On the general nature of 21cm-Lyman-$\alpha$ emitters cross-correlations during reionisation

We explore how the characteristics of the cross-correlation functions between the 21cm emission from the spin-flip transition of neutral hydrogen (HI) and early Lyman-$\alpha$ (Ly$\alpha$) radiation emitting galaxies (Ly$\alpha$ emitters, LAEs) depend on the reionisation history and topology and the simulated volume. For this purpose, we develop an analytic expression for the 21cm-LAE cross-correlation function and compare it to results derived from different Astraeus and 21cmFAST reionisation simulations covering a physically plausible range of scenarios where either low-mass ($<10^{9.5}M_\odot$) or massive ($>10^{9.5}M_\odot$) galaxies drive reionisation. Our key findings are: (i) the negative small-scale ($<2$ cMpc) cross-correlation amplitude scales with the intergalactic medium's (IGM) average HI fraction ($\langle\chi_\mathrm{HI}\rangle$) and spin-temperature weighted overdensity in neutral regions ($\langle1+\delta\rangle_\mathrm{HI}$); (ii) the inversion point of the cross-correlation function traces the peak of the size distribution of ionised regions around LAEs; (iii) the cross-correlation amplitude at small scales is sensitive to the reionisation topology, with its anti-correlation or correlation decreasing the stronger the ionising emissivity of the underlying galaxy population is correlated to the cosmic web gas distribution (i.e. the more low-mass galaxies drive reionisation); (iv) the required simulation volume to not underpredict the 21cm-LAE anti-correlation amplitude when the cross-correlation is derived via the cross-power spectrum rises as the size of ionised regions and their variance increases. Our analytic expression can serve two purposes: to test whether simulation volumes are sufficiently large, and to act as a fitting function when cross-correlating future 21cm signal Square Kilometre Array and LAE galaxy observations.