Galileon cosmology

We study the cosmology of a galileon scalar-tensor theory, obtained by covariantizing the decoupling Lagrangian of the Dvali-Gabadadze-Poratti (DGP) model. Despite being local in $3+1$ dimensions, the resulting cosmological evolution is remarkably similar to that of the full $4+1$-dimensional DGP framework, both for the expansion history and the evolution of density perturbations. As in the DGP model, the covariant galileon theory yields two branches of solutions, depending on the sign of the galileon velocity. Perturbations are stable on one branch and ghostlike on the other. An interesting effect uncovered in our analysis is a cosmological version of the Vainshtein screening mechanism: at early times, the galileon dynamics are dominated by self-interaction terms, resulting in its energy density being suppressed compared to matter or radiation; once the matter density has redshifted sufficiently, the galileon becomes an important component of the energy density and contributes to dark energy. We estimate conservatively that the resulting expansion history is consistent with the observed late-time cosmology, provided that the scale of modification satisfies $r_c\gtrsim 15\mathrm{Gpc}$.

Figure 1

Result of numerically integrating the cosmological evolution equations with $r_c=10\mathrm{Gpc}$. The solid curves denote the matter, radiation, cosmological constant and galileon energy density. The dotted line is the galileon energy density as predicted from the analytic solution.

Figure 2

Same as Fig. 1, this time for $r_c=20\mathrm{Gpc}$.

Figure 3

The fractional contributions to the total energy density from matter (${\Omega }_{\mathrm{m}}$), radiation (${\Omega }_{\mathrm{r}}$), cosmological constant (${\Omega }_{\Lambda }$, dashed line) and the galileon field (${\Omega }_{\pi }$, bold dashed line) for $r_c=20\mathrm{Gpc}$, as a function of redshift.

Figure 4

The effective equation of state for the expansion history, defined in terms of the Hubble parameter by $\dot{H}/H^2=-3(1+w_{\mathrm{tot}})/2$, for $\Lambda \mathrm{CDM}$ (solid), and the galileon model with $r_c=10\mathrm{Gpc}$ (dotted), 15 Gpc (dash-dotted) and 20 Gpc (dashed).

Figure 5

The effective BD parameter, ${\omega }_{\mathrm{BD}}^{\mathrm{eff}}$, with $r_c=10\mathrm{Gpc}$ (dotted), 15 Gpc (dash-dotted) and 20 Gpc (dashed).

Figure 6

Comparison of the galileon energy density ($|{\rho }_{\pi }|$) with the effective energy density coming from the modification to the Friedmann equation in the DGP model ($|{\rho }_{\mathrm{DGP}}|$). We have used $r_c=15\mathrm{Gpc}$ in each case and kept the matter, radiation and cosmological constant densities fixed. The densities ${\rho }_{\pi }$ and ${\rho }_{\mathrm{DGP}}$ agree well for the entire expansion history until today, and begin to diverge in the future when $\pi \sim M_{\mathrm{Pl}}$.

Figure 7

Comparison of the perturbation enhancement factor $1/3\beta$ in our galileon model (dotted) with the corresponding factor $1/3{\beta }_{\mathrm{DGP}}$ in DGP cosmology (dashed). As in Fig. 6, we have used $r_c=15\mathrm{Gpc}$ in each case and kept the matter, radiation and cosmological constant densities fixed.

Figure 8

Value of the galileon field at the present time as a function of $r_c$. For each value of $r_c$, the cosmological term is adjusted to keep ${\Omega }_{\mathrm{m}}^{(0)}=0.26$ fixed. The dashed line represents the threshold value of $0.18M_{\mathrm{Pl}}$ from mass estimates—see the bound in (7.3).

Figure 9

Luminosity distance ($d_{\mathrm{L}}$) as a function of redshift for $\Lambda \mathrm{CDM}$ (solid), and our galileon model with $r_c=10\mathrm{Gpc}$ (dotted), $r_c=15\mathrm{Gpc}$ (dash-dotted) and $r_c=20\mathrm{Gpc}$ (dashed). In each case, we have fixed ${\Omega }_{\mathrm{m}}^{(0)}=0.26$.

Figure 10

Percentage difference between the galileon examples and $\Lambda \mathrm{CDM}$.