It is interesting to see which modified gravity theories have successful Newton limits. There are two known mechanisms for satisfying local gravity constraints, (i) the Vainshtein mechanism [602], and (ii) the chameleon mechanism [344, 343] (already discussed in Section 5.2). Both consist of using non-linearities in order to prevent any other fifth force from propagating freely. The chameleon mechanism uses the non-linearities coming from matter couplings, whereas the Vainshtein mechanism uses the self-coupling of a scalar-field degree of freedom as a source for the non-linear effect.

There are several examples where the Vainshtein mechanism plays an important role. One is the massive gravity in which a consistent free massive graviton is uniquely defined by Pauli–Fierz theory [258, 259]. The massive gravity described by the Fierz–Pauli action cannot be studied through the linearization close to a point-like mass source, because of the crossing of the Vainshtein radius, that is the distance under which the linearization fails to study the metric properly [602]. Then the theory is in the strong-coupling regime, and things become obscure as the theory cannot be understood well mathematically. A similar behavior also appears for the Dvali–Gabadadze–Porrati (DGP) model (we will discuss in the next section), in which the Vainshtein mechanism plays a key role for the small-scale behavior of this model.

Besides a standard massive term, other possible operators which could give rise to the Vainshtein mechanism come from non-linear self-interactions in the kinetic term of a matter field . One of such terms is given by

which respects the Galilean invariance under which ’s gradient shifts by a constant [455] (treated in section 13.7.2). This allows a robust implementation of the Vainshtein mechanism in that nonlinear self-interacting term can allow the decoupling of the field from matter in the gravitationally bounded system without introducing ghosts.Another example of the Vainshtein mechanism may be seen in gravity. Recall that in this theory the contribution to the GB term from matter can be neglected relative to the vacuum value . In Section 12.3.3 we showed that on the Schwarzschild geometry the modification of gravity is very small for the models (12.16) and (12.17), because the GB term has a value much larger than its cosmological value today. The scalar-field degree of freedom acquires a large mass in the region of high density, so that it does not propagate freely. For the model (12.16) we already showed that at the linear level the coefficients and of the spherically symmetric metric (12.32) are

where , and are given by Eqs. (12.38) and (12.39), and for our solar system. Of course this result is trustable only in the region for which . Outside this region non-linearities are important and one cannot rely on approximate methods any longer. Therefore, for this model, we can define the Vainshtein radius as For , this value is well outside the region in which solar-system experiments are carried out. This example shows that the Vainshtein radius is generally model-dependent.In metric f (R) gravity a non-linear effect coming from the coupling to matter fields (in the Einstein frame) is crucially important, because vanishes in the vacuum Schwarzschild background. The local gravity constraints can be satisfied under the chameleon mechanism rather than the non-linear self coupling of the Vainshtein mechanism.

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