There is the divorce process, and this provides additional background.
What are Fathers' Rights in Divorce or Paternity in Utah? The Utah Supreme Court says that, if all things are equal, there is no preference for the mother. But from my experience, the courts often seem to not make all things equal. So, you have more of an uphill battle if you want to fight for anything like equal custody. Custody and Parent-time
The Utah Court of Appeals has again addressed this issue, in a new case, Kielkowski v. Kielkowski http://scholar.google.com/scholar_case?case=14449128863980745795&hl=en&as_sdt=6&as_vis=1&oi=scholarr, or http://law.justia.com/cases/utah/court-of-appeals-published/2015/20130225-ca.html)