Devinder Sivia was arrested on Wednesday night at his home in Southmoor, Oxford, after police discovered the body of Professor Steven Rawlings there.
After Dr Sivia left Abingdon police station today, police said the death may be "a matter for a coroner's inquest rather than a criminal court".
A post-mortem examination proved inconclusive and further tests will be carried out over the coming weeks, Thames Valley Police said.
After Oxford University don Dr Sivia was released on police bail today, Prof Rawlings's wife, Linda Rawlings, rallied to support him.
In a statement issued through Thames Valley Police, Mrs Rawlings said: "I do not believe that Steve's death is murder and I do not believe Devinder should be tarnished in this way."
Detective Superintendent Rob Mason, from the force's major crime unit, said: "This is a tragic incident and our investigations are ongoing to establish the cause of death.
"A substantial amount of information is already in the public domain and we can confirm that the two individuals involved have been friends for over 30 years.
"I would emphasise that the police are investigating all potential circumstances that could have led to his death.
"We are mindful that ultimately the death may be a matter for a coroner's inquest rather than a criminal court and I would ask for patience from both the media and the public while we continue our investigation.
"Due to the post-mortem examination results proving inconclusive and further examinations being required, this has necessitated a lengthy bail date."
Dr Sivia was bailed until April 18.
The 49-year-old was seen being driven out of the police station just before midday in the back of a grey Vauxhall Astra estate.
Emergency crews were called to his friend's suburban home at 11.22pm on Wednesday to reports that a man had been injured.
Paramedics attended but married Prof Rawlings, 50, was declared dead at the scene.
Police are refusing to speculate on the circumstances surrounding his death and said earlier that they were "keeping an open mind".
Mr Mason said detectives were liaising closely with Oxford University and it was "apparent that he (Prof Rawlings) was well-respected and liked within the college and colleagues and students have been devastated by the news".
The force was unable to confirm if the two men - who published the book Foundations Of Science Mathematics together in 1999 - had been out socially together in the hours before his death.
Tributes were paid to the St Peter's College academic yesterday.
Vice-chancellor of Oxford University Professor Andrew Hamilton said in a statement: "The entire university community has been profoundly saddened and shocked by the tragic and untimely death of Professor Steve Rawlings.
"Our thoughts are with his family and friends."
Former BBC Radio 4 controller Mark Damazer, now Master at St Peter's College, said Prof Rawlings was a "much-liked and admired tutor" who leaves a wife, Linda.
He said: "Steve was elected to his fellowship in 1994, having initially been employed by the college for two years as lecturer in mathematics.
"He undertook his undergraduate and doctoral studies at the University of Cambridge, and remained there to do post-doctoral work prior to moving to Oxford.
"Steve held his St Peter's fellowship in conjunction with a university lectureship in astrophysics, later becoming professor.
"He was head of the sub-department of astrophysics of the University of Oxford from 2006 to 2010.
"He was one of the lead scientists in the international Square Kilometre Array (Ska) project and also played a prominent role in the redevelopment of the Goonhilly Satellite Earth Station in Cornwall as a radio astronomy facility.
"He was a much-liked and admired tutor and colleague within the college and will be greatly missed. We extend our deepest sympathy to his wife, Linda."
Dr Sivia, of St John's College, has taught "maths for natural sciences" to chemistry and physics undergraduates for a number of years.
His research interests revolve around the Bayesian probability theory - mathematics for inferring, or reasoning, using probability.
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