Without God, morality doesn’t exist.
And it doesn’t need to.
Have you ever had someone try to convince you that you ought to do something? Having failed to convince you, do they resort, in exasperation, to the phrase “it is the right thing to do”?
The right-wrong moral construct is a tool whereby people seek to impose their will on others when persuasion fails. If they can’t convince you, they simply appeal to morality. For theists, this moral authority is God. And while atheists are fond of dismissing God, they essentially declare themselves to be God in their assertion of what is “clearly” right and wrong.
If we set aside this hopeless question of “what is right and what is wrong?” we can get to the nub of what the atheist ethicist pursues. Human behavior can be either awful or wonderful; we want a set of rules that will inhibit the former while promoting and maximizing the latter.
We can have such a set of rules even without moral authority.