The fourth amendment to the United States Constitution and our constitution here in Colorado demands that law enforcement are only allowed to enter your house with a warrant or under specific and limited circumstances.
Without a warrant the police can only enter your house for the following reasons: First, they ask, and you allow them to enter. This is a common tactic, they will try and pressure and bully you into entering your place without a warrant, and without what is called exigent circumstances. If you allow them to enter, the police will then say that you have consented to them entering. Often times law enforcement will tell you that if you do not allow them to enter they will just go and get a warrant. If that is the case, just tell them to go and get the warrant. Police are also allowed to enter your house without a warrant if they are chasing someone who comes into your place, this legal theory is called hot pursuit. The idea behind this is that a dangerous person who may have just committed a crime should not be allowed to enter a house and be free of arrest. Another reason that police are allowed to enter your house is if they believe that you are actively destroying evidence. Ultimately it is important to remember that the default is that the police can not enter your house without a warrant, and all of the exceptions are supposed to narrowly tailored to specific and limited circumstances.