Why should the law care about what I do behind closed doors?

It shouldn't to a point.

Liberty is not boundless. It does have its limits.

You can stand on a public sidewalk and take pictures of my house. You can't walk into my house uninvited and start taking pictures.

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The law shouldn't care (if you are a law abiding citizen) about who you take into the bedroom as long as they are consenting adults. How many guns you own. What you eat for supper. What kind of TV shows you watch. Whether you watch porn or not.

But what if you're raping little girls behind closed doors? Killing gay men? Watching child porn?

The law doesn't care about what you do behind closed doors as long as it is within the bounds of what the law allows.

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Your speech is free. But if it causes malicious harm to someone, you can be sued.

Society sets laws announcing those actions that it deems unacceptable in polite society. If evidence appears that causes a reasonable person to suspect that illegal activity is going on, society should investigate. Of course society might find itself having to jump through hoops by adhering to constitutional law. It cannot just invade your personal space and demand to know what you're up to just because they don't like you.

If evidence arises that you are doing these things behind closed doors, don't you think the government has a moral obligation to investigate?

What are the beliefs of those who think climate change is a conspiracy theory? What do they predict will happen if we do not address it?