Source Density

AnalitycalDensityStruct(z0::Float64 = 0.9/sqrt(2.), zmin::Float64 = 0.001,
zmax::Float64 = 2.5, surfacedensity::Float64 = 30.,
normalization::Float64 = 1.)

This struct contains the parameters of the source galaxy density as given by the official Euclid forecast, whose expression is given by:

\[n(z)\propto\left(\frac{z}{z_0}\right)^2 \exp{\left(-\left(\frac{z}{z_0}\right)^{-3/2}\right)}\]

The parameters contained in this struct are

• $z_{min}$ and $z_{max}$, the minimum and the maximum redshift considered

• $z_0$, the parameter present in the galaxy distribution

• surfacedensity , the value of the galaxy source density integrated between $z_{min}$ and $z_{max}$

• normalization, the value of parameter which multiplies the source dennsity in order to match the correct surface density

ComputeDensityFunction(z::Float64,
AnalitycalDensity::AnalitycalDensityStruct)

This function returns the source density for a given redshift $z$.

NormalizeAnalitycalDensityStruct(densityparameters::AnalitycalDensityStruct)

This function normalize AnalitycalDensityStruct in order to have the correct value of the surface density once integrated.

InstrumentResponseStruct(cb::Float64 = 1.0, zb::Float64 = 0.0,
σb::Float64 = 0.05, co::Float64 = 1.0, zo::Float64 = 0.1,
σo::Float64 = 0.05, fout::Float64 = 0.1)

When we measure the redshift of a galaxy with redshit $z$, we will measure a redshift $z_p$ with a probability given by the following expression:

\[p(z_p|z) = \frac{1-f_{out}}{\sqrt{2 \pi} \sigma_{b}(1+z)} \exp \left( -\frac{1}{2}\left(\frac{z-c_{b} z_{b}-z_{b}}{\sigma_{b}(1+z)}\right)^{2} \right) + \frac{f_{out}}{\sqrt{2 \pi} \sigma_{\mathrm{o}}(1+z)} \exp \left(-\frac{1}{2}\left(\frac{z-c_{o} z_{p}-z_{o}}{\sigma_{o}(1+z)} \right)^{2}\right)\]

This struct contains all these parameters.

ComputeInstrumentResponse(z::Float64, zp::Float64,
InstrumentResponse::InstrumentResponse)

This function computes the probability that we actually measure a redshift $z_p$ if the real redshift is $z$.

ConvolvedDensityStruct(AnalitycalDensity::AnalitycalDensity = AnalitycalDensityStruct(),
InstrumentResponse::InstrumentResponse = InstrumentResponseStruct()
ZBinArray::Vector{Float64} = Array([0.001, 0.418, 0.560, 0.678, 0.789, 0.900, 1.019, 1.155, 1.324, 1.576, 2.50])
DensityNormalizationArray::Vector{Float64} = ones(length(ZBinArray)-1)
DensityGridArray::AbstractArray{Float64, 2} = ones(length(ZBinArray)-1, 300))

In order to take into account the error in the redshift measurement, the source density is convolved with the InstrumentResponseStruct, according to the following equation

\[n_{i}(z)=\frac{\int_{z_{i}^{-}}^{z_{i}^{+}} \mathrm{d} z_{\mathrm{p}} n(z) p \left(z_{\mathrm{p}} \mid z\right)}{\int_{z_{\min }}^{z_{\max }} \mathrm{d} z \int_{z_{i}^{-}}^{z_{i}^{+}} \mathrm{d} z_{\mathrm{p}} n(z) p \left(z_{\mathrm{p}} \mid z\right)}\]

ComputeConvolvedDensityFunction(z::Float64, i::Int64,
ConvolvedDensity::AsbtractConvolvedDensity,
AnalitycalDensity::AnalitycalDensityStruct,
InstrumentResponse::InstrumentResponse)

This function computes the Convolved density function for a single bin at a given redshift $z$.

NormalizeConvolvedDensityStruct(ConvolvedDensity::AsbtractConvolvedDensity,
AnalitycalDensity::AnalitycalDensityStruct,
InstrumentResponse::InstrumentResponse, CosmologicalGrid::CosmologicalGrid)

This function normalizes ConvolvedDensity such that the integrals of the convolved densities are normalized to 1.

ComputeConvolvedDensityFunctionGrid(CosmologicalGrid::CosmologicalGrid,
ConvolvedDensity::AsbtractConvolvedDensity,
AnalitycalDensity::AnalitycalDensityStruct,
InstrumentResponse::InstrumentResponse)

This function computes the convolved density function for all tomographic bins on the $z$-grid provided by CosmologicalGrid.