ALMACAL II: Extreme star-formation-rate densities in a pair of dusty
  starbursts at $z = 3.442$ revealed by ALMA 20-milliarcsec resolution imaging
  We present ALMA ultra-high-spatial resolution ($\sim 20 \, {\rm mas}$)
observations of dust continuum at $920 \, {\rm \mu m}$ and $1.2 \, {\rm mm}$ in
a pair of submm galaxies (SMGs) at $z = 3.442$, ALMACAL-1 (A-1: $S_{\rm 870 \mu
m} = 6.5 \pm 0.2 \, {\rm mJy}$) and ALMACAL-2 (A-2: $S_{\rm 870 \mu m} = 4.4
\pm 0.2 \, {\rm mJy}$). The spectroscopic redshifts of A-1 and A-2 have been
confirmed via serendipitous detection of up to nine emission lines. Our
ultra-high-spatial resolution data reveal that about half of the star formation
in each of these starbursts is dominated by a single compact clump (FWHM size
of $\sim 350 \, {\rm pc}$). This structure is confirmed by independent datasets
at $920 \, {\rm \mu m}$ and $1.2 \, {\rm mm}$. The star-formation rate (SFR)
surface densities of all these clumps are extremely high, $\Sigma_{\rm SFR}
\sim 1200$ to $\sim 3000 \, {M_\odot \, {\rm yr}^{-1} \, {\rm kpc}^{-2}}$, the
highest found in high-redshift galaxies. There is a small probability that A-1
and A-2 are the lensed components of a background source gravitationally
amplified by the blazar host. If this was the case, the effective radius of the
source would be $R_{\rm eff} \sim 40 \, {\rm pc}$, and the de-magnified SFR
surface density would be $\Sigma_{\rm SFR} \sim 10000 \, {M_\odot \, {\rm
yr}^{-1} \, {\rm kpc}^{-2}}$, comparable with the eastern nucleus of Arp 220.
Despite being unable to rule out an AGN contribution, our results suggest that
a significant percentage of the enormous far-IR luminosity in some dusty
starbursts is concentrated in very small star-forming regions. The high
$\Sigma_{\rm SFR}$ in our pair of SMGs could only be measured thanks to the
ultra-high-resolution ALMA observations used in this work, demonstrating that
long-baseline observations are essential to study and interpret the properties
of dusty starbursts in the early Universe.
