Two modified gravity theories with few degrees of freedom

In this talk, I will present two specific modifications of gravity in 
which the number of degrees of freedom is minimized. I will focus on 
their construction, showing how different methods can be used to 
accomplish the same goal: keeping a low number of degrees of freedom. 
The first theory that I will describe is the minimal theory of 
quasidilaton massive gravity, a Lorentz-breaking theory of massive 
gravity + scalar field. As a second example, I will describe a class of 
modified gravity theories that propagate only the usual two 
polarizations of gravitational waves.