Context. Three-dimensional (3D) distributions of the 862 nm diffuse interstellar band (DIB) carrier have been computed based on Gaia parallaxes and DIB catalogues, in parallel with 3D maps of dust extinction density. Three-dimensional maps provide local diagnostics and information on the distribution of structures in addition to line-of-sight (LOS) integrated quantities, and allow us to focus on poorly studied low-extinction areas. They make cross-matching with other catalogues possible through estimates of the DIB and extinction along any given path. Aims. We re-examined the relationships between the density of DIB carriers and the absorption and emission properties of spatially co-located dust. Along with laboratory identifications of carriers, these properties may shed light on the formation and evolution of this organic matter. They may also help to model dust emission and absorption properties in a more detailed way. Methods. We used the 3D maps of 862 nm DIBs and of dust extinction, as well as available DIB equivalent width (EW) catalogues and published measurements of parameters characterizing the dust extinction law and the dust emission. We studied the relationships between the extinction-normalized 862 nm DIB EW and the extinction level, the total-to-selective extinction ratio RV, and the dust far-IR emission spectral index β. We re-visited the link between several DIBs and the UV absorption bump at 220 nm. Results. The ratio of the 862 nm DIB carrier density to the optical extinction density (DIBnorm862) is increasing in low-density clouds, confirming with local values the trend seen in the LOS data. In both cases, the coefficients of a fitted power law fall within the range of those measured towards SDSS high-latitude targets for 20 different bands, ranking this DIB among those with a high increase, above that of the broad 4430 Å DIB. This is consistent with the recent measurement of a larger scale height above the Plane for the 862 nm DIB compared to that of the 4430 Å DIB. Using map-integrated 862 nm DIB EWs and extinctions along the paths to APOGEE targets with published proxies R′V for the total-to-selective extinction ratio, we found that, despite a large scatter, DIBnorm862 is positively correlated with R′V for those stars with low to moderate extinctions (AV = 0.2 to 2–3 mag). Independently, using stars from the 862 nm DIB catalogue located outside the disk and for the same regime of extinction, DIBnorm862 is found to be globally anti-correlated with the Planck opacity spectral index β. This is consistent with the observed anti-correlation between β and R′V. In the light of a recent result on the variability of the carbon/silicate ratio in dust grains as a source of this anti-correlation, it suggests that DIBnorm862 increases with the fraction of carbonaceous to silicate grains in the co-located dust, in agreement with the carbonaceous nature of DIB carriers and recent evidences for a spatial correlation between DIBnorm862 and the fluxes of carbon-rich ejecta of asymptotic giant branch (AGB) stars. At higher extinction both trends disappear, and there is evidence for a trend reversal. Regarding the link between the height of the 220 nm UV absorption bump and extinction-normalized EWs of DIBs, we found that two factors explain the absence of previous clear results: the correlation disappears when we move from σ-type to ζ-type DIBs and/or from single-cloud LOSs to paths crossing multiple clouds distant from each other; ζ-type bands can be used to predict low and high values of the bump height, provided one adds a correcting factor linked to the ambient radiation field (e.g. the 5780/5797 DIB ratio). We show examples of simple models of the bump height based on the 5780 Å band, the 5850 Å band and the 5780/5797 DIB ratio. We also found an anti-correlation between DIBnorm and the width of the bump, which similarly disappears from σ-type to ζ-type DIBs. This suggests that a fraction of the bump is generated outside the dense molecular clouds. Conclusions. There are complex relationships between DIBs and dust; however, massive measurements of DIBs and extinction and the derived 3D maps may provide some constraints on the density, the nature, and the contribution to extinction and emission of the co-located dust. This requires large stellar spectroscopic surveys and space-based measurements of UV extinction.
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