Gauge-ready formulation of the cosmological kinetic theory in generalized gravity theories

TL;DR

This work develops a gauge-ready cosmological perturbation formalism applicable to generalized gravity theories with multiple scalar fields, interacting fluids, and kinetic (Boltzmann) components, unifying the treatment of scalar, vector, and tensor perturbations. It introduces a flexible gauge strategy that keeps the temporal gauge unfixed, enabling straightforward switching between gauges and cross-checks of numerical results, while incorporating conformal transformations and effective-fluid quantities. The authors formulate the relativistic Boltzmann equations for massless and massive particles, including photons with polarization, within this generalized gravity setting, and implement a four-gauge numerical scheme to compute CMB temperature and polarization spectra. The framework is demonstrated by computing CMB anisotropies and providing insights into early-universe dynamics, gravitational waves, and the role of kinetic species, with potential to constrain a wide class of gravity theories and their perturbative signatures.

Abstract

We present cosmological perturbations of kinetic components based on relativistic Boltzmann equations in the context of generalized gravity theories. Our general theory considers an arbitrary number of scalar fields generally coupled with the gravity, an arbitrary number of mutually interacting hydrodynamic fluids, and components described by the relativistic Boltzmann equations like massive/massless collisionless particles and the photon with the accompanying polarizations. We also include direct interactions among fluids and fields. The background FLRW model includes the general spatial curvature and the cosmological constant. We consider three different types of perturbations, and all the scalar-type perturbation equations are arranged in a gauge-ready form so that one can implement easily the convenient gauge conditions depending on the situation. In the numerical calculation of the Boltzmann equations we have implemented four different gauge conditions in a gauge-ready manner where two of them are new. By comparing solutions solved separately in different gauge conditions we can naturally check the numerical accuracy.

Figures (9)

• Figure 1: We present the evolutions of the adiabatic density perturbations in the corresponding comoving gauges $\delta_{(i)v_{(i)}}$ for several components. $(i)$ includes the baryon (dot, black), CDM (dot-long dash, blue), photon (long dash, red), massless neutrino (short-dash, green), and massive neutrino (solid, cyan). The two figures are (a) $k/a_0=0.1 Mpc^{-1}$ and (b) $k/a_0=0.01 Mpc^{-1}$. Also presented is the $\varphi_v$ (dot-short dash, yellow) where $v$ is the collective fluid velocity. The parameters are: $h = 0.65$, $\Omega_{b0} = 0.06$, $\Omega_{C0} = 0.5$, $\Omega_{\gamma 0} = 5.85\times 10^{-5}$, $\Omega_{\nu 0} = 3.99\times 10^{-5}$, and $\Omega_{\nu_m 0}= 0.44$. We consider a flat background with vanishing $\Lambda$. The absolute value of the vertical scale is arbitrary.
• Figure 2: We present the evolutions of $\varphi$ in various comoving gauge conditions based on fixing the velocity variables of the components, i.e., $\varphi_{v_{(i)}}$, and $\varphi_v$: the baryon $\varphi_{v_{(b)}}$ (dot, black), the CDM $\varphi_{v_{(C)}}$ (dot-long dash, blue), the photon $\varphi_{v_{(\gamma)}}$ (long dash, red), the massless neutrino $\varphi_{v_{(\nu)}}$ (short dash, green), and the one based on the collective velocity $\varphi_{v}$ (dot-short dash, yellow). We consider three different scales: $k/a_0=0.001$ (top), $0.1$ and $0.2 Mpc^{-1}$ (bottom). For $k/a_0=0.001 Mpc^{-1}$, the baryon, CDM, and the collective one are overlapped (top), and the photon and massless neutrino are overlapped (bottom). In order to present the behaviors in three scales in one frame, we change the absolute scale of the amplitude arbitrarily. The parameters are: $\Omega_{C0} = 0.94$, $\Omega_{\nu_m 0} = 0$, and the other parameters are the same as in Fig. \ref{['fig:density1']}.
• Figure 3: The evolution of $\varphi_v$ for different scales: $k/a_0= 0.0001$ (top), $0.001$, $0.01$, $0.2$, and $0.5 Mpc^{-1}$ (bottom). The cases of $k/a_0=0.0001$ and $0.001 Mpc^{-1}$ are almost overlapped. The parameters are the same as in Figure \ref{['fig:varphi1']}.
• Figure 4: We present the evolution of density perturbation in the comoving gauge of the massive neutrino, $\delta_{(\nu_m) v_{(\nu_m)}}$, for different scales: $k/a_0 = 0.01$ (top), $0.1$, $0.2$, $0.3$ and $0.4 Mpc^{-1}$ (bottom). Figure (a) considers the massive neutrino dominated model with a parameter $\Omega_{\nu_m 0}=0.94$. Figure (b) considers substantial amount of the CDM with parameters $\Omega_{C0} = 0.5$ and $\Omega_{\nu_m 0}= 0.44$. In Figure (b) we show the evolution of CDM $\delta_{(C)v_{(C)}}$ (dotted, red) as well. The other parameters are the same as in Fig. \ref{['fig:varphi1']}.
• Figure 5: We present the power spectra $\ell ( \ell + 1) C_\ell$ of the scalar-type perturbation: the temperature $C_\ell^{\Theta \Theta}$ (top, black), the polarization $C_\ell^{EE}$ (middle, green), and the cross correlation $C_\ell^{\Theta E}$ (bottom, blue). We take the adiabatic initial condition with the scale-invariant ($n_S = 1$) spectrum. Figure (a) shows the spectra in logarithmic scale, and (b) shows in real scale. We normalize the spectra using $\ell ( \ell + 1) C_\ell^{\Theta \Theta} = 1$ for $\ell =10$. The parameters are: $\Omega_{C0}=0.25$, $\Omega_{\Lambda 0} = 0.69$, $\Omega_{\nu_m 0}=0.$, and the other parameters are the same as in Fig. \ref{['fig:density1']}.