Modified Gravity – State of the Art Beyond 2nd Order Differential Equations and Diffeomorphism Invariance

Question

I am going to do a state of the art on Modified gravity models. I have found a talk that presents the problematic. In particular, it is said the following things :

Modifying General Relativity

1. How to modify GR:

• extra DoF(s): scalar, vector, tensor field(s);
• going beyond the 2nd order differential equations;
• diffeomorphism invariance breaking;
• higher than 4 dimensions;

2. Solar system constraints

• screening mechanisms
  (Chameleon, Symmetron, k-mouflage, Vainshtein)

3. In the following we will focus on theories with

• an extra scalar and dynamical DoF;
• higher order field equations (in spatial derivatives);
• break diffeomorphism invariance;
• 4 dimensions.

a) I woud like to know what means "going beyond the 2nd order differential equations" ? Is it related to the Einstein-Hilbert action with the Ricci scalar (this one contains second derivatives of the metric, that is to say, first derivatives on Christofell symbols ?).

b) Which diffeomorphism invariance is breaking ? I don't understand this sentence.

c) Finally, it is suggested "extra degrees of freedom" but theses extra degree of freedom are applied on the matter Lagrangian, or for example with the f(R) models, or also another models (in other words, are they applied only on the geometric component of Einstein-Hilbert's action … ? )

Any suggestions/remarks to better understand are welcome.

Answer 1

1. Going beyond second order means considering theories that contain terms that contain higher than second order derivatives of the metric in the equations of motions. E.g. a term like $\partial_{\mu} \partial_{\nu} R$, with $R$ is the Ricci scalar, has fourth-order derivatives of the metric $g_{\mu \nu}$
2. Breaking diffeomorphism invariance means what it says: that the theory will no longer be invariant under diffeomorphisms (or general coordinate transformations). For example, theories with background fields or preferred fields break diffeomorphism invariance. This can be either spontaneous or dynamical, but the main point is that the theory is no longer fully covariant in the sense that coordinates would play some physical role.
3. The extra degrees of freedom will originate in the gravitational sector, but can be equivalently expressed as something like a scalar field (e.g., scalar-tensor formulation of $f(R)$ gravity). In the vacuum case $S_m [g, \phi] =0$ there will be more than the two propagating degrees of freedom of usual GR. These will come from the modified gravitational action.

For much more detail about all these questions, I'd recommend reviews like the following https://arxiv.org/abs/1106.2476