How Do The Predictions Of The Inflationary Paradigm For The Tensor-to-scalar Ratio And The Running Of The Spectral Index Of The Scalar Perturbations Change When Incorporating The Effects Of A Non-minimal Coupling Between The Inflaton Field And The Ricci Curvature In A Model-independent Way?

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When a non-minimal coupling between the inflaton field and the Ricci curvature is introduced into the inflationary paradigm, the predictions for the tensor-to-scalar ratio (r) and the running of the spectral index (α_s) are modified as follows:

  1. Tensor-to-Scalar Ratio (r):

    • The tensor-to-scalar ratio is generally reduced compared to minimal coupling scenarios. This is because the non-minimal coupling typically flattens the effective potential, leading to a decrease in the slow-roll parameter ε, which directly affects r (since r = 16ε). Models like Higgs inflation exemplify this, where the non-minimal coupling results in a lower r.

  2. Running of the Spectral Index (α_s):

    • The running, which measures the scale dependence of the spectral index n_s, can be more significant. Non-minimal coupling introduces additional terms in the slow-roll parameters, particularly affecting η and higher-order derivatives. This can lead to a wider range of possible values for α_s, potentially aligning better with observational data.

  3. Slow-Roll Parameters:

    • The slow-roll parameters ε and η are modified due to the non-minimal coupling, altering the dynamics of the inflaton. These modifications affect the predictions for both r and n_s, often resulting in a spectral index closer to observations.

In conclusion, incorporating non-minimal coupling generally lowers the tensor-to-scalar ratio and can lead to a more pronounced running of the spectral index, enhancing the agreement with observational constraints. These changes depend on the coupling strength and specific potential but represent a model-independent shift from minimal coupling scenarios.