NumCosmoHIPertTwoFluids

Perturbation object for a two fluids system.

This object provides the computation of the two fluid system of cosmological perturbations. This problem is described by two fluids with energy density and pressure given respectively by $\bar{\rho}_i$ and $\bar{p}_i$ for $i = 1,2$.

The system is written in terms of the gauge invariant variable $$\zeta \equiv \Psi - \frac{2\bar{K}}{\kappa(\bar{\rho} + \bar{p})} + E\mathcal{V},$$ and the entropy mode $$S = \frac{\kappa\varpi}{x^3 H}(\mathcal{U}_1 - \mathcal{U}_2),$$ where $\mathcal{U}_i \equiv \psi + E\mathcal{V}_i$ and $$\varpi \equiv \frac{(\bar{\rho}_1+\bar{p}_1)(\bar{\rho}_2+\bar{p}_2)}{\bar{\rho}+\bar{p}}.$$

Their momentum are \begin{split} P_\zeta &= \frac{2\bar{D}^2_\bar{K}\Psi}{x^3E}, \ P_S &= \frac{\delta\rho_2}{\bar{\rho}_2+\bar{p}_2} - \frac{\delta\rho_1}{\bar{\rho}_1 + \bar{p}_1}. \end{split}

The equations of motion in their first order form are \begin{align} \zeta^\prime &= \frac{P_\zeta}{m_\zeta} + Y S, \ P_\zeta^\prime &= -m_\zeta\mu_\zeta^2\zeta, \ S^\prime &= \frac{P_S}{m_S} + Y \zeta, \ P_S^\prime &= -m_S\mu_S^2S. \end{align} The mass $m_\zeta$ and the frequency $\mu_\zeta$ are defined by \begin{align} m_\zeta &= \frac{3\Delta_\bar{K}(\bar{\rho} + \bar{p})}{\rho_\text{crit0} N x^3 c_s^2 E^2}, \ \mu_\zeta^2 &= x^2N^2c_s^2k^2, \ m_S &= \frac{x^3}{c_m^2\varpi N}, \ \mu_S^2 &= x^2N^2c_m^2k^2, \ Y &= \frac{c_n^2}{c_s^2c_m^2}\frac{1}{m_\zeta m_S \Delta_\bar{K} N E}. \end{align} where $\bar{\rho} + \bar{p}$ is the background total energy density plus pressure, $E^2 = H^2/H_0^2$ is the dimensionless Hubble function squared (nc_hicosmo_E2()), $c_s^2$ the speed of sound, $N$ is the lapse function that in this case (using $\alpha$ as time variable) is $N \equiv \vert{}E\vert^{-1}$, $\rho_\text{crit0}$ is the critical density today defined by $\rho_\text{crit0} \equiv 3H_0^2/\kappa$ and $$\Delta_\bar{K} \equiv \frac{k^2}{k^2 + \Omega_{k0}}.$$.