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The big bang was actually first proposed by a priest and physicist, Georges Lemaître.
Stephen Hawking's work was mostly around black holes, and their radiation, now called Hawking's radiation.
His book tells the story of physics for the general public, but he didn't work directly in all that is there.
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If I remember correctly, the idea of the Big Bang stems from the "groundbreaking" book A Brief history of Time by Stephen Hawking.
I don't think he intended for his very reasonable theory to lend credence to an origin or beginning of the universe, because that lends credence to the cosmological argument:
A few years ago, I read an article in Discover magazine that reported on his more recent work, essentially, trying to refute his earlier work in A Brief History of Time. He was dabbling in theoretical work "at the outer limits of testability" if I recall the quote correctly.
That's a fancy way of saying he was now making stuff up.