jostpuur said:
I have difficulty imagining quantum effects in the cosmology. So QG would be a theory where the classical limit is different from the usual classical gravity?
Look at this Reuter paper
http://arxiv.org/abs/0706.0174
Typically a QC model will recover the classical (up to current limits of observation) AWAY FROM SINGULARITIES.
The classical model does not work very well near the singularities, and various puzzles have arisen in classical cosmology because of this.
It is exactly there that the quantum model can give different results (near the singularities) and can possibly resolve some problems.
Then there is the question of TESTING. Because the QC model normally replicates GR in largescale lowenergy limit at least approximately up to present limits of observation, one must find SMALL DIFFERENCES to look for----e.g. in the CMB---so that the QC models can eventually be tested and either falsified or shown to be superior to classical.
this is the general picture, there is an exception which you can see in this paper
http://arxiv.org/abs/0705.4398
this paper is preliminary and needs more work. In this paper it is proposed that LQG can actually predict a measurably different history of acceleration WITHOUT any dark energy----by the cumulative effect of many quantum correction terms distributed throughout space.
This idea was just proposed this year and it needs to be followed up and confirmed. If Bojowald follows up on it, then it exposes LQG to falsification if the acceleration history at higher redshift z is measured and if the history FAILS to agree with the prediction derived from LQG.
that is the only late-universe effect that I know. the other stuff is early universe that one can expect to check for in the CMB and structure formation datafor survey and introduction to Reuter's work, here are slides and audio of his June talk at Loops 07
http://www.matmor.unam.mx/eventos/loops07/talks/PL3/Reuter.pdf
http://www.matmor.unam.mx/eventos/loops07/talks/PL3/Reuter.mp3