Modified Gauss-Bonnet gravity with Lagrange multiplier constraint as mimetic theory

In this paper we propose and extensively study mimetic $f({\cal G})$ modified gravity models, with various scenarios of cosmological evolution, with or without extra matter fluids. The easiest formulation is based on the use of Lagrange multiplier constraint. In certain versions of this theory, it is possible to realize accelerated expansion of the Universe or even unified evolution which includes inflation with dark energy, and at the same time in the same theoretical framework, dark matter is described by the theory. This is achieved by the re-parametrization of the metric tensor, which introduces a new degree of freedom in the cosmological equations and leads to appearance of the mimetic "dark matter" component. In the context of mimetic $f({\cal G})$ theory, we also provide some quite general reconstruction schemes, which enable us to find which $f({\cal G})$ gravity generates a specific cosmological evolution. In addition, we also provide the general reconstruction technique for Lagrange multiplier $f({\cal G})$ gravity. All our results are accompanied by illustrative examples, with special emphasis on bouncing cosmologies.