The impact of lunar topography on the 21-cm power spectrum for grid-based arrays : Insights for the Dark-ages EXplorer (DEX)
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$^{-1}$.