Early-time cosmological solutions in Einstein-scalar-Gauss-Bonnet theory

In this work, we consider a generalized gravitational theory that contains the Einstein term, a scalar field, and the quadratic Gauss-Bonnet (GB) term. We focus on the early-universe dynamics, and demonstrate that a simple choice of the coupling function between the scalar field and the Gauss-Bonnet term and a simplifying assumption regarding the role of the Ricci scalar can lead to new, analytical, elegant solutions with interesting characteristics. We first argue, and demonstrate in the context of two different models, that the presence of the Ricci scalar in the theory at early times (when the curvature is strong) does not affect the actual cosmological solutions. By considering therefore a pure scalar-GB theory with a quadratic coupling function we derive a plethora of interesting, analytic solutions: for a negative coupling parameter, we obtain inflationary, de Sitter-type solutions or expanding solutions with a de Sitter phase in their past and a natural exit mechanism at later times; for a positive coupling function, we find instead singularity-free solutions with no big bang singularity. We show that the aforementioned solutions arise only for this particular choice of coupling function, a result that may hint at some fundamental role that this coupling function may hold in the context of an ultimate theory.

Figure 1

The scale factor $a(t)$ in terms of time, for a linear coupling function and $\lambda =0.1$, and for the values $c_1=0.1,0.5,1,2,5$ (from bottom to top).

Figure 2

The scale factor $a(t)$ and scalar field $\varphi (t)$ versus time for the cosmological solutions with $C_1=0$ and $|\lambda |=0.1,0.2,0.5,1,2,10$. For simplicity, the initial values $a_0$ and ${\varphi }_0$ have been normalized to unity.

Figure 3

Cosmological solutions with $C_1>0$ and $\lambda <0$, and for $\nu =0.5,0.8,1,2,3$.

Figure 4

Cosmological solutions with $C_1<0$ and $\lambda <0$, and for $\widehat{\nu }=1,2,3,4,5$ from left to right.

Figure 5

Cosmological solutions with $C_1>0$ and $\lambda >0$, and for $\tilde{\nu }=0.1,0.5,1,1.5,2$ from left to right.

Figure 6

Ratio of the Hubble function for the complete theory with the Einstein term ($H^{\mathrm{EGB}}$) and the Hubble function without the Einstein term ($H^{\mathrm{GB}}$), for $\lambda =-0.1,-0.2,-0.5,-1$ (from top to bottom).

Figure 7

Early-time cosmological solutions in the scalar-GB theory with a general polynomial coupling function $f(\varphi )=\lambda {\varphi }^n$ and for various values of $n$.

Figure 8

An indicative transitory cosmological solution as this follows from the ansatz (64) that parametrizes the weight of the GB term in the theory.