Matter perturbations in Galileon cosmology

TL;DR

This work derives the full linear perturbation equations for a covariant Galileon cosmology with five Galileon terms and analyzes their impact on structure growth and ISW signals. By extracting a subhorizon quasi-static limit, the authors obtain a scale-independent effective gravitational coupling $G_{ m eff}$ and slip parameter $\eta$, and show how matter perturbations obey $\delta_m''+(2+H'/H)\delta_m'-\tfrac{3}{2}(G_{ m eff}/G)\Omega_m\delta_m\approx0$. Numerical results for representative parameters reveal enhanced growth relative to LCDM during the matter era, a time-varying $\Phi_{\rm eff}$ on ISW scales, and a de Sitter point with $G_{ m eff}/G=1/[3(\alpha-2\beta)]$; observational bounds constrain $G_{ m eff}$ to about $0.5G$–$0.72G$ today and yield characteristic growth indices $\gamma<0.4$ in late-time tracking. These features provide distinctive observational handles to distinguish Galileon cosmology from LCDM with future surveys of large-scale structure, CMB ISW, and weak lensing.

Abstract

We study the evolution of matter density perturbations in Galileon cosmology where the late-time cosmic acceleration can be realized by a field kinetic energy. We obtain full perturbation equations at linear order in the presence of five covariant Lagrangians ${cal L}_i$ ($i=1,...,5$) satisfying a Galilean symmetry in the flat space-time. The equations for a matter perturbation as well as an effective gravitational potential are derived under a quasi-static approximation on sub-horizon scales. This approximation can reproduce full numerical solutions with high accuracy for the wavelengths relevant to large-scale structures. For the model parameters constrained by the background expansion history of the Universe the growth rate of matter perturbations is larger than that in the LCDM model, with the growth index $gamma$ today typically smaller than 0.4. We also find that, even on very large scales associated with the Integrated-Sachs-Wolfe (ISW) effect in Cosmic Microwave Background (CMB) temperature anisotropies, the effective gravitational potential exhibits a temporal growth during the transition from the matter era to the epoch of cosmic acceleration. These properties are useful to distinguish the Galileon model from the LCDM in future high-precision observations.

Figures (5)

• Figure 1: The equation of state $w_{\rm DE}$ versus the redshift $z$ for the model parameters $\alpha=1.37$ and $\beta=0.44$ with two different initial conditions: (A) $r_1=0.03$, $r_2=0.003$, and (B) $r_1=0.999$, $r_2=7.0 \times 10^{-11}$. The case (B) corresponds to the tracker solution, whereas in the case (A) the solution approaches the tracker around today. The present epoch ($z=0$) is shown as a dotted line (we also draw the dotted line in other figures).
• Figure 2: Evolution of the perturbations for $\alpha=1.37$ and $\beta=0.44$ with the background initial conditions $r_1=0.03$ and $r_2=0.003$ (corresponding to the case (A) of Fig. \ref{['wde']}). (Left) $\delta_m/a$ versus $z$ for the wave numbers (a) $k=300a_0H_0$, (b) $k=30a_0H_0$, and (c) $k=5a_0H_0$. (Right) $\Phi_{\rm eff}$ versus $z$ for the wave numbers (a) $k=300a_0H_0$, (b) $k=10a_0H_0$, and (c) $k=5a_0H_0$. Note that $\delta_m/a$ and $\Phi_{\rm eff}$ are divided by their initial amplitudes $\delta_m (t_i)/a(t_i)$ and $\Phi_{\rm eff}(t_i)$, respectively, so that their initial values are normalized to be 1. The bold dotted lines show the results obtained under the quasistatic approximation on subhorizon scales. The choice of initial conditions for perturbations is explained in the text.
• Figure 3: Variation of the growth index $\gamma$ of matter perturbations for $\alpha=1.37$ and $\beta=0.44$ with the mode $k=300a_0H_0$. The initial conditions of $r_1$ and $r_2$ for the cases (A) and (B) are the same as those given in the caption of Fig. \ref{['wde']}.
• Figure 4: Evolution of (i) $\delta_m/a$ and (ii) $\Phi_{\rm eff}$ versus $z$ for $\alpha=1.37$ and $\beta=0.44$ with the initial conditions $r_1=0.999$ and $r_2=7.0 \times 10^{-11}$ [corresponding to the case (B) of Fig. \ref{['wde']}]. Both $\delta_m/a$ and $\Phi_{\rm eff}$ are divided by their initial amplitudes. In this case the background cosmological solution is on the tracker from the onset of integration. The solid line shows the evolution of perturbations for the mode $k=5a_0H_0$, whereas the bold dotted line and bold dashed line represent the result derived under the quasistatic approximation on subhorizon scales.
• Figure 5: Evolution of perturbations for $\alpha=0$ and $\beta=0$ with the initial conditions $r_1=0.05$ and $r_2=0.001$. The lines correspond as follows: (A1) $\delta_m/a$ for the mode $k=300a_0H_0$, (A2) $\Phi_{\rm eff}$ for the mode $k=300a_0H_0$, (B1) $\delta_m/a$ for the mode $k=5a_0H_0$, and (B2) $\Phi_{\rm eff}$ for the mode $k=5a_0H_0$. Both $\delta_m/a$ and $\Phi_{\rm eff}$ are divided by their initial amplitudes. The bold dotted line and bold dashed line correspond to the results obtained under the quasistatic approximation on subhorizon scales.