13.7 Special symmetries

Since general covariance alone does not restrict the choice of the Lagrangian function, e.g., for f (R) theory, one can try to shrink the set of allowed functions by imposing some extra symmetry. In particular one can assume that the theory possesses a symmetry on some special background. However, allowing some theories to be symmetrical on some backgrounds does not imply these theories are viable by default. Nevertheless, this symmetry helps to give stronger constraints on them, as the allowed parameter space drastically reduces. We will discuss here two of the symmetries studied in the literature: Noether symmetries on a FLRW background and the Galileon symmetry on a Minkowski background.

13.7.1 Noether symmetry on FLRW

The action for metric f (R) gravity can be evaluated on a FLRW background, in terms of the fields a(t) and R(t), see [125124Jump To The Next Citation Point] (and also [129128415433429604603199200132]). Then the Lagrangian turns out to be non-singular, ℒ(qi, ˙qi), where q1 = a, and q2 = R. Its Euler–Lagrange equation is given by (∂ ℒ∕∂ ˙qi)⋅ − ∂ℒ ∕∂qi = 0. Contracting these equations with a vector function j α (qi), (where 1 α = α and 2 α = β are two unknown functions of the i q), we obtain

( ) ( ) αi d-∂-ℒ-− ∂ℒ-- = 0. → d-- αi∂-ℒ- = LX ℒ. (13.55 ) dt∂ ˙qi ∂qi dt ∂ ˙qi
Here LX ℒ is the Lie derivative of ℒ with respect to the vector field
∂ ( d ) ∂ X = αi(q )--i + --αi(q) ---i. (13.56 ) ∂q dt ∂ ˙q
If LX ℒ = 0, the Noether Theorem states that the function Σ0 = αi(∂ℒ ∕∂q˙i) is a constant of motion. The generator of the Noether symmetry in metric f (R) gravity on the flat FLRW background is
∂ ∂ ∂ ∂ X = α ---+ β ----+ ˙α--- + ˙β ---. (13.57 ) ∂a ∂R ∂ ˙a ∂R˙
A symmetry exists if the equation LX ℒ = 0 has non-trivial solutions. As a byproduct, the form of f (R), not specified in the point-like Lagrangian ℒ, is determined in correspondence to such a symmetry.

It can be proved that such i α do exist [124], and they correspond to

[ ] α = c1a + c2, β = − 3c1 + c2- -f,R--+ --c3--, (13.58 ) a a2 f,RR af,RR
where c1, c2, c3 are constants. However, in order that LX ℒ vanishes, one also needs to set the constraint (provided that c2R ⁄= 0)
3(c a2 + c )f − c aR (c a2 + c )κ2ρ(0) f,R = ---1-----2------3----+ --1------2----r-, (13.59 ) 2c2R a4c2R
where ρ(0r) is the radiation density today. Since now LX = 0, then αi(∂ℒ∕ ∂ ˙qi) = constant. This constant of motion corresponds to
2 ˙ 2 3 α (6f,RRa R + 12f,Ra ˙a) + β (6f,RRa ˙a) = 6μ 0 = constant, (13.60 )
where μ0 has a dimension of mass.

For a general f it is not possible to solve the Euler–Lagrange equation and the constraint equation (13.59View Equation) at the same time. Hence, we have to use the Noether constraint in order to find the subset of those f which make this possible. Some partial solutions (only when μ0 = 0) were found, but whether this symmetry helps finding viable models of f (R) is still not certain. However, the f (R) theories which possess Noether currents can be more easily constrained, as now the original freedom for the function f in the Lagrangian reduced to the choice of the parameters ci and μ0.

13.7.2 Galileon symmetry

Recently another symmetry, the Galileon symmetry, for a scalar field Lagrangian was imposed on the Minkowski background [455Jump To The Next Citation Point]. This idea is interesting as it tries to decouple light scalar fields from matter making use of non-linearities, but without introducing new ghost degrees of freedom [455Jump To The Next Citation Point]. This symmetry was chosen so that the theory could naturally implement the Vainshtein mechanism. However, the same mechanism, at least in cosmology, seems to appear also in the FLRW background for scalar fields which do not possess such a symmetry (see [539Jump To The Next Citation Point351Jump To The Next Citation Point190Jump To The Next Citation Point]).

Keeping a universal coupling with matter (achieved through a pure nonminimal coupling with R), Nicolis et al. [455Jump To The Next Citation Point] imposed a symmetry called the Galilean invariance on a scalar field π in the Minkowski background. If the equations of motion are invariant under a constant gradient-shift on Minkowski spacetime, that is

μ π → π + c + bμx , (13.61 )
where both c and bμ are constants, we call π a Galileon field. This implies that the equations of motion fix the field up to such a transformation. The point is that the Lagrangian must implement the Vainshtein mechanism in order to pass solar-system constraints. This is achieved by introducing self-interacting non-linear terms in the Lagrangian. It should be noted that the Lagrangian is studied only at second order in the fields (having a nonminimal coupling with R) and the metric itself, whereas the non-linearities are fully kept by neglecting their backreaction on the metric (as the biggest contribution should come only from standard matter). The equations of motion respecting the Galileon symmetry contain terms such as a constant, □π (up to fourth power), and other power contraction of the tensor ∇ μ∇ νπ. It is due to these non-linear derivative terms by which the Vainshtein mechanism can be implemented, as it happens in the DGP model [401].

Nicolis et al. [455] found that there are only five terms ℒ i with i = 1,...,5 which can be inserted into a Lagrangian, such that the equations of motion respect the Galileon symmetry in 4-dimensional Minkowski spacetime. The first three terms are given by

ℒ1 = π, (13.62 ) ℒ2 = ∇ μπ ∇μπ, (13.63 ) μ ℒ3 = □ π∇ μπ ∇ π. (13.64 )
All these terms generate second-order derivative terms only in the equations of motion. The approach in the Minkowski spacetime has motivated to try to find a fully covariant framework in the curved spacetime. In particular, Deffayet et al. [205Jump To The Next Citation Point] found that all the previous 5-terms can be written in a fully covariant way. However, if we want to write down ℒ4 and ℒ5 covariantly in curved spacetime and keep the equations of motion free from higher-derivative terms, we need to introduce couplings between the field π and the Riemann tensor [205]. The following two terms keep the field equations to second-order,
ℒ4 = (∇ μπ∇ μπ )[2(□ π)2 − 2(∇ αβπ)(∇ αβπ) − (1∕2)R ∇ μπ∇ μπ], (13.65 ) λ 3 αβ ν ρ μ ℒ5 = (∇ λπ∇ π )[(□ π) − 3□ π(∇ αβπ)(∇ π) + 2(∇ μ∇ π)(∇ ν∇ π)(∇ ρ∇ π ) − 6(∇ μπ)(∇ μ∇ νπ)(∇ ρπ)G νρ], (13.66 )
where the last terms in Eqs. (13.65View Equation) and (13.66View Equation) are newly introduced in the curved spacetime. These terms possess the required symmetry in Minkowski spacetime, but mostly, they do not introduce derivatives higher than two into the equations of motion. In this sense, although originated from an implementation of the DGP idea, the covariant Galileon field is closer to the approach of the modifications of gravity in f (R,𝒢 ), that is, a formalism which would introduce only second-order equations of motion.

This result can be extended to arbitrary D dimensions [202]. One can find, analogously to the Lovelock action-terms, an infinite tower of terms that can be introduced with the same property of keeping the equations of motion at second order. In particular, let us consider the action

∫ pmax S = dDx √−-g ∑ 𝒞 ℒ , (13.67 ) (n+1,p) (n+1,p) p=0
where pmax is the integer part of (n − 1)∕2 (n ≤ D),
( 1 )p (n − 1)! 𝒞(n+1,p) = − -- ----------------2, (13.68 ) 8 (n − 1 − 2p)!(p!)
and
1 ℒ(n+1,p) = − ---------𝜀μ1μ3...μ2n−1ν1...νD−n𝜀μ2μ4...μ2nν1...νD−nπ;μ1π;μ2(π;λπ;λ)p (D − n)! ∏p n−2∏− 2p × Rμ4i− 1μ4i+1μ4iμ4i+2 π;μ2n−1−2jμ2n−2j. (13.69 ) i=1 j=0
Here 1⋅⋅⋅n 𝜀 is the Levi-Civita tensor. The first product in Eq. (13.69View Equation) is defined to be one when p = 0 and 0 for p < 0, and the second product is one when n = 1 + 2p, and 0 when n < 2 + 2p. In ℒ (n+1,p) there will be n + 1 powers of π, and p powers of the Riemann tensor. In four dimensions, for example, ℒ (1,0) and ℒ (2,0) are identical to ℒ1 and ℒ2 introduced before, respectively. Instead, ℒ (3,0), ℒ(4,0) − (1∕4)ℒ(4,1), and ℒ(5,0) − (3∕4 )ℒ(5,1) reduce to ℒ3, ℒ4, and ℒ5, up to total derivatives, respectively.

In general non-linear terms discussed above may introduce the Vainshtein mechanism to decouple the scalar field from matter around a star, so that solar-system constraints can be satisfied. However the modes can have superluminal propagation, which is not surprising as the kinetic terms get heavily modified in the covariant formalism. Some studies have focused especially on the ℒ3 term only, as this corresponds to the simplest case. For some models the background cosmological evolution is similar to that in the DGP model, although there are ghostlike modes depending on the sign of the time-velocity of the field π [158]. There are some works for cosmological dynamics in Brans–Dicke theory in the presence of the non-linear term ℒ3 [539351190] (although the original Galileon symmetry is not preserved in this scenario). Interestingly the ghost can disappear even for the case in which the Brans–Dicke parameter ω BD is smaller than − 2. Moreover this theory leaves a number of distinct observational signatures such as the enhanced growth rate of matter perturbations and the significant ISW effect in CMB anisotropies.

At the end of this section we should mention conformal gravity in which the conformal invariance forces the gravitational action to be uniquely given by a Weyl action [414340]. Interestingly the conformal symmetry also forces the cosmological constant to be zero at the level of the action [413]. It will be of interest to study the cosmological aspects of such theory, together with the possibility for the avoidance of ghosts and instabilities.


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