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These structures have been observed in overdensities of Ly<jats:italic>\u03b1<\/jats:italic>emitters in the MUSE Extremely Deep Field, a 140 h deep MUSE observation located in the<jats:italic>Hubble<\/jats:italic>Ultra-Deep Field. Among the 22 overdense regions identified, five are likely to harbor very extended Ly<jats:italic>\u03b1<\/jats:italic>emission at high significance with an average surface brightness of 5\u2005\u00d7\u200510<jats:sup>\u221220<\/jats:sup>erg s<jats:sup>\u22121<\/jats:sup>cm<jats:sup>\u22122<\/jats:sup>arcsec<jats:sup>\u22122<\/jats:sup>. Remarkably, 70% of the total Ly<jats:italic>\u03b1<\/jats:italic>luminosity from these filaments comes from beyond the circumgalactic medium of any identified Ly<jats:italic>\u03b1<\/jats:italic>emitter. Fluorescent Ly<jats:italic>\u03b1<\/jats:italic>emission powered by the cosmic UV background can only account for less than 34% of this emission at<jats:italic>z<\/jats:italic>\u2004\u2248\u20043 and for not more than 10% at higher redshift. We find that the bulk of this diffuse emission can be reproduced by the unresolved Ly<jats:italic>\u03b1<\/jats:italic>emission of a large population of ultra low-luminosity Ly<jats:italic>\u03b1<\/jats:italic>emitters (&lt; 10<jats:sup>40<\/jats:sup>erg s<jats:sup>\u22121<\/jats:sup>), provided that the faint end of the Ly<jats:italic>\u03b1<\/jats:italic>luminosity function is steep (<jats:italic>\u03b1<\/jats:italic>\u2004\u2a85\u2004\u22121.8), it extends down to luminosities lower than 10<jats:sup>38<\/jats:sup>\u2005\u2212\u200510<jats:sup>37<\/jats:sup>erg s<jats:sup>\u22121<\/jats:sup>, and the clustering of these Ly<jats:italic>\u03b1<\/jats:italic>emitters is significant (filling factor &lt; 1\/6). If these Ly<jats:italic>\u03b1<\/jats:italic>emitters are powered by star formation, then this implies their luminosity function needs to extend down to star formation rates &lt; 10<jats:sup>\u22124<\/jats:sup>\u2006<jats:italic>M<\/jats:italic><jats:sub>\u2299<\/jats:sub>yr<jats:sup>\u22121<\/jats:sup>. These observations provide the first detection of the cosmic web in Ly<jats:italic>\u03b1<\/jats:italic>emission in typical filamentary environments and the first observational clue indicating the existence of a large population of ultra low-luminosity Ly<jats:italic>\u03b1<\/jats:italic>emitters at high redshift.<\/jats:p>","DOI":"10.1051\/0004-6361\/202039887","type":"journal-article","created":{"date-parts":[[2021,2,17]],"date-time":"2021-02-17T20:20:39Z","timestamp":1613593239000},"page":"A107","source":"Crossref","is-referenced-by-count":71,"title":["The MUSE Extremely Deep Field: The cosmic web in emission at high redshift"],"prefix":"10.1051","volume":"647","author":[{"given":"R.","family":"Bacon","sequence":"first","affiliation":[]},{"given":"D.","family":"Mary","sequence":"additional","affiliation":[]},{"given":"T.","family":"Garel","sequence":"additional","affiliation":[]},{"given":"J.","family":"Blaizot","sequence":"additional","affiliation":[]},{"given":"M.","family":"Maseda","sequence":"additional","affiliation":[]},{"given":"J.","family":"Schaye","sequence":"additional","affiliation":[]},{"given":"L.","family":"Wisotzki","sequence":"additional","affiliation":[]},{"given":"S.","family":"Conseil","sequence":"additional","affiliation":[]},{"given":"J.","family":"Brinchmann","sequence":"additional","affiliation":[]},{"given":"F.","family":"Leclercq","sequence":"additional","affiliation":[]},{"given":"V.","family":"Abril-Melgarejo","sequence":"additional","affiliation":[]},{"given":"L.","family":"Boogaard","sequence":"additional","affiliation":[]},{"given":"N. 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