Reconstruction of the scalar-tensor Lagrangian from ΛCDM background and Noether symmetry

We consider scalar-tensor theories and reconstruct their potential U(Phi) and coupling F(Phi) by demanding a background Lambda CDM cosmology. In particular we impose a background cosmic history H(z) provided by the usual. at Lambda CDM parameterization through the radiation (w(eff) = 1/3), matter (w(eff) = 0) and de Sitter (w(eff) = -1) eras. The cosmological dynamical system which is constrained to obey the Lambda CDM cosmic history presents five critical points in each era, one of which corresponding to the standard General Relativity (GR). In the cases that differ from GR, the reconstructed coupling and potential are of the form F(Phi) similar to Phi(2) and U(Phi) similar to F(Phi)(m), where m is a constant relating the two functions F(Phi) and U(Phi): this assumption is necessary to find out the fixed points of the dynamical system. This class of scalar tensor theories is also theoretically motivated by a completely independent approach: imposing maximal Noether symmetry on the scalar-tensor Lagrangian. This approach independently (i) provides the form of the coupling and the potential as F(Phi) similar to Phi(2) and U(Phi) similar to F(Phi)(m), (ii) provides a conserved charge related to the potential and the coupling, and (iii) allows the derivation of exact solutions by first integrals of motion.