The husband of Peaches Geldof "is not in any way under suspicion of any involvement" in her death, police said tonight.
Kent Police said musician Thomas Cohen has not been arrested or interviewed under caution and "there is no plan to do so".
Detective Chief Inspector Paul Fotheringham said he was making the position clear following recent speculation in the media.
Mr Cohen discovered Ms Geldof, 25, dead in a spare bedroom at the family home in Wrotham, Kent, on April 7.
At the opening of her inquest yesterday, Mr Fotheringham revealed that recent heroin use by Ms Geldof "was likely to have played a role" in her death.
The detective also disclosed tonight that drugs paraphernalia was seized from the property on the day she was discovered.
Mr Fotheringham said: "Following recent speculation in the media, I would like to make it clear that Thomas Cohen is not in any way under suspicion of any involvement in Peaches Geldof-Cohen's death or our concurrent investigation into the supply of drugs.
"He has not been arrested or interviewed under caution and there is no plan to do so. The only statement he has provided related to his discovery of Peaches' body, which is a standard requirement.
"We are in regular contact with both sides of the family to keep them apprised of our progress. Our investigation into the supply of drugs is continuing and remains a live investigation.
"Inaccurate reports have also been made suggesting that no drugs paraphernalia was found at the address, with suggestions that the scene had been 'tampered' with prior to police arrival.
"While no detail will be provided around specific items, to prevent further speculation I will confirm that contrary to rumour in the media my officers did seize drugs paraphernalia from the address on 7 April. The coroner has authorised the disclosure of this information.
"Our investigation on behalf of the coroner continues and there will be no further information released at this time."
Ms Geldof's death marks a parallel to the death of her mother, TV presenter and writer Paula Yates, who died from an accidental heroin overdose at her London home aged 41 in 2000.
At yesterday's inquest, Mr Fotheringham described the unsuccessful efforts that musician Mr Cohen made to contact his wife before he found her body.
Four days before she was discovered, Mr Cohen went with the couple's two sons, Astala, 23 months, and 11-month-old Phaedra, to his parents' south-east London home.
Mr Fotheringham told the hearing that this was a normal arrangement so that he and TV presenter and columnist Ms Geldof could concentrate on their work.
During their time apart, Ms Geldof remained at home in Wrotham and in contact with friends and family, appearing to be her normal self.
Mr Fotheringham told the inquest: ''On Thursday April 3, 2014, Thomas Cohen went with his two sons to stay for the weekend at his parents' house in south-east London.
''This is a normal occurrence which enables Peaches and Thomas to complete work in their respective careers.''
He added: ''It is believed that Peaches spent Saturday afternoon (April 5), into the evening, and Sunday morning at her home address alone.
''Throughout this period she maintained telephone contact with family and friends, including contact with Thomas's mother to arrange a family activity, but this was cancelled.
''Peaches maintained contact with family and friends throughout Sunday and at 5pm Thomas's father Keith Cohen took the youngest child, Phaedra, back to Wrotham and spent half an hour with Peaches before leaving the child with her.''
Mr Fotheringham told the inquest the last contact with Ms Geldof is believed to have been at 7.45pm on the Sunday when she had a telephone conversation with a friend.
He went on: ''All of the friends and family who had contact with Peaches during this period described how she seemed her normal self and was making plans for the future, including a family outing for her sons for the following weekend.
''There was no cause for concern.''
Mr Cohen tried to contact his wife the following morning without success, Mr Fotheringham added at the inquest.
He then travelled to the family home in Kent with his mother, Sue, and Astala, arriving at around 1.30pm.
''Thomas entered the property and located Peaches in the spare bedroom,'' Mr Fotheringham said. It was apparent that she was dead.
''She was located on the edge of the bed with one leg hanging down to the floor and the other leg tucked underneath her. She was slumped across the bed,'' Mr Fotheringham said.
Mr Cohen called out to his mother and Phaedra was found in the property, and the emergency services were called.
Police and paramedics arrived and confirmed she was dead. Kent Police started an investigation but the death was ''unexplained with no third party'' involvement.
A post-mortem examination carried out by Dr Peter Jerreat at Darent Valley Hospital in Dartford was inconclusive.
Further tests confirmed the recent taking of heroin. T he inquest was adjourned for a full hearing on July 23.
None of Ms Geldof's family, including her father Bob, were at the hearing, which lasted around 10 minutes.
At her funeral service in Davington, near Faversham, Ms Geldof's body was carried into church in a poignantly-decorated coffin which included a picture of her young family.
Mr Geldof is thought to have led tributes in front of a host of well-known personalities, including the Duchess of York, supermodel Kate Moss and former Rolling Stones bassist Bill Wyman.
The church, St Mary Magdalene and St Lawrence, was where Ms Geldof married musician Mr Cohen in 2012.
It was also where her mother married Mr Geldof in 1986 and where her funeral service was held.
Former Boomtown Rats singer Mr Geldof has previously paid tribute to his daughter alongside his partner, Jeanne Marine, and Ms Geldof's sisters, Fifi Trixibelle, Pixie and Tiger, saying she was the ''wildest, funniest, cleverest, wittiest and the most bonkers of all of us''.