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The impact of lunar topography on the 21-cm power spectrum for grid-based arrays : Insights for the Dark-ages EXplorer (DEX)

S. Ghosh, L. V. E. Koopmans, C. Brinkerink, A. R. Offringa, A. J. Boonstra, S. A. Brackenhoff, E. Ceccotti, J. K. Chege, L. Y. Gao, B. K. Gehlot, L. I. Gurvits, C. Höfer, F. G. Mertens, M. Mevius, S. Munshi, A. Saxena, J. A. Tauber, H. Vedantham, S. Yatawatta, S. Zaroubi

TL;DR

This paper analyzes how lunar surface topography affects a grid-based 21-cm interferometer designed for the Dark Ages, using the 32×32 DEX concept and the SPADE-21cm end-to-end pipeline. By modeling lateral ($xy$) and vertical ($z$) antenna-position offsets against high-resolution lunar topography data, it quantifies biases in the 2D power spectrum and derives practical deployment tolerances, notably $ rac{\sigma_{ ext{xy}}}{ ext{λ}} \\lesssim 0.027$ at 32.5 MHz and site-dependent vertical constraints governed by the fractal roughness parameters $H$ and $\sigma_s$. The study demonstrates, for the Z42 window, that most contamination remains below the 10–30% level relative to the 21-cm signal for perturbations within these tolerances, while highlighting the greater sensitivity to height variations and the necessity of careful site selection (e.g., Mare Ingenii) and window-function choices (Kaiser20). The results inform array configuration, deployment strategy, and calibration needs for robust 21-cm measurements on the lunar farside, advancing toward a practical path for sub-GHz cosmology. The work also establishes a flexible, physics-driven framework for incorporating lunar topography and sky models into forward simulations of lunar-based radio interferometers.

Abstract

The Dark Ages (DA) provides a crucial window into the physics of the infant Universe, with the 21-cm signal offering the only direct probe for mapping out the three-dimensional distribution of matter at this epoch. To measure this cosmological signal, the Dark-ages EXplorer (DEX) has been proposed as a compact, grid-based radio array on the lunar farside. The minimal design consists of a 32 $\times$ 32 array of 3-m dipole antennas, operating in the $7 - 50$ MHz band. A practical challenge on the lunar surface is that the antennas may get displaced from their intended positions due to deployment imprecisions and non-coplanarity arising from local surface undulations. We present, for the first time, an end-to-end simulation pipeline, called SPADE-21cm, that integrates a sky model with a DA 21-cm signal model simulated in the lunar frame and incorporating lunar topography data. We study the effects of both lateral (xy) and vertical (z) offsets on the two-dimensional power spectra across the $7 - 12$ MHz and $30 - 35$ MHz spectral windows, with tolerance thresholds derived only for the latter. Our results show that positional offsets bias the power spectrum by $10 - 30$ per cent relative to the expected 21-cm power spectrum during DA. Lateral offsets within $σ_{xy}/λ\lesssim 0.027$ (at 32.5 MHz) keep the fraction of Fourier modes with strong contamination (> 50 per cent of the signal) to less than 1 per cent, whereas vertical height offsets affect a larger fraction. This conclusion holds for the 21-cm window with $k_\parallel > 0.5$ $h$ cMpc$^{-1}$ over the range of $k_\perp = 0.003 - 0.009$ $h$ cMpc$^{-1}$.

The impact of lunar topography on the 21-cm power spectrum for grid-based arrays : Insights for the Dark-ages EXplorer (DEX)

TL;DR

This paper analyzes how lunar surface topography affects a grid-based 21-cm interferometer designed for the Dark Ages, using the 32×32 DEX concept and the SPADE-21cm end-to-end pipeline. By modeling lateral () and vertical () antenna-position offsets against high-resolution lunar topography data, it quantifies biases in the 2D power spectrum and derives practical deployment tolerances, notably at 32.5 MHz and site-dependent vertical constraints governed by the fractal roughness parameters and . The study demonstrates, for the Z42 window, that most contamination remains below the 10–30% level relative to the 21-cm signal for perturbations within these tolerances, while highlighting the greater sensitivity to height variations and the necessity of careful site selection (e.g., Mare Ingenii) and window-function choices (Kaiser20). The results inform array configuration, deployment strategy, and calibration needs for robust 21-cm measurements on the lunar farside, advancing toward a practical path for sub-GHz cosmology. The work also establishes a flexible, physics-driven framework for incorporating lunar topography and sky models into forward simulations of lunar-based radio interferometers.

Abstract

The Dark Ages (DA) provides a crucial window into the physics of the infant Universe, with the 21-cm signal offering the only direct probe for mapping out the three-dimensional distribution of matter at this epoch. To measure this cosmological signal, the Dark-ages EXplorer (DEX) has been proposed as a compact, grid-based radio array on the lunar farside. The minimal design consists of a 32 32 array of 3-m dipole antennas, operating in the MHz band. A practical challenge on the lunar surface is that the antennas may get displaced from their intended positions due to deployment imprecisions and non-coplanarity arising from local surface undulations. We present, for the first time, an end-to-end simulation pipeline, called SPADE-21cm, that integrates a sky model with a DA 21-cm signal model simulated in the lunar frame and incorporating lunar topography data. We study the effects of both lateral (xy) and vertical (z) offsets on the two-dimensional power spectra across the MHz and MHz spectral windows, with tolerance thresholds derived only for the latter. Our results show that positional offsets bias the power spectrum by per cent relative to the expected 21-cm power spectrum during DA. Lateral offsets within (at 32.5 MHz) keep the fraction of Fourier modes with strong contamination (> 50 per cent of the signal) to less than 1 per cent, whereas vertical height offsets affect a larger fraction. This conclusion holds for the 21-cm window with cMpc over the range of cMpc.

Paper Structure

This paper contains 34 sections, 46 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lunar Topography
  3. Analysis of the Surface Topography
  4. Topography Dataset
  5. Topography Roughness Parameter
  6. Topography Roughness Results
  7. The Dark Ages Explorer
  8. Correlation Architecture
  9. Design Consideration
  10. Array Configuration and spatial resolution
  11. Bandwidth and Spectral Resolution
  12. Location
  13. Antenna position error
  14. Analytical Treatment
  15. Case 1: Perturbation along $xy$-directions
  16. ...and 19 more sections

Figures (15)

  • Figure 1: Left Panel: LROC NAC DTM of the Mare Ingenii, with a spatial resolution of 2 m. The colorbar indicates elevation in metres measured relative to the mean lunar radius ($\sim$ 1737.4 km). Middle Panel: The region selected from the full DTM for this work. Right Panel: The selected region in the middle panel is further divided into four sub-regions (Surface 1: top left, Surface 2: top right, Surface 3: bottom left, Surface 4: bottom right), each spanning an area $\sim$ 0.09 km$^2$.
  • Figure 2: Top Row: The four sub-surfaces that have been detrended by fitting a 2D plane such that the small-scale topographic features are highlighted. Bottom Row: The observed structure functions (or deviograms) of the detrended surfaces. The derived Hurst exponent, $H$ range from 0.898 - 0.936, indicating persistent, and self-affine behaviour. It is observed that at 4 m scale, the RMS deviation of the elevation lie between 0.14 to 0.30 m, while at $\sim$ 175 scale, they reach up to 7 m. Here, the NAC DTM data points are shown in red, and the fitted power-law model in black.
  • Figure 3: Left Panel: Array layout of the DEX configuration used for the simulation. DEX is planned to be a compact array with 1024 antenna elements, arranged on a regular grid, on the farside of the Moon. Middle Panel: The corresponding instantaneous $u\varv$ coverage of the array for a simulated snapshot of 5 minute within the Z42, demonstrating significant redundancy of the array in $u\varv$ sampling. A color gradient is used to indicate the distribution of independent $u\varv$ samples across the full spectral window. The observation was simulated from Mare Ingenii at RA = 23.052060$^\circ$, DEC = -25.696129$^\circ$, corresponding to an observation time of 16:22:30 UTC on July 30, 2040. Right Panel: A histogram of the baseline lengths ($|\mathbf{u}| = \sqrt{u^{2}+v^{2}}$) distribution (in m) with a bin width of 4 m.
  • Figure 4: The fractional difference of the frequency-averaged absolute visibilities on the $u\varv$ plane, showing the effect of antenna position perturbations along the xy direction relative to the unperturbed array. Results are shown for two spectral windows (rows) and increasing perturbation (columns). Concentric circles indicating constant baseline length are drawn on the $u\varv$ plane for illustrative purposes only.
  • Figure 5: The fractional difference of the frequency-averaged absolute visibilities on the $u\varv$ plane, showing the effect of antenna position offsets for the four different surfaces (columns). Results are shown for two spectral windows (rows).
  • ...and 10 more figures