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

\[P(τ,k) = P₀(k) |Δ(τ,k)|²\]

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, ws::AbstractVector; normalization = :Cl, thread = true)

Compute the angular power spectrum

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

for the given ls, using the quadrature weights ws for the wavenumbers ks. If normalization == :Dl, compute $Dₗ = Cₗ l (l+1) / 2π$ instead.

spectrum_cmb(modes::AbstractVector{<:Symbol}, prob::CosmologyProblem, jl::SphericalBesselCache; normalization = :Cl, kinterp = nothing, τquad = nothing, kquad = nothing, 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

• τquad: Quadrature rule for the line-of-sight integral over $τ$; its nodes are mapped linearly to $[τᵢ, τ₀]$. Defaults to default_τquad.
• kquad: Quadrature rule for line-of-sight integration and the integral over $k$; its nodes are mapped linearly to the $k$-range of kinterp. Defaults to default_kquad.
• kinterp: Interpolator that decides which $k$-modes the perturbation ODEs will be solved explicitly for, and then interpolated in-between to the nodes of kquad.
• 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(modes, sol::CosmologySolution, t)

Get the distances modes at the values t of the independent variable from the solution sol. The times t can be a vector of values of the independent variable, or a pair like M.g.z => zs with values of another variable. The modes can be one or a vector of:

• :χ: line-of-sight comoving distance $χ = τ₀ - τ$,
• :M: transverse comoving distance $D_M = \sin(\sqrt{-Ω_{k0}} χ) / \sqrt{-Ω_{k0}}$,
• :A: angular diameter distance $D_A = a D_M$,
• :L: luminosity distance $D_L = D_M / a$,
• :H: Hubble distance $D_H = 1 / H$,
• :V: volume-averaged distance $D_V = (z D_M^2 D_H)^{1/3}$.

Returned distances are dimensionless in units of $c/H₀$. The model must have the variables χ, a and H.

sound_horizon(sol::CosmologySolution[, t])

Get the photon-baryon sound horizon

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

at the time steps of the solution sol, or at the values t of the independent variable. It is integrated with the background as the variable rₛ.

time_drag(sol::CosmologySolution)

Get the value of the independent time variable $t$ at the baryon drag epoch, when the drag optical depth satisfies

\[ κ_d(t) = -∫_t^{t_0} \frac{κ'}{3ρ_b/4ρ_γ} dt' = 1,\]

where $κ$ is the Thomson optical depth and $κ' = dκ/dt$. The sound horizon at the drag epoch $r_d$ is then sound_horizon(sol, time_drag(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, χs::AbstractVector, ks::AbstractVector, jl::SphericalBesselCache, ws::AbstractVector; l_limber = typemax(Int), thread = true, verbose = false) where {T}

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

\[Iₗ(k) = ∫dχ S(χ,k) jₗ(kχ)\]

over the source function values Ss against the spherical Bessel functions $jₗ(x)$ cached in jl, using the quadrature weights ws for the descending conformal distances $χ = τ₀ - τ$. The element Ss[i,j] holds the source function value $S(χᵢ, kⱼ)$. The Limber approximation

\[Iₗ ≈ √(π/(2l+1)) S((l+1/2)/k, k)\]

is used for l ≥ l_limber. Contributions where $|jₗ(x)|$ is below jltol (at small $x ≪ l$) are skipped.