1. Zong-Kuan Guo Department of Physics, Kinki University
Accelerating Cosmologies in the Einstein-Gauss-Bonnet Theory with a Dilaton
Zong-Kuan Guo
Department of Physics, Kinki University
Based on
K. Bamba, Z.-K. Guo and N. Ohta, arXiv: ,
Progress of Theoretical Physics 118 (2007)

2. Abstract
We study cosmological solutions in low-energy effective heterotic string theory, which consists of the Einstein gravity with the Gauss-Bonnet term and a dilaton. We show that the field equations take the form of an autonomous system for flat internal and external spaces, and we derive all the fixed points in the system. We also examine the time evolution of the solutions and the question of whether the solutions can describe (transient) accelerated expansion of our four-dimensional space in the Einstein frame.

3. Motivation
Inflation: Some cosmological puzzles, such as the horizon problem and the flatness problem, can be explained in the inflation scenario. The most important property of inflation is that it can generate irregularities in the Universe, which may lead to the formation of structure. However, so far the nature of the inflaton has been an open question.

4. String theory: It is known that no 4-dimensional solutions of accelerated expansion can be obtained in context of string or M-theory if internal space is a time-independent nonsingular compact manifold without boundary. However, there exist higher-order “quantum correction” terms in the curvature in M-theory and superstrings, which was not taken into account in the no-go theorem.
Gauss-Bonnet term: The simplest such correction is the Gauss-Bonnet term in the low-energy effective heterotic string. Thus it is an attempt to obtain inflationary solutions in the Einstein-Gauss-Bonnet theory with a dynamical dilaton.

5. Our Model The following low-energy effective action: where
a D-dimensional gravitational constant,
a dilaton field,
in terms of the Regge slope parameter,
Gauss-Bonnet term.

6. In the Einstein frame
where
The metric in D-dimensional ( ) space
The external p-dimensional and internal q-dimensional spaces ( ) are chosen to be maximally symmetric, with the signature of the curvature given by , respectively.

7. Field Equations
where

9. and

10. Cosmological Solutions
We only consider flat internal and external spaces, i.e., the cases
Setting
can be chosen by time reparametrization.
We introduce a new time variable
to remove the exponential factors of the dilaton in the field equations.
Field Equations an autonomous system (x, y, z)

11. The Einstein frame is obtained from
Hence, we have
where we have defined the cosmic time and scale factor by
The condition for expansion is
and the condition for acceleration is

12. There are seven fixed points in this system:
The solution space and flow in the case with a dynamical dilaton are shown in the left figure, where
The solid (red) curves correspond to the case
and the dashed (green) curves correspond to the case

13. Conclusion
For flat internal and external spaces , the field equations can be written as an autonomous system. We obtained the fixed points, and analyzed their stability and cosmological properties.
The solutions give enough e-folding number.
The internal space is shrinking.

14. Outlook Examine the cases with nonzero
How to end the inflation (graceful exit) possibly by stringy effects.
Can we get nearly scale-invariant density perturbation?