Genoma mitocondrial

In this section we review the historical discovery and development of galileon theories in the context of the Dvali-Gabadadze-Porrati (DGP) model [51]. DGP begins by assuming that the 3-brane which we inhabit is embedded in a flat, five dimensional space where the extra dimension is infinitely large. This is a departure from more common theories in which extra dimensions are posited to be quite small, such as Kaluza-Klein models, because the motivations are different. Models with small extra dimensions modify the high energy, short distance physics and since we are interested in modifying the low energy, long distance behavior of gravity the large extra dimension scenario is the appropriate one. Both the higher dimensional space and the brane are given Einstein-Hilbert terms, so that gravity propagates in both spaces, while matter is confined to the brane.

We let M be our 3-brane with coordinates xµ, µ _{∈ {}0,1,2,3_} and N be the five di- mensional space with coordinates XA,A _{∈ {}0,1,2,3,5_}. The brane’s position is given by embedding functions XA(xµ) and the bulk metric, GAB, induces a metric on the brane,

gµν ≡ ∂X

A

∂xµ ∂X B

∂xν GAB. The DGP action is defined to be

SDGP = 1 2M 3 5 Z N d5X√₋GR(5)[G] + Z M d4x√₋g −M₅3K[g] +1 2M 2 plR(4)[g] +LM(ψ) . (1.26) where R(5)[G] is the 5D Ricci scalar generated from GAB, R(4)[g] is the 4D Ricci scalar

generated fromgµν,K(g) is the extrinsic curvature of the 3-brane andLM(ψ) is the matter

action. The masses M5 and Mpl are the five and four dimensional Planck masses, respec-

tively. The dynamical variables of the theory are the bulk metric (GAB), coordinates of

the brane in the bulk (XA(xµ)) and matter fields (ψ); the induced metric does not contain independent degrees of freedom as it is derived fromGAB and theXA’s.

DGP is a rich model with robust phenomenology and features and a detailed analysis is outside of the realm of this thesis. Instead, we concentrate on the basic features of the model and a sketch of how the galileon interactions arise. Of primary importance is the transition in the behavior of gravity as we progress to larger and larger distances. This transition is reflected in the graviton propagator derived from (1.26) whose momentum dependence goes as [71] D(p)_∼ −i p2_{+ 2}M53 M2 4 p p2 . (1.27)

The propagator defines a “crossover distance”rc ≡ M2

pl

M3 5 ≡m

−1_{below which gravity appears}

four dimensional and above which gravity appear five dimensional, that is

D(p)_∼ (_−i p2 p≪r−_c1 −i |p| p≫r−c1 . (1.28) In order to set rc to be of order the Hubble radius, we must have M5∼ 10MeV.

In addition to the tensor mode of the graviton, there is also a vector and a scalar mode which descend fromGµ5andh55respectively. A long and detailed analysis [82] demonstrates

that the scalar sector of the effective 4D lagrangian is a derivatively self-coupled

L_{eff ⊃}3M_pl2m2ππ₋M_pl2m(∂π)2π+1

2mπT . (1.29)

The full action depends on tensor, vector and scalar fields_{h′

µν, Nµ′, π}, which are intricate

field redefinitions of the bulk metric perturbations about flat space,_{Hµν, Hµ5, H55}, needed

to diagonalize the action, but in thedecoupling limitwe can focus our attention on the scalar lagrangian (1.29).

In order to justify this focus, we compare the strong coupling scale of scalar interactions and demonstrate that is is parametrically lower than all other interaction scales appearing in the effective DGP lagrangian. More precisely, when we canonically normalize by using the field ˆπ _∼Mplmπ, we see that the scale suppressing the cubic derivative term in (1.29)

is Λ3 ≡M52/Mpl= m2Mpl

1/3

and ˆπ couples to matter with gravitational strength,_{∼ M}πTˆ

pl.

One can deduce that a typical interaction appearing inL_eff is of the form [90]

∼mM_pl2∂ Nˆµ m1/2_M pl !p ∂πˆ mMpl q (h′_µν)s (1.30) with p +q +s _≥ 3 and where ˆhµν and ˆNµ are the canonically normalized4 tensor and

vector fields. Then if we want to study modifications of gravity well outside of a source’s Schwarzschild radius, i.e. where we can take ˆhµν →0, we need only concern ourselves with

interaction of the form (1.30) with q = 0. Inspection of (1.30) shows that the interaction of the type (p, q) is suppressed by the scale

Λ(p,q)_∼mq+p/2−1M_plp+q−2 1 3p/2+2q−3 =Λq₃M₅3/2p+q−3 1 3p/2+2q−3 (1.31) and since m _≪ Mpl we see, as claimed, that the smallest this scale can ever be is Λ3,

corresponding to p = 0, q = 3. Thus, the π self interactions are the least suppressed and

4_{As indicated in (1.30), the normalization forN}

µis not the standard one for vector fields. Rather, the kinetic term forNµ arises from∼Nµ△Nµ where△=√−, and so the canonical normalization sets the dimension ofNµto be [Nµ] =E3/2.

most important. A formal limit makes this statement more precise. By takingM5, Mpl, T → ∞ with Λ3 and T /Mpl held constant, it is clear from (1.31) that only the cubic scalar

interaction survives and in this so-called decoupling limit the lagrangian becomes

L_πˆ = 1 2πˆπˆ− 1 63/2_Λ3 3 (∂πˆ)2πˆ+ 1 2√6Mpl ˆ πT . (1.32) This scalar degree of freedom is known as the “galileon” and we continue its history in the next section.