5.7.3. Beyond general relativity
Rather than introduce an explicit scalar field to drive inflation, some theories modify the gravitational sector of the theory into something more complicated than general relativity. . Examples are
The last two are theories where the gravitational constant may vary (indeed Jordan-Brans-Dicke theory is a special case of scalar-tensor gravity).
However, a clever trick, known as the conformal transformation,  allows such theories to be rewritten as general relativity plus one or more scalar fields with some potential. Often, only one of those fields is dynamical which returns us once more to the original chaotic inflation scenario!
The most famous example is extended inflation.  In its original form, it transforms precisely into the power-law inflation model that we've already discussed, with the added bonus that it includes a proper method of ending inflation. Unfortunately though, this model is now ruled out by observations.  Indeed, models of inflation based on altering gravity are much more constrained than other types, since we know a lot about gravity and how well general relativity works,  and many models of this kind are very vulnerable to observations.