[Paper Review] Probing symmetric teleparallel gravity in the early universe

Avik De, Dalia Saha+2 more|arXiv (Cornell University)|Sep 25, 2022

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates $f(Q)$ gravity in the early universe using the anisotropic Bianchi-I model, showing it fits inflationary parameters well but fails to produce a decelerated radiation-dominated era, undermining its viability as a dark energy-free cosmological model. The study highlights the theory's second-order field equations and geometric formulation via non-metricity, contrasting it with problematic $f(\mathbb{T})$ models.

ABSTRACT

General theory of relativity can be equivalently formulated on a flat space-time associating a torsion-free affine connection of non-vanishing non-metricity scalar $Q$. In this paper, we present an extension of this, viz., the $f(Q)$ theory of gravity, and explore the early evolution of the universe in the background of anisotropic Bianchi-I model. The $f(Q)$ theory in the current setting through its geometric modification is quite successful in explaining the late time accelerated expansion. Here we note that it accommodates latest released constraints on the inflationary parameters by Planck's collaboration group with excellent precession, but fails to produce a viable decelerated expansion in the radiation dominated era.

Motivation & Objective

• To assess the viability of symmetric teleparallel $f(Q)$ gravity in explaining early universe cosmology without dark energy.
• To investigate whether $f(Q)$ gravity can reproduce a decelerated expansion phase during the radiation-dominated era.
• To compare $f(Q)$ gravity with $f(\mathbb{T})$ theories, emphasizing the absence of skew-symmetric terms and Lorentz invariance issues in $f(Q)$.
• To explore the implications of $f(Q)$ gravity in anisotropic spacetime using the Bianchi-I metric under coincident gauge conditions.
• To determine whether $f(Q)$ gravity can serve as a viable alternative to $\Lambda$CDM by avoiding dark sector dependencies.

Proposed method

• Formulates $f(Q)$ gravity using a flat spacetime with non-vanishing non-metricity scalar $Q$, replacing the Levi-Civita connection with a torsion-free, non-metric affine connection.
• Applies the $f(Q)$ action principle in the coincident gauge to derive field equations for the anisotropic Bianchi-I metric.
• Solves the resulting field equations numerically or analytically under the assumption of spatial homogeneity and axial symmetry.
• Imposes physical constraints such as energy conditions and checks for consistency with known inflationary parameters (e.g., scalar spectral index, tensor-to-scalar ratio).
• Compares the evolution of the scale factor and Hubble parameter across epochs: inflation, radiation, and matter dominance.
• Analyzes the behavior of the effective energy density and pressure to assess whether a decelerated radiation phase emerges.

Experimental results

Research questions

• RQ1 Can $f(Q)$ gravity reproduce a decelerated expansion during the radiation-dominated era in the Bianchi-I model?
• RQ2 How well does $f(Q)$ gravity fit observational inflationary parameters such as $n_s$ and $r$?
• RQ3 Does $f(Q)$ gravity avoid the ghost and instability issues present in $f(R)$ and $f(\mathbb{T})$ theories?
• RQ4 What is the role of non-metricity in driving early universe dynamics compared to curvature or torsion?
• RQ5 Is $f(Q)$ gravity compatible with the requirement of a transition from inflation to radiation domination without introducing dark energy?

Key findings

• The $f(Q)$ theory fits recent observational inflationary parameters with excellent precision, indicating strong compatibility with CMB data.
• Despite this success, the model fails to produce a decelerated expansion phase in the radiation-dominated era, a critical requirement for successful Big Bang nucleosynthesis.
• The scalar field in the model becomes constant for $n = \frac{1}{3}$, effectively behaving as dark energy in the matter era, reintroducing the dark energy problem it aimed to avoid.
• The theory exhibits second-order field equations, avoiding the Ostrogradski instability common in $f(R)$ gravity, and maintains local Lorentz invariance.
• The $f(Q)$ framework avoids the skew-symmetric terms and extra degrees of freedom that plague $f(\mathbb{T})$ theories, making it theoretically more robust.
• The study identifies that $f(Q)$ gravity may still require extension to $f(\mathring{R}, Q)$ to resolve unresolved issues in the field equations, suggesting limitations in its current formulation.