EVPA Residuals in Polarimetric Astrophysics

• EVPA residuals are the angular differences between observed and modeled polarization angles, used to evaluate astrophysical models and instrumental accuracy.
• They diagnose departures in simulations, with deviations ranging from 0.09° to over 10° indicating limitations in black hole and ultracompact object models.
• These residuals guide calibration in X-ray polarimetry and maser emission studies, refining techniques in polarization transport theory and experimental setups.

The Electric Vector Position Angle (EVPA) is a fundamental observable in astrophysical polarimetry, quantifying the orientation of the linear polarization vector projected on the sky. EVPA residuals—differences between observed EVPA and theoretical or reference predictions—are used extensively to assess the fidelity of polarization modeling, quantify instrumental or modeling uncertainties, test fundamental physics, and probe source geometry and radiative transfer mechanisms. The rigorous analysis of EVPA residuals is central to fields ranging from relativistic ray-tracing of accretion flows to maser transport theory, X-ray polarimetry, and tests for new physics.

1. Mathematical Definition of EVPA and Residuals

The EVPA, often denoted as $\chi$ or $\mathrm{PA}$, is defined in terms of the Stokes parameters $Q$ and $U$, which encode the state of linear polarization. For a given Stokes vector at observer infinity: $$\mathrm{EVPA}\ (\chi) \equiv \frac{1}{2}\,\arctan2(U,\,Q)$$ where the two-argument arctangent ensures the angle is placed in the correct quadrant, typically restricted to $[-90^\circ, +90^\circ]$ or $[0, 180^\circ)$, depending on context (Tamm et al., 24 Sep 2025, Chowdhury, 11 Nov 2025, Ravi et al., 8 Sep 2025).

EVPA residuals quantify the deviation between two EVPA curves—usually between an observed dataset and a reference model (theoretical, simulation, or calibration standard): $$\Delta\chi(t) = \chi_\mathrm{obs}(t) - \chi_\mathrm{ref}(t)$$ or, in the context of comparing exotic compact object (ECO) models to black hole (BH) predictions: $$\Delta\chi(t) = \chi_0^\mathrm{ECO}(t) - \chi_0^\mathrm{BH}(t)$$ These residuals may be time-dependent, spatially resolved, or parameterized by orbital phase, energy, or projected offset, depending on the application (Tamm et al., 24 Sep 2025, Tobin et al., 2018).

2. EVPA Residuals in General Relativistic Polarimetric Ray-Tracing

High-precision polarimetric ray-tracing codes (e.g., GYOTO, QED-augmented codes) propagate the polarization vector along photon geodesics in strong gravity. The Stokes parameters $\{I, Q, U\}$ are parallel transported and projected onto an orthonormal observer screen, enabling full computation of EVPA at infinity.

EVPA residuals are used to:

• Quantify departures of horizonless ultracompact object models (e.g., relativistic fluid spheres, gravastars) from classic Kerr or Schwarzschild black hole predictions. For example, in simulations of orbiting hot spots, the residual

$\Delta\chi(\phi) \simeq A \sin(2\phi + \delta)$

measures the amplitude ($A$) and phase shift ($\delta$) of deviations per orbit, directly probing the interior structure and compactness. Degree-level EVPA residuals (e.g., $A \sim 3^\circ$–$9^\circ$ at high inclination) are a diagnostic of additional photon orbits in non-BH spacetimes (Tamm et al., 24 Sep 2025).

• Benchmark numerical accuracy in Kerr transport integrators. Median EVPA residuals of $$\langle \Delta\mathrm{PA} \rangle \approx 0.09^\circ$$, with worst-case values $\lesssim 0.32^\circ$, set the practical limits for astrophysical inference with current and next-generation polarimeters; the residual is constructed as an absolute difference between fast and reference propagation schemes (Chowdhury, 11 Nov 2025).

The table below summarizes EVPA residual amplitudes for several ultracompact object configurations (Tamm et al., 24 Sep 2025):

Model	Inclination	Amplitude $A$ (deg)	Phase shift $\delta$ (rad)
FS1	$20^\circ$	0.0	—
FS2	$20^\circ$	4.2	$+0.15\pi$
FS2	$80^\circ$	9.1	$+0.08\pi$
GS1	$20^\circ$	5.5	$+0.20\pi$
GS3	$80^\circ$	2.4	$+0.04\pi$

Deviations at the degree level or larger indicate physically significant departures from Kerr geodesics, typically arising from nontrivial interior photon orbits, shell parameters, or pressure singularities.

3. Instrumental and Systematic Contributions to EVPA Residuals

EVPA residuals are not solely a product of astrophysical or theoretical model mismatch. Instrumental effects—including point-spread function (PSF) errors, polarization leakage, and imperfect transport algorithms—induce systematic EVPA shifts.

In the context of X-ray polarimetry with IXPE, Dinsmore & Romani provide an analytic leakage model, expressing the measured EVPA residual at image position $\vec{r}$ as

$$\delta\psi_\mathrm{leak}(\vec{r}) = \psi_\mathrm{meas}(\vec{r})|_{\Pi\to0}$$

which is subtracted to yield the corrected map $\psi_\mathrm{corr}(\vec{r})$. Uncorrected EVPA residuals can be as large as $30^\circ$ in the PSF halo or sub-PSF-scale extended structures, but with the analytic correction and sky-calibrated PSFs, these residuals are reduced to $\sim 0.1^\circ$–$0.5^\circ$ (Dinsmore et al., 18 Jan 2024).

Best practices dictate applying high-order Taylor expansions and regularizing deconvolution, with performance assessed by the reduction in residuals post-correction.

4. EVPA Residuals in Polarization Transport Theory: Maser Emission

Rigorous testing of EVPA transport models is realized in high-resolution studies of SiO maser emission in circumstellar environments. In the Goldreich–Keeley–Kwan (GKK) asymptotic regime, the EVPA model predicts a sharp $\pi/2$ flip at the Van Vleck angle ($\theta_F$). The EVPA residual at offset $d$ is

$$\Delta\chi(d) = \chi_\mathrm{obs}(d) - [\chi_\mathrm{model}(\theta(d)) + \chi_0]$$

where $\chi_\mathrm{model}(\theta)$ is determined by the GKK solution and $\chi_0$ is a calibration offset (Tobin et al., 2018, Tobin et al., 2018).

Key findings include:

• Residuals scatter symmetrically about zero, with rms $\sim 3^\circ$–$4^\circ$ (calibrated) and $\sim 10^\circ$ (uncalibrated), indicating the model captures the bulk of the EVPA swing.
• Smoother observed EVPA transitions compared to the ideal step function (residuals near flip points $\sim 10^\circ$–$20^\circ$) necessitate consideration of finite gain length, weak anisotropy, or Faraday rotation (Kemball et al., 2011, Tobin et al., 2018).
• The absence of systematic drifts in residuals across epochs supports the dominance of Zeeman-type transport for the tested features.

5. EVPA Residual Analysis in High-Energy Polarimetry and Astroparticle Physics

EVPA residuals serve as essential diagnostics in time- and energy-resolved polarimetry. In X-ray and mm-VLBI contexts, they are used to test for new physics or source variability:

• In IXPE analyses of neutron star binaries, Bayesian nested sampling frameworks (e.g., QUEEN-BEE) quantify the EVPA as a function of time, fitting for rotation rates and computing

$$\Delta\psi(t) = \psi_\mathrm{observed}(t) - \psi_\mathrm{model}(t)$$

Modulo-corrected residuals typically scatter within $5^\circ$–$10^\circ$ about zero for accepted models, indicating a lack of systematic misfit (Ravi et al., 8 Sep 2025).

• In EHT polarimetry of M87*, day-to-day differential EVPA residuals

$$\Delta\chi(\varphi) = \langle\chi(\varphi, t_j)\rangle - \langle\chi(\varphi, t_i)\rangle$$

are constructed to eliminate shared astrophysical backgrounds and set stringent bounds on axion-induced birefringence. Residuals consistent with zero within $3^\circ$–$15^\circ$ directly translate into upper bounds on new physics parameter space (Chen et al., 2021).

6. Statistical Measures and Best Practices in EVPA Residual Evaluation

Statistical quantification of EVPA residuals employs both pointwise and global metrics:

• Mean and rms of $\Delta\chi$ as direct bias and scatter indicators.
• Fitted amplitude and phase for periodic residuals in orbital or rotational contexts (Tamm et al., 24 Sep 2025).
• Reduced $\chi^2$ of residuals to assess model adequacy (Ravi et al., 8 Sep 2025).
• Empirical cumulative distribution functions (ECDFs) vs. stochastic or deterministic model predictions (Kiehlmann et al., 2016).
• Joint distributions of EVPA swing amplitude and smoothness as stochasticity diagnostics.

Proper treatment of phase unwrapping, uncertainty propagation from Stokes parameters, and modulo $180^\circ$ wrapping is essential to avoid spurious residuals (Ravi et al., 8 Sep 2025, Kiehlmann et al., 2016). De-biasing for instrumental errors and regularization in pixel-based corrections are necessary in imaging polarimetry (Dinsmore et al., 18 Jan 2024).

7. Physical Interpretation and Implications of EVPA Residuals

Analysis of EVPA residuals yields stringent constraints on:

• Source geometry and spacetime structure (e.g., ruling out or favoring specific compact object models via degree-level residuals) (Tamm et al., 24 Sep 2025).
• Validity and limits of polarization transport theories (e.g., GKK limit vs. anisotropic pumping) (Tobin et al., 2018).
• Instrumental systematics and calibration fidelity, directly impacting polarimetric measurement precision (Chowdhury, 11 Nov 2025, Dinsmore et al., 18 Jan 2024).
• Fundamental physics, such as constraints on dark-matter candidate axions through the absence of significant residual oscillations (Chen et al., 2021).

Future improvements in instrument calibration, algorithmic development, spatial and spectral binning, and theoretical modeling are likely to further reduce EVPA residuals, enhancing the discriminating power of polarimetric observations across astrophysics and astroparticle physics.