[Paper Review] Oscillating Universe in Horava-Lifshitz Gravity

Kei Maeda, Yosuke Misonoh+1 more|arXiv (Cornell University)|Jun 14, 2010

Cosmology and Gravitation Theories3 citations

TL;DR

This paper investigates isotropic and homogeneous cosmological solutions in generalized Hoýrava-Lifshitz gravity without the detailed balance condition, demonstrating that vacuum spacetimes can exhibit bounce or oscillatory behavior depending on coupling constants, spatial curvature $K$, and cosmological constant $\Lambda$. It proposes a quantum tunneling mechanism from an oscillating universe to an inflationary phase, yielding a macroscopic cyclic universe, with tunneling probabilities evaluated via Euclidean path integrals and shown to be finite but exponentially suppressed for Planck-scale vacuum energy.

ABSTRACT

We study the dynamics of isotropic and homogeneous universes in the generalized Hořava-Lifshitz gravity, and classify all possible evolutions of vacuum spacetime. In the case without the detailed balance condition, we find a variety of phase structures of vacuum spacetimes depending on the coupling constants as well as the spatial curvature $K$ and a cosmological constant $Λ$. A bounce universe solution is obtained for $Λ> 0, K=\pm 1$ or $Λ= 0, K=- 1$, while an oscillation spacetime is found for $Λ\geq 0, K=1$, or $Λ< 0, K=\pm 1$. We also propose a quantum tunneling scenario from an oscillating spacetime to an inflationary universe, resulting in a macroscopic cyclic universe.

Motivation & Objective

• To classify all possible vacuum spacetime evolutions in generalized Hoýrava-Lifshitz gravity without the detailed balance condition.
• To identify conditions under which non-singular bounces or oscillatory spacetimes arise in FLRW cosmology.
• To propose a quantum tunneling mechanism from an oscillating universe to an inflationary phase, enabling a macroscopic cyclic universe.
• To evaluate the Euclidean action and tunneling probability for the transition from oscillatory to expanding spacetime.

Proposed method

• Derivation of the effective Friedmann equation in non-projectable Hoýrava-Lifshitz gravity with arbitrary coupling constants.
• Analysis of phase space structures using the potential $\mathpzc{U}(\tilde{b})$ derived from the Euclidean action and scale factor $\tilde{b}(\tilde{\tau})$.
• Use of normalized variables $\tilde{b} = b / \ell$ with $\ell = \sqrt{3/\Lambda}$ to simplify the dynamics in the $K=1$, $\Lambda>0$ case.
• Evaluation of the Euclidean action $S_E$ via integration over the normalized scale factor, leading to $S_E \propto \int \tilde{b} \sqrt{2\mathpzc{U}(\tilde{b})} \, d\tilde{b}$.
• Introduction of a change of variables $\tilde{b}^2 = \tilde{b}_T^2(1 - k^2 u^2)$ to express the action in terms of elliptic-type integrals.
• Computation of the tunneling probability $P \sim e^{-S_E}$ in the static universe limit, yielding a finite but exponentially small value for Planck-scale vacuum energy.

Experimental results

Research questions

• RQ1 Under what conditions does Hoýrava-Lifshitz gravity with arbitrary couplings produce a non-singular bounce or oscillatory universe?
• RQ2 How does the absence of the detailed balance condition affect the phase structure of vacuum FLRW spacetimes?
• RQ3 Can quantum tunneling from an oscillating universe lead to a macroscopic inflationary phase in this framework?
• RQ4 What is the magnitude of the tunneling probability from an oscillating to an expanding universe in the Euclidean path integral approach?
• RQ5 How do the coupling constants and cosmological constant influence the stability and dynamics of the oscillatory and bounce solutions?

Key findings

• A bounce universe solution is found for $\Lambda > 0$, $K = \pm 1$, or $\Lambda = 0$, $K = -1$, indicating avoidance of the initial singularity.
• An oscillating spacetime emerges for $\Lambda \geq 0$, $K = 1$, or $\Lambda < 0$, $K = \pm 1$, due to periodic scale factor evolution.
• The Euclidean action for the tunneling process is derived as $S_E = \frac{12\pi^2 \ell^2}{\kappa^2} (\tilde{b}_T^2 - \tilde{b}_{\rm max}^2)^2 (\tilde{b}_T^2 - \tilde{b}_{\rm min}^2)^{1/2} \times \int_0^1 \frac{u^2 du}{1 - k^2 u^2} \sqrt{(1 - u^2)(1 - m^2 u^2)}$.
• In the static universe limit, the action simplifies to $S_E = \frac{4\pi^2 \ell^2}{\kappa^2} (1 - \tilde{g}_{\rm r})^{1/4} \left[1 - \frac{(1 + 2\sqrt{1 - \tilde{g}_{\rm r}})^{1/2} (1 - \sqrt{1 - \tilde{g}_{\rm r}})}{\sqrt{3}(1 - \tilde{g}_{\rm r})^{1/4}} \tanh^{-1}k \right]$, with $k^2 = \frac{3\sqrt{1 - \tilde{g}_{\rm r}}}{1 + 2\sqrt{1 - \tilde{g}_{\rm r}}}$.
• The tunneling probability is estimated as $P \sim \exp\left[-(60 - 120) \times \left(\frac{m_{\rm PL}^4}{\rho_{\rm vac}}\right)\right]$, which is finite but exponentially small for Planck-scale vacuum energy, indicating a viable but rare transition.