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Scalar-Induced Gravitational Waves from self-resonant preheating in $α$-attractor models

Daniel del-Corral, Paolo Gondolo, K. Sravan Kumar, João Marto

TL;DR

We address the production of Scalar-Induced Gravitational Waves during self-resonant preheating after $\alpha$-attractor inflation. The approach combines the Mukhanov-Sasaki description of scalar perturbations, Hill/Floquet analysis of resonance, and a second-order tensor equation with a scalar-scalar-tensor source to compute the SIGW spectrum, then relates it to the present energy density under cosmic expansion. By applying the BBN constraint on the integrated GW energy density, we derive lower bounds $\log_{10}(\alpha) > -3.54$ (T-model) and $> -3.17$ (E-model), translating into $r$-bounds $r > 9.61\times10^{-7}$ and $r > 2.25\times10^{-6}$, with the E-model typically more restrictive due to potential asymmetry. The results highlight that SIGWs in the very-high-frequency band can tightly constrain preheating dynamics in single-field inflation and motivate future non-linear studies and high-frequency GW detectors.

Abstract

After the inflationary phase, the universe enters the preheating phase, during which the inflaton field rolls down its potential and oscillates. When the potential significantly deviates from a parabolic shape at its minimum, these oscillations trigger an instability in the scalar perturbations, leading to their amplification. This phenomenon, known as self-resonance, has important implications in cosmology. Notably, since scalar perturbations couple to tensor perturbations at second order in the equations of motion, this amplification results in the production of Gravitational Waves (GWs), referred to as Scalar-Induced Gravitational Waves (SIGWs). In this study, we investigate the production of SIGWs during the preheating phase for a class of inflationary models known as $α$-attractors, characterized by a single parameter $α$. We focus on small values of this parameter, specifically $α\sim O(10^{-1} - 10^{-4})$, where the self-resonance effect is particularly pronounced. We obtain lower bounds on this parameter, $\log_{10}(α)>-3.54$ for the T-model and $\log_{10}(α)>-3.17$ for the E-model, based on the energy density of SIGWs constrained by Big Bang nucleosynthesis, which ultimately translates into lower bounds on the tensor-to-scalar ratio, $r>9.61\times10^{-7}$ for the T-model and $r>2.25\times10^{-6}$ for the E-model. Note that these bounds on $α$ and $r$ are derived within the linear framework of tensor fluctuations at the level of equations of motion, which nevertheless include scalar-scalar-tensor interactions with metric and matter fields. However, fully non-linear approaches, with all higher-order metric fluctuations, would be needed in the future to further validate these conclusions.

Scalar-Induced Gravitational Waves from self-resonant preheating in $α$-attractor models

TL;DR

We address the production of Scalar-Induced Gravitational Waves during self-resonant preheating after -attractor inflation. The approach combines the Mukhanov-Sasaki description of scalar perturbations, Hill/Floquet analysis of resonance, and a second-order tensor equation with a scalar-scalar-tensor source to compute the SIGW spectrum, then relates it to the present energy density under cosmic expansion. By applying the BBN constraint on the integrated GW energy density, we derive lower bounds (T-model) and (E-model), translating into -bounds and , with the E-model typically more restrictive due to potential asymmetry. The results highlight that SIGWs in the very-high-frequency band can tightly constrain preheating dynamics in single-field inflation and motivate future non-linear studies and high-frequency GW detectors.

Abstract

After the inflationary phase, the universe enters the preheating phase, during which the inflaton field rolls down its potential and oscillates. When the potential significantly deviates from a parabolic shape at its minimum, these oscillations trigger an instability in the scalar perturbations, leading to their amplification. This phenomenon, known as self-resonance, has important implications in cosmology. Notably, since scalar perturbations couple to tensor perturbations at second order in the equations of motion, this amplification results in the production of Gravitational Waves (GWs), referred to as Scalar-Induced Gravitational Waves (SIGWs). In this study, we investigate the production of SIGWs during the preheating phase for a class of inflationary models known as -attractors, characterized by a single parameter . We focus on small values of this parameter, specifically , where the self-resonance effect is particularly pronounced. We obtain lower bounds on this parameter, for the T-model and for the E-model, based on the energy density of SIGWs constrained by Big Bang nucleosynthesis, which ultimately translates into lower bounds on the tensor-to-scalar ratio, for the T-model and for the E-model. Note that these bounds on and are derived within the linear framework of tensor fluctuations at the level of equations of motion, which nevertheless include scalar-scalar-tensor interactions with metric and matter fields. However, fully non-linear approaches, with all higher-order metric fluctuations, would be needed in the future to further validate these conclusions.

Paper Structure

This paper contains 9 sections, 68 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scalar perturbations
  3. Tensor perturbations
  4. Energy density of gravitational waves
  5. Numerical strategy
  6. Conclusions
  7. Source term for SIGWs
  8. Power spectrum of SIGWs
  9. Comments on the results from Lattice simulations

Figures (8)

  • Figure 1: Schematic representation of the production of SIGWs during self-resonant preheating. The upper panel shows the evolution of selected comoving scales as a function of the scale factor of the universe, as well as the comoving Hubble radius $(aH)^{-1}$. The suffixes CMB, end, and rh correspond to the CMB pivot scale, the end of inflation, and the beginning of reheating, respectively. The evolution of the Fourier mode of the metric fluctuation $\Phi_k$ in the bottom panel is shown schematically to illustrate the approximate behavior of the perturbations. See the text for details.
  • Figure 2: Sensitivity curves of some GW detectors (in terms of $\Omega_{GW}$) as a function of the frequency in Hertz. Also shown is the BBN bound, taken from Smith:2006nkaMaggiore:1999vm. Solid (dashed) lines represent operative (planned or theorized) GW detectors. The figure is produced from the analysis of Kuroda:2015owvAggarwal:2020olq and references therein. The bottom lines with arrows mark the limits of the standard frequency bands of GWs, as defined in Kuroda:2015owv. The blue and magenta curves labeled as $r=0.028$ and $r=2.14\times10^{-6}$ represent two spectra of GWs computed from the upper and lower bounds on the tensor-to-scalar ratio; see the text for details. These are computed using the method outlined in this work.
  • Figure 3: Normalized $\alpha$-attractor potentials for several values of $\alpha$: $10^{-1}$ (blue), $10^{-2}$ (orange), $10^{-3}$ (green), and $10^{-4}$ (red).
  • Figure 4: Curvature perturbation $\mathcal{R}_{\bm k}$ as a function of the comoving wavenumber $k$ for (a) T-model and (b) E-model. The evaluation is performed from the end of inflation to 2 e-folds after in steps of 0.1 e-folds. The vertical dashed lines mark the scale $k_{\text{end}}$.
  • Figure 5: Total fractional energy density of GWs, eqn. \ref{['eq:total-omega']}, evaluated today for a T-model (dashed lines) and an E-model (solid lines), with different values of $\alpha$ and as a function of the frequency of each mode, expressed in Hz. The red horizontal line represents the BBN bound. Preheating is assumed to last for 5 e-folds.
  • ...and 3 more figures