Peters had admitted manslaughter and changed his plea to murder on the third day of his trial at the Old Bailey.
After her death, he called 999 to report what he had done and the child was rushed to hospital but died the following day.
He had claimed he was hearing voices as well as suffering from depression prior to her murder.
The trial heard Peters had recently ended a two-and-a-half-year affair with a married woman and was worrying about his finances.
He waited until his wife had gone out before he woke Sophia up in bed by tying a cord around her neck and throttling her for up to half an hour.
When she asked him what he was doing, Peters said “sorry” but carried on as she struggled, the court heard.
When officers arrived at his home, Peters calmly told them: “She’s upstairs. I’ve strangled her.”
Paramedics removed the ligature and performed CPR. She was taken to St George’s hospital and placed in intensive care, but later died.
Detective Inspector Helen Rance, who led the investigation, said: “Sophia was an innocent seven-year-old girl, much loved by her mother, brother and friends. She was tragically murdered by the hands of her own father in the most frightening way.”
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In the months before the killing, Peters searched the internet for “serial killers”, “treatment of child killers in prison” and “premeditated murder”, the court heard.
Sophia’s mother – Peters’ third wife – and other family members sat in court as he admitted murder.
Peters will be sentenced at the Old Bailey on 30 April.