Expected performance of air-shower measurements with the radio-interferometric technique

Interferometric measurements with arrays of radio antennas are a powerful and widely used technique in astronomy. Recently, this technique has been revisited for the reconstruction of extensive air showers [1]. This radio-interferometric technique exploits the coherence in the radio emission emitted by billions of secondary shower particles to reconstruct the shower parameters, in particular the shower axis and depth of the shower maximum $X_\mathrm{max}$. The accuracy previously demonstrated on simulations with an idealized detector is very promising. In this article we evaluate the potential of interferometric $X_\mathrm{max}$ measurements using (simulated) inclined air showers with sparse antenna arrays under realistic conditions. To determine prerequisites for the application of the radio-interferometric technique with various antenna arrays, the influence of inaccuracies in the time synchronisation between antennas and its inter-dependency with the antenna density is investigated in detail. We find a strong correlation between the antenna multiplicity (per event) and the maximum acceptable time jitter, i.e., inaccuracy in the time synchronisation. For data recorded with a time synchronisation accurate to within 1 ns in the commonly used frequency band of 30 to 80 MHz, an antenna multiplicity of $> 50$ is needed to achieve an $X_\mathrm{max}$ resolution of $\sigma_{X_\mathrm{max}} \lesssim 20$ g cm$^{-2}$. For data recorded with 2 ns accuracy, already $\gtrsim 200$ antennas are needed to achieve this $X_\mathrm{max}$ resolution. Furthermore, we find no advantage reconstructing $X_\mathrm{max}$ from data simulated at higher observation frequencies, i.e., up to several hundred MHz. Finally, we provide a generalisation of our results from very inclined air showers to vertical geometries.