I would think they should first try to look to see if your car was there. Then, they should try to call you on the phone, maybe call your other family members or your place of business. Maybe check credit or bank card activity. If you flew on a plane, the FAA should know.
They should look for other indicators as well. Are the lights on? Is the television on? Are pets home? Is the mail checked? Is the power meter moving? I suspect that there would be some odor if you were dead.
Personally, I don't think that heresay is grounds for invading your home, unless they tried several other routes beforehand.
Someone said the cops were acting in good faith. There are several viable channels to take before breaking into someone's house supposedly "in good faith".
Also, your neighbor should never have been allowed into your house regardless.