Is there a spatial dipole in $\alpha$ from quasar absorption spectra?
In plain words
Light from distant quasars passes through gas clouds that absorb at wavelengths set by $\alpha$ at that place and time. Some analyses claimed $\alpha$ differs slightly in opposite directions on the sky, while newer instruments see no change.
Precise statement
Many-multiplet analyses of Keck and VLT spectra (Webb et al 2011) reported a dipole in $\Delta \alpha/\alpha$ of order $10^{-5}$ across the sky at about $4\sigma$; ESPRESSO measurements (Murphy et al 2022) found $\Delta \alpha/\alpha = (1.3 \pm 1.3\,\mathrm{stat} \pm 0.4\,\mathrm{sys})\,\mathrm{ppm}$ toward HE0515-4414. Establish whether any spatial or redshift variation of $\alpha$ exists at the $1\,\mathrm{ppm}$ level once wavelength-calibration, isotopic-abundance and blinding systematics are controlled.
What would settle it
A blinded ESPRESSO, ANDES or ELT survey of many absorbers across the sky with sub-ppm per-system precision and laser-comb calibration.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
A 2024 reanalysis claimed a $7\sigma$ difference between two absorbers toward HE0515-4414 if Mg isotopic abundances are terrestrial; this depends on the assumed isotope ratios and is disputed.
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