A note on viability of nonminimally coupled f(R) theory

Abstract

Consistency conditions for nonminimally coupled f(R) theories have been derived by requiring the absence of tachyons and instabilities in the scalar fluctuations. This note confirms these results and clarifies a subtlety regarding different definitions of sound speeds.

Appendix

In this appendix we rederive the action (2) from action (1), again using, as a cross-check, a somewhat different method from [3]. There we kept the matter Lagrangian $\hat{\mathcal {L}}$ general and applied a Legendre transformation method for both the R and the $\mathcal {\hat{L}}$. Here we specify $\mathcal {\hat{L}}=\hat{X}$ and apply a straightforward generalisation of a well-known procedure that recasts a minimally coupled f(R) gravity into a scalar-tensor theory.

So we begin with action (1) where $\mathcal {\hat{L}}=\hat{X}$. Introducing a field $\alpha$ and a Lagrange multiplier $\varphi$ that sets $\alpha =R$ we can write the action equivalently as

$$\begin{aligned} S = \int \mathrm{d}^4x \sqrt{-\hat{g}}\left[ f_1(\alpha ) + f_2(\alpha )\hat{X} + \varphi \left( \hat{R}-\alpha \right) \right] \,. \end{aligned}$$

(15)

We then vary the action with respect to $\alpha$, and solve it from its own equation of motion

$$\begin{aligned} f_1'(\alpha ) + f_2'(\alpha )\hat{X}=\varphi \quad \Rightarrow \quad \alpha ={A}(\hat{X},\varphi )\,, \end{aligned}$$

(16)

and it is then legitimate to plug the solution back into the action, to obtain

$$\begin{aligned} S= & {} \int \mathrm{d}^4x \sqrt{-\hat{g}}\left[ \varphi \hat{R} + 2V(\hat{X},\varphi )\right] \,, \quad \text {where}\nonumber \\ V(\hat{X},\varphi )= & {} \frac{1}{2}\left( f_1({A})-\varphi {A} + f_2({A})\hat{X}\right) \,. \end{aligned}$$

(17)

Under a conformal rescaling $g_{\mu \nu }=\Omega ^2 g_{\mu \nu }$, the relevant quantities transform as

$$\begin{aligned} \sqrt{-\hat{g}}=\Omega ^{-4}\sqrt{-g}\,, \quad \hat{R}=\Omega ^2 R + 6\Omega \Box \Omega -12(\partial \Omega )^2\,, \quad \hat{X}=\Omega ^2 X\,. \end{aligned}$$

(18)

Performing the rescaling with the conformal factor

$$\begin{aligned} \Omega ^2 = \phi = \sqrt{\frac{3}{2}}\frac{1}{\kappa }\log {\varphi }\,, \end{aligned}$$

(19)

we obtain:

$$\begin{aligned} S= & {} \int \mathrm{d}^4x \sqrt{-g}\left[ \frac{R}{2\kappa ^2} + Y + e^{-2\sqrt{\frac{2}{3}}\kappa \phi }\left( f_1(A) - e^{\sqrt{\frac{2}{3}}\kappa \phi }A + e^{\sqrt{-\frac{2}{3}}\kappa \phi }f_2(A)X \right) \right] , \nonumber \\ A= & {} A\left( e^{\sqrt{\frac{2}{3}}\kappa \phi },e^{-\sqrt{\frac{2}{3}}\kappa \phi }X\right) . \end{aligned}$$

(20)

We have then recovered the same relation between actions (1) and (2) as was obtained in [3].