[Paper Review] Generating cosmological perturbations in non-singular Horndeski cosmologies

Y. Ageeva, Pavel Petrov+1 more|arXiv (Cornell University)|Jul 8, 2022

Cosmology and Gravitation Theories61 references4 citations

TL;DR

This paper constructs a non-singular Horndeski bounce model with strong gravity in the past, showing that cosmological perturbations can be generated during an early contracting phase with observable spectra. By ensuring the scalar sound speed is small but finite, the model achieves a small tensor-to-scalar ratio $ r \sim 10^{-3} $, consistent with observations, while remaining in a classically and quantum-mechanically controllable regime despite early-time strong coupling.

ABSTRACT

We construct a concrete model of Horndeski bounce with strong gravity in the past. Within this model we show that the correct spectra of cosmological perturbations may be generated at early contracting epoch, with mild fine-tuning ensuring that the scalar spectral tilt $n_S$ and tensor-to-scalar ratio $r$ are consistent with observations. The smallness of $r$ is governed by the smallness of the scalar sound speed. Arbitrarily small values of $r$ are forbidden in our setup because of the strong coupling in the past. Nevertheless, we show that it is possible to generate perturbations in a controllable way, i.e. in the regime where the background evolution and perturbations are legitimately described within classical field theory and weakly coupled quantum theory.

Motivation & Objective

• To construct a concrete, non-singular Horndeski bounce model with strong gravitational coupling in the remote past.
• To demonstrate that cosmological perturbations can be generated during the early contracting phase without violating classical or quantum field theory.
• To achieve observational consistency by tuning the scalar sound speed to yield a small tensor-to-scalar ratio $ r \approx 10^{-3} $.
• To show that the no-go theorem for perturbations in standard bouncing Horndeski models can be evaded via power-law asymptotics with $ \mathcal{G}_T, \mathcal{F}_T, \mathcal{G}_S, \mathcal{F}_S \to 0 $ as $ t \to -\infty $.
• To ensure the model remains in a weakly coupled, classically valid regime despite early-time strong coupling, by comparing energy scales to the quantum strong coupling scale.

Proposed method

• Constructs a Horndeski model with power-law asymptotics: $ a(t) \propto (-t)^\chi $, $ \mathcal{G}_T, \mathcal{F}_T, \mathcal{G}_S, \mathcal{F}_S \propto (-t)^{-2\mu} $, with $ 2\mu > \chi + 1 $, ensuring convergence of the critical integrals in the no-go theorem.
• Works directly in the Jordan frame, avoiding conformal transformations used in prior works, and maintains $ G_5 = 0 $, $ G_4 = G_4(\phi) $, simplifying the cubic action structure.
• Derives the quadratic and cubic actions for scalar and tensor perturbations using background equations of motion and integration by parts, without field redefinitions.
• Identifies the dominant cubic interaction terms: $ \mathcal{S}^{(3)}_{\zeta\zeta\zeta} \propto \Lambda_\zeta \partial^2\zeta (\partial_i\zeta)^2 $ with $ \Lambda_\zeta = \mathcal{G}_T^3 / (4\Theta^2) $, and similar terms for $ h\zeta\zeta $ and $ hhh $, all involving at most two spatial derivatives.
• Uses conformal time $ \eta \propto -(-t)^{1-\chi} $ to map the tensor action to an Einstein-frame-like form, revealing a power-law inflationary phase for $ \mu > 1 $.
• Evaluates the scalar spectral tilt $ n_S $ and tensor-to-scalar ratio $ r $ by analyzing the mode functions and power spectra in the long-wavelength limit, with $ r \propto c_s^2 $, where $ c_s $ is the scalar sound speed.

Experimental results

Research questions

• RQ1 Can cosmological perturbations be generated in a non-singular Horndeski bounce model with strong gravity in the remote past?
• RQ2 Is it possible to achieve a small tensor-to-scalar ratio $ r \ll 1 $ while remaining in a classically and quantum-mechanically controllable regime?
• RQ3 How does the divergence of the integrals in the no-go theorem depend on the asymptotic behavior of $ \mathcal{G}_T, \mathcal{F}_T, \mathcal{G}_S, \mathcal{F}_S $ as $ t \to -\infty $?
• RQ4 What is the role of the scalar sound speed in determining $ r $, and can it be tuned to match observational bounds?
• RQ5 Can the model avoid pathologies such as superluminality or ghost instabilities while generating scale-invariant perturbations?

Key findings

• The model successfully generates cosmological perturbations during the early contracting phase with a scalar spectral tilt $ n_S \approx 0.96 $, consistent with Planck observations.
• The tensor-to-scalar ratio is suppressed by the scalar sound speed, yielding $ r \sim 10^{-3} $, which is observationally viable and arises naturally from the smallness of $ c_s^2 $.
• Despite the early-time vanishing of $ \mathcal{G}_T, \mathcal{F}_T, \mathcal{G}_S, \mathcal{F}_S $, the model remains in a classically valid regime because the classical energy scale $ \sim |t|^{-1} $ remains below the quantum strong coupling scale.
• The no-go theorem for perturbations in bouncing Horndeski models is evaded due to the convergence of the integrals (1a) and (1b), ensured by the condition $ 2\mu > \chi + 1 $.
• The dominant cubic interaction in the scalar sector is $ \mathcal{S}^{(3)}_{\zeta\zeta\zeta} = \int dtd^3x \, \Lambda_\zeta \partial^2\zeta (\partial_i\zeta)^2 $ with $ \Lambda_\zeta = \mathcal{G}_T^3 / (4\Theta^2) $, which governs the non-Gaussianity and mode evolution.
• The model avoids the need for beyond-Horndeski or DHOST extensions, remaining within the standard Horndeski class while achieving a healthy, controllable bounce with observable perturbation spectra.