Stable exponential cosmological solutions with 3- and l-dimensional factor spaces in the Einstein–Gauss–Bonnet model with a $$\Lambda $$ Λ -term

Abstract

Abstract A D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term $$\Lambda $$ Λ is studied. We assume the metrics to be diagonal cosmological ones. For certain fine-tuned $$\Lambda $$ Λ , we find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters $$H >0$$ H>0 and h, corresponding to factor spaces of dimensions 3 and $$l > 2$$ l>2 , respectively and $$D = 1 + 3 + l$$ D=1+3+l . The fine-tuned $$\Lambda = \Lambda (x, l, \alpha )$$ Λ=Λ(x,l,α) depends upon the ratio $$h/H = x$$ h/H=x , l and the ratio $$\alpha = \alpha _2/\alpha _1$$ α=α2/α1 of two constants ($$\alpha _2$$ α2 and $$\alpha _1$$ α1 ) of the model. For fixed $$\Lambda , \alpha $$ Λ,α and $$l > 2$$ l>2 the equation $$\Lambda (x,l,\alpha ) = \Lambda $$ Λ(x,l,α)=Λ is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals (the example $$l =3$$ l=3 is presented). For certain restrictions on x we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. A subclass of solutions with small enough variation of the effective gravitational constant G is considered. It is shown that all solutions from this subclass are stable.