CMB spectra, lensing, and tensors

CMBSpectra

Angular power spectra. TT, EE, TE are Vectors indexed by ℓ from ℓ_min, in μK² (raw Cℓ, not ℓ(ℓ+1)Cℓ/2π – use D_ℓ for that).

PrimordialMatrix(modes; amp, tilt, k_pivot = 0.05)

Primordial cross-power specification for a set of scalar modes (any of :adiabatic, :cdi, :bi, :nid, :niv). amp and tilt are symmetric n×n matrices over modes; the dimensionless cross power of modes i,j is

𝒫_ij(k) = amp[i,j] · (k/k_pivot)^(tilt[i,j] − 1),

so the diagonal holds each mode's auto-amplitude and tilt (the adiabatic entry is just A_s, n_s) and the off-diagonal holds the cross-correlations. The total harmonic spectrum sums every ordered pair,

C_ℓ^XY = 4π Σ_ij ∫ dk/k 𝒫_ij(k) Θ_ℓ^{X,i}(k) Θ_ℓ^{Y,j}(k).

amp should be positive semi-definite for a physical spectrum; this is not enforced (an unphysical matrix simply yields C_ℓ that can go negative).

SourceFunction

The line-of-sight sources for one mode k, tabulated on a conformal-time grid. S_T is the temperature source, S_E the polarization source (which is just the visibility-weighted quadrupole, g·Π).

D_ℓ(spec, which = :TT)

ℓ(ℓ+1)C_ℓ/2π in μK² – the form the CMB is conventionally plotted in, chosen because a scale-invariant Sachs-Wolfe plateau is flat in it.

cmb_spectra(c, rec, prim::PrimordialMatrix; kwargs...)

Correlated scalar CMB spectra for a mixture of adiabatic and isocurvature modes, combined through the primordial matrix prim (see PrimordialMatrix). Each mode is evolved with its own initial condition, projected to transfer functions Θℓ^{T,i}, Θℓ^{E,i}, and summed over all mode pairs. Flat geometry only. Returns a CMBSpectra (φφ/Tφ empty).

Passing PrimordialMatrix([:adiabatic]; amp=[A_s;;], tilt=[n_s;;]) reproduces the ordinary adiabatic cmb_spectra.

cmb_spectra(c, rec; lmax = 1500, nk = 3000, kmax = nothing, nk_solve = nothing)

Compute Cℓ^TT, Cℓ^EE and C_ℓ^TE.

C_ℓ = 4π ∫ dln k  Δ²_ℛ(k)  Θ_ℓ(k)²

Two grids matter and both are easy to under-resolve:

• k. Θ_ℓ(k) oscillates with period ≈ 2π/η₀ ≈ 4.4e-4 /Mpc, so the k-grid must resolve that, not the smooth envelope. Under-sampling does not blur the spectrum, it aliases – producing spurious wiggles that look like physics. The grid is therefore linear at high k, not logarithmic.
• k_max. A multipole ℓ is sourced by k ≈ ℓ/η₀, so kmax must reach beyond ℓmax/η₀ or the highest multipoles are simply missing power.

Cost is one Boltzmann solve per k, threaded.

nk_solve decouples the two grids. The Bessel-scale oscillation above lives in the projection Θℓ(k) = ∫ dη S(k,η) jℓ(k(η₀−η)); the line-of-sight source S(k,η) itself varies in k only on the sound-horizon scale Δk ≈ 2π/rs ≈ 0.04 /Mpc. Passing `nksolve = ntherefore solves the Boltzmann hierarchy on onlynk-points, interpolates the source tables linearly onto the fullnk-point grid, and projects densely — the exact integral, evaluated on interpolated sources.nksolve = 600carries ~80 solve points per source oscillation and reproduces the all-k result to ≲0.1% while cutting the hierarchy cost ~8×. The defaultnothingsolves at every k; tighten by raisingnksolve` and confirming the spectra stop changing, same as any grid knob.

source_function(c, bg, p; nη = 4000)

Build ST and SE for the solved mode p.

The derivatives d(g vb)/dη and d²(g Π)/dη² are taken numerically on the grid. That is safe here because g vb and g Π are smooth and compactly supported – the visibility function kills them away from last scattering – so there is no cancellation to amplify and no tail to lose.

LensedCMBSpectra

Lensed spectra at every integer ℓ from 2 to lmax, in μK². BB is the lensing B-mode generated from E; the unlensed scalar BB is zero.

lensed_cmb_spectra(c, rec; lmax = 2000, dl_buffer = 500, kernel = :lastscatter, ...)

Unlensed spectra plus Cℓ^φφ in one mode sweep (`cmbspectra(...; lensing = true)), then the correlation-function lensing. Returns(lensed, unlensed)-- theLensedCMBSpectraand theCMBSpectra` carrying φφ and Tφ.

dl_buffer sets how far past lmax the unlensed spectra are computed, because lensing couples each multipole to its neighbours. TT/TE/EE converge with the default 500 across the whole range; lensed BB is a pure off-diagonal EE→BB transfer that draws power from a broad band of ℓ', so its top ~dl_buffer/2 multipoles are under-converged – with buffer 500, BB is trustworthy to about ℓ = lmax − 300 and falls ~20% low at ℓ = lmax itself. Ask for BB out to some ℓ by giving lmax a few hundred above it, or raise dl_buffer. Reaching φφ (and thus lensed BB) to high ℓ also needs a large kmax in the sweep: the lensing kernel maps ℓ to k ≈ ℓ/χ over a range of χ down to small values, so C_ℓ^φφ at ℓ = 2000 wants kmax ≈ 0.7, not the 2.2·lmax/η₀ that suffices for TT.

lensed_from_cls(TT, EE, TE, φφ; lmax_out, nmu = 0)

Lens dense unlensed spectra (vectors indexed 1 ⟺ ℓ = 0, through ℓunlensedmax) with Cℓ^φφ, returning (TT, EE, TE, BB) at ℓ = 2..lmaxout. This is the Challinor-Lewis full-sky correlation-function algorithm; the unlensed input should extend a few hundred multipoles beyond lmax_out.

solve_tensor_perturbations(c, bg, k; ...)

Evolve one primordial gravitational-wave mode and the particle hierarchies that source and are sourced by it. For nonzero curvature this uses the exact curved streaming coefficients, curved hierarchy closure, (k²+2K)h propagation term, and curvature-normalised regular initial mode. No flat radial projection is performed by this function.

tensor_cmb_spectra(c, rec; r=0.01, n_t=-r/8, ...)

Primordial tensor CMB spectra. r is the tensor-to-scalar ratio at k_pivot; the default tilt is the single-field consistency relation but can be supplied independently. The returned spectra are raw C_l in μK² and include primordial tensor TT, EE, BB, and TE (not lensing-generated scalar BB).