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| gboolean | compute-inv-comoving | Read / Write / Construct |
| NcRecomb * | recomb | Read / Write / Construct |
| double | zf | Read / Write / Construct |
This object implements several distances used in cosmology, here we have the following definitions.
$ \newcommand{\RH}{{R_H}} \newcommand{\RHc}{{R^\mathrm{c}_H}} $
The Hubble radius (or scale) is defined as the inverse of the Hubble
function $H(z)$ [nc_hicosmo_H()],
\begin{equation}\label{eq:def:RHc}
\RH = \frac{c}{H(z)}, \qquad \RH_0 = \frac{c}{H_0},
\end{equation}
where $c$ is the speed of light [ncm_c_c()], $z$ is the redshift and
$H_0 \equiv H(0)$ is the Hubble parameter [nc_hicosmo_H0()]. Similarly,
we also define the comoving Hubble radius as
\begin{equation}\label{eq:def:DH}
\RHc(z) = \frac{c}{aH(z)} = \frac{c(1+z)}{a_0H(z)}, \qquad \RHc_0 = \frac{c}{a_0H_0}
\end{equation}
where ${}_0$ subscript means that the function is calculated at the
present time and the redshift $z$ is defined by the expression
$$1 + z = \frac{a_0}{a}.$$
The comoving distance $D_c$ is defined as
\begin{equation}\label{eq:def:dc}
d_c(z) = \RHc_0\int_0^z \frac{dz^\prime}{E (z^\prime)},
\end{equation}
where $E(z)$ is the normalized Hubble function [nc_hicosmo_E()], i.e.,
\begin{equation}\label{eq:def:Ez}
E(z) \equiv \frac{H(z)}{H_0}.
\end{equation}
In this object we will compute the dimensionless version of the distances, for the comoving distance we define \begin{equation}\label{eq:def:Dc} D_c(z) \equiv \frac{d_c(z)}{\RHc_0}. \end{equation} note, however, that $D_c(z)$ coincides with the proper distance today $r(z) \equiv a_0 d_c(z)$ in unit of the Hubble radius, i.e., $D_c(z) = r(z) / \RH_0$. Therefore, both the comoving distance and the proper distance today can be obtained by multiplying $D_c(z)$ by $\RHc_0$ and $\RH_0$ respectively.
The transverse comoving distance $D_t$ and its derivative with respect to
$z$ are given by
\begin{equation}\label{eq:def:Dt}
D_t(z) = \frac{\sinh\left[\sqrt{\Omega_{k0}}D_c(z)\right]}{\sqrt{\Omega_{k0}}},
\qquad \frac{dD_t}{dz}(z) = \frac{\cosh\left[\sqrt{\Omega_{k0}}D_c(z)\right]}{E(z)},
\end{equation}
where $\Omega_{k0}$ is the value of the curvature today [nc_hicosmo_Omega_k0()].
Using the definition above we have that the luminosity and angular diameter distances
are respectively:
\begin{equation}\label{eq:def:Dl}
D_l = (1 + z)D_t(z), \qquad D_A = D_t (z) / (1 + z),
\end{equation}
and the distance modulus is given by
\begin{equation}\label{eq:def:dmu}
\delta\mu(z) = 5\log_{10}(D_l(z)) + 25.
\end{equation}
Note that the distance modulus is defined as
$$\mu(z) = 5\log_{10}[\RH_0D_l(z)/(1\,\text{Mpc})] + 25.$$
Thus, this differs from our definition by a factor of
$5\log_{10}[\RH_0/(1\,\text{Mpc})]$, i.e.,
$$\mu(z) = \delta\mu(z) + 5\log_{10}[\RH_0/(1\,\text{Mpc})],$$
where $\text{Mpc}$ is megaparsec [ncm_c_Mpc()].
We also implement the following distances between $z_1$ and $z_2$: \begin{align}\label{eq:def:Dt12} D_t(z_1, z_2) &= \frac{\sinh\left\{\sqrt{\Omega_{k0}}\left[D_c(z_2)-D_c(z_1)\right]\right\}}{\sqrt{\Omega_{k0}}}, \\ \label{eq:def:DA12} D_A(z_1, z_2) &= D_t (z_1, z_2) / (1 + z_2). \end{align}
NcDistance *
nc_distance_new (gdouble zf);
Creates a new NcDistance object optimized to perform distance calculations
to redshift up to zf
.
NcDistance *
nc_distance_ref (NcDistance *dist);
Increases the reference count of dist
atomically.
void nc_distance_require_zf (NcDistance *dist,const gdouble zf);
Requires the final redshift of at least zf
.
void nc_distance_set_recomb (NcDistance *dist,NcRecomb *recomb);
This function sets recomb
into dist
.
void nc_distance_compute_inv_comoving (NcDistance *dist,gboolean cpu_inv_xi);
Enable/Disable the computation of $z(\xi)$
void nc_distance_prepare (NcDistance *dist,NcHICosmo *cosmo);
This function prepares the object dist
using cosmo
,
such that all the available distances functions can be evaluated,
e.g. nc_distance_comoving().
void nc_distance_prepare_if_needed (NcDistance *dist,NcHICosmo *cosmo);
This function prepares the object dist
using cosmo
if it was changed since last preparation.
void
nc_distance_free (NcDistance *dist);
Atomically decrements the reference count of dist
by one.
If the reference count drops to 0, all memory allocated by dist
is released.
void
nc_distance_clear (NcDistance **dist);
Atomically decrements the reference count of dist
by one.
If the reference count drops to 0, all memory allocated by dist
is released.
Set pointer to NULL.
gdouble nc_distance_hubble (NcDistance *dist,NcHICosmo *cosmo);
Calculate the curvature scale today as defined in Eq $\eqref{eq:def:DH}$ in
units of megaparsec (Mpc) [ncm_c_Mpc()].
gdouble nc_distance_decoupling_redshift (NcDistance *dist,NcHICosmo *cosmo);
The decoupling redshift $z_\star$ corresponds to the epoch of the last scattering surface of the cosmic microwave background photons.
This function computes $z_\star$ using [nc_hicosmo_z_lss()], if cosmo
implements
it, or using Hu & Sugiyama fitting formula Hu (1996)[arXiv],
$$ z_\star = 1048 \left(1 + 1.24 \times 10^{-3} (\Omega_{b0} h^2)^{-0.738}\right) \left(1 + g_1 (\Omega_{m0} h^2)^{g_2}\right),$$
where $\Omega_{b0} h^2$ [nc_hicosmo_Omega_b0h2()] and $\Omega_{m0} h^2$ [nc_hicosmo_Omega_m0h2()]
are, respectively, the baryonic and matter density parameters times the square
of the dimensionless Hubble parameter $h$, $H_0 = 100 \, h \, \text{km/s} \, \text{Mpc}^{-1}$.
The parameters $g_1$ and $g_2$ are given by
$$g_1 = \frac{0.0783 (\Omega_{b0} h^2)^{-0.238}}{(1 + 39.5 (\Omega_{b0} h^2)^{0.763})}
\; \text{and} \; g_2 = \frac{0.56}{\left(1 + 21.1 (\Omega_{b0} h^2)^{1.81}\right)}.$$
gdouble nc_distance_drag_redshift (NcDistance *dist,NcHICosmo *cosmo);
Drag redshift is the epoch at which baryons were released from photons.
If the dist
object constains a NcRecomb object, it calculates the drag
redshift through the recombination history. Otherwise, it computes $z_d$
using the fitting formula given in Eisenstein & Hu (1998) [arXiv],
$$z_d = \frac{1291 (\Omega_{m0} h^2)^{0.251}}{(1 + 0.659 (\Omega_{m0} h^2)^{0.828})}
\left(1 + b_1 (\Omega_{b0} h^2)^{b_2}\right),$$
where $\Omega_{b0} h^2$ [nc_hicosmo_Omega_b0h2()] and $\Omega_{m0} h^2$ [nc_hicosmo_Omega_m0h2()]
are, respectively, the baryonic and matter density parameters times the square
of the dimensionless Hubble parameter $h$, $H_0 = 100 \, h \, \text{km/s} \, \text{Mpc}^{-1}$.
The parameters $b_1$ and $b_2$ are given by
$$b_1 = 0.313 (\Omega_{m0} h^2)^{-0.419} \left(1 + 0.607 (\Omega_{m0} h^2)^{0.674}\right) \;
\text{and} \; b_2 = 0.238 (\Omega_{m0} h^2)^{0.223}.$$
gdouble nc_distance_shift_parameter_lss (NcDistance *dist,NcHICosmo *cosmo);
Compute the shift parameter $R(z)$ [nc_distance_shift_parameter()] at the
decoupling redshift $z_\star$ [nc_distance_decoupling_redshift()].
gdouble nc_distance_comoving_lss (NcDistance *dist,NcHICosmo *cosmo);
Compute the comoving distance $D_c(z)$ [Eq. \eqref{eq:def:Dc}] at the
decoupling redshift $z_\star$ [nc_distance_decoupling_redshift()].
gdouble nc_distance_acoustic_scale (NcDistance *dist,NcHICosmo *cosmo);
Compute the acoustic scale $l_A (z_\star)$ at $z_\star$ [nc_distance_decoupling_redshift()],
\begin{equation*}
l_A(z_\star) = \pi \frac{D_t (z_\star)}{r_s (z_\star)},
\end{equation*}
where $D_t(z_\star)$ is the comoving transverse distance [nc_distance_transverse()]
and $r_s(z_\star)$ is the sound horizon [nc_distance_sound_horizon()] both computed at $z_\star$.
gdouble nc_distance_theta100CMB (NcDistance *dist,NcHICosmo *cosmo);
Compute the $100\theta_\mathrm{CMB}$ angle at $z_\star$ [nc_distance_decoupling_redshift()],
\begin{equation*}
100\theta_\mathrm{CMB} = 100 \times \frac{r_s (z_\star)}{D_t (z_\star)},
\end{equation*}
where $D_t(z_\star)$ is the comoving transverse distance [nc_distance_transverse()]
and $r_s(z_\star)$ is the sound horizon [nc_distance_sound_horizon()] both
both computed at $z_\star$.
gdouble nc_distance_angular_diameter_curvature_scale (NcDistance *dist,NcHICosmo *cosmo);
We define the angular diameter curvature scale $D_a(z_\star)$ as
$$D_a(z_\star) = \frac{E(z_\star)}{1 + z_\star} D_t(z_\star),$$
where $z_\star$ is the decoupling redshift, given by [nc_distance_decoupling_redshift()],
$E(z_\star)$ is the normalized Hubble function [Eq. $\eqref{eq:def:Ez}$] and
$D_t(z_\star)$ is the transverse comoving distance [Eq. $\eqref{eq:def:Dt}$] both computed at $z_\star$.
gdouble nc_distance_r_zd (NcDistance *dist,NcHICosmo *cosmo);
This function computes the sound horizon [nc_distance_sound_horizon()]
at the drag redshift [nc_distance_drag_redshift()].
gdouble nc_distance_r_zd_Mpc (NcDistance *dist,NcHICosmo *cosmo);
Similar as nc_distance_r_zd(), but now in units of $\mathrm{Mpc}$, i.e., $ R_H \times r(z_d)$.
gdouble nc_distance_comoving (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Calculate the comoving distance $D_c (z)$ as defined in Eq. $\eqref{eq:def:Dc}$.
gdouble nc_distance_transverse (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Compute the transverse comoving distance $D_t (z)$ defined in Eq. $\eqref{eq:def:Dt}$.
gdouble nc_distance_dtransverse_dz (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Compute the derivative of $D_t(z)$ with respect to $z$ defined in Eq. $\eqref{eq:def:Dt}$.
gdouble nc_distance_luminosity (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Compute the luminosity distance $D_l(z)$ defined in Eq. $\eqref{eq:def:Dl}$.
gdouble nc_distance_angular_diameter (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Compute the angular diameter $D_A(z)$, defined in Eq. $\eqref{eq:def:Dl}$.
gdouble nc_distance_dmodulus (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Compute the distance modulus $\delta\mu(z)$ defined in Eq. $\eqref{eq:def:dmu}$.
gdouble nc_distance_comoving_volume_element (NcDistance *dist,NcHICosmo *cosmo,gdouble z);
This function computes the comoving volume element per unit solid angle $d\Omega$
given z
, namely, $$\frac{\mathrm{d}^2V}{\mathrm{d}z\mathrm{d}\Omega} = \frac{D_t^2(z)}{E(z)} = \frac{(1 + z)^2 D_a^2(z)}{E(z)},$$
where $E(z)$ is the normalized Hubble function and $D_t$ is the transverse comoving distance (and $D_a$ is the angular diameter distance).
gdouble nc_distance_sigma_critical (NcDistance *dist,NcHICosmo *cosmo,const gdouble zs,const gdouble zl);
Computes the critical surface density,
\begin{equation}\label{eq:def:SigmaC}
\Sigma_c = \frac{c^2}{4\pi G} \frac{D_s}{D_l D_{ls}},
\end{equation}
where $c^2$ is the speed of light squared [ncm_c_c2()], $G$ is the gravitational constant in units of $m^3/s^2 M_\odot^{-1}$ [ncm_c_G_mass_solar()],
$D_s$ ($D_l$) is the angular diameter distance from the observer to the source (lens), and $D_{ls}$ is the angular diameter distance between
the lens and the source.
gdouble nc_distance_sigma_critical_infinity (NcDistance *dist,NcHICosmo *cosmo,const gdouble zl);
Computes the critical surface density,
\begin{equation}\label{eq:def:SigmaC}
\Sigma_c = \frac{c^2}{4\pi G} \frac{D_\infty}{D_l D_{l\infty}},
\end{equation}
where $c^2$ is the speed of light squared [ncm_c_c2()], $G$ is the gravitational constant in units of $m^3/s^2 M_\odot^{-1}$ [ncm_c_G_mass_solar()],
$D_\infty$ ($D_l$) is the angular diameter distance from the observer to the source at infinite redshift (lens), and $D_{l\infty}$ is the angular diameter
the lens and the source.
gdouble nc_distance_luminosity_hef (NcDistance *dist,NcHICosmo *cosmo,const gdouble z_he,const gdouble z_cmb);
Calculate the luminosity distance $D_l$ corrected to our local frame.
gdouble nc_distance_dmodulus_hef (NcDistance *dist,NcHICosmo *cosmo,const gdouble z_he,const gdouble z_cmb);
Calculate the distance modulus [Eq. $\eqref{eq:def:dmu}$] using the frame corrected luminosity
distance [nc_distance_luminosity_hef()].
gdouble nc_distance_shift_parameter (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
The shift parameter $R(z)$ is defined as
\begin{equation*}
R(z) = \frac{\sqrt{\Omega_{m0} H_0^2}}{c} (1 + z) D_A(z) = \sqrt{\Omega_{m0}} D_t(z),
\end{equation*}
where $\Omega_{m0}$ is the matter density paremeter [nc_hicosmo_Omega_m0()],
$D_A(z) = D_{H_0} D_t(z) / (1 + z)$ is the angular diameter distance and
$D_t(z)$ is the tranverse comoving distance [Eq. $\eqref{eq:def:Dt}$].
gdouble nc_distance_dilation_scale (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
The dilation scale is the spherically averaged distance for perturbations along and
orthogonal to the line of sight,
$$D_V(z) = \left[D_{H_0}^2 D_t(z)^2 \frac{cz}{H(z)} \right]^{1/3},$$
where $D_t(z)$ is the transverse comoving distance [Eq. $\eqref{eq:def:Dt}$], $c$ is the speed of light
[ncm_c_c()] and $H(z)$ is the Hubble function [nc_hicosmo_H()].
See Eisenstein et al. (2005) [arXiv].
This function computes the dimensionless dilation scale:
$$D_V^\star(z) = \left[D_t(z)^2 \frac{z}{E(z)} \right]^{1/3} = \frac{D_V(z)}{D_{H_0}},$$
where $E(z)$ is the normalized Hubble function [nc_hicosmo_E2()].
gdouble nc_distance_bao_A_scale (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
This function returns the BAO $A$ parameter.
It is defined as
$$ A \equiv D_V (z) \frac{\sqrt{\Omega_{m0} H_0^2}}{z c},$$
where $\Omega_{m0}$ is the matter density parameter [nc_hicosmo_Omega_m0()], $c$ is the speed of light [ncm_c_c()],
$H_0$ is the Hubble parameter [nc_hicosmo_H0()] and $D_V(z)$ is the dilation scale [nc_distance_dilation_scale()].
See Section 4.5 from Eisenstein et al. (2005) [arXiv].
gdouble nc_distance_sound_horizon (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Compute the sound horizon $r_s$,
\begin{equation}
\theta_s (z) = \int_{z}^\infty \frac{c_s(z^\prime)}{E(z^\prime)} dz^\prime, \quad r_s (z) = \frac{\mathrm{sinn}\left(\sqrt{\Omega_{k0}}\theta_s\right)}{\sqrt{\Omega_{k0}}},
\end{equation}
where $c_s = \sqrt{c^{b\gamma 2}_s(z)}$ is the baryon-photon plasma speed of sound
(see nc_hicosmo_bgp_cs2() for more details)
and $E(z)$ is the normalized Hubble function [nc_hicosmo_E()].
gdouble nc_distance_dsound_horizon_dz (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Calculate the sound horizon [nc_distance_sound_horizon()] derivative with respect to $z$,
$$\frac{d r_s(z)}{dz} = - \frac{c_s(z)}{E(z)}\, ,$$
where $c_s = \sqrt{c^{b\gamma 2}_s(z)}$ is the baryon-photon plasma speed of sound
(see nc_hicosmo_bgp_cs2() for more details)
and $E(z)$ is the normalized Hubble function [nc_hicosmo_E()].
gdouble nc_distance_bao_r_Dv (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
This function computes $r(z_d) / D_V(z)$,
where $r(z_d)$ is given by nc_distance_r_zd() and $D_V(z)$ by nc_distance_dilation_scale().
For more information see Percival et al. (2007) [arXiv].
gdouble nc_distance_DH_r (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Computes the ratio between the Hubble (distance) radius and the sound horizon at the drag epoch, $$\frac{R_H}{r_s(z_d)} = \frac{c}{H(z) r_d}.$$
gdouble nc_distance_DA_r (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Computes the ratio between the angular-diameter distance and the sound horizon at the drag epoch, $$\frac{D_A(z)}{c \, r_s(z_d)}.$$
gdouble nc_distance_Dt_r (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Computes the ratio between the transverse distance and the sound horizon at the drag epoch, $$\frac{D_t(z)}{c \, r_s(z_d)}.$$
gdouble nc_distance_comoving_z_to_infinity (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Computes the comoving distance from z
to infinity.
gdouble nc_distance_transverse_z_to_infinity (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
Compute the transverse comoving distance $D_t (z)$ (defined in Eq. $\eqref{eq:def:Dt}$)
but from z
to infinity.
gdouble nc_distance_transverse_z1_z2 (NcDistance *dist,NcHICosmo *cosmo,const gdouble z1,const gdouble z2);
Compute the transverse comoving distance between $z_1$ and $z_2$, $D_t (z1, z2)$ defined in Eq. $\eqref{eq:def:Dt12}$.
gdouble nc_distance_angular_diameter_z1_z2 (NcDistance *dist,NcHICosmo *cosmo,const gdouble z1,const gdouble z2);
Compute the angular diameter $D_A(z_1, z_2)$, defined in Eq. $\eqref{eq:def:DA12}$.
gdouble nc_distance_inv_comoving (NcDistance *dist,NcHICosmo *cosmo,gdouble xi);
Computes the inverse of $\xi(z)$.
gdouble nc_distance_cosmic_time (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
This function computes the cosmological time, $t(z)$, defined as \begin{equation*} t(z) = \int_z^{\infty} \frac{dx}{(1+x)E(x)}. \end{equation*}
gdouble nc_distance_lookback_time (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
This functions computes the look-back time, $t_{lb}(z)$, defined as
\begin{equation*} t_{lb}(z) = \int_0^{z} \frac{dx}{(1+x)E(x)}. \end{equation*}
gdouble nc_distance_conformal_time (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
This function computes the cosmological conformal time $\eta(z)$ defined as $$ \eta(z) = \int_z^\infty \frac{dz}{E(z)} \, . $$
gdouble nc_distance_conformal_lookback_time (NcDistance *dist,NcHICosmo *cosmo,const gdouble z);
This function computes the conformal look-back time $\eta_{lb}(z)$.
Within the chosen units it becomes the same as the comoving distance [nc_distance_comoving()],
given in Eq. $\eqref{eq:def:dc}$.
Enumeration to define which method to be applied in order to compute the cosmological distances.
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performs a numerical evaluation. |
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uses the method defined by the implementation of NcHICosmo. |
“compute-inv-comoving” property “compute-inv-comoving” gboolean
Whether to compute the inverse comoving function.
Owner: NcDistance
Flags: Read / Write / Construct
Default value: FALSE
“recomb” property“recomb” NcRecomb *
The recombination object, NcRecomb, used by NcDistance.
Owner: NcDistance
Flags: Read / Write / Construct