I have a genomic dataset for tree populations. I would like to compare the current genetic diversity with the genetic diversity i.e. 20,000 years BP. Would bit be possible to do that using coalescent simulations + model testing approach?
In my opinion, it is possible to deduce the demographic process at certain time point in the past, but there is no method to test the genetic diversity in the specific time point. What you could do is to test the possible genetic diversity change with species distribution modeling and range wide population genetic data.
By genetic diversity what do you mean exactly? Allelic richness? Heterozygosity? You cannot do this using a coalescent approach. The coalescent model could help you understand the past history of lineages sampled in the present, but that is not all lineages available in the present nor in the past. You could use a coalescent approach to estimate theta = 4Nemu which could vary over time. You could then construct time forward simulations to predict quantities such as allelic richness. Heterozygosity is then a function of mating system.
If you have sequence data, you could use psmc/msmc to infer the Effective population size, or use a Bayesian analysis sample trees approach (e.g BEAST) for similar purposes.
Here are a couple of papers which could be of some help.
Determinants of genetic diversity -http://www.nature.com/nrg/journal/v17/n7/full/nrg.2016.58.html?WT.feed_name=subjects_molecular-evolution
Alcala, N. & Vuilleumier, S. Turnover and accumulation of genetic diversity across large time-scale cycles of isolation and connection of populations. Proc. R. Soc. B 281, 20141369 (2014).