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##  Appendix A Additional figures

(a)

(b)

Figure 14: The MASSCW priors allow for a quantification of prediction
uncertainty. Here we show both an absolute and a relative measure of spread
for the single-direction specific intensity distributions (i.e. marginals) at
$\nu_{\mathrm{obs}}=150\ \mathrm{MHz}$. For each direction, we discard data
outside the $1-99\%$ percentile range. Top: marginal standard deviation
(absolute uncertainty), which closely resembles the marginal mean of Figure
9(a). Bottom: marginal standard deviation over mean (relative uncertainty),
which reveals an inverted trend. Figure 15: As Figure 10, but now for
specific-intensity-weighted mean redshift instead of specific intensity.

Figure 16: Thermal bremsstrahlung specific intensity prior marginal medians at
$E_{\mathrm{obs}}=1\ \mathrm{keV}$ for three deep LOFAR HBA fields. For merger
and accretion shock synchrotron predictions of the same regions, see Figure
11. Top: Lockman Hole. Middle: Abell 2255. Bottom: Ursa Major Supercluster.
Figure 17: Due to cosmological redshifting, the calculation of the specific
intensity $I_{\nu}$ at observing frequency $\nu_{\mathrm{obs}}$ necessitates
knowing the monochromatic emission coefficient (MEC) $j_{\nu}$ at a range of
emission frequencies $\nu$ (see Equation 8). In this work, we generate
$j_{\nu}$ at emission frequencies $\nu=\nu_{\mathrm{obs}}$ and
$\nu=\nu_{\mathrm{obs}}\left(1+z_{\mathrm{max}}\right)$, where
$z_{\mathrm{max}}$ is the maximum cosmological redshift of the large-scale
structure reconstructions. We do so using the same Gaussian random field
realisation $\mathcal{Z}$ (and thus percentile random field realisation
$\mathcal{P}$), implicitly assuming that shocks retain their MEC percentile
rank over the frequency range
$[\nu_{\mathrm{obs}},\nu_{\mathrm{obs}}\left(1+z_{\mathrm{max}}\right)]$. This
assumption holds in three scenarios assuming simple power-law synchrotron
spectra for shocks (sketched above). Each solid line represents a shock with
constant percentile rank (over the range shown, at least); the graphs are
drawn with logarithmic scaling. Regarding the sign of the spectral index, we
use the convention $j_{\nu}\propto\nu^{\alpha}$. Left: the integrated spectral
index $\alpha$ is a monotonically _increasing_ function of $j_{\nu}$ at
$\nu=\nu_{\mathrm{obs}}$ (‘brighter shocks have flatter spectra’). Middle: the
integrated spectral index $\alpha$ is a monotonically _decreasing_ function of
$j_{\nu}$ at $\nu=\nu_{\mathrm{obs}}\left(1+z_{\mathrm{max}}\right)$
(‘brighter shocks have steeper spectra’). Right: the integrated spectral index
$\alpha$ is a $j_{\nu}$-independent constant (at all frequencies); this is a
limiting case of the previous two.

In this appendix, we list four figures referenced in the main text, but
relegated for structural clarity.

##  Appendix B Single-shock synchrotron MEC - total matter density scaling
relation

In this appendix, we derive a synchrotron power density - total matter density
scaling relation for individual shocks with high upstream Mach numbers in
cluster outskirts and filaments, assuming the Hoeft & Brüggen (2007) model and
$\gamma=\nicefrac{{5}}{{3}}$. Because the power density and MEC of a single
shock are proportional, this immediately also yields the desired single-shock
synchrotron MEC - total matter density scaling relation.

###  B.1 Temperature and the speed of sound

Note that
$\mathcal{M}_{\mathrm{u}}\coloneqq\frac{v_{\mathrm{u}}}{c_{\mathrm{s,u}}}$,
where $v_{\mathrm{u}}$ is the shock velocity relative to the upstream plasma,
and $c_{\mathrm{s,u}}$ is the speed of sound in the upstream plasma. The
Newton–Laplace equation for an ideal gas predicts
$c_{\mathrm{s,u}}\propto\sqrt{T_{\mathrm{u}}}$, where $T_{\mathrm{u}}$ is the
upstream plasma temperature.252525For example, the speed of sound is $10$
times higher in the $10^{8}\ \mathrm{K}$ ICM than in the $10^{6}\ \mathrm{K}$
WHIM, and $100$ times higher in the $10^{9}\ \mathrm{K}$ ICM than in the
$10^{5}\ \mathrm{K}$ WHIM. The upstream Mach number of a shock incident on the
WHIM would therefore be higher than that of a shock incident on the ICM if the
shocks arrive at the same velocity $v_{\mathrm{u}}$ relative to these media.

###  B.2 The filament regime: low magnetic field strengths

One of the prime reasons for pursuing SCW detections is to gauge the unknown
strength of the Universe’s largest magnetic fields. Numerical simulations by
Vazza et al. (2015, 2017) that reproduce the observed magnetic field strengths
in galaxy clusters, predict magnetic field strengths in filaments that depend
strongly on the magnetogenesis scenario considered, ranging between
$10^{-1}-10^{2}\ \mathrm{nG}$. For the purposes of finding a power density -
matter density scaling relation, the relevant quantity to compare the filament
IGM magnetic field strength $B_{\mathrm{IGM}}$ with at cosmological redshift
$z$, is the CMB magnetic field strength $B_{\mathrm{CMB}}\left(z\right)$. As
the CMB is well modelled by a blackbody262626In fact, it is the most accurate
blackbody ever observed!, the CMB magnetic field strength
$B_{\mathrm{CMB}}\left(z\right)$ is derived by equating the electromagnetic
energy density $u_{\mathrm{EM}}$ of a blackbody of temperature $T$ to the
electromagnetic energy density of a magnetic field of magnitude $B$:

$\displaystyle\frac{4\sigma}{c}T^{4}=u_{\mathrm{EM}}=\frac{1}{2\mu_{0}}B^{2},$
(33)

where $\sigma$ is the Stefan–Boltzmann constant and $\mu_{0}$ is the vacuum
permeability. Upon rearranging, and for $T=T_{\mathrm{CMB}}$ and
$B=B_{\mathrm{CMB}}$, we find

$\displaystyle
B_{\mathrm{CMB}}\left(z\right)=\sqrt{\frac{8\mu_{0}\sigma}{c}}T_{\mathrm{CMB}}^{2}\left(z\right).$
(34)

Let $a$ be the scale factor and let $a_{0}$ be its present-day value. As
$T_{\mathrm{CMB}}^{4}\propto u_{\mathrm{EM}}\propto
a^{-4}=a_{0}^{-4}\left(1+z\right)^{4}$ due to the expansion of the Universe,
it follows that $T_{\mathrm{CMB}}\propto 1+z$, and thus

$\displaystyle B_{\mathrm{CMB}}\left(z\right)$
$\displaystyle=\sqrt{\frac{8\mu_{0}\sigma}{c}}T_{\mathrm{CMB}}^{2}\left(0\right)\left(1+z\right)^{2}$
$\displaystyle=B_{\mathrm{CMB}}\left(0\right)\left(1+z\right)^{2}.$ (35)

Using $T_{\mathrm{CMB}}\left(0\right)=2.725\ \mathrm{K}$ yields
$B_{\mathrm{CMB}}\left(0\right)=3.238\ \mathrm{\mu G}$. Thus, under all
plausible scenarios of magnetogenesis, $B_{\mathrm{IGM}}\left(z\right)\ll
B_{\mathrm{CMB}}\left(0\right)\leq B_{\mathrm{CMB}}\left(z\right)$.

###  B.3 Magnetic field strength and baryon density

A scaling relation between $B$ and $\rho_{\mathrm{BM}}$ follows from
considering the conservation of magnetic flux as the Universe expands. The
magnetic flux through a surface is the product of the surface area and the
magnetic field strength (and the cosine of the angle between the surface
normal and the magnetic field). Over time, the surface area increases $\propto
a^{2}$, so that the magnetic field strength must follow $\propto a^{-2}$ if
conservation of magnetic flux is to hold. Finally, as
$\rho_{\mathrm{BM}}\propto a^{-3}$, one obtains
$B\propto\rho_{\mathrm{BM}}^{\nicefrac{{2}}{{3}}}$. Figure 4 of Vazza et al.
(2017) compares this scaling relation with simulated magnetic field strengths
and baryon densities under various scenarios of magnetogenesis.

###  B.4 The power density expression simplifies

The Hoeft & Brüggen (2007) power density folded into Equation 2 simplifies
appreciably if high Mach numbers and low magnetic field strengths are assumed.
In such a regime, $\alpha\approx-1$ and $\Psi\approx 1$, while
$B_{\mathrm{d}}^{2}\ll B_{\mathrm{CMB}}^{2}$. Thus,

$\displaystyle P_{\nu}$ $\displaystyle\propto\rho_{\mathrm{BM,d}}\cdot
T_{\mathrm{d}}^{\frac{3}{2}}\cdot B_{\mathrm{d}}^{2}$
$\displaystyle=\frac{\rho_{\mathrm{BM,d}}}{\rho_{\mathrm{BM,u}}}\rho_{\mathrm{BM,u}}\cdot\left(\frac{T_{\mathrm{d}}}{T_{\mathrm{u}}}T_{\mathrm{u}}\right)^{\frac{3}{2}}\cdot\left(\frac{B_{\mathrm{d}}}{B_{\mathrm{u}}}B_{\mathrm{u}}\right)^{2}.$
(36)

From the Rankine–Hugoniot jump conditions, one can derive that the compression
factor (Figure 1, central panel), as a function of $\mathcal{M}_{\mathrm{u}}$
and $\gamma$, is

$\displaystyle\frac{\rho_{\mathrm{BM,d}}}{\rho_{\mathrm{BM,u}}}\left(\mathcal{M}_{\mathrm{u}},\gamma\right)=\frac{\left(\gamma+1\right)\mathcal{M}_{\mathrm{u}}^{2}}{\left(\gamma-1\right)\mathcal{M}_{\mathrm{u}}^{2}+2}=\left(\frac{B_{\mathrm{d}}}{B_{\mathrm{u}}}\left(\mathcal{M}_{\mathrm{u}},\gamma\right)\right)^{\frac{3}{2}}.$
(37)

The same equations dictate that the temperature increase (Figure 1, bottom
panel), as a function of $\mathcal{M}_{\mathrm{u}}$ and $\gamma$, is

$\displaystyle\frac{T_{\mathrm{d}}}{T_{\mathrm{u}}}\left(\mathcal{M}_{\mathrm{u}},\gamma\right)=\frac{2\gamma\left(\gamma-1\right)\mathcal{M}_{\mathrm{u}}^{2}+4\gamma-\left(\gamma-1\right)^{2}-2\left(\gamma-1\right)\mathcal{M}_{\mathrm{u}}^{-2}}{\left(\gamma+1\right)^{2}}.$
(38)

Thus, for $\mathcal{M}_{\mathrm{u}}\gg 1$,
$\tfrac{\rho_{\mathrm{BM,d}}}{\rho_{\mathrm{BM,u}}}\propto 1$ and
$\tfrac{T_{\mathrm{d}}}{T_{\mathrm{u}}}\propto\mathcal{M}_{\mathrm{u}}^{2}$.
Returning to the power density scaling relation, we find

$\displaystyle P_{\nu}$
$\displaystyle\propto\rho_{\mathrm{BM,u}}\cdot\left(\mathcal{M}_{\mathrm{u}}^{2}\
T_{\mathrm{u}}\right)^{\frac{3}{2}}\cdot B_{\mathrm{u}}^{2}$
$\displaystyle=\rho_{\mathrm{BM,u}}\cdot v_{\mathrm{u}}^{3}\cdot
B_{\mathrm{u}}^{2}.$ (39)

To arrive at the second line, we use the definition of the upstream Mach
number and the upstream sound speed - upstream plasma temperature scaling
relation.

###  B.5 Upstream velocity and total density

We investigate the upstream velocity - total (i.e. dark and baryonic matter)
density relation for three simple geometries. We invoke Gauss’ law for gravity
to find expressions for the gravitational field, derive the gravitational
potential using the fact that the gravitational force is conservative, and
equate, for a test particle, the loss in gravitational potential energy to the
gain in kinetic energy. We assume that the structures considered have hard
edges and are equidense (with total matter density
$\rho\coloneqq\rho_{\mathrm{BM,u}}+\rho_{\mathrm{DM}}$) within. Upon impact,
the velocity of a test particle starting from rest at a distance $d$, and
falling onto an isolated spherical galaxy cluster with radius $R$, is

$\displaystyle v_{\mathrm{u}}=\sqrt{\frac{8\pi
G}{3}\rho\left(1-\frac{R}{d}\right)}\ R.$ (40)

Upon impact, the velocity of a test particle starting from rest at a distance
$d$, and falling onto an isolated cylindrical filament with radius $R$, is

$\displaystyle v_{\mathrm{u}}=\sqrt{4\pi G\rho\ln{\frac{d}{R}}}\ R.$ (41)

Upon impact, the velocity of a test particle starting from rest at a distance
$d$, and falling onto an isolated, thick planar sheet of half-thickness $R$,
is

$\displaystyle v_{\mathrm{u}}=\sqrt{8\pi G\rho R\left(d-R\right)}.$ (42)

In all three cases, $R$ determines the size of the structure types. Assuming
no relation between total density $\rho$ and $R$, we find
$v_{\mathrm{u}}\propto\sqrt{\rho}$, irrespective of the geometry.

###  B.6 Baryon density and dark matter density

Structure formation theory predicts that after decoupling, and if gas pressure
is ignored, $\rho_{\mathrm{BM,u}}\propto\rho_{\mathrm{DM}}$. _Including_ gas
pressure, this proportionality is expected to remain valid on large scales
only. For filaments in particular, Figure 6 of Gheller et al. (2016) shows
that the baryon fraction in filaments remains close to
$f_{\mathrm{cosmic}}\coloneqq\Omega_{\mathrm{BM},0}\left(\Omega_{\mathrm{BM},0}+\Omega_{\mathrm{DM},0}\right)^{-1}\left(=0.167\right)$
over four orders of magnitude of total baryonic mass, and for redshifts from
$1$ to $0$; this is consistent with
$\rho_{\mathrm{BM,u}}\propto\rho_{\mathrm{DM}}$. Using
$B_{\mathrm{u}}\propto\rho_{\mathrm{BM,u}}^{\nicefrac{{2}}{{3}}}$,
$v_{\mathrm{u}}\propto\sqrt{\rho}$ and
$\rho_{\mathrm{BM,u}}\propto\rho_{\mathrm{DM}}$, the final scaling relation
becomes

$\displaystyle j_{\nu}\propto P_{\nu}\propto\rho^{\frac{23}{6}}.$ (43)

Note that, to obtain a scaling relation for the _total_ (rather than single-
shock) MEC, one should also consider how the shock number density relates to
the total matter density. The total MEC - matter density and the single-shock
MEC - matter density scaling relation exponents are only the same when no such
relationship exists.

##  Appendix C Ray tracing in the cosmological setting

Projecting a ray’s 4D null geodesic in a pure
Friedmann–Lemaître–Robertson–Walker (FLRW) metric onto 3D comoving space
results in a straight line. This follows readily from the FLRW metric in
hyperspherical coordinates

$\displaystyle\mathrm{d}s^{2}=-c^{2}\mathrm{d}t^{2}+\left(\frac{a(t)}{a_{0}}\right)^{2}\left(\mathrm{d}r^{2}+S_{k}^{2}\left(r\right)\mathrm{d}\Omega^{2}\right),$
(44)

where $\mathrm{d}s^{2}$ is the spacetime line element, $c$ is the speed of
light in vacuo, $t$ is physical time since the Big Bang, and $a\left(t\right)$
is the scale factor (with $a_{0}\coloneqq a\left(t_{0}\right)$ being its
present-day ($t=t_{0}$) value). Also, $r$ is the radial comoving distance, $k$
is the Universe’s Gaussian curvature (with SI units $\mathrm{m}^{-2}$),
$S_{k}\left(r\right)\coloneqq r\ \mathrm{sinc}\left(r\sqrt{k}\right)$ is the
transverse comoving distance and
$\mathrm{d}\Omega^{2}\coloneqq\mathrm{d}\theta^{2}+\cos^{2}\theta\mathrm{d}\phi^{2}$.
(The sinc function follows the mathematical (unnormalised) convention.)
Finally, let the location of present-day Earth be the spatial origin. An
initially radial ($\mathrm{d}\Omega^{2}=0$) null ($\mathrm{d}s^{2}=0$)
geodesic thus satisfies

$\displaystyle c\mathrm{d}t=\ \pm\frac{\mathrm{d}r}{1+z},$ (45)

regardless of $k$. As time progresses, such rays maintain their direction and
only change in $r$; this justifies considering the path $\mathcal{L}$ of a
light ray with direction $\hat{r}$ in comoving space as the set of points
$\mathcal{L}\left(\hat{r}\right)\coloneqq\\{r\ \hat{r}\in\mathbb{R}^{3}\ |\
r\in\mathbb{R}_{\geq 0}\\}$.

##  Appendix D Observer’s specific intensity

Our aim is to derive an expression for the specific intensity in direction
$\hat{r}$ at observing frequency $\nu_{\mathrm{obs}}$. We follow Chapter 12 of
Peacock (1999), but generalise to arbitrary $\Lambda$CDM models (by allowing
$\Lambda\neq 0$), and recast the results in terms of the MEC instead of the
emissivity. As in Appendix C, consider a FLRW metric with arbitrary Gaussian
curvature $k$. A comoving volume element $\mathrm{d}V_{\mathrm{c}}$ and the
corresponding proper volume element $\mathrm{d}V_{\mathrm{p}}$ at comoving
radial distance $r$ and cosmological redshift $z=z\left(r\right)$ that cover a
solid angle $\mathrm{d}\Omega$ on the sky are given by

$\displaystyle\mathrm{d}V_{\mathrm{c}}$
$\displaystyle=S_{k}\left(r\right)^{2}\ \mathrm{d}\Omega\mathrm{d}r,$ (46)
$\displaystyle\mathrm{d}V_{\mathrm{p}}$
$\displaystyle=S_{k}\left(r\right)^{2}\
\mathrm{d}\Omega\mathrm{d}r\left(1+z\right)^{-3}.$ (47)

Recall that we have defined $j_{\nu}$ as the _proper_ MEC, and assume that the
filament IGM radiates isotropically. The luminosity density
$\mathrm{d}L_{\nu}$ of the volume, seen in direction $\hat{r}$ and at
_emission_ frequency $\nu=\left(1+z\right)\nu_{\mathrm{obs}}$, then equals

$\displaystyle\mathrm{d}L_{\nu}\left(\hat{r},\nu\right)=4\pi j_{\nu}\left(r\
\hat{r},z,\nu\right)\mathrm{d}V_{\mathrm{p}}.$ (48)

The corresponding observer’s flux density $\mathrm{d}F_{\nu}$ of the volume in
direction $\hat{r}$ at _observing_ frequency $\nu_{\mathrm{obs}}$ is

$\displaystyle\mathrm{d}F_{\nu}\left(\hat{r},\nu_{\mathrm{obs}}\right)=\frac{\mathrm{d}L_{\nu}\left(\hat{r},\nu\right)}{4\pi
S_{k}\left(r\right)^{2}\left(1+z\right)},$ (49)

and so the observer’s specific intensity $\mathrm{d}I_{\nu}$ in direction
$\hat{r}$ at observing frequency $\nu_{\mathrm{obs}}$ is

$\displaystyle\mathrm{d}I_{\nu}\left(\hat{r},\nu_{\mathrm{obs}}\right)\coloneqq\frac{\mathrm{d}F_{\nu}\left(\hat{r},\nu_{\mathrm{obs}}\right)}{\mathrm{d}\Omega}$
(50) $\displaystyle=\frac{4\pi j_{\nu}\left(r\
\hat{r},z,\nu\right)S_{k}\left(r\right)^{2}\mathrm{d}\Omega\mathrm{d}r}{4\pi
S_{k}\left(r\right)^{2}\left(1+z\right)\mathrm{d}\Omega\left(1+z\right)^{3}}=\frac{j_{\nu}\left(r\
\hat{r},z,\nu\right)\ \mathrm{d}r}{\left(1+z\right)^{4}}.$ (51)

We neglect absorption so that the specific intensity of the ray only
accumulates as the ray travels through LSS to the observer: the Universe is
mostly optically thin for $\nu<1\ \mathrm{GHz}$; we assume this holds
perfectly. So, by collecting all contributions along the ray’s path, we obtain
the equivalent of Equation 12.12 of Peacock (1999):

$\displaystyle
I_{\nu}\left(\hat{r},\nu_{\mathrm{obs}}\right)=\int_{0}^{\infty}\frac{j_{\nu}\left(r\
\hat{r},z\left(r\right),\nu_{\mathrm{obs}}\left(1+z\left(r\right)\right)\right)}{\left(1+z\left(r\right)\right)^{4}}\mathrm{d}r.$
(52)

Alternatively, one can express
$I_{\nu}\left(\hat{r},\nu_{\mathrm{obs}}\right)$ as an integral over $z$.
Because

$\displaystyle
r\left(z\right)=\frac{c}{H_{0}}\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{E\left(z^{\prime}\right)},$
(53)

with the dimensionless Hubble parameter
$E\left(z\right)\coloneqq\frac{H\left(z\right)}{H_{0}}$ being
$E\left(z\right)=\sqrt{\Omega_{\mathrm{R},0}\left(1+z\right)^{4}+\Omega_{\mathrm{M},0}\left(1+z\right)^{3}+\Omega_{\mathrm{K},0}\left(1+z\right)^{2}+\Omega_{\Lambda,0}}$
and today’s curvature density parameter being
$\Omega_{\mathrm{K},0}=1-\Omega_{\mathrm{R},0}-\Omega_{\mathrm{M},0}-\Omega_{\Lambda,0}$,
we have

$\displaystyle\mathrm{d}r=\frac{c}{H_{0}}\frac{\mathrm{d}z}{E\left(z\right)}.$
(54)

Combining Equations 52, 53 and 54,

$\displaystyle
I_{\nu}\left(\hat{r},\nu_{\mathrm{obs}}\right)=\frac{c}{H_{0}}\int_{0}^{\infty}\frac{j_{\nu}\left(r\left(z\right)\hat{r},z,\nu_{\mathrm{obs}}\left(1+z\right)\right)}{\left(1+z\right)^{4}E\left(z\right)}\mathrm{d}z.$
(55)

(Barring notational differences, the $\Omega_{\Lambda,0}=0$ (and
$\Omega_{\mathrm{R},0}=0$) limit of this formula is Equation 12.10 of Peacock
(1999).) Finally, we can read off that

$\displaystyle\frac{\mathrm{d}I_{\nu}}{\mathrm{d}z}\left(\hat{r},\nu_{\mathrm{obs}},z\right)=\frac{c}{H_{0}}\frac{j_{\nu}\left(r\left(z\right)\hat{r},z,\nu_{\mathrm{obs}}\left(1+z\right)\right)}{\left(1+z\right)^{4}E\left(z\right)}.$
(56)

##  Appendix E Volume-filling fractions

Table 1: Comparison between cosmic web structure type volume-filling fractions (VFFs) predicted by the cubic unit cell model and those obtained by Forero-Romero et al. (2009) from cosmological simulations for eigenvalue threshold $\lambda_{\mathrm{th}}=1$ and two effective smoothing scales $R_{\mathrm{eff}}$. The VFF ratio columns give the simulation VFFs of the preceding column divided by the cubic unit cell VFFs of the first column. $\begin{array}[]{l | l | l | l | l | l }\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\textrm{cosmic web}&\textrm{cubic unit cells}&\textrm{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\cite[citet]{\@@bibref{Authors Phrase1YearPhrase2}{Forero-Romero12009}{\@@citephrase{(}}{\@@citephrase{)}}}}}&\mathrm{VFF}&\textrm{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\cite[citet]{\@@bibref{Authors Phrase1YearPhrase2}{Forero-Romero12009}{\@@citephrase{(}}{\@@citephrase{)}}}}}&\mathrm{VFF}\\\ \textrm{structure type}&\frac{w_{\mathrm{f}}}{l}=10^{-1}&\lambda_{\mathrm{th}}=1,R_{\mathrm{eff}}=0.88\ h^{-1}\ \mathrm{Mpc}&\textrm{ratio}&\lambda_{\mathrm{th}}=1,R_{\mathrm{eff}}=2.05\ h^{-1}\ \mathrm{Mpc}&\textrm{ratio}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\textbf{{voids}}&72.9\%&76\%&1.04&82\%&1.12\\\ \textbf{{sheets}}&24.3\%&18\%&0.74&14\%&0.58\\\ \textbf{{filaments}}&2.7\%&5\%&1.85&4\%&1.48\\\ \textbf{{clusters}}&0.1\%&0.5\%&5.00&0.28\%&2.80\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$ Figure 18: The cubic unit cell model of Section 3 predicts cosmic web structure type volume-filling fractions (VFFs) that depend on just one parameter: the ratio between the (comoving) filament width and the (comoving) cubic unit cell edge length $\frac{w_{\mathrm{f}}}{l}$.

In this appendix, we compute the volume-filling fractions (VFFs) of the four
canonical structure types (clusters, filaments, sheets and voids) as predicted
by the cubic unit cell geometric model developed in Section 3. A single cubic
unit cell features two typical lengthscales: a large scale ($l_{\mathrm{f}}$),
and a small scale ($w_{\mathrm{f}}$), which can be interpreted as the typical
filament length and width, respectively. The cube obtained by raising the
large scale to the third power, represents a void. Similarly, the three
rectangular cuboids obtained by taking the product of the square of the large
scale, and the small scale, resemble three sheets. The three rectangular
cuboids obtained by taking the product of the large scale, and the square of
the small scale, resemble three filaments. Finally, the cube obtained by
raising the small scale to the third power, resembles a cluster. The natural
volume-filling fractions (VFFs) suggested by this geometry are thus

$\displaystyle\mathrm{VFF}_{\mathrm{c}}\left(\frac{w_{\mathrm{f}}}{l}\right)$
$\displaystyle=\left(\frac{w_{\mathrm{f}}}{l}\right)^{3}$ (63)
$\displaystyle\mathrm{VFF}_{\mathrm{f}}\left(\frac{w_{\mathrm{f}}}{l}\right)$
$\displaystyle=3\left(\frac{w_{\mathrm{f}}}{l}\right)^{2}\left(1-\frac{w_{\mathrm{f}}}{l}\right)$
(64)
$\displaystyle\mathrm{VFF}_{\mathrm{s}}\left(\frac{w_{\mathrm{f}}}{l}\right)$
$\displaystyle=3\frac{w_{\mathrm{f}}}{l}\left(1-\frac{w_{\mathrm{f}}}{l}\right)^{2}$
(65)
$\displaystyle\mathrm{VFF}_{\mathrm{v}}\left(\frac{w_{\mathrm{f}}}{l}\right)$
$\displaystyle=\left(1-\frac{w_{\mathrm{f}}}{l}\right)^{3}$ (66)

See Figure 18. To see that the VFFs sum to 1, we rewrite 1 using the binomial
theorem:

$\displaystyle
1=\left(\frac{w_{\mathrm{f}}}{l}+1-\frac{w_{\mathrm{f}}}{l}\right)^{3}=\sum_{n=0}^{3}\binom{3}{n}\left(\frac{w_{\mathrm{f}}}{l}\right)^{n}\left(1-\frac{w_{\mathrm{f}}}{l}\right)^{3-n},$
(67)

and recognise the VFFs as the four terms in this expansion. The VFFs obtained
from this _one_ -parameter model for $\frac{w_{\mathrm{f}}}{l}=10^{-1}$, are
similar to those retrieved by Forero-Romero et al. (2009) from cosmological
simulations for eigenvalue threshold $\lambda_{\mathrm{th}}=1$ and effective
smoothing scales $R_{\mathrm{eff}}\sim 10^{0}\ \mathrm{Mpc}$ (see Table 1).

##  Appendix F Notation

This paper adopts the SI system of units (for formulae concerning
electromagnetism), and the following symbols. We list dimensionalities in SI
base units with the radian ‘rad’ appended. Current-day quantities are
subscripted with a zero: e.g. $a_{0}$ is today’s scale factor, while $a$ is
the scale factor for arbitrary times. Upstream and downstream quantities are
subscripted with a ‘u’ or ‘d’, respectively: e.g. $T_{\mathrm{u}}$ is the
upstream plasma temperature, whilst $T$ is the general plasma temperature.

$\alpha$ | $\mathrm{1}$ |  integrated synchrotron spectral index
---|---|---
$\alpha_{\mathrm{c}}$ | $\mathrm{1}$ |  typical cluster $\alpha$
$\alpha_{\mathrm{f}}$ | $\mathrm{1}$ |  typical filament $\alpha$
$\bar{\alpha}$ | $\mathrm{1}$ |  MEC-weighted mean $\alpha$
$\beta_{\mathrm{c}}$ | $\mathrm{1}$ |  typical cluster MEC $\left(1+z\right)$ power law exponent
$\beta_{\mathrm{f}}$ | $\mathrm{1}$ |  typical filament MEC $\left(1+z\right)$ power law exponent
$\gamma$ | $\mathrm{1}$ |  adiabatic index
$\theta$ | $\mathrm{rad}$ |  declination (J2000)
$\mu_{0}$ | $\mathrm{kg\ m\ s^{-2}\ A^{-2}}$ |  vacuum permeability
$\nu$ | $\mathrm{s^{-1}}$ |  emission frequency
$\nu_{\mathrm{obs}}$ | $\mathrm{s^{-1}}$ |  observing frequency
$\nu_{\mathrm{ref}}$ | $\mathrm{s^{-1}}$ |  reference frequency (see $\mathcal{C}$)
$\xi_{e}$ | 1 |  electron acceleration efficiency
$\Xi_{\rho}$ | 1 |  CDF of RV $j_{\nu}\ |\ \rho$
$\rho$ | $\mathrm{kg\ m^{-3}}$ |  total matter density
$\rho_{\mathrm{BM}}$ | $\mathrm{kg\ m^{-3}}$ |  baryonic matter density
$\rho_{\mathrm{c}}$ | $\mathrm{kg\ m^{-3}}$ |  critical density
$\rho_{\mathrm{DM}}$ | $\mathrm{kg\ m^{-3}}$ |  dark matter density
$\sigma$ | $\mathrm{kg\ s^{-3}\ K^{-4}}$ |  Stefan–Boltzmann constant
$\phi$ | $\mathrm{rad}$ |  right ascension (J2000)
$\Phi$ | 1 |  CDF of standard normal RV
$\Psi$ | 1 |  as in Hoeft & Brüggen (2007)
$\Omega_{\Lambda}$ | $\mathrm{1}$ |  dark energy density parameter
$\Omega_{\mathrm{BM}}$ | $\mathrm{1}$ |  baryonic matter density parameter
$\Omega_{\mathrm{DM}}$ | $\mathrm{1}$ |  dark matter density parameter
$\Omega_{\mathrm{K}}$ | $\mathrm{1}$ |  curvature density parameter
$\Omega_{\mathrm{M}}$ | $\mathrm{1}$ |  matter density parameter
$\Omega_{\mathrm{R}}$ | $\mathrm{1}$ |  relativistic particle (i.e. photon and neutrino) density parameter
$a$ | $\mathrm{1}$ |  scale factor
---|---|---
$A$ | $\mathrm{m}^{2}$ |  shock surface area
$B$ | $\mathrm{kg\ s^{-2}\ A^{-1}}$ |  proper magnetic field strength
$B_{\mathrm{CMB}}$ | $\mathrm{kg\ s^{-2}\ A^{-1}}$ |  CMB magnetic field strength
$c$ | $\mathrm{m\ s^{-1}}$ |  speed of light in vacuo
$c_{\mathrm{s}}$ | $\mathrm{m\ s^{-1}}$ |  speed of sound
$\mathcal{C}$ | 1 |  typical cluster-to-filament synchrotron MEC ratio at $\nu=\nu_{\mathrm{ref}}$ and $z=0$
---|---|---
$d$ | $\mathrm{m}$ |  initial distance between test particle and equidense cluster centre, filament axis or sheet midplane
$d_{n}$ | $\mathrm{m}$ |  comoving distance to the $n$-th unit cell boundary crossing
$\mathrm{d}\Omega^{2}$ | $\mathrm{rad^{2}}$ |  solid angle element
$\mathrm{d}s^{2}$ | $\mathrm{m^{2}}$ |  spacetime line element
$E$ | $1$ |  dimensionless Hubble parameter
$F_{\bar{Z}}$ | 1 |  CDF of $\bar{Z}$
$G$ | $\mathrm{kg^{-1}\ m^{3}\ s^{-2}}$ |  Newton’s gravitational constant
$h$ | 1 |  Hubble constant divided by $100\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$
$H$ | $\mathrm{s^{-1}}$ |  Hubble parameter
$I_{\nu}$ | $\mathrm{kg\ s^{-2}\ rad^{-2}}$ |  specific intensity (observed)
$j_{\nu}$ | $\mathrm{kg\ m^{-1}\ s^{-2}\ rad^{-2}}$ |  proper (not comoving) MEC
$\mathbf{k}$ | $\mathrm{m}^{-1}$ |  Fourier dual of $\mathbf{r}$
$k$ | $\mathrm{m^{-2}}$ |  FLRW (Gaussian) curvature of the Universe
$l$ | $\mathrm{m}$ |  comoving cubic unit cell edge length
$l_{\mathrm{f}}$ | $\mathrm{m}$ |  comoving filament length
$l_{\mathrm{SE}}$ | $\mathrm{m}$ |  SE kernel lengthscale
$\mathcal{L}$ | - |  light ray path
$m_{r}$ | $\mathrm{1}$ |  Petrosian r-band apparent magnitude
$M$ | 1 |  number of sightlines
$\mathfrak{M}$ | - |  cosmological model parameter tuple
$\mathcal{M}_{\mathrm{u}}$ | $\mathrm{1}$ |  upstream shock Mach number
$n_{e}$ | $\mathrm{m^{-3}}$ |  electron number density
$N$ | 1 |  number of unit cell boundary crossings considered
$p_{\mathrm{c-f}}$ | 1 |  probability to pierce through a cluster, then a filament
$p_{\mathrm{f}}$ | 1 |  probability to pierce through a filament only
$p_{\mathrm{s-v}}$ | 1 |  probability to pierce through a sheet, then a void
$P_{\nu}$ | $\mathrm{kg\ m^{2}\ s^{-2}}$ |  proper power density
$\mathcal{P}$ | 1 |  percentile random field
$\mathbf{r}$ | $\mathrm{m}$ |  comoving position vector
$r$ | $\mathrm{m}$ |  radial comoving distance
$\hat{r}$ | $\mathrm{1}$ |  sky direction unit vector
$\hat{r}_{i}$ | $\mathrm{1}$ |  sky direction unit vector of ray $i$
$R$ | $\mathrm{m}$ |  equidense cluster radius, filament radius or sheet half-width
$\mathcal{R}$ | - |  comoving reconstruction region; subset of $\mathbb{R}^{3}$
$S_{k}$ | $\mathrm{m}$ |  transverse comoving distance
$t$ | $\mathrm{s}$ |  physical time since the Big Bang
---|---|---
$T$ | $\mathrm{K}$ |  proper plasma temperature
$T_{\mathrm{CMB}}$ | $\mathrm{K}$ |  CMB temperature
$u_{\mathrm{EM}}$ | $\mathrm{kg\ m^{-1}\ s^{-2}}$ |  electromagnetic energy density
$v$ | $\mathrm{m\ s^{-1}}$ |  shock or test particle velocity
$V$ | $\mathrm{m}^{3}$ |  shock effective volume
$w_{\mathrm{f}}$ | $\mathrm{m}$ |  typical comoving filament width
$X_{n}$ | 1 |  relative specific intensity contribution of the $n$-th newly-entered unit cell
$X_{nm}$ | 1 |  relative specific intensity contribution of the $n$-th newly-entered unit cell for the $m$-th ray
$\langle y\rangle$ | $\mathrm{m}$ |  shock effective width
$z$, $z_{\mathfrak{M}}$ | $\mathrm{1}$ |  cosmological redshift (under cosmological model $\mathfrak{M}$)
$z_{\mathrm{max}}$ | 1 |  cosmological redshift up to which LSS is considered
$z_{n}$ | $\mathrm{1}$ |  cosmological redshift of $n$-th unit cell boundary crossing
$\bar{z}$ | 1 |  specific-intensity-weighted mean redshift
$\bar{\bar{z}}$ | 1 |  flux-density-weighted mean redshift
$Z$ | 1 |  cosmological redshift RV
$\bar{Z}$ | 1 |  specific-intensity-weighted mean redshift RV
$\mathcal{Z}$ | 1 |  standard normal GRF