Pith. sign in

REVIEW 3 major objections 3 minor 63 references

A space-based detector can reveal gravitational waves from inflationary phase transitions, but reliably measuring the two source parameters requires a signal roughly three times stronger than the detection threshold.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-02 17:40 UTC pith:2QNQUYM6

load-bearing objection A credible Taiji forecast for the InPT secondary background, with good internal consistency, but the headline thresholds are tied to one untested spectral template. the 3 major comments →

arxiv 2603.21762 v2 pith:2QNQUYM6 submitted 2026-03-23 astro-ph.CO hep-ph

Inflationary phase transitions in the early Universe: A Bayesian study with space-based gravitational-wave detectors

Qingyuan Liang , Chen Yang , Haipeng An , Huai-Ke Guo This is my paper
classification astro-ph.CO hep-ph PACS 04.30.-w95.55.Ym98.80.-k
keywords stochastic gravitational-wave backgroundinflationary phase transitionBayesian inferencenested samplingFisher matrixTaiji-like detectortime-delay interferometryparameter recovery
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether a future space-based gravitational-wave observatory, benchmarked on a Taiji-like mission, can detect and characterize the stochastic background produced by a phase transition that happened during cosmic inflation. The authors show that detection is comparatively easy—an absolute signal-to-noise ratio of about 10 suffices—but trustworthy reconstruction of the signal's two free parameters demands an SNR near 33, corresponding to a Bayes factor of roughly e^7.5. They build a realistic simulated dataset that includes the detector's instrumental noise, the astrophysical foreground from unresolved white-dwarf binaries, an extragalactic background, and the three time-delay-interferometry channels, then analyze it with both Fisher-matrix forecasts and full Bayesian nested sampling. The result matters because it separates 'seeing a signal' from 'learning what produced it', giving a concrete benchmark for what a future mission must achieve to probe inflation through stochastic gravitational radiation.

Core claim

The paper establishes that the inflationary phase-transition signal, modeled minimally through an effective amplitude B_ref and reference frequency f_ref, can be detected by a Taiji-like detector at absolute SNR about 10, but reliable parameter recovery requires a more stringent SNR near 33 (ln BF ≈ 7.5). For a fiducial injection with log10 B_ref = −14 and log10(f_ref/Hz) = −3, the analysis yields SNR_a = 118, SNR_r = 66, ln BF = 42.24, and a relative uncertainty of roughly 31% in log10 B_ref. The work also shows that astrophysical foregrounds and backgrounds degrade the precision of the cosmological parameter reconstruction and shift the boundaries among exclusion, detection, and reliable-r

What carries the argument

The central object is the secondary gravitational-wave background arising from curvature perturbations produced by a phase transition during inflation, parameterized by a two-parameter template Ω_InPT(f) = B_ref F(f/f_ref), where F is a broken power-law shape rising as x^3 and falling as x^{−10} with a peak near x ≈ 5. The analysis machinery is a full frequency-domain Bayesian framework using the A, E, and T time-delay-interferometry channels of a Taiji-like detector, with instrumental noise, a Galactic double-white-dwarf foreground, an extragalactic power-law background, and nested sampling for parameter estimation and Bayes-factor computation, cross-checked against Fisher-matrix forecasts.

Load-bearing premise

The broken-power-law spectral shape F(f/f_ref) taken from earlier work for the representative case β/H_inf = 5 is assumed to be the true template, and the secondary gravitational-wave component is assumed to dominate over the primary; if either assumption fails, the SNR contours, Bayes factors, and recovery thresholds in Section 4 all shift.

What would settle it

Recompute the detection and reliable-recovery contours using an alternative spectral shape, for instance with β/H_inf = 10 (different c1 and c2) or with the primary component included; if the SNR=33 contour moves by more than the statistical uncertainties quoted, the thresholds are template-dependent. Observationally, a future measurement showing a high-frequency spectral slope shallower than f^{−10} in the InPT band would disfavor the assumed F(f/f_ref) and require reinterpreting the recovery thresholds.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A Taiji-like mission can detect an inflationary phase-transition background at SNR ≈ 10, meaning that moderately strong signals are not missed even when only a single detector is available.
  • Reliable measurement of the spectral parameters B_ref and f_ref requires SNR ≈ 33 (ln BF ≈ 7.5), so future searches should quote both detection significance and parameter-recovery thresholds, not just a single SNR cutoff.
  • Astrophysical foregrounds and backgrounds, particularly their amplitudes and spectral slopes, directly degrade the precision with which the inflationary signal parameters can be reconstructed.
  • A detected InPT signal with amplitude A_ref below 10^−5 can still be well characterized, and f_ref encodes when the phase transition occurred relative to the end of inflation—about 26 e-folds before the end in the fiducial scenario.
  • Fisher-matrix forecasts and nested-sampling Bayesian results agree well in this regime, supporting the use of Fisher methods for rapid survey scans while reserving full Bayesian analysis for candidate signals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the true InPT spectral shape differs from the assumed F(f/f_ref) template (e.g., for a different β/H_inf value), the SNR=33 threshold and the recovered parameter uncertainties could shift substantially; re-running the analysis with c1 and c2 varied over their allowed range would directly test the robustness of the quoted thresholds.
  • A two-detector network (for instance, a Taiji-like mission plus a LISA-like mission) would allow cross-correlation of independent noise realizations, which could lower the reliable-recovery threshold well below SNR=33; this is a natural extension of the single-detector framework presented here.
  • The same analysis pipeline could be applied to other cosmological stochastic backgrounds, such as those from cosmic strings or primordial black hole formation, where the competition between astrophysical foregrounds and a broken power-law signal is similar.
  • The mapping from B_ref and f_ref to microphysical parameters (latent heat, transition rate β/H_inf, and the e-fold time of the transition) is not explicitly inverted in the paper; doing so would turn the recovery thresholds into direct constraints on inflationary particle physics.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies detectability and parameter reconstruction of a stochastic gravitational-wave background from inflationary phase transitions (InPTs) for a Taiji-like space-based detector. The signal is modeled in a 'minimal, model-independent' form as Omega_InPT(f)=B_ref F(f/f_ref), with the shape F built from the curvature power spectrum for the representative choice beta/H_inf=5. The analysis includes TDI A/E/T channels, instrumental noise, a Galactic double-white-dwarf foreground, and an extragalactic power-law background. Parameter inference is performed with nested sampling and compared with Fisher-matrix forecasts. The central quantitative claims are that detection requires SNR_a >= 10, reliable parameter recovery requires SNR_a ~ 33 (ln BF ~ 7.5), and that for the fiducial injection (log10 B_ref=-14, log10(f_ref/Hz)=-3) the pipeline gives SNR_a=118, SNR_r=66, ln BF=42.24, with ~31% relative uncertainty on log10 B_ref.

Significance. If the assumed signal template and the secondary-dominance assumption are valid, the paper provides a useful benchmark: the pipeline is carefully constructed, the Fisher/nested-sampling agreement in Table 2 is nontrivial, and the SNR/BF scaling in Table 3 is monotonic as expected. The explicit treatment of foreground/background separation in a Bayesian framework is valuable. However, the headline detection and recovery thresholds are conditional on a template shape that is fixed to one beta/H_inf value and on an asserted dominance of the secondary GW component; neither condition is demonstrated. The paper should be regarded as a self-consistency study of the assumed template rather than a robust prediction for InPT signals in general.

major comments (3)
  1. [Sec. 2, Eqs. (2.8)-(2.10)] The text states, immediately before fixing beta/H_inf=5, that 'the shape function is independent of beta/H_inf'. This is internally inconsistent: S(x) in Eq. (2.10) depends on c1 and c2, which the text states depend on beta/H_inf, and F(x) in Eq. (2.8) is a functional of S. Consequently F(f/f_ref), and hence every SNR, Bayes factor, and recovery threshold in Sec. 4 and Table 3, depends on beta/H_inf. The paper must either quantify this sensitivity by repeating the analysis for representative beta/H_inf values, or explicitly restrict all conclusions to the single chosen value. As written, the central quantitative claims are conditional on an untested template choice.
  2. [Sec. 2, Eqs. (2.11)-(2.13)] The decision to model only the secondary GW component rests on the assertion that it is 'parametrically larger' than the primary. The ratio of the two peak amplitudes scales as (1/epsilon^2)(M_Pl/phi_0)^4 (H_inf/beta)(L/rho_inf)^2, which can be order one or smaller for plausible parameter values (e.g., larger beta/H_inf, smaller L/rho_inf, or larger epsilon). The paper should provide the explicit ratio and specify the parameter region in which the secondary dominates; otherwise the signal model omits a potentially comparable primary contribution and the thresholds in Sec. 4 are not supported.
  3. [Sec. 3, after Eq. (3.10); Sec. 4] The central statistics SNR_a and SNR_r are not defined in this manuscript ('not repeated here', reference to [53]). Because the detection and recovery thresholds are quoted in these units, the paper is not self-contained. In addition, the numerical shape F(f/f_ref) is needed to reproduce Fig. 2 and Table 3; if it is taken from Ref. [34], please provide the numerical template or public code. As it stands, the reproducibility of the main results requires consulting two external papers and a template that is fixed by hand.
minor comments (3)
  1. [Header] The title page reads 'PREPARED FOR SUBMISSION TOJHEP' (missing space); typographical error.
  2. [Sec. 4, after Eq. (3.10)] The reference [63] for the 'strong evidence' threshold ln BF ~ 7.5 is a GRB lensing paper; a standard reference for Bayes-factor scales (Jeffreys; Kass & Raftery) would be more appropriate.
  3. [Fig. 2, right panel] The caption does not clearly distinguish which dashed/dot-dashed curve corresponds to enhanced/suppressed astrophysical background versus foreground; please label the curves explicitly or spell this out in the caption.

Circularity Check

0 steps flagged

No significant circularity: the numerical results are conditional injection-recovery forecasts under an assumed InPT template; the unsupported 'model-independent shape' claim is a correctness caveat, not a circular reduction.

full rationale

Strict circularity requires a quantity claimed as a prediction to be equivalent, by construction or by fitted-parameter renaming, to the very input used to produce it. No such step is present here. The paper defines the InPT signal as Ω_InPT(f)=B_ref F(f/f_ref) (Eq. 2.13), injects a fiducial (B_ref,f_ref), and recovers it with the same template; the reported SNR_a=118, SNR_r=66, ln BF=42.24, and the ~31% uncertainty in log10 B_ref are properties of that assumed injection, making this a standard detectability/parameter-estimation forecast rather than a derivation of the signal's existence. The spectral template F and coefficients c1=0.31, c2=0.17 are taken from Ref. [34], and the SNR/NS methodology from Refs. [52,53], all involving overlapping authors; however, Ref. [34] is an independent published derivation testable, e.g., by PTA searches, and Refs. [52,53] are external methodology papers, so self-citation does not by itself make the pipeline circular under the given rules. The text's assertion 'Since the shape function is independent of β/H_inf' is inconsistent with Eqs. (2.8)-(2.10), because F(x) integrates S(vx)S(sqrt(...)x) and S(x) contains c1,c2 which depend on β/H_inf. This means the headline numbers are conditional on the β/H=5 template and on the dominance of the secondary GW component; that is a model-dependence/correctness risk and a reproducibility gap (explicit F and SNR definitions deferred to refs), not a reduction of the conclusions to their inputs. Hence no load-bearing circular step meets the strict quotation standard; score 2 reflects the minor self-citation burden and the unsubstantiated model-independence claim.

Axiom & Free-Parameter Ledger

13 free parameters · 8 axioms · 0 invented entities

The inference itself has 10 free parameters (Table 1), of which B_ref and f_ref are the target InPT parameters and the rest are noise/foreground nuisance parameters. Additionally, the signal template shape (c1,c2,β/H_inf) and the benchmark H_inf are chosen from prior literature rather than derived. The central claim is a forecast conditional on all these inputs.

free parameters (13)
  • B_ref (effective InPT amplitude) = 1e-14 fiducial injection; prior log10 B_ref ∈ (-16,-9)
    Target signal amplitude in Eq. (3.10); absorbs Ω_R A_ref^2 and is fitted in the Bayesian analysis.
  • f_ref (reference frequency) = 1e-3 Hz fiducial; prior log10(f_ref/Hz) ∈ (-5,-1)
    Target signal frequency scale in Eq. (3.10); fitted in the Bayesian analysis.
  • N_acc (acceleration noise amplitude) = 3e-15 fiducial; prior (0,20)e-15
    Instrumental noise parameter in Eq. (3.3); fitted as nuisance.
  • δx (optical metrology noise amplitude) = 8e-12 fiducial; prior (0,20)e-12
    Instrumental noise parameter in Eq. (3.3); fitted as nuisance.
  • A1 (DWD foreground amplitude) = 10^-15.4 fiducial; prior log10 A1 ∈ (-17,-13)
    Galactic foreground parameter in Eq. (3.8); fitted as nuisance.
  • α1 (DWD foreground low-frequency index) = -5.7 fiducial; prior (-10,-3)
    Galactic foreground parameter in Eq. (3.8); fitted as nuisance.
  • A2 (DWD foreground high-frequency amplitude) = 10^-6.32 fiducial; prior log10 A2 ∈ (-10,-2)
    Galactic foreground parameter in Eq. (3.8); fitted as nuisance.
  • α2 (DWD foreground high-frequency index) = -6.2 fiducial; prior (-10,-1)
    Galactic foreground parameter in Eq. (3.8); fitted as nuisance.
  • Ω_ast (extragalactic background amplitude) = 10^-11.5 fiducial; prior log10 Ω_ast ∈ (-15,-8)
    Extragalactic background parameter in Eq. (3.9); fitted as nuisance.
  • ε (extragalactic background spectral index) = 0.667 fiducial; prior (-2,3)
    Extragalactic background parameter in Eq. (3.9); fitted as nuisance.
  • β/H_inf (phase-transition rate parameter) = 5 (chosen representative value)
    Sets the shape-function coefficients c1,c2 via Ref [34]; not fitted here and not varied.
  • c1, c2 (shape-function coefficients) = 0.31, 0.17 (from Ref [34] for β/H_inf=5)
    Define F(f/f_ref) in Eqs. (2.8)-(2.10); chosen from prior literature, not re-derived.
  • H_inf (inflationary Hubble scale for e-fold mapping) = 10^14 GeV (benchmark in Eq. (2.4) context)
    Used to map f_ref to N_e for the '26 e-folds' statement; chosen, not constrained here.
axioms (8)
  • domain assumption Inflation occurred and generated primordial curvature perturbations that seed large-scale structure.
    Sec. 2 opening; standard cosmological paradigm, not proven in this paper.
  • domain assumption The secondary GWs sourced by curvature perturbations dominate over primary bubble-collision GWs over the relevant parameter space.
    Sec. 2, Eqs. (2.11)-(2.12): paper states secondary is 'parametrically larger' and focuses exclusively on it; this is asserted via scaling relations from Ref. [34].
  • domain assumption The spectral shape F(x) with c1=0.31, c2=0.17 (for β/H_inf=5) from Ref [34] correctly describes the InPT SGWB.
    Eqs. (2.8)-(2.10) and (3.10); the shape is taken from prior work co-authored by C. Yang and is not re-derived or varied.
  • domain assumption Taiji noise model (acceleration + optical metrology) and TDI A/E/T response functions with equal, time-independent arm lengths are accurate.
    Eqs. (3.3)-(3.6), taken from Ref [52].
  • domain assumption The astrophysical foreground/background models (broken power-law DWD, power-law extragalactic) are complete and correctly specified.
    Eqs. (3.7)-(3.9); no unmodeled components are considered.
  • standard math Noise is stationary and Gaussian within each 10^6 s segment, and frequency bins are independent.
    Sec. 3, likelihood construction; standard assumption for this type of analysis.
  • ad hoc to paper Uniform log priors over the chosen ranges for the ten model parameters are appropriate.
    Table 1; prior ranges are choices that affect marginal posteriors and Bayes factors.
  • ad hoc to paper SNR thresholds (exclusion=2, detection=10, reliable recovery=33) and ln BF≈7.5 as strong-evidence threshold are valid conventions.
    Sec. 4, following Refs [61-63]; these are literature conventions, not derived here.

pith-pipeline@v1.3.0-alltime-deepseek · 11227 in / 16601 out tokens · 148444 ms · 2026-08-02T17:40:36.010673+00:00 · methodology

0 comments
read the original abstract

Inflationary phase transitions can generate a stochastic gravitational-wave background that probes primordial physics. We study the detectability and parameter reconstruction of such a signal with a space-based gravitational-wave detector. Using a Taiji-like mission as a benchmark, we construct a realistic data-analysis framework that includes instrumental noise, astrophysical foregrounds and backgrounds, and the $A$, $E$, and $T$ time-delay interferometry channels. The target signal is described in a minimal, model-independent form and analyzed using both Fisher-matrix forecasts and Bayesian inference with nested sampling. We quantify detection significance and parameter-recovery thresholds, showing that, while detection is achievable at moderate signal-to-noise ratios, stronger signals provide more reliable parameter reconstruction. These results offer a realistic assessment of the capability of future space-based missions to probe inflationary phase transitions through stochastic gravitational radiation.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

63 extracted references · 52 linked inside Pith

  1. [1]

    Caldwell et al.,Detection of early-universe gravitational-wave signatures and fundamental physics,Gen

    R. Caldwell et al.,Detection of early-universe gravitational-wave signatures and fundamental physics,Gen. Rel. Grav.54(2022) 156 [2203.07972]

    Pith/arXiv arXiv 2022

  2. [2]

    Roshan and G

    R. Roshan and G. White,Using gravitational waves to see the first second of the Universe, Rev. Mod. Phys.97(2025) 015001 [2401.04388]

    Pith/arXiv arXiv 2025

  3. [3]

    Christensen,Stochastic Gravitational Wave Backgrounds,Rept

    N. Christensen,Stochastic Gravitational Wave Backgrounds,Rept. Prog. Phys.82(2019) 016903 [1811.08797]

    Pith/arXiv arXiv 2019

  4. [4]

    Caprini and D.G

    C. Caprini and D.G. Figueroa,Cosmological Backgrounds of Gravitational Waves,Class. Quant. Grav.35(2018) 163001 [1801.04268]

    Pith/arXiv arXiv 2018

  5. [5]

    Kosowsky and M.S

    A. Kosowsky and M.S. Turner,Gravitational radiation from colliding vacuum bubbles: envelope approximation to many bubble collisions,Phys. Rev. D47(1993) 4372 [astro-ph/9211004]

    Pith/arXiv arXiv 1993

  6. [6]

    Garcia-Bellido and D.G

    J. Garcia-Bellido and D.G. Figueroa,A stochastic background of gravitational waves from hybrid preheating,Phys. Rev. Lett.98(2007) 061302 [astro-ph/0701014]

    Pith/arXiv arXiv 2007

  7. [7]

    LiGong, P

    B. LiGong, P. Shi and S.-J. Wang,Gravitational waves originated from the early universe: A review and perspective,Sci. Sin. Phys. Mech. Astro.55(2025) 230405

    2025

  8. [8]

    NANOGRAVcollaboration,The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,Astrophys. J. Lett.951(2023) L8 [2306.16213]

    Pith/arXiv arXiv 2023

  9. [9]

    NANOGRAVcollaboration,The NANOGrav 15 yr Data Set: Observations and Timing of 68 Millisecond Pulsars,Astrophys. J. Lett.951(2023) L9 [2306.16217]. – 12 –

    Pith/arXiv arXiv 2023

  10. [10]

    The dataset and timing analysis,Astron

    EPTA collaboration,The second data release from the European Pulsar Timing Array - I. The dataset and timing analysis,Astron. Astrophys.678(2023) A48 [2306.16224]

    Pith/arXiv arXiv 2023

  11. [11]

    Athron, A

    P. Athron, A. Fowlie, C.-T. Lu, L. Morris, L. Wu, Y . Wu et al.,Can Supercooled Phase Transitions Explain the Gravitational Wave Background Observed by Pulsar Timing Arrays?,Phys. Rev. Lett.132(2024) 221001 [2306.17239]

    Pith/arXiv arXiv 2024

  12. [12]

    LIGO SCIENTIFIC, VIRGOcollaboration,All-sky search for long-duration gravitational wave transients in the first Advanced LIGO observing run,Class. Quant. Grav.35(2018) 065009 [1711.06843]

    Pith/arXiv arXiv 2018

  13. [13]

    Kumar and T

    P. Kumar and T. Dent,Optimized search for a binary black hole merger population in LIGO-Virgo O3 data,Phys. Rev. D110(2024) 043036 [2403.10439]

    Pith/arXiv arXiv 2024

  14. [14]

    KAGRA collaboration,Overview of KAGRA: Detector design and construction history, PTEP2021(2021) 05A101 [2005.05574]

    arXiv 2021

  15. [15]

    LIGO SCIENTIFIC, VIRGO, KAGRA collaboration,Upper Limits on the Isotropic Gravitational-Wave Background from the first part of LIGO, Virgo, and KAGRA’s fourth Observing Run,2508.20721

    Pith/arXiv arXiv

  16. [16]

    LIGO SCIENTIFIC, VIRGO, KAGRA collaboration,Cosmological and High Energy Physics implications from gravitational-wave background searches in LIGO-Virgo-KAGRA’s O1-O4a runs,2510.26848

    arXiv

  17. [17]

    LISA collaboration,Laser Interferometer Space Antenna,1702.00786

    Pith/arXiv arXiv

  18. [18]

    Robson, N.J

    T. Robson, N.J. Cornish and C. Liu,The construction and use of LISA sensitivity curves, Class. Quant. Grav.36(2019) 105011 [1803.01944]

    Pith/arXiv arXiv 2019

  19. [19]

    Rel.26(2023) 5 [2204.05434]

    LISA COSMOLOGYWORKINGGROUPcollaboration,Cosmology with the Laser Interferometer Space Antenna,Living Rev. Rel.26(2023) 5 [2204.05434]

    Pith/arXiv arXiv 2023

  20. [20]

    Hu and Y .-L

    W.-R. Hu and Y .-L. Wu,The Taiji Program in Space for gravitational wave physics and the nature of gravity,Natl. Sci. Rev.4(2017) 685

    2017

  21. [21]

    Ruan, Z.-K

    W.-H. Ruan, Z.-K. Guo, R.-G. Cai and Y .-Z. Zhang,Taiji program: Gravitational-wave sources,Int. J. Mod. Phys. A35(2020) 2050075 [1807.09495]

    Pith/arXiv arXiv 2020

  22. [22]

    Wu,Hyperunified field theory and Taiji program in space for GWD,Int

    Y .-L. Wu,Hyperunified field theory and Taiji program in space for GWD,Int. J. Mod. Phys. A33(2018) 1844014 [1805.10119]

    Pith/arXiv arXiv 2018

  23. [23]

    TIANQINcollaboration,TianQin: a space-borne gravitational wave detector,Class. Quant. Grav.33(2016) 035010 [1512.02076]

    Pith/arXiv arXiv 2016

  24. [24]

    TIANQINcollaboration,The TianQin project: current progress on science and technology, PTEP2021(2021) 05A107 [2008.10332]

    arXiv 2021

  25. [25]

    Luo et al.,The first round result from the TianQin-1 satellite,Class

    J. Luo et al.,The first round result from the TianQin-1 satellite,Class. Quant. Grav.37 (2020) 185013 [2008.09534]

    Pith/arXiv arXiv 2020

  26. [26]

    Weir,Gravitational waves from a first order electroweak phase transition: a brief review,Phil

    D.J. Weir,Gravitational waves from a first order electroweak phase transition: a brief review,Phil. Trans. Roy. Soc. Lond. A376(2018) 20170126 [1705.01783]. – 13 –

    Pith/arXiv arXiv 2018

  27. [27]

    Mazumdar and G

    A. Mazumdar and G. White,Review of cosmic phase transitions: their significance and experimental signatures,Rept. Prog. Phys.82(2019) 076901 [1811.01948]

    Pith/arXiv arXiv 2019

  28. [28]

    Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]

    C. Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]

    Pith/arXiv arXiv 2020

  29. [29]

    Bian et al.,The Gravitational-wave physics II: Progress,Sci

    L. Bian et al.,The Gravitational-wave physics II: Progress,Sci. China Phys. Mech. Astron. 64(2021) 120401 [2106.10235]

    Pith/arXiv arXiv 2021

  30. [30]

    Athron, C

    P. Athron, C. Bal ´azs, A. Fowlie, L. Morris and L. Wu,Cosmological phase transitions: From perturbative particle physics to gravitational waves,Prog. Part. Nucl. Phys.135 (2024) 104094 [2305.02357]

    Pith/arXiv arXiv 2024

  31. [31]

    An, K.-F

    H. An, K.-F. Lyu, L.-T. Wang and S. Zhou,A unique gravitational wave signal from phase transition during inflation*,Chin. Phys. C46(2022) 101001 [2009.12381]

    Pith/arXiv arXiv 2022

  32. [32]

    An, K.-F

    H. An, K.-F. Lyu, L.-T. Wang and S. Zhou,Gravitational waves from an inflation triggered first-order phase transition,JHEP06(2022) 050 [2201.05171]

    Pith/arXiv arXiv 2022

  33. [33]

    An and C

    H. An and C. Yang,Gravitational waves produced by domain walls during inflation,Phys. Rev. D109(2024) 123508 [2304.02361]

    Pith/arXiv arXiv 2024

  34. [34]

    H. An, B. Su, H. Tai, L.-T. Wang and C. Yang,Phase transition during inflation and the gravitational wave signal at pulsar timing arrays,Phys. Rev. D109(2024) L121304 [2308.00070]

    arXiv 2024

  35. [35]

    Caprini, D.G

    C. Caprini, D.G. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni et al., Reconstructing the spectral shape of a stochastic gravitational wave background with LISA, JCAP11(2019) 017 [1906.09244]

    Pith/arXiv arXiv 2019

  36. [36]

    Boileau, N

    G. Boileau, N. Christensen, R. Meyer and N.J. Cornish,Spectral separation of the stochastic gravitational-wave background for LISA: Observing both cosmological and astrophysical backgrounds,Phys. Rev. D103(2021) 103529 [2011.05055]

    Pith/arXiv arXiv 2021

  37. [37]

    Boileau, A

    G. Boileau, A. Lamberts, N. Christensen, N.J. Cornish and R. Meyer,Spectral separation of the stochastic gravitational-wave background for LISA in the context of a modulated Galactic foreground,Mon. Not. Roy. Astron. Soc.508(2021) 803 [2105.04283]

    Pith/arXiv arXiv 2021

  38. [38]

    LISA COSMOLOGYWORKINGGROUPcollaboration,Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation,JCAP10 (2024) 020 [2403.03723]

    Pith/arXiv arXiv 2024

  39. [39]

    Biscoveanu, C

    S. Biscoveanu, C. Talbot, E. Thrane and R. Smith,Measuring the primordial gravitational-wave background in the presence of astrophysical foregrounds,Phys. Rev. Lett.125(2020) 241101 [2009.04418]

    Pith/arXiv arXiv 2020

  40. [40]

    Tinto and S.V

    M. Tinto and S.V . Dhurandhar,Time-Delay Interferometry,Living Rev. Rel.17(2014) 6

    2014

  41. [41]

    Smith, T.L

    T.L. Smith, T.L. Smith, R.R. Caldwell and R. Caldwell,LISA for Cosmologists: Calculating the Signal-to-Noise Ratio for Stochastic and Deterministic Sources,Phys. Rev. D100(2019) 104055 [1908.00546]. – 14 –

    Pith/arXiv arXiv 2019

  42. [42]

    Adams and N.J

    M.R. Adams and N.J. Cornish,Discriminating between a Stochastic Gravitational Wave Background and Instrument Noise,Phys. Rev. D82(2010) 022002 [1002.1291]

    Pith/arXiv arXiv 2010

  43. [43]

    Allen and J.D

    B. Allen and J.D. Romano,Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities,Phys. Rev. D59(1999) 102001 [gr-qc/9710117]

    Pith/arXiv arXiv 1999

  44. [44]

    Gowling and M

    C. Gowling and M. Hindmarsh,Observational prospects for phase transitions at LISA: Fisher matrix analysis,JCAP10(2021) 039 [2106.05984]

    Pith/arXiv arXiv 2021

  45. [45]

    Gowling, M

    C. Gowling, M. Hindmarsh, D.C. Hooper and J. Torrado,Reconstructing physical parameters from template gravitational wave spectra at LISA: first order phase transitions, JCAP04(2023) 061 [2209.13551]

    Pith/arXiv arXiv 2023

  46. [46]

    Boileau, N

    G. Boileau, N. Christensen, C. Gowling, M. Hindmarsh and R. Meyer,Prospects for LISA to detect a gravitational-wave background from first order phase transitions,JCAP02(2023) 056 [2209.13277]

    Pith/arXiv arXiv 2023

  47. [47]

    Lewicki, M

    M. Lewicki, M. Merchand, L. Sagunski, P. Schicho and D. Schmitt,Impact of theoretical uncertainties on model parameter reconstruction from GW signals sourced by cosmological phase transitions,Phys. Rev. D110(2024) 023538 [2403.03769]

    Pith/arXiv arXiv 2024

  48. [48]

    Huang, Z.-C

    F. Huang, Z.-C. Chen and Q.-G. Huang,Detecting cosmological phase transitions with Taiji: sensitivity analysis and parameter estimation*,Chin. Phys.49(2025) 105103 [2504.16712]

    Pith/arXiv arXiv 2025

  49. [49]

    Romano and N.J

    J.D. Romano and N.J. Cornish,Detection methods for stochastic gravitational-wave backgrounds: a unified treatment,Living Rev. Rel.20(2017) 2 [1608.06889]

    Pith/arXiv arXiv 2017

  50. [50]

    O’Shaughnessy, B

    R. O’Shaughnessy, B. Farr, E. Ochsner, H.-S. Cho, C. Kim and C.-H. Lee,Parameter estimation of gravitational waves from nonprecessing black hole-neutron star inspirals with higher harmonics: Comparing Markov-chain Monte Carlo posteriors to an effective Fisher matrix,Phys. Rev. D89(2014) 064048 [1308.4704]

    Pith/arXiv arXiv 2014

  51. [51]

    Porter and N.J

    E.K. Porter and N.J. Cornish,Fisher versus Bayes: A comparison of parameter estimation techniques for massive black hole binaries to high redshifts with eLISA,Phys. Rev. D91 (2015) 104001 [1502.05735]

    Pith/arXiv arXiv 2015

  52. [52]

    Guan, H.-K

    S. Guan, H.-K. Guo, D. Jiao, Q. Liang, L. Wu and Y . Zhang,Measuring Gravitational Wave Spectrum from Electroweak Phase Transition and Higgs Self-Couplings,2511.00996

    Pith/arXiv arXiv

  53. [53]

    Liang, L

    Q. Liang, L. Bian, H.-K. Guo and Y . Wu,Bayesian Analysis of the Complex Singlet Model with Phase Transition Gravitational Waves,2511.21488

    Pith/arXiv arXiv

  54. [54]

    Guth,The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,Phys

    A.H. Guth,The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,Phys. Rev. D23(1981) 347

    1981

  55. [55]

    Linde,A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,Phys

    A.D. Linde,A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,Phys. Lett. B108 (1982) 389

    1982

  56. [56]

    Albrecht and P.J

    A. Albrecht and P.J. Steinhardt,Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,Phys. Rev. Lett.48(1982) 1220. – 15 –

    1982

  57. [57]

    Baumann, P.J

    D. Baumann, P.J. Steinhardt, K. Takahashi and K. Ichiki,Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations,Phys. Rev. D76(2007) 084019 [hep-th/0703290]

    Pith/arXiv arXiv 2007

  58. [58]

    Kohri and T

    K. Kohri and T. Terada,Semianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbations,Phys. Rev. D97(2018) 123532 [1804.08577]

    Pith/arXiv arXiv 2018

  59. [59]

    Tinto and S.V

    M. Tinto and S.V . Dhurandhar,TIME DELAY,Living Rev. Rel.8(2005) 4 [gr-qc/0409034]

    Pith/arXiv arXiv 2005

  60. [60]

    Chen, Q.-G

    Z.-C. Chen, Q.-G. Huang, C. Liu, L. Liu, X.-J. Liu, Y . Wu et al.,Prospects for Taiji to detect a gravitational-wave background from cosmic strings,JCAP03(2024) 022 [2310.00411]

    Pith/arXiv arXiv 2024

  61. [61]

    Boileau, T

    G. Boileau, T. Bruel, A. Toubiana, A. Lamberts and N. Christensen,Gravitational-wave background from extragalactic double white dwarfs for LISA,Astron. Astrophys.702(2025) A246 [2506.18390]

    arXiv 2025

  62. [62]

    J. Chen, C. Liu and Y .-L. Zhang,Circularly polarized gravitational wave background search with a network of space-borne triangular detectors,JCAP05(2025) 050 [2410.18916]

    Pith/arXiv arXiv 2025

  63. [63]

    Wang, L.-Y

    Y . Wang, L.-Y . Jiang, C.-K. Li, J. Ren, S.-P. Tang, Z.-M. Zhou et al.,GRB 200716C: Evidence for a Short Burst Being Lensed,Astrophys. J. Lett.918(2021) L34 [2107.10796]. – 16 –

    Pith/arXiv arXiv 2021