Cosmology

init_cosmology(H0::Real = 70.0, 
               w::Real  = -1.0, 
               Omega_m0::Real = 0.3, 
               Omega_r0::Real = 0.0, 
               Omega_w0::Real = 0.7,
               Omega_k0::Real = 1.0 - Omega_m0 - Omega_r0 - Omega_w0)

A struct to initialize a cosmology with given parameters.

For typical cases, the user provides values of H0, Omega_m0, and Omega_w0 (and Omega_r0 if non-zero). For such a case, the default value of Omega_k0 ensures that the sum of density parameters is equal to 1 (i.e., $\Omega_{m0} + \Omega_{r0} + \Omega_{w0} + \Omega_{k0} = 1$). However, if the user provides all four density parameters, the value of Omega_k0 provided by the user will be used and the universe will not necessarily be spatially flat.

Arguments

• H0::Real = 70.0: Hubble constant in ${\rm \mathbf{km/s/Mpc}}$.
• w::Real = -1.0: Dark energy equation of state parameter.
• Omega_m0::Real = 0.3: Matter density parameter.
• Omega_r0::Real = 0.0: Radiation density parameter.
• Omega_w0::Real = 0.7: Dark energy density parameter.
• Omega_k0::Real = 1.0 - Omega_m0 - Omega_r0 - Omega_w0: Curvature density parameter.

Returns

• init_cosmology::AbstractCosmology: Cosmology object.

scale_factor(z::Real) --> Real

Calculate scale factor, $a$, for a given redshift $z$,

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

Arguments

• z: Redshift.

Returns

• a: Scale factor.

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

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

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Returns

• Ez : Dimensionless Hubble parameter.

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

Calculate Hubble parameter, $H(z)$, for a given redshift $z$,

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Returns

• Hz : Hubble parameter (in ${\rm \mathbf{km/s/Mpc}}$).

hubble_time(H0::Real) --> Real

Calculate the Hubble time, $t_H$,

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

Arguments

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

Returns

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

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

Calculate the age of the Universe, $t_{\rm age})$, at a given redshift $z$,

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Returns

• t_age : Age of the Universe (in ${\rm \mathbf{Gyr}}$).

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

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 : Cosmology object.
• z : Redshift.

Returns

• t_lookback : Lookback time (in ${\rm \mathbf{Gyr}}$).

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

Calculate the critical density $(\rho_c)$ of the Universe at redshift $z$ in given units,

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Keyword Arguments

• unit = :kg_m3: Output units of the critical surface density.
  • :kg_m3 $\Rightarrow{\rm \mathbf{kg/m^3}}$
  • :msun_pc3 $\Rightarrow{\rm \mathbf{M_⊙/pc^3}}$

Returns

• rho_c : Critical density of the Universe.

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

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: Cosmology object.
• z : Redshift.

Returns

• Omega_m : Dimensionless matter density parameter.

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

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: Cosmology object.
• z : Redshift.

Returns

• Omega_r : Dimensionless radiation density parameter.

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

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: Cosmology object.
• z : Redshift.

Returns

• Omega_w : Dimensionless dark energy density parameter.

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

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: Cosmology object.
• z : Redshift.

Returns

• Omega_k : Dimensionless curvature density parameter.

hubble_distance(H0::Real) --> Real

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

Returns

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

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

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: Cosmology object.
• z1 : First redshift.
• z2 : Second redshift.

Returns

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

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

Calculate the comoving transverse 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: Cosmology object.
• z1 : First redshift.
• z2 : Second redshift.

Returns

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

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

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

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Returns

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

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

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: Cosmology object.
• z1 : First redshift.
• z2 : Second redshift.

Returns

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

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

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

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Returns

• mu : Distance modulus.

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

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: Cosmology object.
• z : Redshift.

Returns

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

zs2adis(cosmology::AbstractCosmology, z_d::Real, zs::Real;
        dC = nothing, atol = DIST_ATOL, rtol = DIST_RTOL) --> Real

Distance ratio $a_{\rm dis} = D_{ds}/D_s = D_A(z_d, zs) / D_A(0, zs)$ — the forward map that adis2zs inverts. The $(1+z_s)$ factors cancel, so $a_{\rm dis} = D_M(z_d, zs) / D_M(0, zs)$, built from the radial comoving distances via $χ(z_d, zs) = χ(zs) - χ(z_d)$. Only two 1-D integrals are needed, and $χ(z_d)$ is constant for a fixed cosmology + lens redshift — pass it as dC (radial comoving distance to the lens, meters) to cut each call to a single integral. Valid for flat, open, and closed models and any w.

Arguments

• cosmology: Cosmology object.
• z_d : Lens redshift.
• zs : Source redshift (must satisfy zs > z_d).

Keyword Arguments

• dC = nothing: Precomputed radial comoving distance $χ(z_d)$ (meters).
• atol = DIST_ATOL,
• rtol = DIST_RTOL: Quadrature tolerances.

Returns

• a_dis : Dimensionless distance ratio $D_{ds}/D_s$.

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

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

Arguments

• cosmology: Cosmology object.
• z_d : Lens redshift.
• adis : Distance ratio, $a_{\rm dis}$.

Keyword Arguments

• max_iter = 10000: Maximum number of iterations.
• tol = 1E-6 : Tolerance for the root finding algorithm.

Returns

• z_s : Redshift of the source.

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

Calculate the comving volume element $(dV_C)$ at redshift $z$,

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Returns

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

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

Calculate the total comving volume up to redshift $z$,

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

Arguments

• cosmology: Cosmology object.
• z : Redshift.

Returns

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