The cosmological constant problem is the disagreement in 'measured' values of the cosmological constant and theoretical calculations of its value. The cosmological constant was first proposed by Einstein as a mathematical necessity to rationalise how the universe has resisted collapsing under its own gravity. General relativity, however, predicts the vacuum energy density to be very different to its measured value (i.e. as needed to explain the measured expansion of the universe). There is no generally accepted explanation for this discrepancy, although numerous papers have been written about it.
“Quantum mechanics suggests the vacuum itself should fluctuate imperceptibly. In general relativity, those tiny quantum fluctuations produce an energy that would serve as the cosmological constant. Yet, it should be 120 orders of magnitude too big—big enough to obliterate the universe.”
It was examined in a 1994 paper for the American Journal of Physics (Volume 63, Issue 7) by Ronald J. Adler, Brendan Casey, and Ovid C. Jacob, who called it the 'Vacuum Catastrophe'. The paper described the spectacularly conflicting results of calculations of the energy density of a vacuum and actual 'measured' values.
“When summed up over all modes to some reasonable maximum this leads to an enormous energy density of the vacuum. Such an energy density presents serious problems when gravity is considered, since its gravitational field has not been observed; indeed the non-observation places an upper limit on the energy of the vacuum, which is in extreme conflict with the theoretical estimate: about 120 orders of magnitude!”
Further reading : A 2107 article from Science linking the Cosmological Constant problem to Dark Energy