That is absolutely correct. And the key to deriving it is to use the two unnumbered equations below 10.9.6 in Weinberg's "Gravitation and Cosmology" book.
atyy,
Thanks so much for taking the time to reply!
However, I was referring to the recent book "Cosmology" by Weinberg and not the older "Gravitation and Cosmology" which is what you seem to be referring to to. (For a moment I thought you're a crazy person!)
I think I've figured out...
I am trying to learn gauge issues in cosmology from Weinberg's "Cosmology". However, I am completely stumped at how he gets 5.3.4 from 5.3.3. I was doing great until that point and now I'm going crazy. Any help would be highly appreciated!