I'm trying to understand the connection between the $\Lambda$ from cosmology and the $\Lambda$ from QFT.
Cosmology: The cosmological constant enters the Einstein equations. In the special case of the Friedman universe it enters the Friedman equations. If we consider a spatially flat de Sitter universe, we get that
therefore $\Lambda$ has the mass dimension of 2 ($[c]=0$, $[H_\Lambda]=1$).
QFT: The cut-off parameter $\Lambda$ in momentum space is such that the integration goes till $k\leq\Lambda$. Therefore $\Lambda$ has the mass dimension 1 ([k]=1).
On the other hand, both scales seem to be used interchangeably, for example here: http://www.perimeterinstitute.ca/videos/cosmological-constant To quote:
If we take the idea of the Planck length as a fundamental (minimum) scale and if additionally we impose the Cosmological Constant ($\Lambda$) as and infrared (IR) cut-off parameter.
But how can this work if the cosmological constant and the cut-off parameter have different mass dimensions?