Taiji Resolving Power for the Transverse Scalar Mode of Gravitational Waves

Abstract: We investigate how small a co-propagating transverse-scalar component can be resolved by Taiji in an already identified bright tensor chirp. Using a source-tracked tensor-null response, we formulate the problem in terms of the minimum resolvable scalar strain fraction and evaluate it over the sky. For a one-year benchmark chirp with tensor signal-to-noise ratio $ρ<em>T=1000$, we find that Taiji can rule out transverse-scalar strain fractions $ε_b\gtrsim0.532\%$ at the all-sky median level. The threshold scales as $ε</em>{b,\min}\proptoρ_T<sup>{-1}$, so brighter tensor events can probe correspondingly smaller scalar fractions. We further find that this sub-percent resolving power remains predictable under small source-parameter mismatches through the associated tensor-leakage structure.

• The paper develops a dynamic tensor-null response method for Taiji that tracks changing detector geometry and chirp frequency while retaining the full unequal-arm A/E covariance.
• The forecast finds a median minimum resolvable scalar strain fraction of 0.532% for a one-year chirp with tensor SNR 1000, with 95.2% of the sky reaching below 1%.
• The analysis shows that accurate source tracking is essential, since sky-position, frequency, chirp-mass, and timing mismatches can leak tensor power into the null channel and mimic a scalar signal.

This paper develops a detector-level forecast for how small a co-propagating transverse-scalar (breathing) gravitational-wave component Taiji can resolve when embedded in an already identified, bright tensor chirp (2608.11852). The work extends the tensor null-response channel (t-NRC) formalism of a prior monochromatic study (Xu et al., 15 Jul 2026) to a source-tracked, chirping signal and couples it to an unequal-arm $A/E$ tensor normalization, yielding a quantitative answer: for a one-year benchmark chirp with tensor SNR $\rho_T=1000$, Taiji can rule out scalar strain fractions $\epsilon_b \gtrsim 0.532\%$ at the all-sky median level.

Dynamic tensor-null response and the scalar-fraction statistic

The construction uses the six one-way laser links of the Taiji heliocentric constellation, evaluated along the time-dependent orbit with first-generation TDI Sagnac variables. At each epoch, tensor-null coefficients are defined in the three-dimensional Sagnac channel space via the antisymmetric cross product of the $+$ and $\times$ response vectors, so that the combination annihilates both tensor polarizations while generally retaining a breathing response. Because the coefficients are recomputed along the evolving chirp track, $\mathbf a_t(t)=\mathbf a_t[f(t),t]$, the cancellation follows both the detector geometry and the gravitational-wave frequency as they change over the observation. Epochs where the two tensor response vectors become nearly linearly dependent are excluded via a conditioning cut $q_t \geq 0.05$, which retains 99.6% of the one-year track for the benchmark.

The tensor amplitude is normalized independently with the standard Michelson-type $A/E$ channels, without imposing the equal-arm approximation $S_{AE}=0$, so the full Hermitian covariance of the flexing constellation is retained. Combining the breathing information accumulated in the null channel with the polarization-averaged tensor information in the $A/E$ network gives the central statistic:

$\rho_T=1000$0

with an operational scalar-channel threshold $\rho_T=1000$1. This formulation cleanly separates the brightness of the identified event ($\rho_T=1000$2) from the detector's relative resolving power ($\rho_T=1000$3), and implies the scaling $\rho_T=1000$4 for fixed source track, sky position, and observation time. The authors are explicit that $\rho_T=1000$5 is a strain-amplitude fraction, not an energy fraction, an SNR fraction, or a theory coupling, and that the forecast is secondary-noise-limited under the assumption that laser-frequency noise is suppressed by TDI.

All-sky resolving power for the benchmark chirp

The benchmark waveform is a leading-order quasi-circular inspiral with $\rho_T=1000$6, evolving from $\rho_T=1000$7 mHz to approximately 53 mHz over one year. These parameters were selected by a conditional scan of the $\rho_T=1000$8 plane at a reference sky direction inherited from the earlier detector-response study, and the authors show in an appendix that the preferred waveform parameters shift with sky direction (chirp mass varies most strongly, while preferred initial frequencies remain in the tens-of-mHz band). The conditional minima at three tested directions all remain at the few-$\rho_T=1000$9 level, supporting the benchmark as representative of the sensitivity scale, though not as a sky-independent optimum.

For the fixed benchmark track at $\epsilon_b \gtrsim 0.532\%$0, the all-sky percentiles of the minimum resolvable scalar fraction are:

Statistic	Value
10th percentile	0.385%
Median	0.532%
90th percentile	0.819%
Best sky direction	0.306%

Coverage is substantial: 95.2% of the sky reaches $\epsilon_b \gtrsim 0.532\%$1, and 40.7% reaches below 0.5%. Because the threshold scales as $\epsilon_b \gtrsim 0.532\%$2, the median improves to 0.266% at $\epsilon_b \gtrsim 0.532\%$3 and 0.177% at $\epsilon_b \gtrsim 0.532\%$4; at the latter value, 92.7% of the sky probes below 0.3%. A monochromatic appendix calculation confirms that the favorable response lies in a broad window at a few $\epsilon_b \gtrsim 0.532\%$5 Hz, which the benchmark chirp traverses, and that the window survives the conditioning cut, indicating it is not a numerical artifact of the null construction. For observations beyond one year, the absolute strain requirement scales as $\epsilon_b \gtrsim 0.532\%$6 under a repeated-annual-rate approximation, though the scalar-fraction forecast at fixed $\epsilon_b \gtrsim 0.532\%$7 is governed by the relative information rates rather than the observation time alone.

Robustness to source-parameter mismatch

The sub-percent reach assumes exact knowledge of the tensor source parameters. Errors in the assumed sky position or chirp track break the tensor cancellation and leak tensor power into the null channel; at $\epsilon_b \gtrsim 0.532\%$8, a residual tensor response at the few-$\epsilon_b \gtrsim 0.532\%$9 level already produces leakage SNR comparable to $+$0. One-parameter scans at the median representative sky give the displacements at which worst-polarization leakage reaches this reference level:

Mismatched parameter	Tolerance
Ecliptic latitude $+$1	8.39 arcmin
Ecliptic longitude $+$2	8.78 arcmin
Initial frequency $+$3	474 ppm
Chirp mass $+$4	0.361%
Track time origin $+$5	1.14 days

The authors emphasize these are leakage tolerances, not parameter-estimation uncertainties, and should not be read as five independent cuts on an allowed parameter region. A joint local analysis makes this concrete: the polarization-averaged leakage is well described by a quadratic form whose normalized metric has eigenvalues $+$6, $+$7, $+$8, $+$9, and $\times$0, so only four leakage directions are locally resolved. The nearly null eigenvector is a correlated shift of $\times$1 and $\times$2, a degeneracy that follows directly from the chirp parameterization, where both quantities modify the constant part of the frequency track. The allowed mismatch region is therefore strongly anisotropic, with weak coupling between sky coordinates and intrinsic chirp parameters. Validation in an appendix shows the local metric reproduces exact leakage recomputations to the sub-$\times$3 level, with zero-mismatch residuals at $\times$4.

Limitations and open questions

Several limitations are conceded explicitly. The benchmark uses a leading-order Newtonian chirp and is a detector benchmark, not an astrophysically preferred source population. The forecast assumes first-generation TDI responses with exact laser-noise suppression, whereas a flexing constellation strictly requires second-generation TDI; the authors note the two have equivalent GW sensitivity. Most significantly, the leakage metric propagates a specified parameter error into tensor contamination but does not determine whether parameter estimation for an actual event will achieve the required tracking accuracy — that requires a Fisher or Bayesian analysis with a consistent tensor waveform model, which the paper deliberately leaves open. Two further open problems are stated concretely: a systematic optimization of waveform parameters against sky-averaged or population-weighted scalar-fraction criteria, and the connection of the detector-level reach to specific source models such as gravitational quantum field theory, where the scalar amplitude is set by source dynamics rather than being a free phenomenological parameter.

Conclusion

The paper quantifies Taiji's resolving power for a co-propagating transverse-scalar polarization in an identified bright tensor chirp: a few-$\times$5 strain fraction at $\times$6, improving linearly with tensor SNR and covering most of the sky at the sub-percent level. The result is conditional on accurate source tracking, for which the paper provides explicit leakage tolerances and their correlated joint structure. It remains an open question whether real parameter estimation can meet those tolerances, and which astrophysical sources in specific beyond-GR theories would produce scalar fractions in the probed range.