Power Spectrum

Power spectrum components $C_{\ell}$ are encoded as Vector{T}. Healpix.jl implements functions to perform sht operations on power spectra, e.g. to obtain a map or a set of 'Alm', as well as writing/reading a power spectrum from a FITS file.

The functions ending with ! are mutating functions, which means that they assume that the result must be saved in a preallocated variable; they are space- and time-efficient and should be used when you want your code to be performant, or when you plan to apply the same operation several times (e.g., in a Monte Carlo simulation).

Loading and saving power spectrum components

Healpix.jl implements functions to read/write the components $C_{\ell}$ from/to a FITS files.

Healpix.readClFromFITS — Function

readClFromFITS{T <: Real}(f::CFITSIO.FITSFile, t::Type{T}; col_num = 2) -> Vector{T}
readClFromFITS{T <: Real}(fileName::String, t::Type{T}; col_num = 2) -> Vector{T}

Read a set of C_ℓ coefficients from a FITS file.

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Healpix.writeClToFITS — Function

writeClToFITS(f::CFITSIO.FITSFile, Cl::Vector{T}) where {T <: Real}
writeClToFITS(fileName, Cl::Vector{T}; overwrite = true) where {T <: Real}

Write a set of C_ℓ coefficients to a FITS file.

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Converting different power spectrum representation

It's often useful to represent, especially for plotting it, the power spectrum in the form of $D_{\ell}$. Healpix.jl implements a couple of functions to convert a power spectrum from/to such a representation.

Healpix.cl2dl — Function

cl2dl(cl::AbstractVector{T}, lmin::Integer) where {T <: Real}

Convert a set of $C_{\ell}$ to $D_{\ell}$ power spectrum, where $D_{\ell} = \ell (\ell + 1) C_{\ell} / 2\pi$. The first components are set to zero if not present.

Arguments:

cl::AbstractVector{T} : Array of C_ℓ components
lmin::Integer : minimum l in the representation of the C_ℓ power spectrum

Returns:

Vector{T} : Array of D_ℓ power spectrum components

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Healpix.dl2cl — Function

dl2cl(dl::AbstractVector{T}, lmin::Integer) where {T <: Real}

Convert a set of $D_{\ell}$ to $C_{\ell}$ power spectrum, where $C_{\ell} = 2\pi D_{\ell} / \ell (\ell + 1)$. The first components are set to zero if not present. The monopole component is set to zero in any case to avoid Inf values.

Arguments:

dl::AbstractVector{T} : Array of D_ℓ components
lmin::Integer : minimum l in the representation of the Dℓ power spectrum

Returns:

Vector{T} : Array of C_ℓ power spectrum components

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Synthesizing harmonic coefficients from power spectrum

Generate a Alm instance with a random set of $a_{\ell m}$ coefficients. Each harmonic coefficient $a_{\ell m}$ is a realization of a gaussian distribution with zero mean and $C_{\ell}$ variance.

Healpix.synalm! — Function

synalm!(cl::Vector{T}, alm::Alm{ComplexF64, Vector{ComplexF64}}, rng::AbstractRNG) where {T <: Real}
synalm!(cl::Vector{T}, alm::Alm{ComplexF64, Vector{ComplexF64}}) where {T <: Real}

Generate a set of $a_{\ell m}$ from a given power spectra $C_{\ell}$. The output is written into the Alm object passed in input.

Arguments:

cl::AbstractVector{T}: The array representing the power spectrum components $C_{\ell}$,

starting from $\ell = 0$.

alm::Alm{Complex{T}}: The array representing the spherical harmonics coefficients $a_{\ell m}$

we want to write the result into.

rng::AbstractRNG : (optional) the RNG to be used for generating the $a_{\ell m}$. It allows

to set the seed beforehand guaranteeing the reproducibility of the process.

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Healpix.synalm — Function

synalm(cl::Vector{T}, lmax::Integer, mmax::Integer, rng::AbstractRNG) where {T <: Real}
synalm(cl::Vector{T}, lmax::Integer, mmax::Integer) where {T <: Real}
synalm(cl::Vector{T}, lmax::Integer, rng::AbstractRNG) where {T <: Real}
synalm(cl::Vector{T}, lmax::Integer) where {T <: Real}
synalm(cl::Vector{T}, rng::AbstractRNG) where {T <: Real}
synalm(cl::Vector{T}) where {T <: Real}

Generate a set of $a_{\ell m}$ from a given power spectra $C_{\ell}$. The output is written into a new Alm object of given lmax.

Arguments:

cl::AbstractVector{T}: The array representing the power spectrum components $C_{\ell}$,

starting from $\ell = 0$.

lmax::Integer: the maximum $ℓ$ coefficient, will default to length(cl)-1 if not specified.
mmax::Integer: the maximum $m$ coefficient, will default to lmax if not specified.
rng::AbstractRNG : (optional) the RNG to be used for generating the $a_{\ell m}$. It allows

to set the seed beforehand guaranteeing the reproducibility of the process.

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Generating a map from power spectrum

Synthesize a set of Alm through 'synalm' and generates a map from it through 'alm2map'.

Healpix.synfast! — Function

synfast!(cl::Vector{T}, map::HealpixMap{T, RingOrder}, lmax::Integer, rng::AbstractRNG) where {T <: Real}
synfast!(cl::Vector{T}, map::HealpixMap{T, RingOrder}, lmax::Integer) where {T <: Real}
synfast!(cl::Vector{T}, map::HealpixMap{T, RingOrder}, rng::AbstractRNG) where {T <: Real}
synfast!(cl::Vector{T}, map::HealpixMap{T, RingOrder}) where {T <: Real}

Generate a map from a given power spectra $C_{\ell}$. The result is saved into the HealpixMap passed in input.

Arguments:

cl::AbstractVector{T}: The array representing the power spectrum components $C_{\ell}$.
map::HealpixMap{T, RingOrder}: the map that will contain the result.
lmax::Integer: the maximum $ℓ$ coefficient, will default to length(cl)-1 if not specified.
rng::AbstractRNG : (optional) the RNG to be used for generating the $a_{\ell m}$. It allows

to set the seed beforehand guaranteeing the reproducibility of the process.

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Healpix.synfast — Function

synfast(cl::Vector{T}, nside::Integer, lmax::Integer, rng::AbstractRNG) where {T <: Real}
synfast(cl::Vector{T}, nside::Integer, lmax::Integer) where {T <: Real}
synfast(cl::Vector{T}, nside::Integer, rng::AbstractRNG) where {T <: Real}
synfast(cl::Vector{T}, nside::Integer) where {T <: Real}

Generate a HealpixMap with given Nside, from a given power spectra $C_{\ell}$.

Arguments:

cl::AbstractVector{T}: The array representing the power spectrum components $C_{\ell}$.
nside::Integer: nside of the map that will contain the result.
lmax::Integer: the maximum $ℓ$ coefficient, will default to length(cl)-1 if not specified.
rng::AbstractRNG : (optional) the RNG to be used for generating the $a_{\ell m}$. It allows

to set the seed beforehand guaranteeing the reproducibility of the process.

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Computing the power spectrum from a map

Compute the (cross-) power spectrum of one (or two) 'HealpixMap'.

Healpix.anafast — Function

anafast(map::HealpixMap{Float64, RingOrder, AA}; lmax=nothing, mmax=nothing, niter::Integer = 3) where {T <: Real,AA <: AbstractArray{T,1}} -> Vector{Float64}
anafast(map₁::HealpixMap{Float64, RingOrder, AA}, map₂::HealpixMap{Float64, RingOrder, AA}; lmax=nothing, mmax=nothing, niter::Integer = 3) where {T <: Real,AA <: AbstractArray{T,1}} -> Vector{Float64}

Computes the power spectrum of a Healpix map, or the cross-spectrum between two maps if map2 is given. No removal of monopole or dipole is performed. The input maps must be in ring-ordering.

Arguments:

map₁::HealpixMap{Float64, RingOrder, AA}: the spherical harmonic coefficients of the first field
map₂::HealpixMap{Float64, RingOrder, AA}: the spherical harmonic coefficients of the second field

Returns:

Array{T} containing $C_{\ell}$, with the first element referring to ℓ=0.

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