Walker's coming-of-age novel is set against a gloomy scientific backdrop the Earth's rotation has begun to slow down

If you want to know whether this sentence is correctly punctuated, it is not. There is a semicolon missing after the word backdrop.

Walker's coming-of-age novel is set against a gloomy scientific backdrop; the Earth's rotation has begun to slow down.

You need to separate these two sentences using a semicolon.