This paper gives the optimal solutions to several robust fault detection problems such as H_-/H_infin, H₂/H_infin and H_infin /H_infin problems for linear time varying systems in time domain, which extends the previous results on linear time invariant systems in frequency domain. It is shown that all three problems have the same optimal detection filter and the filter is a simple observer by solving a standard differential Riccati equation. Finally, an example is given to illustrate our results.