interferometry Gravitational Wave Detection, and Pulsar Timing Array Detection\n- 1.12 Strain Power Spectral Density Amplitude, Characteristic Strain, and Normalized Gravitational Wave Energy Spectral Density', '2 Basic Principles of Astrodynamical Missions and Space Gravitational Wave Detection': '- 2.1 Gravitational and Orbital Observations/Experiments in the Solar System\n- 2.2 Gravitomagnetic Effect and the Measurement of Solar Angular Momentum and Galactic Angular Momentum', '3 Spacecraft Doppler Tracking and Pulsed Laser Ranging': '- 3.1 Spacecraft Electromagnetic Wave Doppler Tracking and Its Gravitational Wave Response\n- 3.2 Pulsed Laser Ranging\n- 4 Pulsar Timing Array Gravitational Wave Detection\n- 5 Optical Clock and Atom Interferometry Gravitational Wave Detection\n- 6 Optical Path Noise, Inertial Sensor/Accelerometer Noise (Core Noise), and Frequency Noise Effects in Space Laser Interferometric Gravitational Wave Detectors', '7 Space Optical Interferometric Gravitational Wave Detection, Delay Lines, and Time-Delay Interferometry': '- 7.1 Stellar Interferometers and Gravitational Wave Detection\n- 7.1.1 Detection of Micro-Hertz and Sub-Micro-Hertz Low-Frequency Gravitational Waves\n- 7.1.2 Detection of Ultra-Low-Frequency (10 fHz-300 pHz) Gravitational Waves\n- 7.2 Generalized Michelson Interferometry and Time-Delay Interferometry', '9 Scientific Goals of Space Gravitational Wave Detectio n': 'This section reviews and discusses the scientific objectives of proposed and ongoing mid-frequency (decihertz) and low-frequency (millihertz, microhertz) space gravitational wave detection missions and projects (references 45, 50, 51, 78, 80, 81, 128). LISA was approved earlier this year, scheduled for launch in 2035. Further research over the next decade on scientific objectives and data analysis will be highly beneficial for preparing space gravitational wave detection missions.', '10 Basic Orbital Configurations, Angular Resolution, and Multiple-Formation Configurations': 'In the first section, we will first discuss the orbital configurations around Earth. Sections two and three  provide  an  overview  of  basic  orbits  akin  to  LISA  and  ASTRO-GW.  Sections  four  and  five respectively  discuss  their  angular  resolution  and  configurations  with  multiple  formations.  These fundamental configurations will be used in section eleven for numerical design and orbital optimization for both types of missions.', '11 Orbital Design and Optimization Using Ephemerides': '- 11.1 CGC Ephemerides\n- 11.2 Numerical Orbital Design and Optimization for LISA and Taiji\n- 11.3 Orbital Optimization for ASTROD-GW\n- 11.3.1 CGC 2.7.1 Ephemerides\n- 11.3.2 Initial Selection of Spacecraft Conditions\n- 11.3.3 Optimization Methods and Post-Optimization Characteristics', '12 Deployment of Earth-Like Solar Orbit Formations': '- 13 Time-Delay Interferometry (TDI)\n- 14 Payload\n- 15 Summary and Outlook', 'Translation of the Chinese Abstract': 'Space-based  gravitational  wave  detection  is  based  on  the  astrodynamical  equations  derived  from gravitational theory to detect changes in distance between spacecraft/celestial bodies and/or their state changes caused by gravitational waves. The fundamental method involves using electromagnetic waves (including radio waves, microwaves, light waves, X-rays, gamma rays, etc.) for Doppler tracking and comparing  to  the  stable  frequency  standards  (sources)  at  both  the  transmitting  and  receiving  ends. Examples  include  microwave  Doppler  tracking,  optical  clock  gravitational  wave  detection,  atom interferometry gravitational wave detection, and laser interferometry gravitational wave detection. If the frequency  sources  at  both  ends  are  not  sufficiently  stable,  a  generalized  dual-path  Michelson interferometer  based  on  Doppler  tracking  combinations  is  needed.  Currently,  the  main  space-based  \ngravitational  wave  detectors  under  construction  or  planning  are  laser  interferometers,  which  cover medium frequency (0.1-10 Hz) and low-frequency (millihertz 0.1-100 mHz and microhertz 0.1-100 μHz) gravitational  wave  detection  bands.  This  article  reviews  the  current  status  and  prospects  of  these gravitational wave detection methods.', '1. Introduction': 'Space laser interferometric gravitational wave detectors are fundamentally based on the generalized Michelson interferometer principle. Michelson interferometry splits the wave front of a light beam into two parts, creating two beams that travel along different paths and interfere when they recombine at a same location. Since its invention in the nineteenth century, Michelson interferometry has been a highly precise measurement method, requiring that the path length difference between the two paths be less than the  coherence  length.  Gravitational  waves  affect  the  optical  paths  differently,  and  space  laser interferometric  gravitational  wave  detection  utilizes  Michelson  interferometry  to  detect  this  path difference to measure the incoming gravitational waves.  \nThis article reviews and discusses the sensitivity required for space gravitational wave detection using generalized Michelson interferometry, as well as the demands on path (orbit) design and core noise (laser metrology and inertial sensing). It also explores the potential for detecting various gravitational wave sources. Space-based gravitational wave detection essentially measures the changes in distance between  spacecraft/celestial  bodies  caused  by  gravitational  waves  arriving  in  the  solar  system.  To distinguish the effects of various factors on distance changes, astronomical dynamical equations are used, with the measurements implemented by space missions focused on astronomical dynamics.  \nCurrently, the main space-based gravitational wave detectors under construction or planning are laser interferometers using generalized Michelson interferometry, covering medium frequency (0.1-10 Hz)  and  low-frequency  (millihertz  0.1-100  mHz  and  microhertz  0.1-100  μHz)  gravitational  wave detection  bands.  Generalized  Michelson  interferometry  can  be  considered  a  combination  of  Doppler tracking methods. Microwave Doppler tracking has been realized and constrains the strength of arriving gravitational waves. The basic principle behind detecting very low-frequency (nanohertz 0.3-100 nHz) background  gravitational  waves  with  pulsar  timing  arrays  is  based  on  statistical  Doppler  tracking methods.  \nOptical clock gravitational wave detection and atom interferometry gravitational wave detection methods have development and competitive potential,  as  their  principles  are  also  based  on  Doppler tracking. Space starlight interferometers have the potential to detect gravitational waves too. This article briefly  reviews  and  discusses  the  current  research  status  and  prospects  of  these  gravitational  wave detection methods.', '1.1 Historical Background': 'The  detection  of  gravitational  waves  is  inseparable  from  the  development  of  astronomical observation and astrometry. Before Galileo Galilei (1564-1642) [1] and others used telescopes to observe  \ncelestial bodies and stars in 1609, humans observed the positions and movements of celestial bodies with the naked eye and mechanical instruments (such as armillas (armillary spheres) and simplified armillas). Guo Shoujing (1231-1316) invented twelve new instruments, including the simplified armilla and the giant gnomon, to observe celestial phenomena. He also participated in the compilation of the "Shoushi Calendar" (1276-1280), edited by Xu Heng (1209-1281) et al., completing an accurate and applicable calendar of the time ([Ming] Song Lian. Yuan History · Astronomical Records [2]). Tycho Brahe (15461601) established Uraniborg and Stjerneborg observatories on the island of Hven in the Øresund and continuously improved and invented many new instruments for long-term observation of the stars [3]. His assistant, Johannes Kepler, used this observational data to develop the three laws of planetary motion (Kepler\'s laws) [4]. Kepler\'s laws, Galileo\'s law of motion on inclined planes [5], and the development of astronomy in the first half of the 17th century led to the formulation of Newton\'s three laws and his system of the world. During this period, Cassini and Picard [6] in Paris in 1671, and Richer in South America measured the distance to Mars using triangulation, determining the scale of the solar system. In 1676, Rømer discovered the finite speed of light by observing the time differences of Io\'s eclipses at different times and calculating the changes in distance between Jupiter and Earth, thus estimating the speed of light [7]. This was a highlight in the history of astrometry.  \nThe completion of Newton\'s system of the world [8] and the extensive astronomical observations of the 17th and 18th centuries further propelled the development of astronomy. The discovery and study of solar spectrum lines in the early 19th century marked the beginning of astrophysics [9]. In the field of astrometry during this period: Halley predicted in 1705 that the great comet observed in 1680 would return in 1758. Indeed, on December 25,  1758, German amateur astronomer Johann Georg Palitzsch observed its return [10]. This comet was later named Halley\'s Comet, which will next return in 2061. In 1781, William Herschel discovered Uranus. Continued observations revealed discrepancies between its orbit and Newtonian mechanics calculations. In 1846, Le Verrier predicted the presence of another planet causing  orbital  perturbations  at  a  specific  location.  That  same  year,  German  astronomers  Galle  and d\'Arrest discovered a new planet within one degree of the predicted position, which was named Neptune [10]. This was the peak of Newtonian system development. In 1859, Le Verrier discovered that Mercury\'s perihelion advanced at a rate of 38" per century faster than calculated by Newton\'s law of universal gravitation  [11,12].  This  discrepancy,  known  as  the  anomaly  in  Mercury\'s  perihelion  precession, prompted many new gravitational theories in the latter half of the 19th century [12].', '1.2 The Difference Between the Gravitational Wave Amplitude Predicted by General Relativity and the Detectable Precision of the Technology at That Time Historical Background': "In  1915, Einstein's General Theory of Relativity ultimately explained the anomaly in Mercury's perihelion  precession  [12-14],  and  theoretically  predicted  the  existence  and  radiation  intensity  of gravitational waves [14,15]. However, at that time, Einstein estimated the intensity of gravitational wave radiation to be extremely small compared to the detectable levels, asserting that while gravitational wave radiation exists, it would be technically impossible to measure it.  \nIn  1910, the astronomical community determined the size and mass of exceptionally dim white dwarfs [16,17]. Binary white dwarfs in the Milky Way galaxy can produce millihertz gravitational waves, and the foreground radiation formed by the superposition of these waves is a significant target for current space-based gravitational wave detection. This also represented an existing source of gravitational waves at the time. The characteristic strain caused by this radiation in the solar system would be approximately 10 -19 -10 -20 , resulting in a millihertz variation of about 10 -19 -10 -20 radians for celestial bodies from our line of sight to the solar system. At that time, astronomical observations did not employ adaptive optics, and angle observations were limited by atmospheric seeing, approximately 1 arcsecond (about 5×10 -6 radians), a difference of 14-15 orders of magnitude from the detectable precision. Thus, it was indeed technically impossible to measure gravitational waves in the 1910s [18].  \nAfter  a  century  of  technological  development  and  persistent  efforts  by  the  gravitational  wave research community, including Weber's experiments using resonant bars to detect gravitational waves, the LIGO and Virgo teams finally detected gravitational waves from a black hole merger in 2015 using the LIGO detectors with 4 km arms [19,20]. Subsequently, in 2017, gravitational waves from a neutron star merger were detected using the LIGO and Virgo detectors, with the Virgo detector having 3 km arms [21]. The strain amplitude was around 10 -20 .  \nIn 1915, the precision of measuring strain with a Michelson interferometer was about 10 ppm, which was 15 orders of magnitude less precise than the necessary detection level. Due to the influence of ground and atmospheric mass variations and vibrations, detecting millihertz gravitational waves on the ground or underground is almost impossible. To detect millihertz gravitational waves produced by binary white dwarfs  or  gravitational  waves  from  supermassive  black  hole  mergers,  it  is  necessary  to  conduct observations in space.", '1.3 The Origin of Space Gravitational Wave Detection': "In 1957, the Soviet Union launched humanity's first satellite, marking the beginning of the space age. The invention of lasers in the 1960s, along with the implementation of satellite laser ranging and lunar laser ranging during the same decade, allowed for astronomical ranging precision to reach meterlevel  and  later  centimeter-level,  millimeter-level,  and  currently  progressing  towards  sub-millimeter accuracy [22,23]. These advancements, combined with developments in radio and microwave ranging, significantly improved the precision of lunar and planetary ephemerides [24-26]. Regarding astronomical angular  observations,  adaptive  optics  technology  has  been  realized  on  the  ground  [27],  and  stellar interferometers  have  been  constructed  [28,29].  In  space,  the  proposed  Space  Interferometry  Mission (SIM) and the successful Gaia mission [30,31] have achieved precisions at sub-milliarcsecond and submicroarcsecond (sub-µas) levels, respectively. Very Long Baseline Interferometry (VLBI) has achieved angular precision of 1 µas, moving towards sub-microarcsecond accuracy [32,33].  \nThe development of lunar laser ranging and the 1970s research on drag-free space flight, combined with the proposal of ground-based kilometer-scale laser gravitational wave detection concepts, prompted the development of space-based laser interferometry concepts for gravitational wave detection between  \n1977 and 1989 [34-40]. The first public proposals for space-based laser interferometers for gravitational wave detection were presented at the following two conferences: (i) at the Active Optical Devices and Applications conference held in Washington in 1980 by Decker, Randall, Bender, and Faller [35]; (ii) at the Second International Conference on Precision Measurement and Fundamental Constants (PMFC-II) held  in  Gaithersburg  from  June  8-12,  1981, by  Faller  and  Bender  [36,37]. This  pioneering  proposal [36,37] introduced the concept of a gravitational wave mission using laser interferometry in space. It mentioned  two  essential  elements:  drag-free  spacecraft  to  compensate  for  disturbances  and  laser interferometry to enhance measurement sensitivity.  \nIn 1989, Faller, Bender, and their colleagues [40] selected the spacecraft formation orbit shown in Figure 1 as the mission orbit for their space-based laser gravitational wave observatory (LAGOS: LAser Gravitational-wave Observatory in Space). The formation consists of three spacecraft trailing the Earth's orbit by about 30 degrees, forming a triangular configuration with 1 million-kilometer arms. The central spacecraft orbits the sun with a period of one year, and the plane of the three spacecraft forms  a 60degree angle with the ecliptic, creating an isosceles right triangle. The end spacecraft receive 530 nm wavelength, 1 W laser beams emitted by the central spacecraft. After receiving the weak light, they use phase-locked local lasers (equivalent to amplification) to retransmit it back to the central spacecraft, generating interference. If a gravitational wave passes through, the interference fringes will change and can be detected (Figures 1 and 2). The orbits of the end spacecraft have an inclination of 1/300 and an eccentricity  of  √3/300  relative  to  the  central  spacecraft's  circular  orbit  with  a  radius  of  1  million kilometers and a one-year period (refer to Section 10.2). The expected strain sensitivity in the 10⁻³-10⁻¹ Hz range is 1×10⁻²¹/√Hz. The primary scientific objectives are to detect continuous gravitational waves from numerous binary star systems and burst gravitational waves from the galaxy formation era.  \nFigure 1 (LAGO's) Laser heterodyne gravitational wave antenna. The figure is from ref. [40].  \n<!-- image -->  \nFigure 2 (LAGO's) Laser heterodyne gravitational wave antenna showing spacecraft detail. The figure is from ref. [40].  \n<!-- image -->  \nBender and Faller,  who  had  dedicated  their  time  to  lunar  laser  ranging  and  measuring  free-fall acceleration  using  interferometry,  naturally  proposed  this  space-based  gravitational  wave  detection experiment. In fact, in the interferometric measurement of Earth's gravitational acceleration, the motion of a test mass freely falling within a vacuum drop vessel can be considered as drag-free motion in space [41]. The discrepancies observed in the 1990 comparison of absolute gravimeters at the International Bureau of Weights and Measures (BIPM) could be partially explained by corrections related to the finite speed of light in interferometric measurements [42]. In spacecraft tracking, the finite speed of light is always  accounted  for.  The  test  masses  in  gravitational  wave  missions  and  gravimeters  can  both  be regarded as free-falling within the solar system, tracked using celestial dynamics equations. Thus, we see  a  connection  between  space  geodesy,  Galileo's  equivalence  principle  (universality  of  free-fall) experiments,  and  gravitational  wave  detection  missions.  An  example  of  this  connection  is  the contribution of the LISA (Laser Interferometer Space Antenna) space laser interferometric gravitational wave detection team to the continuous laser ranging gravity gradient measurement of the GRACE followon space mission [43], and how developing this sub-project contributes to LISA technology. Another example is the adjustment of the Taiji-1 technology demonstrator satellite for observing the Earth's geoid after completing its primary mission [44].", '1.4 LISA (Laser Interferometic Space Antenna for Gravitational Wave Detection)': 'A significant advancement in space-based gravitational wave detection was the selection of LISA (Laser Interferometer Space Antenna, with an arm length of 5 million kilometers) by the European Space Agency (ESA) in 1993 for an M3 assessment study, and its subsequent inclusion as the third cornerstone mission of "Horizon 2000 Plus" [45]. After 2000, LISA became a joint ESA-NASA mission until NASA withdrew in 2011. In 1998, LISA Pathfinder was chosen as the second mission in ESA\'s Small Missions for Advanced Research in Technology (SMART) program, aimed at developing and testing extremely precise drag-free spacecraft technology. In commemoration of the centenary of General Relativity, ESA launched LISA Pathfinder on December 3, 2015, from the European Spaceport in Kourou, French Guiana,  \nusing a Vega rocket, successfully demonstrating the drag-free technology necessary for gravitational wave observation [46,47].  \nSimultaneously, ESA sponsored a common bus technology reference study for fundamental physics, completed  in  2008  [48].  In  2011,  the  LISA  team  proposed  the  New  Gravitation-wave  Observatory (NGO)/eLISA (evolved LISA) [49] to accommodate budget changes, reducing the target arm length from 5 million kilometers to 1 million kilometers, which received favorable evaluations. In November 2013, ESA announced the selection of scientific themes for the L2 and L3 mission opportunities-L2\'s "The hot  and  energetic  universe"  and  L3\'s  "The  gravitational  universe"  (ESA  Media  Relations Office/Communication  Department.  The  hot  and  energetic  Universe  and  the  search  for  elusive gravitational waves,  \nhttp://www.esa.int/Our\\_Activities/Space\\_Science/ESA\\_s\\_new\\_vision\\_to\\_study\\_the\\_invisible\\_Univers  \ne).  At  that  time,  a  potential  launch  opportunity  for  ESA\'s  L3  mission  was  in  2034.  Following  the successful launch of the ESA-NASA collaborative project LISA Pathfinder in December 2015 and the achievement of its technical objectives in 2016, ESA officially selected LISA as its third large-class space mission (L3) in June 2017 [50]. NASA agreed to participate, contributing 20% of the mission\'s total cost. The LISA science team then selected an arm length of 2.5 million kilometers (2.5 Gm), with the detection formation consisting of three spacecraft forming a precise equilateral triangle within 1% accuracy  (Figure  3),  capable  of  performing  multiple  Time  Delay  Interferometry  (TDI)  [50].  The definition study report was completed in September 2023 [51].  \nFigure 3 (Color online) Schematic of LISA/LISA-type orbit configuration in Earth-like solar orbit. The figure is from ref. [45].  \n<!-- image -->  \nOn January 25 of this year, ESA\'s Science Programme Committee approved the LISA space mission [52]. This marks the first time a scientific effort to detect and study gravitational waves from space has received formal adoption, recognizing that the mission concept and technology are sufficiently advanced. The committee has authorized the construction of the instruments and spacecraft, with work set to begin in January 2025, once a European industrial contractor is selected. LISA is scheduled to launch in 2035  \non an Ariane 6 rocket and will take a year to transition to its scientific orbit, with the science phase commencing in  2036. The  LISA  space  mission  aims  to  detect  gravitational  waves  produced  by  the inspiral and merger of numerous binary black hole systems with masses ranging from 10 4 to 10 7 M ☉ , enabling the study of the co-evolution of black holes and galaxies. It will also utilize the characteristic distance  scale  of  binary  black  hole  gravitational  wave  sources  to  accurately  measure  the  luminosity distances of cosmic objects, aiding in the verification of cosmic expansion rates determined by other methods. By observing the distribution of numerous binary white dwarfs and neutron stars within the Milky Way, LISA will enhance the galactic structure model obtained from ESA\'s Gaia astrometric space mission and test the MOdified Newtonian Dynamics (MOND) theory. Additionally, LISA will explore gravitational waves from the early universe and test General Relativity and cosmological theories.', '1.5 ASTROD: Astrodynamical Space Tests of Relativity Using Optical Devices and the ASTRODGW Laser Interferometric Space Gravitational Wave Detection Mission': "The  overall  concept  of  the  Astrodynamical  Space  Test  of  Relativity  using  Optical  Devices (ASTROD) involves using a set of spacecraft in drag-free flight within the solar system. These spacecraft would use optical devices to measure distances between each other, aiming to detect the gravitational field of the solar system, measure related solar system parameters, test relativistic gravity, observe solar oscillations (lowl g-modes, f-modes, and p-modes), and detect gravitational waves. The foundational ASTROD scheme was proposed in 1993 and has since been in the stages of concept, simulation, and laboratory research [53-70].  \nIn 1996, the concept of ASTROD I (also called Mini-ASTROD) was proposed. This involved a mission using two-way ranging between ground stations and the ASTROD I spacecraft to test general relativity and map the gravitational field of the solar system [53]. Simulation studies indicated that for the  Eddington  parameter  γ,  the  precision  of  solar  system  relativistic  gravity  tests  could  reach  10 -9 , improving current test precision by 3 to 4 orders of magnitude [71-75].  \nIn early 2009, responding to the Chinese Academy of Sciences' call for gravitational wave mission research,  the  ASTROD-GW  (ASTROD  optimized  for  gravitational  waves)  mission  concept  was proposed. This concept involved three spacecraft operating near the Sun-Earth Lagrange points L3, L4, and L5, using 260 Gm arm-length laser interferometry to detect gravitational waves [76-81], as shown in Figure 4 [78]. To achieve sensitivity in the direction of the ecliptic pole, the spacecraft formation must have  some  inclination.  The  schematic  diagram  of  the  ASTROD-GW  orbital  configuration  with inclination is shown in Figure 5 [82].  \nThe arm length of the ASTROD-GW space mission is 100 times that of LISA, providing two orders of magnitude better sensitivity at low frequencies. This will enable the detection of gravitational waves produced by the spin and merger of binary black holes with masses ranging from 10 5 to 10 10 M ☉ , allowing for further study of the co-evolution of black holes and galaxies. The observation of characteristic scales of  binary  black  hole  gravitational  wave  sources  will  allow  for  more  accurate  measurements  of  the luminosity  distances  of  cosmic  objects,  helping  to  verify  and  improve  the  measurements  of  cosmic  \nexpansion determined by LISA and other methods. Additionally, ASTROD-GW will explore microhertz gravitational waves from the early universe and provide a broader range of tests for general relativity and cosmology.  \nBefore the ASTROD-GW scheme was proposed, the concept of Super-ASTROD was introduced in 1996  [53].  This  concept  involved  three  spacecraft  in  triangular  formation  in  Jupiter-like  orbits,  one spacecraft  near  the  Sun-Earth  Lagrange  point  L1,  and  another  in  a  Jupiter-like  orbit  with  a  large inclination.  In  2008,  dual  scientific  goals  for  gravitational  wave  detection  and  cosmological model/relativistic  gravity  testing  were  studied  [83].  With  the  proposal  of  ASTROD-GW,  the  basic gravitational  wave  detection  configuration  for  Super-ASTROD  involved  placing  three  of  its  4-5 spacecraft near the Sun-Jupiter Lagrange points L3, L4, and L5. The arm length of the Super-ASTROD space mission is five times that of ASTROD, offering five times better sensitivity at low frequencies. This would enable the detection of the spin and merger of larger black holes, further exploring the cosmic gravitational wave background radiation and testing cosmological theories.  \nFigure 4 Schematic of ASTROD-GW orbit configuration [78].  \n<!-- image -->  \nFigure 5 (Color online) Schematic of ASTROD-GW orbit configuration with inclination [82]. 3D view with the scale of the vertical axis multiplied tenfold. Its projection on the ecliptic plane looks basically like Figure 4.  \n<!-- image -->", '1.6 DECIGO and BBO Laser Interferometric Space Gravitational Wave Detection Missions': 'DECIGO  (DECI-Hz  Gravitational  Observatory),  a  sub-hertz  interferometer  gravitational  wave observatory, was proposed in 2001 [84]. Its objective is to detect background gravitational waves from the early universe in the mid-frequency observational band between the high-frequency band of groundbased detectors and the low-frequency band of other space-based gravitational wave detectors at that time. DECIGO will employ the Fabry-Perot method similar to ground-based interferometers but with an arm length of 1000 km and a finesse of 10. As a successor to LISA, the United States proposed BBO (Big Bang Observer) with an arm length of 50,000 km [85], also aiming for similar goals. As depicted in Figure 6, DECIGO/BBO has two potential configurations: one with 3 spacecraft in a group of 4 groups and another with 4 spacecraft in a group of 3 groups, totaling 12 spacecraft each. They will directly measure  the  stochastic  background  of  gravitational  waves  through  correlation  analysis  [86].  Their primary goal is to detect primordial gravitational waves. Following the detection of gravitational waves by ground-based detectors, the DECIGO team proposed B-DECIGO as a precursor to DECIGO. It is designed  as  a  cluster  of  three  spacecraft  separated  by  100  km,  derived  from  a  simplified  design  of DECIGO. The mirrors have a diameter of 0.3 m, mass of 30 kg, finesse of 100, and sufficient noise sensitivity to detect many mid-frequency gravitational wave sources. B-DECIGO represents a feasible planning phase between the first and second stages [87].  \nFigure 6 (Color online) Two schematic configurations of BBO and DECIGO in Earth-like solar orbits.  \n<!-- image -->', '1.7 Taiji and TianQin Laser Interferometric Space Gravitational Wave Detection Missions': 'In 2015, following the successful direct detection of gravitational waves from binary black hole mergers three times using the aLIGO twin 4 km ground-based detectors, gravitational wave detection became a hot topic. The successful launch of LISA Pathfinder and achievement of its technical goals [46] further fueled interest in space-based gravitational wave detection. In 2016, the Taiji team advanced the ALIA-descope space gravitational wave detection plan [88] proposed in 2014, featuring an arm length of 3 million kilometers and using a LISA-like orbit. This initiative was formally named the Taiji mission  \n[89],  capable  of  joint  observations  with  LISA  to  enhance  scientific  objectives  [90-93],  particularly improving the localization of gravitational wave sources. The orbital configuration is illustrated in Figure 7 [93].  \nFigure 7 (Color online) Configuration of the LISA-Taiji network. The LISA constellation follows the Earth by 20°, while the Taiji constellation leads the Earth by 20°. The figure is from ref. [93].  \n<!-- image -->  \nThe TianQin team proposed the TianQin mission with an arm length of 170,000 kilometers, using a geocentric orbit [94]. The orbital configuration is illustrated in Figure 8 [94,95]. TianQin primarily operates in the millihertz frequency band and can participate in networked joint observations to enhance scientific objectives [96,97]. In 2018, the Taiji team launched the Taiji-1 technology satellite [98], and the  TianQin  team  launched  the  TianQin-1  technology  satellite  [99],  completing  the  first  phase  of technical development and testing. Both teams are actively engaged in conceptual and technical research to complete their definition study reports.  \nFigure 8 (Color online) Schematic of TianQin (S/C1, S/C2 & S/C3) orbit configuration. The direction to J0806 is shown. The figure is from refs. [94,95].  \n<!-- image -->', '1.8 Mid-Frequency Gravitational Wave Detection Planning': 'Due to theoretical gaps in black hole mass distribution, in the first decade of this century, many believed that sources of astrophysical gravitational waves from black hole mergers in the mid-frequency  \nband  were  extremely  rare.  Consequently,  there  was  a  push  to  detect  low-amplitude  primordial gravitational waves in this frequency band, leading to the proposal and study of two mission conceptsDECIGO [84] and BBO (Big Bang Observer) [85]. Because of the high sensitivity required for these mission concepts, they fall under the category of second or third-generation space-based gravitational wave missions. Their  most  sensitive  detection  band  lies  between  the  35-500  Hz  gravitational  waves detected  on  the  ground  in  2015  and  the  millihertz  gravitational  waves  that  LISA  aims  to  detect, specifically in the sub-hertz mid-frequency band (0.1 Hz - 10 Hz).  \nThe  LIGO-Virgo  detections  of  binary  black  hole  mergers  often  observed  sources  with  masses around 30 solar masses, whose gravitational waves generated from pre-merger inspiral traverse the entire mid-frequency band. In 2019, LIGO-Virgo detected an event with a total mass of about 150 solar masses [100], further enhancing expectations for detectable sources in the  mid-frequency gravitational  wave range.  \nPrior to 2015, methods for mid-frequency gravitational wave detection included the aforementioned DECIGO and BBO space-based laser interferometer gravitational wave detection concepts, along with various ground-based gravitational wave detection concepts [101]. Post-2016, further research has been conducted on these detectors, as well as proposals for various timely detectors [102,103].  \nFor the latest plans and developments in lunar-based gravitational wave detectors, please refer to the website of the 1st Sino-European Workshop on Lunar-Based Gravitational Waves Detection, April 27-30, 2024, Beijing: https://swlgwd2024.scimeeting.cn/en/web/index/22418\\_1878717\\_\\_.  \nRegarding  space-based  laser  interferometer  detection,  projects  like  AMIGO  (Astrodynamical Middle-frequency Interferometer GW Observatory) [104-106] and B-DECIGO [87] are crucial. AMIGO and B-DECIGO aim to network observations and detect intermediate-mass black hole binaries. AMIGO is technically classified as a first-generation middle-frequency laser interferometer gravitational wave detector.', '1.9 Joint Observations with Gravitational Wave Detector Networks': "The network collaboration of LIGO-Virgo has significantly enhanced the precision of gravitational wave source localization, promoting the realization of multi-messenger observations.  \nSpace-based  gravitational  wave  detectors  participating  in  network  observations  offer  several advantages, including rapid localization of gravitational wave sources, in-depth joint tests of relativistic gravity,  polarization  testing  of  gravitational  waves,  potential  measurement  of  non-zero  cosmic polarization  rotation,  better  determination  of  galactic  gravitational  wave  backgrounds,  and  increased detection  rates  of  stellar-mass  binary  black  hole  systems  and  massive  binary  black  hole  system gravitational wave events, among others.  \nThe Taiji space gravitational wave detection plan, with an arm length of 3 million kilometers, adopts a detection scheme similar to LISA. If joint observations can be conducted during the LISA detection period (after 2036), it will rapidly improve the localization precision of low-frequency gravitational wave sources in the 0.1 mHz to 0.1 Hz frequency range [90-93], facilitating multi-messenger observations.  \nUsing LISA-Taiji as an example, possible network configuration choices are illustrated (Figure 9) [92,93]: (i) LISA-Taiji-c, where Taiji-c is co-located with LISA and coplanar. Due to its good optical overlap with LISA, this configuration is most sensitive for random gravitational wave observations. (ii) LISA-Taiji-p (baseline configuration), where Taiji-p is offset from Earth by about 20 degrees and the formation plane is inclined +60 degrees relative to the ecliptic plane, similar to LISA. (iii) LISA-Taijim, where Taiji-m is offset from Earth by about 20 degrees and the formation plane is inclined -60 degrees relative to the ecliptic plane.  \nFigure 9 (Color online) Diagrams of the LISA and alternative Taiji mission orbit deployments. The left plot shows the LISA mission, which trailing the Earth by approximately 20° and is inclined by +60° with respect to the ecliptic plane. The Taiji-p leads the Earth by ∼ 20° and is inclined by +60°. The Taiji-c is co-located and coplanar with LISA. The right panel shows the LISA and Taiji-m, which lead the Earth by ∼ 20° and -60° inclined. The angle between the LISA and Taiji-m constellation planes is ∼ 71°, and the angle between LISA and Taiji-p formation planes is ∼ 34.5°. The figure is from refs. [92,93].  \n<!-- image -->  \nCurrently, the LISA-Taiji-p configuration serves as the baseline configuration. The Taiji team will timely decide on the final configuration to adopt.  \nIn linearized phenomenological models or template-based least squares (Gaussian, Kalman, Fisher matrix,  etc.)  parameter  estimation  using  independent  detector  networks,  the  order  or  sequence  of observational data does not affect the final results. For example, in a network composed of detectors A, B, and C: (i) A observes and obtains data block A1 during time interval [ t A1, T A1], and data block A2 during time interval [ t A2, T A2]; (ii) B observes and obtains data block B1 during time interval [ t B1, T B1], and data block B2 during time interval [ t B2, T B2]; (iii) C observes and obtains data block C1 during time interval  [ t C1, T C1],  and  data  block  C2  during  time  interval  [ t C2, T C2].  The  least  squares  parameter estimation method does not differentiate whether the total data blocks are used or estimated in different orders, or even sequentially in any order; the final results will be the same.  \nDue to similar sensitivity targets between LISA and Taiji, treating LISA and Taiji independently would increase sensitivity by approximately 1.4 times. However, with a LISA-Taiji network, there are additional observable quantities related to relative positions and orientations, crucial for determining the angular position and direction of gravitational sources. With these, angular positions and directions can be  rapidly  determined,  similar  to  the  LIGO-Virgo-KAGRA  team's  capabilities  in  ground-based  \ngravitational  wave  observations.  In  a  space-based  LISA-Taiji  network,  precise  determination  of  the relative  distances  between  LISA  and  Taiji,  and  their  directions,  are  crucial  observable  quantities  for angular position and direction determination. These improve the estimation of other parameters quickly. LISA and Taiji-p/Taiji-m will be separated in orbit by approximately 100 Gm or about 40 degrees, which is 30-40 times the size of LISA and Taiji individually.  \nMost  observable  gravitational  wave  sources  are  (quasi-)periodic.  For  these  sources,  the  sky localization area over two months can be improved by two orders of magnitude compared to a single detector. After six months or a year, a single LISA or Taiji detector rotates around its orbit, while these sources are (quasi-)periodic, so a single LISA or Taiji is equivalent to multiple detectors in different positions and times, making it possible to detect (quasi-)periodic sources with good sky positioning. The network will only have a greater than 2-fold improvement in solid angle positioning resolution (2 1/2  2 1/2  the  enhancement  factor  due  to  the  relative  distance  information  between  LISA  and  Taiji). Nevertheless, early rapid improvement in network positioning resolution is crucial for multi-messenger observations.  \nThe  LIGO-Virgo  network  and  the  LISA-Taiji  network  are  examples  of  simultaneous  and  cofrequency band observations. In addition to rapid localization, simultaneous or non-simultaneous multiband observations increase the number of observable events. If phenomenological/theoretical templates are  robust,  observing  at  different  frequencies,  either  simultaneously  or  non-simultaneously,  will effectively improve parameter estimation. For example, using a single binary star merger template across millihertz  (mHz),  decihertz  (deci-Hz),  and  decahertz  (deca-Hz)  frequencies,  the  space-based gravitational wave detector network (LT-AMIGO-ET-CE) comprising LISA, TAIJI, AMIGO, ET, and CE  can  improve  parameter  estimation  accuracy  by  two  orders  of  magnitude  (Table  1)  [107].  Joint observations by LT-AMIGO-ET-CE enhance the localization of gravitational wave sources by two to three orders of magnitude, with the precision of determining the position of most detectable binary black hole gravitational wave sources at Δ90% ranging from 7 × 10 -7 to 2 × 10 -3 deg 2 (median Δ90% is 4.6 × 10 -5 ),  facilitating  the  identification  of  electromagnetic/neutrino  sources  associated  with  gravitational wave  sources  and  promoting  active  multi-messenger  astronomy  observations.  The  results  of  multimessenger enhanced observations will strengthen the distinguishability of various gravitational wave source models, advancing understanding of stellar formation, black holes, and galaxy co-evolution. The multi-band capability of LISA-TAIJI-AMIGO-ET-CE to locate a large proportion of binary black hole systems within a sky area of 10 -3 deg 2 , with a relative error in luminosity distance measurement of σdL/dL ≤ 0.001, where the host galaxy can likely be directly determined through gravitational wave observations. Together with redshift measurements, precision in determining cosmic model parameters and the Hubble constant  can  reach  one  thousandth  or  even  better,  significantly  advancing  our  understanding  of  the universe and dark energy.  \nIn  this  simulation  study  [107],  to  investigate  the  prospects  of  multi-band  gravitational  wave observations of binary black hole systems evolving from the inspiral phase to the final merger phase, four different models were considered, taking into account different formation channels and constraints  \nfrom  LVK  O3  observations  [108]:  The  local  merger  rate  density R (0)  in  the  first  three  models  is normalized to the latest constraints provided by LVK after its first three observation runs [108]. Based on the redshift evolution of merger rate density in each model, the primary mass distribution of binary black  hole  systems,  and  the  distribution  of  mass  ratios,  simulated  binary  black  hole  systems  were generated using Monte Carlo methods. Considering the uncertainty of LVK's constraints on the local merger rate density, each model generated 100 simulated samples, including three different R (0) values: 19.1, 10.6, and 27.5 Gpc -3 yr -1 , representing the median and 90% confidence interval of the constrained local merger rate density. The four models are: (i) EMBS model: Assumes all binary black hole systems originate from the evolution of massive binary stars in galactic fields (referred to as the EMBS [Evolution of Massive Binary Stars in galactic field] channel) [109]. (ii) Dynamical model: Assumes all binary black hole systems form through dynamical interactions in environments like dense star clusters (referred to as the dynamical channel, using the A023 GC channel model from reference [110]). (iii) Mixed model: Considers a combination of the EMBS formation channel and the dynamical formation channel, with 75% of binary black hole systems originating from the EMBS channel and the remaining 25% from the dynamical  channel.  (iv)  GWTC-3  model:  Based  on  the  latest  observational  constraints  from  LVK collaboration [108]. The redshift-dependent merger rate density scales proportionally with (1 + z ) κ , where at low redshifts (i.e., z ≤ 1), κ = 2.7 +1.8 -1.9.  \nTable 1 Median values of the distributions of parameter estimation uncertainties for multiband BBHs a) [107]  \na) The first column indicates the name of the gravitational wave detectors or their different combinations. The second column shows the median value of the localization accuracy ΔΩ90% distribution for 100 multi-band binary black hole systems. The third, fourth, and last columns respectively display the median values of the distribution for the relative error in luminosity distance measurement σ d L/ d L, the relative error in the chirp mass ratio Δ M c/ M c, and the relative error in the symmetric mass ratio σ η. For the lower and upper bounds of the 68% confidence interval for each of these quantities, please refer to reference [107].  \nFigure 10 depicts the evolutionary trajectories of characteristic amplitudes of simulated binary black hole systems (mock BBHs) across sensitivity spectra of various missions.  \nIn this simulation, it is assumed that LISA, Taiji, and AMIGO begin observing simultaneously and each has a continuous observation period of 4 years, although this may differ from actual future scenarios. If AMIGO begins observing after the observation periods of LISA/Taiji have ended, some binary black hole systems that merge during the LISA/Taiji observation period would not be observable by AMIGO before it begins. However, some binary black hole systems detected in the low-frequency band with merger timescales  greater  than  4  years  could  evolve  into  the  mid-frequency  band  and  eventually  be observable by AMIGO.  \nFigure 10 (Color online) Evolution tracks of characteristic amplitudes for the mock BBHs which are detectable by LISA-Taiji-AMIGO (i.e. S/N ratios ρLISA-Taiji ≥5 and ρAMIGO ≥5, blue lines), by LISA-Taiji (i.e.  ρLISA-Taiji  ≥8,  yellow  lines),  and  by  AMIGO (i.e. ρAMIGO ≥8, magenta lines), respectively. All the BBHs plotted here are assumed to be observed for a continuous period of 4 yr. Orange, pink, sky blue, green,  and  purple  curves  represent  the  sensitivity  curves  for  LISA,  Taiji,  AMIGO,  CE,  and  ET, respectively [107].  \n<!-- image -->  \nSimulation studies of detectable binary black hole systems observed by different gravitational wave detectors at different frequency bands serve as demonstrations and planning for multi-band gravitational wave observations.", '1.10 Micro-Hertz Gravitational Wave Detection Planning, U.S. Astro2020 APC White Paper, and ESA Post-LISA Space Gravitational Wave Detection Preliminary Solicitation White Paper': 'Last  year  the  pulsar  timing  array  detected  background  gravitational  waves  in  the  nanohertz frequency band, while LISA and Taiji aim to detect gravitational waves in the millihertz low-frequency band (0.1 mHz - 100 mHz). Between these two bands lies the low-frequency microhertz band (0.1 μHz - 100 μHz) of gravitational waves. For the detection of low-frequency microhertz gravitational waves, there are astrodynamical space laser interferometric gravitational wave detection concepts as described in Section 1.5 [76-82]. Their primary objective is to detect gravitational waves produced by the merger and coalescence of binary black holes with masses ranging from 10 5 to 10 10 M ☉ , studying the coevolution of black holes and galaxies, and accurately measuring distances to cosmic objects using binary black hole gravitational wave sources.  \nThe experimental technology development required for space low-frequency microhertz band (0.1 μHz - 100 μHz) laser interferometric gravitational wave detectors has matured to a level comparable to LISA  and  Taiji:  their  strain  sensitivity  is  limited  by  inertial  sensor/accelerometer  noise.  Below  1 millihertz,  astronomical  unit  (AU)-scale  arm  length  gravitational  wave  detectors  are  two  orders  of magnitude more sensitive compared to Gm-scale arm length detectors like LISA/Taiji. Constraints from  \nconfusion backgrounds still exist above 0.1 millihertz, but below 0.01 millihertz, AU-scale gravitational wave detectors are not constrained by these. The sky resolution of AU-scale arm length gravitational wave detectors is on average a factor of (S/N) 2 better than Gm-scale arm length detectors for source localization, i.e., four orders of magnitude. In such finely resolved sky regions, averaging less than one galaxy,  enabling  clear  identification  of  host  galaxies  of  gravitational  wave  sources  through  multimessenger observations, with redshift measurements becoming a routine observable. Cosmodynamics can  be  established  through  observations,  providing  detection  evidence  for  "dark  energy"  and  "dark matter" and their evolution.  \nThe European Space Agency (ESA) and NASA collaborated on LISA Pathfinder (LISA Precursor), successfully launched in December 2015, achieving its technical objectives by 2016. In June 2017, ESA formally designated LISA as its L3 large space mission [50], and after NASA\'s participation, the United States solicited the Astro2020 APC White Paper and ESA solicited the LISA Space Gravitational Wave Detection White Paper.  \nIn 2019, the Baker et al. [112] white paper proposed two possible space gravitational wave schemes, one of which has sensitivity to the gravitational wave\'s most sensitive frequency band (1 mHz - 1 Hz) higher by an order of magnitude than LISA\'s sensitivity to the same frequency band, and the other has sensitivity to the gravitational wave\'s most sensitive frequency band in microhertz (0.1 - 100 μHz); the former is illustrated by the ALIA [113], and the latter is illustrated by the Folkner low-frequency space mission concept [114], and suggested as a third possibility such as gLISA [115-118] scheme. The two space gravitational  wave  schemes correspond to the  AMIGO and  ASTROD-GW space gravitational wave space detection plans proposed in China.  \nThe European Space Agency has called for white papers on space missions from 2035 to 2050. Four white papers [119, 120, 121, 122] on gravitational wave astronomy and gravitational wave detection primarily discuss and propose scientific objectives, and briefly propose corresponding possible space mission concepts. The first white paper concerns mid-frequency gravitational wave observations [119]; the second white paper concerns low-frequency gravitational wave observations in millihertz [120]; the third white paper concerns low-frequency microhertz gravitational wave observations [121]; the fourth white paper concerns high angular resolution gravitational wave astronomy [122], with high-resolution angular observations and high signal-to-noise ratio observations both capable of providing high angular resolution. These four white papers propose three gravitational wave detection schemes: (i) Decihertz Observatories (DOs) [119], corresponding to our AMIGO; (ii) ESA AMIGO [120], which correspondig to a more sensitive mission after the solar-orbit Taiji mission; (iii) μAres [121], with three spacecraft on the Martian orbit. Recently, ESA has made a mission concept study on LISAmax [123] relying on three spacecraft in Earth-like orbit, corresponding to our ASTROD-GW.  \nThere is also a related white paper [124] on detecting dark matter near the solar system, using space laser gravitational wave detection technology, which is also within the scope considered by ESA.  \nIn the previous section, we saw an example of multi-band gravitational wave observation simulation using LISA-TAIJI-AMIGO-ET-CE, simulating the observation of a group of stellar-mass binary black  \nhole inspirals and mergers statistically inferred by the discoveries by ground-based gravitational wave (GW) observatories, demonstrating the importance and effectiveness of such observations (simulations). Such observations (simulations) can be applied to any other combination of mid-frequency gravitational waves, such as PTA-LISAmax-LT or PTA-ASTROD-GW-LT, etc.  \nTable 2 lists the orbital configurations, arm lengths, orbital periods, planned number of spacecraft, acceleration  noise  requirements,  laser  ranging  noise  requirements,  and  gravitational  wave-sensitive frequency bands for space-based laser interferometric gravitational wave detection missions that have been approved, launched technology demonstration satellites, or are currently under study.  \nTable 2 A compilation of laser-interferometric GW Mission Proposals  \na) gLISA [115] was formed by merging two independent but similar proposals-GADFLI and GEOGRAWI.', '1.11 Doppler Tracking, Optical Clock Gravitational Wave Detection, Atom Interferometry Gravitational Wave Detection, and Pulsar Timing Array Detection': 'The above discussion pertains to laser interferometric gravitational wave detection using dual-path generalized Michelson interferometry based on Doppler tracking combinations. The principle of Doppler tracking  for  measuring  gravitational  waves  is  illustrated  in  Figure  11  [125].  Single-frequency electromagnetic waves emitted from ground-based tracking stations are received by Doppler tracking on spacecraft, which then relay the signals back to ground stations for tracking and recording time. The  \nintegrated frequency difference corresponds to a phase difference, which in turn corresponds to a time delay.  If  a  gravitational  wave  passes  through,  the  time  at  which  electromagnetic  waves  are  received changes, allowing the detection of gravitational waves. The key lies in whether the clocks at the recording time can detect the time difference caused by gravitational waves.  \nFigure 11 Early diagram for tracking geometry. The figure is from ref. [125].  \n<!-- image -->  \nIf the stability of the spacecraft clock/frequency source is insufficient, round-trip timing must be used, known as two-way ranging. If ground-based hydrogen maser is used, the precision can reach 10 -15 to 10 -16 ,  corresponding to the accuracy achieved in current microwave Doppler tracking gravitational wave detections. See Section 3 for details.  \nTo  date,  the  primary  alternative  approach  has  been  using  dual-path  generalized  Michelson interferometry based on Doppler tracking combinations, ensuring that the optical path length difference is below the requirement of frequency noise. See Section 7 for details.  \nThe stability and accuracy of optical clocks have reached levels of 10 ,  making  them  an  option for  astronomical  unit-scale  microhertz  gravitational  wave  detectors.  The  development  of  atomic interferometers has also made them feasible for decihertz gravitational wave detectors. See Section 5 for details.  \nMillisecond pulsars emit stable pulses that can serve as clocks; the arrival time of these pulses at receiving stations is affected by gravitational waves passing through the line of sight. Although the initial phase of pulsar pulses is unknown, a group of millisecond pulsars can statistically detect gravitational wave events and backgrounds. See Section 4 for details.  \nFrom  the  above  discussion,  we  understand  that  the  fundamental  equation  for  space-based gravitational wave detection is the Doppler tracking equation (see Section 3).', '1.12 Strain Power Spectral Density Amplitude, Characteristic Strain, and Normalized Gravitational Wave Energy Spectral Density': 'When discussing gravitational wave sensitivity and signal strength, three commonly used plotting methods include: characteristic strain hc ( f ) vs. frequency f , square root of power spectral density [ Sh ( f )] 1/2 vs.  frequency f ,  and  normalized  gravitational  wave  energy  density  Ωgw  vs.  frequency f .  For  detailed definitions of these quantities, please refer to references [126,127].  \nFigures  12-14  respectively  depict  the  relationship  between  gravitational  wave  detector  strain sensitivity and strain power spectral density amplitude [ Sh ( f )] 1/2 vs. frequency for the low-frequency and mid-frequency  ranges,  as  well  as  plots  of  characteristic  strain hc ( f )  vs.  frequency  and  normalized gravitational wave energy density Ωgw vs. frequency. The relationships of these 3 ordinates are as follows:  \nhc ( f ) = f 1 / 2 [ Sh ( f )] 1/2 ; Ωgw( f ) = (2π 2 /3 H 0 2 ) f 3 Sh ( f ) = (2π 2 /3 H 0 2 ) f 2 hc 2 ( f ). (1)  \nThe  detailed  explanations  and  interpretations  of  Figures  12-14  can  be  found  in  the  following sections and in references [126]. A significant portion of these figures is derived from the corresponding low-frequency and mid-frequency sections of Figure 2-4 in reference [126], which itself draws from the cited literature, particularly [127].  \nThis review updates and expands upon a review of space-based gravitational wave detection written in commemoration of the centenary of the theory of general relativity in 2016 [128]. The progress made over these eight years has been extensive and robust.', '2.1 Gravitational and Orbital Observations/Experiments in the Solar System': "From  a  historical  perspective,  with  the  improvement  in  accuracy  and  precision,  the  orbits  and gravitational observations/experiments within the solar system have always been crucial resources for the development of fundamental physical laws. This holds true for both the Newtonian world system and the development of Einstein's general relativity [129-131]. With significant improvements in orbit and gravitational measurements, we are currently in an era of tremendous progress in testing and developing these fundamental laws. The gravitational field within the solar system is determined by three factors: the dynamic distribution of matter within the solar system, the dynamic distribution of matter outside the solar  system  (Galactic  and  cosmic),  and  gravitational  waves  propagating  through  the  solar  system. Different  relativistic  gravitational  theories  and  cosmologies  make  different  predictions  for  the  solar system's gravitational field. Therefore, precise measurements of the solar system's gravitational field not only enable gravitational wave observations, determine the distribution of matter within the solar system, and assess observable (testable) effectson our galaxy and the universe, but also serve to test these  \nFigure 12 (Color online) Strain power spectral density amplitude (asd) vs. frequency for various GW detectors and GW sources. The black lines show the inspiral, coalescence and oscillation phases of GW emission from various equal-mass black-hole binary mergers in circular orbits at various redshift: Solid line, z = 1; dashed line, z =  5;  long-dashed line z = 20. See text for more explanation. CSDT: Cassini Spacecraft Doppler Tracking; SMBH-GWB: Supermassive Black Hole-GW Background.  \n<!-- image -->  \nrelativistic  theories  and  cosmologies.  To  measure  the  solar  system's  gravitational  field,  we measure/monitor the distances between different natural and/or artificial celestial bodies. In the solar system, the equations of motion for celestial bodies or spacecraft are given by the celestial mechanics equations:  \na = a N + a 1PN + a 2PN + a Gal-Cosm + a GW + a nongrav, (2)  \nwhere a is the acceleration of celestial bodies or spacecraft, a N is the acceleration caused by Newtonian gravitational forces within the solar system, a 1PN is the acceleration due to first-order post-Newtonian effects within the solar system, a 2PN is the acceleration due to second-order post-Newtonian effects within  \nFigure 13 (Color online) Characteristic strain h c vs. frequency for various GW detectors and sources. The black lines show the inspiral, coalescence and oscillation phases of GW emission from various equal-mass black-hole binary mergers in circular orbits at various redshift: Solid line, z = 1; dashed line, z = 5; long-dashed line z = 20. See text for more explanation. CSDT: Cassini Spacecraft Doppler Tracking; SMBH-GWB: Supermassive Black Hole-GW Background.  \n<!-- image -->  \nthe solar system, a Gal-Cosm is the acceleration due to Galactic and cosmic gravitational forces, a GW is the acceleration due to gravitational waves, and a nongrav is the acceleration due to all other non-gravitational sources  [78].  The  distances  between  spacecraft  primarily  depend  on  solar  system  gravity  (including gravitational oscillations caused by the Sun) and incoming gravitational waves, and are also affected by gravitational waves generated by solar oscillations and the motion of the Milky Way. Predicting these distances  as  a  function  of  time  relies  on  correct  gravitational  theories  and  cosmologies.  Accurate measurements of these distances over time will help determine the reasons for distance variations and validate the correct gravitational theory/cosmology. Ideally, a  fleet of drag-free spacecraft navigating within the solar system using optical (or other sensitive) devices to measure mutual distances would map  \nCSDT  \nFigure 14 (Color online) Normalized GW energy spectral density ΩGW vs. frequency for various GW detectors and GW sources. The black lines show the inspiral, coalescence and oscillation phases of GW emission from various equal-mass black-hole binary mergers in circular orbits at various redshift: Solid line, z = 1; dashed line, z =  5;  long-dashed line z = 20. See text for more explanation. CSDT: Cassini Spacecraft Doppler Tracking; SMBH-GWB: Supermassive Black Hole-GW Background.  \n<!-- image -->  \nthe solar system's gravitational field, measure relevant solar system parameters, test relativistic gravity, observe solar lowl g, f, p mode oscillations, and detect gravitational waves [78, 132]. In practice, certain orbital configurations are conducive to testing relativistic gravity; others are suitable for measuring solar system parameters; and still  others are advantageous  for detecting gravitational  waves.  These orbital configurations are integral parts of mission designs for various purposes [78, 132].  \nTo  test  relativistic  gravity,  spacecraft  need  to  enter  inner  solar  orbits  where  the  gravitational influence of the Sun is stronger, or send signals passing near the edge of the Sun to experience a stronger gravitational effect. An example of such a mission concept is ASTROD I, which used continuous laser  \nranging with 1 mm precision around superior solar conjunction to measure Shapiro delays of light and improve experimental errors in relativistic parameters [63, 71-73].  \nThe BepiColombo spacecraft, a joint mission of the European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA), launched on October 20, 2018 (https://en.wikipedia.org/wiki/BepiColombo; http://sci.esa.int/bepicolombo/) [133, 134]. On October 1, 2021, it made its first flyby of Mercury and is set to arrive in orbit around Mercury on December 5, 2025 (https://www.esa.int/Science\\_Exploration/Space\\_Science/BepiColombo).  One  of  its  radio  science objectives is to test relativistic gravity. While in orbit around Mercury, it will measure the motion of Mercury's center of mass with accuracy several orders of magnitude higher than radar ranging to the surface of Mercury. This presents a good opportunity to measure the advance of Mercury's perihelion and Shapiro time delay, and improve post-Newtonian parameters by orders of magnitude [135].  \nTo measure or improve solar and planetary parameters, spacecraft need to approach the object under study  or  have  extremely  high  sensitivity.  Typical  examples  include  the  NEAR  Near-Earth  Asteroid Rendezvous mission [136] and the MESSENGER Mercury probe [137].  \nCurrent activities in space gravitational wave detection primarily focus on the development and implementation  of  space-based  laser  interferometric  gravitational  wave  detectors.  Given  that  current laser  stability  reaches  only  10 -16 ,  which  does  not  meet  the  requirements of space  gravitational  wave detection at 10 -21 , it is necessary to use generalized Michelson interferometry methods to ensure that the optical path difference between two beams of light remains below a certain threshold.  \nSpace-based  laser  interferometric  gravitational  wave  detection  is  extremely  sensitive  within  its detection frequency range and holds great potential for measuring gravitational magnetic effects. In the next section, we will discuss possible methods for measuring gravitational magnetic effects.", '2.2 Gravitomagnetic Effect, and the Measurement of Solar Angular Momentum and Galactic Angular Momentum': 'When the charge is stationary, it produces an electric field; when it moves, it produces a magnetic field.  When  matter  is  stationary,  it  produces  a  gravitational  (gravitoelectric)  field;  when  it  moves,  it produces a gravitomagnetic field. Gravitomagnetic field can drag local reference frames with/around it it, which is an important aspect of relativistic gravity and also a manifestation of gravitomagnetism. After Lense-Thirring published their paper [138], more than 100 years of theoretical research and experimental efforts have resulted in the astrodynamical measurements of the Gravity Probe B and the LAGEOSLARES mission achieving a precision of Earth\'s gravitomagnetic field at the level of 2% [139, 140, 141, 142]. Planned/proposed astrodynamical missions within the solar system would potentially measure the solar gravitomagnetic effect separately from other target quantities and further improve the experimental measurement  accuracy  of  the  gravitomagnetic  effect.  A  specific  series  proposal  for  astrodynamical missions is the ASTROD series --- ASTROD I, ASTROD (ASTROD II), ASTROD-GW, and SuperASTROD [72, 73, 74, 75, 78, 81, 82, 83]. LISA [45] and TAIJI [98] are two ongoing astrodynamical missions in the solar system, with their main goals being the detection of gravitational waves. However,  \ndue to their precision requirements, they have the potential to test solar system relativistic gravity and measure the Lense-Thirring effect. This potential is discussed here. In space astrodynamical missions, the Sagnac effect in the gravitational field is much larger than Lense-Thirring\'s, which must be clarified first before separate measurements can be made. In rotating platforms/frames, light circulating in one direction and the opposite direction takes different time intervals, which is commonly referred to as the Sagnac effect. For split light beams, the phase shift Δφ derived from Sagnac interference fringes is:  \n φ  (8  /  c )   A , (3)  \nwhere ω is the angular rotation speed of the platform/frame, A is the oriented area of the loop, and λ is the wavelength of light. The phase shift is proportional to the dot product of the angular rotation speed and the oriented area of the frame. Converted into a time difference Δ t for two light paths, equation (3) transforms into  \n t  4   A / c 2 . (4)  \nFor mobile formations, in a gravitational field (or where the gravitational field can be neglected in calculation),  the  time  difference  in  light  propagation  between  closed  and  reverse  paths  can  also  be referred to as the Sagnac effect.  \nTable 3 lists the numerical optical path differences for the Sagnac-α, β, γ and Sagnac-α2, β2, γ2 Time Delay Interferometry (TDI) configurations of spacecraft formations during scientific simulation missions of LISA and Taiji (refer to Section 7.2 (48) formula and explanation), as well as the average optical path differences for α, β, γ at different starting points [145]. The 10 th row of Table 3 shows the Sagnac effect calculated analytically under a triangular formation with an angle of 60 degrees to the ecliptic plane. The calculation is as follows (Ni W-T, Ciufolini I, Wang G. LISA, Taiji, Sagnac effect, Lense-Thirring effect, the solar angular momentum and the Galactic angular momentum, in preparation).  \nIn the reference frame where the center of mass of the LISA/Taiji formation rotates around the sun, if the LISA/Taiji formation is stationary, the Sagnac time difference between the two optical paths of Sagnac-α obtained from formula (4) is  \n t 1  4 Ω  A / c 2 = 3 1/2 Ω( L 2 / c 2 )cos 60  + O( e ) = (3 1/2 /2)Ω( L 2 / c 2 ) + O( e ), (5)  \nwhere e is the eccentricity of the spacecraft orbit, see Section 10.1.  \nThe additional delay caused by the LISA/Taiji formation rotating clockwise about its center is  \n t 2  4 Ω  A / c 2 = -3 1/ 2 Ω ( L 2 / c 2 ) cos 0  = -3 1/2 Ω ( L 2 / c 2 ) + O( e ).                                                (6)  \nHence ， the total Sagnac time difference is  \n t Sagnac =  t 1 +  t 2 = -(3 1/2 / 2 c 2 )Ω ( L 2 / c 2 ) + O( e ). (7)  \nTable 3 Compilation of the RMS (root mean square) path length differences together with their Sagnac part and Lense-Thirring part  of  first  generation  Sagnac-α,  Sagnac-β  and  Sagnac-γ,  and  second  generation  Sagnac-α2,  Sagnac-β2  and  Sagnac-γ2  TDI configurations for LISA and Taiji a) [145,153]  \na) χ is a parameter with a value of approximately 1, determined by Lense-Thirring measurements. λ\' is the angle between the orbital  plane  normal  and  the  direction  of  the  Sun\'s  angular  momentum.  χ\'  is  a  parameter  determined  by  Lense-Thirring measurements, with a value of around or greater than 1. λ" is the angle between the orbital plane normal and the direction of the Galaxy\'s angular momentum.  \nFor LISA, L LISA = 2.5 Gm, Ω = 1.9909865788600 × 10 -7 Hz, we have  \n t Sagnac-LISA = -(3 1/2 / 2 c 2 )Ω( L LISA 2 / c 2 ) + O( e ) = -1.199054 × 10 -5 s × [1 + O( e )].                     (8)  \nFor Taiji, L Taiji = 3 Gm, Ω = 1.9909865788600 × 10 -7 Hz, we have  \n t Sagnac-Taiji = -(3 1/2 / 2 c 2 )Ω( L TAIJI 2 / c 2 ) + O( e ) = -1.726638 × 10 -5 s × [1 + O( e )].                      (9)  \nThe next approximation needs to consider the linear correction term for eccentricity e . Shown on the right of Figure 9, if the formation plane is selected at an angle of -60 ° to the ecliptic plane, the sign of Δ t 2 changes, and the Sagnac time difference becomes -3 times that of formulas (8) and (9).  \nIn rotating bodies\' gravitational fields, local inertial frames are rotated by the rotating body. If the rotational speed of the rotating body is of the order of Ω, the local inertial frame is rotated at a speed of the order ( U / c 2 ) Ω, where U is the Newtonian potential of the rotating body at that point. This effect is a type  of  gravitomagnetic  effect,  commonly  known  as  frame  dragging  or  the  aforementioned  LenseThirring effect.  \nThe  Lense-Thirring  effect  is  very  important  in  astrophysics  and  in  the  determination  and implementation of ultra-precision astronomical reference frames. The Gravity Probe B and Lageos-Lares laser ranging experiments have measured Earth\'s Lense-Thirring effect to 2%. Several authors have also proposed methods to measure the Lense-Thirring effect caused by the Sun (e.g., references [146]) and proposed the possibility of measuring the Lense-Thirring effect of the Milky Way (e.g., references [147]).  \nIn a quasi-Minkowski coordinate system, for weak field situations, as a first approximation, we express the metric gαβ as  \ngαβ = ηαβ + hαβ, | hαβ | << 1 , ηαβ = diag(1, -1, -1, -1) , (10)  \nh00 = -2 U/c 2  -2( GM  / c 2 ) , (11)  \nh0i = -2( G / c 3 ) ( J  x ) i / r 3 . (12)  \nSince hαβ is a small quantity in the solar system\'s gravitational field (< 4 × 10 -6 ), we expand and linearize hαβ , obtaining the linear (weak field) approximation of light propagation equations under the physical metric (10-12). Let r = r ( t ) = ( x ( t ), y ( t ), z ( t )) be the trajectory of light. Light propagation follows the geodesic of metric gαβ , with trajectory r ( t ) satisfying  \n0 = ds 2 = gαβdx α dx β = (1 + h 00) c 2 dt 2 + 2 h0icdx i dt - (  ij -hij ) dx i dx j . (13)  \nUnder the post-Minkowski approximation, we represent dx i /dt as  \ndx i / dt = ( dx i / dt ) (0) i + O( h ) = cn (0) i + cn (1) i + O( h 2 ), with ∑ i ( n (0) i ) 2 = 1, (14)  \nwhere cn (0) i are constants, cn (1) i are functions of the trajectory, and are of order O( h ). Substituting (14) into (13), we solve for | d r / dt |, obtaining the light propagation equation of order O( h ):  \n| d r / dt | = c [1 + (1/2) h 00 + h0in (0) i + (1/2) hijn (0) i n (0) j + O( h 2 )]. (15)  \nThus, the light travel time Δ t TT (time delay) between two observers (spacecraft) is  \n∆ t TT = (1/ c ) ∫| d r | [1 - (1/2) h 00 h0in (0) i - (1/2) hijn (0) i n (0) j + O( h 2 )].                                             (16)  \nFor the solar system with quadrupole moment J 2 and angular momentum J [= ( J x, J y, J z)],  \ng 00 = 1 - 2 U / c 2 + 2 βU 2 / c 4 + O( v 5 / c 5 ), g 0 i = -2( G / c 3 ) ( J  x ) i / r 3 + O( v 5 / c 5 ), gij = - (1 + 2 γU / c 2 ) δij + O( v 4 / c 4 ). (17)  \nChoosing the z-axis along the initial light propagation direction from z 1 to z 2, i.e., n (0) i = (0, 0, 1), we obtain  \n∆ t TT = ∫ dt = (1/ c )∫ dz [1 + ½(1 + γ) U + h0in (0)i + O( h 2 )] = ∆ t N + [(1 + γ)/2] ∆ tS GR + ∆ tL-T GR + O( h 2 )] = (1/ c ) ( z 2 -z 1) + (1 + γ) ( GM ☉ / c 3 ) ln {[( z 2 2 + b 2 ) 1/2 + z 2]/[( z 1 2 + b 2 ) 1/2 + z 1]} + (2/ c 3 ) G N J cos  \'  {2/b -1 / ( z 2 2 +b 2 ) 1/2 - 1 / ( z 1 2 +b 2 ) 1/2 } + O( h 2 ), ( z 1 < 0, z 2 > 0),        (18)  \nwhere the first term is the Newtonian light travel time Δ t N (Römer[7] delay), the second term is the  \nrelativistic Shapiro time delay Δ t S GR [148,149], and the third term is the Lense-Thirring effect on light travel time ΔtTT-LT GR . b is the minimum distance of the light trajectory from the Sun (impact parameter), and λ\' is the angle between the orbital plane normal and the solar angular momentum direction.  \nThe gravitatomagnetic field h 0 i effect in equation (12), namely the Lense-Thirring effect on light travel time, is  \n∆ t TT-LT = (1/ c )∫| d r | [ h0in (0) i + O( h 2 )] = (1/ c )∫ d r  h + O( h 2 ).                                                            (19)  \nHere h = ( h0i ) is a 3-vector. According to Stokes\' theorem, for a closed path  \n∆ t TT-LT = (1/ c ) ∫ S  h  d S + O( h 2 ),                                                                                              (20)  \nwhere, S denotes an oriented surface enclosed by oriented closed paths. For local space networks like LISA/TAIJI, the integral of Δ t TT-LT can be approximated using the inner product of ∇ × h at the center of the detection array and the oriented surface area S :  \n∆ t TT-LT = (1/c)  h  S = (1/c) |  h | S cos  \',                                                                              (21)  \nwhere λ\' is the angle between ∇ × h at the center of the detection array and the oriented surface area S , and S represents the absolute value of S .  \nSeveral  determinations  of  the  solar  rotational  inertia  and  angular  momentum  obtained  through helioseismology observations are approximately around 2×10 41 kg m 2 s -1 . Here, we use the parameter χ to represent an approximately determined value close to 1, i.e., J sun = 2χ×10 41 kg m 2 s -1 (a modern value for solar spin angular momentum is 1.8838×10 41 kg m 2 s -1 [150], and Allen\'s Astrophysical Quantities, 3rd  edition,  page  161  gives  a  value  of  1.63×10 41 kg  m 2 s -1 ).  Due  to  the  deviation  of  solar  angular momentum from the vertical direction relative to the ecliptic plane by 7°, for missions like LISA and Taiji, λ\' varies approximately between 60°±7° with an annual cycle.  \nFor TAIJI, aside from factor of cos the cos λ\' , Δ t TT-LT = 0.758 nm = 2.52 as, for LISA, Δ t TT-LT=0.526 nm = 1.76 as. The Lense-Thirring effect in time delay interferometry (TDI) involves twice the time (optical path) difference in Sagnac-α2, Sagnac-β2, Sagnac-γ2 (positive loop minus negative loop, see Section 7.2). Thus, for LISA, it amounts to 3.52 as (1.05 nm), and for TAIJI, it amounts to 5.04 as (1.52 nm) (converted to strain amplitude of 0.4-0.5 × 10 -18 at a 0.03 μHz annual frequency). These numerical values are listed in Table 3, sixth row from the bottom.  \nFor the inner Galaxy angular momentum J inner Galaxy , a benchmark model is  \nJ inner Galaxy = M inner Galaxy  R sun-to-Galaxy  V solar-system-relative-to-Galaxy.                                                    (22)  \nChoosing M inner Galaxy = mass inside Sun\'s orbit around Galactic Center = 2  10 11 M sun    (Galactic total mass:  10 12 M sun; Galactic virial mass: 1.3  0.3  10 12 M sun) ； R sun-to-Galaxy = 8.2 kpc    (8.2  0.1  \nkpc, Bland-Hawthorn and Gerhard 2017 [151]) ； V solar-system-relative-to-Galaxy  =  200 km/s    (250 km/s, Bland-Hawthorn and Gerhard 2017 [151]), we make a conservative estimate of J inner Galaxy to be  \nJ inner Galaxy =  \' 2  10 11 M ☉  2  10 30 kg/ M ☉  8.2 kpc  3.086 × 10 19 m/kpc  200 km/s =  \' 2  10 67 kg m 2 /s . (23)  \nBecause  the  Solar  System  is  very  close  to  the  Galactic  equatorial  plane,  the  Galactic gravitational field is approximately  \n h  2( G / c 3 ) ( J / r 3 ) = 0.0626  10 -28 m -1   \',                                                                        (24)  \ntherefore, on the Earth\'s orbit around the Sun, the ratio of the Galactic to Solar gravitational field is  \n0.0626  10 -28 m -1   \' / (0.292 x 10 -27 m -1   ) = 0.0214  \' /  = (1/47) (  \' /  ).                      (25)  \nBecause  the  normal  of  the  LISA  and  Taiji  spacecraft  formation  plane  and  the  Solar  angular momentum angle λ \' varies between 60°±7° annually, the gravitational magnetic effect also has an annual variation with a frequency of 0.03 μHz. LISA and Taiji have an inertial-sensing noise requirement of 3 fm s -2 Hz -1/2 at 1 mHz, to achieve an accuracy of 0.4-0.5 × 10 -18 @ 1 mHz, requiring approximately 1 year. At 100 μHz, it is 10 -15 fm  s -2 Hz -1/2 ,  requiring  approximately 10  years. However, at 0.03  μHz LISA/Taiji has no requirment, extraneous optimistic estimate with LISA/Taiji method, it might reach ∼ 1 nm s -2 Hz -1/2 , requiring approximately 10 7-8 years. In the fourth international workshop on gravitational magnetic  and  large-scale  rotation  measurements  (GRM  2023:  IV  International  Workshop  on Gravitomagnetism and Large-Scale Rotation Measurement, June 14-16, 2023, Pisa) the consensus is that with the current LISA and Taiji space mission plans, the detection of solar gravitomagnetic effects is not forseen [152, 153] (papers being prepared: Ni W-T, Ciufolini I, Wang G. LISA, TAIJI, Sagnac effect, Lense-Thirring Effect, the Solar angular momentum, and the Galactic angular momentum.).  \nEquation (18) affects the change ∆ t TT-LT in light propagation time due to gravitomagnetic influence:  \n∆ t TT-LT = (2/ c 3 ) G N J cos  \'  {2/b -1 / ( z 2 2 +b 2 ) 1/2 - 1 / ( z 1 2 +b 2 ) 1/2 } + O( h 2 ), ( z 1 < 0, z 2 > 0), （ 26 ）  \nFor the propagation of laser beams between the ASTROD-GW spacecraft, where b=0.5 AU, z 1=-0.866 AU, z 2=0.866 AU. From this equation, the time ∆ t TT-LT affected by the solar gravitomagnetic field is calculated as:  \n∆ t TT-LT = 0.0439 ps = 13.17  m/ c .                                                                                                        (27)  \nFor Sagnac-α, Sagnac-β, Sagnac-γ: Δ t TT-LT = 0.0439 ps × 6 =0.263 ps =70.9 μm/c.  \nTable  4  lists  the  path  differences  in  time-delay  interferometry  (TDI)  configurations  for  various orbital inclinations during the ASTROD-GW scientific mission [refer to Section 7.2], rows 3-6, and the average path differences for different starting points α, β, γ [82]. Rows 7, 8, 9 of Table 4 respectively list the Sagnac effect of the formation under various orbital inclinations, contributions of orbital inclination, and Lense-Thirring effect, and rows 10-13 list the relevant noise of ASTROD-GW. For detailed analysis and discussion of whether the solar angular momentum and galactic angular momentum can be measured, please refer to [153, 154].  \nFor laser/light passing near the edge of the Sun with b ≈ R ☉ , if z1 = -0.866 AU, z2 = 0.866 AU. From this (26) equation, the duration affected by the solar gravitomagnetic field is calculated as:  \n∆ t TT-LT = 9.42 ps = 2.825 mm/ c .                                                                                                            (28)  \nThe accuracy of pulse laser ranging can reach 1 mm or better. In the ASTROD-GW mission concept, if  ground stations add dual-wavelength dual-unidirectional laser ranging with spacecraft near L3, the gravitomagnetic  effect  of  the  Sun  can  be  measured,  and  thus  the  solar  angular  momentum  can  be determined.  In  the  concepts  of  ASTROD-I  (Mini-ASTROD)  and  ASTROD  (ASTROD-II)  space missions, lasers will pass near the edge of the Sun, enabling measurement of the solar gravitomagnetic effect and hence the determination of solar angular momentum [72, 73 and references therein].  \nTable 4 Compilation of the RMS path length differences together with their Sagnac part and Lense-Thirring part of first generation Sagnac-α, Sagnac-β and Sagnac-γ TDI configurations for ASTROD-GW of various degrees of formation inclination (0°, 0.5°, 1°, 1.5°, 2°, 2.5°, and 3°) with respect to the ecliptic plane. χ is a parameter of value ≈ 1 to be determined by ASTROD-GW LenseThirring measurementa) [82,153,154]  \na) χ is a parameter with a value of approximately 1, determined by Lense-Thirring measurements. λ\' is the angle between the orbital plane normal and the direction of the Sun\'s angular momentum. χ\' is a parameter determined by Lense-Thirring measurements, with a value around or greater than 1. λ" is the angle between the orbital plane normal and the direction of the Galaxy\'s angular momentum. Upcoming paper: Ni W-T, Ciufolini I, Wang G. LISA, Taiji, Sagnac effect, Lense-Thirring effect, the solar angular momentum, and the Galactic angular momentum.', '3.1 Spacecraft Electromagnetic Wave Doppler Tracking and Its Gravitational Wave Response': 'Radio  Doppler  tracking  of  spacecraft  in  space  missions  can  be  used  to  constrain/detect  the magnitude of low-frequency gravitational waves. Such gravitational wave detection experiments involve ground-based Doppler tracking radio antennas and distant spacecraft. Doppler tracking measures changes in their relative distances. Through these measurements, the size of gravitational waves between test masses can be detected/constrained. In 1967, Braginsky and Gertsenshtein [155] first proposed the use of  Doppler  data  from  spacecraft  tracking  for  gravitational  wave  searches.  In  1971,  Anderson  [156] utilized existing data for this purpose. Davis [157] derived Doppler tracking gravitational wave responses under certain special conditions in 1974; Estabrook and Wahlquist [125] derived a general formula in 1975 for the effects of gravitational waves crossing the line of sight of a spacecraft on Doppler tracking frequency measurements (also see [158]).  \nIn spacecraft Doppler tracking, highly stable clocks on Earth are used as references to control singlefrequency radio wave transmissions to the spacecraft (uplink). When the transponder on the spacecraft receives the single-frequency radio wave, it phase-locks or frequency-offset locks its local oscillator and coherently retransmits the local oscillator signal to ground stations (downlink).  \nThe one-way Doppler response y ( t ) is defined as:  \ny ( t ) ≡ δν/ ν 0 ≡ ( ν 1( t ) -ν 0)/ ν 0,                                                                                                                    (29)  \nwhere ν 0  is  the  frequency of  the transmitted signal and ν 1  is  the  frequency of the received signal. In current experimental/observational scenarios, gravitational wave sources are very distant from us, and upon arrival, can be treated as plane waves with very small amplitudes. For small-amplitude plane waves propagating along the z-direction in general relativity, the spacetime metric is:  \nds 2 = dt 2 - ( δij + hij ( ct -z )) dx i dx j ,          | hij | << 1,                                                                                  (30)  \nwhere Latin indices run from 1 to 3, and repeated indices imply summation. Estabrook and Wahlquist [125, 158] derived the one-way and two-way Doppler responses for plane gravitational waves in the weak-field approximation (Equations (10)) in a transverse-traceless gauge. For spacecraft S/C2 receiving signals from S/C1, the one-way Doppler response formula, using the notation from Armstrong, Estabrook, and Tinto [159], is:  \ny ( t ) = (1 k · n ) [ Ψ ( t - (1 + k · n ) L ) - Ψ( t )],                                                                                          (31)  \nwhere k = ( k i ) = ( k 1 , k 2 , k 3 ) is the unit vector in the direction of gravitational wave propagation, n = ( n i ) = ( n 1 , n 2 , n 3 ) is the unit vector along the spacecraft 1 to spacecraft 2 link, and L is the path length of the Doppler link. The function Ψ ( t ) is defined as:  \n]}.                                                                                                        (32)  \nAs noted and derived by Tinto and da Silva Alves [160], the Doppler response formulas (31) and (32) are also valid for any gravitational wave solution in the form of Equation (10) of any gravitational metric theory.  \nFrom the one-way Doppler response, the two-way and multi-link responses can be derived. The two-way Doppler response is derived as follows. The local oscillator transmits a beam aimed at the uplink spacecraft, and produces beats with signals returned from spacecraft and measures the phase. The phase (and frequency) measured as a function of time is analyzed. The Doppler response for a single link is given by (31). The two-way Doppler tracking [125, 158] response is given by:  \ny ( t ) = - (1 k · n ) Ψ( t ) - 2( k · n ) Ψ( t - (1 + k · n ) L ) + (1 + k · n ) Ψ( t - 2 L ).                                        (33)  \nThe three terms in equation (33) correspond respectively to the amplitude of gravitational waves when receiving Doppler tracking signals on Earth, when spacecraft retransmit, and when tracking signals are  transmitted  from  Earth.  Microwave  Doppler  tracking  involves  an  ultra-stable  oscillator  on  the spacecraft.  Its  precision  is  not  as  good  as  ground-based  hydrogen-maser  clocks,  so  the  accuracy  of microwave Doppler tracking depends on ground-based hydrogen-maser clocks (or other clocks used on the ground).  \nDoppler tracking of Viking S/C (S-band, 2.3 GHz) [161], Voyager I S/C (S-band uplink + coherent retransmission S-band and X-band (8.4 GHz) downlink) [162], Pioneer 10 (S-band) [163], and Pioneer 11 (S-band) [164] has been used for gravitational wave detection and provided constraints on the lowfrequency gravitational wave background.  \nThe latest measurements come from the Cassini spacecraft Doppler tracking (CSDT). Armstrong, Iess, Tortora, and Bertotti [165] used the Cassini multi-link radio system during the 2001-2002 flybys to improve observational constraints on the isotropic background of low-frequency gravitational waves. The Cassini multi-link radio system simultaneously receives signals from the X and Ka bands on two uplinks and transmits three downlink signals, where the X band is coherent with the X-band uplink, the Ka band is coherent with the X-band uplink, and the Ka band is coherent with the Ka-band uplink. Armstrong et al. [165] used the Cassini multi-link radio system with higher frequencies and advanced tropospheric calibration systems to reduce the impact of major noise sources (plasma and tropospheric scintillation)  to  levels  below  other  noises.  The  data  obtained  were  used  to  construct  upper  limits  of isotropic background intensity in the frequency range of 1 µHz to 1 mHz [165]. The characteristic strain upper limit curve labeled CSDT in Figure 13 is a smoothed version of curve 4 in reference [165]. The corresponding  CSDT curves in  Figure  12  for  strain  spectral  density  amplitude  and  in  Figure  14  for normalized spectral energy density were converted using formula (1). The minima on these curves are: (i)  at  several  frequencies  in  the  range  of  0.2-0.7  mHz  [ Sh ( f )] 1/2 <  8  ×  10 -13 ;  (ii)  at  a  frequency  of approximately 0.3 mHz, hc ( f ) < 2 × 10 -15 ; (iii) at a frequency of 1.2 μHz, Ωgw( f ) < 0.03.  \nIf equipped with starborne optical clocks, the gravitational wave sensitivity of spacecraft Doppler tracking can be improved by at least 1-2 orders of magnitude [166].  \nIn spacecraft radio tracking, the frequency of the received signal is tracked. Its integral is the phase. In spacecraft radio ranging, the phase of the received signal is measured. The derivative of the phase is the frequency. For coherent retransmission, apart from determining a constant addition, the accumulated measured phase is essentially relative distance.', '3.2 Pulsed Laser Ranging': "Another method for measuring range and distance is using pulse timing. This is what satellite laser ranging and lunar laser ranging do. For ranging through the Earth's atmosphere, the optimal method to find atmospheric delay is to use two colors (two wavelengths) to measure the atmospheric delay and subtract  the  effect.  The  ranging  accuracy  of  dual-color  (two-wavelength)  satellite  laser  ranging  has reached sub-millimeter levels. With the implementation of the new generation of lunar laser ranging [167, 168], the precision of lunar distance measurements has also reached millimeter levels and is approaching sub-millimeter levels. The laser link time transfer (T2L2) event timer on the Jason 2 satellite has achieved a starborne timing accuracy of 3 ps (0.9 mm) [169, 170]. Based on these advancements, the technological capability of single-directional ranging systems across the entire solar system can achieve millimeter or even sub-millimeter precision. With this level of accuracy and an expanded range of 20 astronomical units, the ability to probe the fundamental laws of spacetime and measure the solar system's gravitational field  will be significantly enhanced [72-75]. For 1 millimeter in 20 astronomical units, the fractional uncertainty is 3×10 -16 .  Achieving this fractional uncertainty level requires matched laser stability and clock accuracy; this precision has been achieved in the laboratory and will be realized in space. One of the goals of ASTROD I [71-75, 128] using onboard precise clocks is to enhance the gravitational wave sensitivity of the Cassini spacecraft Doppler tracking by an order of magnitude. In fact, optical clock stability and quasi-precision have already reached the 10 -19 level. When space optical clocks achieve this level, pulse laser ranging combined with drag-free technology will become an important alternative for microhertz low-frequency gravitational wave detection.  \nThe  fundamental principles of spacecraft Doppler tracking, spacecraft laser ranging, space laser interferometry,  and  pulsar  timing  arrays  (PTAs)  for  gravitational  wave  detection  are  similar.  In  the development  of  gravitational  wave  detection  methods,  spacecraft  Doppler  tracking  and  pulse  laser ranging have played important roles. The methods using space laser interferometry and pulsar timing arrays have become two important approaches for detecting gravitational waves. The following sections will discuss pulsar timing gravitational wave detection, clock and atomic interferometry gravitational wave detection, and core noise in space laser interferometry gravitational wave detection.", '4 Pulsar Timing Array Gravitational Wave Detection': "In  1967,  pulsars  were  discovered.  In  1978,  Sazhin  proposed  using  pulsar  timing  to  detect gravitational waves from binary supermassive stars with masses between 10 8 and 10 10 M ☉ , which have  \norbital periods of years [171]. In 1979, Detweiler analyzed published data from pulsar pulse emission and gave an upper limit on the energy density of random gravitational waves with a period of 1 year as the critical density of the universe [172]. In 1982, Backer et al. discovered stable millisecond pulsars [173], increasing the precision of timing measurements.  \nWhen gravitational waves pass through the line of sight to pulsar observations, the arrival time of pulses is affected. This effect can be used to observe gravitational waves. The phase when pulsar pulses arrive at radio observatories can be observed, but their initial time and phase are unknown and must be statistically averaged over initial phases. In 1983, Hellings and Downs derived the correlation c HD of timing residuals from two pulsars with an observing angle Θ for an isotropic random gravitational wave background, which depends only on Θ and not on the individual angular positions of the two pulsars on the celestial sphere. The formula is [174]:  \nc HD = 1/2 + (3/4) (1 - cos Θ ) [ln (1 - cos Θ ) - 1/6].      (34)  \nThis  formula  is  known  as  the  Hellings-Downs  formula,  and  the  curve  it  describes  is  called  the Hellings-Downs curve, as shown by the black dashed line in  Figure 15. Because the  background is assumed to be isotropic, the correlation should only depend on the angle and not on the individual angular positions of the pulsars. Because gravitational waves in general relativity are quadrupolar, the formula takes the above form. At that time, it was used to constrain the energy density of gravitational waves in the frequency range of 4-10 nHz to less than 1.4 × 10 -4 of the critical density of the universe. It also constrained the precision and stability of pulsars as timing signals.  \nFigure  15  (Color  online)  Bayesian  reconstruction  of  normalized  interpulsar  correlations,  modeled  as  a  cubic  spline  within  a variable-exponent power-law model. The violins plot the marginal posterior densities (plus median and 68% credible values) of the correlations at the knots. The knot positions are fixed and are chosen on the basis of features of the Hellings-Downs curve (also shown as a dashed black line for reference): they include the maximum and minimum angular separations, the two zero-crossings of the Hellings-Downs curve, and the position of minimum correlation. See section 3 of ref. [185] for more details. The figure is from ref. [185].  \n<!-- image -->  \nIn  1990,  Foster  and  Backer  [175]  proposed  using  an  array  of  stable  millisecond  pulsars  for:  (i) providing a long-term standard; (ii) detecting disturbances  in the Earth's orbit; (iii) detecting cosmic background  radiation.  The  millisecond  pulsar  array  subsequently  became  the  primary  method  for detecting nanohertz gravitational waves. In 1996 and 2002, McHugh et al. [176] and Lommen [177], respectively, derived upper limits on the gravitational wave background in the frequency range of 4-40 nHz as Ωgw ≤ 10 -7 , and at 6 × 10 -8 Hz as Ωgw ≤ 4 × 10 -9 from timing observations of millisecond pulsars. In 2002, the National Observatory of CAS also proposed constructing a 50 m radio telescope array to detect  extremely  low-frequency  gravitational  waves  or  provide  better  constraints  using  an  array observing approximately 10 millisecond pulsars [178].  \nBy 2015, there were three major pulsar timing arrays (PTA) engaged in long-term observations: the European  Pulsar  Timing  Array  (EPTA,  http://www.epta.eu.org/),  the  North  American  Nanohertz Observatory for Gravitational Waves (NANOGrav, http://nanograv.org/), and the Parkes Pulsar Timing Array (PPTA, http://www.atnf.csiro.au/research/pulsar/ppta/) [179]. Reference [179] discusses pulsar timing arrays and their sensitivity.  \nTable  5  lists  the  upper  limits  on  isotropic  stochastic  backgrounds  given  by  EPTA,  PPTA,  and NANOGrav in 2015 [180-182]. These constraints assume a relationship between the characteristic strain hc ( f ) of the gravitational wave background and frequency:  \nhc ( f ) = A yr [ f / (1 yr -1 )] α .      [α = - (2/3)]                                                                                                (35)  \nUpper limits On the isotropic stochastic background from 3 arrays [180-182] pulsar timing  \nIf α takes values other than -(2/3), the constraints on A yr are of the same order of magnitude. These constraints  have  ruled  out  predictions  from  many  models  at  the  time,  excluding  most  models  of supermassive black hole formation.  \nAfter improvements in timing accuracy, there are now six pulsar timing arrays (PTAs): the addition of the Chinese Pulsar Timing Array (CPTA), the Indian Pulsar Timing Array (InPTA) [183], and the South  African  Pulsar  Timing  Array  (SAPTA)  [184].  The  International  Pulsar  Timing  Array  (IPTA, http://www.ipta4gw.org/)  is  an  alliance  of  regional  PTAs  aimed  at  achieving  better  sensitivity  to gravitational waves. Currently, PPTA, EPTA, NANOGrav, and InPTA are formal members of IPTA, while SAPTA and CPTA are observers within IPTA. In recent years, these six pulsar timing arrays have significantly improved the accuracy of gravitational wave detection.  \nBy 2015, the resolution had nearly reached the level of detecting gravitational waves. Systematic errors  need  to  be  clarified  first.  Following  improvements  in  solar  system  ephemerides,  NANOGrav, EPTA/InPTA, PPTA, and CPTA submitted four papers (arXiv numbers: 2306.16213 to 2306.16216) on  \nJune 28, 2023, reporting the discovery of  nanohertz background gravitational  waves [185-188]. The compiled  amplitudes  of  the  nanohertz  background  gravitational  waves  from  their  observations  and analyses are presented in Table 6. The theoretical models in Table 6 correspond to Equations (34) and (35). The angle-dependent observational analysis results from NANOGrav are shown in Figure 15 [185].  \nPPTA and CPTA observations [185-188]  \nIf all contributions from supermassive binary black holes (SMBHBs) evolve purely through circular orbital energy loss to gravitational radiation, then the resulting gravitational wave background (GWB) spectrum can be well described by Equation (35), namely the simple f -2/3 characteristic strain power-law [189]. However, gravitational wave background signals not arising from this mechanism may also exist in  the  nanohertz  band.  These  signals  include  those  from  inflationary  primordial  gravitational  waves [190,191],  gravitational  waves  induced  by  scalar  fields,  vector  fields  [192],  and  various  processes induced  by  cosmological  phase  transitions  [193-197]  (such  as  vacuum  bubble  collisions  after  phase transitions, sound waves, turbulence), as well as various defects such as cosmic strings [198-200] or domain walls [201,202] decay. Establishing these requires more precise observations and careful model analyses.  \nLooking ahead over the next hundred years, we [126] adopt and extrapolate estimates from Moore, Taylor, and Gair [203]. The sensitivity of Pulsar Timing Arrays (PTAs) to single-frequency gravitational waves primarily depends on timing precision, including the modeled timing residuals (root-mean-square deviation of time residuals). The bandwidth depends on the sampling frequency and the duration of data span. For observations every Δ t time interval over an observation span of T , the bandwidth f is [1/ T , 1/Δ t ]. The sensitivity hc ( f ) has a frequency dependence linearly related to f :  \nhc ( f ) = B yr ( f / yr -1 ), (1/ T ) < f < (1/  t ).                                                                                                  (36)  \nWe  assume B yr  is  proportional  to  timing  precision,  inversely  proportional  to  the  duration  of observations, and inversely proportional to the number of pulsars in the PTA. Moore, Taylor, and Gair [203] set a canonical PTA (MTG canonical PTA), assuming a random distribution of 36 pulsars in the sky, observed every two weeks with a timing precision of 100 ns over 5 years; the sensitivity of this canonical PTA is (36) with B yr = 4 × 10 -16 , roughly equivalent to the OPEN 1 simulated dataset in the IPTA Data Challenge (http://ipta4gw.org/?page\\_id=89). In Table 7, we estimate the sensitivity of IPTA,  \nFAST107, and SKA108 under observation spans of 20 years, 50 years, and 100 years, listing the basic simulation assumptions for IPTA, FAST, and SKA.  \nTable Sensitivities of IPTA FAST and SKA to monochromatic GWs [126]  \nRough estimated sensitivity curves for IPTA, FAST, and SKA are [126]:  \nhc ( f ) = 1  10 -16 ( f / yr -1 ), 1.58  10 -9 Hz < f < 8.27  10 -7 Hz, for IPTA-20,                                              (37)  \nhc ( f ) = 1.5  10 -17 ( f/ yr -1 ), 6.34  10 -10 Hz < f < 8.27  10 -7 Hz, for FAST-50,                                            (38)  \nhc ( f ) = 1.5  10 -18 ( f/ yr -1 ), 3.17  10 -10 Hz < f < 8.27  10 -7 Hz, for SKA-100.                                          (39)  \nWe note the corresponding sensitivity curves plotted in Figures 2, 3, and 4 of reference [126]. The Ωgw( f ) sensitivity of SKA for single-frequency gravitational waves reaches 10 -22 around a frequency of approximately 3.17 × 10 -10 Hz. These curves are denoted as IPTA-20, FAST-50, and SKA-100.  \nGlobal 24-hour observations of gravitational waves from J1713+0747 provide upper limits in the frequency range of 10 -5 -10 -3 Hz [204]; solid lines in Figures 12-14 represent limits for random directions, while dashed lines indicate limits for pulsar directions [126].", '5 Optical Clock and Atom Interferometry Gravitational Wave Detection': "In Section 3, we discussed spacecraft Doppler tracking and pulse laser ranging, using microwave atomic clocks as time and frequency references. In the previous section, pulsar array gravitational wave detection  was  discussed,  utilizing  pulsar  regularity  in  pulse  intervals  as  the  basis  for  time.  With  the improvement in the accuracy and stability of optical clocks, optical clocks can also serve as time and frequency standards between two spacecraft (or between ground and a spacecraft) [126].  \nIn addition to using optical clocks instead of hydrogen maser clocks for microwave Doppler tracking in gravitational wave detection, there are currently two specific proposals for optical clock-based Doppler tracking systems. One such system uses a strontium optical lattice clock, as shown in Figure 16 [205]. Another system known as the Interplanetary Network of Optical lattice clocks (INO) also employs optical lattice clocks, as depicted in Figure 17 [206].  \nOptical clocks have advanced to unprecedented levels of stability, precision, and sensitivity [207209],  with  uncertainties  now  reaching  below  10 -18 and  approaching  the  10 -19 level  [207].  Further  \nminiaturization is underway, and optical clocks are becoming competitive with laser interferometry for arm lengths equal or greater than astronomical unit scales.  \nFigure  16  (Color  online)  Proposed  optical-clock  gravitational  wave  detector.  The  detector  consists  of  two  identical  drag-free spacecraft, A and B, separated from each other by a distance d along the x axis. Each satellite contains a free-floating reference mass, an ultrastable laser, and a strontium optical lattice clock. A mirror is mounted on the free mass and is used to define the standing wave of light forming the optical lattice and confining the Sr atoms. Some of the laser light from spacecraft A (orange, dashed line) is sent to spacecraft B. The light first passes through an acousto-optic modulator driven at frequency f A, which offsets the frequency of the light reaching photodiode 2B in spacecraft B and enables the phase locking of laser B to laser A through heterodyne detection. Vibrations and thermal drifts of the optics on each satellite can be corrected locally by feeding back on the beat notes at 2 f A;B on photodiodes 1A;B. Light from laser B (blue, dotted line) is sent back to satellite A to verify the phase lock, to maintain  pointing  stability,  and  to  enable  operation  in  the  reverse  mode,  with  laser  A  locked  to  laser  B.  A  plus-polarized gravitational wave propagating along the z axis induces relative motion between the two free masses, which can be detected using a clock comparison measurement protocol. The figure is from ref. [205].  \n<!-- image -->  \nFigure 17 (Color online) A planned location of the spacecraft: Lagrangian points L 1, L 4, and L 5 of the Sun-Earth orbit. The L 1 is at 1/100 AU from the Earth, while L 4 and L 5  form equilateral/isosceles triangle with the Sun/the Earth, respectively; the distance between L 1L 4 ( L 5) is 1 AU, while that of L 4L 5 is 3 1/2 AU. Two-frequency radio or light will be used for communication between spacecraft. The inset explains that the solar panel of the spacecraft is separated as a parasol from the main body, in order to prevent acceleration noise due to solar wind. The figure is from ref. [206].  \n<!-- image -->  \nSince the successful demonstration of AIs from the teams of Pritchard [210], Bordé [211], and Zhu [212] in 1991, atomic interferometry (AI) technology has been in rapid and promising development phase. It  has  made  significant  contributions  to  precision  measurements  and  fundamental  physics.  Atomic interferometric gravimeters now rival optical interferometric gravimeters. From August 21 to September 29, 2023, the preliminary draft results of the 11th International Comparison of Absolute Gravimeters (ICAG  2023)  showed  that  the  atomic  interferometric  gravimeter  WAG-H5-2  from  the  Innovation Academy of Precision Measurement Science and Technology (APM) of CAS, exhibited consistency with the FG5 from JILA at the Table Mountain Geophysical Observatory (TMGO), Boulder, Colorado, with uncertainties comparable [213].  \nIn 2011, Stanford University proposed using atomic interferometers as an alternative method on the LISA band for detecting gravitational waves [214-218]. However, achieving LISA-like sensitivity with this  proposal  requires  further  precision  improvements  and  addressing  several  challenges  [216,  217]. Since  the  proposal,  significant  efforts  have  been  made  in  laboratory  research  and  development, demonstrating high technological content and application potential.  \nIn 2015, the SYRTE at the Paris Observatory launched the first phase of its MIGA (Matter-wave laser Interferometric Gravitation Antenna) project, constructing a 200-meter optical cavity in the LSBB (Laboratoire  Souterrain  à  Bas  Bruit)  underground  laboratory  in  Rustrel  for  studying  atomic interferometers [219]. In the project's second phase, MIGA will focus on scientific experiments and data analysis to explore the spatiotemporal structure of the local gravity field in the LSBB area. Meanwhile, MIGA is also evaluating the potential future applications of atomic interferometers in the intermediate frequency band (0.1-10 Hz) for gravitational wave detection.  \nCurrently,  there  are  five  large  fully-funded  prototype  projects  under  construction:  a  10-meter fountain project in Stanford, USA [220] and the MAGIS-100 project at FNAL [221] in the US; the MIGA project in France [222]; the VLBAI project in Hannover, Germany [223]; and the AION-10 project in Oxford, UK [224], potentially with a 100-meter site in Boulby, under investigation at CERN; a 10-meter fountain  project  [225]  and  the  ZAIGA  project  in  China  [226].  These  projects  will  demonstrate  the feasibility  of  long-arm  atomic  interferometric  detection,  paving  the  way  for  future  ground-based kilometer-scale experiments. Discussions are already underway for constructing multiple kilometer-scale detector projects, including the ELGAR project in Europe [227], the MAGIS-km project at the Sanford Underground Research Facility (SURF) in the US [221], the AION-km project at the STFC Boulby Facility in the UK [224], and the advanced ZAIGA project in China [226]. The goal is to have at least one  kilometer-scale  detector  operational  around  2035.  These  kilometer-scale  experiments  will systematically  explore  the  intermediate  frequency  band  of  gravitational  waves  and  potentially  detect ultra-light  dark  matter,  while  demonstrating key technologies required for future space-based atomic interferometric detection missions like AEDGE [228], as illustrated in Figure 18. The ultimate goal of atomic  interferometer  gravitational  wave  detection  is  missions  such  as  AEDGE,  where  atomic interferometry serves as the detection frequency standard.  \nFigure 18 (Color online) The GW strain sensitivities and benchmark signals from BH binaries of different masses at different redshifts. The coloured dots indicate the times before mergers at which inspirals could be measured. The figure is from ref. [228].  \n<!-- image -->  \nUnlike photons, atoms come in many types. A significant advantage of atomic interferometry for detecting  ultra-light/zero-mass  physical  fields  and  dark  matter  is  its  wide  applicability:  it  can  detect various potential physical fields and dark matter with different interactions. For example, scalar fields [229, 230, 231], pseudo-scalar fields [232, 233] (like axions [234, 235, 236]), vector fields [237], or dark photons [238, 239].  \nRegarding  intermediate-frequency  gravitational  wave  detection,  apart  from  the  space-based schemes discussed earlier and AEDGE, there are (i) AIGSO [240], which uses a 10-kilometer arm length atomic interferometer, with challenging technological development; (ii) various proposals using optical clocks, which could be prototyped first. Atomic interferometry can be applied as a gravity gradiometer, offering an option for separating gravitational wave signals.", '6 Optical Path Noise, Inertial Sensor/Accelerometer Noise (Core Noise), and Frequency Noise Effects in Space Laser Interferometric Gravitational Wave Detectors': 'In the context of space-based laser interferometric gravitational wave detectors, there are two core noises:  optical  metrology  (length)  noise  and  inertial/accelerometer  noise.  Regarding  laser  frequency noise, its effect on the combined signal in the generalized Michelson configuration depends on the optical path difference of the two paths.  \nIn space, due to the large distances involved, at the receiving end near the spacecraft, it is necessary to phase-lock the local laser oscillator to lock in the propagation (amplification) of the weak incoming light beam to another spacecraft or back; this measures the final interferometric phase difference between the two paths. The interferometric phase noise δφinterference at the final receiving satellite is given by:  \nδ φ interference = 2π · δν ( f ) · Δ L / c + phase locking noise + timing noise + accumulated other noises along the two paths,                                                            (40)  \nwhere δν ( f ) is the frequency noise of the laser source at frequency f , Δ L is the optical path difference between the two chosen paths, and c is the speed of light. Therefore, to reduce the interferometric phase noise δφ interference,  we  must reduce the frequency noise of the laser source or decrease the optical path difference between the two paths, or both.  \nAfter  traveling  over  long  distances,  the  light  received  by  the  telescope  on  another  spacecraft  is greatly attenuated and needs to be amplified to transmit it to the next spacecraft. The amplification is achieved by phase-locking the local laser with the incoming weak laser either in phase or with a known frequency  offset.  Thus,  the  phase  information  is  continuously  propagated  under  conditions  of  phase coherence or known frequency offset. The phase records measured by the phase meter are tagged with time labels, commonly using pseudo-random codes, for subsequent propagation identification in data analysis. The method of obtaining the interferometric signal at the receiving satellite with time-tagged tracking  analysis  is  known  as  Time-Delayed  Interferometry  (TDI).  To  meet  the  requirements  of interferometric phase noise δφ interference, the TDI configuration chosen for each space gravitational wave mission must satisfy the optical path difference conditions set by Equation (40).  \nAs shown in Figure 19, simple TDIs include the first-generation Michelson TDI, typically referred to as X, Y, and Z TDI; the second-generation Michelson TDI, known as X1, Y1, and Z1 TDI; and the original (zeroth-generation) Michelson combinations, referred to as X0, Y0, and Z0 TDI. Discussions on various combinations of TDI are found in Section 7.2. Both phase locking and time tagging introduce noise.  Other  noises  include  Tilt-to-Length  (TTL)  noise,  among  others.  Without  orbital  propulsion corrections, changes in arm lengths due to the motion of the solar system can reach 1% for missions similar  to  LISA,  and  0.01-0.1%  for  missions  similar  to  ASTROD-GW.  Typically,  appropriate  TDI configurations (combinations) are selected to meet the requirement that the interferometric phase noise caused  by  laser  frequency  noise δν ( f )  is  smaller  than  the  core  noises  (optical  metrology  noise  and inertial/accelerometer noise) when discussing detection sensitivity and detection targets. For example, for a laser source with laser frequency noise δν ( f )  of  30  Hz/√Hz, X, Y, and Z TDIs do not meet the requirements of LISA and ASTROD-GW, while X1, Y1, and Z1 TDIs do meet the requirements of LISA and ASTROD-GW.  \nTable 2 lists the core noises of current laser interferometric gravitational wave space mission projects and plans. Inertial/accelerometer noise determines the precision of strain measurement reference points, while optical metrology noise determines the precision of the measurements themselves. Laser frequency noise occurs during the measurement process, but after selecting appropriate TDI configuration combinations for measurement results, it is mitigated below the core noises.  \nThe primary source of optical metrology noise is the shot noise in the long-arm laser link:  \n.                                                                                                          (41)  \nThis equation refers to the RF noise caused by the detection of lasers from adjacent spacecraft. The higher the optical power P received, the lower the RF noise. Using parameters estimated for LISA [50], this  value  is  6.9  μcycles·Hz -1/2 ,  which  converts  to  an  optical  metrology  noise  of  7.34  pm·Hz -1/2 for λ=1.064 μm. With other metrology noises sufficiently small, the requirement 10 (15) pm·Hz -1/2 listed in Table 2 is feasible. Similar derivations of requirements apply to other space gravitational wave detection missions.  \nIn December 2015, LISA Pathfinder was successfully launched, reducing the amplitude spectrum of inertial/accelerometer noise to [46,243]:  \n[ P a( f )] 1/2  3 fm s -2 Hz -1/2 [1 + (10 -4 Hz/ f ) 2 + 16 (2 × 10 -5 Hz/ f ) 10 ], (2 × 10 -5 Hz  f  1 Hz).      (42 ）  \nFigure 19 (Color online) The S/C layout-time delay diagrams for the zeroth-generation TDI channel Michelson X0, the firstgeneration  TDI  channel  Michelson-X,  and  the  second-generation  TDI  channel  Michelson-X1.  The  vertical  lines  indicate  the trajectories of S/C in the time direction ( ① -③ indicate S/C i , i =1, 2, 3), and the ticks on each y -axis show the value of time delay with respect to the TDI ending time τ =0. The blue solid and red dashed lines show the paths of TDI channels, and indicate two groups of interfered laser beams. The figure (Wang diagrams) is adapted from refs. [241,242].  \n<!-- image -->  \nAll space gravitational wave detection plans are based on this, each setting its specific goals. For instance, LISA adopts a slightly relaxed criterion:  \n[ P a,LISA( f )] 1/2  3 fm s -2 Hz -1/2 [1 + (0.4 mHz / f ) 2 ] 1/2 [1 + ( f /8 mHz) 4 ] 1/2 , (1×10 -4 Hz  f  1 Hz),    (43)  \nApart from the low-frequency red relaxed factor [1 + (0.4 mHz/f) 2 ] 1/2 , it also includes a blue relaxed factor [1 + (f/8 mHz) 4 ] 1/2 . Similar considerations apply to the Taiji project. For discussions on other space gravitational wave missions, refer to Section 8.', 'Interferometry': 'There  are  two  methods  for  space  optical  interferometric  gravitational  wave  detection:  one  uses Michelson  stellar  interferometers,  and  the  other  uses  generalized  Michelson  time-delay  laser interferometers. The following two subsections discuss these methods. Section 3 discusses the potential development of continuously tunable fiber delay lines for use in both stellar interferometers and spacebased time-delay interferometers.', '7.1 Stellar Interferometers and Gravitational Wave Detection': "In 1868, Fizeau [243, 244] proposed placing an aperture with two narrow slits in front of a telescope to use interferometry to overcome the broadening of stellar images caused by atmospheric disturbances and  measure the angular size of stars.  In  1873-74,  Stephan  [245],  following  Fizeau's  idea,  observed several bright stars using the 80 cm Foucault reflector at the Marseille Observatory and a double-slit aperture,  and  indeed  saw  interference  fringes.  When  the  distance d between  the  double  slits  of  the aperture increased to 1.22 λ / θ , where θ is the angular diameter, the visibility of the fringes should have been zero. However, when the slit distance was increased to 65 cm, the interference fringes remained clearly visible, leading Stephan to conclude that the angular diameters θ of these bright stars were all less than 160 milliarcseconds (approximately 1.22 λ / d , where λ is the wavelength).  \nIn 1890, Michelson [246] used a stellar interferometer [247], the principle of which is schematically shown in Figure 20, to measure the angular sizes of celestial bodies. For a spherical star with an angular diameter θ , the angular diameter measured when the interference fringes disappear (1.22 λ / d ) corresponds to the measured angular diameter. In 1921, Michelson and Pease installed a 6-meter interferometer in front of the 2.5-meter Hooker telescope at the Wilson Observatory and measured the angular diameter of Betelgeuse (α Orionis) to be 0.47 arcseconds [250], based on its parallax value of 0.018 arcseconds, thereby determining a diameter of 3.84 × 10 8 km (2.58 AU), marking the first measurement of a star's diameter.  \nFigure 20 Schematic of the original Michelson stellar inferterometer. When the fringe visibility vanishes, the 1.22 λ / d is the  \n<!-- image -->  \napparent angular diameter. (https://www.wikiwand.com/en/Michelson\\_stellar\\_interferometer).  \nIn  1954  at  Jodrell  Bank,  after  completing  a  radio  stellar  intensity  interferometer  together  with students Jennison and das Gupta [248], Hanbury Brown and theoretical physicist Twiss established the mathematical foundation of optical intensity interferometry [249]. In 1955, Hanbury Brown and Twiss confirmed  their  concept  in  the  laboratory.  Subsequently,  Hanbury  Brown  gathered  and  assembled equipment to measure the angular diameter of Sirius. Borrowing a 1.5-meter searchlight used during World War II, they replaced the lamp with a photomultiplier tube to receive and record starlight from Sirius.  They  amplified  the  fluctuations  in  anode  current,  and  linked  telescopes  modified  from  two searchlights with a baseline separation of up to 10 meters to measure the current fluctuations. Over a cold period of four months, they accumulated 18 hours of integrated data, measuring Sirius' angular diameter to be 6.82±0.52 milliarcseconds (mas), marking the first direct measurement of the angular diameter of a main-sequence star [250].  \nThe successful use of searchlights led to the establishment of a 10-meter prototype stellar intensity interferometer, prompting further development of an instrument dedicated to measuring main-sequence stars. After much effort, the Department of Scientific and Industrial Research (DSIR) in the UK partnered with  the  University  of  Sydney  and  the  University  of  Manchester  to  construct  the  first  astronomical instrument  in  Australia  designed  to  measure  the  diameters  of  numerous  stars-the  Narrabri  Stellar Intensity Interferometer (NSII). The NSII consisted of a large circular track with detector separations ranging from 10 to 188 meters. Operating from 1963 to 1974, it measured the angular diameters of 32 stars.  Alongside  measurements  by  Michelson  and  Pease,  it  preliminarily  completed  systematic measurements of the angular diameters of main-sequence stars and giants [251, 252].  \nIn  1974,  Labeyrie  [253, 254]  completed  a  starlight  interferometer  with  a  baseline  of  12  meters, composed of a pair of separated 0.25-meter telescopes. Interference fringes produced by Vega (α Lyr) in the 500-750 nm spectral range were clearly visible, allowing determination that Vega's angular diameter was less than 5 mas (further confirming observations by Hanbury Brown et al. [255] using the Narrabri Stellar  Intensity  Interferometer,  which  measured  Vega's  483.5  nm  optical  angular  diameter  to  be 3.56±0.40 mas), marking the beginning of modern long baseline stellar interferometry.  \nHere, examples such as the Mark III stellar interferometer [256, 257] at Mt. Wilson Observatory in California  and  the  Space  Interferometry  Mission  (SIM)  [30]  are  briefly  mentioned  to  illustrate advancements. Astronomical  observations require tracking celestial bodies, and Michelson interferometry requires that the optical path lengths of two interfering light beams are equal or differ within the coherence length. For ground-based astronomical facilities, variations in optical path length due to Earth's rotation need to be compensated for. The fringe tracking mechanism of Labeyrie et al.'s stellar interferometer involves moving optical devices on orbiting platforms between two astronomical telescopes to ensure that the optical path difference between two beams is equal or within the coherence length. The tracking method of the Mark III stellar interferometer uses a maneuverable variable optical delay line, as shown in Figure 21. Optical path difference compensation can also be achieved using continuously adjustable fiber delay lines, as detailed in Section 7.3.  \nFigure 21 (Color online) Schematic diagram of a current Michelson stellar inferterometer using the delay line.  \n<!-- image -->  \nThe  Mark  III  stellar  interferometer  operated  from  1987  to  1992,  conducting  measurements  of celestial positions, stellar angular diameters, and binary star orbits within the wavelength range of 450800 nm. The Mark I and Mark II were experimental setups that paved the way for the development of Mark III technology. Mark III consisted of three alt-azimuth telescopes (N, S, E) mounted on pedestals, forming two baselines NS and ES each of 12 meters for measuring celestial positions. Interferometric measurements  with  multiple  baseline  telescopes  can  synthesize  imaging.  Mark  III  also  featured  two telescopes in orbit, movable between 11 stations, with an effective aperture of 7.5 cm (sufficiently small), obviating  the  need  for  adaptive  optics  at  Mt.  Wilson.  The  United  States  Navy  Precision  Optical Interferometer (NPOI) [28, 29] is a follow-up to Mark III, comprising a Y-shaped three-arm imaging interferometer with arm lengths of 250 m and baselines up to 437 m, making it the highest-resolution optical imaging instrument for ground-based astronomical observations today. NPOI is a complex system capable  of  achieving  milliarcsecond  precision  in  astronomical  angle  measurements.  The  angular resolution of ground-based stellar interferometers is constrained by atmospheric disturbances, with limits in the tens of microarcseconds.  \nAnother successor project of Mark III, the Space Interferometry Mission (SIM) [30], also known as SIM Lite (formerly SIM PlanetQuest), aimed primarily at detecting Earth-sized planets in the habitable zones of nearby stars outside the  solar system. SIM experienced multiple delays and was ultimately canceled in 2010.  \nIn Section 7.1.1, we discuss methods and conceptual tasks for detecting low-frequency gravitational waves  using  stellar  interferometers.  Gravity  fields  and  gravitational  waves  permeate  the  universe, affecting both the light sources observed by stellar interferometers and the interferometers themselves. In extremely precise measurements, the influence of gravitational waves on the light paths and detectors must be considered.  \nGravitational waves with periods longer than the observational timespan produce a simple apparent motion  pattern  in  the  sky  [258].  Therefore,  accurately  measuring  the  proper  motion  of  quasi-stellar objects will be a method for detecting ultra-low-frequency (10 fHz - 300 pHz) gravitational waves. We discuss this detection method in Section 7.1.2.", '7.1.1 Detection of Micro-Hertz and Sub-Micro-Hertz Low-Frequency Gravitational Waves': 'If gravitational waves pass through the path of starlight reaching a stellar interferometer, they can affect  the  measurements  of  the  angles  of  starlight  sources  made  by  the  interferometer.  The  spectral changes in the measured celestial angles can reveal information about gravitational waves.  \nIn 2021, Park et al. [259] proposed a method to detect microhertz low-frequency gravitational waves using a stellar interferometer. This method involves two coherent light beams from distant stars detected by  two  spacecraft  at  the  ends  of  a  space-based  long  baseline,  suitable  for  amplitude  or  intensity interferometric measurements, as shown in Figure 22. Park et al. analyzed a stellar interferometer using visible light (550 nm) from the Crab Nebula received by spacecraft separated by 1 Gm (Gigameter), considering shot noise and possible acceleration noise achievable. They suggested that the sensitivity for detecting gravitational waves around 250 μHz could be comparable to that of LISA. In 2022, Fedderke et  al.  [260]  further  proposed  and  analyzed  methods  for  using  a  stellar  interferometer  to  detect  submicrohertz gravitational waves (10 nHz - 1 μHz) from planetary and white dwarf sources. These methods are still in early stages of discussion and require further research and analysis.  \nFigure 22 (Color online) A classical method with a three satellites configuration (left) and an intensity interferometer configuration of two satellites (right) for stellar interferometry of detecting gravitational waves. The figure is from ref. [259].  \n<!-- image -->', '7.1.2 Detection of Ultra-Low-Frequency (10 fHz-300 pHz) Gravitational Waves [126]': 'The gravitational wave periods at 10 fHz are approximately 10 14 seconds, or about 3.16 million years. At 300 pHz, the gravitational wave period is 3.3 × 10 9 seconds, roughly equivalent to 100 years. Gravitational waves with periods longer than the observation time span create a simple apparent proper motion pattern in the sky [258]. Therefore, precisely measuring the apparent proper motion of quasars  \nand similar astrophysical objects can be a method to detect ultra-low-frequency gravitational waves (10 fHz - 300 pHz). Gwinn et al. [261] used this method to constrain the normalized energy density spectrum of stochastic gravitational waves, with frequencies below 2 × 10 -9 Hz and above 3 × 10 -18 Hz (including the ultra-low-frequency range), to be less than 0.11 h -2 (95% confidence level) times the critical density of  our  universe.  Using  the  Hubble  constant value obtained from Planck 2018, H 0  =  (67.4  ±  0.5)  km s -1 Mpc -1 [262, 263], gives h = 0.674, leading to a constraint of 0.24 according to its corresponding critical density value (labeled as QA [Quasar Astrometry] in Figure 4 of reference [126]). Long baseline optical interferometers/space-based stellar interferometers capable of microarcsecond and nanoarcsecond (nas) astrometry measurements are technically feasible [264]. Using such interferometers, precise astrometric measurements  of  quasar  proper  motions  can  be  improved  by  four  orders  of  magnitude,  achieving accuracy on the order of nas/yr. In terms of sensitivity, measuring or constraining Ωgw(f) can achieve a precision of 2.4 × 10 -9 or better (labeled as QAG [Quasar Astrometry Goal] in Figure 4 of reference [126]).  \nUsing equation (1), we obtain the constraint on the characteristic strain hc ( f ):  \nhc(f) < 4.2  10 -19 (Hz/f), for 3  10 -18 Hz < f < 2  10 -9 Hz.                                                                          (44)  \nWith a four-order improvement in angular resolution, the sensitivity reaches:  \nhc(f) = 4.2  \n 10 -23 (Hz/f), for 3  10 -18 Hz < f < 2  10 -9 Hz.                                                                          (45)  \nCurves (44) and (45) are plotted in Figure 2 of reference [126], labeled as QA and QAG. Using equation (1), Ωgw( f ) can be converted to [ Sh ( f )] 1/2 and plotted in Figure 3 of reference [126].', '7.2 Generalized Michelson Interferometry and Time-Delay Interferometry': "In the Michelson interferometer, the wavefront is split into two parts to enter two different paths, and then the two wavefronts are recombined for interference. For white light, Michelson must precisely match  the  lengths  of  the  two  optical  paths  to  produce  interference  fringes.  The  original  Michelson interferometer used white light interference: the white light Michelson interferometer requires the two arms to be nearly equal in length, differing by no more than one wavelength. The earliest Michelson interferometer  was  used to compare the  speed of light in  different directions, such as in the  famous Michelson-Morley  experiment  and  in  comparisons  of  the  meter  prototype  and  its  replicas.  In  1925, Michelson and Gale [265] used a carbon arc light source with good coherence to measure the Sagnac effect  around a rectangular plot of 2010 feet × 1113 feet (latitude: 41°46'), obtaining 0.230 ± 0.005 interference fringes and an error in measuring the Earth's rotation speed of 2%.  \nAfter the invention of lasers, the coherence length became longer. People could construct unequal arm Michelson interferometers. Although the interference accuracy is not limited by frequency standards (such as clocks), it is still constrained by coherence length (frequency noise) [equation (40)]. Another  \nconfiguration  of  the  Michelson  interferometer  is  the  Mach-Zehnder  interferometer.  Doppler  tracking requires  measuring phase of beat to be generated by the local  oscillator as a function of time.  After appropriate consideration of heterodyne, equations (31) and (32) remain applicable.  \nBecause  the  deviation  of  the  electromagnetic  wave  velocity  in  plasma  from  vacuum  velocity is inversely proportional to the square of the frequency, time uncertainties caused by solar wind or ionized gas in radio and microwave propagation are smaller in the Ka band (32 GHz) than in the X band (8.4 GHz), and smaller in the X band than in the S band (2.3 GHz). This is one of the motivations for using the  Ka  band  and  X  band  for  Doppler  tracking  by  the  Cassini  spacecraft  to  achieve  better  noise performance. Another motivation is the shorter wavelength, which improves measurement accuracy. At optical frequencies, wavelengths are more than four orders of magnitude smaller, and plasma effects are eight  orders  of  magnitude  smaller.  Therefore,  optical  methods  began  to  be  used  in  the  field  of gravitational  wave  detection  when  higher  optical  path  measurement  sensitivity  was  required.  As sensitivity improves, we need to suppress various noise below the target sensitivity level. This requires (i) reducing acceleration noise and achieving drag-free technology; (ii) minimizing laser noise as much as possible. LISA Pathfinder demonstrated basic drag-free technology [46, 47]. To reduce laser noise requirements, laser stability is needed. The best method is to achieve absolute stability; for example, by locking to iodine molecular lines. However, the stability currently achievable is not yet sufficient to meet the  required  sensitivity  of  10 -21 strain  levels.  To  reduce  laser  noise  requirements,  the  delayed interferometry measurement method (TDI) was developed.  \nFor space-based laser interferometric gravitational wave antennas, arm lengths vary according to the dynamics of the solar system's orbit. To achieve the required sensitivity, laser frequency noise effects must be suppressed below core noise levels (such as optical path noise, acceleration noise, etc.) using the selected TDI configuration. As in equation (40), better matching of optical path lengths results in better elimination of laser frequency noise and makes it easier to achieve the desired sensitivity. With precise matching,  laser  frequency  noise  is  almost  completely  eliminated,  as  in  the  original  Michelson interferometer.  \nFor interferometers with unequal arm lengths (paths), interference can only occur when the path length difference is less than the coherence length. For example, the coherence of a single-mode narrow linewidth (1 kHz) Nd:YAG laser exceeds 100 kilometers and can be used for interferometry with path differences less than 100 kilometers; for interferometric phase accuracy requirements, better matching of the two path lengths is needed, as in LISA's 25 meters and Taiji's 30 meters. The LISA and Taiji gravitational wave detection space missions involve three spacecraft moving along the solar orbit, with nominal arm lengths of 2.5 million kilometers and 3 million kilometers respectively. Orbital motion causes arm length differences of about 1%, approximately 20,000 to 30,000 kilometers, which prevents interference.  \nTDI  design  ensures  that  the  optical  path  difference  between  the  dual  split  laser  beams  is  close enough to produce interference, as shown in Figure 23, where beams are split into two paths (input to  \noutput)  labeled  as  1  and  2,  with  their  beat  frequencies  compared  by  the  phase  meter  receiving  and recording signals from the local oscillator.  \nFigure 23 (Color online) Laser light is divided into two beams ABC (1) and ADEFC (2) from input A to output C. At each link, the phase of beat with the local laser oscillator is received and recorded in the phase meter to produce phase differences.  \n<!-- image -->  \nThere are two methods to discuss interference noise:  \n(I) According to equation (40), the noise of the interferometric signals in arms 1 and 2 includes: (i) Frequency noise corresponding to the optical path difference, i.e., the first term of equation (40); (ii) Noise embedded during the propagation and amplification processes of the laser in arm 1 and arm 2. From the TDI configuration and the phase data structure at each connection point and receiving end, the interference signal can be directly explored. In principle, there is no need to use precise timing data in the link, but familiarity with the link structure is required (see Section 13).  \n(II) Consider the laser frequency noise effects (significant) at each node (A, B, C, D, E, F), along with  the  noise  embedded  during  the  laser  propagation  and  amplification  processes  in  arms  1  and  2, including timing noise. After canceling out the large number of frequency noise effects, the total noise is the interference noise at the receiving end.  \nThe second method is commonly used in LISA to date, as seen in references [159, 266-282]. The first method is implicit in references [53, 55, 82, 105, 128, 145, 241, 242], based on the idea of splitting a laser into two beams for continuous propagation, without directly considering the large number of laser frequency noises in different optical paths.  \nThe first method can determine the optical path difference and the local time of laser pseudo-code signal transmission in drag-free spacecraft (or standard equal-arm formation). The conversion between local  time  and  the  inherent  time  of  the  solar  system  center  can  be  determined  by  laser  pseudo-code ranging and its stability.  \nThese  two  methods converge, as explained in Section 13. Since the laser and the laser-carrying pseudo-codes arrive at each spacecraft simultaneously, the time determined by the pseudo-code at each spacecraft is identical, i.e., the time recorded by the spacecraft clock or ultra-stable oscillator (the local time of the spacecraft). Therefore, the requirements for spacecraft clocks or ultra-stable oscillators can be  relaxed.  Pseudo-code  ranging  can  be  verified  by  Ka-band  microwave  communication  distance  \nmeasurement, spacecraft orbit determination, and planetary ephemeris checks. Moreover, this data can also be used to refine planetary ephemerides.  \nTDI  was  first  studied  in  the  concept  research  of  the  ASTROD  mission  [53,  55].  In  deep  space interferometric  measurements,  long  distances  are  always  involved.  Due  to  the  distance,  the  laser undergoes significant attenuation when it reaches the receiving spacecraft. Amplification is required to transmit the laser back or to another spacecraft. This process involves phase-locking the local laser to the incoming weak laser and transmitting the local laser back or to another spacecraft. Liao et al. [59, 60] demonstrated phase locking of local oscillators with 2 pW incoming laser light in the laboratory. Dick et al. [283] achieved phase locking for incoming weak lasers with 40 fW power. These developments meet the feasibility requirements for e-LISA/NGO [49] and ASTROD-GW [81] power demands. Subsequent experiments  further  improved  feasibility  [284,  285,  286,  287],  with  Sambridge  et  al.  [287]  recently demonstrating phase tracking of sub-fW lasers, averaging slip times exceeding 1000 seconds.  \nIn  the  1990s,  two  TDI  configurations  [53,  55]  were  used  in  concept  studies  of  ASTROD interferometric  measurements,  with  optical  path  differences  calculated  using  Newtonian  dynamics. These two TDI configurations are the unequal-arm Michelson TDI configuration and the Sagnac TDI configuration used for the three-spacecraft formation flight. The principle is to split two laser beams into paths 1 and 2, and interfere at their respective ends. For the unequal-arm Michelson TDI configuration (X configuration),  as  shown  in  Figure  19,  one  laser  beam  starts  from  spacecraft  1  (S/C1),  points  to spacecraft 2 (S/C2), and is received by S/C2, where it optically phase-locks the local laser; then the phase-locked laser beam points back to S/C1 and optically phase-locks another local laser in S/C1; and so on, returning to S/C1 via path 1:  \nPath 1: S/C1 → S/C2 → S/C1 → S/C3 → S/C1.                                                                                  (46)  \nThe second laser beam also starts from S/C1 but follows path 2:  \nPath 2: S/C1 → S/C3 → S/C1 → S/C2 → S/C1.                                                                                  (47)  \nIt returns to S/C1 and interferes coherently with the first laser beam. If the two paths have exactly the same optical path length, laser frequency noise cancels out; if the difference in optical path length is small,  laser  frequency  noise  is  largely  canceled  out.  In  the  Sagnac  TDI  configuration  (Sagnac-α configuration), the two paths are:  \nPath 1: S/C1 → S/C2 → S/C3 → S/C1, Path 2: S/C1 → S/C3 → S/C2 → S/C1.                                (48)  \nIn the Sagnac-α2 configuration, the two paths I and II are I: Path 1 + Path 2 and II: Path 2 + Path 1. Since then,  similar  calculations  have  been  carried  out  for  the  original  LISA  [45],  eLISA/NGO  [49], LISA-like space gravitational wave detection concepts with an arm length of 2 × 10 6 km, zero inclination  \nASTROD-GW [77, 78], non-zero inclination ASTROD-GW [81], and current LISA and Taiji (using relativistic ephemerides calculations) [82, 145, 288, 289, 290, 291].  \nSince  the  proposal  of  the  first  and  second  generations  of  TDI  in  1999  [159,  292],  delayed interferometric  measurements  in  LISA  have  been  extensively  studied.  The  first-generation  TDI compensates for static unequal arm length differences, while the second-generation TDI compensates to some extent for motion. A configuration set (46) and (47) represents the Armstrong, Estabrook, and Tinto meaning of the first-generation TDI X configuration [159]. As with the TDI X configuration, if the laser emits from S/C 2, points to S/C 3 and S/C 1, it is TDI Y; if the laser emits from S/C 3, points to S/C 1 and S/C 2, it is TDI Z. TDI X, TDI Y, and TDI Z are channels available for data analysis. TDI X, TDI Y, and TDI Z form a basis in a linear space, linearly transforming into a base set that are uncorrelated in data analysis (under the equal-arm length approximation) [159, 292]:  \nA = Z - X; E = (X - 2Y + Z) / (6 1/2 ); T = (X + Y + Z) / (3 1/2 ).                                                              (49)  \nFor the triangular formation of three spacecraft, there are still Relay-U, Beacon-P, and Monitor-D [277], as shown in Figure 24 [241, 242, 293].  \nNumerical TDI is discussed in Section 13. For many other aspects of TDI, readers can refer to the review [292].  \n<!-- image -->  \n<!-- image -->  \n<!-- image -->  \nFigure 24 (Color online) The S/C layout-time delay diagrams for the first generation TDI channels Michelson-X, Relay-U, BeaconP, and Monitor-D. The vertical lines indicate the trajectories of S/C in the time direction ( ① -③ indicate S/C i , i =1, 2, 3), and the ticks on each y -axis show the value of time delay with respect to the TDI ending time τ =0. The t rel is the time with respect to the starting time t 0 at starting S/C of TDI. To avoid the crossing caused by TDI paths at noninteger delay times and show the paths clearly, extra trajectory lines are plotted for S/C2 (dotted green lines) and S/C3 (dashed orange lines). The blue lines show the paths of TDI channels, the solid line and dashed line indicate two groups of interfered laser beams. The figure (Wang diagrams) is from refs. [241,242,293].  \n<!-- image -->", '7.3 Continuously Tunable Fiber Delay Lines and Gravitational Wave Detection': 'Due to the precision requirements of detection and the limited coherence length of lasers, for the Taiji and LISA detectors, the requirements for the optical path difference between the two paths should be less than 30 m and 25 m respectively. The three-spacecraft formations of the Taiji and LISA detectors are influenced by gravitational forces such as those from the Sun and planets, causing arm lengths to vary by up to 1% over time. The variation in arm length for the Taiji detector can reach 30,000 km, and for the LISA detector, it can reach 25,000 km. Simple Michelson interferometric paths do not meet the requirement.  For  the  first-generation  Delayed  Michelson  double-interferometric  paths  X,  Y,  Z,  the optical path differences can reach 370 m for the Taiji detector and 240 m for the LISA detector, both of which do not meet the conditions of being less than 30 m and 25 m [145]. For the second-generation Delayed Michelson interferometric fourfold paths X1, Y1, Z1, the optical path differences are less than 5 m for the Taiji detector and less than 3 m for the LISA detector, both of which meet the requirements [145].  \nUsing continuously adjustable fiber delay lines in spacecraft can compensate for the optical path differences in the first-generation Delayed Michelson interferometric double paths X, Y, Z of the Taiji and LISA detectors, thereby meeting their requirements. The principles of manufacturing continuously adjustable  fiber  delay  lines  are  now  described  according  to  reference  [54].  This  method  can  also  be applied  to  other  space  gravitational  wave  detection  missions  such  as  LISAmax,  TianQin,  AMIGO, ASTROD-GW, Super-ASTROD, etc.  \nFor spaceborne astronomical interferometers, an all-fiber delay line is an ideal choice [54, 300]. A schematic configuration of the fiber delay line is shown in Figure 24. The same configuration can also be used for ground-based astronomical interferometers and space gravitational wave detectors [54, 294]. As  shown  in  Figure  25,  we  can  use  EOMs  (Electro-Optic  Modulators)  or  PZTs  (Piezoelectric Transducers), or both, to achieve a change in optical path length from 0 to δ. The difference in total path length between the lower and upper paths in loop k (k = 1, 2, 3, ..., n) is 2 k -1 δ . To continuously change the optical delay from 0 to δ, we can apply voltage to EOMs and/or PZTs. For reaching δ to 2 δ , we can quickly turn off the voltage, switch the laser path of the first loop from the upper path to the lower path, and then increase the  voltage applied to EOMs and/or PZTs, similar to what is done in laser length measurement [295] and fiber optic length measurement [296]. When the optical path difference reaches 2δ, to change the optical delay continuously from 2δ to 3δ, we can quickly turn off the voltage applied to EOMs and/or PZTs simultaneously, switch the laser of the first loop to the upper path, switch the laser of the second loop to the lower path, and then increase the voltage applied to EOMs and/or PZTs again, like a binary counting circuit [295]. Through n loops, we can continuously change the optical delay to a total length of 2 n δ within the switching time precision. The voltage applied to EOMs and PZTs can be quickly turned on and off within 1 millisecond. In 1996, Tsinghua University was developing a timeadjustable directional coupler with a rapid adjustment time of 1 ms in the laboratory. Below is a report on the progress at that time.  \nFigure 25 (Color online) A schematic configuration of fibre delay lines. The figure is from refs. [54,55].  \n<!-- image -->  \nUsing silicon wafers as polishing substrates [297], we have improved the side polishing technique, simultaneously polishing more than eight optical fibers, achieving high sensitivity (85-90 dB) in droplet testing. Utilizing these side-polished optical fibers (semi-couplers) and liquid crystals, reference [298] manufactured  polarizers  with  high  extinction  ratios.  Due  to  this  improved  polishing  technique,  the effective interaction length of the semi-couplers manufactured in reference [298] has been quadrupled compared to those in reference [299]. Following the methods of Digonnet and Shaw [299], reference [298]  produced  mechanically  adjustable  fiber-directional  couplers,  achieving  close  to  100%  power tunability in single-mode fiber couplers.  \nTo meet the adjustable speed requirements of delay lines, the first step involves applying a voltage to  change  the  refractive  index  of  the  matching  liquid  crystal  between  two  semi-couplers  [294].  The expected  response  time  is  1  ms.  This  time  scale  is  much  shorter  than  characteristic  atmospheric turbulence time scales and is sufficient for delay lines in ground-based interferometers. The second step involves  planning  research  on  delay  lines  made  from  mechanically  adjustable  directional  devices, followed by the development of electrically adjustable directional couplers. Subsequently, we plan to create a prototype of a continuously variable all-fiber delay line with a 1 ms time scale and study its characteristics.', '8 Detection Sensitivity and Noise Requirements': "This  section  discusses  the  detection  sensitivity  density  spectrum  of  various  space  laser interferometric gravitational wave detectors and their noise requirements.  \nIn the frequency domain, for an incident gravitational wave  \nh ( f ) = h + ( f ) e+ + h × ( f )e × ,                                                                                                                      (50)  \nthe response h ( f ) of a space-based gravitational wave detector is given by  \nh ( f ) ≡ F + ( θ , φ , ψ , f ) h + ( f ) + F × ( θ , φ , ψ , f ) h × ( f ),                                                                                (51)  \nwhere F + ( θ , φ , ψ , f ) and F × ( θ , φ , ψ , f ) are complex response functions that depend on the spatial angles ( θ , φ ) of the gravitational wave source and its polarization angle ψ .  \nThe  strain  power  spectral  density  sensitivity  of  the  space-based  gravitational  wave  detector  is defined by the effective noise power spectral density:  \nSn ( f ) = Pn ( f )/ R 2 ( f ), （ 52 ）  \nwhere R ( f ) is the average response function of the detector to the spatial direction and polarization of the gravitational  wave  signal.  The  gravitational  wave  response  function R 2 ( f )  relates  the  power  spectral density of the incident gravitational wave signal to the power spectral density of the recorded signal in the detector.  \nFor  any  available  TDI  configuration,  the  average  response  function R TDI( f )  can  be  numerically computed from equations (31) and (32) by calculating the phase difference between the two paths. For TDI X0, X, A, and E configurations, their average response functions R X0 2 ( f ), R X 2 ( f ) , R A 2 ( f ) and R E 2 ( f ) can be approximated as follows according to [300, 301, 302]:  \nR X0 2 ( f ) ≈ 0.3/(1+0.6 x 2 ), R X 2 ( f ) ≈ (16 x 2 sin 2 x ) R X0( f ) ≈ (16 x 2 sin 2 x )·0.3/(1+0.6 x 2 ), R A 2 ( f ) = R E 2 ( f ) ≈ (9/20) (16 x 2 sin 2 x ) / (1+0.6 x 2 ),                                                                                  (53)  \nwhere x = 2πfLx = 2π fL . At low frequencies, the gravitational wave response of X/A/E channels scales wiyh x 4 , and the impact of 1% arm length variation in space missions on the gravitational wave response is minimal. At low frequencies, the response of TDI T channel to gravitational waves correlates with arm length differences [302]:  \nR T 2 ( f )  0.8 x 4 [( L 12 -L 23) 2 +( L 12 -L 13) 2 +( L 13 -L 23) 2 ].                                                                            (54)  \nFigure 26 illustrates the average response functions of LISA and Taiji X, A, E, and T TDI channels. The response of the T channel varies with arm length changes, where the dark gray region corresponds to a 50% reduction in arm length variation over 400 days, and the combined dark and light gray regions correspond to a 90% reduction in arm length over 400 days. Histograms on the right side of the upper and lower images show the T channel response at frequencies of 0.01 mHz, 0.1 mHz, and 1 mHz [302].  \nOptical measurement noise contributes four times to the Michelson X TDI channel, while test mass acceleration noise contributes sixteen times. By dividing displacement by twice the round-trip distance 2L, we convert it into strain, thus obtaining:  \n4 P op,mission/(2 L mission) 2 = P op,mission/( L mission) 2 .                                                                                            (55)  \nSimilarly,  the  factor  1/(2 L )  is  also  applied  to  path  length  changes  caused  by  gravitational  waves: 16 P acc,mission/(2 L mission) 2 = 4 P acc,mission/( L mission) 2 . The strain power spectral density Pn ( f ) is therefore:  \nPn ( f ) = P 12 op ( f ) + P 21 op ( f ) + P 13 op ( f ) + P 31 op ( f ) + {4 cos 2 ( f / f *)  [ P 21 acc ( f ) + P 31 acc ( f )+] + 4 P 12 acc ( f ) + 4 P 13 acc ( f )}/ [(2π f ) 4 ( L ) 2 ] = 4 P op ( f ) + 8 [1+ cos 2 ( f / f *)]  P acc ( f )/[(2π f ) 4 ( L ) 2 ],                                                                (56)  \nIn the above equations, assuming each detector in each spacecraft has the same noise spectrum P op ( f ) and P acc ( f ). The cos 2 ( f / f * ) term arises from the combination of inertial and accelerometer noise in times t and t -2 L in spacecraft 1.  \nFigure 26 (Color online) The average responses of TDI X, A, E, and T channels for LISA (upper panel) and Taiji (lower panel). The T channel is sensitive to the variances of arm lengths, and the dark grey region shows the best 50% percentile in 400 days, and the dark and light grey areas together show the best 90% percentile. The right panel in each plot shows the histogram of the  T channel's response at frequencies 0.01, 0.1, 1 mHz. The figure is from ref. [302].  \n<!-- image -->  \nP X( f ) = 4 P op ( f )/(4 L 2 ) + 8 [1 + cos 2 ( f / f *)] P acc ( f )/[(2π f ) 4 (2 L ) 2 ] = P op ( f )/ L 2 + 2 [1 + cos 2 ( f / f *)] P acc ( f ) / [(2π f ) 4 ( L ) 2 ].                                                                    (57)  \nThe contribution of noise to the gravitational wave signal in the X channel PXn(f)P\\_{Xn}(f)PXn(f) is approximately:  \nP Xn( f ) ≈ 16 x 2 sin 2 x P X( f ).      [related to R X 2 ( f ) ≈ (16 x 2 sin 2 x ) R X0( f ))].                                                  (58)  \nFrom equations (50), (51), (57), and (58), the strain power spectral density sensitivity of the spacebased gravitational wave detector's Michelson X TDI channel, i.e., its effective noise power spectral density, is given by:  \nS Xn( f ) = P Xn( f )/ R X 2 ( f ) = 16 x 2 sin 2 x P X( f )/(16 x 2 sin 2 xR X0( f )) = P X( f )/ R X0( f ) ≈ { P op,mission/( L mission) 2 + 2[1+cos 2 ( f / f *)] P acc,mission/[(2π f ) 4 ( L mission) 2 }/ [0.3/(1+0.6 x 2 )] = [10/(3 L mission 2 )] [ P op,mission 2 + 2[1+cos 2 ( f / f *)] P acc,mission/(2 π f ) 4 ] × (1+0.6 x 2 ) ≈ [10/(3 L mission 2 )] [ P op,mission 2 + 4 P acc,mission/(2 π f ) 4 ] × (1+0.6 x 2 ).                                                                (59)  \nIf the contribution of frequency noise effects and other noise suppressions can be neglected below the core noise level, the gravitational wave sensitivity given below is excellent for most estimates and analyses [300]:  \nSXn,mission 1/2 ( f )=(20/3) 1/2 (1/ L mission)×[(1+( f /(1.29 f mission) 2 ] 1/2 ×[( P op,mission+4 P acc,mission /(2π f ) 4 )] 1/2 Hz -1/2 , (60)  \nwhere L mission is the arm length of the gravitational wave space mission, and f mission=1/(2π L mission) is the detector's  critical  (characteristic)  frequency.  Equation  (60)  is  an  excellent  approximation  of  the gravitational  wave  sensitivity  averaged  over  sky  positions  and  polarizations  for  the  Michelson interferometer X TDI channel in a nearly equilateral triangle formation. For derivation, please refer to references [300] and the cited references therein. It also serves as an excellent approximation for the classic Michelson interferometer in triangular formation.  \nAs mentioned in Section 1.12 and Figure 12, the strain power spectral density amplitude used in this article corresponds to the gravitational wave sensitivity averaged over sky positions and polarizations for the nearly equilateral triangular formation as given by equation (60).  \nAs  described in Section 6, LISA's requirements for accelerometer noise spectrum amplitude are given by equation (43), i.e., [ P acc,LISA( f )] 1/2  3 fm s -2 Hz -1/2 [1 + (0.4 mHz / f ) 2 ] 1/2 ×[1 + ( f /8 mHz) 4 ] 1/2 , (1 × 10 -4 Hz  f  1 Hz); its requirement for optical ranging noise (optical length noise) is  \n[ P op,LISA( f )] 1/2  10 pm Hz -1/2 [1 + (2 mHz / f ) 4 ] 1/2 . (61)  \nSubstituting  these  two  equations  into  equation  (60), we  obtain  the  strain  power  spectral  density amplitude / gravitational wave sensitivity for the LISA detector:  \nS n,LISA 1/2 ( f ) = (20/3) 1/2 (1/ L LISA)×[(1+( f /(1.29 f LISA)) 2 )] 1/2 × [( P op,LISA+4 P acc,LISA /(2π f ) 4 )] 1/2 Hz -1/2 ,            (62)  \nThe strain  power  spectral  density  amplitude  /  gravitational  wave  sensitivity  for  LISA  as  depicted  in Figure 12 is given by this equation. The coefficient for acceleration noise listed for LISA in Table 2 is the factor 3 fm s -2 Hz -1/2 from equation (43). The laser ranging noise (optical length noise) is the factor 10 pm Hz -1/2 from equation (61). After careful estimation, LISA [51] relaxed this coefficient to 15 pm Hz -1/2 in its Definition Study Report.  \nThe Taiji mission has the same requirements for accelerometer noise spectrum amplitude as LISA, given by equation (43). Its requirement for optical ranging noise is:  \n[ P op,Taiji( f )] 1/2  8 pm Hz -1/2 [1 + (2 mHz / f ) 4 ] 1/2 .                                                                                  (63)  \nThe strain  power  spectral  density  amplitude  /  gravitational  wave  sensitivity  for  the  Taiji  detector  as depicted in Figure 12 is given by equation (60) corresponding to Taiji.  \nAccording to equation (40), the laser frequency noise effect in the X channel is:  \nδφ X  2π δ  ( f ) × δt X , （ 64 ）  \nwhere δ ν (f) is the frequency noise of the laser source at frequency f , and δt X is the optical path delay between the two paths of the X channel. (64) also applies to other TDI channels.  \nFigure 27 shows the core noise and laser frequency noise effects for LISA (upper image) and Taiji (lower image) in the first 400 days of scientific mission. The core noise and laser frequency noise effects of the T channel vary due to changes in arm length.  \nFigure 27 (Color online) The core noise (SN, acceleration noise +optical metrology noise) and the laser frequency noise (LFN) effect for LISA (upper panel) and Taiji (lower panel) in the first 400 days. The dark grey shows the highest laser noise effect in 50% of the first 400 days, and dark grey together with light grey show the laser noise effect 90% percentile in T channel. The dark blue shows the highest core noise in 50% of the first 400 days in T channel, and the dark blue together with light blue show the core noise in 90%. The figure is from ref. [302].  \n<!-- image -->  \nThe strain power spectral density amplitude/gravity wave sensitivity of ALIA is plotted based on its parameters from Table 2 and its corresponding equation (60), excluding red and blue factors.  \nASTROD-GW has requirements for the noise spectral amplitude of its inertial/acceleration sensor:  \n[ S acc,ASTROD-GW( f )] 1/2  3 fm s -2 Hz -1/2 [1+(0.1mHz / f ) 2 ] 1/2 .                                                                    (65 )  \nIts requirements for optical ranging noise are:  \n.                                                                    (66)  \nFigure 28 (Color online) The average sensitivities of TDI X, A, E, and T channels by considering the core noises and laser noise effect. The dark color areas show the best sensitivity in the 50% percentile of the first 400 days for X, A, and T channels, and the dark color together with the light color areas show the best sensitivity in 90% percentile. The sensitivities of T channels for equal-arm configuration are shown by dashed curves [303,304]. The figure is from ref. [302].  \n<!-- image -->  \n10-16  \n10-17  \n10-18  \n10-19  \n10-20  \n10-3  \n10-2  \n10-1  \nThe strain power spectral density amplitude/gravity wave sensitivity of ASTROD-GW, according to  equation (60), is plotted in Figure 12. Folkner's  mission has noise requirements and gravity  wave sensitivity similar to ASTROD-GW. The requirements for the inertial/acceleration sensor noise spectral amplitude  and  optical  ranging  noise  of  aASTROD-GW  are  both  ten  times  smaller  than  those  of ASTROD-GW. LISAmax has noise requirement for inertial/acceleration sensors similar to ASTRODGW, and its optical ranging noise is similar to aASTROD-GW.  \nThe strain power spectral density amplitude/gravity wave sensitivity of DECIGO and B-DECIGO in Figure 12 is plotted according to reference [87]. The Big Bang Observer is similar to DECIGO. The strain power spectral density amplitude/gravity wave sensitivity of AMIGO, b-AMIGO, and e-AMIGO is plotted according to reference [106]. The sensitivity curve of DO is similar to e-AMIGO.  \nSuper-ASTROD has noise requirements for inertial/acceleration sensors similar to aASTROD-GW, and its optical ranging noise requirement is 5 nm/Hz 1/2 . The dashed lines extending to lower frequencies around ~100 µHz in Super-ASTROD, aASTROD-GW, and ASTROD-GW represent the removal of the red factor [1 + (0.1 mHz / f ) 2 ] 1/2 from the requirements for inertial/acceleration sensor noise.  \nThe  strain  power  spectral density amplitude/gravity wave sensitivity of TianQin in Figure 12 is plotted according to reference [99].  \nThe  strain  power  spectral  density  amplitude/gravity  wave  sensitivity  of  gLISA  in  Figure  12  is plotted according to references [115-118].", '9.1 Compact Binaries': "From the understanding of astrophysical binary star evolution and the structure of the Milky Way, it  is  estimated  that  the  Milky  Way  contains  roughly  ten  million  compact  binary  and  multi-star gravitational wave sources. Most of these sources contribute only to the gravitational wave foreground, with only approximately tens of thousands of systems discernible by the millihertz space gravitational wave detectors like LISA or Taiji/TianQin, capable of detecting individual inspiral and merger-generated gravitational waves and studying the structure of the Milky Way (references 45, 49, 51). LISA and Taiji are expected to detect about 10,000 double white dwarf binaries, with the majority in the frequency range of 3-6 mHz (orbital periods of approximately 300-600 seconds) (references 45, 49, 51). These sources belong to the population of normal Type Ia supernovae and special supernova progenitors. For a review of electromagnetic counterparts of compact object gravitational wave mergers, refer to reference [305]. In the 3-6 mHz frequency band, LISA/Taiji are more sensitive compared to microhertz low-frequency gravitational wave detection missions like ASTROD-GW, Folkner's mission, and LISAmax (Figure 12). LISA/Taiji will launch first and, in addition to validating binary stars, these signals can also serve as calibration sources for low-frequency gravitational wave detection missions [306, 307, 308].  \nBelow frequencies of a few millihertz, approximately ten million compact binaries will form the detectable foreground for LISA/Taiji and microhertz low-frequency gravitational wave detectors [111, 309, 310, 311, 312, 313, 314, 315). At these frequencies, microhertz low-frequency gravitational wave  \ndetectors are more sensitive compared to millihertz low-frequency gravitational wave detectors (Figure 12).  More  gravitational  wave  sources  will  be  separately  detected,  and  microhertz  low-frequency gravitational  wave  detectors  will  improve  the  observational  outcomes  of  millihertz  low-frequency gravitational wave detectors like LISA and Taiji/TianQin [111].  \nGround-based gravitational wave detectors have already detected neutron star mergers and over a hundred  stellar-mass  black  hole  mergers.  Based  on  the  inferred  distribution  densities  from  existing observations, it is anticipated that millihertz low-frequency gravitational wave detectors will detect tens to hundreds of events involving neutron stars, stellar-mass black holes, or white dwarf mergers. These events will further enhance our understanding of various merger processes [51].  \nSection 1.9, Table 1, and Figure 10 discuss multi-band observations simulations for millihertz, midfrequency and high-frequency gravitational waves, illustrating that the observation of stellar-mass black hole  mergers  will  achieve  measurement  accuracy  approaching  or  surpassing  one-thousandth  in cosmological model parameters and the Hubble constant and will improve our understanding of physical cosmology and dark energy significantly [107].", '9.2 Massive Black Holes and Their Co-Evolution with Galaxies': "The astronomical community has discovered relationships between the  masses of  supermassive black holes and the masses of their host galaxy nuclei, as well as between the masses of supermassive black holes and the velocity dispersions of their host galaxies. These relationships indicate a connection between central supermassive black holes and the evolution of galaxy structure. Observational evidence suggests that supermassive black holes exist in the majority of local galaxy groups. Newly fueled quasars may arise from the merger of two gas-rich massive galaxies. Space interferometers in the low-frequency range  (100  nHz  -  100  mHz)  for  gravitational  wave  observations  and  pulsar  arrays  in  the  ultra-lowfrequency range (300 pHz - 100 nHz) will become key tools for studying the coevolution of galaxies and supermassive black holes.  \nThe standard theory for the formation of supermassive black holes involves the merger of various supermassive black hole binaries. Using space gravitational wave detectors and pulsar timing arrays, gravitational waves from these supermassive black hole binary mergers can be detected and explored across cosmic distances. Despite different merger tree models and merger tree models with seed black holes,  they  predict  significant  detection  rates  for  low-frequency  space  gravitational  wave  detectors (LISA/Taiji/TianQin and ASTROD-GW/Folkner's space mission/LISAmax) and pulsar timing arrays [49, 179, 316, 317, 318]. Pulsar timing arrays are most sensitive in the 300 pHz - 100 nHz frequency range, LISA & Taiji/TianQin are most sensitive in the 3 mHz - 0.1 Hz frequency range, and microhertz low-frequency gravitational wave detectors are most sensitive in the 100 nHz - 3 mHz frequency range (Figures 12-14, references [126] Figures 2-4). Low-frequency gravitational wave detectors will directly observe  how  supermassive  black  holes  form,  grow,  and  interact  throughout  the  history  of  galaxy formation.  ASTROD-GW/LISAmax  will  detect  stochastic  gravitational  wave  backgrounds  from supermassive black hole binary mergers in the 100 nHz - 100 μHz frequency range. LISA/Taiji can detect  \nthe spin, merger, and oscillation of double supermassive black hole sources with masses of 10 5 - 10 7 solar masses, and the spin of double black hole sources with masses of 10 4 solar masses, at an expected rate of 10 to 1000 detections per year [51]. Similar numbers of sources are expected for microhertz lowfrequency gravitational wave detection missions, but with better angular resolution (see section 10.3), covering gravitational wave event masses ranging from 10 5 to 10 10 solar masses (reference 81). These observational results are crucial for studying the coevolution of galaxies hosting supermassive black holes. Low-frequency gravitational wave detection missions will explore the cosmic horizon beyond z ~ 20.  \nFigures 12-14 depict trajectories of some examples of double supermassive black hole gravitational wave sources' inspirals, mergers, and oscillations on the plots. The black lines represent gravitational wave inspirals, mergers, and oscillations emitted by various equal-mass black hole binary mergers on circular orbits at different redshifts: solid line for z = 1, dashed line for z = 5, and long dashed line for z = 20. The trajectories of 10 6 M ☉ - 10 6 M ☉ double supermassive black hole mergers at z = 1, 10 5 M ☉ - 10 5 M ☉ double supermassive black hole mergers at z = 20, and 10 4 M ☉ - 10 4 M ☉ double supermassive black hole mergers at z = 20 are from reference [319]; others are scaled accordingly. Corresponding curves in Figures 13 and 14 are obtained by the transforming equation (1). Double supermassive black hole merger events have high signal-to-noise ratios for space detectors. Some equal-mass (ranging from 10 2 M ☉ to 10 10 M ☉ ) circular orbit events shown in Figures 12-14 are targets for space detectors, with lower mass events possibly detectable in early stages by future ground-based detectors.  \nBy detecting double  supermassive black hole  merger events and their background  gravitational waves, properties and distributions of supermassive black holes can be inferred and potential population models  tested.  For  decades,  pulsar  timing  arrays  have  been  collecting  data  to  detect  stochastic gravitational wave backgrounds from double supermassive black hole mergers. Modeling of the isotropic stochastic  background  spectrum  density  for  double  supermassive  black  hole  mergers  has  led  many authors to obtain frequency attributes of equation (35), with recent observational results from pulsar timing arrays listed in Table 6. The fit of isotropic random background equation (35) model essentially yields a result of A yr = 2.4 +0.7 -0.6, confirming the existence of stochastic gravitational waves in the ultralow-frequency range at a 4σ uncertainty level, broadly consistent with astrophysical understanding.", '9.3 Extreme Mass Ratio Inspirals (EMRI)': 'Extreme  mass  ratio  inspirals  are  sources  of  gravitational  waves  for  space  gravitational  wave detectors. LISA/Taiji are sensitive to central massive objects in the mass range of 10 4 - 10 7 solar masses. LISA/Taiji is expected to detect gravitational wave events ranging from hundreds to thousands per year [51]; ASTROD-GW is similar or more, particularly to larger central black holes, and it offers better angular resolution (section 10.3).', '9.4 Testing Relativistic Gravity': 'An  important  scientific  goal  of  millihertz  low-frequency  gravitational  wave  detection  is  to  test  \ngeneral relativity and precisely study black hole physics under strong gravity [45, 51]. Microhertz lowfrequency gravitational wave detection, with its higher precision in the frequency range of 100 nHz to 1 mHz, will further advance this goal in many aspects. This includes testing strong-field gravity, precise detection of Kerr spacetime properties, and measuring/constraining the mass of gravitons. Considerations related to these aspects have been discussed in reference [320]. The sensitivity at lower frequencies is crucial for enhancing the precision of various tests [320]. Further research in these areas will be highly valuable.', '9.5 Dark Energy and Cosmology': 'In the context of dark energy, determining the value of w in the dark energy equation of state  \nw = p / ρ (67)  \nis crucial. Here, w is a function of cosmic time, p represents pressure, and ρ denotes the density of dark energy. A cosmological constant for dark energy corresponds to w =-1. From observational cosmology, our universe appears to be nearly flat. In a flat Friedman-Lemaître-Robertson-Walker (FLRW) universe, the luminosity distance d L is given by:  \n(68)  \nwhere H 0 is the Hubble constant, ΩDE is the current dark energy density parameter, and w is the assumed constant state equation of dark energy. For non-constant w and non-flat FLRW universes, similar but more complex expressions can be derived. Here, we present Equation (68) to illustrate the process.  \nBy comparing the observed luminosity distance as a function of redshift z , one can determine the parameter w as a function of redshift and compare it with various cosmological models. Dark energy cosmological  models  can  be  tested  in  this  way.  Supernova  observations  and  gamma-ray  burst observations, together with redshift measurements of luminosity distances, are currently the focus of dark energy detectors. Gravitational wave detectors will provide even more precise results.  \nSpacegravitational wave detectors observing the inspiral of binary supermassive black holes and extreme  mass  ratio  inspirals  are  excellent  tools  for  determining  luminosity  distances.  Along  with redshifts determined from related electromagnetic observations of galaxies or galaxy clusters, these space gravitational wave detectors also serve as dark energy probes. The gravitational waveforms produced by mergers of massive black holes in the co-evolution of galaxies provide accurately calibrated luminosity distances at high redshifts. These binary inspiral signals can serve as standard candles/whistles [321, 322].  With  better  angular  resolution  (Section  10.3),  microhertz  low-frequency  gravitational  wave detection will have better opportunities to identify associated electromagnetic redshifts, thus improving the determination of the dark energy equation of state [80, 81].', '9.6 Primordial (Relic) Gravitational Waves': 'To directly detect primordial (or inflationary) gravitational waves, researchers can use frequencies lower  or  higher  than  the  LISA  band  [78,  323].  This  is  because  these  frequencies  may  have  fewer astrophysical foreground sources to mask the detection [324]. DECIGO [84] and Big Bang Observer [85] search for background gravitational waves at higher frequencies, while ASTROD-GW [78, 323] targets the lower frequency range. Their instrument sensitivity goals aim to reach the critical density level of 10 -17 . The main challenge lies in the magnitude of the foreground and whether it can be distinguished.  \nThe straight line in the lower left corner of Figure 12 corresponds to the upper limit of Ωgw=1×10 -16 for the cosmic microwave background of primordial gravitational waves. For ASTROD-GW, sensitivity can reach this level  when using a 6-spacecraft formation for the stochastic detection of gravitational waves. However, the expected upper limit of massive black hole gravitational wave foregrounds is higher than the sensitivity of 3-spacecraft ASTROD-GW. Detecting this background depends on whether the "foreground" from these massive black hole gravitational waves can be separated by different frequency dependencies or other characteristics distinguishable by the 6-spacecraft ASTROD-GW [80].  \nOther  potential  sources  of  gravitational  waves  within  related  frequency  bands,  such  as  those generated by cosmic strings and phase transitions, are also worth exploring, as detailed in references [50, 51].', '10.1 Geocentric Orbital Configurations': "These  orbits  are  around-Earth  orbits  far  from  lunar  orbits  (either  inside  or  outside).  The  orbital configuration of the OMEGA gravitational wave detection mission [325, 326] serves as an early example. Proposed to NASA as a candidate MIDEX mission in 1998, it was re-proposed in 2011 as a mission concept white paper. The mission consists of six identical spacecraft in a stable Earth orbit at an altitude of 600,000 kilometers, with two spacecraft at each vertex forming an approximately equilateral triangle. These orbits are stable, allowing for a three-year scientific operation plan, extendable if needed. The arms of the triangle are approximately 1 million kilometers (1 Gm) long. The mission formation is located beyond lunar orbit.  \nCurrently and potentially selected space mission concepts using around-Earth orbital configurations actively under study include TianQin, gLISA, and B-DECIGO. Research has been conducted on the orbit of  AMIGO  [105],  which  has  opted  for  a  heliocentric  orbit.  Two  mission  proposals,  GEOGRAWI [116]/gLISA [117] and GADFLI [118], use geostationary Earth orbits. These missions operate with three  \nspacecraft in a near-equilateral triangle formation at a distance of approximately 73,000 kilometers. The orbit  selection  for  B-DECIGO,  whether  it  will  use  a  heliocentric  or  geocentric  orbit,  has  not  been finalized.  \nTianQin is a gravitational wave detection mission in an orbit around Earth's center at a distance of 100,000  kilometers.  Three  spacecraft  form  a  nearly  equilateral  triangle  with  an  arm  length  of  about 170,000 kilometers, orbiting Earth with a period of 44 hours [99].  \nDuring the Earth's orbit around the Sun, there are times when sunlight can shine into the telescope's line  of  sight.  When  the  line  of  sight  passes  through  the  Sun,  a  sunshield  is  needed;  OMEGA  [326] proposed a solution that may also be applicable to other missions of this type. TianQin proposes using two groups of three spacecraft each, alternating observations every three months.", '10.2 Basic Orbital Configuration of LISA-Type Missions': "As shown in Figure 3, the basic LISA-type formation [38,45,293,327-332] orbits around the Sun with a circular orbit radius R (= 1 AU). Since the distance (arm length) L between spacecraft is much smaller than the circular orbit radius of 1 AU, we can consider the spacecraft orbits as perturbed orbits around the circular orbit. The orbital equations for the spacecraft relative to the formation center (center of mass) are known as Euler-Hill equations, Hill equations, or Clohessy and Wiltshire equations. Hill used these equations in the 19th century for lunar theory research [333]. Clohessy and Wiltshire [334], after the start of the space age in 1957, derived and used these equations to design terminal guidance systems for satellite rendezvous in 1960. They defined a coordinate system around the center-of-mass orbit of the formation (either around the Sun or Earth) called the CW coordinate system. The origin of this system lies on the circular reference (formation center) orbit, rotating at the same angular velocity Ω as the reference orbit.  \nTo maintain a perturbed orbit at a first-order distance O( α ) = O( L /(2 R )) from the origin and to remain stationary in the CW coordinate system, the orbital elements eccentricity e and inclination i must satisfy the following relationships:  \ne = 3 -1/2 i + O(α), i =ε ≡ α ≡ L /(2 R ). (69)  \nOne method to form a nearly equilateral triangle configuration with side length or arm length L (1 + O(α)) is to space the orbital nodes by 120 degrees and select true anomaly and argument of perigee such that each  spacecraft  reaches  its  maximum  altitude  (northern  hemisphere)  above  the  ecliptic  plane  at  the apogee (first configuration); another method is to place the apogee at the minimum altitude (southern hemisphere)  below  the  ecliptic  plane  (second  configuration)  [38].  With  these  choices,  the  mission configuration plane forms a 60-degree angle with the ecliptic plane, tangential to the formation center circular  orbit.  For  a  square  configuration,  spacing  the  orbital  nodes  by  90  degrees  suffices;  similar constructions can be made for any regular polygon or planar configuration. The first configuration rotates clockwise; the second configuration rotates counterclockwise. Thus, in the CW coordinate system, there  \nare only two planes at ±60 degrees to the reference orbit plane, where spacecraft (test particles) follow the CW equations, rotating rigidly around the origin at angular velocities of -Ω and Ω.  \nFollowing methods by Dhurandhar et al. [332] and Wang and Ni [289], the basic orbital equations for three spacecraft in a LISA-type configuration can be written. First, for any ellipse in the X-Y plane, the orbital equations are:  \nX = R (cos ψ + e ), Y = R (1 e 2 ) 1/2 sin ψ ,                                                                                                (70)  \nwhere R is the semi-major axis of the ellipse, e is the eccentricity, and ψ is the true anomaly. Define α as the planned arm length L of the orbital configuration relative to twice the average Earth orbit radius R (1 AU), i.e., α = L /(2 R ). Using a heliocentric coordinate system ( X , Y , Z ) and selecting initial time t 0 as a specific epoch in the Julian calendar, with X -axis in the direction of the vernal equinox. Define a set of spacecraft elliptical orbits as:  \n(71) . sin ) (cos , sin ) 1 ( , cos ) (cos f f f 2 1 2 f f f ε e ψ R Z ψ e -R Y ε e ψ R X / + = = + =  \nFor LISA (2.5 Gm arm length), R = 1 AU; e = 0.004811; ε = 0.008333. The relationship between true anomaly ψ f and mean anomaly Ω( t -t 0) is:  \n(72) ). ( sin 0 f f t-t Ω e = +    \nwhere Ω is 2π divided by one sidereal year. ψ f can be solved numerically through iteration. Specifically, define ψ k given by:  \n(73) 3. 2, 1, for , ) 1 ( 120 ) ( sin 0 =  = + k -k -t -t Ω ψ e ψ k k  \nDefine X f k , Y f k , Z f k (k = 1, 2, 3) to be:  \n(74) e R Z e -R Y e R X k k k / k k k . sin ) (cos , sin ) 1 ( , cos ) (cos f 2 1 2 f f      + = = + =  \nDefine φ 0 ≡ ψ E - 10º, where ψ E is the Earth's angle position relative to the X-axis at t 0. Define X f( k ), Y f( k ), Z f( k ), ( k = 1, 2, 3), i.e. X f(1), Y f(1), Z f(1); X f(2), Y f(2), Z f(2); X f(3), Y f(3), Z f(3) to be  \n(75) . , ] ) 1 ( 120 cos[ ] ) 1 ( 120 sin[ , ] ) 1 ( 120 sin[ ] ) 1 ( 120 cos[ f ) ( f 0 f 0 f ) ( f 0 f 0 f ) ( f k k k k k k k k Z Z + kY + kX Y + k- Y + kX X =  +  =   =      \nThe basic orbits for three spacecraft (for one-body central problem) are:  \n(76) ). , , ( ), , , ( ), , , ( ) 3 ( f ) 3 ( f ) 3 ( f S/C3 ) 2 ( f ) 2 ( f ) 2 ( f S/C2 ) 1 ( f ) 1 ( f ) 1 ( f S/C1 Z Y X Z Y X Z Y X = = = R R R  \nBoth LISA and TAIJI are space missions of LISA type located at an average distance of 1 AU from the Sun. The initial design positions can be obtained by selecting t = t 0. The initial design velocities can be obtained by calculating the position derivative at time t = t 0. For example, choosing t 0 = JD2461853.0 (March  22,  2028,  12:00:00),  the  initial  design  configurations  (positions  and  velocities)  of  the  three spacecraft of LISA at this time in J2000.0 ecliptic coordinate at the vernal equinox can be computed as described  in  Section  11,  followed  by  optimization  of  arm  length  and  velocity.  The  final  design configuration of the spacecraft is listed in Table 8, column three, from reference [145]. The initial design configuration of TAIJI's three spacecraft at this time in J2000.0 ecliptic coordinate at the vernal equinox can be calculated and listed in Table 9, column three, from reference [145]. Starting from these initial conditions, the orbital configuration can be optimized using planetary and lunar ephemeris as shown in Section 11.2, and the optimized final spacecraft initial state column is listed in the fifth column of Table 4. For other options in different periods (for example, in the period of 2036 approaching the time when the TAIJI program reaches the scientific orbit), the approach is the same.and so forth for other choices at different periods.  \nTable 8 Initial conditions of three S/C for optimized LISA with arm length 2.5 Gm at epoch JD2461853.0 (2028-Mar-22 nd 12:00:00) in J2000 equatorial (Earth mean equator and equinox) solar-system-barycentric coordinate system. The table is extracted from [145].", '10.3 Basic Orbital Configuration of ASTROD-GW': "ASTROD-GW's  basic  configuration  consists  of  three  spacecraft  located  near  the  Sun-Earth Lagrange points L3, L4, and L5, orbiting the Sun in nearly circular orbits, forming an equilateral triangle as shown in Figure 4 [76-81]. In the restricted three-body problem of the Earth-Sun-spacecraft system, the dominant force on the spacecraft comes from the Sun. Since the Earth-Sun orbit is elliptical, the Lagrange points are not stationary in the Earth-Sun rotating frame. The motion of test particles near L3,  \nL4, and L5 deviates from circular orbits by approximately O(e), where e (=0.0167) is the eccentricity of Earth's orbit around the Sun. However, spacecraft can be placed in halo orbits near each Lagrange point, largely compensating for the non-stationary motion of the Lagrange points to maintain nearly circular orbits around the Sun.  \nTable 9 Initial states (conditions) of three S/C of Taiji at epoch JD2461853.0 (2028-Mar-22nd 12:00:00) for our initial choice (third  column) and after optimizations (fifth column) in the J2000 equatorial (Earth mean equator and equinox) solar-systembarycentric coordinate system [145]  \nThe circular orbits of spacecraft near L3, L4, and L5 are stable or quasi-stable within 20 years (their orbits relative to their respective Lagrange points are also stable or quasi-stable, with deviations from circular orbits of the order O( e 2 ) AU). The deviation of the spacecraft triangle from an equilateral triangle is of the order O( e 2 ) in terms of arm length. For non-precession configurations, angular resolution faces an issue of image ambiguity. To address this problem, it is necessary to tilt the orbits relative to the ecliptic. When the spacecraft orbits have an inclination λ (in radians) relative to the ecliptic plane, the change in arm length is approximately O( λ 2 ). Therefore, to match these two effects (achieving O(10 -4 )), λ should be on the order of O(1°).  \nThis section derives the inclined analytical orbits of spacecraft in the solar gravitational field and explains angular resolution and how to eliminate image ambiguity. In Section 11, we will use planetary ephemerides  to  design  and  optimize  orbit  configurations  and  see  that  perturbations  from  all  planets except Earth are on the order of O(10 -4 ). Since L3, L4, and L5 are actually stable within 20 years, the Earth's influence has been considered. Therefore, a suitable tilted circular orbit can serve as our base orbit, with deviations from the optimized actual orbit configuration on the order of O(10 -4 ).  \nFor Super-ASTROD [83], three spacecraft can also be placed near the Sun-Jupiter orbital plane with a small inclination angle at Sun-Jupiter L3, L4, and L5 points, while one or two additional spacecraft have a larger inclination.  \nFor ASTROD-EM [105], three spacecraft will be placed near the Earth-Moon L3, L4, and L5 points. For  spacecraft  dynamics,  we  solve  the  restricted  four-body  (Earth,  Moon,  Sun,  and  spacecraft  with negligible gravitational field) problem. The results are not ideal.  \nIn the initial plan, ASTROD-GW's orbit selection is on the ecliptic plane, with an inclination λ = 0. The angular resolution of gravitational wave sources on the celestial sphere has ecliptic plane image ambiguity. Although resolution is good in most sky areas, it is poor near the ecliptic poles. After 2010, we redesigned ASTROD-GW's basic orbit with a small inclination angle to address these issues, while keeping arm length variations within the range of O(10 -4 ) [80, 81].  \nAccording to [80, 81], the basic idea is that if ASTROD-GW spacecraft orbits are slightly inclined by an angle λ, appropriately designed interferometric planes are also tilted by a similar angle. As the ASTROD-GW formation evolves, interferometric planes can be designed to precess on the ecliptic. Thus, there is no ambiguity in the angular position of gravitational wave sources near the polar regions, and angular  resolution  is  not  poor  near  the  poles  (see  the  next  subsection).  There  is  also  a  half-year modulation effect when measuring the angular momentum of the Sun and the Galaxy, which is beneficial.  \nThe following reviews the derivation of the formation (configuration) of ASTROD-GW spacecraft orbits tilted relative to the ecliptic plane. First, consider the circular orbits of spacecraft in the Newtonian gravitational-central problem (one-body central problem) in spherical coordinates ( r , θ , φ ):  \nr = a , θ = 90°, φ = ωt + φ 0,                                                                                                                    (77)  \nwhere a , ω , and φ 0 are constants. For the spacecraft considered here, we have a = 1 AU, ω = 2π/ T 0, T 0 = 1 sidereal year, and φ 0 is the initial phase in the coordinates under consideration. (For Super-ASTROD spacecraft, a = 5 AU, ω = 2π/ T 0, T 0 = 11 sidereal years; for µAries spacecraft, a = Mars orbit radius, ω = 2π/ T 0, T 0 = Mars year). The spacecraft orbit in Cartesian coordinates at time t is:  \nx = a cos φ = a cos( ωt + φ 0); y = a sin φ = a sin( ωt + φ 0); z = 0.                                                              (78)  \nNow, the orbit is transformed into an orbit with inclination angle λ , and the orbital plane intersects the xy plane (ecliptic plane) at the line φ = Φ0 in the xy plane. The active transformation matrix is", 'Th new spacecraft orbit is': 'For the three orbits with an inclination angle of λ (in radians), choose:  \nS/C I: Φ0(I) = 270°, φ 0(I) = 0°;  \nS/C II: Φ0(II) = 150°, φ 0(II) = 120°; S/C III: Φ0(III) = 30°, φ 0(III) = 240°.                                                                                                  (81)  \nDefine  \nξ ≡ 1 - cos λ = 0.5 λ 2 + O( λ 4 ),                                                                                                              (82)  \nFrom formulas (80) and (81), we obtain  \n(i) S/C I orbit  \n(ii) S/C II orbit', '(iii) S/C III orbit': 'One can easily check that [( x I ) 2 + ( y I ) 2 + ( z I ) 2 ] 1/2 = [( x II ) 2 + ( y II ) 2 + ( z II ) 2 ] 1/2 = [( x III ) 2 + ( y III ) 2 + ( z III ) 2 ] 1/2 = a hold fpr consistency.  \nCalculate the arm vectors V II-I = r II -r I , V III-II = r III -r II 和 V I-III= r I -r III  \n:  \na[-(3/2)cosot +aS[(342 /4)sinot + (3/4)cosot] VIa[-(3/2)sinot + [4)cosot] 86)  \nasinot- (3V2 /2)asinot (87) asim sinot  \na[(3/2)cosot \\_ (3/4) cosot] a[(3/2)sinct + [4)cosot] 88) (342 (3V2  \nThe closure relation V II-I + V III-II + V I-III = 0 is checked. The arm lengths are calculated to be  \n| V II-I| = 3 1/2 a [(1 ξ /2) 2 + sin 2 λ sin 2 ( ωt - 60°)] 1/2 , | V III-II| = 3 1/2 a [(1 ξ /2) 2 + sin 2 λ sin 2 ( ωt )] 1/2 , | V I-III| = 3 1/2 a [(1 ξ /2) 2 + sin 2 λ sin 2 ( ωt + 60°)] 1/2 .                                                                            (89)  \nThe fractional arm length variation is within (1/2) sin 2 λ which is about 10 -4 for λ around 1°.  \n(83)  \nThe cross-product vector N ( t ) ≡ V III-II × V I-III is normal to the orbit configuration plane and has the following components:  \n(90)  \nThe normalized unit normal vector n is then:  \n(91)  \nThe geometric center V c of the ASTROD-GW spacecraft configuration is  \n(92)  \nASTROD-GW configuration consists of 3 sets of 2-arm interferometers. The geometric centers of these 3 interferometers are approximately 0.25 astronomical units from the Sun. Reference [82] considers planetary perturbations and uses planetary ephemerides to numerically simulate and optimize orbital configurations  with  inclinations  of  0.5°,  1°,  1.5°,  2°,  2.5°,  and  3°.  In  Section  11.3,  we  use  the  1° inclination case as an example to explain. When the LISA configuration orbits the Sun, it resembles multiple detector arrays distributed along a 1 AU orbit, suitable for long-term observation of gravitational wave sources. Although the ASTROD-GW configuration extends up to 1.73 AU, its operation around the Sun also approximates multiple detector arrays on a 1 AU orbit.', '10.4 Angular Resolution': "Considering coherent gravitational wave sources' angular resolution. Taking the LISA mission as an example, LISA's detector configuration modulates azimuthal angles by 2π radians and inclination angles by 1.05 radians (60°) due to its orbital motion around the Sun, causing the antenna (for plane waves) response pattern to sweep across the sky once per year. The antenna response is not isotropic, but for single-frequency gravitational wave sources, LISA's average linear angular resolution over a year differs by less than a factor of 3 across all directions [45]. This holds true for all LISA-like gravitational wave detectors. Angular resolution is essentially inversely proportional to the strain signal-to-noise ratio. If LISA's inclination is 0.017-0.052 radians (1-3°), the angular resolution degrades by a factor of 30-10 (approximately the ratio of 1.05 radians sine to 0.03-0.1 radians sine); solid angle localization in the celestial sphere deteriorates by the square of this factor. Away from the polar regions ( θ >> 0.017-0.052 radians), the spherical degree localization in the celestial sphere becomes sin² θ .  If  the  signal-to-noise ratio decreases by a factor of p , the linear angular resolution worsens by p times. Compared to LISA, ASTROD-GW has lower sensitivity above 1 mHz, which worsens angular resolution due to these factors. In the 100 nHz-1 mHz range, ASTROD-GW has better sensitivity than LISA by a factor of 104. Hence, polar resolution is similar to LISA's for ASTROD-GW, while in other sky regions, linear resolution is approximately increased by 104 × sin θ (increasing by 104 times, but solid angle decreasing sin² θ [linear  \nangular  resolution  decreases  sin  θ]).  Although  there  is  a  slight  dependence  on  the  configuration inclination λ , beyond a factor of 3, ASTROD-GW's average antenna pattern is 52 × sin θ times better than LISA's in regions away from the poles. From Equation (91), the average time required is half a year rather than a full year [80, 81].  \nFor more complex sources, such as chirping gravitational waves from binary black holes, fitting is required for accuracy in parameter estimation. However, the trend in parameter accuracy remains the same: it is inversely proportional to the strain signal-to-noise ratio for similar cases.  \nFor Super-ASTROD, which has a strain signal-to-noise ratio five times better than ASTROD-GW in  the  low-frequency  region,  the  angular  resolution  is  also  five  times  better.  For  polar  resolution, ASTROD inclination strategy can be applied. However, due to Super-ASTROD's one or two S/C off the ecliptic, this may not be necessary. For ASTROD-EM, with a lunar orbit inclined about 5° to the ecliptic, and its intersection moving period is 18.61 years, the Lagrange point of the Earth-Moon also moves. Depending on the mission's time and duration, a slightly inclined orbit may not be necessary [105].  \nMost  Earth-orbiting  gravitational  wave  missions  suffer  from  dipole  smearing  and  poor  polar resolution. However, this is  not a major issue as  we only  need to consider two polarities to identify electromagnetic counterparts, and the polar regions are only a small part of the sky.", '10.5 Six/Twelve Spacecraft Formations': 'To detect relic gravitational waves more sensitively using correlated detection, the proposals for Big Bang Observer [85] and DECIGO [84] suggest placing 12 spacecraft in Earth orbit, divided into 3 groups spaced 120 degrees apart on the orbit. Two groups have 3 spacecraft each arranged in a triangular configuration similar to LISA, and the third group has 6 spacecraft arranged in two LISA-like triangles forming a star-shaped structure (Figure 6). Another configuration is to have each group with 4 spacecraft forming an approximately square configuration (also tilted relative to the ecliptic by 60 °) (Figure 6).  \nTo detect background or relic gravitational waves more sensitively, two triangular ASTROD-GW arrays  (i.e.,  6-S/C  constellation)  are  needed  for  correlated  detection.  The  second  near-triangular formation can again be placed near L3, L4, and L5, but with corresponding S/C separated from the first formation by 1 × 10⁶ km to 5 × 10⁶ km [81].', '11 Orbit Design and Optimization Using Ephemerides': "Despite  the  Sun's  dominance  in  the  solar  system,  other  planets  and  celestial  bodies  influence spacecraft  orbits,  especially  Jupiter,  Venus,  and  Earth.  Ephemerides  are  essential  for  orbit  design. Currently,  there  are  three  complete  fundamental  ephemerides  for  the  solar  system:  DE  ephemerides (Development  Ephemerides)  [24],  EPM  ephemerides  (Ephemerides  of  Planets  and  Moon)  [25],  and INPOP ephemerides (Intégrateur Numérique Planétaire de l'Observatoire de Paris) [26]. Any of these ephemerides can be used for orbit design and optimization. For ease of numerical handling, we usually use  the  CGC  (Center  for  Gravitation  and  Cosmology)  ephemeris  framework,  combined  with  initial conditions obtained from the DE ephemerides at a certain epoch, and subsequently evolve using postNewtonian approximations.", '11.1 CGC Ephemerides': "In  1998,  simulations  and  parameter  determinations  for  ASTROD  were  started  [335,  336].  We compiled a post-Newtonian ephemeris of the solar system, including the Sun's quadrupole moment, the eight planets, Pluto, the Moon, and the three largest asteroids. We refer to this working ephemeris as CGC  1.  We  used  this  ephemeris  as  a  base  and  added  random  terms  to  simulate  noise,  generating simulated  range  data.  For  the  ASTROD  mission,  we  used  Kalman  filtering  to  determine  the  fitting accuracy of relativistic gravitational parameters and solar system parameters over 1050 days.  \nTo better assess the measurement of Ġ/G accuracy, we also need to monitor the masses of other asteroids. For this, we considered all known 492 asteroids with diameters greater than 65 km to obtain an improved ephemeris framework-CGC 2-and calculated the disturbances these 492 asteroids cause to the ASTROD spacecraft [337, 338].  \nIn constructing the CGC ephemeris framework, we used the parametrized post-Newtonian (PPN) framework from Brumberg's book [339], with parameters β and γ parameterizing the post-Newtonian framework and its solar system barycenter metric and dynamic equations (where gauge parameter α is set to zero). These equations were used to construct numerical ephemerides for the eight planets, Pluto, the Moon, and the Sun (PPN parameters γ = β = 1, Sun quadrupole moment parameter J ₂ = 2 × 10⁻⁷). Initial positions and velocities at 0:00 on June 10, 2005 were taken from the DE403 ephemeris. Evolution was solved using a 4th-order Runge-Kutta method with a step size h = 0.01 days. In reference [336], the evolution of 11 bodies was expanded to 14 bodies, including the three largest asteroids Vesta, Juno, and Ceres (CGC 1 ephemeris). Due to the Sun's quadrupole axis being slightly inclined to the ecliptic plane (7°), this tilt was neglected in the CGC 1 ephemeris. In CGC 2 ephemeris, disturbances from an additional 489 asteroids were added.  \nIn the first optimization of the ASTROD-GW orbit [340-342], we used the CGC 2.5 ephemeris, considering only the three largest asteroids but adding Earth's precession and nutation; also considering solar and Earth's second harmonic to fourth harmonic. In subsequent simulations, we added perturbations from an additional 349 asteroids, termed CGC 2.7 ephemeris [289-291]. The differences in Earth's orbit evolution compared to DE405 within 3700 days from JD2461944.0 (June 21, 2028) are shown in figure 5  of  reference  [81].  The  difference  in  Earth-Sun  radial  distance  is  less  than  approximately  200  m. Differences in latitude and longitude of the Earth are less than 1 mas.", '11.2 Numerical Orbit Design and Orbit Optimization for LISA and Taiji': "The mission orbit formation configuration for LISA is shown in Figure 3. Initially, the arm length was chosen to be 5 Gm, but due to possible budget constraints, it was later reduced to 1 Gm. After the LISA  Pathfinder  met  all  requirements,  it  was  changed  to  2.5  Gm,  with  the  project  being  officially approved on January 25th of this year. The arm lengths of the three spacecraft must be kept as equal as possible,  and  the  relative  Doppler  velocities  between  any  two  spacecraft  must  be  less  than  10  m/s. Various previous works have analyzed and numerically studied different arm length configurations for LISA [327-332]. For LISA-like space gravitational wave detection missions, we follow Dhurandhar et  \nal.'s  analysis  procedure  [332]  in  Section  10.2  for  initial  selection.  Through  this  orbital  selection, numerical orbits are computed and optimized using CGC ephemerides, similar to what was done for the ASTROD-GW orbit design (see the next subsection). The final optimized initial conditions for LISA are listed in Table 3 of Section 10.2, column 3. The final optimized initial conditions for Taiji are listed in Table 4 of Section 10.2, column 5; Figure 29 illustrates changes in the spacecraft configuration within 2200 days, including arm length, line-of-sight velocities, angle between the two arms, and the angle between the spacecraft and the Sun.  \nFigure 29 Variations of the arm lengths, velocities in the line-of-sight direction, formation angles and angle between barycenter of the S/Cs and Earth in 2200 d for the Taiji S/C configuration with initial conditions given in column 5 (after optimization) of Table 9.  \n<!-- image -->", '11.3 Orbit Optimization of ASTROD-GW': 'The goal of orbit optimization for the ASTROD-GW mission is to equalize the three arm lengths of the ASTROD-GW formation as much as possible and reduce the relative line-of-sight velocities between the  three  pairs  of  spacecraft.  In  our  initial  optimization,  the  start  time  for  the  scientific  phase  of  the mission was chosen as noon on June 21, 2025 (JD2460848.0), and optimization was carried out using CGC 2.5 ephemerides for 3700 days [340-342]. The preparation time for the mission may be longer, with the possibility of extending the mission to exceed 10 years in duration. In subsequent optimizations [290, 291, 343],  we began optimization from noon on June 21, 2028 (JD2461944.0) using CGC 2.7 ephemerides for 20 years, which includes more asteroids than CGC 2.5. In both optimizations, the orbital configurations are set within the ecliptic plane with an inclination angle λ = 0. As the basic configuration of ASTROD-GW transitions to inclined precession orbits, we have redesigned and optimized our orbital configurations.  The  orbital  configurations,  starting  from  noon  on  June  21,  2035  (JD2464500.0),  are worked out using CGC 2.7.1 for 10 years, with inclinations of 0.5°, 1°, 1.5°, 2°, 2.5°, and 3° [82].  \nIn this section, we outline the design and optimization methods for the precession orbit formation (configuration)  according  to  [82],  using  CGC  2.7.1  with  an  inclination  of  1°  as  an  example.  The differences between CGC 2.7.1 and CGC 2.7 are detailed in Section 11.3.1. In Section 11.3.2, we review  \nhow the initial conditions of the spacecraft are selected as the starting point for numerical optimization. In  Section  11.3.3,  we  discuss  optimization  methods,  present  optimized  results  in  tabular  form,  and illustrate the characteristics of the orbits after optimization.', '11.3.1 CGC 2.7.1 Ephemerides': "In the CGC 2.7.1 ephemerides framework, in addition to Vesta, Ceres, and Pallas, we selected 340 asteroids from the Lowell database. The masses of these 340 asteroids are provided by Lowell data [344] rather than estimated based on classification as in CGC 2.7 [288, 289, 291]. The orbital elements of these asteroids are also updated from the Lowell database.  \nOver the period of 10 years starting from June 21, 2035, the differences in Earth's heliocentric distance between CGC 2.7.1 and DE430 calculations are within 150 m, while differences in longitude and latitude are  within 1.4  mas and 0.45 mas, respectively. These differences do not affect our TDI computation results.", '11.3.2 Preliminary Selection of Spacecraft Initial Conditions': "The right ascension of the Earth, R.A., at JD2464500 (June 21, 2035, 12:00:00) according to DE430 ephemerides is 17h57m45.09s, or 269.438°. The initial positions of the three spacecraft are derived by selecting ωt as 89.44° in Equation (81) (where φ = ωt + φ 0), and velocities are obtained by computing the derivative of t from Equation (80). S/C1 near the Lagrange point L3 is partially shadowed by the Sun within the Earth's line of sight (Figure 30, upper image), which obstructs communication with ground stations. To avoid this shadowing, we rotate forward the initial angles Φ0 and φ 0 for this instance (with an inclination of 1.0°) by 2.0°. The orbit of S/C1 is depicted in the lower image of Figure 30. The initial selections of the initial states of the three S/Cs under this scenario are listed in Table 10, column 3.  \nFigure 30 (Color online) S/C1 view from Earth before rotating the initial conditions by an angle (upper diagram) and after rotating by an angle 2.0° (lower diagram) for the case of inclination angle 1.0° (The rotation of 2.0° of S/C1 in orbit is equivalent  to a rotation of 1.0° viewed from Earth). The figure is from ref. [82].  \n<!-- image -->  \nTable 10 Initial states of $ C s for the configuration With an inclination angle of 1 after period optimization and after all optimizations in J2OOO equatorial solar-system-barycentric coordinate system [82]", '11.3.3 Optimization Methods and Post-Optimization Characteristics': "Our  optimization  approach  involves  modifying  initial  velocities  and  heliocentric  distances  to achieve (i) as balanced as possible arm lengths of the ASTROD-GW formation and (ii) reducing the relative Doppler velocities between the three pairs of spacecraft as much as possible.  \nDuring the actual optimization process, we use the following equation to modify the orbital mean period:  \nV new = V prev + Δ V  [1 - (1/3)(Δ T / T )] V prev.                                                                                          (93)  \nFor the case with an inclination of 1°, we calculate the initial selections of the three spacecraft orbits listed in Table 10, column 3, using CGC 2.7.1 ephemerides. The 10-year mean periods for the three S/Cs are 365.256 days (S/C1), 365.267 days (S/C2), and 365.266 days (S/C3). We adjust the initial velocities using Equation (93) to set the mean periods of S/C1, S/C2, and S/C3 to 365.255 days, 365.257 days, and 365.257 days, respectively. The initial conditions after this step are listed in Table 10, column 4. In the next step, we adjust the spacecraft eccentricity to near-circular using the following equations:  \nRnew = Rprev + Δ R  (1 - Δ R / R ) Rprev , Vnew = Vprev + Δ V  (1 - Δ R / R ) Vprev .                      (94)  \nHere, R is  the  spacecraft's  initial  heliocentric  distance.  Adjusting  ±(Δ R / R )  modifies  the  eccentricity without altering the orbital period. Through iterative optimization, the final optimized initial conditions are listed in Table 10, column 5. Figure 31 illustrates the initial and final optimized configuration of the spacecraft with a 1° inclination over 10 years, showing (a) arm lengths, (b) differences in arm lengths, (c) formation angles, (d) line-of-sight velocities, (e) inclination of the formation plane normal relative to the ecliptic plane normal, and (f) direction angles of the formation plane normal.  \nFigure 31 The variation of armlengths (a), difference of armlengths (b), angles between arms (c), velocities in the measure direction (d), inclination of the unit normal n of the ASTROD-GW formation (e) and azimuthal angle of n (f) in 10 years for the case with the nominal formation inclination angle 1°.  \n<!-- image -->  \nFor inclinations of 0.0°, 0.5°, 1.5°, 2°, 2.5°, and 3°, the optimization process is similar to that of 1.0°, and the results can be found in [82].", '12 Deployment of Earth-like Solar Orbit Formations': 'Extensive studies have been conducted on the deployment of spacecraft to Earth orbit, Earth-Moon Lagrange point halo orbits, and Sun-Earth L1 and L2 Lagrange point halo orbits. Here, we briefly discuss the methods for deploying spacecraft to various positions in Earth-like solar orbits. References [81, 345] provide preliminary designs for the transfer orbits of spacecraft from separation from the launch vehicle to near the L3, L4, and L5 points of the mission orbits.  \nIn the mission study of ASTROD I, the spacecraft obtains suitable delta-V before the final stage of separation from the launcher in low Earth orbit (LEO) to enter solar orbits via a ballistic (geodesic) Venus flyby. Employing a similar strategy, ASTROD-GW spacecraft can be directly launched into a designated  \nHohmann transfer orbit or near Venus flyby orbit. Thus, the major delta-V required for each spacecraft to reach its destination occurs near the destination, propelling the spacecraft to stay near the designated Lagrange points (or other Earth-centric solar orbit points). In Table 11, rows 2-4, we list the types of transfer orbits, transfer times, delta-V values for transfers, and specific impulse mass ratios for the three ASTROD-GW S/C. These estimates apply to any mission spacecraft deployed to the same positions.  \nTable 11 . Estimated Delta-V and Propellant Mass Ratio for Solar transfer of S/C  \nThe specific impulse mass ratios for spacecraft 1, 2, and 3 are approximately 0.5-0.55, 0.280, and 0.47, respectively. Relative to 500 kg dry mass, the total masses of the three ASTROD-GW spacecraft at  different  positions  in  the  solar  orbit  range  from  1111-1266  kg,  723  kg,  and  1035  kg  (including propellant and propulsion modules with a propellant mass ratio of 10% of dry mass).  \nEstimations  have been made for deploying spacecraft to other positions in solar orbits, listed in Table 11, rows 5-7. The basic setup includes: (i) spacecraft propelled by efficient propulsion modules (with a specific impulse Isp of 320s) and a propulsion module mass ratio of 10% of total propellant mass, suitable  for  high  delta-V  operations  upon  reaching  the  destination;  (ii)  separation  of  the  propulsion module upon achieving the target state.  \nFurther optimization studies are underway for the deployment of formations with a 20-year mission duration  and  inclinations  relative  to  the  ecliptic  plane  similar  to  LISA  and  ASTROD-GW  missions, following separation from the launcher [346].', '13 Time Delay Interferometry (TDI)': 'In spacecraft formation flight without drag, maintaining precise formation is challenging, and laser frequency stability is limited. As analyzed in Section 6, the noise in interferometric signals scales with laser frequency noise and path differences. To meet noise requirements, selecting appropriate paths for time delay interferometry (TDI) becomes crucial. Further discussions on first and second-generation TDI are provided in Section 7.2. Section 8 reviews the sensitivity of TDI to gravitational waves. Based on precise  orbit  calculations  and  optimizations  from  the  previous  section,  variations  in  arm  lengths  are  \nestimated to be tens of thousands of kilometers for LISA and Taiji, and up to hundreds of thousands of kilometers  for  ASTROD-GW.  TDI  indeed  presents  a  natural  solution.  Another  approach  is  using equilateral configurations like DECIGO and B-DECIGO, necessitating methods to maintain propulsion acceleration without affecting accelerometer sensitivity, see references [347] for more details.  \nFrom measurements to data suitable for theoretical work, it is divided into the following three steps according to reference [51]: L0: Raw data processed onboard the spacecraft. L0.5: Processed on the ground and publicly available. L1: Processed from L0.5 data into time delay interferometry measurement data, used for scientific purposes, forming L2 individual gravitational wave source waveforms, signals, and their parameters, and can be compiled into an L3 catalog.  \nWhen tracking lasers, pseudo-random codes [348] are commonly used. In recent analyses, reference [348] demonstrated the use of data structure and less stringent observational constraints/analysis of Taiji simulated data  for  distance  measurement,  successfully  replacing  pseudo-random  code  methods. This means  that  even  without  using  pseudo-random  codes,  it  is  possible  to  extract  and  track  from  data structures without strict timing requirements or the need for separate timing if there is strong computing power. This implies that interference signals are embedded in the data stream already.  \nThere are still many areas for further research regarding TDI configurations and data structures, such as optimizing configurations/combinations for higher frequencies [349,350].', '14 Payload': "Space-based gravitational wave detection primarily measures changes in the distance between two S/C (or celestial bodies) as gravitational waves pass through. The two S/C (or celestial bodies) must undergo geodesic motion (or such motion can be inferred). Due to the weakness of gravitational waves, distance  measurements  must  be  super-sensitive.  Typical  implementations  (missions)  consist  of  three spacecraft  forming  an  almost  equilateral  triangle  formation.  Three  spacecraft  use  interferometry  to measure distances among each other. Each spacecraft carries a payload consisting of two mass blocks, two telescopes, two lasers, a weak light detection and processing system, a laser stabilization system, and a drag-free system. Precision/optical clocks or absolute laser stabilization systems and absolute laser metrology systems can be used for the lower-frequency part of the space gravitational wave band or for higher accuracy.  \nWeak light phase locking and processing : This is crucial for solar orbit missions. For ASTRODGW, located at a distance of 260 Gm (1.73 AU), and other microhertz gravitational wave detection space missions,  it  is  necessary  to  lock  local  lasers  to  incoming  light  at  100  fW  for  amplification  and manipulation. For 100 fW (λ = 1064 nm) weak light, there are 5 × 10 5 photons per second. This is suitable for  100  kHz  frequency  tuning.  For  LISA,  85  pW  weak  light  locking  is  required,  which  has  been successfully demonstrated (Section 7.2). Sambridge et al. [287] were able to phase track sub-fW lasers with  average  slip  times  exceeding  1000  seconds,  a  major  breakthrough.  Frequency  tracking, modulation/demodulation, and encoding are the key focuses, making it a mature experimental technique  \nthat will benefit future microhertz gravitational wave detection. This is also crucial for deep space optical communication.  \nDrag-free  system  design  and  development :  The  drag-free  system  consists  of  high-precision accelerometers/inertial  sensors  for  detecting  non-drag-free  motion  and  micropropulsion  systems  for feedback  to  maintain  spacecraft  drag-free.  LISA  Pathfinder  successfully  demonstrated  drag-free technology in the frequency range above 100 μHz, meeting not only the requirements of LISA Pathfinder but also those of LISA. This success paved the way for all proposed space missions in 2017 (Table 2). However, more work is needed for the lower frequency range (100 nHz to 100 μHz). We discussed frequency sensitivity spectra and red factors in Section 6. To suppress red factors, position sensor noise must be flat to 100 nHz, requiring small gravitational accelerations generated by the spacecraft on the test  masses  and  capable  of  modeling  them  to  required  levels  for  low  frequencies.  The  self-gravity acceleration needs to be stabilized or measured in real time. For this, absolute laser metrology systems can be used to monitor the position of major mass distributions in S/C. To completely eliminate or exceed this factor, optical sensing and optical feedback control may be required. A laser metrology system based on absolute laser measurements is proposed to reduce noise for ASTROD's accelerometer/inertial sensor design, especially in the low-frequency range. In addition, ASTROD also needs to monitor the positions of spacecraft components for easy modeling [56,57].  \nMicropropulsion systems : Micropropulsion is needed for drag-free feedback control. Field emission electric  propulsion  (FEEP)  systems  have  high  thrust-to-weight  ratios  and  are  good  candidates  for micropropulsion systems. FEEP systems exhibit good sensitivity in the μN range. The main issue with FEEP is its lifespan. Cold gas thrusters have emerged as an alternative due to technical issues encountered in FEEP technology development. The GAIA mission carries cold gas thrusters used for AOCS (Attitude and Orbit Control System) [351]. MICROSCOPE [352] (https://microscope.cnes.fr/en/MICROSCOPE/GP\\_mission.htm) and LISA Pathfinder are equipped with cold  gas  thrusters  based  on  GAIA  technology.  Compared  to  FEEP,  the  main  drawback  of  cold  gas thrusters is the higher mass required per delta-V. The total mission duration is limited by the quantity of propellant stored in tanks. Therefore, if FEEP technology can be used later, it would be the preferred  \nchoice.  \nLaser  system :  Nd  lasers  with  non-planar  ring  oscillators  pumped  by  laser  diodes  are  available, outputting 2 W of power. Frequency noise must be suppressed to 30 Hz/Hz -1/2 in the relevant frequency range. The laser is the source of measurement, and a more stable laser system can simplify and improve the measurement and analysis processes, aiding in achieving higher precision. Recently, commercialized optical  clocks  with  an  accuracy  of  10 -15 have  become  available  [353-355].  Developing  space  laser systems with frequency noise of 3 Hz/Hz -1/2 or 0.3 Hz//Hz -1/2 should be explored experimentally.  \nLaser  frequency  standards/clocks :  Space  optical  clocks  and  optical  frequency  comb  synthesizer technology  are  essential  for  achieving  and  simplifying  the  sensitivity  goals  of  low-frequency gravitational  wave  detection  missions.  Another  use  of  optical  clocks  and  optical  frequency  comb synthesizers is calibrating optical metrology for missions like ASTROD-GW. This is crucial for laser  \nmetrology inertial sensors and monitoring internal distances within spacecraft, correcting local gravity variations caused by thermal effects, etc. All these measurements use lasers as standards. They require calibration using optical frequency standards or absolute stable laser frequencies based on atomic or molecular lines. The advent of space optical clocks and optical combs may simplify the experimental design of missions like ASTROD-GW.  \nCurrently, laboratory optical clocks [207,209] have achieved fractional errors at the 10 -19 level. Such clock accuracy or higher will be used for extremely sensitive measurements of gravity, motion, and inertial  navigation.  The  use  of  such  clocks  will  facilitate  the  detection  of  microhertz  low-frequency gravitational waves and inspire the demand for redesigned low-frequency microhertz space gravitational wave detection schemes. Exploring these schemes in more detail is worthwhile.  \nAbsolute laser metrology system : Using ultra precision laser frequency standards/clocks, an absolute laser metrology system can be constructed to monitor the position of various parts of the spacecraft for gravity modeling.  \nRadiation Monitor : A small radiation detector on the spacecraft will monitor the test mass charging of  inertial/drag-free  sensors.  This  radiation  monitor  can  also  be  used  to  measure  solar  high-energy particles (SEP) and galactic cosmic rays (GCR) in the fields of solar and galactic physics, and applied to space weather [356, 357].", '15 Summary and Outlook': 'The first artificial satellite, Sputnik, was launched in 1957. On December 3, 2015, the gravitational wave technology verification mission, LISA Pathfinder, was launched. This mission successfully tested and demonstrated drag-free technology, meeting the requirements of not only LISA Pathfinder but also those of LISA. This success paved the way for all proposed space gravitational wave missions (Table 2). Space gravitational wave missions are expected to be launched in eleven years. Weak light phase locking has been validated in the laboratory, and weak light technology is rapidly advancing. With the first direct detection of gravitational waves by LIGO and the success of the LISA Pathfinder mission, the future of space gravitational wave detection is bright.  \nThe  primary  scientific  goals  of  space  gravitational  wave  detectors  are  to  detect  (i)  gravitational waves from massive black hole mergers; (ii) extreme mass ratio inspirals; (iii) intermediate mass black holes; (iv) galactic compact binaries; (v) relic gravitational wave background; and (vi) to test theories of gravity. From Figures 12-14, it is evident that the signal-to-noise ratio (S/N) for double massive black hole mergers detected by gravitational waves is very high, and lower frequency bands (100 nHz - 100 μHz) are required for detecting larger-scale  mergers  with  high  S/N.  Therefore,  longer  arms  have  an advantage. In exploring the co-evolution of black holes and galaxies, long-arm missions complement PTA effectively. Long-arm missions with better angular resolution are also more effective in determining the state equation of dark energy, testing relativistic gravity, and potentially exploring inflation physics. Deploying S/C to any position on a heliocentric orbit takes less than 1.8 years with a specific impulse mass ratio of less than 0.55, within the practical range achievable by launch vehicles.  \nNow  is  the  time  to  seriously  consider  the  second  generation  of  space-based  gravitational  wave detectors. Laboratory optical clocks have achieved an error level of 10 -19 and continue to improve. This level of precision in clocks will be used for space applications. This advancement benefits the laser pulse ranging schemes for long-arm configurations. 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