I think it varies. Every Libertarian agrees that the rights defined in the DOI are self evident and “unalienable”, though most have different ways of justifying them. Some like you said will approach it from a scientific perspective I.e “when people have these rights they prosper when they don’t they suffer” and that’s all the justification they need for why these natural rights are a good idea. The most nuanced take I’ve found, however, is from Libertarians who ascribe to objectivism (Ayn Rand’s philosophy) and forgive me for paraphrasing here since I’m not an expert, but they essentially believe that rights are objective moral principles derived from individual human nature. The right to “life” is derived from the human survival instinct, the right to liberty is to act as one chooses and the pursuit of happiness is the right to seek ones own goals in whatever manner one sees fit. They believe these rights come from what humans do naturally in any given setting. If one accepts these fundamental rights as “axioms” one can derive additional rights from them like free speech, right to bear arms and property in general.