The increasing interest in H_ index in fault detection has led to some recent development in time domain investigation. This paper at first presents a necessary and sufficient condition of the H_ index in finite time horizon for linear time-varying systems. It is shown that the feasibility of the H_ index, which is given when the H_ index is greater than a predefined value, is equivalent to the existence of a certain backward differential Riccati equation under a constraint. The square system as a special case is discussed. The corresponding results are extended to systems with unknown initial condition with a modified definition of the H_ index. The condition for weighted index to emphasize the signal effect at different time instant is also derived by transformation. The result is also compared with the famous bounded real lemma. The connection with the linear time-invariant (LTI) case is also discussed. Finally, examples are given to illustrate the main results. (c) 2013 Elsevier Ltd. All rights reserved.