Non-linear clustering during the cosmic Dark Ages and its effect on the 21-cm background from minihaloes

Hydrogen atoms inside virialized minihaloes (with Tvir ⩽ 104 K) generate a radiation background from redshifted 21-cm line emission the angular fluctuations of which reflect clustering before and during reionization. We have shown elsewhere that this emission may be detectable with the planned Low-Frequency Array (LOFAR) and Square Kilometer Array (SKA) in a flat cold dark matter universe with a cosmological constant (ΛCDM). This is a direct probe of structure during the ‘Dark Ages’ at redshifts z ≳ 6 and down to smaller scales than have previously been constrained. In our original calculation, we used a standard approximation known as the ‘linear bias’. Here we improve upon that treatment by considering the effect of non-linear clustering. To accomplish this, we develop a new analytical method for calculating the non-linear Eulerian bias of haloes, which should be useful for other applications as well. Predictions of this method are compared with the results of ΛCDM N-body simulations, showing significantly better agreement than the standard linear bias approximation. When applied to the 21-cm background from minihaloes, our formalism predicts fluctuations that differ from our original predictions by up to 30 per cent at low frequencies (high-z) and small scales. However, within the range of frequencies and angular scales at which the signal could be observable by LOFAR and SKA as currently planned, the differences are small and our original predictions prove robust. Our results indicate that while a smaller frequency bandwidth of observation leads to a higher signal that is more sensitive to non-linear effects, this effect is counteracted by the lowered sensitivity of the radio arrays. We calculate the best frequency bandwidth for these observations to be δνobs ∼ 2 MHz. Finally we combine our simulations with our previous calculations of the 21-cm emission from individual minihaloes to construct illustrative radio maps at z = 9.