The lawsuit is not demanding that states recognize polygamous marriage. Instead, the lawsuit builds on a 2003 United States Supreme Court decision, Lawrence v. Texas, which struck down state sodomy laws as unconstitutional intrusions on the “intimate conduct” of consenting adults. It will ask the federal courts to tell states that they cannot punish polygamists for their own “intimate conduct” so long as they are not breaking other laws, like those regarding child abuse, incest or seeking multiple marriage licenses.

Mr. Brown has a civil marriage with only one of his wives; the rest are “sister wives,” not formally wedded. The Browns are members of the Apostolic United Brethren Church, a fundamentalist offshoot of the Mormon Church, which gave up polygamy around 1890 as Utah was seeking statehood.

Making polygamous unions illegal, they argue, violates the due process and equal protection clauses of the 14th Amendment, as well as the free exercise, establishment, free speech and freedom of association clauses of the First Amendment.