Appendix A — Notation and conventions

A.1 Units and state

Symbol / quantity	Convention
a, z	Scale factor and redshift, a=1/(1+z)
tau	Dimensionless supercomoving expansion clock, Equation 6.4
rho, p, E, v, B	MHD code quantities, not unqualified cgs values
psi	Mixed-GLM divergence-cleaning scalar
gamma	Gas adiabatic index
rho_b(z)	Physical mean baryon density used for cgs conversion
B0_ng	Comoving magnetic amplitude in nG at the declared normalization epoch/band
B_phys	Physical field in G, Equation 6.5
v_unit	MHD_CHEM_VUNIT, cm/s
T	Gas temperature, K; T_CMB=2.725(1+z) K
x_HII	Hydrogen ionization fraction; not the total electron fraction
xe_total	Electrons per hydrogen nucleus including the selected helium model
HII, H2I, HeII arrays	Stored species payloads; interpret mass weighting and centered/neutral storage before reading as fractions
R	3*rho_b*c^2/(4*rho_gamma_energy) in the photon model
lambda_gamma	Physical Thomson mean free path; divided by physical box in code closure
q	k_phys*lambda_gamma, dimensionless

Coordinates are periodic in a unit code cube for the production runner. Cell centers are (i-1/2)/N; Fourier integer modes use 2*pi*n/L, with the forward transform convention exp(-i*k*x) and inverse exp(+i*k*x). RFFT storage keeps the first spatial axis as a half spectrum. A fourth transform dimension in the photon workspace is a channel, not a spatial coordinate.

A.2 Wavenumbers and spectra

The physical TOML box is L_h=cosmology.box_ckpc_h, a comoving length in ckpc/h. Its length in ckpc is L_h/h. Magnetic kcut, kmax, and the spectrum’s numeric k follow the driver’s coordinate box L_s=MHD_BOXSIZE (default 1), not an implicit unit inferred from a TSV filename. To convert coordinate length to comoving Mpc without h:

u_L=\frac{10^{-3}L_h}{hL_s}\ {\rm cMpc},\qquad k_{\rm cMpc^{-1}}=k_{\rm input}/u_L,\qquad P_{\rm cMpc^3}=P_{\rm input}u_L^3. \tag{A.1}

Field-amplitude factors remain: magnetic/force/velocity spectra retain code amplitudes unless separately converted. For dimensionless density contrast, power has only coordinate-volume units.

The estimator uses normalized Fourier amplitudes and P=V*mean(|fhat|^2) over occupied modes, summing vector components where applicable. The radial k value is the occupied-mode mean, not necessarily the geometric bin center. mu=abs(k_axis)/|k|; isotropic reductions weight each angular bin by Nmodes.

\Delta^2(k)=\frac{k^3P(k)}{2\pi^2}. \tag{A.2}

This combination is invariant under the coordinate-length conversion above. Keep empty bins as missing. State windowing, finite-volume/CIC effects, physical band limits, shell averaging, and any additional smoothing when comparing power.

A.3 Normalization and diagnostics

Brms=sqrt(mean(Bx^2+By^2+Bz^2)). Reference-band normalization and total-band normalization are different as described in initial conditions. divBdx_over_brms uses the centered-grid divergence and grid spacing divided by measured magnetic RMS. It is a numerical diagnostic, not a proof of exact continuum solenoidality. For constrained-transport runs divb_face_dx_over_brms uses the exact face divergence (round-off level); divBdx_over_brms stays the centred cell-field quantity and is not constrained to zero.

delta_b_rms and delta_dm_rms measure density contrasts; they are not raw density standard deviations in g/cm3. Density PDFs are volume-weighted per log10(rho/reference density). Pressure/tension power requires a cross term, Equation 5.1. The photon energy ledger tracks resolved perturbation energy; algebraically eliminated higher-multipole heat is outside that ledger.

A.4 Phase comparisons

Keep signed complex amplitudes when comparing linear references. Sine and cosine ICs differ by phase; taking absolute values can hide a sign/clock error. A magnetic sine forcing produces quadratic 2k terms; compare the observable’s mode, not automatically the magnetic fundamental. Match cell centers, polarization, guide-field orientation, exact output epochs, and peculiar-velocity units before forming a relative error.