F(R) Gravity - Equivalent Formalism

Equivalent Formalism

Under certain additional conditions we can simplify the analysis of f(R) theories by introducing an auxiliary field Φ. Assuming f′′(R) ≠ 0 for all R, let V(Φ) be the Legendre transform of f(R) so that Φ = f′(R) and R = V′(Φ). Then, one obtains the O'Hanlon (1972) action

We have the Euler-Lagrange equations

Eliminating Φ, we obtain exactly the same equations as before. However, the equations are only second order in the derivatives, instead of fourth order.

We are currently working with the Jordan frame. By performing a conformal rescaling

we transform to the Einstein frame:

after integrating by parts.

Defining

,

and substituting

This is general relativity coupled to a real scalar field: using f(R) theories to describe the accelerating universe is practically equivalent to using quintessence.