For those who want to do it with an Activity, and not by using a service (there can be situations when this is necessary), I did it by starting the Activity the usual way, and then in the Activity's onCreate I sent it to the background by calling moveTaskToBack(true) .
See Sending a running application to background programmatically
If you use this method, you should add an extra info to the intent by which you start the Activity, so that it knows in onCreate that it must send itself to the background. You do this by calling the Intent's putExtra method. In the Activity's onCreate, you get the calling Intent with getIntent, then retrieve the info you put there with getStringExtra, for instance (I used a string extra info). See the documentation for the Intent class.
Update august 2018: Note, however, that on some devices this can make your Activity blink (that is, briefly appear) on the screen before it's sent to the background. It appears that the system treats the notification to send the task to the background way too late. If you call the method in onCreate, it should have plenty of time to know about it before it draws the Activity window. IMO this is a system flaw, it shouldn't happen like this but it does.
So, anyway, if this is a concern and if you can, you'd probably better do it with a service.