NumCosmoHIPertAdiab

Perturbation object for adiabatic mode only.

This object provides the computation of the adiabatic mode for the cosmological perturbations. It solves the equation of motion for the gauge invariant variable (see [Vitenti (2013)][XVitenti2013] for notation and details) $$ \zeta \equiv \Psi - \frac{2\bar{K}}{\kappa(\bar{\rho} + \bar{p})} + H\mathcal{V}. $$ Its conjugated momentum is give by \begin{split} P_\zeta &= \frac{2\bar{D}^2_\bar{K}\Psi}{x^3H}, \end{split}

The equations of motion in their first order form are \begin{align} \zeta^\prime &= \frac{P_\zeta}{m_\zeta}, \ P_\zeta^\prime &= -m_\zeta\mu_\zeta^2\zeta. \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, \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 $$ \Delta_\bar{K} \equiv \frac{k^2}{k^2 + \Omega_{k0}}. $$.