Systematic effect induced by misalignment in a Reflective Polarization Modulator for CMB, and application to the LiteBIRD case

Technical Summary: Systematic Effect Induced by Misalignment in a Reflective Polarization Modulator for CMB, and Application to the LiteBIRD Case

Problem Statement
The LiteBIRD space mission aims to measure Cosmic Microwave Background (CMB) polarization with unprecedented precision to detect primordial B-modes and constrain the tensor-to-scalar ratio, $r$. A critical component of the mission is the rotating Half-Wave Plate (HWP), used to modulate polarization signals and mitigate systematic errors. This paper investigates a specific systematic effect arising in reflective HWP configurations: a constant misalignment between the HWP's rotation axis and its optical axis. Even if the HWP is physically ideal (plane-parallel), this misalignment mimics the effect of a physical wedge found in transmissive HWPs. The authors term this the "wedge-like effect." This misalignment induces a time-dependent pointing error that rotates synchronously with the HWP, potentially introducing spurious polarization signals and biasing the measurement of $r$.

Methodology
The authors utilize the LiteBIRD Simulation (LBS) framework to perform end-to-end simulations tracking the propagation of the wedge-like effect from Time-Ordered Data (TOD) to reconstructed CMB maps and angular power spectra.

• Simulation Setup: The study employs a simplified "CMB-only" sky model, excluding instrumental noise, the dipole, and astrophysical foregrounds. This simplification is intended to isolate the deterministic imprint of the wedge effect without the degeneracies introduced by component separation or noise.
• Configuration: The analysis focuses on a reflective HWP configuration for the Medium-to-High Frequency Telescope (MHFT). The wedge angle ($\alpha$) is modeled as a constant tilt of the rotation axis relative to the optical axis.
• Data Processing: Simulations are run with varying wedge angles ($\alpha$) and detector configurations (ranging from 1 to 6 detectors, organized in orthogonal pairs). The map-making process assumes no prior knowledge of the wedge; the pointing reconstruction uses the nominal scanning strategy, while the TOD is generated with the wedge-induced pointing offset.
• Analysis: The authors analyze the resulting residual maps, angular power spectra ($C_\ell$), and the bias introduced to the tensor-to-scalar ratio ($\Delta r_{wedge}$). They employ likelihood functions to estimate $r$ and fit the B-mode spectra to distinguish between primordial tensor modes and lensing-like contamination.

Key Contributions and Results

1. Characterization of the Wedge-Like Effect: The paper demonstrates that a constant misalignment angle $\alpha$ creates a circular pointing error pattern on the sky with an angular diameter of $4\alpha$. This results in a redistribution of sky hits and introduces spurious polarization signals.
2. Impact on Power Spectra: The wedge-like effect generates spurious B-mode power. Crucially, the spectral shape of this contamination mimics gravitational lensing B-modes rather than primordial tensor modes. Consequently, the effect cannot be fully absorbed by simply rescaling the primordial tensor amplitude ($r$) in a single-parameter fit.
3. Bias on the Tensor-to-Scalar Ratio ($r$):
   • The wedge effect leads to a systematic overestimation of $r$ if not accounted for.
   • By evaluating the total error $\Delta r_{wedge}$, the authors establish a constraint on the maximum allowable wedge angle. For a mission duration of 1 day with 2 detectors, the maximum allowable wedge angle to keep systematic errors below the LiteBIRD budget ($\Delta r_{max} = 5.7 \times 10^{-6}$) is found to be $\alpha_{max} = 12.7$ arcmin.
   • For an HWP diameter of 500 mm, this corresponds to a mechanical edge displacement of approximately 2 mm.
4. Scaling with Detector Count: The study finds that the contamination level decreases as the number of orthogonal detector pairs increases. With 6 detectors, the allowable wedge angle relaxes to $\alpha_{max} = 22.7$ arcmin. This suggests that the wedge-induced pointing errors partially cancel out when combining signals from multiple detectors.
5. Comparison with Transmissive HWPs: The authors compare the reflective case to the transmissive case (where wedge angle $\psi$ is defined by non-parallel surfaces). They derive an equivalence relation $\alpha_{eq} \approx \frac{n-1}{2}\psi$, where $n$ is the refractive index. For a sapphire transmissive HWP ($n \approx 3.1$), the equivalent reflective tilt is roughly 4.2 arcmin for the same pointing deviation. The paper notes that meta-material HWPs with lower effective refractive indices could offer a modest relaxation of these requirements.

Significance and Claims
The paper claims to provide a proof-of-concept demonstration of how a specific optical misalignment in reflective HWPs propagates through the LiteBIRD data analysis pipeline. Its primary significance lies in:

• Establishing Alignment Requirements: Quantifying the mechanical tolerance ($\alpha_{max}$) required for the HWP rotation axis alignment to ensure the systematic error remains within mission specifications.
• Identifying the Nature of Contamination: Highlighting that the wedge-like effect produces a "spurious lensing" signal rather than a primordial tensor signal, which complicates the separation of the true $r$ signal from systematics.
• Demonstrating Mitigation via Redundancy: Showing that increasing the number of detectors reduces the impact of the wedge effect, providing a design insight for future CMB experiments.

Limitations and Future Work
The authors explicitly acknowledge the limitations of their study:

• Idealized Conditions: The analysis assumes perfectly matched detectors within orthogonal pairs and neglects noise and foregrounds. In a realistic scenario, detector mismatch could lead to significant Intensity-to-Polarization ($I \to P$) leakage, which is not captured here.
• Static Assumption: The study assumes a constant wedge angle. The authors note that potential "wobbling" of the reflective HWP could introduce time-dependent variations in the wedge angle, which may have non-negligible impacts not addressed in this work.
• Calibration: The analysis assumes the wedge angle is unknown during map-making. The impact of imperfect calibration (where the estimated angle differs from the true angle) is left for future investigation.

In conclusion, the paper asserts that precise optical alignment is critical for future CMB polarization experiments like LiteBIRD and that the wedge-like effect must be carefully managed to avoid biasing the measurement of primordial gravitational waves.