Cosmology in $R^2$-gravity: Effects of a Higher Derivative Scalar Condensate Background

A well known extension of Einstein General Relativity is the addition of an $R^2$-term, which is free of ghost excitations and in the linearized framework, reduces Einstein General Relativity and an additional higher derivative scalar. According to \cite{Chakraborty:2020ktp}, the above scalar sector can sustain a Time Crystal-like minimum energy state, with non-trivial time dependence. Exploiting previous result that the scalar can sustain modes with periodic time dependence in its lowest energy, we consider this condensate as a source and study the Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) cosmology in this background. The effect of the $R^2$-term is interpreted as a back reaction. A remarkable consequence of the condensate is that, irrespective of open or close geometry of the Universe, for an appropriate choice of parameter window, the condensate can induce a decelerating phase before the accelerated expansion starts and again, in some cases, it can help to avoid the singularity in the deceleration parameter (that is present in conventional FLRW Cosmology).