Cosmological aspects of the unimodular-mimetic $f(\mathcal{G})$ gravity

In this work we introduce and study the unimodular-mimetic $f(\mathcal{G})$ gravity, where unimodular and mimetic constraints are incorporated through corresponding Lagrange multipliers. We present field equations governing this theory and discuss their main properties. By using the reconstruction scheme, we obtain quadratic unimodular-mimetic $f(\mathcal{G})=A\mathcal{G}^2$ gravity capable of describing hybrid expansion law and the power law evolution. Furthermore, we employ an inverted reconstruction technique in order to derive specific $f(\mathcal{G})$ function that reproduces the Hubble rate of symmetric bounce. The unimodular-mimetic $f(\mathcal{G})=A\mathcal{G}^2$ is also shown to be compatible with the BICEP2/Keck and Planck data. To this end, we incorporate updated constraints on the scalar-to-tensor ratio and spectral index, utilizing a perfect fluid approach to the slow-roll parameters. Through an analysis of that kind, we demonstrate that the theoretical framework presented here can indeed characterize inflation that agrees with the observational findings. Consequently, the introduced extension appears to have potential to describe and encompass a wide spectrum of cosmological models.