Dark energy and cosmological solutions in second-order string gravity

TL;DR

The paper investigates cosmological evolution in a $D$-dimensional, string-inspired gravity framework that includes second-order curvature corrections, a dilaton or modulus scalar, and a barotropic fluid. By analyzing fixed-scalar, linear-dilaton, and logarithmic-modulus scenarios, it derives de Sitter and inflationary solutions, conducts phase-space stability studies, and examines observational constraints on late-time acceleration. A key finding is that Gauss–Bonnet gravity with no matter generally cannot account for current acceleration, whereas coupling between matter and the scalar field or modulus dynamics can lead to stable de Sitter phases and can avoid big rip singularities even in phantom regimes. The results illuminate how higher-curvature corrections interact with scalar fields and matter to shape the fate of the universe, offering pathways to reconcile string-inspired corrections with cosmological observations.

Abstract

We study the cosmological evolution based upon a $D$-dimensional action in low-energy effective string theory in the presence of second-order curvature corrections and a modulus scalar field (dilaton or compactification modulus). A barotropic perfect fluid coupled to the scalar field is also allowed. Phase space analysis and the stability of asymptotic solutions are performed for a number of models which include ($i$) fixed scalar field, ($ii$) linear dilaton in string frame, and ($iii$) logarithmic modulus in Einstein frame. We confront analytical solutions with observational constraints for deceleration parameter and show that Gauss-Bonnet gravity (with no matter fields) may not explain the current acceleration of the universe. We also study the future evolution of the universe using the GB parametrization and find that big rip singularities can be avoided even in the presence of a phantom fluid because of the balance between the fluid and curvature corrections. A non-minimal coupling between the fluid and the modulus field also opens up the interesting possibility to avoid big rip regardless of the details of the fluid equation of state.

Figures (7)

• Figure 1: The phase space plot for $d=4$, $\xi=1$, $F=1$, $w=-0.9$, $c_{1}=-1$ and $c_{2}=-1$ when the scalar field $\phi$ is fixed. In this case the de Sitter fixed point $(x_c, y_c, z_c)= (1.22, 0, 0)$ is a stable attractor.
• Figure 2: The phase portrait (plot of $y$ versus $x$) of the system in the linear dilaton case with $d=3$, $c_{1}=2$, $\lambda=1/4$, $a_4=-1$ and $c_{2}=-1$. Trajectories starting anywhere in the phase space converge at $(x_c, y_c, z_c)= (0.62, 0, 0)$ for a constant value of $v =1.40$. The fixed point corresponds to a stable de Sitter solution.
• Figure 3: Evolution of $\omega_{1}$ and $\omega_2$ for $c_2=0$, $d=3$, $z=0$, and $\xi_{0}=2$ without a barotropic fluid. We find that the system approaches the low-curvature solution (\ref{['grsol']}) characterized by constant $\omega_{1}$ and $\omega_{2}$ ($\omega_{1}=1/3$ and $\omega_{2}=2/3$).
• Figure 4: Variation of $\omega_{1}$ and $\omega_2$ for $c_2=0$, $d=3$, and $\xi_{0}=2$ in the presence of a cosmological constant given by $\Lambda= 48\tilde{\delta}$. The asymptotic behavior of $H$ and $\dot{\phi}$ is characterized by $\beta=-1$, $\omega_{1}=1$ and $\omega_{2}=4$ as estimated analytically.
• Figure 5: Evolution of $H$ and $\rho$ with $\xi_{0}=-2$, $w=-1.1$ for (a) $Q=0$ and (b) $Q=-5$. We choose initial conditions as $H_{i}=0.2$, $\phi_{i}=2.0$ and $\rho_{i}=0.1$. The curves (a1) and (b1) represent the evolution of $H$ for $Q=0$ and $Q=-5$, respectively, while the curves (a2) and (b2) show the evolution of $\rho$ for corresponding $Q$. We find that the big rip singularity is avoided when the negative coupling $Q$ is present.