The Paths of Gravity in Galileon Cosmology

Abstract

Galileon gravity offers a robust gravitational theory for explaining cosmic acceleration, having a rich phenomenology of testable behaviors. We explore three classes of Galileon models -- standard uncoupled, and linearly or derivatively coupled to matter -- investigating the expansion history with particular attention to early time and late time attractors, as well as the linear perturbations. From the relativistic and nonrelativistic Poisson equations we calculate the generalizations of the gravitational strength (Newton's constant), deriving its early and late time behavior. By scanning through the parameters we derive distributions of the gravitational strength at various epochs and trace the paths of gravity in its evolution. Using ghost-free and stability criteria we restrict the allowed parameter space, finding in particular that the linear and derivative coupled models are severely constrained by classical instabilities in the early universe.

Figures (10)

• Figure 1: [Left panel] The redshift evolution of $\bar{H}^{2}$ (solid red), $8\pi G \rho_{m}/3H_{0}^{2}$ (dotted blue), $8\pi G \rho_r/3H_{0}^{2}$ (dot-dashed magenta) and $8\pi G \rho_{\pi}/3H_{0}^{2}$ (long dash black) for the uncoupled Galileon, with parameters $c_{2} = -27.3$, $c_{3}=-12.8$, $c_{4} = -1.7$, $c_{5} = -1$. Here, the scalar field energy density slowly grows relative to matter at early times, as expected from the analytic results. The $c_{5}$ term dominates the $\pi$ dynamics up to $z\sim 10$, after which we observe a turnaround to approach the late time de Sitter point. [Right panel] The effective equation of state of the Galileon field. During radiation domination $w=-1/4$, in agreement with the analytic results; the model currently accelerates the expansion and approaches a late time de Sitter point.
• Figure 2: [Left panel] The redshift evolution of viable linear coupled Galileons is restricted to be near that of the equivalent uncoupled Galileon, except at late times. The late time behaviour of $\bar{H}^{2}$ for the linearly coupled model (dotted black) is compared to the uncoupled model $\bar{H}^{2}$ (solid red) with the parameter choices $(c_{2},c_{3},c_{4},c_{5},c_{0}) = (-5.9,-2,-1,-1,0.05)$ and $(-6.0,-2,-1,-1, 0)$ respectively ($c_{2}$ is adjusted to ensure $\Omega_{\pi,0} =0.76$ in both cases). Note the Hubble parameter in the linear coupling case no longer asymptotes to a constant value. [Right panel] The effective equation of state for the linearly coupled $\pi$ field as a function of redshift. At early times $w$ resembles the uncoupled case, e.g. Fig. \ref{['fig:expunc']}, but deviates at later times and in the future has $w$ close to but not exactly $-1$.
• Figure 3: [Left panel] The evolution of $\bar{H}^{2}$ (solid red), $8\pi G \rho_m/3H_{0}^{2}$ (dotted blue), $8\pi G \rho_r/3H_{0}^{2}$ (dot-dashed magenta) and $8\pi G \rho_{\pi}/3H_{0}^{2}$ (long dash black) for the derivatively coupled Galileon, with $c_{2} = -9.1$, $c_{3} = -1$, $c_{4}=c_{5} =0$. [Right panel] The equation of state of the $\pi$ field as a function of redshift. The numerical evolution exhibits the analytic attractor results.
• Figure 4: The thawing of gravity away from its general relativistic strength, $G^{(\phi)}_{\rm eff}/G_{\rm N} - 1$ (solid red), $G^{(\psi)}_{\rm eff}/G_{\rm N} - 1$ (long dash black), and $G^{(\phi+\psi)}_{\rm eff}/G_{\rm N} - 1$ (short dash blue) for model parameters $(c_{2},c_{3},c_{4},c_{5}) = (-15.3,-5.73,-1.2,-1)$. All three functions share a common redshift dependence at early times and a common future asymptote, as discussed in the text.
• Figure 5: [Left panel] Dark red shaded region shows the area violating the no-ghost condition in the $C$-$D$ plane, fixing $c_5=0$. The straight lines show the cuts through the plane corresponding to special values $c_n=0$ as labeled. The light cyan shaded region shows the Laplace instability $c_s^2<0$, and the thin "bubbly" red region shows the superluminal condition $c_s^2>1$ (see Appendix \ref{['sec:applapl']}). $C>0$ is also ruled out. [Right panel] For the $c_G=0$ case the no-ghost region lies between the solid black lines. These boundaries correspond to the limits $G_{\rm eff,dS}/G_N=\pm1/2$; the blue dashed lines give $G_{\rm eff,dS}/G_N=\pm1$, and the red dotted lines give $G_{\rm eff,dS}/G_N=\pm5$. Positive values are for the upper line of each set. $G_{\rm eff,dS}$ passes through infinity for $4D-3C=14$. Note $G_{\rm eff,dS}/G_N$ cannot lie between $-1/2$ and $1/2$. Further constraints include positivity of scalar field energy density at high redshift, limiting the parameter space to the left of the long dashed, diagonal cyan line when the $c_5$ term dominates, and the Laplace inequality (\ref{['eq:unclaplace']}), with shaded magenta regions obeying $0\le c_s^2\le 1$.