What's the causal link between Peak Oil and recession/depression, and how strong is it?

One possibility: rising oil prices cause inflation, and that causes recession. This is a plausible argument: that PO would cause inflation, inflation would cause central banks to tighten credit, and tightened credit would cause recession.

But, that wasn't the case from 2004-08. Inflation never rose about 4%, and core inflation rose much less. Interest rates rose at a modest rate on a historical basis, but not because of oil prices - the Federal Reserve made an explicit decision to not worry about the impact of oil prices on the general price level, assuming that they'd come back down eventually (as they partly have). The Fed plans to continue this policy.

So, what's the causal link between PO and recession/depression?

James Hamilton showed one: that oil shocks caused fear, uncertainty and doubt among car buyers, who put off purchases, thus reducing overall capital investment, thus reducing aggregate demand, causing recession.

"..., the technological costs associated with trying to reallocate specialized labor or capital could result in a temporary period of unemployment as laid-off workers wait for demand for their sector to resume. Bresnahan and Ramey (1993), Hamilton (2009b), and Ramey and Vine (2010) demonstrated the economic importance of shifts in motor vehicle demand in the recessions that followed several historical oil shocks."

page 26 http://econweb.ucsd.edu/~jhamilton/handbook_climate.pdf

The problem: this is a short term effect. If oil prices stay high, they’ll switch to buying more fuel efficient vehicles and car sales will rise again. Again, this is what we say from 2011 to 2014: oil prices stayed high, and yet car sales recovered to historically very high levels.

Other research at the St. Louis Fed, which I can show if  desired,  shows that oil shocks only cause short term recessions.

So, what's the causal link between PO and recession/depression?

Hamilton ascribes more importance to it than most, and he thinks that the 2004-08 oil shock (in which prices rose roughly 5x) shaved roughly 2% off of US GDP, cumulatively. That's not TEOTWAWKI.