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Primary Beam Chromaticity in HIRAX: I. Characterization from Simulations and Power Spectrum Implications

This paper characterizes the chromaticity of the HIRAX primary beam using a flexible, physics-based modeling technique and demonstrates that accounting for azimuthal variations up to the octupolar order is sufficient to minimize foreground leakage and bias in 21cm power spectrum measurements.

Original authors: Ajith Sampath, Devin Crichton, Kavilan Moodley, H. Cynthia Chiang, Eloy De Lera Acedo, Simthembile Dlamini, Sindhu Gaddam, Kit M. Gerodias, Quentin Gueuning, N. Gupta, Pascal Hitz, Aditya Krishna Kari

Published 2026-01-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ajith Sampath, Devin Crichton, Kavilan Moodley, H. Cynthia Chiang, Eloy De Lera Acedo, Simthembile Dlamini, Sindhu Gaddam, Kit M. Gerodias, Quentin Gueuning, N. Gupta, Pascal Hitz, Aditya Krishna Karigiri Madhusudhan, Shreyam Parth Krishna, V. Mugundhan, Edwin Retana-Montenegro, Benjamin R. B. Saliwanchik, Mario G. Santos, Anthony Walters

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Listening to the Universe's Echo

Imagine the universe is a giant concert hall. For billions of years, it has been humming a very quiet, specific note: the "21cm" signal from neutral hydrogen gas. This signal is the key to understanding Dark Energy, the mysterious force pushing the universe apart and making it expand faster.

The HIRAX telescope is a massive radio microphone array in South Africa designed to listen to this faint hum. However, there is a problem: the concert hall is incredibly noisy. There are loud, bright sources (like our own Milky Way galaxy and distant radio galaxies) that are millions of times louder than the tiny hydrogen signal we want to hear.

The Problem: The "Rainbow" Lens

To hear the quiet signal, astronomers try to subtract the loud noise. They know the loud noise is "smooth" (it doesn't change much as you tune the radio dial), while the hydrogen signal has a specific pattern.

However, the telescope itself has a flaw. Think of the telescope's dish not as a perfect mirror, but as a lens that changes color depending on the light hitting it. In radio terms, this is called chromaticity.

The Analogy: Imagine looking through a pair of glasses that are perfectly clear in the center but have a weird, rippling rainbow pattern on the edges. If you look at a smooth, white wall (the noise), the rainbow pattern on the glasses makes the wall look like it has ripples and colors.
The Consequence: When the telescope looks at the smooth radio noise, its "rainbow glasses" (the beam) accidentally paint ripples onto the data. These ripples look exactly like the hydrogen signal we are trying to find. This is called "leakage." The noise leaks into the quiet zone, hiding the real signal.

What This Paper Did: Mapping the Glasses

The authors wanted to figure out exactly what these "glasses" look like so they can mathematically remove the ripples. They didn't just look at the center of the view (the main beam); they looked at the edges and the far corners (the sidelobes).

The Simulation: They built a virtual version of the HIRAX telescope on a computer using physics software. They simulated how the dish reacts to radio waves across a wide range of frequencies (400 to 800 MHz).
The "Ripple" Discovery: They found that the telescope's main view isn't a smooth, perfect circle. It has a "ripple" effect (like a stone dropped in a pond) caused by the reflection between the dish and the feed antenna. This ripple changes as the frequency changes, making the noise look even more like a signal.
The New Tool (Zernike Transforms): To describe this complex shape, they used a new mathematical toolkit.
- Old Way: Imagine trying to describe a complex cloud shape using only a few simple circles. You'd miss the details.
- New Way: They used a flexible set of shapes (called Zernike Transforms) that can stretch, twist, and ripple to match the telescope's actual shape perfectly, including the weird edges.

The Experiment: How Much Detail Do We Need?

The team ran a test to see how much detail they needed to capture to stop the noise from leaking. They tried different levels of complexity:

Level 1 (Simple): Just the main center circle. Result: The noise still leaked through because the edges were wrong.
Level 2 (Better): The center circle plus the first few "ripples" (sidelobes). Result: Much better, but still some leakage.
Level 3 (Best): The center circle plus the ripples and the fact that the shape isn't perfectly round (it's slightly lopsided).

The Key Finding:
They discovered that to stop the noise from leaking, they needed to model the telescope's view out to about 60 degrees from the center (a very wide view) and include the "lopsided" (azimuthal) features.

The "Octupole" Rule: They found that modeling up to the fourth-order variation (which they call "octupolar") was enough. Think of this as needing to describe the shape not just as a circle, but as a circle with four gentle bumps. Once you include these bumps, the leakage stops, and the "clean window" for the signal opens up.

Why This Matters

If you don't model the telescope's "rainbow glasses" correctly, you can't tell the difference between the noise and the signal. You might think you've found Dark Energy, but you've actually just found a reflection in your own glasses.

This paper provides the blueprint for how to draw those glasses accurately. It tells the HIRAX team exactly how much detail they need to measure (using drones to map the beam in real life) to ensure that when they finally listen to the universe, they are hearing the truth, not a distortion.

Summary in One Sentence

This paper proves that to hear the faint whisper of the early universe, we must mathematically map the complex, colorful, and slightly lopsided "shape" of the telescope's view, extending far beyond the center, to prevent the loud background noise from masquerading as the signal we seek.

Technical Summary: Primary Beam Chromaticity in HIRAX: I. Characterization from Simulations and Power Spectrum Implications

Problem Statement
The Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX) aims to constrain dark energy by mapping Baryon Acoustic Oscillations (BAO) via 21cm intensity mapping. A critical challenge in this endeavor is the presence of foregrounds (galactic and extragalactic) that are orders of magnitude brighter than the cosmological signal. While foregrounds are spectrally smooth, the frequency-dependent (chromatic) response of the telescope's primary beam causes these smooth foregrounds to "leak" into the clean signal window in Fourier space. This leakage is exacerbated by the complex spatial and spectral structures of the beam's sidelobes and mainlobe. Previous studies have modeled beams up to two or three sidelobes, but significant contamination can arise from sources in farther sidelobes, and existing formalisms often lack the capability to characterize chromaticity in wide-field interferometers extending far into the sidelobes.

Methodology
The authors present a physically motivated beam modeling technique designed to analytically represent the full Stokes I primary beam of HIRAX, extending beyond the mainlobe.

Electromagnetic Simulations: The study utilizes electromagnetic (EM) simulations performed in the CST Studio Suite. The model simplifies the HIRAX dish-and-feed configuration (removing legs and cables to reduce computational cost) while retaining the effective diameter of ~4.5 m and the f/0.21 focal ratio. Simulations cover the 400–800 MHz bandwidth with 81 channels.
Analytical Basis (Zernike Transforms): To model the beam, the authors employ a flexible basis derived from the Fourier transform of Zernike polynomials, termed "Zernike Transforms" (ZT). Unlike traditional image-space modeling that focuses on the mainlobe, this basis allows for the separation of radial and azimuthal structures. The model is defined as a sum of real-valued basis functions $X^m_n(\theta, \phi)$ , where $n$ represents the radial order and $m$ the azimuthal order.
Fitting Procedure: The simulated beams are fitted channel-by-channel using a numerical optimization routine. The authors utilize a weighted least-squares approach, up-weighting sidelobes to prevent over-fitting to the mainlobe. The fitting determines coefficients $\alpha^m_n$ and a frequency-dependent scaling parameter $\theta_0$ .
Power Spectrum Analysis: To quantify the impact of beam modeling errors, the authors simulate mock visibilities for a 256-element HIRAX array using the Global Sky Model (GSM) and a Gaussian random field for the 21cm signal. They perform model subtraction using various beam approximations (naive Gaussian, fitted Gaussian, and ZT models with varying radial and azimuthal truncations) to recover the 21cm power spectrum. The fidelity of recovery is assessed using a "naive power spectrum transfer function" ( $T$ ), defined as the ratio of the recovered power spectrum to the true signal power spectrum.

Key Contributions and Results

Beam Characterization: The study characterizes the chromaticity of the HIRAX mainlobe and sidelobes. The mainlobe exhibits a "beam-ripple" structure (sinusoidal variation in beamwidth and amplitude) caused by reflections between the feed and dish surface, which introduces spectral structure that mixes with foregrounds.
Modeling Efficacy: The Zernike Transform model successfully captures complex sidelobe structures. The authors find that modeling up to the octupolar azimuthal order (fourth-order angular variation, $m_{max}=4$ ) is sufficient to minimize leakage outside the wedge with minimal bias.
Impact on Power Spectrum:
- Mainlobe-only models: Models that ignore sidelobes or rely on simple Gaussian fits (even those capturing mainlobe chromaticity) result in significant foreground leakage and residual ripple structures in the power spectrum, particularly at scales $k \sim 0.25$ and $0.5 \, \text{Mpc}^{-1}$ .
- Radial vs. Azimuthal Modes: Increasing the radial order ( $n$ ) is essential to capture the extent of the sidelobes. However, the study indicates that capturing up to $m_{max}=2$ (quadrupolar modes) is largely sufficient to avoid major issues regarding azimuthal asymmetry, though higher orders contribute to the fit.
- Leakage Mitigation: As the beam model complexity increases (approaching the "ground truth" simulated beam), the leakage of foreground power into the clean signal window decreases, and the transfer function approaches unity.
Limitations: The authors note that the current simulations assume noise-free visibilities, perfect redundancy, and uniform beam responses. They explicitly state that cross-coupling effects, polarization leakage (Stokes Q, U, V), and the impact of closely packed dishes on beam shape are not addressed in this work.

Significance and Claims
The paper claims to be the first in a two-part series addressing primary beam chromaticity in HIRAX. Its primary significance lies in establishing a formalism for modeling the full beam, including far sidelobes and their chromaticity, which is vital for high-precision 21cm intensity mapping.

The authors modestly claim that their work:

Provides a necessary analytical basis for understanding beam behavior extending far into sidelobes, a gap in previous 21cm literature.
Demonstrates that neglecting sidelobe chromaticity and azimuthal asymmetries leads to significant foreground leakage and bias in power spectrum estimation.
Sets specific goals for future drone-based beam mapping campaigns, suggesting that modeling up to the octupolar azimuthal order and capturing a substantial radial extent is required to minimize systematic errors.
Lays the groundwork for a subsequent paper that will apply this modeling framework to real drone-mapped beam data and propose specific foreground mitigation strategies.

The paper does not claim to have solved the foreground subtraction problem entirely but rather identifies beam modeling as a critical systematic that must be controlled to achieve the survey's cosmological goals. It emphasizes that while transfer functions can theoretically debias results, accurate characterization of the beam response remains the prerequisite for any successful foreground cleaning or avoidance strategy.