Cosmology

The Cosmology module controls the cosmological parameters and various derived quantities. We can initialize a cosmology using init_cosmology. For examples on how to use this module and various functions within it, see Example - 1.

Initialization

LensFactory.Cosmology.init_cosmology — Type

init_cosmology(H0::RV=70.0, w::RV=-1.0, Omega_m0::RV=0.3, Omega_r0::RV=0.0, Omega_w0::RV=0.7)

An Abstract type to initialize a cosmology with given parameters.

Arguments

H0::RV: Hubble constant in ${\rm \mathbf{km/s/Mpc}}$.
w::RV: Dark energy equation of state parameter.
Omega_m0::RV: Matter density parameter.
Omega_r0::RV: Radiation density parameter.
Omega_w0::RV: Dark energy density parameter.

Returns

Omega_k0: Curvature density parameter.

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Basics

LensFactory.Cosmology.scale_factor — Function

scale_factor(z::RV) --> RV

Calculate scale factor $(a)$ at redshift $z$,

\[a = \frac{1}{1+z}.\]

Arguments

z::RV: Redshift.

Returns

a::RV: Scale factor.

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LensFactory.Cosmology.Ez — Function

Ez(cosmology::AbstractCosmology, z::RV) --> RV

Calculate dimensionless Hubble parameter $(E)$ at redshift, $z$,

\[E(z) = \sqrt{ Ω_{m0} (1+z)^2 + Ω_{r0} (1+z)^4 + Ω_{k0} (1+z)^2 + Ω_{w0} (1+z)^{3(1+w)} }.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

Ez::RV: Dimensionless Hubble parameter.

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LensFactory.Cosmology.hubble_parameter — Function

hubble_parameter(cosmology::AbstractCosmology, z::RV) --> RV

Calculate Hubble parameter $(H)$ at redshift $z$ in ${\rm \mathbf{km/s/Mpc}}$,

\[H(z) = H_0 \: E(z).\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

H::RV: Hubble parameter.

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Time

LensFactory.Cosmology.hubble_time — Function

hubble_time(H0::RV) --> RV

Calculate Hubble time $(t_H)$ in ${\rm \mathbf{Gyr}}$,

\[t_H = \frac{1}{H_0}.\]

Arguments

H0::RV: Hubble constant in ${\rm \mathbf{km/s/Mpc}}$.

Returns

tH::RV: Hubble time in ${\rm \mathbf{Gyr}}$.

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LensFactory.Cosmology.age — Function

age(cosmology::AbstractCosmology, z::RV) --> RV

Calculate age $(t_{\rm age})$ of the Universe at redshift $z$ in ${\rm \mathbf{Gyr}}$,

\[t_{\rm age}(z) = \int_z^\infty \frac{1}{(1+z')H(z')} dz'.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

age::RV: Age of the Universe in ${\rm \mathbf{Gyr}}$.

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LensFactory.Cosmology.lookback_time — Function

lookback_time(cosmology::AbstractCosmology, z::RV) --> RV

Calculate lookbak time $(t_L)$ to a given redshift $z$ in ${\rm \mathbf{Gyr}}$,

\[t_L(z) = \int_0^z \frac{1}{(1+z')H(z')} dz'.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

tL::RV: Lookback time in ${\rm \mathbf{Gyr}}$.

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Density parameters

LensFactory.Cosmology.rho_cz — Function

rho_cz(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the critical density $(\rho_c)$ of the Universe at redshift $z$ in ${\rm \mathbf{kg/m^3}}$,

\[\rho_c(z) = \frac{3 H^2(z)}{8 π {\rm G} }.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

rho_c::RV: Critical density in ${\rm \mathbf{kg/m^3}}$.

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LensFactory.Cosmology.Omega_mz — Function

Omega_mz(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the dimensionless matter density parameter $(\Omega_{m})$ at redshift $z$,

\[Ω_{m}(z) = Ω_{m}(0) \: (1+z)^3 \left( \frac{H0}{H(z)} \right)^2.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

Omega_m::RV: Dimensionless matter density parameter.

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LensFactory.Cosmology.Omega_rz — Function

Omega_rz(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the dimensionless radiation density parameter $(\Omega_{r})$ at redshift $z$,

\[Ω_{r}(z) = Ω_{r}(0) \: (1+z)^4 \left( \frac{H_0}{H(z)} \right)^2.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

Omega_r::RV: Dimensionless radiation density parameter.

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LensFactory.Cosmology.Omega_wz — Function

Omega_wz(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the dimensionless dark energy density parameter $(\Omega_{w})$ at redshift $z$,

\[Ω_{w}(z) = Ω_{w0} (1+z)^{3(1+w)} \left( \frac{H_0}{H(z)} \right)^2.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

Omega_w::RV: Dimensionless dark energy density parameter.

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LensFactory.Cosmology.Omega_kz — Function

Omega_kz(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the dimensionless curvature density parameter $(\Omega_{k})$ at redshift $z$,

\[Ω_{k}(z) = Ω_{k0} (1+z)^2 \left( \frac{H_0}{H(z)} \right)^2.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

Omega_k::RV: Dimensionless curvature density parameter.

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Distances

LensFactory.Cosmology.hubble_distance — Function

hubble_distance(H0::RV) --> RV

Calculate the Hubble distance (i.e., size of the observable Universe) $(D_H)$ in ${\rm \mathbf{meters}}$,

\[D_H = \frac{\rm c}{\rm H_0}.\]

Arguments

H0::RV: Hubble constant in ${\rm \mathbf{km/s/Mpc}}$.

Returns

D_H::RV: Hubble distance in ${\rm \mathbf{meters}}$.

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LensFactory.Cosmology.comoving_distance_radial — Function

comoving_distance_radial(cosmo::AbstractCosmology, z1::RV, z2::RV) --> RV

Calculate the comoving radial distance $(D_C)$ between $z_1$ and $z_2$ in ${\rm \mathbf{meters}}$. The formula is,

\[D_C = D_H \int_{z_1}^{z_2} \frac{dz'}{E(z')}.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z1::RV: Redshift.
z2::RV: Redshift.

Returns

D_C::RV: Comoving radial distance in ${\rm \mathbf{meters}}$.

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LensFactory.Cosmology.comoving_distance_transverse — Function

comoving_distance_transverse(cosmo::AbstractCosmology, z1::RV, z2::RV) --> RV

Calculate the comoving radial distance $(D_M)$ between, $z_1$ and $z_2$ in ${\rm \mathbf{meters}}$,

\[D_M = \begin{cases} D_H \frac{1}{\sqrt{\Omega_k}} \sinh \left[ \sqrt{\Omega_k} \: D_C / D_H \right], & \text{if } \Omega_k > 0, \\ D_C, & \text{if } \Omega_k = 0, \\ D_H \frac{1}{\sqrt{|\Omega_k|}} \sin \left[ \sqrt{|\Omega_k|} \: D_C / D_H \right], & \text{if } \Omega_k < 0. \\ \end{cases}\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z1::RV: Redshift.
z2::RV: Redshift.

Returns

D_M::RV: Comoving radial distance in ${\rm \mathbf{meters}}$.

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LensFactory.Cosmology.luminosity_distance — Function

luminosity_distance(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the luminosity distance $(D_L)$ to redshift $z$ in ${\rm \mathbf{meters}}$,

\[D_L(z) = (1+z) D_M(z).\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

D_L::RV: Luminosity distance in ${\rm \mathbf{meters}}$.

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LensFactory.Cosmology.angular_diameter_distance — Function

angular_diameter_distance(cosmology::AbstractCosmology, z1::RV, z2::RV) --> RV

Calculate the angular diameter distance $(D_A)$ between redshifts $z_1$ and $z_2$ in ${\rm \mathbf{meters}}$,

\[D_A(z_1, z_2) = \frac{D_M(z_1, z_2)}{1+z_2}.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z1::RV: Redshift.
z2::RV: Redshift.

Returns

D_A::RV: Angular diameter distance in ${\rm \mathbf{meters}}$.

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LensFactory.Cosmology.distance_modulus — Function

distance_modulus(cosmo::AbstractCosmology, z::RV) --> RV

Calculate the distance modulus $(\mu)$ to a given redshift $z$,

\[\mu = 5 \log\left( \frac{D_L}{\rm pc} \right) - 5.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

mu::RV: Distance modulus.

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LensFactory.Cosmology.angular_scale — Function

angular_scale(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the angular size in ${\rm \mathbf{Kpc}}$ for $1''$ on sky at redhsift, $z$,

\[d = 1'' \times \left( \frac{D_A}{\rm{kpc}} \right).\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

d::RV: Angular size in ${\rm \mathbf{Kpc}}$.

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LensFactory.Cosmology.adis2zs — Function

adis2zs(cosmology::AbstractCosmology, z_d::RV, adis::RV; max_iter::Int64=10000, tol::Float64=1e-6) --> RV

Calculate the source redshift ($z_s$) from the distance ratio ($a_{\rm dis}$), using Bi-section method.

Arguments

cosmology::AbstractCosmology: Cosmology object.
z_d::RV: Lens redshift.
adis::RV: Distance ratio, $a_{\rm dis}$.
max_iter::Int64=10000: Maximum number of iterations.
tol::Float64=1e-6: Tolerance for the root finding algorithm.

Returns

z_s::RV: Redshift of the source.

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Volumes

LensFactory.Cosmology.comoving_volume_element — Function

comoving_volume_element(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the comving volume element $(dV_C)$ at redshift $z$ in ${\rm \mathbf{Gpc^3}}$,

\[dV_C = D_H \frac{D_M^2(z)}{E(z)}.\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

dV_C::RV: Comving volume element in ${\rm \mathbf{Gpc^3}}$.

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LensFactory.Cosmology.comoving_volume — Function

comoving_volume(cosmology::AbstractCosmology, z::RV) --> RV

Calculate the total comving volume up to redshift $z$ in ${\rm \mathbf{Gpc^3}}$,

\[V_C = \begin{cases} \frac{4π}{2} \frac{D_H^3}{Ω_k} \left[ \frac{D_M}{D_H} \sqrt{1+Ω_k \left(\frac{D_M}{D_H}\right)^2} - \frac{1}{\sqrt{|Ω_k|}} {\rm arcsinh}\left( \sqrt{|Ω_k|} \frac{D_M}{D_H} \right) \right], & \text{if } Ω_k > 0, \\ \frac{4π}{3} D_M^3, & \text{if } Ω_k = 0, \\ \frac{4π}{2} \frac{D_H^3}{|Ω_k|} \left[ \frac{D_M}{D_H} \sqrt{1+Ω_k \left(\frac{D_M}{D_H}\right)^2} - \frac{1}{\sqrt{|Ω_k|}} \arcsin\left( \sqrt{|Ω_k|} \frac{D_M}{D_H} \right) \right], & \text{if } Ω_k < 0. \\ \end{cases}\]

Arguments

cosmology::AbstractCosmology: Cosmology object.
z::RV: Redshift.

Returns

com_vol::RV: Comving volume up to redshift $z$ in ${\rm \mathbf{Gpc^3}}$.

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