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As the title suggests, can the current observational techniques detect the 3D large-scale distribution of the baryonic gas, rather than just the gas within groups or clusters?

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Yes!

The cosmic web

Matter in the Universe is distributed not uniformly, but in the so-called cosmic web. This large-scale structure consists of sheets and filament of dark matter, baryonic gas, and galaxies, intercepted by huge voids of significantly lower density. Where filaments meet is where we see the largest clusters of galaxies.

This description is a general theoretical prediction of cosmic structure formation. Galaxies have been observed lying in such structures since the 1980's (Tully 1986; Kopylov et al. 1988; Geller & Huchra 1989).

Baryons in absorption

The gas between the galaxies, however, is more elusive. It has been observed indirectly through the so-called Lyman α forest, about which I wrote previously here, here, and here. Briefly, the mechanism behind this phenomenon is that, as light from a bright background source (usually a quasar) travels through the cosmic web toward us, an absorption line is created for each absorbing cloud at the wavelength corresponding to the hydrogen Lyman α line at the redshift of the absorbing gas.

This effect creates a "forest" of absorption lines in the spectrum of the background source, and the density and distribution of absorption lines then tells you the density and distribution of filaments. This is only in 1D, of course, along the line of sight to the source, but given enough sources, you can eventually work out a 3D distribution. The interest lies not in the actual 3D distribution, however, but in a statistical description of the distribution, usually analyzed through its power spectrum.

Baryons in emission

Detecting the gas in emission is more challenging, because it is so dilute. In the vicinity of galaxy overdensities, the gas can be "lit up" by ionizing radiation from the galaxies or quasars: hydrogen is ionized, and when it recombines, it emits Lyman α. Usually more importantly, Lyman α emitted from the central sources can be scattered toward the observed.

Gas lit up in this way manifests itself as so-called Lyman α blobs, and shows you the "knot" part of the cosmic web. But recently, more filamentary structures have been observed to larger distances from the overdensities (Umehata et al. 2019). And even more recently, Bacon et al. (2021) reported on the detection of filamentary structures in more typical environments. Both of these observations were conducted using the integral field spectrograph MUSE mounted on the Very Large Telescope. This instrument gives you an image, but for each pixel, you also have a spectrum, allowing you so measure e.g. redshifts. The Bacon et al. observations required 140 hours of exposure time, highlighting the difficulties of observing something so faint.

3dweb 3D distribution of filaments, seen in Lyman α. The image itself shows you the 2D distribution, and the spectrum allows you to measure the redshift and hence distance along the line of sight, providing you with the 3rd dimension. Note though that the redshift also reflects "real" motion (i.e. not the expansion of the Universe), so the exact distance should be taken with a grain of salt. Image credit: Umehata et al. 2019.

Future complimentary observations

In addition to Lyman α, at low redshifts the web could also be observed with upcoming facilities through the hydrogen 21 cm line (Braun 2004), synchrotron radiation (Oei et al. 2021), and even X-rays (Bregman et al. 2009). I don't know so much about this though.

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