One H2 molecule per ten million H-atoms reveals sub-pc scale cold overdensities at z~4

We present the detection and analysis of H2 absorption at z = 4.24 towards the bright quasar J0007-5705, observed with the Very Large Telescope as part of the ESPRESSO QUasar Absorption Line Survey (EQUALS). The high resolving power, R~120000, enables the identification of extremely weak H2 lines in several rotational levels at a total column density of N(H2)~2x10^14 cm^-2, among the lowest ever measured in quasar absorption systems. Remarkably, this constitutes the highest-redshift H2 detection to date. Two velocity components are resolved, separated by only 3 km/s: a narrow (b~1.7 km/s) and a broader (b~6.2 km/s) component. Modelling the rotational population of H2 yields density of log nH/cm^-3 ~ 2.8 with temperature of ~40K (typical of the cold neutral medium) for the narrow component and log nH/cm^-3 ~ 1.4 , T~600K for the warmer, more turbulent component under a moderate ultraviolet (UV) field, suggesting at least several Mpc distance from the quasar. This system reveals the existence of tiny (down to ~0.01 pc), cold overdensities in the neutral medium. Their detection among only 7 damped Lyman-alpha systems in EQUALS suggests that they may be widespread yet usually remain undetected. H2 provides an exceptionally sensitive probe of these structures: even a minute molecular fraction produces measurable Lyman-Werner absorption lines along the extremely narrow optical beam -- the size of the quasar's accretion disc -- when observed at sufficiently high spectral resolution. High-resolution spectroscopy on extremely large telescopes may routinely detect and resolve such structures in the distant Universe, when 21-cm absorption will trace the collective contribution of many cold cloudlets toward larger radio background sources.