Astroparticle Physics

8.4 The Friedmann Equation from General Relativity 179sphere and test galaxy could be anywhere in the universe,the equation for R applies to any pair of galaxies sufficientlyfar apart that one can regard the intervening matter as beinghomogeneously distributed.The Friedmann equation can also be applied to the earlyuniverse, before the formation of galaxies. It will hold evenfor an ionized plasma as long as the universe is electricallyneutral overall and one averages over large enoughdistances. The scale factor R in that case represents the distancebetween any two elements of matter sufficiently farapart such that gravity is the only force that does not cancelout.early universe8.4 The Friedmann Equationfrom General Relativity 2“Since relativity is a piece of mathematics,popular accounts that try to explainit without mathematics are almost certainto fail.”Eric RogersAs mentioned above, the Friedmann equation can also beobtained from general relativity. Even though this approachis mathematically demanding, this elegant method will besketched in the following. More detailed presentations aregiven in the literature [8, 9].In general relativity the scale factor R(t) enters as a factorin the metric tensor g µν . This relates the space-time intervalds 2 to changes in coordinates x µ bymetric tensords 2 = g µν dx µ dx ν , (8.17)where µ, ν = 0, 1, 2, 3 and summation over repeated indicesis implied. The most general form of the metric tensorfor an isotropic and homogeneous universe is given by theRobertson–Walker metric,ds 2 = dt 2[dr− R 2 2 ((t)1 − kr 2 + r2 dθ 2 + sin 2 θ dφ 2)] , (8.18)2 This section is mathematically demanding and should beskipped by those readers, who are not familiar with tensors.Robertson–Walker metric