The calculated mass of the universe ranges anywhere from 10^52 kg to 10^60 Since different methods were used, the disparity between the numbers is so great.
The matter content of the Universe are Atoms(~4.5%), Dark matter(~23%) and Dark energy(~72.5%).
The prediction of Dark Matter followed directly from the fact that the gravitational lensing of background objects and the slight increase of angular speed of stars in galaxies surrounding, via which they predicted the ratio of Mass of Atoms and Dark matter.
Now the concept of Dark energy came from the observation that the observable universe is expanding at an ever increasing rate. Whereas Einstein, wanting the universe static, gave the cosmological constant; the scientists nowadays think it was nothing but a correspondence for Dark energy. There are certain other evidences of it's presence. WMAP gives us the estimated values of their abundance.
Now the mass of the observable universe (total mass of atoms) is calculated in 3 ways:-
1. The estimated critical density being 9.30×10^−27 kg/m^3, and volume of observable universe being 3.6×10^80 m^3, total mass comes 3.35×10^54 kg. Therefore the mass of atoms comes about 1.5×10^53 kg.
2. Based on stellar density.
Taking the Sun's mass(2 × 10^30 kg) as mean stellar mass and calculating the number of stars from the stellar density, about a star per billion of ly^3, multiplied by the observable volume; gives the mass of about 3×10^52 kg.
It is a very rough estimate as it excludes intergalactic gas which is ~10 times that of stellar mass.
3. Based on steady-state universe
Sir Fred Hoyle's method for the calculation:-
m=4/3×π×(c/H)^3×ρ where HH is the Hubble Constant. This evaluates to
c32GHc32GH
which approximates to 8×105^2kg