Linear gas mass density: a unifying descriptor of Cosmic Filaments

Cosmic filaments connect the nodes of the large-scale web and are thought to harbour a substantial fraction of the low-z baryon budget, yet a physically motivated framework for organising their diverse gas properties has been lacking. Here we show that the linear mass density $Λ$ = M_gas/L_fil, measured within the entropy radius r_K (the peak of the radial gas entropy profile) plays this role, analogous to mass in halo physics. Using TNG300-1 at z=0, we characterise the structural, thermodynamic, and chemical properties of 5637 filaments spanning more than two decades in $Λ$. Filaments form a continuous thermodynamic sequence with $Λ$, ranging from long, thin, cool, metal-poor structures to shorter, wider, hotter, and more enriched ones, with core temperatures spanning two orders of magnitude across the full WHIM window. The ratio of mass- to volume-weighted temperature reveals a shift in heat distribution: low-$Λ$ filaments have their hotter gas preferentially associated with denser material, whereas in high-$Λ$ filaments the hotter gas is diffuse. Well-defined scaling relations connect $Λ$ to temperature, electron density, pressure, entropy, and metallicity, with the T-$Λ$ scaling qualitatively consistent with gravitational heating. We present the first measurement of the gaseous linear mass density function, revealing a broad distribution with a flat low-$Λ$ regime and a steep exponential decline at high $Λ$, analogous to the halo mass function. The integrated Compton-y parameter $Y^{\rm int}$ scales systematically with $Λ$, providing a tSZ proxy accessible to current surveys. Finally, galaxy richness per unit length correlates tightly with $Λ$ (Pearson r=0.80), enabling $Λ$-based classification of filaments from spectroscopic surveys alone and guiding targeted searches for the missing baryons in the cosmic web.