ИСТИНА

The $F(R)$ gravity models, in which $F$ is an arbitrary differential function of the Ricci scalar $R$, are popular generalizations of the general relativity. The conformal transformation of the metric allows to obtain from $F(R)$ gravity models to the General Relativity models with a minimally coupled standard scalar field, called models in the Einstein frame. In the case of the F(R) gravity models with scalar fields, one gets chiral cosmological models in the Einstein frame. This transformation is possible if and only if the first derivative of $F_{,R}\equiv\frac{dF(R)}{dR}>0$. So, solutions, on which the function $F_{,R}$ changes its sign, have no analogues in the Einstein frame. At the same time, such smooth solutions exist in the isotropic and homogenous universe. It has known that F(R) gravity models without scalar fields have anisotropic instabilities associated with the crossing of the hypersurface $F_{,R}=0$. Therefore, the solutions in the spatially flat Friedmann--Lemaitre--Robertson--Walker (FLRW) metric can be smooth, whereas solutions in the Bianchi I metric should have singularities. We consider a pure $R^2$ model ($F(R)=F_0R^2$, where $F_0$ is a positive constant) with a massless scalar or phantom scalar field. The general solution in the case of the FLRW metric has been found in the paper V.R. Ivanov and S.Yu. Vernov, Eur. Phys. J. C 81, 985 (2021). Also, smooth particular solutions, on which $F_{,R}=F_0R$ changes its sign, have been found. In the recent paper (V.R. Ivanov and S.Yu. Vernov, arXiv:2301.06836), we study this $R^2$ gravity model with a scalar field in the Bianchi I metric. We have shown that the evolution equations have a singular point at $R = 0$ if the anisotropy is not equal to zero. So, we do not lose smooth solutions if put an additional condition $R>0$. Using this condition, we get the corresponding Einstein frame model by a conformal metric transformation, find a general solution for this model and get the corresponding solutions for the initial $R^2$ model by an inverse conformal transformation. By this way, we have found Bianchi I solutions for $R^2$ with a scalar field. We also analyzed which types of solutions can exist in the case of the phantom scalar field only. The general solution in the Einstein frame has been found in terms of elementary functions. This general solution gives explicitly the general solution for the initial $R^2$ model in a parametric time. Solutions in the cosmic time for this model have been constructed in quadratures.