[Paper Review] Intermediate inflation under the scrutiny of recent data

Sérgio del Campo|arXiv (Cornell University)|Apr 7, 2014

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper investigates intermediate inflation models using the Hamilton-Jacobi approach with flow equations to compute observational parameters like the scalar spectral index, its running, and tensor-to-scalar ratio. It finds that these models are strongly disfavored by recent Planck and BICEP2 data, lying outside the 95% credible region in the $n_s/r$ plane for all parameter values, and suffer from a lack of a physical minimum in the scalar potential.

ABSTRACT

We use the flow equations to determine the different hierarchy Hubble parameters as a function of the number of e-folds for intermediate models in single-field inflation. The obtained expressions allow us to determine at second order in the hierarchy Hubble parameters different observational parameters. We distinguish the scalar spectral index, its running and the tensor-to-scalar ratio, among others. Recently, it has been noticed that measurements released by Planck, combined with the WMAP large-angle polarization are in tension with this sort of model. Here, we show in detail why this occur. The conclusions do not change even when the recent BICEP2 data are included.

Motivation & Objective

• To analyze intermediate inflation models beyond the slow-roll approximation using the Hamilton-Jacobi framework.
• To compute second-order observational parameters—such as the scalar spectral index $n_s$, its running $\alpha_s$, and tensor-to-scalar ratio $r$—using hierarchy flow equations.
• To test the viability of intermediate inflation against the latest cosmological data from Planck and BICEP2.
• To identify fundamental flaws in the intermediate inflation model, particularly the absence of a minimum in the scalar potential.
• To assess whether similar issues affect related models like logamediate inflation.

Proposed method

• Employing the Hamilton-Jacobi approach, where the Hubble parameter $H(\phi)$ serves as a generating function for the inflaton potential.
• Deriving hierarchy Hubble parameters via flow equations that relate higher-order parameters to derivatives of the first Hubble parameter with respect to e-folds.
• Expressing key observational parameters ($n_s$, $\alpha_s$, $r$) up to second order in the hierarchy parameters using analytical flow equation solutions.
• Using the relation $r(n_s) \approx \frac{8\beta}{\beta - 2}(1 - n_s)$ to map model predictions in the $n_s/r$ plane.
• Comparing model predictions with 68% and 95% credible regions from Planck + WMAP large-angle polarization and BICEP2 data.
• Analyzing the scalar potential structure to assess whether it allows for a natural end of inflation.

Experimental results

Research questions

• RQ1 How do intermediate inflation models perform when analyzed beyond the slow-roll approximation using flow equations?
• RQ2 To what extent do intermediate inflation predictions for $n_s$ and $r$ align with current Planck and BICEP2 observational constraints?
• RQ3 What are the implications of the scalar potential lacking a minimum for the physical viability of intermediate inflation?
• RQ4 Why do intermediate inflation models fail to satisfy the 95% credible region in the $n_s/r$ plane despite being analytically well-defined?
• RQ5 Are similar issues expected in related models such as logamediate inflation under the single-field assumption?

Key findings

• Intermediate inflation models, when analyzed via second-order flow equations, predict values of $n_s$ and $r$ that lie entirely outside the 95% credible region of Planck + WMAP and BICEP2 data.
• The model is disfavored for all values of the parameter $f$ (or equivalently $\beta$), with curves lying beyond the 95% confidence contour in the $n_s/r$ plane.
• Even when including BICEP2 data, the model remains inconsistent with observations, as the predicted $r(n_s)$ relation falls outside the combined data contours.
• The scalar potential in intermediate inflation does not possess a minimum for finite $\phi$, approaching zero only as $\phi \to \infty$, which prevents a natural end of inflation.
• The absence of a potential minimum implies that an additional scalar field (e.g., a curvaton) is required to end inflation, undermining the single-field assumption.
• The model's failure is not due to approximation errors but stems from fundamental structural issues in the potential, suggesting broader problems for similar models like logamediate inflation.