Randa Kamblawi, 34, was discovered in her Nicolson Street home on Tuesday lunchtime.
A 45-year-old man is due to appear at Edinburgh Sheriff Court tomorrow after being arrested last night.
The victim attended Edinburgh's Central Mosque, where prayers were said for her last night.
Imam Sajjad Asim said he did not know Ms Kamblawi well, but added: "Everyone who attends our mosque is part of our Muslim brotherhood, and so she was our sister. We would like to extend our sympathies to the family, and our prayers for our dead sister. I will pray to Allah that she will rest in peace in Jannah (paradise]."
Ms Kamblawi was a married housewife who was studying English on a college course.
Originally from Sudan, she moved to the Capital around seven years ago and had a large circle of friends and family in the city.
She had two children, Mohammed, five, and Tagwa, three, who are now being cared for by relatives.
Friends of Ms Kamblawi have spoken of their disbelief.
Mustafa Lotfi, owner of Edinburgh Telecom on Nicolson Street, also hails from Sudan and said: "Randa was a good person, and a good mother to her two kids. We're all very shocked by what has happened."
And Saleh Safi, 54, who runs the wholefoods shop below the flat, said he was in a state of shock.
Chief Inspector Tony Beveridge said: "I would like to extend our sympathies to Randa Kamblawi's family and friends at this most difficult of times."
A post-mortem examination carried out yesterday afternoon found that Ms Kamblawi died after suffering "severe" head wounds.
Officers said tests were continuing and forensic examinations of Ms Kamblawi's flat continued last night.
She was found in the bath by police who called at the flat at around 1.15pm on Tuesday.
Police contacted the Sudanese Embassy following the murder and were working to inform the victim's relatives in Sudan.
Islamic law calls for the bodies of deceased to be buried as soon as possible.
Chief Insp Beveridge added: "We have a protocol with the Central Mosque, which we are working to. There is an understanding that bodies cannot be released for burial straight away in a murder investigation.
"The defence has the right to an independent post-mortem (examination] so we cannot release the body until circumstances allow it."
Twenty police officers are involved in the investigation, with the inquiry team operating out of Portobello police station.