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Table F.1.
Symbols, SI base units, and descriptions of all quantities used in this work. Quantities are ordered alphabetically by their symbols, with Greek symbols preceding Roman ones.
| symbol | SI base units | description |
|---|---|---|
| α | 1 | integrated synchrotron spectral index |
| αc | 1 | typical cluster α |
| αf | 1 | typical filament α |
 |
1 | MEC-weighted mean α |
| βc | 1 | typical cluster MEC (1+z) power law exponent |
| βf | 1 | typical filament MEC (1+z) power law exponent |
| γ | 1 | adiabatic index |
| θ | rad | declination (J2000) |
| μ0 | kg m s−2 A−2 | vacuum permeability |
| ν | s−1 | emission frequency |
| νobs | s−1 | observing frequency |
| νref | s−1 | reference frequency (see 𝒞) |
| ξe | 1 | electron acceleration efficiency |
| Ξρ | 1 | CDF of RV jν | ρ |
| ρ | kg m−3 | total matter density |
| ρBM | kg m−3 | baryonic matter density |
| ρc | kg m−3 | critical density |
| ρDM | kg m−3 | dark matter density |
| σ | kg s−3 K−4 | Stefan–Boltzmann constant |
| ϕ | rad | right ascension (J2000) |
| Φ | 1 | CDF of standard normal RV |
| Ψ | 1 | as in Hoeft & Brüggen (2007) |
| ΩΛ | 1 | dark energy density parameter |
| ΩBM | 1 | baryonic matter density parameter |
| ΩDM | 1 | dark matter density parameter |
| ΩK | 1 | curvature density parameter |
| ΩM | 1 | matter density parameter |
| ΩR | 1 | relativistic particle (i.e. photon and neutrino) density parameter |
| a | 1 | scale factor |
| A | m2 | shock surface area |
| B | kg s−2 A−1 | proper magnetic field strength |
| BCMB | kg s−2 A−1 | CMB magnetic field strength |
| c | m s−1 | speed of light in vacuo |
| cs | m s−1 | speed of sound |
| 𝒞 | 1 | typical cluster-to-filament synchrotron MEC ratio at ν = νref and z = 0 |
| d | m | initial distance between test particle and equidense cluster centre, filament axis or sheet midplane |
| dn | m | comoving distance to the n-th unit cell boundary crossing |
| dΩ2 | rad2 | solid angle element |
| ds2 | m2 | spacetime line element |
| E | 1 | dimensionless Hubble parameter |
 |
1 | CDF of  |
| G | kg−1 m3 s−2 | Newton’s gravitational constant |
| h | 1 | Hubble constant divided by 100 km s−1 Mpc−1 |
| H | s−1 | Hubble parameter |
| Iν | kg s−2 rad−2 | specific intensity (observed) |
| jν | kg m−1 s−2 rad−2 | proper (not comoving) MEC |
| k | m−1 | Fourier dual of r |
| k | m−2 | FLRW (Gaussian) curvature of the Universe |
| l | m | comoving cubic unit cell edge length |
| lf | m | comoving filament length |
| lSE | m | SE kernel lengthscale |
| ℒ | - | light ray path |
| mr | 1 | Petrosian r-band apparent magnitude |
| M | 1 | number of sightlines |
| 𝔐 | - | cosmological model parameter tuple |
| ℳu | 1 | upstream shock Mach number |
| ne | m−3 | electron number density |
| N | 1 | number of unit cell boundary crossings considered |
| pc − f | 1 | probability to pierce through a cluster, then a filament |
| pf | 1 | probability to pierce through a filament only |
| ps − v | 1 | probability to pierce through a sheet, then a void |
| Pν | kg m2 s−2 | proper power density |
| 𝒫 | 1 | percentile random field |
| r | m | comoving position vector |
| r | m | radial comoving distance |
 |
1 | sky direction unit vector |
 |
1 | sky direction unit vector of ray i |
| R | m | equidense cluster radius, filament radius or sheet half-width |
| ℛ | - | comoving reconstruction region; subset of ℝ3 |
| Sk | m | transverse comoving distance |
| t | s | physical time since the Big Bang |
| T | K | proper plasma temperature |
| TCMB | K | CMB temperature |
| uEM | kg m−1 s−2 | electromagnetic energy density |
| v | m s−1 | shock or test particle velocity |
| V | m3 | shock effective volume |
| wf | m | typical comoving filament width |
| Xn | 1 | relative specific intensity contribution of the n-th newly-entered unit cell |
| Xnm | 1 | relative specific intensity contribution of the n-th newly-entered unit cell for the m-th ray |
| ⟨y⟩ | m | shock effective width |
| z, z𝔐 | 1 | cosmological redshift (under parameters 𝔐) |
| zmax | 1 | cosmological redshift up to which LSS is considered |
| zn | 1 | cosmological redshift of n-th unit cell boundary crossing |
 |
1 | specific-intensity–weighted mean redshift |
 |
1 | flux-density–weighted mean redshift |
| Z | 1 | cosmological redshift RV |
 |
1 | specific-intensity–weighted mean redshift RV |
| 𝒵 | 1 | standard normal GRF |
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