Permitting an officer to enter your home is the equivalent of waiving your Fourth Amendment right against unreasonable searches and seizures. Without a warrant, police officers absolutely cannot enter your home without your permission or an emergency circumstance* that could justify their entry.*Exigent Circumstances A judge may uphold an officer’s warrantless search or seizure if "exigent circumstances" exist. Exigent circumstances were described by one court as "an emergency situation requiring swift action to prevent imminent danger to life or serious damage to property, or to forestall the imminent escape of a suspect or destruction of evidence."
WARNING: If you ever face a real-life police encounter where the officer is urging you to consent to a search, do not try to figure out whether or not the search is legally permissible. You must assume that the search is not legally permissible and that the search will only be legal if you consent. If an officer is in fact legally allowed to search you, you have nothing to lose by refusing to consent.