In Canada, as I understand it, the only excuse for entering with neither permission or a warrant is when reasonable grounds exist to believe that someone's life would otherwise be in danger (mere potential bodily harm might suffice, I am not sure). The key is that the police (or other persons) have to have reasonable grounds to believe that the circumstance is a genuine emergency, and they will have to defend that decision in court if the homeowner takes the matter there.
As a general principle, no one gets to enter someone's home if they are not invited. A warrant or a genuine emergency are the only exceptions here, and I expect in most democracies.