[Paper Review] Non-metricity with bounday terms: $f(Q,C)$ gravity and cosmology

Avik De, Tee‐How Loo+1 more|arXiv (Cornell University)|Aug 1, 2023

Cosmology and Gravitation TheoriesPhysics and Astronomy20 references9 citations

TL;DR

The paper formulates f(Q,C) gravity in symmetric teleparallel geometry by including the boundary term C with the non-metricity scalar Q in the action, derives the general field equations, and explores FRW cosmology for three connection types, yielding a geometrical dark-energy sector and possible matter–dark-energy interactions, including phantom-divide crossing in a specific model.

ABSTRACT

We formulate $f(Q,C)$ gravity and cosmology. Such a construction is based on the symmetric teleparallel geometry, but apart form the non-metricity scalar $Q$ we incorporate in the Lagrangian the boundary term $C$ of its difference form the standard Levi-Civita Ricci scalar $\mathring R$. We extract the general metric and affine connection field equations, we apply them at a cosmological framework, and adopting three different types of symmetric teleparallel affine connections we obtain the modified Friedmann equations. As we show, we acquire an effective dark-energy sector of geometrical origin, which can lead to interesting cosmological phenomenology. Additionally, we may obtain an effective interaction between matter and dark energy. Finally, examining a specific model, we show that we can obtain the usual thermal history of the universe, with the sequence of matter and dark-energy epochs, while the effective dark-energy equation-of-state parameter can be quintessence-like, phantom-like, or cross the phantom-divide during evolution.

Motivation & Objective

• Introduce and motivate f(Q,C) gravity by incorporating the boundary term C alongside Q in the Lagrangian.
• Derive the metric and affine connection field equations in symmetric teleparallel geometry.
• Apply the theory to cosmology and obtain modified Friedmann equations for three symmetric teleparallel connections.
• Identify the effective geometrical dark-energy sector and possible matter–dark energy interactions.
• Demonstrate a specific model yielding standard cosmic history and phantom-divide crossing.

Proposed method

• Propose the action S = ∫ (1/2κ) f(Q,C) √(-g) d^4x and perform variations to obtain field equations (metric and connection).
• Compute Q and C in symmetric teleparallel settings and derive covariant metric field equations (including f_Q and f_C terms).
• Define the effective stress-energy T^{eff}_{μν} and show how the usual Einstein tensor relates to T^{eff} in f(Q,C) gravity.
• Specialize to FRW with three symmetric teleparallel connections, deriving the corresponding Friedmann equations (3.7–3.12 out of context).
• Extract explicit expressions for the effective dark-energy density and pressure ρ_DE, p_DE for each connection type (Eqs. 4.10–4.11, 4.20–4.21, 4.28–4.29).
• Discuss energy conservation and possible matter–dark-energy exchanges via the connection field equations (Eqs. 3.17–3.20, 4.22, 4.30).

Figure 1 : The evolution of the effective dark energy density parameter $\Omega_{DE}$ and of the matter density parameter $\Omega_{m}$ , as a function of the redshift $z$ , for $f(Q,C)$ cosmology with Type II non-vanishing connection ( 4.12 ) with condition ( 4.17 ), with $\gamma_{0}=1$ in units whe

Experimental results

Research questions

• RQ1 How does including the boundary term C in the Q-based gravity modify cosmological dynamics compared to f(Q) or f(R)-based theories?
• RQ2 What are the resulting Friedmann equations and cosmological phenomenology for the three symmetric teleparallel connections in FRW spacetime?
• RQ3 Can f(Q,C) gravity produce an effective geometrical dark-energy sector and permit interactions with matter?
• RQ4 Under what conditions can the effective dark-energy equation of state cross the phantom divide (w_DE = -1)?

Key findings

• f(Q,C) gravity yields an effective geometrical dark-energy sector in FRW cosmology.
• For Connection Type I the Friedmann equations acquire ρ_DE and p_DE given by equations (4.10)–(4.11).
• Connection Type II introduces an effective interaction in the continuity equation (4.22) through the γ-dependent terms.
• Connection Type III yields a different but related ρ_DE, p_DE structure (4.28–4.29) and a corresponding interaction term (4.30).
• A concrete example with Type II and γ(t) = γ0/a^3(t) yields standard cosmic history and a w_DE that can be quintessence-like, phantom-like, or cross the phantom divide (Figures 1–2).

Figure 2 : The evolution of the effective dark-energy equation-of-state parameter $w_{DE}$ given in ( 4.33 ), as a function of the redshift $z$ , for $f(Q,C)$ cosmology with with Type II non-vanishing connection ( 4.12 ) with condition ( 4.17 ), with $\gamma_{0}=1$ in units where $\kappa=1$ . We hav