[Paper Review] The probability of inflation in Loop Quantum Cosmology

William Nelson|ArXiv.org|Aug 24, 2007

Noncommutative and Quantum Gravity Theories17 references3 citations

TL;DR

This paper investigates the probability of sufficient slow-roll inflation in loop quantum cosmology (LQC), extending classical results that show exponential suppression of inflationary onset. Using effective LQC equations and a canonical measure, it finds that quantum gravity corrections do not overcome this suppression except for highly unnatural parameter choices, implying single-field inflation remains exponentially unlikely even in the quantum regime.

ABSTRACT

The probability of there being sufficient inflation to solve the fine-tuning associated with the horizon and flatness problems has recently been shown to be exponentially small, within the context of classical general relativity. Here this result is extended by considering loop quantum gravity effects, that are significant at small scales. In addition to accounting for high-energy departures from classicality, it is shown that, in contrast to the classical case, the loop quantum cosmological probability measure is naturally finite, at least in some well defined region. It is also shown that these loop quantum gravity corrections can overcome the classical suppression of the probability only for extremely unnatural choices of ambiguity parameters, implying that single field, slow-roll inflation is exponentially unlikely.

Motivation & Objective

• To assess whether loop quantum cosmology (LQC) corrections alleviate the exponential suppression of inflationary onset found in classical general relativity.
• To determine if the probability of achieving sufficient e-foldings (N ≳ 60) in single-field slow-roll inflation is enhanced by quantum gravity effects in LQC.
• To evaluate the robustness of the classical probability measure in the effective LQC framework, particularly regarding finiteness and dependence on ambiguity parameters.
• To compare the LQC probability measure with classical results and assess whether quantum corrections can render inflation a generic outcome.

Proposed method

• The study employs effective continuum equations derived from loop quantum cosmology (LQC), incorporating quantum corrections at high energies.
• It applies a canonical measure based on late-time equivalence of universes, as proposed in Ref. [21], to compute inflation probability in the LQC framework.
• The analysis uses the Hamiltonian constraint and holonomy corrections from LQC to model the dynamics of scalar field inflation in a homogeneous, isotropic universe.
• The probability is computed as a function of e-foldings N, with attention to the behavior of key terms like $[1 - (n-m+1)rac{q}{3D}rac{ ho D}{ ho q}]^{-1}$ and $\beta^2 \propto D_s^{\frac{m(\alpha+1)-\alpha(n+1)}{2\alpha}}$.
• The derivation accounts for ambiguity parameters (m, n, α) arising from the quantization procedure, particularly the Barbero-Immirzi parameter and regularization choices.
• The results are evaluated for standard $V(\phi) \sim \phi^2$ and $V(\phi) \sim \phi^4$ inflationary potentials, focusing on phenomenological relevance.

Experimental results

Research questions

• RQ1 Does loop quantum cosmology suppress the exponential probability suppression of inflation found in classical general relativity?
• RQ2 Can quantum gravity corrections in LQC significantly increase the likelihood of achieving $N \gtrsim 60$ e-foldings of slow-roll inflation?
• RQ3 What is the dependence of the inflation probability on the ambiguity parameters (m, n, α) in the LQC framework?
• RQ4 Is the probability measure in LQC finite and well-defined, particularly in contrast to the classical case?
• RQ5 For which choices of parameters does the LQC probability overcome the classical exponential suppression?

Key findings

• The probability of achieving $N \gtrsim 60$ e-foldings of single-field slow-roll inflation remains exponentially suppressed in loop quantum cosmology, scaling as $\exp(-3N)$, similar to the classical case.
• The LQC probability measure is finite and well-defined in the relevant region, in contrast to the classical case where such measures can diverge.
• Quantum corrections from LQC do not overcome the classical exponential suppression unless the ambiguity parameters satisfy $\frac{m(\alpha+1)-\alpha(n+1)}{2\alpha} \gtrsim 10^{110}$, which is considered physically unnatural.
• The term $\beta^2 \propto D_s^{\frac{m(\alpha+1)-\alpha(n+1)}{2\alpha}}$ can become large for sufficiently large exponents, but only for extreme parameter choices.
• Even in the LQC regime, the factor $[1 - (n-m+1)\frac{q}{3D}\frac{\partial D}{\partial q}]^{-1}$ does not significantly enhance the probability for $N \gtrsim 22.5$, due to potential zeros during inflation.
• The results imply that single-field slow-roll inflation is not a generic outcome in the effective LQC framework, challenging its naturalness despite quantum gravity corrections.