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
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
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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