1. Introduction
2. Palatini inflation with a non-minimal coupling
2.1. Calculating the inflationary parameters
1. Electroweak regime If $| \xi ({\phi }^{2}-{v}^{2})| \ll 1$, φ ≈ χ and VJ(φ) ≈ VE(χ). Thus, in this limit, the inflationary predictions for the non-minimal coupling case are approximately the same with the ones for the minimal coupling case. | |
2. Induced gravity limit [36] In this limit (ξv2 = 1, F(φ) = ξφ2), Z(φ) = ξφ2 and using equation ( $ \begin{eqnarray}\phi =v\exp \left(\chi \sqrt{\xi }\right),\end{eqnarray}$ here we set χ(v) = 0. | |
3. Large-field limit If φ2 ≫ v2 during inflation, we have $ \begin{eqnarray}\phi \simeq \displaystyle \frac{1}{\sqrt{\xi }}\sinh \left(\chi \sqrt{\xi }\right),\end{eqnarray}$ in the Palatini formulation. Using equation ( |
2.2. Different reheating scenarios
1. High-N scenario ωr = 1/3, this case corresponds to the assumption of instant reheating. | |
2. Middle-N scenario ωr = 0 and the temperature of reheating is taken as Tr = 109 GeV, while computing ρr using the SM value for the usual number of relativistic degrees of freedom values for g* = 106.75. | |
3. Low-N scenario ωr = 0, same as middle-N scenario. But in this case, the reheat temperature Tr = 100 GeV. |
Figure 1. The figure illustrates that ns − r predictions for different ξ values and v = 0.01 for various reheating cases as described in the text for Higgs potential in the Palatini formalism. The points on each curve represent ξ = 10−2.5, 10−2, 10−1.5, 10−1, and 1, top to bottom. The pink (red) contour corresponds to the 95% (68%) CL contours based on the data taken by the Keck Array/BICEP2 and Planck collaborations [10]. |
3. Quadratic potential
Figure 2. The figure shows that ns − r predictions for quadratic potential in the Palatini formalism for high-N scenario for the different ξ values that are mentioned in the text. The pink (red) contour correspond to the 95% (68%) CL contour given by the Keck Array/BICEP2 and Planck collaborations [10]. |
Figure 3. For quadratic potential in the Palatini formalism for high-N scenario, the figures show that m and α values as functions of ξ. |
4. Higgs potential
Figure 4. For Higgs potential in the Palatini formalism in the cases of φ > v and high-N scenario, in the top figures display changing ns and r values for different ξ cases as function of v and the bottom figure shows that ns − r predictions for selected ξ values. The pink (red) contour correspond to the 95% (68%) CL contour given by the Keck Array/BICEP2 and Planck collaborations [10]. |
Figure 5. For Higgs potential in the Palatini formalism, the change in α and A as a function of v is plotted for different ξ values in the cases of φ > v and high-N scenario. |
Figure 6. For Higgs potential in the Palatini formalism in the cases of φ > v and low-N scenario, in the top figures display changing ns and r values for different ξ cases as function of v and the bottom figure shows that ns − r predictions for selected ξ values. The pink (red) contour correspond to the 95% (68%) CL contour given by the Keck Array/BICEP2 and Planck collaborations [10]. |
Figure 7. For Higgs potential in the Palatini formalism, the change in α and A as a function of v is plotted for different ξ values in the cases of φ > v and low-N scenario. |
Figure 8. For Higgs potential in the Palatini formalism in the cases of φ < v and high-N scenario, in the top figures display changing ns and r values for different ξ cases as function of v. The bottom figures show that ns − r predictions for selected ξ values, left panel: ξ > 0 and ξ = 0 cases, right panel: ξ < 0 cases. The pink (red) contour correspond to the 95% (68%) CL contour given by the Keck Array/BICEP2 and Planck collaborations [10]. |
Figure 9. For Higgs potential in the Palatini formalism, the change in α and A as a function of v is plotted for different ξ values in the cases of φ < v and high-N scenario. |
Figure 10. For Higgs potential in the Palatini formalism in the cases of φ < v and low-N scenario, in the top figures display changing ns and r values for different ξ cases as function of v. The bottom figures show that ns − r predictions for selected ξ values, left panel: ξ > 0 and ξ = 0 cases, right panel: ξ < 0 cases. The pink (red) contour correspond to the 95% (68%) CL contour given by the Keck Array/BICEP2 and Planck collaborations [10]. |
Figure 11. For Higgs potential in the Palatini formalism, the change in α and A as a function of v is plotted for different ξ values in the cases of φ < v and low-N scenario. |
Figure 12. For Higgs potential in the Palatini formalism, the change in ns, r, α and A as a function of v is plotted for different ξ values and for φ > v in the induced gravity limit which corresponds to ξv2 = 1. |
5. Hilltop potentials
Figure 13. For hilltop potentials in the Palatini formalism, top figures display that ns, r values as functions of ξ for τ = 0.01 and different μ values in the cases of φ < v and high-N scenario. The bottom figure displays ns − r predictions based on range of the top figures ξ values for τ = 0.01. The pink (red) line corresponds to the 95% (68%) CL contour given by the Keck Array/BICEP2 and Planck collaborations [10]. |
Figure 14. For hilltop potentials in the Palatini formalism, the figure shows that α values as functions of ξ for τ = 0.01 and different μ values in the cases of φ < v and high-N scenario. |
Figure 15. For hilltop potentials in the Palatini formalism, top figures display that ns, r values as functions of ξ for τ = 0.01 and different μ values in the cases of φ < v and low-N scenario. The bottom figure displays ns − r predictions based on range of the top figures ξ values for τ = 0.01. The pink (red) line corresponds to the 95% (68%) CL contour given by the Keck Array/BICEP2 and Planck collaborations [10]. |
Figure 16. For hilltop potentials in the Palatini formalism, the figure shows that α values as functions of ξ for τ = 0.01 and different μ values in the cases of φ < v and low-N scenario. |