One H_2 molecule per ten million H atoms reveals sub-parsec-scale cold overdensities at z~4

We present the detection and analysis of H_2 absorption at z = 4.24 towards the bright quasar ) ≈ 2 , , which is among the lowest ever measured in quasar absorption systems. Remarkably, this constitutes the highest redshift H_2 detection to date. Two velocity components are resolved that are separated by only 3,̨ms: a narrow (b ∼ 1.7,̨ms) and a broader (b ≃ 6.2,̨ms) component. Modelling the rotational population of H_2 yields a density of łog which was observed with the Very Large Telescope as part of the ESPRESSO QUasar Absorption Line Survey (EQUALS). The high resolving power of R ≈ 120,000 enables the identification of extremely weak H_2 lines in several rotational levels at a total column density of N( ̊m H_2 14 cm^ -2 ̊m H /cm -3 ∼ 2.8and temperature of ∼40,K (typical of the cold neutral medium) for the narrow component and łog n_ ̊m H /cm^ -3 ∼ 1.4 ,T in the neutral medium. Their detection among only seven damped Lyman-α systems in EQUALS suggests that they may be widespread yet usually remain undetected. H_2 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, where 21-cm absorption traces the collective contribution of many for the warmer, more turbulent component under a moderate ultraviolet (UV) field, suggesting at least a several-megaparsec distance from the quasar. This system reveals the existence of tiny (down to ∼ 0.01 pc), cold overdensities cold cloudlets towards larger radio background sources.