No, it doesn't account for the expansion of the universe. Not at all. Not even close. Add the cosmological constant back in to GR and you get a *static* universe, per Einstein's 1919 paper...something that Edwin Hubble conclusively disproved in the 1960's with his direct observations of our expanding universe.
All of this back and forth nonsense with GR and String Theories just goes to show that our current crop of scientists will try *anything*, no matter how absurd, in order to avoid saying that either GR or QM is bunk. Putting the CC back into GR, 4 decades after Hubble, now *that* is funny!
Yet one of those two theories *is* bunk. They are mutually exclusive, after all.