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Quintessential inflation studied through Semiclassical Methods

Jordan Zambrano, Miguel Agama, Marcos Garzón, Werner Bramer-Escamilla, Clara Rojas, Teófilo Vargas

TL;DR

This study connects inflation and late-time acceleration through quintessential inflation in the α-attractor framework, solving scalar and tensor perturbations with semiclassical methods. It compares the uniform-approximation and third-order phase-integral approaches against slow-roll and full numerics to compute $P_S(k)$, $P_T(k)$, and observables $n_S$ and $r$, highlighting that the third-order phase-integral yields the best scalar-spectrum accuracy while slow-roll best handles tensors. By contrasting with Starobinsky inflation, the work shows both models align with Planck constraints, with α-attractor yielding a smaller tensor-to-scalar ratio $r$. The paper also provides analytic background fits $a_{ m fit}(t)$ and $\varphi_{ m fit}(t)$ to facilitate semiclassical analyses in quintessential-inflation scenarios, advancing practical tools for precision cosmology.

Abstract

In this work, we solved the scalar and tensor perturbation equations numerically and using the improved uniform approximation method together with the third-order phase-integral method, for the $α$-attractor inflationary model. This inflationary model has become very important because it allows us to describe the initial accelerated expansion of the universe in the inflationary epoch, and the current accelerated expansion with the same potential that depends on one scalar field $\varphi$. Once the equations for the scalar and tensor power spectra are found, we calculate the observables: the scalar-to-tensor ratio $r$, and the scalar spectral index $n_S$, concluding that semiclassical methods give excellent results compared to numerical integration. We also compare both observables in the $α$-attractor and the Starobinsky inflationary model.

Quintessential inflation studied through Semiclassical Methods

TL;DR

This study connects inflation and late-time acceleration through quintessential inflation in the α-attractor framework, solving scalar and tensor perturbations with semiclassical methods. It compares the uniform-approximation and third-order phase-integral approaches against slow-roll and full numerics to compute , , and observables and , highlighting that the third-order phase-integral yields the best scalar-spectrum accuracy while slow-roll best handles tensors. By contrasting with Starobinsky inflation, the work shows both models align with Planck constraints, with α-attractor yielding a smaller tensor-to-scalar ratio . The paper also provides analytic background fits and to facilitate semiclassical analyses in quintessential-inflation scenarios, advancing practical tools for precision cosmology.

Abstract

In this work, we solved the scalar and tensor perturbation equations numerically and using the improved uniform approximation method together with the third-order phase-integral method, for the -attractor inflationary model. This inflationary model has become very important because it allows us to describe the initial accelerated expansion of the universe in the inflationary epoch, and the current accelerated expansion with the same potential that depends on one scalar field . Once the equations for the scalar and tensor power spectra are found, we calculate the observables: the scalar-to-tensor ratio , and the scalar spectral index , concluding that semiclassical methods give excellent results compared to numerical integration. We also compare both observables in the -attractor and the Starobinsky inflationary model.

Paper Structure

This paper contains 26 sections, 84 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The $\alpha$--attractor Inflationary Model
  3. Background equations
  4. Numerical solution
  5. Solution into the Slow--roll approximation
  6. Fit of thes Scale Factor $a(t)$
  7. Fit of the Scalar Field $\varphi(t)$
  8. Comparison of the plots for the scale factor $a(t)$ and the scalar field $\varphi(t)$ calculated numerically and with the slow--roll approximation
  9. Comparison of the plots of the scale factor $a(t)$ and the scalar field $\varphi(t)$ calculated numerically and with the fit expression
  10. Statistical comparison of the scale factor $a(t)$ and the scalar field $\varphi(t)$ calculated numerically, with the slow--roll approximation, and with the fit expression
  11. Equation of perturbations
  12. Solution to the equations of perturbations
  13. Numerical Integration
  14. Slow--roll approximation
  15. Uniform approximation method
  16. ...and 11 more sections

Figures (10)

  • Figure 1: $\alpha$--attractor inflationary potential for $n=124$, $\alpha=10^{-2}$, and $\lambda=2.2399 \times 10^{-66}$.
  • Figure 2: Scale factor $a(t)$ for the $\alpha$--attractor inflationary model until $t=7 \times 10^7$. Purple dashed--line represents the numerical solution, and light blue solid line represents the slow--roll approximation.
  • Figure 3: Scalar field $\varphi(t)$ for the $\alpha$--attractor inflationary model until $t=7 \times 10^7$. Blue dashed--line represents the numerical solution, and light blue solid represents the slow--roll approximation.
  • Figure 4: Scale factor $a(t)$ for the $\alpha$--attractor inflationary model until $t=7 \times 10^7$. The magenta dashed--line line represents the numerical solution, and light blue solid line represents the fit model.
  • Figure 5: Scalar field $\varphi(t)$ for the $\alpha$--attractor inflationary model until $t=7 \times 10^7$. Blue dashed--line line represents the numerical solution, and light blue solid line represents the fit model.
  • ...and 5 more figures