[Paper Review] Novel couplings between nonmetricity and matter

Francisco S. N. Lobo, Tiberiu Harko+3 more|arXiv (Cornell University)|Jan 3, 2019

Cosmology and Gravitation Theories4 citations

TL;DR

This paper introduces a novel class of gravity theories by nonminimally coupling the nonmetricity scalar $ Q $ to the matter Lagrangian within a metric-affine framework, leading to nonconserved energy-momentum and an extra force. The key contribution is a cosmologically viable extension of symmetric teleparallel gravity that generates effective dark energy-like dynamics through $ f_1(Q) $ and $ f_2(Q) $ couplings, enabling late-time accelerated expansion without standard dark energy components.

ABSTRACT

We present a novel theory of gravity, namely, an extension of symmetric teleparallel gravity. This is done by introducing a new class of theories where the nonmetricity $Q$ is coupled nonminimally to the matter Lagrangian. This nonminimal coupling entails the nonconservation of the energy-momentum tensor, and consequently the appearance of an extra force. We also present several cosmological applications.

Motivation & Objective

• To extend symmetric teleparallel gravity by introducing nonminimal couplings between nonmetricity and matter fields.
• To explore the physical consequences of nonconserved energy-momentum tensors arising from such couplings.
• To construct cosmological models that reproduce late-time acceleration without invoking dark energy.
• To establish theoretical consistency and phenomenological viability of the proposed $ f(Q) $-gravity extension.

Proposed method

• Formulates an action principle with two functions: $ f_1(Q) $ for the nonmetricity part and $ f_2(Q)L_M $ for the nonminimally coupled matter Lagrangian.
• Derives gravitational field equations via independent variations with respect to the metric and affine connection in the metric-affine formalism.
• Introduces the superpotential $ P^{ ho}_{\mu\nu} $ and traces $ Q_\alpha $, $ \tilde{Q}_\alpha $ to express nonmetricity invariants.
• Derives the effective energy-momentum tensor and identifies an extra force $ \mathcal{F}^\lambda $ arising from nonconservation of $ T_{\mu\nu} $.
• Applies the formalism to Friedmann-Robertson-Walker (FRW) cosmology, deriving generalized Friedmann equations with effective energy density $ \rho_{\rm eff} $ and pressure $ p_{\rm eff} $.
• Expresses the deceleration parameter $ q $ and dark energy equation of state $ w $ in terms of $ f_1(Q) $, $ f_2(Q) $, and their derivatives.

Experimental results

Research questions

• RQ1 How does nonminimal coupling between nonmetricity $ Q $ and matter Lagrangian $ L_M $ affect energy-momentum conservation?
• RQ2 What are the dynamical consequences of such a coupling in cosmological spacetimes?
• RQ3 Can this framework reproduce late-time cosmic acceleration without standard dark energy?
• RQ4 What is the structure of the extra force acting on matter in this theory?
• RQ5 How do the effective energy density and pressure in the generalized Friedmann equations depend on the coupling functions $ f_1(Q) $ and $ f_2(Q) $?

Key findings

• The energy-momentum tensor is nonconserved due to the nonminimal $ Q $-matter coupling, leading to an extra force $ \mathcal{F}^\lambda $ with components $ \mathcal{F}^\lambda_{\mathcal{T}} $ and $ \mathcal{F}^\lambda_{\mathcal{H}} $.
• For a perfect fluid with $ L_M = p $, the extra force $ \mathcal{F}^\lambda_{\mathcal{T}} $ vanishes identically, simplifying the dynamics.
• In FRW cosmology, the generalized Friedmann equations take the form $ 3H^2 = \rho_{\rm eff} $ and $ 2\dot{H} + 3H^2 = -p_{\rm eff} $, with $ \rho_{\rm eff} $ and $ p_{\rm eff} $ depending on $ f_1(Q) $, $ f_2(Q) $, and $ F = f_1'(Q) + 2f_2'(Q)L_M $.
• The effective equation of state parameter $ w = p_{\rm eff}/\rho_{\rm eff} $ is expressed in terms of $ \dot{F} $, $ f_1 $, $ f_2 $, $ \rho $, and $ p $, enabling dark energy-like behavior.
• In vacuum ($ \rho = p = 0 $), the deceleration parameter becomes $ q = -1 + 12\dot{F}H/f_1 $, showing that specific forms of $ f_1 $ and $ f_2 $ can yield de Sitter expansion.
• The theory supports a late-time de Sitter phase for appropriate choices of $ f_1(Q) $ and $ f_2(Q) $, indicating potential for a gravitational origin of cosmic acceleration.