Fundamental axioms in LCDM What are the axioms (if any) behind the LCDM model of cosmology?
NB: axioms, not postulates (e.g., inflation)
The fundamental assumptions of LCDM cosmology are:


General relativity is valid on cosmological scales.
Universe is dominated by "cold" dark matter (origin, composition unknown).
The metric of the universe is given by the  Friedmann–Lemaître–Robertson–Walker metric.


Reference: https://en.wikipedia.org/wiki/Lambda-CDM_model
 A combination of 1 and 2 helped explain large scaled structure formation, which is smaller masses merging to become bigger systems while 3 helped explain the expansion of space depending on the matter density.

The lambda parameter comes from invoking general relativity. Although Einstein added it to support his own view of a static universe (disproved a decade later by Hubble's observations), it was later re-interpreted as a negative energy density of space (see Harvey, Alex 2012 https://arxiv.org/pdf/1211.6338.pdf and/or Carroll, Sean 2001 https://arxiv.org/abs/astro-ph/0004075). 

Hope this helped.