Testing the Transverse Scalar Mode of Gravitational Quantum Field Theory with Taiji and LISA

Abstract: Space-based gravitational-wave (GW) detectors, including LISA and Taiji, offer unprecedented access to regimes where alternative theories of gravity may deviate from General Relativity (GR). Gravitational Quantum Field Theory (GQFT) provides a novel framework in which the Poincaré-type inhomogeneous spin symmetry of Weyl-type fermions in the Standard Model is elevated to a gauge symmetry. Within this construction, the fundamental gravitational field is identified with a gravigauge field which behaves as a Goldstone-type bi-covariant vector field. Unlike GR, GQFT predicts additional polarization states: one transverse scalar (breathing) mode and two vector modes. In this work, we focus on the transverse, isotropic scalar mode and investigate its detectability with Taiji. To isolate this mode, we employ the null-response channel (NRC), a specific interferometric combination designed to suppress contributions from other polarizations. We implement an analytical, dynamic orbital model to realistically simulate a triangular constellation. We compute the response functions and sensitivity curves for various interferometric channels, compare them with the standard Michelson channel, and demonstrate the effectiveness of the NRC approach. Our results show that the NRC provides a reliable, waveform-independent criterion for testing non-GR polarizations, and we anticipate that it will serve as a valuable tool for probing gravitational theories in future space-based GW missions.

• The paper introduces a null-response, waveform-independent strategy to distinguish energy distributions unique to Gravitational Quantum Field Theory from these generated by General Theory.
• The authors identify the polarizations of general relativity--transverse scalar modes
• Breathing-mode observability improves by 4 orders of magnitude at low frequencies with realistic dynamic orbits.

The paper develops a waveform-independent interferometric strategy for testing Gravitational Quantum Field Theory (GQFT) against General Relativity (GR) using space-based gravitational-wave detectors such as Taiji and LISA. GQFT, formulated by Wu and collaborators, elevates a Poincaré-type inhomogeneous spin symmetry of Weyl fermions to a gauge symmetry on a flat Minkowski base space. Its fundamental field is a gravigauge field $\chi^a_\mu(x)$ — a Goldstone-type bi-covariant vector identified with the massless graviton — whose composite $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$ reproduces the metric of GR. Linearization of the theory yields five transverse polarizations: the two tensor modes of GR, two vector modes tied to spin-gauge dynamics, and one extra transverse scalar breathing mode $\Psi$. Because current space-borne detectors measure only geodesic-deviation effects, only three geometric polarizations are observable; this work targets the breathing mode.

Null-response channels as polarization discriminants

The methodological core is the null-response channel (NRC). Any first-generation TDI combination can be written as a linear combination of the Sagnac channels $\alpha$, $\beta$, $\gamma$ with frequency-dependent coefficients $a_i$. Since the SNR is a positive-definite functional, it vanishes identically if and only if $$a_1\tilde{\alpha}_p + a_2\tilde{\beta}_p + a_3\tilde{\gamma}_p = 0$$ for every suppressed polarization $p$. The authors verify that for the full set of GQFT polarizations the response matrix is full rank, so no channel nulls all modes simultaneously. However, because the breathing mode is scalar and cannot be exchanged with tensor modes, two dedicated channels exist:

• Tensor-mode NRC (t-NRC): coefficients given by the cross product $$\mathbf{a}^t = \tilde{\mathbf{A}}_+ \times \tilde{\mathbf{A}}_\times$$, nulling plus and cross while retaining sensitivity to the breathing mode.
• Breathing-mode NRC (b-NRC): coefficients $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$0, nulling the breathing mode while retaining tensor response.

Because these coefficients are constructed purely from the Sagnac responses to monochromatic plane waves, the criterion is independent of any assumed waveform $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$1. A signal appearing in t-NRC after a detection in the Michelson $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$2 channel would constitute evidence for non-tensorial polarization, i.e., physics beyond GR.

Static constellation results

Using an analytical single-link response evaluated in the spacecraft frame with fixed geometry, the authors compute polarization-averaged response functions $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$3 and sensitivity curves $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$4, where the noise PSD accounts for correlations among Sagnac channels using Taiji noise parameters ($\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$5 m, $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$6). In the low-frequency regime the responses exhibit power-law behavior:

Channel	Observable mode	Low-frequency scaling
t-NRC	breathing	$\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$7
b-NRC	tensor	$\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$8

The steep $\chi_{\mu\nu} = \eta_{ab}\chi^a_\mu\chi^b_\nu$9 scaling reflects the isotropy of the breathing mode: its nearly uniform projection across arms forces cancellation to high order in TDI combinations. Consequently, NRC configurations are less sensitive than Michelson $\Psi$0 at low frequencies, but in the band approaching 0.05–1 Hz they discriminate polarizations without significant sensitivity loss. The paper proposes an operational diagnostic: the ratio of SNRs between NRC and $\Psi$1 channels follows distinct curves depending on whether the signal contains tensor or breathing content, providing a quantitative classification tool once a source is detected.

Dynamic constellation results

The analysis is then extended to realistic heliocentric orbits including second-order eccentricity corrections ($\Psi$2), following the Rubbo–Cornish–Poujade parametrization with arm lengths $\Psi$3 m that vary at order $\Psi$4. Orbital motion breaks the isotropy that suppresses the scalar response, and the effect is substantial:

• Breathing mode in t-NRC: scaling improves from $\Psi$5 to $\Psi$6, and the low-frequency sensitivity improves by four orders of magnitude relative to the static case.
• Tensor mode in b-NRC: scaling remains $\Psi$7, with response varying by roughly 20% compared to the static configuration.

This asymmetry carries a direct implication: orbital dynamics must be included in any realistic data-analysis pipeline targeting scalar polarizations, since neglecting them would underestimate the detectability of the breathing mode by orders of magnitude. Time-domain illustrations confirm the qualitative picture — under t-NRC the tensor contribution vanishes exactly while a faint residual breathing signal survives, whereas under b-NRC the GR and GQFT responses become identical, which the authors interpret as a consistency check of the tensor sector within GQFT.

Limitations and open questions

Several caveats bound the results. First, the two vector modes of GQFT couple to spin-related antisymmetric tensors and are not addressed here; their detectability requires incorporating spin-gravity coupling effects beyond geodesic deviation. Second, the analysis uses first-generation TDI throughout, justified by the claim that second-generation channels yield equivalent GW sensitivities; whether the larger algebraic structure of second-generation TDI admits NRCs with broader or purer mode suppression remains unexplored. Third, the numerical results use monochromatic plane waves at selected sky positions, and the sky-averaged sensitivity curves assume specific Taiji noise realizations; extension to realistic waveforms and LISA-specific noise budgets is left open. Finally, the four-order-of-magnitude enhancement relies on the second-order orbital model, so the robustness of the conclusion under higher-order orbit modeling or full numerical ephemerides has not been established.

Conclusion

This paper establishes a systematic, waveform-independent procedure for distinguishing GR from GQFT via polarization content, built on analytically constructed null-response channels over the Sagnac basis. Its principal quantitative findings are the $\Psi$8 versus $\Psi$9 scaling transition of the breathing-mode response in t-NRC when orbital dynamics are included, the associated four-order-of-magnitude low-frequency sensitivity gain, and the demonstration that NRC-based SNR ratios provide a practical discriminator without high-frequency sensitivity loss. The framework applies directly to Taiji and LISA data analysis and defines a concrete observational test: absence of signal in t-NRC constrains the breathing mode, while presence signals physics beyond GR.