Eisenstein–Hu Transfer Function

Eisenstein–Hu Transfer Function

This page describes the Eisenstein & Hu (1998) fitting function for the matter transfer function implemented by NcTransferFuncEH. The full fit, including baryon acoustic features, is reproduced here; the zero-baryon (“no-wiggle”) variant is implemented separately by NcTransferFuncEHNoBaryon.

Reference: Eisenstein & Hu (1998).

Decomposition

The transfer function is split into baryon and cold-dark-matter contributions, \[ T(k) = \frac{\Omega_b}{\Omega_m} T_b(k) + \frac{\Omega_c}{\Omega_m} T_c(k), \] where \(\Omega_b\), \(\Omega_m\), and \(\Omega_c\) are the baryon, total matter, and cold-dark-matter density parameters today, and \(T_b(k)\) and \(T_c(k)\) are the baryon and CDM weights.

Cold Dark Matter Term

\[ T_c(k) = f\,\widetilde{T}_0(k, 1, \beta_c) + (1-f)\,\widetilde{T}_0(k, \alpha_c, \beta_c), \] with \[ f = \frac{1}{1 + (ks/5.4)^4}, \] \[ \widetilde{T}_0(k, \alpha_c, \beta_c) = \frac{\ln\left(e + 1.8\,\beta_c\,q\right)}{\ln\left(e + 1.8\,\beta_c\,q\right) + C q^2}, \qquad C = \frac{14.2}{\alpha_c} + \frac{386}{1 + 69.9\,q^{1.08}}, \] and the dimensionless wavenumber \[ q = \frac{k}{13.41\,k_{\rm eq}}. \] The shape parameters \(\alpha_c\) and \(\beta_c\) are fit by \[ \alpha_c = a_1^{-\Omega_b/\Omega_m}\,a_2^{-(\Omega_b/\Omega_m)^3}, \] \[ a_1 = (46.9\,\Omega_m h^2)^{0.670}\left[1 + (32.1\,\Omega_m h^2)^{-0.532}\right], \] \[ a_2 = (12.0\,\Omega_m h^2)^{0.424}\left[1 + (45.0\,\Omega_m h^2)^{-0.582}\right], \] \[ \beta_c^{-1} = 1 + b_1\left[(\Omega_c/\Omega_m)^{b_2} - 1\right], \] \[ b_1 = 0.944\left[1 + (458\,\Omega_m h^2)^{-0.708}\right]^{-1}, \qquad b_2 = (0.395\,\Omega_m h^2)^{-0.0266}. \]

Baryon Term

\[ T_b(k) = \left[\frac{\widetilde{T}_0(k, 1, 1)}{1 + (ks/5.2)^2} + \frac{\alpha_b}{1 + (\beta_b/ks)^3}\,\mathrm{e}^{-(k/k_{\rm Silk})^{1.4}}\right]\frac{\sin(k\tilde{s})}{k\tilde{s}}. \]

The sound horizon scale is \[ s = \int_0^{z_d} c_s (1+z)\,\mathrm{d}t = \frac{2}{3k_{\rm eq}}\sqrt{\frac{6}{R(z_{\rm eq})}}\,\ln\left(\frac{\sqrt{1+R(z_d)} + \sqrt{R(z_d) + R(z_{\rm eq})}}{1 + \sqrt{R(z_{\rm eq})}}\right), \] where \(z_d\) is the drag redshift, \(c_s\) is the baryon-photon plasma speed of sound, and \[ R(z) \equiv 31.5\,\Omega_b h^2\left(\frac{T_{\rm CMB}}{2.7}\right)^{-4}\left(\frac{z}{10^3}\right)^{-1}. \]

The Silk damping scale is approximated by \[ k_{\rm Silk} = 1.6\,(\Omega_b h^2)^{0.52}(\Omega_m h^2)^{0.73}\left[1 + (10.4\,\Omega_m h^2)^{-0.95}\right]\,\mathrm{Mpc}^{-1}, \] and the remaining baryon parameters are \[ \alpha_b = 2.07\,k_{\rm eq}\,s\,(1 + R_d)^{-3/4}\,G\!\left(\frac{1 + z_{\rm eq}}{1 + z_d}\right), \] \[ G(y) = y\left[-6\sqrt{1+y} + (2+3y)\ln\left(\frac{\sqrt{1+y} + 1}{\sqrt{1+y} - 1}\right)\right], \] \[ \tilde{s}(k) = \frac{s}{\left[1 + (\beta_{\rm node}/ks)^3\right]^{1/3}}, \qquad \beta_{\rm node} = 8.41\,(\Omega_m h^2)^{0.435}, \] \[ \beta_b = 0.5 + \frac{\Omega_b}{\Omega_m} + \left(3 - 2\frac{\Omega_b}{\Omega_m}\right)\sqrt{(17.2\,\Omega_m h^2)^2 + 1}. \]

Background Scales

\[ k_{\rm eq} = (2\,\Omega_m H_0^2\,z_{\rm eq})^{1/2}, \qquad z_{\rm eq} = 2.5\times10^4\,\Omega_m h^2\left(\frac{T_{\rm CMB}}{2.7}\right)^{-4}, \] \[ R(z_d) = 31.5\,\Omega_b h^2\left(\frac{T_{\rm CMB}}{2.7}\right)^{-4}\left(\frac{z_d}{10^3}\right)^{-1}. \]

Zero-Baryon (No-Wiggle) Variant

NcTransferFuncEHNoBaryon implements the Eisenstein & Hu (1998) zero-baryon (“no-wiggle”) form, which smooths over the baryon acoustic oscillations while retaining the broadband shape. Writing \(\omega_b = \Omega_b h^2\), \(\omega_m = \Omega_m h^2\), and \(\theta = T_{\rm CMB}/2.7\), the transfer function is \[ T_0(k) = \frac{L_0}{L_0 + C_0\,q^2}, \qquad L_0 = \ln(2e + 1.8\,q), \qquad C_0 = 14.2 + \frac{731}{1 + 62.5\,q}. \] The dimensionless wavenumber uses an effective shape parameter \(\Gamma_{\rm eff}\) rather than a sharp sound-horizon feature, \[ q = \frac{k\,\theta^2}{\Gamma_{\rm eff}}, \qquad \Gamma_{\rm eff} = \omega_m\left[\alpha_\Gamma + \frac{1 - \alpha_\Gamma}{1 + (0.43\,k s)^4}\right], \] with the approximate sound horizon and shape suppression \[ s = \frac{44.5\,\ln(9.83/\omega_m)}{\sqrt{1 + 10\,\omega_b^{3/4}}}\ \mathrm{Mpc}, \] \[ \alpha_\Gamma = 1 - 0.328\,\ln(431\,\omega_m)\,\frac{\omega_b}{\omega_m} + 0.38\,\ln(22.3\,\omega_m)\left(\frac{\omega_b}{\omega_m}\right)^2. \] Here \(k\) is in \(\mathrm{Mpc}^{-1}\).

API Reference

See NcTransferFuncEH for the full class reference, and NcTransferFuncEHNoBaryon for the zero-baryon variant. The cosmology enters through the NcHICosmo density parameters nc_hicosmo_Omega_b0, nc_hicosmo_Omega_m0, and nc_hicosmo_Omega_c0, the drag redshift nc_distance_drag_redshift, and the plasma sound speed nc_hicosmo_bgp_cs2.