Gauge-Invariant Initial Conditions and Early Time Perturbations in Quintessence Universes

TL;DR

This work addresses the initial conditions and early-time evolution of cosmological perturbations in a radiation-dominated universe containing photons, baryons, CDM, neutrinos, and a tracking quintessence field. It introduces a gauge-invariant matrix formulation, recasting the perturbation dynamics as a matrix equation and analyzing the dominant modes via the eigenstructure of the evolution operator. The key result is that there are four dominant modes—adiabatic, CDM isocurvature, baryon isocurvature, and neutrino isocurvature—and quintessence does not add an independent dominant mode. CMB calculations show that non-adiabatic initial conditions are strongly constrained by observations, while quintessence does not introduce new dominant isocurvature content, providing constraints on quintessence models and their early-universe behavior.

Abstract

We present a systematic treatment of the initial conditions and evolution of cosmological perturbations in a universe containing photons, baryons, neutrinos, cold dark matter, and a scalar quintessence field. By formulating the evolution in terms of a differential equation involving a matrix acting on a vector comprised of the perturbation variables, we can use the familiar language of eigenvalues and eigenvectors. As the largest eigenvalue of the evolution matrix is fourfold degenerate, it follows that there are four dominant modes with non-diverging gravitational potential at early times, corresponding to adiabatic, cold dark matter isocurvature, baryon isocurvature and neutrino isocurvature perturbations. We conclude that quintessence does not lead to an additional independent mode.

Figures (2)

• Figure 1: Gauge-invariant energy density perturbation $\Delta_{q}$ and quintessence field fluctuation $X$ as simulated (straight and dashed-dotted lines), compared to the analytic solution of Equations (\ref{['adiabatic_mode']}) and (\ref{['eqn::Xana2']}) (dashed and dotted lines) as a function of conformal time $\tau$ for adiabatic initial conditions. Radiation and matter equality corresponds to $\tau = 109 \rm Mpc$. Shown is the mode for $k = 0.1\,\textrm{Mpc}^{-1}$ and the cosmological parameters have been $\Omega_{b}^0 h^2 = 0.022,\ h=0.7,\ \Omega_{\rm m}^0 = 0.3,\ \Omega_{\rm q}^0=0.7$.
• Figure 2: CMB Temperature spectra as a function of multipole $l$ in an early quintessence cosmology. The pure adiabatic (AD), CDM isocurvature (CI), neutrino isocurvature (NI) mode and three different combinations of these dominant modes are plotted. For comparison with experimental data we also give the WMAP measurements of the CMB Spergel:2003cb. The spectrum of the pure baryon isocurvature mode is essentially identical to that of the pure CDM isocurvature mode. All spectra have been normalized to the same power at $l=10$.