Inflation with Non-Minimal Coupling: Metric vs. Palatini Formulations (0803.2664v2)

Abstract: We analyze non-minimally coupled scalar field theories in metric (second-order) and Palatini (first-order) formalisms in a comparative fashion. After contrasting them in a general setup, we specialize to inflation and find that the two formalisms differ in their predictions for various cosmological parameters. The main reason is that dependencies on the non-minimal coupling parameter are different in the two formalisms. For successful inflation, the Palatini approach prefers a much larger value for the non-minimal coupling parameter than the Metric approach. Unlike the Metric formalism, in Palatini, the inflaton stays well below the Planck scale whereby providing a natural inflationary epoch.

• The paper demonstrates that metric and Palatini formalisms yield distinct inflationary parameter dependencies due to their treatment of the affine connection.
• It finds that the Palatini approach requires a much larger non-minimal coupling ζ, resulting in sub-Planckian inflaton values and extremely low tensor-to-scalar ratios.
• These contrasting predictions offer testable implications for cosmic observables, paving the way for refined models in inflationary cosmology.

Analysis of Non-Minimal Coupling in Cosmic Inflation: Metric vs. Palatini Approaches

The comparative examination of scalar field theories in the context of cosmic inflation through non-minimal coupling forms the crux of the paper by Bauer and Demir, titled "Inflation with Non-Minimal Coupling: Metric vs. Palatini Formulations." This analysis distinguishes itself by contrasting the predictions of the Metric and the Palatini formalisms, unveiling the differences in their implications for cosmological parameters, specifically within an inflationary framework.

Analytical Framework

In both the Metric and Palatini formulations, the inflationary dynamics are driven by a non-minimally coupled scalar field. Fundamentally, these approaches differ in how the affine connection is treated: the Metric formalism adopts the Levi-Civita connection outright, while the Palatini formalism allows the connection to emerge dynamically from the scalar field's coupling to curvature. This distinction becomes crucial in determining the dependency on the non-minimal coupling parameter, denoted by $\zeta$.

Principal Findings

• Coupling Parameter Dependency: A significant divergence is noted in the preferred value of $\zeta$ for successful inflation between the two formalisms. The Palatini approach necessitates a considerably larger $\zeta$ compared to the Metric formalism. This stems from the divergent treatment of the affine connection, which directly influences the scalar field's dynamic scale and inflationary predictions.
• Inflationary Predictions: In the Metric formulation, $\zeta$ is approximately $4.9 \times 10^4 \sqrt{\lambda}$, while the Palatini approach requires $\zeta \approx 1.45 \times 10^{10} \lambda$. This indicates a stark contrast in how each formulation aligns with empirical data for $e$-foldings and spectral indices.
• Behavior of the Inflaton: The inflaton field, crucial for driving inflation, experiences different roles. In the Metric formalism, the inflaton $\psi$ must attain super-Planckian values, leading to concerns regarding quantum corrections and effective field theory validity. Conversely, the Palatini formalism maintains the inflaton values well below the Planck scale, suggesting a natural ease with which this model fits within known quantum gravity constraints.
• Observational Implications: Despite both formalisms predicting a similar spectral index ($n \approx 0.97$), they diverge in the expected tensor-to-scalar ratio. The Palatini approach forecasts extremely low tensor perturbations (close to $r = 10^{-14}$), a point of potential falsifiability given adequate future observational precision.

Implications and Future Directions

This analysis suggests that the choice between the Metric and Palatini formulations bears significant implications for theoretical and observational cosmology. The starkly different predictions for both $\zeta$ and inflaton values mean that Palatini's framework might be a more natural fit within the current understanding of high-energy physics, particularly if future observations discount significant tensor perturbations.

Theoretical exploration is warranted to further elucidate the impact of scalar fields in cosmological scenarios, potentially broadening the landscape of viable inflation models. Additionally, extending this dual approach analysis to other cosmological phenomena could provide deeper insights into the fundamental nature of spacetime and field interactions.

Conclusion

In summary, the paper furnishes a meticulous account of how the Metric and Palatini formulations, when endowed with non-minimal coupling, yield divergent cosmic inflationary landscapes. The Palatini formalism, with its smaller values for inflationary parameters and consequently low tensor perturbation predictions, could offer a more congruent path when juxtaposing theories against cosmological observations and the limitations of effective field theories. The research paves the way for further empirical testing and theoretical refinement in the exploration of scalar field dynamics within different gravitational paradigms.