Lorentzian Vacuum Transitions in $f(R)$ gravity

Abstract

We study Lorentzian vacuum transition probabilities between two minima of a scalar field potential within the framework of $f(R)$ gravity. The analysis extends the previously considered WKB expansion of the Wheeler-DeWitt equation to modified gravity theories, up to second order. We apply the general method for homogeneous and isotropic FLRW universes, with zero and positive spatial curvature, for any $f(R)$ model. For the flat case we obtain analytic expressions for the transition probabilities for any model if we assume a constant Ricci scalar; this assumption has been considered in previous studies, in the Euclidean approach, from symmetry arguments. On the other hand, we also obtain explicit solutions without this assumption for power-law $f(R)=R^{1+n}$ models. Moreover, in the positive curvature scenario, we obtain that the assumption of a constant Ricci scalar is not consistent, but we are able to find analytical solutions in approximated regimes. In all cases we have found that the general behavior of the probabilities already found for Einstein Gravity is preserved, including the prediction of a non-singular initial state due to quantum corrections, even though the probabilities increase or decrease in a model dependent way.

Figures (10)

• Figure 1: Semiclassical contribution to the transition probability for $f(R)=R^{1+n}$ for the GR Result $n\to 0$ (Dashed line), $n=1/3$ (Red line), $n=3/2$ (Purple line), $n=5/2$ (Yellow line), $n=3$ (Blue line), $n=4$ (Gray line) and $n\to \infty$ (Dotted line) . With the parameters $V_{B}=5$, $V_{A}=10$, $\operatorname{Vol}(X)=1$ and $T_{0}=1$.
• Figure 2: First Quantum Correction to the transition probability $\Gamma_0 + \Gamma_1$ for $f(R)=R^{1+n}$ using the GR result $n \to 0$ (Dashed line), $n=1/3$ (Red line), $n=3/2$ (Purple line), $n=5/2$ (Yellow line), $n=3$ (Blue line), $n=4$ (Gray line) and $n\to\infty$ (Dotted line). With the parameters $V_{B}=5$, $V_{A}=10$, $T_{0}=1$, $T_{1}=0.1$, and $T_{1}^{f(R)}=0.05$.
• Figure 3: Second Quantum Correction to the transition probability $\Gamma_0 + \Gamma_1+\Gamma_2$ for $f(R)=R^{1+n}$ using $n=0.05$ (Dashed line), $n=1/3$ (Red line), $n=3/2$ (Purple line), $n=5/2$ (Yellow line), $n=3$ (Blue line), $n=4$ (Gray line) and $n\to\infty$ (dotted line) and the GR result (Black dashed line). With the parameters $V_{B}=5$, $V_{A}=10$, $T_{0}=1$, $T_{0}^{f(R)}=1$, $T_{1}=0.1$, $T_{1}^{f(R)}=0.05$, $T_{2}=10^{-3}$, and $T_{2}^{f(R)}=10^{-5}$.
• Figure 4: Semiclassical contribution for the transition probability for the models $f(R)=R^2$ (Red line), $f(R)=R+\beta R^2$ (Blue line) and GR (Gray line), with parameters $V_B = -5$, $V_A = -1$, $\beta=10^{3}$, $T_0 = 1$, and $T_0^{f(R)} = 0.5$.
• Figure 5: First quantum correction for the transition probability $\Gamma_{0}+\Gamma_{1}$ for the models $f(R)=R^2$ (Red line), $f(R)=R+\beta R^2$ (Blue line) and GR (Gray line) with parameters$V_B = -5$, $V_A = -1$, $T_0 = 1$, $T_0^{f(R)} = 0.5$, $T_1=0.1$ and $T_1^{f(R)}=0.5$.