Santa Fe sheriff Adan Mendoza told Sky News' US partner NBC News it appeared the couple could have been dead for "several days, possibly even up to a couple weeks" before being found.
An investigator noted people may not show signs of poisoning if they are exposed to gas leaks or carbon monoxide, but the local utility company tested gas lines in and around the home after the bodies were found and did not find any sign of problems, the warrant said.
The sheriff's office said another oddity was the fact the couple were found in separate rooms and that there was no obvious sign of a gas leak.
After they found the couple, the deputies searched the rest of the property and found nothing else "out of place", according to the warrant.
The warrant pointed to the fact that the property's front door was open, as well as two of the couple's dogs being found alive while another was dead.
