Did the Illuminati invent the big bang theory?
The Illuminati seem to take the brunt for a lot of things these days, but the answer to your question is; No, the Illuminati did not invent the big bang theory.
If the Illuminati didn't, who did?
His work isn't as well known as those of Einstein or Hubble, mostly due to the fact that he worked in Europe rather than the United States, but his work changed the course of cosmology. Indeed, Hubble and many other scientists further progressed the idea of the big bang theory following Lemaitre's early work.
But what about the Illuminati?
The original Illuminati were the Bavarian Illuminati, and were a secret society founded in 1776. They opposed superstition, religious influence, and state power (amongst other things).
Today, the Illuminati refers to groups that claim to be connected to the Bavarian Illuminati or other secret societies. These groups, however, are not confirmed as being genuine or even real for that matter.
If you're interested in some pretty freaky Illuminati theories, have a look at this!