Global portraits of nonminimal inflation: Metric and Palatini formalism

In this paper, we study the global phase space dynamics of single nonminimally coupled scalar field inflation models in the metric and Palatini formalisms. Working in the Jordan frame, we derive the scalar-tensor general field equations and flat Friedmann-Lemaître-Robertson-Walker cosmological equations and present the Palatini and metric equations in a common framework. We show that inflation is characterized by a “master” trajectory from a saddle-type de Sitter fixed point to a stable node fixed point, approximated by slow-roll conditions (presented for the first time in the Palatini formalism). We show that, despite different underlying equations, the fixed point structure and properties of many models are congruent in metric and Palatini formalisms, which explains their qualitative similarities and their suitability for driving inflation. On the other hand, the global phase portraits reveal how even models which predict the same values for observable perturbations differ, both to the extent of the phase space physically available to their trajectories, as well as their past asymptotic states. We also note how the slow-roll conditions tend to underestimate the end of inflationary accelerated expansion experienced by the true nonlinear “master” solution. The explicit examples we consider range from the metric and Palatini induced gravity quintic potential with a Coleman-Weinberg correction factor to Starobinsky, metric, and Palatini nonminimal Higgs, as well as second-order pole and several nontrivial Palatini models.

Figure 1

Cosmological phase portraits of the Coleman-Weinberg model (60) with $\xi =1$, $\lambda =0.129$, ${\varphi }_0=0.5$ (${\mathrm{\Phi }}_0\approx 0.199M_{\mathrm{Pl}}$), $\mathrm{\Lambda }=0$ in the metric formalism (a) and Palatini formalism (b). Green background stands for superaccelerated, light green accelerated, white decelerated, and yellow superstiff expansion, while gray covers the unphysical region. Orange trajectories are heteroclinic orbits between the fixed points, and the dashed curve marks the path of slow-roll approximation.

Figure 2

Cosmological phase portraits of observationally equivalent models: (a) metric Starobinsky model (97) with $\beta =40000$, (b) metric Higgs model (106) with $\xi =100$, $\lambda =0.129$, $v=0$, (c) minimally coupled pole inflation model (111) with $\alpha =1$, $\lambda ={10}^{-4}$, (d) nonminimally coupled Palatini model with $\xi =100$, $\lambda ={10}^{-5}$, and $\alpha =\sqrt{2/3}$ (in all cases $M_{\mathrm{Pl}}=1$). The green background stands for superaccelerated, light green accelerated, white decelerated, and yellow superstiff expansion, while gray covers the unphysical region. Orange trajectories are heteroclinic orbits between the fixed points, and the dashed curve marks the path of slow-roll approximation.

Figure 3

Continuation of cosmological phase portraits of observationally equivalent models: (a) Palatini noncanonical Higgs model (120) with $\alpha =1$, $\lambda =1.29\times {10}^{-5}$ (b) Palatini induced gravity model (123) with $\lambda =9.5\times {10}^{-11}$, $n=2/3$, ${\mathrm{\Phi }}_0=\sqrt{2/3}$. Green background stands for superaccelerated, light green accelerated, white decelerated, and yellow superstiff expansion, while gray covers the unphysical region. Orange trajectories are heteroclinic orbits between the fixed points, and the dashed curve marks the path of slow-roll approximation.

Figure 4

Cosmological phase portraits of (a) the Palatini Higgs model (106) with $\xi =100$, $\lambda =0.129$, $\mathrm{\Lambda }=0$ and (b) induced gravity in the metric formalism model with $\lambda =9.5\times {10}^{-11}$, $n=2/3$, ${\mathrm{\Phi }}_0=\sqrt{2/3}$. Green background stands for superaccelerated, light green accelerated, white decelerated, and yellow superstiff expansion, while gray covers the unphysical region. Orange trajectories are heteroclinic orbits between the fixed points, and the dashed curve marks the path of slow-roll approximation.