[Paper Review] Dynamical Chern-Simons modified gravity and Friedmann-Robertson-Walker metric

C. Furtado, J. R. Nascimento+2 more|arXiv (Cornell University)|May 11, 2010

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper investigates the consistency of the Friedmann-Robertson-Walker (FRW) metric in dynamical Chern-Simons modified gravity, where the Chern-Simons coefficient $Θ$ is a dynamical scalar field. It shows that the modified gravity model supports accelerated cosmic expansion similar to the Chaplygin gas model and introduces a new density component $\Omega_{\Theta}$, with a numerically estimated value of $\Omega_{0,\Theta} \approx 2.2 \times 10^{-6}$, which is subdominant but non-negligible in the cosmological energy budget.

ABSTRACT

We study the conditions for the consistency of the Friedmann-Robertson-Walker (FRW) metric with the dynamical Chern-Simons modified gravity. It turns out to be that in this situation the accelerated expansion of the Universe takes place, with the time dependence of the scale factor turns out to be similar to the case of presence of the Chaplygin gas. Also we found that this modification changes the total density of the Universe and therefore gives a nontrivial impact to a cosmological scenario.

Motivation & Objective

• To assess whether the Friedmann-Robertson-Walker (FRW) metric is a consistent solution in dynamical Chern-Simons modified gravity, where the Chern-Simons coefficient $\Theta$ is a dynamical scalar field rather than a constant.
• To determine the cosmological impact of the dynamical $\Theta$ field on the expansion history of the Universe.
• To estimate the contribution of the $\Theta$ field to the total energy density of the Universe using observational constraints on the Hubble parameter, matter density, and dark energy.

Proposed method

• Formulate the action for dynamical Chern-Simons modified gravity, including the Einstein-Hilbert term, a dynamical Chern-Simons term proportional to $\Theta \, {}^{*}RR$, and a kinetic term for $\Theta$, with $\Theta$ treated as a dynamical scalar field.
• Derive the modified Einstein equations and the scalar field equation of motion, with the Cotton tensor $C_{\mu\nu}$ coupling to the Pontryagin term $^{*}RR$.
• Apply the FRW metric ansatz to compute the relevant geometric quantities: Christoffel symbols, Riemann and Ricci tensors, and Ricci scalar.
• Compute the Cotton tensor components for the FRW metric and show that $C^{\mu\nu} = 0$, implying the modified Einstein equations reduce to $G_{\mu\nu} = 8\pi G T_{\mu\nu}$, with the energy-momentum tensor including contributions from $\Theta$.
• Derive the modified Friedmann equation including a new term $\Omega_{\Theta}$, which arises from the dynamical $\Theta$ field, and express the Hubble parameter in terms of $\Omega_{0,d}$, $\Omega_{0,r}$, $\Omega_{0,\Lambda}$, and $\Omega_{0,\Theta}$.
• Use observational constraints (e.g., $H_0 \approx 2.27 \times 10^{-18}\,\text{s}^{-1}$, $t_0 \approx 12.5 \times 10^9\,\text{years}$, $\Omega_{0,d} \approx 0.3$, $\Omega_{0,\Lambda} \approx 0.7$) to numerically solve for $\Omega_{0,\Theta}$.

Experimental results

Research questions

• RQ1 Does the Friedmann-Robertson-Walker metric remain a consistent solution in dynamical Chern-Simons modified gravity with a dynamical $\Theta$ field?
• RQ2 How does the dynamical $\Theta$ field affect the expansion rate and the form of the Friedmann equation?
• RQ3 What is the numerical value of the density parameter $\Omega_{0,\Theta}$ associated with the $\Theta$ field, given current cosmological observations?
• RQ4 How does the time evolution of the scale factor in this model compare to that of the Chaplygin gas model?
• RQ5 Is the contribution of $\Omega_{0,\Theta}$ significant enough to alter the dominance of dark matter and dark energy in the present Universe?

Key findings

• The FRW metric is a consistent solution in dynamical Chern-Simons modified gravity, as the Cotton tensor vanishes for this metric, allowing the modified Einstein equations to reduce to standard form with an effective energy-momentum tensor.
• The modified gravity model supports accelerated expansion of the Universe, with the scale factor's time dependence resembling that of the Chaplygin gas model, indicating a similar cosmological evolution.
• A new contribution to the total energy density arises from the dynamical $\Theta$ field, quantified by the density parameter $\Omega_{\Theta}$, which modifies the standard Friedmann equation.
• Using observational constraints on the Hubble parameter, age of the Universe, and matter/dark energy densities, the paper numerically estimates $\Omega_{0,\Theta} \approx 2.2 \times 10^{-6}$.
• This value is subdominant compared to dark matter ($\Omega_{0,d} \approx 0.3$) and dark energy ($\Omega_{0,\Lambda} \approx 0.7$), but non-negligible in the context of modified gravity phenomenology.
• The contribution $\Omega_{0,\Theta}$ is smaller than the radiation density parameter $\Omega_{0,r} \approx 8.4 \times 10^{-5}$, confirming that the $\Theta$ field does not disrupt the current dominance of dark matter and dark energy.