Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges

TL;DR

This work presents a complete linear-theory framework for scalar cosmological perturbations in the synchronous and conformal Newtonian gauges, incorporating CDM, baryons, photons, massless neutrinos, and massive neutrinos with a detailed phase-space treatment. By solving the coupled Einstein–Boltzmann–fluid equations in both gauges and implementing accurate initial conditions on super-horizon scales, it provides high-precision predictions for matter transfer and CMB anisotropies, including polarization and helium recombination, for CDM and CDM+HDM models with varying $\Omega_\nu$. The authors deliver first-rate angular power spectra for CDM+HDM models, demonstrate gauge-consistency and the physical interpretation of super-horizon behavior, and supply publicly available code for broader use. This work significantly improves the precision and reliability of CMB and large-scale-structure predictions in models with massive neutrinos and complex matter components.

Abstract

This paper presents a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. It differs from others in the literature in that we give, in both gauges, a complete discussion of all particle species that are relevant to any flat cold dark matter (CDM), hot dark matter (HDM), or CDM+HDM models (including a possible cosmological constant). The particles considered include CDM, baryons, photons, massless neutrinos, and massive neutrinos (an HDM candidate), where the CDM and baryons are treated as fluids while a detailed phase-space description is given to the photons and neutrinos. Particular care is applied to the massive neutrino component, which has been either ignored or approximated crudely in previous works. Isentropic initial conditions on super-horizon scales are derived. The coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric and density perturbations are then solved numerically in both gauges for the standard CDM model and two CDM+HDM models with neutrino mass densities $\onu=0.2$ and 0.3, assuming a scale-invariant, adiabatic spectrum of primordial fluctuations. We also give the full details of the cosmic microwave background anisotropy, and present the first accurate calculations of the angular power spectra in the two CDM+HDM models including photon polarization, higher neutrino multipole moments, and helium recombination. The numerical programs for both gauges are available at http://arcturus.mit.edu/cosmics/ .

Figures (6)

• Figure 1: Top: The angular power spectrum $C_l$ of the photon anisotropy (including polarization) in the CDM (solid), the $\Omega_\nu=0.2$ CDM+HDM (dashed), and the $\Omega_\nu=0.3$ CDM+HDM (dotted) models. Middle: The ratio of $C_l$ for the CDM+HDM models relative to the CDM (dashed for $\Omega_\nu=0.2$; dotted for $\Omega_\nu=0.3$). Bottom: The fractional difference in $C_l$ with and without polarization for the three models.
• Figure 2: The scalar metric perturbations $\phi(k,\tau)$ and $\psi(k,\tau)$ in the conformal Newtonian gauge in the $\Omega_\nu=0.2$ CDM+HDM model. The 31 curves from left to right correspond to 31 values of $k$ between 10.0 Mpc$^{-1}$ and 0.01 Mpc$^{-1}$. The labels $\tau_{\rm nr}$, $\tau_{\rm eq}$ and $\tau_{\rm rec}$ indicate, respectively, the time the 4.7 eV neutrinos become non-relativistic, the matter-radiation equality time, and the recombination time.
• Figure 3: Evolution of the density fields in the synchronous gauge (top panels) and the conformal Newtonian gauge (bottom panels) in the $\Omega_\nu=0.2$ CDM+HDM model for 3 wavenumbers $k$= 0.01 (Fig. \ref{['fig:del']}a), 0.1 (Fig. \ref{['fig:del']}b) and 1.0 (Fig. \ref{['fig:del']}c) Mpc$^{-1}$. In each figure, the five lines represent $\delta_c, \delta_b, \delta_\gamma, \delta_\nu$ and $\delta_h$ for the CDM (solid), baryon (dash-dotted), photon (long-dashed), massless neutrino (dotted), and massive neutrino (short-dashed) components, respectively.
• Figure 4: The $z=0$ power spectra for the pure CDM (solid), the $\Omega_\nu=0.2$ CDM+HDM (dashed), and the $\Omega_\nu=0.3$ CDM+HDM (dotted) models. In the mixed models, the upper curve is for the CDM component and the lower one is for the HDM. The COBE-compatible quadrupole moment of $Q_{\rm rms-PS}=17.6\,\mu\,K$ is used.
• Figure :