Stability Level

Functions

Properties

Object Hierarchy

    GObject
    ╰── NcTransferFunc
        ╰── NcTransferFuncEH

Includes

#include <numcosmo/lss/nc_transfer_func_eh.h>

Description

This objects implements the Eisenstein-Hu fitting function for the transfer function. See Eisenstein and Hu (1998) [arXiv] for more details.

The transfer function is divided into a sum of two picies, \begin{equation*} T(k) = \frac{\Omega_b}{\Omega_m} T_b(k) + \frac{\Omega_{c}}{\Omega_m} T_c(k) \, , \end{equation*} where $\Omega_b$, $\Omega_m$ and $\Omega_{c}$ are the baryons (nc_hicosmo_Omega_b0), matter (nc_hicosmo_Omega_m0) and cold dark matter (nc_hicosmo_Omega_c0) density parameters today, respectevely. $T_b(k)$ and $T_c(k)$ are the weights from baryons and cold dark matter to the transfer function $T(k)$.

The cold dark matter term is defined as, \begin{equation*} T_c(k) = f \, \widetilde{T}_0(k, 1, \beta_c) + (1-f) \, \widetilde{T}_0(k, \alpha_c, \beta_c) \end{equation*} and $$ f = \frac{1}{1+ (ks/5.4)^4} \, ,$$ with $$\widetilde{T}_0(k, \alpha_c, \beta_c) = \frac{\ln\left( e + 1.8 \beta_c \, q \right)}{\ln\left( e + 1.8 \beta_c \, q \right) + C q^2} $$ and $$ C = \frac{14.2}{\alpha_c} + \frac{386}{1+ 69.9 \, q^{1.08}} \, .$$ The paremeter $q$, is defined as, $$ q = \frac{k}{13.41 k_{eq}} \, .$$ $\alpha_c$ and $\beta_c$ are fit by, $$\alpha_c = a_1^{-\Omega_b/\Omega_m} \, a_2^{-(\Omega_b/\Omega_m)^3} \, ,$$ $$ a_1 = (46.9 \, \Omega_m h^2)^{0.670} \left[ 1 + (32.1 \, \Omega_m h^2)^{-0.532} \right] \, ,$$ $$ a_2 = (12.0 \, \Omega_m h^2)^{0.424} \left[ 1 + (45.0 \, \Omega_m h^2)^{-0.582} \right] \, ,$$ $$ \beta_c^{-1} = 1 + b_1 \left[ (\Omega_c/\Omega_m)^{b_{2}} -1\right] \, ,$$ $$ b_1 = 0.944 \left[ 1+ (458\, \Omega_m h^2)^{-0.708} \right]^{-1} \, , $$ $$ b_2 = (0.395 \, \Omega_m h^2)^{-0.0266} \, .$$

The baryon term is defined as, \begin{equation*} T_b(k) = \left[ \frac{\widetilde{T}_0(k, 1, 1)}{1 + (ks/5.2)^2} + \frac{\alpha_b}{1+ (\beta_b/ks)^3}\mathrm{e}^{-(k/k_{\mathrm{Silk}})^{1.4}} \right] \, \frac{\sin (k \tilde{s})}{k \tilde{s}} \, . \end{equation*} where, $s$ is the sound horizon scale, given by, $$ s = \int_0^{z_d} c_s (1+z) \mathrm{d}t = \frac{2}{3k_{eq}}\sqrt{\frac{6}{R(z_{eq})}} \ln \left( \frac{\sqrt{1+R(z_d)} + \sqrt{R(z_{d}) + R(z_{eq})}}{1 + \sqrt{R(z_{eq})}} \right) \, ,$$ with $z_d$ as the drag redshift (nc_distance_drag_redshift) and $c_s$ is the baryon-photon plasma speed of sound (see nc_hicosmo_bgp_cs2). The quantity $R$ is defined as, $$ R(z) \equiv 31.5 \, \Omega_b h^2 \, \left( \frac{T_{\mathrm{CMB}}}{2.7} \right)^{-4} \left( \frac{z}{10^3} \right)^{-1} \, .$$ The Silk damping scale is well fit by the approximation, $$ k_{Silk} = 1.6 (\Omega_b h^2)^{0.52}(\Omega_m h^2)^{0.73} \left[ 1 + (10.4 \, \Omega_m h^2)^{-0.95} \right] \, \mathrm{Mpc}^{-1} \, ,$$ and $$ \alpha_b = 2.07 \, k_{eq} \, s (1+ R_d )^{-3/4} G \left( \frac{1+ z_{eq}}{1+z_d} \right) \, ,$$ $$ G(y) = y \left[-6 \sqrt{1+y} + (2+3y) \ln \left( \frac{\sqrt{1+y} + 1}{\sqrt{1+y} - 1} \right) \right] \, ,$$ $$ k_{eq} = (2 \Omega_m H_0^2 z_{eq})^{1/2} \, ,$$ $$ z_{eq} = 2.5 \times 10^4 \Omega_m h^2 \, (T_{\mathrm{CMB}} / 2.7)^{-4} \, ,$$ $$ R(z_d) = 31.5 \, \Omega_b h^2 (T_{cmb} / 2.7)^{-4} (z_d/10^3)^{-1} \, ,$$ $$ \tilde{s}(k) = \frac{s}{\left[ 1 + (\beta_{node}/ks)^3 \right]^{1/3}} \, ,$$ $$\beta_{node} = 8.41 \, (\Omega_m h^2)^{0.435} \, , $$ an finally the last parameter of the fitting function, $$\beta_b = 0.5 + \frac{\Omega_b}{\Omega_m} + \left( 3 -2 \frac{\Omega_b}{\Omega_m} \right) \sqrt{(17.2 \, \Omega_m h^2)^2 +1} \, .$$

With those relations in hand it is possible to evaluate the transfer function from Eisenstein and Hu (1998) fitting formula.

Functions

NcTransferFunc *
nc_transfer_func_eh_new (void);

void
nc_transfer_func_eh_set_CCL_comp (NcTransferFuncEH *tf_eh,
                                  gboolean CCL_comp);

Property Details

  “CCL-comp”                 gboolean