Linear Perturbations In Galileon Gravity Models

(1)We discussed in the previous chapter that, before undergoing dedicated studies of nonlinear structure formation in modified gravity models, one should first learn about their goodness-of-fit to the data from the CMB, SNIa and BAO. In this chapter, we take the first steps towards using these data to constrain the Covariant Galileon gravity model [2, 3] by studying its linear perturbation theory predictions.In the Galileon model, the deviations from GR are mediated by a scalar field phi, dubbed the Galileon, whose Lagrangian density is invariant under the Galilean shift symmetry. partial derivative(mu)phi -> partial derivative(mu)phi + b(mu) (where b(mu) is a constant vector), in flat spacetime. Such a field appears, for instance, as a brane-bending mode in the decoupling limit of the four-dimensional boundary effective action of the DGP braneworld model [4-6] which was proposed before the Galileon model. However, despite being theoretically appealing, the self-accelerating branch of the DGP model is plagued by the ghost problems (energy states unbounded from below) [7-10]. Taking the decoupling limit of the DGP model as inspiration, it was shown in Ref. [2] that in four-dimensional Minkowski space there are only five Galilean invariant Lagrangians that lead to second-order field equations, despite containing highly nonlinear derivative self-couplings of the scalar field. The second-order nature of the equations of motion is crucial to avoid the presence of Ostrogradski ghosts [11]. Furthermore, the structure of these five Lagrangians is such that their classical solutions receive no quantum corrections to any loop order in perturbation field theory [12], i.e., the theory is non-renormalizable. This means that the theory is an effective field theory whose classical solutions can be trusted up to the energy scale above which a quantum completion of the theory becomes inevitable. In Ref. [3, 13], it was shown how these Lagrangians could be generalised to curved spacetimes. These authors concluded that explicit couplings between the Galileon field derivatives and curvature tensors are needed to keep the equations of motion up to second-order (see Ref. [14] for a recent discussion about how such couplings are not strictly needed). Such couplings however break the Galilean shift symmetry which is only a symmetry of the model in the limit of flat spacetime. The couplings of the Galileon field to the curvature tensors and to itself in the equations of motion change the way in which particle geodesics responds to the matter distribution, which is why the Galileon model falls under the category of modified gravity.Since the equations of motion are kept up to second order, it means that the Galileon model is a subclass of the more general Horndeski theory [15-17]. The Horndeski action is the most general single scalar field action one can write that yields only second order field equations of motion of the metric and scalar fields. Besides the Galileon model, it therefore encompasses simpler cases such as Quintessence (cf. Eq. (1.9)) and f (R) (cf. Eq. (1.11)) models as well as other models which also involve derivative couplings of the scalar field that have recently generated some interest such as Kinetic Gravity Braiding [18-20], Fab-Four [21-25], K-mouflage (cf. Eq. (1.14)) and others [26-28]. An important difference between the Galileon model studied here and some other corners of Horndeski's general theory is that in the Galileon model there are no free functions.In this chapter, we start by presenting the action and field equations of the Covariant Galileon model. We then derive the fully covariant and gauge invariant linearly perturbed field equations and solve them with a modified version of the CAMB code. We analyse the model predictions for the CMB temperature, CMB lensing and linear matter power spectra, and also for the time evolution of the lensing potential. We do this for a limited number of parameter values to illustrate how the model predictions are obtained. A main goal of this chapter is to build some intuition for the contraint results of Chap. 3.