Abstract
Context. Methanol is an important tracer to probe physical and chemical conditions in the interstellar medium of galaxies. Methanol is also the most sensitive target molecule for probing potential space-time variations of the proton-electron mass ratio, μ, a dimensionless constant of nature. Aims. We present an extensive study of the strongest submillimeter absorption lines of methanol (with rest frequencies between 300 and 520 GHz) in the z = 0.89 molecular absorber toward PKS 1830-211, the only high-redshift object in which methanol has been detected. Our goals are to constrain the excitation of the methanol lines and to investigate the cosmological invariance of μ based on their relative kinematics. Methods. We observed 14 transitions of methanol, five of the A-form and nine of the E-form, and three transitions of A-13CH3OH, with ALMA. We analyzed the line profiles with a Gaussian fitting and constructed a global line profile that is able to match all observations after allowing for variations of the source covering factor, line opacity scaling, and relative bulk velocity offsets. We explore methanol excitation by running the non local thermal equilibrium radiative transfer code RADEX on a grid of kinetic temperatures and H2 volume densities. Results. Methanol absorption is detected in only one of the two lines of sight (the southwest) to PKS 1830-211. There, the excitation analysis points to a cool (∼10 - 20 K) and dense (∼104 - 5 cm-3) methanol gas. Under these conditions, several methanol transitions become anti-inverted, with excitation temperatures below the temperature of the cosmic microwave background. In addition, we measure an abundance ratio A/E = 1.0 ± 0.1, an abundance ratio CH3OH/H2 ∼ 2 × 10-8, and a 12CH3OH/13CH3OH ratio 62 ± 3. Our analysis shows that the bulk velocities of the different transitions are primarily correlated with the observing epoch due to morphological changes in the background quasar's emission. There is a weaker correlation between bulk velocities and the lower level energies of the transitions, which could be a signature of temperature-velocity gradients in the absorbing gas. As a result, we do not find evidence for variations of μ, and we estimate Δμ/μ=(-1.8 ± 1.2) × 10-7 at 1-σ from our multivariate linear regression. Conclusions. We set a robust upper limit |Δμ/μ| < 3.6 × 10-7 (3σ) for the invariance of μ at a look-back time of half the present age of the Universe. Our analysis highlights that systematics need to be carefully taken into account in future radio molecular absorption studies aimed at testing Δμ/μ below the 10-7 horizon.