[Paper Review] Issues on the cosmological constant

Jun’ichi Yokoyama|ArXiv.org|May 19, 2003

Cosmology and Gravitation Theories8 references3 citations

TL;DR

This paper addresses the cosmological constant problem by proposing a theory with degenerate perturbative vacua where the small observed dark energy density arises from quantum tunneling via instantons, separating the hierarchy problem (Problem I) from the smallness of vacuum energy (Problem II). The model predicts a tiny vacuum energy density proportional to $ m^4 e^{-S_0} $, with the smallness explained by a large instanton action $ S_0 \approx 120\ln 10 + 4\ln(m/M_G) $, naturally avoiding fine-tuning.

ABSTRACT

Some issues of the cosmological constant or dark energy are briefly reviewed. There are an increasing number of observations that constrain the equation of state of dark energy more stringently and favor the time-independent cosmological constant. Then a plausible model of dark energy would be a theory with degenerate perturbative vacua in which its origin is explained by a nonperturbative effect so that, unlike quintessence, k-essence etc., it is separable from the perturbative problem why its amplitude is smaller than the Planckian density by a factor of $\order (10^{-120})$.

Motivation & Objective

• To address the cosmological constant problem by distinguishing between the hierarchy problem (Problem I) and the smallness of observed dark energy (Problem II).
• To propose a mechanism where the observed vacuum energy arises from quantum tunneling between degenerate perturbative vacua.
• To provide a framework that naturally explains the smallness of dark energy without introducing fine-tuned parameters.
• To reconcile observational constraints favoring a cosmological constant with a dynamical quantum origin of vacuum energy.

Proposed method

• Assumes the existence of two or more degenerate perturbative vacua $|+\rangle$ and $|-\rangle$ with zero vacuum energy after solving Problem I.
• Introduces quantum tunneling between these vacua via an instanton with Euclidean action $ S_0 $, leading to energy eigenstates $ |S\rangle $ and $ |A\rangle $.
• Calculates the true vacuum energy as $ \rho_{\rm v} = -m^4 e^{-S_0} $ for the ground state and $ \rho_{\rm v} = m^4 e^{-S_0} $ for the first excited state.
• Derives the required instanton action $ S_0 = 120\ln 10 + 4\ln(m/M_G) $ to match the observed dark energy density $ \sim 10^{-120} M_G^4 $.
• Requires $ m \gtrsim M_G $ to ensure the metastability of the perturbative vacuum over cosmological timescales.
• Uses the superposition of vacua to explain the observed finite dark energy density with 50% probability in the current vacuum state.

Experimental results

Research questions

• RQ1 How can the small observed value of the cosmological constant be explained without fine-tuning?
• RQ2 Can the cosmological constant problem be separated into two distinct issues: the hierarchy problem (why vacuum energy is zero) and the smallness problem (why it is not zero but tiny)?
• RQ3 What nonperturbative mechanism can generate a tiny but finite vacuum energy density from degenerate perturbative vacua?
• RQ4 Is the observed dark energy consistent with a time-independent cosmological constant, and how does this constrain dynamical dark energy models?
• RQ5 What are the long-term cosmological consequences of a stable, positive cosmological constant?

Key findings

• Observational data increasingly favor a time-independent cosmological constant with equation of state $ w \approx -1 $, disfavoring dynamical models like quintessence.
• The model explains the smallness of dark energy not through small parameters but through a large instanton action $ S_0 \approx 120\ln 10 + 4\ln(m/M_G) $, which naturally suppresses the vacuum energy.
• The observed dark energy density $ \sim 10^{-120} M_G^4 $ is reproduced when $ S_0 $ is tuned to this value, with $ m \gtrsim M_G $ ensuring long-lived metastable vacua.
• The model separates Problem I (vanishing vacuum energy) from Problem II (small nonzero value), offering a unified framework that avoids fine-tuning.
• The future of the Universe under a stable $ \Lambda $-dominated de Sitter phase includes an event horizon at ~5.1 Gpc, limiting observable galaxies to $ z < 1.8 $, and eventual isolation of only the Local Group.
• Extragalactic astronomy and cosmology will be effectively impossible beyond ~100 billion years, emphasizing the urgency of current research.