Abstract class for density profile functions.

This abstract class describes the radial matter density profile in real space. Each implementation must provide, at least, the dimensionless 3D density: \begin{equation}\label{def:dlrho} \hat\rho(x) \equiv \frac{\rho(x r_s)}{\rho_s}, \quad \rho(r) = \rho_s \hat\rho\left(\frac{r}{r_s}\right), \end{equation} where $\rho(r)$ is the actual density profile, $\rho_s$ is the profile scale and $r_s$ the scale radius. This function corresponds to the virtual function nc_halo_density_profile_eval_dl_density().

Parametrization

The two parameters $\rho_s$ and $r_s$ are described by the fundamental parametrization in terms of $M_\Delta$ and the concentration $c_\Delta$ given a mass defined by $\Delta$ and a background density $\rho_\mathrm{bg}$. These quantities are provided by the halo mass summary object, which is passed to the profile object at construction time.

Computing $r_s$

The mass-radius relation defined in terms of the background density is \begin{equation}\label{eq:mrr} M_\Delta = \frac{4\pi}{3}r_\Delta^3\Delta\,\rho_\mathrm{bg}, \end{equation} which implicitly defines $r_\Delta$. The concentration $c_\Delta$ is then defined by \begin{equation}\label{def:cDelta} c_\Delta \equiv \frac{r_\Delta}{r_s}. \end{equation} Consequently, the scale radius $r_s$ can be computed from $M_\Delta$ and $c_\Delta$ using \begin{equation}\label{def:r_s} r_s = \frac{1}{c_\Delta}\left(\frac{3M_\Delta}{4\pi\Delta\,\rho_\mathrm{bg}}\right)^{1/3} = \frac{r_{s0}}{(\Delta\,\rho_\mathrm{bg})^{1/3}}, \qquad r_{s0} \equiv \frac{1}{c_\Delta}\left(\frac{3M_\Delta}{4\pi}\right)^{1/3}. \end{equation} We split the expression of $r_s$ in a constant part $r_{s0}$ and a redshift dependent (time-dependent) part $(\Delta\,\rho_\mathrm{bg}(z))^{-1/3}$.

Note that, the parameter $r_s$ can be computed directly from $(M_\Delta,\; c_\Delta)$, given the mass definition, without referring to $\hat\rho(x)$.

Computing $\rho_s$

Now, applying the mass definition $M_\Delta$ in terms of the radius $r_\Delta$ to our profile results in \begin{equation}\label{eq:def:Mr} M_\Delta = \int_0^{r_\Delta}4\pi r^2\rho(r)\mathrm{d}r = 4\pi r_s^3 \rho_s \int_0^{c_\Delta}x^2\hat\rho(x)\mathrm{d}x = 4\pi r_s^3 \rho_s I_{x^2\hat\rho}(c_\Delta), \end{equation} where we defined \begin{equation}\label{def:Ix2_dld} I_{x^2\hat\rho}(c_\Delta) \equiv \int_0^{c_\Delta}x^2\hat\rho(x)\mathrm{d}x. \end{equation} This integral can be implemented through the virtual method nc_halo_density_profile_eval_dl_spher_mass(), otherwise it will be computed numerically using nc_halo_density_profile_eval_dl_density(). This same mass can be obtained from the background density using mass-radius relation \eqref{eq:mrr}, consequently \begin{equation}\label{def:rho_s} \rho_s = \frac{c_\Delta^3\Delta\,\rho_\mathrm{bg}}{3I_{x^2\hat\rho}(c_\Delta)}. \end{equation} The only redshift dependency (time-dependency) here comes from the value of $\rho_\mathrm{bg}(z)$, for this reason it is convenient to define a constant quantity \begin{equation}\label{def:rho_s0} \rho_{s0} \equiv \frac{\rho_s}{\Delta\,\rho_\mathrm{bg}} = \frac{c_\Delta^3}{3I_{x^2\hat\rho}(c_\Delta)}. \end{equation}

2D projection

The surface density obtained from the projection of the density profile along the line-of-sight is given by \begin{align} \Sigma(R) &= \int_{-\infty}^\infty\rho(\sqrt{R^2+z^2})\mathrm{d}z, \ &= 2\rho_s\int_{0}^\infty\hat\rho(\sqrt{R^2/r_s^2 + z^2/r_s^2})\mathrm{d}z, \ \label{eq:def:hatSigma} &= r_s\rho_s\hat\Sigma(R / r_s), & \hat\Sigma(X) &\equiv 2\int_{0}^\infty\hat\rho(\sqrt{X^2 + u^2})\mathrm{d}u. \end{align} In the equation above we obtain the 2D projection $\Sigma(R)$ in terms of its dimensionless version $\hat\Sigma(X)$, where $X = R / r_s$. The user can implement the method nc_halo_density_profile_eval_dl_2d_density() providing $\hat\Sigma(X)$ directly or rely on the numerical implementation.

Mass on the cylinder of radius $R$

Using the 2D projection $\Sigma(R)$ one computes the total mass inside an infinite cylinder of radius $R$ using \begin{align} \overline{M}(R) &= \int_0^R\Sigma(R’)2\pi R’\mathrm{d}R’ = 2\pi r_s^3\rho_s \hat{\overline{M}}(<R/r_s), \ \label{eq:def:cylmass} \hat{\overline{M}}(X) &\equiv \int_0^X\hat\Sigma(X’)X’\mathrm{d}X’. \end{align} Here it is possible to implement the function $\hat{\overline{M}}(X)$ through the method nc_halo_density_profile_eval_dl_cyl_mass() or to use the default numerical implementation.

Numerical computation

If the implementation (i.e., a particular radial profile implementation of this abstract class) does not provide any of the functions: nc_halo_density_profile_eval_dl_spher_mass(), nc_halo_density_profile_eval_dl_2d_density(), nc_halo_density_profile_eval_dl_cyl_mass(), they will be computed numerically integrating the density $\hat{\rho}$ (nc_halo_density_profile_eval_dl_density()). These functions will be prepared to be computed inside the interval $(X_i,\,X_f)$ defined by NcHaloDensityProfile:lnXi and NcHaloDensityProfile:lnXf and using the relative tolerance NcHaloDensityProfile:reltol. See the following functions to control this behavior: nc_halo_density_profile_set_reltol(),

nc_halo_density_profile_set_lnXi(),

nc_halo_density_profile_set_lnXf().

Units

Distance: $[r] = \mathrm{Mpc}$; Mass: $[M_\Delta] = \mathrm{M}\odot$; Density: $[\rho] = \mathrm{M}\odot \, \mathrm{Mpc}^{-3}$; Surface mass density: $[\Sigma] = \mathrm{M}_\odot \, \mathrm{Mpc}^{-2}$.