A geometric multimessenger consistency test of radiative and near-zone gravity with LISA and SKA

Technical Summary: A Geometric Multimessenger Consistency Test of Radiative and Near-Zone Gravity

Problem Statement
Compact binary pulsars offer a unique opportunity to test General Relativity (GR) by observing the same system through two independent messengers: precision radio timing and low-frequency gravitational waves (GWs). While existing tests often focus on orbital decay ($\dot{P}_b$) or post-Keplerian parameter consistency within a single theory, this paper proposes a distinct geometric consistency test. The core problem addressed is whether the orbital inclination ($\iota$) inferred from the radiative sector (via GW tensor polarization) agrees with the inclination inferred from the near-zone sector (via the Shapiro delay in radio timing). In GR, these are functions of the same physical angle and must agree; a discrepancy would indicate either unmodeled systematics or a fundamental mismatch between the near-zone metric and the radiative field, potentially signaling modified gravity.

Methodology
The authors define a null residual quantity, $\varepsilon(t_0) = s_{\text{Shapiro}}(t_0) - s_{\text{GW}}(t_0)$, where $s = \sin \iota$. The analysis employs a Fisher matrix approach to estimate the attainable precision ($\sigma_\varepsilon$) for representative systems observable by the Square Kilometre Array (SKA) and the Laser Interferometer Space Antenna (LISA).

1. Radio Timing (Near-Zone):

   • The Shapiro shape parameter $s$ is extracted using a seven-parameter timing model ($\theta = \{s, r, x, P_b, T_0, e, \omega\}$) based on the Damour–Taylor parametrization.
   • The analysis incorporates a conservative radio baseline: a time-of-arrival (TOA) uncertainty of $\sigma_{\text{TOA}} = 1\,\mu\text{s}$ and an effective number of independent measurements $N_{\text{eff}} = 10^4$. This accounts for radiometer noise, jitter, dispersion measure variations, and profile evolution.
   • Robustness checks include the inclusion of the Einstein delay ($\gamma_E$) and the treatment of eccentricity priors to ensure the Shapiro constraint is not artificially inflated by omitted terms.

2. Gravitational Waves (Radiative Sector):

   • The inclination is inferred from the ratio of tensor polarization amplitudes ($R_{\text{GW}} = A_\times / A_+$), which depends solely on $\iota$ for a slowly evolving binary.
   • The LISA signal-to-noise ratio (SNR) is calculated using a sky-averaged sensitivity curve including the Galactic foreground.
   • The inclination uncertainty is approximated using the Takahashi–Seto polarimetric Fisher scaling. To account for unmodeled covariances (e.g., distance-inclination degeneracy, foreground subtraction residuals), the authors introduce a covariance-inflation factor $\kappa_{\text{LISA}}$ (baseline $\kappa=1$, stress-tested up to $\kappa=3$).

3. System Selection:

   • Four benchmark systems are analyzed: a Double Neutron Star (DNS), a Pulsar–White Dwarf (PSR+WD), a hypothetical Pulsar–Black Hole (PSR+BH), and a double-pulsar-like system.
   • The analysis assumes a common epoch $t_0$, requiring the propagation of radio and GW ephemerides to a shared reference time, accounting for potential orbital precession.

Key Contributions and Results
The paper provides a quantitative forecast for the precision of the geometric consistency test $\varepsilon$:

• Attainable Precision: For a favorable DNS system at 5 kpc observed for four years, the residual uncertainty is estimated at $\sigma_\varepsilon \simeq 4 \times 10^{-3}$. For a hypothetical, high-SNR PSR+BH system, the precision improves to $\sigma_\varepsilon \simeq 9 \times 10^{-4}$.
• Limiting Factors: The study identifies that the test is primarily limited by GW polarimetry, not radio timing. Once the Shapiro delay is measured with sufficient precision (which the conservative radio baseline achieves), further improvements in radio timing yield diminishing returns for the null residual. The uncertainty scales almost linearly with the LISA covariance inflation factor $\kappa_{\text{LISA}}$.
• System Viability:
  • DNS: Serves as the robust astrophysical benchmark.
  • PSR+BH: Illustrates the potential reach if such a high-mass companion is discovered, though it requires a chirping template due to orbital evolution.
  • PSR+WD & Double-Pulsar-like: These serve as negative controls; despite excellent radio timing, they are poor candidates for this specific test because their GW signals are too weak or the frequency is too low for LISA polarimetry to be informative.
• Theoretical Interpretation: The authors explicitly avoid mapping $\varepsilon$ to specific scalar-tensor or vector-tensor coupling constants. Instead, they frame the result as a phenomenological bound on "tensor-amplitude bias" or "near-zone metric bias." They note that if phase dephasing (e.g., from dipole radiation) is large, phase-based tests would detect the inconsistency before this amplitude-geometry test becomes relevant.

Significance and Claims
The paper positions this test as a complementary null relation to standard binary pulsar bounds. Its significance lies not in surpassing existing constraints on scalar-tensor coupling (which are often tighter via $\dot{P}_b$), but in testing a different projection of theory space: the consistency between the near-zone metric and the radiative field geometry.

The authors claim that this work defines a quantitative target for future joint Bayesian analyses. They emphasize that the geometric definition of the residual is robust and theory-independent, while its interpretation requires specific waveform models. The primary practical conclusion is that future efforts to constrain modified gravity using this method should prioritize improvements in LISA polarimetry, foreground subtraction, and source localization, rather than solely focusing on increasing radio timing precision beyond the current conservative baseline. The paper concludes by outlining a roadmap for a full Bayesian implementation using Time-Delay Interferometry (TDI) channels and realistic timing likelihoods.