A VLT/MUSE galaxy survey towards QSO Q1410: looking for a WHIM traced by BLAs in inter-cluster filaments

Cosmological simulations predict that a significant fraction of the low-$z$ baryon budget resides in large-scale filaments in the form of a diffuse plasma at temperatures $T \sim 10^{5} - 10^{7}$ K. However, direct observation of this so-called warm-hot intergalactic medium (WHIM) has been elusive. In the $Λ$CDM paradigm, galaxy clusters correspond to the nodes of the cosmic web at the intersection of several large-scale filamentary threads. In previous work, we used HST/COS data to conduct the first survey of broad HI Ly$α$ absorbers (BLAs) potentially produced by WHIM in inter-cluster filaments. We targeted a single QSO, namely Q1410, whose sight-line intersects $7$ independent inter-cluster axes at impact parameters $<3$ Mpc (co-moving), and found a tentative excess of a factor of ${\sim}4$ with respect to the field. Here, we further investigate the origin of these BLAs by performing a blind galaxy survey within the Q1410 field using VLT/MUSE. We identified $77$ sources and obtained the redshifts for $52$ of them. Out of the total sample of $7$ BLAs in inter-cluster axes, we found $3$ without any galaxy counterpart to stringent luminosity limits ($\sim 4 \times 10^{8}$ L$_{\odot}$ ${\sim} 0.01$ L$_{*}$), providing further evidence that these BLAs may represent genuine WHIM detections. We combined this sample with other suitable BLAs from the literature and inferred the corresponding baryon mean density for these filaments in the range $Ω^{\rm fil}_{\rm bar}= 0.02-0.04$. Our rough estimates are consistent with the predictions from numerical simulations but still subject to large systematic uncertainties, mostly from the adopted geometry, ionization corrections and density profile.