Inflationary models with a quadratic relationship between the parameters of cosmological perturbations

TL;DR

The paper develops a reconstruction framework that determines the scalar-field potential $V(\phi)$ from a prescribed evolution of the field with the number of e-folds, $\phi(N)$, within Einstein gravity and slow-roll dynamics. By expressing $H(N)$, $\varepsilon(N)$, and $\delta(N)$ in terms of $\phi(N)$, the authors reconstruct $V(\phi)$ and demonstrate a concrete logarithmic example that yields a quadratic perturbation relation $r=4(1-n_S)^2$. They compute exact background and perturbation quantities, showing the model can satisfy Planck constraints for $n_S$ and $r$ with a consistent $\Delta N$ between horizon crossing and inflation end, and they compare their exact-results to ML-derived optimal potentials, highlighting differences due to exact versus approximate slow-roll treatments. The work emphasizes the need to use exact cosmological solutions when validating inflationary models against observations and discusses potential extensions to modified gravity theories and gravitational waves implications.

Abstract

An approach to construct cosmological inflation models on the basis of a certain dependence of the scalar field evolution on the e-folds number is considered. The reconstruction of the model background parameters according to the kind of specific connection between the parameters of cosmological perturbations is proposed. By the slow-roll approximation, as functions of the e-fold number, the scalar field potential and the Hubble parameter are determined. The logarithmic dependence between the scalar field evolution and the number of e-folds is investigated as the example. The derived model was found to obey a quadratic law between the parameters of cosmological perturbations. On the basis of suggested approach, a new cosmological inflation model with an exponential potential is proposed. This complies with existing observational constraints for both background and perturbation spectrum parameters. A comparison of obtained results with the solutions obtained by the symbol regression is made. The restrictions for arbitrary parameters in the model with logarithmic potential are also considered. The proposed form of the scalar field potential is consistent with one of the optimal potentials obtained when using machine learning methods to analyze the correspondence of inflationary models to observational data and gravitational waves contribution to the anisotropy and polarization of the CMB.