Stable exponential cosmological solutions with three different Hubble-like parameters in EGB model with a $$\Lambda $$Λ -term

Eur. Phys. J. C (2020) 80:543 https://doi.org/10.1140/epjc/s10052-020-8107-3 Regular Article - Theoretical Physics Stable exponential cosmological solutions with three different Hubble-like parameters in EGB model with a -term K. K. Ernazarov1, V. D. Ivashchuk1,2,a 1 Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow 117198, Russian Federation 2 Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Street, Moscow 119361, Russian Federation Received: 30 March 2020 / Accepted: 2 June 2020 / Published online: 16 June 2020 © The Author(s) 2020 Abstract We consider a D-dimensional Einstein-GaussBonnet model with a cosmological term  and two non-zero constants: α1 and α2 . We restrict the metrics to be diagonal ones and study a class of solutions with exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H = 0, h 1 and h 2 , obeying m H + k1 h 1 + k2 h 2 = 0 and corresponding to factor spaces of dimensions m > 1, k1 > 1 and k2 > 1, respectively (D = 1 + m + k1 + k2 ). We analyse two cases: i) m < k1 < k2 and ii) 1 < k1 = k2 = k, k = m. We show that in both cases the solutions exist if α = α2 /α1 > 0 and α > 0 satisfies certain restrictions, e.g. upper and lower bounds. In case ii) explicit relations for exact solutions are found. In both cases the subclasses of stable and non-stable solutions are singled out. For m > 3 the case i) contains a subclass of solutions describing an exponential expansion of 3-dimensional subspace with Hubble parameter H > 0 and zero variation of the effective gravitational constant G. The case H = 0 is also considered. 1 Introduction In this paper we consider D-dimensional Einstein-GaussBonnet (EGB) model with a -term. To some extent this model is unique among the other higher-dimensional extensions of General Relativity (GR) with second order in curvature terms. The reason is the following one: the equations of motion for this model are of the second order (in derivatives) like it takes place in the Einstein gravity. It is well known that the so-called Gauss-Bonnet term appeared in (super)string theory as a first order correction (in α  ) to the (super)string effective action (e.g. heterotic one) [1–4]. a e-mail: (corresponding author) Currently, EGB gravitational model in diverse dimensions and its modifications, see [5–30] and Refs. therein, are rather popular objects for studying in cosmology. They are used for possible explanation of accelerating expansion of the Universe (i.e. solving the dark energy problem), which follow from supernova (type Ia) observational data [31–33]. One may expect that the second order form of the equations of motion for these models will lead us to solutions which are in some sense close to those coming from GR and its higher dimensional extensions (e.g. avoiding the ghosts branches at least). The D-dimensional EGB model is a particular case of the Lovelock model [34]. The equations of motion for the Lovelock model have also at most second order derivatives of the metric (as it takes place in GR). We note that at present there exist several modifications of Einstein and EGB actions which correspond to F(R), R + f (G), f (R, G) theories (e.g. for D = 4), where R is scalar curvature and G is GaussBonnet term. These modifications are under intensive studying devoted to cosmological, astrophysical and other applications, see [28–30] and references therein. In this paper we restrict ourselves to diagonal metrics and study (mainly) a class of cosmological solutions with exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H = 0, h 1 and h 2 , corresponding to factor spaces of dimensions m > 1, k1 > 1 and k2 > 1, respectively, with a restriction imposed: S1 = m H + k1 h 1 + k2 h 2 = 0, and D = 1 + m + k1 + k2 . This restriction forbids the solutions with constant volume factor. We note that in generic anisotropic case with Hubblen like parameters h 1 , . . . , h n obeying S1 = i=1 h i  = 0 (n = D − 1) the number of different real numbers among h 1 , . . . , h n should not exceed 3 [25]. Here we study two cases: i) m < k1 < k2 and ii) 1 < k1 = k2 = k, k = m. We show that in both cases the solutions exist only if α = α2 /α1 > 0,  > 0 and 123 543 Page 2 of 15 Eur. Phys. J. C (2020) 80:543  obeys certain restrictions, e.g. inequalities of the form: 0 < λ∗ (m, k1 , k2 ) < α < λ∗∗ (m, k1 , k2 ). We note that in superstring inspired models α is positive and corresponds to Regge slope parameter α  which is inverse proportional to the tension of the (super)string; non-zero -terms appear for non-critical superstrings. The solutions under consideration are reduced to solutions of polynomial master equation of fourth order or less, which may be solved in radicals for all m > 1, k1 > 1 and k2 > 1. In the case ii) 1 < k1 = k2 = k, k = m we present explicit exact solutions for Hubble-like parameters. Here we use our previous results from refs. [23,25] in studying the stability of the solutions under consideration. In Sect. 5 we single out (for both cases i) and ii)) the subclasses of stable and non-stable solutions. In Sect. 6 we present as an example a subclass of solutions (for the case i)) describing an exponential expansion of 3-dimensional subspace with Hubble parameter H > 0 and zero variation of the effective gravitational constant G (in Jordan frame) which was obtained in Ref. [26] for fixed value of  (depending upon m, k1 , k2 and α > 0). We note that earlier Ref. [27] was dealing with exponential cosmological solutions in the EGB model (with a -term) with two non-coinciding Hubble-like parameters H > 0 and h obeying S1 = m H + lh 1 = 0 and corresponding to mand l-dimensional factor spaces (m > 2, l > 2). In this case there were two sets of solutions obeying: a) α > 0,  < α −1 λ+ (m, l) and b) α < 0,  > |α|−1 λ− (m, l), with λ± (m, l) > 0. Thus, the case of two (non-coinciding) Hubble-like parameters from Ref. [27] drastically differs from the case of three (non-coinciding) Hubble-like parameters which is studied in this paper. 2 The cosmological model The action of the model reads   d D z |g|{α1 (R[g] − 2) + α2 L2 [g]}, S= (2.1) M where g = g M N dz M ⊗ dz N is the metric defined on the manifold M, dim M = D, |g| = | det(g M N )|,  is the cosmological term, R[g] is scalar curvature, L2 [g] = R M N P Q R M N P Q − 4R M N R M N + R 2 is the standard Gauss-Bonnet term and α1 , α2 are nonzero constants. We consider the manifold M = R × M1 × . . . × Mn 123 (2.2) with the metric g = −dt ⊗ dt + n  i Bi e2v t dy i ⊗ dy i , (2.3) i=1 where Bi > 0 are arbitrary constants, i = 1, . . . , n, and M1 , . . . , Mn are one-dimensional manifolds (either R or S 1 ) and n > 3. The equations of motion for the action (2.1) give us the set of polynomial equations [23] E = G i j v i v (...truncated)