Cosmological Perturbation Theory in the Synchronous vs. Conformal Newtonian Gauge

TL;DR

The paper develops a complete linear framework for scalar cosmological perturbations in both the synchronous and conformal Newtonian gauges, deriving a precise gauge transformation and the coupled Einstein–Boltzmann–fluid equations for CDM, baryons, photons, and neutrinos in an $oxed{Ω=1}$ universe. It demarcates how isentropic perturbations evolve outside the horizon in the two gauges, provides analytic initial conditions, and demonstrates numerical results for a CDM+HDM model to illustrate horizon-entry and free-streaming effects. The conformal Newtonian gauge is highlighted as particularly convenient for avoiding gauge artifacts and for implementing neutrino sampling in N-body simulations, while the synchronous gauge remains a traditional reference frame; together, they furnish a robust, parallel toolkit for studying linear perturbations and initial conditions in mixed dark-matter cosmologies. The work thus enables accurate modeling of large-scale structure formation and informs the generation of HDM initial conditions for simulations.

Abstract

We present a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. We first derive the transformation law relating the two gauges. We then write down in parallel in both gauges the coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric perturbations and the density fluctuations of the particle species. The particle species considered include cold dark matter (CDM), baryons, photons, massless neutrinos, and massive neutrinos (a hot dark matter or HDM candidate), where the CDM and baryon components are treated as fluids while a detailed phase-space description is given to the photons and neutrinos. The linear evolution equations presented are applicable to any $Ω=1$ model with CDM or a mixture of CDM and HDM. Isentropic initial conditions on super-horizon scales are derived. The equations are solved numerically in both gauges for a CDM+HDM model with $Ω_{\rm cold}=0.65,$ $Ω_{\rm hot}=0.3$, and $Ω_{\rm baryon}=0.05$. We discuss the evolution of the metric and the density perturbations and compare their different behaviors outside the horizon in the two gauges. In a companion paper we integrate the geodesic equations for the neutrino particles in the perturbed conformal Newtonian background metric computed here. The purpose is to obtain an accurate sampling of the neutrino phase space for the HDM initial conditions in $N$-body simulations of the CDM+HDM models.

Figures (2)

• Figure 1: The scalar metric perturbations $\phi(k,\tau)$ (Fig. 1a) and $\psi(k,\tau)$ (Fig. 1b) in the conformal Newtonian gauge as a function of $\tau$. The 41 curves from left to right correspond to 41 values of $k$ between 100.0 Mpc$^{-1}$ and 0.01 Mpc$^{-1}$. The labels $\tau_{\rm nr}$, $\tau_{\rm eq}$ and $\tau_{\rm rec}$ indicate the time 7 eV neutrinos become non-relativistic, the matter-radiation equality time, and the recombination time, respectively.
• Figure 2: Evolution of the density fields in the synchronous gauge (top panels) and the conformal Newtonian gauge (bottom panels) for 3 wavenumbers $k$= 0.01 (Fig. 2a), 0.1 (Fig. 2b) and 1.0 (Fig. 2c) Mpc$^{-1}$. In each figure, the five lines represent $\delta_c, \delta_b, \delta_\gamma, \delta_\nu$ and $\delta_h$ for the CDM (solid curve), baryon (dash-dotted), photon (long-dashed), massless neutrino (dotted), and massive neutrino (short-dashed) components, respectively.