Cosmological Constant Problem and Renormalized Vacuum Energy Density in Curved Background

TL;DR

This paper investigates the origin of vacuum energy density in curved spacetime, clarifying that it arises from quantum effects rather than RG running, and provides bounds on scalar field masses based on cosmological data.

Contribution

It rigorously demonstrates that vacuum energy density in curved spacetime is due to quantum effects, not RG running, and refines the understanding of its origin in cosmology.

Findings

Vacuum energy density is due to quantum effects in curved spacetime.

Derived an upper bound on scalar field mass: m  M_{Pl}.

Clarified the physical interpretation of vacuum energy in cosmology.

Abstract

The current vacuum energy density observed as dark energy ${ \rho }_{ \rm dark }\simeq 2.5\times10^{-47}\ {\rm GeV^{4}}$ is unacceptably small compared with any other scales. Therefore, we encounter serious fine-tuning problem and theoretical difficulty to derive the dark energy. However, the theoretically attractive scenario has been proposed and discussed in literature: In terms of the renormalization-group (RG) running of the cosmological constant, the vacuum energy density can be expressed as ${ \rho }_{ \rm vacuum }\simeq m^{2}H^{2}$ where $m$ is the mass of the scalar field and rather dynamical in curved spacetime. However, there has been no rigorous proof to derive this expression and there are some criticisms about the physical interpretation of the RG running cosmological constant. In the present paper, we revisit the RG running effects of the cosmological constant and investigate the renormalized vacuum energy density in curved spacetime. We demonstrate that the vacuum energy density described by ${ \rho }_{ \rm vacuum }\simeq m^{2}H^{2}$ appears as quantum effects of the curved background rather than the running effects of cosmological constant. Comparing to cosmological observational data, we obtain an upper bound on the mass of the scalar fields to be smaller than the Planck mass, $m \lesssim M_{\rm Pl}$.