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 :

1. Modifying General Relativity

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.

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.