A baby boy, who suffocated in his cot on New Year's Day, was alone in the house in Manurewa, Auckland. He was five weeks old.
Police said on Thursday his mother was next door drinking with family and friends, and had left the baby sleeping alone in his cot.
Detective Inspector Jim Gallagher said police are liaising with the coroner and their legal team, to see if there is any criminal culpability.
But they are not suggesting there is any intent to charge anyone at this stage. He said the family is devastated by the death.
The Fire Service has also helped the investigation with specialist equipment to test the temperature in the house, as it was very hot when officers arrived.