Reconstruction of the scalar–tensor Lagrangian from aΛCDM background and Noether symmetry

Abstract

We consider scalar–tensor theories and reconstruct their potentialU(Φ) and couplingF(Φ) by demandinga background ΛCDM cosmology. In particular we impose a background cosmic historyH(z) provided by the usualflat ΛCDM parameterizationthrough the radiation (weff = 1/3), matter (weff = 0)and de Sitter (weff = −1) eras. The cosmological dynamical system which is constrained to obey theΛCDM cosmic history presents five critical points in each era, one of whichcorresponding to the standard General Relativity (GR). In the cases thatdiffer from GR, the reconstructed coupling and potential are of the formF(Φ)∼Φ2 andU(Φ)∼F(Φ)m, wherem is a constant relatingthe two functions F(Φ) and U(Φ): this assumption is necessary to find out the fixed points of the dynamical system. Thisclass of scalar tensor theories is also theoretically motivated by a completely independentapproach: imposing maximal Noether symmetry on the scalar–tensor Lagrangian. Thisapproach independently (i) provides the form of the coupling and the potential asF(Φ)∼Φ2 and U(Φ)∼F(Φ)m, (ii) provides a conserved charge related to the potential and the coupling, and (iii) allowsthe derivation of exact solutions by first integrals of motion.