Primordial spectra and inflation

The primordial hook

The Boltzmann solver runs at unit primordial curvature, so the primordial spectrum multiplies transfer functions linearly and can be swapped without re-solving anything. Three levels of generality:

The default power law $\Delta^2_\mathcal{R}(k) = A_s (k/k_\star)^{n_s-1}$ from the A_s, n_s, k_pivot parameters.

An arbitrary spectrum: cosmology(primordial = f) accepts any callable $k \mapsto \Delta^2_\mathcal{R}(k)$ — features, steps, running, broken scale invariance, a table — and every observable (CMB, lensing, $P(k)$, induced gravitational waves) picks it up through the same hook.

Correlated multi-mode spectra: PrimordialMatrix specifies auto- and cross-power for any set of adiabatic and isocurvature modes, $\mathcal{P}_{ij}(k) = A_{ij} (k/k_\star)^{n_{ij}-1}$, and the corresponding cmb_spectra method evolves each mode's transfer functions and sums every ordered pair — fully correlated isocurvature phenomenology, not just pure modes.

Inflation

inflaton_spectrum computes the spectrum instead of positing it. The background trajectory of a canonical field in a user potential $V(\phi)$ (reduced Planck units) is integrated exactly to the end of inflation, and each comoving mode's Mukhanov–Sasaki equation is solved from Bunch–Davies initial conditions deep inside the horizon through freeze-out — no slow-roll truncation anywhere, so any feature the potential imprints appears in $\Delta^2_\mathcal{R}(k)$ with its true shape. Tensors are evolved the same way for $\Delta^2_t(k)$ and $r$. For the workhorse $m^2\phi^2$ test the exact evolution lands on the slow-roll attractor predictions to 0.03% in $n_s$.

An effective sound speed $c_s(N)$ (the EFT-of-inflation quadratic action, $z = a\sqrt{2\varepsilon}/c_s$) can be supplied for models with non-trivial kinetic structure. The solver refuses $c_s > 1$ — superluminal propagation is a causality violation, whatever the literature it appears in — and warns when $c_s$ drops low enough to raise strong-coupling concerns. Sharp transitions in $c_s$ or the potential are handled by the exact integration, which makes the code suited to studying how transition width affects the resulting features: observables from "instantaneous" transitions can depend strongly on the smoothing assumed, and here that sensitivity can be quantified rather than assumed away.

The returned InflatonSpectrum is callable and plugs directly into cosmology(primordial = s) and into the induced-gravitational-wave pipeline (see Scalar-induced gravitational waves).

Reference

Cosmic.InflatonSpectrum — Type

InflatonSpectrum

Computed primordial spectrum. Callable: s(k) returns the dimensionless curvature power Δ²ℛ(k) at physical wavenumber k (1/Mpc); `tensorpower(s, k)returns Δ²_t(k). Derived summary numbers:ns,r,As` at the pivot.

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Cosmic.inflation_background — Method

inflation_background(V, dV; φ0, φ0_dot=nothing, N_max=400, npts=6000)

Integrate the exact single-field background for potential V (with derivative dV), starting from field value φ0 on the slow-roll attractor (initial φN = −Vφ/V unless φ0_dot is given), until inflation ends at ε₁ = 1.

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Cosmic.inflaton_spectrum — Method

inflaton_spectrum(V, dV; φ0, N_pivot=55, k_pivot=0.05,
                  decades_below=4, decades_above=4, per_decade=20)

Compute the primordial curvature and tensor spectra from potential V (Planck units, dV its derivative), starting the background at field value φ0. The pivot scale k_pivot (1/Mpc) is placed N_pivot e-folds before the end of inflation. Returns an InflatonSpectrum suitable for cosmology(primordial = ...).

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