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.
