Why didn't the Achaemenid Empire invade the Arabian Peninsula?

Because it is a vast desert without any obvious resources or attraction. Empires don't invade places just to add colored areas to a map, like a computer game, or an old board game like Risk. They take places over to accumulate wealth and power. read more

The Achaemenid Empire (/ ə ˈ k iː m ə n ɪ d /; c. 550–330 BC), also called the First Persian Empire, was an empire based in Western Asia, founded by Cyrus the Great. Ranging at its greatest extent from the Balkans and Eastern Europe proper in the west to the Indus Valley in the east, it was larger than any previous empire in history, spanning 5.5 million square kilometers. read more

So the Achaemenids never invaded the Arabian Peninsula because they had a close friendship with the Arabs. This is exactly the answer I was looking for, I really appreciate it. But the Arabs were still presenting the Empire with Talons even though they were not a province. read more