High-resolution international LOFAR observations of 4C~43.15 -- Spectral ages and injection indices in a high-z radio galaxy

Radio sources with steep spectra are preferentially associated with the most distant galaxies, the $\alpha-z$ relation, but the reason for this relation is an open question. The spatial distribution of spectra in high-z radio sources can be used to study this relation, and low-frequency observations are particularly important in understanding the particle acceleration and injection mechanisms. However, the small angular sizes of high-z sources together with the inherently low resolution of low-frequency radio telescopes until now has prevented high angular resolution low-frequency observations of distant objects. Here we present subarcsecond observations of a $z = 2.4$ radio galaxy at frequencies between $121$ MHz and $166$ MHz. We measure the spatial distribution of spectra, and discuss the implications for models of the $\alpha-z$ relation. We targeted 4C 43.15 with the High Band Antennas (HBAs) of the \textit{International LOFAR Telescope} (ILT) with a range of baselines up to $1300\ \mathrm{km}$. At the central frequency of $143$ MHz we achieve an angular resolution of $\sim 0.3''$. By complementing our data with archival \textit{Very Large Array} (VLA) data we study the spectral index distribution across 4C 43.15 between $55\ \mathrm{MHz}$ and $8.4\ \mathrm{GHz}$ at resolutions of $0.4''$ and $0.9''$. With a magnetic field strength of $B = 5.2$ nT and fitted injection indices of $\alpha^\mathrm{north}_\mathrm{inj} = -0.8$ and $\alpha^\mathrm{south}_\mathrm{inj} = -0.6$, fitting a Tribble spectral ageing model results in a spectral age of $\tau_\mathrm{spec} = 1.1 \pm 0.1$ Myr. We conclude that our data on 4C 43.15 indicates that inverse Compton losses could become comparable to or exceed synchrotron losses at higher redshifts and that inverse Compton losses could be a viable explanation for the $\alpha-z$ relation.