Well-defined f(Q) gravity, reconstruction of FLRW spacetime and unification of inflation with dark energy epoch

We formulate the convenient and mimetic f(Q) gravities in terms of the metric and four scalar fields with the Q being the non-metricity scalar. As a result, it is shown that the obtained field equations are well-defined. By using the field equations, we show how the f(Q) models which realise any given FLRW spacetime may be reconstructed. We construct convenient f(Q) gravity and mimetic f(Q) gravity, which describe the inflation and dark energy epochs and even unify the inflation and dark energy. Moreover, a radiation-dominated epoch and early dark energy may be easily included in the above theories as is explicitly shown. It is shown that the inflationary spectral index ns and the tensor-to-scalar ratio r are consistent with Planck data for some models of f(Q) gravity. Corresponding asymptotic forms of f(Q) and mimetic f(Q) are given. Finally, some remarks on possible reconstruction for spherically symmetric spacetime are given.