Exact cosmological solutions with nonminimal derivative coupling

TL;DR

The paper analyzes a scalar field nonminimally coupled to curvature through derivative terms with action $S=\int d^4x\sqrt{-g}\{ R/(8\pi) - [g_{\mu\nu}+\kappa G_{\mu\nu}]\phi^{,\mu}\phi^{,\nu} \}$, focusing on the special choice $-2\kappa_1=\kappa_2\equiv\kappa$ that reduces the field equations to second order. In a spatially flat FRW background, the authors derive exact cosmological solutions showing that late-time evolution is universal: $a(t)\sim t^{1/3}$ and $\phi(t)\sim \ln t$, independent of $\kappa$. The sign of $\kappa$ crucially affects early-time behavior: for $\kappa<0$ the universe has an initial singularity with $a(t)\sim (t-t_i)^{2/3}$ and regular $\phi$, while for $\kappa>0$ the early universe undergoes quasi-de Sitter expansion with $H=(3\sqrt{\kappa})^{-1}$ and $\phi(t)\sim e^{-t/\sqrt{\kappa}}$, followed by a graceful exit to the late-time regime. The results imply inflation and exit can be achieved without a fine-tuned potential, providing exact solutions that illuminate the role of nonminimal derivative coupling.

Abstract

We consider a gravitational theory of a scalar field $φ$ with nonminimal derivative coupling to curvature. The coupling terms have the form $κ_1 Rφ_{,μ}φ^{,μ}$ and $κ_2 R_{μν}φ^{,μ}φ^{,ν}$ where $κ_1$ and $κ_2$ are coupling parameters with dimensions of length-squared. In general, field equations of the theory contain third derivatives of $g_{μν}$ and $φ$. However, in the case $-2κ_1=κ_2\equivκ$ the derivative coupling term reads $κG_{μν}φ^{,mu}φ^{,ν}$ and the order of corresponding field equations is reduced up to second one. Assuming $-2κ_1=κ_2$, we study the spatially-flat Friedman-Robertson-Walker model with a scale factor $a(t)$ and find new exact cosmological solutions. It is shown that properties of the model at early stages crucially depends on the sign of $κ$. For negative $κ$ the model has an initial cosmological singularity, i.e. $a(t)\sim (t-t_i)^{2/3}$ in the limit $t\to t_i$; and for positive $κ$ the universe at early stages has the quasi-de Sitter behavior, i.e. $a(t)\sim e^{Ht}$ in the limit $t\to-\infty$, where $H=(3\sqrtκ)^{-1}$. The corresponding scalar field $φ$ is exponentially growing at $t\to-\infty$, i.e. $φ(t)\sim e^{-t/\sqrtκ}$. At late stages the universe evolution does not depend on $κ$ at all; namely, for any $κ$ one has $a(t)\sim t^{1/3}$ at $t\to\infty$. Summarizing, we conclude that a cosmological model with nonminimal derivative coupling of the form $κG_{μν}φ^{,mu}φ^{,ν}$ is able to explain in a unique manner both a quasi-de Sitter phase and an exit from it without any fine-tuned potential.

Figures (2)

• Figure 1: Plots of $\alpha(t)$ for negative $\kappa=-1;-10;-100$ (from right to left). The thick curve corresponds to $\kappa=0$ (no derivative coupling).
• Figure 2: Plots of $\alpha(t)$ for positive $\kappa=1;10;100$ (from right to left). The thick curve corresponds to $\kappa=0$ (no derivative coupling).