Observable quantities

spectrum_primordial(k, h, As, ns=1.0; kp = 0.05*(L100/h))

Compute the primordial power spectrum

\[P₀(k) = 2π² Aₛ (k/kₚ)^{nₛ-1} / k³\]

with spectral amplitude As, spectral index ns and pivot scale wavenumber kp at the wavenumber(s) k. All wavenumbers are in units of $H₀/c$, and the default pivot scale is 0.05/Mpc.

spectrum_matter([modes,] prob::CosmologyProblem, k[, τ]; kwargs...)

Compute the matter power spectrum

\[P(k,τ) = P₀(k) |Δ(k,τ)|²\]

of the total gauge-invariant overdensity

\[Δ = (∑ₛρₛΔₛ) / (∑ₛρₛ)\]

for one or more modes at wavenumbers k and conformal time(s) τ from the problem prob. The problem is solved for the given $k$, and the matter power spectrum is saved at the given $τ$.

• modes must be :c (CDM), :b (baryons), :h (massive neutrinos), :m (matter; equivalent to $c+b+h$), a vector thereof, or unspecified to use :m.
• k must be a vector of wavenumbers in units of $H₀/c$.
• τ must be a single or a vector of conformal times, or unspecified to use $τ = τ₀$ today.
• kwargs... are keyword arguments that are forwarded to solve(prob, k; kwargs...).

spectrum_matter([modes,] sol::CosmologySolution, k[, τ]; kwargs...)

Compute the matter power spectrum in the same way, but interpolate between wavenumbers and times already stored in the solution sol.

spectrum_matter_nonlinear(sol::CosmologySolution, k)

Compute the nonlinear matter power spectrum from the cosmology solution sol at wavenumber(s) k using halofit implemented in MatterPower.jl.

spectrum_cmb(ΘlAs::AbstractMatrix, ΘlBs::AbstractMatrix, P0s::AbstractVector, ls::AbstractVector, ks::AbstractVector; normalization = :Cl, thread = true)

Compute the angular power spectrum

\[Cₗᴬᴮ = (2/π) ∫\mathrm{d}k \, k² P₀(k) Θₗᴬ(k,τ₀) Θₗᴮ(k,τ₀)\]

for the given ls. If normalization == :Dl, compute $Dₗ = Cₗ l (l+1) / 2π$ instead.

spectrum_cmb(modes::AbstractVector{<:Symbol}, prob::CosmologyProblem, jl::SphericalBesselCache; normalization = :Cl, kinterp = nothing, Δkτ0 = 2π/4, xs = cosgrid(0.0, 1.0; length=1200), l_limber = 11, bgalg = default_bgalg(prob), bgreltol = 1e-7, bgabstol = 1e-7, bgopts = (), ptalg = default_ptalg(prob), ptreltol = 1e-5, ptabstol = 1e-5, ptopts = (), thread = true, verbose = false, kwargs...)

Compute angular CMB power spectra $Cₗᴬᴮ$ at angular wavenumbers ls from the cosmological problem prob. The requested modes are specified as a vector of symbols in the form :AB, where A and B are T (temperature), E (E-mode polarization) or ψ (lensing). The spectra are of dimensionless temperature fluctuations relative to the present photon temperature $T_{γ0}$; multiply by $T_{γ0}^2$ to get dimensionful spectra. Returns a matrix of $Cₗ$ if normalization is :Cl, or $Dₗ = l(l+1)/2π$ if normalization is :Dl.

Precision parameters

• xs: Grid of $(τ-τᵢ)/(τ₀-τᵢ)$ specifying the $τ$-points that will be sampled in line-of-sight integration (also if the independent variable is not $τ$), or an integer number of points interpolated from the background time steps.
• kinterp: Interpolator that decides which $k$-modes the perturbation ODEs will be solved explicitly for, and then interpolated in-between to a finer grid set by Δkτ0.
• Δkτ0: Grid spacing to use when integrating over $k$ to project to $ℓ$-space.
• l_limber: Use Limber approximation for lensing line-of-sight integrals with equal or greater $ℓ$.
• bgalg/ptalg, bgreltol/ptreltol, bgabstol/ptabstol: ODE algorithms and tolerances for the background/perturbation stages.
• bgopts/ptopts: extra options for the background/perturbation ODE solves.

Examples

using SymBoltz
M = ΛCDM()
pars = parameters_Planck18(M)
prob = CosmologyProblem(M, pars)

ls = 10:10:1000
jl = SphericalBesselCache(ls)
modes = [:TT, :TE, :ψψ, :ψT]
Dls = spectrum_cmb(modes, prob, jl; normalization = :Dl)

spectrum_cmb(modes::AbstractVector, prob::CosmologyProblem, jl::SphericalBesselCache, ls::AbstractVector; kwargs...)

Same, but compute the spectrum properly only for jl.l and then interpolate the results to all ls.

correlation_function(sol::CosmologySolution; N = 2048, spline = true)

Compute the two-point correlation function in real space by Fourier transforming the matter power spectrum of sol with N points the FFTLog algorithm implemented in TwoFAST. Returns N radii and correlation function values (e.g. r, ξ).

variance_matter(sol::CosmologySolution, R)

Compute the variance $⟨δ²⟩$ of the linear matter density field with a top-hat filter with radius R in units of $c/H₀$. Wraps the implementation in MatterPower.jl.

stddev_matter(sol::CosmologySolution, R)

Compute the standard deviation $√(⟨δ²⟩)$ of the linear matter density field with a top-hat filter with radius R in units of $c/H₀$.

distance_luminosity(χ, a, h, Ωk0 = 0)

Compute luminosity distances (in meters)

\[d_L = \frac{c}{H_0} \frac{r}{a}, \quad \mathrm{where} \quad r = \chi \, \frac{\sin\left(\sqrt{-Ω_{k0}} \, \chi\right)}{\sqrt{-Ω_{k0}} \, \chi},\]

from conformal lookback times χ, scale factors a, Hubble parameter h and curvature density Ωk0.

sound_horizon(sol::CosmologySolution)

Cumulatively integrate the sound horizon

\[ rₛ(τ) = ∫_0^τ dτ cₛ = ∫_0^τ \frac{dτ}{√(3(1+3ρ_b/4ρ_γ))}\]

to the time steps of the solution sol.

source_grid(
    prob::CosmologyProblem, Ss, τs, ks[, bgsols];
    bgalg = default_bgalg(prob), bgreltol = 1e-7, bgabstol = 1e-7, bgopts = (),
    ptalg = default_ptalg(prob), ptreltol = 1e-5, ptabstol = 1e-5, ptopts = (),
    thread = true, verbose = false
)

Compute and evaluate source functions $S(τ,k)$ with symbolic expressions Ss on a grid with conformal times τs and wavenumbers ks from the problem prob. Returns a matrix of size (Nτ, Nk), where each element is a vector of length NS = length(Ss) holding all source values at that (τ, k) point.

The algorithms bgalg/ptalg, tolerances bgreltol/ptreltol and bgabstol/ptabstol and extra options bgopts/ptopts are passed to the background/perturbation ODE solves. If the background solutions bgsols are passed, they are used directly and the background options are not accepted.

los_integrate(Ss::AbstractMatrix{T}, ls::AbstractVector, τs::AbstractVector, ks::AbstractVector, jl::SphericalBesselCache; l_limber = typemax(Int), thread = true, verbose = false) where {T}

For the given ls and ks, compute the line-of-sight integrals

\[Iₗ(k) = ∫dτ S(k,τ) jₗ(k(τ₀-τ))\]

over the source function values Ss against the spherical Bessel functions $jₗ(x)$ cached in jl. The element Ss[i,j] holds the source function value $S(τᵢ, kⱼ)$. The Limber approximation

\[Iₗ ≈ √(π/(2l+1)) S(τ₀-(l+1/2)/k, k)\]

is used for l ≥ l_limber.