Pith. sign in

REVIEW 2 major objections 5 minor 2 cited by

Alternative LISA-TAIJI networks: Detectability of the Parity Violation in Stochastic Gravitational Wave Background

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read LISA-TAIJIm, with TAIJI tilted opposite to LISA, is about ten times more sensitive than LISA-TAIJIp to circular polarization in the stochastic gravitational wave background at low frequencies.

desk verdict A credible, well-scoped forecast that LISA-TAIJIm beats LISA-TAIJIp by ~10x for V-mode SGWB detection at low frequencies; worth refereeing and citing, with minor caveats about noise approximations and code availability. read the letter →

arxiv 2412.18420 v2 pith:YSFSDMW2 submitted 2024-12-24 gr-qc

Ju Chen , Chang Liu , Yun-Long Zhang , Gang Wang This is my paper · ORCID
classification gr-qc PACS 04.30.-w04.80.Nn
keywords stochasticgravitationalwavebackgroundparityviolationcircularpolarizationoverlapreductionfunctionLISA-TAIJInetworktime-delayinterferometryFisherinformationmatrixpower-lawintegratedsensitivity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Parity violation in gravity would leave a circularly polarized component in the stochastic gravitational wave background (SGWB), and this paper asks whether the planned LISA and TAIJI missions can detect it by flying together. The comparison is between two network geometries: LISA-TAIJIp, in which both triangular constellations are tilted at +60 degrees to the ecliptic, and LISA-TAIJIm, in which TAIJI is tilted at -60 degrees. The central claim is that the opposite-inclination network is about one order of magnitude more sensitive to the circular-polarization (V) component at lower frequencies, because the misaligned detector planes avoid the cancellation that suppresses chiral signals. The paper quantifies this with signal-to-noise forecasts for power-law, single-peak, and broken power-law spectra and with Fisher-matrix parameter estimation, concluding that LISA-TAIJIm is the more promising configuration for testing parity-violating gravity in the millihertz band.

What carries the argument

The central object is the effective overlap reduction function for circular polarization, $\Gamma^V_{\mathrm{eff}}$, defined from the four A/E channel pairs of the two triangular detectors after the intensity component is projected out. It combines the per-pair overlap reduction functions $\Gamma^I_\kappa$ and $\Gamma^V_\kappa$ with their cross-term, so it measures how much of the V signal survives in the cross-correlation once the intensity I is removed. The hybrid Relay time-delay interferometry (TDI) scheme, a laser-noise-cancelling combination of delayed arm-link measurements, supplies the channel responses and noise power spectral densities used in the calculation; the opposite inclination of the TAIJIm constellation is what keeps $\Gamma^V_{\mathrm{eff}}$ from vanishing at low frequencies.

What would settle it

Recompute the V-mode effective overlap reduction function at $f\lesssim 1\,\mathrm{mHz}$ with full numerical orbits, time-varying arm lengths, and second-generation TDI, and check whether the LISA-TAIJIm to LISA-TAIJIp sensitivity ratio remains near ten; if the gap closes, the central claim is contradicted.

Watch

Extended reading notes

Core claim

The paper's core claim is that the detectability of parity violation in the isotropic SGWB is set by the V-mode effective overlap reduction function $\Gamma^V_{\mathrm{eff}}$, and that this quantity is much larger for LISA-TAIJIm than for LISA-TAIJIp at frequencies below about a millihertz. In the low-frequency limit the V-mode overlap reduction function for LISA-TAIJIp becomes negligible, so its polarization SNR saturates as the lower frequency cutoff is decreased, while LISA-TAIJIm retains a substantial correlation. Consequently, the power-law integrated sensitivity to the V component improves by roughly one order of magnitude, and for all three spectral models LISA-TAIJIm yields higher SNR over most of the amplitude-polarization parameter space and tighter constraints on the polarization fraction $\Pi$.

Load-bearing premise

The order-of-magnitude improvement rests on the simplified equal-arm-length noise model for the hybrid Relay TDI channels; if realistic LISA and TAIJI orbits change the low-frequency V-mode overlap reduction functions, the claimed gap could shrink.

Editorial extensions

If this is right

  • Mission planners would prefer the -60 degree TAIJI inclination when the science goal is detecting circular polarization in the millihertz band.
  • The LISA-TAIJIm network could reach a V-mode sensitivity comparable to its intensity sensitivity near 2 mHz, allowing both the background's total power and its chirality to be characterized in the same observation.
  • For power-law, single-peak, and broken power-law spectra, LISA-TAIJIm outperforms LISA-TAIJIp over most of the amplitude-polarization plane, so the choice of orbit affects not only detection but also parameter estimation.
  • Fisher-matrix forecasts indicate that the polarization fraction $\Pi$ would be measured with smaller uncertainty by LISA-TAIJIm, while spectral-shape parameters would be constrained to similar precision by either network.
  • The result gives a concrete target: a 3-year joint LISA-TAIJI observation with $\rho_{\rm thr}=10$ could probe the V component at fractional energy densities otherwise inaccessible to a single planar detector.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A mission-level corollary the paper does not spell out: TAIJI's inclination should be treated as a science variable, with the -60 degree option favored for parity-violation science at little loss of intensity sensitivity.
  • The paper assumes a constant polarization fraction $\Pi$; applying the same effective-overlap-reduction-function machinery to frequency-dependent $\Pi$, as predicted by some axion and Chern-Simons models, could reveal whether any spectral shape favors the opposite-inclination network even more strongly.
  • The same comparison could be run for other pairs of future space-based detectors, using the effective-overlap-reduction-function ratio as a pre-launch ranking criterion for chiral sensitivity.
  • Because propagation birefringence also converts parity violation into circular polarization, the improved V sensitivity of LISA-TAIJIm would strengthen constraints on parity-violating gravity from propagation as well as from generation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper compares two LISA-TAIJI network geometries for detecting a circularly polarized (parity-violating) stochastic gravitational-wave background: LISA-TAIJIp, in which both constellations are inclined at +60 degrees, and LISA-TAIJIm, in which TAIJI is inclined at -60 degrees. The authors compute effective overlap reduction functions for the intensity (I) and circular-polarization (V) components using hybrid Relay TDI and the SATDI package, construct power-law integrated sensitivity curves, evaluate SNRs for power-law, single-peak, and broken power-law spectra, and forecast parameter uncertainties with the Fisher information matrix. The central claim is that LISA-TAIJIm is roughly an order of magnitude more sensitive to the V component at low frequencies and provides better constraints on the polarization fraction.

Significance. If the result holds, it gives a concrete, actionable input to mission planning: choosing the -60 degree TAIJI inclination would improve the joint network's ability to detect a circularly polarized SGWB in the mHz band. The comparison is mostly geometric and does not rely on fitted parameters; the SNR and Fisher derivations follow standard references, and the use of SATDI for Relay TDI is a strength. The main caveat is the robustness of the predicted low-frequency suppression of the LISA-TAIJIp V-mode overlap reduction function to realistic orbit and TDI effects, which is the basis for the headline order-of-magnitude improvement.

major comments (2)
  1. [Section IV.A, Eq. (30), Fig. 2] The headline claim of approximately one order of magnitude higher V-mode sensitivity is a ratio in which the LISA-TAIJIp effective V-mode ORF is a small residual arising from the near-parallel-plane cancellation at low frequencies. The manuscript states in Section I that the evaluation uses 'realistic orbits' and a more robust TDI scheme, but Section II derives the noise PSD under the equal-arm-length assumption, Eq. (1), and the response model used for the ORFs is not documented. Because a modest absolute change in the suppressed LISA-TAIJIp ORF could substantially change the ratio, please clarify whether Fig. 2 uses full time-varying orbits with arm-length mismatch and constellation precession, and quantify how the low-frequency V-ORF for LISA-TAIJIp changes under such effects. If the current calculation is restricted to the equal-arm idealized geometry, the abstract and conclusion should say so explicitly.
  2. [Section V.C, Eqs. (38)-(39), Fig. 6] The Fisher-matrix comparison includes LISA-TAIJIp at fiducial parameters for which the total network SNR is low. The text says the fiducial SNRs are approximately 40, 25, and 50, but the subsequent paragraph and Fig. 5 indicate that the LISA-TAIJIp SNR is substantially lower than that of LISA-TAIJIm and, for the power-law model, is close to the rho_thr = 10 threshold used elsewhere. Since Eq. (39) requires rho >> 1 for reliable Fisher information forecasts, the quoted 1-sigma uncertainties for LISA-TAIJIp in Fig. 6 may not be valid in that regime. Please report the per-network SNR values at the fiducial points, and either restrict the FIM comparison to configurations with sufficient SNR or state the SNR values so the reader can judge the validity of the forecast.
minor comments (5)
  1. [Fig. 2 and Fig. 3 captions] The color assignments for the two networks are inconsistent: Fig. 2 says LISA-TAIJIp and LISA-TAIJIm are orange and blue, while Fig. 3 says they are blue and orange. Please unify the captions and the text describing the curves.
  2. [Section V.B] There is a typo, 'adpot', in the sentence introducing the broken power-law parameter range; it should be 'adopt'.
  3. [Fig. 5 caption] The caption contains the duplicated phrase 'spectrum spectrum'; please remove the repetition.
  4. [Eq. (33)] The power-law integrated sensitivity curve is defined as an envelope over all spectral indices alpha, but the paper does not state the grid or range of alpha used in the numerical maximization; please specify it.
  5. [Abstract and Section VI] The phrase 'sensitivity ... approximately one order of magnitude greater' refers specifically to the low-frequency part of the PLI sensitivity curve; the integrated SNR improvement for the spectral models studied is more modest, as seen in the SNR-ratio plots. Please make this distinction explicit in the abstract and conclusion so readers do not interpret the claim as a tenfold improvement in total detection SNR.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed sensitivity comparison is computed from independent ORF and TDI inputs, not from fitted parameters or a self-citation chain.

full rationale

The central claim (abstract; Sec. VI) is a numerical comparison of PLI sensitivities for LISA-TAIJIp and LISA-TAIJIm obtained from Eq. (32) using the effective overlap reduction functions of Eq. (30). These ORFs are computed by explicit sky integrals of detector response functions (Eq. 15) with the hybrid Relay TDI response and noise PSD stated in Section II (Eqs. 1-3). No parameter entering the final sensitivity ratio is fitted to the quantity being compared. The signal spectra in Eqs. (34)-(36) and the polarization fraction Pi are fiducial inputs taken from external literature, and the SNR and Fisher results are derived from those inputs rather than used to define them. Self-citations [26], [32], and [39]-[45] provide the orbital configurations, earlier long-wavelength formulas, the TDI scheme, and the SATDI code, but the present calculation is self-contained: it recomputes the ORFs and sensitivity curves for the two networks. The near-zero low-frequency V-mode ORF for LISA-TAIJIp is a geometric consequence of the +60/+60 parallel-plane configuration, not a normalization choice or a redefinition of the output. Therefore, no equation is equivalent to its input by construction, no fitted parameter is renamed a prediction, and no load-bearing uniqueness claim is imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends on standard SGWB correlation formalism, the assumed orbital geometries, the equal-arm-length TDI noise model, and the chosen analysis settings (e.g., Tobs = 3 years and SNR threshold 10). These are domain assumptions and analysis choices, not free parameters fitted to data; no new physical entities are introduced.

assumptions (4)
  • domain assumption The SGWB is stationary and isotropic, with only the I and V Stokes parameters nonzero (Q and U vanish).
    Stated in Section III, following [30,49]; this is the standard assumption for a primordial isotropic background, and Q/U are argued not to contribute after averaging over directions.
  • domain assumption Detector noises are Gaussian, stationary, and mutually uncorrelated across channels and detectors.
    Used in Section IV to obtain the cross-correlation variance in Eq. (20); this is the standard noise model for stochastic background searches.
  • domain assumption The TDI A and E channels have identical noise PSDs given by the equal-arm-length expression in Eq. (1).
    The paper explicitly adopts the equal arm-length approximation in Section II, citing the full hybrid Relay TDI expressions from [40] and the SATDI code for numerical evaluation.
  • ad hoc to paper The polarization fraction Pi(f) = V(f)/I(f) is constant over frequency.
    Stated in Section V.A as a simplification; this affects the Fisher-matrix and SNR forecasts for specific models but not the central ORF comparison.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Alternative LISA-TAIJI networks: Detectability of the Parity Violation in Stochastic Gravitational Wave Background." pith.science (2026). https://pith.science/paper/YSFSDMW2

@misc{pith2026241218420,
  author       = {Pith},
  title        = {Pith review of: Alternative LISA-TAIJI networks: Detectability of the Parity Violation in Stochastic Gravitational Wave Background},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YSFSDMW2}},
  note         = {Machine review of arXiv:2412.18420}
}
abstract

The detection of parity violation in the isotropic stochastic gravitational wave background (SGWB) will serve a crucial probe for new physics, particularly in parity-violating theories of gravity. The joint observations by the planned space-borne gravitational wave detectors, LISA and TAIJI, will offer a unique opportunity to observe such effects in the millihertz (mHz) band. This study evaluates the detectability of parity violation in the SGWB using two network configurations: LISA-TAIJIp and LISA-TAIJIm. The former configuration consists of LISA (inclined at $+60^\circ$ relative to the ecliptic plane) and TAIJIp (also inclined at $+60^\circ$), while the latter network pairs LISA with TAIJIm (inclined at $-60^\circ$). Our analysis demonstrates that the sensitivity of the LISA-TAIJIm network to parity violation in the SGWB is approximately one order of magnitude greater than that of the LISA-TAIJIp network at lower frequencies. To quantify the performance of the two networks, we evaluate the signal-to-noise ratios for different spectral shapes, including power-law, single-peak, and broken power-law models, and estimate parameter determination using the Fisher information matrix. The results confirm that LISA-TAIJIm outperforms LISA-TAIJIp in detecting the SGWB with circular polarization components, offering a superior opportunity to test parity-violating gravitational constraints on various mechanisms in the mHz band.

Figures

Figures reproduced from arXiv: 2412.18420 by the authors.

Figure 1
Figure 1. FIG. 1. Configurations of alternative LISA-TAIJI networks. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The effective overlap reduction functions (top) and average sensitivities (bottom) of the LISA-TAIJI networks for the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The power-law integrated sensitivity curves for the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The SNRs for the SGWB with power-law (top), single-peak (middle), and broken power-law (bottom) spectrum models [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The SNR ratios of the two LISA-TAIJI networks for the SGWB with power-law (upper left), single peak (upper right), [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Corner plots for the parameter uncertainties of SGWB estimated from Fisher information matrix. The upper left plot [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Measuring gravitational wave spectrum from electroweak phase transition and Higgs self-couplings

    hep-ph 2025-11 unverdicted novelty 5.0 of 10

    Using simulated Taiji data, the authors show that a stochastic gravitational-wave signal from an electroweak phase transition in the singlet-extended Standard Model can constrain the Higgs cubic and quartic self-couplings.

  2. Beyond general relativity: gravitational waves in non-minimally coupled theories

    gr-qc 2025-10 conditional novelty 5.0 of 10

    A generalized propagation parameterization for gravitational-wave strains is extended to O(H²) and O(H′), then mapped to Kalb-Ramond, axion-dilaton–Chern-Simons–Gauss-Bonnet, and U(1) dark-photon models.

Reference graph

Works this paper leans on

107 extracted references · 12 canonical work pages · cited by 2 Pith papers

  1. [1]

    (34) The fiducial signal is defined with Ω 1 = 4 .446 × 10−12 and α1 = 2 /3 at the reference frequency fc = 1 mHz [71]

    Power-Law Model [61–70]: ΩPL = Ω1 f fc α1 . (34) The fiducial signal is defined with Ω 1 = 4 .446 × 10−12 and α1 = 2 /3 at the reference frequency fc = 1 mHz [71]. The power-law spectrum is very common in cosmo- logical processes. Parity violation can arise in scenarios involving pseudo-scalar inflatons or modified gravity [72– 75]. Measurements of such p...

  2. [2]

    (35) The typical parameters are set to ∆ 2 = 0 .2, Ω 2 = 1 × 10−11, and fc = 3 mHz [83, 84]

    Single Peak Model [78–82]: ΩSP = Ω2 exp " − (log10(f /fc))2 ∆2 2 # . (35) The typical parameters are set to ∆ 2 = 0 .2, Ω 2 = 1 × 10−11, and fc = 3 mHz [83, 84]. Single-peak SGWB spectrum can also arise in cosmological processes, with parity-violating GWs potentially generated during in- flation through mechanisms such as dynamical Chern- Simons gravity [85, 86]

  3. [3]

    (36) The fiducial parameters are assumed to be α2 = 3, α3 = −4, ∆ = 2, with an amplitude Ω 3 = 1 × 10−9 and reference frequency fc = 10 mHz, see e.g

    Broken Power-Law Model [87–95]: ΩBPL = Ω3 f fc α2 " 1 + 0.75 f fc ∆#(α3−α2)/∆ . (36) The fiducial parameters are assumed to be α2 = 3, α3 = −4, ∆ = 2, with an amplitude Ω 3 = 1 × 10−9 and reference frequency fc = 10 mHz, see e.g. [92, 93, 95, 96]. The broken power-law spectrum is commonly produced during first-order phase transitions, see e.g. [11]. A pro...

  4. [4]

    Maleknejad, Axion Inflation with an SU(2) Gauge Field: Detectable Chiral Gravity Waves, JHEP 07, 104, arXiv:1604.03327 [hep-ph]

    A. Maleknejad, Axion Inflation with an SU(2) Gauge Field: Detectable Chiral Gravity Waves, JHEP 07, 104, arXiv:1604.03327 [hep-ph]

    arXiv

  5. [5]

    Christensen, Stochastic Gravitational Wave Back- grounds, Rept

    N. Christensen, Stochastic Gravitational Wave Back- grounds, Rept. Prog. Phys. 82, 016903 (2019)

    2019

  6. [6]

    Callister, L

    T. Callister, L. Jenks, D. Holz, and N. Yunes, A 12 New Probe of Gravitational Parity Violation Through (Non-)Observation of the Stochastic Gravitational- Wave Background, (2023), arXiv:2312.12532 [gr-qc]

    arXiv 2023

  7. [8]

    C. S. Machado, W. Ratzinger, P. Schwaller, and B. A. Stefanek, Gravitational wave probes of ax- ionlike particles, Phys. Rev. D 102, 075033 (2020), arXiv:1912.01007 [hep-ph]

    arXiv 2020

  8. [9]

    Satoh, S

    M. Satoh, S. Kanno, and J. Soda, Circular Polarization of Primordial Gravitational Waves in String-inspired In- flationary Cosmology, Phys. Rev. D 77, 023526 (2008), arXiv:0706.3585 [astro-ph]

    arXiv 2008

Show all 107 references
  1. [10]

    Adshead and M

    P. Adshead and M. Wyman, Chromo-Natural Inflation: Natural inflation on a steep potential with classical non-Abelian gauge fields, Phys. Rev. Lett. 108, 261302 (2012), arXiv:1202.2366 [hep-th]

    2012 arXiv

  2. [11]

    C. S. Machado, W. Ratzinger, P. Schwaller, and B. A. Stefanek, Audible Axions, JHEP 01, 053, arXiv:1811.01950 [hep-ph]

    arXiv

  3. [12]

    Califano, R

    M. Califano, R. D’Agostino, and D. Vernieri, Parity vi- olation in gravitational waves and observational bounds from third-generation detectors, Phys. Rev. D 109, 104062 (2024), arXiv:2311.02161 [gr-qc]

    2024 arXiv

  4. [13]

    Maleknejad, Gravitational ABJ Anomaly, Stochas- tic Matter Production, and Leptogenesis, (2024), arXiv:2412.09490 [hep-ph]

    A. Maleknejad, Gravitational ABJ Anomaly, Stochas- tic Matter Production, and Leptogenesis, (2024), arXiv:2412.09490 [hep-ph]

    2024 arXiv

  5. [14]

    Kahniashvili, G

    T. Kahniashvili, G. Gogoberidze, and B. Ratra, Polar- ized cosmological gravitational waves from primordial helical turbulence, Phys. Rev. Lett. 95, 151301 (2005)

    2005

  6. [15]

    Roper Pol, S

    A. Roper Pol, S. Mandal, A. Brandenburg, T. Kah- niashvili, and A. Kosowsky, Numerical simulations of gravitational waves from early-universe turbulence, Phys. Rev. D 102, 083512 (2020), arXiv:1903.08585 [astro-ph.CO]

    2020 arXiv

  7. [16]

    Omiya and N

    H. Omiya and N. Seto, Measuring the maximally al- lowed polarization states of the isotropic stochastic gravitational wave background with the ground-based detectors, Phys. Rev. D 107, 124027 (2023)

    2023

  8. [17]

    T.-C. Li, T. Zhu, W. Zhao, and A. Wang, Power spec- tra and circular polarization of primordial gravitational waves with parity and Lorentz violations, JCAP 07, 005, arXiv:2403.05841 [gr-qc]

    arXiv

  9. [18]

    Martinovic, C

    K. Martinovic, C. Badger, M. Sakellariadou, and V. Mandic, Searching for parity violation with the LIGO-Virgo-KAGRA network, Phys. Rev. D 104, L081101 (2021)

    2021

  10. [19]

    Jiang and Q.-G

    Y. Jiang and Q.-G. Huang, Upper limits on the polar- ized isotropic stochastic gravitational-wave background from advanced LIGO-Virgo’s first three observing runs, JCAP 02, 026

  11. [20]

    Belgacem and M

    E. Belgacem and M. Kamionkowski, Chirality of the gravitational-wave background and pulsar-timing ar- rays, Phys. Rev. D 102, 023004 (2020)

    2020

  12. [21]

    Seto, Gravitational Wave Background Search by Correlating Multiple Triangular Detectors in the mHz Band, Phys

    N. Seto, Gravitational Wave Background Search by Correlating Multiple Triangular Detectors in the mHz Band, Phys. Rev. D 102, 123547 (2020)

    2020

  13. [22]

    Orlando, M

    G. Orlando, M. Pieroni, and A. Ricciardone, Measuring Parity Violation in the Stochastic Gravitational Wave Background with the LISA-Taiji network, JCAP 03, 069

  14. [23]

    Kato and J

    R. Kato and J. Soda, Probing circular polarization in stochastic gravitational wave background with pulsar timing arrays, Phys. Rev. D 93, 062003 (2016)

    2016

  15. [24]

    S. V. Dhurandhar, K. Rajesh Nayak, S. Koshti, and J. Y. Vinet, Fundamentals of the LISA stable flight for- mation, Class. Quant. Grav. 22, 481 (2005)

    2005

  16. [25]

    Amaro-Seoane et al

    P. Amaro-Seoane et al. (LISA), Laser Interferome- ter Space Antenna, (2017), arXiv:1702.00786 [astro- ph.IM]

    2017 arXiv

  17. [26]

    Colpi et al., LISA Definition Study Report, (2024), arXiv:2402.07571 [astro-ph.CO]

    M. Colpi et al., LISA Definition Study Report, (2024), arXiv:2402.07571 [astro-ph.CO]

    2024 arXiv

  18. [27]

    Ruan, Z.-K

    W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, Taiji Program: Gravitational-Wave Sources, arXiv e- prints 1807, arXiv:1807.09495 (2018)

    2018 arXiv

  19. [28]

    Y. Zhao, Y. Lu, C. Yan, Z. Chen, and W.-T. Ni, Multi- band gravitational wave observations of stellar binary black holes at the low to middle and high frequencies, Monthly Notices of the Royal Astronomical Society522, 2951 (2023), arXiv:2306.02636 [astro-ph]

    2023 arXiv

  20. [29]

    W.-H. Ruan, C. Liu, Z.-K. Guo, Y.-L. Wu, and R.-G. Cai, The LISA–Taiji network, Nature Astronomy4, 108 (2020)

    2020

  21. [30]

    Wang, W.-T

    G. Wang, W.-T. Ni, W.-B. Han, P. Xu, and Z. Luo, Alternative LISA-TAIJI networks, Phys. Rev. D 104, 024012 (2021)

    2021

  22. [31]

    Chen, C.-S

    J. Chen, C.-S. Yan, Y.-J. Lu, Y.-T. Zhao, and J.-Q. Ge, On detecting stellar binary black holes via the LISA- Taiji network, Research in Astronomy and Astrophysics 21, 285 (2021)

    2021

  23. [32]

    J. Chen, C. Liu, and Y.-L. Zhang, Parity-violating Gravitational Wave Background Search with a Net- work of Space-borne Triangular Detectors, (2024), arXiv:2410.18916 [gr-qc]

    2024 arXiv

  24. [33]

    Cai, Z.-K

    R.-G. Cai, Z.-K. Guo, B. Hu, C. Liu, Y. Lu, W.-T. Ni, W.-H. Ruan, N. Seto, G. Wang, and Y.-L. Wu, On networks of space-based gravitational-wave detectors, Fund. Res. 4, 1072 (2024), arXiv:2305.04551 [gr-qc]

    2024 arXiv

  25. [34]

    Seto, Prospects for direct detection of circular po- larization of gravitational-wave background, Phys

    N. Seto, Prospects for direct detection of circular po- larization of gravitational-wave background, Phys. Rev. Lett. 97, 151101 (2006)

    2006

  26. [35]

    Seto and A

    N. Seto and A. Taruya, Measuring a Parity Violation Signature in the Early Universe via Ground-based Laser Interferometers, Phys. Rev. Lett. 99, 121101 (2007)

    2007

  27. [36]

    Tinto, D

    M. Tinto, D. A. Shaddock, J. Sylvestre, and J. W. Armstrong, Implementation of time-delay interferome- try for LISA, Phys. Rev. D 67, 122003 (2003), arXiv:gr- qc/0303013 [gr-qc]

    2003

  28. [37]

    Wang and W.-B

    G. Wang and W.-B. Han, Alternative LISA-TAIJI net- works: Detectability of the isotropic stochastic gravi- tational wave background, Phys. Rev. D 104, 104015 (2021)

    2021

  29. [38]

    J. W. Armstrong, F. B. Estabrook, and M. Tinto, Time-Delay Interferometry for Space-based Gravita- tional Wave Searches, Astrophys. J. 527, 814 (1999)

    1999

  30. [39]

    F. B. Estabrook, M. Tinto, and J. W. Armstrong, Time- delay analysis of LISA gravitational wave data: Elim- ination of spacecraft motion effects, Phys. Rev. D 62, 042002 (2000)

    2000

  31. [40]

    Wang, Time delay interferometry with minimal null frequencies, Phys

    G. Wang, Time delay interferometry with minimal null frequencies, Phys. Rev. D 110, 042005 (2024), arXiv:2403.01490 [gr-qc]

    2024 arXiv

  32. [41]

    Vallisneri, Geometric time delay interferometry, Phys

    M. Vallisneri, Geometric time delay interferometry, Phys. Rev. D 72, 042003 (2005), [Erratum: Phys. Rev. D 76, 109903(2007)], arXiv:gr-qc/0504145 [gr-qc]

    2005 arXiv

  33. [42]

    Tinto and S

    M. Tinto and S. V. Dhurandhar, Time-delay interfer- 13 ometry, Living Rev. Rel. 24, 1 (2021)

    2021

  34. [43]

    Wang, Time-Delay Interferometry for ASTROD-GW (2011), arXiv:2406.14173 [gr-qc]

    G. Wang, Time-Delay Interferometry for ASTROD-GW (2011), arXiv:2406.14173 [gr-qc]

    2011 arXiv

  35. [44]

    T. A. Prince, M. Tinto, S. L. Larson, and J. W. Arm- strong, The LISA optimal sensitivity, Phys. Rev. D 66, 122002 (2002)

    2002

  36. [45]

    Wang, Enhancing noise characterization with robust time delay interferometry combination, Phys

    G. Wang, Enhancing noise characterization with robust time delay interferometry combination, Phys. Rev. D 110, 064085 (2024), arXiv:2406.11305 [gr-qc]

    2024 arXiv

  37. [46]

    Wang, W.-T

    G. Wang, W.-T. Ni, W.-B. Han, and C.-F. Qiao, Al- gorithm for time-delay interferometry numerical simu- lation and sensitivity investigation, Phys. Rev. D 103, 122006 (2021), arXiv:2010.15544 [gr-qc]

    2021 arXiv

  38. [47]

    Muratore, D

    M. Muratore, D. Vetrugno, and S. Vitale, Revisitation of time delay interferometry combinations that suppress laser noise in LISA, Class. Quant. Grav. 37, 185019 (2020), arXiv:2001.11221 [astro-ph.IM]

    2020 arXiv

  39. [48]

    Allen and J

    B. Allen and J. D. Romano, Detecting a stochastic background of gravitational radiation: Signal process- ing strategies and sensitivities, Phys. Rev. D 59, 102001 (1999)

    1999

  40. [49]

    Wang, SATDI: Simulation and Analysis for Time- Delay Interferometry, (2024), arXiv:2403.01726 [gr-qc]

    G. Wang, SATDI: Simulation and Analysis for Time- Delay Interferometry, (2024), arXiv:2403.01726 [gr-qc]

    2024 arXiv

  41. [50]

    Z. Luo, Z. Guo, G. Jin, Y. Wu, and W. Hu, A brief analysis to Taiji: Science and technology, Results Phys. 16, 102918 (2020)

    2020

  42. [51]

    Maggiore, Gravitational Waves: Volume 1: Theory and Experiments , 1st ed

    M. Maggiore, Gravitational Waves: Volume 1: Theory and Experiments , 1st ed. (Oxford University PressOx- ford, 2007)

    2007

  43. [52]

    J. D. Romano and N. J. Cornish, Detection methods for stochastic gravitational-wave backgrounds: A uni- fied treatment, Living Rev. Rel. 20, 2 (2017)

    2017

  44. [53]

    Seto, Quest for circular polarization of gravitational wave background and orbits of laser interferometers in space, Phys

    N. Seto, Quest for circular polarization of gravitational wave background and orbits of laser interferometers in space, Phys. Rev. D 75, 061302 (2007)

    2007

  45. [54]

    Belgacem, F

    E. Belgacem, F. Iacovelli, M. Maggiore, M. Mancarella, and N. Muttoni, The spectral density of astrophysical stochastic backgrounds, (2024), arXiv:2411.04028 [gr- qc]

    2024 arXiv

  46. [55]

    Seto and A

    N. Seto and A. Taruya, Polarization analysis of gravitational-wave backgrounds from the correlation signals of ground-based interferometers: Measuring a circular-polarization mode, Phys. Rev. D 77, 103001 (2008)

    2008

  47. [56]

    Thrane and J

    E. Thrane and J. D. Romano, Sensitivity curves for searches for gravitational-wave backgrounds, Physical Review D 88, 124032 (2013)

    2013

  48. [57]

    N. J. Cornish and S. L. Larson, Space missions to de- tect the cosmic gravitational wave background, Class. Quant. Grav. 18, 3473 (2001), arXiv:gr-qc/0103075

    2001 arXiv

  49. [58]

    N. J. Cornish, Detecting a stochastic gravitational wave background with the Laser Interferometer Space An- tenna, Phys. Rev. D 65, 022004 (2002)

    2002

  50. [59]

    Seto, Measuring Parity Asymmetry of Gravitational Wave Backgrounds with a Heliocentric Detector Net- work in the mHz Band, Phys

    N. Seto, Measuring Parity Asymmetry of Gravitational Wave Backgrounds with a Heliocentric Detector Net- work in the mHz Band, Phys. Rev. Lett. 125, 251101 (2020)

    2020

  51. [60]

    Auclair et al

    P. Auclair et al. (LISA Cosmology Working Group), Cosmology with the Laser Interferometer Space An- tenna, Living Rev. Rel. 26, 5 (2023), arXiv:2204.05434 [astro-ph.CO]

    2023 arXiv

  52. [61]

    Binetruy, A

    P. Binetruy, A. Bohe, C. Caprini, and J.-F. Dufaux, Cosmological Backgrounds of Gravitational Waves and eLISA/NGO: Phase Transitions, Cosmic Strings and Other Sources, JCAP 06, 027, arXiv:1201.0983 [gr-qc]

    arXiv

  53. [62]

    Caprini and D

    C. Caprini and D. G. Figueroa, Cosmological Back- grounds of Gravitational Waves, Class. Quant. Grav. 35, 163001 (2018), arXiv:1801.04268 [astro-ph.CO]

    2018 arXiv

  54. [63]

    Kuroyanagi, T

    S. Kuroyanagi, T. Chiba, and T. Takahashi, Probing the Universe through the Stochastic Gravitational Wave Background, JCAP 11, 038, arXiv:1807.00786 [astro- ph.CO]

    arXiv

  55. [64]

    Giovannini, Production and detection of relic gravi- tons in quintessential inflationary models, Phys

    M. Giovannini, Production and detection of relic gravi- tons in quintessential inflationary models, Phys. Rev. D 60, 123511 (1999), arXiv:astro-ph/9903004

    1999 arXiv

  56. [65]

    M. S. Turner, M. J. White, and J. E. Lidsey, Ten- sor perturbations in inflationary models as a probe of cosmology, Phys. Rev. D 48, 4613 (1993), arXiv:astro- ph/9306029

    1993

  57. [66]

    Giovannini, Gravitational waves constraints on postinflationary phases stiffer than radiation, Phys

    M. Giovannini, Gravitational waves constraints on postinflationary phases stiffer than radiation, Phys. Rev. D 58, 083504 (1998), arXiv:hep-ph/9806329

    1998 arXiv

  58. [67]

    P. J. E. Peebles and A. Vilenkin, Quintessential in- flation, Phys. Rev. D 59, 063505 (1999), arXiv:astro- ph/9810509

    1999

  59. [68]

    Tashiro, T

    H. Tashiro, T. Chiba, and M. Sasaki, Reheating after quintessential inflation and gravitational waves, Class. Quant. Grav. 21, 1761 (2004), arXiv:gr-qc/0307068

    2004 arXiv

  60. [69]

    Giovannini, Spikes in the relic graviton background from quintessential inflation, Class

    M. Giovannini, Spikes in the relic graviton background from quintessential inflation, Class. Quant. Grav. 16, 2905 (1999), arXiv:hep-ph/9903263

    1999 arXiv

  61. [70]

    R.-G. Cai, S. Pi, and M. Sasaki, Universal infrared scaling of gravitational wave background spectra, Phys. Rev. D 102, 083528 (2020), arXiv:1909.13728 [astro- ph.CO]

    2020 arXiv

  62. [71]

    Auclair et al

    P. Auclair et al. , Probing the gravitational wave back- ground from cosmic strings with LISA, JCAP 04, 034, arXiv:1909.00819 [astro-ph.CO]

    1909 arXiv

  63. [72]

    S. H.-S. Alexander, M. E. Peskin, and M. M. Sheikh- Jabbari, Leptogenesis from gravity waves in models of inflation, Phys. Rev. Lett.96, 081301 (2006), arXiv:hep- th/0403069

    2006

  64. [73]

    Giovannini, Thermal history of the plasma and high- frequency gravitons, Class

    M. Giovannini, Thermal history of the plasma and high- frequency gravitons, Class. Quant. Grav. 26, 045004 (2009), arXiv:0807.4317 [astro-ph]

    2009 arXiv

  65. [74]

    Dom` enech, S

    G. Dom` enech, S. Pi, and M. Sasaki, Induced gravita- tional waves as a probe of thermal history of the uni- verse, JCAP 08, 017, arXiv:2005.12314 [gr-qc]

    2005 arXiv

  66. [75]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Search for the isotropic stochastic background using data from Ad- vanced LIGO’s second observing run, Phys. Rev. D100, 061101 (2019), arXiv:1903.02886 [gr-qc]

    2019 arXiv

  67. [76]

    Seto and A

    N. Seto and A. Taruya, Measuring a Parity Violation Signature in the Early Universe via Ground-based Laser Interferometers, Phys. Rev. Lett. 99, 121101 (2007), arXiv:0707.0535 [astro-ph]. 14

    2007 arXiv

  68. [77]

    Takahashi and J

    T. Takahashi and J. Soda, Chiral Primordial Gravita- tional Waves from a Lifshitz Point, Phys. Rev. Lett. 102, 231301 (2009), arXiv:0904.0554 [hep-th]

    2009 arXiv

  69. [78]

    Magueijo and D

    J. Magueijo and D. M. T. Benincasa, Chiral vacuum fluctuations in quantum gravity, Phys. Rev. Lett. 106, 121302 (2011), arXiv:1010.3552 [gr-qc]

    2011 arXiv

  70. [79]

    Sorbo, Parity violation in the Cosmic Microwave Background from a pseudoscalar inflaton, JCAP 06, 003, arXiv:1101.1525 [astro-ph.CO]

    L. Sorbo, Parity violation in the Cosmic Microwave Background from a pseudoscalar inflaton, JCAP 06, 003, arXiv:1101.1525 [astro-ph.CO]

    arXiv

  71. [80]

    R.-G. Cai, S. Pi, S.-J. Wang, and X.-Y. Yang, Reso- nant multiple peaks in the induced gravitational waves, JCAP 05, 013, arXiv:1901.10152 [astro-ph.CO]

    1901 arXiv

  72. [81]

    S. G. Crowder, R. Namba, V. Mandic, S. Mukohyama, and M. Peloso, Measurement of Parity Violation in the Early Universe using Gravitational-wave Detectors, Phys. Lett. B 726, 66 (2013), arXiv:1212.4165 [astro- ph.CO]

    2013 arXiv

  73. [82]

    Kohri and T

    K. Kohri and T. Terada, Semianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbations, Phys. Rev. D 97, 123532 (2018), arXiv:1804.08577 [gr-qc]

    2018 arXiv

  74. [83]

    Inomata, K

    K. Inomata, K. Kohri, T. Nakama, and T. Terada, En- hancement of Gravitational Waves Induced by Scalar Perturbations due to a Sudden Transition from an Early Matter Era to the Radiation Era, Phys. Rev. D 100, 043532 (2019), [Erratum: Phys.Rev.D 108, 049901 (2023)], arXiv:1904....

    2019 arXiv

  75. [84]

    Flauger, N

    R. Flauger, N. Karnesis, G. Nardini, M. Pieroni, A. Ric- ciardone, and J. Torrado, Improved reconstruction of a stochastic gravitational wave background with LISA, JCAP 01, 059, arXiv:2009.11845 [astro-ph.CO]

    2009 arXiv

  76. [85]

    White, L

    G. White, L. Pearce, D. Vagie, and A. Kusenko, De- tectable Gravitational Wave Signals from Affleck-Dine Baryogenesis, Phys. Rev. Lett. 127, 181601 (2021), arXiv:2105.11655 [hep-ph]

    2021 arXiv

  77. [86]

    K. D. Lozanov and V. Takhistov, Enhanced Gravita- tional Waves from Inflaton Oscillons, Phys. Rev. Lett. 130, 181002 (2023), arXiv:2204.07152 [astro-ph.CO]

    2023 arXiv

  78. [87]

    Caprini, D

    C. Caprini, D. G. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni, A. Ricciardone, and G. Tasinato, Reconstructing the spectral shape of a stochastic grav- itational wave background with LISA, JCAP 11, 017, arXiv:1906.09244 [astro-ph.CO]

    1906 arXiv

  79. [88]

    Kosowsky, M

    A. Kosowsky, M. S. Turner, and R. Watkins, Gravita- tional waves from first order cosmological phase transi- tions, Phys. Rev. Lett. 69, 2026 (1992)

    1992

  80. [89]

    C. Fu, J. Liu, T. Zhu, H. Yu, and P. Wu, Resonance instability of primordial gravitational waves during in- flation in Chern–Simons gravity, Eur. Phys. J. C 81, 204 (2021), arXiv:2006.03771 [gr-qc]

    2021 arXiv

  81. [90]

    Peng, Z.-M

    Z.-Z. Peng, Z.-M. Zeng, C. Fu, and Z.-K. Guo, Gen- eration of gravitational waves in dynamical Chern- Simons gravity, Phys. Rev. D 106, 124044 (2022), arXiv:2209.10374 [gr-qc]

    2022 arXiv

  82. [91]

    Vilenkin, Gravitational Field of Vacuum Domain Walls and Strings, Phys

    A. Vilenkin, Gravitational Field of Vacuum Domain Walls and Strings, Phys. Rev. D 23, 852 (1981)

    1981

  83. [92]

    Hindmarsh, S

    M. Hindmarsh, S. J. Huber, K. Rummukainen, and D. J. Weir, Gravitational waves from the sound of a first order phase transition, Phys. Rev. Lett. 112, 041301 (2014), arXiv:1304.2433 [hep-ph]

    2014 arXiv

  84. [93]

    Kamionkowski, A

    M. Kamionkowski, A. Kosowsky, and M. S. Turner, Gravitational radiation from first order phase tran- sitions, Phys. Rev. D 49, 2837 (1994), arXiv:astro- ph/9310044

    1994

  85. [94]

    Gleiser and R

    M. Gleiser and R. Roberts, Gravitational waves from collapsing vacuum domains, Phys. Rev. Lett. 81, 5497 (1998), arXiv:astro-ph/9807260

    1998 arXiv

  86. [95]

    Grojean and G

    C. Grojean and G. Servant, Gravitational Waves from Phase Transitions at the Electroweak Scale and Beyond, Phys. Rev. D 75, 043507 (2007), arXiv:hep-ph/0607107

    2007 arXiv

  87. [96]

    Martinovic, P

    K. Martinovic, P. M. Meyers, M. Sakellariadou, and N. Christensen, Simultaneous estimation of astrophysi- cal and cosmological stochastic gravitational-wave back- grounds with terrestrial detectors, Phys. Rev. D 103, 043023 (2021), arXiv:2011.05697 [gr-qc]

    2021 arXiv

  88. [97]

    Caprini et al., Science with the space-based interfer- ometer eLISA

    C. Caprini et al., Science with the space-based interfer- ometer eLISA. II: Gravitational waves from cosmologi- cal phase transitions, JCAP 04, 001, arXiv:1512.06239 [astro-ph.CO]

    arXiv

  89. [98]

    Saikawa, A review of gravitational waves from cosmic domain walls, Universe 3, 40 (2017), arXiv:1703.02576 [hep-ph]

    K. Saikawa, A review of gravitational waves from cosmic domain walls, Universe 3, 40 (2017), arXiv:1703.02576 [hep-ph]

    2017 arXiv

  90. [99]

    Caprini et al

    C. Caprini et al. , Detecting gravitational waves from cosmological phase transitions with LISA: an update, JCAP 03, 024, arXiv:1910.13125 [astro-ph.CO]

    1910 arXiv

  91. [100]

    T. L. Smith, T. L. Smith, R. R. Caldwell, and R. Cald- well, LISA for Cosmologists: Calculating the Signal-to- Noise Ratio for Stochastic and Deterministic Sources, Phys. Rev. D 100, 104055 (2019)

    2019

  92. [101]

    We thank helpful discussions with Jing Liu, Yong Tang and Yang Jiang

    and scipy [102], and the plots are made by utiliz- ing matplotlib [103] and GetDist [104]. We thank helpful discussions with Jing Liu, Yong Tang and Yang Jiang

  93. [102]

    K. Ding, C. Fu, B. Xu, and Y.-L. Zhang, Chiral gravi- tational wave background in millihertz from axion-like fields, SCIENTIA SINICA Physica, Mechanica & As- tronomica 54, 270408 (2024)

    2024

  94. [103]

    B. Xu, K. Ding, H. Su, J. Chen, and Y.-L. Zhang, Chi- ral Gravitational Wave Background from Audible Axion via Nieh-Yan Term, (2024), arXiv:2411.08691 [hep-ph]

    2024 arXiv

  95. [104]

    Boileau, N

    G. Boileau, N. Christensen, R. Meyer, and N. J. Cor- nish, Spectral separation of the stochastic gravitational- wave background for LISA: Observing both cosmologi- cal and astrophysical backgrounds, Phys. Rev. D 103, 103529 (2021)

    2021

  96. [105]

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gom- mers, P. Virtanen, D. Cournapeau, E. Wieser, J. Tay- lor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R ´ ıo, M. Wiebe, P. Peterson, P. G´ erard-Marchant, K. She...

    2020

  97. [106]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- terson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. ...

    2020

  98. [107]

    J. D. Hunter, Matplotlib: A 2D Graphics Environment, Comput. Sci. Eng. 9, 90 (2007)

    2007

  99. [108]

    Lewis, GetDist: a Python package for analysing Monte Carlo samples, (2019), arXiv:1910.13970 [astro- ph.IM]

    A. Lewis, GetDist: a Python package for analysing Monte Carlo samples, (2019), arXiv:1910.13970 [astro- ph.IM]

    2019 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.