They say they are not looking for anyone else in connection with the sudden death of Emily Barut, aged 11, whose mother, Bernadette Scully, is a well-known GP in the area.
Dr Scully is in the intensive care unit of the Midland Regional Hospital after being brought there unconscious on Saturday.
Emergency services responded to a call from the family home in Bachelor’s Walk on Saturday night.
The call is understood to have come from Dr Scully’s partner, Andrius Kozlovskis, a Lithuanian-born musician and choir master.
Gardaí arrived and the cottage, close to the town centre, was sealed off for forensic examination.
Emily’s father, Turkish-born Haluk Barut, who is separated from Dr Scully, was at the hospital last night.
He formerly ran the Anatolia Restaurant in Harbour Street in the town.
A postmortem proved inconclusive and results are awaited on toxicology tests on Emily, who had cerebral palsy, used a wheelchair and required 24-hour care.
Emily was also unconscious when she was taken to the hospital at about 9pm on Saturday. She was pronounced dead shortly afterwards and State Pathologist Dr Marie Cassidy carried out the autopsy.
Superintendent John Moloney, who is leading the Garda investigation, said no one else was being sought in connection with the incident. “The postmortem is over and it’s inconclusive. We await toxicology results and the investigation is ongoing into the circumstances of her death,” said Supt Moloney.
Dr Scully, who run a large practice from the Medical Centre in Clonminch, has been living in the house on Bachelor’s Walk since 2003, having returned to Ireland from Australia.
A native of Rhode, Co Offaly, Dr Scully, aged 55, is well regarded by her patients and those who know her.
Emily was last night described as a “bright and engaging child” with an interest in technology.
One local resident who worked in a phone shop in Tullamore said: “I remember her coming into the shop looking for a new phone or an accessory. She always loved to add a bit of bling to her phone.
“The last time I met her she was about eight or nine years of age and she came in for a new phone. Even though her speech was hesitant because of her condition, she was able to make herself understood.
“Unlike most of the other kids coming into the shop, she knew exactly what she wanted. She was a lovely little girl.”
Local people were shocked at the tragedy and few wished to speak about it yesterday. One neighbour, Patrick Longworth, described Dr Scully as a quiet person and said people did not know what could have happened.
Tullamore parish priest Fr Sean Heaney was called to the hospital to administer the last rites to the child. “It is always a very moving thing when a young child dies,” he said yesterday. “It is always a very sad event.”