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 n s 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.