[Paper Review] Computation of non-Gaussianity in loop quantum cosmology

V. Sreenath, Iván Agulló+1 more|arXiv (Cornell University)|Apr 1, 2019

Noncommutative and Quantum Gravity Theories4 citations

TL;DR

This paper computes primordial non-Gaussianity in loop quantum cosmology (LQC) using the dressed metric approach, analyzing second-order perturbations from the quantum bounce. It finds that next-to-leading order corrections to the power spectrum are negligible, and the resulting $f_{\text{NL}}$ remains within observational bounds from Planck, supporting the viability of LQC as a pre-inflationary model with detectable, scale-dependent non-Gaussianity at low multipoles.

ABSTRACT

We summarize our investigations of the second-order perturbations in loop quantum cosmology (LQC). We shall discuss, primarily, two aspects. Firstly, whether the second-order contributions arising from the cosmic bounce, occurring at Planck scale, could be large enough to break the validity of perturbation theory. Secondly, the implications of the upper bounds on primordial non-Gaussianity, arrived at by the Planck collaboration, on the LQC phenomenology.

Motivation & Objective

• To assess whether second-order perturbative corrections from the LQC bounce invalidate standard perturbation theory.
• To evaluate whether the non-Gaussianity generated during the quantum bounce is consistent with Planck satellite constraints on primordial non-Gaussianity.
• To explore new phenomenological predictions of LQC, particularly scale-dependent $f_{\text{NL}}$ effects at low multipoles.
• To constrain the scalar field value at the bounce using observational bounds on $f_{\text{NL}}$.

Proposed method

• Uses the dressed metric approach to compute the bispectrum of curvature perturbations in LQC, treating perturbations as quantum fields on an effective FLRW background.
• Derives the interaction Hamiltonian at third order in perturbations, sourced by the effective background dynamics of the quantum bounce.
• Expresses the curvature bispectrum in terms of inflaton perturbation bispectra and the $a/z$ ratio, linking to observable power spectra.
• Applies Cauchy’s residue theorem to evaluate time-integrated contributions to the bispectrum, yielding an exponential spectral dependence $e^{-\alpha k_t/k_{\text{LQC}}}$.
• Computes the first-order correction to the power spectrum using the bispectrum, quantifying its relative magnitude via $\Delta\mathcal{P}_{\mathcal{R}}/\mathcal{P}_{\mathcal{R}}$.
• Performs numerical and analytical comparisons to validate the spectral dependence and correction size, confirming sub-dominance of higher-order terms.

Experimental results

Research questions

• RQ1 Is the second-order correction to the power spectrum in LQC sub-dominant, ensuring the validity of perturbation theory?
• RQ2 Can the non-Gaussianity generated during the LQC bounce produce $f_{\text{NL}}$ values detectable by Planck, particularly at low multipoles?
• RQ3 What constraints can be placed on the scalar field value at the bounce ($\phi_B$) given Planck’s $f_{\text{NL}}$ bounds and the expected scale of non-Gaussianity?
• RQ4 How does the bounce's curvature affect the amplitude and scale dependence of $f_{\text{NL}}$?
• RQ5 Can the oscillatory features in $f_{\text{NL}}$ relax the conservative bounds on $\phi_B$?

Key findings

• The first-order correction to the power spectrum, $\Delta\mathcal{P}_{\mathcal{R}}/\mathcal{P}_{\mathcal{R}}$, is bounded by $10^{-4}$, confirming that higher-order corrections are negligible and perturbation theory remains valid.
• The dimensionless non-Gaussianity parameter $f_{\text{NL}}$ remains within observational bounds, with $f_{\text{NL}} \leq 10^4$, ensuring consistency with Planck data.
• For $\rho_{\text{B}} = 1\,M_{\text{Pl}}^4$, the scalar field at the bounce is constrained to $7.46\,M_{\text{Pl}} \leq \phi_{\text{B}} \leq 7.82\,M_{\text{Pl}}$, based on the requirement that bounce imprints appear at observable scales.
• The spectral dependence of the bispectrum is found to be $e^{-\alpha k_t/k_{\text{LQC}}}$, matching numerical results and confirming analytical predictions.
• Variations in $\phi_B$ and $\rho_{\text{B}}$ only shift the scale or amplitude of $f_{\text{NL}}$, without altering the qualitative behavior, indicating robustness of the results.
• The oscillatory nature of $f_{\text{NL}}$ at low multipoles suggests that conservative bounds on $\phi_B$ may be relaxed in a more detailed analysis.