I will suggest naming intermediate computations and avoiding use of the percent operator %
.
As for what went wrong, it involves use of QuantityVariable
and I do not know if it is a bug or a misuse. The gist is that at least in some cases you will need to explicitly evaluate the second argument. The minor variation of your code below seems to give a plausible result.
mm = mUmaximum /. Solve[Sqrt[
c^5/((4 Pi)^2*G blackbodyEmissivity BoltzmannConstant *
CBRTemp^4)] == (2 G mUmaximum)/
c^2 + (4 G \[Pi] mUmaximum)/c^2 - (
4 G \[Pi] Cos[(c^3 t)/(G mUmaximum)] mUmaximum)/c^2 /.
t -> 0,
mUmaximum] /. {t -> 0,
BoltzmannConstant -> Quantity[1, "StefanBoltzmannConstant"],
blackbodyEmissivity ->
Quantity[1, "Joules"/("Meters"*"Meters"*"Seconds")],
G -> Quantity[1, "GravitationalConstant"],
c -> Quantity[1, "SpeedOfLight"],
CBRTemp -> Quantity[2.73, "Kelvins"]};
qu = QuantityUnit[First[mm]];
ud = UnitDimensions[qu]
(* Out[531]= {{"MassUnit", 1/2}, {"TimeUnit", 3/2}} *)
dc = DimensionalCombinations[
{QuantityVariable["mUmx", Evaluate@ud]},
QuantityVariable["mss", "Mass"],
IncludeQuantities -> "PhysicalConstants",
GeneratedParameters -> None
];
ne = First[Rest[dc]]
(* Out[534]= (Quantity[1, ("SpeedOfLight")^(9/4)/(
"GravitationalConstant")^(3/4)]) Sqrt[QuantityVariable[
"mUmx",{{"MassUnit", 1/2}, {"TimeUnit", 3/2}}]] *)